[    {        "title": "2204.11595v1.Ultrafast_racetrack_based_on_compensated_Co_Gd_based_synthetic_ferrimagnet_with_all_optical_switching.pdf",        "content": "Ultrafast racetrack based on compensated Co/Gd-based\nsynthetic ferrimagnet with all-optical switching\nPingzhi Li1*, Thomas J. Kools1, Bert Koopmans1and Reinoud Lavrijsen1\n1*Department of Applied Physics, Eindhoven University of Technology, Eindhoven, 5612\nAZ, Netherlands.\n*Corresponding author(s). E-mail(s): p.li1@tue.nl;\nKeywords: synthetic ferrimagnet, angular momentum compensation, current-induced domain wall motion,\nall-optical switching\nSpin-orbitronics [1, 2] and single pulse\nall-optical switching (AOS) [3] of magne-\ntization are two major successes of the\nrapidly advancing \feld of nanomagnetism\nin recent years, with high potential for\nenabling novel, fast and energy-e\u000ecient\nmemory and logic platforms. Fast current-\ninduced domain wall motion (CIDWM) [4]\nand single shot AOS [5] have been individu-\nally demonstrated in di\u000berent ferrimagnetic\nalloys. However, the stringent requirement\nfor their composition control [5, 6] makes\nthese alloys challenging materials for wafer\nscale production [3]. Here, we simultane-\nously demonstrate fast CIDWM and energy\ne\u000ecient AOS in a synthetic ferrimagnetic\nsystem based on multilayered [Co/Gd] 2.\nWe \frstly show that AOS is present in\nits full composition range. We \fnd that\ncurrent-driven domain wall velocities over\n2000 m/s at room temperature, achieved by\ncompensating the total angular momentum\nthrough layer thickness tuning. Further-\nmore, analytical modeling of the CIDWM\nreveals that Joule heating needs to be\ntreated transiently to properly describe the\nCIDWM for our sub-ns current pulses. Our\nstudies establish [Co/Gd]-based syntheticferrimagnets to be a unique materials plat-\nform for domain wall devices with access to\nultrafast single pulse AOS.\nResearch in spintronics over the last decades\nhas demonstrated a possibility for further evo-\nlution of solid-state electronic application plat-\nforms beyond CMOS. The racetrack memory [7],\nan envisioned novel data storage device concept\nbased on using magnetic domain walls (DWs) [8],\nutilizes frontier spintronic mechanisms to func-\ntion as an ultra-dense on-chip memory with an\noperation speed comparable to low-level cache\n[9]. The e\u000eciency of racetrack memory relies\non the fast CIDWM in a material with per-\npendicular magnetic anisotropy. The maximum\nvelocity was signi\fcantly enhanced over the years\nby combining spin-orbit torques (SOTs) and the\nDzyaloshinskii-Moriya interaction (DMI) [10{12]\nwith synthetic antiferromagnets (SAFs), which\nresulted in reported DW velocities close to 750\nm/s [13].\nDespite the large improvement, the energy e\u000e-\nciency is still limited due to the weak strength\nof the antiferromagnetic (AF) coupling. There-\nfore, the materials platform of rare earth (RE)-\ntransition metal (TM) compounds garnered con-\nsiderable attention, promising faster CIDWM due\nto the much stronger direct AF coupling than the\n1arXiv:2204.11595v1  [cond-mat.mes-hall]  25 Apr 20222 Article Title\nindirect exchange coupling [14] utilized in SAFs.\nFurthermore, the SOTs used to drive the DW\npromise to be highly e\u000ecient in the RE-TM sys-\ntems as a result of the long spin coherence length\n[15]. Cosequently, high velocity CIDWM has been\nreported in Co-Gd-based ferrimagnetic alloy sys-\ntems [4, 16] when the angular momentum in the\nmagnetic material is compensated, being at least\na factor of three faster than that of the previously\nreported SAFs.\nBesides the e\u000ecient CIDWM, single pulse all-\noptical switching (AOS) of the magnetization\n[5] in the RE-TM systems has obtained signi\f-\ncant attention thanks to its sub-picosecond [17]\nenergy e\u000ecient [3, 18, 19] magnetization switch-\ning enabled by the ultrafast angular momentum\ntransfer upon laser excitation [20]. This can be\nuseful as a new generation of ultrafast magnetic\nmemory, as well as a data bu\u000ber between electron-\nics and integrated photonics [3, 21, 22]. Recently,\na synthetic ferrimagnetic system based on a Pt/-\nCo/Gd [19, 23] layered structure has shown high\nrobustness [24] for such a hybrid integration.\nThese kinds of synthetic ferrimagnets have\nsome distinct advantages over RE-TM alloys. For\ninstance, AOS is not limited by the exact compo-\nsition [25]. They also withstand thermal annealing\n[26] and o\u000ber easier magnetic composition control\nat wafer scale than the alloy system, as well as bet-\nter access to interface engineering. Therefore, it\nhas been proposed that such a materials platform\nhas high potential to realize a hybrid integration\nof DW memory in photonic platforms to further\nenhance their storage density [21, 27].\nSo far, the CIDWM of Co/Gd bilayers [27,\n28] has been investigated. However, the highest\nreported velocity, achieved at cryogenic conditions\n[28], was several times lower than that reported\nin alloys [4], in part due to large net angu-\nlar momentum, low compensation temperature as\nwell as DW pinning e\u000bects. In this report, we\ntherefore propose a materials platform based on\nthe [Co/Gd] 2synthetic ferrimagnet capable of\naccommodating both e\u000ecient CIDWM at room\ntemperature (RT) and single-pulse AOS, with\ncompatibility for wafer scale production as well as\nrobustness for engineering.\nThe magnetostatic and AOS properties of\nthe [Co/Gd] 2materials platform under investi-\ngation in this work are shown in Fig. 1. ATa/Pt/[Co/Gd] 2/TaN multilayer is deposited on\na Si/SiO 2substrate, where the \frst Gd layer is\nwedged (see Fig 1.a and Methods) to tune the\nnet magnetic moment. Due to the proximity e\u000bect\nby the Co, a net magnetization is induced in the\nGd which drops o\u000b steeply away from the Co\ninterface [29]. Compared to the bilayer Co/Gd\n[23, 28], we double the magnetic volume of the Co\nin [Co/Gd] 2, while tripling the number of inter-\nfaces at which magnetization is induced in the Gd.\nThis allows us to more easily compensate both\nthe angular momentum and magnetic moment of\nthe system at RT by varying the individual layer\nthicknesses.\nWe measured hysteresis loops by polar MOKE\nat RT across the thickness range of Gd between\n0-1.6 nm. In Fig. 1.b we plot the MOKE sig-\nnal intensity, de\fned as the di\u000berence in signal\nbetween positive and negative saturation, as well\nas the coercivity obtained from these hysteresis\nloops. The loops corresponding to the colored dots\nin Fig. 1.b are presented in Fig. 1.c.\nThe 100% remanence indicates a well-de\fned\nperpendicular magnetic anisotropy as well as tight\nexchange coupling between the layers. Impor-\ntantly, at the Gd thickness of 0.97 nm, the MOKE\nsignal switches sign (see black curve in Fig. 1.b).\nThis can be observed as well in Fig. 1.c as an\ninverted hysteresis loop. At this thickness the\nmagnetization is compensated, i.e. the Co and\nGd magnetization cancel each other. This is fur-\nther evidenced by the divergence in coercivity,\nand con\frmed by superconducting quantum inter-\nference device measurements (see Sup. A). Thus,\nupon increasing the thickness of Gd, the mag-\nnetic balance is shifted from being Co-dominated\nto Gd-dominated.\nWith respect to the magnetostatics, we note\nthat the magnetization and angular momentum\ncompensation thickness in these [Co/Gd]-based\nferrimagnets are similar but not identical due to\nthe di\u000berent Land\u0013 e g-factors of Co ( \u00182.2) and Gd\n(\u00182.0). Therefore, when discussing compensation\nwithout further speci\fcation in the remainder of\nthis text, we refer to angular momentum compen-\nsation as this is the relevant condition for e\u000ecient\nCIDWM [4, 28, 30].\nTo investigate whether our 4-layer Co/Gd sys-\ntem is all-optically switchable, and how it depends\non compensation conditions, we performed AOSArticle Title 3\na) b) c)\nCo(0.6)Gd(0-1.6)Co(0.6)Gd(1.5)TaN(4)\nPt(4)\nTa(4)d)\n1234567\nCo-dominated\nGd-dominated\nFig. 1 a): Schematic illustration of the layer stack used in our study (thickness in nm). b): Result of static MOKE study,\nwhere the MOKE signal (black) is de\fned by the di\u000berence in signal intensity between saturation at positive and negative\napplied \feld. Coercivity (red) was measured at a \fxed scanning speed of 10 mT/s (red). c): Hysteresis loops measured at the\nsample thicknesses marked by the coloured dots in b). d): Threshold \ruence for all-optical switching across the wedge shown\nin a). Inset shows toggle switched domains at two sides of the compensation thickness for varying amount of subsequent\npulses.\nexperiments by illuminating the wedge with sin-\ngle femtosecond laser pulses with varying pulse\nenergy, and examined the sample under a polar\nKerr microscope. We \fnd that single pulse AOS is\npresent at every thickness on the [Co/Gd] 2wedge.\nAn example of the resulting toggle switched\ndomains around the compensation boundary is\nshown in the inset of Fig. 1.d, which can be\nobserved from the contrast inversion (see Sup. B\nfor more images). This distinct feature that the\nAOS occurs both in the deep Co-dominated and\nGd-dominated composition is unlike that of alloys\n[18] and Co/Tb multilayers [31], in which AOS\nis only present within \u00061.5 % of the composition\nwindow. This shows the \rexibility of our system\nin terms of AOS engineering.\nWe further plot the threshold \ruence as a func-\ntion of Gd thickness, calculated from the switched\narea and the corresponding pulse energy [23],\nin Fig. 1.d. We \fnd that the threshold \ruence\ndepends signi\fcantly on the material composition,\nand is at a minimum around the compensation\npoint. Initially, the threshold \ruence decreases\nmonotonously with Gd thickness up to 1 nm, with\na reduction of the threshold \ruence of more than\n50% reaching 0.9 mJ/cm2, which corresponds to\n25 fJ for a 502nm2domain. Such a switching\nenergy is an order of magnitude lower than that\nof typical SOT and spin-transfer torque switching\nof a ferromagnet [32].\nAlthough a full dissection of the exact mech-\nanism behind the thickness-dependence of the\nthreshold \ruence is beyond the scope of this work,\nwe note that previous studies have shown thatreducing the Curie temperature of layered sys-\ntems can lead to a reduction of the threshold\n\ruence [6, 19, 23, 25]. In our case, as the Gd layer\nincreases, the Curie temperature of the total stack\nis expected to drop, because the mutual exchange\nstabilization of the two Co layers is weakened,\nleading to a reduction of threshold \ruence.\nWith compensation and AOS at RT con\frmed,\nin the remainder of this work we demonstrate\nand explain the DW velocities of over 2000 m/s,\nas achieved in [Co/Gd] 2. In the case of such an\nAF coupled system with DMI, two contributions\nmainly drive the coherent motion of the DW: the\nDMI-torque and the exchange coupling torque.\nThese two torques give rise to spin dynamics\nupon excitation by the spin Hall-e\u000bect-induced\nspin accumulation, and both lead to DW motion\nalong the current direction (for positive DMI and\nspin Hall angle [11, 33]).\nIt is well understood that the exchange cou-\npling torque only drives the DW motion e\u000eciently\nnear the compensation point [13, 28]. Adding\nadditional AF coupled layers to a ferromagnet\ncan facilitate this, however, can also increase the\nangular momentum of the DW to be translated.\nNonetheless, following the discussion in Sup. C, we\nstill expect a signi\fcant increase of the DW veloc-\nity in our study, as the enhancement of exchange\ncoupling torque due to compensation signi\fcantly\noverwhelms the e\u000bect of added magnetic inertia.\nFor this reason, we experimentally determined\nthe DW velocity from the CIDWM in [Co/Gd] 2\nas a function of lower Gd thickness. This study\nis performed in the same stack showcased in Fig.\n1.a. The wedged thin \flm was patterned into4 Article Title\na)\nb)c)\nd) e) f)\n0.4 ns\nFig. 2 a): Example image from DW motion experiments performed by di\u000berential polar Kerr microscopy, where the setup\nis illustrated schematically as a pulse generator and a 6 GHz bandwidth oscilloscope. The DW position after applying\nconsecutive pulses is shown. b): The measured DW velocity as a function of current density for three Gd layer thicknesses.\nc): The DW velocity as a function of Gd layer thickness for various current densities, where 10 % error bars are not shown.\nHere we marked the quanti\fed magnetization compensation thickness tMobtained from polar MOKE measurements as\nwell as the corresponding angular momentum compensation thickness tAbased on the estimation from magnetometry\nmeasurement. d): Compensation thickness as obtained from polar MOKE measurements of the [Co/Gd] 2at various ambient\ntemperatures. e): The temporal pro\fle of the current pulse used in the experiment (normalized by the peak current density),\nand its subsequent temperature rise (as simulated with COMSOL) at 3.6 TA/m2. f): Result of the theoretical 1-D modeling:\nthe DW velocity as a function of Gd thickness, where the magnetization pro\fle as well as its temperature dependence is in\nline with experimenes, while the current pulse and temperature pro\fle follow the pro\fle given in d). The velocity maximum\nis marked with a grey dot. The vertical dashed line indicates the angular momentum compensation thickness.\nmicro-structures as shown in Fig. 2.a (for details\nsee Methods and Sup. Fig. E4.b). The magnetic\ndomains were imaged by polar Kerr microscopy\n(see Fig. 2.a). Short current pulses of around 1 ns\nduration with varying amplitudes are injected into\nthe structures to determine the DW velocity (see\nSup. D). We observe that the DW moves coher-\nently along the current direction (see Fig. 2.a).\nThis con\frms a left-handed N\u0013 eel-type DW in our\nsystem as a result of a positive DMI, and a posi-\ntive spin Hall angle [11, 33], as is characteristic of\nthe Pt/Co interface [11].In Fig. 2.b we compare the resulting DW veloc-\nity as a function of peak current density for a\nCo-dominated (0.69 nm), compensated (0.93 nm)\nand Gd-dominated stack (1.28 nm), considering\nthe magnetic composition at RT. We demonstrate\nthat DW velocities of over 2000 m/s can be\nachieved at the highest current densities, at least\nthree times larger than that observed in SAFs [13].\nFurthermore, we notice from the crossing point\nbetween the red and black curve that the Gd\nthickness at which the highest velocity is observed\nbecomes larger with increasing current density. We\nattribute this to the e\u000bect of Joule heating inducedArticle Title 5\nby the current pulses. Earlier studies show that the\nGd magnetization is quenched more severely dur-\ning Joule heating than that of Co [28, 29, 34, 35].\nThe consequence is that at higher temperature rel-\natively more Gd than at RT is needed to achieve\ncompensation. This fact is con\frmed for our stacks\nby the upward shift of the compensation thickness\nwith increasing sample temperature using polar\nMOKE (see Fig. 2.d). We \fnd that this hypothesis\nof a Joule-heating-induced shift of the compensa-\ntion thickness explains the CIDWM experiments\nwell. To understand this, it useful to consider two\nregimes in Fig. 2.b in more detail.\nForJ < 2:3 TA/m2, before saturation e\u000bects\nstart to occur and for limited heating, we observe\na steady rise of the velocity, consistent with earlier\nwork on CIDWM in materials with AF coupling.\nAs expected, the compensated sample is driven\nmost e\u000eciently. We visualize this further by plot-\nting the DW velocity as a function of Gd thickness\nacross the Co and Gd-dominated regimes, as is\nshown in Fig. 2.c. Here, we observe that the DW\nvelocity in this range of current densities spikes\nat the RT compensation thickness, further prov-\ning the origin of the velocity increase is due to the\nenhanced exchange coupling torque.\nIn contrast, for J >2:3 TA/m2, the DW veloc-\nity saturates both for the Co-dominated and the\ncompensated sample. This is typical behaviour of\nthe CIDWM of a ferromagnet [11, 13, 28, 33, 36],\nwhich is limited by the DMI-torque. Contrar-\nily, the Gd-dominated sample shows a consistent\nvelocity rise typically associated with a compen-\nsated system [13]. This is consistent with the\nhypothesis that the increased Joule heating shifts\nthe compensation point to larger Gd thicknesses\nduring the pulse, and is supported by the velocity\npeak shift observed in Fig. 2.c.\nIn order to quantify the e\u000bect of the change\nin compensation thickness due to Joule heating\non the CIDWM, we \frst numerically investigate\nthe temperature transient for the current pulses\napplied in the experiment (see Sup. E). In Fig.\n2.e we plot a typical current density pro\fle and\nthe corresponding modelled temperature change\nin the device. We observe a delay of \u00180:4 ns\nbetween the peak current density, and the peak\ntemperature. This is a consequence of the factthat the pulse duration approaches the character-\nistic time scale of the heat transport between the\nsample and the substrate (See Sup.E).\nImportantly, around the peak current den-\nsity where the driving torque exerted on the DW\nis largest, the sample temperature is changing\nrapidly from 0 to 50 % of maximum temperature\nwithin 0.4 ns. Therefore, the net angular momen-\ntum changes rapidly during the application of\nthe pulse. This makes assigning an e\u000bective tem-\nperature during DW motion like previous studies\n[4, 28, 35, 37] inaccurate.\nTherefore, we incorporated the transient cur-\nrent density pro\fle, and the corresponding tem-\nperature rise into an analytical 1-dimensional\nmodel of the CIDWM (see Sup. F), where we\nintroduce a thickness and temperature dependent\nmagnetization pro\fle following our SQUID mea-\nsurements (see Sup. A). The resulting modelled\nDW velocity as a function of thickness and current\ndensity is plotted in Fig. 2.f.\nWe \fnd that the velocity peak shift with\nincreased current density as shown in Fig. 2.b is\nwell described by our model. Furthermore, we also\n\fnd that the rapidly changing angular momentum\nduring the current pulse gives rise to a broad-\nening of the velocity peak compared to the case\nwith constant temperature previously described in\n[4, 28](see Sup. G).\nThese results suggest that in general transient\ntreatment of the temperature pro\fle is neces-\nsary to properly address the DW dynamics for\nultrashort current pulses. Furthermore, to bene\ft\nfrom large exchange coupling torque at elevated\ntemperature, which is a likely scenario in high-\nspeed electronics, pushing the composition into\nthe Gd-dominated regime is required.\nIn summary, we have presented a materi-\nals platform of a synthetic ferrimagnet based\non [Co/Gd] 2which can be e\u000bectively tuned\nnear the compensation point via layer thickness,\nand exhibits e\u000ecient single pulse AOS. We also\nexperimentally and numerically showed that fast\nCIDWM with velocities over 2000 m/s can be\nachieved with the help of angular momentum\ncompensation. Finally, we addressed the transient\nJoule heating e\u000bect during CIDWM, which gives\nimportant insight into our experimental obser-\nvations and potential technological implementa-\ntion. Our results therefore lay the foundation\nfor an e\u000bective materials platform that exhibits6 Article Title\nboth ultrafast CIDWM and AOS, paving the way\nfor synthetic ferrimagnets to become the new\nparadigm of ferrimagnetic spintronics and pro-\nviding a jumping board for further integration\nbetween photonics and spintronics.\nMethods. Magnetic \flms were deposited on Si\nsubstrates with a 100 nm thermally oxidized SiO 2\nlayer by D.C. magnetron sputtering in a system\nwith a base pressure of \u00185\u000210\u00009mBar. A\nthickness wedge is created with the help of a\nmoving wedge shutter during sputtering. From\nthese thin \flms, nanostrips were fabricated using\nelectron-beam lithography and lift-o\u000b. The gold\ncontacts were made by wire bonding. The out-\nof-plane component of the magnetization (M z)\nof the nanostrips was measured by polar Kerr\nmicroscopy. An in-plane \feld of 200 mT along\nthe current direction, combined with a 3 TA/m2\ncurrent pulse was used to nucleate domains.\nBoth MOKE and Kerr Microscopy characteriza-\ntion were performed using a 700 nm continuous\nwave laser in the polar con\fguration. Here, we\nchose the wavelength of the laser light (around 700\nnm) such that the detected signal is only sensitive\nto the magneto-optical response from the Co layer\nand Kerr rotation is maximized.\nSupplementary information. All supplemen-\ntary information except for digital \fles are pre-\nsented in the Appendix.\nContribution. P.L. and T.J.K. contributed\nequally to this work in terms of design and con-\nduct of the project. B.K. and R.L. supervised the\nproject. All authors contributed to the writing of\nthe manuscript.\nAcknowledgments. This project has received\nfunding from the European Union's Horizon\n2020 research and innovation programme under\nthe Marie Sklodowska-Curie grant agreement\nNo.860060. This work is also part of the Gravita-\ntion programme `Research Centre for Integrated\nNanophotonics', which is \fnanced by the Nether-\nlands Organisation for Scienti\fc Research (NWO).\nCompeting Interests. Authors declare no\ncon\ricts of interest.Appendix A SQUID\ncharacterization\nIn order to characterize the magne-\ntostatic properties of the sample, a\nseries of 6 samples of composition\nTaN(4)/Pt(4)/Co(0.6)/Gd( tGd)/Co(0.7)/Gd(1.5)\n/TaN(4), with tGd= [0:2;0:4;::;1:2;1:4] nm were\ngrown. Using VSM-SQUID, out-of-plane hystere-\nsis loops were measured, which are shown in Fig.\nA1.a. The decreasing moment when going from\na middle Gd thickness of 0.4 to 1.4 nm, and the\ndiverging coercivity at the minimum of the mea-\nsured moment are indicative of a compensated\nsystem. This can also be seen in the plot of the\nmagnetic moment normalized by the sample area\nin Fig. A1.b. The temperature dependence of the\nmagnetization was also investigated, and a typical\nresult is shown in Fig. A1.c. In order to estimate\nthe Curie temperature of the stack and obtain an\nexpression for the magnetization Mtot, we follow\nearlier work on 3d-4f ferrimagnetic alloys in \ft-\nting a critical exponent to the data of the form\n[30, 38] given by:\nMtot=MCo(T)\u0000MGd(T) =\nMCo0\u0012\n1\u0000T\nTC\u0013\u0018Co\n\u0000MGd0\u0012\n1\u0000T\nTC\u0013\u0018Gd\n;\n(A1)\nwhereMGd0andMCo0and\u0018Gd,\u0018Coare the\nmagnetization at zero temperature and the critical\nexponent of Co and Gd, respectively. We assume\nMCo0= 1:4 MA/m, and \fnd that the critical\nexponents\u0018Co= 0.50,\u0018Gd= 0.72, and TC= 450 K\ndescribe the temperature dependence of the mag-\nnetic moment well (Fig. A1.c). We note that the\nmagnetization in the Gd is an ill de\fned quantity,\nsince the Gd transitions from partially magnetized\ndue to the proximity e\u000bect, to truly ferromag-\nnetic at low temperature. We therefore estimate\nthe value of MGd0by considering the known thick-\nness dependence of the magnetic moment in the\nstackmtot. This quantity is calculated as the sum\nof that of the individual layers as follows:\nmtot=mCo1+mGd1+mCo2+mGd2\n=MCo(T) (tCo1+tCo2)\u0000Article Title 7\na) b) c)\nFig. A1 VSM-SQUID characterization of the out-of-plane moment of the Co(0.6)/Gd(x)/Co(0.7)/Gd(1.5) material system.\na): Hysteresis loops across the compensation point at room temperature. b): Room temperature magnetic moment per unit\narea as a function of middle Gd thickness. c): Typical temperature dependence of the magnetic moment in a Co-dominated\nquadlayer sample. The red line indicates the best \ft of equation A1.\nMGd(T) (2tGd1+tGd2);(A2)\nwhere:\nmCo1=MCo(T)tCo1\nmCo2=MCo(T)tCo2\nmGd1= 2MGd(T)tGd1\nmCo1=MGd(T)tGd2: (A3)\nHere,tCo1,tCo2,tGd1,tGd2are the thicknesses of\nthe bottom (coming from the substrate) and top\nCo and Gd layers, respectively. We \fnd the red-\ncurve in Fig. A1 b, which represents equation (A2)\nfor the stack dimensions used in the experiment,\nwhich follows the SQUID data reasonably well by\nchoosingMGd0= 0.66 MA/m. It should be noted\nhere that this is about a factor of 2 lower than\nvalues typically found in CoFeGd alloys [30]. We\nargue that this is a consequence of the inhomoge-\nneous proximity induced magnetization in the Gd,\nwhich decays when moving away from the Co/Gd\ninterface, and is therefore concentrated at the\ninterface. In order to describe the magnetization\nfully, an intermixing pro\fle and proximity-induced\nmagnetization pro\fle needs to be assumed and\nsubstantiated, which is beyond the scope of this\nwork. We nonetheless choose this simple approach\nof homogeneous magnetization in the Gd, as the\nintra-layer exchange present in the Gd is expected\nto be strong enough to have the total angular\nmomentum in the Gd layer respond coherently to\nthe excitation of the DW, making the total angu-\nlar momentum present in the DW the relevant\nparameter and not the exact way it is distributed.Appendix B All optical\nswitching in\n[Co/Gd] 2\nIn addition to the meta-data of all-optical switch-\ning as shown in the main text (Fig. 1.d, repeated\nin Fig. B2.b), in Fig. B2.a we present a typical\nexcerpt of the polar Kerr microscope images used\nto obtain the threshold \ruence as a function of\nGd-thickness.\nAppendix C Velocity\nenhancement\nexpectation\nTo argue why a signi\fcant velocity enhancement is\nexpected regardless of the added magnetic volume\nfrom the multiple magnetic layers, it is instructive\nto discuss the ratio between the expected DMI-\ntorque-driven DW velocity in an uncompensated\nsamplevucto the exchange coupling torque-driven\nDW velocity in a compensated sample vc. Here we\ndiscuss this using an analytical approach based on\na 1-D model of DW motion [13, 28, 39] (see also\nsection F), the ratio is given by:\nvc\nvuc=~J\u0012SH\u0001\n2e\u000bDAuc\nAc: (C4)\nHere, we consider the fact that vuc, is limited\nby the strength of the DMI, following [4, 36]:\nvuc=\u0019D\n2P\nnjAnj;=\u0019D\n2Auc; (C5)8 Article Title\na)\nb)\n200 μm\n1.50\n1.30 \n1.05 \n0.96 \n0.80 \n0.50 \n0.30 \n0.10 Gd thickness (nm)\n180 160 140 120 100 80 60 40\nPulse enery (nJ)\nFig. B2 (a) Di\u000berential Kerr images (background taken\nat a Gd thickness of 0.1 nm) taken at di\u000berent locations\non the Gd wedge sample, which was illuminated by single\nlaser pulses with varying pulse energy. The inverted back-\nground contrast around 1 nm is due to the transition from\nCo-dominated to Gd-dominated. b): Calculated threshold\n\ruence as a function of Gd layer thickness. The threshold\n\ruence below and above compensation is calculated with\nthe average of 4 neighboring rows.\nwhereDis the surface DMI energy density, and A n\nandAucare the total angular momentum of the\nnthlayer and the full stack, respectively. On the\nother hand, vcis for strong enough exchange cou-\npling limited by the amount of angular momentum\ntransferred to the DW via the spin Hall-e\u000bect, and\ncan for a multilayered system be expressed as:\nvc=~J\u0012SHE\u0001\n4e\u000bP\nnjAnj=~J\u0012SHE\u0001\n4e\u000bA c:; (C6)where ~is the reduced Planck constant, Jis the\ncurrent density, \u0012SHE is the spin Hall angle for\nanti-damping-lik torque, eis the electron charge, \u000b\nis the Gilbert damping parameter, Acis the total\nangular momentum of the full stack, and \u0001 is the\nDW width, which shows no dependence on the\nDMI.\nTo estimate the expected velocity enhance-\nment, we consider equation (C4) for the simple\ncase where Auc=Ac= 1=2. This corresponds to\nthe situation where one antiferromagnetically cou-\npled layer is added to a magnetic system, which\nexactly compensates the angular momentum in\nthe original system. Within this simple model,\nand using equation (C4), the condition for DW\nvelocity enhancement then becomes:\nvc\nvuc=~J\u0012SH\u0001\n4e\u000bD>1: (C7)\nWe \fnd that for the parameters used in the\nsimulations (see Sup. F), a net gain in DW veloc-\nity can already be obtained at a current density\nof 0.28 TA/m2. Therefore, the exchange cou-\npling torque in our system can be expected to\nresult in a much higher velocity compared with\nthe uncompensated case where the DMI-torque\ndominates regardless of the inevitable increase in\nmagnetic volume for the synthetic ferrimagnets\nunder discussion.\nAppendix D Domain wall\nvelocity charac-\nterization\nIn this section, we discuss the experimental details\nof the characterization of the current-induced DW\nvelocity.\nWe \frst discuss the principle of the measure-\nment. To magnetically saturate the sample, we\n\frst reverse the magnetization in the wire with\nrespect to the rest of the device by a combined\naction of an intense current pulse and an in-\nplane \feld along the current direction. Then, N\ncurrent pulses with amplitude J, and pulse dura-\ntion \u0001t, displace the DW by a distance \u0001 L,\nwhich is recorded by polar Kerr microscopy. The\nresulting DW velocity, vDW, is then calculated as\nvDW=\u0001L\nN\u0001t. We applied multiple pulses of the\nsame amplitude to allow the DW to travel enough\ndistance from the start to the end of the wire toArticle Title 9\nminimize the errors of the length measurement\nand statistical errors caused by variation between\npulses as a result of wire edge roughness and other\npotential fabrication imperfections. For the case of\nhigh current density or high velocity, this process\nis repeated several times for better averaging due\nto lowNbeing needed to translate the DW along\nthe full wire length.\nThe method mentioned above requires the\nideal case of a current pulse with a square pro-\n\fle. Next, we discuss the pulse pro\fle used in our\nexperiment, based on which we de\fne our e\u000bec-\ntive pulse width and amplitude. The amplitude\nand duration of the pulse needs to be calibrated\ncarefully in order to clearly de\fne the DW veloc-\nity. In our measurement set-up, we terminate\nthe device with a 6 GHz bandwidth oscilloscope\n(characteristic impedance 50 \n), allowing us to\ncharacterize the electrical pulse pro\fle directly\nduring the measurements. The resulting voltage\nversus time pro\fle obtained from the oscilloscope,\nis then converted to current density values versus\ntime based on the DC resistance and the geom-\netry of the devices. Here, we have to note that\nin our calculations, only the thickness of Pt was\nconsidered in our calculation (assuming no cur-\nrent \rowing in both the ferromagnetic layers and\nTa). Such an assumption is made because Pt is\nknown to be more conductive than the other met-\nals used in the stack [28]. Therefore, the energy\ne\u000eciency of CIDWM in our study is a conservative\nestimate, since the current is likely also \rowing\nin the other metallic layers. To avoid dramatic\nheat e\u000bects induced by long current pulses, which\nmight screen much of the physics, we used ultra-\nshort pulses down to 1 ns in our study. The pulse\namplitude can reach up to 4 \u00021012A/m2without\nfully demagnetizing the sample. A typical mea-\nsurement of the current pulse pro\fle is shown in\nFig. D3.a. In our measurement, we de\fne the pulse\namplitude to be the peak value (1.8 \u00021012A/m2in\nthis case). The pulse exhibits a spike shape, which\nrequires us to de\fne an e\u000bective pulse width. We\nwe de\fne the e\u000bective pulse width in our mea-\nsurement by taking the full width half maximum\n(0.8 ns) of the active area (see gray region in\nFig. D3.a) of the pulse obtained by subtracting\nthe pulse shape by a strong, experimentally deter-\nmined, pinning threshold (0.5 TA/m2). Thus, the\ntime needed for the current pulse to pull up andthe decay of the pulse below the pinning thresh-\nold is not considered, since in this region the DW\nis not displaced or to a limited extent. To de\fne\nthe pinning threshold, we applied over 500 current\npulses when the DW is at various positions of the\nwire. The pinning threshold is then de\fned as the\ncurrent density where no detectable movement of\nthe DW is found. A further reason for using short\npulses is the fact that the temperature rise during\npulse time can be easily suppressed by the heat\ndissipation and capacity of the substrate since\nthe heat front is still in the propagation regime.\nThis also leads to a transient delay of temperature\nrise (see the next sections for its in\ruence on the\nCIDWM).\nTo justify the e\u000bective pulse parameters,\nwe compare the measurement result with that\nobtained from a longer pulse (L) and a middle\nlong pulse (M) shown in Fig. D3.b, of which the\nshape is closer to a square pulse. As seen from\nFig. D3.b, the L and M pulses both have an over-\nshoot and a slanted decay at the end of the pulse.\nWe compensate for this e\u000bect by subtracting the\nDW displacement induced by a shorter pulse from\nthat of a longer pulse. Since the di\u000berence between\nthese pulse shapes resembles a square pulse shape\nwith known pulse amplitude and width, the DW\nvelocity can be more accurately calculated. How-\never, in order to neglect the contribution from\nthe part of the pulse that was not subtracted, it\nrequires the thermal e\u000bect to be low (since the ini-\ntial spike contributes heat as well). Thus, it is only\napplicable for low current density.\nNow, we introduce a reference pulse (R)\n(shown in red), which resemble the spike of L and\nM, such that its di\u000berence with L and M will\nresults in a square pulse with a duration longer\nthan 1 ns as shown in Fig. D3.b. We calculated the\nDW velocity for the samples used in our experi-\nment (shown in Fig. 2) based on the pulse time\ndi\u000berence between L and M (L-M), L and R (L-R),\nas well as M and R (M-R), which are plotted in\nFig. D3.c (current amplitude used as shown in Fig.\nD3.b). We compare the obtained results with that\ncalculated based on the short pulse (S) used in\nthe experiment shown in the main paper (see blue\ncurve in Fig. D3. b), which di\u000bers in amplitude\nfrom the pulse pro\fle in Fig. D3. a.\nWe observe that the DW velocities using these\napproaches are in the same order of magnitude,\nfollowing the same general dependence on Gd10 Article Title\na) b) c)\nFig. D3 a): The pulse pro\fle of a typical pulse used in the DW velocity measurements. The strong pinning region which\nwas excluded during the pulse width estimation was marked as brown, the active region from where the e\u000bective pulse width\nis obtained is mark as gray. The pulse width and amplitude obtained for this characteristic pulse pro\fle are marked in the\nplot. b): The pulse pro\fle of a short pulse (S) used for the experimental results shown in the main text, a middle long pulse\n(M), a long pulse (L) and a reference pulse (R). The estimated time di\u000berence between those pulses are given and will be\nused to calculate the DW velocity. c): The DW velocity obtained from a short pulse, and other di\u000berential pulse method.\nthickness. The DW velocity calculated using S is\nfound to be systematically lower. Here we choose\nto be conservative instead of trying to compensate\nfor this discrepancy, as \fnite thermal activation\nis expected to increase the DW velocity by reduc-\ning the DW friction [11, 28, 40], as is the case for\nlong pulses. So far, little study has been spent on\nsuch an issue in the \row regime of the CIDWM,\nalthough the absolute magnitude of enhancement\nwas once shown to be 20 % for a SAF [41] for 40 K\nof temperature increase. So we make a compromise\nby taking a large but safe margin avoiding over-\nestimation of the DW velocity as well as possible\nover-claim (for large velocity).\nAppendix E COMSOL\nsimulations of\nJoule heating\nIn this section, we present our study on the tem-\nperature rise during the application of the current\npulses. This is a crucial ingredient for the anal-\nysis of the experimentally observed DW motion,\nas well as design considerations for future appli-\ncations. Here, we based our study on COMSOL\nsimulations, which employs multiphysics modeling\nincluding thermal transport and electrical con-\nduction. In our model, we de\fned the geometry\nbased on the real physical dimensions of the device\nused in this study (see Fig. E4.b) and the mate-\nrial parameters either from the COMSOL material\nlibrary or from our measurements (see Table E1),\nwhich takes into account temperature-dependente\u000bects. We model the metallic stack as a dou-\nble layer consisting of a 8 nm non-conductive\nlayer, and a 6 nm conductive layer, of which\nthe conductivity is obtained from RT DC mea-\nsurements, while the density, heat capacity and\nthermal conductivity of the two layers are kept the\nsame.\nTable E1 Parameters used in COMSOL\nsimulation for Joule heating induced by the\ncurrent pulse.\nParameter @ RT Value Unit\nThickness of the Si substrate 0.5 mm\nThickness of SiO 2 100 nm\nDiameter of wire bond 50 \u0016m\nHeat Capacity of Si Substrate 678 J/(kg \u0001K)\nDensity of Si 2320 kg/m3\nThermal Conductivity of Si 134 W/(m \u0001K)\nHeat Capacity of SiO 2Substrate 730 J/(kg \u0001K)\nDensity of SiO 2 2200 kg/m3\nThermal Conductivity of SiO 2 1.4 W/(m \u0001K)\nThickness of conductive metal \flm 6 nm\nThickness of non-conductive metal \flm 8 nm\nElectrical Conductivity of Metal 0.893 106\u0001S/m\nHeat Capacity of the Metal 133 J/(kg \u0001K)\nDensity of the metal 21450 kg/m3\nThermal Conductivity of the Metal 71.6 W/(m \u0001K)\nRelative permittivity 10000 1\nMinimum Mesh Dimension 0.5 nmArticle Title 11\nElectrical stimuli are applied through a circu-\nlar region (yellow disks in Fig. E4.b), of which\nthe size is close to the physical size of the electri-\ncal contacting wire bonds in our experiments. The\nwire is signi\fcantly heated compared with the rest\nof the metal structure owing to its large current\ndensity. Air convection is present only at the top\nsurface to emulate the experimental conditions.\nFollowing these notions, we computed the temper-\nature rise relative to RT as a result of a current\npulse similar to the one used in the experiment,\nas indicated in Fig. E4.a. In order to describe\nthis pulse pro\fle, we \ft an empirical function J(t)\nconsisting of the sum of three Gaussians:\nJ(t) =3X\ni=1Ai\nwip\n\u0019=2exp \n\u00002(t\u0000tc,i)2\nw2\ni!\n:(E8)\nThe best-\ft parameters are summarized in\ntable F3. The same pro\fle is used to de\fne\nthe voltage pulse that is applied to the system\nin COMSOL. The resulting temperature rise for\nvarious peak current densities is plotted in Fig.\nE4.c. One important observation here, is that the\nheating of the magnetic layer is delayed by approx-\nimately 0.4 ns with respect to the current pulse.\nConsequently, when the SOT is maximized at the\npeak of the current pulse, the angular momen-\ntum is still rapidly changing. Since the exchange\ncoupling torque e\u000eciency is linked to the angu-\nlar momentum balance between the Co and Gd,\nthis transient heating leads to a transient DW\nmobility.\nFor Joule heating, the expected temperature\nrise depends quadratically on the current den-\nsity. In order to describe the temperature rise for\nany value of the peak current density J0, we \ft\nan asymmetric sigmoidial function to the tem-\nperature rise at various current densities given\nby:\nT(t) =y0+b01\u0000exp\u0010\nt\u0000tc\nl2\u0011\n1 + exp\u0010\n\u0000t\u0000tc\nl1\u0011: (E9)\nThe resulting \fts for various current densities are\nshown in Fig. E4.d. During this \ft all parameters\nexcept the amplitude of the peak b0were taken\nconstant. The found amplitude, which describesthe heating is then plotted as a function of cur-\nrent density in Fig. E4.e. We \ft a simple quadratic\nfunction to this quantity:\nb0(J0) =k0J2\n0; (E10)\nwhich is shown as the red line. The correspondence\nbetween the \ft and the temperature rise modelled\nby COMSOL suggests that Joule heating plays the\ndominant role in our system.\nIn summary, based on the best \ft to COMSOL\nsimulations and the quadratic dependence of the\ntemperature change on current density, we made\nan estimation of the temporal pro\fle of the tem-\nperature rise in our sample. This asymmetric sig-\nmoidal pro\fle with the quadratic prefactor given\nin equation E9 and E10, will be used to dynam-\nically describe the change in temperature in the\nnumerical 1-dimensional simulations of CIDWM\nin our [Co/Gd] 2samples.\nTable E2 Summary of the \ftting parameters\nin equations E8 and E9, found to best describe\nthe experimental current pro\fle and the\nmodeled temperature rise due to Joule\nheating, respectively.\nParameter Value Unit\ntc,1 -2.85 ns\nw1 0.42 ns\nA1 0.54 TA m\u00002ns\ntc,2 -3.29 ns\nw2 1.23 ns\nA2 0.29 TA m\u00002ns\ntc,3 -2.29 ns\nw3 0.59 ns\nA3 0.31 TA m\u00002ns\ny0 293.15 K\ntc 2.15 ns\nl1 0.11 ns\nl2 1.94 ns\nk0 20.7 K m4TA\u0000212 Article Title\na) b)\nc) e)\nSiO2 (100 nm)\nSi (500 μm)200 μm 600 μm 100 μm \n50 μm 50 μm 2 μm \n10 nm \nd)\nFig. E4 Joule heating analysis and current pulse de\fnition for the model. a): Fit of three Gaussians (eq. E8) to a typical\nexperimental current pulse pro\fle. b): Schematic of the device geometry used in the experiment, as well as in COMSOL. c):\nTransient temperature modeled using COMSOL for the experimental current pulse pro\fle at various peak current densities.\nd): Fits of equation E9 to the temperature pro\fle at various peak current densities J0. e): Parabola \ft to the prefactor b0\nin eq. E9 obtained from the analysis from d).\nAppendix F 1-dimensional\nmodel of\ncurrent induced\ndomain wall\nmotion\nIn this section the analytical 1D-model for\ncurrent-induced DW motion in the [Co/Gd] 2mag-\nnetic system is described. This model is an exten-\nsion to earlier work by Bl asing et al. [28], who\nintroduced the model to describe the DW dynam-\nics of a Co/Gd bilayer. We will \frst summarize\nthe main points of this model, noting that a full\ndiscussion, can be found elsewhere [41].\nThe model discussed below describes the\ndynamics of an up-down DW in a nanowire device\n[length (x-direction)\u001dwidth (y-direction)\u001d\nheight (z-direction)]. The orientation of the spins\nin the DW will be discussed in terms of spherical\ncoordinates \u0012and\u001e, which describe the polar and\nazimuthal angles of spins in the system, respec-\ntively. We de\fne \u0012= 0 and\u0012=\u0019to be the\npositive and negative out-of-plane ( z) direction\nrespectively, and \u001e= 0 and\u001e=\u0019to be pointingto the positive and negative x-direction, respec-\ntively. Within the DW, the spins rotate gradually\nfrom\u0012= 0 to\u0012=\u0019. In equilbrium, the DW angle,\nwhich we de\fne as the azimuthal angle of the spin\nin the middle of the DW will point along \u001e=\u0019\n(-x) due to the counterclockwise DMI.\nIn order to then derive the equations of\nmotion of the spins in the DW in the presence\nof non-conservative forces, like Gilbert damping\nand current-induced torques like the spin transfer\ntorque (STT) and spin Hall e\u000bect (SHE) torque,\nthe Rayleigh-Lagrange equation is solved:\n@L\n@pi\u0000d\ndt@L\n@_pi\u0000@F\n@_pi: (F11)\nHerepirefers to the free variables in the system,\nLandFdescribe the Lagrangian and the dissi-\npation function, respectively. Within our model\nof [Co/Gd] 2there are \fve free variables: The\nazimuthal angles \u001efor the separate layers, and the\nDW position q, from which the DW velocity _ qis\nderived:\n_q=\u0000\u0001\nsin\u0012_\u0012: (F12)\nHere it is assumed that the DW positions are\ntightly coupled, such that qis always the same forArticle Title 13\nthe di\u000berent layers. The quantities LandFare\nde\fned as:\nL=Z1\n\u00001!DW\u0000Tdx; (F13)\nand\nF=Z1\n\u00001P\u000b+PSTT+PSHEdx; (F14)\nrespectively, where P\u000b,PSTTandPSHE, are the\ndissipation functions for Gilbert Damping, STT\nand SHE respectively, Tis the \"kinetic energy\"\nof the DW which we take identical to Blaesing et\nal. [41], and \fnally !DWis the energy density of\nthe DW. In order to de\fne !DW,P\u000b,PSTTand\nPSHE, it is useful to introduce some compact nota-\ntion for parameters corresponding to a particular\nmagnetic layer. We will refer to parameters corre-\nsponding to speci\fc magnetic layers through the\nsubscripts 1, 2, 3 and 4, where the numbering\nrefers to the layer in the Pt/[Co/Gd] 2structure\nstarting with layer 1 from the bottom upwards\n(i.e. layer 1 is the Co layer interfaced with the Pt\netc.). Using this notation !DWis given by:\n!DW=D1\n\u0001cos\u001e1sin\u0012\n+4X\ni=1Kitisin2\u0012+4X\ni=1Aex,iti\n\u00012sin2\u0012\n+3X\ni=1Jex\u0000\ncos2\u0012\u0000cos (\u001ei\u0000\u001ei+1) sin2\u0012\u0001\n:(F15)\nHere,Dis the DMI constant, \u0001 is the DW width,\nKiis the anistropy energy density, tiis again the\nthickness,Aexis the exchange sti\u000bness, and Jex\nis the exchange coupling strength. The DMI is\nassumed to only interact with the bottom Co layer\n(i= 1) as it originates from the Co/Pt inter-\nface, and no DMI at the Co/Gd interface has yet\nbeen reported. The \fnal term, describing the anti-\nferromagnetic exchange coupling has been chosen\nsuch that negative Jexpromotes antiferromagnetic\ncoupling.\nNext, the dissipation functions are de\fned as:P\u000b+PSTT+PSHE=4X\ni=1\u0000\u000bi;mi\u0001\n\ri\u0010\n_\u001ei2+ _q2=\u00012\u0011\n+4X\ni=1\u001diuimi\n\u0001\risin2\u0012_\u001ei\u0000\fiuimi\n\u00012\risin2\u0012_q\n\u0000~\u0012SHJ\n2esin\u0012\u0012cos\u001e1_q\n\u0001+ sin\u001e1cos\u0012_\u001e1\u0013\n;\n(F16)\nwhere\u000bis the Gilbert damping coe\u000ecient, \ris the\ngyromagnetic ratio, \fis the parameter describ-\ning the non-adiabatic component to the STT, ~is\nthe reduced Planck constant, \u0012SHis the spin-Hall\nangle,Jis the current density, eis the electron\ncharge, and \u001diis a variable that is equal to 1 and\n-1 in a Co layer ( i= 1;3) and Gd layer ( i= 2;4)\nrespectively. Finally, uiis given by:\nui=~Pi\ridi\n2emiJ; (F17)\nwherePiis the degree of polarization of the\ncharge current. It should be noted that the spin-\nHall current is in this work modelled to only\ninteract with the Co layer interfaced with the Pt.\nRecent work has indicated that the spin coherence\nlength in ferrimagnetic multilayers can be larger\nthan the thickness of the individual layers in this\nstudy [15]. Hence, it is possible that part of the\nspin current injected along the z-direction can also\ninteract with the other magnetic layers when it is\n(partially) transmitted through the \frst Co layer.\nIn order to then solve equation (F11) for the\n\fve degrees of freedom in our system, we evalu-\nate the integrals in equations (F13) and (F14) by\nassuming a typical Bloch pro\fle of the polar angle\n\u0012as a function of x:\n\u0012(x) = 2 tan\u00001\u0012\nexp\u0014x\u0000q(t)\n\u0001\u0015\u0013\n: (F18)\nThe resulting equations of motion in absence of an\napplied \feld for q,\u001e1,\u001e2,\u001e3,\u001e4are then \fnally\ngiven by:\n_q=\u0001\u0010P4\ni=1mi\u000bi\n\ri\u0001\u0011\u0012\n\u00004X\ni=1miui\fi\n\ri\u0001\n\u0000\u0019~\u0012SHJ\n4ecos\u001e1\u00004X\ni=1\u001dimi_\u001ei\n\ri\u0013\n(F19)14 Article Title\n_\u001e1=u1\n\u000b1\u0001+D1\u0019\r1sin\u001e1\n2m1\u000b1\u0001\n\u0000Jex\r1sin\u001e1\u0000\u001e2\nm1\u000b1+_q\n\u000b1\u0001(F20)\n_\u001e2=\u0000u2\n\u000b2\u0001+Jex\r2sin\u001e2\u0000\u001e3\n\u000b2m2\n\u0000Jex\r2sin\u001e1\u0000\u001e2\n\u000b2m2+_q\n\u000b2\u0001(F21)\n_\u001e3=u3\n\u000b3\u0001\u0000Jex\r3sin\u001e3\u0000\u001e2\nm3\u000b3\n\u0000Jex\r3sin\u001e4\u0000\u001e3\n\u000b3m3+_q\n\u000b3\u0001(F22)\n_\u001e4=u4\n\u000b4\u0001\u0000Jex\r4sin\u001e4\u0000\u001e3\n\u000b4m4+_q\n\u000b4\u0001:(F23)\nWe extended the model described earlier by\nBlaesing in three ways. First, we implement\nthe thickness and temperature-dependence of the\nmagnetization based on the SQUID measurements\ndiscussed in section A, in order to account for the\ndi\u000berent thicknesses of the samples investigated in\nthe experiment. Second, the temperature of the\nsample, contrary to earlier work, is also taken as\na dynamic parameter since we \fnd that for the\nultrashort current pulses investigated in the main\ntext, the sample is still heating up rapidly at the\npeak of the current pulse, when the driving torque\nis largest. Finally, we trivially extended the mod-\neled magnetic system from two to four coupled\nmacrospins.\nIn order to arrive at the results presented in the\nmain text, we numerically solve these \fve coupled\ndi\u000berential equations. For this we use the tempo-\nral current density pro\fle Jas given in equation\nE8 and table E2, taking into account the exper-\nimentally observed threshold current density of\n0.5 TA/m2(see Sup. D). The magnetization and\nits temperature dependence are implemented as\ndescribed in section A. All other physical parame-\nters used in the model are summarized in table F3.\nIn order to \fnally characterize the velocity fromthe model, we divide the DW displacement by the\nsame e\u000bective pulse width of 0.8 ns used to cal-\nculate the experimental domain wall velocity (see\nSup. D).\nWe \fnally note that for the realistic mate-\nrial parameters obtained from literature, the 1-D\nmodel generally predicts lower DW velocities than\nthe experiments. However, an increase of the DW\nwidth from 8 to 12 nm (both reasonable for\nthese kind of ferrimagnetic systems [47, 48]) could\nalready address this discrepancy in velocity with-\nout impacting the qualitative dependence of the\nDW velocity on the Gd thickness and current\ndensity.\nAppendix G Velocity peak\nbroadening\ndescribed by\ndynamic\ntemperature\nTo further substantiate the claim that dynamic\nsimulation of the temperature is important to\nproperly interpret the CIDWM experiments in our\n[Co/Gd] 2multilayers, we make a more elaborate\ncomparison between simulation with a dynamic\ntemperature pro\fle, and a constant temperature.\nFor this we use the same simulations as pre-\nsented in other parts of this work, and the same\ntemperature pro\fle for the dynamic temperature\nsimulations. For the constant temperature simu-\nlations we take the temperature to be 85% of the\nmaximum temperature of the dynamic tempera-\nture pro\fle, as this gives rise to a similar shift of\nthe thickness at which the peak velocity occurs.\nA typical comparison of the modelled DW veloc-\nity between static and dynamic temperature can\nbe seen in Fig. G5 for peak current densities J0of\n1.6, 2.6 and 3.6 TA/m2. It can be seen that for low\ncurrent densities the resulting velocity pro\fles are\nalmost identical, whereas for high current densi-\nties the velocity is more sharply peaked around the\ncompensation thickness. The main e\u000bect at high\ncurrent density of the dynamic temperature pro-\n\fle is a simultaneous lowering of the peak velocity\nand a broadening of the velocity peak, both of\nwhich can be understood from the magnetic com-\nposition changing rapidly throughout the peak ofArticle Title 15\nTable F3 Parameters used in 1D-modeling of CIDWM in [Co/Gd] 2.\nParameter Unit Co 1 Co2 Gd1 Gd2\ng - 2.2(a)2.2(a)2.0(b)2.0(b)\n\r 1011rad/s/T 1.93(c)1.93(c)1.76(c)1.76(c)\n\u000b - 0.1(d)0.1(d)0.1(e)0.1(e)\n\u0001 nm 8(f)8(f)8(f)8(f)\nJex mJ/m2-0.9(g)-0.9(g)-0.9(g)-0.9(g)\n\f - -0.5(h)-0.5(h)-0.5(h)-0.5(h)\nP - 0.1(i)0.1(i)0.1(i)0.1(i)\n\u0012SH - 0.08(j)0(k)0(k)0(k)\nD pJ/m 0.25(l)0(m)0(m)0(m)\naFrom [42, 43].\nbFrom [44].\ncCalculated using the layer's respective Land\u0013 e g-factor as \r=g\u0016B\n~.\ndFrom [11].\neAssumed similar to that of Co [45].\nfSimilar to [4, 28]. Assumed constant throughout the layers due to strong\nexchange.\ngFrom [46].\nhEstimate obtained from [16].\niEstimate obtained from [16].\njEstimate obtained from [47, 48].\nkSpin current interaction with the 4f moments in Gd is very small [49, 50].\nlChosen such that the simulations \ft the _ qvs.Jexperimental data.\nmNo DMI has been reported at the Co/Gd interface.\na) b) c)\nFig. G5 Comparison of using dynamic and static temperature in the 1D model of CIDWM. The constant temperature is\nchosen to be 85% of the maximum temperature from the COMSOL simulations at a given peak current density, as it leads\nto a comparable shift of the velocity peak. a): 1D CIDWM simulations comparing the velocity for the two temperature\npro\fles at 1.6, 2.6 and 3.6 TA/m2. b): Normalized DW velocity for the 1D CIDWM simulation with the constant and\ndynamic temperature pro\fles, compared with the normalized velocities found from the experiments at peak current density\nJ0= 2:6 TA/m2. c): Same as b), but for J0= 3:6 TA/m2.\nthe current density. As the degree of compensa-\ntion changes, the resulting translation of the DW\nbecomes more of an average over many di\u000berent\nmagnetic compositions which have varying degrees\nof DW mobility.In the experiments a similar phenomenon can\nbe observed, and is qualitatively shown in Fig.\nG5.b and G5.c. Again, for the lower current den-\nsity (J0= 2:6 TA/m2, \fg. G5.b), the heating\ne\u000bect within the temporal width of the current\npulse is limited, and the velocity with thickness is16 Article Title\nwell described by both the constant and dynamic\ntemperature. A drastic change is observed how-\never for large current densities ( J0= 3:6 TA/m2,\n\fg. G5.c), where we \fnd that the dynamic tem-\nperature pro\fle describes the trend of the exper-\niment well. Contrarily the constant temperature\napproximation does not work anymore beyond\npotentially pinpointing the shift in the velocity\npeak.\nReferences\n[1] S. K. Kim, G. S. D. Beach, K.-J. Lee, T. Ono,\nT. Rasing, and H. Yang, \\Ferrimagnetic spin-\ntronics,\" Nature Materials , vol. 21, no. 1,\npp. 24{34, 2022.\n[2] B. Dieny, I. L. Prejbeanu, K. Garello,\nP. Gambardella, P. Freitas, R. Lehndor\u000b,\nW. Raberg, U. Ebels, S. O. Demokritov,\nJ. Akerman, A. Deac, P. Pirro, C. Adel-\nmann, A. Anane, A. V. Chumak, A. Hirohata,\nS. Mangin, S. O. Valenzuela, M. C. 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J.ˇSaf´ arik University, Park Angelinum 9, 041 54 Koˇ sice, Slov akia\n(Dated: July 16, 2018)\nAbstract\nMixed spin-1/2 and spin-1 Ising ferrimagnets on a triangula r lattice with sublattices A, B and\nC are studied for two spin value distributions ( SA,SB,SC) = (1/2,1/2,1) and (1 /2,1,1) by Monte\nCarlo simulations. The non-bipartite character of the latt ice induces geometrical frustration in\nboth systems, which leads to the critical behavior rather di fferent from their ferromagnetic coun-\nterparts. We confirm second-order phase transitions belong ing to the standard Ising universality\nclass occurring at higher temperatures, however, in both mo dels these change at tricritical points\n(TCP) to first-order transitions at lower temperatures. In t he model (1 /2,1/2,1), TCP occurs on\nthe boundary between paramagnetic and ferrimagnetic ( ±1/2,±1/2,∓1) phases. The boundary\nbetween two ferrimagnetic phases ( ±1/2,±1/2,∓1) and (±1/2,∓1/2,0) at lower temperatures is\nalways first order and it is joined by a line of second-order ph ase transitions between the paramag-\nnetic and the ferrimagnetic ( ±1/2,∓1/2,0) phases at a critical endpoint. The tricritical behavior\nis also confirmed in the model (1 /2,1,1) on the boundary between the paramagnetic and ferrimag-\nnetic (0,±1,∓1) phases.\nPACS numbers: 05.50.+q, 64.60.De, 75.10.Hk, 75.30.Kz, 75.50.Gg\nKeywords: Mixed-spin system, Frustrated Ising ferrimagnet, Tr iangular lattice, Monte Carlo simulation,\nTricritical point, Critical endpoint\n1I. INTRODUCTION\nMixed-spin Ising systems have been mostly investigated as possible m odels of some\ntypesofferrimagneticandmolecular-basedmagneticmaterials. Th eusedapproachesinclude\nanexacttreatmentinspecialcases1–5,mean-fieldapproximation6,7,effective-fieldtheorywith\ncorrelations8–14, Monte Carlo simulations15–22and some other methods23–28. The main focus\nwere their phase diagrams as well as technologically interesting comp ensation behavior with\npossibility to achieve zero total magnetization by tuning of tempera ture below the critical\npoint. Most of the studies considered the simplest models consisting of two sublattices one\nof which is occupied with spins S= 1/2 and the other with S= 1. Such a mixed-spin model\ncan be described by the Hamiltonian\nH=−J/summationdisplay\n/angbracketlefti,j/angbracketrightσiSj−D/summationdisplay\njS2\nj, (1)\nwhereσi=±1/2 andSj=±1,0 are spins on different sublattices, ∝angbracketlefti,j∝angbracketrightdenotes the sum\nover nearest neighbors, J <0 is a antiferromagnetic exchange interaction parameter and D\nis a single-ion anisotropy parameter. Negative values of the parame terDfavor nonmagnetic\nstates with Sj= 0 and positive values magnetic states with Sj=±1.\nDue to persisting ambiguities majority of the investigations focused on the simplest lat-\ntices, i.e., the square in two and cubic in three dimensions. We note tha t a long standing\ncontroversy regarding the critical and compensation behaviors e ven for the most studied\ncase of the model on a square lattice was solved only recently by Mon te Carlo simulation\nthat has convincingly shown22that there are neither tricritical nor compensation points, as\nhad been suggested by some previous approximative approaches6,8,9,28. On the other hand,\nin the same study the presence of both the tricritical point and a line of compensation\npoints was confirmed in the three-dimensional model on a simple cubic lattice. This finding\nmight suggest that the increased dimensionality is responsible for th e appearance of the tri-\ncritical and compensation behaviors. Nevertheless, our recent s tudy on a triangular lattice\nferromagnet29demonstrated that the tricritical point can also appear in a two-dim ensional\nlattice as long as the coordination number is sufficiently high. The effec t of the coordination\nnumber in the presence of bond disorder on tricritical behavior of a two dimensional system\nwas also recently studied in a random Blume-Capel model on a triangu lar lattice30.\nWe point out that the previous studies were performed on bipartite lattices, in which\n2A BC(a)\nAC(b)\nB\nFIG. 1: (Color online) Mixed-spin S= (SA,SB,SC) models on a triangular lattice consisting of\nsublattices A, B and C, with (a) S= (1/2,1/2,1) mixing - model I and (b) S= (1/2,1,1) mixing\n- model II. Small and large circles denote spin-1/2 and spin- 1 sites, respectively.\ncase the sign of the exchange interaction is irrelevant to the therm odynamic and critical\nproperties of the model in the absence of an external field. On the other hand, the present\nmixed-spin model is considered on a non-bipartite triangular lattice, in which case the sign\nof the exchange interaction matters. Namely, in contrast to the f erromagnetic case, the\nferrimagnetic interaction will induce geometrical frustration, whic h can be expected to have\nsome impact on the critical behavior. As shown in Fig. 1, the lattice co nsists of three\nsublattices A, B and C, occupied with spins S= (SA,SB,SC). This allows to further study\nthe model in two different mixing modes. We can consider a mixed-spin S= (1/2,1/2,1)\nmodel I, as schematically depicted in Fig. 1(a), in which one sublattice is occupied with\nspinS= 1 sites and the remaining two sublattices with spin S= 1/2 sites. Thus, each\nspin-1 site is surrounded by z= 6 nearest neighbors with spin S= 1/2. The other way\nof the spin-mixing is realized in a S= (1/2,1,1) model II, shown in Fig. 1(b), which is\nobtained when the spin-1/2 and spin-1 sites in the model I are swapp ed. The two models\nwere shown to display qualitatively different critical behaviors even f or the ferromagnetic\nexchange interactions29.\nThe goal of the present study is to examine effects of the geometr ical frustration on the\ncritical behavior of the above defined ferrimagnetic mixed-spin sys tems, to determine their\nphase diagrams and to confront them with their ferromagnetic cou nterparts as well as the\npure spin-1/2 and spin-1 antiferromagnetic systems.\n3100102104106−1−0.8−0.6−0.4−0.200.20.40.60.81\nMCSe/|J|                                  mO\n  \nmodel II: ms3, kBT/|J|=1.0model I: ms1, kBT/|J|=0.7\nmodel I: e/|J|, kBT/|J|=0.7\nmodel II: e/|J|, kBT/|J|=1.0\nFIG. 2: (Color online) Time evolutions of the internal energ y per site e/|J|and the staggered mag-\nnetizations ms1,ms3(see definitions below), starting from thedisordered phase at the temperatures\nkBT/|J|= 0.7 in model I and 1.0 in model II, for D/|J|= 0 and L= 120.\nII. MONTE CARLO SIMULATION\nIn order to study the behavior of various thermodynamic quantitie s in the parameter\nspace and to determine the phase diagrams we use Monte Carlo (MC) simulations with the\nMetropolis update rule and employ the periodic boundary conditions. We consider lattices\nwith the size L×L, withLranging from 24 up to 120. We perform N= 2×105up\nto 106MCS (Monte Carlo sweeps), the first 20% of which are used to bring t he system\nto equilibrium and then discarded, and the remaining data are used to estimate thermal\naverages and statistical errors. In order to demonstrate that the used MCS is sufficient to\nensure equilibrium conditions, in Fig. 2 we present MC time evolutions of some relevant\nquantities, such as the order parameters ms1(model I) and ms3(model II) and the internal\nenergy per site e/|J|, starting from the disordered phase. The simulations are perform ed\nin both models for the largest of the considered lattice sizes L= 120, which is the most\ndifficult to equilibrate, at the temperatures in the vicinity of the resp ective critical points\nforD/|J|= 0. The plots show that in these cases the equilibrium is reached in less than\n104MCS.\nThe phase boundaries are roughly determined from the maxima of so me thermodynamic\nfunctions, such as the specific heat, for a selected fixed value of L. We chose L= 48, as\n4a compromise value above which the specific heat maxima positions do n ot change con-\nsiderably and the phase diagrams can be determined in a relatively wide parameter space\nin a reasonable computational time. In the region where the critical line as a function of\nthe single-ion anisotropy parameter Dis more or less horizontal it is convenient to obtain\ntemperature dependencies of the calculated quantities at a fixed v alue ofD. In such a\ncase simulations start from the paramagnetic phase using random in itial configurations with\nthe temperature gradually decreased and a new simulation starting from the final config-\nuration obtained at the previous temperature. On the other hand , if the phase boundary\nshape changes to vertical we obtain variations of the quantities as functions of the single-ion\nanisotropy parameter Dat a fixed temperature. Then simulations start from appropriately\nchosen states (i.e., not necessarily random), expected in the cons idered region of the param-\neter space. Following the above described approach we ensure tha t the system is maintained\nclose to the equilibrium in the entire range of the changing parameter and thus considerably\nshortens thermalization periods. In order to estimate statistical errors, we perform three\nindependent simulations at all considered parameter values.\nAt some selected points of the phase boundaries we perform a more thorough finite-\nsize scaling (FSS) analysis in order to determine more precisely the loc ation of the critical\npoints and the corresponding critical exponents. In such a case w e perform more extensive\nsimulations using up to N= 107MCS and apply the reweighing techniques31. For more\nreliableestimationofstatistical errors, inthiscase weusedtheΓ-m ethod32. Having obtained\nthe maxima of the relevant quantities, we apply the linear fitting proc edure for logarithms of\ndata with errors, following the method in York et al.33. In order to assess the quality of the\nfitting, asameasureofgoodnessoffitweevaluatedanadjustedc oefficient ofdetermination34\nof the linear fit R2. The critical points and the exponents are then extracted from t he FSS\nanalysis, using the linear sizes L= 24,48,72,96 and 120.\nWe calculate the following quantities: the internal energy per spin e=∝angbracketleftH∝angbracketright/L2, the\nrespective sublattice magnetizations per site mX, (X = A, B or C), as order parameters on\nthe respective sublattices, which for the model I are given by\nmA(B)= 3∝angbracketleft|MA(B)|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈A(B)σi/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (2)\nmC= 3∝angbracketleft|MC|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈CSi/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (3)\n5and for the model II by\nmA= 3∝angbracketleft|MA|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈Aσi/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (4)\nmB(C)= 3∝angbracketleft|MB(C)|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈B(C)Si/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (5)\nwhere∝angbracketleft···∝angbracketrightdenotes thermal average. Based on the ground-state consider ations (see below),\nfortheidentified orderedphaseswe additionallydefinethefollowingo rder parametersforthe\nentire system, which take values between 0 in the fully disordered an d 1 in the fully ordered\nphase. For the model I we introduce two order parameters (stag gered magnetizations per\nsite)ms1andms2given by\nms1=∝angbracketleft|Ms1|∝angbracketright/L2=/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle2/summationdisplay\ni∈Aσi+2/summationdisplay\nj∈Bσj−/summationdisplay\nk∈CSk/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2, (6)\nand\nms2=∝angbracketleft|Ms2|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ni∈Aσi−/summationdisplay\nj∈Bσj/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/L2. (7)\nFor the model II we define the order parameter ms3as\nms3=∝angbracketleft|Ms3|∝angbracketright/L2= 3/angbracketleftBig/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj∈BSj−/summationdisplay\nk∈CSk/vextendsingle/vextendsingle/vextendsingle/angbracketrightBig\n/2L2. (8)\nUnlike in ferrimagnetic systems on bipartite lattices, the three subla ttices in the present\nsystem facilitate spin arrangements in such a way that the total ne t magnetization is always\nzero and, therefore, of no practical use.\nFurther, we calculate the susceptibilities pertaining to the respect ive order parameters\nO=MX(X = A, B, C) and also O=Msi(i= 1,2 or 3)\nχO=∝angbracketleftO2∝angbracketright−∝angbracketleftO∝angbracketright2\nNOkBT, (9)\nthe specific heat per site c\nc=∝angbracketleftH2∝angbracketright−∝angbracketleftH∝angbracketright2\nNOkBT2, (10)\nwhereNOis the number of sites on the (sub)lattice on which Ois defined. Further, we\ndefine the logarithmic derivatives of ∝angbracketleftO∝angbracketrightand∝angbracketleftO2∝angbracketrightwith respect to β= 1/kBT,\nD1O=∂\n∂βln∝angbracketleftO∝angbracketright=∝angbracketleftOH∝angbracketright\n∝angbracketleftO∝angbracketright−∝angbracketleftH∝angbracketright, (11)\n6D2O=∂\n∂βln∝angbracketleftO2∝angbracketright=∝angbracketleftO2H∝angbracketright\n∝angbracketleftO2∝angbracketright−∝angbracketleftH∝angbracketright, (12)\nand finally the fourth-order Binder cumulant UOcorresponding to the quantity O\nUO= 1−∝angbracketleftO4∝angbracketright\n3∝angbracketleftO2∝angbracketright2. (13)\nThe above standard thermodynamic quantities (9-10), as well as t he less traditional ones\ndefined by Eqs. (11-13), serve to obtain estimates of the respec tive critical exponents by FSS\nanalysis35,36. In particular, we use the following scaling relations, applied to the ma ximum\nvalues of the following functions:\nχO,max(L)∝LγO/νO, (14)\ncmax(L)∝Lα/νO, (15)\nD1O,max(L)∝L1/νO, (16)\nD2O,max(L)∝L1/νO, (17)\nwhereαis the critical exponent of the specific heat and νO,γOare the critical exponents\nof the correlation length and susceptibility, respectively, pertainin g to the quantity O. In\ncase of a first-order phase transition, the above quantities (14- 17) are expected to scale as\n∝Ld, whered= 2 is the system dimension. The order parameter cumulant, defined by\nEq. (13), can also serve for a simple yet relatively precise location of the phase transition\npoint as a point at which the cumulant curves obtained for different s ystem sizes intersect\nat a universal value, e.g., UO(Tc) = 0.611 for a two-dimensional Ising model37.\nIII. RESULTS\nA. Ground state\nLet us first identify all the possible ground states (GS) for entire r ange of the single-ion\nanisotropy parameter D. Considering the lattice system consisting of three interpenetrat ing\nsublatticesA,BandC,asschematicallydepictedinFig.1, theHamilton iansoftherespective\nmodels I and II can be defined as\nHI=−J/parenleftBig/summationdisplay\ni∈A,j∈Bσiσj+/summationdisplay\ni∈A,k∈CσiSk+/summationdisplay\nj∈B,k∈CσjSk/parenrightBig\n−D/summationdisplay\nk∈CS2\nk, (18)\n7TABLE I: Ground state configurations and the respective ener gies for different ranges of the single-\nion anisotropy parameter.\nModel I II\nD/|J| State Energy e/|J|State Energy e/|J|\n(−∞,−3/2) FRI\n2: (±1/2,∓1/2,0) −1/4 P: (0 ,0,0) 0\n(−3/2,∞) FRI\n1: (±1/2,±1/2,∓1)−3/4−D/3|J|FRII: (0,±1,∓1)−1−2D/3|J|\nHII=−J/parenleftBig/summationdisplay\ni∈A,j∈BσiSj+/summationdisplay\ni∈A,k∈CσiSk+/summationdisplay\nj∈B,k∈CSjSk/parenrightBig\n−D/parenleftBig/summationdisplay\nj∈BS2\nj+/summationdisplay\nk∈CS2\nk/parenrightBig\n.(19)\nFocusing on a triangular elementary unit cell consisting of the spins SA,SB,SC, from\nthe Hamiltonians (18) and (19) one can obtain expressions for the r educed energies per\nspine/|J|of different spin arrangements as functions of D/|J|. Then the ground states\nare determined as configurations corresponding to the lowest ene rgies for different values of\nD/|J|, as tabulated in Table I. There are two long-range order (LRO) fer rimagnetic (FR)\nstates FRI\n1and FRI\n2in the model I and one LRO ferrimagnetic state FRIIand one disordered\nparamagnetic (P) phase in the model II. We note that while in FRI\n2zero means nonmagnetic\nstates (Sj= 0) of spins on sublattice C, in FRIIzero means magnetic states ( σi=±1/2) of\nspins on sublattice A but equally in states +1 /2 and−1/2, thus giving zero net sublattice\nmagnetization. The critical value of the single-ion anisotropy param eter separating the\nrespective phases is the same for both models Dc/|J|=−3/2.\nB. Monte Carlo\n1. Model I: S= (1/2,1/2,1)\nPhase boundaries between the disordered paramagnetic and the r espective ordered ferri-\nmagnetic phases are determined from the specific heat maxima for a fixedL= 4848and the\nnature of the ordered phases is established from the introduced o rder parameters ms1and\nms2. TheseareshowninFigs.3(a)and3(b)forselectedvaluesofthep arameter D/|J|below,\nclose to and above the critical value Dc/|J|. All the specific heat curves show pronounced\nsharp peaks, signifying phase transitions to the low-temperature ferrimagnetic states. How-\never, the (pseudo)transition temperatures appear to be a nonm onotonic functions of D/|J|.\n80.20.30.40.50.60.70.80.900.511.522.5\n  \nkBT/|J|c(a)\nD/|J|=0 D/|J|=−2D/|J|=−1.48\n0.20.30.40.50.60.70.80.900.10.20.30.40.50.60.70.80.91\nkBT/|J|ms1, ms2\n  \n(b)\nms1, D/|J|=0\nms2, D/|J|=0\nms1, D/|J|=−2ms2, D/|J|=−2ms1, D/|J|=−1.48\nms2, D/|J|=−1.48\nFIG. 3: (Color online) Temperature dependencies of (a) the s pecific heat and (b) the order param-\neters, for selected values of D/|J|and a fixed lattice size L= 48. In (a) additionally the results\nforL= 72 are shown (squares).\nMore specifically, the transition temperature for D/|J|=−1.48, i.e., close to the critical\nvalueDc/|J|, is lower than for the other two values below and above Dc/|J|. Moreover, the\ncorresponding specific heat maximum has a spike-like shape and its ma gnitude is one order\nhigher then the other maxima, which is typical for a first-order pha se transition. In order\nto support our claim, that the peaks’ positions do not significantly c hange above L= 48,\nin Fig. 3(a) we also included the results obtained for L= 72. This will also become evi-\ndent later on in the respective phase diagrams in which for some selec ted points the phase\ntransition temperatures estimated for L= 48 will be compared with those determined for L\nextrapolated to infinity. The order parameters depicted in Fig. 3(b ) demonstrate that the\ntransition for D/|J|=−2 is to the state ( ±1/2,∓1/2,0), characterized by a finite values\nofms2, while for D/|J|=−1.48 and 0 the system tends to the state ( ±1/2,±1/2,∓1),\ncharacterized by a finite values of ms1. The discontinuous behavior of the order parameter\nforD/|J|=−1.48 corroborates the first-order nature of the transition.\nBy FSS analysis at two representative values of D/|J|= 0 and −2, selected on either\nside of the critical value Dc/|J|=−3/2, we confirmed that the disorder-to-order phase\ntransitions to both ( ±1/2,±1/2,∓1) and (±1/2,∓1/2,0) ferrimagnetic phases are indeed\nsecond order and belong to the standard Ising universality class. T he slopes of the fitted\n93 3.5 4 4.5 522.533.544.555.566.5\nln(L)  \n γM\ns1/νM\ns1 = 1.756 ± 0.006\n 1/νM\ns1 = 0.989 ± 0.011\n 1/νM\ns1 = 0.988 ± 0.013ln(χM\ns1,max), ln(D1M\ns1,max), ln(D2M\ns1,max)ln(χM\ns1,max)\nln(D1M\ns1,max)\nln(D2M\ns1,max)(a)\n3 3.5 4 4.5 51234567\nln(L)ln(χM\ns2,max), ln(D1M\ns2,max), ln(D2M\ns2,max)\n  \n γM\ns2/νM\ns2=1.753 ± 0.003\n 1/νM\ns2 = 0.982 ± 0.008 1/νM\ns2 = 0.985 ± 0.008ln(χM\ns2,max)\nln(D1M\ns2,max)\nln(D2M\ns2,max)(b)\n0.745 0.75 0.7550.520.540.560.580.60.620.64\nkBT/|J|UM\ns1\n  \nL=24\nL=48\nL=72\nL=96\nL=120(c)\n0.362 0.364 0.366 0.368 0.370.3720.450.50.550.60.65UM\ns2\nkBT/|J|  \nL=24\nL=48\nL=72\nL=96\nL=120(d)\n−1−0.5 00.511.522.50.20.250.30.350.40.450.50.550.60.65\nL1/ν\nM\ns1(T−Tc)/TcUM\ns1\n  \nL=24\nL=48\nL=72\nL=96\nL=120(e)\n−1−0.5 00.511.522.50.20.250.30.350.40.450.50.550.60.65\nL1/ν\nM\ns2(T−Tc)/TcUM\ns2\n  \nL=24\nL=48\nL=72\nL=96\nL=120(f)\nFIG. 4: (Color online) FSS analysis of the critical exponent s ratios 1 /νOandγO/νO(a,b), the\nfourth-order cumulant UOtemperature dependencies for different L(c,d) and the UOdata collapse\nanalysis (e,f), for O=Ms1atD/|J|= 0 (left column) and for O=Ms2atD/|J|=−2 (right\ncolumn). The coefficients of determination R2for the respective fits (from top to bottom) are\n0.9999, 0.9998, 0.9997 in (a) and 0.9999, 1.0000, 1.0000 in ( b).\n10−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.100.20.40.60.81\nD/|J|ms1\n  \nD/|J| down\nD/|J| upkBT/|J|=0.150.20.250.30.35\nFIG. 5: (Color online) Order parameter ms1as a function of the increasing ( ⊲) and decreasing ( ⊳)\nsingle-ion anisotropy parameter D/|J|at various temperatures and L= 48. The double-headed\narrows mark the hysteresis widths.\ncurves in Figs. 4(a) and 4(b) represent ratios of the critical expo nents 1/νOandγO/νO,\nfollowing from the scaling relations (14)-(17), where O=Msi(i= 1,2). We also checked\nthat for both D/|J|= 0 and −2 the specific heat maxima follow the logarithmic scaling\ncmax=c0+c1ln(L), as expected for the Ising universality class in two dimensions (not\nshown). Having performed the FSS analysis, these two critical poin ts can be determined\nwith a higher accuracy from the Binder cumulant crossing method38, as an intersection of\nthe Binder parameter UOcurves for different lattice sizes LandO=Msi(i= 1,2) (see\nFigs.4(c)and4(d)). Thecriticaltemperaturesweredetermined askBTc/|J|= 0.7485±0.001\natD/|J|= 0 and kBTc/|J|= 0.3663±0.001 atD/|J|=−2. Also the critical values of\nthe Binder cumulants UO(Tc) = 0.61137confirm the Ising universality class. Furthermore, in\nFigs. 4(e),4(f)we show thatfor thecritical temperatures deter mined by theBinder cumulant\ncrossing method the data for different lattice sizes indeed collapse o n a single curve. The\napparent first-order character of the phase transition at D/|J|=−1.48 will be discussed\nbelow.\nLet us now examine the transition between the two low-temperatur e ferrimagnetic phases\nFRI\n1and FRI\n2. Since the expected phase boundary is almost vertical to the x-ax is, instead\nof the temperature dependencies of various thermodynamic func tions it is more convenient\nto look into their single-ion parameter dependencies at a fixed tempe rature. By plotting\n11−0.25 −0.2 −0.15 −0.1 −0.0500.511.522.533.544.55x 105\ne/|J|  \nL=24\nL=48\nL=96\nL=120(a)\n3 3.5 4 4.5 5123456789\nln(L)ln(cmax)                ln( χM\ns1,max)\n  \nD/|J|=−1.47\nD/|J|=−1.47D/|J|=−1.48, αt/νt=1.60D/|J|=−1.4825, α/ν=2D/|J|=−1.48, γt/νt=1.85(b)\nD/|J|=−1.4825, γ/ν=2\n3 3.5 4 4.5 51234567\nln(L)  \n γM\ns1/νM\ns1 = 1.744 ± 0.008\n 1/νM\ns1 = 0.996 ± 0.009\n 1/νM\ns1 = 0.999 ± 0.007ln(χM\ns1,max), ln(D1M\ns1,max), ln(D2M\ns1,max)ln(χM\ns1,max)\nln(D1M\ns1,max)\nln(D2M\ns1,max)(c)\nFIG. 6: (Color online) (a) Energy distributions for D/|J|=−1.47 and different L. The respective\ntemperatures are tuned by the reweighing technique to achie ve approximately equal peak heights.\n(b) FSS analysis of the susceptibility χMs1and the specific heat c, forD/|J|=−1.47,−1.48 and\n−1.4825. For D/|J|=−1.48and−1.4825thelog-log plotsarerespectivelyfittedtotheexactva lues\nof the tricritical exponents ratios γt/νt= 1.85,αt/νt= 1.60 and the exponent 2, corresponding\nto the system volume. (c) FSS analysis of χMs1,D1Ms1,D2Ms1forD/|J|=−1.46, with the\ncoefficients of determination R2for the respective fits (from top to bottom) 0.9999, 1.0000, 0 .9999.\nthe order parameters as increasing and decreasing functions of D/|J|one can observe their\ndiscontinuous character and the appearance of hysteresis loops , the widths of which increase\nwith decreasing temperature. This behavior is demonstrated in Fig. 5 for the order parame-\nterms1andL= 48. Such behaviorsignalsfirst-orderphasetransitions. Theyse em topersist\n124 4.5 5 5.5 6−0.500.51\n105 MCSsublattice magnetizations\n  \nm1\nm2\nm3(a)\n3.5 4 4.5 5 5.5−1−0.8−0.6−0.4−0.200.20.40.60.81\n105 MCSsublattice magnetizations\n  \nm1\nm2\nm3(b)\nFIG. 7: (Color online) Time evolution of the sublattice magn etizations at the points (a)\n(D/|J|,kBT/|J|) = (−1.487,0.33) and (b) ( −1.47,0.35), forL= 48.\neven to higher temperatures at which the hysteretic behavior in no t apparent any longer.\nThe highest temperatures at which we still could observe some signs of first-order phase\ntransitions, such as bimodal energy distribution, was kBT/|J|= 0.35 and the corresponding\nvalue ofD/|J|=−1.47 (see Fig. 6(a)).\nNevertheless, the energy barrier separating the two peaks upon initial increase seems\nto decrease for larger L, which indicate that the phase transition may not be truly first\norder. Indeed, when we checked whether the specific heat and su sceptibility scale with the\nsystem volume, as it should be in case of a first-order transition, we found that within\nthe used lattice sizes the linear ansatz could not be established for D/|J|=−1.47, as\nshown in Fig. 6(b). We note that such a behavior that can lead to misin terpretation of a\nsecond-order transition as first order was also observed in the Blu me-Capel model as well\nas in the frustrated J1−J2Ising antiferromagnet on a square lattice39,40. Clearly first-order\nscaling is observed only at D/|J|=−1.4825. Nevertheless, fairly good linear fits were also\nachieved in the case of D/|J|=−1.48 with the exponents that are between 2 and the exact\ntricritical values (1.85 for γ/νand 1.6 for α/ν)41,42, suggesting that the system is close to\nthe tricritical point. On the other hand, for D/|J|=−1.46, the FSS presented in Fig. 6(c)\nclearly indicates a second-order phase transition. Thus, we rough ly estimate the tricritical\npoint at ( Dt/|J|,kBTt/|J|) = (−1.47±0.01,0.35±0.01).\nIt is interesting to notice that the phase transition at this point is no longer between\n13−3 −2 −1 0 1 200.10.20.30.40.50.60.70.80.9\nD/|J|kBTc/|J|P\nCEFR1I\nFR2ITCP\nfirst order\nFIG. 8: (Color online) Phase diagram of the model I in ( kBT/|J|−D/|J|) parameter space. The\nempty circles represent thephasetransition temperatures kBTc/|J|between theparamagnetic state\nP and the ferrimagnetic states FRI\n1(±1/2,±1/2,∓1) and FRI\n2(±1/2,∓1/2,0), estimated from the\nspecific heat peaks for L= 48, the empty triangles mark the hysteresis widths at first- order\ntransitions between the phases FRI\n1and FRI\n2with the expected phase transition boundary marked\nby the dash-dot line. The filled symbols at finite temperature s show more precise values obtained\nfrom the FSS analysis and the Binder cumulant crossing, wher e the diamond is the tricritical point\n(TCP), the hexagon is the critical endpoint (CE) and the squa re at (Dc/|J|,kBTc/|J|) = (−3/2,0)\nrepresents the exact value of the GS transition point.\nthe two ferrimagnetic phases FRI\n1and FRI\n2but between the phase FRI\n1and the param-\nagnetic phase. This is in line with the above observation of the first-o rder-like features\nof the phase transition at D/|J|=−1.4825 between the paramagnetic and FRI\n1phases.\nThe fact that there are first-order phase transitions between F RI\n1and FRI\n2as well as FRI\n1\nand paramagnetic phases can be verified by looking at the relevant o rder parameters. For\ndemonstration, in Figs. 7 we show segments of sublattice magnetiza tion time series ob-\ntained at ( D/|J|,kBT/|J|) = (−1.487,0.33) (Fig. 7(a)) and ( −1.47,0.35) (Fig. 7(b)). In\nboth cases we can see discontinuous switching between two phases , however, in either\npoint those phases are different. Namely, at ( −1.487,0.33) the sublattice magnetizations\n(mA,mB,mC) switch between the values characteristic for the states ( ±1/2,∓1/2,0)\nand (±1/2,±1/2,∓1), while at ( −1.47,0.35) they switch between the values characteris-\n14tic for the states (0 ,0,0) and (±1/2,±1/2,∓1). This finding implies the existence of a\ncritical endpoint (CE) at which the second-order transition bound ary between the para-\nmagnetic and FRI\n2phases joins the above discussed first-order transition line at abo ut\n(Dce/|J|,kBTce/|J|) = (−1.485±0.005,0.335±0.005).\nThe resulting phase diagram is presented in Fig. 8. The empty circles r epresent the\npseudo-critical points determined from the specific heat maxima at L= 48 and the empty\ntriangles mark the metastable branches of the first-order phase transitions obtained from\nthe order parameter hysteresis loops. The filled circles and square s at finite temperatures\nrepresent respectively second- and first-order transition point s obtained from the FSS anal-\nysis and the filled diamond marks approximate location of the tricritica l point. The filled\nsquare at ( Dc/|J|,kBTc/|J|) = (−3/2,0)shows the exact locations ofthe ground-statephase\ntransition. The expected first-order phase transition boundary , marked by the dash-dot line,\nis obtained by a simple linear interpolation between the estimated tricr itical and the exact\nGS transition points and only serves as a guide to the eye. We note th at in this highly sup-\npressed mixed-phase region the first-order phase boundaries ca n be located quite precisely,\nfor example, by multicanonical MC simulations43,44. However, in our case, we are only in-\nterested in approximate location of the phase boundaries. Then, c onsidering the exact value\nof the GS transition point, the estimate location of the tricritical po int and assuming no\nanomalous behavior, such as reentrance, the low-temperature p art of the phase boundary\nmust be practically vertical to the D/|J|axis.\n2. Model II: S= (1/2,1,1)\nAlso for the model II the phase boundary as a function of the single -ion anisotropy\nparameter D/|J|is estimated from the specific heat maxima for L= 48, except for D/|J|\nclose to the critical value of Dc/|J|=−3/2, where the boundary becomes almost vertical.\nThe only identified LRO phase is the ferrimagnetic phase FRII: (±1/2,±1,∓1), present\nforD > D c, and the phase transition is second order complying with the standa rd Ising\nuniversality class. This is illustrated in Fig. 9(a) for a selected value of D/|J|= 0, in which\nwe show that the obtained ratios of the critical exponents 1 /νMs3andγMs3/νMs3are in a\ngood agreement with the 2D Ising universality class values 1 and 7 /4, respectively. We also\nchecked the consistency of the specific heat critical exponent va lueα= 0 by verifying the\n153 3.5 4 4.5 522.533.544.555.5\nln(L)ln(χM\ns3,max), ln(D1M\ns3,max), ln(D2M\ns3,max)\n  \nln(χM\ns3,max)\nln(D1M\ns3,max)\nln(D2M\ns3,max)(a)\nγM\ns3/νM\ns3 = 1.757 ± 0.006\n1/νM\ns3 = 1.000 ± 0.013\n1/νM\ns3 = 1.001 ± 0.015\n1.055 1.061.065 1.071.075 1.080.580.60.620.640.660.68\nkBT/|J|UM\ns3\n  \n0 0.02 0.041.061.081.11.121.141.161.18\nL−1  \nkBTmax,χ\nM\ns3/|J|\nkBTmax,D1\nM\ns3/|J|\nkBTmax,D2\nM\ns3/|J|\nL=24\nL=48\nL=72\nL=96\nL=120(b)\n−1−0.500.511.520.20.250.30.350.40.450.50.550.60.65\nL1/ν\nM\ns3(T−Tc)/TcUM\ns3\n  \nL=24\nL=48\nL=72\nL−96\nL=120(c)\nFIG. 9: (Color online) (a) FSS analysis of the critical expon ent ratios 1 /νMs3andγMs3/νMs3, (b)\nthe fourth-order cumulant UMs3temperature dependencies for different L, and (c) the UMs3data\ncollapse analysis at D/|J|= 0. In (a) the coefficients of determination R2for the respective fits\n(from top to bottom) are 0.9998, 0.9999 and 0.9999. Inset in ( b) shows an alternative way of the\ncritical temperature estimation from the FSS analysis.\nlogarithmic scaling (not shown). The critical temperature for D/|J|= 0 estimated by the\nBinder cumulant method (Fig. 9(b)) and FSS analysis (inset in Fig. 9(b )) takes the value\nkBTc/|J|= 1.0635±0.0015 and the cumulant curves for different Lintersect at the universal\nvalue ofUMs3(Tc) = 0.611.\nOn approach to the critical value Dc/|J|=−3/2 the phase boundary rapidly drops and\n16−1.6 −1.55 −1.5 −1.45 −1.4 −1.3500.10.20.30.40.50.60.70.80.91\nD/|J|ms3kBT/|J|=0.3\nkBT/|J|=0.05kBT/|J|=0.2\nFIG. 10: (Color online) Order parameter ms3as a function of the increasing ( ⊲) and decreasing\n(⊳) single-ion anisotropy parameter D/|J|at various temperatures and L= 48.\nbecomes almost vertical. Therefore, in order to locate the critical temperatures in this re-\ngion, it is more convenient to measure the physical quantities at a fix ed temperature as\nfunctions of the parameter D/|J|. At sufficiently low temperatures the measured quantities\nshow some properties typical for first-order phase transitions. Namely, as the anisotropy\nparameter D/|J|is decreased and increased at the fixed temperature the sublattic e mag-\nnetizations, the order parameter ms3and the internal energy show discontinuities at some\nvalues of D/|J|, as demonstrated in Fig. 10 for ms3. Nevertheless, it is interesting to notice\nthat, in contrast to the strongly hysteretic behavior of the mode l I at the transition between\nthe two ferrimagnetic phases, no apparent hysteresis can be obs erved in the present model.\nDiscontinuous character of the transition reflected in bimodality of the relevant observables,\nsuch as the internal energy, disappears at higher temperatures but is still evident at temper-\natures slightly above kBT/|J|= 0.2. In Fig. 11(a) it is demonstrated for D/|J|=−1.48 and\nthe temperatures kBT/|J| ≈0.215 tuned by the reweighing technique for each Lto achieve\ndistributions with the two modes of about the same heights. The inse t gives us some idea\nabout the characteristic tunneling times between the coexisting ph ases forL= 48. However,\nsimilar tothesituationinthemodelIpresented inFig.6, for D/|J|=−1.48thespecific heat\nand staggered susceptibility maxima do not scale with volume. Such a s caling is observed\nonly at slightly lower temperatures for D/|J|=−1.4825 (at least for L≥72), as shown in\nFig. 11(b). At higher temperatures the second-order phase tra nsition with standard Ising\n17−0.02−0.015 −0.01−0.005 00.005 0.0100.511.522.53x 105\ne/|J|  \nL=48\nL=72\nL=96\nL=1208 9−505x 10−3\n105 MCSe/|J|(a)\n3 3.5 4 4.5 5−20246810\nln(L)ln(cmax)             ln( χs3,max)\n  \nD/|J|=−1.47\nD/|J|=−1.48D/|J|=−1.47D/|J|=−1.48\nD/|J|=−1.4825, α/ν=2(b)\nD/|J|=−1.4825, γ/ν=2\n3 3.5 4 4.5 50123456\nln(L)ln(χM\ns3,max), ln(D1M\ns3,max), ln(D2M\ns3,max)\n  \nln(χM\ns3,max), γI/νI=1.75\nln(D1M\ns3,max), 1/νI=1.00ln(D2M\ns3,max), 1/νI=1.00(c)\nFIG. 11: (Color online) (a) Energy distributions for D/|J|=−1.48 and different L. The respective\ntemperatures are tuned by the reweighing technique to achie ve approximately equal peak heights.\nTime evolution of the internal energy shown in the inset demo nstrates tunneling between the\ncoexisting phases for L= 48. (b) FSS analysis of the susceptibility χMs3and the specific heat\nc, forD/|J|=−1.47,−1.48 and−1.4825. For D/|J|=−1.4825 the log-log plots are fitted to\nthe exponent 2, corresponding to the system volume. (c) FSS a nalysis of χMs3,D1Ms3,D2Ms3for\nD/|J|=−1.46withthefittedIsingvaluesofthecritical exponentsrati osγI/νI= 1.75,1/νI= 1.00.\ncritical exponents is recovered for D/|J|=−1.46, although again larger system sizes are\nrequired to reach the linear asymptotic regime (Fig. 11(c)). Thus, the tricritical point in\nthe model II is roughly located at ( Dt/|J|,kBTt/|J|) = (−1.47±0.01,0.27±0.04).\nThe above results can be summarized into the phase diagram shown in Fig. 12. As in\n18−1.5−1−0.500.511.5200.20.40.60.811.21.4\nD/|J|kBTc/|J|P\nTCPFRII\nfirst order\nFIG. 12: (Color online) Phase diagram of the model II in ( kBT/|J|−D/|J|) parameter space. The\nempty circles represent the phase transition temperatures kBTc/|J|between the paramagnetic P\nand the ferrimagnetic phase FRII(0,±1,∓1) estimated from the specific heat peaks for L= 48,\nthe filled circle shows a more precise value obtained from the FSS analysis and Binder cumulant\ncrossing, the filled diamond is the tricritical point and the filled squares represent a first-order\ntransition point determined from FSS analysis at D/|J|=−1.48 and the exact value of the GS\ntransition point at Dc/|J|=−3/2. The empty triangles mark the first-order transition relat ed\ndiscontinuities in the order-parameter ms3in theD/|J|increasing ( ⊲) and decreasing ( ⊳) processes.\nthe phase diagram of the model I, the empty circles represent the pseudo-critical points\ndetermined from the specific heat maxima at L= 48 and the empty triangles mark the first-\norder phase transitions located from the jumps in the order param eter loops obtained by\nincreasing and decreasing of the single-ion parameter D/|J|. The filled circle at D/|J|= 0,\nthe square at D/|J|=−1.48 and the filled diamond represent respectively second-order,\nfirst-order and tricritical points, obtained from the FSS analysis. As in Fig. 8, the filled\nsquare at ( Dc/|J|,kBTc/|J|) = (−3/2,0) represents the exact value of the ground-state\nphase transition point.\nIV. CONCLUSIONS\nWe have studied the mixed spin-1/2 and spin-1 Ising ferrimagnets on a triangular lattice\nwithsublattices A, BandC,intwo mixing modes: ( SA,SB,SC) = (1/2,1/2,1)(model I)and\n19(SA,SB,SC) = (1/2,1,1)(model II). The pure spin-1/2 and spin-1 Ising antiferromagne ts on\na triangular lattice show respectively no LRO45and partial LRO for some range of a single-\nion anisotropy parameter at low temperature with quasi-LRO of the Berezinskii-Kosterlitz-\nThouless type at higher temperatures46. In comparison with these models in the present\nferrimagnetic models the frustration is partially accommodated by d ifferent ferrimagnetic\nspin arrangements and thus their critical behavior is rather differe nt from the pure systems.\nOn the other hand, the net magnetization of both ferrimagnetic mo dels is always zero and\nthus they canshow no compensation points. Therefore, fromthis point ofview, the behavior\nof the present mixed-spin models is typical for antiferromagnets r ather than ferrimagnets.\nAs for the critical properties, the model I shows two ferrimagnet ic phases FRI\n1:\n(±1/2,±1/2,∓1) and FRI\n2: (±1/2,∓1/2,0), which may be compared with the ferromag-\nnetic case displaying two ferromagnetic phases ( ±1/2,±1/2,±1) and (±1/2,±1/2,0)29. We\nnote that on bipartite lattices the thermodynamic behavior of the s ystems with ferrimag-\nnetic and ferromagnetic interactions is the same and thus the phas e diagrams would be\nidentical. However, there are substantial differences between th e two phase diagrams for the\ntriangular lattice ferromagnetic and ferrimagnetic models. First of all, due to frustration\nthe transition temperatures for the ferrimagnetic case are signifi cantly reduced and collapse\nwith the ferromagnetic boundary only in the large negative D/|J|limit, when the partial\nfrustration inducing magnetic states on the C-sublattice are comp letely suppressed and the\ncritical temperature for either case tends to the exact spin-1/2 Ising value on a honeycomb\nlatticekBTc/|J|= 0.379747. The frustration is further increased close to the boundary be-\ntween the two ferrimagnetic phases FRI\n1and FRI\n2, which is reflected in the depression in the\norder-disorder phase boundary that is absent in the ferromagne tic model. Nevertheless, the\nfrustration did not seem to affect the standard Ising values of the critical exponents. We\nnote that besides the above presented points of D/|J|=−2 and 0, we also performed the\nFSS analysis in this region of an increased frustration at the P-FRI\n1branch of the phase\ndiagram for D/|J|=−1.6 (not shown) but did not find any deviation larger than statis-\ntical errors from the standard values. However, the most consp icuous difference from the\nferromagnetic case is the presence of the strongly discontinuous phase transition between\nthe ferrimagnetic phases ( ±1/2,±1/2,∓1) and (±1/2,∓1/2,0), which is completely absent\nbetween the ferromagnetic phases ( ±1/2,±1/2,±1) and (±1/2,±1/2,0). This can be ex-\nplained by the fact that in the ferrimagnetic case dramatic changes occur when the two\n20spin-1/2 sublattices A and B, forming a connected honeycomb back bone, switch their mag-\nnetizations between 1 /2 and−1/2 and the spin-1 sublattice C between magnetic ±1 and\nnonmagnetic 0 states. On the other hand, in the ferromagnetic ca se nothing happens in\nsublattices A and B and in sublattice C the change between magnetic ±1 and nonmagnetic\n0 states occurs only gradually in the isolated (mutually directly nonint eracting) spins.\nThe model II has been shown to display only one ordered ferrimagne tic state FRII:\n(0,±1,∓1) with no LRO on sublattice A and an antiferromagnetic LRO on the re maining\nsublattices B and C forming a honeycomb lattice. Similar phase occurs in the pure spin-\n1 triangular antiferromagnet for the single-ion anisotropy parame ter−3/2< D/|J|<0,\nhowever, it is destabilized for D/|J|>046. On the other hand, in the present model II\nthe increasing D/|J|stabilizes the ferrimagnetic phase FRIIand the critical temperature for\nD/|J| → ∞tends to kBTc/|J|= 1.518847, i.e., the exact value of the spin-1/2 Ising model\non a honeycomb lattice when the spin states ±1 are considered in the Hamiltonian instead\nof±1/2. 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Prog. Phys. 30, 615 (1967).\n48Hence, in fact these are only pseudo-critical points\n23"    },    {        "title": "2011.02001v1.An_attempt_to_simulate_laser_induced_all_optical_spin_switching_in_a_crystalline_ferrimagnet.pdf",        "content": "arXiv:2011.02001v1  [cond-mat.mtrl-sci]  3 Nov 2020An attempt to simulate laser-induced all-optical spin swit ching in a crystalline\nferrimagnet\nG. P. Zhang∗, Robert Meadows, and Antonio Tamayo\nDepartment of Physics, Indiana State University, Terre Hau te, IN 47809, USA†\nY. H. Bai\nOffice of Information Technology, Indiana State University, Terre Haute, Indiana 47809, USA\nThomas F. George\nDepartments of Chemistry & Biochemistry and Physics & Astro nomy\nUniversity of Missouri-St. Louis, St. Louis, MO 63121, USA\n(Dated: November 5, 2020)\nInterest in all-optical spin switching (AOS) is growing rap idly. The recent discovery of AOS in\nMn2RuGa provides a much needed clean case of crystalline ferrim agnets for theoretical simulations.\nHere, we attempt to simulate it using the state-of-the-art fi rst-principles method combined with the\nHeisenberg exchange model. We first compute the spin moments at two inequivalent manganese\nsites and then feed them into our model Hamiltonian. We emplo y an ultrafast laser pulse to switch\nthe spins. We find that there is a similar optimal laser field am plitude to switch spins. However, we\nfind that the exchange interaction has a significant effect on t he system switchability. Weakening\nthe exchange interaction could make the system unswitchabl e. This provides a crucial insight into\nthe switching mechanism in ferrimagnets.\nPACS numbers: 75.78.Jp, 75.40.Gb, 78.20.Ls, 75.70.-i\nCentral to the magnetic storage device is the writ-\ning/reading speed of magnetic bits in a storage medium.\nTraditionally, these operations are mostly driven by an\nexternal magnetic field. A full-optical driven spin manip-\nulation could break the speed barrier of several hundred\npicoseconds set by the Zeeman interaction and magnetic\ndipole-dipole interaction. In 1996, Beaurepaire et al.[1]\nshowed that when they shone a 60-fs pulse on the ferro-\nmagnetic nickel thin film, they found a sharp decrease in\nthe Kerr signal within 1 ps. This finding received imme-\ndiate attention worldwide, and a new research field, fem-\ntomagnetism, wasborn[2,3]. Researchintensified, andis\nfar beyond the scope of the original research interest. In\n2007, Stanciu et al.[4] showed that a left-circularly po-\nlarized laser pulse can switch an up-spin to down, while a\nright-circularly polarized laser pulse can switch a down-\nspinup. Thisremarkablepropertyrepresentsaninterest-\ningnew magnetic phenomenononan ultrafasttime scale,\nalthough their compound, GdFeCo, is not new. GdFeCo\nhas been used in traditional magneto-optical recording\n(see the references cited in [5, 6]). However, being to\nable to switch spins on a picosecond time scale optically\nis new, and has raised the possibility for a real applica-\ntion. However,itisunclearhowthelaserpulsecanswitch\nspins directly. Ostler et al.[7] further showed that if the\nlaser intensity is increased above a certain level, regard-\nless of laser helicity, each pulse can flip spins from one\ndirection to another deterministically. They argued that\n∗Author to whom correspondence should be addressed.\n†Electronic address: guo-ping.zhang@outlook.com.there is a threshold intensity that one has to exceed to\nchange from all-optical helicity dependent spin switch-\ning to all-optical helicity independent switching, but the\nactual picture is more complicated [8, 9].\nFor a long time, GdFeCo was the only material that\nshows AOS. Soon, many more materials were found [10–\n14]. However, these materials are mostly amorphous,\nwhich introduces an uncertainty in theoretical simula-\ntions and represents a formidable task. In 2017, Vomir\net al.[15] reported the first observation of AOS in a\nPt/Co/Pt ferromagnetic stack, but the switching is not\ncomplete. Very recently, Banerjee et al.[16] showed\nsingle-pulse all-optical toggle switching of magnetization\nin Mn2RuGa. Mn 2RuGa is a ferrimagnetic Heusler com-\npound, with a cubic structure. Two Mn atoms are not\nequivalent, and have different spin moments. They are\nantiferromagnetically coupled. As shown before [17], fer-\nrimagnets have a big advantage over ferromagnets and\nantiferromagnets. This offers an ideal theoretical model.\nIn this paper, we investigate all-optical switching in\nMn2RuGa. Different from prior studies, we compute the\nspin moments at two Mn sites using the first-principles\ndensity functional theory. These spin moments are fed\ninto the Heisenberg exchange model with both spin-orbit\ncoupling and a harmonic potential [18, 19]. We find that\nnot any arbitrary laser field amplitude can switch spins.\nThere is a narrowwindow of opportunity where the spins\nat two Mn sites can be switched into their respective op-\nposite directions. Because of the strong spin moments\nat two Mn sites, its switching is very stable. Quite dif-\nferent from other systems, we find that if we reduce the\nexchange interaction, the spins precess strongly at both\nMn sites, very much like a regular antiferromagnet, in-2\nstead of a ferrimagnet. These strong spin oscillation oc-\ncur at both Mn 1and Mn 2sites. Their oscillation period\nis inverselyproportionalto the laser field amplitude, sim-\nilar to the Rabi frequency in a two-level system. For a\nweak exchange interaction case, regardless of the magni-\ntude ofthe laserfield amplitude, the spin switchingis not\nobserved. This points out an entirely different scenario\nfrom ferromagnetic cases [19] and demagnetization [20].\nThe rich picture that is found here revealsa crucial effect\nof the effect of exchange interaction on AOS, and should\nmotivate further experimental and theoretical investiga-\ntions in the future.\nMn2RuGa is a Heusler ferrimagnet, with a stoichio-\nmetric composition of X2YZand space group F¯43m.\nTwo Mn atoms, Mn 1and Mn 2, are situated at (4 a)\nand (4c), which are magnetically inequivalent [21, 22].\nTheir spins are antiferromagnetically coupled. To prop-\nerlyinvestigatemagneticpropertiesofMn 2RuGa, weem-\nploythe density functional theoryusing the full-potentialaugmented plane wave method as implemented in the\nWien2k code [23, 24]. Our first-principles calculation\nshows that Mn 1has a spin moment of 3.17232 µBand\nMn2has -2.30765 µB. This agreeswith priorcalculations\n[21, 22]. So each cell has a net spin moment of 1.02394\nµB, close to unity, which is consistent with the nature of\na stoichiometric half-metal [25]. The spin moments on\nRu and Ga are very small, and will be ignored below.\nIt is not often recognized that the large spin mo-\nments on Mn atoms are advantageous since the spin-\norbit torque is proportional to the spin moment [18, 19].\nSinceitisnotpossibletosimulateall-opticalspinreversal\nat the first-principles level, in the following we will feed\nthese two spin moments into our Heisenberg-exchange\ncoupled harmonic model [17, 19, 26, 27], and limit our-\nselves to a small system with 101 lattice sites along the x\naxis and yaxis, respectively, with two monolayers along\nthezaxis. Our Hamiltonian is\nH=/summationdisplay\ni/bracketleftbiggp2\ni\n2m+V(ri)+λLi·Si−eE(r,t)·ri/bracketrightbigg\n−/summationdisplay\nijJexSi·Sj, (1)\nwhere terms from the left to right are respectively the\nkinetic energy operator of the electron, the potential en-\nergyoperator,the spin-orbitcoupling, the interactionbe-\ntween the laser and system, and the exchange interaction\nbetween spins. Our exchange parameter Jexis still time-\nindependent, although prior studies have shown that the\nexchange interaction itself could be affected by the elec-\ntricfield[28,29] .λisthe spin-orbitcouplingconstant, Li\nandSiare the orbital and spin angular momenta at site\ni, respectively, and pandrare the momentum and posi-\ntion operators of the electron, respectively. We choose a\nspherical harmonic potential V(ri) =1\n2mΩ2r2\niwith sys-\ntem frequency Ω. E(r,t) is the laser field. This model is\nthe only magnetic field-free model currently available to\nsimulate spin reversal, while the commonly used model\nemploys an effective magnetic field [7], which should be\navoided. It represents a small step towards a complete\nmodel.\nIn order to compute the spin change, we solve the\nHeisenberg equation of motion [20] for each spin oper-\nator at every site under laser excitation [17]. Figure 1(a)\nshows the spin zcomponent at two Mn sites as a func-\ntion of time. We employ a laser pulse of 60 fs, with a\nfield amplitude of 0.017 V /˚A. We see that the spin at\nthe Mn 1site starts from the positive zaxis (see the cir-\ncles). Upon laser excitation, it switches over the negative\nzaxis, while the spin at the Mn 2site switches up from\nits−zdirection. This is consistent with the experimen-\ntal observation [16]. The strong spin moment stabilizes\nthe entire switching process. However, not any arbitrarylaser field amplitude can lead to faithful switching. Fig-\nure 1(b) shows how the final spin changes with the laser\nfield amplitude. The dependence is highly nonlinear. If\nwe use a weak laser pulse, there is little change in spins\nat both sites. But if the laser field is too strong, the\nspins overturn toward the xyplane, so there is no spin\nreversal either. We find that the optimal field amplitude\nis 0.017 V /˚A, whose result is shown in Fig. 1(a). In this\nregard, Mn 2RuGa is pretty much similar to other ferri-\nmagnets where there is an optimal amplitude [17, 19].\nMicroscopically, the real situation is more complicated.\nTo this end, there is no generic understanding of spin\nswitching in both ferromagnets [14, 15] and ferrimagnets\n[4]. It has been often argued that the angular momen-\ntum exchange between two spin sublattices in ferrimag-\nnetic GdFeCo [30] is the key to AOS. Theoretically, this\nmomentum exchange picture is interesting, but such mo-\nmentum exchange between sublattices, if it exists, occurs\nall the time through the exchange interaction, with or\nwithout the laser. In other words, it must be something\nextra due to the laser that switches the spin. In GdFeCo,\narigoroustesting isdifficult because itis amorphous,and\nit is difficult to tell whether a model system really repre-\nsents a true GdFeCo sample. This brings ambiguity to a\ntheoretical simulation. Mn 2RuGa removes this ambigu-\nity completely.\nAs a first test, we investigate the effect of the exchange\ninteraction on AOS. We reduce the exchange interaction\nfrom 0.1 eV to 0.001 eV. From prior studies, we know\nsuch a reduction does not constitute a major issue for3\n0 0.01 0.02 0.03 0.04\nA0(V/Å)−2−1012Sz(h− )−400 −200 0 200 400 600Time (fs)\n−2−1012Sz(h− ) Mn1\nMn2(a)\n(b)\nFIG. 1: (a) The zcomponent ofthe spins at the Mn 1and Mn 2\nsites as a function of time at the optimal laser field amplitud e.\nHere, thelaser amplitudeis0.017 V /˚A, andthepulseduration\nis 60 fs. The empty circles denote the spin at the Mn 1site,\nwhile the boxes refer to the spin at the Mn 2site. We see there\nis a clear spin reversal upon laser excitation. (b) Dependen ce\nof theSzas a function of the laser field amplitude A0. The\nempty circles refer to the spin at Mn 1site, and the boxes refer\nto the spin at Mn 2site. A weak laser field does not reverse\nthe spins, but a too strong laser field can not either. There is\na narrow window that one can switch spins.\n0 0.010.020.030.04\nA0(V/Å)4006008001000Period (fs)\n0 0.010.020.030.04\nA0(V/Å)−1012Sz(h− )−500 0 500 1000 1500 2000Time (fs)\n−2−1012S(h− )Sx\nSy\nSz(a)\n(b) (c)\nFIG. 2: (a) Spin precession at Mn 1site under a reduced\nexchange interaction. Here we choose J= 0.001 eV. The rest\nof parameters are the same as those in Fig. 1. The solid,\ndotted, and dashed lines denote the x,y, andzcomponents\nof the spin, respectively. (b) The spin oscillation period d e-\ncreases with laser field amplitude A0. (c) At any of the laser\nfield amplitudes, spin reversal is not found. Only a strong\noscillation is noticed. The solid line is the time-average o f the\nspin, and the dotted and dashed lines refer to the maximum\nand minimum spin values, respectively. The effect of the ex-\nchange interaction on spin reversal is much more pronounced\nin Mn 2RuGa than in other materials.demagnetization in a system with a small spin moment\n[20]. Figure 2(a) shows the spin change at Mn 1(the sit-\nuation is similar at Mn 2), where the solid, dotted, and\ndashed lines denote the x,y, andzcomponents, respec-\ntively. Both the laser duration and amplitude are exactly\nthe same as those in Fig. 1. We see that there is a strong\noscillation in all these three components. We note in\npassing that these three components must obey the op-\nerator permutation [8], where they can not be considered\na linear reversal [31]. In principle, we need to cut off the\nsimulation around 1 ps, after which we need to introduce\ndamping, but to demonstrate the high accuracy of our\ncalculation, we do not use the damping. These strong\noscillations resemble a pure antiferromagnetic case. The\nspins at two neighboring sites are out of phase and re-\nmain antiferromagnetically coupled, even upon laser ex-\ncitation. The laser pulse essentially initiates the spin dy-\nnamics, and the exchange interaction takes over, without\nswitching the spins. For this reason, the angular momen-\ntum exchange picture for AOS can not explain this even\nin the same ferrimagnet. The period of the oscillation\nis not determined by the exchange interaction and spin\nmoment alone. Figure 2(b) shows that as we increase the\nlaser field amplitude, the period becomes shorter. The\nsmall fluctuation at the largest amplitudes is due to the\nperiodsamplingbecausetheoscillationisnotstrictlyhar-\nmonic. This laser-field dependence of the oscillation pe-\nriod is very similar to the Rabi period dependence. For\nall the field amplitudes that we investigate, we do not\nsee a case where the spins are reversed. Figure 2(c) illus-\ntrates the average (solid line), maximum (dotted line),\nand minimum (dashed line) of the final spin. We see the\naverage spin never becomes negative (the initial spin is\nalong the + zaxis). The maximum and minimum values\nshow the limits of spin. Our results point out an impor-\ntant fact: In a ferrimagnet, the effect of the exchange\ninteraction is far more complicated than thought.\nNow, we have two cases: One shows AOS, and the\nother does not. We can directly check whether the prior\ncriteria proposed by Mentink et al.[30] apply to them.\nTheir argument is based on a two-spin system, so for the\npure exchange interaction, the spins at two sublattices\nmust obey the scalar form of spins, ∂S1/∂t=−∂S2/∂t,\nwith the extra term from demagnetization. In our sys-\ntem, each spin is coupled with more than four neighbor-\ningspins, sowetaketwoneighboringspinsasanexample.\nFor the above nonswitchable case (Fig. 2), we find that\nS1x+S2xandS1y+S2yare not constant, so they do not\nobey∂S1/∂t=−∂S2/∂t. OurS1x+S2xdecreasesfrom 0\nto about −1¯hwith oscillations, while S1y+S2yincreases\nfrom 0 to about +1¯ hat the same rate. For our switch-\nable case (Fig. 1), ∂S1/∂t=−∂S2/∂tis not fulfilled\neither. Instead, we find that our result obeys the vector\nformS1(t)/|S1(0)|=−S2(t)/|S2(0)|. This shows that\nthe simple argument based on a two-spin model is not\napplicable to our realistic case. We plan to investigate\nthis issue further in a much larger system.\nIn conclusion, we have carried out a joint first-4\nprinciples density functional theory and model simula-\ntion of all-optical spin reversal in Mn 2RuGa. We are\nable to find a case that the spins can be switched with-\nout employing a magnetic field. The system also shows\nan optimal laser electric field amplitude, with the same\nprofile like those in other systems. The spins at Mn 1site\nare switched from the + zaxis to−zaxis, while those at\nMn2site are switched from the −zaxis to + zaxis. This\nisfullyconsistentwiththeexperimentalfindings[16]. We\nfind that the exchange interaction has a significant effect\non the switching. When we reduce the exchange to 0.001\neV, we find the system becomes unswitchable. It behaves\nlike a regular antiferromagnet. Our finding is expected\nto motivate further theoretical and experimental investi-\ngations in the future.\nAcknowledgments\nThis work was solely supported by the U.S. De-\npartment of Energy under Contract No. DE-FG02-06ER46304. Part of the work was done on Indiana\nState University’s Quantum Cluster and High Perfor-\nmance computers. This research used resources of the\nNational Energy Research Scientific Computing Center,\nwhich is supported by the Office of Science of the U.S.\nDepartment of Energy under Contract No. DE-AC02-\n05CH11231. Our calculations also used resources of the\nArgonne Leadership Computing Facility at Argonne Na-\ntional Laboratory, which is supported by the Office of\nScience of the U.S. Department of Energy under Con-\ntract No. DE-AC02-06CH11357.\nAvailability of data. 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The spin-orbit torque induced effective field is further quantified  in the domain wall \nmotion regime . A divergent behavior that scales with the inverse of magnetic moment is confirmed close \nto the compensation point, which is consistent with angular momentum conservation . Moreover , we also \nquantify the Dzyaloshinskii -Moriya interaction  energy in  the Ta/ Co1-xTbx system  and we find that  the \nenergy density increases as a function of the Tb concentration . The demonstrated spin -orbit torque \nswitching, in combinatio n with the fast magnetic dynamics and minimal net magnetization of \nferrimagnetic alloys, promises spintronic devices that are faster and with higher density than traditional \nferromagnetic systems.  \n  \n  2 \n I. Introduction  \n There has been great interest recently in using antiferromagnet ically coupled  materials as opposed \nto ferromagnet ic materials (FM) to store information . Compared with FM, antiferromagnet ically coupled  \nsystems exhibit  fast dynami cs, as well as immunities  against perturbation s from external magnetic field s, \npotentially ena bling  spintronic devices with higher speed and density  [1,2] . Rare earth (RE) – transition  \nmetal (TM) ferrimagnetic alloys are one potential candidate material for realizing such devic es. Inside \nRE-TM alloys, the moments o f TM elements (such as Fe, Co, Ni) and the RE elements (e.g., Gd, Tb , Ho, \netc) can be aligned with anti -paralle l orientations due to the exchange interaction between the f and d \nelectrons  [3]. By varying the relative concentrations of the two species, one can reach compensation \npoints where the net magnet ic moment or angular momentum goes  to zero  [4–7]. Moreover, because of \nthe different origins of magnetism in the two species, transport related properties are dominated by TM in \nthese alloys, providing a way to read out the magnetic state even in a compensated system. In this work, \nwe show that by uti lizing the current induced spin -orbit torque, one can switch magnetic moments in \nTa/Co 1-xTbx bilayer films. Particularly, we found that effective fields generated from the spin -orbit torque  \nscaled with the inverse of magnetization and reached maximum when the composition approaches  the \nmagnetic  compensation point. The large effective spin -orbit torque and the previously demonstrated fast \ndynamics  [8,9]  in these ferrimagnetic systems provide a prom ising platform for high speed spintronic \napplications.  \n \nII. Characterization  of Magnetic Properties  \n Spin-orbit torque  (SOT) originates from spin -orbit interaction  (SOI)  induced spin generation in \nthe bulk (i.e., the spin Hall effect)  [10,11] or the surface (i.e., the Rashba -Edelstein effect)  [12,13]  of solid \nmaterials. So far , SOTs have proven to be an efficient method of controlling the ferromagnetic state of \nnanoscale  devices  [14–17]. Recently, it was demonstrated that one can utilize SOT to  switch TM-\ndominant Co FeTb ferrimagnet s with perpendicular magnetic anisotropy (PMA)  [18]. It is therefore \ninteresting to ask what the relationship is between the chemical composition of RE -TM alloys and the 3 \n SOT effi ciency, and whether or not one can switch compensated ferrimagnets using SOT.  To answer \nthese questions, we grew a series of Ta(5)/Co 1-xTbx(t)/Ru(2 ) (thickness in nm) films using  magnetron \nsputtering. The Co1-xTbx alloys were deposited by co -sputtering Co and Tb sources with different \nsputtering powers. The concentration  of Tb x was calculated from the deposition rates  and varied between \n0.1 and 0.3 , while the layer thickness t ranges from 1.7 nm to 2.6 nm.  The magnetic properties of the \ndeposited films were examined using vibra ting sample magnetometry (VSM), and PMA was observed for \nall samples [Fig. 1 (a)].  Furthermore, the magnetic moment goes through zero and the coercive fields \nreach their maximum  around x ≈ 0.22, which is consistent with the room temperature magnetic moment \ncompensation point xcM  reported earlier  [6,7] . The dependence of the magnetic moment on the Tb \nconcentration is summarized in Fig. 1(c), which agrees well with t he trend  line calculated by assuming an \nanti-parallel alignment between the Co and Tb moment  (dashed lines).  The samples were then patterned \ninto Hall b ars with dimensions  4 x 44 μm2 [Fig. 1(d)]. The anomalous Hall resistance ( RAH) vs magnetic \nfield H curves measured from these samples are plotted in Fig. 1(b) and summarized in Fig. 1(e). The \npolarity of the hysteresis loops changes sign across xcM, consistent with previous studies where RAH is \nshown to be dominated by the momentum of the Co sublattice  [19,20] .  \n \nIII. Current -Induced Switching  \nFig. 2 (a) illustrates the current -induced magnetic switching for the series of samples. During \nthese measurements, in -plane magnetic fields of ±2000 Oe were applied in the current flowing direction \n[y axis in Fig. 1 (d )]. Previous studies showed that an in -plane field is necessary to ensure dete rministic \nmagnetic switching of  PMA films , as it can break the symmetry between two equivalent final states  [21–\n26]  .  As shown in Fig. 2(a ), the current -induced switching shows opposite p olarities under the positive \nand negative applied fields, consistent with the model  of SOT induced switching  [26]. Furthermore, under \nthe same in -plane field, the switching polarity changes sign as the samples go from being Co -dominant to \nTb-dominant . This phenomen on can be explained by considering a macro spin model as shown in Fig. \n2(b). In SOT switching,  the Slonczewski torque  [27] is proportional to 𝒎̂×(𝝈̂×𝒎̂), where 𝒎̂ is the unit 4 \n vector along the magnetic moment direction and 𝝈̂ is the orientation of electron spins generated from SOI  \n[along the 𝒙̂ direction in Fig. 1(d )]. Because the torque is an even  function of the local magnetic moment \n𝒎̂, effects from both sublattices in the RE -TM alloy add constructively  [1,28] . At equilibrium positions, \n𝒎̂ and 𝝈̂ are perpendicular to each other , and it is usually conve nient to use an effective field  [22] 𝐻𝑆𝑇∝\n𝝈̂×𝒎̂ to analyze  the SOT effect on magnetic switching. The equilibrium position of 𝒎̂ can then be \ndetermined by balancing the anisotropy field 𝐻𝑎𝑛, the applied in -plane field 𝐻𝑦, and 𝐻𝑆𝑇. As shown in \nFig. 2(b ), when 𝐻𝑦 and 𝝈̂ are given by the illustrated directions, the final orientation of the Co  sublattice  \nmagnetic moment will be close to  the +𝒛̂ direction f or the Co dominant sample and  close to  the −𝒛̂ \ndirection for the Tb dominant sample, giving rise to opposite Hall voltages.  Based upon this analysis, the \ncurrent induced switching should exhibit the same polarity change across the compensation point as the \nmagnetic field induced switching. We note that all of the samples follow this rule except for  the \nCo0.77Tb0.23 sample, where the field switching data had determined it to be Tb -dominant  but the current \ninduced switching corresponds to a Co -dominant  sample. A careful study on this sample reveals that this  \nchange  simply arises  from Joule heating induced temperature change. In RE-TM alloys , the magnetic \nmoments of the RE atom s have stronger temperature  dependence compared with TM atoms . \nConsequently , when the temperature increases, a higher RE concentration is ne cessary  to achieve the \nsame magnetic moment compensation point  [3–9,29] . By measuring the RAH  vs H curves of the \nCo0.77Tb0.23 sample under different applied currents , we found that the polarity of the field induced \nswitching did change si gn when the current density is higher than 2 × 107 A∙cm-2, suggest ing that the \nsample underwent a  (reversible) transition from Tb -dominant to Co -dominant . \n \nIV. Quantitative Determination of Spin-Orbit -Torque Efficiency  \n The critical current of SOT induced switching in a multi -domain sample is influenced by defect -\nrelated factors such as domain nucleation and domain wall  (DW)  pinning  [24]. Therefore, the SOT \nefficiency cannot  be simply extracted using the switching current values determined in Fig. 2 ( a). To 5 \n quantify the SOT in our samples, we measured the SOT induced effective  field in the DW motion regime \nby comparing it with the applied perpendicular field, using  the approach developed by C. F. Pai et \nal. [25]. It has been shown that in PMA films with N éel DWs , the Slonczewski term acts on the DW as an \neffective perpendicular magnetic field and induces DW motion [Fig. 3(g)]  [21–25].Therefore, by \nmeasuring the current induced shift in the RAH vs Hz curves, one can determine the magnitude of the SOT.  \nFig. 3 (a) and (b) show  typical  field induced switching curves  for a Co-dominant  sample Co 0.82Tb0.18 and a \nTb-dominant sample Co0.75Tb0.25. Under the  applied current of ±3 mA  and in -plane field of 2000 Oe, the  \ncenters of hysteresis loops  are offset from zero , with opposite values for opposite current directions. The \ncurrent dependence of the offset fiel ds are summarized in Fig. 3 (c ) and (d) for 𝐻𝑦=0 and ±2000 Oe, \nwhere a linear relationship between the offset field and the applied current is obtained. In these plots, the \nratio between the offset field 𝐻𝑧𝑒𝑓𝑓 and current density 𝐽𝑒 curve represents the efficiency of the SOT at the \nTa/RE -TM interface, defined as 𝜒 ≡𝐻𝑧𝑒𝑓𝑓\n𝐽𝑒.  𝜒 as a function of  applied  𝐻𝑦 for Co 0.82Tb0.18 and Co 0.75Tb0.25 \nsamples are plotted in Fig. 3 (e ) and (f), respectively . 𝜒 grows linearly in magnitud e for small values of \n𝐻𝑦, until reaching the saturation efficiency 𝜒𝑠𝑎𝑡 at a large  in-plane field 𝐻𝑦𝑠𝑎𝑡. The evolution of 𝜒 as a \nfunction of 𝐻𝑦 comes from the chirality change of the DWs in the sample. It is known that because of the \nDzyalosh inskii -Moriya interaction  (DMI) mechanism  at the heavy metal/magnetic metal interface  [30] or \ninside the bulk of RE -TM alloy  [31], stable Néel DW with spontaneous chiralities are formed.  Under zero \n𝐻𝑦, the DWs do not favor either sw itching polarities, leading to a zero offset field. As 𝐻𝑦 increases,  the \nDMI induced effective field 𝐻𝐷𝑀𝐼 is partially canceled, and DWs start to move in directions that facilitate \nmagnetic switching.  𝐻𝑦𝑠𝑎𝑡 therefore represents the minimum  field that is required to completely overcom e \n𝐻𝐷𝑀𝐼 and 𝜒𝑠𝑎𝑡 represents the maximum efficiency of the SOT .  \n Fig. 4 (a) illustrates the dependence of 𝜒𝑠𝑎𝑡 on x in Co1-xTbx samples. It can be seen that 𝜒𝑠𝑎𝑡 \ndiverges near xCM, with the larges t value occurring for the sample with smallest magnetization.  This result \nis consistent with the spin torque theory, where the ratio between the SOT effective field and applied 6 \n charge current is 𝜒𝑠𝑎𝑡=(𝜋/2)(𝜉ℏ/2𝑒𝜇0𝑀𝑠𝑡)  [25,32]. Here 𝜉=𝐽𝑆𝐽𝑒⁄ represents the effective spin Hall \nangle,  ℏ\n2𝑒𝐽𝑠 is the spin current density,  ℏ is Planck’s constant,  𝜇0 is the vacuum permeability , and 𝑀𝑠 is \nthe saturation magnetization. Note that this model of spin torque is based upon the conservation of total \nangular momentum. Previously it has been suggested to utilize ferrimagnetic materials with minimized \n𝑀𝑠 to increase the efficiency of spin torqu e induced switching  [33]. However, it was not verified if an \nefficient spin absorption could be achieved at the surface of a ferrimagnet material with antiparallel \naligned  sublattices. Moreover, because of the mixture between the spin angular momentum and orbital \nangular momentum in RE -TM alloys, there have been debates over the conservation of total angular \nmomentum in these systems  [34]. Our experiment al results provide clear evidence on the strong \nefficiency of spin orbit torques in antiferromagnetically  coupled materials. Within the experimenta l \naccuracy, we found that the effective field from the SOT does follow the simple trend given by 1/𝑀𝑆𝑡 \n[dashed lines in Fig. 4(a)], reflecting total angular momentum conservation. 𝜉 in our samples is \ndetermined  to be ~0.0 3, smaller than previously repo rted values from Ta/magnetic layer devices, possibly  \ndue to the relatively smaller spin-mixing conductance at the Ta/CoTb interface  [35].  \nIn addition to  the magnetic moment compensation point xcM, inside RE -TM systems there  also \nexists an  angular momentum  compensation point  xcJ due to the different g factors associated with spin and \norbit al angular moment um. For our Co 1-xTbx system, using the g factors of Co (~2.2) and Tb (~1.5) \natoms  [36,37]  , along with the relation 𝐽𝐶𝑜(𝑇𝑏)= 𝑀𝐶𝑜(𝑇𝑏)/𝛾𝐶𝑜(𝑇𝑏), where 𝛾𝐶𝑜(𝑇𝑏)= −𝑔𝐶𝑜(𝑇𝑏)𝜇𝐵/ℏ (𝜇𝐵 \nbeing the Bohr magneton, 𝛾𝐶𝑜(𝑇𝑏) the gyromagnetic ratio, and 𝐽𝐶𝑜(𝑇𝑏) the total angular mo mentum per \nunit volume), we determine xcJ  to be ~17%, which is within the range of the studied samples and lower \nthan xcM. Previously  it was demonstrated that ultrafast field-driven magnetic dynamics could be excited \naround xcJ [8,9] . According to  Landau -Lifshiz -Gilbert equation of a ferr imagnetic system  [27,29,38] , the \nspin torque term leads to  𝑑𝒎̂\n𝑑𝑡~−𝛾𝑒𝑓𝑓ℏ𝐽𝑠\n2𝑒𝜇0𝑀𝑆𝑡(𝒎̂×𝝈̂×𝒎̂) , where 𝛾𝑒𝑓𝑓=(𝑀𝐶𝑜−𝑀𝑇𝑏)/(𝐽𝐶𝑜−𝐽𝑇𝑏) is \nthe effective gyromagnetic ratio  and 𝑀𝑆=𝑀𝐶𝑜−𝑀𝑇𝑏 [8,9] . When 𝐽𝐶𝑜−𝐽𝑇𝑏 approaches zero, the time 7 \n evolution of 𝒎̂ diverg es if 𝐽𝑠 remains finite at xcJ. As observed in Fig. 4(a), 𝜒𝑠𝑎𝑡 remains roughly \nunchanged across xcJ, suggesting that similar to the field -driven experiment, SOT could also be used as an \nefficient drive force for achieving fast dynamics at this concentration. Finally, we find the switching \npolarity keeps the same sign  across xcJ, differ ing from the current induced switching of CoGd spin valves \nstudied in Ref.  [29], where a sw itching polarity reversal was observed between xcJ and xcM. This \ndifference is due to the presence of different switching mechanisms: in SOT induced switching of PMA \nfilms, the two  competing  torques are the field torque 𝛾𝑒𝑓𝑓𝑴×𝑯𝑒𝑓𝑓 and the spin tor que. Because the two \nterms have the same pre -factor 𝛾𝑒𝑓𝑓 and are only functions of 𝑴, 𝑯𝑒𝑓𝑓, and 𝝈̂,  under the same applied 𝝈̂ \nand 𝑯𝑦, the orientation of 𝒎̂ will remain the same (Fig. 2b ), regardless of the sign of  𝛾𝑒𝑓𝑓. In contrast, \nthe anti -damping switching of spin valves changes polarity for regions with 𝛾𝑒𝑓𝑓<0, as explained in \nRef. [29].  \n \nV. Measurement of the Dzyaloshinskii -Moriya I nteraction  Energy  \n The in -plane field needed for saturating the SOT, 𝐻𝑦𝑠𝑎𝑡, is plotted against the Tb concentration in \nFig. 4(b). First of all, we notice that 𝐻𝑦𝑠𝑎𝑡 is largest  near xcM. This  result  is consistent with fact that  the \neffective  DMI field  [32] 𝐻𝐷𝑀𝐼 =𝐷/𝑀𝑠𝑡𝜇0∆, where 𝐷 is the DMI energy density and ∆ is the DW width, \nwould become divergent when 𝑀𝑠 approaches zero . Secondly, 𝐻𝑦𝑠𝑎𝑡 is generally larger for the Tb-\ndominant  sample s than the Co -dominant ones. For example, the sample with the highest Co \nconcentration,  Co0.87Tb0.13, shows  𝐻𝑦𝑠𝑎𝑡~100 Oe, which is close to the  reported saturation field of Ta/FM \nstacks  [25]. However, in the Tb dominant sample Co0.71Tb0.29, which has similar magnetic moment, 𝐻𝑦𝑠𝑎𝑡 \nis found to be ~1500 Oe. In the inset of Fig. 4(b) we plot 𝐻𝑦𝑠𝑎𝑡𝑀𝑠𝑡, which increases roughly linearly as a \nfunction of x. By calculating the DW width ∆ =√𝐴𝐾𝑢⁄ using the determined anisotropy energy 𝐾𝑢 =  6.4 \n× 104 J∙m-2 from Tb - and Co -dominant samples  and the reported exchange stiffness  [39] A ~ 1.4 × 10-11 8 \n J/m, we get D in the range of 0.05~0.66 pJ/m. The increasing DMI energy with increasing Tb \nconcentration can  be explained by  the strong  spin-orbit coupling and large deviation from the free \nelectron g factor  [30] in the Tb atoms. The generation of magnetic textures such as chiral DWs and \nmagnetic skyrmion s  [40] relies on the competition between the DMI energy and other magnetostatic \nenergies. Therefore, the tunable DMI through chemical composition provides a useful a knob for \ncontrolling magnetic phases.  \n \nVI. Conclusion  \n To summarize, we demonstrated SOT induced switching in Co1-xTbx thin films with a wide range \nof chemical compositions. 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Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura,“Real -\nspace observation of a two -dimens ional skyrmion crystal,” Nature 465, 901 (2010).  \n \n  11 \n  \n \nFig. 1 (a) Out of plane magnetization curves  of Co 1-xTbx films . (b) RAH as a function of perpendicular \nmagnetic field. (c ) Magnetic  momen ts of Co1-xTbx alloys as a function of Tb concentration . (d) Schematic \nof the device geometry for  RAH measurement. (e) RAH as a function of Tb concentration .  \n \n \n \n \n \n \n \n \n \n \n12 \n  \n \nFig. 2 (a) Current induced SOT switching of Co 1-xTbx for in -plane fields of ± 2000 Oe. The current \ndensity inside Ta is calculated based on the conductivity of Ta thin films.  (b) Schematic of effective fields \nin a ferrimagnetic system. Fields acting on the moment  consist of the in -plane field  Hy, the anisotropy \nfield Han, and the SOT field  HST. \n \n  \n13 \n  \n \nFig. 3 (a), (c),(e) Measurements on Co-dominant  sample Co 0.82Tb0.18. (b),(d), (f) Measurements on Tb-\ndominant  sample Co 0.75Tb0.25. (a),(b ) RAH  vs. applied perpendicular field under  a DC current of ±3 mA. \n(c),(d ) SOT effective field as a function of applied current density  under in -plane fields of ±2000, and 0 \nOe. (e ),(f) SOT efficiency vs.  Hy. Efficiency saturates at the field  Hsaty. (g) SOT induced DW motion in \nthe ferrimagnetic system for Tb-dominant  and Co-dominant  films , showing that the effective \nperpendicular field has the same sign in both cases.  \n14 \n  \n \nFig. 4 (a ) Saturation efficiency and (b) in-plane saturation field for differe nt Co 1-xTbx films. Both the \nsaturation efficiency  and in-plane saturation field  are largest near the magnetic moment compensation \npoint.  The dashed line in (a) shows the trend calculated from χsat ~1/ Mst. Inset of (b) plots the product of \nthe in -plane sat uration field and magnetic moment M st, indicating an increase in DMI energy density D \nwith increasing Tb concentration.  \n \n \n \n"    },    {        "title": "2009.10222v1.Magneto_Elastic_Coupling_to_Coherent_Acoustic_Phonon_Modes_in_Ferrimagnetic_Insulator_GdTiO__3_.pdf",        "content": "arXiv:2009.10222v1  [cond-mat.str-el]  21 Sep 2020Magneto-Elastic Coupling to Coherent Acoustic Phonon\nModes in Ferrimagnetic Insulator GdTiO 3\nD. Lovinger,1E. Zoghlin,2P. Kissin,1G. Ahn,3K. Ahadi,2P. Kim,1M.\nPoore,1S. Stemmer,2S. J. Moon,3S. D. Wilson,2and R. D. Averitt1\n1Department of Physics, University of California, San Diego , La Jolla, CA, 92093\n2Materials Department, University of California, Santa Bar bara, CA, 93106\n3Department of Physics, Hanyang University, Seoul, South Ko rea, 04763\n(Dated: September 23, 2020)\nIn this work we investigate single crystal GdTiO 3, a promising candidate material for Floquet en-\ngineering and magnetic control, using ultrafast optical pu mp-probe reflectivity and magneto-optical\nKerr spectroscopy. GdTiO 3is a Mott-Hubbard insulator with a ferrimagnetic and orbita lly ordered\nground state ( TC= 32 K). We observe multiple signatures of the magnetic phase transition in\nthe photoinduced reflectivity signal, in response to above b and-gap 660 nm excitation. Magnetic\ndynamics measured via Kerr spectroscopy reveal optical per turbation of the ferrimagnetic order\non spin-lattice coupling timescales, highlighting the com petition between the Gd3+and Ti3+mag-\nnetic sub-lattices. Furthermore, a strong coherent oscill ation is present in the reflection and Kerr\ndynamics, attributable to an acoustic strain wave launched by the pump pulse. The amplitude of\nthis acoustic mode is highly dependent on the magnetic order of the system, growing sharply in\nmagnitude at TC, indicative of strong magneto-elastic coupling. The drivi ng mechanism, involving\nstrain-induced modification of the magnetic exchange inter action, implies an indirect method of\ncoupling light to the magnetic degrees of freedom and emphas izes the potential of GdTiO 3as a\ntunable quantum material.\nI. INTRODUCTION\nTherare-earthtitanates (unit formulaRTiO 3, whereR\nis a rare-earth ion) are a class of complex materials with\nstrongly correlated spin, orbital, and lattice degrees of\nfreedom. They are3 d1compounds with a single d-orbital\nelectron occupying the Ti3+t2gorbital, whose degener-\nacy is broken by strong crystal field splitting [1]. This\npresents an opportunity to study a strongly correlated\nsystem in relative simplicity, which nonetheless exhibits\nrich physics and interesting properties. The perovskiteti-\ntanates, for example, are Mott-Hubbard (MH) insulators\nwith interconnected orbital and spin order [2–6]. Of par-\nticular interest is the complex magnetic phase diagram\nfor this class of materials, with a magnetic ground state\nthat varies from ferrimagnetic to antiferromagnetic as a\nfunction of the rare-earthion size and subsequent change\nin Ti-O-Ti bond angle [1, 7]. Various theories have at-\ntempted to explain the magnetic order in titanates [5, 8–\n10], all of which highlight the need to consider the roles\nof structure and electronic correlation to understand the\ncomplexity embodied in the magnetic phase diagram.\nCommon to all descriptions of magnetism in the per-\novskite titanates is the defining role of the lattice and\nits distortion. It has been argued, for example, that the\ndegree of GdFeO 3distortion and changes to the Ti-O-\nTi bond angle directly modify the exchange interaction\nwhich, in turn, determines the magnetic order [7, 11].\nMore recent results emphasize the importance of orbital\norder in determining the magnetic order. In particular,\nthe direct coupling between the orbital order and lat-\ntice, ratherthantheorthorhombicdistortion, contributes\nmost strongly to the ground state [12–14]. Whether itis particular structural distortions or more generalized\nJahn-Teller distortions, and regardless of the role of or-\nbital ordering, it is clear that magnetic order in titanates\nis highly dependent on the lattice.\nIn this work we study GdTiO 3(GTO), a titanate with\nan orthorhombic perovskite-type unit cell and relatively\nlarge GdFeO 3-type distortion. GTO lies just within the\nferromagnetic (FM) region of the phase diagram. The\nproximity to the FM-AFM transition makes GTO partic-\nularly sensitive to the effect of structural changes on the\nmagnetism [12]. Below the critical temperature TC=\n32 K it is ferrimagnetically (fM) ordered: the Ti3+spins\nare aligned ferromagnetically along the c-axis and cou-\npled antiferromagnetically to the Gd sublattice [3, 8, 12].\nThe magnetism saturates at 6 µB(7µBGd – 1µBTi) in\na relatively small field of ∼0.1 T, with no discernable hys-\nteresis [15]. The magnetocrystalline anisotropy is small,\nwith the a-axis as the hard magnetization axis and the\nb-cplane nearly isotropic. The fM order is accompanied\nand mediated by ( yz, zx, yz, zx )-type orbital order, a re-\nsult of inter-atomic hybridization between the t 2gand eg\norbitals [1].\nThepresentworkonGTOis motivated notonlyby the\nrelative simplicity of the system and rich interconnected\norder, but also the potential for Floquet engineering and\nultrafast control of magnetism. Liu et al.explored the\nMott insulating titanates as a candidate for tuning the\nspin-orbital Floquet Hamiltonian and subsequent modi-\nfication of the spin exchange interaction using light [16].\nMeanwhile, Khalsa et al.suggest direct excitation of a\nGTO mid-IR active phonon mode to transiently mod-\nify the exchange interaction and switch the ground state\nfrom FM to AFM on ultrafast timescales [17]. A similar2\n0 2 4 6 \nPhoton Energy (eV )0123451()( -1 cm -1 )\n10 \n50 \n150\n3001 2 3 \nPhoton Energy (eV) 120160200240\n0 2 4 6 \nPhoton Energy (eV )0123456n( )1 2 3 \nPhoton Energy (eV) 2.12.22.310 3(a) (b)\nTi 1 t2g  LHBTi 2 t2g UHB\nO 2pEFU\n1.88 eV(c)\nT (K) \n10 \n50 \n150\n300T (K) 1.88 eV\nFIG. 1. (a) Optical conductivity and (b) index of refraction of GTO/LSAT thin film as a function of photon energy and\ntemperature. Red arrows indicate the 1.88 eV pump/probe ene rgy. The weak feature in σ1at 2 eV corresponds to the MH\ngap, while the steep feature near 5 eV arises from O 2pto Ti3dand Gd 4fcharge transfer transitions. (c) Depiction of 1.88 eV\nlaser excitation, corresponding to intersite Ti 3d-3dtransition across the MH gap.\nexperiment utilizing phononic control has been proposed\nby Guet al.in other titanates [18].\nWhile the conditions of our experiment lie outside\nthe regimes discussed above, we do observe strong cou-\npling between light, the lattice, and the sample mag-\nnetism. Time-resolved pump-probe and magneto-optical\nKerr effect (MOKE) measurements tuned to ∼1.88 eV,\njust above the bandgap, allow us to measure the evo-\nlution of photoexcited states on femtosecond – picosec-\nond timescales. We observe multiple signatures of the\nmagnetic phase transition in the photoinduced reflectiv-\nity signal, as well as optical perturbation of the fM order\non spin-lattice coupling timescales in the MOKE signal.\nIn addition, an acoustic phonon mode is present in both\nsignals, whose amplitude is highly coupled to the mag-\nnetic order. This implies strong magneto-elasticcoupling\nthrough transient, strain-induced modification of the ex-\nchange interaction, connecting the lattice and magnetic\ndegrees of freedom and indicating that the exchange in-\nteraction is tunable on ultrafast timescales.\nII. METHODS\nSingle crystaland thin film samplesofGdTiO 3werein-\nvestigated. The photoinduced reflectivity signal in both\nis extremely similar and the following work, except for\nthe measurement of the optical constants, was performed\non a single crystal sample. For comparison, the thin\nfilm time-resolved reflectivity data is presented in SM\nI [19]. GdTiO 3thin films ( ∼20 nm) were grown on\na (001)(La 0.3Sr0.7)(Al0.65Ta0.35)O3(LSAT) substrate by\nhybrid molecular beam epitaxy [20]. GdTiO 3bulk sin-\ngle crystals were grown by high pressure laser floating\nzone method [21]. A small fraction of the crystal rodwas cut and polished to optical quality, with bc-axis in\nplane and a-axis out of plane. Powder X-ray diffraction\nmeasurements indicate extremely high quality crystals\nwith no notable impurity peaks and lattice parameters\nat 5.393, 5.691, 7.664 ˚A fora,b,c-axis [21], well matched\nto literature values [12]. Magnetization measurements in-\ndicate no visible hysteresis and a saturation moment of\n6µB/FU.\nTo determine the optical conductivity and index of re-\nfraction, frequency-dependent reflectivity spectra R(ω)\nin the photon energy region between 3 meV and 85\nmeV were measured by using a Bruker VERTEX 70v\nFourier transform spectrometer. The GdTiO 3thin film\nwas mounted in a continuous liquid helium flow cryostat.\nWe used two spectroscopic ellipsometers (IR-VASE Mark\nII and M-2000, J. A. Woollam Co.) for obtaining the\ncomplex dielectric constants ǫ(ω) =ǫ1(ω) +iǫ2(ω) in\nthe energy range from 60 meV to 0.75 eV and 0.75 eV\nto 6.4 eV, respectively. The optical conductivity of the\nGdTiO 3film was obtained by two-layermodel fit employ-\ning Drude-Lorentz oscillators for optical response of each\nlayer [22].\nUltrafast optical pump-probe reflectivity measure-\nments (∆R/R) are performed using a 1040 nm 200 kHz\nSpectra-Physics Spirit Yb-based hybrid-fiber laser cou-\npled to a non-colinear optical parametric amplifier. The\namplifierproduces ∼20fs pulsescenteredat660nm (1.88\neV), which are split, cross-polarized (pump s-polarized,\nprobep), and used asdegeneratepump and probebeams.\nThe pump is aligned along the b-axis of the GTO crystal.\nThis excitation corresponds to an intersite Ti 3d–3d\ntransition across the Mott-Hubbard gap, shown in Fig.\n1(c). A moderate pump fluence of ∼100µJ/cm2is used\nto minimize sample heating ( ∼4 K at 10 K), ensuring we\nare in the linear excitation regime.3\nTime-resolved magneto-optical Kerr spectroscopy is\nused to probe the magnetization dynamics. The same op-\ntical system described above is used here, including laser\nenergy, fluence, and magneto-optical cryostat (Quan-\ntum Design OptiCool). The photoinduced Kerr rotation\n(∆θK) is measured using balanced photodiodes in the po-\nlar Kerr geometry at near-normalincidence, in a continu-\nously variable external magnetic field (0 – 7 T), with the\npump polarized along the b-axis and Kerr probe polar-\nized along the c-axis of the crystal. The magnetic field is\napplied normal to the sample surface, along the a-axis of\nthe crystal, resulting in a Kerr signal proportional to the\nout-of-plane z-component of the photoinduced change in\nmagnetization ∆ Mz. In order to eliminate non-magnetic\ncontributions to the signal and ensure we are measur-\ning genuine spin dynamics, we take the difference of the\nKerrsignalatvariouspositive andnegativeapplied fields:\n∆θK= ∆θ(+M)−∆θ(−M) (see SM IV for details)\n[19, 23–27].\nIII. EXPERIMENTAL RESULTS\nThe temperature-dependent optical conductivity of\nthin-film GdTiO 3is shown in Fig. 1(a) for photon ener-\ngies ranging from 3 meV to 6.5 eV, and for temperatures\nfrom 10 K to 300 K. A weak feature is present, centered\nat 2 eV, corresponding to the Mott-Hubbard gap. Apart\nfrom weak thermal broadening with increasing temper-\nature, the peak at 2 eV is nearly temperature indepen-\ndent. While early studies of GTO measured a MH gap\nof 0.2 – 0.7 eV [28], more recent photoluminescence and\nDFT/DFT+U results place the gap closer to 1.8 – 2 eV\n[29]. Thesmallpeakintheopticalconductivityspectrum\nat 2 eV measured here supports these recent findings. At\nmuch higher energies we observe a significant increase in\nthe optical conductivity. The features near 5 eV corre-\nspond to O 2pto Ti3dand Gd 4ftransitions. Fig. 1(b)\nshows the index of refraction in the same energy range,\nwhich remains relatively constant as a function of tem-\nperature.\nThe time-dependent photoinduced change in reflectiv-\nity ∆R/Rfor a GdTiO 3single crystal is shown in Fig.\n2(a), for all measured temperatures between 10 – 295\nK (legend on Fig. 2(b)). The black lines represent ex-\nponential fits to the data as described below. The pho-\ntoinduced change in ∆ R/Ris positive; following laser\nexcitation a non-equilibrium electron population is estab-\nlished in ∼500 fs, which then exchanges energy and equi-\nlibrates with the spin and lattice subsystems throughvar-\nious pathways, each with a characteristic timescale. This\nis visible as the slower, multi-component exponential re-\nlaxation. As the temperature is decreased from 295 K\nthe signal amplitude increases, recovery dynamics slow,\nandtwoadditionalfeaturesemerge. The firstisa delayed\nrise time, correspondingto a further departure from equi-\nlibrium in the first ∼15 ps, emerging below T= 100 K.0 200 400 600 800 1000 \nTime (ps) 010 20 R/R 10 -3 \n0 5 10 15 \nTime (ps) 0510 R/R 10 -3 0 500 1000\nTime (ps) 0510 15 20 R/R 10 -3 \n10 K\nslow e-ph \ns-l OO \n100 K0 100 200 300\nTemperature (K) 0510 15 20 25 Peak R/R (500 ps) 10 -3 \n295\n200\n100\n80 \n60 \n45 \n40 \n35 \n30 \n25 \n20 \n10 (a) (b)\n(c)T (K) \nFIG.2. (a)Photoinduceddifferentialreflectivitysignal∆ R/R\nat all measured temperatures, from 10 K to 295 K, taken on\na single crystal GdTiO 3sample. The black curves are expo-\nnential fits to the data, of the form given in Eq. 1. The\nblue star indicates the data curve taken at TC. (b) ∆R/R\nvalues at 500 ps, an approximation of the peak signal at all\ntemperatures. The red line is a power law fit, commonly seen\nin systems undergoing a second-order magnetic phase transi -\ntion. (c) Representative pump-probe scans at 10 K and 100\nK, indicating the various timescales involved in the recove ry\nprocess (see Eq. 1).\nSecond, there is a crossover point visible at delay times\nof∼200 ps where recovery dynamics flatten and reverse\ndirection to become an additional rise time. This occurs\nprecisely as the ferrimagnetic ordering temperature TC\n= 32 K is crossed (marked by a blue star), indicating\nthat magnetization dynamics manifest in the differential\nreflectivity signal.\nTo further investigate the temperature dependence of\nthe reflectivity signal, we plot the peak signal amplitude\n(at 500 ps) in Fig 2(b). The behavior here is distinctive,\nnot uncommon in materials undergoing a second-order\nmagnetic phase transition. The red curve represents a\npower-law fit to the data, of the form A=A0t−w, where\nwis the critical exponent and tis the reduced tempera-\nturet=|T−TC|\nTC. The value of wdepends upon the sym-\nmetry and universality class of the magnetic transition.\nOurfit producesacriticalexponent w= 1.28±0.02. This\nvery nearly matches the critical behavior predicted by\ndynamical scaling theory for the 3D Ising model, which\nyieldsw≈1.32[30–32](furtherdetailedinSMII)[19,30–\n35]. Whilethisisanindirectmethodofmeasuringcritical\ndynamics and is not intended to be a rigorous analysis,4\n050 100 150   s-l  (ps)   \n-6 -4 -2 0\nAmplitude s-l 10 -3\n 50\n 60\n 80\n100\n150\n250 10\n 20\n 25\n 30\n 35\n 400 100 200 300 \nTemperature (K) \n0\nTime (ps) -3 -2 -1 01R/R Residual 10 -4 \n0 50 100 150\nTime (ps) -2 -1 01R/R Res. 10 -4 (a)\n(b)\n10 20 30 40 50 T (K) \nFIG. 3. (a) Time constant (black) and amplitude (red) of the\nspin-lattice coupling term, extracted from exponential fit s to\nthe ∆R/Rdata. The vertical gray section indicates the fM\ntransition region TC= 32 K, where the lifetime is too long\nto measure (see 30 K curve marked by a blue star in Fig.\n2(a)). The reddashed line depicts zero amplitudeand clarifi es\nthe crossover region. (b) Coherent acoustic phonon respons e,\nisolated by subtracting the exponential fits from the ∆ R/R\ndata. The inset shows the ∆ R/Rresidual to a longer delay\ntime of 150 ps, where higher frequency components emerge.\nit is clear that the peak reflectivity follows power-law be-\nhavior as expected at a magnetic phase transition. The\nresults suggests that there is indeed a magnetic contribu-\ntion in the ∆ R/Rsignal. Additionally, the qualitative\nform of the peak amplitude vs temperature follows that\nof the temperature dependent magnetic susceptibility in\nbulk GTO [15], and the magnetization M in films [21, 36].\nWhile by no means conclusive, the universal scaling be-\nhavior and agreement with thermal magnetization does\nstrongly suggest that the ∆ R/Rsignal measured, par-\nticularly at longer times (500+ ps), is sensitive to spin\ndynamics.\nTosubstantiatetheseclaims, wequantitativelyanalyze\nthe full time-dependent response. Below 100 K, the dy-\nnamics can be fit by a sum of four exponentials with a\nconstant offset, of the form:∆R/R(t) =Ae−phe−t/τe−ph+AOOe−t/τOO+\nAs−le−t/τs−l+Aslowe−t/τslow+C,(1)\nshown as black lines in Fig. 2(a). Not listed is an ad-\nditional error function term, which describes the initial\nstep-like rise dynamics at t= 0. A visual representation\nofthevarioustimescalesisshowninFig. 2(c)fortwotem-\nperatures. After excitation the dynamics follow a general\ntrend; there is a very fast initial recovery, τe−phon the\norder of ∼500 fs, followed by an intermediate term τOO\non the order of 2 – 8 ps, both of which are clearly visible\nin the inset of Fig. 2(c). Note that τOOis an additional\nrise time which vanishes at higher temperatures, the full\ndynamics fitting to only 3 exponentials (i.e. above TC).\nThis is followed by a slower term τs−lon the order of\n100’s of picoseconds, and a final much slower recovery\nτslow. A careful inspection of the reflectivity data also in-\ndicates the presence of small oscillations about the black\nfitted curves, which we discuss below.\nThese measured timescales are well separated and can\nbe attributed to distinct physical processes. The initial\npump pulse excites an intersite Ti 3d-3dtransition. This\ndirectly creates a population of hot carriers which ther-\nmalize via electron-electron (e-e) scattering, then subse-\nquently exchange energy with the lattice, orbital, and\nspin degreesof freedom. We focus on the spin-lattice cou-\npling process here, with a full discussion of the remaining\nprocesses and time constants in SM III [1, 11, 19, 37–45].\nThe most relevant component of the ∆ R/Rsignal\nis the third fitted exponential, τs−l, attributed to spin-\nlattice coupling and shown in Fig. 3(a). This term has a\ncharacteristic timescale of 10 – 140 ps, excluding the re-\ngion at the magnetic phase transition temperature TC=\n32K where the lifetime growstoo long to accuratelymea-\nsure. This critical region is visible as a flattening of the\n∆R/R recoveryat 30 K, indicated by the blue star in Fig.\n2(a). Before the onset of this third recovery term τs−l,\nthe ∆R/R signal reveals dynamics indicative of electron-\nlattice equilibration. It follows that this longer lifetime\nis related to equilibration of the spin subsystem with the\nlattice. The time constant measured, on the order of 100\nps in the magnetic phase, is consistent with spin-lattice\ncoupling in other magnetic insulators [46–48]. The char-\nacteristic time is relatively constant in the paramagnetic\nphase until 150 −200 K, where it begins to slowly in-\ncrease. This corresponds to the onset temperature ( ∼180\nK) of spin-spin coupling between the Gd3+and Ti3+ions\n[8]. Closer to 100 K τs−lfurther increases, indicating the\nonset of short-range fM spin correlations. This is also\napparent in the increase in amplitude at this tempera-\nture. Finally, as TCis crossed (dark gray region) we see\nevidence of the second-order ferrimagnetic phase transi-\ntionasthetimeconstantdivergesandamplitudeswitches\nsign. The now-negative amplitude implies an additional\nrise time in the signal; as energy is transferred to spins\nand the ferrimagnetic order is disrupted, the system is5\n-2 0246 (rad) 10 -4 \n-2 0246\n0 500 1000 \nTime (ps)-2 0246 (rad) 10 -4 \n0 500 1000\nTime (ps) 0246810 10 -4 \n 10\n 20\n 25\n 30\n 35\n 40\n 50\n 60\n 80\n25010 -4 H = 0.1 T 0.25 T\n0.5 T1 TT (K) \nFIG. 4. Time-resolved Kerr dynamics at various magnetic\nfields, recorded as the difference between the MOKE signal\nin opposing field directions ∆ θ= ∆θ(+H)−∆θ(−H). This\nis a measure of the photoinduced change in the out-of-plane\nmagnetization ∆ Mz. Red arrows indicate the crossover to\nnegative values of ∆ θ.\nbrought further out of equilibrium. In the paramagnetic\nphase there is no long-range spin order to disrupt, such\nthat spin-lattice thermalization manifests as a simple re-\ncovery to equilibrium. The critical behavior, amplitude\nreversal, timescale, and temperature dependence of the\nτs−lcomponent all suggest that we are measuring spin-\nlattice coupling on a timescale of ∼100 ps, and that it is\nhighly sensitive to the onset of magnetic order.\nThe final interesting feature of the ∆ R/Rdata is a\nslow coherent oscillation, prominent at early times. By\nsubtracting the exponential fits at each temperature we\ncan extract the oscillatory component, plotted in Fig.\n3(b). The result is peculiar – we observe a low-frequency\nphonon mode which grows in amplitude and becomes\nchirped, slowing down and redshifting as it propagates.\nThe oscillationperiod (on the orderof20ps), suggestsan\nacoustic strain wave launched by the pump pulse which\npropagates through the crystal [49]. The probe beam re-\nflected from the sample surface interferes with a portion\nreflected from the strain wave boundary, resulting in an\noscillatory signal. The temperature dependence of this\nmode is striking – the amplitude is relatively constant at\nhigh temperatures, then grows sharply precisely at the\nfM phase transition temperature. Though it appears to\nbe an acoustic mode, it is also clearly coupled to the\nmagnetic order. This suggests strong magneto-acoustic\ncoupling, tying the dynamics of the magnetic subsystem\nto the transiently strained lattice.\nTo gain further insight into the magnetization dynam-\nics and the influence upon acoustic phonon propaga-tion, we utilize time-resolved magneto-optical Kerr effect\n(MOKE) spectroscopy. Fig. 4presents the photoinduced\nKerr rotation ∆ θ, proportional to the change in out-of-\nplane (a-axis) magnetization ∆ Mz, for all temperatures\nand four fields between 0.1 – 1 T. For details of the anal-\nysis, see SM IV [19, 23–27]. Additional static Kerr rota-\ntion measurements are presented in SM V [19]. At lower\nfield strengths, the Kerr signal reveals a quick rise in\nthe photoinduced out-of-plane magnetization ∆ Mz, fol-\nlowed by a reduction and change in sign of ∆ Mz. This\ncan be interpreted asa pump-induced increaseand subse-\nquent decrease in the net out-of-plane magnetic moment,\nbut not necessarily a reversal of the total magnetic mo-\nment. Therearetwoprimarycomponentsto the Kerrsig-\nnal, one positive (growing in ∼100 ps), and one negative\n(growing in slower, ∼100 – 500 ps). These dynamics are\nslow and long-lived, as expected in magnetic insulators\nlikeGTOduetothelocalizednatureofquasiparticles[48].\nTo describethe temperature dependence ofthe signal, we\nfocus on lower field strengths H= 0.1−0.5 T. At high\ntemperature, in the paramagnetic phase, the Kerr signal\nis weak and indicates the lack of long-range magnetic or-\nder. As the temperature is lowered there is an increase in\nthe photoinduced rotation, with a clear negative signal\nemerging below TC. This negative component is largest\nand appears at earlier delays right at the transition tem-\nperature ( TC= 32 K). With decreasingtemperature, the\ncrossover to negative values of ∆θoccurs at later times.\nWell below TC, in the strongly ordered phase, the signal\nremains positive at all time delays.\nWe also observe a significant field dependence in the\ndata. The maximum signal amplitude at all tempera-\ntures increaseswith increasing field. In addition, the neg-\nativeamplitude componentismostpronouncedat0.25T,\ndecreasing in amplitude at higher fields and vanishing en-\ntirely by 1 T. At this high field, we note that the photoin-\nduced magnetization dynamics look qualitatively similar\nto the photoinduced reflectivity signal ∆ R/Rshown in\nFig.2(a). In the ∆ R/Rdata, the measured signal is\nprimarily the result of Ti sublattice dynamics due to the\n1.9 eV intersite Ti-Ti excitation, and is dominated by Ti\nspin dynamics: the spin-lattice and spin relaxationterms\n(τs−landτslow). It follows that the MOKE signal mea-\nsured at 1 T is primarily a measure of Ti spin dynamics\ndue to its similarity with the ∆ R/Rsignal. Fits to the 1\nT MOKE data support this, yielding a component with\na timescale of 100 – 200 ps and a very similar tempera-\nture dependence when compared to τs−lextracted from\nthe ∆R/Rdata (see SM VI for details) [19]. As the field\nis lowered from 1 T, the magnetization dynamics must\nbe increasingly influenced by the Gd spins. The ferri-\nmagnetic nature of GTO, with two competing magnetic\nsublattices, is key to understanding the observed behav-\nior as we now discuss.6\n 50 \n 60 \n 80 \n100 \n150 \n250  10 \n 20 \n 25 \n 30 \n 35 \n 40 \nTime (ps)-2 -1 012 Residual (rad) 10 -5 Gd \nTi \n  MZMZ > 0 MZ < 0 \nMZ > > 0 \nMMZ << 00\nMMZ >>>> 0\nT (K) (a)\n(b)\n0 10 20 30 40 50 H = 0.1 T\nH = 1 Tt < s-lt ≥ s-l \nt = 0\nHz\na-axis\nFIG. 5. (a) Schematic depiction of the spin dynamics. Pho-\ntoexcitation directly perturbs the Ti spins, which fluctuat e\nand decrease their projection along the applied field H (para l-\nlel toa-axis, +z) int < τs−lps, increasing the MOKE signal.\nAt longer times: if H and/or magnetic order is weak, induced\nspin fluctuations and the AFM exchange coupling lowers the\nprojection of Gd spins along z. Conversely, if H is larger\nthan the exchange field, at 1 T, Gd does not reorient and no\nnegative component of the signal is observed. (b) Coherent\nacoustic phonon response, isolated by subtracting the expo -\nnential fits from the time-resolved Kerr data (at 1 T applied\nfield). The dynamics appear similar to the ∆ R/Rresidual,\nimplying a common origin which we attribute to a coherent\nstrain wave launched by the pump pulse. The appearance\nof this signal in the Kerr response indicates coherent acous tic\nphonon manipulation of the magnetic order, presumably from\nexchange modulation.\nIV. DISCUSSION\nGTO is ferrimagnetic, the Ti and Gd sublattices cou-\npled via an AFM exchange interaction. Gd spins have a\nsignificantly larger magnetic moment than Ti, 7 µBvs1µBrespectively [15]. Below TC, at zero field, the two\nsublattices are aligned into fM domains such that there is\nnomacroscopicmoment. As the applied field Halongthe\na-axis is increased, spins are rotated to form long-range\ncollinearfMorder,withtheGdsublatticealignedparallel\nto H and Ti anti-parallel. In a field of only 0.1 T satu-\nration is approached, with spins slightly canted from the\na-axis/H and a net magnetization of M≈5µB. With in-\ncreasing field, canting and spin fluctuations are reduced,\nincreasing the net moment along H. As we approach 1 T,\nspin fluctuations are minimized and the magnetization\nbecomes saturated at M≈6.0µB[15]. The interac-\ntion of these two competing magnetic sublattices after\nphotoexcitation will depend on the temperature and ap-\nplied field and is illustrated in Fig. 5(a). The 1.9 eV\npump pulse directly excites the Ti sublattice, increas-\ning Ti spin fluctuations on timescales t < τs−l. This\ncauses partial reorientation and a decrease in the projec-\ntion of Ti spins along the Gd moment, corresponding to\na rapid increase of ∆ Mzand a rise in the MOKE signal\n(i.e. the Ti sublattice magnetization oriented along -zis\ndecreased, leading to an overall increase in the net mag-\nnetization in the + zdirection due to the ferrimagnetic\norder). Various pathways exist which may perturb the\nTi spins on such timescales, including spin-orbit coupling\n[50, 51] (orbital order is disrupted in t <8 ps, see SM III)\n[1, 11, 19, 40–45], and exchange modification, discussed\nbelow. Subsequently, energy is transferredto the Gd sub-\nlattice through spin-lattice thermalization on a timescale\nt≥τs−l. The spin-lattice coupling timescale of 100 –\n200 ps measured from fits to the ∆ R/Rand MOKE data\ncorresponds to the timescale on which the MOKE signal\nchanges sign, indicating the delayed contribution of Gd\nspins to the signal. Such ultrafast magnetic sublattice\ndynamics, albeit with different mechanisms, have been\ndiscussed in a variety of materials, including those with\nsimilar rare-earth/transition metal correlations [52, 53].\nThe behavior that follows is field-dependent. At low\nfields, at times on the order of τs−l, the additional heat\ntransfer and the strong AFM exchange coupling between\nGd spins and partially-reorientedTi spins causesa reduc-\ntion in the Gd moment along the field direction. This is\nseen as the negative component, decreasing the signal\non spin-lattice timescales until the net ∆ Mzis negative.\nAt higher field strengths, the applied field locks Gd mo-\nments in place parallel to the field direction, minimizing\nfluctuations. After photoexcitation, the net magnetiza-\ntion only increases as the anti-parallel Ti spins fluctuate\nand partially reorient. This is true also at low tempera-\ntureswherethemagneticorderismorefirmlyestablished,\nand explains why ∆ Mzgoes negative only in the weakly\nordered state near TC.\nFinally, we cannot discount the possibility of direct\nphoto-induced modification of the exchange interactions.\nWhile the simplest explanation of the MOKE signal in-\nvolves only heating and spin-lattice coupling, the over-\nall heating is small (no more than ∼4 K at the lowest\ntemperatures at the fluence used). It is therefore not un-7\n024\n0 50 100\nFrequency (GHz) 012FFT Amplitude  (10 -7 )\n800 nm 10 \n 20 \n 25 \n 30 \n 35 \n 40 \n 50 \n 60 \n 80 \n100 \n295 T (K) \nFFT Amplitude  (10 -6 )\n0 50 100012\nFrequency (GHz) 012\n40 ps 60 ps\n0 50 100\nFrequency (GHz) TC = 32 K\n0 100 200 300 \nTemperature (K) 01FFT (norm.) 4  GHz\n20 GHz\n50 GHz\n80 GHz(a)\n(b)(c)\n(d) (e)660 nm\nFIG. 6. FFT amplitude of the ∆ R/Rresidual, taken at pump/probe wavelength of 660 nm (a) and 80 0 nm (b). The red dashed\nlines indicate the approximate peak positions of the 660 nm F FT. Note the redshift to lower frequency at higher wavelengt h.\n(c) The integrated FFT amplitudes at various frequencies as a function of temperature, normalized. (d) The FFT limited t o\nthe first 40 ps and (e) 60 ps of the ∆ R/Rdata. The higher frequency mode emerges only after 40 ps.\nreasonable to consider more direct electronic changes to\nthe system. The exchange interaction in the titanates is\nhighly dependent on the Ti-O-Ti bond angle and degree\nof GdFeO 3distortion, as well as the orbital order and\noccupation [7, 11]. GTO in particular lies on the cusp of\nthe AFM-FM phase boundary, making it especially sus-\nceptible to changes in these parameters. Photoexcitation\ndirectly disrupts the orbital occupation, which could af-\nfect the octahedral distortion and thus the spin exchange\ninteraction. This in turn would provide the drive for re-\norientation of Ti spins and change in M zat timescales\nt < τs−l, and for subsequent perturbation of Gd spins\nthrough exchange coupling with Ti. Further calculations\nof the energy scales of the Ti-Gd exchange field and cor-\nresponding timescales are required to confirm this.\nTo compare the magnetic dynamics to the ∆ R/Rre-\nsponse, we fit the MOKE data to a series of exponentials\nsimilar to Eq. 1 and subtract the fits. Once again, a\nslow coherent oscillation is revealed, shown in Fig. 5(b)\nfor the data taken at 1 T. The similarity of the oscilla-\ntory Kerr signal to the oscillation in ∆ R/Ris striking –\nboth phonon modes have the same frequency, same time-\ndependent redshift, and same temperature dependence,\nwith the amplitude growing rapidly at TC. We rule out\nthe possibility of a magnon – at lower fields there is no\nchange in the frequency of oscillation as we would ex-\npect from coherent spin precession (SM VII) [19]. The\namplitude is highly field-dependent however, becoming\nmuch smaller at lower fields. These observations suggestthat the oscillatory mode in the MOKE signal, neces-\nsarily a magnetic phenomenon due to the nature of the\nmeasurement technique, has the same origin as the os-\ncillatory mode in the ∆ R/Rsignal. This is consistent\nwith our interpretation of an acoustic strain wave with\nstrong magneto-elastic coupling. This mechanism has\nbeen studied in a variety of ferromagnetic systems, and\ninvolves elastic stress modifying the magnetic anisotropy,\nwhich exerts a torque on the spins and alters the net\nmagnetization [54–56].\nTo quantify the acoustic phonon response, we show in\nFig.6(a) the FFT of the full ∆ R/Rresidual, taken from\nFig.3(b) (inset). The oscillatory mode with ∼20 ps pe-\nriod featured in Fig. 3(b) appears as a strong peak at\n∼50 GHz. In this region of interest, it is apparent that\nthere are additional higher frequency modes in addition\nto the 50 GHz mode. The temperature dependence is\nalso clear; while the FFT amplitude is nearly constant\nat high temperatures, it grows rapidly upon approaching\nTC= 32 K and a higher frequency peak at ∼80 GHz\nemerges. This again suggests coupling to the magnetic\norder. Fig. 6(b) applies the same FFT analysis to data\ntaken at an increased pump/probe wavelengthof 800 nm.\nThe features are similar, but exhibit a clear redshift as\nindicated by the red dashed lines. This behavior is con-\nsistent with an acoustic strain wave since it arises (for\n∆R/R) from interference of the probe with itself. The\nphonon frequency is wavelength dependent, its form is8\ngiven by:\nf= 2nv/λ, (2)\nwherenis the index of refraction, vis the sound veloc-\nity, and λis the probe wavelength [49]. As we observe,\na higher probe wavelength results in a lower frequency\nacoustic phonon. Using the measured index of refraction\nnin Fig.1(b) wecanalsoestimatethe sound velocity. At\nthe lower frequency peak near 50 GHz we obtain a sound\nvelocity of 7 .2×103m/sand 7.7×103m/sfor a 660 and\n800 nm probe, respectively. This is a very reasonable\nrange for acoustic propagation in solid materials. These\nresults, and the fact that the oscillation frequency does\nnot depend on magnetic field, confirms our classification\nof the phonon mode as an acoustic strain wave.\nTomorecloselyexaminethelinktomagnetism,weplot\nthe integrated FFT amplitudes for all frequency peaks in\nFig.6(c). The normalized curves show a striking trend;\nthe amplitude is nearly constant at high temperatures,\nbut sharply increases at or very near to the magnetic or-\ndering transition. The temperature dependence of the\nFFT amplitudes follows the magnetic order parameter\nandisremarkablysimilartothedivergenceoneexpectsat\na second-order magnetic phase transition. This indicates\nthe presence of magneto-elastic coupling. The acoustic\nattenuation of sound waves near magnetic phase tran-\nsitions is well studied, and literature suggests that the\nattenuation follows power law behavior, similar to our re-\nsult [54]. In the vicinity of TC, energy density and spin\nfluctuations play the primary role in attenuation. This\nbehavior has been studied in a wide range of magneto-\nelasticallycoupledmaterials, including Ni [54], CoF 2[57],\nand MnF 2[58].\nA final interesting feature to note is shown in Fig. 6(d-\ne), comparing an FFT of the ∆R/Rdata limited to the\nfirst 40 ps (d) and to the first 60 ps (e) of the scan. This\nanalysis reveals that the high frequency component at\n80 GHz begins to emerge only after 40 ps, which is also\nvisible in the time-domain data (Fig. 3(b) inset). This\ntimescale is similar to the spin-lattice coupling timescale\nmeasured in both ∆ R/Rand MOKE, which ranges from\n∼50 – 150 ps. We have also discussed the spin dynam-\nics following photoexcitation, where Ti spins are imme-\ndiately perturbed and Gd follows after exchange path-\nway alterations and spin-lattice thermalization. Given\nthe similar timescales, we suggest that the emergence of\nthe 80 GHz mode indicates the onset of Gd spin dynam-\nics. Roughly 50 ps after photoexcitation the Gd spin sub-\nsystem begins thermalizing and fluctuating. This damps\nthe acoustic oscillation and changes the magnetic back-\nground. The nowhigherenergyofthe Gd spins altersthe\nspin-phonon and magnetostrictive interaction strengths,\nresulting in a change to the magnetically-coupled elas-\ntic parameters of the lattice and a subsequent shift in\nphonon frequency.\nA microscopic description of magneto-elastic coupling\ninvolves a transient modification of the exchange interac-\ntion. As the acoustic wave propagates it modulates thedistance between lattice sites and spins. This in turn pro-\nduces a periodic modification of the exchange interaction\nbetween neighboring spins, coupling the acoustic wave to\nthe magnetic order parameters. The result is an attenua-\ntion of the acoustic wave in the high-temperature phase\nwhere spin fluctuations are large, lessening as spin cor-\nrelations increase in the low temperature ordered phase.\nThe same mechanism decreases acoustic attenuation, in-\ncreasing the phonon amplitude, in an applied magnetic\nfield as observed in our MOKE signal. This has been\ndescribed by an approximate analytical theory [58–60],\nwhich generally predicts maximal acoustic damping at\nthe critical point and a MHz frequency shift in the or-\ndered phase. We observe that the damping is consis-\ntently large throughout the high temperature paramag-\nnetic phase, and we do not observe such a frequency shift\nwith temperature. In our experiment, however, a MHz\nfrequency shift is too small to be observed, and the dy-\nnamics at picosecond timescales are strongly coupled to\nout-of-equilibrium degrees of freedom that will affect the\nacousticwavepropagationandattenuationinotherunan-\nticipated ways.\nThephonon behaviorweobserveundoubtedly suggests\na strong coupling of the lattice to the magnetic order\nin GdTiO 3. Furthermore, the mechanism implies tran-\nsient exchange modification on an ultrafast timescale.\nThese conclusions are not without precedence. Ultra-\nfast magneto-elastic coupling has been demonstrated by\nBigotet al., for example, in Ni thin films [61], with exper-\niments going so far as to control the magnetic precession\nthrough acoustic pulses [62]. Kimel et al.have shown\noptical quenching of magnetic order through phonon-\nmagnon coupling in FeBO 3[48] and Nova et al.have\nshownthatMid-IRandTHzexcitationresonantwithspe-\ncific lattice modes is able to drive collective spin preces-\nsion [63]. Our work represents another potential method\nof using light to indirectly alter the magnetic degrees of\nfreedom on ultrafast timescales, through coupling to an\nacoustic phonon mode.\nV. CONCLUSION\nWe have used a multi-modal approach, consisting\nof time-resolved photoinduced reflectivity and magneto-\noptical Kerr (MOKE) spectroscopy, to study magneto-\nelastic coupling in the ferrimagnetic insulator GdTiO 3.\nWe observe multiple, clear signatures of the ferrimag-\nnetically ordered phase at TC= 32 K in both signals,\nand measure spin-lattice thermalization timescales τs−l\non the order of 100 picoseconds, as might be expected in\na magnetic insulator.\nFrom the MOKE signal we observe long-lived spin dy-\nnamics and optical perturbation of the ferrimagnetic or-\nder. This includes a change in sign of the photoinduced\nmagnetization on the same timescale as spin-lattice cou-\npling. The ferrimagnetic nature of GTO, with two mag-9\nnetic sublattices coupled antiferromagnetically, is respon-\nsible. Photoexcitation at 660 nm directly perturbs the\nTi moments, increasing fluctuations and causing a par-\ntial reorientation and decrease in the projection of Ti\nspins along the Gd moment. This is measured as an\nincrease in the MOKE signal. Heat is then transferred\ntothe Gdsubsystem throughspin-latticecoupling, which\nwhen combinedwith the AFM exchangeinteractionleads\nto a reduction of the Gd magnetic moment along the z-\ndirection, lowering the net magnetization. Modified ex-\nchange pathways likely also play a role in the delayed\nreorientation of Gd spins on these timescales. The data\nshows that (a) there is a delayed response of the Gd ions\ntotheopticalexcitationand(b)thatspin-latticecoupling\nand the AFM exchange interaction facilitates this.\nIn both the reflectivity and MOKE signals, a clear\ncoherent acoustic phonon is present. This strain wave\nlaunched by pump is intimately tied to the sample mag-\nnetism, with an amplitude that grows sharply at TCand\nclosely follows the magnetic order parameter. As the\nacoustic wave propagates it periodically alters the dis-\ntance between local spins, modifying the exchange inter-\naction. In this way, the lattice parameters are coupled to\nthe magnetic order, which causes an attenuation of the\nacoustic mode near and above TC, where spin fluctua-\ntions are large. This represents a laser-induced modifica-\ntion of the exchange interaction on ultrafast timescales\nthroughcouplingtoanacousticphononmode. Whilethe-\nory exists to describe magneto-elastic coupling, it is not\nparticularly well-suited to the experiment and timescalesmeasured here. A deeper theoretical understanding of\nthe mechanisms at work would be instrumental in quan-\ntifying our results and motivating further studies. This\nwork also suggests that more controlled excitation may\nbe of interest in transiently controlling the properties of\nmaterials. An experiment ofthis naturehas alreadybeen\nproposed to modify the exchange interaction in GTO, us-\ning a resonant mid-IR pulse to directly excite specific\nphonon modes [17]. The work performed here indicates\nthe potential for GTO, and likely other titanates, as tun-\nable magnetic materials, and highlights the need for fur-\nther investigations of this nature on the road to coherent\ncontrol of materials on ultrafast timescales.\nVI. ACKNOWLEDGEMENTS\nWe thank Leon Balents for helpful discussions and\nassistance with interpretation of the data. This work\nwas supported primarily by ARO Award W911NF-16-\n1-0361 and additional support was provided by the W\nM Keck Foundation (SDW). 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Malinowski 2, M. Hehn 2, F. Pressacco 3, M. \nSilly 3, F. Sirotti 4, C. Boeglin 1 and N. Bergeard 1 \n1 Université de Strasbourg, CNRS, Institut de Physiqu e et Chimie des Matériaux de Strasbourg, \nUMR 7504, F-67000 Strasbourg, France.  \n2 Institut Jean Lamour, CNRS - Université de Lorraine , 54011 Nancy, France.  \n3 Synchrotron SOLEIL, L'Orme des Merisiers, Saint-Aub in, 91192 Gif-sur-Yvette, France  \n4 Physique de la Matière Condensée, Ecole Polytechni que, CNRS, 91128 Palaiseau, France  \n \nAbstract : \nWe have investigated the laser induced ultrafast dy namics of Gd 4f spins at the surface \nof Co xGd 100-x alloys by means of surface-sensitive and time-reso lved dichroic resonant Auger \nspectroscopy. We have observed that the laser induc ed quenching of Gd 4f magnetic order at \nthe surface of the Co xGd 100-x alloys occur on a much longer time scale than that  previously \nreported in “bulk sensitive” time-resolved experime nts. In parallel, we have characterized the \nstatic structural and magnetic properties at the su rface and in the bulk of these alloys by \ncombining Physical Property Measurement System (PPM S) magnetometry with X-ray \nMagnetic Circular Dichroism in absorption spectrosc opy (XMCD) and X-Ray Photoelectron \nspectroscopy (XPS). The PPMS and XMCD measurements give information regarding the \ncomposition in the bulk of the alloys. The XPS meas urements show non-homogeneous \ncomposition at the surface of the alloys with a str ongly increased Gd content within the first \nlayers compared to the nominal bulk values. Such la rger Gd concentration results in a reduced \nindirect Gd 4f spin-lattice coupling. It explains t he “slower” Gd 4f demagnetization we have \nobserved in our surface-sensitive and time-resolved  measurements compared to that previously \nreported by “bulk-sensitive” measurements.  \nIntroduction : 2 \n The discovery of deterministic helicity-independent  all-optical spin switching (HI-\nAOS) [OST12] has driven intensive investigations on  the laser induced ultrafast spin dynamics \nin ferrimagnetic RE-TM alloys [RAD11, LOP13, GRA13,  BER14, RAD15, HIG16, HEN19].  \nThe element- and time-resolved experiments have sho wn that the RE 4f and TM 3d spin \ndynamics occur on distinct time-scales, which is co nsidered as the key ingredient for HI-AOS \n[IIH18]. They have also shown that the characterist ic times associated with these ultrafast \ndynamical processes depend on the temperature and o n the alloy composition [LOP13, FER17, \nCHE19, REN20, FER21, FER23]. Recently, Hennecke et al have also reported on transient \nmagnetization in-depth gradients that appear in GdF e alloys excited by femtosecond laser \npulses [HEN22]. These seminal works call for furthe r systematic experimental and theoretical \ninvestigations to reveal the correlations between t he static magnetic properties of these \nferrimagnetic alloys and their unusual laser induce d ultrafast spin dynamics [ATX14]. Such \nknowledge is of prime importance to identify the mo st appropriate materials for technological \napplications. In parallel, previous experimental wo rks have revealed that amorphous RE-TM \nferrimagnetic alloys show lateral [GRA13] and in de pth non-homogeneous composition \n[HEB16, HAL18, INY23] as well as RE segregation at interfaces [SHE81, BER17]. On the \nbasis of these previous observations, one can legit imately wonder whether the laser induced RE \n4f spin dynamics at the surface of the alloy is aff ected by the composition discrepancies in \nrespect with that in the bulk [FER23]. Disparate sp in dynamics at the surface and in the bulk \nwould be detrimental for technological applications , such as data storage devices that require \nthin magnetic layers [SEN21].  \n In this work, we have investigated the laser induc ed ultrafast dynamics of Gd 4f spins \nat the surface of Co xGd 100-x alloys for bulk compositions x = 80 and x = 65 by means of surface-\nsensitive time-resolved dichroic resonant Auger spe ctroscopy (TR-DRAS) [BEA13, SIL17]. \nWe have observed that the laser induced quenching o f Gd 4f magnetic order at the surface of \nthe Co xGd 100-x alloys occur on a much longer time scale than that  previously reported in bulk \nsensitive time-resolved experiments [LOP13, RAD15, FER23]. In parallel, we have \ncharacterized the structural and magnetic static pr operties at the surface and in the bulk of these \nalloys by combining Physical Property Measurement S ystem (PPMS) magnetometry with X-\nray Magnetic Circular Dichroism in absorption spect roscopy (XMCD) [NAK99] and X-Ray \nPhotoelectron spectroscopy (XPS). The PPMS measurem ents confirm the nominal bulk \ncomposition of the alloys. The XPS measurements evi dence the non-homogeneous composition \nat the surface of the alloys with a strongly increa sed Gd content within the first layers [SHE81, 3 \n BER17, BAL16, HAS18, INY23]. As a consequence, we i nfer that the larger Gd concentration \nin these top layers compared to that of the nominal  bulk composition allows to explain the \n“slower” Gd 4f demagnetization we have observed in our surface-sensitive and time-resolved \nmeasurements compared to that previously obtained w ith  “bulk-sensitive” techniques [FER23].      \nExperimental : (methods, techniques, and materials studied)  \nThe Co xGd 100-x(20) alloy layers were grown by DC co-sputtering on  \n[Ta(5)/Cu(20)/Ta(5)] x5  multilayers deposited on Si substrates (units in n m).  The layers were \ncapped with an Al(5) layer to prevent degradation o f the alloy in the ambient atmosphere \n[BER17]. This Al(5) layer is partly oxidized during  exposure to air, but its thickness ensures \nthat a metallic Al layer remains in contact with th e Co xGd 100-x(20) alloys. Therefore, the actual \nstructure of the samples is Al 2O3(~3)/Al(~2)/Co xGd 100-x(20)/ [Ta(5)/Cu(20)/Ta(5)] 5/Si. It is \nworth mentioning that the samples were deposited on  a 1” Si wafer which was cleaved after \ndeposition to ensure identical samples for the vari ous experimental techniques. In this study, \nwe have investigated the static and dynamics magnet ic properties of alloys with the nominal \ncomposition Co 80 Gd 20 and Co 65 Gd 35 respectively. The bulk compositions of the alloys w ere \ncontrolled by tuning the deposition rates of the cr ucibles. The actual alloy compositions were \nverified by recording the temperature dependence of  magnetization by mean of Physical \nProperty Measurement System (PPMS) cryostat with Vi brating Sample Magnetometer (VSM) \noption head (figure 1). Indeed, we have compared th e temperature of magnetic compensation \nor the Curie temperature extracted from these measu rements to the tabulated values in literature \n[TAO74, HAN89] to estimate the composition.   \nThe CoGd magnetic properties were also characterize d by mean of X-ray Magnetic \nCircular Dichroism in absorption spectroscopy (XMCD ) [NAK99]. We have recorded the X-\nray Absorption Spectra (XAS) at the Gd M 4,5  edges in the total electron yield acquisition mode  \n(figure 3). These measurements were performed in th e main UHV chamber on the TEMPO \nbeamline at synchrotron SOLEIL by using circularly polarized soft X-ray [POL10]. The X-ray \nbeam impinges the sample at an angle of αX = 44° in respect with the samples normal (figure \n2) since the Co xGd 100-x alloys display in-plane magnetic anisotropy with a  preferential axis \n[TAY76, BER17]. The data acquisitions were performe d in the remnant magnetic state after \nsaturation along the in-plane preferential axis by a magnetic pulse of 300 ms duration and 200 \nOe maximum amplitude. The XMCD spectra were obtaine d by recording the XAS spectra with \nopposite directions of the external magnetic field (figure 3a and 3c) [BER17]. The coercive \nfield of the Co 65 Gd 35 alloy was below 200 Oe for temperatures (T) ranging  between 80 K to 4 \n 300 K, while the coercive field of the Co 80 Gd 20 alloy was below 200 Oe for T > 200K. Since \nwe have restricted ourselves to qualitative analysi s, we haven’t corrected the displayed spectra \nfor the saturation effects [NAK99]. Prior to XMCD m easurements, the alloys were sputtered in \nthe preparation chamber of the TEMPO beamline to pa rtly remove the Al capping layer to \nincrease the signal. We have kept a metallic Al lay er to prevent the oxidization during the \nmeasurements [BER17].  \nAmorphous RE-TM ferrimagnetic alloys are known to d isplay in-depth compositional \ngradients [HEB16, HAL18, INY23] or RE segregation a t the surface [SHE81, BER17]. The \ncomposition at the surface of the alloys was thus i nvestigated by mean of X-Ray Photoelectron \nSpectroscopy (XPS). The measurements were performed  in the main chamber of the TEMPO \nbeamline by using the SES2002 photoelectron analyze r. Prior to XPS measurements, the alloys \nwere sputtered in-situ with Ar+ ions to completely remove the Al 2O3(~3)/Al(~2) capping layer. \nHowever, we have regularly monitored the thickness of the remaining capping layer as a \nfunction of sputtering time in order to minimize th e etching of the alloys. We have set the \nphoton energy to 700 eV and we have recorded the XP S spectra of the Al 2p, Gd 4f, Gd 4d as \nwell as the Co 3d and Co 3p core-levels (figure 4a and 4b). The sample’s normal was aligned \nwith the entrance of the photoelectron analyzer whi le the X-ray beam impinged the samples at \nan angle of αX = 44° in respect with the sample’s normal (figure 2). We have repeated these \nmeasurements at photon energies of 400 eV and 1000 eV. This protocol allows varying the \nsurface sensitivity of core-level XPS since the ele ctron inelastic mean free path depends on the \nkinetic energy of photoelectrons and thus on the ph oton energy (table 1) [JAB11]. As a \nconsequence, the larger photon energies allow probi ng deeper into the layer. In figure 5, we \nshow the Gd 4f and Co 3p core-levels as a function of the photon energy for the Co 80 Gd 20  (a) \nand Co 65 Gd 35  (b)  alloys, respectively. A Shirley-like background was  subtracted from the \nexperimental XPS core-level peaks, while the peak’s  area was normalized by the \nphotoionization cross-section of the materials [YEH 85]. We have then normalized the spectra \nwith the height of the Gd 4f peak for direct compar ison. In order to quantify the Gd excess at \nthe surface of the alloy in respect with the nomina l composition, we define R as the ratio \nbetween the Gd 4f and Co 3p core-level peak’s area.  The dependence of R on the photon energy \nis depicted in figure 6a. We have developed an elem entary model to provide a qualitative \ndescription of the alloy profiles [PAC23]. We have considered that the XPS signal of the specie \nx coming from the i th  layer /g1827/g3051/g3036 is given by /g1857/g1876/g1868 /g4672 /uni2211 /g4672/g2879/g2869\n/g3030/g3042/g3046 /g4666/g3087/g4667./g4666/g3030/g3004/g3042 /g4666/g3037/g4667∗/g3090/g3004/g3042/g2878/g3030/g3008/g3031 /g4666/g3037/g4667∗/g3090/g3008/g3031 /g4667/g4673/g3036\n/g3037/g2880/g2868 /g4673 with c x(j) \nand λx the concentration of the specie x in the layers j (on top of the layer i) and the 5 \n photoelectron inelastic mean free path (IMFP) respe ctively. The total XPS signal of specie x is \nthen given by /g1827/g3051=/uni2211 /g1829/g3051/g3036/g1827/g3051/g3036 /g3015\n/g3036/g2880/g2868  with N the alloy thickness and R is given by /g3002/g3256/g3279 \n/g3002/g3252/g3290 .  We have used \na genetic algorithm in order to determine the profi le which gives the best match between the \ncalculated dependence of R on photon energy and the  experimental values (figure 6a). \nConsidering the limited inelastic mean free path of  photoelectrons in CoGd alloys (table 1), we \nhave set the composition to the nominal value as gi ven by VSM measurements (figure 1) for \nthicknesses above 5 nm.       \nThe laser induced ultrafast Gd 4f spin dynamics was  recorded by using TR-DRAS \n[BEA13, SIL17] at the TEMPO Beamline of synchrotron  SOLEIL [POL10] by using both the \nhybrid (figure 7) and low-alpha (figure 8) filling mode [SIL17]. This technique offers surface \nsensitivity, element specificity, the sensitivity t o 4f magnetic order and time-resolutions of 60 \nps or 12 ps in the hybrid and low-alpha filling mod e, respectively. In our experiments, the \nenergy of the circularly polarized X-ray pulses mat ches the Gd M 5 absorption edge \n(E hν=1192.7eV) while the Auger photoelectrons (E KE =1184.5eV) are collected by a Scienta \n2002 photoelectron analyzer equipped with a delay l ine [BER11]. The kinetic energies of Auger \nphotoelectrons allow separating them from the photo electrons generated by laser absorption \n[SIR14]. The intensity of the generated Auger photo electrons (in e-/sec) is proportional to the \nX-ray absorption in the Co xGd 100-x layers. Since the X-ray absorption of circularly p olarized X-\nray in magnetic materials depends on the magnetizat ion, the Auger photoelectron yield depends \non the magnetization, as depicted in figures 7(a,b)  and figures 8(a,b). The difference between \nthe Auger photoelectrons collected for two opposite  directions of the magnetic field (H+ and \nH-) is proportional to the magnetization [SIL17]. T he variation in the amplitude of the signal \nproportional to the magnetization is indicated by t he blue M arrows in the figures. The TR-\nDRAS experiments have consisted in recording the Au ger photoelectrons distribution for both \nmagnetic field helicities as a function of the dela y between the IR pump and the X-ray probe \npulses by using either the hybrid (time resolution ~60 ps) or the low alpha (time resolution \n~12ps) modes. In figures 7(a,b) and 8(a, b), we dis play the Auger photoelectron yield for two \nmagnetic field helicities at negative delay (a) and  t=100 ps after laser excitation (b) for hybrid \nand low-alpha modes, respectively. Then, the photoe lectron Auger yields are integrated over \nthe investigated kinetic energy range. The differen ce of the integrated signal between both \nmagnetic field helicities as a function of the pump -probe delay (figure 7c and 8c) is labelled \nTR-DRAS in the following. The laser frequency was s et to 141 kHz, which is six times lower \nthan the frequency of the isolated electron bunch i n the SOLEIL filling pattern. The thick 6 \n Ta/Cu/Ta buffer layer beneath the Co xGd 100-x alloys allows to enhance the heat dissipation \nduring the pump-probe experiments, ensuring moderat e temperature elevation by DC-heating. \nThe laser beam impinges the sample at an angle of 6 7° in respect with the sample’s normal \n(figure 2). For each delay step, a pulsed magnetic field of 300 ms duration, 200 Oe amplitude \nis applied along the magnetic easy axis to saturate  the alloys. The acquisition is then performed \nat remanence (H ext  = 0 Oe), which prevents perturbation of the photoe lectron trajectories \ntowards the analyzer. The detection scheme allows c ollecting separately the photoelectron \npulses generated by every isolated X-ray pulses, wh ich means that between two successive laser \nexcitations, 1 X-ray pulse probes the excited magne tic state of the alloy while the next 5 pulses \nprobe the non-excited magnetic state of the alloy. Thus, we have direct evidences that the \nsamples are not damaged during the acquisition and that the magnetization of the alloy is \nrestored to its equilibrium state between two succe ssive excitations even in absence of external \nmagnetic field.  It also allows normalizing the photoelectron Auger yield coming from the \nexcited magnetic state by the photoelectron Auger y ield coming from the non-excited magnetic \nstate to increase significantly the signal to noise  ratio [SIL17]. The time-resolved experiments \nwere carried out for the Co 65 Gd 35 and Co 80 Gd 20 alloys (figures 7 and 8). We have not fully \nremoved the Al protective layer during the time-con suming pump-probe experiments to avoid \noxidization of the top layers [BER17]. The cryostat  temperatures were set to 80 and 200 K \nwhile the laser fluences were 2.1 and 2.8 mJ/cm² re spectively. The higher temperature for the \nCo 80 Gd 20 alloy was needed to reach a coercive field below 20 0 Oe, while the larger laser fluence \nwas selected to obtain similar demagnetization ampl itudes. We have restricted the laser fluence \nto 2.8 mJ/cm²  because we observed traces of degradation at the su rface of the alloys for laser \nfluence above 3 mJ/cm². Furthermore, we have restri cted our investigations to moderate \nexcitation (~50 % demagnetization) for direct compa rison with previous bulk-sensitive XMCD \nexperiments performed in pure Gd [WIE11, ESC14] and  CoGd alloys [FER23]. As a \nconsequence, for the Co 65 Gd 35 (Co 80 Gd 20 ) alloy, the equilibrium temperature was below \n(above) the temperature of magnetic compensation [F ER17]. The laser induced dynamics of \nGd 4f spins in the Co 65 Gd 35 alloy was also recorded on a broader delay range by  using the \nhybrid mode (figure 7). Such measurements allow to determine accurately the characteristic \nrecovery times in order to lower the uncertainties in fitting the TR-DRAS data obtained in the \nlow-alpha mode. For the Co 80 Gd 20  alloy, the delay range investigated in the low-alp ha operation \nmode was sufficient to estimate the characteristic recovery times.     \nExperimental results and discussion : 7 \n  The net magnetization of the alloys as a function of temperature measured by PPMS-\nVSM magnetometry and XMCD spectroscopy are plotted in figure 1 and figure 3 respectively. \nWe observe that the magnetization of the Co 65 Gd 35 alloy goes to zero at T ~ 370 K (figure 1), \nas expected from tabulated data [TAO74, HAN89]. We also observe that the magnetization of \nthe Co 80 Gd 20 alloy crosses zero at T = 170 K (figure 1), which i s the temperature of magnetic \ncompensation (T comp ). This value for T comp  is also consistent with the nominal composition as  \nattested by tabulated data [HAN89]. The opposite si gns of the XMCD signal at the Gd M 5 edges \nfor the Co 80 Gd 20 alloy (figure 3b) and the Co 65 Gd 35 alloy (figure 3d) confirm that the XMCD \nspectra were acquired above (below) the temperature  of magnetic compensation for the \nCo 80 Gd 20  (Co 65 Gd 35 ) alloy. The characterization of the magnetic propert ies by mean of PPMS-\nVSM magnetometry and XMCD spectroscopy show that th e actual composition of the alloy in \nthe bulk matches the nominal composition.  \nAfter ensuring that the alloys have the correct com position in the bulk, we focused on \nthe electronic and structural properties at the sur face of the alloy by means of XPS (figure 4). \nIn figure 5a and 5b, we observe that the height of the Co 3p peak increases in respect with that \nof the Gd 4f peak when the photon energy increases for both samples. It demonstrates that the \nsurface of the alloys is less concentrated in Co th an the bulk. Our model gives a qualitative \nagreement with the experimental variation of R (fig ure 6a). The profile which corresponds to \nthe calculated variation of R with photon energy is  depicted in figure 6b. In the case of Co 80 Gd 20  \nalloy, the simulated profile shows a segregated Gd layer on top of an almost homogeneous \nsample with almost the nominal composition as previ ously reported for a comparable alloy \ncomposition [BER17]. However, the simulated profile  of the Co 65 Gd 35  alloy shows a \npronounced Gd compositional gradient at the surface  of the alloy and thus, a sizable increase \nof the Gd contents over almost 2 nm in respect with  the nominal composition. A more accurate \ndetermination of the profile given for layers deepe r than 2 nm should be confirmed by other \nexperimental techniques with higher probing depth. Our observations based on XPS \ndemonstrate that the alloys are non-homogeneous and  that these non-homogeneity’s \ndiscrepancies depend on the alloy composition.              \nThe TR-DRAS measurements show a quenching of the Gd  4f magnetic order which is \nfollowed by a recovery for both alloys (Figure 7c, 8c). However, our data show different \ndynamical responses to laser excitation for the Gd sublattices in both alloys. Indeed, we observe \nthat the maximum demagnetization amplitudes are rea ched at delays d max  = 28 ps and d max  = \n100 ps for the Co 80 Gd 20  and the Co 65 Gd 35 alloys respectively (figure 8). The data were adjus ted 8 \n by exponential functions convoluted with a Gaussian  function to extract the characteristic \ndemagnetization ( τM) and recovery ( τR) times (solid lines in figures 7c and 8c) [BOE10, LOP12, \nBER14]. We have estimated τM = 12 ps and τR = 55±10 ps for the Co 80 Gd 20 alloy and τM = \n37±10 ps and τR = 400±60 ps for the Co 65 Gd 35 alloy. For the Co 80 Gd 20 alloy, we have set a lower \nlimit of 12 ps for τM during the fit processing which is given by the ex perimental time resolution. \nFor the Co 65 Gd 35 alloy, the recovery time ( τR = 400±60 ps) was extracted from the data acquired \nin the hybrid mode (figure 7c).  \nThe faster recovery for the Co 80 Gd 20  alloy ( τR = 55±10 ps) measured at T = 200K \ncompared to the Co 65 Gd 35 alloy ( τR = 400±60 ps) measured at T = 80K is probably partl y caused \nby the temperature dependent thermal conductivity i n RE-TM alloys as shown by Hopkins et \nal. in FeCoGd alloys [HOP12]. Such temperature depe ndent recovery for Gd 4f spins in CoGd \nalloys has been reported recently [FER23]. Furtherm ore, the article published by Ferté et al. \nallows explaining that the characteristic demagneti zation time τM < 12 ps for the Co 80 Gd 20 alloy \nis shorter than that for the Co 65 Gd 35  alloy ( τM = 37±10 ps), because of the different Gd \ncomposition of the alloys. Indeed, in CoGd alloys, the indirect Gd 4f spin-lattice coupling is \nenhanced in respect with pure Gd layers by the Gd 4 f – Gd 5d intra-atomic exchange coupling, \nthe Co 3d – Gd 5d inter-atomic exchange coupling an d the Co 3d spin-lattice coupling [REN19, \nHEN19, FER23]. By reducing the Co concentration at the surface of the alloy, the indirect Gd \n4f spin-lattice coupling is also reduced. However, the laser induced dynamics of Gd 4f spins on \ntop of the Co 65 Gd 35 alloy shows similarities with that reported by Wiet struk et al. for pure Gd \nlayer in low-alpha mode at BESSY [WIE11]. Indeed, a lthough the time-resolution of our TR-\nDRAS experiment and the statistic do not allow for resolving the two-step demagnetization, it \nis worth noticing that the demagnetization times we  extracted from our experimental data ( τM \n= 37±10 ps) is similar to the characteristic times of the “slow” demagnetization process of pure \nGd [WIE11] rather than that reported by means of bu lk-sensitive TR-XMCD in Co 72 Gd 28  alloy \n(τM ~ 4 ps [FER23]). This observation is thus qualitat ively consistent with the rich Gd content \nat the surface of the Co 65 Gd 35  alloys, which induces disparate Gd 4f spin dynamic s compared \nto that expected within the bulk. It is worth notic ing that Graves et al. have shown that lateral \ncomposition gradient also influence the spin dynami cs in RE-TM alloys [GRA13]. However, \nthe amplitude in composition non-homogeneities they  have reported are much lower than that \nwe estimate at the surface of our layers. In the ca se of the Co 80 Gd 20  alloy, any discussion \nregarding the characteristic demagnetization time f or the Gd 4f sublattice and its comparison \nwith previous bulk-sensitive measurements [LOP13, R AD15] are hampered by the time-9 \n resolution of our experiment. Our works show that a n accurate description of the Gd 4f spin \ndynamics in ferrimagnetic alloys requires particula r attention to their structural properties. Our \nwork also calls for further in-depth resolved exper iments in RE-TM alloys with nanometer \nspatial resolution combined with sub-picosecond tim e resolutions either at large scale facilities \n[JAL17] or by using table-top X-ray sources [HEN22] .      \n  \nConclusions:  \nIn this work, we propose an experimental investigat ion on the structural and static \nmagnetic properties as well as on the laser induced  ultrafast Gd 4f spin dynamics at the surface \nof Co xGd 100-x ferrimagnetic alloys. We have shown that the surfa ce of the CoGd alloys is Gd-\nrich compared to the bulk and these non-homogeneiti es are composition dependent. Indeed, the \nGd migration at the surface is more pronounced in a lloys that display a larger Gd content in the \nbulk. We have used surface sensitive time-resolved dichroic resonant Auger spectroscopy to \nshow that the Gd 4f spin dynamics is much slower at  the surface of Co 65 Gd 35  alloy compared \nto that previously reported in the bulk of comparab le alloys by means of time-resolved XMCD \nmeasurements. This discrepancy can be explained by the larger Gd composition on the top of \nthe alloys which results in a reduced indirect Gd 4 f spin-lattice coupling. However, our \npreliminary work gives a partial view of the Gd 4f spin dynamics because of the limited time-\nresolution and calls for further in-depth and time- resolved experiments. Among the point to be \nclarified, we hint that the exchange coupling betwe en the Gd-rich surface and the bulk of the \nalloy may also play a role on the Gd 4f spin dynami cs by modifying the effective Curie \ntemperature (T Curie ). T Curie  is known to be a key parameter to determine the ch aracteristic \ndemagnetization times in the framework of the micro scopic 3 temperatures model [KOO10]. \nThese issues have to be addressed for future techno logical applications which require a fine \ntuning of the characteristic demagnetization times [REM20] or a downsizing of the \nferrimagnetic alloy layers [LAL17, SEN21].  \nAcknowledgements: \n \nWe acknowledge SOLEIL for provision of synchrotron radiation facilities in using the beamline \nTEMPO. This project has received fundings from the European Union’s Horizon 2020 research and \ninnovation program under the Marie Skłodowska-Curie  grant agreement number 847471, the French \nnational agency for research ANR-20-CE42-0012-01 an d from the Région Grand Est. The competence 10 \n center Magnetism and Cryogenics, Institut Jean Lamo ur, is acknowledged for magnetometry \nmeasurements. \n \nReferences:  \n \n[OST12]  Ostler et al. Nature Commun. 3, 666 (2012)  \n[RAD11]  Radu et al, Nature. 472, 205 (2011)  \n[LOP13] V. López-Flores, N. Bergeard, V. Halté, C. Stamm, N. Pontius, M. Hehn, E. Otero, E. \nBeaurepaire, and C. Boeglin, Phys. Rev. B 87, 21441 2 (2013)  \n[GRA13]  Graves et al. Nature Materials. 12, 293 (2013)  \n[BER14] Bergeard et al. Nature Communications 5, 34 66 (2014) \n[RAD15] I. Radu, C. Stamm, A. Eschenlohr, F. Radu, R. Abrudan, K. Vahaplar, T. Kachel, N. \nPontius, R. Mitzner, K. Holldack, and A. Föhlisch,  SPIN 5, 1550004 (2015)  \n[HIG16]  Higley et al. 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B 96, 220411(R) (2 017)  \n \nTable 1: Mean free paths of the different core-level photoel ectrons Co3d, Co3p, Gd4f and Gd \n4d as a function of the incident photon energy as g iven by [JAB11].  \n \n \nPhoton energy \n(eV)  Mean free path λ (Å)  \nλCo 3d  λCo 3p  λGd 4f  λGd 4d  \n400  6.6  6.2  9.8  7.3  12 \n 700  8.5  8.2  14.7  12.5  \n1000  10  9.7  19.4  17.3  \n \nFigures:  \n \n \nFigure 1: Magnetization as a function of temperatur e measured by mean of VSM magnetometry for the Co 80 Gd 20  (black \nfilled squares) and Co 65 Gd 35  (red empty circles) alloys. The values are normali zed by the magnetization at T = 80K.  \n \n \nFigure 2: Geometry of the experiment where the exte rnal magnetic field can be applied in the plane of the films. Incidence \nangles of the X rays and IR pump are defined by α X (=44°) and α IR  (=67°) in respect with the surface normal. \n \n13 \n  \nFigure 3: X-ray absorption spectra for two opposite  helicities of the external magnetic field (black a nd red curves) at the Gd \nM4,5  absorption edges for the Co 80 Gd 20  alloy at T = 220K (a) and for the Co 65 Gd 35   alloy at T = 80 K (c). (b) The XMCD spectra \nat the Gd M 4,5  absorption edges for the Co 80 Gd 20  alloy at T = 220K and T = 300K. (d) The XMCD spect ra at the Gd M 4,5  \nabsorption edges for the Co 65 Gd 35  alloy at T = 80K, T = 200K and T = 300K.   \n \n \nFigure 4:  X-ray photoelectron spectra taken at 700 eV photon energy showing the Co 3p , 3d, Gd 4f ,4d and the Al  2p core-\nlevels for the alloys (a) Co 80 Gd 20  and (b) Co 65 Gd 35 .  \n \n \nFigure 5: X-ray Photoelectron spectra of the Co 3p and Gd 4f core-levels for (a) Co 80 Gd 20  and (b) Co 65 Gd 35  at the photon \nenergies of 400 eV (black spectra), 700 eV (red spe ctra) and 1000 eV (green spectra). The spectra have  been subtracted by a \n14 \n Shirley function and normalized by the photoionizat ion cross section [JAB11]. All spectra have been no rmalized to 1 at the \nGd4f. \n \n \nFigure 6: (a) Comparison between the experimental ( symbols) and the calculated (solid lines) energy de pendence of the ratio \nR between the Gd 4f and Co 3p core levels. (b) Simu lated Gd composition profile at the surface of the Co 80 Gd 20  (black line) \nand the Co 65 Gd 35  (red line) alloys \n \nFigure 7: Auger electron yield for two magnetic fie ld helicities at negative (a) and positive (b) dela ys for the Co 65 Gd 35 alloy \nmeasured by using the hybrid filling mode (time res olution 60 ps). (c) Normalized magnetic circular di chroism as a function \nof the pump-probe delay for the Co 65 Gd 35 alloy (red filled circle) at T = 80 K. The solid li nes are exponential fit as described in \ntext. \n15 \n  \nFigure 8: Auger electron yield for two magnetic fie ld helicities at negative (a) and positive (b) dela ys for the Co 65 Gd 35 alloy \nmeasured by using the low-α filling mode (time-reso lution 12 ps). (c) Normalized magnetic circular dic hroism as a function of \nthe pump-probe delay for the Co 65 Gd 35 (red filled circle) at T = 80 K and the Co 80 Gd 20 at T = 200 K (black empty squares) \nalloys. The solid lines are exponential fit as desc ribed in text. \n \n \n"    },    {        "title": "1409.8430v2.Predicting_a_Ferrimagnetic_Phase_of_Zn2FeOsO6_with_Strong_Magnetoelectric_Coupling.pdf",        "content": " \n1 \n Predicting a Ferrimagnetic  Phase of Zn 2FeOsO 6 with Strong  Magnetoelectric Coupling  \nP. S. Wang ,1 W. Ren,2 L. Bellaiche,3 and H. J. Xiang*1 \n \n1Key Laboratory of Computational Physical Sciences (Ministry of Education), State Key \nLaboratory of Surface Physics, Collaborative Innovation Center of Advanced Microstructures , \nand Department of Physics, Fudan University, Shanghai 200433, P. R. China  \n2Depa rtment of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, P. R. China  \n3Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, \nFayette ville, Arkansas 72701, USA  \nAbstract  \nMultiferroic  materials, in which ferroelectric and magnetic ordering coexist, are of fundamental \ninterest for the development of novel memory devices that allow for electrical writing and non -\ndestructive magnetic readout operation. The great challenge is to create roo m temperature \nmultife rroic materials with strongly coupled ferroelectric and ferromagnetic (or ferrimagnetic) \nordering s. BiFeO 3 has been the most heavily investigated  single -phase multiferroic to date due to \nthe coexistence  of its magnetic order and ferroe lectric order at room temperature. However, there \nis no net magnetic moment in the cycloidal ( antiferromagnetic -like) magnetic state  of bulk \nBiFeO 3, which severely limits its realistic applications in electric field controlled spintronic  \ndevices. Here, we predict that double perovskite Zn 2FeOsO 6 is a new multiferroic with properties \nsuperior to BiFeO 3. First, there are strong ferroelectricity and strong ferrimagnetism at room \ntemperature in Zn 2FeOsO 6. Second, the easy -plane of the spontan eous magnetization can be \nswitched by an external electric field, evidencing the strong magnetoelectric coupling existing in  \n2 \n this system. Our results suggest that ferrimagnetic 3d -5d double perovskite may therefore be \nused to achieve voltage control of mag netism in future spintronic devices.  \nPACS:  75.85.+t,  75.50.Gg, 71.15.Mb, 71.15.Rf  \n \n \n The highly efficient control of magnetism  by an  electric field in a  solid may widen the \nbottle -neck of the state -of-the-art spin -electronics (spintronics) technology, such  as magnetic \nstorage  and magnetic random -access memory. Multiferroics  [1-7], which show simultaneous \nferroelectric and magnetic ordering s, provide an ideal platform for the electric field control of \nmagnetism because of the coupling  between their dual order parameters. For realistic \napplications, one need s to design/discover room temperature multiferroic materials with strong \ncoupled ferroelectric and ferromagnetic (or ferrimagnetic) ordering.  \n Perovskite -structure bismuth ferrite (BiFeO 3) is currently the most  studied room \ntemperature single -phase multiferroic, mostly because its large  polarization and high \nferroelectric  Curie temperature (~820 ° C) make it appealing  for applications in ferroelectric non -\nvolatile memories and high temperature  electronics. Bulk BiFeO 3 is an antiferromagnet with \nNé el  temperature T N ≈ 643 K  [8]. The Fe magnetic moments order almost in a checkerboard G -\ntype manner with a cycloidal spiral  spin structure in which the antiferromagnetic (AFM) axis \nrotates through  the crystal with an incommensurate long -wavelength period  [9]. This spiral spin \nstructure leads to a cancellation of  any macroscopic magnetization. The magnetic properties of \nBiFeO 3 thin films  were found to be  markedly different from those of the bulk: The  spiral spin \nstructure seems to be suppressed and a weak magnetization appears  [10]. Nevertheless, the \nmagnetization is too  small  for many applications  [11]. In addition, an interesting low -field  \n3 \n magnetoelectric (ME) effect at  room temperature was discovered in Z -type hexaferrite \nSr3Co2F24O41 by Kitagawa et al.  [12]. Unfortunately, the electric polarization (about 20 µC/m2) \ninduced by the spin order  is too low  [13]. \nIn searching for new multiferroic compounds, the double perovskite system was proposed \nas a promising candidate  [14, 15]. The double perovskite structure A 2BB′O6 is derived from the \nABO 3 perovskite structure. The two cations B and B ′ occupy the octahedra l B sites of perovskite  \nwith the rock salt ordering. Double perovskite Bi 2NiMnO 6 was successfully synthesized under \nhigh-pressure, which displays the multiferroic behavior with a high ferroelectric transition \ntemperature (485 K) but a low ferromagnetic transition  temperature (14 0 K) [14]. Polar LiNbO 3 \n(LN) -type Mn 2FeMO 6 (M=Nb, Ta) compounds were prepared at 1573 K under  7 GPa  [16]. \nUnfortunately, the magnetic ground state of Mn 2FeMO 6 is AFM with a rather low Né el  \ntemperature  (around 80 K). Very recently, LN -type polar magnetic Zn 2FeTaO 6 was obtained via \nhigh pressure and temperature synthesis  [17]. The AFM magnetic transition temperature (T N∼22 \nK) for Zn2FeTaO 6 is also low. In a pioneering work, Ležaić  and Spaldin proposed to design \nmultiferroics based on 3d -5d ordered  double perovskites  [18]. They found that Bi 2NiReO 6 and \nBi2MnReO 6 are insulating and exhibit a robust ferrimagnetism that persists above room  \ntemperature. Although coherent heteroepitaxy strain may stabilize the R3 ferroelectric ( FE) state, \nfree-standing bulk  of Bi2NiReO 6 and Bi 2MnReO 6 unfortunately take the non -polar P21/n \nstructure as the ground state. The magnetic properties of non -polar double perovskites \nCa2FeOsO 6 [19] and Sr 2FeOsO 6 [20] were also theoretically investigated.  Recently, Zhao et al.  \npredicted  that double perovskite superlattices R 2NiMnO 6/La 2NiMnO 6 (R is a rare -earth ion) \nexhibit an  electrical polarization and strong ferromagnetic order near room temperature  [21].  \n4 \n However, the ME coupling in these superlattices appears to be weak  and the polariz ation to be \nsmall . \nIn this work, we predict that double perovskite Zn 2FeOsO 6 takes the FE LN -type structure \nas the ground state through a global structure searching. Similar to Bi 2NiReO 6 and Bi 2MnReO 6, \nZn2FeOsO 6 exhibit a strong ferrimagnetism at room  temperature. Importantly, there is a rather \nstrong magnetic anisotropy with the easy -plane of magnetization perpendicular to the FE \npolarization  due to the presence of the significant 3d -5d Dzyaloshinskii -Moriya  (DM) interaction . \nThis suggests that the swi tching between the 71 °or 109°FE domains by the electric field will \ncause the rotation of the magnetic easy -plane. Our work therefore indicates that Zn 2FeOsO 6 may \nbe a material of choice  for realizing voltage control of magnetism at room temperature.  \nIt is well-known that there  are several lattice instabilities  including ferroelectric distortion s \nand oxygen octahedron rotation s in perovskite materials . We now examine how double -\nperovskite Zn 2FeOsO 6 distort s to lower the total energy. For this purpose, we per form a global \nsearch for the lowest energy structure based on the genetic algorithm (GA) specially designed for \nfinding the optimal structural distortion  [22]. We repeat the simulations three times. All three \nsimulations consistently show that the polar rhombohedral structure with the R3 space group \n[shown in Fig. 1( b) and ( c)] has the lowest energy for Zn 2FeOsO 6. Similar to Zn 2FeTaO 6, \nZn2FeOsO 6 with the R3 structure is based on the R3c LN -type structure. Previous experiments \nshowed that double perovskite  structure A 2BB′O6 may adopt other structures, such as the P21/n  \n[23], \n3R[24], and C2  structure s [25]. Our density functional theory ( DFT ) calculations show \nthat the R3 phase of Zn2FeOsO 6 has a lower energy than the P21/n, \n3R , and C2 structures by  \n0.22, 0.09, 0.45 eV/f.u., respectively. This strongly suggest s that double perovskite Zn 2FeOsO 6  \n5 \n adopts the R3 structure as its  ground state. This can be understood by using the tolerance factor \ndefined for the LN -type ABO 3 system. It was shown  that when the tolerance factor \n(\n2( )AO\nR\nBOrrt\nrr\n , where \nAr , \nBr and \nOr  are the ionic radii  of the A -site ion, B -site ion and O ion ) \nis smaller than 1, the polar \n3Rc  structure is more stable than the \n3Rc  ABO 3 structure due to the \nA-site instability  [26]. In the case of Zn 2FeOsO 6, the average tolerance factor ( 0.75) is smaller \nthan 1, which suggests that the \n3R  structure is more stable than the non -polar \n3R  structure.  \nNote that phonon calculation s shows that the \n3R  state of Zn2FeOsO 6 is dynamically stable  [27]. \nAdditional tests indicate that the Fe and Os ions tend to order in a rock salt manner (i.e., double \nperovskite  configuration ) to lower the Coulomb interaction  energy [27,28]. \nOur electronic structure calculation shows that the R3 phase of Zn2FeOsO 6 in the \nferrimagnetic  state is insulating  [27]. The density of states plot shows that the Fe majority 3d \nstates are almost fully occupied, while the minority states are almost empty. This suggests that \nthe Fe ion takes the high -spin Fe3+ (d5) valence state. It is also clear that the Os ion takes the \nhigh-spin Os5+ (d3) valence state, which contra sts with the case of Ba2NaOsO 6 where the Os \natom takes a 5d1 valence electron  configuration  [29]. This is also consistent with the total \nmagnetic moment of 2 µB/f.u. for the ferrimagnetic state from the collinear spin -polarized \ncalculation . Through the four -state mapping approach  which is able to deal with  spin in teractions \nbetween two different atomic types [30], we compute the symmetric exchange parameters to find \nthat the magnetic ground state of R3 Zn 2FeOsO 6 is indeed ferrimagnetic. The Fe -Os \nsuperexchange interactions mediated by the corner -sharing O ions [J 1 and J 2, see Fig. 1( c)] are \nstrongly AFM (J 1 = 31.68  meV , J2 = 29.62  meV ). Here, the spin interaction parameters are  \n6 \n effective by setting the spin values of Fe3+ and Os5+ to 1. Our results seem  to be in contradiction  \nwith the Goodenough -Kanamori rule which predicts a ferromagnetic interaction between a d5 ion \nand a d3 ion since the virtual electron transfer from a half -filled σ -bond e orbital on the d5 ion to \nan empty e orbital on the d3 ion dominates the antiferromag netic π -bonding t -electron transfer  \n[31]. This discrepancy is because the <Fe -O-Os angle  (about 140 °) in Zn 2FeOsO 6 is much \nsmaller than 180 °(which is the case for which the Goodenough -Kanamori rule applies ). The \nAFM super -superexchange interaction [J 3 and J 4, see Fig. 1( c)] between the Os ions is much \nweaker (J 3 =7.25 meV , J4 =3.24 meV ) than J 1 and J 2, but not negligible. Since the Fe 3d orbitals \nare much more localized than Os 5d orbitals, the super -superexchange interactions between the \nFe ions are negligible. The dominance of the AFM Fe -Os exchange interactions suggests that the \nmagnetic ground state of R3 Zn 2FeOsO 6 should be ferri magnetic, which is also confirmed by the \nMonte Carlo (MC) simulations to be discussed below.  \nThe magnetic anisotropy in R3 Zn 2FeOsO 6 is investigated by including spin -orbit coupling \n(SOC) in the DFT calculation. The DFT+U+SOC calculation shows that the magnetic moments \ntend to lie in the hexagonal ab -plane. Interestingly, when magnetic moments are in the ab -plane, \nthere is a spin canting between the Fe 3d moments and Os 5d moments. We perform a test \ncalculation to explicitly illustrate this point. We fix the Fe spin moments along the x -direction, \nand compute the total energies as a function of the angle ( α) between Fe and Os  moments. Fig. \n2(a) shows that the total energy has a minima at αmin≈174°  and the total energy curve is \nasymmetric with respect to the axis of α = 180°. This is solely due to the SOC effect since the \ntotal energy from the DFT+U calculations has a minima at  α = 180°.  The deviation of α min by 6°  \nfrom 180°  implies that the magnetic moment also acquires a weak component along the y -axis,  \n7 \n which can be naturally understood by the universal law [32] . As shown in Fig. 2(b), the \nferrimagnetic state with the moments  along the c -axis is higher in energy by 0.55 meV/f.u. than \nthe ferrimagnetic state with the Fe and Os moments aligned oppositely in the ab -plane ( α = 180° ). \nThe canting of the spins ( α = 174° ) further lowers the total energy by 0.97 meV /f.u.. Therefore, \nthe spin canting is extremely important for the easy -plane behavior.  \nTo understand the magnetic anisotropy in R3 Zn 2FeOsO 6, we consider the general spin \nHamiltonian  [30]: \n2\nspin ij i j ij i j i iz i j i j iH J S S D (S S ) A S       \n , which includes symmetric \nexchange interactions, antisymmetric DM interactions, and single -ion anisotropy. For the \nmagnetic structure in which the magnetic unit cell only contains one Fe ion and one Os ion, the \ntotal spin interaction energy can be written as:  \n \nz 2 z 2\nspin 1 2 Fe Os 0i Fe Os Fe Fe Os Os\ni 1,6E 3(J J ) ( ) A (S ) A (S )\n         S S D S S , \nwhere the DM interaction vectors D0i are shown in the insert of Fig. 2(a), A Fe and A Os are the \neffective single -ion anisotropy parameters. Here, the DM interaction s between two different \nkinds of atoms are considered [33-35]. Our calculations show that AFe = 0.89 meV and A Os = -\n0.34 meV, suggesting an overall weak easy -plane behavior resulting from the single -ion \nanisotropy term. The exchange interaction between the Os ions is omitted since it is a constant \nfor such magnetic structure. Our analysis shows  that the DM interaction results in the spin \ncanting, and subsequently contributes dominantly to the easy -plane anisotropy [27].  \nBy performing the parallel tempering  (PT) MC simulation using  the full spin Hamiltonian \n(containing symmetric exchange interactions, antisymmetric DM interactions, and single -ion \nanisotropy), we obtain the thermodynamic behavior of the magnetism in the R3 phase of  \n8 \n Zn2FeOsO 6 [36]. As shown in Fig. 3, there is a peak at 39 4 K in the specific heat curve, \nindicating that there is a magnetic  phase transition at 394  K. The total spin moment (M ab) in the \nab-plane is also plotted as a function of temperature. We can see that the in -plane total spin \nmoment increase s rapidly near the magnetic phase transition point. This suggests that the in -\nplane  total spin moment can be chosen as the order parameter, and the paramagnetic phase \ntransforms to the ferrimagnetic phase with moments in the  ab-plane at 39 4 K. We notice that the \nin-plane total spin moment is larger than 2 μB/f.u. at low temperature. This  is a consequence of \nthe spin canting resulting from the DM interactions. Our MC simulation suggests that R3 \nZn2FeOsO 6 is a room temperature ferrimagnet.  Note also that t est calculations show that the \nferrimagnetic transition temperatures of Zn 2FeOsO 6 with the low -energy phases \n3R  and P21/n \nare 344 K and 356 K, respectively. This indicates that the effect of the ferroelectric displacement, \noxygen octahedron tilts, and the associated magnetoelectric coupling on the ferrimagnetic \ntransition temperature is rather weak  and does not prevent Zn2FeOsO 6 from having a magnetic \ntransition temperature above 300K . \nSimilar to the LiNbO 3 case where the paraelectric structure has the \nR3c  symmetry, the \nparaelectric state corresponding to the FE R3 Zn2FeOsO 6 can be chosen to be \nR3 . Fig ure 4(a) \nshows the double well potential for Zn2FeOsO 6. We can see that the barrier between the two FE \nstate with opposite electric polarizations along the pseudo -cubic direction [111] is 0.09 eV/f.u., \nwhich is smaller than  the value (about 0.20 eV/f.u.) in the case of PbTiO 3 [37]. Taking the \nR3  \nstructure as the reference structure, the electric polarization of R3 Zn2FeOsO 6 is computed to be \n54.7 μC/cm2, which  is much higher than that (2 5.0 μC/cm2) in bulk tetragonal BaTiO 3 [38]. Thus, \nR3 Zn2FeOsO 6 should be a switchable ferroelectric with a high electric polarization.  Our tests  \n9 \n based on an effective model Hamiltonian and molecular dynamics simulations show that the \nferroelectric transition temperature of Zn2FeOsO 6 is well above room temperatu re (see Part 9 and \n10 of [27] ). \nLet us now discuss the possible ME coupling in double -perovskite Zn2FeOsO 6. Because of \nthe cubic symmetry of the perovskite structure, there are eight symmetrically equivalent <111> \ndirections. In rhombohedral  Zn2FeOsO 6, the spontaneous electric polarization is directed along \none of  the eight <111> axes of the perovskite structure. Thus, in the sample of double -perovskite \nZn2FeOsO 6, there might occur eight different FE domains [see Fig. 4( b)]. Our above calculations \nshow that the easy -plane of mag netization is always perpendicular to the direction of the electric \npolarizatio n. Although a 180°  switching  of the ferroelectric polarization should not affect the \nmagnetic state, a 71°  or 109°  switch of the FE domains by the electric field will change the  \norientation  of the easy -plane of magnetization, as shown schematically in Fig. 4( c). This could \nbe a promising  route to manipulate the orientation of the ferrimagnetism by an electric field. A \nsimilar ME coupling mechanism in BiFeO 3 thin films has been de monstrated experimentally by \nZhao et al. , who showed that the AFM  plane can be switched by an electric field  [39]. Note that \nmagnetoelectric effects can be classified in to two different types : one for which changing the \nmagnitude of the polarization  affects the magnitude of the magnetization (energy of the form\n22PM\n) and one for which changing the direction of  \nP\n changes the direction of  \nM\n (energy of \nthe form  \nPM\n ). In Zn 2FeOsO 6, the first type of ME effect is weak, while the second type of ME \neffect is strong.   \nWe now compare Zn2FeOsO 6 with the classic multiferroic BiFeO 3. First, they adopt similar \nrhombohedral  structures. Second, both compounds have high electric al polarizations. Third, both  \n10 \n compounds are room temperature multiferroics. Fourth, the ME coupling mechanism is rather \nsimilar in that the magnetic easy -plane can be manipulated by electric field. However, the \nmagnetic ground state of R3 Zn2FeOsO 6 is dramatically different from BiFeO 3. Zn2FeOsO 6 has a \nferrimagnetic ground state, while BiFeO 3 is AFM. And the magnetic anisotropy in Zn2FeOsO 6 is \nstronger than that (0.2 meV when U(Fe) = 5 eV [40]) in BiFeO 3 because of the strong SOC \neffect of the 5d Os ion. These desirable properties  make  Zn2FeOsO 6 suitable for realizing \nelectric -field control of magnetism at room temperature.  \nThe reason why we practically propose d Zn2FeOsO 6 as a possible multiferroic is two -fold. \nFirst, polar Zn 2FeTaO 6 has already been synthesized under high -pressure, as mentioned above. \nSecond, it was experimentally  showed that Ca 2FeOsO 6 crystallizes into an ordered double -\nperovskite  structure with a space gro up of P21/n under high -pressure  and high -temperature, and \nCa2FeOsO 6 presents a long -range  ferrimagnetic transition above room temperature (T c ∼320 K)  \n[24]. Therefore, we expect that Zn 2FeOsO 6 is synthesizable and would most likely exhibit both \nferroelectricity and ferrimagnetism at room temperature.  \nWork at Fudan was supported by NSFC, FANEDD, NCET -10-0351, Research Program of \nShanghai Municipality and MOE, the Special Funds for Major State Basic Research, Program \nfor Professor of Special Appoin tment (Eastern Scholar), and Fok Ying Tung Education \nFoundation.  L.B. thanks the Department of Energy, Office of Basic Energy Sciences, under \ncontract ER -46612 . \ne-mail: hxiang@fudan.edu.cn  \nReferences  \n[1] S.-W. Cheong  and M. Mostovoy,  Nature Mater.  6, 13 (2007).   \n11 \n [2] R. Ramesh  and N. Spaldin , Nature Mater.  6, 21 (2007).  \n[3] S. Picozzi  and C. Ederer,  J. Phys. : Condens. Matter  21, 303201 (2009) .  \n[4] K. F.  Wang,  J.-M. Liu, and Z. Ren, Adv. Phys.  58, 321 (2009) .  \n[5] J. van den Brink  and D. Khomskii,  J. Phys .: Condens . Matter  20, 434217 (2008).  \n[6] Y .  Tokura  and S. Seki,  Adv. 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Matter 24, 312201  (2012) ] states that the weak component of the magnetization has a \ndirection that is given by the cross -product of the oxygen octahedral  tilting vector (that is lying \nalong the c -direction in the hexagonal setting of the R3 phase) with the G -type antiferromagnetic \nvector (that is chosen here to be along the x -axis).  Note that this law also implies that choosing a \nG-type antiferromagnetic v ector being along the same direction as the ox ygen octahedral tilting \nvector (that is, the c -axis) should not produce any weak component of the magnetization \nperpendicularly to the G -type antiferromagnetic vector.  This is precisely what we further fou nd \nin the simulations when the Fe and Os moments are chosen to lie (in antiparallel fashion) along \nthe c -axis.  \n[33] A. Fert and P. M. Levy, Phys. Rev. Lett. 44, 1538 (1980).  \n[34] M. Heide, G. Bihlmayer, S.  Blügel, Physica B 404, 2678 (2009) . \n[35] Y . A. Izyumov, Sov . Phys. Usp . 27, 845 (1984) . \n[36] P. S. Wang and H. J. Xiang, Phys. Rev. X 4, 011035 (2014).  \n[37] R. E.  Cohen,  Nature (London)  358, 136 (1992); C. Ederer and N. A.  Spaldin, Phys. Rev. B  \n74, 024102 (2006).  \n[38] J. B. Neaton  and K. M.  Rabe , Appl.  Phys. Lett . 82, 1586 (2003).   \n14 \n [39] T. Zhao , A. Scholl ,  F. Zavaliche , K. Lee, M. Barry , A. Doran , M. P.  Cruz , Y . H.  Chu, C. \nEderer, N. A.  Spaldin, R. R.  Das, D. M.  Kim, S. H.  Baek , C. B.  Eom , and R. Ramesh , Nature \nMater.  5, 823 (2006).  \n[40] C. Ederer  and N . A. Spaldin , Phys. Rev. B 71, 060401 (R) ( 2005 ). \n \nFIG. 1(color online) . (a) The pseudo -cubic structure of double perovskite  Zn2FeOsO 6. (b) The \npolar rhombohedral structure of R3 Zn 2FeOsO 6.  (c ) The Fe -Os superexchange interactions \nmediated by the corner -sharing O ions (J 1 and J 2) and t he AFM super -superexchange interaction \n(J3 and J 4) between the Os ions . (d) The ferrimagnetic s tructure  with the Fe and Os spin moments \naligned in the ab -plane.   \n \n15 \n  \nFIG. 2(color online).  (a) The total energies as a function of the angle α [see the insert ] from the \ndirect DFT+U+SOC calculations  (small circles ). The total energy has a minima at αmin≈174° . \nThe total energy curve can be  described rather well by the formula \nzz\nspin 1 2 01 04E 3(J J )cos 3(D D )sin    \n (blue line) . The insert shows t he DM interaction \nvectors D0i (i =1, 6) between a Fe ion and six Os ions  and t he definition of the angl e α between \nthe Fe spin and Os spin in the ab -plane.  (b) Relative energies between different spin \nconfigurations. The ferrimagnetic state with the moments along the c -axis is higher in energy by \n0.55 meV/f.u. than the ferrimagnetic state with the Fe and Os moments aligned oppositely in the \nab-plane ( α = 180° ), and t he canting of the spins ( α = 174° ) further lowers the total energy by \n0.97 meV /f.u.. \n \n16 \n  \nFIG. 3(color online).  The specific heat and t he total in -plane spin moment (M ab) as a function of \ntemperature from the PTMC simulations. The specific heat  curve indicates that the ferrimagnetic \nCurie temperature (Tc) is 39 4 K. The in -plane total spin moment increases rapidly near T c. \n \n \n \n \n \n \n17 \n  \nFIG. 4(color online).  (a) The total energy as a function of  the electric polarization  for Zn2FeOsO 6. \nIt displays the  double well potential with an energy barrier of 0.09 eV/f.u.. (b) Eight possible \norientations of the FE polarization vector ( P) in the sample of  double -perovskite Zn2FeOsO 6. (c) \nIllustration of the ME coupling in Zn 2FeOsO 6. A 71°  or 109°  switch of the FE domains  by the \nexternal electric field will be associated with the reorientation  of the easy -plane of magnetization.  \n \n"    },    {        "title": "1805.10607v1.Prediction_of_new_multiferroic_and_magnetoelectric_material_Fe3Se4.pdf",        "content": "Prediction of new multiferroic and magnetoelectric material Fe 3Se4 \nDeobrat Singh1, Sanjeev K. Gupta2,*, Haiying He3 and Yogesh Sonvane1,* \n1Advance Materials Lab, D epartment of Applied Physics, S. V. National Institute of Technology,  \nSurat 395007, India.  \n2Computational Materials and Nanoscience Group, D epartment of Physics and Electronics,  \nSt. Xavier’s College, Ahmedabad 380009, India.  \n3Department of Physics and Astronomy, Valparaiso University, Valparaiso, Indiana 46383, USA . \n \nAbstract  \nNowdays , multiferroic materials with magnetoelectric coupling have many  real-world  applications in \nthe fields of novel m emory devices. It is challenging is to create multiferroic materials with strongly  \ncoupled ferroelectric and ferrimagnetic orderings at room te mperature . The single crystal of ferric \nselenide ( Fe3Se4) shows  type-II multiferroic due to the coexistence of ferroelectric as well as magnetic \nordering at room temperature.  We have  investigated  the lattice instability , electronic structure , \nferroelectric , ferrimagnet ic ordering and transport  properties  of ferroelectric metal  Fe3Se4. The density \nof states shows considerable hybridization of Fe -3d and Se -4p states near the Fermi level confirming its \nmetallic behavior. The magnetic moments of Fe cations follow  a type -II ferr imagnetic and ferroelectric \nordering with a calculated total magnetic moment of 4.25\nB  per unit cell (Fe 6Se8). The strong \ncovalent bonding nature of Fe -Se leads to its ferroelectric properties. In addition, the symmetry analysis \nsuggests that tilting of Fe sub -lattice with 3d -t2g orbital ordering is due to the Jahn -Teller (JT) \ndistortion. This study provides further insight in the development of spintronics related technology \nusing  multiferroic materials.  \n \n*Corre sponding author s: sanjeev.gupta@sxca.edu.in   (Dr. Sanjeev Gupta)  \nyas@phy.svnit.ac.in  (Dr. Yogesh Sonvane)  \n \n  Introduction  \nRecent year, m ultiferroic and magnetoelectric materials are becoming more and more indispensable  for \nmany forms of current multi-functional technology, such as highly sensitive magnetic field sensor, \nfilter  transducers, filters, phase shifters, memory devices and oscillator s1-4. For practical applications, \nwe needs to discover multiferroic materials at room temperature which is strongly coupled ferroelectric \nand ferrimagnetism ordering. In addition to their important polarization properties, ferroelectrics are \nalso pyroel ectric5, where i t develops  a voltage across the material upon heating, while  in the case of \npiezoelectric  a voltage is developed in response to strain across the material. These properties allow \nferroelectric  materials  to be utilized in many device applications, including non -volatile memory6, \nthermal detectors7, piezoelectric applications8, and energy harvesters9. The coexistence of magnetism \nand ferroelectricity in a material is called multiferroicity, which is  of even  great er technological and \nfundamental importance. This has added o ther potential applications . For instance , it can be utilized in \ndata storage system s, which count  both magnetic and electr onic state s of the compound to store \ninformation, and in the magneto -elect ronic device s using multiferroic  thin films10. \n Some transition metal compounds, such as BiMnO 3 and BiFeO 3 with magnetic Mn3+ and Fe3+ ions, \nare ferroelectric11, owing  to the intricate interplay between  spin, charge, orbital and lattice degree of \nfreedom  in th ese material s12. The mechanisms behind ferroelectricity , however, are  not fully \nunderstood because of the complicated structures of early ferroelectrics ( i.e. BaTiO 313). The compound \nCaMnO 3 is also an interesting counter example which shows ferroelectricity and magnetic ordering due \nto the  d3 configuration of Mn atom s14. It is worthy to be  noted that in general the ferroelectricity \nbehavior require s nearly  unoccupied  subshell  orbitals of the transition metal cations , while partial ly \nfilling d orbitals are required to have magnetic moment s. This dilemma largely hinders the coexistence \nof ferroelectricity and magnetism in a material and explains the scarcity  of multiferroic  materials15. \n Recently,  Bishwas  et al.,16 studied the  synthesized manganese doped iron selenide  nanostructures \nin an attempt t o increase the energy product . Another research group Shao -jie Li et al.,17 obtained  high \nCurie temperature and coercivity performance of Fe 3-xCrxSe4 nanostructures which can be utilized in \nthe alternative low -cost hard -magnetic materials . Gen Long et al .,18 demonstrated that  at low \ntemperatures, Fe 3Se4 nanostructures exhibit giant coercivity. It was proposed that t his unusual ly large \ncoercivity originates from the large magneto -crystalline anisotropy of the monoclinic structure of \nFe3Se4 with ordered Fe vacancies . The ferromagnetic material  BaFe 2Se3 is of particular interest because \nit breaks the parity symmetry of the crystal  structure and displ ays exchange striction effects19. Indeed, \nthe iron displacements in the crystal structure are prominent, as reve aled by neutron studies20, 21. Dong \net al.22 show ed that the first nearest -neighbor distances between Fe (↑) and Fe (↑) [or Fe (↓) and Fe (↓)] at 0 K become 3.14 Å, much larger  than the Fe (↑) and Fe (↓) distance 2.88 Å. On the other hand, this \nexchange striction is not sufficient to induce Ferroelectric (FE) and Polarization (P) since it breaks \nparity but not space -inversion s ymmetry.  Most ferroelectric materials are transition metal oxides having \nvacant d subshells. In this respect the  two configurations are not so different , but study of the difference \nin filling of the d -orbitals as theoretically expected for ferroelectricity and magnetism makes these two \nordered states mutually exclusive.  According to Giovannetti et. al. [24], the ferroelectric and metallic \nstate are found in LiOsO 3 material.  Interestingly, however, i t has been  suggested that several other \nmeta llic transi tions could be \"ferroelectric\"23 such as the transition in LiOsO 324 due to the appearance of \na polar axis.  \nIn this work, we focus on the  Fe3Se4  material derived from the monoclinic  phase. We have study \nthe lattice instabilities including ferroelectric distortions in monoclinic Fe 3Se4 materials. We have \nperformed first principles calculations within the spin -polarized density functional theory (DFT) \nframework to obtain the  electronic band structure, the role of electron correlations in the metallic state ,  \nmagnetic properties  of the Fe 3Se4 material.   \nMethodology  \nIn th is work,  all calculations are based on the  density functional theory (DFT)25 as implemented in \nthe Quantum  Espresso (QE) package26. The Kohn -Sham equation s were solved using  the Perdew -\nBurke -Ernzerhof (PBE) e xchange -correlation functionals27 formulated within the  generalized grad ient \napproximation (GGA) scheme28. We have included sixteen valence electrons for Fe (3s2, 3p6\n, 3d6\n, 4s2) \nand six valence electron s for Se (4s2\n, 4p4) in our calculation s. The kinetic energy cut-off is set to  60 Ry \nwhich yielded good convergence in results of energies and ground state structural parameters. For \nstructural optimization we have used t he conjugate gradient algorithm29. The lattice parameters and the \ninternal coordinates of atoms were optimized with in the space group of I12/m1 (monoclinic ) with the \ncriteria  for force and pressure below  10−3 a.u. A Monkhorst -Pack k -point mesh30 of 9×9×11 was used \nfor electronic band structure and density of state calculations , while  the spontaneous polarization w as \ncalculated using  the Berry phase method31 with a k-point mesh of 5×5×7 . All the calculations were \nspin-polarized . Scalar relativistic Troullier -Martins ultra-soft pseudopotential s32 were employed  with \nnon-linear core correc tions33. \n \nResult s and Discussion  \nStructural and electronic properties  \n \nAs a benchmark test for the approach used in this work, we have investigate d the structural , electronic and magnetic properties of the monoclinic Fe3Se4 phase . Fe3Se4 adopts a normal spinel \nstructure  as shown in Fig. 1  and the calculated lattice parameters are listed in Table 1. The Fe3+ ions are \nlocated in the octahedral sites of the monoclinic l attice formed by Se2- anions forming magnetic \nordering . The calculated structural parameter s are in very good agreement with previous reported \nvalues (Table 1).  \n \nTable  1. Optimized  structural lattice constants of monoclinic Fe3Se4. \nReferenc es a (Å)  b (Å)  c (Å) β (º) \nPresent  6.100  3.520  11.030  91.15  \nExp34,35,16 6.208, 6.167, 6.159  3.525, 3.537, 3.493  11.2832, 11.170, 12.730  92, 92 , - \n \n \nFigure 1. (A) Relaxed  structure of monoclinic Fe3Se4 (Fe in red and Se in yellow) ; (B) distribution of \nelectric dipole moments ( black arrows) of Fe atoms due to partial ionic displacement along the a \ndirection).  \n  \n \nFigure 2. The electronic band structure of monoclinic Fe3Se4 with up spin (left side) and down spin \n(right side) shown separately.  \n \nThe electronic band structure of monoclinic Fe 3Se4 is plotted  in Figure 2 . The Fermi level EF is set \nto zero. It is clearly shown that there are bands crossing  at EF, thereby  the band gap is zero and Fe 3Se4 \nis metallic in nature.  This is further confirmed from the density of states ( DOS ) plots (see  Figure S1 in \nESI), which demonstrate  asymmetric -spin  and  -spin DOS. A careful look of the projected density \nof states (PDOS) reveals ferrimagnetic spin order ing between the iron and selenium atom s. The \nselenium atoms carried a small magnetic moment (0.14 \nB unit cell ) due to charge transfer and \ngeometric distortion . Therefore , monoclinic Fe3Se4 shows metallic behavior in the ferrimagnetic (FM) \nstate.  \nIn order to check whether the metallic behavior is an artifact of neglecting the spin -orbit coupling  \n(SOC) , we have further calculated electronic band structure of Fe 3Se4 with the inclusion of SOC \ninteraction  (shown in Figure S2 in ESI) . The spin -orbit energy splitting  is large r at points of high -\nsymmetry . For instance, t he spin -orbit splitting at gamma point  is about  14 meV and 30 m eV in valence \nband maximum (VBM) and conduction band minimum (CBM) , respectively.  Since  the monoclinic \nFe3Se4 lattice has inversion symmetry, due to the Kramers degeneracy36, each energy band line is at \nleast doubl y degenerate for both spin state s. Our results show that  the inclusion of the SOC interaction \ninfluences only three p -energy bands of Se atom s and five d -energy bands of Fe  atom s and the metallic \nbehavior remains unchanged . \n \n \nFigure 3. PDOS of Fe ions in relaxed Fe 3Se4. The numbering of Fe is the same as in Figure 1(B).  Fe-1. \nFe-2 and Fe -5 forms one layer , while  Fe-3, Fe -4 and Fe -6 forms another . \n \n The origin of the metallic behavior of the monoclinic Fe 3Se4 can also be understood from the \nPDOS plots (Figure S1 in ESI). It is clear that the spin -up and spin -down states determining the FM \nmetallicity mainly originate from the edge Fe -3d and edge Se -4p orbitals.  Fe atoms contribute more to \nDOS than the Se atoms at Fermi level and the majority of the density of states near the Fermi level for \nFe3Se4 is attributed to the Fe -3d states. The Se -3p bands overlap wit h the Fe -3d bands in the -7 eV to 3 \neV energy range, representing a hybridization of the Se -3p and Fe -3d states to form the covalent \nbonding while Se -3p and Se -3s orbitals also have small contribution to the magnetic property of Fe 3Se4. \nThe difference of t he spin -up band and spin -down band of Fe -3d orbitals show that they carry very \nlarge magnetic  moment in Fe 3Se4. Further, the spin polarization is negative near Fermi because of the \nelectronic DOS of spin -down electrons is larger than that of spin -up electr ons.  \n Furthermore, we have identified the orbital ordering (OO) state in Fe 3Se4 as shown in projected \nDOS in Figure 3 with five electrons in 3d states of Fe ions in the 3D -coordinates ( xyz) with the x and y \naxes pointing to the crystal [\n101 ] directions and z axis directed to the crystal c-axis. In the minority \nspin channel, four Fe ions (Fe -1, Fe -2, Fe -3 and Fe -4) show t 2g bands right below the Fermi level ( EF) \ndown to1.6 eV with reduced DOS at the EF. All Fe ions in the unit cell of bulk Fe 3Se4 can be \ncategorized  into two groups according to the orbital characters : spin-up/down e g bands fully \nunoccupied and spin -down t 2g bands partially occupied, respectively. Bands of Fe -1, Fe -2, Fe -3 and Fe -\n4 [Figure 3(A, B, C and D)] are of predominate d x2\n-y2and d z2 characters, while bands of Fe -5 and Fe -6 \n[Figure 3(E) and 3(F)] are of mainly d xy, dz2 and d x2\n-y2 orbitals.   \nThese result s demonstrate that the configuration  of -spin 3dt 2g OO state s in the Fe sub -lattice , \nwhere a  -spin electron of each Fe has occup ied a mixed t 2g state. The mixing of orbitals is a \ncombination of two of the t 2g states  with the third empty t 2g states37. In addition, it is  found that the -\nspin electrons of Fe from 3dt 2g OO state occupying, respectively, the canted d xy and d yz orbitals of Fe -1 \nto Fe -4 and d yz of Fe-5 and Fe -6 sublattice , as they  contribute more at the Fermi level . The OO pattern \nis clearly seen in the distribution of charge density on each Fe3+ with the distribution on Fe -1, Fe -2, Fe -\n3 and Fe -4 belonging  to one state, while with Fe -5 and Fe -6 belonging to another state.  \n \nSymmetry analysis  \nAccording to  the ionic model37, we have acquired  a metallic ground state in consistence . It is also \nshown that the tilting 3dt 2g OO states  on the Fe sub -lattice is  strongly  related to the John -Teller (JT) \ndistortions37. The main contributions in electronic structure were utilized for Fe ions to investig ate the correlation effects in 3d -electrons.  The OO states are generally found in 3de g manganite frameworks \nwhere the helpful Jahn -Teller (JT) mutilations are huge because of the solid hybridization between the \neg and O -2p electrons38. The t2g OO states with higher degeneracy and moderately weaker JT distortion  \nis additionally found in confined 3d frameworks magnetite39. In the ionic model, the five 3d electrons \nof the Fe3+ ion possess the t 2g triplet degenerate  and leave the higher e g doublet degenerate  vacant, \nunder the octahedral crystal  field (see Figure S4 in ESI) . As per Hund's rule , Fe3+ is in the high spin  \nstates with the spin  arrangement of (\n3\n2gt\n2\n2gt  ), giving rise to a magnetic moment of 4.25\nB unit \ncell: and it indicates metallic ground  state with the majority -spin at the Fermi level accountable for \nthe conductivity . Assess ing three 3d electrons, the spin  up t 2g states  is moderately confined with a \nsuppressed bandwidt h and energy of lower band, while the -spin t2g band is pushed upwards \nmarginally [se e ESI in Figure S 1]. Besides, the -spin t2g band is thus completely occupied and at E F \nthere is no band gap , giving rise to a metallic ground state in consistence with the valence setup (\n3\n2gt\n2\n2gt\n ) of Fe from ion ic model as display ed in Figure S 3 in ESI . \nThe tilting of FeSe 6 octahedral site (Fig. 1  (A)) is to accommodate  two sorts of distortion of the Se \nlattice : the two Se ions at the tip of the octahedron deform  along the z-axis (Fe-Se top bond length = \n2.44 Å), while two ions of the coplanar Se move upwards and two  ions move  downwards along the z-\naxis. The JT distortions  with Fe-Se bond length (2.46 Å , between Fe -1 and Se ) along the y-axis and \nextended bond length (2.63 Å) along the x-axis (Figure  1) split the triply degenerate  t2g orbitals into the \nlower d xz and higher d yz and d xy orbitals . The JT distortions  could  bring down the Coulomb interaction \nbetween the ions of Se  and the Fe d xz states , which is a combination  of d xy and d yz states  as appeared in \nFigure S 4 in ESI . Moreover , the d xz states  with intermediate  directions between nearest Se anions would \nadditionally stabilize the lattice distortion.  \n It is well known tha t, if we include the spin -orbit coupling (SOC) interaction then the observed \norbital -ordering in  SrRuO 3 would be destroyed ; otherwise if we do not include SOC interaction then \nthe polyhedral crystal field makes the cubic harmonics (break ing the inversion symmetry)40 a nature \nbasis set resulting in lower t 2g and higher e g bands37. The Se octahedral site also break parity in each \nsite as Figure 1  shows that Se -5 is above the ladder’s plane, but the next Se -7 is below, and the \ndistances of Se -5 and Se -7 to the iron ladder plane should be the same in magnitude and opposite in \nsign (“antisym metric”). However, the OO  introduces a fundamental modification in the symmetry.  \nNow the blocks m ade of four Fe (↑) [or four Fe (↓)] are no longer identical to pair of two Fe (↑) and \ntwo Fe (↓). Then, the heights of  Se-5 and Se-7 do not require  to be antisymmetric anymore; their \ndistances to the  xz plane can become different. A  similar mechanism works for the edge Se's, e.g., Se -1 and Se -5. As a result, the atomic positions of Se break the space inversion symmetry, creating a local \nFerroelectric (FE) Polarization (P) pointing perpendicular to the iron yz plane (almost along the x- axis).  \nTo clarify more of the features of  OO states,  we have calculated charge density distribution (0.66e/ \nÅ3) and  depicted in Figure S5 in ESI  which corresponds  to the -spin t2g bands below the Fermi level ( -\n1.80-0.0 eV) [see  ESI in Figure S1]. It has been observed that electron de nsities corresponding t o -\nspin electrons are totally different for each atom  present in the zigzag whereas when we compare both \nthe spins,  electron densities  are partially  different from  each other. This suggests that in case of Fe 3Se4, \nspin down electrons present at unsaturated edge Fe -atoms are responsible for conduction as also \nconfirmed through corresponding band structure and DOS.  The charge contour  of real space \ndistribution of spin  dependent  electron densities suggests that  the possibility  of metallic spin  \npolar ization in Fe 3Se4 structures is totally attributed  to the localization  of unpaired electrons at \nunsaturated Fe atoms  which are situated at face s of the unit cell in Figure S 5(A) and S5(B) in ESI .  \n \n The yellow  spheres show  that the Se anions attract more electrons while, the Fe cations lose more \n3d electrons in Fe 3Se4, shown by bright magenta lobes pointing along the Fe directions. By contrast, the  \ndensity difference is weak but also exists in the Se ladder plane. There is a dark yellow sphere \nintroduce d as a Se atom, with negative value: this suggests that the outmost electrons of Fe are more \nextended ( delocalized), also supporting the covalent scenario for Fe 3Se4. \n \nMagnetic and ferroelectric properties  \n \nThe calculated total magnetic moment per unit cell o f monoclinic Fe3Se4 with 14 atoms per unit cell \n(6-Fe atom and 8-Se atom) is 4.25\nB . This value is consistent  with the previous DFT calculations \n(4.34\nB /unit cell)35, but significantly higher than  the reported  experimental value  for \nFe3Se4 nanostructures  (2.2\nB /unit cell )35. This was attributed to the  spin fluctuation and the long -range \nordering as measured by experiments, which normally would be smaller than the calculated value at 0 \nK.  \nWe have calculated spontaneous polarization (Ps)  along  each of the directions  x, y and z, for the \nferroelectric phase apply ing the Berry phase approach for bulk Fe3Se4. The direction of Ps in Fe 3Se4 \nsingle crystal lies in the monoclinic ac plane and the magnitude of the segment of spontaneous \npolarization along the a-axis of the monoclinic un it cell. The spontaneous polarization (Ps) in Fe 3Se4 \nhas a value of ~44.60 μC/cm2 and the direction of Ps vector makes an angle 91.15° with the major \nsurface (normal to c-direction). The calculated values of three component s of the spontaneous polarization  vectors are P x= 23.14 μC/cm2, Py= 6.20 μC/cm2and P z=37.62 μC/cm2, respectively.  The \nmagnitude of spontaneous polarization of Fe 3Se4 is in very good agreement with previously reported  \nexperimental value41. Roy et al .,42 provided the spontaneous polarization of ferroelectric bismuth \ntitanate  with a value of 42.83  μC/cm2 which  is fairly comparable  to our calculated value for \npolarization . Another study by Ravindran  et al.,43 gave the spontaneous polarization of BiFeO 3 which is \nalso within the reported agreement between theoretical and experimental values. Therefore, it is clear \nthat the calculated values of spontaneous polarization using GGA fall in the range of 43 -68 μC/cm2. \nDielectric  and transport properties  \n Furthe rmore, we have calculated the frequency dependent optical properties including dielectric \nfunction, absorption  coefficient  using DFT within the random phase approximation (RPA)44. The \nfrequency dependent dielectric function can be written as  \n12 ( ) ( ) ( ) i       . Where\n1() and \n2()\nare the real and imaginary parts of the complex dielectric function, respectively. \n2()  is \ndetermined by summation over electronic states and \n1()  is obtained using the Kramers –Kronig (KK) \nrelationship45. \n The real and imaginary parts of the complex dielectric function \n  versus frequency for bulk \nFe3Se4 are presented in Figure (4A and 4B). The real (\n1 ) and imaginary (\n2 ) part follow the general \ntendency of a metal which can be well explained by the classical Drude theory. In Figure 4A and 4B, \nwith increasing in frequency, the\n1 decreases while \n2 initially increases and decreases above the \n1.5x1014 Hz up to 2.0x1014 Hz region, as expected in a metal46. The variation of dielectric constant \n  \nand loss tangent is shown in Figure 4C and 4D. With increasing frequency, the \n  decreases and \ntan\ninitially increases at certain frequencies and then it is decreases in all the polarization direction (in -\nplane and out -of-plane). The loss tangent measures the loss -rate of power in an oscillatory dissipative \nsystem47. It is clearly seen that the variation \ntan executes  in the same trend as \n2 . Since \n2  is lower \nthan \n1 , then the energy loss  of the materials  is relatively low. This suggests that the material  possesses \ngood optical qualities due to lower energy losses and lower  scattering of the incident radiation47. Our \ntheoretical results of the dielectric function of bulk Fe 3Se4 are in excellent agreement with previously \nreported experimental results48 except \n2 . But according to other theoretical investigation our results is \nexpected in a metal46. \n  \nFigure 4. (A) Real and  (B) imaginary  part of the complex dielectric constant, (C) magnitude of the \ncomplex dielectric function  and (D) loss tangent  verses  frequency at 1014 Hz at room temperature (300 \nK) of bulk Fe 3Se4. (E) negative value of real part of complex dielectric function is a function of photon \nenergy.  \n The real part of complex dielectric function \n1  is shown in Figure 4(E). In Fig ure 4(E) , we plot \nthe real parts of dielectric functions  in all the polarisation direction  for the materials. Character of this \nmaterial exhibit metallic behavior, i.e. Drude peaks at low energy due to intraband contribution and the \nreal part of ε(ω) crossing from negative value to positive value with increasin g frequency  and it’s \nshows o scillatory behavior upto 15 eV . The strong anisotropy in the real part of complex dielectric \nfunction is also quite obvious. Remarkably, because of this anisotropy, the real part of complex \ndielectric function, \n1  in x, y and z -direction , changes sign at a frequency which is different from \neach other , leading to an extended frequency window in which the each components have different \nsigns shown in shaded region with gray color in Figure 4(E) . Such type of sign difference is the \ncharacteristic feature in real part of dielectric function is known as indefinite media49. Generally, \nindefinite media are mostly in artificially assembled structures  which require complicated fabrication \nprocess and usua lly have high dissipation. This theoretical results suggest that crystalline solid Fe 3Se4 \nin the bulk form  would just be indefinite materials for a frequency range spanning the near infrared.  \n \nFigure 5. The variation of (A) thermal conductivity\n , (B) heat capacity \npC  as a function of \ntemperature.  \n The thermal conductivity ( ) of a material  originates from two fundamental sources: (i) electrons \nand hole s transporting heat (κ e) and (ii) phonons trave lling through the lattice (κ l). Most of the \nelectronic term (κ e) is straight  forwardly associated with the electrical conductivity which can be well \nexplained by  Wiedemann –Franz hypothesis50. From Figure 5(A), the thermal conductivity  of Fe3Se4 is \nalmost linearly increased from 50 K to 400 K , and at room temperature the thermal conductivity is 4.56  \nW/ (m K). The thermal conductivity is  dominated  by the lattice contribution  as it is about 1.0 W/m K at \nT=50 K and it increases to 5.57  W/mK at T=400 K which is  higher  than the values observed in  most  \nthermoelectric materials50, 51. Since  the heat flow in a material is directly proportional to its thermal \nconductivity , the heat flow in bulk Fe 3Se4 will be  higher  as well .  \n       Specific heat estimation  is one of  the most reliable techniques for exploring temperature reliance of \nmaterials.  The heat capacity \npC  of bulk Fe 3Se4 increas es with temperature as shown in Figure 5(B ),  \ndemonstrat ing a similar behavior  as the thermal conductivity . In addition, the variation of \npC  with \ntemperature follow s the well-known  Debye form and shows excellent fit at lower  temperatures.  Our \ntheoretical results are in  good agr eement with experimental results48. Furthermore, the estimated \nelectric conductivity as a function of temperature of Fe 3Se4 reveals nearly a linear behavior  with a \npositive slope  as plotted  in Figure 6. This could be ascribed to the way that the electrical conductivity \nwas evaluated based on  DFT by assuming a constant scattering time  \n14( 10 ) s  . In general,  in ca se of \nmetallic system s, the electrical conductivity decreases with respect to temperature owing to the \noccurance of more  collisions  among electrons and between electrons  and phonons  which shorten the \nmean free path of charge carriers. I n our case , similar behavior are found in  its metallic system , the \nelectrical conductivity of Fe 3Se4 continuously decreases  with increasing tem perature. This is likely due \nto the enhanced delocalization of hybrid orbitals.   \n \nFigure 6. The variation of electronic conductivity\n  as a function of temperature.   \n \n \n \n \n \n \n \n \n \n \n \nFigure 7. The absorption coefficient \n  in the unit of 105/cm of bulk Fe 3Se4 for all the polarization  \ndirection in the electric field. (A) The absorption coefficient up to 15 eV and (B) absorption coefficient \nup to 40 eV beyond 28 eV the absorption will be almost zero . \nThe absorption coefficient \n  in all the polarization directions of the electric field (depicted in \nFigure 7) show  two main peaks in all polarizations. The first main peak oc curs at energy around 5  eV \nthat is related to π electron plasmon ic peak s. And other  peak occurs around 15  eV that is associated  to \nπ+σ electron plasmon ic peak s. Moreov er, at the energy range of 10 -20 eV, the value of absorption for \nall cases is very high. \nConclusion s \nIn conclusion s, we investigated the  novel multiferroic material shows  ferroelectric, ferrima gnetism \nand electronic structure in monoclinic Fe3Se4 material.   The nature of orbital ordering , and its  close \nconnection to the JT distortion is unwound, which is responsible for ferroelectric like instability . Fe3Se4 \ntakes  a ferromagnetic ordering with a  total magnetic moment of 4.25\nB per unit cell (6 Fe atom and 8 \nSe atom) . The spontaneous polarization (44.60 μC/cm2) provides evidence of its ferroelectric behavior. \nOur study suggests that the monoclinic phase of Fe 3Se4 material possesses the unusual dual behavior of \nmetallicity and ferroelectricity . The stabilization of the ferroelectric structure in Fe 3Se4 coexisting with \nmetallic conductance is the consequence of a decoupling between the metallic electrons in the t 2g \nelectrons from the soft phonons which break the inversion symmetry . The development of multiferroic \nmaterial are usefull applications in spintronics -related technologies for ultrahigh -density memory and \nquantum -computer devices is underway.  \n \nAcknowledgem ents \nHelpful discussion with Drs. Pankaj Poddar and Mousumi Sen is acknowledged . S. K. G. \nacknowledges the use of high performance computing clusters at IUAC, New Delhi and YUVA, \nPARAM  II, Pune to obtain the partial results presented in this paper. S . K. G and Y. A. S  also thank the \nScience and Engineering Research Board (SERB), India for the financial support (grant nu mbers.: \nYSS/2015/001269  and EEQ/2016/000217, respectively ). D. 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This manuscript version is made available under the CC -BY-NC-ND 4.0 license \nhttp://creativecommons.org/licenses/by -nc-nd/4.0/  \nAbstract:  \n The magnetic orientations and switching fields of a CoCrPt -Ru-CoCrPt synthetic ferrimagnet with \nperpendicular magnetic  anisotropy have been studied in the temperature  range from 2 K to 300 K. It was \nfound that two sets of magnetic transitions occur in the CoCrPt -Ru-CoCrPt ferrimagnet across this \ntemperature range . The first set exhibits three magnetic transitions in the 50  K – 370 K range, whereas  the \nsecond involves only two transitions  in the 2 K and 50 K range. The observed magnetic  hysteresis curves of \nthe synthetic ferrimagnet are explained  using the energy diagram technique  framework  pioneered  by Koplak \net al.  [1] which accurately describes the competition between interlayer exchange  coupling  energy, Zeeman \nenergy, and anisotropy energy in the system. In this work we expand the framework to include synthetic \nferrimagnets ( SFMs ) comprising higher perpendicular magnetic anisotropy materials and large (4X) interlayer \nexchange coupling energies  which are promising for the development of ultrafast (ps) magnetic switching free \nlayers in MTJ structures .  Furthermore, we apply the analysis  to predict SFM magnetic hysteresis curve s in a \ntemperature regime that includes  temperature  extrema that  a synthet ic ferrimagnet would be expected to \nreliably operate at, were it to be utilized as a free layer in a memory  or sensor  spintronic device.  \nKeywords:  \nSynthetic ferrimagnet,  Magnetic films , CoCrPt , Interlayer e xchange coupling, Perpendicular magnetic \nrecording, Magnetic switching , spintronics, magnetic tunnel junctions  \nIntroduction:  \nSynthetic ferrimagnet ( SFM ) trilayers consist of two  antiparallel  ferromagnetic  (FM) films separated by \na thin non-ferromagnetic  metallic interlayer. For the case of identical FM layers, if the films are dissimilar in \nthickness, the SFM structure will exhibit a net magnetic moment  (uncompensated ferrimagnet) . The  interlayer  \nexchange coupling energy  (IECE)  of the SFM  varies with the interlayer thickness in an oscillatory fashion  [2] \nand it has been attributed to various physical processes that include  dipolar magnetostatic interaction s and  2 Ruderman -Kittel -Kasuya -Yosida (RKKY) coup ling. First observed  by Gr ünberg et al.  [3], films exhibiting  \nantiferromagnetic coupling  were utilized  shortly thereafter  in magnetic sen sor devices based on the  giant \nmagnetoresistance (GMR) observed in Fe/Cr antiferromagnet s tructures  [4][5]. More recently, SFMs  and \nantiferromagnets have been utilized in magnetic tunnel junctions (MTJs) to provide exchange bias to the \nreference layer (typically a single, magnetically hard ferromagnet) such that its magnetization is “fixed” [6,7] . \nSFM trilayers or antiferromagnets have been considered to replace the recording or reference layer of the MTJ \n[8,9] . When utilized in an MTJ device the strength  of coupling can determine if the SF M is acting as a reference \nor free layer . The coupling strength is derived from  measuring the magnetic field required for ov ercoming the \nIECE which renders the magnetization orientation of the individual layers to be parallel.  It is noted that free \nlayer  SFM structures incorporated into MTJ devices  have demonstrated lower critical switching current s than \nsingle FM free layers  with out negatively  affecting  thermal stability  [10]. Additionally , we have  proposed that \nSFM free layers can exhibit ultrafast switching speeds down to the picosecond time regime [11]. \nMost  MTJ devices utilize  CoFeB as the FM electrode due to high tunneling magnetoresistance \nmeasured when used with MgO tunneling barrier s [12]. However, the  maximum  thickness of CoFeB exhibiting \nperpendicular magnetic anisotropy  (PMA ) is limited to around 1 .5 nm [13]. The magneti zation of CoFeB is also \nrelatively high, which increases the charge current needed for spin -transfer torque switching. CoCrPt is a \nmaterial of interest for MTJ applications  due to its low magnetization and its large anis otropy  [14], resulting in  \nlower switching current s, improve d thermal stability , and the use of thicker FM layers with concomitant \nprocess control improvements. In addition, the SFM configuration circumvents the materials -restrictive low \nmagnetic damping requirement for selection of the FM thin film for MTJ devices [11]. \n It is essential to tailor the  IECE and switching properties of the S FM structure  for use in memory \ndevices . However, the magnetic properties of the SFM are temperature dependent ,  and memory devices \ncould  be expected to operate  under extreme conditions  within the range of 2 00 K to 370  K. In this paper  the \nIECE of CoCrPt -Ru-CoCrPt trilayer structures has been  investigated from  2 K to 300  K. It has been observed  by \nKoplak et al . [1] that with decreasing temperature , the hysteresis loops of S FMs vary dramatically , and these \nauthors  developed  a formalism  to describe  the observed changes in the hysteresis loop as a function of \ntemperature in a CoFeB -Ta-CoFeB synthetic antiferromagnet . They employ an  energy balance approach that \nincludes the Zeeman energy, the IECE, and energy barriers for switching arising from the effective magnetic \nanisotropy energy.  It was found that t he two main  parameters  controlling the switching behavior with \ndecreasing temperature is the ratio of the magnetic moments of the two constituent ferromagnetic layers  as \nwell as the energy barrier  for switching  of each film, which is temperature dependent . In this paper th e energy \ndiagram technique introduced by Koplak et al. is used to describe the  magnetic  transitions measured  in a  3 CoCrPt -Ru-CoCrPt SFM as a function of  temperature.  This is compared  with their results on the CoFeB -Ta-\nCoFeB S FM structure.  Predictions are also  made for the magnetic transitions of the CoCrPt -Ru-CoCrPt SFM at \n200 K to 370  K to exemplify  the practical use of the energy diagram technique  for assessing the robustness of \na potential sensor or memory device employing a SFM read layer.  \n \nMaterials and Methods : \nAll films were deposited  without substrate heating or bias in a magnetron sputter system with  a base \npressure < 10-7 Torr. The films were grown on oxidized silicon (100) substrates, with dimensions ~ 5mmx5mm. \nThese small coupons were obtained by manual dicing of larger (2” diameter) Si wafers. Thus, the actual \ndimensions of each sample exhibited small variations. Area measurem ents for each coupon were performed \nwith optical microscopy with a reference scale. The error in surface area determination is  ~8%. The thin film \nstructure consisted of the following: Ta(5  nm)/Ru(10  nm)/CoCrPt( 1.7 nm)/Ru( X nm)/CoCrPt( 1.3 nm)/Ru(5  nm). \nThe CoCrPt sputtering target has a nominal composition of Co 70Cr18Pt12. The Ta/Ru seed layer was used to \npromote CoCrPt growth  with its basal (002) plane parallel to the thin film plane (c -axis out of plane) . Magnetic \nhysteresis loops were collected using a Quantum Design MPMS -3 superconducting quantum interference \ndevice (SQUID) magnetometer with 10-8 emu sensitivity.  All samples were mounted in the magnetometer \nusing a plastic straw attachment. Measurements at 370  K could not be performed using this method d ue to \nerrors on account of warping of the straw mount at this temperature.  All magnetic hysteresis loops presented \nin this work were performed  with the substrate aligned perpendicular to the direction of the applied magnetic \nfield.  \nResults and Discussion:  \n \nFigure 1 . Left: Schematic representation of  the film stack  cross -sectio n: red arrows indicat e the direction of \n                                                     \n                   \n               \n                             \n                            \n           \n   4 magnetization of the constituent FM layers at remanence . Right:  Hysteresis curves of the S FM structures with \ntwo different Ru interlayer thicknesses  measured at 300  K. \n As shown in Fig. 1, the IECE of the CoCrPt -Ru-CoCrPt S FM can be tailored by varying the Ru interlayer \nthickness. The IECE per unit area of a S FM with dissimilar FM layers was esti mated  by Koplak et al.  using  the \nexpression:  JEX = -HBm2/S. Here H B is the bias field which  indicates the center of the outer loop . In Fig. 2a-b, H B \nis labeled and measured by finding the center of the outer loops  at which point the SFM becomes saturated. \nAt 2 K (Fig. 2c), there is no outer loop unlike in Fig. 2a -b. In this case a minor loop must be taken to locate the \nHB field, shown by the red curve in Fig. 2c . Peng et al. [15] performed similar minor loop measur ement to \ndetermine  the exchange coupling field in CoCrPt (18 nm)/Co(1 nm)/Ru(0.9 nm)/CoPtCr(8 nm) A FM structures \nas the magnetic measurements of these samples exhibited no outer loops present in the major loop of the \nhysteresis curve. A comparable measureme nt was later performed by Bandiera et al. [9], where a minor loop \nwas measured to deduce the exchange coupling field in a SFM structure comprising Co/Pt multilayers , Co \ninterlayers, Ru  and CoFeB. Both studies demonstrate that the exchange coupling field ca n be obtained by   the \nreversal of the softer magnetic layer in the SFM. Peng et al. measured this minor loop in the first quadrant of \nthe hysteresis loop, while Bandiera et al. measure d the minor loop in the third quadrant . \nIn the J EX  equation, m2 is the magnetic moment from the thinner magnetic layer  (m1 being the moment from \nthe thicker layer) , and S is the surface area of the film. The measured surface areas, S, are determined to be \n0.21 ± 0.01  cm2 and 0.09 ± 0.01 cm2 for the hyst eresis loops shown in Fig. 1 for the SFMs with 0.5 nm and 0.8 \nnm Ru spacer layers, respectively. The corresponding JEX values are calculated -0.07 erg/cm2 and -0.03 erg/cm2 \nfor Ru=0.5  nm and 0.8  nm respectively.  We note that the magnitude of IECE stronlgy depends not only on the \nRu interlayer thickness, but also on the magnetic and structural properties of the Ru/CoCrPt interface. In \nparticular, Peng et al. [15] in their study of CoCrPt(18 nm)/Ru(0.9 nm)/CoCrPt(8 nm) AFM system, \ndemonstrated that the additi on of a Co(1  nm) interlayer between Ru and CoPtCr, incremented  J EX from 0.13 \nerg/cm2 to 0.8 erg/cm2. Generally, higher IECE is desirable for memory applications , including novel devices \nsuch as a double MTJ containing two S FM reference layers and a S FM free layer discussed in [11] which is \npredicted to switch in ps time scales.    5  \nFigure 2 . Hysteresis curves of CoCrPt(1.7)/Ru(0.8)/CoCrPt 1.3) S FM at a) 300 K, b) 50 K, and c) 2 K. The \nswitching behavior from 50-300 K includes three magnetic transitions while two transitions  are observed  at 2 \nK. In Figs. 2 a and 2b, the center of the o uter loops is indicated by H B. In Fig. 2c, the HB indicates the center of \nthe minor loop (red curve) associated with the switching of the thinner magnetic layer.  \n An SFM has four possible magnetic configurations ( ↑↑, ↑↓, ↓↑, ↓↓), where the left and right \narrows indicate the bottom  (thicker)  and top  (thinner)  FM layers, respectively  as shown in Fig. 2.  The number \nof transitions found in the hysteresis loop at a given temperature depend on the IECE, the Zeeman energy, and \nthe energy barrier for magnetic reversal. From the literature, it is also evident  that the  magnetic  field \nsweeping rate  influences the magnetic switching behavior  [16]. However, in this work  each  data  point of the \nhysteresis curve  is collected once the applied field has stabilized.  Also, each hysteresis curve is collected once \nthe sample temperature has fully stabilized . \n Koplak et al.  found for the CoFeB -Ta-CoFeB S FM three types of  hysteresis loop s over the temperature \nrange studied. In the 180 K-300 K range,  three magnetic transitions were  observed  for Type I hysteresis  (↑↑-\n↑↓, ↑↓-↓↑, ↓↑-↓↓). Two transitions are present  between  120 K-170 K for Type II hysteresis  (↑↑-↑↓, \n↑↓-↓↓) and 2  K-110 K (↑↑-↓↑, ↓↑-↓↓) for Type III hysteresis . These transitions  are observed  when \nthe applied magnetic  field is swept from positive to negative . The reverse transitions are encountered  when  \nthe field is swept from negative to positive. In the case of t he CoCrPt -Ru-CoCrPt system , we observe  three \ntransitions ( ↑↑-↑↓, ↑↓-↓↑, ↓↑-↓↓) in the  50 K to 300 K  range (Fig. 2a -b). Whereas t he number of \nmagnetic  transitions  reduces to two  when the sample is cooled down to 2 K (Fig. 2c) , (↑↑-↑↓, ↑↓-↓↓) \nsimilar to the hysteresis loop measured  in the  120 K-170 K range in the CoFeB -Ta-CoFeB system. The third set \nof magnetic transitions (↑↑-↓↑, ↓↑-↓↓) that are reported  for the CoFeB -Ta-CoFeB S FM (Type III \nhysteresis) are not observe d for the CoCrPt -Ru-CoCrPt S FM.  To understand the  differences in magnetic \ntransitions between the SFM system here reported (CoCrPt -Ru-CoCrPt) and that studi ed by Koplak et al.  \n(CoFeB -Ta-CoFeB) , we provide a comparative analysis  based on the magnetic properties of the constituent \nlayers and on the higher IECE  (4X at 300  K) provided by the Ru in our study.  An important contribution  of this \n       6 study is the extension of  the energy balance formulation  developed by Koplak et al.  to SFM structures \ncomprising higher perpendicular magnetic anisotropy ferromagnets that are critical for thermally stable single \nnm scale magnetic structures. Table 1 summarizes the types of hysteresis curves observed in both the CoCrPt -\nRu-CoCrPt and the CoFeB -Ta-CoFeB SFMs as a function of temperature . \n \nIn Table 1, the  hysteresis types and the associated magnetic transitions observed in the CoCrPt -Ru-CoCrPt SFM \nare shown in the second column  and they  are compared to the  CoFeB -Ta-CoFeB SFM results reported  by \nKoplak et al . [1] in the third column . The indicated magnetic transitions occur when the magnetic field is swept \nfrom positive saturation to negative saturation. The bold arrows represent  the magnetic moment,  m1, of the \nthicker magnetic layer. The temperature range where each type of hysteresis curve is observed are provided  \nunder the heading of the different SFM structures.  Three magnetic transitions are reported by Koplak et al. for \nthe CoFeB -Ta-CoFeB, while two magnetic transitions are observed for the CoCrPt -Ru-CoCrPt SFM.  \nHere we analyze the magnetic transitions exhibited by the SFM with a 0.8 nm Ru spacer (Fig. 1) using \nthe energy diagram technique. This technique relies on a simple e nergy balance (Eq. 1) which contains the \nIECE (EEX), Zeeman energy (E Z), and the potential barriers E eff1 and E eff2 (corresponding to the 1.7 nm and 1.3 \nnm CoCrPt, respectively).  \nEq. 1)  𝑬𝑻𝒐𝒕𝒂𝒍 =𝑬𝑬𝑿+𝑬𝒁+𝑬𝒆𝒇𝒇𝟏 +𝑬𝒆𝒇𝒇𝟐 \nThe IECE, EEX, is proportional to the surface area of the sample and can be estimated from  |𝐸𝐸𝑋|=𝐻𝐵∙\n𝑚2. Here HB represents the bias field, which is measured by locating the center of the minor loop of the softer \nmagnet  as previously described . In the case of the f irst type of switching shown in Fig s. 3a and 3 b, there are \nthree loops: the center field of the outer loops is HB and indicates the strength of IECE. The potential  energy  \nbarrier  separates the two perpendicular orientations of magnetization . Notably the hysteresis curves are \nmeasured along the easy -axis, therefore,  Eeff1 and E eff2 are not equal to the anisotropy energy determin ed \nfrom  the hard axis hysteresis. In a hysteresis loop  with three transitions  (Figs. 3a and 3 b), E eff1 can be \n                            \n                           \n                 \n   \n                 \n     ,     ,                      \n                  \n     ,                   \n                   \n     ,              7 estimated from  the coercive field of the outer loops as 𝐻𝐶−𝑜𝑢𝑡𝑒𝑟 =𝐸𝑒𝑓𝑓 1\n2∙𝑚1. Then E eff2 can be calculated from  the \ncoercive field of the inner loop expressed by 𝐻𝐶−𝑖𝑛𝑛𝑒𝑟 =𝐸𝑒𝑓𝑓 1+𝐸𝑒𝑓𝑓 2\n2(𝑚1−𝑚2). These estimat es arise  from the equations \nderived by  Koplak et al.  describing  the possible magnetic transition s. The Zeeman energy, E Z, is proportional to \nthe applied magnetic field and can be expressed as 𝐸𝑍=−(𝑚1+𝑚2)∙𝐻.  \nThe second type of hysteresis curve shown in Fig. 3c has no outer loops, which are needed to estimate \nEEX. However, one can still calculate H B and thus E EX by measuring the minor loop as s hown in Fig. 2c. This is \nobtained by switching the softer, thinner m 2 magnetic layer after the SFM has been saturated [9,15] . This \nmethod of measuring the bias field advances the application of  the energy balance framework to SFM \nstructures where there are no outer loops in the hysteresis curve . HB, labeled in Fig. 2c, was determined to be \n-1798 Oe. The minor  loop  in Fig. 2c is  measured by saturating the SFM to the ↓↓ orientation , sweeping the \nmagnetic field to just beyond the ↓↓-↓↑ magnetic transition, and then saturating the SFM back to the ↓↓ \norientation.  This indicates an IECE of -0.11 erg/cm2 for the S FM at 2 K. After E EX is obtained, E eff1 and E eff2 can \nbe calculated using equations 𝐻↑↑−↑↓=2|𝐸𝐸𝑋|−𝐸𝑒𝑓𝑓 2\n2𝑚2 and 𝐻↑↓−↓↓=−2|𝐸𝐸𝑋|+𝐸𝑒𝑓𝑓 1\n2𝑚1, respectively.  The resulting \nenergy diagram (Fig. 3c) is consistent with the transition fields of the minor loop and the satu rated hysteresis \nloop.  Eeff1 and E eff2 are plotted at each temperature in Fig. 5a. We note that , Eeff1 is larger than E eff2 until the \ntemperature is lowered to 2 K , most likely due to the changing m 1/m 2 ratio as temperature is decreased.  It is \nnoted that the error associated with the E eff parameters is around ± 1x10-5 erg so the observed difference in \ntheir values is significant. A similar behavior was observed by Koplak et al . which is observed at 100 K  and is \nattributed  to the changing ratio of m 1 to m 2 as temperature is decreased.  \n \n \nFigure 3 . Energy diagrams of the CoCrPt(1.7)/Ru(0.8)/CoCrPt(1.3) S FM at a) 300 K, b) 50 K, and  c) 2 K. The solid \nlines indicate the total energy , excluding the energy barriers, while the dashed lines include the temperature -\ndependent energy barrier  term, E eff. The hysteresis curves are shown in each pane  with corresponding \n      \n      \n        \n        \n   8 magnetic moments on the secondary axis.  The red hysteresis loop in c) displa ys the minor loop measured  to \ndetermine  HB. Dashed arrows indicate  the energies associated with the minor loop and the corresponding \ntransitions.  \nThe energy diagrams shown in Fig. 3 describe the magnetic transitions occurring for each temperature. \nSolid li nes indicate the energy of the S FM system with zero E eff, i.e. when there are  no energy barriers to \novercome. The solid lines are thus , the addition of the Zeeman energy and the IECE, with a y -intercept equal \nto the IECE. The dashed lines represent the total energy of the system after the E eff1 and E eff2 are included, as \ndescribed by  Eq. 1. As the magnetic field is swept, the magnetic orientation present  is the one with the lowest \nenergy. Shown in Fig. 3a, the blue solid line represents the  ↑↑ orientation as the field is lowered from +2 T. If \nthe energy barrier for reversal of each layer is zero, the ↑↑-↑↓ will occur at the intersection of  the blue  \nsolid line and the orange solid line representing the total energy of the ↑↓ orientation. At 300  K, the Eeff \nenerg ies for m 1 and m 2 are negligibly low such that the magnetic transition s occur approximately at the solid \nline intersections.  \nThe potential barriers become larger  as the temperature is decreased to 2 K (Fig. 5a). Since the E eff \nterms are not field -dependent, they shift the dashed lines up along the y -axis.  For a magnetic transition to \noccur , the magnetic field must be changed such that the energy barrier between the solid and dashed line is \ncrossed. As seen in Fig.  3b, the ↑↑-↑↓ transition no longer occurs at the intersection of the solid blue and \norange lines, but at the point where the potential barrier of another orientation energy is crossed. As the \npotential barriers increase with lower temperature, certain tra nsitions are prohibited from occurring due to \nthe existence of lower energy states from other magnetic orientations. At 2  K (Fig. 3c ), the ↑↓-↓↑ transition \ndoes not occur as it does  at 300  K and 50  K (Fig. 3a -b) since the potential barrier of the ↓↓ state is lower in \nenergy than the ↓↑ state.   \nThe third set of magnetic transitions  (↑↑-↓↑, ↓↑-↓↓), or Type III hysteresis, occurs when the \ncondition 𝐸𝑒𝑓𝑓 1<𝐸𝑒𝑓𝑓 2∙𝑚1\n𝑚2−2|𝐸𝐸𝑋|∙𝑚1−𝑚2\n𝑚2 is satisfied [1]. The CoCrPt -Ru-CoCrPt SFM studied does not \nmeet this requirement and does not show this set of transitions even down to 2  K. Compared to the CoFeB -Ta-\nCoFeB  SFM reported by  Koplak et al . [1], which has an IECE at 300  K of EEX/S = -0.01 erg/cm2, the CoCrPt -Ru-\nCoCrPt SFM has an IECE at 300  K of E EX/S = -0.04 erg/cm2. The CoFeB -Ta-CoFeB SFM and the CoCrPt -Ru-CoCrPt \nSFM have m 1/m 2 ratio s of 1.38 and 1.79, respectively . At 300  K the effective anisotropy energy barriers for \nboth systems are: E eff1/S = 4.0x10-3 erg/cm2 and E eff2/S = 2.5x10-3 erg/cm2 for CoFeB -Ta-CoFeB, E eff1/S = 1.7 x10-3 \nerg/cm2 and E eff2/S = 0.73x10-3 erg/cm2 for the CoCrPt -Ru-CoCrPt. The energy barriers for the CoCrPt -Ru-\nCoCrPt are lower for E eff1 and E eff2 by a factor of 2.3 and 3.4, respectively. This disparity in E eff causes the right  9 side of the inequa lity to be lower, which explains why the third set of magnetic transitions are absent  in the \nCoCrPt -Ru-CoCrPt SFM.   \nFigure 4 illustrates the dependence of the left and right side s of the inequality 𝐸𝑒𝑓𝑓 1<𝐸𝑒𝑓𝑓 2∙𝑚1\n𝑚2−\n2|𝐸𝐸𝑋|∙𝑚1−𝑚2\n𝑚2 on the IECE for both the CoCrPt and CoFeB SFM systems. The plot shows the energies \ncalculated at 100  K, since CoFeB -Ta-CoFeB shows the third set of magnetic transitions at this temperature. At a \ngiven IECE, the third set of magnetic transitions should be observed  whe n the right side of the equation is \nlarger than E eff1. When plotted as a function of E EX, the right side of the inequality is a line whose  slope is \ndetermined  by the m 1/m 2 ratio and the intercept by  the product of  Eeff2 and m 1/m 2. The dependence of the \nright side of the inequality on m 1/m 2 is also illustrated in Fig. 4a -b. As m 1/m 2 approaches 1, the slope  and the \ny-axis intercept  of the right side of the inequality  is lower ed. Since it has been shown that fast spin transfer \ntorque switching can be achiev ed with a low m 1/m 2 ratio [11], this analysis is important in understanding the \ntype of hysteresis curve s that will be present in the SFM when tailoring the ratio of magnetic moments.  It is \nevident from Fig. 4a that the magnetic switching behavior of the CoCrPt -Ru-CoCrPt SFM will not exhibit the \nthird type of magnetic switching for either stronger or weaker IECE as the intercept of the right side of the \ninequality is lower than E eff1. \nAs mentioned earlier, the SFM can be used as a replacement for a single FM layer in a memory device. \nSuch devices are expected to operate successfully over  a wide range of temperatu res. Therefore, it is \nimportant to predict the behavior of the SFM at any temperature. The energy diagram technique can be used \nto predict the transition fields of the SFM if the temperature dependence of E eff, EEX, and the magnetization, \nm, are  known. Fig ure 5a-c show s the temperature dependence of E eff, EEX, and m, respectively.  Eeff, Eeff1, and \nEeff2 are  proportional to 𝑚𝑛(𝑛+1)\n2 at lower temperatures (<150  K), while at higher temperatures the potential \nenergy barriers are proportional to 𝑚𝑛 similar to the temperature dependence of the magnetic anisotropy \nobserved for  other materials  [17,18] . Here, m is the magnetic moment and n is the exponent  of the magnetic \nanisotropy function (n=2 is typical for uniaxial anisotropy). Fig. 5a shows the fit for E eff2 based on the 𝑚𝑛(𝑛+1)\n2 \nproportionality. Good agreement is observed  with the fit until around 150  K, above which there are significant \ndifference s between the meas ured E eff and the fit.  At higher temperatures,  Eeff2 is proportional to  𝑚𝑛 as \npredicted in [17]. Eeff1 shows a similar temperature dependence as Eeff2  and the same  exponent for the  \nmagnetic anisotropy function  fits the data , however  the data point at 50  K significantly deviates from the \ntrend. The parameters E EX and m change linearly  in this temperature range . The m values are later \nextrapolated to 370  K using this linear fit. This is appropriate since this temperature is well below the Curie  10 temperature reported for Co 70Cr18Pt12 (~ 673 K [19]), around which the temperature dependence of m would \nsignificantly depart from the linear trend . It is noted in Fig. 5c that the m 1/m 2 ratio across the temperature \nrange studied is larger than the value of 1.3 expected from the nominal layer thickness of 1.7  nm and 1.3  nm. \nAs an example , at 2  K and 300  K the ratio is 1.38 and 1.7, respectively. The discrepancy may result from the \nmanner in which the magnetic moments are determined for the results presented in Fig. 5c, namely, they are \nestimated from the magnetization measurements of the SFM hysteresis loops at saturation and remanence. It \nis feasible that at remanence, contributions from the thin film multi -domain magnetic state may influence the \nmagnitude of the measured magnetization. In addition, whereas the film thicknesses and moment s of the \nconstituent layers were determined in  samples  grown on Ta(5  nm)/Ru(10  nm)  underlayers, in  the SFM, m2  is \ngrown on thin Ru , therefore its crystalline quality may differ from m 1.  This could also result in difference s in \nmagnetic properties of the layers as a function of temperature.  \nFitting the trends seen in Fig. 5 allows one to predict the behavior at 2 00 K and 370  K, the temperature \nextrema that a spintronic sensor or memory device could potentially expected to ope rate reliably.   The energy \ndiagrams for the CoCrPt(1.7)/Ru(0.8)/CoCrPt(1.3) SFM at these two temperatures are shown in Fig. 6. The \nenergy diagram in Fig. 6a is interpolated from the parameters depicted in Fig. 5, while the energy diagram in \nFig. 5b is extr apolated from fitted parameters in Fig 5.   These energy diagrams are constructed using the fitted \nparameters from Fig. 5. Both energy diagrams in Fig. 6a and 6b show magnetic transitions corresponding to \nthe type I regime [1]. The transition field for ↑↑-↑↓ (where m 1 reversal occurs) is predicted to be  1400 Oe \nand 950 Oe at 2 00 K and 370 K, respectively.  In Fig. 6a the hysteresis diagram measured at 200  K is shown and \nagrees with the predicted transitions from the energy diagram.  The hysteresis curve at 370  K was not be \nacquired due to errors introduced by deformation of the straw  sample mounting arrangement employed for \nall other measurements as discussed in the materials and methods section . \nThe CoCrPt -Ru-CoCrPt SFM system provides significant advantages over CoFeB -Ta-CoFeB as free layers \nin MTJ structures. These derive from the lower saturation magnetization and its superior magnetic anisotropy \n[14]. The anisotropy in CoCrPt is largely determined by its magnetocrystalline anisotropy as opposed to \ninterface anisotropy for the case of CoFeB. Therefore, Co CrPt  films exhibit PMA for film thicknesses up to 15 \nnm. Thus, the m 1/m 2 ratio in the films can be controlled more precisely, allowing for more tunability of the \nSFM properties. This is of particular interest for the implementation of ps magnetic switching employing SFM \nstructur es as proposed by Camsari et al. [11]. Said SFMs require larger IECE than that provided by Ta layers in \norder to achieve ps magnetic switching. Therefore, it is important to extend the energy diagram technique  11 developed by Koplak et al . to this material st ructure to understand its behavior at temperature regimes of \ninterest for practical utilization of spintronic memory devices exploiting these SFM s. \n \nFigure 4 . The left and right sides of the ine quality  𝐸𝑒𝑓𝑓 1<𝐸𝑒𝑓𝑓 2∙𝑚1\n𝑚2−2|𝐸𝐸𝑋|∙𝑚1−𝑚2\n𝑚2 plotted as a function of \nEEX for  the a) CoCrPt -Ru-CoCrPt and  b) CoFeB -Ta-CoFeB SFM.  The black solid and dashed lines labeled “Right – \nm1/m 2” represents the right side of the inequality at different magnetic ratios. The left side of the inequality is \nrepre sented by the red curve and is labeled E eff1. The black squares shown in a) represent the observed IECE of \nthe CoCrPt -Ru-CoCrPt SFM at 100  K. \n \nFigure 5. a) Eeff1, Eeff2, and E eff (Total) plotted versus temperature.  b) |E EX| plotted versus temperature.  c) The \nmagnetic moments, m 1 and m 2 of the 1.7  nm and 1.3  nm thick CoCrPt layers, respectively, plotted versus \ntemperature.  \n    \n       12  \nFigure 6. The energy diagram for the CoCrPt(1.7)/Ru(0.8)/CoCrPt(1.3) SFM at 2 00 K and 370  K. The energy \ndiagram in a) is interpolated from the parameters de rived in Fig. 5, while the energy diagram in b) is \nextrapolated from fitted parameters in Fig 5.  The t ransit ion fields are indicated  by the vertical dashed lines.  \nThe hysteresis curve measured at 200  K is shown in a).  \nConclusions:  \nMTJ devices incorporating SFM structures as free layers are expected operate reliably over a wide \ntemperature range that under extrem e conditions could range from 200 K to 370 K. Therefore, it is important \nto determine the temperature dependence of their magnetic transitions to engineer key material properties \nsuch as IECE and the energy barriers associated with magnetization reversal. In this paper the IECE of \nCoCrPt/Ru/CoCrPt SFMs has been investigated from 2 K to 300 K. Building on previous work by Koplak et al. \n[1], this work further elucidates the temperature -dependence of the potential barrier for magnetization \nreversal. The magnit ude of  H B was derived from minor loop measurements, as the hysteresis curves exhibited \nno outer loops, a procedure reported also in refs. [9,15]. H B is needed to estimate IECE and details of the \nenergy diagram technique first described by Koplak et al. [1]. In this  stduy, we observe two types of hysteresis \ncurves in CoCrPt/Ru/CoCrPt SFMs: one above 50 K with three subloops and the other at 2 K with two magnetic \ntransitions. Type III hysteresis, seen in the CoFeB -Ta-CoFeB SFM system in Koplak et al., is not observed in the \nCoCrPt/Ru/CoCrPt SFM, due to the large E eff1 which prevents the Eeff1<Eeff2∙m1\nm2−2|EEX|∙m1−m2\nm2  inequality \nfrom being satisfied. E eff1 corresponds to the energy barrier between magnetization directions present in the \nCoCrPt film.  \nThe utilization of  the energy diagram technique to understand magnetic transitions in CoCrPt /Ru/CoCrPt SFMs \nenabled us to predict  the magnetic orientations present in th is SFM system at any given temperature. Future \nwork will include studies in SFM structures comprising diff erent ferromagnetic materials , alternative  exchange \n        \n      \n        \n       13 coupling interlayers such as Rh to increment the magnitude of  IECE , as well as the incorporation of nanoscale \ninterlayer  magnetic materials between the FM constituent layers and the non -magnetic spacer t o enhance \nIECE, as demonstrated by Peng et al [15] will also be investigated . In addition, we plan to extend the \ntemperature range investigated in this study and to utilize synchrotron structural and magnetic \ncharacterization probes to better characterize their structural and magnetic properties.  \n Funding:  \nThis research was funded in part by Dr. Marinero’s startup funds from Purdue University School of Materials \nEngineering and in part by the Office of Naval Research ( Award# N00014 -18-1-2481) . \nReferences:  \n[1] O. Koplak, A. Talantsev, Y. Lu, A. Hamadeh, P. Pirro, T. Hauet, R. Morgunov, S. Mangin, Magnetization \nswitching diagram of a perpendicular synthetic ferrimagnet CoFeB/Ta/CoFeB bilayer, J. Magn. Magn. \nMater. 433 (2017) 91 –97. https://doi.org/10.1016/j.jmmm.2017.02.047.  \n[2] S.S.P. Parkin, N. More, K.P. Roche, Oscillations in Exchange Coupling and Magnetoresistance in Metallic \nSuperlattices Structures: Cu/Ru, Co/Cr and Fe/Cr, Phys.  Rev. Lett. 64 (1990) 2304 –2308. \nhttps://doi.org/10.1103/PhysRevLett.64.2304.  \n[3] P. Grünberg, R. Schreiber, Y. Pang, M.B. Brodsky, H. Sowers, Layered Magnetic Structures: Evidence for \nAntiferromagnetic Coupling of Fe Layers across Cr Interlayers, Phys. Re v. Lett. 57 (1986) 2442 –2445. \nhttps://doi.org/10.1103/PhysRevLett.57.2442.  \n[4] M.N. Baibich, J.M. Broto, A. Fert, F.N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, J. \nChazelas, Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Superlattice s, Phys. Rev. Lett. 61 (1988) \n2472–2475. https://doi.org/10.1103/PhysRevLett.61.2472.  \n[5] G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Enhanced magnetoresistance in layered magnetic \nstructures with antiferromagnetic interlayer exchange, Phys. Rev. B. 3 9 (1989) 4828 –4830. \nhttps://doi.org/10.1103/PhysRevB.39.4828.  \n[6] S. Parkin, X. Jiang, C. Kaiser, A. Panchula, K. Roche, M. Samant, Magnetically engineered spintronic \nsensors and memory, Proc. IEEE. 91 (2003) 661 –679. https://doi.org/10.1109/JPROC.2003.811807.  \n[7] S.S.P. Parkin, C. Kaiser, A. Panchula, P.M. Rice, B. Hughes, M. Samant, S.H. Yang, Giant tunnelling \nmagnetoresistance at room temperature with MgO (100) tunnel barriers, Nat. Mater. 3 (2004) 862 –867.  14 https://d oi.org/10.1038/nmat1256.  \n[8] D. Watanabe, Et Al., Interlayer exchange coupling in perpendicularly magnetized synthetic ferrimagnet \nstructure using CoCrPt and CoFeB, J. Phys. Conf. Ser. 200 (2010) 72104. https://doi.org/10.1088/1742 -\n6596/200/7/072104.  \n[9] S. Bandiera, R.C. Sousa, Y. Dahmane, C. Ducruet, C. Portemont, V. Baltz, S. Auffret, I.L. Prejbeanu, B. \nDieny, Comparison of synthetic antiferromagnets and hard ferromagnets as reference layer in magnetic \ntunnel junctions with perpendicular magnetic anisotr opy, IEEE Magn. Lett. 1 (2010). \nhttps://doi.org/10.1109/LMAG.2010.2052238.  \n[10] J. Hayakawa, S. Ikeda, K. Miura, M. Yamanouchi, Y.M. Lee, R. Sasaki, M. Ichimura, K. Ito, T. Kawahara, R. \nTakemura, T. Meguro, F. Matsukura, H. Takahashi, H. Matsuoka, H. Ohno,  Current -induced \nmagnetization switching in MgO barrier magnetic tunnel junctions with CoFeB -based synthetic \nferrimagnetic free layers, in: IEEE Trans. Magn., 2008: pp. 1962 –1967. \nhttps://doi.org/10.1109/TMAG.2008.924545.  \n[11] K.Y. Camsari, A.Z. Pervaiz, R . Faria, E.E. Marinero, S. Datta, Ultrafast Spin -Transfer -Torque Switching of \nSynthetic Ferrimagnets, IEEE Magn. Lett. 7 (2016) 1 –5. https://doi.org/10.1109/LMAG.2016.2610942.  \n[12] S. Ikeda, J. Hayakawa, Y. Ashizawa, Y.M. Lee, K. Miura, H. Hasegawa, M. Tsu noda, F. Matsukura, H. \nOhno, Tunnel magnetoresistance of 604% at 300 K by suppression of Ta diffusion in CoFeBMgOCoFeB \npseudo -spin-valves annealed at high temperature, Appl. Phys. Lett. 93 (2008). \nhttps://doi.org/10.1063/1.2976435.  \n[13] S. Ikeda, K. Miura,  H. Yamamoto, K. Mizunuma, H.D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, \nH. Ohno, A perpendicular -anisotropy CoFeB – MgO magnetic tunnel junction, Nat. Mater. 9 (2010) 721 –\n724. https://doi.org/10.1038/nmat2804.  \n[14] D. Watanabe, S. Mizukami, M. O ogane, H. Naganuma, Y. Ando, T. Miyazaki, Fabrication of MgO -based \nmagnetic tunnel junctions with CoCrPt perpendicularly magnetized electrodes, J. Appl. Phys. 105 (2009) \n126–129. https://doi.org/10.1063/1.3062816.  \n[15] Y. Peng, J.G. Zhu, D.E. Laughlin, Int erfacial Co nanolayers for enhancing interlayer exchange coupling in \nantiferromagnetic interlayer exchange coupling media, J. Appl. Phys. 91 (2002) 7676 –7678. \nhttps://doi.org/10.1063/1.1452262.  \n[16] R.B. Morgunov, E.I. Kunitsyna, A.D. Talantsev, O. V. Kopl ak, T. Fache, Y. Lu, S. Mangin, Influence of the  15 magnetic field sweeping rate on magnetic transitions in synthetic ferrimagnets with perpendicular \nanisotropy, Appl. Phys. Lett. 114 (2019). https://doi.org/10.1063/1.5096951.  \n[17] J.B. Staunton, L. Szunyogh,  A. Buruzs, B.L. Gyorffy, S. Ostanin, L. Udvardi, Temperature dependence of \nmagnetic anisotropy: An ab initio approach, Phys. Rev. B - Condens. Matter Mater. Phys. 74 (2006) \n144411. https://doi.org/10.1103/PhysRevB.74.144411.  \n[18] B.D. Cullity, C.D. Graham , Introduction to Magnetic Materials, John Wiley & Sons, Inc., Hoboken, NJ, \nUSA, 2008. https://doi.org/10.1002/9780470386323.  \n[19] M.F. Doerner, T. Yogi, D.S. Parker, S. Lambert, B. Hermsmeier, O.C. Allegranza, T. Nguyen, Composition \nEffects in High Densit y CoPtCr Media, IEEE Trans. Magn. 29 (1993) 3667 –3669. \nhttps://doi.org/10.1109/20.281263.  \n "    },    {        "title": "2402.00783v1.Element_specific_and_high_bandwidth_ferromagnetic_resonance_spectroscopy_with_a_coherent__extreme_ultraviolet__EUV__source.pdf",        "content": "Element -specific and high -bandwidth ferromagnetic resonance spectroscopy  \nwith a coherent , extreme ultraviolet (EUV) source  \n \nMichael Tanksalvala1, Anthony Kos1, Jacob Wisser1, Scott Diddams2,3,4, Hans T. Nembach4,5, and Justin M. \nShaw1 \n \n1Applied Physics Division, National Institute of Standards and Technology, Boulder, CO 80305  \n2Time & Frequency Division, National Institute of Standards and Technology, Boulder, CO 80305  \n3Electrical, Computer & Energy Engineering, University of Colorado , Boulder, Colorado 80309, USA  \n4Department of Physics, University of Colorado, Boulder, Colorado 80309, USA  \n5Associate, Physical Measurement Laboratory, National Institute of Standards and Technology, Boulder, Colorado \n80305, USA  \n \nWe developed and appl ied a tabletop,  ultrafast , high -harmonic generation  (HHG)  source  to measure the \nelement -specific ferromagnetic resonance (FMR) in ultra -thin magnetic alloys and multilayers  on a n \nopaque Si substrate . We demonstrate a continuous wave bandwidth of 6 2 GHz, with promise to extend \nto 100  GHz or higher . This laboratory -scale instrument  detects the FMR using ultrafast, extreme \nultraviolet (EUV) light,  with photon energies spanning the M -edges of most relevant magnetic elements. \nAn RF  frequency c omb generator  is used to produce a microwave excitation that is intrinsically \nsynchronized to the EUV  pulses  with a timing jitter of 1.4 ps or better . We apply this system to measure \nthe dynamics in a multilayer system as well as Ni -Fe and Co -Fe alloys. Since this instrument operates in \nreflection -mode, it is a mile stone toward measuring and imaging the dynamics of the magnetic state and \nspin transport of active devices  on arbitrary and opaque substrates . The higher bandwidth also enables \nmeasurements of materials with high magnetic anisotropy, as well as ferrimagnets, antiferromagnets,  \nand short -wavelength (high wavevector) spinwaves in nanostructures  or nanodevices . Furthermore, the \ncoherence and short wavelength of the EUV will enable extending these studies using dynamic \nnanoscale lensless imaging techniques such as coherent diffractive imaging, ptychography, and \nholography.   \n \nIntroduction  \n \nMagnetic multilayer and device structures  are the fundamental building blocks of data storage  and \nmagnetic random access memories (MRAM) , as well as emerging  spintronic , magnonic  and spin -\norbitronic  devices.1–5 As these technologies pursue faster operational speeds and push further into the \ndeep nanoscale, new demands are placed on nanoscale characterization to evaluate their performance \nand elucidate the underlying physics  and materials science  that drives these technologies . This is \nespecially true for dynamic phenomena , which ha ve more  limitations on the  tools that are available  for \nsuch studies . Ferromagnetic  resonance spectroscopy (FMR) is the fundamental tool for measuring these \nphenomena ; by driving spin precession at  around  the Larmor frequency and fitting the precession \namplitude and phase to material -dependent terms in the Landau -Lifshitz equation, FMR  provides the \nability to measure magnetic properties  and dynamics  in thin -film systems and devices . The spin \nprecession  is usually measured as a change in the microwave s absorbed, transmitted or reflected from \nthe sample6,7or through transport phenomena in fully fabricated nanodevices.8 This approach exploits \nthe inductive coupling between the sample and waveguide  and consequently  averages over the thickness of the sample; therefore, in more complex heterostructures FMR elucidates sample -averaged \nproperties, and measuring magnetic dynamics at individual layers or interfaces becomes difficult.  \n \nFor over a decade, x -ray detected ferromagnetic resonance spectroscopy (XFMR) has served as an \nenhance ment to  FMR by detecting the spin precession with  an element -specific probe (x -rays or extreme \nultraviolet light). Higher -energy photons in the extreme ultraviolet (EUV) or x -ray — as opposed to \noptical or microwave wavelength — provide element -specific information through direct access to \natomic core level transitions .  In this technique , x-ray magnetic circular dichroism (XMCD) and FMR are \ncombined by exciting the sample with a radio frequency (RF) or microwave source that is generated \ndirectly from the master clock of the synchrotron, and is therefore synchronized to the x -ray pulses.9,10  \nThe element -specific response of the sample is recorded by tuning the x -ray energy to a specific atomic \ncore transition and measuring XMCD spectr a at various time delays.  The ability to measure element -\nresolved FMR has been important for measuring magnetic systems , since it gives the ability to  decouple \nthe magnetic dynamics  of different layers or sublattices within a sample if the magnetic layers contain \ndifferent magnetic elements.11–16  Such studies have uncovered the details of phenomena such as \ninterlayer exchange10,17,18 and the presence of spin currents in nonmagnetic  layers  that can be used to  \nFigure 1 . Experiment Layout : (a) Schematic of XFMR system.  Once the EUV light is generated, the \nremaining driving IR light is mostly removed and the EUV is focused onto the sample.  After reflecting \nfrom the sample, it reflects from a grating and is focused onto the camera.  In par allel, microwaves \nare generated from a photodiode that measures a small portion of the laser oscillator beam, and are \nthen delivered to the sample to excite it near ferromagnetic resonance. (b) Simplified schematic \nshowing minimu m microwave components necessary to generate EUV -synchronized microwaves for \nXFMR.  More complete schematics are in Appendix B .  (c) Example data showing the ability to \nindividually monitor the spin dynamics of different elements or layers within a magnetic sample.  \nidentify contributions to spin -pumping .19–23  Furthermore, XFMR has be en integrated into a scanning \ntransmission x -ray microscopy (STXM) modality to provide such dynamic measurement with nanoscale \nresolution.24  \n \nOne technical limitation of synchrotron -based, time -resolved measurements is the maximum  frequency \nthat can be achieved.  This limitation primarily comes from the pulse width of the electron bunches , \nwhich typically ranges from 50 –100 ps  and limits measurement frequencies to only a few GHz.  The pulse \nwidth  can be shortened  by using  low-alpha mode of operation, which reduces the pulse width to about \n7–10 ps.25,26  This mode of operation has been used to demonstrate XFMR bandwidths up to \napproximately 11 GHz;10,27,28 however, such bandwidths are of limited utility for many technological \nmaterials that have high magnetic anisotropy , short -wavelength modes that result from confinement,  \nand/or many materials with opposing magnetic  moments such as antiferromagnets.29–33  Furthermore, \nextension to many non -magnetic phenomena, such as the measurement of phonon modes, also requires \nhigher frequencies.34  While pump -probe approaches at femtosecond slicing and free electron laser \nfacilities can measure ultrafast dynamics in the time domain, th ey do not comprise a resonant excitation \nof a particular mode.  In other words, an ultrafast pulsed excitation will excite every mode and frequency \nin the system, which can make it challenging to image, isolate or separate a specific excitation — \nespecially if the excitation of interest is not dominant.   \n \nAdvances in tabletop light sources , such as high -harmonic generation (HHG)  sources , open new \nopportunities to advance laboratory -based  measurement capability  to compliment the capability of \nsynchrotrons .  HHG sources have been used to measure  ultrafast  processes including magnetization \ndynamics , owing to the exceptionally small pulse width approaching the attosecond regime.35,36  In these  \nstudies , the combination of ultrafast  and element -specific capability was critical to elucidate and \nseparate  many phenomena including superdiffusive spin currents, ultrafast  magnon generation, optically \ninduced spin -transfer processes, and exchange coupling between elements in an alloy.36–44  These studies  \ntypically rely on a n optical  pump -probe approach , with  short pulse duration s (10 to 100 fs) and intrinsic \ntiming synchronization of the pump and probe naturally enabl ing measurements with  high temporal \nresolution.   \n \nIn this paper, we report on development of the first  tabletop -scale XFMR instrument  (shown in Fig. 1) , \nand demonstrate the highest -frequency  XFMR to date, at 17 GHz.  This instrument leverages the \nfemtosecond pulse duration of the EUV probe  to provide an intrinsic frequency limit exceeding 10 THz , \nwith a practical limitation around 100  GHz for some samples .  Currently, i t generates a phase -stable \nmicrowave excitation up to 60 GHz by using an RF frequency comb generator driven by a photodiode at \nthe output o f the laser oscillator .  The practical frequency limitation is currently set by the timing jitter of \nthe microwaves (<1. 4 ps, or ~100 GHz), as well as the limited precession cone angle exhibited by most \nsamples at higher frequencies.  \n \nAn important characteristic of our approach is that there is no constraint on the  substrates that can be \nused, since measurements can be performed in either transmission or reflection geometries.  This is \nespecially advantageous  when combined with imaging  devices since special fabrication on membranes \nor specialized substrates such as MgO for fluorescence detection are not needed .  In transmission , either \nmagnetic linear dichroism (MLD) or magnetic circular dichroism (MCD) can be used.45–49  Since  the shorter a bsorption length of EUV relative to soft -x-rays limit s the thickness of the sample in which the \nEUV can probe  (10~1000 nm generally,  e.g. ~340 nm for 50–70 eV light in Si) ,50 the demonstration in this \npaper is done in reflection and uses the transverse magneto -optic Kerr effect (T -MOKE), which is \nroutinely employed with linearly polarized EUV sources.36 \n \nFinally , the short -wavelength photons that span the EUV to soft -x-ray region of the spectrum are useful \nfor nanoscopic  imaging , given the wavelength of the photons .  In our case, the high c oherence further \nenables  multiple lensless imaging modalities such as coherent diffractive imaging (CDI), ptychography, \nand holography.51–53  Of import is that ptychography and other CDI techniques can be used in both \nreflection and transmission.51  Recent work also combined XFMR with x -ray reflectometry to resolve the \ndynamic response as a function of depth.54  Since EUV reflectometry combined with ptychography has \nalready been demonstrated with HHG sources, this work opens up new possibilities for full 3D spatial \nimaging of high -bandwidth dynamics in nanostructures.55 \n \nThe rest of the paper is structured  as follows: First, we outline  how we generate EUV light and use it to \nprobe  magnetic moments with element -specificity .  Next, we describe how we generate high -frequency \nmicrowaves that are phase -synchronized to our EUV pulses.  Then, we describe how we combine these \nto perform XFMR measurements.  Finally , we show XFMR results on three samples: permalloy  (8.5 GHz) , \nCoFe  (17 GHz) , and a Ni/Fe multilayer  (8.5 GHz, with layer -specific dynamics) .   \n \nExperiment Methods  \n \nThe setup used to generate the phase -stable microwave excitation and EUV  probe is shown in Fig. 1 .  It \ninvolves two parallel components  that are each  seeded by a Ti:sapphire laser oscillator  (15 fs, ~8 2 MHz, \n~400 mW) .  The first is the microwave part that generates high -frequency microwaves (up to >60 GHz) t o \nexcite the sample, and that are synchronized to the EUV probe with a controllable phase shift.  The \nsecond  is the optical half that ultimately focuses femtosecond -duration extreme ultraviolet (EUV) lig ht \npulses onto the sample to probe the precession of spins therein.  Finally, the EUV light that reflects from \nthe sample is collected in a grating -based spectrometer.  \n \nExtreme Ultraviolet Light Probes Element -Specific Magnetic Moment  \n \nWe use a regenerative Ti:sapphire laser amplifier to produce 3 kHz, ~1 mJ, 35 fs pulsed light with 800 nm \nwavelength.  We use this to perform high harmonic generation (HHG) by focusing it  into a hollow -core \nglass capillary filled with 300 –700 Torr Neon gas .  This  produces a spectrum of discrete harmonics with \nphoton energies spanning 35 –70 eV , thus span ning  the M -edges of most magnetic elements.  Moreover, \nthese harmonics are separated by 3 eV , are pulsed at a 3 kHz rate with a time duration shorter than the \ndriving laser ( typically  < 10 fs) , and have a linear polarization matching that of the driving  infrared ( IR) \nlight .  We use horizontally polarized light so that the beamline can be in t he plane of the optical table  \neven after reflecting from the sample around Brewster’s angle for the EUV . \n Once the EUV is generated, we attenuate the driving IR using  two mirrors oriented near Brewster’s angle  \nfor the IR , and further extinguish  the IR by absorbing it in thin metal  foil(s) (usually 0.5 μm Al).  Then, the \nEUV passes through a variable -diameter aperture, and is focused onto the sample using  a 6° angle of \nincidence ( AOI), grazing -incidence toroidal mirror.  This mirror is placed 2 m from the glass capillary and \nhas a focal length of 750 mm, resulti ng in a spot size of approximately  40 μm diameter [the width of the \ncenter conductor of the coplanar waveguide sample s, Fig 2 (c)].  Since we are using the T -MOKE \ngeometry [see Fig. 2 (b)], the sample is oriented near Brewster’s angle, specifically at 50° from grazing  \nsince this angle maximizes signal from many of our samples (see Fig. 4).  We can energize a vertically -\noriented coil magnet for static T -MOKE  measurements  or a horizontally -oriented electromagnet for our \nXFM R studies .  Both magnetic fields are parallel to the sample plane.   By reversing the direction of the \nmagnetic field or (for XFMR) by toggling the RF on/off with a fixed magnetic field, we can observe the \nelement -specific magnetic state of the sample as the variation of the spectral peak in tensity at the \nabsorption edge of each element.  In both cases , the projection of the magnetic moment onto the \nvertical direction is the detected magnetic signal  — for XFMR, th e signal will oscillate sinusoidally with \nthe spins’ precession .  Finally, the reflected EUV strikes a 400  mm toroid in a 4f imaging configuration \n(magnification of 1) and a 500 grooves/mm blazed grating (period: 2 μm) oriented in the conical \ngeometry, and is collected on a scientific EUV CCD detector.  This results in a spectrum  with separated \npeaks , shown in Fig. 2 (a). \n \nGeneration of synchronized >60 GHz microwaves  \n  \nFigure 2. Probing Element -Specific Magnetization : (a) Example EUV spectra after reflecting from a gold \nsample (relatively flat spectral response) and from a magnetic film (permalloy).  The dips in reflectivity \nindicate the absorption edges of the elements present (Fe and Ni).  The red and blue regions i ndicate the \nposition and width of the Fe and Ni absorption edges.  (b) Picture of the sample chamber, which is \nconfigured in the transverse magneto -optic Kerr effect (T -MOKE) geometry.  A coil magnet placed a bove \nour sample (indicated, but not shown) enables us to measure the static magnetic signal from our \nsample.  The higher -field electromagnet has an in -plane, horizontal axis; during XFMR, the spins precess \nabout the horizontal axis and the projection of th e resulting magnetic moment onto the vertical \ndirection is the measured signal.  (c) Picture of a sample.  Each sample is a lithographically patterned 50 \nΩ coplanar waveguide with magnetic layer(s) patterned on top of the center conductor.  The center \ncond uctor is 40 μm wide, and matches the size of the EUV focus.  An additional, much larger region of \nthe same magnetic layer is patterned far from the waveguide for verification of static signals.  \nThe pulse width of the EUV is <30 fs, which sets an intrinsic frequency limitation well above 10 THz.  \nTherefore, the practical frequency limitation for XFMR with this instrument is set by the  signal -to-noise \nratio (SNR, which is sample -dependent and decreases at higher frequency due to smaller precession \ncone angle s) and the timin g jitter between the microwave  excitation and the EUV probe , which has a \nstandard deviation of  roughly  1.3 ps, as seen in Fig. 3 .   \n \nTo maximize the SNR, we adopt an on/off measurement scheme, in which every other measured \nspectrum is made without  the microwave signal applied.  By subtracting the spectrum with the \nmicrowaves off from that with the microwaves on, the effect of longer -timescale fluctuations of the EUV \nintensity can be minimized .  Since our exposure times were 5 seconds, the effects of fluctuations on this \ntimescale or shorter are not suppressed.  We partially correct these shorter -term instabilities by  \nFigure 3. Using YIG -Tuned filters as a broadband phase shifter : (a,b) Timing jitter of 17 GHz and 51 \nGHz microwaves, measured using a 40 GHz sampling oscilloscope.  (c) Spectrum analyzer trace \nshowing that the neighboring harmonics are suppressed by >30 dB, resulting in a pure, narrow -\nlinewidth, sine wave excitation.  (d) Imparted phase shift as a function of YIG control current, \nshowing that t he same control current provides greater phase shift at higher frequencies.  \nassuming that they are perfectly correlated  across all the harmonic peaks (including those with no \nmagnetic signal) and dividing  all peaks by the frame -to-frame fluctuations of the peak  intensitie s far \naway from the magnetic absorption edges .  This is only approximately correct, but consistently improves \nour data quality.  \n \nTo minimize the timing jitter  between the microwave excitation and the EUV probe , we generate the \nmicrowaves directly from the IR laser.  In particular, w e use a similar approach to that taken at \nsynchrotron facilities, wherein an RF frequency comb generator (FCG) produces  harmonics of  the master \nclock that times the el ectron bunches.9,19  In this case, we adopt the Ti -sapphire oscillator itself as the \nmaster  clock  (81.6  MHz).  A portion of the oscillator beam  is directed onto a photodiode, and the \nresulting electrical signal drive s an RF frequency comb generator  that produces harmonics up to at least \n18 GHz (> 220th harmonic).   Since the oscillator is simultaneously used to seed the regenerative amplifier , \nthe EUV pulses are intrinsically synchronized  to the microwave radiation .   \n \nAn upper bound of the short -term phase jitter can be made by measuring the microwave signal with a \nsampling oscilloscope that is triggered on a fast photodiode (rise time: 30 ps) that views the 3 kHz laser \nthat drives the HHG process.  Shown in Fig. 3 (b) is a measurement of 5 1 GHz microwaves made in this \nway, which shows a timing jitter of 1. 4 ps.  Due to the limited bandwidth of the available microwave \ncomponents, we used different configurations for 8.5 GHz, 17 GHz, and 62 GHz frequencies  (see Fig. S2) .  \nImportantly, the timing jitter was roughly the same (<1. 4 ps) for each configuration, even when we used \nadditional components like frequency multipliers.   The actual jitter may be lower  than 1. 4 ps, owing to \nthe fact that smaller values are beyond the measurement bandwidth of the oscilloscopes.  Furthermore, \nthe tim ing jitter may be further improved by use of a high -speed photodiode as input to the FCG, instead \nof the current one which has 1 ns rise time.   Considering this upper limit of timing jitter, there is no \nfundamental limitation to achieve measurement frequencies of 100 GHz or higher.  \n \nWhile we need to excite FMR with a single frequency, t he FCG outputs a large series of spectral lines that \nspan several hundred  harmonics of the fundamental 82 MHz repetition rate of the oscillator .  As shown \nin the simplified schematic diagram in Fig. 1 (b) and in the more complete diagrams in Appendix B, we \nuse a  pair of YIG filters as a narrow -band filter to  isolate  a single frequency or “tooth” of the  frequency  \ncomb  [suppressing other frequencies by more than 30 dB, as shown in Fig. 3(b)].  In addition to filtering \nout undesired frequencies , the YIG filters shift the phase of the signal as they are detuned from their \nresonance; we exploit this property  for use as a self-contained broadband  microwave phase shifter  that \nrequires no additional components (see Appendix A).  Of course, the total phase range is limited by the \nwidth of the YIG resonance.  Using two YIGs in tandem ( instead of one ) gives a larger range of access ible \nphase shifts  while keeping the amplitude relatively flat.  F or the frequencies investigated here,  using two \nYIG filters enables  the phase to be continuously shifted over a range of at least 270 degrees (8.5 GHz), \n540 degrees (17 GHz), and with the range generally increasing in proportion to frequency  [see Fig. 3 (d)]. \n \nWith our current setup , we are able to vary the base frequency of the FCG from 8.5 GHz to 12 GHz.  This \nlimitation is simply set by the bandwidth of the microwave components (including YIG filters, amplifiers, \ncirculators, high -speed switch, etc.) that we had available on hand ; however, standard , commercially \navailable products can be used for other /wider  bands , extending both to lower and higher frequency \nranges.   In our current setup, h igher frequencies  are achieved by inserting  frequency multipliers (2x  and 3x) that enable us to  extend the output frequencies to values exceeding  60 GHz  (see Appendix B).  The \nadvantage to this approach is that most of the circuit remains unchanged  (i.e., operates at its base \nbandwidth)  since the frequency multiplier is inserted just before the sample , requiring only one higher  \nfrequency amplifier and components after the multiplier .  Figure 3 (a, b) shows an example of a 17 GHz \nsignal generated by frequency doubling 8.5 GHz, and a 51 GHz signal generated by using both a \nfrequency doubler and a frequency tripler.   A final note is that , by introducing a circulator and a lock -in \namplifier [shown in Fig. 1 (b)], we are able to perform  in situ inductive FMR measurements  on the \nsample .  This not only provides information about the optimal fields to use during the XFMR \nmeasurement, but  also allows us to moni tor the sample for an y possible damage , heating,  or other \nchanges during the measurement.  \n \nXFMR Procedure  \n \nAfter we have generated femtosecond -duration EUV light in the appropriate range of photon energies \n(45–70 eV) and generated microwaves that are phase -synchronized to those EUV pulses, we can perform \nXFMR  using the following steps .  We first set the excitation microwave frequency (e.g., to 17 GHz) by \ncoarsely tuning the YIG filter  current s to select a tooth of the RF frequency comb.  Fine -tuning the \ncurrent s about this point will control the phase shift of the microwaves.  To quantify this in situ, we use a \nsampling oscilloscope, which is triggered on the 800 nm laser that drives HHG, to measure the \nmicrowaves, and sweep the YIG voltages independently to determine the phase shift per volt (approx. \n600 deg/ mA at 8.5 GHz)  and the accessible range of phase shifts .   \n \nAfter setting the frequency and calibrating the phase -shifting YIG control current s, we use  our in situ \ninductive FMR setup  to determine the magnetic fields  to perform XFMR  (i.e., resonance field and \nlinewidth) , as well as the proper microwave power to maximize our signal but suffer no effects from \nheating  and ensure that we stay within the linear regime of the magnetization dynamics .  Th e inductive \nFMR trace  will show a linear combination of the real and imaginary parts of the microwave reflected \nfrom the sample.  It is easiest to interpret the imaginary part , which we achieve by use of RF phase \nshifting components.  \n \nOnce we know the fields over which to perform XFMR, we send EUV light onto the sample.  We can then \nmeasure and optimize ou r system using  a static T -MOKE spectrum obtained by energizing a small coil \nmagnet above our sample.  By switching the polarity of the current, the magnetic state of the sample will \nreverse and this will write itself onto the harmonic spectrum; in other words , the harmonic peaks at the \nphoton energies within the Fe M -edge will change intensity proportionally to the magnetic state of the  \nFe.  By measuring the distance between the grating and the camera and then counting the number of \npixels between the specular beam and the harmonic peaks of interest, we can estimate the photon \nenergy of each harmonic peak and determine which element is m agnetized  by using the grating \nequation ( 𝜆=𝑇sin(𝜃)/𝑞, where λ is the wavelength, T is the period of the grating, θ is the diffraction \nangle, and q is the diffraction order ).  To improve the SNR o f our static T -MOKE measurements, we \naverage several frames ( typically 20 frames per magnetic field polarity ) and generally use an exposure \ntime of 5 seconds  for each frame while 4x4 binning.  To avoid condensation issues on the camera sensor  \ndue to operating  in only high vacuum conditions , we do not cool the sensor.    \n Finally, if we fix the applied magnetic field and \nthe YIG currents , then each EUV pulse will \nimpinge on the sample at a consistent phase \nof the applied microwaves, and therefore at \nthe same precessional phase of the spins \nwithin the sample.  We again use 5 sec ond \nexposure times  for each frame  and average \nover many frames ( typically  100 with \nmicrowaves applied and 100 without).  A \ndifference spectrum is obtained by recording \nalternating spectra with the microwaves \nturned on/off and subtracting them from \neach other .  As a result, the difference \nspectrum is proportional to the projection of \nthe magnetization on to the vertical axis at a \nparticular phase in the precession ; as the \nphase is shifted through 360 °, this measure s a \nsinusoidal response.   This process  can be \nrepeated  at a series of magnetic  fields to map \nout the full , element -resolved,  field -swept \nferromagnetic resonanc e peak .  Notably, the \nprocess of recording spectra with the \nmicrowaves alternately on and off mitigates \nthe effects of long -term intensity drift of the \nEUV source.  \n \nSample Fabrication  \n \nSample s were fabricated using optical \nlithography  combined with  lift-off processes \non thermally oxidized Si wafers.  The first step \nwas to pattern the waveguide structure.  This \nconsist s of a coplanar waveguide (CPW) \nstructure that is designed to mate with a \ncommercial high -frequency end-launcher on \none end. This connection is designed to \nmaintain a 50 Ohm impedance, minimizing \nreflections or losses across that transition  \nto/from the end launcher .  The CPW then tapers down to a center conductor width of 40 μm in most \ncases, but other widths are also used .  To form the CPW, 5 nm Ti and 100 nm of Au was evaporated and \nlifted off.  A second lithograph y step was used to define the magnetic lay er of interest.  This was \npatterned on top of the narrow section of the CPW , as shown in Figure 2(c).  The magnetic material was \nthen deposited using magnetron sputtering .  In all cases, a 3 nm layer of TaO x was first deposited prior to \ndepositing  the magnetic layer  in order to prevent electrical contact between the magnetic layer and the  \nFigure 4. Samples and selection of measurement \nangle:   The three magnetic sample structures are \nshown schematically in (a -c).  On the left are the \nstructures that were deposited on top of the gold \ncenter conductor, and on the right are the simulated \nmagnetic signal vs. photon energy and incidence \nangle for e ach sample.  We selected 50 ° from grazing \nfor its sensitivity to our permalloy sample.  The other \nsamples would have had higher signal at steeper \nincidence angles.  \nCPW.  This also prevent s any spin -\npumping losses  when the magnetic layer \nis driven to FMR .  After lift -off of the \nmagnetic layer, the wafer was diced using \na diamond saw.   \n \nIn this study, we focus on three magnetic \nsamples , whose structures are \nsummarized in Figure 4.  First, we \nperformed XFMR on permalloy at 8.5 \nGHz.  This sample could not be used to \ndemonstrate XFMR at frequencies above \n11 GHz due to the magnetic field  \nlimitation of our current setup  (max  μ0 H = \n180 mT).  We instead focus on a Co25Fe75 \nsample to achieve higher -frequency XFMR \nmeasurements  at 17 GHz .  The increased \nsaturation magnetization of µ0Ms = 2.4 T \nincreases the FMR frequency to more \nthan 17 GHz for the magnetic fields that \nwe have available.56  Finally, to show the \nability to measure independ ently the  \ndynamics  of different layers with in a \nmultilayer sample, we measured a \nNi/TaO x/Fe multilayer . \n \nXFMR  Results:  Permalloy at 8.5 GHz  \n \nAs a first demonstration , we measure d \nthe XFMR response of a 10 nm thick \nNi80Fe20 (permalloy) sample.  The \nindividual spectra at the microwave phase \ngiving the  maximum magnetic signal are \nshown in Figure 5 (a).  Two spectra are \nshown: o ne each with  the microwaves \nturned on and turned  off.  There is a small \nbut measurable  difference between the \ntwo spectra  that measures spin \nprecession within the sample;  this signal \nis significantly  enhanced at the expected photon energies corresponding to the M -edges of Ni and Fe , \nand varies sinusoidally with the phase of the applied microwave excitation .  Th e difference between the \ntwo curves is related to the  magnetic asymmetry , which  is calculated as (Ion - Ioff)/(Ion + Ioff), or the \ndifference between the intensiti es of the spectra when the rf field is on Ion and when it is off, Ioff divided \nby the sum  of the intensities .  The magnetic asymmetry is show n in the bottom half of Figure 5(a). For  \nFigure 5. Element -resolved XFMR on permalloy, 8.5 GHz:  \n(a) The top plot shows the raw spectra, and the bottom \nplot shows an example XFMR asymmetry spectrum. \nThere are peaks in the magnetic asymmetry at the M -\nedges of Ni and Fe, indicating the amplitude of the \ndynamic response.  (b,c) show the measured spin \nprecession vs. time for Fe and Ni.  (d,e) show the XFMR \ntraces of the individual elements.  Overlaid are the \ninductive FMR measured in situ  and the simultaneous fit \nof amplitude and phase.  \nthe sake of visualization, we suppress  the noise in the asymmetry spectra in the region s between the \nharmonics where there is little to no signal ; we do so by  adding a large value to the denominator  when \nthe intensity is below a threshold  that is set to a value just above the minimum counts in between the \nharmonics . This minimizes the effect of amplifying  noise in the spectral region where there is little to no \nsignal  and/or prevent a divide by zero .  It is important to note that this is not used in the quantitative \nfitting of the spectra, but solely  to improve visualization .  \n \nThe amplitude of the asymmetry should vary sinusoidally at the microwave driving frequency .  To \ncapture the amplitude and phase of the response for each element , the magnetic spectra  are recorded \nat several values of the rf phase.  The amplitudes at the Fe and Ni M -edges  are plotted as a function of \nthe phase in Figures 5(b) and 5(c), respectively.  A sinusoid is then fit to the data , using the frequency \nmeasured  by a spectrum analyzer  during the experiment .  This process is repeated for all values of the \napplied  bias field — a few examples are also shown in Figures 5(b) and 5(c).  It is clear that , as expected,  \nboth the amplitude and phase of these  sinusoidal behaviors change as the magnetic field is also \nchanged.  Figure 5(d) and 5(e) are  element -resolved XFMR plots.  That is to say, they show  the amplitude \n(black circle) and phase (blue open tria ngle) obtained from the sinusoidal fits  of the Fe and Ni peaks data  \nfor several values of the applied magnetic bias field.  The amplitude  A and phase  φ are simultaneously fit \nto the following Lorentzian and sigmoidal arctangent function s, respectively :  \n \n𝐴=𝑦0+(2𝐴0\n𝜋)𝑤\n√4(𝐻−𝐻0)2+𝑤2 \n \n𝜑=𝜑0+180°\n𝜋(arctan(𝐻−𝐻0\n𝑤)+𝜋\n2) \n \nwhere, y0 and φ0 are offsets, A0 is the amplitude  of the resonance,  w is the linewidth, H is the magnetic \nfield,  and H0 is the resonance field .  The simultaneous fit is indicated in the figure  as solid lines .  The \nidentical response between  the Fe and Ni is expected  based on the strong exchange coupling and \nstandard models for FMR.  This provides rudimentary validation of the approach .  Furthermore, t he \nquality of the simultaneous fit highlights the agreement between  the measured and expected behavior \nas the external bias field is swept th rough the ferromagnetic resonance.   To confirm the  measurement of \nthe FMR, we overlay  as a red line  the field -swept FMR spectrum that we obtain via the in situ  inductive \nmethod by measuring the amplitude of the reflected microwaves from the CPW sample structure.  Good  \nagreement is seen between the XFMR and inductive FMR methods.   The small difference in linewidth \nbetween the inductive FMR and XFMR could be due to the fact that XFMR is probing a localized area of \nthe sample, whereas the inductive approach is probing the entire sample which will likely have more \ninhomogeneity.   This is especially true given the fact that the current sample was fabricated in a lift -off \nprocess on a thick Au layer.  This produces edge regions with both line edge roughness and thickness \nvariation , coupled wit h the fact that the Au underlayer itself induces additio nal film roughness.  Finally, \nwe note that since our EUV beam is focused in a roughly -Gaussian spot with a peak intensity near the \ncenter of the waveguide, o ur EUV measurements intrinsically probe the center region of the waveguide \nmore than the edges . \n  \nXFMR Results: Co 25Fe75 at 17 GHz  \n \nThe second sample that we studied is Co25Fe75; due to its high M s, we were able to measure XFMR at 17 \nGHz even with  magnetic fields as low as µ0H ≈ 150 mT.  Notably, at 50° from grazing , our setup  has a low  \ncontrast for the magnetic signal at the Co M-edge as shown in Figure 4(b).  This limitation is solely due to \nthe current incidence angle of our system , and can be overcome by changing this angle to 52°.  \nRegardless, we are still  able to measure the dynamics of each element , as shown in Figure 6 (a) .  Broadly \nspeaking, the Co and Fe behave the same, as expected; however, without more careful data analysis, \nquantitative XFMR traces cannot be shown  for Co and Fe independently  (e.g., to compare phase shifts)  \ndue to the proximity of the Co (57 –62 eV)  and Fe (50 –55 eV)  M-edges .   \n \nThe inductive FMR trace shown in Figure 6(c) shows a double -resonance , which we suspect is due to \ninhomogeneity in the sample.  This material is known to have large inhomogeneity when deposited on \nrough surfaces or when non -ideal seed and capping layers are used.57  The XFMR trace also shows this  \ndouble peak  structure , and the locations of the two resonances agree in both methods .  As expected, \nthe phase in the XFMR is not a clean arctan due to superposition of the two resonances.  The relative \namplitudes of the two peaks are not the same in the XFMR and inductive FMR , which is likely due to the \nfact that the EUV probes only a small region (~60 μm diameter), whereas inductive FMR probes the \nentire structure.   We do not think this comes from uncertainty in our XFMR data; i n addition to \nestimating the error bars from the quality of the sinusoidal fits (i.e., the variance in the data), we tested \nthe repeatability of the XFMR measurement by measuring  at 150 mT  both at the beginning and end of \nthe measurement [included in Fig. 6(c)] .  Within their respective error bars, these measurements at 150 \nmT agree and demonstrate the stability and robustness of the measurement system.    \nFigure 6. High -Bandwidth XFMR on Co 25Fe75, 17 GHz:  (a) The top plot shows the raw spectra, and the \nbottom plot shows an example XFMR magnetic asymmetry.  The Co and Fe absorption edges are \nclose to each other, so the dynamics are more difficult to separate quantitatively. (b) The spin \nprecession of the Fe  atoms.  (c) The XFMR trace of the Fe, with the inductive FMR overlaid.  \n \nXFMR Results: Ni/Fe Multilayer at 8.5 GHz, showing element -specific dynamics  \n \nThe third and final sample in this study is a multilayer sample with Ni and Fe layers that are separated by \nan insulating  3 nm TaO x layer.  The magnetic isolation of the Fe and Ni layers should yield independent \ndynamics of the layer except for some potential dipolar coupling.   Figure 7(b) shows the in situ  inductive \nFMR response  of the sample taken at 8.5 GHz , where in two peaks are observed.  The high amplitude and \nnarrow peak near  40 mT is presumably due to the Fe layer since Fe has higher M s and lower damping \nrelative to Ni.58  The broad and low amplitude peak near  120 mT is likely from the Ni, by the same \nargument.  Indeed, XFMR measurements at 40 mT reveal that the  precession amplitude of the Fe layer is \nsignificantly higher than that of the Ni.  Given the broad linewidth of the Ni peak  and the potential for \ndipolar coupling between the layers, it is not surprising to see a non -zero precession angle for Ni as well.   \nRepeating this measurement at μ0 H ~125 mT shows the opposite behavior , where the precession  \namplitude of the Ni  signal is significantly more than that of the Fe.   Furthermore, in both cases there is a \nnotic eable phase shift between the precession of the two elements .  Such a phase shift is to be expected \nsince the resonance fields (and therefore phase shift through the resonance) are at different locations.  \nAny dipolar coupling would also induce a relative phase shift.  Measurements taken  at more fields across \nthe resonance s would in principle enable us to determine the amount and type of coupling between the  \nFigure 7. Element -Resolved Dynamics in a Magnetic Multilayer, 8.5 GHz:  (a) The top plot shows the \nraw spectra for static T -MOKE.  The second row shows the magnetic asymmetry from the static \nspectrum, which shows contributions at both the Ni and Fe edges.  The bottom two plots are \nrepresentative XFMR magnetic asymmetry spectr a taken at the Ni and Fe resonant fields.  The fact that \nthere is no response of the Fe atoms at the Ni resonance field highlights the ability of XFMR to \nmeasure element -resolved dynamic s.  (b) We measured XFMR at the Fe resonance and at the Ni \nresonance, and the measured element -resolved precession is shown in the rightmost plots.  \nlayers.  Measuring such phase shifts in structures where the coupling is more complicated (e.g., from \nspin currents or weak exchange) allows quantification of such interactions .  This highlights the ability to \nseparate dynamics within different layers or sublattices in scientifically interesting or industry -relevant \nsamples . \n \nDiscussion and Outlook  \n \nThis work shows that high -bandwidth , element -specific  measurements of dynamic magnetic phenomena \ncan be achieved in a laboratory.   Despite the fact that no  reference beam was used to correct for \nintensity flu ctuations and drift, small deviations of the magnetization with cone angles below 5 degree s \nwere easily observed at high -frequency.  Going forward, many straightforward improvements can be \nmade  to increase the sensitivity  and SNR , allowing more subtle phenomena to be observed  along with \nsignificantly smaller cones angles  and allow for more field steps across the resonance .  The latter will be \nparticularly important as measurements are pushed to higher frequencies.   Here, we discuss  several  \nimprovements that can be made to  greatly improve performance . \n \nFirst, a  reference beam that provides  reference spectr a of the EUV light incident on the sample can be \nused to normalize and divide out  intensity fluctuations  on both short and long timescales .  The absence \nof this in the current setup  represent s one of  our biggest barriers to achiev ing higher SNR.  The frame -to-\nframe  RMS intensity is approximately 5% for 5 sec exposures,  and there is also similar drift on longer \ntime scales.   Furthermore, the fluctuations of each harmonic are not necessarily correlated  (though our \nanalysis assumes that they are) .  Such normalization is critical at synchrotrons since the beam intensity \nalso varies significantly over time.  Recently, the  use of a reference  beam  in HHG systems used to \nmeasure spin dynamics was shown to increase SNR by  almost an order of magnitude .59–61 \n \nSecond, reflection -geometry measurements as shown here can be significantly improved by \nincorporating  measurements at multiple angles .39,55  This can be seen in Figure 4, which shows a \ncalculation of the T-MOKE contrast  for the samples used in this study.  This calculation was performed \nusing the scattering matrix approach described in Ref. [62].  While the 50° angle of incidence (measured \nfrom grazing incidence) used here is near the maximum -contrast point for the Ni 80Fe20 sample, we would \nachieve better SNR and better -separated Ni and Fe peaks in the Co25Fe75 and Ni/Fe multilayer sample s if \nthe scattering angle was changed to 52 degrees.   Incorporating the ability to change the incidence angle \neasily will enable us to optimize the signal for general samples, as well as improve the depth -sensitivity \nof our measur ement. \n \nThird, while we have shown adequate synchronization of the EUV pulses to 62 GHz microwave  fields, the \ntiming jitter we measure supports extension to significantly higher frequencies  by using commercially \navailable  components.  Furthermore, it may be possible to improve the timing jitter by replacing the \nphotodiode  that is used for the  input to the  frequency comb generator .  We found that using a sharper \nrise time on the input to the frequency comb generator reduced the timing jitter.   Since t he current \nphot odiode  still has a  relatively long  rise time (low bandwidth) of 1 ns , which  depending on the \nelectronic noise present  can produce additional timing jitter , we anticipate that using a photodiode with \na shorter rise time will further improve the timing jitter .  Also, i n addition to using multipliers , \nsynchronized 100+ GHz tones with higher signal -to-noise ratio c an be generated by driving an electro optic  modulator with a 10 or 20  GHz harmonic of the 80 MHz laser repletion rate .63  We further \nnote that our current Ta -sapphire oscillator is passive and lacks any active frequency stabilization.  \nFemtosecond -level timing jitter between optical and electronic signals is accessible  with techniques that \nemploy stabilization of the underlying Ti:sapphire frequency comb mode spacing and carrier -envelope \noffset frequency .63–65 \n \nFinally, as already mentioned, this technique will be extended to include an imaging modality whereby \nthe coherence of the EUV will be exploited .51,53  Lensless imaging techniques such as coherent diffractive \nimaging, ptychography, and holography have been used with EUV and x -ray light to image \nnanostructures  and magnetic textures  with resolution smaller than the illumination spot size and \napproaching the diffraction limit .66,67  Furthermore, these techniques provide quantitative measurement \nof the phase shift imparted by the sample, which provides a dramatic improvement in contrast to certain  \nsample features , such as topography and oxidation state .55  Combining such a technique with the XFMR \ninstrument developed in this paper will enable us to perform  dynamic, in situ  measurements of \nfunctioning devices and elucidate the underlying physics.  \n \nSummary  \n \nWe have demonstrated the ability to bring element -specific, x -ray detected ferromagnetic resonance \nspectroscopy (XFMR) to a laboratory setting.  We did so by combining a high -harmonic generation (HHG) \nlight source , which generates 35 –70 eV light, with an RF frequency comb generator, and were able to \ngenerate phase -stable microwaves (timing jitter < 1. 4 ps) above 60 GHz that we can use for XFMR \nmeasurements at the M -edge of most magnetic elements.   The instrument can perform measurements \nin transmission or r eflection; we showed reflection -mode measurements here that use the transverse \nmagneto -optic Kerr effect (T -MOKE) geometry for magnetic contrast even on samples with  opaque \nsubstrates (e.g., thick silicon).  We used our instrument to perform high -frequency XFMR measurements \non three samples: permalloy (8.5 GHz), Co 25Fe75 (17 GHz), and a Ni/Fe multilayer sample that showed \ndifferent dynamics in the Ni and Fe layers  (8.5 GHz) .   \n \nThis system provides the capability in the near future to measure element -, layer -, or sublattice -specific \ndynamics in industry -relevant and scientifically interesting magnetic thin films.   Moreover, by introducing \nvariable -angle measurements into this system, we will enhance the SNR and depth -sensitivity of the \ntechnique to be able to monitor precisely the spin transport and accumulation across interfaces.   Finally , \nthe spatial coherence of the source, combined with the ability to synchronize electrical pulses to the EUV \nprobe, lays the foundation  for performing these measurements in -operando on individual or arrays of \nfunctional spintronic devices.  \n \nAppendix  A: Tuning the YIG Pair   \nWe are able to tune the pair of YIG filters by using a pair of  part-per-million  (PPM )-stabl e current sources \n(one per YIG).   In particular, we drive the YIGs with roughly 200  mA, and changing the current by 1  µA \nshifts the phase by roughly a degree at 8.5 GHz  [shown both in Fig. 3(d) and Fig. S(1a)] .  One YIG alone \ndoes not provide 360° phase shift at 8.5 GHz (or below) before the transmitted amplitude starts to \ndecline.  By using two YIGs and  sweeping the currents together, we are able to shift the microwave phase over a full cycle while maintaining a relatively constant microwave amplitude.  Fig. S1 shows the \namplitude and phase of the transmitted microwaves as the YIG filters are tuned together (vertical) and \napart (horizontal).  \n \n \nAppendix B: Complete RF Generator Schematic  \n \nFigure S2 shows the full schematic for the microwave generator that was used to generate 8.5–62 GHz.  \nWe use frequency multipliers to achieve frequencies above 13 GHz.  Above 54 GHz , we encounter \nsignificant losses in our system since we exceed the bandwidth of several components , as well as the 40 \nGHz sampling oscilloscope.  Despite this fact, we are still able to evaluate the timing  stability  and test the \nsynchronization of microwaves to  the EUV pulses up to  62 GHz.  Figure S2 shows that we can still \nmeasure the synchronization our EUV pulses up to frequencies of 62 GHz, albeit with increased noise \ndue to microwave losses.  Additionally , the final bandpass filter  fails to suppress the 4 1 GHz harmonic \nalso output by the tripler  when generating 62 GHz microwaves, leading to some beating of the signal.  \nHowever, these are not fundamental limitations since higher bandwidth components are commercially \navailable.   \nFigure S1. Tuning the YIG filter pair:  (a) The amplitude of the transmitted microwaves is roughly \nmaintained over the passband of the filters.  The dashed purple arrow shows an example range of \napproximately 360 ° phase shift at 8.5 GHz. (b) The phase is shifted if the YIG filter currents are \nchanged together.  \n  \nAcknowledgments  \nThe authors are grateful to Henry Kapteyn, Margaret  Murnane , and Tom Silva for valuable discussions \nand advice , and to Dmitriy  Zusin and Christian Gentry for developing the multilayer reflectivity \ncalculating code .  MT and JW ackno wledge funding support from the National Research Council (NRC)  \nFigure S2. Complete RF Generation Circuit Schematic for up to 62 GHz:  (a) The full circuit schematic for \ngenerating synchronized microwaves is shown.  The frequency -doubling and frequency -tripling \nelectronics are omitted when generating lower frequencies.  (b -d) Traces measured with our 40 GHz \nbandwidth sampling scope when the circuit is generating 40 GHz, 57 GHz, and 62 GHz microwaves, \nrespectively.  In (b), the trigger pulse (measured by a high -speed, real -time oscilloscope) is overlaid.  \nNote that the be ating at 62 GHz is due to exceeding the rated bandwidth of multiple components, most \nnotably the final bandpass filter passes 41 GHz in addition to the desired 62 GHz.  \nPost -Doctoral Fellowship program.  HTN acknowledges support through the NIST cooperative agreement   \n70NANB18H006  with the University of Colorado Boulder . \n \nReferences  \n \n1 D. Edelstein, M. Rizzolo, D. Sil, A. Dutta, J. DeBrosse, M. Wordeman, A. Arceo, I. C. Chu, J. Demarest, E. \nR. J. 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Petrovic1,3\n1Department of Condensed Matter Physics and Materials Scien ce,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n2Department of Condensed Matter Physics, Weizmann Institut e of Science, Rehovot 7610001, Israel\n3Department of Materials Science and Chemical Engineering,\nStony Brook University, Stony Brook, NY 11790, USA\n(Dated: January 18, 2021)\nWe carried out a comprehensive study of electronic transpor t, thermal and thermodynamic prop-\nerties inFeCr 2Te4single crystals. Itexhibits bad-metallic behavior andano malous Hall effect(AHE)\nbelow a weak-itinerant paramagentic-to-ferrimagnetic tr ansition Tc∼123 K. The linear scaling be-\ntween the anomalous Hall resistivity ρxyand the longitudinal resistivity ρxximplies that the AHE\nin FeCr 2Te4is most likely dominated by extrinsic skew-scattering mech anism rather than intrinsic\nKL or extrinsic side-jump mechanism, which is supported by o ur Berry phase calculations.\nINTRODUCTION\nThe anomalous Hall effect (AHE) in metals is linked\nto an asymmetry in carrier paths and the effects of spin-\norbit interaction. This is typically observed in ferromag-\nnets since an electric current induces a transversevoltage\ndrop in zero magnetic field which is proportional to mag-\nnetization[1,2]. Spin-orbitcouplinginthe ferromagnetic\nbands leads to anomalous carrier velocities and intrinsic\nAHE [3]. The intrinsic Kaplus-Luttinger (KL) mecha-\nnism can be reinterpreted as a manifestation of Berry-\nphase effects on occupied electronic Bloch states [4, 5].\nThe extrinsic mechanisms involving skew-scattering and\nside-jump mechanisms can also give rise to the AHE and\nare induced by asymmetric scattering of conduction elec-\ntrons [6, 7]. In recent years it has been shown that the\nAHEvelocitiesarisefromthetopologicalBerrycurvature\nwhich generate an effective magnetic field in momentum\nspace in varieties of Dirac materials with noncollinear\nspin configuration [8–12].\nFeCr2Ch4(Ch = O, S, Se, Te) materials show rich cor-\nrelatedelectronphysics. FeCr 2O4spinelshowsacomplex\nmagnetic phase diagram with a ferrimagnetic (FIM) and\nmultiferroic order below 80 K, a strong spin-lattice cou-\npling and orbital order due to the Jahn-Teller distortion\n[13–17]. FeCr 2S4isa multiferroicferrimagnetbelow Tc=\n165 K with large changes of resistivity in magnetic field\n[18–22]. FeCr 2Se4orders antiferromagnetically with TN\n= 218 K in an insulating state despite with larger ligand\nchalcogenatom[23–25]. FeCr 2S4andFeCr 2Se4havesim-\nilar electronic structure with nearly trivalent Cr3+and\ndivalent Fe2+states, and there is a strong hybridization\nbetween Fe 3 d- and Ch p-states [26]. FeCr 2Te4shows no\nsemiconducting gap and a FIM order below Tc= 123 K\n[27, 28].\nIn this work, we performed a comprehensive study of\nelectronic and thermal transport properties in FeCr 2Te4\nsinglecrystals. TheAHEobservedbelow Tcisdominated\nby the skew-scattering mechanism, i.e., by the Bloch\nstate transport lifetime arising from electron scatteringby impurities or defects in the presence of spin-orbit ef-\nfects, and is smaller than the intrinsic AHE revealed by\ndensity functional calculations.\nEXPERIMENTAL AND COMPUTATIONAL\nDETAILS\nSingle crystals growth and crystal structure details\nare described in ref.[28]. Electrical and thermal trans-\nport were measured in quantum design PPMS-9. The\nlongitudinal and Hall resistivity were measured using\na standard four-probe method. In order to effectively\neliminate the longitudinal resistivity contribution due to\nvoltage probe misalignment, the Hall resistivity was ob-\ntained by the difference of transverse resistance mea-\nsured at positive and negative fields, i.e., ρxy(µ0H) =\n[ρ(+µ0H)−ρ(−µ0H)]/2. Isothermal magnetization was\nmeasured in quantum design MPMS-XL5.\nWe performed density functional theory (DFT) cal-\nculations with the Perdew-Burke-Ernzerhof (PBE) [29]\nexchange-correlation functional that is implemented in\nthe Vienna ab initio simulation package(VASP) [30]. We\nadoptedtheexperimentalcrystalstructurewiththe ferri-\nmagnetism (parallelto the lattice vector c) [28]. The cut-\noff energy for the plane wavebasis is 300eV. A k-mesh of\n10×10×10wasused in the Brillouin zone sampling. The\nspin-orbit coupling was included. The intrinsic anoma-\nlousHallconductivity(AHC) andSeebeckcoefficientwas\ncalculated in a tight-binding scheme based on the maxi-\nmally localized Wannier functions [31].\nRESULTS AND DISCUSSIONS\nFigure 1(a) shows the temperature-dependent heat ca-\npacityCp(T) for FeCr 2Te4. A clear anomaly around 123\nK corresponds well to the paramagnetic (PM)-FIM tran-\nsition. The high temperature Cp(T) approaches the Du-\nlong Petit value of 3 NR≈172 J mol−1K−1, whereR2\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s52/s54/s56/s49/s48/s48 /s49/s50/s48 /s50/s52/s48 /s51/s54/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s52/s54\n/s48 /s53/s48 /s49/s48/s48/s52/s46/s48/s52/s46/s53/s53/s46/s48/s67\n/s112/s32/s40/s74/s47/s109/s111/s108/s45/s75/s41\n/s84/s32/s40/s75/s41/s40/s97/s41\n/s51/s78/s82/s49/s50/s51/s32/s75/s83/s32/s40 /s86/s47/s75/s41\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s67\n/s112/s47/s84/s32/s40/s74/s47/s109/s111/s108/s45/s75/s50\n/s41\n/s84/s50\n/s32/s40/s75/s50\n/s41\n/s40/s99/s41/s120/s120/s32/s40/s49/s48/s45/s52\n/s32/s99/s109/s41\n/s84/s32/s40/s75/s41/s120/s120/s40/s49/s48/s45/s52\n/s99/s109/s41\n/s84/s32/s40/s75/s41\nFIG. 1. (Color online) (a) Temperature-dependent heat ca-\npacityCp(T) for FeCr 2Te4. Inset shows the low temperature\nCp(T)/TvsT2curve fitted by Cp(T)/T=γ+βT2. (b) See-\nbeck coefficient S(T) and (c) in-plane resistivity ρxx(T) for\nFeCr2Te4single crystal. Inset in (c) shows data below 100 K\nfitted by ρ(T) =ρ0+aT3/2+bT2(solid line) in comparison\nwithρ(T) =ρ0+cT2(dashed line).\n= 8.314 J mol−1K−1is the molar gas constant. The\nlow temperature data from 2 to 18 K are featureless and\ncould be fitted by using Cp(T)/T=γ+βT2, where the\nfirst term is the Sommerfeld electronic specific heat co-\nefficient and the second term is low-temperature limit of\nlattice heat capacity[inset in Fig. 1(a)]. The fitting gives\nγ= 61(2) mJ mol−1K−2andβ= 1.7(1) mJ mol−1K−4.\nThe Debye temperature Θ D= 199(1) K can be calcu-\nlated by using Θ D= (12π4NR/5β)1/3, whereN= 7 is\nthe number of atoms per formula unit.\nThe Seebeck coefficient S(T) of FeCr 2Te4is positive in\nthe whole temperature range, indicating dominant hole-\ntype carriers [Fig. 1(b)]. The S(T) changes slope around\nTcand gradually decreases with decreasing temperature.\nAs we know, the S(T) depends sensitively on the Fermi\nsurface. The slope change of S(T) reflects the possible\nreconstruction of Fermi surface passing through the PM-\nFIM transition. At low temperature, the diffusive See-\nbeck response of Fermi liquid dominates and is expected\nto be linear in T. In a metal with dominant single-band\ntransport, the Seebeck coefficient could be described by\nthe Mott relationship,\nS=π2\n3k2\nBT\neN(εF)\nn, (1)\nwhereN(εF) is the density of states (DOS), εFis the\nFermi energy, nis carrier concentration, kBis the Boltz-\nman constant and eis the absolute value of electronic\ncharge [32]. The derived dS/dTbelow 26 K is ∼0.074(1)\nµV K−2. TheS(T) curve is consistent with our calcu-/s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s49/s50/s51/s52\n/s114\n/s120/s121/s32/s40 /s109/s87 /s32/s99/s109/s41\n/s109\n/s48/s72/s32/s40/s84/s41/s32/s56/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s49/s50/s48/s32/s75/s40/s98/s41\n/s109\n/s48/s72/s32/s47/s47/s32/s99\n/s32/s50/s48/s32/s75\n/s32/s52/s48/s32/s75\n/s32/s54/s48/s32/s75/s77/s32/s40 /s109\n/s66/s47/s102/s46/s117/s46/s41\n/s109\n/s48/s72/s40/s84/s41/s32/s50/s48/s32/s75\n/s32/s52/s48/s32/s75\n/s32/s54/s48/s32/s75\n/s32/s56/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s49/s50/s48/s32/s75/s40/s97/s41\n/s109\n/s48/s72/s32/s47/s47/s32/s99\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s84/s32/s40/s75/s41/s82\n/s48/s32/s40/s49/s48/s45/s50\n/s32/s99/s109/s51\n/s47/s67/s41/s40/s99/s41\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s82\n/s115/s32/s40/s99/s109/s51\n/s47/s67/s41\nFIG. 2. (Color online) Out-of-plane field dependence of (a) d c\nmagnetization M(µ0H) and (b) Hall resistivity ρxy(µ0H) for\nFeCr2Te4at indicated temperatures. (c) Temperature depen-\ndence of ordinary Hall coefficient R0(left axis) and anoma-\nlous Hall coefficient Rs(right axis) fitted from the ρxyvsµ0H\ncurves using ρxy=R0µ0H+RsM.\nlations based on Boltzmann equations and DFT band\nstructure [see below in Fig. 4(b)]. The electronic specific\nheat is:\nCe=π2\n3k2\nBTN(εF). (2)\nFrom Eq. (1), thermopower probes the specific heat per\nelectron: S=Ce/ne. The units are V K−1forS, J K−1\nm−3forCe, and m−3forn, respectively. It is common to\nexpressγ=Ce/TinJK−2mol−1units. Inordertofocus\non theS/Ceratio, we define a dimensionless quantity\nq=S\nTNAe\nγ, (3)\nwhereNAis the Avogadro number. This gives the num-\nber of carriers per formula unit (proportional to 1 /n)\n[33]. The obtained q= 0.10(1) indicates about 0.1 hole\nper formula unit within the Boltzmann framework [33].\nFigure 1(c) shows the temperature-dependent in-plane\nresistivity ρxx(T) of FeCr 2Te4, indicating a metallic be-\nhavior with a relatively low residual resistivity ratio\n[RRR = ρ(300K)/ ρ(2K) =1.7]. Aclearkinkis observed\natTc, correspondingwell to the PM-FIM transition. The\nrenormalized spin fluctuation theory suggests that the\nelectrical resistivity shows a T2dependence for itinerant\nferromagnetic system [34]. In FeCr 2Te4, the low temper-\nature resistivity fitting gives a better result by adding an3\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s50/s52/s54/s56/s49/s48/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s51/s46/s52/s48 /s45/s51/s46/s51/s53 /s45/s51/s46/s51/s48/s45/s53/s46/s53/s53/s45/s53/s46/s53/s48/s45/s53/s46/s52/s53/s45/s53/s46/s52/s48\n/s65 /s120/s121/s32/s40\n/s99/s109/s41/s45/s49\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s40/s100/s41/s40/s97/s41\n/s83\n/s72/s32/s40/s49/s48/s45/s50\n/s32/s86/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s110/s32/s40/s49/s48/s50/s49\n/s32/s99/s109/s45/s51\n/s41\n/s84/s32/s40/s75/s41/s108/s111/s103/s32/s65 /s120/s121\n/s32/s40 /s99/s109/s41\n/s108/s111/s103/s32\n/s120/s120/s32/s40 /s32/s99/s109/s41/s32/s61/s32/s49/s46/s49/s40/s50/s41/s40/s99/s41\nFIG. 3. (Color online) Temperature dependence of the carrie r\nconcentration (a) and the anomalous Hall conductivity σA\nxy=\nρA\nxy/(ρ2\nxx+ρ2\nxy) (b). Scaling behavior of the anomalous Hall\nresistivity (c) and the coefficient SH=µ0Rs/ρ2\nxx(d).\nadditional T3/2term that describes the contribution of\nspin fluctuation scattering [35].\nρ(T) =ρ0+aT3\n2+bT2, (4)\nwhereρ0is the residual resistivity, aandbare constants.\nThe fitting yields ρ0= 366(1) µΩ cm,a= 1.00(3) ×10−1\nµΩ cm K−1, andb= 2.8(3)×10−3µΩ cm K−2, indicating\ntheT3/2term predominates. This means the interaction\nbetween conduction electrons and localized spins could\nnot be simply treated asa small perturbation to a system\nof free electrons, i.e., strong electron correlation should\nbe considered in FeCr 2Te4.\nFigure 2(a) shows the isothermal magnetization mea-\nsured at varioustemperatures below Tc. All the M(µ0H)\ncurves rapidly increase in low field and change slowly in\nhigh field. Field dependence of Hall resistivity ρxy(µ0H)\nfor FeCr 2Te4at the corresponding temperatures are de-\npicted in Fig. 2(b). All the ρxy(µ0H) curves jump in low\nfield and then become linear-in-field in high field, indi-\ncating an AHE in FeCr 2Te4crystal. In general, the Hall\nresistivity ρxyin ferromagnets is made up of two parts,\nρxy=ρO\nxy+ρA\nxy=R0µ0H+RsM, (5)\nwhereρO\nxyandρA\nxyare the ordinary and anomalous Hall\nresistivity, respectively [36–39]. R0is the ordinary Hall\ncoefficientfromwhichapparentcarrierconcentrationand\ntype can be determined ( R0= 1/nq).Rsis the anoma-\nlous Hall coefficient. With a linear fit of ρxy(µ0H) in\nFIG. 4. (Color online) (a) Crystal structure, Brillouin zon e\n(BZ),andelectronic structureofFeCr 2Te4. Theredvectorsin\nthe crystal structure represent the directions of the magne tic\nmoments on Fe and Cr. The high symmetric k-paths in the\nBZ are shown. The x, y, and z directions of the Cartesian co-\nordinate are along the lattice vectors a, b, and c, respectiv ely.\nThe Fermi energy is set tozero. Calculated (b)Seebeck coeffi-\ncientSand (c) anomalous Hall conductivity σxyof FeCr 2Te4.\nThe calculated Sin low temperature shows good agreement\nwith the experiment. The σxyat the Fermi level (zero) is ∼\n127 (Ω cm)−1, much larger than the measured value of 22 (Ω\ncm)−1.\nhigh field, the slope and intercept corresponds to R0and\nρA\nxy, respectively. Rscan be obtained from ρA\nxy=RsMs\nwithMstaken from linear fit of M(µ0H) curves in high\nfield. The temperature dependence of derived R0andRs\nis plotted in Fig. 2(c). The value of R0is positive, in line\nwith the positive S(T), confirming the hole-type carries.\nThe derived Rsgradually decreases with decreasing tem-\nperature. Its magnitude is about two orders larger than\nthat ofR0.\nThe derived carrier concentration nis shown in Fig.\n3(a). The n∼0.5×1021cm−3at 20 K corresponds to ∼\n0.04 holes per formula unit, comparable to the value es-\ntimated from q. Taken into account a weak temperature-\ndependent ρ(T) [Fig. 1(c)], the estimated n∼1.11×1021\ncm−3from 484 µΩ cm near 100 K points to a mean free\npathλ∼0.44 nm. This is comparable to the lattice pa-\nrameters and is close to the Mott-Ioffe-Regel limit [40].\nThe AHC σA\nxy(≈ρA\nxy/ρ2\nxx) is plotted in Fig. 3(b). Theo-\nretically, intrinsic contribution of σA\nxy,inis of the order of\ne2/(hd), where eis the electronic charge, his the Plank\nconstant, and dis the lattice parameter [41]. Taking\nd≈V1/3∼4.3˚A,σA\nxy,inis estimated ∼900 (Ω cm)−1,4\nmuch larger than the obtained values in Fig. 3(b). Ex-\ntrinsic side-jump contribution of σA\nxy,sjis usually of the\norder of e2/(hd)(εSO/EF), where εSOandEFis spin-\norbital interaction energy and Fermi energy, respectively\n[42]. The value of εSO/EFis generally less than 10−2for\nmetallic ferromagnets. As we can see, the σA\nxyis about\n22 (Ω cm)−1at 20 K and exhibits a moderate tempera-\nture dependence. This value is much smaller than σA\nxy,in\n∼900 (Ω cm)−1, which precludes the possibility of in-\ntrinsic KL mechanism. Based on the band structure, as\nshown in Fig. 4, we obtained the intrinsic AHC as127(Ω\ncm)−1,whichismuchlargerthanthemeasuredvaluetoo.\nThe extrinsic side-jump mechanism, where the potential\nfield induced by impurities contributes to the anomalous\ngroup velocity, follows a scaling behavior of ρA\nxy=βρ2\nxx,\nthe same with intrinsic KL mechanism. The scaling be-\nhaviorof ρA\nxyvsρxxgivesα∼1.1(2) by using ρA\nxy=βρα\nxx\n[Fig. 3(c)], which also precludes the possibility of side-\njump and KL mechanism with α= 2. It points to that\nthe skew-scattering possibly dominates, which describes\nasymmetric scattering induced by impurities or defects\nand contributes to AHE with α= 1. Furthermore, the\nscaling coefficient SH=µ0Rs/ρ2\nxx=σA\nxy/Ms[Fig. 3(d)]\nis weaklytemperature-dependent and iscomparablewith\nthose in traditional itinerant ferromagnets, such as Fe\nand Ni (SH∼0.01−0.2V−1) [43, 44]. It is proposedthat\nthe FIM in FeCr 2Te4is itinerant ferromagnetism among\nantiferromagnetically coupled Cr-Fe-Cr trimers [28]. In\na noncomplanar spin trimer structures the topologically\nnontrivial Berry phase is induced by spin chirality rather\nthan spin-orbit effect, resulting in chirality-induced in-\ntrinsic AHE [45–48]. Our result excludes such scenario\nin Cr-Fe-Cr trimers in FeCr 2Te4[28].\nCONCLUSIONS\nIn summary, we studied the electronic transport prop-\nerties and AHE in FeCr 2Te4single crystal. The AHE be-\nlowTc= 123 K is dominated by extrinsic skew-scattering\nmechanism rather than the intrinsic KL or extrinsic side-\njump mechanism, which is confirmed by our DFT calcu-\nlations. The spin structure of Cr-Fe-Cr trimers proposed\nfor FeCr 2Te4is of interest to check by neutron scattering\nexperiments on powder and single crystals in the future.\nACKNOWLEDGEMENTS\nWork at Brookhaven National Laboratory (BNL) is\nsupportedbytheOfficeofBasicEnergySciences, Materi-\nals Sciences and Engineering Division, U.S. Department\nof Energy (DOE) under Contract No. DE-SC0012704.\nB.Y. acknowledges the financial support by the Willner\nFamily Leadership Institute for the Weizmann Institute\nof Science, the Benoziyo Endowment Fund for the Ad-vancement of Science, Ruth and Herman Albert Schol-\nars Program for New Scientists, the European Research\nCouncil (ERC Consolidator Grant No. 815869, “Nonlin-\nearTopo”).\n∗Present address: Los Alamos National Laboratory,\nMS K764, Los Alamos NM 87545, USA\n[1] E. H. Hall, Proc. Phys. Soc. Lond. 4, 325 (1880).\n[2] Y. Onose, N. Takeshita, C. Terakura, H. Takagi and Y.\nTokura, Phys. Rev. B 72, 224431 (2005).\n[3] R. Karplus and J. M. 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Allred,5Stuart Brown,4Turan Birol,3and Ni Ni1,\u0003\n1Department of Physics and Astronomy and California NanoSystems Institute,\nUniversity of California, Los Angeles, CA 90095, USA\n2Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\n3Department of chemical engineering and materials science,University of Minnesota, MN 55455, USA\n4Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA\n5Department of Chemistry and Biochemistry, University of Alabama, Tuscaloosa, AL 35487, USA\nA combined study of transport, thermodynamic, neutron di\u000braction, nuclear magnetic resonance\nmeasurements and \frst principles calculation were performed for \f-V2PO 5single crystal. It was\nshown to be a semiconductor with a band gap of 0.48 eV, undergoing a charge ordering (unusual\nV2+and V3+) phase transition accompanied by a tetragonal to monoclinic structural distortion at\n610 K and a paramagnetic to ferrimagnetic phase transition at 128 K with a propagation vector\nofk= 0. The easy axis is in the monoclinic acplane pointing 47(9)\u000eaway from the monoclinic a\naxis. This collinear ferrimagnetic structure and anisotropic isothermal magnetization measurements\nsuggest weak magnetic anisotropy in this compound. The \frst principles calculations indicate that\nthe intra-chain interactions in the face-sharing VO 6chains dominate the magnetic hamiltonian and\nidentify the \u0000+\n5normal mode of the lattice vibration to be responsible for the charge ordering and\nthus the structural phase transition.\nI. INTRODUCTION\nCharge ordering(CO), the long-range ordering of tran-\nsition metal ions with di\u000berent oxidization states, is a\nprominent feature in mixed valent 3 dtransition metal\noxides [1]. Due to the strong Coulomb interaction in the\ncharge ordering state, a high symmetry to low symmetry\nstructural distortion can occur, accompanied with the\nsudden enhancement in the electrical resistivity arising\nfrom the charge localization. The competition between\nthis charge disproportionation and the exchange inter-\nactions among magnetic transition metal ions has led to\nemergent phenomena, such as colossal magnetoresistance\nin RE 1\u0000xAxMnO 3(RE = rare earth, A = alkaline earth)\n[2, 3], superconductivity in \f-Ag 0:33V2O5[4], etc.\nThe vanadium phosphorus oxide system (V-P-O) has\ndistinct structural stacking and variable valences of\nvanadium, providing a great avenue to investigate the\nstructure-property relationship, enriching our under-\nstanding on the competition of CO and various exchange\ninteraction. The fundamental building blocks of the V-\nP-O system consist of VO 6octahedra or VO 4tetrahedra\nlinked by PO 4tetrahedra with valence P5+. Rich 3d\nvanadium magnetism and valences have been observed.\nFor example, VPO 4with V3+ions, containing one di-\nmensional chains of edge-sharing VO 6octahedra, under-\ngoes an incommensurate antiferromagnetic (AFM) phase\ntransition at 26 K and then a commensurate AFM phase\ntransition at 10.3 K [5]. \u000b-VO(PO 3)2with one dimen-\nsional chains of corner-sharing VO 6octahedra, is AFM\nat 1.9 K with valence V4+[6]. Mixed valence V3+and\nV4+antiferromagnetically couple together below 5 K in\n\u0003Corresponding author: nini@physics.ucla.eduV2(VO)(P 2O7)2, where segments of edge-sharing VO 4\ntetrahedra and VO 6octahedra exist [7]. Alternating V4+\nspin-chain model can be used to describe the magnetism\nin (VO) 2P2O7with corner and edge-sharing VO 6octa-\nhedra ladders [8{12].\nIn this article, we investigated \f-V2PO 5. In the tetrag-\nonal phase in Ref. [13] (Fig. 3(b)), it contains chains of\nface-sharing VO 6octahedra linked by PO 4tetrahedra.\nThese chains are stacked in layers along the caxis, run-\nning alternately along the aorbaxis in the adjacent\nlayers (Fig. 3b). This material is intriguing in three as-\npects. Firstly, the valence analysis with P5+and O2\u0000in-\ndicates remarkably low valence V2:5+in this compound,\nwhich may suggest possible CO of V3+and very uncom-\nmon valence V2+[14]. Secondly, the face-sharing VO 6\noctahedra in the building block is very rare for vana-\ndium oxides, implying unusually strong intra-chain in-\nteraction between V ions. What's more, although the\nparallel chains in each layer do not share any oxygen,\nthe perpendicularly-running chains in the neighboring\nlayers are corner-sharing. As a result, the other im-\nportant magnetic interaction is the inter-chain interac-\ntion between the corner-sharing V ions in neighboring\nlayers. Thirdly, a recent \frst-principles calculation sug-\ngested that the \f-V2PO 5is a ferromagnetic (FM) topo-\nlogical Weyl and node-line semimetal without any trivial\nband at the Fermi level [15], being a great material plat-\nform to study the emergent phenomena in FM topological\nsemimetals.\nDespite of these remarkable aspects we discussed\nabove, neither physical properties nor possible structural\ndistortion has been investigated for \f-V2PO 5, therefore,\nwe performed a combined study of the single crystalline\n\f-V2PO 5by x-ray and neutron di\u000braction as well as\nNMR, transport and thermodynamic measurements. We\ndiscovered that \f-V2PO 5is a semiconductor with a bandarXiv:1712.09973v1  [cond-mat.str-el]  28 Dec 20172\ngap of 0.48 eV. Upon cooling, a charge ordering phase\ntransition accompanied with a tetragonal to monoclinic\nstructural phase transition occurs at 610 K, followed by\na long range ferrimagnetic phase transition below 128 K.\nII. EXPERIMENTAL METHODS\nPrecursor\f-V2PO 5powder was made by solid state\nreaction. V 2O5powder and phosphorus chunks were\nweighed according to the stoichiometric ratio 1 : 1 and\nsealed in a quartz tube under vacuum. The ampule was\nslowly heated up to 600\u000eC and dwelled for 2 hours, and\nthen was increased to 1000\u000eC and stayed for 2 days be-\nfore it was quenched in water. The resultant \f-V2PO 5\n(\u00182 g) powder and iodine \rakes (10 mg / cm3) were\nloaded into a 15-cm long quartz tube and sealed under\nvacuum. Single crystals of \f-V2PO 5were then grown\nby chemical vapor transport method [13]. The hot end\nwas set at 1000\u000eC and the cold end was set at 900\u000eC.\nAfter two weeks, quite a few sizable three dimensional\nsingle crystals (\u00184 mm\u00024mm\u00022mm) were found at the\ncold end. The inset of Fig. 1(a) shows a \f-V2PO 5single\ncrystal against 1 mm scale.\nThroughout the paper, abplane is the plane where\nchains locate in. The (hkl) Tmeans the peak indexed\nin the tetragonal structure while (hkl) Mmeans the peak\nindexed in the monoclinic structure.\nMagnetic properties were measured in a Quantum\nDesign (QD) Magnetic Properties Measurement System\n(MPMS3). A single crystal around 20 mg with a pol-\nishedabsurface and a single crystal with as-grown (011)\nsurface were used. Temperature dependent heat capac-\nity was measured in a QD Dynocool Physical Properties\nMeasurement System (Dynoccol PPMS) using the relax-\nation technique at zero \feld. To enhance the thermal\ncontact and lower the measurement time, the \f-V2PO 5\nsingle crystal was ground into powder and then mixed\nwith silver powder according to the mass ratio of 1 : 1\n. The heat capacity of \f-V2PO 5was then obtained by\nsubtracting the heat capacity of silver [16]. Below 200\nK, the two wire ETO method was used for the electric\nresistivity measurement in PPMS. From 200 K to 400 K,\nthe electric resistivity was measured with standard four-\npoint method while above 400 K, it was measured in a\nhomemade high temperature resistivity probe.\nSingle crystal neutron di\u000braction was performed at\nthe HB-3A four-circle di\u000bractometer equipped with a\n2D detector at the High Flux Isotope Reactor(HFIR) at\nORNL. Neutron wavelength of 1.546 \u0017A was used from a\nbent perfect Si-220 monochromator [17]. The pyrolytic\ngraphite (PG) \flter was used before the sample to re-\nduce the half- \u0015neutrons. Representational analysis with\nSARAh [18] was run to search for the possible magnetic\nsymmetries. The nuclear and magnetic structure re\fne-\nments were carried out with the FullProf Suite[19]. Pow-\nder X-ray di\u000braction measurements were performed using\na PANalytical Empyrean di\u000bractometer (Cu K \u000bradia-tion). Using the Fullprof suit[19], Rietveld re\fnement\nwas carried out to re\fne the powder X-ray di\u000braction\ndata with the crystal structure determined by single crys-\ntal neutron di\u000braction.\nNuclear magnetic resonance (NMR) measurement was\ndone under a \fxed magnetic \feld of approximately 8.5 T,\napplied along the direction perpendicular to the abplane,\nwhere the chains locate in. The spectra were collected\nby performing an optimized \u0019=2-\u001c-\u0019spin-echo pulse se-\nquence. The spin-lattice relaxation time T1was mea-\nsured by integration of the phase corrected real part of\nthe spin echo using the saturation-recovery technique cite\nand spin echo decay time T2was measured by altering \u001c\nin the sequence. Spin-lattice relaxation time T1is ob-\ntained by the magnetization recovery \ftting to a single\nexponential form.\nFirst principles Density Functional Theory calcula-\ntions were performed to compare the energies of di\u000berent\nmagnetic con\fgurations. We used PAW as implemented\nin VASP with PBEsol exchange correlation functional\n[20{22]. A 8\u00028\u00028 k-point grid and energy cut o\u000b of\n500 eV ensures convergence in the primitive cell with 4\nformula units.\nIII. EXPERIMENTAL RESULTS\nA. Magnetic, transport and thermodynamic\nproperties\nFigure 1 (a)-(c) show the anisotropic magnetic prop-\nerties of\f-V2PO 5. Figure 1(a) presents the temperature\ndependent M=H taken atH= 1 kOe from 2 K to 250\nK in zero-\feld-cooled (ZFC) warming and \feld-cooled\nmode with Hparallel and perpendicular to the abplane.\nThe sharp upturns of the curves and the bifurcation in\nZFC and FC data for both directions below TIindicate\nthe existence of ferromagnetic component. The smooth\nZFC and FC curves suggest no other magnetic transi-\ntion below TI. Comparing with the other V-O-P mate-\nrials, the magnetic transition temperature is quite high\n[5{7], suggesting strong exchange interactions. Figure 1.\n(b) shows the temperature dependent M=H (blue) and\nH=M (black) measured from 300 K to 1000 K with H\n// (0 1 1) TatH= 10 kOe. Firstly, we see a subtle but\ndiscernable enhancement in M=H at the characteristic\ntemperature TII= 610 K, suggesting a possible phase\ntransition here. Secondly, upon cooling, linear Curie-\nWeiss behavior can be clearly seen from 1000 K to 500 K\ninH=M . By \ftting H=M from 1000 K to 500 K using the\nCurie-Weiss formula H=M =C=(T\u0000\u0012cw), whereCis the\nCurie constant and \u0012cwis the Weiss temperature, we ob-\ntained\u0016eff= 3.7(2)\u0016B/V and\u0012cw=\u0000900 K. The \u0016eff\nis larger than the one of V2:5+but comparable to the\none ofV2+. The large negative \u0012cwwithj\u0012cw=TIj\u00187.2,\nsuggests strong antiferromagnetic interaction. Thirdly,\ntheH=M fromTIto 500 K shows a crossover concave\nbehavior with temperature.3\nFIG. 1. (a) ZFC and FC M=H vs.TunderH= 1 kOe with H==ab andH?abfrom 2 K to 250 K. Inset: picture of \f-V2O5P\nsingle crystal against 1-mm scale. (b) H=M andM=H vs.TunderH=10 kOe with H==(011) T2 K to 1000 K. The red line\nis the Curie-Weiss \ft. (c) Isothermal M(H) curves at 2 K with H==ab andH?ab. Inset:M(H) curves at 300 K and 800\nK along with H==(011) T. (d) Speci\fc heat Cpvs.Tfrom 2 K to 200 K. The red line is the \ftting curve by Debye model.\nInset: Temperature dependence of magnetic entropy. (e) Resistivity \u001avs.Tfrom 130 K to 760 K. The red line emphasizes\nthe transition at TII. Inset:\u001avs. 1=T. Red line: the \ftting curve using the thermal excitation model. (f) The Arrott plot at\nvarious temperatures from 122 K to 137 K.\nFigure 1 (c) shows the anisotropic \feld dependent mag-\nnetizationM(H) taken at 2 K with H==ab andH?ab.\nThe crystal orientation was determined by x-ray di\u000brac-\ntion (Fig. S1) [23]. Clear hysteresis can be observed\nin both directions, con\frming the existence of ferromag-\nnetic component. Both curves show very similar shape\nand magnitude, suggesting weak magnetic anisotropy in\nthis system. The coercive \felds are around 1.3 kOe for\nboth. The remanent moments are 0.22 \u0016B/V forH?ab\nand 0.25\u0016B/V forH==ab and the saturation moments\nare 0.27\u0016B/V forH?aband 0.31\u0016B/V forH==ab ,\nwhich are so much smaller than the saturation moment\nof V2+(d3) and V3+(d2) ions, suggesting that instead\nof ferromagnetism, this material is likely ferrimagnetic or\ncanted antiferromagnetic below TI. The inset of Fig. 1(c)\nshows theM(H) curves taken at 300 K and 800 K with\nH// (0 1 1) T. Both curves are linear with the applied\nmagnetic \feld without hysteresis. The Arrott plot ( M2\nvs.H=M ) has been widely used to determine the fer-\nromagnetic phase transition temperature [24, 25], where\nthe curve of M2vs.H=M passes through the origin of\nthe plot at the transition temperature. To determine the\nvalue ofTI, isothermal M(H) curves are measured from\n122 K to 137 K. The Arrott plot are calculated and shown\nin Fig. 1(f), which suggests that TI\u0018128 K.\nFigure 1 (d) shows the temperature dependent Cp=T\ndata (blue) taken from 2 K to 200 K. A heat capacityanomaly featuring a second order phase transition ap-\npears around 128 K, accompanying with the magnetic\nphase transition observed in Fig. 1 (a). To estimate the\nmagnetic entropy, Debye model is used to \ft the heat\ncapacity to provide the non-magnetic background. The\n\ftted curve is shown in red in Fig. 1(d) and the \ftted\nDebye temperature is \u0002 D= 620 K. By subtracting the\nnon-magnetic background from \ftting, we obtained the\nmagnetic entropy as SM= 5.6 J/mol V-K2. This value is\nsigni\fcantly smaller than RLn4 of V2+andRLn3 of V3+,\nbut rather approximate RLn2. This may be caused by\nthe strong V-O covalency which lowers the moment size\nof V or a result of the entropy release above 128 K due\nto the chain structure and strong intrachain interaction\n[26, 27].\nFigure 1(e) shows the resistivity ( \u001a) of the\f-V2PO 5\nsingle crystals vs. temperature from 130 K up to 760 K.\nInstead of the semi-metal suggested by the theoretical\nprediction [15], it is a semiconductor. A semiconductor\nto semiconductor phase transition is discernable at 610\nK, which con\frms the possible phase transition at TII\nsuggested by the subtle susceptibility increase shown in\nFig. 1 (b). \u001avs. reciprocal temperature is plotted in\nthe inset of Fig. 1(d). By \ftting the data between 280\nK to 130 K with the thermal excitation model \u001a(T) =\n\u001a(0)exp(Eg=2kBT), the estimated gap size of \f-V2PO 5\nis 0.48 eV. The gap value is similar to 0.45-0.57 eV of the4\nVanadium phosphate glass [28].\nB. Structural phase transition and charge ordering\nBased on the transport, magnetic and heat capacity\nmeasurements, we have shown that \f-V2PO 5has one\nphase transition at 610 K and the other magnetic phase\ntransition at 128 K. To investigate the nature of these\ntwo phase transitions, single crystal neutron di\u000braction\nand NMR measurements are performed, which are sum-\nmarized in Fig. 2.\nFigure 2 (a) and (c) show the order parameter plot\nof the (1 1 4) Tneutron peak up to 650 K and (1 1 0) T\nneutron peak up to 450 K, respectively and Fig. 2 (b)\npresents the rocking curve scan for the (1 1 4) Tpeak. It is\ntwinned structure below 610 K, so we keep the tetragonal\nindex for convenience. The order parameter plot of the\nrelative stronger peak (1 1 4) T(Fig. 2(a)) indicates two\nphase transitions occurring at 610 K and 128 K, respec-\ntively, which is consistent with the order parameter plot\nof the (1 1 0) Tpeak (Fig. 2(c)) and (1 0 1) Tpeak (Fig.\nS2) [23]. The full data were collected at 4.5 K, 300 K,\nand 650 K to cover all three phase regions. At 650 K, the\ndata can be well \ftted in I41/amd symmetry (Table I).\nSince both (1 1 0) Tand (1 1 4) Tpeaks are symmetry dis-\nallowed re\rections in the tetragonal I41/amd structure,\nthe fact that we observed these peaks at room temper-\nature (Fig. 2(b)) suggests possible structural/magnetic\nphase transitions at 610 K.\nTo identify if the phase between 128 K and 610 K has\na magnetic component, phosphorus-31 NMR (31P-NMR)\nmeasurements were carried out. These are summarized\nin Fig. 2(e)-(f). In Fig. 2(d), we report the31P-NMR\nspectra at various temperatures from 170 K to 300 K.\n31P nuclear spin I= 1=2, and all sites are equivalent\nforB?ab. At 300 K, the spectra shift from the Lar-\nmor frequency by Ks= 0:493\u00060:002%, where the mean\nand uncertainty were calculated using the gaussian \ft-\nting. This value is on the same order of another VPO\nsample with V3+and about twice as much as that with\nV4+[29, 30], where a similar shift and broadening were\nobserved. The observations are interpreted as evidence\nfor no long range magnetic ordering in the intermediate,\ncharge-ordered phase. Below 190 K, a minor absorption\npeak at 146.4 MHz is resolved. Since it accounts for\nonly 3% of the total spin intensities, we expect the signal\nto be extrinsic due to the sites at twin boundaries and\nexclude it in our analysis. From the slope of K- \u001fplot\nshown in Fig.2 (e), we estimate the hyper\fne coupling\nconstant to be Acc= (10:23\u00060:87) kOe/\u0016B, which is\ntransferred from the unpaired electrons from the second\nnearest neighbors of P [31]. Figure 2(f) shows the tem-\nperature dependence of the spin-lattice relaxation time\nT1and spin-spin relaxation time T2. The relatively short\nand constant T1is consistent with a paramagnetic phase\nfrom 150 k to 300 K [32]. Meanwhile, T2starts drop-\nping rapidly below 210 K as the system approaches the\nFIG. 2. (a) The (1 1 4) Tneutron peak intensity vs. T. (b)\nThe (1 1 4) Tneutron peak intensity vs. !. (c) The (1 1 0) T\nneutron peak intensity vs. T. (d) P NMR frequency spectra.\nThe spectra are conserved after corrections for T 2. (e) Knight\nshift K and the peak width \u0001F obtained from (a) vs. mag-\nnetic susceptibility \u001f. Dashed line shows a linear \fttings of\nK. (f) Spin-lattice relaxation time T1and spin-spin relaxation\ntimeT2vs. T.\n128 K transition. This behavior is associated with slow\nlongitudinal \ructuations, which are likely related with\nthe onset of the 128 K magnetic phase transition. This\nis also consistent with the broadening observed in Fig.\n2(d) which appears as the magnetic correlation develops\nupon cooling. However, since 1/ T2is on the order of a\nfew to hundreds of KHz, we conclude that most of our\nbroadening, which is on the order of several MHz, is from\nthe inhomogeneous internal \feld.\nSince NMR shows that the phase transition at 610 K\nis not of magnetic origin, the phase transition at 610\nK should be a structural distortion. By including four\ntwinned structure domains, the room temperature neu-\ntron data can be well \ftted with the C2/cmonoclinic\nsymmetry, suggesting a tetragonal to monoclinic phase\ntransition at 610 K. Using this crystal structure, we re-\n\fned our powder X-ray di\u000braction taken at room tem-\nperature and obtained a very good \ft as shown in Fig.5\nFIG. 3. (a) The experimental and re\fned powder X-ray\ndi\u000braction patterns for \f-V2O5P at 300 K. Black: experi-\nmental pattern. Red: re\fned pattern. green: the di\u000berence\nbetween the experimental and re\fned patterns. Black ticks:\nthe Bragg peak positions in the monoclinic structure. Inset:\nenlarged view from 55.5\u000eto 58\u000e. (b)(c): The crystal structure\nof\f-V2O5P at 650 K (b) and 300 K (c).\n3(a). The detailed crystal structures at 4.5 K, 300 K\nand 650 K are summarized in Table I. The high temper-\nature tetragonal and low temperature monoclinic struc-\ntures are visualized in Fig. 3(b) and (c), respectively. In\nthe monoclinic phase, the monoclinic caxis is 121.45\u000e\nfrom theabplane which is the plane where the chains\nsit in. The unique V site in the tetragonal structure sep-\narates into V1 and V2 sites with the V1 atoms and V2\natoms alternately locating along each chain direction (V1\nand V2 sites are labeled in Fig. 3(c)), as a result, the av-\nerage bond length of VO 6octahedra on V1 site increases\nwhile that on V2 site decreases. Bond-valence analysis\nof the monoclinic crystal structure assigns charges of 2.0\nand 2.9 toV1andV2respectively, which is a smoking gun\nproof of the charge order [33, 34].\nC. Ferrimagnetic structure below 128 K\nThe magnetic order onsets at 128 K while the charge\norder continues to develop below the magnetic transi-TABLE I. The crystal structure of the \f-V2OPO 4phase at\n4.5 K, 300 K and 650 K, respectively.\n\f-V2PO 5at 4.5 K monoclinic C2/c\na= 7.563 \u0017A b=7.563 \u0017A c=7.235 \u0017A\f= 121.51\u000e\nRF2=0.0691 wRF2=0.0841 R F=0.044 \u001f2=19.2\nsite x/a y/b z/c\nV1 0 1/2 0\nV2 1/4 1/4 0\nO1 0.066(3) 0.749(2) 0.632(2)\nO2 0.317(2) 0.496(2) 0.598(1)\nO3 0 0.652(2) 1/4\nP 0 0.121(2) 1/4\n\f-V2PO 5at 300 K monoclinic C2/c\na= 7.570 \u0017A b=7.570 \u0017A c=7.232 \u0017A\f= 121.56\u000e\nRF2=0.0734 wRF2=0.0956 R F=0.0473 \u001f2=25.1\nsite x/a y/b z/c\nV1 0 1/2 0\nV2 1/4 1/4 0\nO1 0.065(3) 0.749(2) 0.630(2)\nO2 0.318(2) 0.496(2) 0.599(1)\nO3 0 0.652(2) 1/4\nP 0 0.119(3) 1/4\n\f-V2PO 5at 650K Tetragonal I41/amd\na= 5.357(2) \u0017A b= 5.357(2) \u0017A c=12.373(4) \u0017A\nRF2=0.0707 wRF2=0.0888 R F=0.0421 \u001f2=5.12\nsite x/a y/b z/c\nV1 1/4 1/4 1/4\nP1 0 3/4 1/8\nO1 0 -0.012(1) 0.193(4)\nO2 0 3/4 5/8\nFIG. 4. (a)-(c): Three magnetic structure models showing\nferrimagnetism with di\u000berent easy axis. Model (a) is collinear\nsuggesting weak magnetic anisotropy while Models (b) and\n(c) are noncollinear indicating strong magnetic anisotropy.\nModel (a) is the magnetic structure of \f-V2PO 5.6\ntion. Since no observed sharp change can be determined\nby the structure re\fnement at 4 K (see Table I), no fur-\nther structural phase transition below 128 K is discern-\nable. The magnetic propagation vector is k=0, which\nmeans that the magnetic scattering signal appears on\ntop of the nuclear Bragg peaks. To determine the mag-\nnetic structure more precisely, the magnetic signals were\nextracted by subtracting the data measured just above\n128 K from that at 4 K. During the procedure, to se-\nlect peaks which are insensitive to the thermal displace-\nments and charge ordering, we compared the data mea-\nsured at 300 K and 450 K and only selected a peak if\nthe change of its intensity is much smaller than the ex-\ntracted magnetic intensity. The selected re\rections are\nlisted in Table SI [23]. Since (1 1 0) T, (1 1 4) T, (1 0\n1)T, (0 0 2) Tand (0 0 4) Tpeaks were measured with a\nlong counting time and also tracked upon warming, they\nare highly reliable as indicated by their small error bars\nshown in the Table SI. We then performed the represen-\ntational analysis that determines the symmetry-allowed\nmagnetic structures for a second-order magnetic tran-\nsition. It yielded two magnetic symmetries, C2=cand\nC20=c0. Only the C20=c0can \ft our data (see Table SI).\nThe obtained magnetic structure is ferrimagnetic. Spins\non all V2+(V1 sites) atoms are parallel and so do the\nspins on all V3+(V2 sites) atoms while these two spin\nsublattices are antiparallel to each other. Since it is un-\nlikely for the V2+(d3) to be in a low spin state due to\nthe longer V-O bond length on V1 site and thus weaker\ncrystal electric \feld, the moment MV1> M V2. Figure\n4 shows the ferrimagnetic structure with three possible\neasy axis assignments where MV1> M V2. The calcu-\nlated peak intensity and goodness of \ft are summarized\nin Table SI. The model in the left panel of Fig. 4(a)\nis our pick for the \f-V2PO 5which gives the best \ft of\nthe data as shown in the right panel of Fig. 4(a). The\nMV1= 1:4(1)\u0016BandMV2= 1:2(1)\u0016B. The easy axis is\ninacMplane and 47(9) degrees away from aMtowards\ncM. The magnetic structure is collinear, suggesting weak\nmagnetic anisotropy. This is indeed consistent with the\nanisotropic M(H) measurements shown in Fig. 1(c). The\nother two models shown in Fig. 4(b) and (c) are with the\neasy axis along the chain direction (Fig. 4(b)) or perpen-\ndicular to the chain direction on the abplane (Fig. 4(c)).\nThese two magnetic models are non-collinear with strong\nmagnetic anisotropy. Since the goodness of \ft for these\ntwo latter models are poor, they are not the right mag-\nnetic structure for \f-V2PO 5.\nIV. DISCUSSION\nIt is of particular interest to ask if the charge order is\nsolely responsible for the reduction in the crystal symme-\ntry, or rather if it is a secondary order parameter to some\nother electronic phase transition. In order to preclude\nthis possibility and elucidate the nature of the charge or-\ndering transition at 610 K, we performed a group theo-TABLE II. Energies of di\u000berent magnetic con\fgurations from\n\frst principle.\nIntra-chain Inter-chain Energy (meV/f.u.)\nFerrimagnetic Ferromagnetic 0\nFerrimagnetic Antiferromagnetic 9\nFerromagnetic Ferromagnetic 60\nFerromagnetic Antiferromagnetic 71\nretical analysis of the lattice distortion using the Isotropy\nSoftware Suite.[35] The distortion from the high temper-\nature tetragonal structure ( I41=amd ) to the low temper-\nature monoclinic structure ( C2=c) can be caused by two\nnormal modes of lattice vibration, \u0000+\n5and \u0000+\n4. \u0000+\n5re-\nduces the symmetry from I41=amd toC2=c, and is the\nonly candidate for the primary structural order parame-\nter, whereas \u0000+\n4by itself reduces the symmetry to Fddd .\nC2cis a subgroup of Fddd , and as a result, \u0000+\n4is most\nlikely a secondary order parameter that is not important\nin the energetics of the phase transition. At the same\ntime, the charge order itself, which is a di\u000berentiation\nof the neighboring V ions in the same chain, transforms\nas the \u0000+\n5irreducible representation does for the high\nsymmetry structure. Figure 5(a) shows the displacement\nof the oxygen atoms in the VO 6face-sharing chain due\nto the \u0000+\n5normal mode. This mode breaks the symme-\ntry between the V ions that are symmetry equivalent at\nI41=amd , and decreases the V-O bond length for V2while\nincreasing it for V1(Fig. 5(b)) as expected in a charge\nordering transition. We therefore conclude that to re-\nduce the symmetry to the monoclinic phase, the charge\norder, by itself, is su\u000ecient and no other magnetic or\nelectronic mechanisms are necessary. This is consistent\nwith the fact that the TCOis almost 5 times of the Tmag.\nWe also note that \u0000+\n5is a Raman active mode, and as a\nresult, signature of the charge ordering transition should\nbe visible in the Raman spectrum of \f-V2PO 5.\nWe performed the \frst principles calculation using\nDFT+U with U = 4 eV to correct for the underestima-\ntion of the on-site coulomb interaction on the V ion [36].\nOur DFT calculations predict magnetic moments of 2.6\n\u0016Band 1.8\u0016Brespectively inside the V1andV2spheres,\nbut the band structure (Fig. 5(c)) shows no partially\n\flled bands. This signals strong hybridization between\ntheVand theOions. DFT gives a magnetic moment\n0.5\u0016BperVincluding the interstitials. This is a strong\noverestimation compared to the experimental value. The\nreason of this is likely the DFT+U's tendency to overes-\ntimate the ordered moments when there are dynamical\n\ructuations present, and it is possible that a more ad-\nvanced \frst principles method (such as the Dynamical\nMean Field Theory) can reproduce the experimentally\nobserved value of the local moments.\nTo understand the magnetic order, we calculated the\nenergies of phases with di\u000berent magnetic orders from\nDFT, as listed in Table II. The lowest energy phase is\npredicted to have ferrimagnetic intra-chain order, where7\nFIG. 5. (a) A sketch of the Oxygen anion displacements due\nto the \u0000+\n5mode which is responsible for charge ordering. (b)\nThe average V\u0000Obond length as a function of the \u0000+\n5normal\nmode amplitude. (c) DFT band structure in the ferrimagnetic\nstate. Majority and minority spin bands are shown in red and\nblue respectively.\nthe moments of the neighboring V1andV2ions on the\nsame chain are aligned anti-parallel, and ferromagnetic\ninter-chain order, so that, for example, the magnetic mo-\nments of all V1ions are parallel. This observation is in\nline with the experimental observation. The energy cost\nof having a magnetic phase where the di\u000berent chains\nhave antiparallel moments is bout 9-10 meV per formula\nunit, whereas the energy cost of having spins on the same\nchain parallel is 60 meV per formula unit. This suggests\nthat the intra-chain interactions between the V1andV2\nions in the face-sharing VO 6chains are the dominant\nterm in the magnetic hamiltonian.\nTo understand the crossover behavior in H/M shown\nin in Fig. 1(b), we calculated the energetics of di\u000ber-\nent magnetic phases in the high temperature tetrago-\nnal structure ( I41=amd ) with the same parameters (not\nshown), and found that similar couplings apply to mag-netic moments in that structure too. This explains the\ncross-over: Above the charge ordering temperature, the\nmagnetic moments on all V ions are equal, and its Curie-\nWeiss behaviour is that of an antiferromagnet, with a\nnegative Curie temperature. However, charge ordering\nmakes the moments unequal, and as a result, below 610\nK the Curie-Weiss behaviour is that of an ferrimagnet,\nwhich has a positive Curie temperature like a ferromag-\nnet.\nTo address if there is non-trivial topology in this com-\npound, the DFT band structure in Fig. 5(c) is calcu-\nlated in the ferrimagnetic ground state. Unlike the DFT\nband structure calculated in the ferromagnetic phase and\nthe tetragonal structure without U [15], there are no\nband crossing at the Fermi level. And more importantly,\nthe top of the valence and the bottom of the conduc-\ntion bands have opposite spin directions. This obser-\nvation precludes any possibility of topological phases in\n\f-V2PO 5.\nV. CONCLUSION\nIn conclusion, we have grown and characterized \f-\nV2PO 5single crystals. A tetragonal to monoclinic struc-\ntural phase transition at 610 K and a paramagnetic to\nferrimagnetic phase transition are revealed by transport,\nmagnetic, speci\fc heat, single crystal neutron di\u000braction\nand NMR measurements. Below 610 K, the single V site\nat the high temperature tetragonal phase distorts into\ntwo alternating V sites, leading to the increase of V-O\nbond length of one V site but the decrease of the other\nV site. Our \frst principles calculation shows that this\ndistortion was caused by the \u0000+\n5normal mode of lat-\ntice vibration. Accompanied with the distortion, charge\nordering of V2+and V3+is undoubtedly suggested by\nthe Bond-valence analysis. Below 128 K, the spins order\nparallel on each sublattice of V sites while the spins on\none sublattice order antiparallel to the other. With the\neasy axis being in the monoclinic acplane and 47(9)\u000e\naway from the monoclinic atowardscaxis., this gives a\ncollinear ferrimagnetic structure with the moment to be\n1.4(1)\u0016Bon V2+site and 1.2(1) \u0016Bon V3+site, suggest-\ning weak magnetic anisotropy and dominant role of the\nintra-chain V-V interaction in magnetism. No non-trivial\ntopology is suggested by our \frst principles calculation.\nACKNOWLEDGMENTS\nWork at UCLA (JX, CWH, RC, NN) was supported\nby NSF DMREF program under the award NSF DM-\nREF project DMREF-1629457. Work at ORNL HFIR\nwas sponsored by the Scienti\fc User Facilities Division,\nO\u000ece of Science, Basic Energy Sciences, U.S. Depart-\nment of Energy. Work at UMN was supported by NSF\nDMREF program under the award NSF DMREF project\nDMREF-1629260. 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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. CR acknowledges support of NSF award DMR 1808190 and \n1954606.  \n \nCONFLICT OF INTEREST  \nThe authors have no conflicts to disclose.  \n \nDATA AVAILABILITY  \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request.  \n \n    \n43 \n REFERENCES  \n(1) Shirsath, S. E.; Cazorla, C.; Lu, T.; Zhang, L.; Tay, Y. Y.; Lou, X.; Liu, Y.; Li, S.; Wang, \nD. Interface -Charge Induced Giant Electrocaloric Effect in Lead Free Ferroelectric Thin -\nFilm Bilayers. Nano Lett.  2019 , 20 (2), 1262 –1271.  \n(2) Shirsath, S. E.; Wang, D.; Zhang, J.; Morisako, A.; Li, S.; Liu, X. Single -Crystal -like \nTextured Growth of CoFe2O4 Thin Film on an Amorphous Substrate: A Self -Bilayer \nApproach. ACS Appl. Electron. Mater.  2020 , 2 (11), 3650 –3657.  \n(3) Lu, Q.; Li, Y.; Peng, B.; Tang, H.; Zhang, Y.; He, Z.; Wang, L.; Li, C.; Su, W.; Yang, Q.; \nothers. 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Ralph1, 2\n1)Cornell University, Ithaca, New York 14853, USA\n2)Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853,\nUSA\n(Dated: 9 October 2018)\nWe analyze the structural and magnetic characteristics of (111)-oriented lutetium\niron garnet (Lu 3Fe5O12) \flms grown by molecular-beam epitaxy, for \flms as thin as\n2.8 nm. Thickness-dependent measurements of the in- and out-of-plane ferromagnetic\nresonance allow us to quantify the e\u000bects of two-magnon scattering, along with the\nsurface anisotropy and the saturation magnetization. We achieve e\u000bective damping\ncoe\u000ecients of 11 :1(9)\u000210\u00004for 5.3 nm \flms and 32(3) \u000210\u00004for 2.8 nm \flms,\namong the lowest values reported to date for any insulating ferrimagnetic sample of\ncomparable thickness.\na)Electronic mail: clj72@cornell.edu\n1arXiv:1609.04753v1  [cond-mat.mes-hall]  15 Sep 2016Insulating ferrimagnets are of interest for spintronic applications because they can possess\nvery small damping parameters, as low as 10\u00005in the bulk.1They also provide the potential\nfor improving the e\u000eciency of magnetic manipulation using spin-orbit torques from heavy\nmetals2,3and topological insulators,4,5because ferrimagnetic insulators will not shunt an\napplied charge current away from the material generating the spin-orbit torque. Making\npractical devices from ferrimagnetic insulators will require techniques capable of growing\nvery thin \flms (a few tens of nm and below) while maintaining low damping. Much of the\nprevious research in this \feld has focused on yttrium iron garnet (Y 3Fe5O12, YIG) grown\nby pulsed-laser deposition or o\u000b-axis sputtering,6{10but YIG is just one in a family of rare\nearth iron garnets with potentially useful properties.11Here we examine the magnetic and\nstructural properties of thin, (111)-oriented \flms of lutetium iron garnet (Lu 3Fe5O12, LuIG)\ngrown by an alternative method, molecular-beam epitaxy (MBE).12We \fnd that MBE is\ncapable of providing sub-10-nm \flms with very low values of damping, rivaling or surpassing\nother deposition techniques. We are able to grow LuIG \flms down to 2.8 nm, or 4 layers\nalong the interplanar spacing d111(0.71 nm),11,13while retaining high crystalline quality.\nWe report in- and out-of-plane ferromagnetic resonance measurements as a function of \flm\nthickness, demonstrating reduced two-magnon scattering compared to previous work. We\nachieve e\u000bective damping coe\u000ecients as low as 11 :1(9)\u000210\u00004for 5.3 nm LuIG \flms and\n32(3)\u000210\u00004for 2.8 nm \flms, which can be compared to the best previous report for very\nthin YIG, 38\u000210\u00004for a 4 nm \flm.6\nAs an iron garnet, LuIG has ferrimagnetic properties similar to YIG. The magnetic\nmoments in both materials arise from their Fe3+ions, which interact via super-exchange\nthrough oxygen atoms.11,14In bulk samples, LuIG has a slightly higher room-temperature\nsaturation magnetization (1815 Oe) than YIG (1760 Oe).11,14,15The bulk lattice parameters\nfor LuIG (12.283 \u0017A) and YIG (12.376 \u0017A) di\u000ber by 0.75%.16,17Both materials can be grown\non isostructural gadolinium gallium garnet (Gd 3Ga5O12, GGG) substrates, which have a\ncubic lattice parameter of 12.383 \u0017A. The resulting mismatch causes biaxial tensile strain\nwith a maximum value of 0.81% and 0.07% for LuIG and YIG, respectively. High-quality\nYIG \flms have been grown previously using o\u000b-axis sputter deposition10,18{21and pulsed-\nlaser deposition (PLD).6,22{28The best reported damping values for thin YIG \flms grown\nby PLD to date include 2 :3\u000210\u00004for a 20 nm \flm,63:2\u000210\u00004for a 10 nm \flm treated\nwith a post-growth etching procedure,29and 0:7\u000210\u00004for a 20 nm \flm treated with a post-\n2growth high-temperature anneal.30For o\u000b-axis sputtering, the best reported values include\n6:1\u000210\u00004for a 16 nm \flm,2112:4\u000210\u00004for a 10.2 nm \flm,19and 0:9\u000210\u00004for a 22 nm \flm\nwith a post-growth high-temperature anneal.20Previous measurements of \flms thinner than\n10 nm recorded signi\fcant two-magnon scattering,6,19and much larger damping parameters\nof 38\u000210\u00004for a 4 nm \flm and 16 \u000210\u00004for a 7 nm \flm.6\nHere we report the growth of epitaxial LuIG \flms with thicknesses from 2.8 to 40 nm\nby reactive MBE on (111) GGG substrates. (We study LuIG, rather than YIG, primarily\nbecause Lu is available within our MBE chamber.) Our substrates are prepared by anneal-\ning at 1300 °C for 3 hr in an air furnace to produce well-de\fned unit-cell steps and smooth\nterraces (see Supplementary Information (SI)). During growth, we simultaneously co-supply\nLu and Fe with an accuracy of \u00065%, to achieve the stoichiometric atomic ratio of Lu:Fe=3:5.\nWe use distilled ozone (O 3) at a background pressure of 1 :0\u000210\u00006Torr as the oxidant. The\ngrowth temperature is 950 to 970 °C, achieved by radiatively heating the backside of the\nGGG substrates, which are coated with 400 nm of Pt to enhance thermal absorption. The\nquality of crystal growth is monitored using in-situ re\rection high-energy electron di\u000brac-\ntion (RHEED) along both the [1 \u001610] and [11 \u00162] in-plane azimuthal directions. The RHEED\nintensity oscillations (Fig. 1(a)) indicate layer-by-layer growth,31with an oscillation period\ncorresponding to the d444spacing, which is a quarter of a single LuIG layer ( d111= 0:71 nm)\nalong the (111)-orientation. We also observe sharp RHEED features and clear Kikuchi lines\nduring growth, as seen in Fig. 1(b,c) for a 10 nm \flm, demonstrating that our \flms are of\nhigh crystalline quality. These features are not observed if the \rux drifts more than \u00065%,\nor if the growth temperature is less than 900 °C.\nWe quantify the strain state and verify the crystalline quality with four-circle X-ray\ndi\u000braction (XRD) measurements. The normalized rocking curves for \flms with di\u000berent\nthicknesses (except the 2.8 nm \flm), overlaid in Fig. 2(a), all have full-width at half-\nmaximum (FWHM) values that are less than 0.004 °, limited by the GGG substrate. This\nindicates that our \flms are commensurately strained, and are at the maximal strain state\nof 0.81% set by the lattice mismatch with the substrate. While the rocking curve measure-\nments on the 2.8 nm \flm lack su\u000ecient signal-to-noise for analysis, the thicker \flms suggest\nthat the strain state is also commensurate for this \flm. The surfaces of the \flms are char-\nacterized by atomic force microscopy. Figure 2(b) shows the 2.8 nm \flm, with a measured\nsurface roughness of 0.26 nm (RMS) over a 5 µm x 5 µm scan area. This indicates that the\n3surface quality is substrate limited, which we observe for all thicknesses. Figure 2(c) shows\nthe\u0012=2\u0012XRD patterns of the LuIG thin \flms for all thicknesses grown. The visible Laue\noscillations con\frm thickness measurements we make with the RHEED intensity oscillations\nand \rux calibrations. Low-angle X-ray re\rectively (XRR) determines the \flm thicknesses\nas 2.84(1), 5.33(2), 9.94(2), 20.16(3) and 40.37(10) nm, which we nominally report as 2.8,\n5.3, 10, 20, and 40 nm.\nThe magnetic properties of the MBE-grown LuIG \flms are characterized by measuring\nthe frequency and thickness dependence of ferromagnetic resonance (FMR). The samples\nare placed, LuIG-side down, on a broadband coplanar waveguide so that the Oersted \feld\nof the waveguide excites FMR at GHz frequencies.32We measure the FMR spectra at \fxed\nfrequency by sweeping the applied magnetic \feld, oriented either in-plane (IP) parallel to the\ncoplanar waveguide or out-of-plane (OOP). For the IP measurements, we position the \flm so\nthat the applied magnetic \feld is always along the [11 \u00162] crystal orientation. The measured\nsignal corresponds to the derivative absorption, which we detect via the voltage from a\ndetector diode. We achieve optimal sensitivity using lock-in ampli\fcation by modulating\nboth the input power and the applied \feld. All of the FMR measurements are performed\nat room temperature. Further details of the FMR apparatus are described in the SI.\nFigure 2(d) shows the IP-FMR response at 5 GHz for LuIG samples with di\u000berent thick-\nnesses. Two trends are apparent as the \flm thickness is reduced: (i) the resonance position\nshifts to higher \felds and (ii) the linewidth increases substantially. Below we show that\nboth of these e\u000bects can be explained by two-magnon scattering.33{35We focus \frst on the\nbehavior of the resonance \felds. We have measured the IP-FMR resonances for each \flm\nthickness at frequencies from 1 to 10 GHz. The evolution as a function of frequency is shown\nin Fig. 3(a) and as a function of thickness in Fig. 3(b).\nIn the presence of two-magnon scattering, the IP resonance \feld Hk\nrpredicted by the\nKittel equation in the thin-\flm limit takes the form33,36\nHk\nr(f;t) =s\u00124\u0019M e\u000b(t)\n2\u00132\n\u0000\u00122\u0019f\nj\rj+ \u0001Hr(t)\u00132\n\u00004\u0019M e\u000b(t)\n2;(1)\nwithfthe excitation frequency, tthe \flm thickness, \u0001 Hra renormalization shift associated\nwith two-magnon scattering, and \rthe gyromagnetic ratio. We measured j\rj=2\u0019= 2:77(2)\n4MHz/Oe based on the frequency dependence of the OOP resonance \feld H?\nr(see SI). The\ne\u000bective anisotropy \feld 4 \u0019M e\u000bis expected to depend on the \flm thickness, because it\ncontains contributions from both bulk demagnetization and surface anisotropy:\n4\u0019M e\u000b= 4\u0019Ms+2Ks\nMst: (2)\nHereMsis the saturation magnetization and Ksis the surface anisotropy energy. The renor-\nmalization shift produced by two-magnon scattering can be related to the surface anisotropy\nas33,36\n\u0001Hr(t) =r\u00122Ks\nMst\u00132\n; (3)\nwhereris a parameter characterizing the strength of two-magnon scattering.\nWe performed a global least-squares \ft of Eqs. (1)-(3) to all the data in Fig. 3 using three\n\ftting parameters r, 4\u0019Ms, andKs. As shown by the lines in Fig. 3, we \fnd excellent \fts\nassuming that all three parameters are independent of \flm thickness, obtaining the values\nr= 4:9(2)\u000210\u00004Oe\u00001, 4\u0019Ms= 1609(1) Oe, and Ks=\u00008:52(8)\u000210\u00003erg/cm2. We also\nattempted to \ft the data without the two-magnon contribution (i.e., with the constraint r=\n0 Oe\u00001), but we found signi\fcant discrepancies for the 2.8 \flm, especially at low frequencies\n(see SI). The non-zero value of rimplies that the two-magnon mechanism is active. For our\n2.8 nm \flm, the renormalization shift is \u0001 Hr= 110 Oe, similar to that found in a 2.7 nm\nNiFe \flm.36This is the \frst report of the renormalization shift in iron garnets. The value\nof 4\u0019Msdetermined by the \ft is signi\fcantly lower than the bulk LuIG value of 1815 Oe.15\nThis reduction is qualitatively consistent with the tensile strain in our \flms from the GGG\nsubstrate. The tensile strain is expected to enhance the antiferromagnetic super-exchange\ninteraction between the two inequivalent Fe3+lattices in the LuIG and therefore reduces the\noverall saturation magnetization.11,14The negative sign that we \fnd for Ksindicates that\nthe surface anisotropy reduces the e\u000bective demagnetization \feld 4 \u0019M e\u000bcompared to the\nbulk value. The magnitude of Ksis relatively weak, however (e.g., more than two orders\nof magnitude smaller than Ksfor annealed CoFeB).37With our values for 4 \u0019MsandKs,\nonly for extremely thin LuIG \flms, <0:8 nm, might the magnetic anisotropy be turned\nperpendicular to the sample plane. For any thickness above this, 4 \u0019M e\u000bfavors in-plane\nmagnetization.\n5Next we consider the FWHM linewidths (\u0001 H) of the IP FMR resonances for our LuIG\n\flms as a function of thickness and FMR frequency. The linewidths of our samples are\nsu\u000eciently narrow that small inhomogeneities in the \flms can result in overlapping but dis-\ntinguishable resonances, as has often been seen previously in measurements on thin garnet\n\flms.6,25,38To make an accurate determination of the intrinsic linewidths, we \ft each mea-\nsured curve to the sum of multiple (2 in this analysis) Lorentzian derivative curves with their\nwidths constrained to be identical (see SI for details). This procedure produces values for\nthe linewidth that are consistent with the results for \flms that can be cleaved into samples\nsu\u000eciently small to isolate a single resonance (see SI).\nFigure 4(a) shows the measured frequency dependence of the linewidth for each of our\n\flms. We observe a linear dependence on frequency up to \u00188 GHz. At higher frequencies,\nthe linewidths deviate from linearity, most obviously for the 2.8 and 5.3 nm \flms. This high-\nfrequency curvature is qualitatively consistent with the e\u000bect of two-magnon scattering, as\nobserved previously in PLD-grown YIG \flms.6Using the expression32\n\u0001H(f) =4\u0019\u000bf\nj\rj+ \u0001H0; (4)\nwe can de\fne an e\u000bective Gilbert damping parameter, \u000b, for each value of \flm thickness\nbased on linear \fts to the data below 8 GHz (Fig. 4(b)). The line shown in Fig. 4(b) is a \ft\nto a phenomenological form\n\u000b=\u000bG+\u000b2M\u0012A\nt2+B\nt\u0013\n; (5)\nwith\u000bG= 0:9(6)\u000210\u00004,A\u000b2M= 125(45)\u000210\u00004nm2, andB\u000b2M= 36(11)\u000210\u00004nm.\nOur damping values are among the best reported for any garnet \flm, and for the \frst\ntime extend the viable thickness of low-damping thin \flms well below 10 nm. We measure\n\u000b= 11:1(9)\u000210\u00004for 5.3 nm LuIG \flms and 32(3) \u000210\u00004for 2.8 nm \flms. We speculate\nthat our MBE growth procedure minimizes the amount of surface roughness and other\ndefects even for very thin LuIG \flms, compared to other deposition techniques, and thereby\nprovides a reduced level of two-magnon scattering. Similar MBE growth procedures may\nalso allow the production of sub-10-nm \flms made from YIG and other garnets, assisting in\nthe development of a wide variety of spintronic devices incorporating these materials.\n6ACKNOWLEDGMENTS\nWe acknowledge F. Guo for helpful discussion on two-magnon theory. This work was\nsupported by the National Science Foundation (DMR-1406333) with partial support from\nthe Cornell Center for Materials Research (CCMR), part of the NSF MRSEC program\n(DMR-1120296). We made use of the Cornell Nanoscale Facility, a member of the National\nNanotechnology Coordinated Infrastructure (NNCI), which is supported by the NSF (ECCS-\n1542081) and also the CCMR Shared Facilities. The work of H.P. and D.G.S. is supported\nby the Air Force O\u000ece of Scienti\fc Research under award number FA9550-16-1-0192.\nREFERENCES\n1M. Sparks, Ferromagnetic-Relaxation Theory (McGraw-Hill, New York, 1964).\n2I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Au\u000bret,\nS. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n3L. Liu, C.-F. C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science\n336, 555 (2012).\n4A. R. Mellnik, J. S. Lee, A. Richardella, J. L. 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(b,c) Kikuchi lines in the RHEED image taken along both [1 \u001610] and [11 \u00162]\nazimuthal directions.\n11(nm)\n0.81.0\n0.6\n0.4\n0.2\n0.0\nc da b\n2 μmFIG. 2. (a) X-ray di\u000braction (XRD) rocking curves for all of the LuIG thicknesses above 2.8 nm\nindicate commensurate growth and consistent strain. (b) Representative atomic force microscopy\nimage of the surface of the 2.8 nm \flm, showing a RMS roughness of 0.26 nm over 5 µm x 5 µm\nscan size, which indicates the roughness is substrate limited. (c) \u0012=2\u0012XRD scans of LuIG thin\n\flms grown on (111) GGG substrates as a function of \flm thickness. The asterisk marks the 444\nGGG substrate re\rection. (d) Normalized derivative-absorption FMR spectra of the correspond-\ning samples taken at 5 GHz show narrow linewidths that decrease for increasing thickness. The\nresonance position also depends on the thickness.\n12a bFIG. 3. (a,b) In-plane FMR resonance \felds of each LuIG sample (a) as a function of frequency for\ndi\u000berent sample thicknesses and (b) as a function of thickness for di\u000berent frequencies. The solid\nlines in (a) and (b) represent simultaneous \fts to Eq. (1) with the 3 \ftting parameters r, 4\u0019Ms,\nandKs.\n1311.1(9)31.5(3.0)\n6.2(4)\n3.4(8)1.8(4)this worka\nbFIG. 4. (a) Frequency dependence of the FMR linewidth, for LuIG \flms of di\u000berent thickness.\nThe linewidths are \ft to straight lines up to 8 GHz, after which the linewidths start to roll o\u000b,\nfollowing the signature of two-magnon scattering. (b) Thickness dependence our measured values\nof magnetic damping (black squares). The line depicts the phenomenological form of Eq. (5).\nPreviously-reported results for damping in thin YIG \flms are shown for \flms deposited by PLD\n(open blue symbols) PLD and o\u000b-axis sputtering (open red symbols). Open triangles represent\npost-processed \flms.\n14"    },    {        "title": "1703.07515v1.Fast_domain_wall_motion_induced_by_antiferromagnetic_spin_dynamics_at_the_angular_momentum_compensation_temperature_of_ferrimagnets.pdf",        "content": "Fast domain wall motion induced by antiferromagnetic spin dynamics at the angular momentum \ncompensation temperature of ferrimagnets  \n \nKab-Jin Kim1,2†★, Se Kwon Kim3†, Takayuki Tono1, Se-Hyeok Oh4,Takaya Okuno1, Woo Seung Ham1, \nYuushou Hirata1, Sanghoon Kim1, Gyoungchoon Go5, Yaroslav Tserkovnyak3, Arata Tsukamoto6, \nTakahiro Moriyama1, Kyung -Jin Lee4,5,7★, and Teruo Ono1★ \n \n \n \n1Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, 611 -0011, Japan  \n2 Department of Physics, Korea Advanced Institu te of Science and Technology, Daejeon 34141, Korea  \n3Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA  \n4Department of Nano -Semiconductor and Engineering, Korea University, Seoul 02841, Korea  \n5Department of Mat erials Science & Engineering, Korea University, Seoul 02841, South Korea  \n6College of Science and Technology, Nihon University, Funabashi, Chiba 274 -8501, Japan  \n7KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, \nSouth Korea  \n \n★ Correspondence to: kabjin@scl.kyoto -u.ac.jp, kj_lee@korea.ac.kr , ono@scl.kyoto -u.ac.jp  \n  Antiferromagnetic spintronics is an emerging research field which aims to utilize \nantiferromagnets as core elements in spintronic devices1,2. A central mo tivation toward this \ndirection is that antiferromagnetic spin dynamics is expected to be much faster than ferromagnetic \ncounterpart because antiferromagnets have higher resonance frequencies than ferromagnets3. \nRecent theories indeed predicted faster dynam ics of antiferromagnetic domain walls (DWs) than \nferromagnetic DWs4-6. However, experimental investigations of antiferromagnetic spin dynamics \nhave remained unexplored mainly because of the immunity of antiferromagnets to magnetic fields. \nFurthermore, this  immunity makes field -driven antiferromagnetic DW motion impossible despite \nrich physics of field -driven DW dynamics as proven in ferromagnetic DW studies. Here we show \nthat fast field -driven antiferromagnetic spin dynamics is realized in ferrimagnets at t he angular \nmomentum compensation point TA. Using rare -earth–3d-transition metal ferrimagnetic compounds \nwhere net magnetic moment is nonzero at TA, the field -driven DW mobility remarkably enhance s \nup to 20 km s−1T−1. The collective coordinate approach generalized for ferrimagnets7 and atomistic \nspin model simulations6,8 show that this remarkable enhancement is a consequence of \nantiferromagnetic spin dynamics at TA. Our finding allows us to inve stigate the physic s of \nantiferromagnetic spin dynamics and highlights the importance of tuning of the angular \nmomentum compensation point of ferrimagnets, which could be a key towards ferrimagnetic \nspintronics.  Encoding information using magnetic DW motion  is essential for future magnetic memory \ndevices , such as racetrack memor ies9,10 . High-speed DW motion is a key prerequisite for making the \nracetrack feasible. However, velocity breakdown due to the angular precession  of DW , referred to as the \nWalker breakdown11, generall y limits the functional performance in ferromagnet -based DW devices. \nRecently, it was reported  that the DW speed boosts up  significantly in antiferromagnets  due to the  \nsuppression of  the angular precession4-6. However, the immunity  of antiferromag nets to m agnetic fields \nyields  notorious difficulties  in creating, manipulating , and detecting  antiferromagnetic DW s, com pared to \nferromagnetic one s. One possibility  to avoid these difficulties  is offered by the synthetic \nantiferromagnets12, where the net magnetic moment can be controlled by tuning the thickness of two \nferromagnetic layers coupled antiferromagnetically. However, they still suffer from the field -immunity  \nwhen the net magnetic moment approaches zero , preventing the study of antiferromagnetic DW \ndynami cs. Here we  show that magnetic field -controlled antiferromagnetic spin dynamics can be achieved  \nby employing ferrimagnets.  \nThere  are a class of ferrimagnet s, rare earth ( RE)–transition metal ( TM) compound s, where the \nspins of two inequivalent  sublattices  are coupled antiferromagnetically. Because of different Land é g-\nfactors  between RE and TM elements , these ferrimagnet s have two special temperatures below the Curie \ntemperature : the magneti sation compensation temperature, TM, at which the two magnetic momen ts \ncancel each other , and the angular momentum compensa tion temperature , TA, at which the net angular \nmomentum vanishes13-15. In particular, the existence of TA in ferrimagnets provides a framework to \ninvestigate antiferromagnetic spin dynamics. It is beca use the time evolution of the state of a magnet is \ngoverned by the commutation relation of the angular momentum, not of the magnetic moment . As a \nresult,  the nature of the dynamics of the ferrimagnets is expected to change  from ferromagnetic to \nantiferroma gnetic as approaching the angular momentum compensation point  TA. Furthermore, the net \nmagnetic moment of ferrimagnets is nonzero at TA and can thus  couple to an external magnetic field , \nopen ing a new possibility of field -driven antiferromagnetic spin dyna mics.  In order to pursuit this possibility, w e first describe distinguishing  features of ferrimagnetic DWs \nnear TA based on the collective coordinate approach7. A ferrimagnetic DW effectively acts as an \nantiferromagnetic DW  around TA. The low -energy dynami cs of a  DW in quasi one -dimensional magnets \nis generally described by two collective coordinates, its position X and angle Φ, which capture  the \ntranslational  and spin -rotational degrees of freedom  of the DW , respectively.  In ferromagnets, they are \ngyrotropically coupled by the Berry phase  that is proportional to the net spin density , and the motion of a  \nDW slows down  severely above a certain critical stimulus, engendering the phenomenon of the Walker \nbreakdown11. In antiferromagnets, on the other hand,  the dynamics of  X and Φ are independent owing to \nvanishing the net spin density7. The DW dynamics in antiferromagnets  is thus free from  the Walker \nbreakdown  and can be fast in a broad range of external driving forces compar ed to that in ferromagnets.   \nIn the following, we explain  a theory for the field-driven DW dynamics  in ferrimagnets  in high \nfields , which agrees with  our experi ment al results  as discussed  later. The DW velocity  can be derived  as \nfollows  by invoking the energy conservation and the gyrotropic coupling between the two collective \ncoordinates  X and Φ (see Supplementary  Information for microscopic derivations) . For a sufficiently \nstrong external field , the anisotropy  energy can be neglected and the time derivatives of X and Φ can be \nconsidered to approach  constant values16, \nV X and \n . The energy -dissipation rate caused by \nthese dynamics is given by \n2 2 2\n22 112   V sαsα P Α , where \nΑ is the cross sectional area of \nthe magnet, \n is the domain -wall width, \n1α and \n1sare the Gilbert damping constant and the spin angular \nmomentum density of one sublattice, respectively, and \n2α and \n2s are for the other sublattice. Invoking the \nconservation of the total energy, the rate of energy dissipation can be equated to the decreasing rate of the \nZeeman energy induced by the translation motion of the domain wall, which yields the equation, \nVHM M V sαsα Α Α2 12 2 2\n22 11 2 2   \n, where \n1M  and \n2M are th e magnetization of the \ntwo sublattices, and \nH is an external field. Note that the net magnetization, \n2 1M M , does not vanish at \nTA due to the difference in the Land é g-factors  of two sublattice atoms , which is essential to drive a DW with an external field.  In addition, when there is finite net spin angular momentum \n2 1ss , e.g., away \nfrom TA, the angular and linear velocities are related by the gyrotropic coupling whose strength is \nproportional to the net  angular momentum \n2 1ss17. Balancing the  gyrotropic force on Φ with the  \ndissipative force yields  \nVss sαsα2 1 22 11  . Solving the two aforementioned equations for \nV\nand \n , we obtain  \n \n H\nss s s)M M(s sV2\n2 12\n22 112 1 22 11\n \n\n,      \n\n H\nss s s)M M(ss\n2\n2 12\n22 112 1 2 1\n\n   (1) \nAs the system approaches the angu lar momentum compensation point \n02 1ss , the domain wall \nspeed  \nV  increases, whereas the precessional frequency \n  decreases. At T = TA, X and Φ are completely \ndecoupled  and the pure translational  dynamics  of the DW  is obtained , implying that the ferrimagnetic DW \neffectively acts as antiferromagnetic DW  and its motion is driven by a magnetic field at T = TA.  \nIn order to prove the above theoretical prediction, we investigate DW dynamics  in ferrimagneti c \nGdFeCo compounds. Figure 1  shows a schematic illustration of our sample. SiN(5 nm)/Gd 23Fe67.4Co9.6 \n(30 nm)/SiN(5 nm) films are  deposited on intrinsic Si substrate by magnetron sputtering . GdFeCo is a \nwell-known RE– TM ferrimagnet ic compound , in which RE and TM moment s are coupled \nantiferromagnetically18. The relative mag netic moment s of RE and TM can be easily controlled by \nvarying the composition or temperature, so that TM and TA can be easily designed  in RE –TM \nferrimagnet s. The GdFeCo film is then patte rned into micro wires  with 5 μm wi dth and 65  μm length  \nusing electron beam lithography and Ar ion milling . A Hall bar is designed to detect the DW motion via \nthe anomalous Hall effect (AHE) voltage , VH.  \nWe first characterise the magnetic properties of the GdFeCo  microstrips . Figure 2 a shows the \nhysteresis loop s of GdFeCo microstrips  at various temperatures . The  AHE resistance , RH (RH = VH /I), is \nmeasured by sweeping the  out-of-plane magnetic field , BZ. Square hysteresis loops are clearly observed, \nindicating that GdFeCo has a perpend icular magnetic anisotropy. The coercivity field, BC, and the \nmagnitude of the Hall resistance  change , ΔRH \n  Z Z B R B R R H H H , are extracted from the hysteresis loop s and summarised in Fig. 2b. BC increases with increasing temperature , but a sudden drop  \nis observed at T = 220 K.  A sign change of RH is observed at the same temperature.  This is a typical \nbehaviour of ferrimagnet s at the magnetisation  compensation temperature TM19. As T approaches TM, the \nnet magnetic moment converges to zero , and thus , a large r magnetic field is required to obtain  a \nsufficiently high Zeeman energy to switch the magnetisati on. Thus, BC diverges at TM. The s ign change of \nRH represents additional  evidence of TM. The magneto -transport properties of GdFeCo are known to be \ndominated by FeCo mome nts because  the 4 f shell, which is respons ible for the magnetic properties of Gd, \nis located far below the Fermi energy level20. Thus, the sign change of RH indicates a change in  the \nrelative direction of the FeCo  moments  with respect t o the magnetic field, which occur s at TM. At T < TM, \nthe Gd moment dominates over the FeCo moment so that the Gd moment  aligns alo ng the magnetic field \ndirection . However,  at T > TM, the FeCo moment  is dominant and thus  aligns along the magnetic field \ndirection. Therefore, Fig. 2b allows us to identify TM for our GdFeCo sample , which  is approximately 220 \nK. \nAlthough TM can be easily determined by magnetisation  or magneto -transport  measurement s, it is \ngenerally not easy to determine  TA because TA is not related to the net magnetisation  but rather to the  \nangular momentum of the system. For GdFeCo, the net magneti sation  \nM and angular momentum \nA  are \nwritten as follows13,14,21. \nFeCo Gd M M M\n  and \nFeCo FeCo Gd Gd FeCo Gd   / M / M A AA \n  , where \n FeCoGdM\n and \n FeCoGdA are the magnetic moment and angular momentum  of the Gd (FeCo) sub -lattices , \nrespectively , and \nB g FeCoGd FeCoGd  is the gyromagnetic ratio of Gd (FeCo ), where \nB  is the Bohr \nmagneton and \n  is the reduced Plank constant. According to the literature s22–24, \nFeCog  (~2.2) is slightly \nlarger than  \nGdg  (~2) owing to the spin -orbit coupling of FeCo and zero orbital angular momentum of the \nhalf-filled 4 f shell of Gd ; therefore,  TA is expected to be higher than TM in GdFeCo.  \nBased  on above consideration , we measure the field-driven  DW speed  at T > TM using a real-time \nDW detection method25–27. We first saturate the magnetisation  by applying a large  negative field (| B| > \n|BC|) and then switch the field to the positive direction . Thi s positive field is a DW driving field , Bd, and should be  smaller than  BC (|Bd| < |BC|). Next , we inject a current pulse into the electrode to create a DW by \na current -induced Oersted field , as shown in Fig.  1. The created DW propagate s along the wire due to the  \npresence of  Bd, and the DW motion is detected at the Hall bar by monitoring the change s in VH. Here , the \nchange s in VH are recorded by an oscilloscope  such that nanosecond time -resolution can be achieved . The \nDW speed  can be calculated from the arri val time and the travel distance (60 μm) of the DW.  The details \nof the measureme nt scheme are explained in the Method  section .  \nFigure 3 a shows the DW speed  as a function of Bd at several temperatures  above TM. The DW \nvelocity increases linearly with field for all temperatures. Such a linear behaviour can be described by\n0 dB B v\n. Here , μ is referred to as the DW mobility and B0 is the correction  field, which  generally \narises from imperfection s of the sample  or complexit ies of the internal DW structure28,29. Figure 3b  shows  \nthe DW  velocity as a function of temperature for several bias fields. The DW velocity shows a sharp peak \nas expected near T = TA based on Eq. (1). The DW mobility μ is estimat ed from the linear fit in Fig. 3 a \n(dashed lines) and is plotted as a fu nction of  the temperature in Fig. 3 c. Starting from T = 260  K, which is \nslightly higher than TM, μ increases steeply , reach ing its maximum at T = 310  K, and then decreases with \na further  increase in  temperature. The peak mobility is as high as 20 km ·s−1·T−1 at T = 310  K. These \nexperimental results are in agreement  with the analytical expression Eq. (1), which  predicts a Lorentz ian \nshape of the DW velocity near T = TA with the width \n22 11 sαsα~ . Such a consistency between \nexperiment and theory manifests that the ferrimagnetic DW indeed acts as an antiferromagnetic DW  at T \n= TA.  \nWe next perform atomic spin model simulations  based on the atomistic Landau -Lifshitz -Gilbert \n(LLG) equation5,8 (see Method section for details)  to verify the experimental result and theor etical \nprediction . We employ a set of the reduced magnetic moments around s1−𝑠2=0 as shown in Table 1. \nThe total number of set is 9  in which the index  5 corresponds to  the temperature at TA. We assume that a \nDW is of Bloch -type with perpendicular mag netic anisotropy along the z axis. Fig ure 4a shows the DW \nvelocity as a function of the external field BZ, applied along the z axis. Velocities increase  linearly with BZ for BZ  > 10 mT . The numerical results (circular symbols) are in excellent agreement w ith the analytic \nsolution (solid lines) for the DW velocity for high fields in Eq. (1). The inset of Fig.  4a shows the DW \nvelocity in low field regime s (BZ  < 10 mT) . The Walker breakdown occurs in this regime  except for the \ncase of TA. The vertical dashed  lines represent the Walker breakdown field. Fig ure 4b shows the DW \nvelocity as a  function of  δs=s1−𝑠2. At δs=0, the DW velocity  is the highest , which agrees with the \nexperimental result  and theory . This good agreement also supports that field -driven antiferromagnetic \nspin dynamics is realized in ferrimagnets at T A. \n To date , the angular momentum compensation  point and its effect on the magnetisation  dynamics  \nhave  often been overlooked in studies of ferrimagnet s. Laser-induced magnetisation  switching18,30,31 and \nmagnetic DW motion32–33, which have bee n major research themes  of ferrima gnets, have mostly been \nstudied without identifying  TA. It has been investigated in the context of magnetic resonance  or \nmagnetisation  switching by current  around TA 14,15,34, but a clear identification of spin dynamics at T = TA \nhave  been remained elusive15. On the other hand, our results  clearly show that the antiferromagnetic spin \ndynamics is achievable at T = TA. Moreo ver, such antiferromagne tic spin dynamics  can be controlled by \nmagnetic field due to the finite ma gnetic moment at T = TA, which opens a way of studying field-driven \nantiferromagnetic spin dynamics. Furthermore, the fact that field-driven DW speed  exhibits a sharp and \nnarrow peak at TA provides  a sim ple but accurate  method to determine TA, which has not been possible.  \nWe also achieve a fast DW speed near room temperature , opening a possibility for ultra -high speed \ndevice operation at room temperature. We expect that s uch findings are also advantageous  for current -\ninduced DW motion in ferrimagnets. A low threshold current density , more than one order of magnitude \nsmaller than that of ferromagnets,  has already been demonstrated  in ferrimagnets32–33. Therefore, by \ntuning TA, one could obtain  high-speed an d low power consumed  spintronic  device s using ferrimagnet s, \nwhich could even be superior to ferromagnetic system s. To conclude, o ur work suggests that  revealing \nand tailoring TA, which has not been paid much attention , is crucial for controlling ferrimagne tic \nmagnetisation  dynamics, and therefore could be a key for realising ferrimagnet ic spintronics .  Method  \nFilm  preparation and device fabrication.  The studied samples are amorphous thin films of \nGd 23Fe67.4Co9.6 of 30 nm thickness which have been deposited  by magnetron sputtering. To avoid \noxidation of the GdFeCo layer, 5 nm Si 3N4 were used as buffer and capping layers, respectively. The \nfilms  exhibit an out -of-plane magnetic anisotropy . GdFeCo micro strips  with a 100 nm -wide Hall bar \nstructure were fabricat ed using electron beam lithography and Ar ion milling process. A negative tone \nelectron beam resist (maN -2403) was used for lithography at a fine resolution (~5 nm). For current \ninjection, Ti(5 nm)/Au(100 nm) electrodes were stacked on the wire. To make an  Ohmic contact, the \nSi3N4 capping layer was removed  by weak ion milling before electrode deposition.  \nExperimental setup  for field -driven domain wall motion . A pulse generator (Picosecond 10, 300B) \nwas used to generate a current pulse to  create the DW. 100m A and 10ns current pulse is used to create the \nDW. For field -driven DW motion, 1mA dc current (corresponding current density is 7  109 A·m−2) was \nflowed along the wire to generate anomalous Hall voltage, VH. Yokogawa 7651  was used as a current \nsource. The VH at the Hall cross was recorded by the oscilloscope  (Textronix 7354)  through the 46 dB \ndifferent ial amplifier. Low temperature probe sta tion was used for measuring the DW motion in a wide \nrange of temperature.  \nDW detection technique . We used a time -of-flight measurement of DW propagation to obtain a DW \nspeed  in a flow regime. The procedure for measuring the DW speed  is as follows . First, a large out -of-\nplane magnetic field Bsat = −200 mT is applied to reset the magnetisation . Next, a drive field Bd, in the \nrange of | BP| < |Bd| < |BC|, is applied in the opposite direction. Here, BP is the pinning field of DW motion \nand BC is the coercive field of  the sample . Since the  Bd is smaller than the BC, the drive field does not \nreverse the magnetisation  or create DWs. N ext, a current pulse (100 mA, 1 0 ns) is injected by a pulse \ngenerator  to create  a DW next to the contact line  through current -induced Oersted field . As soon as the \nDW is create d, the  Bd pushes the DW because the Bd is larger than the BP. Then  the DW propagates along \nthe wire and passes through the Hall cross  region . When the DW passes through the Hall cross, the Hall \nvoltage changes abruptly because the magnetisation  state of th e Hall cross reverses as a result of the DW passage. This Hall signal change is recorded by the oscilloscope through the 46dB differential amplifier. \nWe refer to this as a ‘signal trace’. Since the detected Hall voltage change includes a large background \nsignal, we subtract the background from the ‘signal trace’ by measuring a ‘reference trace’. The reference \ntrace is obtained in the same manner as the signal trace, except that the saturation field direction is \nreversed ( Bsat = +200 mT). In this reference t race, no DW is nucleated , so that only the electronic noise \ncan be detected in the oscilloscope  in the reference trace . To obtain a sufficiently high signal -to-noise \nratio, we averaged the data from 5 repeated measurements .  \nAtomic spin model simulation . We adopt the atomistic model simulation s ince the ferrimagnet consists \nof two magnetic components, i.e., RE and TM on an  atomic scale . The one -dimensional Hamiltonian of \nferrimagnet is described by ℋ=𝐴𝑠𝑖𝑚∑𝑺𝒊∙𝑺𝒊+𝟏 𝑖 −𝐾𝑠𝑖𝑚∑(𝑺𝒊∙𝒛̂)2𝑖 +𝜅𝑠𝑖𝑚∑(𝑺𝒊∙𝒚̂)2𝑖 , where 𝑺𝒊 is the \nnormalized magnetic moment at lattice site  𝑖. The odd number of 𝑖 represents a site for TM, and the even \nnumber of 𝑖 represents a site for RE. 𝐴𝑠𝑖𝑚,𝐾𝑠𝑖𝑚,𝜅𝑠𝑖𝑚 denote the exchange, easy -axis anisotropy along the \nz axis, and hard -axis anisotropy, respectively. We sol ve the atomistic LLG equation  𝜕𝑺𝒊\n𝜕𝑡=−𝛾𝑖𝑺𝒊×\n𝑯𝒆𝒇𝒇 ,𝒊+𝛼𝑖𝑺𝒊×𝜕𝑺𝒊\n𝜕𝑡, where   𝑯𝒆𝒇𝒇 ,𝒊=−1\n𝑀𝑖𝜕ℋ\n𝜕𝑺𝒊  is the effective field, γi=gi 𝜇𝐵ℏ⁄ is the gyromagnetic \nratio, and 𝑀𝑖 is the magnetic moment for site 𝑖. We use parameters as 𝐴𝑠𝑖𝑚=7.5𝑚𝑒𝑉 ,𝐾𝑠𝑖𝑚=\n0.3𝑚𝑒𝑉,𝜅𝑠𝑖𝑚=−0.8𝜇𝑒𝑉, damping constant  𝛼𝑇𝑀=𝛼𝑅𝐸=0.004, the lattice constant is 0.4 nm, and \nLand é g-factors for each site are gTM=2.2 and gRE=2. References  \n1. MacDonald, A. H. & Tsoi, M. Antiferromagnetic metal spintronics. Phil. Tra ns. R. Soc. A  369, 3098 –\n3114 (2011).  \n2. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics . Nat. Nanotech . \n11, 231-241 (2016 ). \n3. Keffer, F. & Kittel, C. Theory of antiferromagnetic resonance. Phys. Rev . 85, 329 –337 (1952).  \n4. Gomon ay, O. , Jungwirth, T.  & Sinova , J. High Antiferromagnetic Domain Wall Velocity Induced by \nNéel Spin -Orbit Torques . Phys. Rev. Lett.  117, 017202  (2016).  \n5. Shiino, T. et al. Antiferromagnetic Domain Wall Motion Driven by Spin -Orbit Torques. Phys. Rev. \nLett. 117, 087203 (2016).  \n6. Tveten, E.G.,  Qaiumzadeh, A., &  Brataas, A.  Antiferromagnetic Domain Wall Motion Induced by \nSpin Waves . Phys. Rev. Lett . 112, 147204 (2014 ). \n7. Tveten, E.G.,  Qaiumzadeh, A., Tretiakov,  O. A. & Brataas, A.  Staggered Dynamics in \nAntiferromagnets  by Collective Coordinates. Phys. Rev. Lett . 110, 127208 (2013) . \n8. Evans , R.F.L. et al. Atomistic spin model simulations of magnetic nanomaterials. J. Phys.: \nCondensed Matter , 26, 103202 (2014) . \n9. Yamaguchi, A. et al. Real -space observation of current -driven domain wall motion in submicron \nmagnetic wires. Phys. Rev. Lett . 92, 077205 (2004).  \n10. Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic domain -wall racetrack memory. Science  320, \n190–194 (2008).  \n11. Schryer, N. L. &Wa lker, L. R. The motion of 180 ° domain walls in uniform dc magnetic fields. J. \nAppl. Phys . 45, 5406 -5421 (1974).  \n12. Yang, S. -H., Ryu, K. -S. & Parkin, S. Domain -wall velocities of up to 750  m s−1 driven by exchange -\ncoupling torque in synthetic antiferromagnets . Nature Nanotech . 10, 221 –226 (2015) . \n13. Wangness, R. K. Sublattice effects in magnetic resonance. Phys. Rev . 91, 1085 -1091 (1953).  14. Stanciu, C. D. et al. Ultrafast spin dynamics across compensation points in ferrimagnetic GdFeCo: \nThe role of angular momentum compensation. Phys. Rev. B  73, 220402(R) (2006).  \n15. Binder, M. et al. Magnetization dynamics of the ferrimagnet CoGd near the compensation of \nmagnetizat ion and angular momentum. Phys. Rev. B  74, 134404 (2006).  \n16. Clarke, D. J. , Tretiakov, O. A. , Chern, G. -W., Bazaliy, Ya. B. & Tchernyshyov , O. Dynamics of a \nvortex domain wall in a magnetic nanostrip: Application of the collective -coordinate approach. Phys. \nRev. B  78, 134412 (2008) . \n17. Thiele, A.A. Steady -State Motion of Magnetic Domains . Phys. Rev. Lett . 30, 230  (1973).  \n18. Radu, I.  et al. Transient ferromagnetic -like state mediating ultrafast reversal of antiferromagnetically \ncoupled spins, Nature , 472, 205 -208 (20 11). \n19. Okuno, T. et al. Temperature dependence of magnetoresistance in GdFeCo/Pt heterostructure, Appl. \nPhys. Express  9, 073001 (2016).  \n20. Tanaka, H., Takayama, S. & Fujiwara, T. Electronic -structure calculations for amorphous and \ncrystalline Gd 33Fe67alloys. Phys. Rev. B  46, 7390 -7394 (1992).  \n21. Tsuya, N. Microwave resonance in ferrimagnetic substance. Prog. Theoret. Phys . 7, 263 -265 (1952).  \n22. Kittel, C. On the Gyromagnetic Ratio and Spectroscopic Splitting Factor of Ferromagnetic \nSubstances. Phys. Rev . 76, 743 (1949 ). \n23. Scott, G. G. Review of Gyromagnetic Ratio Experiments. Rev. Mod. Phys . 34, 102 (1962).  \n24. Min, B. I. and Jang, Y. -R. The effect of the spin -orbit interaction on the electronic structure of \nmagnetic materials. J. Phys. Condens. Matter  3, 5131 (1991).  \n25. Yoshim ura, Y. et al. Soliton -like magnetic domain wall motion induced by the interfacial \nDzyaloshinskii –Moriya interaction. Nat. Phys . 12, 157 -161 (2016).  \n26. Tono, T.  et al. Chiral magnetic domain wall in ferrimagnetic GdFeCo wires. Appl. Phys. Express  8, \n073001 (2 015).  \n27. Kim, K. -J. et al. Observation of asymmetry in domain wall speed under transverse magnetic field. \nAPL Mater.  4, 032504  (2016).  28. Ono, T. et al. Propagation of a domain wall in a submicrometer magnetic wire. Science  284, 468 –470 \n(1999).  \n29. Volkov, V. V. & B okov, V. A.  Domain Wall Dynamics in Ferromagnets.  Physics of the Solid State  \n50, 199 -228 (2008).  \n30. Alebrand, S. et al. Light -induced magnetization reversal of high -anisotropy TbCo alloy films. Appl. \nPhys. Lett.  101, 162408 (2012).  \n31. Stanciu, C. D. et al. Subpi cosecond Magnetization Reversal across Ferrimagnetic Compensation \nPoints . Phys. Rev. Lett.  99, 217204  (2007). \n32. Ngo, D -T., Ikeda, K. & Awano, H. Direct Observation of Domain Wall Motion Induced by Low -\nCurrent Density in TbFeCo Wires. Appl. Phys. Express  4, 093002 (2011).  \n33. Awano, H., Investigation of domain wall motion in RE -TM magnetic wire towards a current driven \nmemory and logic. J. Magn. Magn. Mater . 383, 50-55 (2015).  \n34. Jiang, X., Gao., L. Sun J. Z. & Parkin S. S. P. Temperature Dependence of Current -Induce d \nMagnetization Switching in Spin Valves with a Ferrimagnetic CoGd Free Layer. Phys. Rev. Lett.  97, \n217202 (2006).  Acknowledgements  \nThis work was partly supported by JSPS KAKENHI Grant Numbers 15H05702, 26870300, 26870304, \n26103002, 25220604 , 2604316  Collaborative Research Program of the Institute for Chemical Research, \nKyoto University, and R & D project for ICT Key Technology of MEXT from the Japan Society for the \nPromotion of Science (JSPS). KJK acknowledges support from the KAIST start -up funding.  SKK and YT \nacknowledge the support from the Army Research Office under Contract No. 911NF -14-1-0016 . K.-J.L. \nacknowledges support from Creative Materials  Discovery Program through the National Research \nFoundation of Korea  (NRF -2015M3D1A1070465) . \n \nAuthor contri butions  \nK.-J.K., T.M. , and T.O. planed  the study. A.T. grew and optimi sed the GdFeCo film. T.T. fabricated the \ndevice  and performed  the experiment  with the guide of K. -J.K.. T.Okuno , W.-S.H., Y.H. , and  S.K. helped \nthe experiment . S.-K.K., K.-J.L., and Y.T.  provide theory.  S.-H.O., G.G., and K. -J.L. performed the \nnumerical  simulation. K.-J.K., S.-K.K., K. -J.L., T.M., and T.O analysed the result s. K.-J.K., S.-K.K., K. -\nJ.L., T.M.,  and T.O. wrote the manuscript.  \n \nAdditional  Information  \nSupplementary Information  is available in the online version of the paper. Reprints and permissions \ninformation is available at www.nature.com/reprints . Correspondence and requests for materials should \nbe addressed to K.-J.K, K.-J.L. and T. O.  \nCompeting financial interests  \nThe authors declare no competing financial interests.  \n  Figure Legends  \nFigure 1| Schematic illustration of Device structure . Schematic illustration of GdFeCo \nmicrowire . The inset  shows  schematic illustration of two spin sub-lattices  below and above the \nmagnetisation  compensation temperature, TM. Blue and red arrows indicate Gd and FeCo moments, \nrespectively.  \nFigure 2| Identification of magnetisation  compensation temperature TM. a, Anomalous Hall \neffect resistance RH as a function of perpendicular magnetic field BZ for several temperatures as denoted \nin the figure. b,. Coercive field, BC, and the magnitude of the Hall resistance change , ΔRH \n  Z Z B R B R R H H H\nwith respective to the temperature. The region shaded in red indicates the \nmagnetisation  compensation temperature TM \n \nFigure 3| Field -driven domain wall  (DW)  dynamics across the angular momentum \ncompensation temperature TA. a. DW speed  v as a function of driving field Bd for several \ntemperatures as denoted in the figure. Dashed lines are best fits based on \n0 dB B v . b. DW speed  v \nas a function of temperature T for several driving fields as denoted in the figure.  c. DW mobility μ as a \nfunction of temperature T. The red and blue shaded regions in b and c indicate the magnetisation  \ncompensation temperature, TM, and angular momentum compensation temperature, TA, respectively.  \n \nFigure 4| Simulation results of ferri magnetic domain wal l (DW) a. DW speed as a function of \nthe out -of-plane field BZ for various indices (see Table 1). Symbols are numerical results whereas solid \nlines are Eq. (1). Inset shows low field regimes, where vertical dotted lines indicate the Walker \nbreakdown fields.  b. Computed DW speed as a function of of δs=s1−𝑠2 at various values of BZ.  \n \n   \n \n \nFig.1  \n  \n \n \n \n \n \nFig.2  \n  \n \n \n \n \nFig.3  \n  \n \n \n \n \n \n \n \nFig.4  \n \n \n \n  \n \nTABLE 1. Parameters used in the numerical simulation.  \nIndex  1 2 3 4 5 6 7 8 9 \n𝑴𝑭𝒆𝑪𝒐 (𝒌𝑨/𝒎) 1120  1115  1110  1105  1100  1095  1090  1085  1080  \n𝑴𝑮𝒅(𝒌𝑨/𝒎) 1040  1030  1020  1010  1000  990 980 970 960 \n𝜹𝒔(𝟏𝟎−𝟕𝑱∙𝒔 𝒎𝟑⁄ ) -1.24 -0.93 -0.62 -0.31 0 0.31 0.62 0.93 1.24 \n \n "    },    {        "title": "2008.03062v2.Cavity_magnon_polariton_based_precision_magnetometry.pdf",        "content": "Cavity magnon polariton based precision magnetometry\nN. Crescini,1, 2, a)C. Braggio,2, 3G. Carugno,2, 3A. Ortolan,1and G. Ruoso1\n1)INFN-LNL, Viale dell’Università 2, 35020 Legnaro (PD), Italy\n2)Dipartimento di Fisica e Astronomia, Via Marzolo 8, 35131 Padova, Italy\n3)INFN-Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy\n(Dated: 2 October 2020)\nA photon-magnon hybrid system can be realized by coupling the electron spin resonance of a magnetic material to a\nmicrowave cavity mode. The quasiparticles associated with the system dynamics are the cavity magnon polaritons,\nwhich arise from the mixing of strongly coupled magnons and photons. We illustrate how these particles can be used\nto probe the magnetization of a sample with a remarkable sensitivity, devising suitable spin-magnetometers which\nultimately can be used to directly assess oscillating magnetic fields. Specifically, the capability of cavity magnon\npolaritons of converting magnetic excitations to electromagnetic ones, allows for translating to magnetism the quantum-\nlimited sensitivity reached by state-of-the-art microwave detectors. Here we employ hybrid systems composed of\nmicrowave cavities and ferrimagnetic spheres, to experimentally implement two types of novel spin-magnetometers.\nAmong the most studied types of hybrid systems, an im-\nportant role is played by photon-magnon hybrid systems\n(PMHSs)1,2. These yielded remarkable results in the study\nof light-matter interaction3, and in the last decades emerged\nas promising constituents for new quantum technologies as\nwell4–6. PMHSs have different forms, as they are built with\nmiscellaneous building blocks, but the underlying physics is\nsimilar. In a magnetic field B0, a spin can change its quantum\nstate from\u00001=2 to a +1=2 by absorbing a spin-1 boson, like a\nphoton, and vice versa by emitting one. In this sense, a quanta\nof spin excitation with energy ¯hwm=mBB0can be effectively\ndescribed as a quasiparticle, known as magnon, which can\nturn into a photon of the same energy ¯hwc7. This recipro-\ncal conversion is quantified by the interaction strength gcm,\nknown as vacuum Rabi splitting, which is the rate at which\nmagnons are converted into photons and vice versa. When gcm\nis much larger than the damping rates of the magnon gmand\nof the photon gc, the system is in the strong coupling regime,\nand the quasiparticles arising from this mixing are known as\ncavity magnon polaritons (CMPs)8,9.\nPMHSs are widely investigated for advancing quantum\ninformation science. In this field their importance lies in\nbuilding quantum memories10–16, in converting microwaves\nto optical photons17–21, or in quantum sensing, where the de-\ntection of single magnons was recently demonstrated22–24.\nCMP recently found new applications in the field of non-\nHermitian physics25–27, where they already yielded outstand-\ning results28. Exceptional points, spots of the system’s pa-\nrameter space highly sensitive to external stimulations, can\nbe probed with PMHSs29,30, and new configurations may be\ndesigned to access more exotic phenomena and study their\napplications31,32. The potential of hybrid systems was also\nshown in many other applications of quantum physics33–37.\nA distinguished physical realisation of this model can\nbe obtained by hybridising the microwave photons of\na resonant cavity with the magnons of a ferrimagnetic\ninsulator38–42. Such scheme was implemented with multi-\nple purposes, for example to develop new quantum tech-\na)Electronic mail: nicolo.crescini@phd.unipd.itnologies with qubits10,43, or for microwave-to-optical photon\nconversion20,21, making it an established platform of hybrid\nmagnonics.\nIn the devices described in this letter, we employ cop-\nper cavities as a photonic resonator and Yttrium Iron Garnet\n(YIG) spheres as magnetic material (see Fig. 1a). YIG has the\nexceptionally high electron spin density of 2 \u00021028spin=m3\nalready at room temperature, and a linewidth as narrow as\n1 MHz. This latter value is matched to the one of a typical\ncopper cavity and, thanks to the chosen spherical shape, is not\naffected by geometric demagnetization. Being employed in a\nnumber of microwave and rf devices, YIG is among the most\nwell-known ferrites, and hence is readily available. The mag-\nnetic sample is placed inside the cavity, where the rf magnetic\nfield is maximum for the selected cavity mode, and is mag-\nnetised with a static field B0perpendicular to the cavity one.\nIn this way, the Kittel mode of magnetisation couples to the\nmicrowave cavity photons, and the system exhibits the typi-\ncal anticrossing dispersion relation, of which an example is\nshown in Fig. 1b. The coupling strength depends on the work-\ning frequency, on the microwave mode volume, and on the\nnumber of spins involved40, but it is normally large enough to\nlet the photon (magnon) oscillate into magnon (photon) many\ntimes before being dissipated.\nThis feature of CMP to be a mixed state of microwaves and\nspin excitations allows one to extract information on magnons\nby monitoring photons. In the presence of a strong coupling,\nthe signal transduction is efficient, i. e. without signal loss, as\na spin excitation is more likely converted to a photon and de-\ntected, than it is to be dissipated due to the PMHS losses (see\nFig. 1c for a schematic diagram). Amongst other techniques to\nmeasure spin-waves4, the use of CMP is a particularly simple\napproach which exploits the sensitivity of microwave technol-\nogy and transfers it to the detection of magnons. The strong\ncoupling makes the energy stored in a cavity dependent on\nthe one in the material, so an antenna coupled to the elec-\ntromagnetic field of the cavity gives a simple access to the\nfeatures of the spin system44. Nowadays electronics is ex-\ntremely developed, and the detection of electromagnetic radi-\nation has been brought to the standard quantum limit of linear\namplifiers. At microwave frequencies, Josephson Parametric\nAmplifiers (JPA) were demonstrated to be the best devices toarXiv:2008.03062v2  [quant-ph]  30 Sep 20202\nCavity\nMagnetic sample\nMagnetic field\nMagnet(a) (b)\nCavity modeKittel mode\n(c)\nExternal \ninputReadoutPhotons\nMagnonsStrong coupling\nFIG. 1. Schematic representation of a typical PMHS (a), anticrossing\ncurve (b), and diagram of a spin-magnetometer working principle (c).\nPart (a) represents a PMHS consisting in a YIG sphere housed in a\nmicrowave cavity under a static magnetic field. Plot (b) is measured\nwith a .5 mm-diameter YIG sphere in a 14 GHz copper cavity, the\ncolour scale is in logarithmic arbitrary units, where blue to yellow is\nlow to high transmission, and the dashed lines show the uncoupled\ncavity and Kittel modes.\nmeasure tiniest amounts of power45. Thanks to CMPs, such\nprecision can be shifted to a magnetic measurement, as the\nelectromagnetic power in the cavity is highly dependent on\nthe magnetisation of the sample when the coupling strength\nlargely exceeds the system dissipations gcm\u001dgm;gc. It fol-\nlows that, under these conditions, the quantum-limited read-\nout of a JPA can be exploited to detect spin excitations.\nAt microwave frequencies, measuring a sample’s magne-\ntization becomes increasingly difficult because of technolog-\nical limitations and fundamental problems, like for example\nradiation damping46–48. In free space, radiation damping con-\nsists in the magnetic dipole emission of a magnetised sample\nwhich, at GHz frequencies, drastically decreases the coher-\nence time, limiting the experimental sensitivity. This effect is\navoided in PMHSs, as the sample is housed in a resonant cav-\nity which removes the damping by inhibiting the phase space\nof the emission49.\nFor all their characteristics, PMHSs emerge as an outstand-\ning platform for precision magnetic measurement, which are\nof interest for a broad range of applications as well as for ap-\nproaching fundamental physics issues. Hereafter, we describe\ntwo types of spin-magnetometers which can be designed with\nhybrid systems, detail their design and report on their opera-\ntion. We notice that a high occupation number of the modes\npermits to treat them as classical oscillators, which is often the\ncase throughout this work, so we rely on a classical treatment\nof the fields. These devices are originally meant to measure\ntiniest oscillation of a sample’s magnetization, related for ex-\nample to a Dark Matter Axion field49,50, but can be used to\nassess many other physical phenomena.\nTransverse spin-magnetometer (TSM). - Let’s now focus\non a hybrid system like the one of in Fig. 1, where a magne-\ntised YIG sphere is placed in a microwave cavity. If an oscil-\nlating electromagnetic, or pseudo-electromagnetic, field b1isoriented perpendicularly to the static field, its quanta can be\nabsorbed by the hybrid magnetic mode. As the magnetization\nvector Mprecesses over the static field, an excitation lying on\nthe precession plane can resonantly interact with it, and the\nsystem evolves according to Bloch equations\ndM\ndt=g(M\u0002b1)?+M\nTs; (1)\nwhere g= (2p)28GHz =T is the electron gyromagnetic ratio,\nandTsis the system relaxation time. The driven magnetization\nresulting from Eq. (1) is\nM(t) =gmBnsTscos(w1t); (2)\nwhere w1is the frequency of b1. In a steady state, the power of\nb1is absorbed, re-emitted by the magnetization, and rapidly\nconverted into photons thanks to the strong coupling.\nThe optimal experimental condition is an antenna critically\ncoupled to the cavity, which in steady state can extract up to\nhalf of the power deposited by the external field, resulting in\nP1=gmBNsw1b2\n1Ts; (3)\nwhere Nsis the number of spins of the hybrid system, and\nthe field frequency w1is on resonance with one of the hybrid\nmodes. To calculate the magnetic sensitivity of the TSM, in\nEq. (3) we substitute the deposited power P1(in Watts) with\nthe power sensitivity of the readout electronics sP(in Watts\nper unit of bandwidth), and recast the equation to isolate the\nmagnetic field. We obtain the sensitivity of the TSM\nsb1=rsP\ngmBNsw1Ts; (4)\nin Tesla per unit of bandwidth, which is the field detectable\nin 1 second integration time with a unitary signal-to-noise ra-\ntio. Eq. (4) also shows that the spin-magnetometer sensitiv-\nity increases for larger spin-number and longer hybrid system\ncoherence times. This suggests the use of high quality-factor\ncavities and samples to get a long Ts, and of a large volume\nof high spin density magnetic material to increase Ns. In this\nsense, we found a good compromise in YIG. The scalability\nof the PMHS is of fundamental importance to obtain an in-\ncreased sensitivity of the setup, as it is directly related to the\nincrement of Ns. To this aim we design spin-magnetometers\nbased on multi-samples PMHS51,52, embedded in cylindrical\ncavities. To further boost the magnetic sensitivity, we reduce\nsPby operating the device at milli-Kelvin temperatures, to re-\nduce thermal noises and to consent the use of quantum-limited\namplifiers.\nFollowing these directions, we built a TSM whose scheme\nis reported in Fig. 2a. Its PHMS comprises ten YIG spheres,\nall of 2.1 mm-diameter, produced in-house. These are biased\nwith a magnetic field supplied by a superconducting mag-\nnet, with 7 ppm uniformity over the volume containing the\nspheres. We realise the PMHS by placing the spheres along\nthe axis of a cylindrical cavity (33 mm-diameter, 65 mm-\nlength) allowing them to couple with the uniform rf magnetic\nfield of the TM110 mode at 10.7 GHz.3\nThe PMHS has been designed to reduce the effects of the\nmagnetic dipole interaction between different spheres and of\nhigher order magnetostatic modes. By removing the degener-\nacy of the TM110 mode we limit the interference of other cav-\nity modes; this is achieved employing a cavity with a quasi-\ncircular section53,54. To describe this system we used a sec-\nond quantisation model consisting in four coupled harmonic\noscillators. We fit it to the experimental anticrossing curve of\nFig. 2b53. We then operate the magnetometer in the frequency\nband 10.2-10.4 GHz, part of the lower frequency hybrid mode\nrange, as identified by the fit (dashed line in the figure)52. The\noperational range is matched with the working band of our\nJosephson Parametric Converter (JPC), i.e. a JPA formed by\na Josephson ring modulator shunted with four inductances55.\nThe JPC tuning is allowed by a small superconducting coil\nbiased with a constant current, as shown Fig. 2c. The dashed\nlines in Fig. 2c includes the 10.2-10.4 GHz frequency interval,\nshowing that in this range the lower frequency hybrid mode\ncan be monitored with our amplifier. The JPC is screened\nfrom external disturbances with different layers of supercon-\nducting and m-metal shields, and we verified that the solenoid\nproviding the static field is not affecting the resonance fre-\nquencies of the amplifier.\nThe noise temperature and gain of the electronics chain has\nbeen characterised with the injection of microwave signals of\nknown amplitude in an antenna weakly coupled to the cavity.\nThe effective noise temperature results Tn'1K, which sets\nthe noise power per unit of bandwidth sP=kBTn, where kBis\nBoltzmann constant. The contribution of the quantum limit to\nthe noise budget is 0.5 K, and the remaining 0.5 K is consis-\ntent with extra noise added by the second-stage amplifier, by\nthe losses of the wires and by the PMHS thermodynamic tem-\nperature of\u0018100mK52. The spin number and relaxation time\nare obtained by fitting our model to the transmission measure-\nments of Fig. 2b. The measurement of sP, and of NsandTs\nthrough the PMHS spectroscopy, allows us to calculate the\nsensitivity of the TSM using Eq. (4). With the parameters of\nthis setup we obtain a magnetic sensitivity of\nsb1=0:9\u000210\u000018h\u00101K\nTn\u0011\u0010Ns\n1021\u0011\n\u0010w1=2p\n10:4GHz\u0011\u0010Ts\n168ns\u0011i1=2Tp\nHz:(5)\nThat the sensitivity given by Eq. (4) holds if the field to be\ndetected has two characteristics: a coherence time longer than\nTs, and a coherence length long enough to comprise all the Ns\nspins.\nIn particular, this is the case of the field induced by Dark\nMatter axions49,50, which at GHz frequencies satisfies both\nthese conditions. We used this TSM with a fixed band-\nwidth of 5 kHz to search for axions, obtaining a limit on\ntheir effective field of 5 :5\u000210\u000019T with about ten hours of\nintegration52. A TSM has the advantage of being sensitive to\na (pseudo)magnetic field acting on a sample which is within\nthe volume of a resonant cavity. In such a controlled envi-\nronment, external electromagnetic disturbances are unlikely\nto be present, making it an interesting testbed for fundamen-\ntal physics, which are usually not subjected to such screening.\nYIG sphere\nCopper cavity\nSuperconducting magnet\nB-field profile of the \nTM110 modeIsolators JPAReadout\n(b)\n(c)(a)\nHybrid mode\nWorking frequencies[dB]\n[deg]Dipole antenna\nB0 directionFIG. 2. (a) Simplified scheme of the TSM operated for an axion\nsearch52(see text for details). (b) Anticrossing curve of the 10 YIG\nspheres PMHS, where the dashed line indicates the low-frequency\nhybrid mode monitored during the measurement. (c) Phase-current\ndiagram of the JPC mounted in this setup, here the dashed line shows\nthe optimal working points of the amplifier. From plots (b) and (c)\none notes that the 10.2-10.4 GHz band enables both the PMHS signal\ntransduction and the JPA amplification.\nHowever, from the point of view of the TSM possible tech-\nnological employment, this feature is a limitation. In fact, the\nscreening due to the cavity makes it difficult to expose the ma-\nterial to a field which is uniform and coherent over the mag-\nnetic material volume. Hence, the application of this device is\nprobably limited to the search of new physics.\nLongitudinal spin-magnetometer (LSM). - In another pos-\nsible measurement scheme a persistent oscillating B-field is\nparallel to the static one. In this configuration, the sample’s\nmagnetization precesses about a field B0+b2sin(w2t), where\nw2andb2are the oscillating field frequency and amplitude,\nandtis time. To illustrate the experimental arrangement, we\nfirst consider a simplified scheme including only the mate-4\nrial and ignoring the presence of the cavity. The experimen-\ntal scheme is shown in Fig. 3a, where a sphere is surrounded\nby two crossed loops. Loop number 1 is used to excite the\nmaterial, while loop number 2 senses the transmitted rf sig-\nnal, and S21plots are measured. The electron spin resonance\n(ESR) frequency wmof the magnetised sample is modulated\nat the frequency w2\u001cwmby varying the field b2\u001cB0. If\na monochromatic tone is applied on resonance with wm, the\neffect of b2is then to transfer some of the pump power, the\ncarrier, to sidebands at frequencies wm\u0006nw2, as schemati-\ncally shown in Fig. 3a for n=1. In the S21spectrum of this\nsimplified system, the amplitude of the first order sideband\nresults\nz1=pA2\npQb2\n2B0; (6)\nwhere Apis the carrier amplitude and Q=wm=gmthe qual-\nity factor of the ESR. In a standard ESR technique an exter-\nnally applied b2is used to detect the derivative of the ESR\ncurve with a lock-in amplifier. Here we invert such scheme,\nand search for oscillating b2-fields by sensing the presence of\nsidebands. The detection of sidebands is limited by the ef-\nfective noise temperature of the system determining sP, the\npower sensitivity already defined in the case of the TSM. The\namplitude z1is given by Eq. (6) only within the linewidth of\nthe ESR, and drastically reduces for w2>gm. On the other\nhand, when w2<gm, extra noise induced by the pump resid-\nual amplitude modulation will increase sP. Moreover, at GHz\nfrequencies, radiation damping broadens the linewidth of the\nESR, reducing Q. As a consequence, in this configuration the\nsensitivity for measuring a b2field is poor and needs some im-\nprovements that can be engineered using PMHSs as follows.\nBy including a cavity one may consider a PMHS’s hybrid\nmode instead of a bare ESR. CMPs are immune from radiation\ndamping thus we can couple the ESR to a microwave cavity\nto improve the detection sensitivity. We call wpone of the\nPMHS resonant frequencies: wpis also modulated by the os-\ncillating field as ¶wp=¶B0'r\u0002g, where 0\u0014r\u00141 is a field-\ndependent coefficient. When wmis equal to the cavity mode\nresonant frequency, r=1=2. The rf electromagnetic field of\nthe PMHS, pumped with a tone on-resonance with one of the\nhybrid modes, i. e. at wp, is phase-modulated through the vari-\nation of the resonant frequency, and therefore produces side-\nbands too. Their amplitude drastically decreases when they\ndepart from the resonance frequency by several linewidths,\nbut in a PMHS, at the frequency of the second hybrid mode,\none sideband does not vanish and hence can be detected (see\nFig. 3b). This device is thus sensitive to fields which are at\nfrequencies w2'2gcm, the splitting of the two hybrid modes.\nThe possibility of detecting the sideband at a frequency much\ndifferent from the pumping one allows us to drastically reduce\nthe noise by heavily filtering the pump noise. A waveguide is\na high pass filter which can cut low frequencies by tens of dB,\nand that we employ to remove the background related to the\npump. If the sideband frequency wp\u0000w2is below the waveg-\nuide cut-off, its amplitude is not filtered but the pump noise\nis (see the electronic scheme in Fig. 3b). By assuming that\nthe carrier noise can be made lower than thermal fluctuations,\n2B0\nb2 sin(ω2t)\nFrequencyS21\n1pumpω\nFrequencyS21\nESR\nFrequencyS21\nPMHSω\nωB0\nWaveguide cutoff(a)\n(b)\n12High-pass filterPMHS transmissionRF magnetic fieldMicrowave cavityMagnet\nm\nm\nmSideband\nreadout\nPump FilterCirculator AmplifierLSM electronics\ncut passb2=0\nb2≠0\nωm+ω2ESR\nωm-ω2\nωp+ω2 ωp-ω2b2 sin(ω2t)\nωpωm\nMicrowave powerYIG sphereFIG. 3. Schematic explanation of the LSM working principles. (a)\nUsual design used for the detection of an ESR in a magnetic material\nconsisting in a spherical sample (grey) surrounded by two loops in\nfree space. The rf is fed into the system by loop antenna 1 and the\noutput is read with the perpendicular loop 2. A pump, shown as a\nblue line in the spectra, is applied on-resonance with the ESR curve\n(green areas in the S21spectra). In absence of other fields the result is\na single peak (left plot), while with a superimposed oscillating field\nthe phase of the carrier is modulated by the shifting of the ESR in-\nduced by b2sin(w2t). Two sidebands, reported in dark red in the right\nplot, appear at wm\u0006w2. (b) In the LSM a microwave tone is applied\nat the frequency of an hybrid mode, while the detection of a side-\nband, on-resonance with the second mode, probes the presence of\nb2-like fields. The picture shows our room-temperature pilot setup,\ncomprising a YIG sphere and a perforated cavity which allows for\nthe calibration of the spin-magnetometer. Numbers 1 and 2 are two\nantennas coupled to the cavity, and the side of the cavity coloured in\nlight orange represent a hole housing a loop used for calibration. See\ntext for further details.\nthe latter becomes the fundamental limitation to the appara-\ntus sensitivity. The magnetic sensitivity can be calculated by\nrephrasing Eq. (6), and substituting z1(the sideband power)5\nwith the readout sensitivity sPto obtain\nsb2=2B0\nprQs\nsP\nA2p; (7)\nwhere, in this case, Qis the quality factor of the hybrid mode.\nFrom Eq. (7) one can see that the carrier power A2\npcan be ar-\nbitrarily increased to improve the LSM magnetic sensitivity,\nassuming that its noise can be reduced consequently. With\nrealistic parameters of our PMHS, one can estimate the sensi-\ntivity of a room-temperature LSM with Eq. (7), resulting in\nsb2=10:4\u0010B0\n0:4T\u0011\u0010Q\n104\u0011s\u0010A2n=kB\n300K\u0011\u0010100mW\nA2p\u0011fTp\nHz;\n(8)\nwhich is already competitive with state-of-the-art mag-\nnetometers like superconducting quantum interference\ndevices (SQUIDs)56–60or spin-exchange relaxation-free\n(SERFs)61–63. Interestingly, sb2does not depend on any\nextensive parameter, in contrast with sb1which relies on the\ntotal number of spins. This means that the LSM can in princi-\nple be miniaturised without compromising its sensitivity, and\nremoving the need of detecting a uniform field over a large\nvolume.\nA room-temperature prototype was devised to test the ac-\ntual functioning of this device, and a scheme of the setup is re-\nported in Fig. 3b. The number of spins in the sphere, together\nwith the shape of the rf-magnetic field of the cavity mode,\nset the magnetometer working frequency w2'(2p)200MHz.\nThe cavity mode and the ESR resonate at 11.5 GHz (corre-\nsponding to B0=0:4T), and their linewidths determine the\noverall quality factor of the hybrid mode Q=2750, which\nis approximately the average of the two. An antenna with\nvariable coupling is connected to the cavity to inject and ex-\ntract power from the hybrid system through a circulator. A\nmicrowave pump on resonance with the high-frequency hy-\nbrid mode at wpis filtered with a waveguide before being in-\njected in the PMHS, obtaining the input power A2\np=0:2mW.\nThe signal to be detected is the PMHS output power of the\nsideband at wp\u0000w2, the lower hybrid mode frequency. At\nwp\u0000w2, the background is mainly thermal thanks to the fil-\ntering waveguide. The extracted signal is amplified with a\nlow noise HEMT before being acquired with a spectrum anal-\nyser, and the whole electronic chain has been characterised\nby injecting calibrated signals. The readout noise results\nsP'4\u000210\u000021W=Hz, mostly due to room temperature ther-\nmodynamic fluctuations, and two orders of magnitude lower\nthan the pump noise, showing that our configuration almost\nremoves the tone-induced background. To calibrate the mag-\nnetic sensitivity of the device, we inject pico-Tesla fields at\n200 MHz using a single loop on one side of the cavity, which\ngenerates a known field parallel to B0on the YIG sphere (see\nFig. 3b). The setup was not optimized, but the expected losses\ndue to imperfect matchings can be measured and accounted\nfor by a factor k=2:1, lowering the LSM sensitivity. With\nthese quantities, from Eq. (7), the expected sensitivity of the\napparatus results ksb2=1:9pT=p\nHz. The prototype was cal-\nibrated with fields ranging from 2 to 14 pT and shows a mea-sured sensitivity of 2 :0\u00060:4pT=p\nHz, compatible with the\nestimated value64. In this setup the loop on the side of the cav-\nity was used for calibration, but in principle, it can be an input\ncoil which transduces a field to the sensitive element of the\nmagnetometer (the magnetic sphere). This signal transduc-\ntion is similar to what is usually done with SQUIDs, where an\ninput coil is coupled to the junctions loop. From Eq. (7), one\nnotices that the LSM magnetic sensitivity benefits from high\nquality factors, low readout noise, and high pump power. The\nformer feature is related to the quality factors of the PMHS,\nwhich should comprise narrow-linewidth cavities and mag-\nnetic materials to improve the magnetometer sensitivity. The\nlatter two essentially depend on the microwave electronics of\nthe setup. Since the sensitivity is size-independent, miniatur-\nisation can be foreseen by using 2D printed resonators and\nsmall quantities of material. Eventually, we mention that\nmultiplexing was also shown to be a viable option in similar\ndevices65–67.\nThe sensitivity of the two PMHS-based magnetometers is\nlimited by the noise of the readout noise temperature, which\nultimately consists in quantum fluctuations68. We foresee\nthe use of broadband Travelling Wave JPA69–71to overcome\nthe standard JPA limitation of being resonant. To overcome\nthe quantum-limit one may rely on single photon or magnon\ncounters, which are unaffected by this issue, rather than on\nlinear amplifiers.\nA downside of both the magnetometers is that their resonant\nnature implies a reduced bandiwdth, limited to the linewidth\nof an hybrid mode. Nevertheless, as the resonant frequencies\nof the hybrid modes can be changed with a tuning of the B0\nfield, the band of both the TSM and LSM can be extended. In\nparticular, for TSMs this changes the hybrid mode frequency\n(see Fig. 2b), and for LSMs is a variation of the vacuum-Rabi\nsplitting 2 gcm. Since the working band of the two magne-\ntometers is controlled by the dynamics of CMPs, the latter\nwas studied in a separate work72.\nIn conclusion, we described and operated two different\ntypes of CMP-based magnetometers, which show an outstand-\ning magnetic sensitivity. The TSM is a device that bene-\nfits from its scalability, which lowers the minimum detectable\nmagnetization oscillations. We believe that it is more suitable\nfor studying fundamental physics, for instance in the search\nfor Axions, where Dark Matter can be described as a wide,\nuniform, and persistent rf field acting on the electron spins.\nThe LSM is a device of simpler application, as it can precisely\ndetect faint magnetic fields localised on a small spin ensem-\nble. Its sensitivity relies on the design and engineering of the\nPMHS, which can be further developed to reach remarkable\nsensitivity improvements. We mention that its usage to search\nfor Axions is immediate, and that a single LSM can scan a\nbroad Axion-mass range by changing the CMP vacuum-Rabi\nsplitting. We showed that the unique features of PMHS makes\nthem suitable to assess fundamental physics problems, and we\nenvision more future applications of these systems as testbeds\nfor precision magnetometry.6\nACKNOWLEDGMENTS\nThe authors would like to acknowledge the contribution of\nthe QUAX collaboration in the development of these devices.\nWe also thank Enrico Berto, Andrea Benato, Fulvio Calaon,\nand Mario Tessaro for the help in the building of the experi-\nmental setups, and in particular for the aid with the mechanics,\ncryogenics, and electronics of the apparatuses. 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Ruoso,\n“Magnon-driven dynamics of a hybrid system excited with ultrafast optical\npulses,” Communications Physics 3, 164 (2020)."    },    {        "title": "1907.01850v1.Dynamic_magnetic_features_of_a_mixed_ferro_ferrimagnetic_ternary_alloy_in_the_form_of_AB__p_C___1_p__.pdf",        "content": "mehmet.bati@erdogan.edu.tr  Dynamic magnetic features of a mixed ferro -ferrimagnetic ternary alloy in the \nform of AB pC1-p \nMehmet Batı1,a   Mehmet Ertaşb \naDepartment of Physics, Recep Tayyip Erdoğan University, Rize, Turkey  \nbDepartment of Physics, Erciyes University, Kayseri, Turkey  \n \nAbstract  \nDynamic magnetic features of a mixed ferro -ferrimagnetic ternary alloy in the form of ABpC1-p, \nespecially . The effect of Hamiltonian parameters on the dynamic magnetic features of the system \nare investigated. F or this aim, an ABpC1-p ternary alloy system was simulated within the mean - \nfield approximation based on a Glauber type stochastic dynamic and for simplicity, A, B and C \nions as SA = 1/2, SB = 1 and SC = 3/2, were chosen respectively.  It was found that in our dynamic \nsystem the critical temperature was always dependent on the concentration ratio of the ternary \nalloy.  \nKeywords:  ABpC1-p ternary alloys; Dynamic magnetic features ; Mean -field approximation, \nGlauber type stochastic  \n \n1- Introduction  \n \nMixed -spin Ising systems provide a good model for studying a ferro -ferrimagnetic ternary \nalloy. Ternary alloy in the form of ABpC1-p composed of Prussian blue analogs [1] have been the \ntopic of much research  because of their interesting magnetic behaviors,  such as  photo induced \nmagnetization, charge -transfer -induced spin tr ansitions, the existence of compensation \ntemperatures and h ydrogen storage capacity [2 -11]. It  has been widely studied in literature by \nvarious methods in equilibrium statistical physics,  such as exact recursion relations (ERR) on the \nBethe lattice, mean -field approximation (MFA), Monte Carlo (MC) simulations and effective -field \ntheory (EFT) with correlations [12 -29]. In the studies, ternary alloy in the form of ABpC1-p has \nbeen modelled w ith different magnitudes of spins, namely A, B and C. For instance, the magnetic \nfeatures of a mixed ferro -ferrimagnetic ternary alloy in the form of ABpC1-p consist of three \ndifferent metal ions with ternary Ising spins (1/2, 1, 3/2) [12 -16]; (3/2, 1, 1/2 ) [17]; (1/2, 1, 5/2) \n[18]; (1/2, 3/2, 5/2) [19], (1, 3/2, 5/2) [20 -23], (3/2, 1, 5/2) [24 -26] and with Ising spin s (3/2, 2, \n5/2) [27 -29]. It is worth noticing that there are many experimental studies on Prussian blue analogs \n[30–36].   \n Generally, dynamic p roperty  investigations of ABpC1-p ternary alloy s are difficult because of \ntheir structural complexity. Despite, the equilibrium magnetic properties of a mixed ferro -\nferrimagnetic ABpC1-p ternary alloy having  been studied in detail, there are only two studies  within \nour best knowledge about the dynamic properties of the ABpC1-p ternary alloy system contain ing \nspin (3/2, 1, 5/2) [37, 38]. The aim of this study was focused on studying the dynamic magnetic \nproperties of a ternary alloy system. Dynamic phas e transitions (DPT) were  obtained and the effect \nof Hamiltonian parameters on the dynamic properties of the system were  investigated. For this \naim, an ABpC1-p ternary alloy system  was simulated  by utilizing the MFA based on Glauber type \nstochastic dynamic and for simplicity, A, B and C ions as SA = 1/2 SB = 1 and SC = 3/2  were chosen, \nrespectively.  It is worth noting  that the DPT has been extensively studied theoretically [39 -45] \nand experimentally [46 -50] for different systems during the last decades.  \nThe paper is organized in the following way: Section II describe s the models and its \nformulations.  Section III presents  the numerical results. Finally, the conclusion is given in Sec. IV.   \n \n2- Model formulation  \n \nA mixed ferro -ferrimagnetic ABpC1-p ternary al loy system was considered on two \ninterpenetrating square sublattices. One of the sublattices only has spins SA = ± 1/2 as the other \nsublattices have spins  SB = ±1, 0 or SC = ± 3/2, ± 1/2. A sketch of the present model can be seen in \nFig. 1. The Hamiltonian  can be written as follows:  \n \n    1 1 , (1)\n          ΗA B C A B C\ni AB j j AC j j i j j j j\nij i jS J S J S h(t) S S S ξ ξ ξ ξ\n  \n \nwhere < ij> shows a summation over all pairs of the nearest -neighboring sites of different \nsublattices and JAB > 0 and JAC < 0 (model the ferro -ferrimagnetic interactions ) are the nearest -\nneighbor exchange constants. h(t) is the oscillating external magnetic field and is described by \nℎ(𝑡)=ℎ0cos(𝑤𝑡), where h0, w and t are the time, amplitude and angular frequency. 𝜉𝑗 is \ndistributed random variables and it takes the value of unity or zero, according to whether site j is \nfilled by an ion of B or C, respectively. So, 𝜉𝑗 is described by  \n \n                                                      \n 1 1 , (2)   j j jP p pξ ξ ξ                  \n    \n where p and (1−𝑝) are the concentration of B and C ions, respectively. A mixed ferro -\nferrimagnetic ABpC1-p ternary alloy system is in contact with an isothermal heat bath at an absolute \ntemperature Tabs and evolves according to the Glau ber-type stochastic process at a rate of 1/ τ. From \nthe master equation associated to the stochastic process, it follows that the average magnetization \nsatisfies the following equation [39 -45], \n \n              \n11 , (3a)2    A A B C\nAB j AC j 0\njjdτ S S tanh β J p S +J S p h cos wtdt\n          \n                  \n\n3\n, (3b)\n21\n  \nA\nAB i 0\ni BB\nA\nAB i 0\nisinhβ J p S +ph cos wt\ndτ S Sdtcoshβ J p S +ph cos wt                                 \n         \n \n 3 1.5 1 0.5 1, (3c)2 1.5 1 2 0.5 1                 CCsinh p βx sinh p βx dτ S Sdt cosh p βx cosh p βx                              \n \nwhere 𝑥=𝐽𝐴𝐶∑𝑆𝑖𝐴\n𝑖 +ℎ0cos(𝑤𝑡). Using  the mean -field theory; the dynamic mean -field \napproximation equations are obtained as follows  \n \n       \n  11 , (4a)2     A A AB AB A AC AC C 0dm m tanh β J z m p+J z m p h cosd                      \n  \n  3\n, (4b)\n21   AB BA A 0\nBB\nAB BA A 0sinhβ J z m p+ph cos dmmd coshβ J z m p+ph cos\n \n \n 3 1.5 1 0.5 1, (4c)2 1.5 1 2 0.5 1                 CCsinh p βy sinh p βy dmmd cosh p βy cosh p βy\n \n \nwhere 𝑦=𝐽𝐴𝐶𝑧𝐶𝐴𝑚𝐴+ℎ0cos(𝜉),  𝜉=𝑤𝑡, Ω=𝜏𝑤 and Ω=2𝜋,  zAB, zBA, zAC and zCA are taken \n4 for  a square lattice.  \n                                                                         𝑀𝑖=1\n𝜏∫𝑚𝑖(𝜉)𝑑𝜉                                                                                   (5) \n  \n where 𝑖=𝐴,𝐵 and 𝐶. In other words, 𝑀𝐴, 𝑀𝐵 and 𝑀𝐶 correspond to the dynamic order parameters \nof the magnetic components A, B and C. The total magnetization of the system is  \n \n                                                                        𝑀𝑇=(𝑀𝐴+𝑀𝐵+𝑀𝐶)\n2                                                                                   (6) \n \nThe physical parameters have been scaled in terms of 𝐽𝐴𝐵. For example, reduced temperature and \nfield amplitude are respectively   defined as 𝑇=𝑘𝐵𝑇𝑎𝑏𝑠\n𝐽𝐴𝐵, and ℎ=ℎ0\n𝐽𝐴𝐵,  throughout the  \n  \n3- Results and Discussion  \n \nThe effects of the concentration ratio p and the exchange interaction ratio 𝑅 (|𝐽𝐴𝐶|\n𝐽𝐴𝐵) on dynamic \nmagnetization and DPT  of the ternary alloy have been examined.  It should be noted that the p = 0 \ncase corresponds to a ferrimagnetic mixed spin -1/2 and spin -3/2 system while for p = 1, \ncorresponds to  a mixed spin -1/2 and spin -1 ferromagnetic system.  The phase diagram of the \nternary alloy in ( 𝑅−𝑇𝐶) and ( 𝑝−𝑇𝐶) planes  are shown  in Fig . 2 and Fig . 3, respectively. In these \nfigures, upper graphs are plotted for ℎ=0.1 and the lower ones are plotted for ℎ=0.5. The \ncritical temperature value (phase transition temperature)  is a little decrease d with increasing h. \n(𝑇𝐶=2.71 for h=0.1, 𝑇𝐶=2.67 for h=0.5)  The r eason for this situation  is that , the higher field \namplitude becomes dominant against the ferromagnetic and antiferromagnetic nearest -neighbor \nbonds.  It can be seen  that from the figures  that 𝑇𝐶 increases as 𝑅 increases and  the 𝑇𝐶 values do \nnot change with 𝑅 for 𝑝= 1.0. Because the system become  an AB alloy, there is no  AC \ninteraction. Therefore t he system becomes independent from 𝑅.  \n \nIn this section, the effects of p and R on the magnetization of a ternary alloy of the type ABpC1-p \nare discussed.   Fig. 4 shows the total magnetization  chancing with scaled temperature  for R=0.5, \n1.0 and R=2.0 values.  It is again seen  from the  figures that a ll the total magnetization curves merge \nat a unique transition temperature for p = 1.0.  For the p =0.0 case, Tc=0 at  R =0.0 (see Fig . 1 and \n2) and the dynamic critical temperature of the system increase with an increasing  of R. For p=0.25, \nthe antiferromagnetic exchange interaction between  the A and C magnetic components becomes \neffective in the system for larger R values . In other words , the 𝐽𝐴𝐶  interaction  becomes dominant \nand because  𝐽𝐴𝐶  is negative,  the 𝑀𝑇  results are negative.  It is noted  that the saturation values of \nmagnetization increase for p=0.25 and decrease for p=0.75 with the increasing of R. A second \norder phase transition occurs in the system for R=0.5 and R=1.0. But for R=2, first the  system gives  \n the first order phase transition and then the second order phase transition occurs.  𝑀𝑇 decreases as \nR increases in the range 0.5 < R <1 at p = 0.5 and after 𝑅>1 (AC interaction is more  dominant), \nas R increases, 𝑀𝑇 orientation increases by changing.  \n \nIn Fig. 5 the results have been depicted for R=1.0.  As expected, t he sign  of the 𝑀𝐴 magnetization  \nis  negative ( 𝑀𝐴=−1/2) 𝑀𝐵=0.0 and 𝑀𝐶=3/2 at T=0 for p=0.0.  For 0 .0<p<1.0 the sign  of \nthe 𝑀𝐶 magnetization is  negative  (𝑀𝐶=−3/2),  𝑀𝐴=0.5 and 𝑀𝐵=1.0 at T=0. Lastly, for \np=1.0 𝑀𝐴=1/2,  𝑀𝐵=1.0  and 𝑀𝐶=0 at T=0.  There are no dynamic magnetic properties \ninvestigation of Ternary alloy for spin (½,1,3/2). However, there is an investigation for spin \n(3/2,1,5/2) [37]. Very interesting behaviors with the variations of scaled temperature, \nconcentrations of the spins and ratio of the bilinear interactions are found in agreement with the \nliterature.  An interesting feature that has been observed in equilibrium systems is the existence of \na special interact ion ratio 𝑅𝐶=|𝐽𝐴C|/𝐽𝐴B where the critical temperature of the system becomes \nindependent of the concentration ratio 𝑝 of the ternary alloy [25]. However,  in the dynamic system \nit has been  observed  that critical temperature  is always dependent  on the  concentration ratio 𝑝 of \nthe ternary alloy.  Theoretically, t his behavior was first  reported  firstly  in literatu re by Vatansever \n[37]. Dynamic behaviors of ternary alloys have not yet been studied experimentally.  Actually, \nthere  are only a few  experimen tal studies  about dynamic phase transition in literature  [51-53], such \nas, time dependent magnetic behavior of uniaxial cobalt films in the vicinity of the dynamic phase \ntransition (DPT) as a function of an oscillating magnetic field experimentally studied  by Berger \nand coworkers  [51].   \n \n4-  Conclusions  \n \nIn this paper , the dynamic magnetic features of the  mixed ferro -ferrimagnetic  ternary alloy of the \ntype ABpC1-p on a square  lattice, composed  of spins 1/2, 1, and 3/2 has been studied . A ternary \nalloy system  was simulated within the mean - field approximation based on Glauber type stochastic \ndynamic s. The concentration ratio was obtained as well as the exchange interaction ratio \ndependence of critical temperature.  It was found that the critical temperature cou ld be controlled \nby the concentration ratio and the exchange interaction ratio. 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Phys. 81 (2018) 054001  \n  \n List of the Figure Captions  \n \nFig. 1: (color online) Schematic  representation of ternary alloy spin model the type of ABpC1-p \ndefined on a square lattice.  A, B and C represented in yellow, blue and red respectively . \nInteraction between A and B sites is ferromagnetic (FM) and A and C antiferromagnetic \n(AFM).   \nFig. 2: Dynamic phase diagrams of the system in a ( 𝑅−𝑇𝐶) plane for the different values of p \nwith ℎ=0.1 and ℎ=0.5. \nFig. 3: Dynamic phase diagrams of the system in a ( 𝑝−𝑇𝐶) plane for the different values of R \nwith ℎ=0.1 and ℎ=0.5. \nFig. 4: (color online) Scaled  temperature dependence of total magnetization for different values of \n𝑝 with 𝑅=0.5 (upper figure),  𝑅=1.0 (middle figure) and  𝑅=2.0 (lower figure) . \nFig. 5: (color online)  Scaled temperature dependence of magnetization ( 𝑀𝐴,  𝑀𝐵  and 𝑀𝐶) for \ndifferent values of 𝑝 with 𝑅=1.0. \n \n \n \n \n \nFig. 1: \n \nSA = 1/2  \n JAB > 0 (FM)  \nJBC < 0 (AFM)  \nSB=1 \nSC = 3/2   \n \nh=0.1\nR0.0 0.5 1.0 1.5 2.0 2.5 3.0TC\n02468\np=0.00 \np=0.25 \np=0.50 \np=0.75 \np=1.00 \nh=0.5\nR0.0 0.5 1.0 1.5 2.0 2.5 3.0TC\n02468\np=0.00 \np=0.25 \np=0.50 \np=0.75 \np=1.00  \n \nFig. 2:   \n  \nh=0.1\np0.0 0.2 0.4 0.6 0.8 1.0TC\n02468\nR=0.0 \nR=0.5\nR=1.0\nR=1.5\nR=2.0\nR=2.5\nR=3.0\nh=0.5\np0.0 0.2 0.4 0.6 0.8 1.0TC\n0246R=0.0 \nR=0.5\nR=1.0\nR=1.5\nR=2.0\nR=2.5\nR=3.0\n \n \nFig. 3:  \n  \nT0 1 2 3 4MT\n0.00.20.40.60.8\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\nR=JAC/JAB=0.5\nR=JAC/JAB=1.0\nT0 1 2 3 4MT\n-0.4-0.20.00.20.40.60.8\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\nR=JAC/JAB=2.0\nT0 1 2 3 4MT\n-0.6-0.4-0.20.00.20.40.60.81.0\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\n \nFig. 4:  \n  \np=0.00\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-1.0-0.50.00.51.01.52.0\nMA\nMB\nMC\np=0.25\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=0.50\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=0.75\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=1.00\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n0.00.20.40.60.81.01.2\nMA\nMB\nMC\n \n \nFig. 5: \n \n "    },    {        "title": "1706.08488v1.Perpendicular_magnetic_anisotropy_in_insulating_ferrimagnetic_gadolinium_iron_garnet_thin_films.pdf",        "content": "Perpendicular magnetic anisotropy in insulating ferrimagnetic gadolinium iron garnet\nthin \flms\nH. Maier-Flaig,1, 2S. Gepr ags,1Z. Qiu,3, 4E. Saitoh,3, 4, 5, 6, 7R. Gross,1, 2, 8\nM. Weiler,1, 2H. Huebl,1, 2, 8and S. T. B. Goennenwein1, 2, 8, 9, 10\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany\n2Physik-Department, Technische Universit at M unchen, Garching, Germany\n3WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Japan\n4Spin Quantum Recti\fcation Project, ERATO, Japan Science and Technology Agency, Sendai, Japan\n5Institute for Materials Research, Tohoku University, Sendai, Japan\n6PRESTO, Japan Science and Technology Agency, Saitama, Japan\n7Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan\n8Nanosystems Initiative Munich, M unchen, Germany\n9Institut f ur Festk oper- und Materialphysik, Technische Universit at Dresden, Dresden, Germany\n10Center for Transport and Devices of Emergent Materials, Technische Universit at Dresden, 01062 Dresden\n(Dated: June 27, 2017)\nWe present experimental control of the magnetic anisotropy in a gadolinium iron garnet (GdIG)\nthin \flm from in-plane to perpendicular anisotropy by simply changing the sample temperature.\nThe magnetic hysteresis loops obtained by SQUID magnetometry measurements unambiguously\nreveal a change of the magnetically easy axis from out-of-plane to in-plane depending on the sam-\nple temperature. Additionally, we con\frm these \fndings by the use of temperature dependent\nbroadband ferromagnetic resonance spectroscopy (FMR). In order to determine the e\u000bective mag-\nnetization, we utilize the intrinsic advantage of FMR spectroscopy which allows to determine the\nmagnetic anisotropy independent of the paramagnetic substrate, while magnetometry determines\nthe combined magnetic moment from \flm and substrate. This enables us to quantitatively evalu-\nate the anisotropy and the smooth transition from in-plane to perpendicular magnetic anisotropy.\nFurthermore, we derive the temperature dependent g-factor and the Gilbert damping of the GdIG\nthin \flm.\nControlling the magnetization direction of magnetic\nsystems without the need to switch an external static\nmagnetic \feld is a challenge that has seen tremendous\nprogress in the past years. It is of considerable interest\nfor applications as it is a key prerequisite to store infor-\nmation in magnetic media in a fast, reliable and energy\ne\u000ecient way. Two notable approaches to achieve this in\nthin magnetic \flms are switching the magnetization by\nshort laser pulses[1, 2] and switching the magnetization\nvia spin orbit torques[3{5]. For both methods, materials\nwith an easy magnetic anisotropy axis oriented perpen-\ndicular to the \flm plane are of particular interest. While\nall-optical switching requires a magnetization component\nperpendicular to the \flm plane in order to transfer angu-\nlar momentum[2], spin orbit torque switching with per-\npendicularly polarized materials allows fast and reliable\noperation at low current densities[3]. Therefore great ef-\nforts have been undertaken to achieve magnetic thin \flms\nwith perpendicular magnetic anisotropy.[6] However, re-\nsearch has mainly been focused on conducting ferromag-\nnets that are subject to eddy current losses and thus of-\nten feature large magnetization damping. Magnetic gar-\nnets are a class of highly tailorable magnetic insulators\nthat have been under investigation and in use in appli-\ncations for the past six decades.[7{9] The deposition of\ngarnet thin \flms using sputtering, pulsed laser deposition\nor liquid phase epitaxy, and their properties are very well\nunderstood. In particular, doping the parent compound\n(yttrium iron garnet, YIG) with rare earth elements is apowerful means to tune the static and dynamic magnetic\nproperties of these materials.[7, 10, 11]\nHere, we study the magnetic properties of a gadolin-\nium iron garnet thin \flm sample using broadband fer-\nromagnetic resonance (FMR) and SQUID magnetome-\ntry. By changing the temperature, we achieve a transi-\ntion from the typical in-plane magnetic anisotropy (IPA),\ndominated by the magnetic shape anisotropy, to a per-\npendicular magnetic anisotropy (PMA) at about 190 K.\nWe furthermore report the magnetodynamic properties\nof GdIG con\frming and extending previous results.[7]\nI. MATERIAL AND SAMPLE DETAILS\nWe investigate a 2 :6µm thick gadolinium iron gar-\nnet (Gd 3Fe5O3, GdIG) \flm grown by liquid phase epi-\ntaxy (LPE) on a (111)-oriented gadolinium gallium gar-\nnet substrate (GGG). The sample is identical to the\none used in Ref. 12 and is described there in detail.\nGdIG is a compensating ferrimagnet composed of two\ne\u000bective magnetic sublattices: The magnetic sublattice\nof the Gd ions and an e\u000bective sublattice of the two\nstrongly antiferromagnetically coupled Fe sublattices.\nThe magnetization of the coupled Fe sublattices shows a\nweak temperature dependence below room temperature\nand decreases from approximately 190 kA m\u00001at 5 K to\n140 kA m\u00001at 300 K.[8] The Gd sublattice magnetization\nfollows a Brillouin-like function and decreases drasticallyarXiv:1706.08488v1  [cond-mat.mtrl-sci]  26 Jun 20172\nfrom approximately 800 kA m\u00001at 5 K to 120 kA m\u00001at\n300 K.[8] As the Gd and the net Fe sublattice magneti-\nzations are aligned anti-parallel, the remanent magneti-\nzations cancel each other at the so-called compensation\ntemperature Tcomp = 285 K of the material.[13] Hence,\nthe remanent net magnetization Mof GdIG vanishes at\nTcomp.\nThe typical magnetic anisotropies in thin garnet \flms\nare the shape anisotropy and the cubic magnetocrys-\ntalline anisotropy, but also growth induced anisotropies\nand magnetoelastic e\u000bects due to epitaxial strain have\nbeen reported in literature.[14, 15] We \fnd that our ex-\nperimental data can be understood by taking into ac-\ncount only shape anisotropy and an additional anisotropy\n\feld perpendicular to the \flm plane. A full determina-\ntion of the anisotropy contributions is in principle pos-\nsible with FMR. Angle dependent FMR measurements\n(not shown) indicate an anisotropy of cubic symmetry\nwith the easy axis along the crystal [111] direction in\nagreement with literature.[16] The measurements sug-\ngest that the origin of the additional anisotropy \feld\nperpendicular to the \flm plane is the cubic magnetocrys-\ntalline anisotropy. However, the low signal amplitude and\nthe large FMR linewidth towards Tcomp in combination\nwith a small misalignment of the sample, render a com-\nplete, temperature dependent anisotropy analysis impos-\nsible. In the following, we therefore focus only on shape\nanisotropy and the additional out-of-plane anisotropy\n\feld.\nII. SQUID MAGNETOMETRY\nSQUID magnetometry measures the projection of the\nmagnetic moment of a sample on the applied magnetic\n\feld direction. For thin magnetic \flms, however, the\nbackground signal from the comparatively thick sub-\nstrate can be on the order of or even exceed the magnetic\nmomentmof the thin \flm and hereby impede the quan-\ntitative determination of m. Our 2:6µm thick GdIG \flm\nis grown on a 500 µm thick GGG substrate warranting a\ncareful subtraction of the paramagnetic background sig-\nnal of the substrate. In our experiments, H0is applied\nperpendicular to the \flm plane and thus, the projection\nof the net magnetization M=m=Vto the out-of-plane\naxis is recorded as M?. Fig. 1 shows M?of the GdIG \flm\nas function of the externally applied magnetic \feld H0.\nIn the investigated small region of H0, the magnetization\nof the paramagnetic substrate can be approximated by\na linear background that has been subtracted from the\ndata. The two magnetic hysteresis loops shown in Fig. 1\nare typical for low temperatures ( T.170 K) and for\ntemperatures close to Tcomp. The hysteresis loops unam-\nbiguously evidence hard and easy axis behavior, respec-\ntively. Towards low temperatures ( T= 170 K, Fig. 1 (a))\nthe net magnetization M=jMjincreases and hence, the\nanisotropy energy associated with the demagnetization\n\feldHshape =\u0000M?[17] dominates and forces the mag-\n−150 −100 −50 0 50 100 150\nµ0H0(mT)-40-2002040M⊥(kA/m)a\n170 K (1)(2)(3)\nMH0z\n−100 mT 10 mTz\n−150 −100 −50 0 50 100 150\nµ0H0(mT)-4-2024M⊥(kA/m)b\n250 K(2)(3)\n10 mTz − HC+ HC\n−100 mTH0M z\n(1)FIG. 1. Out-of-plane magnetization component M?mea-\nsured by SQUID magnetometry. For di\u000berent temperatures,\nmagnetically hard (170K, (a)) and easy (250K, (b)) axis loops\nare observed. The arrows on the data indicate the sweep di-\nrection ofH0. The insets schematically show the magnetiza-\ntion direction MandH0=H0zwith the \flm normal zat\nthe indicated values of H0.\nnetization to stay in-plane. At these low temperatures,\nthe anisotropy \feld perpendicular to the \flm plane, Hk,\ncaused by the additional anisotropy contribution has a\nconstant, comparatively small magnitude. We therefore\nobserve a hard axis loop in the out-of-plane direction:\nUpon increasing H0from\u0000150 mT to +150 mT, Mcon-\ntinuously rotates from the out-of-plane (oop) direction to\nthe in-plane (ip) direction and back to the oop direction\nagain. The same continuous rotation happens for the op-\nposite sweep direction of H0with very little hysteresis.\nFor temperatures close to Tcomp (T= 250 K, Fig. 1 (b)),\nHshape becomes negligible due to the decreasing Mwhile\nHkincreases as shown below. Hence, the out-of-plane di-\nrection becomes the magnetically easy axis and, in turn,\nan easy-axis hysteresis loop is observed: After applying\na large negative H0[(1) in Fig. 1 (a)] MandH0are\n\frst parallel. Sweeping to a positive H0,M\frst stays\nparallel to the \flm normal and thus M?remains con-\nstant [(2) in Fig. 1 (a)] until it suddenly \rips to being\naligned anti-parallel to the \flm normal at H0>+Hc\n[(3) in Fig. 1 (a)]. These loops clearly demonstrate that\nthe nature of the anisotropy changes from IPA to PMA\non varying temperature.3\n0.0 0.5 1.0\nµ0H0(T)0102030ωres/2π (GHz)110K 190K 240Ka\n0.70 0.75µ0H0(T)\n−40040∆S21×103\n110 K0.30.40.5\nµ0H0(T)−0.50.00.5\n∆S21×103\n240 K\n0 50 100 150 200 250 300\nT(K)0.00.5−µ 0Hi(T)b\nHani=−Meff\nHshape =−M⊥\nHk=M⊥−MeffIPA PMA\nRe ImReIm\nFIG. 2. Broadband FMR spectroscopy data reveiling a smooth transition from in-plane to perpendicular anisotropy. (a)\nFMR resonance frequency plotted against H0taken for three di\u000berent temperatures (symbols) and \ft to Eq. (3) (solid lines).\nFor an IPA, a positive e\u000bective magnetization Me\u000b(positivex-axis intercept) is extracted, whereas Me\u000bis negative for a PMA.\n(inset) Exemplary resonance spectra (symbols) at 14 :5 GHz recorded at 110 K and 240 K as well as the \fts to Eq. (1) used\nto determine !res(solid lines). A complex o\u000bset S0\n21has been subtracted for visual clarity, plotted is \u0001 S21=S21\u0000S0\n21.(b)\nAnisotropy \feld Hani=\u0000Me\u000bas a function of temperature (open squares). Prediction for shape anisotropy Hshape based on\nSQUID magnetometry data (solid line) from Ref. 18. The additional perpendicular anisotropy \feld Hk=M?\u0000Me\u000b(red dots)\nincreases to approximately 0 :18 T at 250 K where its value is essentially identical to Hanidue to the vanishing M?.\nIII. BROADBAND FERROMAGNETIC\nRESONANCE\nIn order to quantify the transition from in-plane to per-\npendicular anisotropy found in the SQUID magnetome-\ntry data, broadband FMR is performed as a function of\ntemperature with the external magnetic \feld H0applied\nalong the \flm normal.[19] For this, H0is swept while\nthe complex microwave transmission S21of a coplanar\nwaveguide loaded with the sample is recorded at vari-\nous \fxed frequencies between 10 GHz and 25 GHz. We\nperform \fts of S21to[20]\nS21(H0)j!=\u0000iZ\u001f(H0) +A+B\u0001H0 (1)\nwith the complex parameters AandBaccounting for a\nlinear \feld-dependent background signal of S21, the com-\nplex FMR amplitude Z, and the Polder susceptibility[21,\n22]\n\u001f(H0) =Me\u000b(H\u0000Me\u000b)\n(H\u0000Me\u000b)2\u0000H2\ne\u000b+i\u0001H\n2(H\u0000Me\u000b):(2)\nHere,\ris the gyromagnetic ratio, He\u000b=!=(\r\u00160), and\n!is the microwave frequency and the e\u000bective magneti-\nzationMe\u000b=Hres\u0000!res=(\r\u00160). From the \ft, the res-\nonance \feld Hresand the full width at half-maximum\n(FWHM) linewidth \u0001 His extracted. Exemplary data\nforS21(data points) and the \fts to Eq. (1) (solid lines)\nat two distinct temperatures are shown in the two in-\nsets of Fig. 2 (a). We obtain excellent agreement of the\n\fts and the data. The insets furthermore show that the\nsignal amplitude is signi\fcantly smaller for T= 240 K\nthan for 110 K. This is expected as the signal amplitude\nis proportional to the net magnetization Mof the sam-\nple which decreases considerably with increasing temper-\nature (cf. Fig. 2 (b)). At the same time, the linewidthdrastically increases as discussed in the following section.\nThese two aspects prevent a reliable analysis of the FMR\nsignal in the temperature region 250 K <T < 300 K (i.e.\naround the compensation temperature). Therefore we do\nnot report data in this temperature region. Nevertheless,\nFMR is ideally suited to investigate the magnetic prop-\nerties of the GdIG \flm selectively, i.e. independent of\nthe substrate, for temperatures below Tcomp.\nAs all measurements are performed in the high \feld\nlimit of FMR, the dispersions shown in Fig. 2 (a) are\nlinear and we can use the Kittel equation\n!res=\r\u00160(Hres\u0000Me\u000b) (3)\nto extract\randMe\u000b. It is customary to describe the\nmagnetic anisotropy using Me\u000bwhich can be related to\nan anisotropy \feld Hanialong + zasMe\u000b=\u0000Hani=\nM?\u0000Hkfor positive H0. Here,Haniis given by the\ndemagnetization \feld Hshape =\u0000M?(along\u0000z) and\nthe anisotropy \feld Hkof the additional perpendicular\nanisotropy (along + z). Evidently, Me\u000bcan be deter-\nmined by linearly extrapolating the data to !res= 0.\nThe FMR dispersion and the \ft to Eq. (3) (solid lines) are\nshown for three selected temperatures in Fig. 2 (a). At\n110 K (blue curve) Me\u000bis positive. Therefore, M >H k\nindicating that shape anisotropy dominates, and the \flm\nplane is a magnetically easy plane while the oop direction\nis a magnetically hard axis. At 240 K (red curve) Me\u000bis\nnegative and hence, the oop direction is a magnetically\neasy axis. Figure 2 (b) shows the extracted Me\u000b(T).\nAt 190 K, Me\u000bchanges sign. Above this tempera-\nture (marked in red), the oop axis is magnetically easy\n(PMA) and below this temperature (marked in blue),\nthe oop axis is magnetically hard (IPA). The knowledge\nofM?(T) obtained from SQUID measurements allows\nto separate the additional anisotropy \feld HkfromMe\u000b4\n1.61.82.02.2g\na\n10−310−210−1αb\n0 50 100 150 200 250\nT(K)100030005000∆ω0/2π(MHz)c\nFIG. 3. Key parameters characterizing the magnetiza-\ntion dynamics of GdIG as a function of T:(a)g-factor\ng=\r~=\u0016B,(b)Gilbert damping constant \u000band(c)inhomo-\ngeneous linewidth \u0001 !0=(2\u0019)\n(red dots in Fig. 2 (b)). Hk=M?\u0000Me\u000bincreases\nconsiderably for temperatures close to Tcomp while at\nthe same time the contribution of the shape anisotropy,\nHshape =\u0000M?trends to zero. For T'180 K,Hkex-\nceedsHshape which is indicated by the sign change of\nMe\u000b. Above this temperature, we thus observe PMA. We\nuse the magnetization Mdetermined using SQUID mag-\nnetometry from Ref. 18 normalized to the here recorded\nMe\u000bat 10 K in order to quantify Hk. The maximal value\n\u00160Hk= 0:18 T is obtained at 250 K which is the highest\nmeasured temperature due to the decreasing signal-to-\nnoise ratio towards Tcomp.\nWe can furthermore extract the g-factor and damp-\ning parameters from FMR. The evolution of the g-factor\ng=\r~\n\u0016Bwith temperature is shown in Fig. 3 (a). We ob-\nserve a substantial decrease of gtowardsTcomp. This is\nconsistent with reports in literature for bulk GIG and can\nbe explained considering that the g-factors of Gd and Fe\nions are slightly di\u000berent such that the angular momen-\ntum compensation temperature is larger than the mag-\nnetization compensation temperature.[23] The linewidth\n\u0001!=\r\u0001Hcan be separated into a inhomogeneous con-\ntribution \u0001 !0= \u0001!(H0= 0) and a damping contribu-\ntion varying linear with frequency with the slope \u000b:\n\u0001!= 2\u000b\u0001!res+ \u0001!0: (4)\nClose toTcomp = 285 K, the dominant contribution to\nthe linewidth is \u0001 !0which increases by more than an\norder of magnitude from 390 MHz at 10 K to 6350 MHz\nat 250 K [Fig. 3 (c)]. This temperature dependence of the\nlinewidth has been described theoretically by Clogston\net al.[24, 25] in terms of a dipole narrowing of the in-homogeneous broadening and was reported experimen-\ntally before[7, 16]. As opposed to these single frequency\nexperiments, our broadband experiments allow to sepa-\nrate inhomogeneous and intrinsic damping contributions\nto the linewidth. We \fnd that in addition to the in-\nhomogeneous broadening of the line, also the Gilbert-\nlike (linearly frequency dependent) contribution to the\nlinewidth changes signi\fcantly: Upon approaching Tcomp\n[Fig. 3 (b)], the Gilbert damping parameter \u000bincreases\nby an order of magnitude. Note, however, that due to\nthe large linewidth and the small magnetic moment of\nthe \flm, the determination of \u000bhas a relatively large\nuncertainty.[26] A more reliable determination of the\ntemperature evolution of \u000busing a single crystal GdIG\nsample that gives access to the intrinsic bulk damping\nparameters remains an important task.\nIV. CONCLUSIONS\nWe investigate the temperature evolution of the mag-\nnetic anisotropy of a GdIG thin \flm using SQUID mag-\nnetometry as well as broadband ferromagnetic resonance\nspectroscopy. At temperatures far away from the com-\npensation temperature Tcomp, the SQUID magnetome-\ntry reveals hard axis hysteresis loops in the out-of-plane\ndirection due to shape anisotropy dominating the mag-\nnetic con\fguration. In contrast, at temperatures close to\nthe compensation point, we observe easy axis hysteresis\nloops. Broadband ferromagnetic resonance spectroscopy\nreveals a sign change of the e\u000bective magnetization (the\nmagnetic anisotropy \feld) which is in line with the mag-\nnetometry measurements and allows a quantitative anal-\nysis of the anisotropy \felds. We explain the qualitative\nanisotropy modi\fcations as a function of temperature by\nthe fact that the magnetic shape anisotropy contribu-\ntion is reduced considerably close to Tcomp due to the re-\nduced net magnetization, while the additional perpendic-\nular anisotropy \feld increases considerably. We conclude\nthat by changing the temperature the nature of the mag-\nnetic anisotropy can be changed from an in-plane mag-\nnetic anisotropy to a perpendicular magnetic anisotropy.\nThis perpendicular anisotropy close to Tcomp in combi-\nnation with the small magnetization of the material may\nenable optical switching experiments in insulating fer-\nromagnetic garnet materials. Furthermore, we analyze\nthe temperature dependence of the FMR linewidth and\ntheg-factor of the GdIG thin \flm where we \fnd values\ncompatible with bulk GdIG[7, 25]. The linewidth can\nbe separated into a Gilbert-like and an inhomogeneous\ncontribution. We show that in addition to the previously\nreported increase of the inhomogeneous broadening, also\nthe Gilbert-like damping increases signi\fcantly when ap-\nproachingTcomp5\nV. ACKNOWLEDGMENTS\nWe gratefully acknowledge funding via the priority pro-\ngram Spin Caloric Transport (spinCAT), (Projects GO\n944/4 and GR 1132/18), the priority program SPP 1601(HU 1896/2-1) and the collaborative research center SFB\n631 of the Deutsche Forschungsgemeinschaft.\nVI. BIBLIOGRAPHY\n[1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Physical Review\nLetters 99, 1 (2007).\n[2] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fuller-\nton, Science 345, 1337 (2014).\n[3] K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner,\nA. Ghosh, S. Au\u000bret, O. Boulle, G. Gaudin, and P. Gam-\nbardella, Applied Physics Letters 105, 212402 (2014).\n[4] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Au\u000bret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n[5] A. Brataas, A. D. Kent, and H. Ohno, Nature Materials\n11, 372 (2012).\n[6] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D.\nGan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura,\nand H. Ohno, Nature Materials 9, 721 (2010).\n[7] B. Calhoun, J. Overmeyer, and W. Smith, Physical Re-\nview107(1957).\n[8] G. F. Dionne, Journal of Applied Physics 42, 2142\n(1971).\n[9] J. D. Adam, L. E. Davis, G. F. Dionne, E. F. Schloemann,\nand S. N. Stitzer, IEEE Transactions on Microwave The-\nory and Techniques 50, 721 (2002).\n[10] K. P. Belov, L. A. Malevskaya, and V. I. Sokoldv, Soviet\nPhysics JETP 12, 1074 (1961).\n[11] P. R oschmann and W. Tolksdorf, Materials Research\nBulletin 18, 449 (1983).\n[12] H. Maier-Flaig, M. Harder, S. Klingler, Z. Qiu, E. Saitoh,\nM. Weiler, S. Gepr ags, R. Gross, S. T. B. Goennen-\nwein, and H. Huebl, Applied Physics Letters 110, 132401\n(2017).\n[13] G. F. Dionne, Magnetic Oxides (Springer US, Boston,\nMA, 2009).\n[14] S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, Jour-\nnal of Applied Physics 106, 123917 (2009).\n[15] S. A. Manuilov and A. M. Grishin, Journal of Applied\nPhysics 108, 013902 (2010).\n[16] G. P. Rodrigue, H. Meyer, and R. V. Jones, Journal of\nApplied Physics 31, S376 (1960).\n[17] We use the demagnetization factors of a in\fnite thin \flm:\nNx;y;z= (0;0;1).\n[18] S. Gepr ags, A. Kehlberger, F. D. Coletta, Z. Qiu, E.-J.\nGuo, T. Schulz, C. Mix, S. Meyer, A. Kamra, M. Al-\nthammer, H. Huebl, G. Jakob, Y. Ohnuma, H. Adachi,\nJ. Barker, S. Maekawa, G. E. W. Bauer, E. Saitoh,\nR. Gross, S. T. B. Goennenwein, and M. Kl aui, Nature\nCommunications 7, 10452 (2016).\n[19] The alignment of the sample is con\frmed at low temper-\natures by performing rotations of the magnetic \feld di-\nrection at \fxed magnetic \feld magnitude while recording\nthe frequency of resonance !res. As the shape anisotropydominates at low temperatures, !resgoes through an\neasy-to-identify minimum when the sample is aligned\noop.\n[20] H. Maier-Flaig, S. T. B. Goennenwein, R. Ohshima,\nM. Shiraishi, R. Gross, H. Huebl, and M. Weiler, arXiv\npreprint arXiv:1705.05694 .\n[21] J. M. Shaw, H. T. Nembach, and T. J. Silva, Physical\nReview B 87, 054416 (2013).\n[22] H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider,\nM. J. Carey, S. Maat, and J. R. Childress, Physical\nReview B 84, 054424 (2011).\n[23] R. K. Wangsness, American Journal of Physics 24, 60\n(1956).\n[24] A. M. Clogston, Journal of Applied Physics 29, 334\n(1958).\n[25] S. Geschwind and A. M. Clogston, Physical Review 108,\n49 (1957).\n[26] For the given signal-to-noise ratio and the large\nlinewidth, \u000band \u0001!0are correlated to a non-\nnegligible degree with a correlation coe\u000ecient of\nC(intercept;slope) = \u00000:967."    },    {        "title": "1903.10271v1.Octahedral_tilting_and_emergence_of_ferrimagnetism_in_cobalt_ruthenium_based_double_perovskites.pdf",        "content": "arXiv:1903.10271v1  [cond-mat.mtrl-sci]  25 Mar 2019Octahedral tilting and emergence of ferrimagnetism in coba lt-ruthenium based double\nperovskites\nManjil Das, Prabir Dutta, Saurav Giri and Subham Majumdar∗∗\nSchool of Physical Sciences, Indian Association for the Cul tivation of Science,\n2A & B Raja S. C. Mullick Road, Jadavpur, Kolkata 700 032, INDI A\nRare earth based cobalt-ruthenium double perovskites A 2CoRuO 6(A = La, Pr, Nd and Sm)\nwere synthesized and investigated for their structural and magnetic properties. All the compounds\ncrystallize in the monoclinic P21/nstructure with the indication of antisite disorder between Co\nand Ru sites. While, La compound is already reported to have a n antiferromagnetic state below\n27 K, the Pr, Nd and Sm systems are found to be ferrimagnetic be lowTc= 46, 55 and 78 K\nrespectively. Field dependent magnetization data indicat e prominent hysteresis loop below Tcin\nthe samples containing magnetic rare-earth ions, however m agnetization does not saturate even\nat the highest applied fields. Our structural analysis indic ates strong distortion in the Co-O-Ru\nbond angle, as La3+is replaced by smaller rare-earth ions such as Pr3+, Nd3+and Sm3+. The\nobserved ferrimagnetism is possibly associated with the en hanced antiferromagnetic superexchange\ninteraction in the Co-O-Ru pathway due to bond bending. The P r, Nd and Sm samples also show\nsmall magnetocaloric effect with Nd sample showing highest v alue of magnitude ∼3 Jkg−1K−1at\n50 kOe. The change in entropy below 20 K is found to be positive in the Sm sample as compared\nto the negative value in the Nd counterpart.\nI. INTRODUCTION\nSince the discovery of low field room temperature\nmagneto-resistance in Sr 2FeMoO 61, the double per-\novskites (A 2BB′O6) have been intensely studied2. These\nquaternary compounds can be synthesized with a vary-\ning combinations of cations at the A (alkaline earth or\nrare earth metals) and B/B′(3d, 4dor 5dtransition met-\nals) sites, which provides a scope to access diverse ma-\nterial properties within the similar crystallographic en-\nvironment. Elements with partially filled dlevel at the\nB/B′can give rise to wealth of magnetic ground states\nincluding ferromagnetism, antiferromagnetism, ferrimag-\nnetism as well as glassy magnetic phase. Double per-\novskites are also associated with intriguing electronic\nproperties3such as half-metallic behaviour4, tunneling\nmagneto-resistance5, metal-insulator transition6and so\non.\nIn case of insulating A 2BB′O6double perovskites, the\nmagnetic interaction between B and B′ions is primarily\nB-O-B′superexchange type and Goodenough-Kanamori\nrule can predict the sign of the interaction7. Ideal dou-\nble perovskites have cubic symmetry, but the presence of\ncations with small ionic radii at the A site can distort the\nstructure. Such distortion lowers the lattice symmetry to\ntetragonal or monoclinic, where the BO 6/B′O6octahe-\ndra get tilted through the bending of B-O-B′bond angle.\nIt has been found that for two fixed B and B′, the mag-\nnetic ground state is very much sensitive to this bond\nangle. Doping at the A site can change the bond an-\ngle and henceforth the nature of the ordered magnetic\nstate. For example, substitution of Ca at the Sr site\nof Sr2CoOsO 6drives the system from an antiferromag-\nnetic (AFM) insulator to spin-glass (SG) and eventually\nto a ferrimagnetic (FI) state on full replacement of Sr\nby Ca8,9. There is also report of drastic change in fer-\nrimagnetic coercivity in (Ca,Ba) 2FeReO 6under hydro-static pressure due to the buckling of Fe-O-Re bond10.\nIt has been argued that the magnetic ground state in\nthese insulating systems is an outcome of the compe-\ntition between the interactions along the paths B-O-B′\nand B-O-B′-O-B (and similarly, B′-O-B-O-B′). When\nthe octahedral tilting is minimal, the B-O-B′-O-B type\ninteraction dominates, giving rise to strong AFM corre-\nlations within B sublattice11. However, with increasing\ndistortion, B-O-B′becomes stronger with the simultane-\nous weakening of B-O-B′-O-B coupling, and a simple FI\nstate emerges (provided the magnetic moments at B and\nB′ions are unequal) due to the strong AFM coupling be-\ntween B and B′ions. The intermediate spin-glass state\npossibly arises from these competing interactions.\nRecently, La 2CoRuO 6compound has been shown to\nhave an AFM ground state, which crystallizes in the dis-\ntortedmonoclinicstructure12. Interestingly, the isostruc-\ntural Y 2CoRuO 6shows ferrimagnetism, and turns into a\nspin-glass on La doping at the Y site13. It is therefore\nworthwhile to study the magnetic states of A 2CoRuO 6\nwith different rare-earth atoms at the A site. It is well\nknownthattheionicradiusofrare-earth(inthe3+state)\ndiminishes with increasing atomic number, which can\ntune the lattice distortion. The main goal of the present\nwork is to study the effect of lattice distortion and as-\nsociated change in the magnetic properties with varying\nrare-earth ion at the A site. It is also worthwhile to note\nhow the rare-earth moment is affecting the ground state\nmagnetic properties. In a recent report, the magnetic\nproperties of A 2CoMnO 6compounds were found to get\nstrongly affected by the variation of rare-earth ion at the\nA-site14.2\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s32/s73\n/s111/s98/s115\n/s32/s73\n/s99/s97/s108\n/s32/s73\n/s111/s98/s115/s45/s73\n/s99/s97/s108\n/s32/s66/s114/s97/s103/s103/s32/s112/s111/s115/s105/s116/s105/s111/s110/s115\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121 /s40/s97/s46/s117/s46/s41\n/s50 /s40/s100/s101/s103/s114/s101/s101/s41/s98\n/s99\n/s97/s91/s97/s93 /s91/s98/s93\n/s32/s73\n/s111/s98/s115\n/s32/s73\n/s99/s97/s108\n/s32/s73\n/s111/s98/s115/s45/s73\n/s99/s97/s108\n/s32/s66/s114/s97/s103/s103/s32/s112/s111/s115/s105/s116/s105/s111/s110/s115\n/s32/s32\n/s50 /s40/s100/s101/s103/s114/s101/s101/s41\nFIG. 1. (a) and (b) show powder x-ray diffraction data of LCRO a nd SCRO respectively collected at room temperature.\nThe inset of (a) shows the crystal structure of the sample. Gr een, pink, blue and spheres indicate La, Ru, Co and O atoms\nrespectively. The bottom panel indicates the octahedral ti lt of four compositions.\nII. EXPERIMENTAL DETAILS\nSingle phase polycrystalline A 2CoRuO 6samples (for\nA = La, Pr, Nd and Sm) were synthesized by solid state\ntechnique. Stiochiometric amounts of A 2O3(except Pr-\nsample, where Pr 6O11was used), Co 3O4and RuO 2were\nwell mixed in a agate morter pestle and calcined at 1073\nK for12h and then sinteredat 1473K for24h in the pel-\nlet form with one intermediate grinding. The powder X-\nray diffraction (PXRD) data were obtained in RIGAKU\nX-ray diffractometer with Cu-K αradiation in the range\n15◦to 80◦. Magnetization ( M) measurements were car-\nried out using a SQUID magnetometer (Quantum De-\nsign, MPMS-3) up to 70 kOe. High field magnetic mea-\nsurements (up to 150 kOe) was performed on a vibrat-\ning sample magnetometer from Cryogenic Ltd., UK. The\ntemperature dependence of the electrical resistivity ( ρ)\nwas measured by DC four-probe method in the temper-\nature range between 50 K and 300 K.III. SAMPLE CHARACTERIZATION\nAll four compositions, La 2CoRuO 6(LCRO),\nPr2CoRuO 6(LCRO) Nd 2CoRuO 6(NCRO) and\nSm2CoRuO 6(SCRO), crystallize in monoclinic rock salt\nstructure (space group P21/n)15. For double perovskite\nA2BB′O6, one can define a Goldschmidt tolerance factor\nt=rA+rO √\n2(rB+rO), whererA, andrOare the ionic radii of\nA and O respectively, while rBstands for the average\nradius of B and B′. It has been found that if t <1,\nthere can be distortion from the ideal cubic structure2.\nFor the present case, r3+\nLa= 103.2 pm, r3+\nPr= 99 pm r3+\nNd=\n98.3 pm, r3+\nSm= 95.8 pm, r2+\nCo= 65(74.5) pm for low-spin\n(high-spin), r4+\nRu= 62 pm and r2−\nO= 140 pm, which give\ntin the range of 0.81-0.83 for four samples. Evidently\nfor these compositions, tis significantly lower than unity,\nand this explains the observed monoclinic symmetry\nrather than ideal cubic one. This lower symmetry is\nassociated with the tilting of the (B,B′)O6octahedra.\nIn order to determine the crystallographic parameters\nof our samples, we have performed Reitveld refinement\non the room temperature PXRD data using MAUD\nsoftware package16. The data converges well with3\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48 /s51/s50/s48/s48/s46/s48/s50/s48/s46/s48/s52\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s49/s50/s48/s46/s48/s49/s56/s48/s46/s48/s50/s52\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48 /s51/s50/s48/s48/s56/s49/s54\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48 /s51/s50/s48/s48/s49/s54/s51/s50\n/s50/s52 /s52/s56 /s55/s50/s48/s50/s48/s52/s48\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s50/s52/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48 /s51/s50/s48/s48/s56/s49/s54\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s50/s53/s48/s46/s48/s53/s48/s40/s101/s109/s117/s47/s109/s111/s108/s41/s40/s101/s109/s117/s47/s109/s111/s108/s41/s32/s49/s48/s48/s32/s79/s101\n/s32/s90/s70/s67\n/s32/s70/s67/s32/s90/s70/s67\n/s32/s70/s67\n/s32\n/s84 /s32/s40/s75/s41/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s91/s97/s93/s76/s67/s82/s79\n/s84/s32 /s40/s75/s41/s32/s49/s48/s48/s32/s79/s101\n/s32/s32\n/s84/s32 /s40/s75/s41\n/s78/s67/s82/s79\n/s32/s49/s48/s48/s32/s79/s101/s84/s32 /s40/s75/s41/s40/s101/s109/s117/s47/s109/s111/s108/s41/s32\n/s32\n/s32/s32\n/s91/s99/s93/s84\n/s80 /s84\n/s80/s83/s67/s82/s79\n/s32/s49/s48/s48/s32/s79/s101/s32/s90/s70/s67\n/s32/s70/s67/s32\n/s32\n/s32/s32\n/s91/s100/s93\n/s32/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s84/s32 /s40/s75/s41/s32/s49/s48/s48/s32/s79/s101\n/s32/s53/s48/s48/s32/s79/s101\n/s32/s49/s48/s48/s48/s32/s79/s101\n/s32/s32/s32\n/s32/s32/s49/s32/s107/s79/s101/s84\n/s80\n/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s32/s49/s48/s48/s32/s79/s101/s32/s90/s70/s67\n/s32/s70/s67/s80/s67/s82/s79\n/s91/s98/s93\n/s32/s32\n/s32/s49/s48/s48/s32/s79/s101\n/s32/s32\nFIG. 2. (a)-(d) represent temperature variation of magneti c susceptibility for LCRO, PCRO, NCRO and SCRO respectively\nfor an applied field of 100 Oe both in ZFC and FC protocols. The i nsets of (a) and (b) show the modified Curie-Weiss fit to\nthe susceptibility data of respective samples. The inset of (c) shows the susceptibility of NCRO measured under 1 kOe of fi eld.\nThe inset in (d) shows an enlarged view of the ZFC susceptibil ity of SCRO recorded at different applied fields.\nmonoclinic space group P2 1/n for all the samples along\nwith antisite disorder between Co and Ru sites [Figs. 1\n(a) and (b)]. Our calculations show that there are 1-5%\nantisite defects, i.e., the fractional occupancy of B site\nconsists of 99-95% Co and 1-5% Ru. The antisite defect\nis found to be large in Nd and Sm compounds and it\nis low in case of La and Pr counterparts. The refined\ncrystallographic parameters are depicted in Table 1,\nand they match well with the previous reported data17.\nWe have also added the structural data of Y 2CoRuO 6\n(YCRO) from reference13for comparison. The crystal\nstructure of LCRO, as obtained from the refinement of\nour PXRD data, is shown in the inset of fig. 1 (a). It is\nclearly evident that the (Co,Ru)O 6octahedra are tilted.\nThe average tilting angle is defined as /angbracketleftΨ/angbracketright=1\n2[π−/angbracketleftΦ/angbracketright],\nwhere/angbracketleftΦ/angbracketrightis the average inter-octahedral Co-O-Ru\nangle. Clearly, the tilt angle increases systematically aswe movefrom La to Sm, which is the effect of the gradual\nbending of Co-O-Ru bond (see Table 1). It is interesting\nto note that all the crystallographic parameters vary\nsystematically with the ionic radius for La, Pr, Nd and\nSm samples.\nIV. RESULTS\nIV.1. Magnetic studies\nFigs. 2 (a)-(d) depict the temperature ( T) dependence\nof susceptibility ( χ=M/H) measured under different\nvalues of Hvalues for all the four samples, where both\nzero-field-cooled (ZFC) and field-cooled (FC) measure-\nments were performed. LCRO [fig.2 (a)] shows well de-\nfined peak at the Ne´ el temperature TN= 27 K, indicat-4\nParameters LCRO PCRO NCRO SCRO YCRO\nr3+\nA(˚A) 1.032 0.990 0.983 0.958 0.900\na(˚A) 5.575(2) 5.497(1) 5.440(5) 5.390(2) 5.266\nb(˚A) 5.638(7) 5.689(7) 5.717(7) 5.722(8) 5.711\nc(˚A) 7.886(2) 7.802(4) 7.739(2) 7.685(7) 7.558\nβ(◦) 90.01(1) 89.88(7) 89.99(8) 89.93(2) 90.03\n/angbracketleft∠Co-O-Ru /angbracketright(◦)152.4 150.9 143.4 139.5 141.7\n/angbracketleftΨ/angbracketright(◦) 13.8 14.6 18.3 20.3 19.7\nMag. state AFM FI FI FI FI\nTran. temp. TN= 27 K Tc= 46 K Tc= 55 K Tc= 78 K Tc= 82 K\nHcoer –9 kOe (2 K) 8 kOe (2 K) 22 kOe (2 K) 22.5 kOe (5 K)\nTABLE I. Crystallographic lattice parameters ( a,b,c, andβ), average Co-O-Ru bond angle, average octahedral tilt angl e (Ψ),\nmagnetic state, magnetic transition temperatures and coer civity are depicted for A 2CoRuO 6(A = La, Pr, Nd, Sm and Y). The\nparameters for Y compound are obtained from reference13. The ionic radii of rare-earth ions (in the 3+ state with coor dination\nnumber VI) at the A site are also shown18.\ning an AFM ground state and it matches well with the\nprevious report7,12,19. TheTvariation of susceptibility\n(χ=M/H) of LCRO in the paramagnetic (PM) state\ncannot be fitted with a simple Curie-Weiss law. How-\never, the χ(T) data above 100 K can be fitted well with\na modified Curie-Weiss law, χCW(T) =C/(T−θ)+χ0,\nwherean additional Tindependent term ( χ0) is included.\nHereCis the Curie constant and θis the Curie-Weiss\ntemperature. The effective PM moment, µeff, obtained\nfrom Curie-Weiss fitting, is found to be 6.63 µB/f.u. The\nvalue ofθis−140 K, signifying strong AFM correlations.\nThe FC and ZFC data show weak divergence below TN.\nWe also observed an upward rise in the χ(T) data below\n6 K. The observed value of µeffis higher than expected\nfor a Co2+-high-spin and Ru4+-low-spin states7, which\nwas attributed to extended 4 d-orbitals of Ru19.\nFor PCRO, NCRO and SCRO, the χ(T) data are dras-\ntically different from that of LCRO [figs. 2 (b), (c) and\n(d) respectively], and it isquite eventful. χshowsasharp\nrise below Tc= 46, 55 and 78 K for these magnetic rare-\nearth containing samples respectively. The FC and ZFC\nsusceptibilities show strong irreversibility below Tc. The\ndivergence exists even in measurement at H= 1 kOe [see\ninset of fig. 2 (c)], however the extend of divergence re-\nduces. The point of bifurcation also moves to lower T\nwith increasing H. The ZFC data show a well defined\npeak at a temperature TP, which lies below Tc. Notably,\nwe observe a change in the value of TPwith increasing\nH. TheHdependence of TPis particularly significant\nfor SCRO, where we observe a shift of Tpby 16 K when\nHis changed from 100 Oe to 1 kOe [see inset of fig. 2\n(c)]. Similar shift in the observed peak in ferrimagnetic\nNd2CoMnO 6was also reported previously, which was at-\ntributed to the presenceofferromagnetic(FM) and AFM\nclusters20. TheFCsusceptibility, ontheotherhand, rises\nmonotonically for all the samples with decreasing T.\nThe susceptibility data of PCRO and NCRO can be\nwell fitted with χCW(T) above 100 K, which gives the\nvalues of µeffto be 6.59 and 6.47 µBrespectively. Thevalue of θis -25 (-22) K for Pr(Nd) sample. These val-\nues ofµeffare slightly lower than the expected value of\n6.97 (7.01) µBwith Pr3+(Nd3+), Co2+(high-spin) and\nRu4+(low-spin) states. This mismatch can be caused\nby the presence of antisite disorder. Such disorder is ex-\npected to cause a decrease in the magnetic moment, and\nempirically Mactual=Mobs/(1−2D)21whereDis the\ndegree of disorder between Co and Ru atoms. For exam-\nple, in case of NCRO with D= 0.05 and Mobs= 6.47\nµB,Mactualis found to be 7.18 µB, which is very close\nto the theoretically predicted value.\nFor SCRO, we failed to achieve good fit using χCW(T)\nin the temperature range 100-315 K. The separation be-\ntween ground ( J= 5/2) and first excited ( J= 7/2) mul-\ntiplets in Sm3+is small, and their mixing can be respon-\nsible for the observed non-Curie-Weiss behaviour22.\nThe isothermal MversusHdata are shown in figs. 3\n(a)to(d). ForLCRO,alinear M−Hcurveisobtainedat\n2 K [see fig. 3 (a)], indicating AFM state. On the other\nhand, Pr, Nd and Sm compounds show significant hys-\nteresis with large coercive field ( Hcoer). Non-zero Hcoer\nis observed for these FI systems just below Tc, and it in-\ncreases with decreasing T. The values of Hcoerare found\nto be 9, 8 and 22 kOe at 2 K for PCRO, NCRO and\nSCRO respectively. However, Mdoes not saturate even\nat 70 kOe of field for none the samples. We have also\nrecorded high field magnetization for SCRO, as shown in\nthe inset of fig. 3 (d). The M−Hcurve at 5 K does\nnot fully saturate even at 150 kOe of applied field. Mat-\ntains a value close to 2 µBat the highest H. The value of\nHcoerfor SCRO at 5 K is found to be 10.6 kOe, which is\nsmaller than the value of Hcoerreported for Y 2CoRuO 6\n(∼22.5 kOe) at the same T.\nConsideringnon-zerocoercivityandthepresenceofan-\ntisite disorder,wehaverecorded M−Hhysteresisloopat\n2 K after the sample being field-cooled from room tem-\nperature. In case of inhomogeneous magnetic systems, a\nshift in the hysteresis loop along the field axis may be\nobserved due to the interfacial coupling of two magnetic5\n/s45/s55/s48 /s45/s51/s53 /s48 /s51/s53 /s55/s48/s45/s51/s48/s51/s45/s55/s48 /s45/s51/s53 /s48 /s51/s53 /s55/s48/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52\n/s45/s55/s48 /s45/s51/s53 /s48 /s51/s53 /s55/s48/s45/s49/s46/s54/s48/s46/s48/s49/s46/s54\n/s45/s49/s53/s48 /s45/s55/s53 /s48 /s55/s53 /s49/s53/s48/s45/s50/s48/s50/s45/s55/s48 /s45/s51/s53 /s48 /s51/s53 /s55/s48/s45/s51/s48/s51/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46 /s41\n/s52/s53/s32/s75/s50/s48/s32/s75/s50/s32/s75\n/s32\n/s32/s32/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46 /s41\n/s91/s99/s93/s78/s67/s82/s79/s91/s97/s93/s32/s50/s32/s75\n/s32\n/s32/s32\n/s76/s67/s82/s79\n/s32/s83/s67/s82/s79/s50/s32/s75/s32\n/s91/s100/s93\n/s72 /s40/s107/s79/s101/s41\n/s32/s32\n/s53/s32/s75\n/s32/s32 /s32/s32/s80/s67/s82/s79/s52/s48/s32/s75/s49/s48/s32/s75/s50/s32/s75\n/s91/s98/s93/s32 /s32/s32\nFIG. 3. (a) to (d) show isothermal magnetization data up to fie ld 70 kOe at different temperatures for LCRO, PCRO, NCRO\nand SCRO respectively. The inset of (d) shows the M−Hcurve for SCRO at 5 K for maximum field of 150 kOe.\nphases, and it is referred as exchange bias effect23. Many\ndouble perovskites show exchange bias effect due to the\npresence of antisite disorder24. However, we failed to ob-\nserve such exchange bias in NCRO and SCRO samples,\nwhich possibly rule out the existence of large magnetic\ninhomogeneity in the system.\nIn orderto investigatethe effect ofexternal field on the\nmagnetic state, we have measured magneto-caloric effect\n(MCE) ofthe samples in terms ofentropy-change(∆ SM)\nbyH. In the recent past, MCE has emerged out to be\nan important technique for green refrigeration25. In the\npresent work, we have obtained MCE from our isother-\nmal magnetization data recorded at different constant\ntemperatures. From the theory of thermodynamics,\n∆SM(0→H0) =/integraldisplayH0\n0/parenleftbigg∂M\n∂T/parenrightbigg\nHdH,\nwhere ∆SM(0→H0) denotes the entropy change for thechange in Hfrom 0 to H026. Figs. 4 (a) to (c) show\n∆SM(T,H0) versus Tplot at different values of H0for\nPCRO, NCRO and SCRO samples respectively.\nThe magnitude of MCE is found to be low for all three\nsamples. For the Pr and Nd samples, ∆ SM(T) is mostly\nnegative with its magnitude peaking around 12 and 8\nK (peak magnitude: 1.7 and 2.9 Jkg−1K−1atH0=\n50 kOe) respectively. A broad feature is also observed\nin ∆SM(T,H0) data around 35 K. On the other hand,\nSCRO shows a contrasting behaviour as far as the MCE\nis concerned. ∆ SM(T) for SCRO is positive below 20 K,\nand increases with decreasing temperature (at least for\nH0>10 kOe). ∆ SMattains a value of 3.1 Jkg−1K−1for\nH0= 50 kOe at 2 K.\nAs already discussed, double perovskite systems can\nshow glassy magnetic state27. An intermediate SG state\nis observed when the AFM state is transformed into an\nFI state by suitable A-site doping. In order to investi-6\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s51/s45/s50/s45/s49/s48/s49\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s45/s49/s48/s49/s50/s51\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s50/s45/s49/s48/s49\n/s91/s98/s93\n/s32/s32\n/s84 /s32/s40/s75/s41/s32/s50/s48/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s32/s52/s48/s32/s107/s79/s101\n/s32/s53/s48/s32/s107/s79/s101\n/s91/s97/s93/s78/s67/s82/s79\n/s91/s99/s93/s32\n/s32/s83/s67/s82/s79\n/s32/s32\n/s32/s49/s48/s32/s107/s79/s101\n/s32/s50/s48/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s32/s53/s48/s32/s107/s79/s101/s32 /s32/s83\n/s77/s32/s40/s74/s107/s103/s45/s49\n/s75/s45/s49\n/s41\n/s32/s32/s50/s48/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s32/s52/s48/s32/s107/s79/s101\n/s32/s53/s48/s32/s107/s79/s101/s80/s67/s82/s79\n/s32/s32\nFIG. 4. (a) to (c) respectively show the Tvariation of ∆ SMof PCRO, NCRO and SCRO at different magnetic fields.\ngate the possibility of a glassy state, particularly those\nwhich lie acrossthe AFM-FI boundary, field-cooled-field-\nstopmemorymeasurementwereperformedon Prand Nd\nsamples28,29. In this protocol, the samples were cooled\nin 100 Oe of field down to 2 K with intermediate stops at\nseveral temperatures below Tc. Subsequently, the sam-\nples were heated back in 100 Oe and dc magnetization\nwas measured. We do not observe any anomaly at the\nstopping temperatures during heating, which rules out\nthe possibility of any glassy (spin glass or cluster glass)\nor super paramagnetic state in PCRO and NCRO sam-\nples.\nIV.2. Electrical transport\nLikemanyotherA 2BB′O6compounds, LCRO,PCRO,\nNCRO and SCRO show semiconducting behaviour as ev-\nident from the transport data depicted in figs. 5 (a) to\n(d) respectively. The values of ρat room temperature\n(∼300 K) are found to be 8.86, 8.81, 4.16 and 6.68 Ω-\ncm for La, Pr, Nd and Sm compounds. Our analysis on\ntheρ(T) data indicates that all three compositions show\nMott Variable Range (VRH) hopping conduction, where\nρ(T)∼exp/bracketleftBig/parenleftbigT0\nT/parenrightbig1\n4/bracketrightBig\n. This is also quite common among\ndisordered double perovskite such as Sr 2MnRuO 630or\nSr2CoSbO 631. The VRH type conduction is quite clearly\nvisible from the log ρversusT−1/4plots in the respec-\ntive insets of figs. 5 (a), (b) and (c). While for La and\nNd compounds, the VRH nature is present almost over\nthe full range of temperature (it deviates only below 65\nK), Pr and Sm compounds show VRH conduction only\nin the range 160 to 60 K. The values of the parameter T0\nassociated with the VRH conduction are found to be 6.9\n×107, 5.1×107, 8.8×107and 6.8×107K for La, Pr, Nd\nand Sm compounds respectively.V. DISCUSSIONS\nIt turns out that the magnetic properties of samples\ncontaining magnetic rare-earth are drastically different\nfrom that of LCRO. Naively, one can relate the FI state\nwith the magnetic moment of A site. However, FI state\nis also observed in Y 2CoRuO 6, where A site contains\nnonmagnetic Y3+ions. In order to address the issue, let\nus first discuss various salient observations made on the\nstudied samples.\n1. The Co-O-Ru bond distortion (and consequently,\nthe octahedral tilt) is found to increase as we move\nfrom LCRO to heavier rare-earth containing com-\npounds. This is due to the reduction of ionic radius\nat the A-site (lanthanide contraction), as we pro-\nceed from La to Sm. It has been alreadymentioned\nthat the smaller radius of A-element leads to larger\nB-O-B′bond distortion2.\n2. LCRO with relatively smaller bond distortion\nshows AFM ground state. On the other hand,\nPCRO, NCRO and SCRO show large increase in\nMbelow the magnetic transition at Tc. Isothermal\nmagnetization curves show large hysteresis with\nthe presenceofsignificant remanentmagnetization.\nIsostructural Y 2CoRuO 6shows similar magnetic\nbehaviour and the Co-O-Ru superexchange inter-\naction is found to be AFM in nature leading to a\nFI ground state (Co and Ru sublattices have differ-\nent moment values)13. In analogy with the Y com-\npound, we can conclude that the ground states of\nPCRO, NCRO and SCRO are ferrimagnetic. Sim-\nilar to Y compound, FI state in PCRO and subse-\nquent compounds emerges possibly due to the Co-\nO-Ru bond distortion, rather than A-site magnetic\nmoment formation.\n3. All four studied compounds are found to be semi-\nconductingwithreasonablyhighresistivity( ∼MΩ-\ncm)aroundthemagneticanomalies. Therefore,the\nmagnetic interaction is likely to be mediated by the7\n/s55/s53 /s49/s53/s48 /s50/s50/s53 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s53/s50/s49/s46/s48/s52\n/s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s52/s56/s49/s50/s55/s53 /s49/s53/s48 /s50/s50/s53 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s51/s51/s48/s46/s54/s54\n/s48/s46/s50/s52 /s48/s46/s51/s48 /s48/s46/s51/s54/s52/s56/s49/s50\n/s55/s53 /s49/s53/s48 /s50/s50/s53 /s51/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s52/s56/s49/s50/s55/s53 /s49/s53/s48 /s50/s50/s53 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s50/s56/s48/s46/s53/s54\n/s48/s46/s50/s52 /s48/s46/s51/s48 /s48/s46/s51/s54/s52/s56/s49/s50\n/s83/s67/s82/s79/s78/s67/s82/s79/s91/s99/s93/s32\n/s32/s32/s32/s108/s110/s32\n/s84 /s32/s45/s49/s47/s52\n/s32/s40/s75/s32/s45/s49/s47/s52\n/s41 /s32/s76/s67/s82/s79/s40 /s45/s99/s109/s41\n/s40 /s41/s40 /s45/s99/s109/s41\n/s84 /s32/s45/s49/s47/s52\n/s32/s40/s75/s32/s45/s49/s47/s52\n/s41 /s32/s108/s110/s32\n/s32/s32\n/s91/s97/s93\n/s32\n/s32\n/s32/s32\n/s91/s100/s93\n/s108/s110/s32\n/s32/s32\n/s32/s32\n/s84 /s32/s45/s49/s47/s52\n/s32/s40/s75/s32/s45/s49/s47/s52\n/s41 /s32\n/s32/s32/s80/s67/s82/s79/s91/s98/s93\n/s84 /s32/s45/s49/s47/s52\n/s32/s40/s75/s32/s45/s49/s47/s52\n/s41 /s32/s108/s110/s32\n/s32/s32\n/s32/s32\nFIG. 5. (a) to (d) respectively show resistivity data as a fun ction of temperature for LCRO, PCRO, NCRO and SCRO. The\ninsets show the VRH-type fittings to the data.\nsuperexchange, rather than the double-exchange\nmechanism.\n4. The coercivity associated with M−Hhysteresis\nloop is found to be much higher in case of SCRO,\nhowever it is lower than the value reported for\nY2CoRuO 6. It appears that the coercivity is not\ndirectly connected to the fact whether A site con-\ntains a magnetic (such as Pr, Nd or Sm) or non-\nmagnetic (here Y) ion.\n5. ∆SMversusTplots for PCRO, NCRO and SCRO\nshow further anomaly at around 8-12 K. It is diffi-\ncult to guess the origin of such anomaly. However,\nconsideringthe presence of rare-earthat the A-site,\nit may signify the low- Tordering of the magnetic\nrare-earth ions.\nAs already mentioned, a transition from AFM to FI\nvia a glassy magnetic state has been observed in sev-\neral double perovskites with the bending of the B-O-\nB′bond8–10,13, where the lattice distortion was cre-\nated by systematic doping at the A site or by ap-\nplying hydrostatic pressure. In the present case, wefound similar effect when one rare-earth ion is replaced\nby another one with smaller ionic radius. In analogy\nwith the idea mooted in case of Sr 2−xCaxFeOsO 6and\nSr2−xCaxCoOsO 68,9, the AFM state in LCRO is due to\nthe strong AFM correlation along the long bonds Co-O-\nRu-O-Co and Ru-O-Co-O-Ru when the Co-O-Ru bond\ndistortion is low. Replacement of La by Pr initiates\nstrong bending in Co-O-Ru bond (see Table 1), which\npossibly strengthen the Co-O-Ru superexchangeover the\nmagnetic interaction on longer Co-O-Ru-O-Co and Ru-\nO-Co-O-Ru pathways leading to FI state. The bending\nfurther enhances in Sm compound, and a significantly\nlarge coercive field is observed.\nThe above argument is also supported by the drastic\nchange in the values of paramagnetic Curie temperature\nθinPrandNdcompounds(-25and-22Krespectively)as\ncompared to the antiferromagnetically ordered La coun-\nterpart. This may be an indication of the weakening\nof the exchange interaction along longer Co-O-Ru-O-Co\nand Ru-O-Co-O-Ru pathways. However, θstill remains\nnegative due to the presence of Co-O-Ru AFM interac-\ntion.\nIn case of SrCaCoOsO 6and La 2−xYxCoRuO 6(x≈8\n0.25 to 1.5) SG states are observed8,13, when Ca(Y) is\ndoped at the Sr(La) site, which has been assigned due to\nthe frustration between long (B-O-B′-O-B) and short(B-\nO-B′) exchangepaths. However, wedo notsee anyglassy\nmagnetic state in neither of the Pr and Nd compounds.\nFor La 1.25Y0.75CoRuO 6with tilt angle 15.5◦, a promi-\nnent frequency dispersion is observed in the ac suscep-\ntibility data. On the other hand PCRO with lower tilt\nangle (14.6◦) shows ordered FI state. Possibly, the emer-\ngence of glassy state in doped samples is also connected\nwith the doping induced disorder. It is to be noted that\nLa1.25Y0.75CoRuO 6sample has huge antisite disorder of\n20%, as compared to 1% antisite disorder in PCRO.\nIt is now pertinent to address the role of 4 fmoment\nfrom the rare-earth present at the A site. In case of\nisostructural Er 2CoMnO 6, rare-earth moment orders at\na relatively lower Tthan the Co-Mn ordering tempera-\nture32. From our magnetization data, it is hard to iden-\ntify the ordering of rare-earth moment. The ∆ SMversus\nTdata depicted in fig.4 show peak like anomalies be-\ntween 8 and 12 K in PCRO, NCRO and SCRO , and\nthey can be probable ordering points of rare-earth mo-\nments. In order to shed more light on this issue, we have\ncompared the moments of A 2CoRuO 6(A = Pr, Nd, Sm\nand Y), where the magnetic data of Y 2CoRuO 6is ob-\ntained from reference13. The moments at 5 K ( M5K), on\napplying 50 kOe of field, is found to be 2.3, 2.9, 1.4 and\n0.8µB/f.u. on the virgin line of the M−Hcurve for Pr,\nNd, Sm and Y compounds. Y does not carry any mo-ment, while the total angular momentum J= 4, 9/2 and\n5/2 for Pr3+, Nd3+and Sm3+states respectively. The\nmagnetic moments of these three ions are 3.58, 3.62 and\n0.84µBrespectively. Clearly, the variation of M5Kcor-\nresponds well with the variation of rare-earth moment.\nThis signifies that the A site rare-earth moment remains\nin ordered state at 5 K. It is important to perform a\nneutron diffraction study to ascertain the true magnetic\nstructure of these compounds.\nIn summary we have studied the structural, magnetic\nas well as transport properties of the double perovskites\nLa2CoRuO 6, Pr2CoRuO 6, Nd2CoRuO 6, Sm2CoRuO 6.\nWe observe a systematic change in the magnetic ground\nstate as La is replaced by Pr, Nd and Sm. This matches\nwell with the case of Fe-Os and Co-Os based double per-\novskites,wherelatticedistortiontunesthestrengthofthe\nmagnetic interactions in different exchange pathways.\nVI. ACKNOWLEDGMENT\nThe work is supported by the financial grant from\nDST-SERB project (EMR/2017/001058). MD would\nlike to thank CSIR, India for her research fellowship,\nwhile PD thanks DST-SERB for his NPDF fellowship\n(PDF/2017/001061).\nREFERENCES\n∗ ∗sspsm2@iacs.res.in\n1Kobayashi K -I, Kimura T, Sawada H, Terakura K and\nTokura Y 1989 Nature395677\n2Vasala S and Karppinen 2015 Prog. Solid State Chem. 43\n1\n3Nag A, Jana S, Middey S and Ray S 2017 Ind. J. Phys. 91\n883\n4Serrate D, Teresa J M De and Ibarra M R 2007 J. Phys.:\nCondens. Matter 19023201\n5SarmaDD,RaySugata, TanakaK,KobayashiM,Fujimori\nA, Sanyal P, Krishnamurthy H R and Dasgupta C 2007\nPhys. Rev. Lett. 98157205\n6Kato H, Okuda T , Okimoto Y, Tomioka Y, Oikawa K,\nKamiyama T and Tokura Y 2002 Phys. Rev. B 65144404\n7Dass R I, Yan J -Q and Goodenough J B 2004 Phy. Rev.\nB69094416\n8Morrow R, Yan Jiaqiang, McGuire Michael A, Freeland\nJohn W, Haskel Daniel and Woodward Patrick M 2015\nPhys. Rev. 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Petrovic1,2\n1Condensed Matter Physics and Materials Science Department ,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n2Department of Physics and Astronomy, Stony Brook Universit y, Stony Brook, New York 11790, USA\n3National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA\n4Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA\n(Dated: August 19, 2020)\nWe carried out a comprehensive study of magnetic critical be havior in single crystals of ternary\nchalcogenide FeCr 2Te4that undergoes a ferrimagnetic transition below Tc∼123 K. Detailed critical\nbehavior analysis and scaled magnetic entropy change indic ate a second-order ferrimagentic transi-\ntion. Critical exponents β= 0.30(1) with Tc= 122.4(5) K,γ= 1.22(1) with Tc= 122.8(1) K, and\nδ= 4.24(2) at Tc∼123 K suggest that the spins approach three-dimensional Isi ng (β= 0.325, γ\n= 1.24, and δ= 4.82) model coupled with the attractive long-range intera ctions between spins that\ndecay as J(r)≈r−4.88. Our results suggest that the ferrimagnetism in FeCr 2Te4is due to itinerant\nferromagnetism among the antiferromagnetically coupled C r-Fe-Cr trimers.\nINTRODUCTION\nTernaryACr 2X4(A =transitionmetal, X=S, Se, and\nTe) exhibit a variety of magnetic and electronic proper-\nties. The family includes metallic CuCr 2X4and semi-\nconducting Hg(Cd)Cr 2Se4ferromagnets [1–3], semicon-\nducting Fe(Mn)Cr 2S4ferrimagnets [4–6], and insulating\nZnCr2S4antiferromagnet [7]. The FeCr 2X4compounds\nshow competing spin-orbit and exchange interactions [8].\nFeCr2S4is ferrimagnetic (FIM) insulator below Tc= 165\nK, shows a crossover transition from insulator to metal\nnearTcand colossal magnetoresistance behavior [9–11].\nFeCr2Se4is an insulating antiferromagnet (AFM) with\nTN= 218 K and ferrimagnetic with a small magnetic\nmoment of 0.007 µBbelow 75 K [12–14]. It should be\nnoted that FeCr 2Se4crystallizes in the Cr 3S4-type mon-\noclinic structure described within C2/mspace group, in\ncontrast to the cubic spinel-type of FeCr 2S4. However,\nFeCr2S4and FeCr 2Se4have similar electronic structure\nwith nearly trivalent Cr3+and divalent Fe2+states [15].\nThe magnetic moments of Cr ions are antiparallel to\nthoseofFeionsinFeCr 2X4, andthereisstronghybridiza-\ntion between Fe 3 d-states and X p-states [15].\nFeCr2Te4has not been studied much presumably due\nto the difficulty in sample preparation [16–18]. Demeaux\net al. first grew the single crystals of FeCr 2Te4[16]. The\ncrystal structure was reported as a defective NiAs-type\nwithinP63/mmcspace group, where Fe and Cr occupy\nthe same site with alloying ratio of 1 : 2 and a net oc-\ncupancy of 0.75 [16]. In contrast, Valiev et al. reported\nthat FeCr 2Te4crystalizes in a CoMo 2S4-type structure\nwithinI2/mspace group, in which Fe and Cr are octa-\nhedrally coordinated by six Te [17]. Recently, a series of\npolycrystals FeCr 2Se4−xTexwere synthesized [18]. Sub-\nstitution with Te gradually suppresses the AFM order of\nFeCr2Se4, and leads to a short range ferromagnetic(FM)\ncluster metallic state in polycrystal FeCr 2Te4[18]. In or-der to study the intrinsic physical property, high quality\nsingle crystal is required.\nIn this work, we successfully fabricated single crystals\nof FeCr 2Te4and performed a comprehensive study of the\nstructural and magnetic properties. Our analysis of crit-\nicality around Tcindicates that FeCr 2Te4displays the\n3D-Ising behavior, with the magnetic exchange distance\ndecaying as J(r)≈r−4.88. Our first principles calcula-\ntions suggest that the ferrimagnetism in FeCr 2Te4stems\nfrom the itinerant ferromagnetism among the antiferro-\nmagnetically coupled Cr-Fe-Cr trimers. Since transition-\nmetal chalcogenides represent model systems for explor-\ning local structure-related relationship between the bro-\nken symmetry and d-orbital magnetism [19], detailed lo-\ncal stucture investigation of this system would be highly\ndesirable and would bring important new insights.\nEXPERIMENTAL DETAILS\nSingle crystals of FeCr 2Te4were fabricated by melting\nstoichiometric mixture of Fe (99.99%, Alfa Aesar) pow-\nder, Cr (99.95%, Alfa Aesar) powder, and Te (99.9999%,\nAlfa Aesar) pieces. The starting materials were vacuum-\nsealed in a quartz tube, heated to 1200◦C over 12\nh, slowly cooled to 900◦C at a slow rate of 1◦C/h,\nand then quenched into iced water. The single crystal\nx-ray diffraction (XRD) data were taken with Cu Kα\n(λ= 0.15418 nm) radiation of a Rigaku Miniflex pow-\nder diffractometer. In order to obtain more comprehen-\nsive crystallographic information, the powder XRD mea-\nsurements were performed at the PDF beamline (28-ID-\n1) at National Synchrotron Light Source II (NSLS II)\nat Brookhaven National Laboratory (BNL) using Perkin\nElmer image plate detector. The setup utilized x-ray\nbeam with a wavelength of 0.1666 ˚A, and the sample to\ndetectordistance of1.25m, ascalibrated usinga Ni stan-2\ndard. Sample was cooled with an Oxford Cryosystems\n700 cryostream using liquid nitrogen. Raw data were\nintegrated and converted to intensity vs. scattering an-\ngle using the software pyFAI [20]. The average structure\nwas assessed from raw powder diffraction data using the\nGeneralStructureAnalysisSystemII(GSAS-II) software\npackage[21]. Theelementalanalysiswasperformedusing\nenergy-dispersive x-ray spectroscopy (EDS) in a JEOL\nLSM-6500 scanning electron microscope (SEM). The x-\nray absorption spectroscopy (XAS) measurements were\nperformed at 8-ID beamline of the NSLS II (BNL) in a\nfluorescencemode. Thex-rayabsorptionnearedgestruc-\nture (XANES) and extended x-ray absorption fine struc-\nture (EXAFS) spectra were processed using the Athena\nsoftware package. The AUTOBK code was used to nor-\nmalize the absorption coefficient, and separate the EX-\nAFS signal, χ(k), from the atom-absorption background.\nThe extracted EXAFS signal, χ(k), wasweighed by k2to\nemphasize the high-energy oscillation and then Fourier-\ntransformed in krange from 2 to 10 ˚A−1to analyze\nthe data in Rspace. The x-ray photoelectron spec-\ntroscopy (XPS) experiment was carried out in an ultra-\nhigh vacuum system with base pressures <5×10−9Torr,\nequipped with a hemispherical electron energy analyzer\n(SPECS, PHOIBOS 100) and a twin anode x-ray source\n(SPECS, XR50). Al Kα(1486 eV) radiation was used at\n13 kV and 30 mA. The angle between the analyzer and\nthe x-ray source was 45◦and photoelectrons were col-\nlectedalongthesamplesurfacenormal. TheXPSspectra\nwereanalyzedanddeconvolutedusingtheCasaXPSsoft-\nware. The dc/ac magnetic susceptibility were measured\nin Quantum Design MPMS-XL5 system. The applied\nfield (Ha) was corrected as H=Ha−NM, whereMis\nthe measured magnetization and Nis the demagnetiza-\ntion factor. The corrected Hwas used for the analysis of\nmagnetic entropy change and critical behavior.\nRESULTS AND DISCUSSIONS\nIn the single-crystal XRD pattern (inset to Fig. 1),\nonly (00 l) peaks were observed. Synchrotron powder\nXRD pattern of pulverized crystal of FeCr 2Te4can be\nwell fitted by using a monoclinic structure with the I2/m\nspace group (Fig. 1), confirming main phase of FeCr 2Te4\nwith less than 4% FeTe impurity. The determined lattice\nparameters at 300 K are a= 6.822(2)˚A,b= 3.938(1)˚A,\nc= 11.983(5)˚A, andβ= 90.00(5)◦, close to the reported\nvalues [17, 18]. No structural transition was observed on\ncooling, showing a compression along caxis with c=\n11.867(5) ˚A and a slight expansion in the abplane with\na= 6.825(1)˚A andb= 3.943(1)˚A down to 105 K (Table\nI).\nThe ratio of elements in the single crystal as deter-\nmined by EDS is Fe : Cr : Te = 0.99(2) : 1.90(2) : 4.0(1)\n[Fig. 2(a)], and it is referred to as FeCr 2Te4through-\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48\n/s50 /s113 /s32/s40/s100/s101/s103/s46/s41/s40/s48/s48/s50/s41\n/s40/s48/s48/s52/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s32/s117/s46/s41\nFIG. 1. (Color online) Synchrotron powder x-ray diffraction\n(XRD) data and structural model refinements. The data are\nshown by (+) and ( ◦), structural model fit at 300 K and\n105 K is shown by red and blue solid line, respectively. The\ndifference curves are given by green solid lines, offset for cl ar-\nity. The vertical tick marks represent Bragg reflections of t he\nI2/mspace group andupto4% of FeTe impurity. Insetshows\nsingle crystal XRDpattern of FeCr 2Te4at room temperature.\nout this paper. The information on the valence states of\nFe, Cr and Te atoms can be obtained from the element\ncore-level XPS spectra [Fig. 2(b)]: Fe2+(2p3/2∼710\neV, 2p1/2∼723 eV), Cr3+(2p3/2∼575 eV, 2 p1/2∼586\neV), Te2−(3d5/2∼572eV, 3 d3/2∼583 eV)]. Figure 2(c)\nshows the normalized Cr and Fe K-edge XANES spectra,\nin which a similar prepeak feature is observed, indicating\nsimilar local atomic environment for Fe and Cr atoms.\nThe prepeak feature for Fe K-edge is somewhat weaker\nthan that of Cr K-edge, suggesting a weaker lattice dis-\ntortion in FeTe 6when compared with CrTe 6. The edge\nfeatures are close to the standard compounds with Cr3+\nand Fe2+oxidation states [22, 23], in line with the XPS\nresult.\nThe local environment of Fe and Cr atoms is revealed\nin the EXAFS spectra of FeCr 2Te4measured at room\ntemperature [Figs. 2(d) and 2(e)]. In a single-scattering\napproximation, the EXAFS can be described by [24]:\nχ(k) =/summationdisplay\niNiS2\n0\nkR2\nifi(k,Ri)e−2Ri\nλe−2k2σ2\nisin[2kRi+δi(k)],\nwhereNiis the number of neighbouring atoms at a dis-\ntanceRifromthephotoabsorbingatom. S2\n0isthepassive\nelectrons reduction factor, fi(k,Ri) is the backscattering\namplitude, λis the photoelectron mean free path, δiis\nthe phase shift, and σ2\niis the correlated Debye-Waller\nfactor measuring the mean square relative displacement\nof the photoabsorber-backscatter pairs. The corrected3\n/s53/s46/s57/s56 /s54/s46/s48/s48 /s55/s46/s49/s48 /s55/s46/s49/s50 /s55/s46/s49/s52/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s53/s54/s48 /s53/s56/s48 /s55/s48/s48 /s55/s50/s48\n/s50 /s51 /s52/s48/s49/s50\n/s50 /s51 /s52/s48/s46/s48/s48/s46/s54/s49/s46/s50\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48/s45/s49/s48/s49\n/s50 /s52 /s54 /s56 /s49/s48/s45/s49/s48/s49\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48\n/s40/s97/s41\n/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32 /s109 /s40/s69/s41\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s107/s101/s86/s41/s67/s114/s32/s75/s45/s101/s100/s103/s101/s40/s99/s41\n/s70/s101/s40/s98/s41\n/s40/s100/s41 /s40/s101/s41/s51/s48\n/s50/s48\n/s49/s48\n/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s66/s105/s110/s100/s105/s110/s103/s32/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s70/s101/s32/s50/s112\n/s51/s47/s50\n/s70/s101/s32/s50/s112\n/s49/s47/s50/s84/s101/s32/s51/s100\n/s53/s47/s50\n/s84/s101/s32/s51/s100\n/s51/s47/s50/s67/s114/s32/s50/s112\n/s51/s47/s50\n/s67/s114/s32/s50/s112\n/s49/s47/s50\n/s231/s99 /s40/s82/s41 /s231 /s32/s40/s97/s46/s117/s46/s41\n/s82/s40/s197/s41/s40/s103/s41\n/s67/s114/s32/s45/s32/s84/s101/s50/s46/s55/s48/s32/s197/s231/s99 /s40/s82/s41 /s231 /s32/s40/s97/s46/s117/s46/s41/s107/s101/s86/s49/s48 /s49/s50 /s49/s52/s49/s54 /s49/s56/s50/s48/s56/s54/s52 /s50/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s32/s117/s46/s41/s53/s48/s54/s48\n/s52/s48\n/s70/s101 /s67/s114/s84/s101/s70/s101\n/s67/s114/s84/s101\n/s82/s40/s197/s41/s40/s102/s41\n/s70/s101/s32/s45/s32/s84/s101/s50/s46/s53/s56/s32/s197\n/s107/s50\n/s99 /s40/s107/s41/s32/s40/s197/s45/s50\n/s41\n/s107/s32/s40/s197/s45/s49\n/s41\n/s107/s50\n/s99 /s40/s107/s41/s32/s40/s197/s45/s50\n/s41\n/s107/s32/s40/s197/s45/s49\n/s41\nFIG. 2. (Color online) (a) Crystal structure and results of\nEDS analysis on FeCr 2Te4. Room-temperature Fe-2p, Cr-2p,\nand Te-3d core-level XPS (b) and normalized Cr and Fe K-\nedge XANES spectra (c). Fe and Cr K-edge EXAFS oscilla-\ntions (d,e) and Fourier transform magnitudes (f,g) of EXAFS\ndata measured at room temperature. The experimental data\nare shown as blue symbols alongside the model fit plotted as\nred line. Corresponding first coordination shell of Fe and Cr\nare shown in the insets.\nmain peak in the Fourier transform magnitudes of Fe K-\nedge EXAFS around R∼2.58˚A is clearly smaller than\nthat of Cr K-edge EXAFS at R∼2.70˚A [Figs. 2(f) and\n2(g)]. The different local Fe-Te and Cr-Te bond lengths\nsuggest that the Fe and Cr atoms might occupy different\ncrystallographic sites, ruling out the possibility of NiAs-\ntype structure with the same Fe/Cr sites. Then we focus\non the first nearest neighbors of Fe and Cr atoms rang-\ning from 1.5 to 3.5 ˚A. The main peak corresponds to two\nFe-Te bond lengths of 2.69 ˚A and 2.75 ˚A, and four dif-\nferent Cr-Te bond lengths with 2.58 ˚A, 2.67˚A, and 2.80\n˚A, 2.86 ˚A, respectively, extracted from the model fits\nwith fixed coordination number CN. The peaks above\n3.25˚A are due to longer Fe-Cr, Fe-Te, and Cr-Te bond\ndistances, and the multiple scattering involving different\nnear neighbours of the Fe/Cr atoms.\nFigure3(a)showsthe temperaturedependenceofmag-TABLE I. Average and local structural parameters extracted\nfrom the powder XRD and the EXAFS spectra of FeCr 2Te4.\nCN is coordination number based on crystallographic value,\nR is interatomic distance, and σ2is Debye Waller factor.\n300 K 105 K\na(˚A) 6.822(2) 6.825(1)\nb(˚A) 3.938(1) 3.943(1)\nc(˚A) 11.983(5) 11.867(5)\nβ(◦) 90.00(5) 90.01(15)\natom site x y z\nFe 2 a 0 0 0\nCr 4 i 0.001(3) 0 0.2541(8)\nTe1 4 i 0.328(4) 0 0.3726(4)\nTe2 4 i 0.339(4) 0 0.8785(4)\nbond CN R ( ˚A) ∆R ( ˚A) σ2(˚A2)\nCr-Te1 2 2.58 0.01 0.001\nCr-Te2 1 2.67 0.01 0.001\nCr-Te3 1 2.80 0.11 0.003\nCr-Te4 2 2.86 0.11 0.003\nFe-Te1 4 2.69 0.36 0.02\nFe-Te2 2 2.75 0.24 0.02\nnetization measured in out-of-plane field µ0H= 0.1 T,\nin which χincreases with decreasing temperature and in-\ncreases abruptly near Tcdue to the paramagnetic (PM)-\nFIM transition. The in-plane χ(T) [inset in Fig. 3(a)]\nis much smaller than that in out-of-plane field, indicat-\ning the presence of large magnetic anisotropy with easy\ncaxis. The average susceptibility χave= (2/3)χab+\n(1/3)χcfrom 150 to 300 K can be fitted by the Curie-\nWeiss law χave=χ0+C/(T−θ) [Fig. 3(c)], which yields\nχ0= 0.87(1) emu/mol-Oe, C= 7.91(6) emu-K/mol-Oe\n[µeff= 7.95(9)µBper formulaunit], and θ= 126.6(2)K.\nThe positive value of θindicates dominance of ferromag-\nnetic or ferrimagnetic exchangeinteractions in FeCr 2Te4.\nFor monoclinic FeCr 2Se4with a similar layered struc-\nture, the individual spins of Fe and Cr ions have AFM\ncoupling along the caxis with the distance of 2.956 ˚A,\nwhile FM coupling along the baxis with the distance of\n3.617˚A [25]. At low temperature, the FM interaction\ndominating over the AFM interaction results in a FIM\nground state. For FeCr 2Te4, the enhanced hybridization\nbetween d-orbital of Fe and Cr with p-orbital of Te plays\nan important role in magnetic coupling. Based on our\nfirst-principle calculation (see below), the FIM structure\nwhere Fe and Cr atoms have opposite spin orientations\nare the most stable state in FeCr 2Te4, when compared\nwith the FM and AFM structures, which needs further\nverificationby neutron scatteringexperiment. The bifur-\ncation between ZFC and FC curves [Fig. 3(a)] might be\ndue to strong magnetic anisotropy and/or multidomain\nstructure, which has also been observed in other long-4\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s45/s53/s46/s48 /s45/s50/s46/s53 /s48/s46/s48 /s50/s46/s53 /s53/s46/s48/s45/s52/s45/s50/s48/s50/s52/s32/s90/s70/s67\n/s32/s70/s67/s32/s40/s101/s109/s117/s47/s109/s111/s108/s45/s79/s101/s41\n/s84/s32/s40/s75/s41/s48/s72/s47/s47/s99/s32/s97/s116/s32/s48/s46/s49/s32/s84/s40/s97/s41\n/s48/s72/s47/s47/s97/s98/s32/s97/s116/s32/s48/s46/s49/s84/s32/s90/s70/s67\n/s32/s70/s67/s32/s40/s101/s109/s117/s47/s109/s111/s108/s45/s79/s101/s41\n/s84/s32/s40/s75/s41\n/s97/s118/s101/s32/s40/s109/s111/s108/s45/s79/s101/s47/s101/s109/s117/s41\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s32/s61/s32/s48/s46/s49/s32/s84/s97/s118/s101/s32/s61/s32 /s32/s67/s47/s40/s84/s45 /s41/s109/s39/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s84/s32/s40/s75/s41/s84\n/s99/s32/s61/s32/s49/s50/s52/s32/s75\n/s72\n/s97/s99/s61/s32/s51/s46/s56/s32/s79/s101/s32\n/s102/s32/s61/s32/s52/s57/s57/s32/s72/s122/s40/s100/s41 /s40/s99/s41\n/s32\n/s48/s72/s47/s47/s97/s98\n/s32\n/s48/s72/s47/s47/s99\n/s32/s32/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s48/s72/s32/s40/s84/s41/s84/s32/s61/s32/s50/s32/s75\nFIG. 3. (Color online) Temperature-dependent dc magnetic\nsusceptibility χ(T) in zero-field cooling (ZFC) and field cool-\ning (FC) modes taken at µ0H= 0.1 T for µ0H/bardblc(a) and\nµ0H/bardblab(inset), respectively. (b) 1 /χ(T) taken at µ0H=\n0.1 T along with Curie-Weiss fit from 150 to 300 K. (c) Field\ndependence of magnetization for FeCr 2Te4measured at T= 2\nK. (d) Ac susceptibility real part m′(T) measured with oscil-\nlating ac field of 3.8 Oe and frequency of 499 Hz. The chosen\nexperimental parameters (field and frequency) allow for wel l\ndefined moment and adequate frequency resolution.\nrange FM single crystals, such as U 2RhSi3[26]. The\nmagnetization loops of FeCr 2Te4for both field directions\natT=2Kconfirmsalargemagneticanisotropyandeasy\ncaxis [Fig. 3(c)]. The sudden jumps around µ0H≈ ±1\nTalongthe caxiscanbeascribedtothemagneticdomain\ncreeping behavior, i.e., the magnetic domain walls jump\nfrom one pinning site to another. Then we estimated the\nRhodes-Wohlfarth ratio (RWR) for FeCr 2Te4, which is\ndefined as Pc/PswithPcobtained from the effective mo-\nmentPc(Pc+2) =P2\neffandPsis the saturation moment\nobtained in the ordered state [27–29]. RWR is 1 for a lo-\ncalized system and is larger in an itinerant system. Here\nwe obtain RWR ≈1.69 for FeCr 2Te4, indicating a weak\nitinerant character. To obtain the accurate Curie tem-\nperature Tc, out-of-plane ac susceptibility was measured\nat oscillating ac field of 3.8 Oe and frequency of 499 Hz.\nThe single sharp peak in the real part m′(T) [Fig. 3(d)]\ngives the Tc= 124 K.\nIn the following we discuss the nature of the PM-FIM\ntransition for FeCr 2Te4. The magnetization isotherms\nalong easy caxis were measured at various temperatures\nin the vicinity of Tc[Fig. 4(a)]. We first considered the/s48 /s49 /s50 /s51 /s52 /s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s50/s52/s54/s56\n/s49/s49/s48 /s49/s50/s48 /s49/s51/s48 /s49/s52/s48 /s49/s53/s48 /s49/s54/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s48/s49/s50/s51/s52/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s109\n/s48/s72/s32/s40/s84/s41/s40/s97/s41\n/s84/s32/s61/s32/s49/s48/s48/s32/s75\n/s84/s32/s61/s32/s49/s53/s48/s32/s75\n/s68 /s84/s32/s61/s32/s50/s32/s75/s109\n/s48/s72/s32/s47/s47/s32/s99\n/s77/s50\n/s32/s40/s49/s48/s32/s101/s109/s117/s47/s103/s41/s50\n/s109\n/s48/s72/s47/s77/s32/s40/s84/s45/s103/s47/s101/s109/s117/s41/s40/s98/s41\n/s49/s53/s48/s32/s75/s49/s48/s48/s32/s75/s45/s68 /s83\n/s77/s32/s40/s74/s47/s107/s103/s45/s75/s41\n/s84/s32/s40/s75/s41/s109\n/s48/s72/s32/s40/s84/s41/s40/s100/s41\n/s53/s46/s48\n/s52/s46/s48\n/s51/s46/s53\n/s49/s46/s50/s49/s46/s54/s50/s46/s48/s50/s46/s53/s51/s46/s48/s52/s46/s53\n/s48/s46/s52/s32/s84/s48/s46/s56/s77/s49/s47/s98\n/s32/s40/s49/s48/s52\n/s32/s40/s101/s109/s117/s47/s103/s41/s49/s47 /s98\n/s41\n/s40/s109\n/s48/s72/s47/s77/s41/s49/s47 /s103\n/s32/s40/s84/s45/s103/s47/s101/s109/s117/s41/s49/s47 /s103/s40/s99/s41\n/s49/s48/s48/s32/s75\n/s49/s53/s48/s32/s75/s98 /s32/s61/s32/s48/s46/s51/s50\n/s103 /s32/s61/s32/s49/s46/s50/s49\nFIG. 4. (Color online) (a) Typical initial isothermal magne -\ntization curves from 100 K to 150 K with a temperature step\nof 2 K for FeCr 2Te4single crystal. (b) Arrott plot ( β= 0.5,\nγ= 1) and (c) the modified Arrott plot with the optimum ex-\nponents β= 0.32 andγ= 1.21. (d) Temperature-dependent\nmagnetic entropy change −∆SM(T) at various fields change.\nwell-known Arrott plot [30]. From the Landau theory,\nthe Arrott plot of M2vsH/Mshould appear as paral-\nlel straight lines above and below Tc, and the line passes\nthrough the origin at Tc. It is clear that the mean field\ncritical exponent does not work for FeCr 2Te4, as illus-\ntrated by the set of curved lines shown in Fig. 4(b).\nFor a second-order phase transition, the spontaneous\nmagnetization MsbelowTc, the inverse initial suscepti-\nbilityχ−1\n0aboveTc, and the field-dependent magnetiza-\ntionM(H) atTcare [31–33]:\nMs(T) =M0(−ε)β,ε <0,T < T c, (1)\nχ−1\n0(T) = (h0/m0)εγ,ε >0,T > T c,(2)\nM=DH1/δ,T=Tc, (3)\nwhereε= (T−Tc)/Tcis the reduced temperature, and\nM0,h0/m0andDare the critical amplitudes. In a\nmore general case, the modified Arrott plot ( H/M)1/γ=\naε+bM1/βwith self-consistent method was considered\n[34, 35]. Figure 4(c) presents the final modified Arrott\nplot ofM1/βvs (H/M)1/γwithβ= 0.32 andγ= 1.21,\nshowing a set of quasi-parallel lines at high field region.\nThen we extracted χ−1\n0(T) andMs(T) as the intercepts5\n/s50 /s51 /s52 /s53/s49/s54/s50/s52\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s45/s56 /s45/s52 /s48 /s52 /s56/s48/s49/s50/s51/s52/s49/s48/s56 /s49/s49/s52 /s49/s50/s48 /s49/s50/s54 /s49/s51/s50 /s49/s51/s56/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s84/s32/s40/s75/s41/s77\n/s115/s32/s40/s101/s109/s117/s47/s103/s41/s84\n/s99/s61/s32/s49/s50/s50/s46/s57/s40/s51/s41/s32/s75\n/s32/s61/s32/s48/s46/s51/s51/s40/s50/s41\n/s84\n/s99/s61/s32/s49/s50/s50/s46/s55/s40/s49/s41/s32/s75\n/s32/s61/s32/s49/s46/s50/s48/s40/s49/s41/s40/s97/s41\n/s48/s50/s52/s54/s56/s49/s48\n/s40/s49/s48/s45/s50\n/s32/s84/s45/s103/s47/s101/s109/s117/s41\n/s49/s48/s56 /s49/s49/s52 /s49/s50/s48 /s49/s50/s54 /s49/s51/s50 /s49/s51/s56/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48\n/s84/s32/s40/s75/s41/s77\n/s115/s40/s100/s77\n/s115/s47/s100/s84/s41/s45/s49\n/s32/s40/s75/s41/s84\n/s99/s61/s32/s49/s50/s50/s46/s52/s40/s53/s41/s32/s75\n/s32/s61/s32/s48/s46/s51/s48/s40/s49/s41\n/s84\n/s99/s61/s32/s49/s50/s50/s46/s56/s40/s49/s41/s32/s75\n/s32/s61/s32/s49/s46/s50/s50/s40/s49/s41/s40/s98/s41\n/s48/s51/s54/s57/s49/s50\n/s40 /s100/s84/s41/s45/s49\n/s32\n/s40/s75/s41/s32/s61/s32/s52/s46/s56/s51/s40/s54/s41/s77/s32/s40 /s101/s109/s117/s32/s103/s45/s49\n/s41\n/s48/s72/s32/s40/s84/s41/s84\n/s99/s32/s61/s32/s49/s50/s51/s32/s75\n/s77 /s40/s101/s109/s117/s32/s103/s45/s49\n/s41\n/s48/s72 /s40 /s41/s84/s32/s60/s32/s84\n/s99\n/s84/s32/s62/s32/s84\n/s99/s40/s99/s41/s77/s40\n/s48/s72/s41/s45/s49/s47\n/s49/s48/s45/s51\n/s40\n/s48/s72/s41/s45/s84\n/s99/s32/s61/s32/s49/s50/s51/s32/s75/s40/s100/s41\nFIG. 5. (Color online) (a) Temperature dependence of the\nspontaneous magnetization Ms(left) and the inverse initial\nsusceptibility χ−1\n0(right)withsolid fittingcurves. Insetshows\nlogMvs log(µ0H) collected at Tc= 123 K with linear fitting\ncurve. (b) Kouvel-Fisher plots of Ms(dMs/dT)−1(left axis)\nandχ−1\n0(dχ−1\n0/dT)−1(right axis) with solid fitting curves.\n(c) Scaled magnetization mvs scaled field hbelow and above\nTcfor FeCr 2Te4. (d) The rescaling of the M(µ0H) curves by\nM(µ0H)−1/δvsε(µ0H)−1/(βδ).\non theH/Maxis and the positive M2axis, respectively.\nThe magnetic entropy change can be estimated using the\nMaxwell’s relation [36]:\n∆SM(T,H) =/integraldisplayH\n0/bracketleftbigg∂M(T,H)\n∂T/bracketrightbigg\nHdH. (4)\nFigure 4(d) presents the calculated −∆SMas a function\noftemperature. The −∆SMshowsabroadpeakcentered\nnearTcand the peak value monotonically increases with\nincreasing field. The maximum value of −∆SMreaches\n1.92 J kg−1K−1with a field change of 5 T. There is a\nslight shift of −∆SMpeak towards higher temperature\nwith increasing field, which also excludes the mean field\nmodel [37].\nFigure 5(a) presents the extracted Ms(T) andχ−1\n0(T)\nas a function of temperature. According to Eqs. (1) and\n(2), thecriticalexponents β= 0.33(2)with Tc= 122.9(3)\nK, andγ= 1.20(1) with Tc= 122.7(1) K, are obtained.\nThe exponent βdescribes the rapid increase of the or-\nder parameter below Tc. The exponent γdescribes how\nmagnetic susceptibility diverges at Tc. Here the obtained\nresult describes critical behavior of the net spontaneous\nmagnetization that arises in ferrimagnet. In the Kouvel-\nFisher (KF) relation [38]:\nMs(T)[dMs(T)/dT]−1= (T−Tc)/β, (5)χ−1\n0(T)[dχ−1\n0(T)/dT]−1= (T−Tc)/γ. (6)\nLinear fittings to the plots of Ms(T)[dMs(T)/dT]−1and\nχ−1\n0(T)[dχ−1\n0(T)/dT]−1in Fig. 5(b) yield β= 0.30(1)\nwithTc= 122.4(5) K, and γ= 1.22(1) with Tc=\n122.8(1) K. The third exponent δcan be calculated from\nthe Widom scaling relation δ= 1+γ/β[39]. From βand\nγobtainedwith the modified Arrottplotand the Kouvel-\nFisher plot, δ= 4.6(2) and 5.1(1) are obtained, respec-\ntively, which are close to the direct fit of δ= 4.83(6) tak-\ning into account that M=DH1/δatTc= 123K [inset in\nFig. 5(a)]. The obtained critical exponents of FeCr 2Te4\nare very close to the theoretically predicted values of 3D\nIsing model ( β= 0.325, γ= 1.24, and δ= 4.82) (Table\nII).\nScaling analysis can be used to estimate the reliability\nof the obtained critical exponents and Tc. The magnetic\nequation of state in the critical region is expressed as\nM(H,ε) =εβf±(H/εβ+γ), (7)\nwheref+forT > T candf−forT < T c, respectively,\nare the regular functions. Eq. (7) can be further written\nin terms of scaled magnetization m≡ε−βM(H,ε) and\nscaled field h≡ε−(β+γ)Hasm=f±(h). This suggests\nthat for true scaling relations and the right choice of β,\nγ, andδ, scaled mandhwill fall on universal curves\naboveTcand below Tc, respectively. As shown in Fig.\n5(c), allthe datacollapseon twoseparatebranchesbelow\nand above Tc, respectively. The scaling equation of state\ntakes another form,\nH\nMδ=k/parenleftBigε\nH1/β/parenrightBig\n, (8)\nwherek(x) is the scaling function. From the above equa-\ntion, all the data should also fall into a single curve.\nThis is indeed seen [Fig. 5(d)]; the M(µ0H)−1/δvs\nε(µ0H)−1/(βδ)experimental data collapse into a single\ncurve and the Tclocates at the zero point of the horizon-\ntal axis. The well-rescaled curves confirm the reliability\nof the obtained critical exponents and Tc.\nFurthermore, it is important to discuss the nature as\nwell as the range of magnetic interaction in FeCr 2Te4. In\na homogeneous magnet the universality class of the mag-\nnetic phase transition depends on the exchange distance\nJ(r). In renormalization group theory analysis the inter-\naction decays with distance rasJ(r)≈r−(3+σ), where\nσis a positive constant [41]. The susceptibility exponent\nγis:\nγ= 1+4\nd/parenleftbiggn+2\nn+8/parenrightbigg\n∆σ+8(n+2)(n−4)\nd2(n+8)2\n×/bracketleftBigg\n1+2G(d\n2)(7n+20)\n(n−4)(n+8)/bracketrightBigg\n∆σ2,(9)\nwhere ∆σ= (σ−d\n2) andG(d\n2) = 3−1\n4(d\n2)2,nis the spin\ndimensionality [42]. When σ >2, the Heisenberg model6\nTABLE II. Comparison of critical exponents of FeCr 2Te4with different theoretical models.\nReference Technique Tc− Tc+ β γ δ\nFeCr2Te4 This work Modified Arrott plot 122.9(3) 122.7(1) 0.33(2) 1.2 0(1) 4.6(2)\nThis work Kouvel-Fisher plot 122.4(5) 122.8(1) 0.30(1) 1.2 2(1) 5.1(1)\nThis work Critical isotherm 4.83(6)\n3D Heisenberg 28 Theory 0.365 1.386 4.8\n3D XY 28 Theory 0.345 1.316 4.81\n3D Ising 28 Theory 0.325 1.24 4.82\nTricritical mean field 36 Theory 0.25 1.0 5.0\nTABLE III. The first-principles total energy (in meV) per for -\nmula unit of different magnetic patterns as shown in Fig. 6(a)\nusing the 300 K and 105 K experimental structures. In the\nE-AF-2 pattern, a Cr atom is antiferromagnetically and fer-\nromagnetically aligned with the closest and farthermost of its\nsix nearest Cr neighbors, respectively; the opposite holds in\nthe E-AF-1 pattern. FM means the ferromagnetic configura-\ntion.\nStructure Ferri FM C-AF E-AF-1 E-AF-2\n300 K 8 90 197 31 82\n105 K 0 56 171 72 32\nis valid forthe 3Disotropicmagnet, where J(r) decreases\nfaster than r−5. Whenσ≤3/2, the mean-field model is\nsatisfied, expectingthat J(r) decreasesslowerthan r−4.5.\nFor the 3D-Ising model with d= 3 and n= 1,σ= 1.88\nis obtained, leading to spin interactions J(r) decaying as\nJ(r)≈r−4.88. This calculation suggests that the spin\ninteraction in FeCr 2Te4is close to the 3D Ising localized-\ntype coupled with a long-range ( σ= 1.88) interaction,\nin line with its weak itinerant character. Meanwhile, the\ncorrelation length ( ξ) correlates with the critical expo-\nnentν(ν=γ/σ), where ξ=ξ0[(T−Tc)/Tc]−ν. It gives\nthatν= 0.64(1) and α= 0.08 (α= 2−νd).\nTo get further insight into the magnetism, we per-\nformed first-principles calculations using density func-\ntion theory. We applied the WIEN2K implementation\n[43] of the full potential linearized augmented plane-wave\nmethod in generalized-gradient approximation using the\nPBEsolfunctional [44]. The basis size wasdetermined by\nRmtKmax= 7 and the Brillouin zone was sampled with\n115 irreducible kpoints to achieve energy convergence of\n1 meV. As shown in Table III, we found that for the ex-\nperimental structure refined at 105 K, the ferrimagnetic\nstate where the Cr and Fe atoms have opposite spin ori-\nentationsis56meVperformulaunitlowerintotalenergy\nthan the ferromagnetic phase and 32 meV lowered than\nthe most stable antiferromagnetic structure (i.e., E-AF-\n2) as observed in FeCr 2Se4[45]. A similar trend of the\nresults holds for the the experimental structure refined\nat 300 K (Table III). A weak easy c-axis anisotropy was\nobtained by inclusion of spin-orbit coupling in the calcu-\nFIG. 6. (Color online) (a) The antiferromagnetic structure s\nused in the first-principles total energy calculations. Ato m-\nresolved density of states (b) in the nonmagnetic state and\n(c) in the ferrimagnetic state where the Cr and Fe atoms\nhave opposite spin orientations, which are shown in the rati o\nof two Cr ions to one Fe ion.\nlations, namely the total energy per formula unit is lower\nby less than 1 meV for the magnetizationalong the caxis\nthan along the aorbaxis. Thus, other sources of mag-\nnetic anisotropy such as dipole-dipole interaction are im-\nportant. The calculated atom-resolved density of states\n(DOS) is shown in Figs. 6(b) and 6(c). For the nonmag-\nnetic case, the Cr-derived DOS has a sharp peak at the\nFermi level [Figs. 6(b)], suggesting a strong Stoner insta-\nbility that yields an itinerant ferromagnetism in the Cr\nlayers. This yields the splitting of about 2.8 eV between\nthe spin-majority and spin-minority bands of Cr charac-7\nter with the dramatic reduction of the Cr-derived DOS\nat the Fermi level [Fig. 6(c)], indicative of the localized\nspin picturefor the Cratoms. Whereas, the nonmagnetic\nFe-derived DOS peaks at about 0 .6 eV below the Fermi\nlevel [Fig. 6(b)] and experiences little reduction at the\nFermi level upon entering the magnetic phase [Fig. 6(c)].\nWe infer that the magnetism of the Fe ions is established\nvia antiferromagnetic superexchange with the neighbor-\ning two Cr ions. The Cr magnetic moment within the\natomic Muffin tins is about 2.88 µB, which is close to\nthe nominal Cr3+S= 3/2 state. With the octahedral\ncoordination, the splitting of the five 3 dorbitals between\nthe high-lying egand low-lying t2gorbitals is substantial.\nTheS= 3/2 state of the Cr3+ion (i.e., 3 d3ort3\n2ge0\ngelec-\ntronconfiguration)meansthattheCr t2gorbitalsarehalf\nfilled, rendering a vanishing orbital angular moment and\na negligible spin-orbit coupling effect. The Fe magnetic\nmoment is about 2.80 µB, significantly deviated from the\nnominal high-spin Fe2+S= 2 state, which indicates its\ndual characters with both localized spins and itinerant\nelectrons. This reveals an interesting interplay of Cr and\nFeelectronicstates,whichallowsthespin-majoritybands\nof the system at the Fermi level to be of Fe character\nrather than Cr character [Fig. 6(c)]. We thus picture\nthe FIM in FeCr 2Te4as itinerant ferromagnetism among\nthe antiferromagnetically coupled Cr-Fe-Cr trimers. The\ntrimers centered at the Fe sites form a body-centered\northorhombic lattice of magnetic dipoles with effective\nmoment of 2 µBS= 4µB[Fig. 3(c)]. In addition to the\neffective Heisenberg exchange interaction, the ith trimer\nis coupled with its ten neighboring trimers [Fig. 6(a)]\nvia dipole-dipole interaction ∝ −(Si·rij)(Sj·rij)/|rij|5,\nwhich tends to align the magnetic moments along the\nbond direction rij=ri−rjwherejdenotes one of the\nneighboring trimers and riis the spatial vector of the ith\ntrimer. Since the body-centered orthorhombic structure\nof the trimers is substantially elongated along the caxis,\neasyc-axis magnetic anisotropy has the overall minimum\ndeviation from the neighboring bond directions. We thus\npredict that the 3D Ising-like ferrimagnetism is sensitive\nto changes in the lattice structure, especially the tilting\nof the Cr-Fe-Cr trimers, which will be verified by future\npressure experiments and computer simulations.\nCONCLUSIONS\nIn summary, we systematically investigated structural\nand magnetic properties of stoichiometric FeCr 2Te4that\ncrystallizes in the I2/mspace group. The second-order\nPM-FIM transition is observed at Tc∼123 K. The\ncritical exponents β,γ, andδestimated from various\ntechniques match reasonably well and follow the scaling\nequation. The analysis of critical behavior suggests that\nFeCr2Te4isa3D-Isingsystemdisplayingalong-rangeex-\nchange interaction with the exchange distance decayingasJ(r)≈r−4.88. Combined experimental and theoret-\nical analysis attributes the ferrimagnetism in FeCr 2Te4\nto itinerant ferromagnetism among the antiferromagneti-\ncallycoupledCr-Fe-Crtrimers. Follow-upstudiesoflocal\natomic structure and magnetism using x-rayand neutron\nscatteringaswellashigh-pressuremethodswillbeofpar-\nticular interest for more comprehensive understanding of\nthis system.\nACKNOWLEDGEMENTS\nWork at BNL is supported by the Office of Basic En-\nergy Sciences, Materials Sciences and Engineering Divi-\nsion, U.S. Department of Energy (DOE) under Contract\nNo. DE-SC0012704. This research used the 28-ID-1 and\n8-ID beamlines of the NSLS II, a U.S. DOE Office of Sci-\nence User Facility operated for the DOE Office of Science\nby BNL under Contract No. DE-SC0012704. This re-\nsearch used resources of the Center for Functional Nano-\nmaterials (CFN), which is a U.S. DOE Office of Science\nFacility, at BNL under Contract No. DE-SC0012704.\n[1] T. Kanomata, H. Ido, and T. Kaneko, J. Phys. Soc. Jpn.\n29, 332 (1970).\n[2] H. W. Lehmann, and F. P. Emmenegger, Solid State\nCommun. 7, 965 (1969).\n[3] N. Menyuk, K. Dwight, and R. J. Arnott, J. Appl. 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B 76, 4176 (1975)."    },    {        "title": "1712.05622v1.Effect_of_the_Canting_of_Local_Anisotropy_Axes_on_Ground_State_Properties_of_a_Ferrimagnetic_Chain_with_Regularly_Alternating_Ising_and_Heisenberg_Spins.pdf",        "content": "arXiv:1712.05622v1  [cond-mat.str-el]  15 Dec 2017Vol.XXX (201X) CSMAG‘16 No.X\nEffect of the Canting of Local Anisotropy Axes on Ground-Stat e Properties of a\nFerrimagnetic Chain with Regularly Alternating Ising and H eisenberg Spins\nJ. Torrico,1,∗M.L. Lyra,1O. Rojas,2S.M. de Souza,2and J. Streˇ cka3\n1Instituto de F´ ısica, Universidade Federal de Alagoas, 570 72-970 Maceio, AL, Brazil\n2Departamento de F´ ısica, Universidade Federal de Lavras, 3 7200-000, Lavras-MG\n3Institute of Physics, Faculty of Science, P. J. ˇSaf´ arik University, Park Angelinum 9, 040 01 Koˇ sice, Slov akia\nThe effect of the canting of local anisotropy axes on the groun d-state phase diagram and mag-\nnetization of a ferrimagnetic chain with regularly alterna ting Ising and Heisenberg spins is exactly\nexamined in an arbitrarily oriented magnetic field. It is sho wn that individual contributions of Ising\nand Heisenberg spins to the total magnetization basically d epend on the spatial orientation of the\nmagnetic field and the canting angle between two different loc al anisotropy axes of the Ising spins.\nPACS numbers: 75.10.Pq ; 75.10.Kt ; 75.30.Kz ; 75.40.Cx ; 75. 60.Ej\nIntroduction\nIn spite of a certain over-simplification, a few\nexactly solved Ising-Heisenberg models capture es-\nsential magnetic features of some real polymeric\ncoordination compounds as for instance Cu(3-\nClpy)2(N3)2[1], [(CuL) 2Dy][Mo(CN) 8] [2, 3] and\n[Fe(H2O)(L)][Nb(CN) 8][Fe(L)] [4]. The rigorous solu-\ntions for the Ising-Heisenberg models thus afford an\nexcellent playground for experimental testing of a lot\nof intriguing magnetic properties such as quantized\nmagnetization plateaus, anomalous thermodynamics,\nenhanced magnetocaloric effect, etc. [1–4]\nRecently, it has been verified that the bimetallic coor-\ndination polymer Dy(NO 3)(DMSO) 2Cu(opba)(DMSO) 2\n(to be further abbreviated as DyCu) can be satisfactorily\ndescribed by the spin-1/2 Ising-Heisenberg chain with\nregularly alternating Ising and Heisenberg spins, which\ncapture the magnetic behavior of Dy3+and Cu2+mag-\nnetic ions, respectively [5]. However, a closer inspection\nof available structural data reveals two crystallographi-\ncally inequivalent orientantions of coordination polyhe-\ndra of Dy3+magnetic ions, which regularly alternate\nalong the DyCu chain [6]. Motivated by this fact, we\nwill investigate in the present work the effect of the cant-\ning between two different local anisotropy axes on the\nground-state properties of the spin-1/2 Ising-Heisenberg\nchain with regularly alternating Ising and Heisenberg\nspins in an arbitrarily oriented magnetic field.\nModel and its Hamiltonian\nLet us introduce the spin-1/2 Ising-Heisenberg chain\nschematically illustrated in Fig. 1, in which the Ising\n∗corresponding author; e-mail: jordanatorrico@gmail.comspins with two different local anisotropy axes z1andz2\nregularly alternate with the Heisenberg spins. The local\nanisotropy axis z1(z2) of the Ising spins σ= 1/2 on odd\n(even) lattice positions is canted by the angle α(-α) from\nthe global frame z-axis. Hence, it follows that the angle\n2αdetermines the overall canting between two coplanar\nlocal anisotropy axes z1andz2. The Heisenberg spins\nS= 1/2 are coupled to their nearest-neighbor Ising spins\nthrough the antiferromagnetic coupling J <0 projected\nintothe respectiveanisotropyaxis. Furthermore, we take\ninto account the effect of the external magnetic field B,\nwhose spatial orientation is given by the angle θdeter-\nmining its tilting from the global frame z-axis. Under\nthese circumstances, the spin-1/2 Ising-Heisenberg chain\ncan be defined through the following Hamiltonian\nH=−JN/2/summationdisplay\ni=1(Sz1\n2i−1σz1\n2i−1+Sz2\n2i−1σz2\n2i+Sz2\n2iσz2\n2i+Sz1\n2iσz1\n2i+1)\n−hz1N/2/summationdisplay\ni=1σz1\n2i−1−hz2N/2/summationdisplay\ni=1σz2\n2i\n−hzN/summationdisplay\ni=1Sz\ni−hxN/summationdisplay\ni=1Sx\ni, (1)\nwherehz1=gz1\n1µBBcos(α−θ) andhz2=gz2\n1µBBcos(α+\nθ) determine projections of the external magnetic field B\ntowards the anisotropy axes of the Ising spins on odd\nand even lattice positions, respectively, gz1\n1andgz2\n1are\nthe respective Land´ e g-factors of the Ising spins and µB\nis the Bohr magneton. Similarly, hz=gz\n2µBBcosθand\nhx=gx\n2µBBsinθdetermine two orthogonal projections\nof the external magnetic field for the Heisenberg spins,\nwhereasgz\n2andgx\n2are the respective spatial components\nof the Land´ e g-factors of the Heisenberg spins.\nThetotalHamiltonianofthespin-1/2Ising-Heisenberg\nchaincanberewrittenasthesumofthecellHamiltonians\nH=N/2/summationdisplay\ni=1(H2i−1+H2i), (2)Ferrimagnetic Chain of Alternating Ising and Heisenberg Sp ins 2\nz z z z z\nxyz α α αz1 z1 z1 z2z2/vectorBθ\nσ2i−1σ2iS2i S2i−1\nSi= 1 /2 Heisenberg spins σi= 1 /2 Ising spins α′=−αα′α′\nFIG. 1: (Color online) Schematic representation of a spin\nchain with regularly alternating Ising and Heisenberg spin s.\nThe angle α(-α) determines the canting of the local\nanisotropy axis z1(z2) from the global frame z-axis for odd\n(even) Ising spins so that 2 αis the canting angle between two\ncoplanar anisotropy axes. The angle θdetermines the tilting\nof the magnetic field from the global frame z-axis.\neach of which involves all the interaction and field terms\nof exactly one Heisenberg spin\nH2i−1=−hz1\n2σz1\n2i−1−hz2\n2σz2\n2i−hz\n2i−1Sz\n2i−1−hx\n2i−1Sx\n2i−1,\nH2i=−hz2\n2σz2\n2i−hz1\n2σz1\n2i+1−hz\n2iSz\n2i−hx\n2iSx\n2i.(3)\nIn above, we have introduced the following notation for\nthe effective longitudinal and transverse fields acting on\nthe Heisenberg spins\nhz\n2i−1=Jcosα/parenleftbig\nσz1\n2i−1+σz2\n2i/parenrightbig\n+gz\n2µBBcosθ,\nhz\n2i=Jcosα/parenleftbig\nσz2\n2i+σz1\n2i+1/parenrightbig\n+gz\n2µBBcosθ,\nhx\n2i−1=Jsinα/parenleftbig\nσz1\n2i−1−σz2\n2i/parenrightbig\n+gx\n2µBBsinθ,\nhx\n2i=−Jsinα/parenleftbig\nσz2\n2i−σz1\n2i+1/parenrightbig\n+gx\n2µBBsinθ.(4)\nIt is noteworthy that the cell Hamiltonians (3) commute\nand hence, they can be diagonalized independently of\neach other by performing a local spin-rotation transfor-\nmation following the approach worked out previously [5].\nIn this way, one obtains the full spectrum of the eigenval-\nues, which can be subsequently utilized for the construc-\ntion of the ground-state phase diagram and magnetiza-\ntionprocess. Thefulldetailsofthiscalculationprocedure\nwill be published elsewheretogetherwith a morecompre-\nhensive analysis of the thermodynamic properties.\nResults and discussion\nLet us illustrate a few typical ground-state phase di-\nagrams and zero-temperature magnetization curves for\nthe most interesting particular case with the antiferro-\nmagnetic coupling J <0, equal Land´ e g-factors of the\nIsing spins gz1\n1=gz2\n1= 20 and equal components of the\nLand´ e g-factor of the Heisenberg spins gx\n2=gz\n2= 2,\nwhich nearly coincide with usual values of gyromagnetic\nratioforDy3+andCu2+magneticions, respectively. Un-\nder these circumstances, one finds four different ground\nstates: two ground states CIF 1and CIF 2with the cantedferromagnetic alignment of the Ising spins\n|CIF1/angbracketright=N/2/productdisplay\ni=1| ր/angbracketright2i−1|ψ/angbracketright2i−1| տ/angbracketright2i|ψ/angbracketright2i,(5)\n|CIF2/angbracketright=N/2/productdisplay\ni=1| ւ/angbracketright2i−1|ψ/angbracketright2i−1| ց/angbracketright2i|ψ/angbracketright2i,(6)\nand two ground states CIA 1and CIA 2with the canted\nantiferromagnetic alignment of the Ising spins\n|CIA1/angbracketright=N/2/productdisplay\ni=1| ր/angbracketright2i−1|ψ/angbracketright2i−1| ց/angbracketright2i|ψ/angbracketright2i,(7)\n|CIA2/angbracketright=N/2/productdisplay\ni=1| ւ/angbracketright2i−1|ψ/angbracketright2i−1| տ/angbracketright2i|ψ/angbracketright2i.(8)\nIt is noteworthythat the state vector | ր/angbracketright2i−1(| ւ/angbracketright2i−1)\ncorrespondstothespinstate σz1\n2i−1= 1/2(σz1\n2i−1=−1/2)\noftheodd-siteIsingspins, thestatevector | տ/angbracketright2i(| ց/angbracketright2i)\ncorresponds to the spin state σz2\n2i= 1/2 (σz2\n2i=−1/2)\nof the even-site Ising spins, while each Heisenberg spin\nunderlies a quantum superposition of both spin states\n|ψ/angbracketrighti=1/radicalbig\na2\ni+1(| ↓/angbracketrighti−ai| ↑/angbracketrighti), (9)\nwhich depends on the orientation of its two nearest-\nneighbor Ising spins via ai=hx\ni/[hz\ni−/radicalbig\n(hz\ni)2+(hx\ni)2].\nFIG. 2: (Color online) The ground-state phase diagram in\npolar coordinates for two different canting angles between t he\nlocal anisotropy axes: (a) 2 α=π/6; (b) 2α=π/4. The\nrelative size of the magnetic field µBB/|J|is represented by\nthe radius of the polar coordinates and the angle θdetermines\nits inclination with respect to the global frame z-axis.\nThe overall ground-state phase diagram in polar co-\nordinates is illustrated in Fig. 2 for two different canting\nangles 2αbetween both local anisotropyaxes. The phase\ndiagram has an obvious symmetry with respect to θ= 0\nandπ/2 axes, the former symmetry axis θ= 0 merely\ninterchanges CIA 1↔CIA2, while the latter symmetry\naxisθ=π/2 is responsible for CIF 1↔CIF2interchange.\nWithout loss of generality, our further discussion will be\ntherefore restricted just to the first quadrant θ∈[0,π/2].\nThe coexistence line between CIF 1and CIA 1phases3 Ferrimagnetic Chain of Alternating Ising and Heisenberg Sp ins\nis macroscopically degenerate with the residual entropy\nper Ising-Heisenberg pair S=kBln(2)/2, whereas CIF 1\nand CIF 2phases coexist together at θ=π/2 up to a\ntriple point (diamond symbol) with the residual entropy\nS=kBln[(√\n5 + 3)/2]/2. In addition, the Heisenberg\nspinsarecompletelyfreetoflip atmacroscopicallydegen-\nerate points (blue circles) with the residual entropy S=\nkBln(2) given by the coordinates B=|J|cos(α)/(2µB),\nθ= 0 andB=|J|sin(α)/(2µB),θ=π/2 forα>∼π/9.\nThese highly degenerate points correspond to a novel-\ntype spin frustration ’half ice, half fire’ [7], which origi-\nnates from the difference between Land´ e g-factors being\nresponsible for a fully frozen (ordered) character of the\nIsing spins and fully floppy (disordered) character of the\nHeisenberg spins.\nFinally, the individual contributions of the Ising and\nHeisenberg spins to the total magnetization are depicted\nin Fig. 3 as a function of the magnetic-field strength for\nseveral spatial orientations θof the applied field. As one\ncan see, any deviation of the magnetic field from its lon-\ngitudinal direction θ= 0 destroys the sharp stepwise de-\npendence in the longitudinalprojectionofthe magnetiza-\ntionmz\n2of the Heisenberg spins. In fact, the longitudinal\ncomponent mz\n2is gradually smeared out upon increasing\nofthe tiltingangle θ, whileits transversepart mx\n2risesup\nto its global maximum successively followed by a gradual\ndecline [cf. Fig. 3(a)-(b)]. The most notabledependences\nofthemagnetizationoftheHeisenbergspinscanbefound\nfor greater tilting angles θof the magnetic field, which\ncause an abrupt jump in both components mx\n2andmz\n2\nof the Heisenberg spins due to a sudden reorientation of\nIsing spins at the phase transition from the canted fer-\nromagnetic phase CIF 1to the canted antiferromagnetic\nphase CIA 1(see the curves for θ= 2π/5). The sharp\nstepwise dependence of the magnetization of the Heisen-\nberg spins is afterwards recovered for the special case of\nthe transverse field θ=π/2, for which it appears in the\ntransverse projection mx\n2while its longitudinal part mz\n2\nequals zero. The coexistence of the canted ferromagnetic\nphases CIF 1and CIF 2is manifested through a quasi-\nlinear dependence of mx\n2at low magnetic fields, where\nother contributions mz\n2=mz1\n1=mz2\n1= 0 vanish.\nConclusion\nIn the present work we have examined in detail the\nground-state phase diagram and zero-temperature mag-\nnetization process of the spin-1/2 Ising-Heisenberg chain\nwith two different local anisotropy axes in an arbitrar-\nily oriented magnetic field. It has been shown that the\nphase diagram involvesin total two canted ferromagnetic\nand two canted antiferromagnetic ground states. An-\nother interesting finding concerns with the existence of a\nfewmacroscopicallydegeneratepoints, atwhichaperfect\norder of the Ising spins accompanies a complete disorder/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48\n/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s48/s46/s53/s48/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s109/s120 /s50/s32/s32/s91/s103/s120 /s50\n/s66/s93\n/s32/s32\n/s32/s32 /s32\n/s32 /s32\n/s32\n/s40/s98/s41\n/s32/s32\n/s32/s109/s122 /s50/s32/s32/s91/s103/s122 /s50\n/s66/s93\n/s40/s99/s41\n/s32/s32/s109/s122\n/s49\n/s49/s32/s32/s32/s32/s91/s103/s122\n/s49\n/s49 /s66/s93\n/s66/s47/s124/s74/s124/s40/s97/s41\n/s40/s100/s41\n/s32/s32/s109/s122\n/s50\n/s49/s32/s32/s32/s32/s91/s103/s122\n/s50\n/s49 /s66/s93\n/s66/s47/s124/s74/s124\nFIG. 3: (Color online) Zero-temperature magnetizations ve r-\nsus the relative strength of the magnetic field for the cantin g\nangle 2α=π/4 and several spatial orientations θof the ap-\nplied field: (a)-(b) the transverse mx\n2and longitudinal mz\n2\nprojections of the Heisenberg spins; (c)-(d) the local proj ec-\ntionsmz1\n1andmz2\n1of the Ising spins towards their easy axes.\nof the Heisenberg spins within the so-called ’half ice, half\nfire’ frustrated ground state [7]. It has been also convinc-\ningly evidenced that the canting angle between two local\nanisotropy axes of the Ising spins and the spatial orien-\ntation of the applied magnetic field basically influences\nthe overall shape of the magnetization curves.\nAcknowledgments\nThis work was partially supported by FAPEAL\n(Alagoas State Research agency), CNPq, CAPES,\nFAPEMIG, VEGA 1/0043/16 and APVV-14-0073.\n[1] J. Streˇ cka, M. Jaˇ sˇ cur, M. Hagiwara, K. Minami, Y.\nNarumi, K. Kindo, Phys. Rev. B 72, 024459 (2005).\nDOI:10.1103/PhysRevB.72.024459.\n[2] W. Van den Heuvel, L.F. Chibotaru, Phys. Rev. B 82,\n174436 (2010). DOI:10.1103/PhysRevB.82.174436.\n[3] S. Bellucci, V. Ohanyan, O. Rojas, EPL105, 47012\n(2014). DOI: 10.1209/0295-5075/105/47012\n[4] S. Sahoo, J.P. Sutter, S. Ramasesha, J. Stat. Phys. 147,\n181 (2012). DOI: 10.1007/s10955-012-0460-7.\n[5] J. Streˇ cka, M. Hagiwara, Y. Han, T. Kida, Z. Honda,\nM. Ikeda, Condens. Matter Phys. 15, 43002 (2012). DOI:\n10.5488/CMP.15.43002.\n[6] G. Calvez, K. Bernot, O. Guillou et al., Inorg. Chim. Acta\n361, 3997 (2008). DOI: 10.1016/j.ica.2008.03.040.\n[7] W. Yin, Ch. Roth, A. Tsvelik, arxiv: 1510.00030v2."    },    {        "title": "1807.02445v1.Spin_torque_induced_magnetization_dynamics_in_ferrimagnets_based_on_Landau_Lifshitz_Bloch_Equation.pdf",        "content": "Spin-torque-induced magnetization dynamics in ferrimagnets based on\nLandau-Lifshitz-Bloch Equation\nZhifeng Zhu,1,\u0003Xuanyao Fong,1and Gengchiau Liang1,y\n1Department of Electrical and Computer Engineering,\nNational University of Singapore, Singapore 117576\n(Dated: July 9, 2018)\nA theoretical model based on the Landau-Lifshitz-Bloch equation is developed to study the spin-\ntorque e\u000bect in ferrimagnets. Experimental \fndings, such as the temperature dependence, the peak\nin spin torque, and the angular-momentum compensation, can be well captured. In contrast to\nthe ferromagnet system, the switching trajectory in ferrimagnets is found to be precession free.\nThe two sublattices are not always collinear, which produces large exchange \feld a\u000becting the\nmagnetization dynamics. The study of material composition shows the existence of an oscillation\nregion at intermediate current density, induced by the nondeterministic switching. Compared to the\nLandau-Lifshitz-Gilbert model, our developed model based on the Landau-Lifshitz-Bloch equation\nenables the systematic study of spin-torque e\u000bect and the evaluation of ferrimagnet-based devices.\nI. INTRODUCTION\nFerrimagnets (FiMs) with antiferromagnetic exchange\ncoupled transition-metal (TM) and rare-earth (RE) al-\nloys have attracted considerable attention due to the rich\nphysics1{7and their promise in device applications8{10.\nThe FiMs are expected to have fast spin dynamics like\nantiferromagnets (AFMs), but their magnetic states can\nbe electrically sensed using the tunnel magnetoresistance\n(TMR) e\u000bect due to the \fnite net magnetization ( mnet),\nwhich can be tuned by temperature ( T) or material com-\nposition (X). In addition, the FiMs have large bulk\nperpendicular anisotropy, which o\u000bers an alternative to\nthe ferromagnets (FMs) and enables the scaling down of\nMRAM down to 20 nm8. Furthermore, di\u000berent g factors\nbetween sublattices induce an angular-momentum com-\npensation point, which enables fast domain-wall motion9.\nThe FiMs can be manipulated by magnetic \feld or\nlaser heating2,3,11{15, but an electrical method, such as\nthe spin-transfer torque (STT)1or the spin-orbit torque\n(SOT)4,16, is preferred for electrical characterizations\nand applications. Therefore, it is important to study the\nmagnetization dynamics under spin torque using a model\nwhich can incorporate the e\u000bects of TandX. How-\never, the commonly used theoretical model based on the\nLandau-Lifshitz-Gilbert (LLG) equation1,17{19is limited\nat \fxedTdue to the assumption of a \fxed magnetization\nlength (see Appendix A for detailed analysis of the LLG\nmodel). In contrast, the Landau-Lifshitz-Bloch (LLB)\nmodel has been widely used to describe the magneti-\nzation dynamics at elevated T20, where the T-induced\nmagnetization-length change is taken into account by in-\ncluding a longitudinal relaxation term. To date, the LLB\nequations have been implemented for both FMs20and\nFiMs21, and recently the e\u000bect of spin torque in FMs\nhas also been included22. Starting from the atomistic\nLandau-Lifshitz equation, in this work, we extend the\nLLB model to capture the spin-torque e\u000bect in FiMs.\nThe numerical simulation of current-induced switching in\na FiM/heavy-metal (HM) bilayer is then performed, andwe \fnd the modi\fed LLB model can reproduce salient\nexperimental \fndings, such as the magnetization com-\npensation, the reversal of switching direction across the\nmagnetization-compensation temperature ( TMC)23, and\nthe peak in spin torque at TMC24. In addition, the spin-\ntorque-induced sublattice dynamics in FiMs is studied\nand compared to that in AFMs and FMs. The switching\ntrajectory is found to be precession free, and the sublat-\ntices are not always collinear. Finally, the e\u000bect of Xon\nFiM properties is studied.\nII. THE MODIFIED\nLANDAU-LIFSHITZ-BLOCH EQUATION\nAs shown in Fig. 1(a), the device structure we stud-\nied consists of a FiM (Gd X(FeCo) 1X) deposited on top\nof a HM layer. The magnetizations of sublattices are\nmanipulated by the SOT generated by in-plane electrical\ncurrent. The FiM is treated as a two-sublattice model,\ni.e., Gd and FeCo, which is justi\fed by the experimen-\ntal observation that the magnetizations of Fe and Co are\nparallel up to the Curie temperature ( TC)25. The mag-\nnetization dynamics of each sublattice is captured by the\nmodi\fed LLB equation2,3,22,25{30(see Appendix B for the\nderivation)\n_mv=\rv(mv\u0002HMFA\nv )\u0000\u0000v;k(1\u0000(mv\u0001m0;v)\nm2v)mv\n\u0000\u0000v?mv\u0002(mv\u0002m0;v)\nm2v;(1)\nwhere \u0000 v;k= \u0003 v;NB(\u00180;v)=(\u00180;vB0(\u00180;v)) and \u0000 v;?=\n\u0003v;N[\u00180;v=B(\u00180;v)\u00001]=2 are the coe\u000ecients of longitudi-\nnal and transverse relaxation, respectively. The dimen-\nsionless \feld is given by\n\u00180;v=\f\u0016v(HMFA\nv +HI=\u0015v): (2)\nEq. (1) contains two coupled equations for FeCo and Gd\nidenti\fed by the subscript v, which need to be solvedarXiv:1807.02445v1  [cond-mat.mtrl-sci]  6 Jul 20182\nsimultaneously10. The \frst term on the right hand side\ndescribes the magnetization precession around the mean-\nfree \feld\nHMFA\nv =Hext+HA;v+ (J0;v=\u0016v)mv+ (J0;vk=\u0016v)mk;\n(3)\nwhich consists of the external magnetic \feld Hext, the\ncrystalline anisotropy \feld HA;v= (2Dv=\u0016v)mv;zezwith\ncoe\u000ecientDv, and the exchange coupling between sub-\nlattices with coe\u000ecients J0;vandJ0;vk. The damping\ncoe\u000ecient is given by\n\u0003v;N= 2\rv\u0015v=(\f\u0016v); (4)\n\f= 1=(kBT); (5)\nwhere\u0015vis the damping constant, \rvis the gyromag-\nnetic ratio, \u0016vis the magnetic moment, and kBis the\nBoltzmann constant. As previously mentioned, the T-\ninduced magnetization-length change is described by the\nlongitudinal relaxation term using the Brillouin function\nB(\u0018) =coth(\u0018)\u00001=\u0018; (6)\nand the spin-torque e\u000bective \feld HIis given by\nHI=JS\u0016h=(2etFiMjXM S;v\u0000qMS;kj); (7)\nwhere \u0016his the reduced Planck constant, eis the electron\ncharge,tFiMis the thickness of FiM layer, and MSis the\nsaturation magnetization. The spin current density JSis\nformulated as\nJS=\u0012SH\u001b\u0002JC; (8)\nwhere\u0012SHis the spin-Hall angle, \u001bis the polarization\nof spin current, and JCis the charge current. The equi-\nlibrium magnetization m0;vis calculated via the coupled\nCurie-Weiss equation\nm0;v=B(\u00180;v)\u00180;v=\u00180;v: (9)\nSimilar to the LLB in FM22, the e\u000bect of spin torque only\nenters the two relaxation terms. Furthermore, Eq. (1) re-\nduces to Eq. (4) in Ref.21when the spin torque vanishes,\nor to Eq. (7) in Ref.22when FeCo and Gd are not distin-\nguished. The numerical integration of Eq. (1) proceeds\nusing a fourth-order predictor-corrector method10.\nThe parameters used in the simulation are determined\nas follows [see Fig. 1(c)]: First, the Curie-Weiss equation\n[Eq. (9)] for pure Gd and FeCo is solved independently\nand \ft to the experimental M-Tcurves31to determine\nthe exchange coupling coe\u000ecients JGd= 0:98\u000210\u000021J\nandJFeCo = 1:5\u000210\u000021J. Then, the JGdFeCo =\u00007:63\u0002\n10\u000021J is obtained by solving the coupled Curie-Weiss\nequations of FiM. In addition, a su\u000ecient anisotropy is\nused to ensure the perpendicular magnetization. The\n\u0012SHand\u0015are swept with Dto \ft the experimental\nM-HandM-Jloops16, and a good agreement10with\nthe experimental data is obtained with \u0012SH= 0:003732,\n\u0015=0.07, and D= 3:2\u000210\u000026J.\nFiM\nHMt = 3 nmxyz(a)\nSolve Curie -Weiss equation for \nFeCo and Gdindependently\nfit with experimental \n“M vs T”\nJFeCo, JGdfit experimental \n“TMCvs X”Solve Coupled Curie -\nWeiss equation for FiM\nJFeCoGdD,θSH, αSolve LLB equation\nfit experimental MH, MI loop(c)(b)Atomistic LL equation \nwith spin torque for FiMFokker -Planck equation\nMean field model\nFinal form (Eq. 1)FIG. 1. (a) Schematic view of the device structure consist-\ning of a FiM layer deposited on top of a HM. The FiM in\nthis study is perpendicularly magnetized Gd X(FeCo) 1Xal-\nloy, where Gd and FeCo are antiferromagnetically coupled.\nProcedure of (b) equation derivation, (c) model validation\nand parameter determination. The derivation starts from the\nLandau-Lifshitz equation including the spin torque, followed\nby the corresponding Fokker-Planck equation to account for\nthe statistic behavior, and yields the \fnal form after using the\nmean \feld approximation. This model is validated by com-\nparing with the experimental MvsTtrajectory31,TMCvs\nGd concentration ( X), andM-HandM-I loops16.\nIII. DETERMINISTIC SWITCHING INDUCED\nBY THE SPIN-ORBIT TORQUE\nWe \frst study the SOT-induced deterministic switch-\ning in FiM using the LLB model. As shown in Fig. 2, the\nJCapplied along the xdirection generates spin torque\nacting on the FiM layer due to the spin-hall e\u000bect (SHE)\nor the inverse spin galvanic e\u000bect (ISGE)33{35. How-\never, the magnetization cannot be switched vertically\nsince the spin torque aligns the magnetization to yaxis.\nThis is similar to the perpendicular FM switched by\nin-plane current, where an external \feld along the cur-\nrent direction ( HX) is required to achieve determinis-\ntic switching36{39. The switching in FM system can be\nunderstood as follows: The switching direction is deter-\nmined by HXasL= \u0001m\u0002HX, and the spin torque,\n\u0001m=m\u0002(m\u0002HI), should be su\u000ecient to overcome the\nenergy barrier. Therefore, the switching direction will be\nreversed by reversing either HXor current direction37.\nRecently, by applying HX, the current-induced deter-\nministic switching in the FiM/HM bilayer has also been\ndemonstrated4,16. The measured M-Jloop clearly shows\nan opposite switching direction by reversing the current,\nwhereas the e\u000bect of HXhas not been investigated. In\nthis study, we show that the switching direction is also\nreversed under opposite HX[see Fig. 2], which can be\nexplained using the abovementioned two-torque analysis\ntogether with the exchange coupling between sublattices.\nAs shown in Fig. 2(a), the FiM at T= 300 K is FeCo\ndominant. The positive JCandHXswitch mFeCo from\ndown to up, and concurrently, the exchange interaction\nturns mGdfrom up to down. When the HXis reversed,\nmFeCo is switched from up to down [see Fig. 2(b)], re-\nsulting in an opposite M-Jtrajectory. To con\frm the\nunique role of HX, we have veri\fed that the equilibrium3\n-1 0 10.3\n0\n-0.3\nJC(1011A/m2)mZ_FeCo(a)\n-1 0 1\nJC(1011A/m2)(b)\nHXJC\nHXJC\nFIG. 2. Simulated SOT-induced switching in FiM\nGd21(FeCo) 79under (a) HX= 1 mT, (b) HX=\u00001 mT atT\n= 300 K. The HXonly breaks the symmetry for deterministic\nswitching. The reversal of switching direction under opposite\nHXis similarly explained using the theory in perpendicular\nFM.\nmagnetization is not altered when only HXis applied,\nand no switching event is observed when the current is\nswept with Hext= 0 or HY. Therefore, the SOT-induced\nswitching in FiM is determined by the dominant sublat-\ntice, followed by the reversal of the other sublattice via\nexchange interaction, and the HXonly breaks switching\nsymmetry. It is also worth noting that the maximum\nmZin Fig. 2 is around 0.3, which is an evidence of the\nT-induced magnetization-length reduction with mZ= 1\nde\fned atT= 0 K.\nAlthough SOT and HXhave similar e\u000bects in switch-\ning FiM and perpendicular FM, the time evolutions of\nmagnetization are very di\u000berent as shown in Fig. 3,\ni.e., the switching trajectory of FiM is precession free,\nwhereas it is precessional in FM. In the SOT-switched\nFM with initial mZ= 1, both anisotropy \feld and spin\ntorque align the magnetization to the + zdirection for\nmZ>0, resulting in a larger precession term compared\ntomZ<0, where the anisotropy \feld and spin torque\nare opposite. Consequently, more precession occurs when\nmZ>0 [see Fig. 3(b)]. Similarly, the precession-free tra-\njectory in FiM is attributed to the small precession term.\nAs illustrated using the 3D trajectories in Fig. 3(c), mGd\nandmFeCo are switched to opposite directions. Due\nto the strong exchange coupling, many studies assume\nthey are always collinear. However, as the time evolu-\ntion of each sublattice and their relative angle shown in\nFig. 4(a), a maximum deviation of 0.9 degree is observed\natt= 30 ns. This number is similar to a recent report\nfrom Mishra et al.4, where a cant of one degree is es-\ntimated from the strength of exchange \feld. Since the\nexchange coupling between sublattices is very strong ( >\n100 T40,41), even a very small cant deviates the behavior\nof FiM from FM, which might contribute to the di\u000berent\nmagnetization dynamics shown in Figs. 3(c) and 3(d).\nSimilar noncollinearity between sublattices is also pre-\ndicted in AFM42, with the deviation angle determined by\nthe strength of spin torque. To achieve a large-angle non-\ncollinearity in FiM, recent study shows that a magnetic\n\feld over 5 T is required43. By studying the \feld-induced\nswitching in FiM [see Fig. 4(b)], a similar trajectory is\nTime (ns)0 300\n-0.30.3(a)\nmZ_FeComX_FeComY_FeCom\nx\nyz(c)\nmGdmFeCoinitial\nfinal\n(b)\n(d)\nxyzTime (ns)0 10 20\nHXJCFiM\nHXJCFMFIG. 3. Time evolution of the SOT-induced switching (a) in\nFiM Gd 21(FeCo) 79and (b) in perpendicular FM at T= 300\nK. (c) and (d) are the 3D trajectories corresponding to (a)\nand (b) respectively. The dot lines in (c) are the projections\non thex-yplane. All simulations start from the equilibrium\nstate where mZ= 0.3. This reduced value re\rects the T-\ndependent magnetization, where mZ= 1 is de\fned at T= 0\nK.\n(b)\n0\n-0.40.4(a)\nTime (ns)5 25mZ_GdmZ_FeCo\n45179.9\n179.1\n25 45θ\nTime (ns)5\nm\nTime (ns)5 15 250\n-0.30.3\nHXJCFiM\nFIG. 4. (a) Time evolution of mZfor FeCo and Gd sublattices\natT= 300 K, with inset showing the magnetization angle\nbetween mGdandmFeCo. (b) Field-induced switching in\nGd21(FeCo) 79atT= 300 K, which has similar trajectories\nwith the current-induced switching.\nobserved compared to Fig. 3(a), indicating that a large\nspin-torque e\u000bective \feld would be required to get a large\nangle deviation. However, as discussed in the next sec-\ntion, large spin torque aligns the magnetization to the\nspin direction, hence no switching happens.\nIV. EFFECT OF TEMPERATURE AND\nMATERIAL COMPOSITION\nTandXare often tuned in experiments to control the\nproperties of FiM4,9,23,44. By measuring the M-Hloops\nas a function of T,TMCcan be identi\fed where the co-\nercive \feld ( HC) diverges. However, TMCmay not exist\nin another sample with a di\u000berent X16. In this study,\nthe LLB equation is used to investigate two samples, i.e.,4\nmagnetization magnitude vs Tmnet\nT (K)0 100 200 3000\n-0.10.10.2\nmGdmFeCo\nGd21(FeCo)79\nGd23(FeCo )77\nTime (ns)0 15 30\nmZ_FeCo\n0\n-0.80.80\n-0.80.8\nT = 130 K\nT = 70 K(b) (a)\n(c)\nFIG. 5. (a) E\u000bect of Ton the net magnetization of\nGd21(FeCo) 79(blue square) and Gd 23(FeCo) 77(red trian-\ngle) withTMC= 75 K below which Gd is dominant, where\nthe net magnetization is calculated using mnet= (1 \u0000\nX)mFeCo\u0016FeCo +XmGd\u0016Gdwith\u0016FeCo = 2:217\u0016Band\n\u0016Gd= 7:63\u0016B. Time evolution of mZ;FeCo under spin torque\nat (b)T= 130 K and (c) 70 K.\nGd21(FeCo) 79and Gd 23(FeCo) 77, and we show that the\nexistence of TMCis determined by the demagnetization\nspeed and the relative magnitude of mFeCo andmGd.\nAs reported in our recent study10, both mFeCo andmGd\ndecrease with Tand vanish at the same temperature lo-\ncated between TC;FeCo (1043 K) and TC;Gd (292 K). The\ncommon Curie temperature is induced by the strong ex-\nchange coupling which speeds up the demagnetization\nprocess in FeCo but slows down that in Gd. As shown\nin Fig. 5(a), the Gd 21(FeCo) 79shows FeCo dominant at\nall temperatures, whereas a transition from Gd to FeCo\ndominant is observed in the other sample. At low T,\nGd dominates due to the larger magnetic moment. As\nTincreases, mnetreduces and vanishes at TMC= 75\nK because of the faster demagnetization process in Gd.\nAboveTMC,mnetrises until a peak and then reduces to\nzero atTC. Furthermore, we \fnd that the magnetization\ndynamics near TMC[Fig. 5(c)] is similar to the one at\nhigherT[Fig. 5(b)], which can be understood by notic-\ning the gradual change in e\u000bective \felds such as HAand\nHI. It is only at TMCthat a sudden change occurs, and\nthe e\u000bective \felds diverge.\nAs shown in Fig. 6(a), the competition between mFeCo\nandmGdis also manifested in the T-dependent M-H\nloops23. In addition to the reversal of switching direction,\ntheHCreaches maximum at TMCto overcome the en-\nergy barrier ( E=\u0000M\u0001H). WhenTis further increased\n(i.e.,T > T MC), both mFeCo andmGdreduce, result-\ning in smaller exchange and anisotropy \felds [Eq. (3)]\nand hence a lower HC. Furthermore, we \fnd the TMC\nobtained from the M-Hloops is consistent with the equi-\nlibrium state calculation [Fig. 5(a)], which is another ev-\nidence that the LLB model captures FiM dynamics.\nFor practical reasons, Tis not preferred as the con-\ntrol parameter in device applications, whereas Xcan be\ntuned during the deposition process. The change of X\nshows similar results to that observed in the Tdepen-\ndence. AsXis increased, the FiM changes from FeCo to\nGd dominant, resulting in a reversal of both M-Hand\nM-Jloops4,16. Due to the vanishing mnetatXMC, the\nH (mT)-400 -200 200 400-0.60.6mZ_FeCo\n0-0.60.6\n-0.60.6\n-0.60.6\n-0.60.6\n-0.60.610 K\n40 K\n60 K\n90 K\n120 K\n190 KGddominant\nFeCo dominant(a)\nJC(1011A/m2)-8 8-0.60.6mZ_FeCo\n0-0.60.6\n-0.60.6\n-0.60.6\n-0.60.6\n-0.60.6X = 0.15\nX = 0.16\nX = 0.17\nX = 0.24\nX = 0.25\nX = 0.26FeCo dominant\nGddominant(b)Hcrit, Icritvs TFIG. 6. (a) M-Hloops at di\u000berent Tin Gd 23(FeCo) 77with\nHX= 2 mT. The blue dot line denotes the transition from\nGd to FeCo dominant. The switching-direction reversal and\nthe peak in HCobserved in experiments23are qualitatively\nreproduced. (b) M-JCloops at di\u000berent XwithHX= 2\nmT. Three dynamic regions are identi\fed, e.g., for X= 0.25,\nsuccessful switching happens for 5 :7\u00021011A/m2< J C<\n6:4\u00021011A/m2, oscillation region for 6 :4\u00021011A/m2<JC<\n8\u00021011A/m2, and distorted region for JC>8\u00021011A/m2\n(i.e.,maligns to the ydirection).\nspin torque diverges4,44. To show the capability of LLB\nmodel in capturing these e\u000bects, we have simulated the\nX-dependent current-induced switching at T= 300 K.\nAs shown in Fig. 6(b), the switching direction reverses\natXMC= 0.24 which separates FeCo and Gd dominant\nregions. In both regions, mnetis switched from down\nto up under positive current, indicating that the SOT-\ninduced switching is determined by mnet. This is di\u000ber-\nent with the anomalous Hall e\u000bect (AHE), where RAHE\nis determined by mFeCo. In contrast to the magnetic-\n\feld-induced switching in Fig. 6(a), no clear peak of crit-\nical switching current density ( JCrit) is observed, which\nis attributed to the increase of spin torque near XMC. In-\nterestingly, three dynamics regions are identi\fed in our\nsimulatedM-Jloops. According to the sub\fgure of X\n= 0.25 in Fig. 6(b), mFeCo is successfully switched from\nup to down for 5 :7\u00021011A/m2<JC<6:4\u00021011A/m2.\nWhen the current exceeds 8 \u00021011A/m2, the magneti-\nzation is aligned with \u001bdue to the dominance of spin\ntorque. The magnetization dynamics in these two regions\nare considered as typical behaviors which have also been\nobserved in FM systems45. For the JCin between, how-\never, an unexpected oscillation occurs. To understand\nthis, we have simulated the SOT-induced switching in\nFM using the LLB model, which is realized by setting X\n= 0, and no oscillation is observed. This indicates the im-\nportant role of exchange coupling, which competes with\nHIandHX, and the oscillation is obtained from the\nbalance of all these interactions, which is a unique prop-\nerty in the system of perpendicular FiM switched by an\nin-plane current.5\nV. CONCLUSION\nIn conclusion, we have developed a theoretical model\nbased on the modi\fed LLB equation to systematically\ndescribe the e\u000bect of temperature, material composition,\nand spin torque in FiM. The e\u000bect of spin torque on\nthe magnetization dynamics is studied in a FiM/HM bi-\nlayer, where the switching trajectory is found to be pre-\ncession free due to the small precession \feld. Similar to\nthe spin-torque switching in AFMs, the two sublattices\nare not always collinear, with a maximum angle deviation\nof 0.9 degree in our study. This cant between sublattices\ninduces an oscillation region between the magnetization\nswitching and the fully alignment with spin polarization,\nwhich is a unique behavior in the perpendicular FiM sys-\ntem. Our results of the spin-torque-induced magnetiza-\ntion dynamics can be helpful in understanding experi-\nmental results and in evaluating FiM-based devices.\nACKNOWLEDGMENTS\nThis work at the National University of Singapore\nwas supported by CRP award no. NRF-CRP12-2013-01,\nNUS FRC R263000B52112, and MOE-2017-T2-2-114.\nAPPENDIX A: NUMERICAL SIMULATION OF\nFIM USING THE LLG MODEL\nBefore we study the LLB model, the LLG equation is\nwidely used to qualitatively explain experimental \fnd-\nings in FiMs. As the FiM consists of antiferromagnetic\ncoupled TM and RE sublattices, it is straightforward to\napply one LLG equation to one sublattice19as\n_Ma=\u0000\raMa\u0002(Ha+hMb) +\u000baMa\u0002_ma\n\u0000\rMa\u0002(Ma\u0002Ha\nI);(10a)\n_Mb=\u0000\rbMb\u0002(Hb+hMa) +\u000bbMb\u0002_mb\n\u0000\rMb\u0002(Mb\u0002Hb\nI);(10b)\nwhere the three terms on the right hand side are preces-\nsion, Gilbert damping, and spin torque, respectively, and\nthese two equations are coupled through the exchange\nterms, i.e., hMbandhMa.\nDespite the clear physical picture behind the coupled\nequations, the analytical study of FiM using these equa-\ntions is complicated. Instead, another e\u000bective LLG\nmodel describing the net magnetization1,17,18is often\nused as\n_Meff=\u0000j\reffj(Me\u000b\u0002He\u000b) +\u000beff(Me\u000b\u0002_Me\u000b)=Meff\n\u0000j\reffj(Me\u000b\u0002(Me\u000b\u0002HI));\n(11)\n\reff(T) =MRE(T)\u0000MTM(T)\nMRE(T)=j\rREj\u0000MTM(T)=j\rTMj;(12)\u000beff(T) =\u0015RE=j\rREj2+\u0015TM=j\rTMj2\nMRE(T)=j\rREj\u0000MTM(T)=j\rTMj;(13)\nwhere\reffand\u000beffare the e\u000bective gyromagnetic ra-\ntio and damping constant, respectively. By assuming\na collinear TM and RE, we can demonstrate that the\ntwo LLG models are equivalent19. The e\u000bective LLG\nmodel is \frstly used to qualitatively explain the unex-\npected sign reversal of magnetoresistance (MR) in STT-\nswitched CoGd layer1, where the coexistence of magneti-\nzation and angular-momentum compensation is observed\nfor the \frst time. In addition, this model has been used\nin the ferromagnetic resonance (FMR) analysis18and\nin explaining the ultrafast spin dynamics in GdFeCo17.\nIn addition, the quantitative study of the spin-torque-\ninduced dynamics in FiM using the coupled LLG equa-\ntions has been studied without considering the di\u000berent g\nfactors or damping constant between sublattices4. In this\nstudy, we choose gFeCo = 2.2,gGd= 29,\u000bFeCo = 0.01,\n\u000bGd= 0.02, and perform numerical simulations of both\nmagnetic-\feld and spin-torque switching using the e\u000bec-\ntive LLG model, which is integrated via the fourth-order\nRunge-Kutta method46. The simulation results are then\ncompared with the experimental and the LLB results in\nthe aspect of critical switching current and switching tra-\njectory.\nLimited by the \fxed magnetization length in the LLG\nmodel, the simulations are performed at \fxed Twhich\nis less than TC;Gd due to the requirement of knowing\nbothMS;Gd andMS;FeCo in Eq. (11). Therefore, we \frst\nchooseT= 200 K and study the \feld-induced switch-\ning withtFiM = 2 nm, and FiM radius rFiM = 25 nm.\nTheMSof both sublattices are taken from their FM\ncounterparts at T= 200 K (MS;Gd = 2:02\u0002106A/m31\nandMS;FeCo = 1:73\u0002106A/m47), resulting in a FeCo-\ndominant sample. As the critical switching \feld is inde-\npendent on the damping constant, we are left with one\nfree parameter, i.e., the crystalline anisotropy \feld ( HK).\nThe magnitude of HKis determined by performing re-\nlaxation simulation, where the magnetization is required\nto return to the perpendicular direction, i.e., the magne-\ntization is \frstly tilted away from the zdirection (e.g.,\n70 degree tilt), and then, it is relaxed without any mag-\nnetic \feld or current. As a result, HKhas to be larger\nthan 0.9 T to maintain a perpendicular magnetization.\nNext, the e\u000bect of HKonHCis studied, where HCin-\ncreases linearly with HK. Consequently, the smallest HC\n= 200 mT is obtained using HK= 0.9 T, much larger\nthan the experimental measured HC= 50 mT23. Fur-\nthermore, when the above procedures are repeated at\notherT(0<T < 293K), we \fnd that the sample is al-\nways FeCo dominant, failing to explain the experimental\nobserved Gd-dominant region.\nHowever, the results from the LLG model can \ft the\nexperimental M-Hloop if the restrictions of taking MS\nfrom FM counterparts are removed [see Fig. 7(a)], which\ncan be justi\fed by the di\u000berence in exchange coupling6\nExp.Sim.\nH (mT)-15 0 15-1.501.5 RAHE(arb. unit)(a)\n-1.3 0 1.3-1.301.3 RAHE(arb. unit)\nJC(1011A/m2)(b)\n(d)\nxyz\n (c)\nxyz\nFIG. 7. Experimental RAHE (blue circle)16and simulated\nmZ;FeCo (red triangle) using the e\u000bective LLG model for (a)\nSOT-induced switching, and (b) magnetic-\feld switching with\nMS;Gd = 3:2\u0002105A/m andMS;FeCo = 1:73\u0002106A/m. (c)\nand (d) are the time evolutions of mFeCo correspond to (a)\nand (b), respectively.\nstrength and lattice occupations between FiM and FM.\nWe then simulate the current-induced switching using\nMSandHKobtained from the \ftted M-Hloop, and a\ngood \ftting shown in Fig. 7(b) is obtained with \u000bFeCo=\n0.01,\u000bGd= 0.02 and spin polarization P= 0.4.\nFigs. 7(c) and 7(d) show the trajectories of \feld and\ncurrent switching, respectively. Both of them are similar\nto that in FMs39, which is expected since the e\u000bective\nLLG model treats the FiM as a single magnet. How-\never, the current-induced magnetization dynamics pre-\ndicted by the LLG model [see Fig. 7(d)] is very di\u000berent\nto that in the LLB model as shown in Fig. 3(c), and more\nresults from time-resolved experiments48would be help-\nful to resolve this discrepancy. As a result, although the\nLLG model can reproduce the experimental M-Hand\nM-Jloops by \ftting MS, di\u000berent sets of MSwill be\ngenerated at di\u000berent T, and some of them might be un-\nrealistic. In addition, these MSare not correlated and\ncannot be explained using a uni\fed theory. In contrast,\nthe LLB model can reproduce the experimental results\nfor all temperatures by using one set of parameters10,\nwhich is attributed to its capability of capturing the T-\ndependent magnetization-length change. In this aspect,\nthe LLB model is more suitable in studying FiM proper-\nties.APPENDIX B: DERIVATION OF THE LLB\nEQUATION INCLUDING SPIN TORQUE\nThe derivation of Eq. (1) starts from the lattice-site\natomistic Landau-Lifshitz equation with an additional\nspin-torque term\n_ s=\r[s\u0002(H+\u0010)\u0000\u0015s\u0002(s\u0002H) + (s\u0002(s\u0002HI))];(14)\nH=Hext+ (2D=\u0016)sz\niez+X\nj2neigJijSij=\u0016; (15)\n<\u0010a(t)\u0010b(t0)>= 2\u0015T\u000eab\u000e(t\u0000t0)=(\r\u00160); (16)\nwhere sis the spin angular momentum, \u0010is the thermal\n\feld with the subscript representing di\u000berent Cartesian\ncomponents (i.e., x,y, andz), andtis the time. The\nthree terms on the right hand side of Eq. (14) represent\nprecession, damping, and spin-torque e\u000bect, respectively.\nThe exchange coupling in the last term of Eq. (15) only\nconsiders the in\ruence of nearest neighbors, and Eq. (16)\nindicates that the sublattice spin is uncorrelated with\nrespect to time and other Cartesian components. The\ndirect simulation using Eq. 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Tribollet  - 01/2019 ) \n1 \n  \n \n \nSiC-YiG X band quantum sensor  for \n \nSensitive  Surface Paramagnetic Resonance  \n  \napplied to  chemistry, biology,  physics . \n \n \n \nJérôme TRIBOLLET  \n \nInstitut de Chimie de Strasbourg, Strasbourg University, UMR 7177 (CNRS -UDS),  \n4 rue Blaise Pascal, CS 90032, F -67081 Strasbourg Cedex, France  \nE-mail : tribollet@unistra.fr  \n \n \n \nABSTRACT  \nHere I pr esent the SiC-YiG Quantum Sensor, allowing electron paramagnetic  resonance  (EPR)  \nstudies of monolayer or few nano meter s thick  chemical, biological or physical samples \nlocated on  the sensor surface . It contains  two parts , a 4H-SiC substrate with  many \nparamagnetic silicon vacancies (V2) located  below its surface, and  YIG ferrimagnetic \nnanostripes. S pins sensing properties are based on optically detected double electron -\nelectron spi n resonance under the strong magnetic field gradient of nanostripes. Here I  \ndescribe fabrication, magnetic, optical and spins sensing properties of th is sensor. I show \nthat the target spins sensitivity is at least five order s of magnitude larger than the o ne of \nstandard X band EPR spectrometer, for which it constitutes , combined with a fiber bundle,  a \npowerful upgrade for sensitive  surface EPR . This sensor can determine the target spins \nplanes  EPR spectrum, their positions with a nanoscale precision of +/ - 1 nm , and their 2D \nconcentration  down  to 1/(2 0nm)2.                                     \n \n \n \n \n \n \n \n \n SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n2 \n  \n Electron paramagnetic resonance1 (EPR) investigation of  electron spins localized \ninside, at surfaces, or at interfaces of ultra thin films is highly relevant . In the field s of \nphotovoltaic2 and photochemistry3, EPR is  useful  to study the spins of photo -created \nelectron -hole pair s, their dissociation, and their eventual transport  or chemical reaction \noccurring at some relevant interface . In opto -electronics with  2D semiconductors4, spins of \ndefects limiting device performance can be identified  and quantified by EPR . In magnetic \ndata storage science5 and in spin-based  quantum comput ing science  using  molecules6 \ngrafted,  tethered, encapsulated or physisorbed on a so lid substrate , it is relevant to study by \nEPR the magnetic properties of those molecules , always  modified by their interaction with \nthe substrate7. In solid supp orted heterogeneous catalysis , it is relevant to study spins \ninvolved in catalytic reactions , using EPR8 and eventually spin trapping methods9. In \nstructural biology , it is relevant to study by EPR spin labeled proteins10,11 introduced in \npolymer suppo rted or  tethered lipid bilayers  membranes12,13. In the context of the \ndevelopment of new theranosti c agents for nano medi cine, it is relevant to study  ligand -\nprotein molecular recognition events occurring on surfaces by EPR, using for example, \nbifunctional spin labels14. As various nanotechnologies now allow to produce nanoscale \nthickness  samples , one ne eds to perform sensitive Surface EPR (S -EPR). However, \ncommercial  EPR spectrometers have not enough sensitivity15 for EPR study of those few \nmonolayers  thick  ultra -thin films , particularly when target spins are diluted  and when \nsamples stacking is not poss ible.   \n Home -made EPR experimental setups have been d eveloped recently , in the context \nof quantum sensors16-20 and quantum computers , reaching single spin sensitivity  by \noptically17,18,21, electrically22 or mechanically23 detected EPR . Some of them  achiev ed the \nnanoscale resolution  imaging , when combined with magnetic devices moving  over  \nsurfaces24,25. Other recent advances in the field of inductively detected EPR have also \nconsiderably improved sensitivity, but  at the price of operating home -made microwav e \ndevices at unco nventional millik elvin temperatures26. Thus, c learly, there is today a gap \nbetween performance s of standard  X band EPR spectrometers already used worldwide by \nmost  of chemists, biologists and physicists , and the ones  of the bests  unconvent ional EPR \nsetups  found in just few laboratories worldwide .  \n Here  I present the theory of  a new Optically Detected M agnetic Resonance  (ODMR)  \nbased electron spins Quantum Sensor, allowing to study  target electron spins of ultra thin \nparamagnetic samples  located on the sensor  surface . It has  nanoscale resolution in one \ndimension, a high sensitivity  due to  spins ensemble ODMR , and importantly , is designed as \nan upgrade of  standard  X band pulsed EPR spectrometers . The design  of the magnetic \nproperties of the sen sor is inspired from the one s of the hybrid paramagnetic -ferromagnetic \nquantum computer device27 I previously proposed . However, here, it is adapted to \nconstrain ts of standard  X band  (10 GHz, 0.35  T, 5 mm sample access ) pulsed EPR resonators \nand spectromet ers and thus to fiber bundle based ODMR28,29. The quantum  sensor  contains \ntwo parts. The first  is a 4H-SiC semiconductor substrate containing, just below its surface, \nisolated negatively charged silicon vacancies  (V2) used as quantum coherent  ODMR spin \nprobes20,21,30 ,31,45. The second part is  an ensemble of ferrimagnetic YIG (Yttrium Iron Garnet)  \nnanostripes32 having narrow sp in wave resonances at X band . A fixed spacer fabricated on \nedges adjust the relative distance between the two parts. Next,  I present the fabrication \nmethodology, magnetic and optical properties , and finally spins sensing properties , based on \nPELDOR spectroscopy1,10,11, 18,33, of this SiC-YiG quantum sensor.  SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n3 \n  The quantum sensor device proposed can be obtained by fabricating its two parts \nseparately and then  integrat ing them  (fig.1  a, b). As said in introduction , the first par t of the \nquantum sensor is a 4H -SiC semiconductor sample , in which silicon vacancies spin \nprobes20,21,30,31 ,45 called V 2 are created  just below the 4H -SiC surface, and on which the ultra -\nthin paramagnetic film of i nterest will have to be deposit ed, anchored  or self assembled  \n(fig.1 ). This is necessary  because the spins sensing principle is related to the many long range \ndipolar coupling s that exist between a given singl e V 2 spin probe and the many neighbor \ntarget spins  (fig.1c ), those couplings affecting the spin coherence time  of V 2 spins probes \nand being  revealed by PELDOR spectroscopy1,10,11,18,33. The 4H -SiC sample can be a 4H -SiC \nsubstrate terminated on one side by an isotopica lly purified 4H -SiC grown layer,  having no \nnuclear spins21 and a  very low residual n type doping (< 1014 cm-3) 21. However, a \ncommercially available 4H -SiC substrate with low n doping  and a natural low amount of non-\nzero  nuclear spins is also a good starting point . \n \nfigure 1  : a/ two parts of the Q uantum Sensor : the paramagnetic 4HSiC one , with V2 spins on front side  of \nthe truncated cone shape island (45°),  and a cone shaped dip  (45°)  on back side;  and the ferrimagnetic one, \nwith many identi cal YIG nanostripes on GGG substrate  (only one stripe shown here for clarity, thus not at \nscale) . Also shown  on b/ , their integration  by a spacer  (not at scale)  and introduction in a standard pulsed \nEPR spectrometer microwave cavity, as well as the fiber b undle and the GRIN microlens  (yellow)  used for \nfiber bundle based  ODMR. b/: Zoom showing the many dipolar couplings (dark lines) existing between V2 \nspins  probes  in 4H -SiC and target spins in the sample, used for quantum sensing by OD PELDOR spectroscopy. \nMolecular target spins and V2 probe spins a re here separated by a capping  layer of few nanometers. W eff \nindicate the width along z direction over which the dipolar magnetic field produced by a nearby YIG \nnanostripe can be considered as homogeneous. dx is t he distance between the plane of V2 spins and t he \nplane of target molecular spins considered here.  d1+d2=dx. Orders of magnitude: C 2D,V2= 1/ (30nm)2 et \nC2D,Target= 1/ (5nm)2, dx=10 nm, et d2 =2nm, d1=8 nm, weff =60nm for a  nearby YIG nanostripe  \n(T=100nm/W=500nm ), whose center  is located at a distance x opt=150 nm here from the V2 spin s plane.  \n  The fabrication process of silicon vacancies V2 spins probes  in 4H -SiC that I propose \nhere is described on top of fig.2 . It is based on an implantation -etching approach , combined \nwith SiC sculpting , in order  to define the appropriate photonic structure  for the optical \nexcitation and detection of V 2 spins probes . After cleaning of th e 4H -SiC surface, 5 nm of \nsacrificial SiO 2 are fabricated  on the  surface  of the 4H -SiC substr ate (thickness of 400 µm) . \nThose 5 nm of Si O2 can be obtained, either by slow oxidation of the 4H -SiC surface34 into \nSiO 2 at around 11 50 °C , or by a lower temperature thin film d eposition  method  like by \nPECVD35 or atomic layer deposition ( ALD)36 at 150°C . High temperature oxidation should SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n4 \n advantageously  remove residual V 2 silicon vacancies initially present in the 3D bulk of the \n4H-SiC sample, as V 2 vacancies are annealed out31 at around 700°C . Then, 20 nm of a \nstopping sacrificial layer of zinc oxide (ZnO ) are deposited on top of SiO 2/4H-SiC, by \nsputtering  or by ALD. Then 22 keV As+ ions are implanted in this tri-layer sample at a  dose \ncomprised between 1.6 1012 cm-2 and 1.6 1013 cm-2. The target dose here is around 8.3 \n1012cm-2, which corresponds, accordin g to SRIM  simulations (see SI), to a 2D effective \nconcentration of As+ ions in the first 2 nm of 4H -SiC of C 2D, As+  = 1/(32nm )2. SRIM simulation s \nalso indicate that the concentration of As+ ions rapidly decay with depth in 4H -SiC and is \nalmost zero after t he first 10 nm of 4H -SiC. SRIM simulations also indicate that such \nimplantation of As+ ions produce 1.3 silicon vacancy per As+ ion in those firs t 2 nm of 4H -SiC. \nOne can thus consider  that we obtain a 2D  effective concentration of silicon vacancies V 2 in \nthe first 2 nm of 4H -SiC of C 2D, eff, V2 = 1/(32nm )2. This concentration rapidly decays  to zero in \nthe next few nanometers in 4H -SiC. Then, 4H -SiC micro -sculpting  is performed either by \ndiamond machining37,38, by laser ablation39, by FIB40 or by another mi cromachining method41. \nThe aim is to produce , on front side , a truncated cone shape island with V 2 spins on top, and \non back side , a cone shape dip (cone edge angle of 45° in both cases), both cones sharing the \nsame symmetry axis  and having an optical qual ity surface roughness (fig.2 top) . Then , ZnO is \netched by HC l, and SiO 2 is etched by HF42. This  leads to a  sculpted sample with shallow \nsilicon vacancies created mainly 2 nm below the surface of the 4H-SiC truncated cone shape \nisland . A post implantation -sculpting -etching  annealing , at a temperature inferior to  600-\n700°C, can eventually be performed to remove some unwanted created defects . Then, a  \ntreatment passivate s the truncated cone shape  4H-SiC island surface, like a H+N plasma \ntreatment43 at 400°C, reducing  its surface de nsity of state to 6 1010 cm-2. Then, eventually  \n(not shown on fig.2) , a few nm capping  layer , easy to functio nalize , can be deposited on th is \npassivated 4H -SiC surface , for example using  ALD of silicon oxide  at low temperature36. Then,  \na spacer of appropriate thickness , 200 nm  here , for example a ring shape spacer  made of \nsilicon oxide, is fabricated by standard lithography and deposition , on the edge s of th e top \nsurface of the 4H -SiC or 4H -SiC/SiO 2 island , under which  the V 2 spins prob es were  created . \nThe diameter of this top 4H -SiC island surface is around 900  µm. The  spacer will allow the \nintegration of the two parts of the quantum sensor device by contacting them (fig.1 b). \nFinally, the  few monolayers  paramagnetic film of i nterest can  be created  on top of the \nsensor surface . It is either chemically anchored or physically adsorbed on the sensor surface , \neventually pre -functionalized. Note also that it is possible to first deposit a nanoscale \nthickness solid thin film on the s ensor surfa ce and then to fabricate a spacer on it, with the \nappropriate thickness.  \n The fabrication process of the YIG  ferrimagnetic nanostripes array on the GGG \n(Gadolinium Gallium Garn et) substrate, necessary for the second part of this quantum \nsensor  (fig.2 botto m),  follows  process es recently published32,44. Tho se process es were  \nsuccessful  in producing YIG nanostructured thin films with narrow spin wave resonances  at X \nband32,44. Shortly, t hose process es use a reactive magnetron sputtering system operating at \nroom temperature with a YIG target. The deposition has to be done through a mask \nfabricated on GG G, obtained by electron beam lithography  (fig.2 bottom ). After the YIG \ndeposition  and mask removal , a ther mal treatment at around 750-800°C under air flow or \noxyg en atmosphere, during around 1 or 2 hours , has to be performed32,44.  SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n5 \n \n \nfigure 2 :  Fabrication of the quantum sensor device : 4H -SiC part (top) and YIG/GGG part (bottom) ; see text \nfor details  on the various successive fabrication processes . When possible , and if it is ad vantageous, the \norder of some processes can be modified , as long as the key targeted quantum sensor properties are \nconserved.  \n Quantum sensing16-20,24,25 by optically detected17,18,21 PELDOR spectroscopy1,10,11,18,33 \n(fig.5)  is only possible if the V 2 spins probes created and coherently manipulated at the \nmicrowave probe frequency fs are sufficiently quantum coherent intrinsically, that is without \nany nearby target spin bath, in order to be able to feel the added spin decoherence1,17,18,24  \nproduced by the spin bath  of the sample  of study , when it is  driven at the microwave pump \nfreque ncy fp (fig.5 ). Let us  discuss  firstly the electron spin coherence time expected for the \nspin S=3/2 of a 4HSiC silicon vacancy (V 2) 21,30,31 ,45 created by this fabrication process few \nnanometers below the surface. Nuclear spin bath spectral diffusion21 is small in 4HSiC  which  \ncontains very few non -zero nuclear spins,  and it can be elimi nated by isotopic purification . \nBulk electron spin bath spectral diffusion  is sm all in lightly n -doped 4HSiC and can be \nreduced by chemical purification  and doping control21. Spin-lattice relaxation  should be  quite \ninefficient for V 2 spins probe s, in view of the very long spin coherence time of 100 µs \nobserved already at room te mperat ure for bulk V 2 spins probes21,30,31. Spin decoherence \ninduced by the residual paramagnetic states present at the 4H -SiC passivated surface  is \nnegligi ble for most V 2 spin probes , due to the low 2D residual defect  concent ration after \npassivation43 (6.1010 cm-2). Thus, the dominant intrinsic decoherence process for V 2 spins \nprobes in this quantum sensor device is expected to be  instantaneous diffusion1 in 2D , \noccu rring among the V 2 spins probe s having the same resonant magnetic field , at fixed \nmicrowave probe  frequency and under the strong dipolar magnetic field gradient produced SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n6 \n by the YIG nanostripes . Note that YIG is fully saturated at X band because its saturation \nfield32,44 is Bsat=1700 G and the external B0 field applied for EPR is around 3500 G .      \n \nfigure 3  : YIG nanostripes  magnetic properties assuming  the following d imensions , width W=500 nm , \nthickness T=100 nm, length L=100µm , and Bsat=1700 G . a/ and b/  Electron spin resonance spectrum at X  (9.7 \nHz) and Q (34 GHz) band respectively, showing, in b lue, the YIG nanostripes spin wave resonances and, in red, \nthe shifted paramagnetic resonance of reference g=2.00 electron spin s, placed at x opt=150 nm above  the YIG \nnanostripe center  (x=0) . Paramagnetic and f errimagnetic resonances have linewidth of 1 G h ere. c/  One \ndimensional eigenenergies of the spin waves along  z axis (horizontal lines) represented on top of the \ninhomogeneous effective confining potential  inside  a YIG  nanostripe  saturated along its width  (here \nz*=300+z ) ; z=0 corresponds to the center of the stripe  . d/ z component of the dipolar magnetic field of the \nYIG nanostripe as a function of x (black), as well as its gradient along x (red) multiplied here by 100 for \nclari ty. e/ and f/ Total effective Zeeman splitting at X band (dot line), expres sed in Gauss  (thus divided by (g \nµB), assuming g=2.00 ), as well as its two contributions:  the one of Bdz  to first order in blue, and the one of \nBdx in red to second order, as produced  by the YIG nanostripe , respectively at xopt (e/) and at xopt -10 nm  \n(f/), and both plotted versus z , to show the lateral homogeneity of this effective Zeeman splitting .  \n \n  The figure 3 summarizes the static and dynamic magnetic properties of the YIG \nnanostripes. The fig.  3d shows that the maximum magnetic field gradient in th e x direction, \nperpendicular to the GGG and 4HSiC surfaces, is of around 0.5 G/nm and is obtained at a \ndistance x opt=150 nm from the center of a given YIG nanostripe. That is why the spacer ha s \nto hav e a thickness of xopt + T/2 =200 nm, such that the V 2 spins probes feel the maximum  \nmagnetic field gradient.  The magnetic field gradient produced by such a YIG nanostripe is \nnot rigorously one dimensional along x. However , as I previously explained in the context of \nquantum computing27, locally , around x opt = 150 nm here , and laterally at z=0 +/ - 30 nm \nalong z, detailed calculations  clearly show (fig. 3e) that in this portion of plane  above each \nYIG nanostr ipe, the dipolar magnetic field can be considered as laterally homogeneous with \na precision of 0.1 G. Even in the portion of plane located at around x opt - 10 nm, and laterally \nat z=0 +/ - 30 nm along z, which is a possible  position where target spins could  be found, the \ndipolar magnetic field can be considered as laterally homogeneous with a precision of 0.3 G  SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n7 \n (fig. 3f) . As the V 2 spins probes in 4HSiC have a narrow linewidth21,30,31 ,45,46 of less than 1 G, \nwith a gradient here of 0.5 G/nm, one can thus consider  that all the V 2 spins probes located \nbetween  xopt and xopt-2nm  (fig 1 c), just  below the  4HSiC surface , and with z=0 +/ - 30 nm \nalong z  (weff=60 nm) ,  have the same resonant magnetic field with a precision of around 1 G. \nAs their 2D concentration  obtained by fabrication is 1/(32nm)2, their decoherence time  \nassociated to instantaneous diffusion in 2D is nume rically calculated  to be T ID,2D= 12.5 µs, \nand i s independent of the temperature . Selective microwave pulses1 can thus excite th is V2 \nspins probes  plane , without exciting the other  more diluted V2 spins planes located in the \nnext few nanometers of  4HSiC . The V2 plane - target spins plane distance  is thus  measured \nhere  with a precision of around +/- 1nm.   \n \nfigure 4  : Some optical properties of the quantum sensor described here: a/  ODMR setup: fibers bundle (6+1, \nin blue), GRIN lens (NA=0.5, 0.25 pitch, diam eter: 500 µm, in yellow ) for collimation after the central fiber, \nEPR tube (in gray), and 4H -SiC sculpted sam ple (edges in red, cone angles are 45°); all are inserted inside a \nmicrowave resonator like the MD5 flexline resonator (the YIG part of the sensor , supporting the SiC one, is \nnot shown here for clarity); also shown on a/, static B0z and microwave magnetic field B 1x(t), some near \nsurface V 2 electric dipoles aligned along the c axis of 4H -SiC (in violet, maximum emission along the z axis , \northogonal to the c axis), and some relevant optical rays for geometric optics investigation of the excitation \nand collec tion efficiencies of this new ODMR based setup for quantum sensing. Blue ray is an optical \npumping ray with many TIR on SiC faces. Black rays are also optical pumping rays, but TIR are not shown for \nclarity. Violet rays are photoluminescence rays emitted a t 10° with respect to the horizontal and they are still \ncollected by TIR in lateral fibers (NA=0.44, diameter: 500µm). See also zoom in SI. b/ sec tion view of the fiber \nbundle just above the SiC sample. c/Negatively charged silicon vacancy V 2 energy level scheme, explaining \nthe optical readout cycle and the optical pumping cycle. Level names31,4 5,46: 1: (Ground State, S=3/2, M sz= -\n3/2 ( or +3/2 )), 2:  (Excited State) , 3: (Meta -stable excited state) , 4: (Ground State, S=3/2, M sz= -1/2 ( or +1/2 )), \nk12 is lase r induced optical absorption/emission rate, k 21 is photoluminescence rate, k 23=k32=kISC is the \nintersystem crossing rate, k 34 is a non-radiative  relaxation rate. d/ and e/ : Numerical simulations of \npopulations, based on rate equations, showing the optical pumping31,4 5,46 time necessary to saturate the \npopulation of V 2 spins in the - 1/2 states (green curve) to its maximum value of 0.5 (Note: one can also show \nthat under such OP, the population of V 2 spins in the + 1/2 state also saturates to 0.5, using a similar energy \nlevel scheme and OP/OD cycles). In d/, k23=k32=1 /(17 ns) at 300K31,4 5,46, and in e/, k23=k32=1/(1700 ns) \nassumed at 5K, and for bo th, k21=1/(6ns), k 34=1/(107 ns), k 12sat=2.6 ns-1. Populations shown: N 1 in black, N 2 in \nred, N 3 in blue, N 4 in green. One finds an optical pumping time of around 20 µs at 5K, and 2 µs at 300K, with \nthose parameters.  \n SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n8 \n  \n It must be also noted here that microwa ve driving of any spin wave resonance of the \nYiG nanostripes of the quantum sensor, during the ODPELDOR sequence used for quantum \nsensing, would add unwanted decoherence27 to V 2 spins probes. That is why the \nferrimagn etic insulating YIG nanostripes were ca refully designed here such that there is no \nspectral overlap between their confined spin wave resonances27 (fig. 3 a, b, c), which are \nnarrow in YIG32,44, and the shifted paramagnetic resonances of the V 2 spins probes (fig. 3a, b). \nNote also that according to my previous theoretical calculation27, thermal fluctuations of Y iG \ndo not contribute to decoherence of V 2 spin probes, due to the reduce d saturation \nmagnetization of Yi G compared to the one of Permalloy previously considered in the context \nof quantum co mputing27. Note also that, as instantaneous diffusion is temperature \nindependent and as Y iG is still ferrimagnetic at room temperature, this hybrid Si C-YiG \nquantum sensor can be used in principle between 4K and 300K.  \n \n The ODMR  at X band of the ensemble o f V 2 spins probes used  for sensitive quantum \nsensing ,  is based on efficient optical pumping21,30,31,4 5,46 (fig. 4  a,c,d,e ), as well as  on the \nefficient collection  of V 2 spins probes  photoluminescence21,30,31,4 5,46 (fig. 4 a,b), by means of a \nfiber bundle28,29, a small GRIN microlens (fig . 1a,b and fig. 4 a,b ), and the man y total internal \nreflexion19 (TIR) occuring both in the sculpted 4HSiC sample (n=2.6) and in the optical fibers \n(fig 4 a and see also SI) . All components of this ODMR setup  can be introduc ed inside \nstandard X band pulsed EPR microwave resonator1,29 allowing PELDOR spectroscopy, like the \nMD5 flexline resonat or47, which accept EPR tubes with external diameter up to 5 mm . \n One can show that the  photoluminescence signal Spl, integrated during T  by the \nphotodetector, in the ODPLEDOR sequence (fig. 5a ), is given by ( see SI ):   Spl = S0.(1-f) ,   with \nS0 = pex.pcoll.pdet.(T/ԎV2).(N V2/8) and f, a function that depends on the parameters: 2.t1, 2.t2, \nTid,2D, td, C 2D,T, pB(fpump) (see SI for definitions and details). Note that pB(fpump) is equal to 1 \nwhen fpump equal the target  spins resonant frequency, and 0, when  fpump is far o ff reso nance \nwith the target spin s resonant frequency . In optimal experimental conditions, the Noise N pl \nis dominated by optical shot noise, Npl = (Spl(pB=0))0.5. Thus the \"net signal\" to \"noise\" ratio R \nis given by R=(S pl(pB=1) - Spl(pB=0 )) / Npl  . The detailed sensitivity analysis of this quantum \nsensor (see SI) shows, that in optimal exper imental conditions, one could obtain the 200 \nMHz ODPELDOR spectrum shown on fig. 6b (100 points, one point each 2MHz assumed \nhere) in 1.2 s, with a large signal to no ise ratio R=2600.   \n The numerically simulated (see SI) spins quantum sensing properties, o btained by \nODPELDOR (fig.5a), are shown on fig. 6. The figure 6a presents the shifted field sweep EPR \nspectrum at 9.7 GHz of V 2 spins probes located at x opt= 150 nm f rom YIG nanostripes (in \ngreen) and of two kinds of target spins S=1 located at x opt-dx=145  nm, that is on the sensor \nsurface (in blue and red, see legend for details), as it could be obtained by direct detected \nEPR, if it would be sensitive enough for Surf ace Paramagnetic Resonance. The edge spin \nwave resonance of Y iG nanostripes having the hig hest resonance field at 9.7 GHz has also \nbeen added to this spectrum (in pink). The shifted EPR line of V 2 at highest field is chosen \nhere for ODPELDOR, which means t hat B 0z is set to this field resonance value, and fs is set to \n9.7 GHz, while f pump is sca nned during ODPELDOR (fig. 5a). The figure 6b shows the resulting \nexpected X band ODPELDOR spectrum versus f pump-fs, scanned over around 200 MHz. The \nfigure 6c indica tes how the normalized ODPELDOR net signal to noise ratio (see SI), given by \nR/R opt= 1-VDeer(td, dx, C 2D,T), depends on  1-VDeer, VDeer being  the DEER11 signal, and thus  how \nit depends  on the relative distance dx between spins probes plane and target spins plane, on SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n9 \n the target spin plane concentration C 2D,T , and on time constant td. Thus clearly, this SiC -YiG \nquantum sensor can determine rapidly the target s pins plane EPR spectrum and its 2D \nconcentration down to 1/(20nm)2, with a sufficiently high net sign al to noise ratio, still \nassuming a V 2 spins probes planar concentration of 1/(32nm)2, and an associated \ninstantaneous diffusion decoherence time in 2D of T ID,2D= 12.5 µs.      \n \n \n \n \nfigure 5: a/ X band OD -PELDOR quantum sensing sequence and b/ X band ODMR spin echo decay sequence \nfor characterization of spin coherence time T2 of V 2 spins probes. The spins states -1/2 and +1/2 are prepared \nsimultaneously by optical pumping (laser pulse of 100 µs assumed here). The microwave probe frequency fs, \nand static fie ld B0z, are adjusted to obtain the paramagnetic resonance at this frequency fs with the chos en \noptically pumped EPR transition of V 2 probes spins, either (-3/2 < --> -1/2), or (+1/2 < --> +3/2 ). Both a/ and b/ \ntime resolved ODMR experiments  corresponds nearl y to standard PELDOR and Echo Decay experiment s1, but \nthey start after optical pumping and t hey are complemented by a last + /-Pi/2 pulse in order to transform \ntransverse magnetization Mx  (tSRT - T - twait), into populations of V 2 spins , which have differen t spin \ndependent photoluminescence and relaxation properties  under laser excitation . This  allow s the final optical \ndetection of EPR, the so-called  spins ensemble ODMR , by means for example, of a gated Photomultiplier \ntube (PMT) . As a first approximation  here, and to better understand the hybrid optical -microwave pulses \nsequences , spins states -1/2 and +1/2 are assumed Dark states, while  spins states -3/2 and +3/2 are assumed \nBright photo -luminescent states45,46.     \n Now I compare the sensitivity of this Si C-YiG fiber bundle based ODMR quantum \nsensor with other setups. Firstly, it must be noted that the same ODPELDOR spectrum as the \none of fig.6b could be obtained also in 1.2 s with a quantum sensor having a single V 2 spin \nprobe, assuming identical experimen tal parameters, but at the price of a reduced net signal \nto noise ratio of only R=2  (see SI) . This new spin  ensemble quantum sensor48 is thus 1000 \ntimes more sensitive than a similar single spin-based  quantum sensor . It is thus \nadvantageous in terms of bot h measurement time and sensitivity. Of course, ensemble \nmeasurements imply an additional statistical averaging of target spins plane properties, \nwhich is not present in single spin probe measurements, but such stati stics is often a \nrelevant information, li ke in biology10,11  and in realistic solid state devices5,6. Also, this spins \nensemble quantum sensor has a nanoscale spatial resolution in 1D due to the static gradient SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n10 \n used, but no scanning and thus no 3D imaging capabilities, contrary to some scanning s ingle \nspin sensors. Thus, those two kinds of quantum sensors are quite complementary research\n   \n \n \n \nfigure 6  : Spins sensing properties of the quantum sensor.  a/  The theoretical shifted field sweep EPR \nspectrum at fs= 9.7 GHz of spins S=3/2 of V2  spins probes (giso=2.0028, uniaxial magnetic anisotropy along c \naxis Dc= +35 MHz,  C3V) located at xopt =150 nm  (in green) and of two different ensembles  of anisotropic \nmolecular nanomagnets with target spins S=1  (S1=1, g iso,1=2.0028,D c,1=20MHz,  C3V, and S 2=1, \ngiso,2=2.0028,D c,2=180MHz,  C3V) located at xopt -dx=145 nm here , thus on the sensor surface  (assuming 3nm of \nSiO2 capping layer ). V2 spins and nanomagnets are assumed here to have their  C3V c axis orthogonal to B0z.  \nEPR simulation in a/ performed with Easyspin software.   b/ ODPELDOR spectrum versus fpump -fs, \nassociated to spectrum a/, assuming B0 z is set equal to the highest EPR resonance of V2 on a/. c/ \nDependence of ODPELDOR normalized net signal  to noise ratio  (see SI) , R/Roptimum=1 -V, on the relati ve \ndistance dx between spins probes plane and target spins plane (dx= 5  nm (1/) , 10 nm (2/), or 15 nm  (3/), \nfrom top to bottom), as well as on the target spin plane concentration (C 2D,Target =1/(d2), with d in nm).  Dark \ntrace is for td= 5µs, red trace is for td=3µs, blue trace is for td= 1µs  (see fig.5 for definition of td) . \n \ntools. However, this new quantum sensor based on spins probes ensemble , has not only the \nadvantage  of being much more sensitive  and fast er, but also to be compatible with standard \nX ba nd pulsed EPR spectrometers,  such that it should  be widely used in a soon  future  by \nmany researchers , already using  standard EPR  and who want to improve its performances . \nThe detailed comparison (see SI) of the sensitivity of standard X band direct inducti vely \ndetected EPR (DD -EPR) with the one of this quantum sensor upgrade d EPR (noted here \nQUSU -EPR) , shows that t he sensitivity gain on target spins number is at least of five orders of a/ b/ \nc/ dx = 5 nm  \ndx = 10 nm  \ndx = 15 nm  SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n11 \n magnitude . It thus clearly allows to perform surface EPR  using this quan tum sensor \ncombined  with a commercial X band pulsed EPR spectromete r and  an optical fiber  bundle . \nThis quantum sensor upgraded EPR  spectroscopy should thus open new research directions , \nlike in the field s of surface chemistry  and photovoltaic , in structur al biology  and \nnanomedicine , as well as  in optoelectronics , spintronics  and quantum information \nprocessing .       \n As a last remark, one can note that th is theoretical work, as well as the e xperimental \ndevelopment29 of this hybrid SiC -YiG quantum senso r, can be viewed as intermediate steps \ntowards the future development of an intermediate scale hybrid YiG -SiC spins qubits -based  \nquantum computer, following the guidelines I previously published27. This not scalable \nquantum computer design could however still be very useful for efficient quantum \nsimulations of new potential molecular drugs49. 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Zhu et al., Applied Physics Letters (2017),  110, p 252401.  \n \n45/ Spin and optical properties of silicon vacancies in silicon carbide - a Review.  S.A. \nTarasenko et al., Phys. Status Solidi B (2018), 255, p 1700258.    \n \n46/ Highly efficient optical pumping of spin defects in silicon carbide for stimu lated \nmicrowave emission.  M. Fischer et al., Phys. Rev. Applied (2018), 9, p 54006.  \n \n47/ Exploiting the symmetry of the resonator mode to enhance PELDOR sensitivity.  E. \nSalvado ri et al., Appl. Magn. Reson.  (2015), 46, p 359.  \n \n48/ Subpicotesla diamond m agnetometry .  T. Wolf et al., Phys . Rev. X (2015), 5, p 41001 . \n \n49/ Hardware -efficient variational quantum eigensolver  for small molecules and quantum \nmagnets .  A. Kandala  et al.,  Nature (201 7), 549, p 242.     \n     \n \n \n \n \n \n \n \n \n \n \n \n \n SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n15 \n  \nSUPPLEMENT ARY INFORMATIONS  \n The number of V 2 spins probes having the same resonant magnetic field placed at \nxopt=150 nm  above a given YIG nanostripe  (500 nm*100  nm*100 µm) , and within an \neffective width of W eff =60 nm around z=0 (fig. 1b), is estimated  to be at least equal to 3000 , \ntaking C2D, V2=1/(32nm)2. Assuming the YIG nanostripes are laterally separated by 5  µm, one \nhas an ensemble of around 500 identical YIG nanostripes over the useful  squar e surface of \nthe sensor estimated to be Su= 500 µm*500  µm, taking into account the spac er width. Thus,  \none has around 1.5 106 identical V 2 spins probes  on the sensor  surface  which have the same \nresonant magnetic field at fixed microwave frequency, that means under the strong gradient \nproduced by the  nanostripes . Note also that the surface S*  associated to target spins having \nthe same resonant magnetic field is approximately given by S*= (60 nm*100 µm) * 500 = 0.3 \n10-4 cm2.   \n The ODMR at X band of the ensemble of V 2 spins probes used for quantum sensing,  \nis based on efficient optical pumping21,30,31,4 5,46  (fig. 4a and 4b), as well as, on efficient \nphotoluminescence collection21,30,31,4 5,46  (fig. 4b and 4c) of the V2 spins probes  in the 4H -SIC \nsculpted sample , by means of a fiber bundle  containing seven fibers  (fig. 1a and 4c) and of a \nsmall GRIN  (gradient index)  microlens (fig. 1a) , as described in details below .   \n The central fiber  sends  exciting l ight, for example at 780 nm  or at 805 nm , along an \noptical axe common to the GRIN microl ens and to the cone shape dip of the 4H -SiC substrate  \n(45° is the half angle of the cone) . The GRIN lens, 0.25 pitch plan -plan, allows collimation of \nthe light emerging from the central fiber.  Then , by means of a first refraction  at the \nair(Helium)/SiC  interface  and then by means of the many total internal refle xions (TIR) \noccuring inside the SiC substrate (n=2.6) (fig.4b), the geometric configuration of the 4H -SIC \nsculpted sample allow s many optical rays to excite  the V2 spins located on the useful sensor \nsurface at the top of the truncated cone shape 4H -SIC isl and. This  TIR strategy is inspired \nfrom  a previous one  adopted for sensors fabrica ted with NV centers in diamond19, but with \nhere a different sample design , difficult to implement w ith diamond  technology , because \ndiamond is harder than Si C and diamond  has not a single defect  axis common to all spins \nprobes , like the V 2 center in 4H -SiC (the c axis of 4H -SiC is the only axis for V 2). This new \ndesign  allows  both  optimization of  optical  excitation and of  photoluminescence  collection  in \nthe restricted volume of an EPR tube of less than 5 mm in external diameter , as required for \nusing standard X band pulsed EPR resonator and spectrometer . Note that t he oblique \nincidence of the exciting light  at the senso r surface  (incidence angle of around 29° on sensor \nsurface  with this design ), after the first refraction , provides a non zero  optical electric field \ncomponent parallel to the c axis and thus allows the  efficient V2 electric dipole \nexcitation21,30,31,4 5,46.Here, I also assume that the optical excitation power at 780  nm or at \n805 nm, at the output of the central  fiber , is sufficiently high  to allow  the full saturation of \nthe optical transition , during OD and OP sequences.  It was previously shown46 that the \noptical power necessary to obtain saturation values of optical  V2 spins pumping is inversely \nproportional to their longitudinal spin -lattice relaxation time T1(T), at the temperature T . As \nT1(T) increases up to several tens of second at 5K46, then less than  1 mW at 780 nm spread \nover a 1mm*1mm square sample is suffic ient at 5K for obtaining such optical pumping \nsaturation.  Of course, at room temperature, much more power is required, typically more SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n16 \n than 100 mw46. Thus, from the above considerations, I consider  here  an optical excitation \nefficiency for V 2 spins located on the useful sensor surface of pex=1.  \n The photoluminescence  of excited negatively charged silicon vacancies V2 in 4H -SiC is \nemitted at 915 nm at low temperature (zero phonon line21,30,31,4 5,46 at 5K ). The excited V2 \nelectric dipoles,  aligned along the c  axis of 4H -SiC, emit their photoluminescence \npreferentially in the plane perpendicular to the c axis, which means here, at the horizon tal. \nThe edges at 45° of the truncated 4H-SiC cone shape island  thus allow , by one reflexion , to \ndirect most of the V 2 spins probes photoluminescence  vertically,  towards the six lateral \nfibers, in which it is efficiently propagated by TIR, till the infrared photoluminescence \ndetector . In order to evaluate more quantitatively the collection  efficiency of this fiber \nbundle  based optical setup , defined as the ratio of the collected optical power over the \nemitted optical power  by V 2 dipoles , one can use the classical model of a linear dipole \naligned along the c axis for the V 2 dipole and its emission profile determined using the \nPointing vector  expression . Using geometric optics  (see fig. 4a)  and considering the various \ndimensions of the setup and the relevant refractive index of the materials of the setup \n(nSiC=2.6, n air=1, an d for fibers n glass=1.5 and NA=0.44), one can determin e that almost all rays \nemitted by the V2 dipole s of the useful sensor surface around  the horizontal direction at +/- \n10° (= π/18 radians ), can, after relevant reflexion s (TIR)  on the 4H -SiC sample surfaces,  enter \ninto the lateral optical fibers with a suff iciently small angle such that TIR allows the \npropagation of those rays without loss till the end of the fibers, towards the photodetector.  \nConsidering  the Pointing vector  expression  associated to the V 2 dipole in spherical \ncoordinates , one can approximate  the collection efficiency pcoll by the ratio between the \nemitted PL and  the collected PL, assuming that the PL is collected by the fiber bundle setup  \nwhen Ө is comprised  between  (π/2 - π/18) and (π/2 + π/18).  \npcoll is thus given by the formula :  \npcoll = ( ꭍ sin3(Ө) dӨ, π/2 - π/18, π/2 + π/18) /( ꭍ sin3(Ө) dӨ, 0, π)  \nand thus one finds here pcoll = 0.25 .  \n  The photodetector can be a near infrared sensitive photomultiplier tube with low \ndark counts, or another low noise in frared photodetect ion setup . Here I assume a standard \ninfrared photodetector efficiency pdet=0.01 . Note also that the bundle is divided, outside the \nstandard EPR cryostat (like the CF935 fro m OXFORD for Bruker  EPR resonators ), into a single \nfiber, the central one used for optical excitation, and in to a bundle of the six lateral fibers \ncollecting the photoluminescence, further directed towards the photodetector.  \n \n Now let us evaluate the net s ignal to noise ratio R of this ODPELDOR experiment and \nthen the sensitivity of this YiG-SiC fiber bundle -based  quantum sensor. Starting from the \nDEER experiment expression1,11,33, directly  related to the ODPELDOR experiment shown on \nfig. 5a, and considerin g the optical detection of V 2 spins probes and thus the last additional \nπ/2 microwave pulse, one obtains a  photoluminescence signal expression Spl, integrated  \nduring T by the photodetector , given by:  Spl = S0.(1-f) ,   with S0 = pex.pcoll.pdet.(T/ԎV2).(N V2/8) \nand f, a function that depends on the parameters: 2.t1, 2.t2, Tid,2D, td, C 2D,T, pB. The function f \nis given by  f = exp( -((2.t1 + 2.t2)/ Tid,2D)2/3).( (1-pB) + p B.Vdeer(td, dx, C2D,Target ) ), where Vdeer is \nthe standard DEER signal. It can be numerically computed using the linear approximation \nand shell factorization model11. This model was previously introduced for calculating the  \nstandard DEER time domain signal in the case of a three -dimensional  distributions of spins.  \nHere, this model ha s been adapted to take into account the bidimensional random \ndistribution of the target spins in their well-defined  plane, parallel to the SiC substrate  SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n17 \n surface . The function p B depends on the frequency detuning between the microwave pump \nfrequency and the target spin resonance frequency at f ixed B 0z. Thus , pB=1 on resonance , \nand p B=0 far off resonance  for an appropri ate duration  π microwave pulse . The function p B is \ngiven by  the us ual probability transition formula describ ing the Rabi oscillation  between the \ntwo appropriate spins quantum states  under application of a microwave pulse .  \n \nIn optimal experimental conditions, the Noise N pl is dominated by the optical shot noise, \ngiven by Npl = (Spl(pB=0))0.5. Thus the \"net signal\" to \"noise\" ratio R is given by the formula \nR=(S pl(pB=1) - Spl(pB=0 )) / Npl . Thus, introducing Ropt, the optimal signal to noise ratio, R  is \ngiven, in the general case, by:  R=R opt*(1- VDeer(td, dx, C2D,T)), with Ropt given by the formula: \nRopt = (S0)0.5. exp( -((2.t1 + 2.t2)/ Tid,2D)2/3) / (1- exp( -((2.t1 + 2.t2)/ Tid,2D)2/3) )0.5 . Note here that \nR/R opt = 1-VDeer, that is why 1-VDeer is plotted on fig.6. Note also that Ropt depends o n the spin \ncoherence time T id,2D  of V 2 spins probes  and on the parameters t 1 and t 2 used in the \nODPELDOR exp eriment . R of course  depends on the concentration of target spins  C2D,T.  \n \nNow, assuming a sensor operating with t 1=0.5  µs, t 2=5.75  µs and 2  t1 + 2 t2=Tid,2D =12.5µs, \nand assuming C 2D,T=1/(10nm)2 , ie suf ficiently large such that when td=5  µs, V Deer(td,C 2D,T)=0 \nie 1 - VDeer(td, dx, C2D,T)=1 (fig.6c top black curve), then one finds the simple following \nexpression for the best expected signal to noise ratio: R= (1/e) .(S0)0.5.  With pex=1, pcoll=0.25 , \npdet=0.01 , a V 2 radiative recombina tion time  ԎV2=6ns , and around N V2=1.5.106 V2 spins \nprobes hav ing the s ame resonant magnetic field in the sensor (see  above ), and choosing a \nphotoluminescence integration time per ODPELDOR sequence T =6 µs for example, one finds \napproximately  R=260, for a single \"one shot one point\" ODPELDOR  experiment . The opti cal \nre-pumping time of V 2 spins  is numerically evaluated to T OPump = 20 µs at 5K assuming k ISC (5K) \n=1/ (1700 ns) (see fig. 4e), but the laser pulse is assumed to last 100 µs here for safety, \nconsider ing the unmeasured value of k ISC at 5K (only known is k ISC(300K)  =1/17ns at 300K31, \nsee fig. 4 d). The ODPELDOR microwave pulse s sequence after optical initialization of V 2 spins \nlast around 20  µs, such that the shot repetition time of full ODPELDOR is thus taken here to \nbe T exp=120µs. Both T tot,exp =N shot*Texp and T tot=N shot*T, increase proportionally to N shot, but R \nonly increase proportionally to (Nshot)0.5. Assuming N shot=100 per point  and a 100 points \nODPELDOR spectrum as a function of f pump (1 point each  2 MHz, 200 MHz scanned), one \ncould obtain such  a 200  MHz spectrum ( see fig. 6b) in 1.2 s with a signal to noise ratio \nR=2600, assuming negligible hardware and software delays for chang ing t he pumping \nmicrowave frequency (o therwise, the experimental time is determined by those delays ).  \n \nIt is here also releva nt to compare standard X band direct inductively  detected EPR (DD -EPR) \nsensitivity , with the one of this quantum sensor  upgraded EPR method . Assuming a 2D target \nspins concentration C 2D,T=1/(10nm )2, and estimating the surface S* of target spins seen by V 2 \nspin probes and having the same resonant magnetic field to around S*=0.3 10-4 cm2 (see  \nabove), one finds that around 3. 107 target spins are sensed by the V 2 spins probes in 12 ms \nper point (one point each 2  MHz, 100 shots per point), with R=2600 . As in DD -EPR15 at X \nband one can typically measure 1011 spins at 300K or 109 spins at 3K  in 1 s with R DDEPR  =3 \n(assuming a 1G linewidth for spins and a 1  Hz detection bandwidth) , one finds  that in order \nto obtain R=2600 in 12 ms, one would need 1015 targe t spins at 300K or 1013 target s pins at \n3K with DD -EPR. Th e sensitivity gain on target spins number with this quantum sensor is thus \ncomprised between  5 and 8 orders of magnitude.  Note that the probe spin s sensitivit y is SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n18 \n considerably higher than the target  spins sensitivity, and it could in principle  reach the single \nV2 probe spin sensitivity with long enough  accumulation times . \n \n Below , I also provide some results  (fig. aux. 1)  of the SRIM simulation s of 22 keV As+ \nions implantation in the trilayer Zn0  (20 nm )/SiO 2(5 nm )/4H-SiC (type n <5.1015 cm-3), \nallowing, after etching of ZnO and SiO 2, to produce shallow silicon vacancies around 2 nm \nbelow the surface of 4H -SiC with an average 2D concentration of C 2D,V2= 1/(32nm)2. SRIM \nsimulations  also confirms the advantage of using a trilayer and not just a Zn0/4H -SiC bilayer , \nbecause one can see on fig. aux. 2, that some Zn atoms can reach the SiO 2 layer due to the \nimplantation process  and related collisions  (SiO 2 is furthe r removed by etching ), but not the \nSiC substrate, thus avoiding pollution with the Zn element of the SiC substrate surface , used \nfor quantum sensing with the  silicon vacancies also produced by this implantation  process .  \n180 200 220 240 260 280 300 320 340 360 380 400 4200,04,0x1048,0x1041,2x1051,6x105220 240 260 280 300 320 340 360 380 400 420-0,010,000,010,020,030,040,050,06As+ ions @ 22 kev \n(cm-1)\ndepth (A°) As22kevTrilVSi (per ion As+ \n@ 22 kev and per A°)\ndepth (A°) VSi\n \nfig.A ux.1:  SRIM simulation  of As+ ions implantation at 22 keV in this trilayer  system  (100 000 shots) . \n   SiC-YiG X band quantum sensor .  (J. Tribollet  - 01/2019 ) \n19 \n \n \n \n \nfig.Aux. 2: SRIM simulation of As+ ions implantation at 22 keV in this trilayer system  (here 6000 shots).  \n \n Below,  I also provide  (fig. aux. 3) a zoom of fig. 4a used for the di scussion of \ngeometric optics in the fiber bundle based ODMR setup adapted to the SiC -YiG quantum \nsensor described here.  \n \nfig.Aux. 3: Zoom of the setup for geometric optics analysis.  \n \n  "    },    {        "title": "2307.04669v2.Reversal_of_the_skyrmion_topological_deflection_across_ferrimagnetic_angular_momentum_compensation.pdf",        "content": "Reversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation\nReversal of the skyrmion topological deflection across ferrimagnetic\nangular momentum compensation\nL. Berges,1R. Weil,1A. Mougin,1and J. Sampaio1\nUniversité Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France\n(*Electronic mail: joao.sampaio@universite-paris-saclay.fr)\n(Dated: 6 October 2023)\nDue to their non-trivial topology, skyrmions describe deflected trajectories, which hinders their straight propagation\nin nanotracks and can lead to their annihilation at the track edges. This deflection is caused by a gyrotropic force\nproportional to the topological charge and the angular momentum density of the host film. In this article we present\nclear evidence of the reversal of the topological deflection angle of skyrmions with the sign of angular momentum\ndensity. We measured the skyrmion trajectories across the angular momentum compensation temperature ( TAC) in\nGdCo thin films, a rare earth/transition metal ferrimagnetic alloy. The sample composition was used to engineer the\nskyrmion stability below and above the TAC. A refined comparison of their dynamical properties evidenced a reversal\nof the skyrmions deflection angle with the total angular momentum density. This reversal is a clear demonstration of\nthe possibility of tuning the skyrmion deflection angle in ferrimagnetic materials and paves the way for deflection-free\nskyrmion devices.\nThe discovery of efficient driving of chiral magnetic tex-\ntures by current-induced spin-orbit torques1–3has opened\nthe possibility of energy-efficient and high-performance spin-\ntronic devices4,5, with applications in digital6or neuromor-\nphic7–10computation, ultra-dense data-storage11,12, and sig-\nnal processing13,14. Chiral textures are stable in magnetic thin\nfilms with a significant Dzyaloshinskii-Moriya interaction\n(DMI), typically induced with an adjacent heavy-metal layer\n(e.g. Pt/Co). Additionally, the heavy-metal layer, through the\nspin Hall effect, converts an applied charge current into a spin\ncurrent that drives the magnetic textures by spin orbit torque\n(SOT). Very promising mobility of chiral magnetic domain\nwalls (DW) has been observed1,15, with nonetheless a saturat-\ning mobility at large current densities2. Another archetypal\nchiral magnetic texture is the skyrmion, a small (down to few\ntens of nm) radially symmetric whirling texture. Although\nhighly mobile16–19, their non-trivial topology induces a trans-\nverse deflection of their trajectory, a phenomenon known as\ngyrotropic deflection or skyrmion Hall effect18,20,21. This re-\nduces the velocity in the forward direction and can lead to the\nannihilation of the skyrmion at the edges of the hosting mag-\nnetic track, and is thus highly undesired.\nThe gyrotropic deflection can be mitigated in magnetic sys-\ntems with anti-parallel lattices22, such as antiferromagnets or\nferrimagnets, where the overall angular momentum density of\nthe double skyrmion can be suppressed. In particular, fer-\nrimagnetic alloys of the rare-earth/transition-metal (RETM)\nfamily, where the RE and TM moments are antiferromagnet-\nically coupled23,24, are a promising example. In a previous\nwork by our team, it was shown that skyrmions in GdCo thin\nfilms attained the high-mobility linear regime beyond pinning,\nand that their velocity and deflection followed the predictions\nof the Thiele model25. However, there is still only little ex-\nperimental evidence of the advantages of these systems26,27,\nespecially regarding the control of the gyrotropic deflection.\nIn RETMs, The balance between the moments of differ-\nent nature can be changed with alloy composition or temper-\nature which leads to two points of interest for skyrmions. At\nthe first one, the magnetic compensation temperature TMC, themagnetization of the two sub-lattices are equal, the total mag-\nnetization ( Ms=MTM−MRE) vanishes, and the size of the\nskyrmions is minimal due to the absence of dipolar fields27.\nAs RE and TM have different gyromagnetic ratios ( γREand\nγTM), the total angular momentum density ( Ls=MTM\nγTM−MRE\nγRE)\nwill vanish at a different temperature, the angular compensa-\ntion temperature TAC. Both TMCandTACdepend on composi-\ntion. The reduction and reversal of the total angular momen-\ntum, which is the root cause of magnetic precession, leads to\ninteresting dynamical properties near TAC, such as e.g. the re-\nversal of the deflection angle of chiral domain wall fingers28\nor the precessionless motion of magnetic domains walls29.\nHowever, the reversal of the skyrmion gyrotropic deflection\natTAChas not yet been demonstrated.\nIn this letter, we measure the velocity and deflection angle\nof skyrmions driven by spin-orbit torques in two Pt/GdCo/Ta\nfilms of different composition, above and below their TAC. We\nshow the dependence of the deflection with angular moment\ndensity, and in particular its reversal by changing sample com-\nposition or temperature. A quantitative analysis with a rigid\ntexture model based on the Thiele equation is used to char-\nacterize the role of the material parameters on the skyrmion\ndynamics.\nThe skyrmion dynamics were measured in two samples.\nSample 1 is composed of a film of (Si/SiOx(100))/ Ta(1)/\nPt(5)/ Gd 0.32Co0.68(5)/ Ta(3) and sample 2 of (Si/SiOx(300))/\nTa(3)/ Pt(5)/ Gd 0.3Co0.7(8)/ Ta(5)/ Pt(1) (thicknesses in nm)\nas presented in the insets in Fig. 1a. The samples were pat-\nterned into 10 µm- or 20 µm-wide tracks in order to apply\ncurrent pulses (Fig.1b). The magnetization as a function of\ntemperature was measured by SQUID magnetometry on un-\npatterned samples and is presented in Fig. 1(a). Sample 1\npresents a TMCaround 360 K whereas sample 2 presents a\nTMCaround 200 K. Therefore, at room temperature, sam-\nple 1 is RE-dominated whereas sample 2 is TM-dominated,\nwhere RE or TM domination refers to which sublattice has\nthe higher magnetic moment and therefore aligns with an ex-\nternal magnetic field. It is useful to use the effective ferromag-arXiv:2307.04669v2  [cond-mat.mtrl-sci]  5 Oct 2023Reversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation 2\nFIG. 1. a) Msversus temperature for sample 1 (top panel,\nblack points) and sample 2 (bottom panel, gray points) measured\nby SQUID (Superconducting Quantum Interference Device) magne-\ntometry, and mean-field-computed curves of Ms(solid line) and Ls\n(dashed line) for both samples. Samples stacks are presented in in-\nsets and skyrmion temperature stability regions in colored bands. b)\nTypical magnetic device studied for sample 2. c) Examples of differ-\nential MOKE images obtained in sample 2 at 350 K.\nnet model of ferrimagnets30, which assumes a signed mag-\nnetization and angular momentum density that are positive,\nby convention, when TM-dominated: Ms=|MCo|−MGd|and\nLs=|LCo|−LGd|. The exact determination of the TACis not\nstraightforward. It was therefore deduced for both samples,\nusing the mean field model described in ref.25. The calculated\nLS(T)are shown by the dashed lines in Fig. 1(a), and yield\nTAC=416 K for sample 1 and TAC=260 K for sample 2.\nThese results are consistent with the empirical law described\nin ref.31which gives TACfor GdCo between 40 to 60 K above\ntheTMC.\nThe magnetic textures are observed in each sample as\na function of temperature by magneto-optical-Kerr-effect\n(MOKE) microscopy. A typical differential MOKE image is\npresented in Fig. 1(c). Skyrmions are observed in the tem-perature ranges indicated by the color bands in Fig. 1(a). In\nthese ranges, starting from a saturated state and lowering the\napplied external magnetic field, skyrmions with a core of op-\nposing magnetization will naturally nucleate at small enough\nfield (−30 to 0 mT for an initial saturation at large negative\nmagnetic field). Skyrmions can also be nucleated by applying\nelectrical pulses25,32. A typical phase diagram (versus tem-\nperature and field) of these samples is presented in a previ-\nous work25. In the studied temperature range, sample 1 only\npresents one skyrmion stability range around 290 K, whereas\nsample 2 presents two skyrmion stability ranges, one around\n90 K and a second around 350 K. In sample 1, the skyrmion\nstability range is below TMC(and TAC), where the film LS<0,\nand so these are dubbed RE-dominated skyrmions. In sam-\nple 2, the skyrmions at 90 K are RE-dominated as well, while\nthe skyrmions at 350 K are TM-dominated (above TMCand\nTACwith therefore LS>0). Note that in the MOKE images,\nthe signal is proportional to the Co sublattice, independently\nof the temperature33. Thus, skyrmions with a core Co mo-\nment pointing along the same direction will appear with the\nsame color (black for −zwith our experimental conditions),\nwhether they are RE- or TM-dominated (Fig. 1c).\nOnce skyrmions are nucleated, electrical pulses of 3 to\n10 ns are applied and MOKE images are acquired in order\nto study the skyrmions dynamics. The skyrmion motion\nis tracked over several pulses using a partially-automated\nprocess described in ref.25, and their velocity and deflection\nare calculated considering the pulse duration and the traveled\ndistance. Typical images of skyrmions displacements are\nshown in Fig. 2, in the case of sample 2 at low temperature\nandLs<0 (a) and high temperature and Ls>0 (b). The\naverage skyrmion diameter was similar for the three studied\ncases, 0.86 ±0.28 µm. An example of the observed skyrmion\ndynamics in sample 1 is presented in Fig. 2(c) with a super-\nposition of successive MOKE images where the skyrmion\ncolor refers to the MOKE image number.\nThe skyrmion deflection ( θsk) and velocity ( v) versus\napplied current density ( j) are presented in Fig. 3(a,b) for\nthe three cases: RE-dominated skyrmions in sample 1, and\nRE- and TM-dominated skyrmions in sample 2. Videos of\nsuccessive displacements in both samples are shown in S.I. In\nthe three cases, the velocity shows a clear depinning transition\nabove a current threshold (different for each case), and then\nfollows a linear regime. The mobility in the linear regime\n(i.e.∆v/∆j) is much higher in sample 2 than in sample 1. In\nsample 2, the mobility of TM-dominated skyrmions is slightly\nhigher than RE-dominated skyrmions. These differences in\nmobility will be discussed later. The linear regime extends up\nto 190 m/s in sample 1 and to 450 m/s in sample 2. At highest\nj, skyrmions are nucleated by the pulse, which hinders the\ntracking analysis and thus limits the maximum jthat can\nbe examined. In the linear regime, the deflection angle θsk\nis approximately constant with the current density, and its\nabsolute value is about 40◦for the three cases. The deflection\nangle is clearly reversed between the TM- and RE-dominated\nskyrmions: it is positive for TM-dominated skyrmions (in\nsample 2) and negative for RE-dominated skyrmions (in bothReversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation 3\nFIG. 2. a)-b) Example of three successive MOKE images separated by 10-ns (6 ns) pulses showing the displacement of skyrmions in sample\n2 with at a) at T = 110 K and j=120 GA/m2and b) T = 350 K and 150 GA/m2(field around 0 mT.) The colored circles identify the same\nskyrmions in the three images. As the images in a) were obtained using a cryostat and are of lower resolution, a different temperature from\nthe one used in the dynamical studies (90 K) was used to render the skyrmions larger and more visible. c) Superposition of four consecutive\nMOKE images, in the case of sample 1 with Ls<0, showing the propagation of three skyrmions for 2 and 4 images. The different forces\ndefined in the Thiele equation acting on the skyrmions are sketched around the black dot.\nFIG. 3. a) Averaged skyrmion deflection θsk, and b) velocity, mea-\nsured in sample 1 at 290 K and 2 at 90 K and 350 K. The error bars\nrepresent the standard deviation of the measurements. The dashed\nlines are obtained with the Thiele based model (with fitted θSHE).\nThe color bands represent the expected error due to the experimental\nerror of θSHE, as described in the main text.\nsamples). The deflection also reverses with core polarity,\ni.e. with the Co moment pointing along +z(which appear\nas white skyrmions in the MOKE images; see Supplemental\nMaterials). The θskin the pining regime is measured to belarger than in the flow regime in sample 1, whereas it is lower\nin sample 2. This is perhaps a bias induced by the different\nnucleation protocol used in these measurements. For sample\n1, skyrmions were only nucleated by current pulses, mostly\nnear one of the edges due to the Oersted field25, whereas\nfor sample 2 they were first nucleated homogeneously by\nmagnetic field. As skyrmions can be annihilated at the\nedges, only the skyrmions that deviate towards the center are\naccounted for, which biases the measurement of the mean θsk.\nThe skyrmion dynamics in the linear regime can be quanti-\ntatively analyzed using a rigid-texture formalism based on the\nThiele equation34. It expresses the equilibrium of all forces\napplied on the magnetic texture that reads in our case as:\nFG+FSOT+αDv=0, where FSOTis the SOT force, FGthe\ngyrotropic force and αD is, in general, a tensor describing\nthe dissipation. This formalism can be applied to skyrmions\nin double-lattice systems as presented in refs.25,35. These\nforces are depicted in Fig. 2 c), on a black dot representing\na skyrmion in the case of Ls<0. The norm of the skyrmion\nvelocity |v|and its deflection θskcan be deduced to be:\n|v|=v0p\n1+ρ2(1)\nθsk=arctan (ρ) (2)\nIn the limit of skyrmions larger than the domain wall width\nparameter ∆, the parameters v0andρare:\nv0≈ −π∆\n2Lα¯h jθSHE\n2et(3)\nρ≈∆\n2πRLS\nLαn (4)Reversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation 4\nT<TMC TMC<T<TAC T>TAC\nMs − + +\nLs − − +\ncore magnetic moment +z⊙ − z⊗ − z⊗\ncore cobalt moment −z⊗ − z⊗ − z⊗\npCo −1 −1 −1\nFG·y − − +\nFSOT +x +x +x\nTABLE I. Signs versus temperature of the material parameters,\nof the skyrmion core configuration parameters and of the expected\nforces acting on a skyrmion with negative polarity (black in the\nMOKE images) driven by a positive ( +x) current.\nParameter Sample 1 (290 K) Sample 2 (350 K)\nγ/2π[GHz/T]a18.3 40.8\nαa0.15 0.175\n|MS|[kA/m]b78 125\n|LS|[kg/(ms)]b6.8×10−74.9×10−7\n|Lα|[kg/(ms)]b1.02×10−78.5×10−8\nµ0Hk[mT]c200 60\nKu[kJ/m3]c11.5 13.6\nD[ mJ/m2]a−0.22 −0.14\nA[pJ/m]a4.6 (4.6)d\nθSHEe0.04 0.09\n2R[µm]f0.85±0.28 0 .86±0.28\nTABLE II. Measured parameters used in the model.aγ,α, the ex-\nchange stiffness Aand the DMI strength D(not used in the model)\nwere determined with Brillouin light scattering (BLS) at 290 K (sam-\nple 1) or 350 K (sample 2).bMs(T)(Fig. 1a) was used to determine\nLS=MS/γandLα=LSα.cHkwas measured from hysteresis cy-\ncles and BLS, from which Kuwas deduced.dAwas measured only\non sample 1 and assumed to be the same in sample 2.eθSHEwas\ndetermined by transport measurements using the double-harmonic\ntechnique25,36(see Supplementary Materials).fThe shown variation\nis the standard deviation of the observed radius and not the error of\nthe average value.\nwhere ¯his the Planck constant, ethe fundamental charge,\ntthe magnetic film thickness, θSHE is the effective SHE\nangle in the Pt layer, Lα=LSαthe energy dissipation rate,\nn=pCo4π=±4πthe topological charge of the skyrmion, R\nits radius, and pCo=±1 is the orientation along zof the core\nCo moment. Because Lαis always positive, the sign of the\ndeflection is given by the sign of the product of Ls(positive\nforT>TAC) and pCo. This sign is presented in Table I as\na function of temperature for pCo=−1, which is the case\nshown here (black skyrmions).\nThe parameters needed for the model were measured on\nboth samples (see Table II). Hk(T)was obtained by analyz-\ning hysteresis loops at various temperatures, which yielded a\nvalue for Ku(Ku=µ0HkMs/2−µ0M2\ns/2) with negligible ther-\nmal variation. The domain wall width parameter was calcu-\nlated using ∆=p\nA/Keff, where Keff=µ0HkMs/2 is the effec-\ntive anisotropy. For Sample 2 at 90 K ( MS=135 kA/m), the\nthermal variation of KuandAwas neglected (as it is smallerthan the precision of the other parameters) and the values at\n350 K were used; LS=12.1×10−7kg/(ms) and Lα=10.7\n×10−8kg/(ms) were deduced using a mean-field model as de-\nscribed in ref.25, assuming constant sub-lattice Gilbert damp-\ning parameters ( Lα=αCo|LCo\ns(T)|+αGd|LGd\ns(T)|). The\nskyrmion diameter was taken from the average observed di-\nameter, which is very similar for the three studied cases37.\nThese measured parameters allow to constrain the model\nand obtain curves for the velocity and deflection angle (dashed\nlines in Fig. 3). A constant deflection is predicted, and its\nvalue is obtained with no fitting parameters. The velocity is\npredicted to be linear with j, and its slope is obtained with\na single fitting parameter, θSHE. The fitted values ( θSHE =\n0.03 for sample 1 and 0.09 for sample 2) are consistent with\nthe precision of the measured θSHE(see Table II and Suppl.\nMat.). The model prediction range, calculated with the esti-\nmated error θSHE, is shown in the figure as a color band38.\nAbove the depinning threshold, where the model is expected\nto be valid, it both reproduces the qualitative behavior (con-\nstant deflection and linear velocity) and agrees quantitatively\nwith the experimental data, within the estimated error margin.\nIn particular, the sign of the deflection angle observed in the\nexperiments agrees with Eq (3b) taking into account the LSof\nthe film ( LS<0 for RE-dominated skyrmions and LS>0 for\nTM-dominated skyrmions).\nThe skyrmion mobility, given by the slope of the velocity\nversus j(Fig. 3(b)), is much higher in sample 2 than in sample\n1 (1.80 at 350 K vs 0.6 m ·s−1/GA·m−2, respectively). This\ndifference in mobility cannot be ascribed to a difference in\nskyrmion diameter (see eq. 1), as the two conditions present\nvery similar average sizes (Table II). This large difference has\nmultiple origins. First, the Lαof sample 2 is lower by 20%.\nThe second major cause is the difference of the film stacks, in\nparticular the thickness of the Ta capping layer. The measured\nθSHEis more than twice higher in sample 2 than in sample 1\n(Table II). This can be expected to be due a better passivation\nof the Ta layer in sample 2 which can therefore contribute\nmore to the SOT than the thinner (3 nm) Ta cap of the sample\n1 which is probably fully oxidized.\nFinally, comparing the skyrmion velocity curves for the\ntwo conditions in sample 2 (at 90 and 350 K), it can be seen\nthat both the depinning current and the mobility in the linear\nregime are significantly different. The depinning current is\nhigher at 90 K, which can be attributed by the thermal nature\nof the depinning process39. The difference in mobility is not\ndue to a difference in skyrmion diameter (which again is very\nsimilar in all three studied conditions). It can be expected that\nseveral magnetic parameters vary between 90 and 350 K, but\nthe experimental mobility can be understood by considering\nonly the variation of Lα(Lα(90 K )\nLα(350 K )≈1.25). This result and the\nThiele model suggest that Lαis a more pertinent parameter\nthan αto characterize the role of dissipation in the skyrmion\nmobility. Interestingly, Lαcan be more easily optimized than\nαto increase mobility, by increasing the sample temperature\n(as was the case here) or by decreasing the material’s Curie\ntemperature (all other parameters remaining equal). A recent\nwork39on skyrmions measured at relatively high temperature\nalso seems to point toward such an effect which seems to beReversal of the skyrmion topological deflection across ferrimagnetic angular momentum compensation 5\nan interesting path to increase skyrmion mobility.\nIn conclusion, we observed the propagation of skyrmions\nin the flow regime, i.e., beyond the effects of pinning in two\nGdCo samples, below and above the angular compensation\ntemperature. The observed mobilities were very large, with\na velocity up to 450 m/s. The skyrmion dynamics was stud-\nied in three cases, two in RE-dominated films and one in a\nTM-dominated film. The deflection angle was constant with\ndriving current and its sign was opposite between RE- and\nTM-dominated cases, both when comparing two samples of\ndifferent composition and when comparing two temperatures\n(above and below TAC) in the same sample. This confirms the\nmodulation of deflection angle θskwith LS.\nThese experiments demonstrate the effects of the angular\nmomentum density LSof the host material on the deflection of\nskyrmions. They show that θskcan be reversed in GdCo ferri-\nmagnetic thin films across their angular compensations, either\nby changing the alloy stoichiometry or simply its temperature.\nIn particular, the reversal of sign of θskacross compensation\nstrongly supports that θskshould be zero at angular moment\ncompensation. The engineering of magnetic parameters that\nwas done to produce the two presented skyrmion-hosting sam-\nples could be repeated rather straightforwardly to engineer a\nfilm with stable skyrmions at TACwith no deflection.\nACKNOWLEDGMENTS\nThe authors thank Stanislas Rohart for fruitful discussions,\nand André Thiaville for the study of the sample properties by\nBLS. This work was supported by a public grant overseen by\nthe French National Research Agency (ANR) as part of the\n“Investissements d’Avenir” program (Labex NanoSaclay, ref-\nerence: ANR-10-LABX-0035, project SPICY). Magnetome-\ntry and Anomalous Hall effect measurements were performed\nat the LPS Physical Measurements Platform.\nSUPPLEMENTARY MATERIAL\nSee supplementary material for videos of successive\nMOKE images showing the skyrmion motion for the three\ntemperature regions discussed in the text. Motion of\nskyrmions of opposite polarity (i.e., pCo= +1; white in the\nMOKE images) is also shown for sample 1. A note on the\nanalysis of the images and the selection of relevant textures\nis also included. Experimental data of θSHEare also included\nfor both samples at several temperatures.\nDATA AVAILABILITY STATEMENT\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.1T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. 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Ohno, “Quantitative char-\nacterization of the spin-orbit torque using harmonic Hall voltage measure-\nments,” Physical Review B 89, 144425 (2014).\n37The skyrmions show a large dispersion of diameter (Table II), which we\nestimate to lead to a ±14% dispersion of velocity and to a ±10º dispersion\nof the deflection angle. However, as the values of velocity and deflection\nwere averaged over many skyrmions and many displacements, the error\ndue to the size dispersion is drastically reduced and is neglected.\n38We estimate that the main sources of error of ρandvis the spin Hall effi-\nciency ( θSHE). which was estimated to be 25% (sample 1) and 12% (sample\n2); see supplemental materials.\n39K. Litzius, J. Leliaert, P. Bassirian, D. Rodrigues, S. Kromin, I. Lemesh,\nJ. Zazvorka, K.-J. Lee, J. Mulkers, N. Kerber, D. Heinze, N. Keil, R. M.\nReeve, M. Weigand, B. Van Waeyenberge, G. Schütz, K. Everschor-Sitte,\nG. S. D. Beach, and M. 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For Fe xO+\nycation clusters we also calculate the corresponding vibrat ion spectra and compare\nthem with experiments. We successfully identify Fe 3O+\n4, Fe4O+\n5, Fe4O+\n6, Fe5O+\n7and propose struc-\ntures for Fe 6O+\n8. Within the triangular geometric structure of Fe 3O+\n4a non-collinear, ferrimagnetic\nand ferromagnetic state are comparable in energy. Fe 4O+\n5and Fe 4O+\n6are ferrimagnetic with a resid-\nual magnetic moment of 1 µBdue to ionization. Fe 5O+\n7is ferrimagnetic due to the odd number of\nFe atoms. We compare the electronic structure with bulk magn etite and find Fe 4O+\n5, Fe4O+\n6, Fe6O+\n8\nto be mixed valence clusters. In contrast, in Fe 3O+\n4and Fe 5O+\n7, all Fe are found to be trivalent.\nPACS numbers: 36.40.Cg, 36.40.Mr, 61.46.Bc, 73.22.-f\nIn nano technology there is an ever increasing demand\nfor increasing the density of electronic and magnetic de-\nvices. This continuous downscaling trend drives the in-\nteresttoelectronicandmagneticstructuresatthe atomic\nscale. In essence, two things are required: first, novel\nmaterials and building blocks with exotic physical prop-\nerties. Second, a fundamental knowledge of the physical\nmechanism of magnetism at the sub-nanometer scale.\nAtomicclusters,havinghighlynon-monotonousbehav-\nior as a function of size, are a promising model system to\nstudy the fundamentals of magnetism at the nanoscale\nand below. Such clusters consist of only tens of atoms.\nQuantum mechanics starts to play an essential role at\nthis small scale, adding extra degrees of freedom. Since\nthese clusters are studied in high vacuum, they are com-\npletely isolated from their environment.\nTo use these clusters as a model system, as a starting\npoint, a detailed understanding of the relation between\ntheir geometry and electronic structure is required.\nEvenin thebulk, ironoxidehasawidevarietyofchem-\nical compositions and phases with many interesting phe-\nnomena, such as the Verwey transition in magnetite.1,2\nExperimentsperformedonsmallgasphaseFe xOyclus-\nters beyond the two-atom case are scarce. The structure\nof one and two Fe atoms with oxygen has been studied\nin an argon matrix using infrared spectra.3,4The corre-\nsponding vibration frequencies have been identified using\ndensity functional theory (DFT).\nIron-oxide nanoparticles have been investigated for\ntheirpotentialuseascatalystinchemicalreactions.5Fur-\nthermore, since the iron-oxygen interaction has a funda-\nmental role in many chemical and biological processes,\nthere have been quite some studies, both experimen-\ntal and theoretical, of the chemical properties of Fe xOy\nclusters.6–12\nThe possible coexistence of two structural isomers for\nstoichiometric iron-oxide clusters in the size range n≥5\nwas experimentally measured using isomer separation by\nion mobility mass spectroscopy for FenOnand FenOn+1(n= 2-9).13Furthermore, the formation of Fe xOyclus-\nters has been studied in the size range ( x= 1-52).14\nThe number of theoretical studies is, however, man-\nifold. The magic cluster Fe 13O8was extensively stud-\nied and identified as a cluster with C1but close to D4h\npoint group symmetry.15–19However, also the geometry\nand electronic structure of other cluster sizes have been\nstudied theoretically.15,20–25The prediction of geometric\nstructures requires a systematic search of the potential\nenergy surface to find the global minimum.\nThe majority of theoretical studies were performed\nusing DFT.4,6,9,10,13–17,20–24,26The number of works in\nwhichFe mOnclusterswerestudied with methods beyond\nDFT is very limited and restricted to very small cluster\nsizes. For FeO+its reactivity towards H2was studied\non a wave-function-based CASPT2D level.12For Fe2O2\nthemolecularandelectronicstructurewerecalculatedus-\ning both DFT and wave-function-based CCSD(T) meth-\nods and a7B2uground state was found.25Furthermore,\nRef. 25 reports that B3LYP functional and CCSD(T)\ncalculations give the same energy ordering of different\nstates, although the energy differences are overestimated\nby the B3LYP approach.\nRecently, the structural evolution of (Fe 2O3)n\nnanoparticles was systematically investigated from the\nFe2O3cluster towards nano particles with n= 1328.9,26\nIn the size range of n= 1-10, an interatomic potential\nwas developed and combined with a genetic algorithm in\nsearchofthe lowest-energyisomer. The isomerslowest in\nenergy were further optimized using DFT and the hybrid\nfunctional B3LYP. This way, a systematic prediction of\nthe cluster structure was done for neutral (Fe 2O3)nclus-\nters.\nBecause of its high computational burden, in DFT the\ngeometric structure is often only relaxed into its nearest\nlocal minimum on the potential energy surface (PES).\nThere is no guarantee that this local minimum corre-\nsponds to the global minimum. Almost all previous2\nworksonlyconsidereitherrandomstructuresormanually\nconstructed geometries. However, for increasing cluster\nsize these methods become less successful in finding the\nlowest-energy isomer. Genetic algorithms, in which sta-\nble geometries are used to create new structures, proved\nto be efficient in finding the global energy minimum.27\nThis method has been successfully used for transition-\nmetal oxide clusters.28,29\nIdentification of the geometric cluster structure is a\ndelicate and computationally demanding task. There-\nfore,comparisonwithanexperimentalmethodtoconfirm\nthe theoreticalfindingsis essential. In this work, wecom-\nbine previously reported experimental vibration spec-\ntra30with first-principles calculations and a genetic al-\ngorithm to determine the geometric structure of cationic\nFexO+\nyclusters. Of the nine cluster sizes reported in\nRef. 30, only the geometric structure of Fe 4O+\n6was iden-\ntified. In this work, we will also identify the geomet-\nric, electronic, and magnetic structure of Fe 3O+\n4, Fe4O+\n5,\nFe5O+\n7and propose structures for Fe 6O+\n8.\nI. COMPUTATIONAL DETAILS\nWe employ a genetic algorithm (GA) as is described in\nRef. 27 in combination with DFT to optimize the cluster\nstructures. For this we use the Vienna ab-initio simu-\nlation package ( vasp)31using the projector augmented\nwave (PAW) method.32,33Since the geometry optimiza-\ntion is the most computationally expensive part of the\ngenetic algorithm, we use the PBE+ Umethod34with\nlimited accuracy for the genetic algorithm. For all ob-\ntained isomers low in energy, we reoptimized the geo-\nmetric structure using the hybrid B3LYP functional with\nhigher accuracy and consider different magnetic config-\nurations. We then calculate the vibration spectra and\ncompare them with experimental results.\nWithin the DFT framework, functionals based on the\nlocal density approximation (LDA) or general gradient\napproximation (GGA) fail to describe strongly interact-\ning systems such as transition-metal oxides.35,36Due to\nthe overestimation of the electron self-interaction, they\npredict metallic behavior instead of the (correct) wide-\nband-gap insulator. In an attempt to correct for this\nself-interaction, one can, for example, employ a hybrid\nfunctional, where a typical amount of 20% of Hartree-\nFockenergyisincorporatedintotheexchange-correlation\nfunctional. Especially for the B3LYP functional it has\nbeen shown that this results in good agreement be-\ntween the geometric structure and vibrational spectra\nfor clusters.28,30,37However, hybrid functionals are quite\ncomputationally expensive compared to LDA and GGA\nfunctionals. Therefore, in the genetic algorithm we em-\nploy the GGA+ Umethod to take into account that FeO\nclusters are strongly interacting systems. We use the\nrotational invariant implementation introduced by Du-\ndarev and a plane wave cutoff energy of 300 eV for these\ncalculations.38The differences between GGA and GGA+ Ufor iron-\noxide cluster calculations have been analyzed in Ref. 15.\nThisstudystressestheimportancetogobeyondGGAfor\ntransition-metal oxide clusters calculations. Aside from\nthe well-known difference for the electronic and magnetic\nstructure, it even finds a different lowest energy isomer\nthan GGA for Fe 32O33. In our genetic algorithm cal-\nculations we use an Ueff=U−Jof 3 eV for the Fe\natoms, based on a comparison between B3LYP calcula-\ntions and PBE+ Ucalculations for the smallest cluster,\nFe3O4(see Sec. IIB). For this comparison we also cal-\nculated the mean absolute difference (∆) between the\noccupied Kohn-Sham energies ( Ei) using B3LYP and\nPBE+U:\n∆ =n/summationdisplay\ni=1|EPBE+U\ni−EB3LYP\ni|\nn, (1)\nwherenis the number of occupied Kohn-Sham levels.\nNote that, the binding distances are only weakly depen-\ndent on the used Ueffand our value of 3 eV is close to val-\nues used in other works (e.g., 5 eV15, 3.6 eV20, 3.6 eV39).\nWe used the genetic algorithm as described in detail\nin Ref. 27. New geometries are formed by the Deaven-\nHo cut and splice crossover operation. To determine the\nfitness we used an exponential function. A generation\ntypically consists of 20 clusters. It has been shown that\nthe geometry of Fe xOyclusters only weakly depends on\nthe magnetic degree of freedom.26Therefore, we restrict\nourselves to the ferromagnetic case in our genetic algo-\nrithm.\nFor all obtained isomers low in energy, we reopti-\nmized the geometric structure using the hybrid B3LYP\nfunctional40,53and consider all possible collinear ori-\nentations of the Fe magnetic moments by constraining\nthe difference in majority and minority electrons. All\nforces were minimized below 10−3eV/˚A. Standard rec-\nommended PAWs with an energy cutoff of 400.0 eV are\nused. The clusters are placed in a periodic box of a\nsize between 11 and 17 ˚A, which we checked to be suf-\nficiently large to eliminate inter cluster interactions for\neach cluster size. For the cluster calculations, a single\nk-point (Γ) is used. Since we also consider cationic clus-\nters, a positive uniform background charge is added and\nwe correct the leading errors in the potential.41,42All\nsimulations were performed without any symmetry con-\nstraints. The reported symmetry groups are determined\nafterwardswithin 0.03 ˚A. Forthe density ofstates(DOS)\ncalculations we used a Gaussian smearing of 0.1 eV for\nvisual clarity.\nTo obtain the vibration spectra, the Hessian matrix\nof an optimized geometry is calculated by considering\nfinite ionic displacements of 0.015 ˚A for all Cartesian co-\nordinates of each atom. The vibration frequencies are\nobtained by diagonalization of the Hessian matrix. The\nabsorption intensity Aiis calculated using43,44\nAi= 974.86gi/parenleftbigg∂µ\n∂Qi/parenrightbigg\n, (2)3\nwheregiis the degeneracy of the vibration mode, Qi\nthe mass weighted vibrational mode, µthe electric dipole\nmoment, and974.86anempiricalfactor. Amethodbased\non four displacements for each ion was also tested but\nyielded the same frequencies and absorption intensities.\nZero-point vibrational energies (ZPVE) were calculated\nfor the isomers lowest in energy of which the vibration\nspectra are also shown.\nFor a quantitative comparison between experimen-\ntal and calculated vibrational spectra, we calculate the\nPendry’s reliability factor.45The Pendry’s reliability fac-\ntor is a well-established method in low-energy electron\ndiffraction (LEED) to quantify the agreement in contin-\nuous spectra and has also been applied to vibrational\nspectroscopy.46\nThe experimental used infrared multiphoton dissoci-\nation method (IR-MPD) does not only depend on the\nabsorption cross section of a vibrational mode, but also\non the dissociation cross section. Therefore, we use the\nPendry’s reliability factor to quantify the comparison of\nvibration spectrasince it is mainly sensitive to peak posi-\ntions opposed to a comparison of squared intensity. This\npeak sensitivity is achieved by comparing the renormal-\nized logarithmic derivative of the intensity I(ω):\nY(ω) =L−1(ω)\nL−2(ω)+W2, (3)\nwhereL(ω) =I′(ω)/I(ω) andWis the typicalFWHM of\nthe peaks in the spectra. The Pendry’s reliability factor\nis defined as:\nRP=/integraldisplay/bracketleftbig\nYth(ω)−Yexpt(ω)/bracketrightbig2\nY2\nth(ω)+Y2\nexpt(ω)dω, (4)\nwhere we integrate over the experimental range of fre-\nquencies. RPvalues range from 0 to 2, where 0 means\nperfect agreement, 1 uncorrelated spectra, and 2 per-\nfect anticorrelation. In practice, RPvalues of 0.3 are\nconsidered acceptable agreement within LEED. Y(ω) is\nstrongly dependent on experimental noise and values\nclose to zero, hence, we calculate Yexpt(ω) by fitting the\nexperimental spectrum with multiple Lorentzian peaks\nand extract the corresponding W. The theoretical fre-\nquencies are also convoluted with Lorentzian peaks with\nthe same W.RPis always minimized as function of a\nrigid shift of all theoretical frequencies.\nFor the calculations on magnetite we used the vasp\ncode. We used a Monkhorst grid of 6 ×6×2 and an\nenergy cutoff of 400 eV. We used the rotationally in-\nvariantLSDA+ UimplementationbyLichtenstein et al.47\nwith effective on-site Coulomb and exchange parameters:\nU= 4.5 eV48andJ= 0.89 eV for the Fe ions.\nWe used the monoclinic structure as described in\nRefs. 39,49, and calculated the electron density with 56\natoms in the unit cell. In Ref. 39, the charge and mag-\nnetic moment were calculated by integrating the density\nand spin density in a sphere with a radius of 1 ˚A forFe. This radius appears to be chosen such that compara-\nble valueswith neutronand x-raydiffraction experiments\nwere obtained.\nNote, there is no unambiguous way to define these\nradii in systems consisting of two or more atom types.\nTherefore, we checked the correspondence of our results\nto the earlier reported ones and also performed calcula-\ntionswith alargerradiusof1.3 ˚AforFeand0.82 ˚AforO.\nThis is a reasonable choice for Fe mO+\nnclusters since the\noverlapbetween different spheres is minimal, but most of\nthe intra cluster space is covered.\nII. RESULTS AND DISCUSSION\nA. Magnetite\nEven in the bulk, iron oxide is well known for its wide\nvarietyofphasesandtransitions. Magnetite(Fe 3O4), the\nmost stable phase of FemOn, is for example well known\nfor its Verweytransition.1,2Above the transitiontemper-\natureTV, the structure is a cubic inverse spinel. Upon\ncooling below TV, the conductivity decreases by two or-\nders of magnitude due to charge ordering. Furthermore,\nthe structure changes to monoclinic.\nMagnetite has the formal chemical formula\n(Fe3+\nA[Fe2+,Fe3+]BO4) where tetrahedral Asites\nare occupied by Fe3+andBsites contain both divalent\n(Fe2+) and trivalent (Fe3+) iron atoms. Since magnetite\nis a mixed valence system, it is an excellent reference\nsystem for our cluster calculations to determine their\nvalence state and corresponding magnetic moment.\nTABLE I: Spin moments within atomic spheres of 1.3 ˚A for\nthe Fe ions in monoclinic Fe 3O4. For reference the values\nwithin a sphere of 1.0 ˚A are also shown. A and B labels are\nconsistent with Ref. 39.\nSite Spin moment ( µB) Spin moment ( µB)\nRadius sphere 1 .3˚A 1 .0˚A\nFe3+(A) −4.02 −3.78\nFe2+(B1) 3 .69 3 .45\nFe3+(B2) 4 .15 3 .93\nFe3+(B3) 4 .06 3 .84\nFe2+(B4) 3 .64 3 .40\nIn Table I, the spin moments are shown for the dif-\nferent iron ions. The magnetic moments on the Aand\nBsites are antiparallel creating a ferrimagnetic struc-\nture. Within the atomic spheres of 1.3 ˚A the Fe2+and\nFe3+ions have a distinct magnetic moment of 4.0 µB\nand 3.7µBrespectively. Note the difference of 0.3 µBis\nmuch smaller than the 1 µBatomic value and does not\ndepend on the size of the atomic sphere used in the range\nbetween 1.0 and 1.3 ˚A.4\nB. GGA+U\nTo determine the optimal Ueffin comparison to the\nB3LYP functional for the genetic algorithm, we per-\nformed PBE+ Ucalculations on the neutral Fe 3O4clus-\nter. The results for the electronic DOS are shown in\nFig. 1 and compared with the hybrid B3LYP functional.\n/gl507=B==U8nt=/gl72/gl57\n[/gl721[/gl72S=38st/gl3386=\n[/gl721/gl50fS=f8yi=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=/gl56/gl72/gl73/gl73=B=n/gl72/gl57\n/gl507=B==U8nU=/gl72/gl57\n[/gl721[/gl72S=38sf/gl3386=\n[/gl721/gl50fS=f8ys=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=/gl56/gl72/gl73/gl73=B=D/gl72/gl57\n/gl507=B==U8ni=/gl72/gl57\n[/gl721[/gl72S=38sU/gl3386=\n[/gl721/gl50fS=f8ys=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=/gl56/gl72/gl73/gl73=B=3/gl72/gl57H/gl72/gl81/gl86/gl76/gl87/gl92=/gl82/gl73=/gl54/gl87/gl68/gl87/gl72/gl86/gl507=B==U8sn=/gl72/gl57 /gl56/gl72/gl73/gl73=B=f/gl72/gl57\n[/gl721[/gl72S=38sU/gl3386=\n[/gl721/gl50fS=f8yn=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=\n/gl507=B==U8it=/gl72/gl57 /gl56/gl72/gl73/gl73=B=U/gl72/gl57\n[/gl721[/gl72S=38sU/gl3386=\n[/gl721/gl50fS=f8yn=/gl3386=\n[/gl721/gl503S=38UU=/gl3386=\n[/gl721[/gl72S=38sf/gl3386=\n[/gl721/gl50fS=f8yn=/gl3386=\n[/gl721/gl503S=f8oo=/gl3386=dD/gl47/gl60/gl51\n[/gl72=/gl86\n[/gl72=/gl83\n[/gl72=/gl71\n/gl50=/gl86\n/gl50=/gl83M1M−/gl50/gl48/gl50=/gl62/gl72/gl57/gl64=/gl237fU /gl237t8s /gl237s /gl23738s U 38s s\nFIG. 1: (Color online) The density of states for the hybrid\nB3LYP functional and PBE+ Ufor different values of Ueff.\nThe average inter atomic distances are shown on the right,\nwhere Fe-O 1and Fe-O 2refer to the Fe-O distances between\nbridging O atoms (side) and the capping O atom (center),\nrespectively. The mean absolute difference ∆ [Eq. 1] between\nthe PBE+ Uand B3LYP energy levels is also shown and is\nminimal for Ueff= 3 eV, indicating the best match in DOS.\nThe valence states within -4 and 0 eV are formed by\nhybridized orbitals between the dorbitals of iron and\ntheporbitals of oxygen. For increasing U, the majority\nspindorbitals of Fe decrease in energy, whereas HOMO-\nLUMO gap increases. Note that the HOMO-LUMO gap\nof 1.5 eV for Ueff= 4 eV still is 0.9 eV smaller than the\n2.4 eV gap for B3LYP. Furthermore, for Ueff= 2 and\n3 eV the Fe dDOS features are very similar to those of\nthe B3LYP result. To quantify this we also calculated\nthe mean absolute difference ∆ [Eq. 1] for the occupied\nlevels; the results are shown in Fig. 1. ∆ is minimal\nforUeff= 3 eV, indicating the best DOS correspondence\nto B3LYP. We also show the corresponding bonding dis-\ntances within the cluster, where Fe-O 1and Fe-O 2refer\nto the Fe-O distances between bridging O atoms (side)\nand the capping O atom (center), respectively. Note the\ninteratomicdistancesonlychangeverylittlewithincreas-\ningUeff. ForUeff= 3eV,the bindingdistancesarewithin0.01˚A; furthermore, for Ueff= 3 eV and B3LYP the oc-\ncupieddorbitalsofFeareatcomparableenergieswithre-\nspect to the HOMO level. We therefore used Ueff= 3 eV\nfor our genetic algorithm calculations.\nC. Fe 3O0\n4\nAlthough the possible number of isomers increases\nrapidly with cluster size, for small systems such as Fe 3O4\nthe number of possibilities is still small. In Fe 3O4, the Fe\natomscaneitherformatriangleorachain. Forthe trian-\ngular configuration, two isomers are low in energy. The\nfirst isomer consists of a ring like structure where the O\natomsoccupybridgingstatesandoneOatomcapstheFe\ntriangle as is shown in Fig. 2 (a). In the second isomer,\nthe additional O atom is not located above the center\nbut forms an extra bridge between the two ferromagnetic\n(FM) ordered Fe atoms as is shown in Fig. 2 (b).\n3e/gl72/gl57 3e/gl72/gl57 [Sae/gl80/gl72/gl57\n/gl71\n/gl69\n/gl68/gl40/gl81/gl72/gl85/gl74/gl92e/gl62/gl72/gl57/gl64\n335tSS5tpp5ti\n/gl54/gl83/gl76/gl81e/gl80/gl68/gl74/gl81/gl72/gl87/gl76/gl93/gl68/gl87/gl76/gl82/gl81e/gl62/gl541 /gl37/gl643 p a z µ S3 Sp Sa/gl41/gl72/gl22/gl50/gl23/gl19\nFIG. 2: (Color online) The energy as function of spin mag-\nnetization for different neutral Fe 3O4isomers. The geometric\nfigures on the right show the corresponding geometric struc-\nture. O atoms are shown in red, Fe spin up and Fe spin\ndown are indicated with orange (red) and green (blue) colors\n(arrows), respectively. For the lowest magnetic states the rel-\native energy differences are also shown in black. Isomers (a)\n(black line) and (b)(red line) are equally low in energy with\na ferrimagnetic and ferromagnetic ground state, respectiv ely\n(0 eV). The M= 6µBstate of isomer (a)is 14 meV higher\nin energy.\nFigure 2 shows the energy as a function of spin mag-\nnetic moment for the neutral Fe 3O4cluster with four\ndifferent isomers. For all spin magnetizations, the ge-\nometric structure is optimized and shown on the right\nwith its magnetic structure lowest in energy. In Fig. 2\nandthe restofthiswork, Fespin upandFe spindownare\nindicated with orange (red) and green (blue) colors (ar-\nrows), respectively. O atoms are shown in red. For the\nneutral cluster, the two triangular isomers are equally\nlow in energy with two different magnetic configurations.\nThe difference is smaller than 1 meV and therefore be-5\nyond the accuracyofDFT. In isomer (a), as indicated by\nthe black line in Fig. 2, the magnetic ground state corre-\nsponds to ferromagneticalignment between the magnetic\nmoments on the Fe atoms and a total magnetic moment\nof 14µB. The Fe-Fe distances are 2.51 ˚A, the Fe-O dis-\ntances for the bridging O atoms and capping O atom are\n1.84 and 1.99 ˚A, respectively. Aside from the FM ground\nstate, also the ferrimagnetic state with a spin magneti-\nzation of 4 µBis low in energy and only 14 meV higher\nthan the ferromagnetic state. Note we also considered\na noncollinear magnetic state with M= 0µB, but this\nmagnetic configuration did not turn out to be energeti-\ncally stable.\nIsomer(b)is equally low in energy and shown in red\nin Fig. 2. The magnetic ground state corresponds to a\nferrimagnetic alignment where the two ferromagnetically\naligned Fe atoms have Fe-O-Fe angles of approximately\n90◦.\nWe also considered zero point vibrational energies for\nthe three lowest-energy levels. When we include these\ninto our consideration, the ferromagneticstate, indicated\nby the black line, is lowest in energy, and the M= 4µB\nandM= 6µBstatesare17and19meVhigherin energy,\nrespectively.\nD. Fe 3O+\n4\nFor the cation Fe 3O+\n4cluster we also considered ring\nand chain configurations with different oxygen locations.\nFor all four isomers we calculated all possible different\ncollinear magnetic states. Since an antiferromagnetic\n(AFM) triangle is the most simple example of geometri-\ncally frustrated magnetism, we also considered the non-\ncollinear state with M= 0µBwhere all magnetic mo-\nments have 120◦angles with respect to each other. The\nresults are shown in Fig. 3. For the charged Fe 3O+\n4clus-\nter, the isomer with a Fe triangle where the fourth O\natom caps the triangle is, like in the neutral cluster, low-\nestin energy,asisshowninFig.4. Threemagneticstates\nare low in energy: 0, 5 and 15 µB, with the M= 5µB\nstate being lowest in energy, and the non-collinear 0 µB\nand ferromagnetic 15 µBare 20 meV and 58 meV higher\nin energy respectively.\nThe ferrimagnetic state which is lowest in energy, has\na reduced symmetry ( Cv) with respect to the ferromag-\nnetic state ( C3v) and the antiferromagnetic state. This\ncould indicate a Jahn-Teller distortion, but could also\nbe the result of the inability of DFT to correctly model\nthe antiferromagnetic ground state.50,51However, to dis-\ntinguish between these two cases, methods beyond DFT\nsuch as CASPT2 and CCSD(T) are required and there-\nfore beyond the scope of this work. Note that different\nmagnetic states only lead to minor differences in the vi-\nbrational frequencies.\nInterestingly, the typical classical displacement during\na zero-point vibration in these clusters is of the order\nof 0.03˚A. This is of the same order as the typical dif-ytµ8/gl80/gl72/gl57 yp18/gl80/gl72/gl57 18/gl72/gl57\n/gl40/gl81/gl72/gl85/gl74/gl928/gl62/gl72/gl57/gl64\n11.pt1.t1.otSS.ptS.tS.otp\n/gl54/gl83/gl76/gl818/gl80/gl68/gl74/gl81/gl72/gl87/gl76/gl93/gl68/gl87/gl76/gl82/gl818/gl62/gl541 /gl37/gl641 S i t o B SS Si St/gl71\n/gl69\n/gl68/gl41/gl72/gl22/gl50/gl23/gl14\nFIG. 3: (Color online) Energy of the Fe 3O+\n4isomers as func-\ntion of spin magnetization. Figures on the right indicate th e\ncorresponding structure. The isomer lowest in energy (a)is\na Fe triangle with three bridge O atoms and one O atom cap-\npingthetriangle. Forthisisomer, theferrimagnetic 5 µBstate\nis lowest in energy. The antiferromagnetic 0 µBand ferromag-\nnetic 15 µBstate are 20 and 58 meV higher in energy, respec-\ntively. Note the antiferromagnetic 0 µBstate corresponds to a\nnon-collinear orientation with 120◦angles between the spins.\nFIG. 4: (Color online) The neutral (left) and cation (right)\nFe3O4lowest-energy isomers. FespinupandFespindown are\nindicated with orange (red) and green (blue) colors (arrows ),\nrespectively. O atoms are shown in red. The interatomic\ndistances are shown in black. The neutral and cation cluster\nhaveC3vandCvpoint group symmetry, respectively.\nference in inter atomic distances between different mag-\nneticstates. Therefore,this couldleadtointerestingphe-\nnomena in which, for example, there is a strong coupling\nthrough exchange between vibrations and magnetism.\nThe second triangular isomer of Fe 3O+\n4is 154 meV\nhigherinenergyandalsoconsistsofaringstructure. The\nmagnetic state lowest in energy has a magnetic moment\nof 5µB. The Fe-Fe bonding distances are 2.5 and 3.0 ˚A\nbetween the AFM and FM bonds within the structure.\nThe Fe-O distances vary between 1.7 and 1.9 ˚A. The\nisomer has a C2vpoint group symmetry.\nThe third and fourth isomers consist of a linear chain\nof Fe atoms with two O bridging atoms between each Fe\npair. The two planes can be parallel or perpendicular,6\n/gl41/gl72/gl22/gl50/gl23/gl14\n/gl53/gl51 [ I1pbViio /gl80/gl72/gl57 /gl70\n/gl53/gl51 [ I1rnVasb /gl80/gl72/gl57 /gl69\n/gl53/gl51 [ I1sI/gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81 /gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl809a/gl64oII pII iII aIII abII\nFIG. 5: (Color online) The experimental vibration spectra\nof Fe3O+\n4and the calculated isomers lowest in energy. The\nreported energy differences include ZPVE. The Pendry’s reli -\nability factor [Eq. 4] is also shown for each isomer.\nwhere the latter is lower in energy. Both isomers have a\nmagnetic moment of 5 µB.\nIn Fig. 5, both the experimental and calculated vibra-\ntion spectra for the different isomers are shown. The\nexperimental spectrum consists of three peaks at 540,\n610 and 670 cm−1. The best match is given by isomer\n(a)with calculated vibrations at 505, 630 and 660 cm−1\nand a corresponding lowest- RPfactor of 0 .30, indicat-\ning a reasonable match with the experimental spectrum.\nSince isomer (a)is also the lowest in energy, it is identi-\nfied as the experimentally observed structure.\nE. Fe 4O0/+\n5\nFe4O5also consists of a ring structure in which the\nO atoms occupy the bridging sites and one O atom is\nlocated above the center, as is shown in Fig. 6. The clus-\nter has antiferromagnetic order. However, not all Fe-Fe\nbonds are antiferromagnetic, but also two ferromagnet-\nically aligned bonds are present. Therefore, the cluster\nhas noC2vpoint group symmetry but C2, since Fe-Fe\nand Fe-O distances vary between 2.72-2.74 ˚A and 1.79-\n2.33˚A respectively. The magnetic state with four AFM\nFe-Fe bonds is 308 meV higher in energy.\nFor Fe4O+\n5the isomer lowest in energy consists of the\nsame ring structure but is more symmetry broken, since\nthe O atom above the ring is off-center as is shown in\nFig. 6. Therefore the two Fe-Fe distances are 2.69 and\n3.07˚A, the Fe-O distances vary between 1.76 and 2.01 ˚A.\nThe isomer has Cspoint group symmetry. Two Fe 2O2\nsquares are present within the cluster. Isomer (a)has\na magnetic moment of 1 µBdue to ionization. Interest-\ningly, the ionized cluster has a different magnetic ground\nstate with four AFM Fe-Fe bonds opposed to the neutral\ncluster.\nFIG. 6: (color online) The neutral (left) and cation (right)\nFe4O5lowest energy isomers. The neutral cluster has C2sym-\nmetry, whereas the cation cluster has Cssymmetry.\nIn Fig. 7 (b), we also show the vibration spectrum of\nthe ferromagnetic state of this cluster. The Fe-Fe dis-\ntancesareincreasedto2.74and3.11 ˚A, respectively. The\nferromagneticstructureis514meVhigherin energy. The\nvibration spectrum is similar but slightly shifted to the\nblue due to the increased bonding distances.\n/gl53/gl51 - bIisV pvp /gl80/gl72/gl57 /gl71\n/gl53/gl51 - bIttV ptv /gl80/gl72/gl57 /gl70\n/gl53/gl51 - bItbV tsp /gl80/gl72/gl57 /gl69\n/gl53/gl51 - bIpo/gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81 /gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl802s/gl64obb pbb ibb Wbb sbbb sobb/gl41/gl72/gl23/gl50/gl24+\nFIG.7: (Color online)Theexperimentalandcalculated vibr a-\ntion spectra of Fe 4O+\n5. The isomer shown in (a)is both the\nlowest in energy and RP[Eq. 4] and can therefore be iden-\ntified as the experimentally observed geometrical structur e.\nThe reported energy differences include ZPVE.\nThe second isomer, 459meV higher in energy, is shown\nin Fig. 7(c). This cage-like structure has Cvpoint group\nsymmetry and a magnetic moment of 9 µB. Figure 7 (d)\nshows the third isomer which is 494 meV higher in en-\nergy compared to Fig. 7 (a). The isomer has almost no\nsymmetry ( C1), and consists of a ring where one Fe-Fe\nbond has two bridging O atoms. The Fe-Fe binding dis-\ntances vary between 2.62 and 3.13 ˚A. The isomer has a\nmagnetic moment of 1 µB.\nIn the experimental vibration spectrum of Fe 4O+\n5\nshown in Fig. 7, five vibration frequencies can be ob-7\nserved: 450, 615, 760, 810, and 1070 cm−1. The vibra-\ntion at 1070 cm−1can be identified as a shifted vibration\nin the O 2messenger attached to the cluster-messenger\ncomplex and is therefore omitted in the RPcalculation.3\nThe best fit is given by isomer Fig. 7 (a)withRP= 0.42,\nwhich is also the isomer lowest in energy. The calculated\nfrequencies: 479, 630, 637, 772 and 796 cm−1match all\nwithin 30 cm−1to the experimental spectrum. Also, the\nrelative intensities between different vibrations are very\nsimilar. Although the ferromagnetic order increases the\nbinding distances within the cluster, the changes in the\nvibration spectrum of Fig. 7 (b)are small and therefore\nthestructurecorrespondingtoFigs.7 (a)and7(b)canbe\nidentified as the experimentally observed structure and\nthe IR-MPD method is not able to resolve the magnetic\nstate in this case.\nF. Fe 4O0/+\n6\nIn Ref.30, the Fe4O+\n6cluster was already identified as\nthe structure shown in Fig. 8 (b). The reported magnetic\nstructure was ferrimagnetic with a magnetic moment of\n9µB.\n/gl53/gl51 - bIrvesWn /gl80/gl72/gl57 /gl69\n/gl53/gl51 - bIpW/gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81/gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl804s/gl64obb pbb ibb Wbb sbbb sobb/gl41/gl72/gl23/gl50/gl25/gl14\nFIG. 8: (color online) The experimental and calculated vibr a-\ntion spectra of Fe 4O+\n6for both theprevious and newmagnetic\nground state. The vibration frequencies are very similar bu t\ndiffer in absorption intensity. The M= 1µBstate in (a)is\n187 meV lower in energy.\nIn our calculations a magnetic state lower in energy\nwas found for the same geometric structure for both\nFe4O6and Fe 4O+\n6. In this state Fe 4O6and Fe 4O+\n6have a\nmagnetic moment of 0 and 1 µBrespectively as is shown\nin Fig. 9. These structures are 194 and 187 meV lower\nin energy for Fe4O6and Fe4O+\n6in comparison to the\npreviously reported state.30The antiferromagnetic mag-\nnetic ground state of Fe 4O6was also previously reported\nin Ref. 26. For Fe 4O6we also calculated a noncollinear\nstate where all magnetic moments point towardsthe cen-\nter of mass, such state with M= 0µBis 30 meV higher\nin energy compared to the collinear M= 0µBstate.\nFor the neutral cluster, minima in energy are obtained\nforM= 0, 10, 20 µBcorresponding to flips of atomic/gl41/gl72t/gl50z1\n/gl41/gl72t/gl50z[/gl40/gl81/gl72/gl85/gl74/gl924/gl62/gl72/gl57/gl64\n11.i22.iMM.i\n/gl48/gl68/gl74/gl81/gl72/gl87/gl76/gl93/gl68/gl87/gl76/gl82/gl814/gl62/gl541 /gl37/gl641 M t z µ 21 2M 2t 2z 2µ M1/gl41/gl72t/gl50z15[\nFIG. 9: (color online) Energy as function of magnetization o f\nthe neutral Fe 4O6and cationic Fe 4O+\n6clusters. The magnetic\nground state corresponds to a total spin magnetic moment of\nM= 0 and M= 1µBfor Fe 4O6, and Fe 4O+\n6respectively.\nmagnetic moments of 5 µBfor each Fe atom. Note this\nalso matches with an ionic picture in which the Fe atoms\nin Fe4O6have a Fe3+valence state resulting in an atomic\nmagnetic moment of 5 µB. The corresponding structure\nis shown in Fig. 10. In Ref. 30 is mentioned that the\nsymmetry in the M= 10µBstate is reduced from Tdfor\nthe ferromagnetic state to C3v. In this antiferromagnetic\nground state, the neutral cluster has D2dsymmetry. In\nFe4O+\n6the symmetry is reduced even further to Csas is\nshown in Fig. 10.\nFIG. 10: (Color online) The neutral (left) and cation (right )\nFe4O6lowest energy isomers. The neutral cluster has D2d\nsymmetry, whereas the cation cluster has Cssymmetry.\nFigure 8 shows both calculated and experimental spec-\ntra for Fe4O+\n6. The vibration spectra for the two calcu-\nlated magnetic states in Figs. (a)and 8(b)show very\nsimilar behavior. The RPvalues of isomer Fig. 8 (a)\n(0.48) and Fig. 8 (b)(0.39) are both large and indicate a\nbetter match for isomer Fig. 8 (b). Although the spectra\nfor Figs. 8 (a)and 8(b)are very similar, the ferrimag-\nnetic structure has an extra vibration at 720 cm−1with\nsmall IR absorption. Furthermore, around 550 cm−1,\nvibrations differ slightly in frequency. Since the men-\ntioned differences cannot be experimentally resolved, the\nIR-MPD method is unable to resolve between different\nmagnetic states and another type of experiments such\nas Stern-Gerlach deflection is required to determine the\nmagnetic moment.8\nG. Fe 5O0/+\n7\nThe neutral Fe 5O7cluster has a “basket” geometry\nas is shown in Fig. 11. The magnetic ground state is\nferrimagnetic with a total moment of 4 µBdue to the\nodd number of Fe atoms. The cluster has C2vsymmetry.\nFIG. 11: (Color online) The neutral (left) and cation (right )\nFe5O7lowest-energy isomers. The neutral cluster has C2v\nsymmetry, whereas the cation cluster has no symmetry.\nThe cationic structure of Fe 5O+\n7is very different and\nshown in Fig. 11. Like Fe 4O+\n6, it consists of a cage-like\nstructure. The Fe-Fe distances range from 2.7 to 3.1 ˚A.\nExcept for the triple bound O atom, all O atoms form\nbridges between two Fe atoms. The ground state has a\nmagnetic moment of 5 µB. The second isomer is similar\n/gl53/gl51 o vWc[/gl70 3nWvc /gl72/gl57\n/gl53/gl51 o vWrb/gl69 3rv- /gl80/gl72/gl57\n/gl53/gl51 o vWc[/gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl80mn/gl64uvv rvv cvv ]vv nvvv nuvv /gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81/gl41/gl72/gl24/gl50/gl26/gl14\nFIG. 12: (Color online) The experimental and calculated vi-\nbration spectra of Fe 5O+\n7. The reported energy differences\ninclude ZPVE.\nto the neutral ”basket” structure and is 394 meV higher\nin energy as is shown in Fig. 12 (b). The structure has\nCssymmetry and a magnetic moment of 5 µB. However,\nthe atomic spin moments have a different arrangement\nfor the neutral and cationic state.\nThe third isomer is shown in Fig. 12 (c)and is 1.04 eV\nhigher in energy. It contains two triple bonded O atoms\nand is ferrimagnetic with M= 5µB.\nThe experimental vibration spectrum shown in Fig. 12has eight distinct vibrations at 375, 490, 520, 570, 615,\n710, 780, and 830 cm−1which are best resembled by the\nisomer lowest in energy shown in Fig. 12 (a), although\nthegapbetween615and710cm−1seemstobe underesti-\nmated. Note that this also explains the high- RPfactorof\n0.65 for isomer Fig. 12 (a). Similar to Fe 4O+\n5and Fe 4O+\n6\nthe absorption intensities of vibrations in the range of\n300-500 cm−1are systematically underestimated. The\nindividualvibrationsofisomerFig. 12 (a)areallin agree-\nment within 35 cm−1. Although isomer Fig. 12 (b)has a\nlowerRP= 0.43, the energy difference of 407 meV with\nisomer Fig. 12 (a)is large and isomer Fig. 12 (b)has a\nvibration at 450 cm−1which is not present in the exper-\nimental spectrum and lacks the experimental 375 cm−1\nvibration. Therefore, isomer Fig. 12 (a)can be identified\nas the most probable ground state.\nH. Fe 6O+\n8\nThe isomer lowest in energy found for Fe6O+\n8is shown\ninFig.13andhas Cssymmetrywherethereflectionplane\nis located through Fe atoms 1, 3, and 6. The magnetic\nmoment of this isomer is 1 µB.\nFIG. 13: (Color online) The cation Fe 6O+\n8isomer lowest in\nenergy. The cluster has Cssymmetry.\nThesecondisomerlowinenergyisshowninFig.14 (b).\nIn this isomer no symmetry is present. Compared to the\nlowest found isomer in Fig. 14 (a)it is 413 meV higher in\nenergy and also has a magnetic moment of 1 µB.\nFigure 14 (c)shows the third isomer, which is a dis-\ntorted octahedral of Fe atoms in which the O atoms cap\nthe Fe triangles. The structure is slightly distorted due\nto the AFM order between spins, which lead to slightly\naltered Fe-Fe distances. This isomer is 483 meV higher\nin energy than isomer Fig. 14 (a).\nFigure 14 also shows the corresponding vibration spec-\ntra of the mentioned isomers and the experimental spec-\ntrum. The experimental spectrum has vibrations at 392,\n420, 500, 730 and 763 cm−1. Note that none of the\nprovided isomers match the experimental vibration spec-\ntrum completely. This is also shown by the large- RP\nvalues of 0.56-0.61 for all calculated isomers. The isomer\nlowest in energy Fig. 14 (a)is the best match since it also\nhasvibrationsat 420and500cm−1, but the vibrationsat9\n/gl53/gl51 o a1r]/gl70 eb-u /gl80/gl72/gl57\n/gl53/gl51 o a1[v/gl69 ebvu /gl80/gl72/gl57\n/gl53/gl51 o a1r[/gl68\n/gl40/gl91/gl83/gl17\n/gl58/gl68/gl89/gl72/gl81/gl88/gl80/gl69/gl72/gl85 /gl62/gl70/gl806v/gl64naa baa [aa -aa vaaa vnaa /gl44/gl53 /gl68/gl69/gl86/gl82/gl85/gl83/gl87/gl76/gl82/gl81/gl41/gl72/gl25/gl50/gl27/gl14\nFIG. 14: (color online) The experimental and calculated vi-\nbration spectra of Fe 6O+\n8. The isomer shown in (a)is the low-\nest in energy. The reported energy differences include ZPVE.\n804 and 825 are considerably shifted with respect to 730\nand 763 cm−1. Furthermore, the vibrations at 640, 671,\nand 713 cm−1are not present in the experimental spec-\ntrum. The vibration spectra shown in Figs. 14 (b)and\n14(c)fit even worse. Therefore, we can not successfully\nidentify the Fe 6O+\n8structure.\nNote that our genetic algorithm implementation only\nuses geometry optimization at the DFT level. At cluster\nsizes of Fe 6O+\n8and larger, preselection using empirical\npotentials instead of immediate geometry optimization\nusing DFT might be more efficient in generating possible\nisomers.\nI. Electronic structure\nIn the bulk, iron-oxide materials have many different\ncrystal structures such as hematite, wustite, and mag-\nnetite with all corresponding different electronic struc-\ntures. While in hematite only trivalent Fe3+is present,\nthe mixed valence state (Fe3+\nA[Fe2+,Fe3+]BO4) in mag-\nnetite leads to interesting physical phenomena such as\nferrimagnetic ordering between the sublattices Aand\nBand the Verwey transition in which orbital ordering\nleads to a first-order phase transition in the electrical\nconductivity.1,2\nIn clusters, stoichiometries corresponding to both\nhematite (Fe4O6) and magnetite (Fe3O4, Fe6O8) and\nother combinations (Fe 4O5, Fe5O7) occur. We therefore\nexpect divalent and trivalent Fe cations to be present in\nthe reported clusters. There is no unique method to de-\ntermine the valence state in materials consisting of mul-\ntiple types of elements. We therefore compare both the\nlocal magnetic moments and the local density of states\n(LDOS) for our cluster calculations with bulk magnetite\nresults shown in Section IIA. Since the Fe2+and Fe3+\nfeatures in the LDOS are very similar for different clustersizes, we show the LDOS of Fe 4O+\n5which contains both\nFe2+and Fe3+in Fig. 15. The LDOS for other cluster\nsizes can be found in the Appendix.\n/gl41/gl720/gl502 /gl47/gl39/gl50/gl54 /gl76/gl81/gl87s4/gl39/gl50/gl54 /gl55/gl82/gl87s4/gl39/gl50/gl54/gl41/gl724/gl86\n/gl41/gl724/gl83\n/gl41/gl724/gl71\n/gl504/gl86\n/gl504/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\n2udu2wdw21d12\n/gl41/gl72u\n/gl41/gl72w\n/gl41/gl721\n/gl41/gl720\n/gl40o/gl40/gl43/gl50/gl48/gl504/gl62/gl72/gl57/gl64/gl237- /gl2373 /gl2370 /gl237w d w 0 3\nFIG. 15: (Color online) The total, integrated and local den-\nsity of states of the Fe atoms for the Fe 4O+\n5cluster. The\ntrivalent Fe(1), Fe(2) and Fe(3) all show 3 dlevels at -6 eV\nand small hybridization with O. The divalent Fe(4), however ,\nshows strong hybridization and a single level at E HOMO.\nTable II shows the local spin moments of the clus-\nters: Fe 3O+\n4, Fe4O+\n5, Fe4O+\n6, Fe5O+\n7and Fe 6O+\n8. For\nFe3O+\n4all three Fe atoms have a similar spin moment\nwithin 0.04 µB. A comparison with magnetite suggests\nall Fe atomsare trivalent. This agreeswith an ionic bond\nmodel. Furthermore, this is confirmed by the integrated\nand local density of states shown in Appendix A. The 3 d\npeaksaround-6eVcorrespondto15electrons,indicating\nthe hybridization between Fe and O is small. Note that,\nthe central oxygen atoms O(4) and O(7) are partially\nspin polarized.\nFor Fe4O+\n5, the spin moment of Fe(4) is 0.5 µBlower\nthan the other Fe atoms, indicating three trivalent and a\nsingle divalent atom. The difference is also in agreement\nwith the magnetite results. The Fe(4) also breaks the C2\nsymmetry as is shown in Fig. 6. The local (LDOS) and\nintegrated density of states are shown in Fig. 15. Note\nthat all Fe3+have 3dpeaks around −6 eV and small\nhybridization with O is present, similar to the Fe 3O+\n4\ncluster. The LDOS of the divalent Fe(4) atom however\nshows strong hybridization with O and a single minority\nlevel at E HOMO.\nWhereasFe 4O6onlycontains trivalent Fe,26for Fe4O+\n6\nthis is no longer the case due to ionization. As can be\nseen from Table II, three trivalent Fe atoms are present,\ntogether with a single Fe4+atom. The spin moment is10\nreduced with respect to Fe3+, consistent with a higher\noxidation state than Fe3+.\nIn Fe5O+\n7, only trivalent Fe atoms are present, con-\nsistent with an ionic model and the ionized state of the\ncluster. Fe 6O+\n8, on the other hand, is again a mixed\nvalence cluster where the magnetic moment of Fe(4) is\n0.4µBlower than the other Fe atoms, indicating Fe(4)\nis divalent. This is also consistent with the LDOS shown\nin Appendix A.\nFigure 16 shows the density of states for the different\ncationicclustersandmagnetite. Thecalculatedbandgap\nof 0.2 eV in magnetite is considerably smaller than for\nthereportedclusters: around3eVforFe 3O+\n4andslightly\nsmaller for Fe 4O+\n5and Fe 4O+\n6. Furthermore, whereas\nmagnetite has a t2gorbital of Fe2+just below the Fermi\nenergy,39in the reported clusters Fe4O+\n5and Fe6O+\n8have\nasimilarlevelduetoadivalentFeatom. Notethatthe3 d\norbitals of Fe3+in the clusters are located around 5.5 eV\nbelow the HOMO level, which is 2 eV higher in energy\ncompared to magnetite.\n/gl39/gl72/gl81/gl86/gl76/gl87/gl92n/gl82/gl73n/gl54/gl87/gl68/gl87/gl72/gl86/gl41/gl72g/gl50-6\n/gl41/gl72M/gl50E6/gl41/gl72n/gl86\n/gl41/gl72n/gl83\n/gl41/gl72n/gl71\n/gl50n/gl86\n/gl50n/gl83n\n/gl41/gl723/gl50g6\n/gl41/gl723/gl50M6\n/gl41/gl724/gl5036\n/gl48/gl68/gl74/gl81/gl72/gl87/gl76/gl87/gl72\n/gl408/gl40/gl43/gl50/gl48/gl50n/gl62/gl72/gl57/gl64/gl237- /gl237g /gl2373 /gl237d 7 d 3 g\nFIG. 16: (Color online) The density of states for Fe xO+\nyclus-\nters. For these calculations a smearing of 0.15 eV was used\nfor convenience of the reader. The HOMO level is located at\n0 eV and the small occupation above the HOMO level is due\nto smearing.III. CONCLUSION\nIn this work, we have studied the geometric, elec-\ntronic and magnetic structure of Fe xO+\nyclusters using\ndensity functional theory. For Fe 3O4we compared bind-\ning distances and electronic structure between the hybrid\nB3LYP functional, and different Ueffin the PBE+ Ufor-\nmalism. We found the best match for Ueff= 3 eV. Using\nthe PBE+ Uformalism and a genetic algorithm, many\npossible isomers were considered. For isomers low in en-\nergy, all different magnetic configurations were further\ngeometrically optimized. Finally, for the cationic clus-\nters we calculated the vibration spectra and compared\nthem with experiments to identify the geometric struc-\nture of Fe 3O+\n4, Fe4O+\n5, Fe4O+\n6, Fe5O+\n7and Fe 6O+\n8. All\ncationic clusters with an even number of Fe atoms have\na small magnetic moment of 1 µBdue to ionization. Fur-\nthermore, comparison with bulk magnetite reveals that\nFe4O+\n5, Fe4O+\n6and Fe 6O+\n8are mixed valence clusters.\nIn contrast, in Fe3O+\n4and Fe5O+\n7all Fe are found to be\ntrivalent.\nIV. 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Figures 17, 18, 19, and 20 show the\ntotal, integrated and local density of states of Fe3O+\n4,\nFe4O+\n6, Fe5O+\n7, and Fe 6O+\n8, respectively. Of these clus-\nters, Fe3O+\n4and Fe5O+\n7are pure trivalent and the LDOS\ncontains 3 dpeaks at -6 eV and small hybridization be-\ntween Fe and O. Fe 4O+\n6contains a single tetravalent Fe\natom, with a similar LDOS compared to Fe3+. The ion-\nized electron is not removed from the 3d levels at -6 eV,\nbut from the hybridized levels with oxygen, as can be\nseen from the integrated density of states. Fe 4O+\n5and\nFe6O+\n8contain a single divalent Fe atom, which has a\ndistinct LDOS, in which there are no peaks around -6 eV12\nbut strong spin polarized hybridization with oxygen and\nasingleoccupiedminoritylevelattheHOMOlevel. Even\nin bulk magnetite, as is shown in Fig. 21, the same fea-\ntures between divalent and trivalent Fe atoms exist.\n/gl41/gl72u/gl50w /gl47/gl39/gl50/gl54 /gl76/gl81/gl87a3/gl39/gl50/gl54 /gl55/gl82/gl87/gl68/gl793/gl39/gl50/gl54/gl41/gl723/gl86\n/gl41/gl723/gl83\n/gl41/gl723/gl71\n/gl503/gl86\n/gl503/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\n5psp5dsd5\n/gl41/gl72p\n/gl41/gl72d\n/gl41/gl72u\n/gl40o/gl40/gl43/gl50/gl48/gl503/gl62/gl72/gl57/gl64/gl237ps /gl2372 /gl2371 /gl237w /gl237d s d w 1\nFIG.17: (Color online)Thetotal, integratedandlocal dens ity\nof states of the Fe 3O+\n4cluster.\n/gl41/gl720/gl502 /gl47/gl39/gl50/gl54 /gl76/gl81/gl87s4/gl39/gl50/gl54 /gl55/gl82/gl87s4/gl39/gl50/gl54/gl41/gl724/gl86\n/gl41/gl724/gl83\n/gl41/gl724/gl71\n/gl504/gl86\n/gl504/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\nudu5wdw51d15\n/gl41/gl72u\n/gl41/gl72w\n/gl41/gl721\n/gl41/gl720\n/gl40o/gl40/gl43/gl50/gl48/gl504/gl62/gl72/gl57/gl64/gl237E /gl2372 /gl2370 /gl237w d w 0 2\nFIG. 18: (Color online) The total, integrated, and local den -\nsity of states of the Fe 4O+\n6cluster. Fe(1) is tetravalent as is\nshown in Table I.\n/gl47/gl39/gl50/gl54/gl41/gl722/gl504 /gl76/gl81/gl87sO/gl39/gl50/gl54 /gl55/gl82/gl87sO/gl39/gl50/gl54/gl41/gl72O/gl86\n/gl41/gl72O/gl83\n/gl41/gl72O/gl71\n/gl50O/gl86\n/gl50O/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\nudwd1d0d2d\n/gl41/gl72u\n/gl41/gl72w\n/gl41/gl721\n/gl41/gl720\n/gl41/gl722\n/gl40o/gl40/gl43/gl50/gl48/gl50O/gl62/gl72/gl57/gl64/gl237E /gl2373 /gl2370 /gl237w d w 0 3\nFIG. 19: (Color online) The total, integrated, and local den -\nsity of states of the Fe 5O+\n7cluster.13\nTABLE II: The spin moment for Fe xO+\nyclusters. The atom numbers correspond to the atom numbers sh own in Figures 4,6,10,\n11, and 13. The spin moment is calculated using atomic sphere s of 1.3 and 0.82 ˚A for Fe and O, respectively.\nCluster Spin moment [ µB]\n1 2 3 4 5 6 7 8\nFe3O+\n4 Fe −3.84 3 .88 3 .88\nO 0 .56 0 .00 0 .00 0 .22\nFe4O+\n5 Fe 3 .89 −3.84 3 .89 −3.40\nO −0.05 0 .13 0 .20 −0.05 0 .20\nFe4O+\n6 Fe −3.22 3 .85 3 .85 −3.79\nO 0 .01 0 .54 0 .01 −0.25 0 .00 0 .00\nFe5O+\n7 Fe 3 .85 3 .87 3 .89 −3.83 −3.80\nO 0 .01 0 .10 0 .03 0 .51 −0.09 0 .05 0 .12\nFe6O+\n8 Fe 3 .80 −3.84 3 .85 −3.47 −3.84 3 .88\nO 0 .01 0 .51 0 .01 0 .01 −0.10 0 .17 −0.10 0 .0114\n/gl41/gl723/gl505 /gl47/gl39/gl50/gl54 /gl76/gl81/gl87s6/gl39/gl50/gl54 /gl55/gl82/gl87s6/gl39/gl50/gl54/gl41/gl726/gl86\n/gl41/gl726/gl83\n/gl41/gl726/gl71\n/gl506/gl86\n/gl506/gl83\n/gl88/gl83\n/gl71/gl82/gl90/gl81\nudwd1d0d2d\n/gl41/gl72u\n/gl41/gl72w\n/gl41/gl721\n/gl41/gl720\n/gl41/gl722\n/gl41/gl723\n/gl40o/gl40/gl43/gl50/gl48/gl506/gl62/gl72/gl57/gl64/gl2375 /gl2373 /gl2370 /gl237w d w 0 3\nFIG. 20: (Color online) The total, integrated, and local den -\nsity of states of the Fe 6O+\n8cluster. All Fe atoms are trivalent\nexcept for Fe(4), which is divalent./gl48/gl68/gl74/gl81/gl72/gl87/gl76/gl87/gl72/gl47/gl39/gl50/gl54 /gl55/gl82/gl873g/gl39/gl50/gl54/gl41/gl722.S/gl36T\n/gl41/gl72).S/gl37AT\n/gl41/gl722.S/gl37)T\n/gl41/gl722.S/gl372T\n/gl41/gl72).S/gl37BT\n/gl40F/gl40/gl41g/gl62/gl72/gl57/gl64/gl237- /gl2374 /gl237B /gl237) ( ) B 4\nFIG. 21: (Color online) The total and local density of states\nof the different Fe atoms in magnetite. The numbering is\nconsistent with Table I. Fe2+and Fe3+have a similar LDOS\nto clusters although the symmetry is very different."    },    {        "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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(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": "2005.03965v3.Sublattice_magnetizations_of_ultrathin_ferrimagnetic_lamellar_nanostructures_between_cobalt_leads.pdf",        "content": "arXiv:2005.03965v3  [cond-mat.mes-hall]  12 Apr 2023SPIN\nVol. 1, No. 1 (2022) 1–11\n©World Scientific Publishing Company\nSublattice magnetizations of ultrathin ferrimagnetic lam ellar\nnanostructures between cobalt leads\nVinod Ashokan*\nDepartment of Physics,\nDr. B. R. Ambedkar National Institute of Technology,\nJalandhar (Punjab) 144 027, India\nashokanv@nitj.ac.in\nA. Khater\nDepartment of Physics, Le Mans University, 72085 Le Mans, Fr ance;\nDepartment of Theoretical Physics, Jan Dlugosz University, Czestochowa, Poland\nM. Abou Ghantous\nScience Department, American University of Technology,\nFidar Campus, Halat, Lebanon\nIn this work we model the salient magnetic properties of the alloy lame llar ferrimagnetic nanos-\ntructures [ Co1−cGdc]ℓ′[Co]ℓ[Co1−cGdc]ℓ′between Cosemi-infinite leads. We have employed the\nIsing spin effective field theory (EFT) to compute the reliable magnet ic exchange constants for\nthe pure cobalt JCo−Coand gadolinium JGd−Gdmaterials, in complete agreement with their\nexperimental data. The sublattice magnetizations of the CoandGdsites on the individual hcp\natomic (0001) planes of the Co−Gdlayered nanostructures are computed for each plane and\ncorresponding sites, by using the combined EFT and mean field theor y (MFT) spin methods.\nThe sublattice magnetizations, effective site magnetic moments, an d ferrimagnetic compensa-\ntion characteristics for the individual hcp atomic planes of the embe dded nanostructures, are\ncomputed as a function of temperature, and for various stable eu tectic concentrations in the\nrangec≤0.5. The theoretical results for the sublattice magnetizations and the local magnetic\nvariablesof these ultrathin ferrimagneticlamellar nanostructured systems, between cobalt leads,\nare necessary for the study of their magnonic transport proper ties, and eventually their spin-\ntronic dynamic computations. The method developed in this work is ge neral and can be applied\nto comparable magnetic systems nanostructured with other mate rials.\nKeywords : effective field theory; mean field theory; sublattice magnetization ; exchange constant;\ncobalt-gadolinium alloy; ferrimagnetic nanojunction.\n1. Introduction\nThe multilayered lamellar nano-magnetic nanos-\ntructurehas madea tremendousprogress in prepar-\ning and analyzing the physical properties of nano-\nmagnetic layered nanostructures and magneticnano-junctions. These systems have a panoply of\nindustrial and technological applications in the ar-\neas of spin wave magnonics1;2;3;4, and spintron-\nics5;6;7. However, the study of nano-magnetic\nlamellar multilayered nanostructure and nanojunc-\n∗Corresponding author\n12Vinod Ashokan; A. Khater and M. Abou Ghantous\ntion composite of rare earth-transition metal alloy\nsystems are still in its infancy. A fundamental and\nintriguing interest associated with such rare earth-\ntransition metal systems is to understand the phe-\nnomenawhichmay ariseduetothedecreaseoftheir\nsize when surface and quantum mechanical effects\ncome into play. The atomic interfaces and sublat-\ntice magnetization in these nanostructures are be-\ncome critical components for the physics of embed-\nded nano-junctions.\nThe nano-magnetic properties of ferrimag-\nnetic alloy nanostructure of rare earth-transition\nmetal has been reported in the literature\n8;9;10;11;12;13;14. The nano fabrication tech-\nnique has made the experimental progress to real-\nize the thin multilayered nanostructures with novel\nphysical properties and promising applications in\nmagnonic devices. In this work we study in par-\nticular the Co-Gd rare earth-transition metal alloy\nsystem; the transition metal Coand rare earth Gd\nare ferromagnetic with their Curie temperatures of\n119.4 and 25.2 meV respectively. Nano-magnetic\nnanojunctions build from Co-Gd alloys in diverse\nmultilayer formats can present hence very useful\nproperties for technological applications at room\ntemperature. In this respect, the bulk Co 1−cGdcal-\nloy materials with different alloy concentrations c,\nhave been studied intensively in the past for diverse\napplications in sensors,magneto optical devices and\nmagnetic storage elements15;16.\nA greater understanding of lamellar Co/Gd\nmultilayers nanostructures has been achieved\n9;11;12. It is noted that when we form an amor-\nphous alloy Co1−cGdcin these systems there is a\nstrong asymmetric spontaneous diffusion of Cointo\ntheGdplane occurs, and interfaces for various con-\ncentration. The experimental techniques made pos-\nsible to control the interdiffusion for some stable\neutectic concentration c≤0.5 while preserving the\nferrimagnetic structure of the multilayer systems.\nThe properties of multilayers presenting alloy in-\nterfaces, with few lamellar atomic planes thick sig-\nnificantly depends on the degree of material inter-\ndiffusion9;11. Such interdiffusion may play crucial\nrole in determining the nano-magnetic properties\nof the multilayers systems8, because the individ-\nual planes are having different exchange couplings\nat their interfaces from the bulk. This has also been\nreported earlier by model calculations which show\nthat nano-atomic scale magnetic alloyed interfaces\ncan significantly modify the nano-magnetic proper-ties of multilayer systems17;18;19. The prepara-\ntion of alloy like and composition stable nanojunc-\ntions composed of CoandGdbetween cobalt leads\nis hence possible in principle experimentally due to\nthe method of controlled interdiffusion process.\nIt should be noted that there has been at-\ntempts in the past to model the magnetic proper-\nties of bulk and layered cobalt-gadolinium systems\n13;20;21usingthe MFT method. These model cal-\nculations have been performed by adjusting in gen-\neral the MFT results to fit the experimental data,\nusing the cobalt spin SCoas a fitting parameter. In\nsome of the calculations where SCois assigned its\nfundamental value, the overall fit function for the\nmagnetization with temperature does not give bet-\nter agreement with the experimental data. Further-\nmore, theasymmetricchoice of nearestneighbor ex-\nchange constants for cobalt JCo−Coand gadolinium\nJGd−Gdis made in these references to reduce the\nnumber of adjustable parameters, but without giv-\ning any fundamental justification. To add to this\ncomplex situation, there is a wide array of exper-\nimental values of exchange constant for the cobalt\nJCo−Coand gadolinium JGd−Gdare available in the\nliterature from different types of measurements22,\nwhich does not help to clarify the situation for ad-\nvanced modeling.\nIn our previous work23, we computed the\nballistic and scattering transport properties of\nspin waves (SW) incident from cobalt leads, on\nto the embedded ultrathin ferrimagnetic cobalt-\ngadolinium [ Co1−cGdc]ℓnanojunction systems be-\ntween the leads. The nanojunction [ Co1−cGdc]ℓis\nprinciple a randomly disordered alloy with varied\nhcp atomic palnes ℓbetween matching hcp planes\nof theColeads, at known stable concentrations\nc≤0.5 for this nano-alloy system. To be able to\ncarry out these computations it was necessary to\ncompute the sublattice magnetizations and mag-\nnetic exchange constants in this system24.\nIn the present work we have modeled the\nsublattice magnetizations and magnetic exchange\nconstants of the alloy layered nanostructures\n[Co1−cGdc]ℓ′[Co]ℓ[Co1−cGdc]ℓ′sandwiched between\nsemi-infinite cobalt leads, at concentrations c≤\n0.5. These triple-nanostructure systems are more\ncomplex than the previously studied single-\nnanostructure systems24. The present work is\nhence motivated by the objective to present a\nmore complex computational modeling of the\nsalient nano-magnetic properties of the triple-Sublattice magnetizations of nano-magnetic layered mater ials3\nnanostructures for fundamental interest, and the\nballistic transport and scattering of spin waves in-\ncident from cobalt leads on complex ferrimagnetic\ncobalt-gadolinium nanojunction systems25. Such\ncomplex systems have as it turns out a richer and\nwider range of spin wave filtering properties. The\ncomplex embedded triple-nanostructures are de-\nnoted symbolically henceforth by [ ℓ′ℓℓ′] for conve-\nnience, corresponding to the alternating alloy and\npure nanostructures. The basal hcp (0001) atomic\nplanesofthe ...Co][ℓ′ℓℓ′][Co...nanojunctionsystems\nare normal to the direction of the c-axis itself con-\nsidered to be along the direction of the leads.\nThe alloy layered [ ℓ′ℓℓ′] nanostructures under\nconsideration are ultrathin ∼1.5 nm composite\ncobalt-gadolinium alloy systems sandwiched be-\ntween Co leads, and are hence different from bulk\nalloy and multilayer systems8;21;13. Also, due to\nthe absence of first principle calculations for Co-\nCo and Co-Gd exchange in the alloy layered [ ℓ′ℓℓ′]\nnanostructures.There is hence effectively a need for\nreliable data for the exchange and sublattice mag-\nnetizations in these systems to be able to develop\nmodelingstudiesfor the ...Co][ℓ′ℓℓ′][Co...nanojunc-\ntions which are key elements for ballistic spin wave\ntransport in magnonic devices23;26;27;28. This\nneed has motivated our EFT calculations to deter-\nmine such data with no fitting parameters, using\nbasic values SCo= 1 and SGd= 7/2 as the spin\nreferences at absolute 0 K.\nThe structure of the paper is as follows. In sec-\ntion 2, the EFT Ising spin method with experimen-\ntal data is computed for the reliable JCo−Coand\nJGd−Gdexchange for the purecrystalline cobalt and\ngadolinium materials. These are then attributed to\nnearest neighbor Co−CoandGd−Gdinterac-\ntions in the ...Co][ℓ′ℓℓ′][Co...nanojunction systems\nfor eutectic stable concentrations c≤0.5. The com-\nbined EFT and MFT methods are presented in sec-\ntion 3, to compute sublattice magnetizations for the\ncobalt and gadolinium sites on the individual hcp\nbasal atomic planes of the alloy layered [ ℓ′ℓℓ′] lamel-\nlar nanostructures as a function of temperature,\nwith thicknesses [2′22′] and [3′33′], and for differ-\nent alloy concentrations c. The sublattice magne-\ntizations and corresponding ferrimagnetic compen-\nsation temperatures are shown in this section. The\noverall discussions and conclusions are presented in\nsection 4.2. EFT modeling of pure CoandGd\nsystems\nTheEFT modeling incorporates the contribution of\nthe single site spin correlations to the order param-\neter, and hence known to be superior to the MFT.\nWe use it in the present work to model and calcu-\nlate the exchange constant for CoandGdcrystals\nover their ordered nano-magnetic phase, by com-\nparing the EFT magnetization results and Curie\ntemperatures with the experimental data29. The\nexchange for CoandGdcrystals are then calcu-\nlated by using our EFT constitutive relations30\nkTc/zJS(S+ 1) = 0 .3127 and kTc/zJS(S+ 1) =\n0.3162, valid for cobalt and gadolinium, respec-\ntively. The EFT exchange JCo−CoandJGd−Gdare\ningoodagreementwiththemeanvaluesofexchange\nconstant obtainedfromextensiveexperimental data\n22forCoandGd. The calculated EFT magne-\ntization and exchange results for cobalt are then\nused for the calculations of sublattice magnetiza-\ntions of the alloy layered ...Co][ℓ′ℓℓ′][Co...nanos-\ntructure between Co leads, by using the MFT\nmethod20;21;13. The schematic representation\nfor the consecutive planes of cobalt-gadolinium\nVCA alloy and pure cobalt plane of nanojunction\n[Co1−cGdc]2[Co]2[Co1−cGdc]2between crystalline\ncobalt leads are shown in Fig.1.\nThe Ising spin Hamiltonian Hin the absence\nof local spin anisotropy and Zeeman effects may be\nexpressed as,\nH=−J/summationdisplay\n<i,j>Siz.Sjz, (1)\nwhere,/angb∇acketlefti,j/angb∇acket∇ightrepresents sum over nearest neighbors\nin the crystal, and Jis the nearest neighbors mag-\nnetic exchange constant that induces spin order\nalong a selected z-axis. The coordination number\nofCoandGdisz= 12, and present negligible\nanisotropy in the bulk material as compared to\ntheexchange. TheHamiltonian Eq.(1)forcomputa-\ntional purposes may be written in more useful form\nas,\nH=/summationdisplay\ni/summationdisplay\nj(−JSjz)Siz≡/summationdisplay\niHi(x)≡/summationdisplay\ni−x Siz,\n(2)\nThe thermodynamic canonical averages for any\ndesiredspin operator “Op” may be calculated by\nthe effective filed theory method using the Van der\nWearden’s (VdW) operator exp( JSz∇). TheMath-\nematica code formulation for any given characteris-4Vinod Ashokan; A. Khater and M. Abou Ghantous\nFig. 1. Schematic representation for the consecutive plane s of cobalt-gadolinium VCA alloy and pure cobalt plane of nan o-\njunction [ Co1−cGdc]2[Co]2[Co1−cGdc]2between crystalline cobalt leads. The hcp crystal c-axis is normal to the symmetry\n(0001) atomic planes.\ntic function fOp(x) of the spin system may be ex-\npressed as\nfOp(x) =Tr(Op.MatrixExp[ −Hi(x)/kT])\nTr(MatrixExp[ −Hi(x)/kT]),(3)\nwhere the Van der Wearden’s (VdW) operator for\ncobalt with S= 1 is given as31,\nexp(JSz∇) =S2\nzcosh(J∇)+Szsinh(J∇)+1−S2\nz.\n(4)\nThe differential operator ∇=∂/∂xoperate with\nproperty <exp(a∇)> fOp(x)|x→0=fOp(x+\na)|x→0=fop(a). The EFT method calcula-\ntions are discussed in detail in the earlier works\n31;32;33;34. With the help of Matrix quantum\nmechanics all required averages can be evaluated\nwith symbolic and numerical procedures.\nThe thermodynamic canonical averages are\nrepresented by <Op>=<(exp(JSz∇))z>\nfOp(x)|x→0. The canonical averages are confined to\nsingle-sitespinvariables σ=< Sz>andq=< S2\nz>\nwith reference to their basic spin values S= 1 for\nCoandS= 7/2 forGdatT= 0K. In the com-\nputational procedure desired decoupling approxi-\nmation is used, which is equivalent to neglecting\nthe site-site correlations < SizSjz>but preserv-\ning the single-site correlations < SizSiz>. Further-\nmore, this decoupling approximation makes EFT\napproachquiteeffectivetocomputethesalient mag-\nnetic properties of the systems.The comparison between the normalized mag-\nnetizations for the CoandGdcrystals calculated\nusing EFT method, and by using the EFT consti-\ntutive relations30,kTc/zJS(S+ 1) = 0 .3127 for\nCo(S= 1,z= 12), and kTc/zJS(S+1) = 0 .3162\nforGd(S= 7/2,z= 12), with their experimental\nmeasurements29;35;36;37, yield their respective\nCurie temperatures, that is 119.4 meV and 25.2\nmeV24. It is striking to note that for Cowith\nspinS= 1, the EFT calculated magnetization and\nCurie temperature agree with the experimentally\nobserved mean value of exchange constant JCo−Co\n= 15.9 meV22. Similarly, the hcp Gdwith the\nspinS= 7/2, the EFT calculated magnetization\nand Curie temperature agree with the correspond-\ning experimentally measured exchange JGd−Gd=\n0.42meV.29.TheEFTcalculated exchange JCo−Co\nand spin variable σCofor sites on the cobalt leads\nseed from the interfaces inwards, to calculate the\nsublattice magnetization using MFT of embedded\nalloy layered [ ℓ′ℓℓ′] nanostructure.\n3. Sublattice magnetizations of the\nalloy layered [ℓ′ℓℓ′]ferrimagnetic\nnanostructures between cobalt\nleads\nInthepresentcomputationalmodelingthealloyhcp\natomic planes of the layered [ ℓ′ℓℓ′] nanostructure\nsandwiched between cobalt leads, are modeled asSublattice magnetizations of nano-magnetic layered mater ials5\ncrystalline atomic planes. On their hcp lattice there\nis a random homogeneous distributions of Coand\nGdatoms. Any random site is considered to have\nthe usual six nearest neighbors in its hcp (0001)\nbasal plane, and another six neighbors on the two\nadjacent planes. The system is made of alternat-\ning hcp (0001) atomic planes, and the structural\nmorphology of the two interfaces between the leads\nand the layered nanostructure are abrupt and crys-\ntalline. The advent of advanced experimental tech-\nniques for Co/Gdmultilayer systems,12;11, per-\nmits minimizing the interface roughness, and the\ncontrol of the atomic interdiffusion towards stable\neutectic compositions c= 0.1 to 0.5.\nTo calculate the sublattice magnetization for\nthe individual basal atomic hcp planes of the al-\nloy layered [ ℓ′ℓℓ′] nanostructures by MFT, the Bril-\nlouin’s functions are used to calculate initially\nthe different spin variables σ(n′)\nαfor then′th lay-\nered atomic plane. To simplify the notation we\nsystematically call σ(n′)\nαas the thermodynamic\ncanonical spin variable so that σ(n′)\nα=Sα.BS≡\nBα(Sα,T,H(n′)\nα), where Sαrepresents the funda-\nmental atomic spin and αnamely for the Co and\nGd atoms, and BSis the Brillouin function. The\nthermodynamic canonical spin variable is given as,\nσ(n′)\nα=\n2Sα+1\n2Coth/parenleftigg\n2Sα+1\n2SαH(n′)\nα\nkT/parenrightigg\n−1\n2Coth/parenleftigg\n1\n2SαH(n′)\nα\nkT/parenrightigg\n(5)\nwhereH(n′)\nαrepresents the molecular field energy\nfor the element αin the atomic plane n′due to its\ninteraction with its z= 12 nearest neighbors. The\nkTrepresentsthermalenergyandtheeffective mag-\nnetic moment per site is ¯M(n′)in then′th plane, is\ngiven as in units of Bohr magnetons by\n¯M(n′)/µB= (1−c)g(n′)\nCoσ(n′)\nCo+cg(n′)\nGdσ(n′)\nGd.(6)\ng(n′)\nαare the g factors for the alloy element on the\nn′th plane. The magnetization for the n′th atomic\nplane is calculated by multiplying ¯M(n′)by the\nnumber of sites per unit volume for the atomic\nplane.3.1.Alloy layered\n[Co1−cGdc]2[Co]2[Co1−cGdc]2\nferrimagnetic nanostructure\nbetween cobalt leads\nConsider in this subsection the embedded layered\n[2′22′] nanostructure between cobalt leads. For the\nCoandGdatomintheferrimagneticalloyed atomic\nplanes are found with the respective probabilities\n(1−c) andc. The molecular field energy usingMFT\nfor aCoatom on the 1st hcp basal plane of the lay-\nered nanostructure at the interface with the cobalt\nlead, may hence be expressed as\nH(1)\nCo= (3σ(B)\nCoJcc)+6[(1−c)σ(1)\nCoJcc+cσ(1)\nGdJcg]\n+3[(1−c)σ(2)\nCoJcc+cσ(2)\nGdJcg]. (7)\nThe exchange interactions are denoted by the\nsimplified notation JCoCo≡Jcc,JGdGd≡Jgg, and\nJCoGd≡Jcg. Equally, the molecular field energy for\naGdatom on the 1st hcp basal plane of the al-\nloy layered nanostructure at the interface with the\ncobalt lead, is\n(a)\nCo\nGdc=0.1 c=0.5\nc=0.5 c=0.1\nstep 0.1\nmeV2|22|\n020406080100120-3-2-101\nkTσCo1σGd16Vinod Ashokan; A. Khater and M. Abou Ghantous\nCo\nGd(b)\nc=0.1\nc=0.5c=0.5\nc=0.1\nmeVstep 0.1\n2|22|\n020406080100120-3-2-101\nkTσCo2σGd2\nFig. 2. Calculated spin variables σCoandσGd, forCoand\nGdsites on the 1st (a) and 2nd (b) hcp basal (0001) planes,\nof the alloy layered [ Co1−cGdc]2[Co]2[Co1−cGdc]2ferrimag-\nnetic nanostructure between cobalt leads, for different all oy\nconcentrations c, as a function of kTin meV. The down (up)\narrows in each figure correspond to the trend of the σspin\nvariations for the Co(Gd) sites with cstep changes.\nH(1)\nGd= (3σ(B)\nCoJcg)+6[(1−c)σ(1)\nCoJcg+cσ(1)\nGdJgg]\n+3[(1−c)σ(2)\nCoJcg+cσ(2)\nGdJgg]. (8)\nIn the present formulation, the seeding spin\nvalue for the lead Coatom at the interface with the\nalloy layered nanostructure is represented by σ(B)\nCo,\nwhich is obtained singularly from the EFT calcula-\ntions described in detail in section 2.\nIn contrast, the molecular field for a Coatom\non the 2nd hcp basal plane of the layered nanos-\ntructure inwards from the 1st, is\nH(2)\nCo= 3[(1−c)σ(1)\nCoJcc+cσ(1)\nGdJcg]+6[(1−c)σ(2)\nCoJcc\n+cσ(2)\nGdJcg]+3σ(3)\nCoJcc (9)\nSimilarly, the corresponding molecular field for a\nGdatom on the 2nd hcp basal plane of the layered\nnanostructure, is\nH(2)\nGd= 3[(1−c)σ(1)\nCoJcg+cσ(1)\nGdJgg]+6[(1−c)σ(2)\nCoJcg\n+cσ(2)\nGdJgg]+3σ(3)\nCoJcg (10)M1/\u0001Bc=0.1\nc=0.5\nM2/\u0000B\n020406080100120-1.0-0.50.00.5\nkTM1,2(persite) / μB\nFig. 3. Calculated magnetic moments per site for sites on\nthe 1st (solid curves) and 2nd (dotted curves) hcp basal\n(0001) planes ofthe alloy layered ferrimagnetic nanostruc ture\n[Co1−cGdc]2[Co]2[Co1−cGdc]2between cobalt leads. They\npresent small differences only at the high temperature kT\nend of the ordered phase. The down arrows follow the varia-\ntion trend for the magnetic moments per site with the cstep\nchanges, on the 1st (solid arrow) and 2nd (dotted arrow) hcp\nbasal (0001) planes.\nThehcp basal planes of the pure[ Co]2layer be-\ntween the alloy layers [ Co1−cGdc]2, are designated\nrespectively as the 3rd and 4th atomic planes. Us-\ning the symmetry properties of the layered [2′22′]\nnanostructure, we note that σ(3)\nCo≡σ(4)\nCo. The corre-\nsponding molecular field for a Coatom on the 3rd\nhcp basal plane is hence\nH(3)\nCo= 3[(1−c)σ(2)\nCoJcc+cσ(2)\nGdJcg]+9σ(3)\nCoJcc(11)\nThe above equations can be put into matrix\nform\n\nH(1)\nCo\nH(1)\nGd\nH(2)\nCo\nH(2)\nGd\nH(3)\nCo\n=\nA1\nA2\n0\n0\n0\n+\nx1x2x3x4x5\ny1y2y3y4y5\nu1u2u3u4u5\nv1v2v3v4v5\nz1z2z3z4z5\n\nσ(1)\nCo\nσ(1)\nGd\nσ(2)\nCo\nσ(2)\nGd\nσ(3)\nCo\n\n(12)Sublattice magnetizations of nano-magnetic layered mater ials7\nand the coefficients matrix is identical to\nx1x2x3x4x5\ny1y2y3y4y5\nu1u2u3u4u5\nv1v2v3v4v5\nz1z2z3z4z5\n\n≡\n2βJcc6cJcgβJcc3cJcg0\n2βJcg6cJggβJcg3cJgg0\nβJcc3cJcg2βJcc6cJcg3Jcc\nβJcg3cJgg2βJcg6cJgg3Jcg\n0 0 βJcc3cJcg9Jcc\n(13)\nwhereA1= 3σ(B)\nCoJcc,A2= 3σ(B)\nCoJcgandβ=\n3(1−c). Using Eqs.(7) to (11), and the spin vari-\nablesσ(n′)\nαformat given by Eq.(5), it follows that\nEq.(12) represents a nonlinear equations and to be\nsolved for the spin variables\nM1/μB\nM2/μB\nM3/μB meVc=0.1\nc=0.5\n3|33|\n020406080100120-1.0-0.50.00.5\nkTM1,2,3(persite) / μB\nFig. 4. Calculated magnetic moments per site for the 1st,\n2nd, and 3rd hcp basal atomic planes of the alloy layered\n[Co1−cGdc]3[Co]3[Co1−cGdc]3ferrimagnetic nanostructure.\nThe magnetic moments per site on the 1st and 3rd hcp planes\n(discontinuous and continues curves, respectively) are qu ite\nsimilar throughout the temperature range of the ordered\nphase. They differ significantly from the magnetic moments\non the 2nd hcp plane (continues curves). The down arrows\ncorrespond to the trend of the magnetic moment variations\nwithcstep changes.\n\nσ(1)\nCo\nσ(1)\nGd\nσ(2)\nCo\nσ(2)\nGd\nσ(3)\nCo\n=\nBCo(SCo,T,H(1)\nCo)\nBGd(SGd,T,H(1)\nGd)\nBCo(SCo,T,H(2)\nCo)\nBGd(SGd,T,H(2)\nGd)\nBCo(SCo,T,H(3)\nCo)\n(14)Solving the above equations numerically, we\ncalculate the spin variables σ(1)\nCo,σ(1)\nGd,σ(2)\nCo,σ(2)\nGdand\nσ(3)\nCoas a function of temperature, for any given al-\nloy concentration c. Note that the symmetry of the\nsystem imposes the following equalities for the spin\nvariables\n/parenleftigg\nσ(1)\nCo\nσ(1)\nGd/parenrightigg\n≡/parenleftigg\nσ(6)\nCo\nσ(6)\nGd/parenrightigg\n,/parenleftigg\nσ(2)\nCo\nσ(2)\nGd/parenrightigg\n≡/parenleftigg\nσ(5)\nCo\nσ(5)\nGd/parenrightigg\n,\n/parenleftig\nσ(3)\nCo/parenrightig\n≡/parenleftig\nσ(4)\nCo/parenrightig\n(15)\nThe calculated spin variables σ(n′)\nCo,σ(n′)\nGdfor the\nnominal n′= 1 andn′= 2 hcp basal planes are pre-\nsented inFig.2, as afunctionof temperatureandfor\neutectic concentrations c= [0.1,0.5] in steps of 0.1.\nFurther, Eq.(6), and g(n′)\nCo≡gCo= 2.2,g(n′)\nGd≡\ngGd= 2 for all n′, yield the magnetic moments per\nsite on the 1st and 2nd hcp basal planes as a func-\ntion of temperature. These are presented for com-\nparison in Fig.3, where a small interesting differ-\nence is observed at the high temperature kTend\nof the ordered phase. Compensation temperatures\nkTcomp<21 meV, are observed for the ferrimag-\nnetic hcp planes for eutectic stable concentrations\nin the range 0 .23< c <0.5.\n3.2.Alloy layered\n[Co1−cGdc]3[Co]3[Co1−cGdc]3\nferrimagnetic nanostructure\nbetween cobalt leads\nThe layered [3′33′] nanostructures under considera-\ntion are symmetric about the origin here taken as\nthe hcp plane n= 0. In this system the symmetry\nproperties to be used are\n/parenleftigg\nσ(1)\nCo\nσ(1)\nGd/parenrightigg\n≡/parenleftigg\nσ(9)\nCo\nσ(9)\nGd/parenrightigg\n,/parenleftigg\nσ(2)\nCo\nσ(2)\nGd/parenrightigg\n≡/parenleftigg\nσ(8)\nCo\nσ(8)\nGd/parenrightigg\n,\n/parenleftigg\nσ(3)\nCo\nσ(3)\nGd/parenrightigg\n≡/parenleftigg\nσ(7)\nCo\nσ(7)\nGd/parenrightigg\n,and/parenleftig\nσ(4)\nCo/parenrightig\n≡/parenleftig\nσ(6)\nCo/parenrightig\n(16)\nSimilarly, as in the previous case [2′22′], the\nequivalent molecular field energy results can be cast\nhere in matrix form as8Vinod Ashokan; A. Khater and M. Abou Ghantous\n\nH(1)\nCo\nH(1)\nGd\nH(2)\nCo\nH(2)\nGd\nH(3)\nCo\nH(3)\nGd\nH(4)\nCo\nH(5)\nCo\n=\nA1\nA2\n0\n0\n0\n0\n0\n0\n\n+\n2βJcc6cJcgβJcc3cJcg0 0 0 0\n2βJcg6cJggβJcg3cJgg0 0 0 0\nβJcc3cJcg2βJcc6cJcgβJcc3cJcg0 0\nβJcg3cJgg2βJcg6cJggβJcg3cJgg0 0\n0 0 βJcc3cJcg2βJcc6cJcg3Jcc0\n0 0 βJcg3cJgg2βJcg6cJgg3Jcg0\n0 0 0 0 βJcc3cJcg6Jcc3Jcc\n0 0 0 0 0 0 6 Jcc6Jcc\n\n×\nσ(1)\nCo\nσ(1)\nGd\nσ(2)\nCo\nσ(2)\nGd\nσ(3)\nCo\nσ(3)\nGd\nσ(4)\nCo\nσ(5)\nCo\n(17)\nwhereA1(kT) = 3MCo(kT)JCo−CoandA2(kT) =\n3MCo(kT)JCo−Gd. This yields the new irreducible\nvariables\n\nσ(1)\nCo\nσ(1)\nGd\nσ(2)\nCo\nσ(2)\nGd\nσ(3)\nCo\nσ(3)\nGd\nσ(4)\nCo\nσ(5)\nCo\n=\nBCo(SCo,T,H(1)\nCo)\nBGd(SGd,T,H(1)\nGd)\nBCo(SCo,T,H(2)\nCo)\nBGd(SGd,T,H(2)\nGd)\nBCo(SCo,T,H(3)\nCo)\nBGd(SGd,T,H(3)\nGd)\nBCo(SCo,T,H(4)\nCo)\nBCo(SCo,T,H(5)\nCo)\n(18)c=0.47\n2|22|3|33|\nmeV(a)\n020406080100120-2.0-1.5-1.0-0.50.00.5\nkTM1(persite) / μB\n3|33|\n2|22|c=0.47 (b)\nmeV\n020406080100120-2.0-1.5-1.0-0.50.00.5\nkTM2(persite) / μB\nFig. 5. Calculated effective magnetic moments per site for\nthe nominal: (a) 1st alloyed, n′= 1, and (b) 2nd alloyed,\nn′= 2, hcp basal planes,\nfor the alloy layered [ Co0.53Gd0.47]2[Co]2[Co0.53Gd0.47]2\nand [Co0.53Gd0.47]3[Co]3[Co0.53Gd0.47]3nanostructures be-\ntween cobalt leads.\nThe above nonlinear equations can be solve nu-\nmerically to obtain the spin variables and magnetic\nmomentspersite,ontheindividualhcpbasalplanes\nof the alloy layered magnetic [3′33′] nanostructure.\nFig.4 presents the calculated results for the mag-\nnetic moments on the 1st, 2nd, and 3rd hcp atomic\nplanes for this system. The magnetic moments per\nsite for the 1st and 3rd hcp planes (discontinuous\nand continues curves, respectively) are quite similar\nthroughout the temperature range of the ordered\nphase. Together, they differ significantly from theSublattice magnetizations of nano-magnetic layered mater ials9\nmagnetic moments per site for the 2nd hcp plane,\nthroughout the temperature range of the ordered\nphase.\nAs defined, the fundamental atomic spins are\nS= 1 forCo, andS= 7/2 forGd, where we con-\nsider the spins to be up for Coand down for Gd,\nin the ferrimagnetic alloy. Note that the Gdfun-\ndamental spin is 3.5 times that of Co. However,\nthe magnetic exchange Jggbetween GdandGdis\nweaker than Jccbetween CoandCo. By studying\nequations7,8,9,and10, onecanseethatthemolec-\nular fields for CoandGdsites vary with tempera-\ntureandcanchangesigns.Thecompetitionbetween\nthese molecular fields along the entire temperature\nrange determines the magnetizations per site, M;\nin the low temperature regime they increase then\nreach a maximum value before losing their magne-\ntized phase. See figures 3 and 4.\nThe computational model is general and can be\nextended to treat individual hcp atomic planes of\nalloy layered [ Co1−cGdc]p[Co]q[Co1−cGdc]rnanos-\ntructures, with r,p,q≥1. This procedure may\nbe generalized to larger layered nanostructures be-\ntween semi-infinite Coleads. It has been observed\nthat while increasing r,p,qthe results for the sub-\nlattice magnetization properties in the core atomic\nplanes tend to limiting solutions.\nFig. 6. Calculated spin variables σ(n′)\nCoand their detailed\nvariations as a function of temperature on the nominal n′\ncobalt hcp basal planes inside the alloy layered ferrimag-\nnetic nanostructures [ Co0.53Gd0.47]2[Co]2[Co0.53Gd0.47]2\nand[Co0.53Gd0.47]3[Co]3[Co0.53Gd0.47]3incomparison with\nthe temperature variation of the spin variable σ(B)\nCoon the\ncobalt leads; see details in the text.3.3.Alloy layered\n[Co0.53Gd0.47]ℓ′[Co]ℓ[Co0.53Gd0.47]ℓ′\nferrimagnetic nanostructure\nbetween cobalt leads\nThis alloy layered magnetic nanostructure between\ncobalt leads, at the characteristic eutectic concen-\ntrationc= 0.47, is particularly interesting since\nCo/Gdmagnetic multilayers at the same composi-\ntion have been reported to be very stable,12. We\nhave appliedhencethe EFT- MFT model approach\nto deduce the spin variables σ(n′)\nCoandσ(n′)\nGd, and\nthe effective magnetic moments per site, for the in-\ndividual hcp basal planes of layered ferrimagnetic\nnanostructures between cobalt leads, as a function\nof temperature, eutectic concentration, and thick-\nnessesℓ= 2 and 3. The integer n′numbers the hcp\nplanes from 1 to 6 for ℓ= 2, and from 1 to 9 for\nℓ= 3.\nThe effective magnetic moments per site calcu-\nlated in the units of Bohr magnetons for the alloyed\nnominal 1st, n′= 1, and 2nd, n′= 2, hcp basal\nplanes for the alloy layered ferrimagnetic nanos-\ntructures [ Co0.53Gd0.47]2[Co]2[Co0.53Gd0.47]2and\n[Co0.53Gd0.47]3[Co]3[Co0.53Gd0.47]3arepresentedin\nFig.5. It is observed that the computed effective\nmagnetic moments per site as a function of tem-\nperature on the nominal n′= 1 hcp basal planes\ndo not vary significantly with increased thickness of\nnanostructure,see Fig.5(a). Thiscan beunderstood\nclearly since the corresponding matrix elements in\nEq.(17) do not change significantly with increasing\nthickness. In contrast, it is observed that the ef-\nfective magnetic moments per site on the nominal\nn′= 2 hcp basal planes do vary significantly with\nincreasing thickness, see Fig.5(b). This is expected\nphysically owing to the changes of the correspond-\ning effective molecular fields for CoandGdsites.\nThe observed variations start at ≈15 meV and per-\nsist for higher temperatures, including room tem-\nperature ≈26 meV.\nIt is also interesting to compute spin vari-\nablesσ(n′)\nCo, and their detailed variations as a func-\ntion of temperature on the nominal n′cobalt\nhcp basal planes inside the alloy layered ferri-\nmagnetic nanostructures, in comparison with the\nspin variable σ(B)\nCoon the cobalt leads as a\nfunction of temperature. This is done for the\n[Co0.53Gd0.47]ℓ′[Co]ℓ[Co0.53Gd0.47]ℓ′layered nanos-\ntructure between cobalt leads, for thicknesses ℓ= 2\nand 3. The calculated results are presented in Fig.6.10Vinod Ashokan; A. Khater and M. Abou Ghantous\nTable 1. Spin variable values σGd=< SGd>for theGdsites on atomic planes 1, 2, .., ℓ, of the\n2′22′and 3′33′nanojunction systems between cobalt leads, are given for st able eutectic compositions\nc≤0.5, at room temperature T=300K, using the theoretical EFT- MFT combined method. The spin\nvariable σCo=< SCo>is≈1 at room temperature for the Cosites throughout the system.\nConcentrations Spin variables σGd\n2′22′3′33′\nc [Co1−cGdc]2,L[Co1−cGdc]2,R[Co1−cGdc]3,L [Co1−cGdc]3,R\n0.1 -1.10 -1.10 -1.10 -1.10 -1.10 -1.08 -1.10 -1.10 -1.08 -1. 10\n0.2 -1.03 -1.03 -1.03 -1.03 -1.03 -1.00 -1.03 -1.03 -1.00 -1. 03\n0.3 -0.97 -0.97 -0.97 -0.97 -0.97 -0.91 -0.97 -0.97 -0.91 -0. 97\n0.4 -0.91 -0.91 -0.91 -0.91 -0.90 -0.81 -0.90 -0.90 -0.81 -0. 90\n0.5 -0.83 -0.83 -0.83 -0.83 -0.82 -0.71 -0.82 -0.82 -0.71 -0. 82\nAs is physically expected, the σ(B)\nCois≥σ(n′)\nCofor all\npureCohcp planes n′inside the layered thicknesses\nℓ, at all temperatures of the ordered ferrimagnetic\nphase. Also as expected, σ(5)\nCois≥σ(4)\nCofor the al-\nloy layered [ Co0.53Gd0.47]3[Co]3[Co0.53Gd0.47]3fer-\nrimagnetic nanostructure between cobalt leads, and\nboth are greater or equal to the σ(3)\nCoof the\n[Co0.53Gd0.47]2[Co]2[Co0.53Gd0.47]2layered nanos-\ntructure. Note that the n′= 3 pure cobalt plane\nfor the [2′22′] layered nanostructure is equivalent\nnominally to the n′= 4 pure cobalt plane for the\n[3′33′] layered nanostructure. The results confirm a\nphysical trend which is expected, and which would\nlead to limiting values with increasing thickness of\nthe layered ferrimagnetic structure between cobalt\nleads.\nWe emphasize that the basic physical variables,\nsuch as the exchange and sublattice magnetizations\nforCoandGdsitesfortheembeddedlayerednanos-\ntructures between cobalt leads, are necessary ele-\nments for the computations of the spin-dynamics\nof magnetic nanojunctions in the field of magnon-\nics, as for the ballistic magnon transport25. In Ta-\nble 1 we present an example for the calculated spin\nvariables < SCo>≈1 throughout the system, and\n< SGd>on theidentified atomic planes1, 2, .., ℓ,of\nthe layered ...Co][2′22′][Co...and...Co][3′33′][Co...\nnanostructures between cobalt leads, at room tem-\nperature T=300K and stable eutectic compositions\nc≤0.5, using the EFT-MFT combined method.\n4. Summary and conclusions\nIn this work, we model the salient sub-lattice mag-\nnetic properties of the alloy layered ferrimagnetic\n[Co1−cGdc]ℓ′[Co]ℓ[Co1−cGdc]ℓ′nanostructures be-\ntween magnetically ordered cobalt leads. In par-\nticular, sublattice magnetizations of the CoandGdsites on the individual hcp (0001) basal planes\nof the alloy layered lamellar nanostructures by\nusing EFT-MFT combined method. The effective\nmagnetic moments per site and sublattice mag-\nnetizations are plotted as a function of tempera-\nture and thicknesses of the lamellar nanostructure.\nThe computational model is general and can repre-\nsents other composite magnetic elements andlamel-\nlar nanostructures. The calculated magnetic ex-\nchange, and spin variables for gadolinium < SGd>\nand cobalt < SCo>, site on the identified hcp\n(0001) basal planes of the alloy layered ferrimag-\nnetic [Co1−cGdc]ℓ′[Co]ℓ[Co1−cGdc]ℓ′nanostructures\nbetween cobalt leads, are very important quanti-\nties for the self-consistent analysis of quantum spin\ndynamics system and the coherent magnon ballis-\ntic transport across such nanostructures. The cal-\nculated results are also important fo the applica-\ntions in the fields of magnonics. The Ising EFT\nmethod serves to determine the magnetic exchange\nconstants for CoandGdsites of purecrystals, char-\nacterized by their fundamental quantum spins, by\ncomparing EFT with the experimental data. By\nseeding the MFT results on the alloy lamellar ferri-\nmagnetic nanostructure by the EFT computations\nof the cobalt leads from the interface inwards, the\nsublattice magnetizations for the CoandGdsites\nin the nanostructure are computed.\nReferences\n1. A.A. Serga, A.V. Chumak, A. Andre, G.A. Melkov,\nA.N. Slavin, S.O. Demokritov, and B. Hillebrands,\nPhys. Rev. Lett. 99, (2007) 227202\n2. T. Schneider, A.A. Serga, B. Leven, B. Hillebrands,\nR.L. Stamps, and M.P. Kostylev, Appl. Phys. Lett.\n92, (2008) 022505\n3. V. V. Kruglyak, S. O. Demokritov and D. Grundler,\nJ. Phys. D: Appl. Phys. 43, (2010) 264001Sublattice magnetizations of nano-magnetic layered mater ials11\n4. K. Lee and S. Kim, J. Appl. Phys. 104, (2008)\n053909\n5. S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. 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Bon\fglio\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands\nK. Rode, K. Siewerska, J. Besbas, G. Y. P. Atcheson, P. Stamenov, and J.M.D. Coey\nCRANN, AMBER and School of Physics, Trinity College Dublin, Ireland\nA.V. Kimel, Th. Rasing, and A. Kirilyuk\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands and\nFELIX Laboratory, Radboud University, Toernooiveld 7c, 6525 ED Nijmegen, The Netherlands\nHere we study both static and time-resolved dynamic magnetic properties of the compensated\nferrimagnet Mn 2RuxGa from room temperature down to 10 K, thus crossing the magnetic compen-\nsation temperature TM. The behaviour is analysed with a model of a simple collinear ferrimagnet\nwith uniaxial anisotropy and site-speci\fc gyromagnetic ratios. We \fnd a maximum zero-applied-\n\feld resonance frequency of \u0018160 GHz and a low intrinsic Gilbert damping \u000b\u00180:02, making it a\nvery attractive candidate for various spintronic applications.\nI. INTRODUCTION\nAntiferromagnets (AFM) and compensated ferrimag-\nnets (FiM) have attracted a lot of attention over the last\ndecade due to their potential use in spin electronics1,2.\nDue to their lack of a net magnetic moment, they are\ninsensitive to external \felds and create no demagnetis-\ning \felds of their own. In addition, their spin dynamics\nreach much higher frequencies than those of their ferro-\nmagnetic (FM) counterparts due to the contribution of\nthe exchange energy in the magnetic free energy3.\nDespite these clear advantages, AFMs are scarcely\nused apart from uni-directional exchange biasing rela-\ntively in spin electronic applications. This is because\nthe lack of net moment also implies that there is no\ndirect way to manipulate their magnetic state. Fur-\nthermore, detecting their magnetic state is also compli-\ncated and is usually possible only by neutron di\u000braction\nmeasurements4, or through interaction with an adjacent\nFM layer5.\nCompensated, metallic FiMs provide an interesting al-\nternative as they combine the high-speed advantages of\nAFMs with those of FMs, namely, the ease to manipu-\nlate their magnetic state. Furthermore, it has been shown\nthat such materials are good candidates for the emerging\n\feld of All-Optical Switching (AOS) in which the mag-\nnetic state is solely controlled by a fast laser pulse6{8.\nA compensated, half-metallic ferrimagnet was \frst en-\nvisaged by van Leuken and de Groot9. In their model\ntwo magnetic ions in crystallographically di\u000berent po-\nsitions couple antiferromagnetically and perfectly com-\npensate each-other, but only one of the two contributes\nto the states at the Fermi energy responsible for elec-\ntronic transport. The \frst experimental realisation of\nthis, Mn 2RuxGa (MRG), was provided by Kurt et al.10.\nMRG crystallises in the XAHeusler structure, space\ngroupF\u001643m, with Mn on the 4 aand 4csites11.\nSubstrate-induced bi-axial strain imposes a slight tetrag-\nonal distortion, which leads to perpendicular magneticanisotropy. Due to the di\u000berent local environment of\nthe two sublattices, the temperature dependence of their\nmagnetic moments di\u000ber, and perfect compensation is\ntherefore obtained at a speci\fc temperature TMthat\ndepends on the Ru concentration xand the degree of\nbiaxial strain. It was previously shown that MRG ex-\nhibits properties usually associated with FMs: a large\nanomalous Hall angle12, that depends only on the mag-\nnetisation of the 4 cmagnetic sublattice13; tunnel magne-\ntoresistance (TMR) of 40 %, a signature of its high spin\npolarisation14, was observed in magnetic tunnel junc-\ntions (MTJs) based on MRG15; and a clear magneto-\noptical Kerr e\u000bect and domain structure, even in the ab-\nsence of a net moment16,17. Strong exchange bias of a\nCoFeB layer by exchange coupling with MRG through\na Hf spacer layer18, as well as single-layer spin-orbit\ntorque19,20showed that MRG combined the qualities of\nFMs and AFMs in spin electronic devices.\nThe spin dynamics in materials where two distinct\nsublattices are subject to di\u000bering internal \felds (ex-\nchange, anisotropy, . . . ) is much richer than that of a\nsimple FM, as previously demonstrated by the obers-\nvation of single-pulse all-optical switching in amorphous\nGdFeCo21,22and very recently in MRG8. Given that the\nmagnetisation of MRG is small, escpecially close to the\ncompensation point, and the related frequency is high,\nnormal ferromagnetic resonance (FMR) spectroscopy is\nunsuited to study their properties. Therefore, we used\nthe all-optical pump-probe technique to characterize the\nresonance frequencies at di\u000berent temperatures in vicin-\nity of the magnetic compensation point. This, together\nwith the simulation of FMR, make it possible to deter-\nmine the e\u000bective g-factors, the anisotropy constants and\ntheir evolution across the compensation point. We found,\nin particular, that our ferrimagnetic half-metallic Heusler\nalloy has resonance frequency up to 160 GHz at zero-\feld\nand a relatively low Gilbert damping.arXiv:1909.09085v1  [cond-mat.mtrl-sci]  19 Sep 20192\nFIG. 1. Net moment measured by magnetometry and coercive\n\feld measured by static Faraday e\u000bect. The upturn of the net\nmoment below T\u001850 K is due to paramagnetic impurities\nin the MgO substrate. TMis indicated by the vertical dotted\nline. As expected the maximum available applied \feld \u00160H=\n7 T is insu\u000ecient to switch the magnetisation close to TM.\nII. EXPERIMENTAL DETAILS\nThin \flm samples of MRG were grown in a `Sham-\nrock' sputter deposition cluster with a base pressure of\n2\u000210\u00008Torr on MgO (001) substrates. Further infor-\nmation on sample deposition can be found elsewhere23.\nThe substrates were kept at 250\u000eC, and a protective\n\u00183 nm layer of aluminium oxide was added at room tem-\nperature. Here we focus on a 53 nm thick sample with\nx= 0:55, leading to TM\u001980 K as determined by SQUID\nmagnetometry using a Quantum Design 5 T MPMS sys-\ntem (see FIG. 1). We are able to study the magneto-\noptical properties both above and below TM.\nThe magnetisation dynamics was investigated using an\nall-optical two-colour pump-probe scheme in a Faraday\ngeometry inside a \u00160Hmax= 7 T superconducting coil-\ncryostat assembly. Both pump and probe were produced\nby a Ti:sapphire femtosecond pulsed laser with a cen-\ntral wavelength of 800 nm, a pulse width of 40 fs and\na repetition rate of 1 kHz. After splitting the beam in\ntwo, the high-intensity one was doubled in frequency by\na BBO crystal (giving \u0015= 400 nm) and then used as\nthe pump while the lower intensity 800 nm beam acted\nas the probe pulse. The time delay between the two was\nadjusted by a mechanical delay stage. The pump was\nthen modulated by a synchronised mechanical chopper\nat 500 Hz to improve the signal to noise ratio by lock-in\ndetection. Both pump and probe beams were linearly\npolarized, and with spot sizes on the sample of 150 µm\nand 70 µm, respectively. The pump pulse hit the sample\nat an incidence angle of \u001910\u000e. After interaction with\nthe sample, we split the probe beam in two orthogonally\npolarized parts using a Wollaston prism and detect the\nchanges in transmission and rotation by calculating the\nFIG. 2. Comparison of hysteresis loops obtained by Faraday,\nAHE, and magnetometry recorded at room temperature. The\ntwo former were recorded with the applied \feld perpendicular\nto the sample surface, while for the latter we show results for\nboth \feld applied parallel and perpendicular to the sample.\nsum and the di\u000berence in intensity of the two signals.\nThe external \feld was applied at 75\u000eto the easy axis of\nmagnetization thus tilting the magnetisation away from\nthe axis. Upon interaction with the pump beam the mag-\nnetisation is momentarily drastically changed24and we\nmonitor its return to the initial con\fguration via remag-\nnetisation and then precession through the time depen-\ndent Faraday e\u000bect on the probe pulse.\nThe static magneto-optical properties were examined\nin the same cryostat/magnet assembly.\nIII. RESULTS & DISCUSSION\nA. Static magnetic properties\nWe \frst focus on the static magnetic properties as\nobserved by the Faraday e\u000bect, and compare them to\nwhat is inferred from magnetometry and the anomalous\nHall e\u000bect. In FIG. 2 we present magnetic hysteresis\nloops as recorded using the three techniques. Due to the\nhalf metallic nature of the sample, the magnetotrans-\nport properties depend only on the 4 csublattice. As the\nmain contribution to the MRG dielectric tensor in the\nvisible and near infrared arises from the Drude tail16,\nboth AHE and Faraday e\u000bect probe essentially the same\nproperties (mainly the spin polarised conduction band of\nMRG), hence we observe overlapping loops for the two\ntechniques. Magnetometry, on the other hand, measures\nthe net moment, or to be precise the small di\u000berence\nbetween two large sublattice moments. The 4 asublat-\ntice, which is insigni\fcant for AHE and Faraday here\ncontributes on equal footing. FIG. 2 shows a clear di\u000ber-\nence in shape between the magnetometry loop and the3\nFIG. 3. Time resolved Faraday e\u000bect recorded at T= 290 K\nin applied \felds ranging from 1 T to 7 T. After the initial\ndemagnetisation seen as a sharp increase in the signal at t\u0018\n0 ps, magnetisation is recovered and followed by precession\naround the e\u000bective \feld until fully damped. The lines are\n\fts to the data. The inset shows the experimental geometry\nfurther detailed in the main text.\nAHE or Faraday loops. We highlight here that the ap-\nparent `soft' contribution that shows switching close to\nzero applied \feld, is not a secondary magnetic phase, but\na signature of the small di\u000berences in the \feld-behaviour\nof the two sublattices. We also note that this behaviour\nis a result of the non-collinear magnetic order of MRG.\nA complete analysis of the dynamic properties therefore\nrequires knowledge of the anisotropy constants on both\nsublattices as well as the (at least) three intra and in-\nter sublattice exchange constants. Such an analysis is\nbeyond the scope of this article, and we limit our anal-\nysis to the simplest model of a single, e\u000bective uniaxial\nanisotropy constant Kuin the exchange approximation\nof the ferrimagnet.\nB. Dynamic properties\nWe now turn to the time-resolved Faraday e\u000bect and\nspin dynamics. Time-resolved Faraday e\u000bect data were\nrecorded at \fve di\u000berent temperatures 10 K, 50 K, 100 K,\n200 K and 290 K, with applied \felds ranging from 1 T to\n7 T.\nFIG. 3 shows the \feld-dependence of the Faraday ef-\nfect as a function of the delay between the pump and\nthe probe pulses, recorded at T= 290 K. Negative de-\nlay indicates the probe is hitting the sample before the\npump. After the initial demagnetisation, the magneti-\nsation recovers and starts precessing around the e\u000bec-\ntive \feld which is determined by the anisotropy and the\napplied \feld. The solid lines in FIG. 3 are \fts to the\ndata to extract the period and the damping of the pre-cession in each case. The \ftting model was an expo-\nnentially damped sinusoid with a phase o\u000bset. We note\nthat the apparent evolution of the amplitude and phase\nwith changing applied magnetic \feld is due to the quasi-\nresonance of the spectrum of the precessional motion\nwith the low-frequency components of the convolution\nbetween the envelope of the probe pulse and the phys-\nical relaxation of the system. The latter include both\nelectron-electron and electron-lattice e\u000bects. A rudimen-\ntary model based on a classical oscillator successfully re-\nproduces the main features of the amplitude and phase\nobserved.\nIn two-sublattice FiMs, the gyromagnetic ratios of the\ntwo sublattices are not necessarily the same. This is par-\nticularly obvious in rare-earth/transition metal alloys,\nand is also the case for MRG despite the two sublat-\ntices being chemically similar; they are both Mn. Due\nto the di\u000berent local environment however, the degree\nof charge transfer for the two di\u000bers. This leads to two\ncharacteristic temperatures, a \frst TMwhere the mag-\nnetic moments compensate, and a second TAwhere the\nangular momenta compensate. It can be shown that for\nthe ferromagnetic mode, the e\u000bective gyromagnetic ratio\n\re\u000bcan then be written25\n\re\u000b=M4c(T)\u0000M4a(T)\nM4c(T)=\r4c\u0000M4a(T)=\r4a(1)\nsubscripti= 4a;4cdenotes sublattice i,Mi(T)\nthe temperature-dependent magnetisation, and \rithe\nsublattice-speci\fc gyromagnetic ratio. \re\u000bis related to\nthe e\u000bective g-factor\nge\u000b=\re\u000bh\n\u0016B(2)\nwherehis the Planck constant and \u0016Bthe Bohr magne-\nton.\nThe frequency of the precession is determined by the\ne\u000bective \feld, which can be inferred from the derivative\nof the magnetic free energy density with respect to M.\nFor an external \feld applied at a given \fxed angle with\nrespect to the easy axis this leads to the Smit-Beljers\nformula26\n!FMR =\re\u000bvuut1\nM2ssin2\u001e\"\n\u000e2E\n\u000e\u00122\u000e2E\n\u000e\u001e2\u0000\u0012\u000e2E\n\u000e\u0012\u000e\u001e\u00132#\n(3)\nwhere\u0012and\u001eare the polar and azimuthal angles of the\nmagnetisation vector, and Ethe magnetic free energy\ndensity\nE=\u0000\u00160H\u0001M+Kusin2\u0012+\u00160M2\nscos2\u0012=2 (4)\nwhere the terms correspond to the Zeeman, anisotropy\nand demagnetising energies, respectively, and Msis the\nnet saturation magnetisation. It should be mentioned\nthat the magnetic anisotropy constant Kuis related to\nM, which is being considered constant in magnitude, via\nKu=\f\u00160M2\ns=2,\fa dimensionless parameter.4\nFIG. 4. Observed precession frequency as a function of the\napplied \feld for various temperatures. The solid lines are \fts\nto the data as described in the main text.\nBased on Eqs. (1) through (4) we \ft our entire data set\nwith\re\u000bandKuas the only free parameters. The exper-\nimental data and the associated \fts are shown as points\nand solid lines in FIG. 4. At all temperatures our simple\nmodel with one e\u000bective gyromagnetic ratio \re\u000band a\nsingle uniaxial anisotropy parameter Kureproduces the\nexperimental data reasonably well. The model systemat-\nically underestimates the resonance frequency for inter-\nmediate \felds, with the point of maximum disagreement\nincreasing with decreasing temperature. We speculate\nthis is due to the use of a simple uniaxial anisotropy in\nthe free energy (see Eq. 4), while the real situation is\nmore likely to be better represented as a sperimagnet. In\nparticular, the non-collinear nature of MRG that leads\nto a deviation from 180\u000eof the angle between the two\nsublattice magnetisations, depending on the applied \feld\nand temperature.\nFrom the \fts in FIG. 4 we infer the values of ge\u000band\nthe anisotropy \feld \u00160Ha=2Ku=Ms. The result is shown\nin FIG. 5. The anisotropy \feld is monotonically increas-\ning with decreasing temperature as the magnetisation\nof the 4csublattice increases in the same temperature\nrange. We highlight here the advantage of determining\nthis \feld through time-resolved magneto-optics as op-\nposed to static magnetometry and optics. Indeed the\nanisotropy \feld as seen by static methods is sensitive to\nthe combination of anisotropy and the netmagnetic mo-\nment, as illustrated in FIG. 1, where the coercive \feld\ndiverges as T!TM. In statics one would expect a di-\nvergence of the anisotropy \feld at the same temperature.\nThe time-resolved methods however distinguish between\nthe net and the sublattice moments, hence better re\rect-\ning the evolution of the intrinsic material properties of\nthe ferrimagnet.\nThe temperature dependence of the anisotropy con-\nstants was a matter for discussion for many years27,28.\nFIG. 5. E\u000bective g-factor,ge\u000b, and the anisotropy \feld\nas determined by time-resolved Faraday e\u000bect. ge\u000b, orange\nsquares, increases from near the free electron value of 2 to 4\njust belowTM, while the anisotropy \feld, blue triangles, in-\ncreases near-linearly with decreasing temperature. A M3\ft,\nred dashes line, of the anisotropy behaviour shows the almost-\nmetallic origin of it, indicating the dominant character of the\n4c sublattice.\nWritten in spherical harmonics the 3 danisotropy can\nbe expressed as, k2Y0\n2(\u0012) +k4Y0\n4(\u0012)29wherek2/\nM(T)3andk4/M(T)10. The experimental measured\nanisotropy is then, K2(T) =ak2(T)+bk4(T), withaand\nbthe contributions of the respective spherical harmonics.\nFIG. 5 shows that a reasonable \ft of our data is ob-\ntained with M(T)3which means, \frst, that the contri-\nbution of the 4thorder harmonic can be neglected, and\nsecond, that the contribution of the TMand 2ndsublat-\ntice is negligible, indicating the dominant character of\nthe 4c sublattice.\nIn addition, we should note here that the high fre-\nquency exchange mode was never observed on our exper-\niments. While far from TMits frequency might be too\nhigh to be observable, in the vicinity of TM, in contrast,\nits frequency is expected to be in the detection range.\nMoreover, given the di\u000berent electronic structure of the\ntwo sublattices, it is expected that the laser pulse should\nselectively excite the sublattice 4c, and therefore lead to\nthe e\u000bective excitation of the exchange mode. We argue\nthat it is the non-collinearity of the sublattices (see sec-\ntion III A) that smears out the coherent precession at\nhigh frequencies.\nThe e\u000bective gyromagnetic ratio, ge\u000b, shows a non-\nmonotonic behaviour. It increases with decreasing Tto-\nwardsTM, reaching a maximum at about 50 K before\ndecreasing again at T= 10 K. We alluded above to\nthe di\u000berence between the magnetic and the angular mo-\nmenta compensation temperatures. We expect that ge\u000b\nreaches a maximum when T=TA30, here between the\nmeasurement at T= 50 K and the magnetic compensa-\ntion temperature TM\u001980 K.5\nFIG. 6. Intrinsic and anisotropic broadening in MRG across\ntheTM. The inset shows the evaluation process of the two\ndamping parameters. A linear \ft is used to evaluate intercept\n(anisotropic broadening) and slope (intrinsic damping) of the\nfrequencies versus the inverse of the decay time. The data\npoint are obtained from the \ft of time-resolved Faraday e\u000bect\nmeasurements (an example is shown in Fig.4).\nFrom XMCD data11, we could estimate spin and or-\nbital moment components of the magnetic moments of\nthe two sublattices, what allowed us to derive the ef-\nfective g-factors for the sublattices as g4a= 2:05 and\ng4c= 2:00. In this case we expect the angular momentum\ncompensation temperature TAto be below TM, opposite\nto what is observed for GdFeCo21. Given this small dif-\nference however, TAandTMare expected to be rather\nclose to each other, consistent with the limited increase\nofge\u000bacross the compensation points.\nWe turn \fnally to the damping of the precessional mo-\ntion of Maround the e\u000bective \feld \u00160He\u000b. Damping is\nusually described via the dimensionless parameter \u000bin\nthe Landau-Lifshiz-Gilbert equation, and it is a measure\nof the dissipation of magnetic energy in the system. In\nthis model, \u000bis a scalar constant and the observed broad-\nening in the time domain is therefore a linear function of\nthe frequency of precession31{33. We infer \u000b0, the total\ndamping, from our \fts of the time-resolved Faraday e\u000bect\nas\u000b0= (\u001cd)\u00001, where\u001cdis the decay time of the \fts. We\nthen, for each temperature, plot \u000b0as a function of the\nobserved frequency and regress the data using a straight\nline \ft. The intrinsic \u000bis the slope of this line, while the\nintercept represents the anisotropic broadening.\nFIG. 6 shows the intrinsic damping \u000band the\nanisotropic broadening as a function of temperature.\nAnisotropic broadening is usually attributed to a vari-\nation of the anisotropy \feld in the region probed by the\nprobe pulse34. For MRG this is due to slight lateral vari-\nations in the Ru content xin the thin \flm sample. Such a\nvariation leads to a variation in e\u000bective TMandTAand\ncan therefore have a large in\ruence on the broadening asa function of temperature. Despite this, the anisotropic\nbroadening is reasonably low in the entire temperature\nrange above TM, and a more likely explanation for its\nrapid increase below TMis that the applied magnetic\n\feld is insu\u000ecient to completely remagnetize the sam-\nple between two pump pulses. As observed in Fig.5, the\nanisotropy \feld reaches almost 4 T at low temperature,\ncomparable to our maximum applied \feld of 7 T. The\nintrinsic damping \u000bis less than 0.02 far from TM, but\nincreases sharply at T= 100 K. We tentatively attribute\nthis to an increasing portion of the available power be-\ning transferred into the high-energy exchange mode, al-\nthough we underline that we have not seen any direct\nevidence of such a mode in any of the experimental data.\nIV. CONCLUSION\nWe have shown that the time-resolved Faraday e\u000bect\nis a powerful tool to determine the spin dynamic proper-\nties in compensated, metallic ferrimagnets. The high spin\npolarisation of MRG enables meaningful Faraday data to\nbe recorded even near TMwhere the net magnetisation\nis vanishingly small, and the dependence of the dynamics\non the sublattice as opposed to the net magnetic prop-\nerties provides a more physical understanding of the ma-\nterial. Furthermore, we \fnd that the ferromagnetic-like\nmode of MRG reaches resonance frequencies as high as\n160 GHz in zero applied \feld, together with a small in-\ntrinsic damping. This value is remarkable if compared\nto well-known materials such as GdFeCo which, at zero\n\feld, resonates at tens of GHz21or [Co/Pt] nmultilay-\ners at 80 GHz35but with higher damping. We should\nhowever stress that, in the presence of strong anisotropy\n\felds, higher frequencies can be reached. Example of that\ncan be found for ferromagnetic Fe/Pt with \u0019280 GHz\n(Ha= 10T)36, and for Heusler-like ferrimagnet (Mn 3Ge\nand Mn 3Ga) with\u0019500 GHz (Ha= 20T)37,38. Never-\ntheless, the examples cited above show a considerably\nhigher intrinsic damping compared to MRG. In addi-\ntion, it was recently shown that MRG exhibits unusu-\nally strong intrinsic spin-orbit torque20. 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Yildirim,\net al. , Applied Physics Letters 109, 032403 (2016)."    },    {        "title": "1111.3430v1.Frustration_Induced_Ferrimagnetism_in_Heisenberg_Spin_Chains.pdf",        "content": "arXiv:1111.3430v1  [cond-mat.str-el]  15 Nov 2011Typeset with jpsj3.cls <ver.1.1> Full Paper\nFrustration-Induced Ferrimagnetism in Heisenberg Spin Ch ains\nTokuro Shimokawa∗and Hiroki Nakano†\nGraduate School of Material Science, University of Hyogo, K amigori, Hyogo 678-1297, Japan\n(Received October 15, 2019)\nWe study ground-stateproperties of the Heisenberg frustrate d spin chain with interactions\nup to fourth nearest neighbors by the exact-diagonalization meth od and the density matrix\nrenormalizationgroupmethod. We find that ferrimagnetismis realize dnot onlyin the caseof\nS=1/2 but also S=1 despite that there is only a single spin site in each unit cell determine d\nfrom the shape of the Hamiltonian. Our numerical results suggest t hat a “multi-sublattice\nstructure” is not required for the occurrence offerrimagnetism in quantum spin systems with\nisotropic interactions.\nKEYWORDS: quantum spin chain, frustration, ferrimagnetis m, DMRG, exact diagonalization\nFerrimagnetism is a fundamental phenomenon in the field of ma gnetism. One of the most\ntypical examples of ferrimagnetism is the ( S,s) = (1,1/2) mixed spin chain with a nearest-\nneighbor antiferromagnetic (AF) interaction.1)In this system, the so-called Lieb-Mattis-type\nferrimagnetism2,3)is realized in the ground state because two different spins are arranged\nalternately in a line owing to the AF interaction. This syste m includes two spins in a unit\ncell of the system. In other known ferrimagnetic cases of qua ntum spin systems except the\nS= 1/2 Heisenberg frustrated spin chain studied in ref. 4, the sit uation that the system has\nmorespinsthanonein each unit cell hasbeenthesame. Until o urrecent study4)demonstrated\nthe occurrence of ferrimagnetism in the ground state of the S= 1/2 Heisenberg frustrated\nspin chain despite the fact that a unit cell of the chain inclu des only a single spin, namely,\nit has no sublattice structure, it had been unclear whether t he “multi-sublattice structure”\nis required for the occurrence of the ferrimagnetism in a qua ntum spin system composed of\nisotropic interactions. The Hamiltonian examined in ref. 4 is given by\nH=J/summationdisplay\ni[Si·Si+1+1\n2Si·Si+2]\n−J′/summationdisplay\ni[Si·Si+3+1\n2(Si·Si+2+ASi·Si+4)], (1)\nwhere the real constant Ais fixed to be unity. Here, Siis theS= 1/2 spin operator at the\n∗E-mail address: rk09s002@stkt.u-hyogo.ac.jp\n†E-mail address: hnakano@sci.u-hyogo.ac.jp\n1/6J. Phys. Soc. Jpn. Full Paper\nsitei. The numerical study of this system clarified the existence o f the ferrimagnetic ground\nstate when the controllable parameter J′/Jis changed. In addition, research confirmed that\nthere are two types of ferrimagnetic phases: the phase of the Lieb-Mattis (LM) type and the\nphase of the non-Lieb-Mattis (NLM) type, which has been foun d in several frustrated spin\nsystems.5–8)\nThe purpose of this study is to confirm that the above example i s not a special or rare\ncase by investigating other models. In this study, we discus s the ground state of Hamiltonian\n(1) not only in the case of S= 1/2, but also in the case of Sibeing an S= 1 spin operator.\nMoreover, we focus on the case of A= 0.4, which is different from A= 1. Note that energies\nare measured in units of J; we setJ= 1 hereafter.\nWe employ two reliable numerical methods, i.e., the density matrix renormalization group\n(DMRG) method9,10)and the exact-diagonalization (ED) method. Both methods ca n give\nprecise physical quantities for finite-size clusters. The D MRG method is very powerful for\na one-dimensional system under the open-boundary conditio n. On the other hand, the ED\nmethod does not suffer from the limitation posed by the shape of the clusters; there is no\nlimitation of boundary conditions, although the ED method c an treat only systems smaller\nthan those that the DMRG method can treat. Note that, in the pr esent research, we use the\n“finite-system” DMRG method.\nIn the present study, two quantities are calculated. One is t he lowest energy in each\nsubspace divided by Sz\ntotto determine the spontaneous magnetization M, whereSz\ntotis the\nzcomponent of the total spin. We obtain the lowest energy E(N,Sz\ntot,J′) for a system size\nNand a given J′. For example, the Sz\ntotdependence of E(N,Sz\ntot,J′) in a specific case of J′\nis presented in the inset of Fig. 1(a). This inset shows the re sults obtained by our DMRG\ncalculations of the system of N= 72 with the maximum number of retained states ( MS) of\n600, and a number of sweeps ( SW) of 10. One can find the spontaneous magnetization Mfor\nagivenJ′as the highest Sz\ntotamong those at the lowest common energy. (See thearrowhead i n\nthe inset.) The other quantity is the local magnetization in the ground state for investigating\nthe spin structure of the highest- Sz\ntotstate. The local magnetization is obtained by calculating\n/angbracketleftSz\ni/angbracketright, whereSz\niis thez-component of the spin at the site iand/angbracketleftO/angbracketrightdenotes the expectation\nvalue of the physical quantity Owith respect to the state of interest.\nFirst, let us show the results of the J′dependence of M/Msin Fig. 1, where Msis the\nsaturated magnetization. Irrespective of S= 1/2 orS= 1, we find the nonmagnetic phase\n(M/Ms= 0) and ferromagnetic phase( M/Ms= 1). Between thetwo phases, wealso findthree\nregions: the regions of 0 < M/M s<1/3,M/Ms= 1/3, and 1/3< M/M s<1. ForS= 1/2,\n2/6J. Phys. Soc. Jpn. Full Paper\n\tB\n\tC\n 0 5 10 15 Stot z–43–42.5–42E nergy S=1/2 N=72 \nDMR G \nMS=600, SW=10 \nJ’ =2.2 \n0 1 2 \nJ’ 00.51M/M sN=24 DMR G \nN=48 DMR G \nN=72 DMR G S=1 A=0.40 1 2 \nJ’ 00.51M/M s\nN=24 E D periodic\nN=24 E D open\nN=72 DMR G S=1/2 A=0.4\nFig. 1. (Color) (a) J′dependence of the normalized magnetization M/Msin the ground state in the\ncase ofS= 1/2 withA= 0.4. In the inset of (a), the lowest energy in each subspace divided by\nSz\ntotis shown. Results of the DMRG calculations are presented when the s ystem size is N= 72\nforJ′= 2.2. The arrowhead indicates the spontaneous magnetization Mfor a given J′;Mis\ndetermined to be the highest Sz\ntotamong the values with the lowest common energy. (b) J′\ndependence of M/Msin the ground state in the case of S= 1 with A= 0.4.\none can see that the region of 0 < M/M s<1/3 is much narrower than the distinctly existing\nregion of NLM ferrimagnetism4)in the case of S= 1/2 withA= 1. The width of the present\nregion for A= 0.4 seems to vanish in the limit of N→ ∞. One finds that the occurrence\nof the NLM ferrimagnetism in Hamiltonian (1) requires a four th-neighbor interaction with A\nthat is larger than the specific value between A= 0.4 andA= 1. The width of the region of\nM/Ms= 1/3 in both cases of S= 1/2 withA= 0.4 andS= 1 with A= 0.4 seems to survive\nin the limit of N→ ∞. The region of 1 /3< M/M s<1 is presumably considered to merge\nwith the ferromagnetic (FM) phase in the thermodynamic limi t. The reason for this is that\nthis region appears only near M/Ms= 1 and that M/Msin this region becomes progressively\nlarger with increasing N. In addition, we cannot confirm this region in the calculatio ns within\nN≤30 of the S= 1/2 system under the periodic-boundary condition irrespecti ve of the\nvalues of A. The issue of whether or not the region of 1 /3< M/M s<1 survives should be\nclarified in future studies; hereafter, we do not pay further attention to this issue.\n3/6J. Phys. Soc. Jpn. Full Paper\n\tB\n \tC\n0 0.02 0.04\n1/N12J’ J’ 4\nJ’ 3\nJ’ 2\nJ’ 1\n0 0.02 0.04\n1/N00.511.5Width of phase 0<M/M s<1/3 \nM/M s=1/3 \nFig. 2. (a) Size dependences of the boundaries of the regions in the case ofS= 1 with A= 0.4. The\nresults presented are those of N= 24,36,48,60, and 72 from the DMRG calculations. (b) Size\ndependence of the width of each region in the case of S= 1 with A= 0.4. The width of the region\nof 0< M/M s<1/3 and that of M/Ms= 1/3 are defined as |J′\n2−J′\n1|and|J′\n3−J′\n2|, respectively.\n0 20 40 60\ni–1 –0.5 00.51<S iz>J’ =2.1, M=24 \nFig. 3. Local magnetization /angbracketleftSz\ni/angbracketrightunder the open-boundary condition: for J′= 2.1 in the case of\nS= 1 with A= 0.4 from the DMRG calculation for N= 72. The site number is denoted by i,\nwhich is classified into i= 3n−2, 3n−1, and 3n, wherenis an integer. Squares, circles, and\ntriangles mean i= 3n−2, 3n−1, and 3n, respectively.\nNext, we study the size dependences of the phase boundaries i n the case of S= 1 with\nA= 0.4 depicted in Fig. 2(a). We present results of four boundarie s:J′=J′\n1between the\nnonmagnetic phase and the region of 0 < M/M s<1/3,J′=J′\n2between the regions of\n0< M/M s<1/3 andM/Ms= 1/3,J′=J′\n3between the regions of M/Ms= 1/3 and\n1/3< M/M s<1, andJ′=J′\n4between the region of 1 /3< M/M s<1 and the FM phase. To\nconfirm the behavior up to the thermodynamic limit, we also ex amine the N−1dependences\nof the two widths of the regions of M/Ms= 1/3 and 0< M/M s<1/3 in Fig. 2(b). Although\nthe width of the region of M/Ms= 1/3 decreases with increasing N, this dependence shows\na behavior that is convex-downwards for large sizes; the wid th seems to converge to 0.3.\nTherefore, the phase of M/Ms= 1/3 definitely survives in the limit of N→ ∞. On the\nother hand, the width of 0 < M/M s<1/3 obviously disappears in the limit of N→ ∞. An\nappropriate tuning of the parameters in Hamiltonian (1) of t heS= 1 system might cause the\n4/6J. Phys. Soc. Jpn. Full Paper\nNLM ferrimagnetism; such parameter sets should be searched for in future studies.\nFinally, we examine the local magnetization /angbracketleftSz\ni/angbracketrightin the phase of M/Ms= 1/3 in the\ncase ofS= 1 with A= 0.4. In Fig. 3, we present our DMRG result of /angbracketleftSz\ni/angbracketrightof the system\nofN= 72. We confirm the up-down-up spin behavior, and this spin st ructure is consistent\nwithM/Ms=1/3 in the parameter region near approximately J′= 2.1 in Fig. 1(b). Thus, this\nphase is considered to be the LM-type ferrimagnetic phase.\nIn summary, we study the ground-state properties of a frustr ated Heisenberg spin chain by\nthe ED and DMRG methods. Despite the fact that this system con sists of only a single spin\nsiteineach unitcell determinedfromtheshapeof theHamilt onian, theLM-typeferrimagnetic\nground state is realized in a finite region not only in the case ofS= 1/2 but also of S= 1.\nThe present models showing ferrimagnetism indicate that a “ multi-sublattice structure” is\nnot required for the occurrence of ferrimagnetism in quantu m spin systems with isotropic\ninteractions as a general circumstance.\nWe are grateful to Professor Y. Hasegawa for his critical rea ding of the manuscript. This\nwork was partly supported by Grants-in-Aid (Nos. 20340096, 23340109, and 23540388) from\nthe Ministry of Education, Culture, Sports, Science and Tec hnology (MEXT) of Japan. This\nwork was partly supported by a Grant-in-Aid (No. 22014012) f or Scientific Research and\nPriority Areas “Novel States of Matter Induced by Frustrati on” from the MEXT of Japan.\nSome of the calculations were carried out at the Supercomput er Center, Institute for Solid\nState Physics, University of Tokyo. Exact-diagonalizatio n calculations in the present work\nwere carried out based on TITPACK Version 2 coded by H. Nishim ori. DMRG calculations\nwere carried out using the ALPS DMRG application.11)\n5/6J. Phys. Soc. Jpn. Full Paper\nReferences\n1) T. Sakai and K. Okamoto: Phys. Rev. B. 65(2002) 214403.\n2) E. Lieb and D. Mattis: J. Math. Phys. 3(1962) 749.\n3) W. Marshall: Proc. Roy. Soc. A 232(1955) 48.\n4) T. Shimokawa and H. Nakano: J. Phys. Soc. Jpn. 80(2011) 043703.\n5) S. Yoshikawa and S. Miyashita: J. Phys. Soc. Jpn. 74(2005) Suppl. 71.\n6) K. Hida: J. Phys.: Condens. Matter 19(2007) 145225.\n7) H. Nakano, T. Shimokawa, and T. Sakai: J. Phys. Soc. Jpn. 80(2011) 033709.\n8) T. Shimokawa and H. Nakano: J. Phys.: Conf. Ser. 320(2011) 012007.\n9) S. R. White: Phys. Rev. Lett. 69(1992) 2863.\n10) S. R. White: Phys. Rev. B. 48(1993) 10345.\n11) A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A. Feiguin, L. G amper, E. Gull, S. Gurtler, A.\nHonecker, R. Igarashi, M. Korner, A. Kozhevnikov, A. Lauchli, S. R. Manmana, M. Matsumoto,\nI. P. McCulloch, F. Michel, R. M. Noack, G. Pawlowski, L. Pollet, T. Pru schke, U. Schollwock, S.\nTodo, S. Trebst, M. Troyer, P. Werner, and S. Wessel: J. Magn. M agn. Mater. 310(2007) 1187\n(see also http://alps.comp-phys.org).\n6/6"    },    {        "title": "2204.14010v4.Entangling_mechanical_vibrations_of_two_massive_ferrimagnets_by_fully_exploiting_the_nonlinearity_of_magnetostriction.pdf",        "content": "Entangling mechanical vibrations of two massive ferrimagnets by fully exploiting\nthe nonlinearity of magnetostriction\nHang Qian,1Zhi-Yuan Fan,1and Jie Li1,\u0003\n1Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation,\nand Zhejiang Province Key Laboratory of Quantum Technology and Device,\nDepartment of Physics, Zhejiang University, Hangzhou 310027, China\n(Dated: December 14, 2022)\nQuantum entanglement in the motion of macroscopic objects is of significance to both fundamental studies\nand quantum technologies. Here we show how to entangle the mechanical vibration modes of two massive\nferrimagnets that are placed in the same microwave cavity. Each ferrimagnet supports a magnon mode and a\nlow-frequency vibration mode coupled by the magnetostrictive force. The two magnon modes are, respectively,\ncoupled to the microwave cavity by the magnetic dipole interaction. We first generate a stationary nonlocal\nentangled state between the vibration mode of the ferrimagnet-1 and the magnon mode of the ferrimagnet-2.\nThis is realized by continuously driving the ferrimagnet-1 with a strong red-detuned microwave field and the en-\ntanglement is achieved by exploiting the magnomechanical parametric down-conversion and the cavity-magnon\nstate-swap interaction. We then switch o \u000bthe pump on the ferrimagnet-1 and, simultaneously, turn on a red-\ndetuned pulsed drive on the ferrimagnet-2. The latter drive is used to activate the magnomechanical beamsplitter\ninteraction, which swaps the magnonic and mechanical states of the ferrimagnet-2. Consequently, the previously\ngenerated phonon-magnon entanglement is transferred to the mechanical modes of two ferrimagnets. The work\nprovides a scheme to prepare entangled states of mechanical motion of two massive objects, which may find\napplications in various studies exploiting macroscopic entangled states.\nI. INTRODUCTION\nEntanglement of mechanical motion has been first demon-\nstrated in two microscopic trapped atomic ions [1], then in two\nsingle-phonon excitations in nanodiamonds [2], and recently\nin two macroscopic optomechanical resonators [3–5]. In the\npast two decades, a number of theoretical proposals have been\ngiven in optomechanics [6–35] for preparing the entanglement\nbetween massive mechanical resonators, utilizing the cou-\npling between optical and mechanical degrees of freedom by\nradiation pressure. In particular, the application of reservoir\nengineering ideas [36–40] to optomechanics [15, 16, 20, 22]\nhas led to a significant and robust mechanical entanglement\nand the entanglement has been successfully demonstrated in\nthe experiment [4].\nExploring novel physical platforms that could prepare\nquantum states at a more massive scale is of great signifi-\ncance for the study of macroscopic quantum phenomena [41],\nthe boundary between the quantum and classical worlds [42],\nand gravitational quantum physics [43, 44], etc. Recently, the\nmagnomechanical system of a large-size ferrimagnet, e.g., yt-\ntrium iron garnet (YIG), that has a dispersive magnon-phonon\ncoupling has shown such a potential [45–58]. The magnon\nand mechanical vibration modes are coupled by the nonlin-\near magnetostrictive interaction, which couples ferrimagnetic\nmagnons to the deformation displacement of the ferrimag-\nnet [59–62]. The magnomechanical Hamiltonian takes the\nsame form as the optomechanical one [63] (by exchanging\nthe roles of magnons and photons), which allows us to pre-\ndict many optomechanical analogues in magnomechanics. By\ncoupling the magnomechanical system to a microwave cav-\n\u0003jieli007@zju.edu.cnity, they form the tripartite cavity magnomechanical system.\nThe magnetostriction has been exploited in cavity magnome-\nchanics to generate macroscopic entangled states of magnons\nand vibration phonons of massive ferrimagnets [45, 47, 48,\n51, 52, 56, 58], as well as nonclassical states of microwave\nfields [64, 65]. Therefore, the magnetostrictive nonlinearity\nbecomes a valuable resource for producing various quantum\nstates of microwave photons, magnons, and phonons. These\nnonclassical states may find potential applications in quan-\ntum information processing [66, 67], quantum metrology [65]\nand quantum networks [68]. Despite of the aforementioned\nmany proposals and very limited experiments [60–62] in cav-\nity magnomechanics, there is so far only one protocol [51] for\nentangling two mechanical vibration modes of macroscopic\nferrimagnets. The protocol [51], however, relies on an exter-\nnal entangled resource and transfers the quantum correlation\nfrom microwave drive fields to two vibration modes. There-\nfore, designing more energy-saving protocols without using\nany external quantum resource is highly needed.\nAlong this line, we present here a scheme for generating\na nonlocal entangled state between the mechanical vibrations\nof two ferrimagnets, without the need of any quantum driving\nfield. Two ferrimagnets are placed in a microwave cavity and\neach ferrimagnet supports a magnon mode and a mechanical\nmode. The cavity mode couples to two magnon modes by\nthe magnetic dipole interaction, and the magnon modes cou-\nple to their local vibration modes by magnetostriction, respec-\ntively. We show that the entanglement between the vibration\nmodes of two ferrimagnets can be achieved by fully exploit-\ning the nonlinear magnetostriction interaction (i.e., exploit-\ning both the magnomechanical parametric down-conversion\n(PDC) and state-swap interactions) and by using the common\ncavity field being an intermediary to distribute quantum cor-\nrelations. The mechanical entanglement is established by two\nsteps. We first generate steady-state entanglement between thearXiv:2204.14010v4  [quant-ph]  13 Dec 20222\nmechanical mode of the ferrimagnet-1 and the magnon mode\nof the ferrimagnet-2. We then activate the magnomechani-\ncal state-swap interaction in the ferrimagnet-2, which trans-\nfers the magnonic state to its locally interacting mechanical\nmode. Consequently, the two mechanical modes get nonlo-\ncally entangled.\nThe remainder of the paper is organized as follows. In\nSec. II, we introduce the general model of the protocol that is\nused in the two steps. We then show how to prepare a station-\nary nonlocal magnon-phonon entanglement with a continuous\nmicrowave pump in Sec. III, and how to transfer this entan-\nglement to two mechanical modes with a pulsed microwave\ndrive in Sec. IV. Finally, we discuss and conclude in Sec. V.\nII. THE MODEL\nThe protocol is based on a hybrid five-mode cavity mag-\nnomechanical system, including a microwave cavity mode,\ntwo magnon modes, and two mechanical vibration modes,\nas depicted in Fig. 1. The magnon modes are embodied by\nthe collective motion of a large number of spins (i.e., spin\nwave) in two ferrimagnets, e.g., two YIG spheres [60–62] or\nmicro bridges [69, 70]. They simultaneously couple to the mi-\ncrowave cavity via the magnetic dipole interaction. This cou-\npling can be strong thanks to the high spin density in YIG [71–\n73]. The mechanical modes refer to the deformation vibration\nmodes of two YIG crystals caused by the magnetostrictive\nforce. Due to the much lower mechanical frequency (rang-\ning from 100to 102MHz) than the magnon frequency (GHz)\nin the typical magnomechanical systems [60–62, 69], the vi-\nbration phonons and the magnons are coupled in a dispersive\nmanner [70, 74, 75]. The Hamiltonian of the system reads\nH=~=!aaya+X\nj=1;2\u0012\n!mjmy\njmj+!bj\n2\u0010\np2\nj+q2\nj\u0011\u0013\n+X\nj=1;2\u0012\ngj\u0010\namy\nj+aymj\u0011\n+G0jmy\njmjqj\u0013\n+i\nk\u0010\nmy\nke\u0000i!0kt\u0000mkei!0kt\u0011\n:(1)\nThe first (second) term describes the energy of the cavity\nmode (magnon modes), of which the frequency is !a(!mj)\nand the annihilation operator is a(mj) with the commutation\nrelation [ a;ay]=1\u0010\u0002mj;my\nj\u0003=1\u0011\n. The magnon frequency\n!mjis determined by the external bias magnetic field Hjvia\n!mj=\rHj, where the gyromagnetic ratio \r=2\u0019=28 GHz=T.\nThe third term denotes the energy of two mechanical vibra-\ntion modes with frequencies !bj, and qjandpj([qj;pj]=i)\nare the dimensionless position and momentum of the vibra-\ntion mode j, modeled as a mechanical oscillator. The cou-\npling gjis the linear cavity-magnon coupling rate, and G0j\nis the bare magnomechanical coupling rate. For large-size\nYIG spheres with the diameter in the 100 \u0016m range [60–\n62], G0jis typically in the 10 mHz range [75]. It, however,\ncan be much stronger for micron-sized YIG bridges [69, 70].\nNevertheless, the e \u000bective magnomechanical coupling can be\nsignificantly enhanced by driving the magnon mode with a\nFIG. 1: (a) Sketch of the system. Two YIG crystals are placed near\nthe maximum magnetic fields of a microwave cavity. Each YIG crys-\ntal is in a uniform bias magnetic field, and supports a magnon mode\nand a mechanical vibration mode. The two magnon modes couple to\nthe same cavity field. Two drive fields are applied successively to the\ntwo magnon modes in the two steps of the protocol. Note that though\nYIG spheres are adopted in the sketch, the YIG crystals can be a\nnonspherical structure, e.g., micron-sized YIG bridges [69, 70]. (b)\nInteractions among the subsystems. The magnon mode mj(j=1;2)\ncouples linearly to the cavity mode awith the coupling strength gj,\nand couples dispersively to the mechanical mode bjwith the e \u000bective\nmagnomechanical coupling rate Gj. A nonlocal phonon-magnon ( b1-\nm2) entanglement is created in the first step with a continuous drive\non the magnon mode m1, and the entanglement is transferred to the\ntwo mechanical modes in the second step by using a pulsed drive on\nthe magnon mode m2.\nstrong microwave field [45]. The driving Hamiltonian is de-\nscribed by the last term, and the corresponding Rabi frequency\n\nk=p\n5\n4\rpNkBk(k=1 or 2) [45], with Bk(!0k) being\nthe amplitude (frequency) of the drive magnetic field, and Nk\nbeing the total number of spins in the kth crystal. We re-\nmark that the model di \u000bers from the one used in Ref. [47]\nby including a second mechanical mode, which brings in a\nsignificant amount of additional thermal noise to the system.\nMore importantly, the present work aims to entangle two low-\nfrequency (in MHz) mechanical modes. This is much more\ndi\u000ecult to prepare than the entanglement of two GHz magnon\nmodes studied in Ref. [47].\nIn what follows, we adopt a two-step procedure to prepare\nthe two mechanical modes in an entangled state, and in each\nstep, we apply a single drive field on either magnon mode\nm1orm2. This avoids the complex Floquet dynamics in our\nhighly hybrid system caused by simultaneously applying mul-\ntiple pump tones [15, 16, 20, 22]. We first generate a nonlo-\ncal entangled state between the mechanical mode b1and the3\nFIG. 2: Mode and drive frequencies used in the first step. When\nthe magnon mode m1is (the cavity and magnon mode m2are) reso-\nnant with the blue (red) mechanical sideband of the drive field with\nfrequency!01, the nonlocal phonon-magnon entanglement Eb1m2is\nestablished.\nmagnon mode m2by continuously driving the magnon mode\nm1. After the system enters a stationary state, we then turn\no\u000bthe drive on m1and, simultaneously, turn on a red-detuned\ndrive on the magnon mode m2to activate the magnomechan-\nical state-swap interaction m2$b2. This operation transfers\nthe quantum correlation shared between b1andm2to two me-\nchanical modes, thus establishing a quantum correlation (i.e.,\nentanglement) between the two mechanical modes.\nIII. STATIONARY NONLOCAL MAGNO-MECHANICAL\nENTANGLEMENT\nIn the first step, we aim to entangle the mechanical mode\nb1and the magnon mode m2. This can be realized by driving\nthe magnon mode m1with a strong red-detuned microwave\nfield [45], see Fig. 2. It was shown that a genuine tripar-\ntite magnon-photon-phonon entangled state can be produced\nwithout involving the second YIG crystal [45]. By using the\npartial result that the mechanical mode b1and the cavity a\nare entangled, and by coupling the cavity to the second nearly\nresonant magnon mode m2(which have a beamsplitter inter-\naction realizing the state-swap operation a$m2), the two\nmodes b1andm2thus get entangled. This is confirmed by the\nnumerical results presented in this section.\nIt should be noted that since the strong drive is applied on\nthefirst YIG crystal, the e \u000bective (magnomechanical) cou-\npling to the mechanical mode b2in the second YIG crystal is\nmuch smaller than that in the first YIG crystal, G2\u001cG1, so\nthe presence of the second mechanical mode, or not, will not\nappreciably a \u000bect the entanglement dynamics analysed above.\nBecause in the next step, the coupling to the second mechani-\ncal mode must be turned on ( G02>0), including this coupling\nin the model also in the first step means no additional opera-\ntion (e.g., adjusting the direction of the bias magnetic field\nto activate or inactivate the coupling G02[47, 60]) has to be\nimplemented between the two steps. Another reason is that,\nas will be shown in Sec. IV, the entanglement between b1-m2\nshould be transferred to the two mechanical modes as soon as\npossible because it rapidly decays when the drive in the first\nstep is switched o \u000b.\nIn the frame rotating at the drive frequency !01, the quan-\ntum Langevin equations (QLEs) describing the system dy-namics are given by\n˙a=\u0000(i\u0001a+\u0014a)a\u0000ig1m1\u0000ig2m2+p\n2\u0014aain;\n˙mj=\u0000(i\u0001mj+\u0014mj)mj\u0000iG0jmjqj\u0000igja+ \n j+q\n2\u0014mjmin\nj;\n˙qj=!bjpj;\n˙pj=\u0000!bjqj\u0000\rbjpj\u0000G0jmy\njmj+\u0018j;\n(2)\nwhere \u0001a=!a\u0000!01,\u0001mj=!mj\u0000!01, and\u0014a,\u0014mjand\rbj\nare the dissipation rates of the cavity, magnon and mechanical\nmodes, respectively. The Rabi frequency \nj= \n 1\u000ej1(j=\n1;2) implies only one drive field applied on the magnon mode\nm1.ainandmin\njare the input noise operators a \u000becting the cav-\nity and magnon modes, whose non-zero correlation functions\narehain(t)ainy(t0)i=[Na(!a)+1]\u000e(t\u0000t0),hainy(t)ain(t0)i=\nNa(!a)\u000e(t\u0000t0),hmin\nj(t)miny\nj(t0)i=[Nmj(!mj)+1]\u000e(t\u0000t0)\nandhminy\nj(t)min\nj(t0)i=Nmj(!mj)\u000e(t\u0000t0). The Langevin force\noperator\u0018jis accounting for the mechanical Brownian mo-\ntion, which is autocorrelated as h\u0018j(t)\u0018j(t0)+\u0018j(t0)\u0018j(t)i \u0019\n\rbj[2Nbj(!bj)+1]\u000e(t\u0000t0), where we consider a high qual-\nity factor Qb=!b=\rb\u001d1 for the mechanical oscillators\nto validate the Markov approximation [76]. Here, Nk(!k)=h\nexp(~!k\nkBT)\u00001i\u00001(k=a;mj;bj) are the equilibrium mean ther-\nmal photon, magnon, and phonon numbers, respectively, at\nthe environmental temperature T, with kBbeing the Boltz-\nmann constant.\nThe strong drive field leads to large amplitudes of the\nmagnon modes and cavity mode due to the magnon-cavity\ncoupling,jhmjij;jhaij \u001d 1. This allows us to linearize the\nnonlinear QLEs (2) by writing each mode operator as a classi-\ncal average plus a fluctuation operator with zero mean value,\ni.e., O=hOi+\u000eO(O=a;mj;qj;pj), and by neglect-\ning small second-order fluctuation terms. Substituting the\nabove mode operators into Eq. (2) yields two sets of lin-\nearized Langevin equations, respectively, for classical aver-\nages and fluctuation operators. By solving the former set of\nequations in the time scale where the system evolves into a\nstationary state, we obtain the solution of the steady-state av-\nerages, which are hpji=0,hqji=\u0000G0jjhmjij2=!bj,hai=\n\u0000i\u0000g1hm1i+g2hm2i\u0001=(i\u0001a+\u0014a), and\nhm1i=\n1(i\u0001a+\u0014a)\ng2\n1+(i˜\u0001m1+\u0014m1)(i\u0001a+\u0014a)\u0000g2\n1g2\n2\ng2\n2+(i˜\u0001m2+\u0014m2)(i\u0001a+\u0014a);\nhm2i=\u0000\n1(i\u0001a+\u0014a)g1g2\ng2\n1+(i˜\u0001m1+\u0014m1)(i\u0001a+\u0014a)\ng2\n2+(i˜\u0001m2+\u0014m2)(i\u0001a+\u0014a)\u0000g2\n1g2\n2\ng2\n1+(i˜\u0001m1+\u0014m1)(i\u0001a+\u0014a);\n(3)\nwhere ˜\u0001mj= \u0001 mj+G0jhqjiis the e \u000bective magnon-drive\ndetuning, which includes the magnetostriction induced fre-\nquency shift. Typically, this frequency shift is negligible (be-\ncause of a small G0[60–62]) with respect to the optimal de-\ntuning used in this work, i.e., j˜\u0001mj\u0000\u0001mjj \u001c j \u0001mjj \u0019!bj.\nTherefore, in what follows we can safely assume ˜\u0001mj\u0019\u0001mj.4\nThe set of the linearized QLEs for the system fluctuations\ncan be written in the matrix form\n˙u(t)=Au(t)+n(t); (4)\nwhere u(t)=\u0002\u000eX(t);\u000eY(t);\u000ex1(t);\u000ey1(t);\u000ex2(t);\u000ey2(t);\u000eq1(t);\n\u000ep1(t);\u000eq2(t);\u000ep2(t)\u0003Tis the vector of the quadrature fluc-\ntuations, and n(t)=hp2\u0014aXin(t);p2\u0014aYin(t);p\n2\u0014m1xin\n1(t);p\n2\u0014m1yin\n1(t);p\n2\u0014m2xin\n2(t);p\n2\u0014m2yin\n2(t);0; \u0018 1(t);0; \u0018 2(t)iT\nis the vector of the input noises entering the system,\nwith\u000eX=\u0010\n\u000ea+\u000eay\u0011\n=p\n2,\u000eY=i\u0010\n\u000eay\u0000\u000ea\u0011\n=p\n2,\n\u000exj=\u0010\n\u000emj+\u000emy\nj\u0011\n=p\n2, and\u000eyj=i\u0010\n\u000emy\nj\u0000\u000emj\u0011\n=p\n2,\nand the drift matrix Ais given by\nA=0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@\u0000\u0014a\u0001a 0 g1 0 g2 0 0 0 0\n\u0000\u0001a\u0000\u0014a\u0000g1 0\u0000g2 0 0 0 0 0\n0 g1\u0000km1 \u0001m1 0 0\u0000Re[G1]0 0 0\n\u0000g10\u0000\u0001m1\u0000\u0014m1 0 0\u0000Im[G1]0 0 0\n0 g2 0 0 \u0000\u0014m2 \u0001m2 0 0\u0000Re[G2]0\n\u0000g20 0 0 \u0000\u0001m2\u0000\u0014m2 0 0\u0000Im[G2]0\n0 0 0 0 0 0 0 !b1 0 0\n0 0\u0000Im[G1]Re[G1] 0 0\u0000!b1\u0000\rb1 0 0\n0 0 0 0 0 0 0 0 0 !b2\n0 0 0 0 \u0000Im[G2]Re[G2] 0 0\u0000!b2\u0000\rb21CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (5)\nwith Gj=ip\n2G0jhmjibeing the e \u000bective magnomechanical\ncoupling rate, which is generally complex.\nBecause of the linearized dynamics of the system and the\nGaussian nature of the input noises, the steady state of the\nsystem’s quantum fluctuations is a five-mode Gaussian state.\nThe state can be completely described by a 10 \u000210 covariance\nmatrix (CM)Vwith entries defined as Vi j=hui(t)uj(t0)+\nuj(t0)ui(t)i=2 (i;j=1;2;:::; 10). The CMVcan be obtained\nby directly solving the Lyapunov equation\nAV+VAT=\u0000D; (6)\nwhere the di \u000busion matrix Dis defined as Di j=\nhni(t)nj(t0)+nj(t0)ni(t)i=\u00022\u000e(t\u0000t0)\u0003and given by D =\ndiag[\u0014a(2Na+1);\u0014a(2Na+1);\u0014m1(2Nm1+1);\u0014m1(2Nm1+1);\n\u0014m2(2Nm2+1);\u0014m2(2Nm2+1);0;\rb1(2Nb1+1);0;\rb2(2Nb2+1)].\nWhen the CM of the system fluctuations is obtained, one\ncan then extract the state of the interesting modes by remov-\ning inVthe rows and columns related to the uninteresting\nmodes and study their entanglement properties. We use the\nlogarithmic negativity [77] to quantify the Gaussian bipartite\nentanglement, of which the definition is provided in the Ap-\npendix.\nIn Fig. 3, we show relevant steady-state bipartite entan-\nglements generated in the first step. The blank areas are\nthe parameter regimes in which the system is unstable, re-\nflected by the fact that the drift matrix Ahas at least one pos-\nitive eigenvalue. We have employed the following parame-\nters [60–62, 69, 70]: !a=2\u0019=10 GHz,!b1=2\u0019=17 MHz,\n!b2=2\u0019=12 MHz,\rb=2\u0019=100 Hz,\u0014a=2\u0019=1 MHz,\u0014m=\u0014a,\ng1=2\u0019=5 MHz, g2=2\u0019=1 MHz, G01=2\u0019'G02=2\u0019=10\nHz, and T=10 mK. The amplitude of the drive magnetic\nfield B1=4:8\u000210\u00004T, corresponding to the drive power\nP1'1:1 mW for two YIG micro bridges (approximated as\nFIG. 3: Nonlocal phonon-magnon entanglement Eb1m2versus (a) \u0001a\nand\u0001m1, and (b) \u0001aand\u0001am2\u0011!a\u0000!m2. (c) The phonon-cavity\nentanglement Eb1aand (d) the mechanical entanglement Eb1b2versus\n\u0001aand\u0001m1. In (d) the whole purple region denotes Eb1b2=0. We\ntake\u0001am2=0:9\u0014ain (a), (c) and (d), and \u0001m1=0:95!b1in (b). The\nblank areas denote the unstable region where the stability condition\nis not fulfilled. See the text for other parameters.\ntwo cuboids) with dimensions of 13.7 \u00023\u00021\u0016m3and 16.4\n\u00023\u00021\u0016m3. This milliwatt power might lead to heating in-\nduced temperature rise. To avoid the possible heating e \u000bect,\none should improve the bare coupling G0jto reduce the power.\nSince the mechanical resonance frequency mainly depends on\nthe length and thickness of the beam [69], the sizes of the\nbeams are chosen corresponding to the mechanical frequen-5\ncies we adopted. To determine the power, we have used the\nrelation between the drive magnetic field B1and the power P1,\ni.e.,B1=p\n2\u00160P1=(lwc) [70], with \u00160being the vacuum mag-\nnetic permeability, cthe speed of the electromagnetic wave\npropagating in vacuum, and land wthe length and width\nof the YIG cuboid. The nonlocal phonon-magnon entangle-\nment Eb1m2(Fig. 3(a)-(b)) is the result of the phonon-cavity\nentanglement Eb1a(Fig. 3(c)) and the cavity-magnon ( a-m2)\nbeamsplitter (state-swap) interaction. The phonon-cavity en-\ntanglement Eb1ais obtained by using the results of Ref. [45],\nwhich studies a tripartite magnon-photon-phonon system (i.e.,\nthem1-a-b1subsystem here). As shown in Ref. [45], a red-\ndetuned strong drive field on the magnon mode m1(with\n\u0001m1\u0019!b1) cools the hot mechanical mode b1(a precondi-\ntion for preparing quantum states) and, simultaneously, acti-\nvates the magnomechanical PDC interaction, which produces\nthe magnomechanical entanglement Eb1m1. The latter process\noccurs only when the pump field is su \u000eciently strong, such\nthat the weak coupling condition jG1j\u001c!b1for taking the\nrotating-wave (RW) approximation to obtain the cooling in-\nteraction/my\n1b1+m1by\n1is no longer satisfied, and the counter-\nRW terms/my\n1by\n1+m1b1(responsible for the PDC interac-\ntion) start to play the role. The magnomechanical entangle-\nment is partially transferred to the phonon-cavity subsystem\nwith Eb1a>0 (Fig. 3(c)) when \u0001a\u0019\u0000!b1[45]. The opti-\nmal detunings \u0001m1\u0019 \u0000\u0001a\u0019!b1imply that the cavity and\nthe magnon mode m1are respectively resonant with the two\nmechanical sidebands of the drive field, see Fig. 2.\nSince the cavity-magnon ( m2) state-swap operation works\noptimally when the two modes are resonant. The entangle-\nment Eb1m2is maximized for a nearly zero cavity-magnon de-\ntuning \u0001am2\u00190, as confirmed by Fig. 3(b). The detuning\n\u0001am2up to several cavity linewidth will significantly hinder\nthe transfer of the entanglement. The fact that the entangle-\nment Eb1m2is transferred from the entanglement Eb1acan also\nbe seen from the complementary distribution of the entangle-\nment in Figs. 3(a) and 3(c), indicating the entanglement flow\namong the subsystems switched on by the beamsplitter cou-\npling.\nIt is worth noting that in the first step the two mechani-\ncal modes are not entangled, as confirmed by Eb1b2=0 in\nFig. 3(d) in a wide range of parameters. This is mainly be-\ncause the magnon mode m2is driven by the nearly resonant\ncavity field, which is not the condition for cooling the me-\nchanical mode b2, or realizing the state-swap interaction be-\ntween the two modes m2andb2. Instead, a red-detuned mi-\ncrowave field should be used to drive the magnon mode m2\nto realize the state-swap operation m2$b2, such that the\nphonon-magnon entanglement Eb1m2can be transferred to the\ntwo mechanical modes, as will be discussed in the next sec-\ntion.\nIV . ENTANGLEMENT BETWEEN TWO MECHANICAL\nMODES\nIn this section, we show how to transfer the phonon-\nmagnon entanglement Eb1m2prepared in the first step to the\nFIG. 4: Mode and drive frequencies used in the second step. A red-\ndetuned pulsed drive with frequency !02is used to activate the mag-\nnomechanical state-swap operation m2$b2, which transfers the\nphonon-magnon entanglement Eb1m2generated in the first step to the\ntwo mechanical modes.\ntwo mechanical modes. To this end, we apply a red-detuned\npulsed microwave drive on the magnon mode m2(see Fig. 4)\nto activate the local magnomechanical state-swap interaction\nm2$b2. The pulsed drive should be fast as the entangle-\nment Eb1m2will quickly decay as soon as the continuous drive\nis turned o \u000bin the first step. To simplify the model, we use\na flattop microwave pulse to drive the magnon mode and then\nthe model and the treatment used in Sec. III for a continuous\ndrive are still valid for the case of a flattop pulse drive. Di \u000ber-\nently, we shall solve the dynamical solutions rather than the\nsteady-state solutions.\nThe QLEs remain the same as in Eq. (2), except that all\nthe detunings are redefined with respect to the frequency !02\nof the pulsed drive and the Rabi frequency is associated with\nthe pulsed drive, i.e., \u0001a=!a\u0000!02,\u0001mj=!mj\u0000!02, and\n\nj= \n 2\u000ej2. The dynamics of the quantum fluctuations of the\nsystem can still be described by Eq. (4) but with dynamical\nhmji(t) and thus Gj(t) in the drift matrix, due to a pulsed drive.\nThe dynamical CM V(t) can be obtained by [29]\nV(t)=M(t)V0M(t)T+Zt\n0ds M (s)DM(s)T; (7)\nwhere tis the duration of the pulsed drive, M(t)=eRt\n0A(\u001c)d\u001c,\nandV0is the CM of the initial state of the system when si-\nmultaneously turn on (o \u000b) the drive on the magnon mode m2\n(m1), which is obtained in the first step by solving the Lya-\npunov equation (6). When the dynamical CM is achieved, we\ncan then study the dynamics of the entanglement.\nFigure 5(a) shows the mechanical entanglement Eb1b2is\nmaximized at the detuning \u0001m2\u0019!b2, which is the optimal\ndetuning for realizing the magnomechanical state-swap inter-\naction, by which the previously generated phonon-magnon\nentanglement Eb1m2is transferred to the mechanical modes.\nNote that Fig. 5(a) is plotted at the optimal pulse duration tmax\nfor a given detuning, yielding a maximal Eb1b2. Figure 5(b)\nshows Eb1b2versus the pulse duration tfor the optimal detun-\ning. As is shown, there is a time window for the presence of\nthe entanglement. The mechanical entanglement emerges a\nshort while after the phonon-magnon entanglement dies out\n(c.f. the inset of Fig. 5(b)). When Eb1b2reaching its maxi-\nmum at tmax, we then turn o \u000bthe pulse drive to decouple the\nmechanics from the rest of the system to protect the entan-\nglement. The two mechanical oscillators then evolve almost\nfreely, and their entanglement is a \u000bected by their local thermal\nbaths, which lasts for a much longer time due to a small me-6\nFIG. 5: Mechanical entanglement Eb1b2versus (a) detuning \u0001m2at\nthe optimal pulse duration tmax; (b) pulse duration tat the optimal\ndetuning \u0001m2=!b2. The inset shows that the phonon-magnon en-\ntanglement Eb1m2dies out around t'0:01\u0016s. (c) Eb1b2versus time\nwhen the drive is turned o \u000battmax. Soon after tmax, the two mechani-\ncal oscillators evolve freely with the only coupling to their local ther-\nmal baths. (d) Eb1b2versus bath temperature at the optimal detuning\nand pulse duration. We take \u0001a=\u00000:95!b1,\u0001m1=0:95!b1(as\ndefined in the first step), and \u0001am2=0:9\u0014ain all plots. The other\nparameters are the same as in Fig. 3.\nchanical damping and a low bath temperature. This is clearly\nshown in Fig. 5(c) (c.f. Fig. 5(b)). The mechanical entan-\nglement is robust against bath temperature and the maximal\nentanglement survives up to T'100 mK, as illustrated in\nFig. 5(d). We have used a drive power P2=1:3 mW, giving\nthe amplitude of the drive magnetic field B2=4:8\u000210\u00004T.\nLastly, we discuss how to detect the mechanical entangle-\nment. The entanglement can be verified by measuring the CM\nof the two mechanical modes. The mechanical quadratures\ncan be measured by coupling the deformation displacement to\nan optical cavity that is driven by a weak red-detuned light to\ntransfer the mechanical state to the optical field. By homo-\ndyning two cavity output fields one can then obtain the CM of\nthe mechanical modes [5, 78].V . CONCLUSION AND DISCUSSION\nWe present a protocol to entangle two mechanical vibra-\ntion modes in a cavity magnomechanical system. The pro-\ntocol contains two steps by applying successively two drive\nfields on two magnon modes to activate di \u000berent functions of\nthe nonlinear magnetostrictive interaction, namely, the mag-\nnomechanical PDC and state-swap operations. We show that\nthe entanglement between two mechanical vibration modes of\ntwo YIG crystals can be achieved by fully exploiting the above\nmagnetostrictive functions and the cavity-magnon state-swap\ninteraction. We remark that our protocol is valid for any mag-\nnomechanical system of ferrimagnets or ferromagnets, spher-\nical [60–62] or nonspherical structures [69, 70], as long as\nthey possess the dispersive coupling between magnons and\nphonons. 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Using combined measurements of a. c. and d. c. magnetization, we estab lish that Ga \nsubstitution for Fe in YBaFe 4O7 leads to an evolution from a geometrically frustrated spin \nglass (for x = 0) to a cationic disorder induced spin glass (x = 0.7 0). We also find  an \nintermediate narrow range of doping where the samples are clearl y phase separated having \nsmall ferrimagnetic clusters embedded in a spin glass matrix. The origin of the ferrimagnet ic \nclusters  lies in the change in symmetry  of the samples  from cubic to hexagonal (and a \nconsequent lifting of the geometric al frustration) as a result of  Ga doping.  We also show the \npresence of exchange bias and domain wall pinning in these samples. The cause of both these \neffects can be traced back to the inherent ph ase separation present in the  samples.  \n \n \n \n \n \n \n \n \nKeywords : “114” oxides, magne tic frustration , phase separation .   \n \n \n \n \n* Corresponding author: Dr. Tapati Sarkar  \ne-mail: tapati.sarkar @ensicaen.fr  \nFax: +33 2 31 95 16 00   \nTel:  +33 2 31 45 26 32   2 Introduction  \n \n The recent studies of the “114” cobaltites (Ln,Ca) 1BaCo 4O7 [1 – 5] and ferrit es \n(Ln,Ca) 1BaFe 4O7 [6 – 8] have generated a lot of interest in the scientific community because \nof their complex magnetic, electronic  and thermoelectric properties  [9]. These cobaltites and \nferrites have the same basic structure, and are closely related to  spinels and barium \nhexaferrites by their close packing of “O 4” and “BaO 3” layers.  This close packing forms a 3 -\ndimensional framework [Fe 4O7]\n (or [Co 4O7]\n) consisting of corner -sharing FeO 4 (or CoO 4) \ntetrahedra , with the lanthanide elements occupying the octahedral sites of this framework.  The \ntriangular geometry of the cobalt (or iron) sublattices ( Fig. 1 ) plays a dominant role in their \nmagnetic properties. It was indeed shown that for hexagonal LnBaCo 4O7 cobaltites ( Fig. 1 \n(a)), there exists a strong com petition between the 1 D magnetic ordering along the \n  \ndirection in the “Co 5” trigonal bipyramids, and the magnetic frustration in the (001) plane \nbuilt up of “Co 3” triangles [10, 11]. In fact, the magnetic frustration can be lifted by an \northorhombic dist ortion of the structure. This is illustrated by the concomitant structural and \nmagnetic transitions that appear at low temperature in these cobaltites [1, 2, 12], and by the \nferrimagnetic structure of CaBaCo 4O7 [13]. Similarly, the “114” ferrites exhibit a  competition \nbetween 1 D magnetic ordering and 2 D magnetic frustration, as has been shown for the \nhexagonal phases CaBaFe 4O7 [6], and for CaBa Fe4-xLixO7 [14]. But importantly, the “114” \nferrites differ from the “114” cobaltites by the fact that the LnBaFe 4O7 oxides exhibit a cubic \nstructure [7]. Though the latter is closely related to the hexagonal structure, the iron sublattice \nis very different ( Fig. 1 (b) ), consisting of “Fe 4” tetrahedra instead of “Fe 5” bipyramids and \n“Fe 3” triangles. No structural tra nsition appears at low temperature, and consequently, the \ncubic ferrites exhibit a spin glass behaviour due to a perfect  geometrical frustration.  Further, \nthe LnBaFe 4O7 series exhibit s an oxidation state disorder. Unlike the case of CaBaCo 4O7 [13], \nno char ge ordering is observed in LnBaFe 4O7, and this disorder is also important for the \nobserved glassiness.  \nRecently, we showed that the substitution of a divalent cation, Zn2+, for iron in \nYBaFe 4O7, allowed the hexagonal symmetry to be stabilized at the detrim ent of the cubic one \n[15]. Paradoxically, it was observed that the substitution of this diamagnetic cation for Fe2+ \ninduces ferrimagnetism, in contrast to the spin glass behaviour of the undoped phase \nYBaFe 4O7. In fact, a competition between ferrimagnetism  and magnetic frustration was \nobserved for the hexagonal phase YBaFe 4-xZnxO7. This was interpreted as the effect of two  3 antagonist phenomena: the partial lifting of the geometrical frustration due to the appearance of \nthe hexagonal symmetry inducing a 1 D magnetic ordering, and the existence of cationic \ndisordering favouring the glassy state.  \n Bearing in mind that the Fe2+:Fe3+ ratio is a crucial factor governing  the magnetic \nproperties of iron oxides, it must be emphasized that the substitution of Zn2+ for Fe2+ increases \nthe average valence of iron , i.e. the Fe2+:Fe3+ ratio decreases from 3 in the spin glass phase \nYBaFe 4O7 to 2.6 – 1.5 in the solid solution YBaFe 4-xZnxO7 when x changes from 0.4 to 1.5 \n[15]. In order to further understand the role of the ave rage valence of iron in the magnetic \nproperties of these ferrites, we have investigated the possibility of substitution of a diamagnetic \ncation such as gallium for Fe3+ in the YBaFe 4O7 structure. In the present study of the ferrite \nYBaFe 4-xGaxO7, we show t hat the introduction of gallium in the structure stabilizes the \nhexagonal symmetry, similar to the zinc substitution, but differently from the latter, the lifting \nof the geometrical frustration induces the formation of ferrimagnetic clusters embedded in a \nspin glass matrix, which tend to disappear as the gallium content increases, leading to a pure \nspin glass for higher Ga content, with a higher T g compared to YBaFe 4O7. \n \nExperimental  \n \nPhase -pure samples of YBaFe 4-xGaxO7  [x = 0.4 0 – 0.70] were prepared by solid state \nreaction technique. The precursors used were Y 2O3, BaFe 2O4, Ga 2O3, Fe 2O3 and metallic Fe \npowder. First, the precursor BaFe 2O4 was prepared from a stoichiometric mixture of BaCO 3 \nand Fe 2O3 annealed at 1200°C for 12 hrs in air. In a second step, a stoichiometric mixture of \nY2O3, BaFe 2O4, Ga 2O3, Fe 2O3 and metallic Fe powder was intimately ground and pressed in \nthe form of rectangular bars. The bars were then kept in an alumina finger, sealed in silica \ntubes under vacuum and annealed at 1100°C for 1 2 hrs. Finally, the samples were quenched to \nroom temperature in order to stabilize the “114” phase.  \nThe X -ray diffraction patterns were registered with a Panalytical X’Pert Pro \ndiffractometer under a continuous scanning mode in the 2\n  range 10° - 120° an d step size \n 2\n \n= 0.017°.  The cationic composition was confirmed by means of Energy Dispersive X -Ray \nSpectroscopy  (EDS) technique using a Scanning Electron Microscope (ZEISS Supra 55). The \nd. c. magnetization measurements were performed using a superconduc ting quantum \ninterference device (SQUID) magnetometer with variable temperature cryostat (Quantum \nDesign, San Diego, USA). The a. c. susceptibility, \n ac(T) was measured with a PPMS from \nQuantum Design with the frequency ranging from 10 Hz to 10 kHz. H ac was kept fixed at 10  4 Oe, while H dc was varied from 0 Oe to 2000 Oe. All the magnetic properties were registered on \ndense ceramic bars of dimensions ~ 4 \n  2 \n 2 mm3. \n \nResults and discussion  \n \n \n Similar  to Zn substitution, Ga substitution also favours the forma tion of the hexagonal \nphase at the expense of the cubic one. Nevertheless, the homogeneity range of the hexagonal \nYBaFe 4-xGaxO7 solid solu tion is significantly different  (0.40 \n  x \n 0.70) vis – à – vis that of \nYBaFe 4-xZnxO7 [15]. The cubic symmetry of YBaF e4O7 is retained for 0 \n  x \n 0.20, whereas \nthe domain 0.20 < x < 0.40 is biphasic, corresponding to a mixture of the cubic and hexagonal \nphases.  On the other hand,  for x > 0.70 , several impurity phases appear , namely Y 2O3 and \nGa2O3. The cationic compositio n of the single phase obtained for the range 0.40 \n  x \n 0.70 \nusing EDS analysis are also shown in Table 1 .  \n  \nStructural characterization  \n \n In Fig. 2, we show the X -ray diffraction (XRD) pattern s of the two end members, (a) \nYBaFe 3.6Ga0.4O7 and (b) YBaFe 3.3Ga0.7O7 as representative example s. As stated before, t he \nsample s are seen to stabilize in the hexagonal symmetry with the space group P63mc. The \nRietveld analysis of the lattice structure was done using the FULLPROF refinement program \n[16] and the fit s are also shown in Fig. 2. All the samples in the range x = 0.4 0 – 0.70 were \nseen to stabilize in the same hexagonal symmetry.  \nThe extracted cell parameters have been tabulated in Table 1 . The ionic radius of Fe3+ (0.49 \nÅ) is very similar to that of Ga3+ (0.47 Å). As can be seen from the extracted cell parameters \nshown in Table 1, a increases very slightly as x increases (an increase of only ~ 0.08 % as x \nincreases from 0.4 to 0.7), while c shows a slight decrease (~ 0.11 %). This causes the cell \nvolume to re main practically unchanged as a function of doping in accordance with the \nsimilar ionic radii of Fe3+ and Ga3+. \n \nD. C. magnetization studies  \n \n In the “114” ferrites, it has been established earlier [8, 15] that ferrimagnetism is \ninherently linked with the cross -over from cubic to hexagonal symmetry. The doping -induced \ntransition to the hexagonal symmetry involves a partial lifting of the 3D geometrical  5 frustration, which is the root cause of the appearance of ferrimagnetism. Thus, we restrict our \ndiscussion  of the magnetic data to the YBaFe 4-xGaxO7 samples exhibiting hexagonal symmetry \n(0.4 \n  x \n 0.7). We note here that the cubic samples (x < 0.2) are spin glasses similar to the \nundoped YBaFe 4O7, and will not be discussed further.  \n The temperature dependence  of d. c. magneti c susceptibility (\n dc = M/H)  was registered \naccording to the standard zero field cooled (ZFC) and field cooled (FC) procedures. A \nmagnetic field of 0.3 T was applied during the measurements. The measurements were done in \na temperature rang e of 5 K to 300 K. The \n ZFC(T) and \n FC(T) curves of all the samples are \nshown in Fig. 3. \n Undoped YBaFe 4O7 is a spin glass with T g = 50 K [ 7]. The ZFC \n dc versus T curve for \nYBaFe 4O7 shows a pure cusp -like shape [ 7], typical of canonical spin glasses.  A close look at \nFig. 3 reveals that the YBaFe 4-xGaxO7 series of samples shows two different kinds of low \ntemperature M ZFC(T) curves vis -à-vis the shape of the curves. While the \n dc(T) curve of the \nhighest substituted  sample (x = 0.7 0) is very similar to that o f canonical spin glasses (with a \npeak at ~ 50 K and a gradual decrease of the magnetization value below 50 K),  for the lowest \ndoped sample (x = 0.4 0), there is a sharp drop in the susceptibility  value below the temperature \nat which \n ZFC reaches its maximum  value (75 K). The susceptibility  value drops sharply till ~ \n50 K (marked by a black arrow in Fig. 3 (a)), below which the decrease in \n ZFC is more \ngradual.  We note here that the measuring field that we have chosen (0.3 T) is smaller than the \ncoercive fiel d of the YBaFe 3.6Ga0.4O7 sample at T = 5 K (data shown later in Fig. 5 ). Thus, it is \nquite possible that the sharp drop in the susceptibility value occurs at the temperature where the \ncoercive field of the sample becomes smaller than 0.3 T. However, follow ing this argument, \nwe should have obtained similar sharp drops in the \n ZFC(T) curves for the x = 0.5 and x = 0.6 \nsamples also, as the coercive fields of the x = 0.5 and x = 0.6 samples at T = 5 K are also larger \nthan 0.3 T. Instead , it is observed that the  sharp drop in the \n ZFC(T) curve seen in the x = 0.40 \nsample is reduced to small kinks in the \n ZFC(T) curves for the x = 0.50 and x = 0.60 samples.  \nMore importantly, a study of the temperature dependence of the coercive field (H C) of the x = \n0.4  sample (d ata not shown here) reveals that H C becomes smaller than 0.3 T  at ~ 17 K (i.e. at \na temperature much below 50 K).  This suggests  that this feature is not a simple effect of the \ncoercive  field, rather it may have  a more complex origin.  \nAnother possibility is  that this sudden decrease in \n ZFC(T) is due to domain wall \npinning effects, which  has, in fact,  been observed  previously  in manganites [1 7 – 19]. Due to \npinning, the domains would not  freely rotate below the pinning temperature unless a high  6 enough extern al field is present to overcome the pinned state. Upon zero field cooling, the \ndomains would be  pinned into random orientations. Whe n a low field is applied (0.3 T  in this \ncase), the pinning effect still dominates over the effect of the applied magnetic fi eld, and the \nmagnetization is lower than what would be expected in the absence of pinning. However, the \npinned domain walls can be thermally activated by increasing the temperature. This could be \nthe cause of the  visible jump in the \n ZFC(T) curve at the te mperature where the pinning effects \nare overcome by temperature (~ 50 K).  As can be seen in Fig. 3, the pinning effect gradually \ndecreases as the doping concentration is increased (the sharp drop in the \n ZFC(T) curve seen in \nthe x = 0.4 0 sample is reduced to small kinks in the \n ZFC(T) curves for the x = 0.5 0 and x = \n0.60 samples, and completely vanishes for the x = 0.7 0 sample).  This indicates  that the domain \nwall pinning is more prominent for small doping and vanishes for higher doping. This is \ncounter -intuitive if the pinning is thought to arise due to the presence of Ga in the lattice. Thus, \nthe fact that the domain wall pinning decreases with an increase in the doping concentration \nleads us to believe that this pinning does not arise from the disorder in  the system. Rather, it \nhas a more complex origin, which we discuss later.  \n Before we proceed further, we perform some additional measurements to make sure \nthat the sharp drop in the \n ZFC(T) curve observed in the lowest doped sample is indeed due to \ndomain  wall effects , and not arising from some additional (antiferro) magnetic transition in the \nsample.  Thus, we subject the x = 0.4 0 sample to a degaussing experiment [1 7], wherein the \nsample was initially cooled from 300 K down to 5 K in a zero external magne tic field. At 5 K, \na large magnetic field (5 T) was applied. The magnetic field was then reduced to zero, and the \nsample was degaussed at 5 K by cycling a field of reducing intensity so that the remanent \nmagnetization of the sample was reduced to zero. A m agnetic field of 0.3 T was then applied, \nand the \n dc(T) curve was recorded while warming the sample, in the same wa y as ZFC \nmagnetization is recorded. The results are shown in Fig. 4. We find that the sudden sharp drop \nobserved in the normal ly obtained  ZFC curve ( Fig. 4 (a)) vanishes when the sample is \nsubjected to a high enough magnetic field, and then degaussed  (Fig. 4 (b)). This experiment, \nthus, provides supplementary evidence that the sudden drop in magnetization seen below 75 K \nis not due to any kind of (antiferro) magnetic transition  in the sample , but is probably  \nassociated with domain wall pinning effects. The high magnetic field (5 T) to which the sample \nwas subjected was sufficient for domain wall displacements thereby destroying  the pinning. In \nfact, a ZFC magnetization recorded under a high enough field of 5 T (see inset of Fig. 4 (a)) \ndoes not show any sharp drop in the magnetization of the sample.   7 The d. c. magnetization M(H) curves of all the samples  registered at T = 5 K  are shown \nin Fig. 5. The virgin curves of the M(H) loops are represented by black circles while the rest of \nthe M(H) loops are shown by red lines. The first notable point is that for higher Ga substitution \n(x = 0.7 0), the M(H) loop is narrow and S – shaped ( Fig. 5 (d)), which is quite typical of spin \nglasses  and superparamagnets . On the other hand, for lower Ga substitution, the samples have \nlarger loops with higher values of the c oercivity and remanent magnetization, which keep \nincreasing as the doping concentration is decreas ed. This indicates the presence of a higher \ndegree of magnetic ordering in the lower doped samples as compared to the higher doped ones.  \n Another feature  which strongly supports the presence of domain wall pinning  is that the \nvirgin curve of the x = 0.4 0 sample lies slightly outside the main M(H) loop ( Fig. 5 (a)). This \nunusual feature of the virgin curve lying outside the main hysteresis loop has earlier been \nassociated with irreversible domain wall motion in spinel oxides [ 20]. We also note that the \nvirgi n curve starts to shift inside the main M(H) loop as the doping concentration (x) is \nincreased, and for the x = 0.7 0 sample, the entire virgin curve lies inside the main M(H) loop \n(Fig. 5 (d)). We again note that the domain wall pinning is more prominent i n the samples with \nlower doping concentration.  \n In Fig. 6, we once again show the d. c. magnetization M(H) curves of all the samples \nregistered at T = 5 K, but in three different modes: (i) normal ZFC mode, (ii) FC mode with a \nmagnetic field of 2 T  and (iii) FC mode with a magnetic field of H = - 2 T. In the ZFC mode, \nthe samples were cooled from 300 K to 5 K in zero external magnetic field, following which M \nversus H curves were registered. In the FC mode, on the other hand, the samples were cooled \nfrom 30 0 K to 5 K in the presence of an external magnetic field (H = 2 T  or – 2 T), and then M \nversus H curves were registered.  For the highest substituted  sample (x = 0.7 0), all three  M(H) \ncurves overlap each other ( Fig. 6 (d)). However, for lower doping concent ration, the field \ncooled M(H) loops exhibit shifts both in the field as well as in the magnetization axes.  This is \nthe exchange bias phenomenon [ 21, 22] that results from exchange interaction between \nferromagnetic and antiferromagnetic materials. In our YB aFe 4-xGaxO7 samples with low Ga \nconcentration, the observed exchange bias can be explained in terms of interfacial exchange \ncoupling between the coexisting ferrimagnetic cluster glass and the disordered spin glass -like \nphases.  This exchange bias effect ari sing from the inherent phase separation in the YBaFe 4-\nxGaxO7 samples is similar to that seen in some disordered manganites [ 23]. As can be seen \nfrom Fig. 6, the exchange bias effect keeps decreasing as the doping concentration is increased, \nand as stated b efore, it completely disappears for the doping concentration x = 0 .70. We can \nexplain this observation by considering that for lower Ga concentration, the samples consist of  8 coexisting ferrimagnetic clusters embedded in a spin glass -like matrix, but as the  doping \nconcentration by the diamagnetic cation (Ga) is increased, the ferrimagnetic clusters are \nprogressively reduced and we ultimately get a homogeneous spin glass (x = 0.7 0). The absence \nof phase separation in the x = 0.7 0 sample, thus, results in an a bsence of the exchange bias \neffect. The fact that the YBaFe 4-xGaxO7 samples with lower Ga concentration are intrinsically \nphase separated, while the x = 0.7 0 sample is not, also affords us an alternative explanation for \nthe domain wall pinning effects seen  in the lower doped samples. As stated previously, the fact \nthat the domain wall pinning is seen in the lower doped samples and not in th e x = 0.7 0 sample \nmeans that it cannot arise from the disorder in the system. Rather, we believe that the pinning \narise s from an interplay between the two magnetic phases in the phase separated samples. Such \na domain wall pinning process arising from the interplay between two coexisting magnetic \nphases has been seen earlier in intermetallic alloys [ 24]. We also note that a part from the \nexchange bias effect in the lower doped samples, field cooling also results in an overall \nincrease in the coercivity and remanence magnetization values. This can be interpreted as an \nincrease in the volume fraction of the magnetically ordered  phase when the samples are cooled \nin the presence of an external magnetic field.  Since field cooling improves the remanence in \nthe lower doped samples, hence it was important to register M -H curves after field cooling  \nwith positive as well as negative coo ling fields and check whether the M -H loops shift in \nopposite directions in order to confirm that there is indeed a genuine exchange bias effect in \nthe lower doped samples.  \n \nA. C. magnetic susceptibility studies  \n \n The measurements of the a. c. magnetic sus ceptibility \n ac(T, f, H) were performed at \ndifferent frequencies ranging from 10 Hz to 10 kHz, and different external magnetic field s \n(Hdc) ranging from 0 T to 0.2 T using a PPMS facility. The amplitude of the a.  c. magnetic \nfield was ~ 0.001 T In Fig. 7 and Fig. 8, we show the temperature dependence of the real (in -\nphase) component of the a. c. susceptibility  in the temperature range 10 K – 160 K  of the \nlowest doped sample (x = 0.4 0) and the highest doped sample (x = 0.7 0) respectively, with a  \nmeasuring f requency of 10 kHz and in zero magnetic field ( Hdc = 0).  \nFrom Fig. 7, it is clear that the \n '(T) curve of the x = 0.4 0 sample shows two features, \none at T = 86 K (marked by a black arrow), and the second at T = 44 K (marked by a red \narrow). While the high  temperature feature at 86 K is a clear peak, the low temperature one at \n44 K is a shoulder like feature and is more clearly evidenced in the imaginary (out -of-phase)  9 component of the a. c. susceptibity (shown in inset (a) of Fig. 7). Repeating the measure ments \nusing four measuring frequencies (ranging from 10 Hz – 10 kHz) reveals that both the features \nare frequency dependent (this is shown in inset (b) of Fig. 7). The x = 0.7 0 sample, on the other \nhand, shows only one feature (at T = 60 K) in the \n '(T) an d \n''(T) curves, shown in the main \npanel and inset (a) of Fig. 8 respectively. Inset (b) of Fig. 8 shows the \n '(T) curves measured \nusing four different frequencies, and reveals that this peak  at 60 K is also strongly frequency \ndependent.  We note that the i maginary part of \n  for YBaFe 3.6Ga0.4O7 is ~ 8 % of the real part \nof \n . This is commonly found for systems where the spin domains are relatively large. \nYBaFe 3.6Ga0.4O7 can thus be described as a cluster glass. On the other hand, the imaginary part \nof \n for YBaFe 3.3Ga0.7O7 is significantly smaller (about 2.4% of the real part of \n ) which \nindicates that YBaFe 3.3Ga0.7O7 is closer to a canonical spin glass.  \n Although all the features in the a. c. susceptibility curves of the two samples described \nabove have a si ngle commonality , in that they are all strongly frequency dependent, but the \nnature a nd origin of these peaks can be quite  different.  Specifically, we need to establish the \nnature of the low temperature shoulder in the x = 0.4 sample at T ~ 50 K. Since it occurs close \nto the temperature where we ha ve evidenced domain wall pinning from  the d. c. magnetic data, \nit is tempting to attribute this low temperature shoulder in the a. c. susceptibility data to the \nsame phenomenon. However, it is also possible that t his low temperature feature is a \nsuperparamagnetic effect of the ferrimagnetic clusters.  To investigate this , we perform further \nmeasurements of \n '(T) and \n ''(T) of the two limiting samples  (x = 0.4 0 and x = 0.7 0) in the \npresence of different external magn etic fields H dc ranging from 0 to 0.2 T. The results for the \nYBaFe 3.3Ga0.7O7 sample are shown in Fig. 9. It is seen that both \n ' and \n '' are strongly \nsuppressed by the magnetic field.  The peak also shows a continual shift towards lower \ntemperature as the e xternal magnetic field is increased  (see the black arrows in Fig. 9). This is \ntypical of the behaviour of a spin glass freezing temperature under the influence of magnetic \nfield. \n In Fig. 10, we show the results for the YBaFe 3.6Ga0.4O7 sample. It is seen t hat w hile the \nhigh temperature peak was significantly suppressed in the presence of external magnetic field, \nthe low temperature peak was largely unaffected relative to the case H dc = 0. This rules out the \nscenario of superparamagnetism being responsible f or this low temperature feature, and \nconfirms that the 50 K anomaly arises due  to enhanced domain wall pinning,  signatures of \nwhich have been observed and commented upon earlier in the d. c. magnetization \nmeasurements also.  We also note that the high tempe rature peak shifts towards higher  10 temperature as the external magnetic field is increased (see the black arrow in Fig. 10 (a)). \nThis is not expected for a pure  spin glass freezing transition, where the peak should shift \ntowards lower temperature as the mag netic field is increased. We believe that this anomaly \narises because the x = 0.4 0 sample is not a pure spin glass, rather it is a phase separated sample \nconsisting of ferrimagnetic clusters embedded in a spin glass matrix.  \n \nConclusion  \n \n These results show  that the substitution of Ga3+ for Fe3+ in YBaFe 4O7 induces a \nstructural transition from cubic to hexagonal, similar to the substitution of Zn2+ for Fe2+ in this \ncompound. Though the two types of substitutions induce a lifting of the geometrical frustratio n \nthrough a change of the structure, the effect of these diamagnetic cations upon the magnetic \nproperties is different. A strong ferrimagnetic component is induced by zinc substitution [15], \nwhereas Ga substitution leads to the formation of ferrimagnetic c lusters embedded in a spin \nglass matrix, essentially leading to phase separation in the samples. The difference originates \nfrom the opposite evolution of the Fe3+:Fe2+ ratio as the substitution rate increases in the two \ncases. Both Fe3+ and Fe2+ exhibit th e high spin configuration since they have a tetrahedral \ncoordination in these ferrites. Thus, the magnetic moment induced by the eg2t2g3 Fe3+ cations \nshould be much higher than that induced by the eg3t2g3 Fe2+ cations. Hence, an increase in the \nFe3+:Fe2+ ratio should favour stronger magnetic interactions.  In the case of Zn2+ doping, the \nFe3+:Fe2+ ratio increases , thereby  favouring the appearance of ferrimagnetism. On the other \nhand, for Ga3+ doping, the Fe3+:Fe2+ ratio decreases , thereby inducing only weak \nferrimagnetism and cluster formation . In both series, YBaFe 4-xGaxO7 and YBaFe 4-xZnxO7, a \ndilution effect is observed with an increase in the doping concentration. As a consequence, \nferrimagnetism is weakened for higher concentrations in the Zn – phase. In the Ga – phase, the \nferrimagnetic clusters are magnetically coupled by exchange interactions mediated through the \nsurrounding spin glass matrix. For higher Ga concentrations, the exchange coupling between \nthe ferrimagnetic clusters becomes less efficient, ultimately leading to the formation of a pure \nspin glass phase for x = 0.70, which is similar to the pristine sample (x = 0), but with a slightly \nhigher T g (60 K). The Ga – substituted phase also differs from the Zn – phase by the presence \nof exchange bias  and domain wall pinning. The cause of both these effects can be traced back \nto the inherent phase separation present in the samples.  \n \n  11 Acknowledgements  \n \nWe acknowledge the CNRS and the Conseil Regional of Basse Normandie for financial \nsupport in the frame  of Emergence Program  and N°10P01391 . V. P. acknowledges support by \nthe ANR -09-JCJC -0017 -01 (Ref: JC09_442369).  \n \nReferences  : \n \n      [1]     Martin Valldor and Magnus Andersson, Solid State Sciences , 2002 , 4, 923  \n      [2]     Martin Valldor, J. Phys.: Con dens. Matter ., 2004 , 16, 9209  \n      [3]     A. Huq, J. F. Mitchell, H. Zheng, L. C. Chapon, P. G. Radaelli, K. S. Knight and P.  \n               W. Stephens, J. Solid State Chem ., 2006 , 179, 1136  \n      [4]     D. D. Khalyavin, L. C. Chapon, P. G. Radaelli, H. Zheng and J. F. Mitchell, Phys.  \n                Rev. B , 2009 , 80, 144107  \n      [5]     V. Caignaert, V. Pralong, A. Maignan and B. Raveau, Solid State Communications ,  \n                2009 , 149, 453  \n      [6]     B. Raveau, V. Caignaert, V. Pralong, D.  Pelloquin and A. Maignan, Chem. Mater .,  \n                2008 , 20, 6295  \n      [7]     V. Caignaert, A. M. Abakumov, D. Pelloquin, V. Pralong, A. Maignan, G. Van  \n                Tendeloo and B. Raveau, Chem. 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Sarkar, V. Pralong , V. Caignaert and B. Raveau , Chem. Mater ., 2010, 22, 2885  \n      [16]  J. Rodriguez -Carvajal, An Introduction t o the Program FULLPROF 2000; Laboratoire  \n               Léon Brillouin, CEA -CNRS: Saclay, France (2001)  \n      [17]   P. A. Joy and S. K. Date, J. Magn. Magn. Mater ., 2000 , 220, 106  \n      [18]   C. R. Sankar and P. A. Joy, Phys. Rev. B , 2005 , 72, 024405  \n      [19]   T. Gao, S. X. Cao, K. Liu, B. J. Kang, L. M. Yu, S. J. Yuan and J. C. Zhang, Journal  \n                of Phys: Conf. Series , 2009 , 150, 042038  \n      [20]   P. A. Joy and S. K. Date, J. Magn. Magn. Mater ., 2000 , 210, 31 \n      [21]   R. L. Stamps,  J. Phys. D, 2000 , 33, R247  \n      [22]   W. H. Meiklejohn and C. P. Bean , Phys. Rev ., 1956 , 102, 1413  \n      [23]   S. Karmakar, S. Taran, E. Bose, B. K. Chaudhuri, C. P. Sun, C. L. Huang and H. D.  \n                Yang, Phys. Rev. B , 2008 , 77, 144409  \n      [24]   A. Bracchi, K. Samwer, S. Schneider and J. F. Löffler, Appl. Phys. Lett ., 2003 , 82,  \n                721 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  13 Table caption s \n \nTable 1 : Cell parameters as obtained from the Rietveld refinement of X -ray powder diffraction \ndata.  \n \nFigure  captions  \n \nFigure 1:  Schematic representation of (a) hexagonal LnBaCo 4O7 and (b) cubic YBaFe 4O7 \n(adapted from Ref. 8). For details, see text.  \n \nFigure 2:  X-ray diffraction pattern along with the fits for (a) YBaFe 3.6Ga0.4O7 and (b) \nYBaFe 3.3Ga0.7O7. \n \nFigure 3: Temperature dependence of the magnetic susceptibility (\n dc = M/H) collected \naccording to zero field cooling (ZFC) and field cooling (FC) processes for YBaFe 4-xGaxO7 (a) \nx = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = 0.70, measured at B = 0.3 T.  \n \nFigur e 4: \nZFC(T) curves of YBaFe 3.6Ga0.4O7 recorded (a) in the ZFC mode without \ndegaussing, and (b) after applying a magnetic field of 5 T and degaussing the ZFC sample (see \ntext for details). The inset in (a) shows \n ZFC(T) recorded under a magnetizing field o f 5 T.  \n \nFigure 5:  M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K . The virgin curves are shown in black circles, while the rest of the \nhysteresis loops are shown in red lines.  \n \nFigure 6:  M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K, measured after zero field cooling (red open circles), field cooling in \na field of 2 T (black lines) and field cooling in a field of - 2 T (blue line s).  \n \nFigure 7 : Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.6Ga0.4O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.   14  \nFigure 8 : Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.3Ga0.7O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \nFigure 9 : The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.3Ga0.7O7 as a function of temperature. The driving frequency was \nfixed a t f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \nFigure 10 : The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.6Ga0.4O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  15 Table 1  \n \nDoping \nconcentration \n(x)  \nCrystal \nsystem \n(Space \ngroup)  \n  \nUnit cell parameters  \n c / a \n2 x as \nobtained \nfrom EDS \nanalysis  a (Å)  c (Å)  \n0.40 Hexagonal \n(P63mc)  \n6.320 (1) \n 10.383 (1) 1.6428 3.02 0.43 (2)  \n0.50 Hexagonal \n(P63mc)  \n6.322 (1) \n 10.376 (1) 1.6413 3.15 0.50 (1)  \n0.60 Hexagonal \n(P63mc)  \n6.323 (1) \n 10.37 4 (1) 1.6407 2.94 0.59 (1)  \n0.70 Hexagonal \n(P63mc)  \n6.325 (1)  \n 10.37 2 (1)  1.6398 3.46 0.74 (3)  \n \n \n \n \n \n \n \n \n \n \n \n \n  16 \n \n \nFig. 1 . Schematic representation of (a) hexagonal LnBaCo 4O7 and (b) cubic YBaFe 4O7 \n(adapted from Ref. 8). For details, s ee text.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \n \nFig. 2. X-ray diffraction pattern along with the fit s for (a) YBaFe 3.6Ga0.4O7 and (b) \nYBaFe 3.3Ga0.7O7. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  18 \n \n \nFig. 3. Temperature dependence of the magnetic susceptibility (\n dc = M/H) collected according \nto zer o field cooling (ZFC) and field cooling (FC) processes for  YBaFe 4-xGaxO7 (a) x = 0.40, \n(b) x = 0.50, (c) x = 0.60 and (d) x = 0.70, measured at H = 0.3 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  19 \n \n \nFig. 4. \nZFC(T) curves of YBaFe 3.6Ga0.4O7 recorded (a) in the ZFC mode  without dega ussing , \nand (b) after applying a magnetic field of 5 T and degaussing the ZFC sample (see text for \ndetails). The inset in (a) shows \n ZFC(T) recorded under a magnetizing field of 5 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n  20 \n \n \nFig. 5. M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K . The virgin curves are shown in black circles, while the rest of the \nhysteresis loops are shown in red lines.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  21 \n \n \nFig. 6. M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50,  (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K, measured after zero field  cooling (red open circles), field cooling in \na field of 2 T (black lines)  and field cooling in a field of - 2 T (blue lines) .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  22 \n \n \nFig. 7. Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.6Ga0.4O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibi lity, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \n \n \n \n \n \n \n \n \n \n \n  23 \n \n \nFig. 8. Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.3Ga0.7O7 as a f unction of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \n \n \n \n \n \n \n \n \n \n \n  24 \n \n \nFig. 9. The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.3Ga0.7O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and  Hac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n  25 \n \n \nFig. 10. The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.6Ga0.4O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n "    },    {        "title": "1210.6700v1.Dirac_half_metal_in_a_triangular_ferrimagnet.pdf",        "content": "arXiv:1210.6700v1  [cond-mat.str-el]  24 Oct 2012APS/123-QED\nDirac half-metal in a triangular ferrimagnet\nHiroaki Ishizuka1and Yukitoshi Motome1\n1Department of Applied Physics, University of Tokyo, Hongo, 7-3-1, Bunkyo, Tokyo 113-8656, Japan\n(Dated: September 13, 2021)\nAn idea is proposed for realizing a fully spin-polarized Dir ac semimetal in frustrated itinerant\nmagnets. We show that itinerant electrons on a triangular la ttice exhibit the Dirac cone dispersion\nwithhalf-metallic behavior inthepresence ofathree-subl attice ferrimagnetic order. The Diracnodes\nhavethesamestructureasthoseofgraphene. Byvariational calculation andMonteCarlosimulation,\nwe demonstrate that the ferrimagnetic order with the Dirac n ode spontaneously emerges in a simple\nKondo lattice model with Ising anisotropy. The realization will be beneficial for spintronics as a\ncandidate for spin-current generator.\nPACS numbers: 71.10.Fd, 73.22.Pr, 72.25.-b\nMassless Dirac fermions show substantially different\nnaturefromordinaryelectrons. The peculiarnatureorig-\ninates in the characteristic energy dispersion —the nodal\nstructure with linear dispersion often referred to as the\nDirac cone. While the Dirac fermions were originally\nintroduced in the relativistic quantum theory, recent dis-\ncovery of graphene [1, 2], a single layer sheet of graphite,\nhascarvedoutanewdirectionoftheirstudyincondensed\nmatter systems [3, 4]. In graphene, two Dirac cones ap-\npear in the energy dispersion of πelectrons, which are\nat theKandK′points in the Brillouin zone for the\ntwo-dimensional honeycomb lattice. The Fermi level of\nthis two dimensional conductor comes right at the nodal\npoints, and the low-energy Hamiltonian is well approx-\nimated by the Weyl equation [5]. Various remarkable\nelectronic and transport properties of graphene mainly\nowe to these Dirac cones in the band structure.\nThe extraordinary nature of Dirac fermions in\ngraphene has also attracted a great interest from appli-\ncation to electronics [3]. From the viewpoint of such po-\ntential applications, it is of great interest to control the\ncharacteristic band structure. Furthermore, it is also de-\nsired to control the electronic spin degree of freedom for\nthe application to spintronics [6]. However, there is not\nso much flexibility in graphene, as the Dirac cone is a di-\nrect consequence of the honeycomb lattice geometry and\nthe relativistic spin-orbit interaction is very weak.\nIn this Letter, we propose an alternative solution for\nmanipulating the spin degree of freedom by seeking pos-\nsible emergence of Dirac fermions from itinerant mag-\nnets. We show that itinerant electrons coupled to a\nwell-known ferrimagnet on a triangular lattice give rise\nto the Dirac nodes in their band structure, similar to\nthose of graphene. The resultant masslessDirac fermions\nare spin-polarized, and they are stable in a wide range\nof the spin-charge coupling including typical values in\nsolids. We demonstrate that, by an unbiased Monte\nCarlo (MC) simulation as well as a variational calcu-\nlation, such Dirac half-metal with ferrimagnetic order\nspontaneously emerges in a minimal Kondo-lattice type\nmodel. The results strongly suggest the possibility ofrealizing the exotic electronic state in transition-metal\nand rare-earth compounds, which generally retain much\nhigher controllable degrees of freedom than graphene.\nSuch a new family will not only add a member to the\nknown list of Dirac electrons in solids [7–10], but also\nbring a completely new aspect by the spin polarization.\nIn a half-metal, the electric current is perfectly spin-\npolarized as the low energy excitations only exist for the\nmajority spin [6]. This nature works as a spin-current\ngenerator by filtering out the minority-spin electrons.\nThus, our proposal opens a new frontier for the applica-\ntion of Dirac massless fermions, especially for spintron-\nics [11].\nLet us first discuss a naive, rather trivial approach\nto achieve a Dirac half-metal. We here consider a\nsingle-band ferromagnetic Kondo lattice model (double-\nexchange model) on a honeycomb or kagome lattice [see\nFigs. 1(a) and 1(b)]. The model consists of the nearest-\nneighbor hopping of electrons and the exchange inter-\naction between the electron spin and localized moment,\nwhose Hamiltonian is given by\nH=−t/summationdisplay\n/angbracketlefti,j/angbracketright,σ(c†\niσcjσ+H.c.)−J/summationdisplay\niσi·Si.(1)\nHere,ciσ(c†\niσ) is the annihilation(creation) operatorof an\nitinerant electron with spin σ=↑,↓atith site,σiandSi\nrepresent the itinerant and localized spin, respectively; t\nis the transfer integral and Jis the onsite Kondo cou-\npling. Hereafter we take t= 1 andJ >0.\nIn this model, when Jis sufficiently large compared\nto the bandwidth at J= 0, a ferromagnetic order is\nstabilized by the double-exchange mechanism in a wide\nrange of electron filling n=/summationtext\niσ/angbracketleftc†\niσciσ/angbracketright/2N, whereN\nis the system size [12, 13]. In the ferromagnetic phase,\nthe band is split into two by the large exchange coupling\naccording to the spin component, and each band has ex-\nactly the same form as that for the noninteracting case\nJ= 0. Hence, in principle, the Dirac half-metal arises\nfor the honeycomb and kagome lattices, as the nonin-\nteracting bands on these lattices have the Dirac nodes.\nHowever, these situations are very difficult to realize in2\n(a)\nFIG. 1. (color online). Schematic pictures of (a) a hon-\neycomb ferromagnet, (b) kagome ferromagnet, and (c) three-\nsublattice triangular ferrimagnet. The arrows at each site\nrepresent localized spins.\nsolids as neither such a strong exchange interaction nor\nthe honeycomb and kagome structures is easily realized\nin magnetic compounds.\nAs a more realistic approach, here we propose a sim-\nple, but rather nontrivial route to the half-metallic Dirac\nfermion systems. Let us consider the model in Eq. (1) on\na triangular lattice, and the situation in which a three-\nsublattice collinear ferrimagnetic order with up-up-down\nspin configuration is realized —see Fig. 1(c). By treating\nthe localized moments as classical spins with |Si|= 1,\nthe band structure is easily calculated by the exact diag-\nonalization of the Hamiltonian. The lower three bands\nof the totally six bands are shown in Fig. 2; the two red\nbands are of up spins, and the blue band is of down spin\n(the other upper three bands have the similar form with\nopposite spins).\nThe band structure has a notable feature at the energy\nε=−J; the two up-spin bands touch with each other\nat theKandK′points in the Brillouin zone to form\na Dirac-type point node with linear dispersion, and the\ndown-spinbandhas the bandtop atthe samepoints with\nan ordinary parabolic dispersion. See also the enlarged\nfigure in Fig. 2(b) and the energy dispersion along the\nsymmetric lines in Fig. 2(c).\nInthissituation, whentheelectronfillingisat n= 1/3,\nthe two lower bands are fully occupied while the remain-\ningbands(includingtheupperthree)areunoccupied; theFermi level is located at the nodes where the three bands\nmeet. Asthedown-spinbandhasanenergygap,thehalf-\nmetallic Dirac electrons are obtained by electron doping\nto the unoccupied up-spin band. Although hole doping\nhides the Dirac nature as the down-spin parabolic band\nis doped at the same time, the situation is avoided by in-\ntroducing an additional antiferromagnetic exchange cou-\npling between the neighboring sites, J′/summationtext\n/angbracketlefti,j/angbracketrightσi·Sj[14].\nA finiteJ′>0 shifts the down-spin band to the lower\nenergy and isolates the half-metallic Dirac nodes ener-\ngetically, as demonstrated in Figs. 2(d) and 2(e). Hence,\nthe simple ferrimagnetic order on the triangular lattice\nrealizes the peculiar Dirac half-metallic state near 1/3\nfilling.\nThe Dirac nodes have essentially the same structures\nas those in graphene. Under the ferrimagnetic order, the\nHamiltonian is written as\nH=/summationdisplay\nk\n−Jσz\nAτkτ∗\nk\nτ∗\nk−Jσz\nBτk\nτkτ∗\nk(J+6J′)σz\nC\n.(2)\nHere, the upper tworowscorrespondto the sites with the\nup localized moment ( A,Bsublattices) and the bottom\nrow is for the down one ( Csublattice) in the three-site\nunit cell. In Eq. (2), σzis thezcomponent of the Pauli\nmatrix for itinerant electrons, kis the wave vector, and\nτkis the Fourier transform of the hopping term given by\nτk=−t[eikx+ei/parenleftBig\n−kx\n2+√\n3\n2ky/parenrightBig\n+ei/parenleftBig\n−kx\n2−√\n3\n2ky/parenrightBig\n]. By using\nthek·pperturbation around the KandK′points in the\nBrillouin zone [5] and by expanding the result up to the\nfirst order in terms of tκx/Jandtκy/J(κis the relative\nwave vector measured from KandK′points), we end up\nwith the low-energy Hamiltonian which is factorized into\ntwo parts. One is a 2 ×2 Hamiltonian for the up-spin\nhoneycomb subnetwork of the AandBsublattices, and\nthe other is a localized state at the down-spin sites in the\nCsublattice. The former is given by\nHDirac\nk±=/parenleftbigg\n−J3\n2it(κx±iκy)\n−3\n2it(κx∓iκy)−J/parenrightbigg\n,(3)\nwhere the sign ±corresponds to the KandK′points.\nThis has an equivalent form to that of graphene.\nIt is worthy to note that the Dirac nodes are formed\nimmediately by switching on J. However, when Jis\nsmall, the low-energy physics at n= 1/3 is not char-\nacterized solely by the massless Dirac fermions because\nthere is a band overlap at the energy of the Dirac\nnodes. The band overlap comes from the second lower\nband for up spin, which has an energy minimum at\nk= (2π/3,0) points and its threefold symmetric points\nfor smallJ; the minimum energy is given by ε(2π\n3,0)=\nt/2−/radicalig\n(J+3J′−t/2)2+2t2. In order for the Dirac\nnodes to be isolated at the Fermi level, this energy\nshould be higher than that at the KandK′points,3\nKM Γ Γ01\n-1\n-2\n-3\n-4\n-5\n-6\n-7(c)\nε(a)\n2\n0\n-2\n-4\n-6\n0\n2π\n32π\n3-\n04π\n3√3-ε\nkx\nkyK\nMΓ\n-8\nK'\n4π\n3√3(b)\nε\nkykx-1.0\n-1.5\n-2.0\n-2.5\n-3.0\n00(d)\nε\nkykx-1.0\n-1.5\n-2.0\n-2.5\n-3.0\n00\n(e)\nKM Γ Γ01\n-1\n-2\n-3\n-4\n-5\n-6\n-7ε2π\n34π\n3√32π\n34π\n3√3\nFIG. 2. (color online). Band structures of the model in Eq. (1 ) under the three-sublattice ferrimagnetic order at J= 2. (a)\nThe overall band structure of the three lower-energy bands a tJ′= 0, (b) the enlarged view near the Fermi level ε=−Jat\nn= 1/3 in the first quadrant, and (c) the cut along the symmetric lin es. (d) and (e) show the results at J′= 0.05. The arrows\nindicate the spins for each band. In (a), the gray hexagon on t he basal plane shows the first Brillouin zone for the magnetic\nsupercell. The dashed line in (e) indicates the Fermi level i n the MC simulation shown in Fig. 4.\nεK=−(J+3J′). Hence, the Dirac nodes are energeti-\ncally isolated and play a decisive role when the condition\n(J+3J′)/t>1 is satisfied. This condition is important\nbecausethenecessary JandJ′aremuchsmallerthanthe\nnoninteracting bandwidth 9 t, and it is indeed satisfied in\nwide range of materials.\nSo far, we assumed the presence of three-sublattice fer-\nrimagnetic order. In the following, we show that such\norder is indeed stable in the Kondo-lattice type model\nas Eq. (1). We here simplify the model by assuming the\nlocalized moments are the Ising spins taking the values\nSi=±1.\nFirst, we investigate the ground state phase diagram\nnearn= 1/3 by a variational calculation. We com-\npare the ground state energy of the two-sublattice stripe\nphase and three-sublattice ferrimagnetic phase appeared\nin the previous study [15], in addition to the ferromag-\nnetic phase. The results at J= 2 are shown in Fig. 3 for\nJ′= 0 and 0.05. AtJ′= 0, the ground state in the plot-\nted rangeis dominated by the ferrimagneticphaseas well\nas the stripe phase. The different phases are separated\nbyphaseseparation. As showninFig. 3(b), the introduc-\ntion of small J′largely stabilizes the ferrimagnetic phase\nnearn= 1/3 as well as the stripe phase. This is because\nthe itinerant electron spins are polarized parallel to the\nlocalized spins in the ground state, leading to an energy\ngain (loss) by the antiferromagnetic J′for the two states\n(the ferromagnetic state).\nWe next examine the stability of the ferrimagnetic or-\nderatfinite temperaturesbyanunbiasedMC simulation.\nFor the simulation, a standard algorithm for fermionnJ\n012345678\n0.260.280.30 0.38 0.320.340.36\nnJ\n2-sub stripe\n0.260.280.30 0.38 0.320.340.36012345678(a)\nFIG. 3. (color online). Ground state phase diagram obtained\nby variational calculation at (a) J′= 0 and (b) J′= 0.05.\nThe schematic picture of magnetic structure in each phase\nis shown. The white region indicates the electronic phase\nseparation (PS)andthedottedvertical lines indicate n= 1/3.4\nMxy[1/√2 ]\n|Mz|[x√3]\nN =12x12\nN =12x18\nN =18x18\n0.00.20.40.60.81.01.2\n(a)Tc\nTN =12x12\nN =12x18\nN =18x18\n0.00.20.40.60.8\n0.00 0.05 0.10 0.15 0.20 0.25(c)\nψχz[x5] N =12x12\nN =12x18\nN =18x18\n020406080100120140160180(b)\nχxyTKT\nFIG. 4. (color online). MC results for (a) the pseudo mo-\nmentsMxyand|Mz|, (b) corresponding susceptibilities χxy\nandχz, and (c) azimuth parameter ψ. The data are calcu-\nlated atn= 0.34.\nsystems coupled to classical fields is used [16]. In this\nmethod, the trace overthe fermions in the partitionfunc-\ntion is calculated by the exact diagonalization, while the\ntrace over classical spin configurations is computed by\na classical MC method using the Metropolis dynamics.\nThe phase transition to ferrimagnetic phase is detected\nby using two parameter [17]. One is the pseudo-moment\ndefined by\n˜Sm=\n2√\n6−1√\n6−1√\n6\n01√\n2−1√\n21√\n31√\n31√\n3\n\nSi\nSj\nSk\n,(4)\nwheremis the index for the three-site unit cells, and\n(i,j,k) denote the three sites in the mth unit cell be-\nlonging to the sublattices ( A,B,C), respectively. We\nmeasure the summation M= (3/N)/summationtext\nm˜Smand the\nsusceptibility. The other is the azimuth parameter ψ\ndefined by ψ= (˜Mxy)3cos6φM, whereφMis the az-\nimuth angle of Min thexyplane and ˜Mxy= 3M2\nxy/8\n(M2\nxy=M2\nx+M2\ny). The ferrimagnetic ordering is sig-\nnaled byMxy→2/radicalbig\n2/3,|Mz| →1/√\n3, andψ→1 at\nlow temperature T→0, respectively [15, 18, 19].\nFigure 4 shows the MC results at J= 2 andJ′= 0.05\nin the slightly electron doped region to n= 1/3 [see alsoFig.2(e)]. Theresultsindicatetwosuccessivephasetran-\nsitions atTKT= 0.192(15) and at Tc= 0.108(9). The\ntransition temperatures are estimated by extrapolating\nthe peak of susceptibilities χxyandχzasN→ ∞. The\ntransition at TKTis considered as a Kosterlitz-Thouless\ntype with the growth of quasi-long-range order [15]. On\nthe other hand, the phase transition at Tcis a three-\nsublatticeferrimagneticordering. TheMC resultand the\nabove analysis for the ground state consistently indicate\nthat the three-sublattice ferrimagnetic order is stabilized\nin the vicinity of n= 1/3 in the wide range of parame-\nters forJandJ′, spontaneously giving rise to the Dirac\nhalf-metal.\nAs such ferrimagnetic order was indeed observed in\nseveralinsulatingmagnets[20,21], ourresultsinthemin-\nimal model will stimulatethe hunt forDirachalf-metal in\ntransition-metal and rare-earth compounds. The present\nresults will be qualitatively robust even when extend-\ning the model to more realistic situation. For instance,\ntheferrimagneticstateremainsstablewhenincluding the\ntransverse components of localized spins, at least, in the\npresence of the Ising anisotropy. Multi-band effect may\nbe avoided under a particular crystal field; for instance,\nthed-electrona1gorbital isolated by a strong trigonal\nfield is a good candidate for the realization. Interlayer\ncoupling, however, may open a gap at the Dirac nodes.\nNevertheless, a straightforward stacking of layers or suf-\nficiently isolated layers in a controlled thin film will be\npromising to preserve the massless nature.\nThe authors thank Y. Matsushita, A. Shitade, and\nY. Yamaji for helpful comments. H.I. is supported\nby Grant-in-Aid for JSPS Fellows. This research\nwas supported by KAKENHI (No.19052008, 21340090,\n22540372, and 24340076), Global COE Program “the\nPhysical Sciences Frontier”, the Strategic Programs for\nInnovative Research (SPIRE), MEXT, and the Compu-\ntational Materials Science Initiative (CMSI), Japan.\n[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nY. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A.\nFirsov, Science 306, 666 (2004).\n[2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nI. V. Katsunelson, I. V. Grigorieva, S. V. Dubonos, and\nA. A. Firsov, Nature 438, 197 (2005).\n[3] A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183\n(2006).\n[4] For a recent review, see A. H. Castro Neto, N. M. R.\nPeres, K. S. Novoselov, and A. K. Geim, Rev.Mod. Phys.\n81, 109 (2009).\n[5] G. W. Semenoff, Phys. Rev. 53, 2449 (1984).\n[6] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. Daughton, S. von Molnar, M. L. Roukes, A. Y.\nChtchelakanova, amd D. M. Treger, Science 294, 1488\n(2001).\n[7] M. H. Cohen and E. I. Blount, Phil. Mag. 5, 115 (1960).5\n[8] T. Konoike, K. Uchida, and T. Osada, J. Phys. Soc. Jpn.\n81, 043601 (2012).\n[9] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[10] S. Ishibashi, K. Terakura, and H. Hosono, J. Phys. Soc.\nJpn.77, 053709 (2008).\n[11] D. Pesin and A. H. MacDonald, Nature Mater. 11, 409\n(2012).\n[12] C. Zener, Phys. Rev. 82, 403 (1951).\n[13] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675\n(1955).\n[14] An off-site Kondo coupling may exist generally in the\nKondo lattice systems, although the magnitude is much\nsmaller than the onsite one and the sign depends on\nthe orbital nature of itinerant and localized electrons.\nThe antiferromagnetic superexchange coupling between\nneighboring localized spins, given by JAF/summationtext\n/angbracketlefti,j/angbracketrightSi·Sj,\nmay also exist, but it neither modifies the band structure\nnor harms the stability of the ferrimagnetic state.[15] H. Ishizuka and Y. Motome, Phys. Rev. Lett. 108,\n257205 (2012).\n[16] S. Yunoki,J. Hu, A.L.Malvezzi, A.Moreo, N.Furukawa,\nand E. Dagotto, Phys. Rev. Lett. 80, 845 (1998).\n[17] In principle, the ferrimagnetic ordering can be detect ed\nby the spin structure factor. In the finite-size MC cal-\nculations for the current model, however, it is useful to\nemploy the pseudo-moment and its azimuth parameter\nfor distinguishing it from a three-sublattice partial diso r-\nder and Kosterlitz-Thouless type quasi long-range order.\nSee also Ref. [15, 18, 19].\n[18] H. Takayama, K. Matsumoto, H. Kawahara, and K.\nWada, J. Phys. Soc. Jpn. 52, 2888 (1983).\n[19] S. Fujiki, K. Shutoh, S. Inawashiro, Y. Abe, and S. Kat-\nsura, J. Phys. Soc. Jpn. 55, 3326 (1986).\n[20] M. Tanaka, H. Iwasaki, K. Siratori, and I. Shindo, J.\nPhys. Soc. Jpn. 58, 1433 (1989).\n[21] J. Iida, M. Tanaka, Y. Nakagawa, S. Funahash, N.\nKimizuka, and S. Takekawa, J. Phys. Soc. Jpn. 62, 1723\n(1993)."    },    {        "title": "1010.4872v1.High_spin_polarization_in_epitaxial_films_of_ferrimagnetic_Mn3Ga.pdf",        "content": "High spin polarization in epitaxial films of ferrim agnetic Mn 3Ga \n \nH. Kurt,* K. Rode. M. Venkatesan, P. S. Stamenov an d J. M. D. Coey \n \nSchool of Physics and CRANN, Trinity College, Dubli n 2, Ireland \n \n* kurth@tcd.ie \n \nPACS number(s): 75.76.+j, 75.70.-i, 75.60.Ej, 75.50 .-y  \n \nAbstract \n \nFerrimagnetic Mn 3Ga exhibits a unique combination of low saturation magnetization \n(Ms = 0.11 MA m -1) and high perpendicular anisotropy with a uniaxial  anisotropy \nconstant of Ku = 0.89 MJ m -3. Epitaxial c-axis films exhibit spin polarization as high \nas 58%, measured using point contact Andreev reflec tion. These epitaxial films will \nbe able to support thermally stable sub-10 nm bits for spin transfer torque memories.  \n \n \n \n \n \n \n \n There is a revival of interest in Heusler alloys, m otivated by the extraordinary \nvariety of electronic ground states that can be ach ieved by varying the number of \nvalence electrons 1, 2, composition 3-5 and atomic order in these materials. The Heusler \nfamily include semiconductors, metals and half meta ls 6, as well as ferromagnets, \nantiferromagnets and superconductors, and even perh aps compensated ferrimagnetic \nhalf metals 7 and topological insulators 8, 9. Thin films of half-metallic Heuslers with a \nhigh Curie temperature, such as Co 2MnSi, have been successfully used in magnetic \ntunnel junctions 10-13  and spin valves 14 . The magnetization of these films lies in-plane, \nbut tetragonally-distorted Heuslers could offer hig h perpendicular anisotropy, \nnecessary for thermally-stable sub-10 nm tunnel jun ctions and spin valves. Here we \npresent epitaxially grown tetragonal Mn 3Ga films, that exhibit a spin polarization of \nup to 58%, together with uniaxial anisotropy ( K1 = 0.89 MJ m -3) and low \nmagnetization ( Ms = 0.11 MA m -1), a combination of properties that may be ideal fo r \ntiny perpendicular spin-torque switchable elements that will allow for scalable \nmagnetic memory and logic.  \n \nThe general composition of the L21 cubic Heusler alloys is X 2YZ. In the \nperfectly-ordered state, illustrated in Fig. 1a, th e Y and Z atoms occupy two \ninterpenetrating face-centred cubic lattices, where  each is octahedrally coordinated by \nthe other, and the X atoms form a simple cubic latt ice, where they are tetrahedrally \ncoordinated by both Y and Z atoms, at the corners o f a cube. The X–Y , X–X, Y–Y \nbond lengths are √3a0/4, a0/2 and a0/√2, respectively, where a0 is the cubic lattice \nparameter. The material we discuss here, Mn 3Ga, forms two stable crystal structures. \nThe high temperature hexagonal D019  phase is a triangular antiferromagnet, easily \nobtained by arc melting 15, 16 . The tetragonal D022  phase is a ferrimagnet, usually obtained by annealing the hexagonal material at 350 -400 °C for 1-2 weeks 15, 17, 18 . The \nspin polarization at the Fermi level has been calcu lated to be 88 % for the tetragonal \nphase, which has been suggested as a potential mate rial for spin torque applications 19 . \nTo this end, thin films with appropriate magnetic p roperties were needed. \n \nThe D022  structure is a highly-distorted tetragonal variant  of the L21 Heusler \nunit cell, which has been stretched by 27% along c axis. The unit cell is outlined on \nFig. 1a by the red line; lattice parameters of the unit cell (space group I4/ mmm ), \nwhich contains two Mn 3Ga formula units, are a = 394 pm and c = 710 pm. As a result \nof the tetragonal structure, both 4 d tetrahedral X-sites and 2b octahedral Y-sites are \nsubject to strong uniaxial ligand fields, which lea d to uniaxial anisotropy at both Mn \npositions. Imperfect atomic order results in some M n population of the 2 a Z sites, \nwhich are similarly distorted.  \n \nGenerally, the Mn moment and Mn-Mn exchange in meta llic alloys are very \nsensitive to the inter-atomic distances. Widely-spa ced Mn atoms with a bond length ≥ \n290 pm tend to have a large moment, of up to 4 µB, and couple ferromagnetically 20 . \nNearest-neighbour Mn atoms have much smaller moment s, and couple \nantiferromagnetically when the bond length is the r ange 250 – 280 pm. As a result, the \nmagnetic order is often complex. αMn, for example has a large cubic cell ( a = 890 \npm) with four different manganese sites having mome nts ranging from 0.5 to 2.6 µB, \nand a complex noncollinear antiferromagnetic struct ure with TN = 95 K. The \nmagnetism of tetragonal Mn 3Ga is simpler. The Y sublattice is ferromagneticall y \ncoupled, as expected from the long Y-Y bonds (391, 450 pm), but there is a strong \nantiferromagnetic X-Y intersublattice interaction ( bond length 264 pm) which leads to ferrimagnetic order. This overcomes the X-X interac tions, which are \nantiferromagnetic in-plane (bond length 278 pm) and  ferromagnetic along the c-axis \n(bond length 355 pm). The antiferromagnetic X-X int eractions are frustrated by the \nstrong antiferromagnetic X-Y coupling, and the X su blattice remains ferromagnetic, \nwith its moment aligned antiparallel to that of the  Y sublattice. Different occupancies \nand magnetic moments in X and Y sites leads to a co mpensated ferrimagnetic \nstructure with alternating spins on atomic planes a long c axis (Fig. 1b). Imperfect \natomic ordering of Mn and Ga may, however, produce local deviations from collinear \nferrimagnetism. The Curie temperature of Mn 3Ga is high. It would be much greater \nthan 770 K were it not for the transition to the an tiferromagnetically ordered \nhexagonal D019 structure at this temperature 15, 19 .  \n \nFig. 1 (a) The cubic unit cell of the L21 Heusler alloys. The tetragonal unit cell \nof the D022  structure is indicated by the red line. (b) In Mn 3Ga, it is stretched along \nthe c-axis. The X and Y positions are occupied by Mn, an d the Z position is occupied \nby Ga. The magnetic couplings in X and Y sublattice s are antiferromagnetic and \nferromagnetic respectively, but the stronger antife rromagnetic intersublattice coupling \nbetween X and Y sites frustrates the X site moments  leading to a ferrimagnetic structure. \n \nWe have grown oriented films of Mn 3Ga on MgO (001) substrates with various buffer \nlayers by dc magnetron sputtering from a Mn 3Ga target in a chamber with 2.0 × 10 -8 \nTorr base pressure. The Mn 3Ga growth rate was approximately 1nm/min. The \nsubstrate temperature was varied from 250-375 °C. Buffer layers were 10-30 nm of \noriented Pt (001), Cr (001) or MgO (001). Pt seed l ayers were prepared by pulsed \nlaser deposition at 500 °C substrate temperature under 40 µbar oxygen partia l \npressure. The films grow with (001) orientation ( c-axis normal to plane) on all three. \nMost of our stacks were made by dc-magnetron sputte ring; Mn 3Ga films grown by \npulsed laser deposition turned out to have the hexa gonal D019  structure. X-ray \ndiffraction was carried out using a Bruker D8 Disco very diffractometer with a \nmonochromated Cu K α1 source.  Magnetization measurements were made usin g a \nQuantum Design MPMS SQUID and PPMS vibrating sample  magnetometers. \n \nStructural, magnetic and spin polarization data for  representative films are \nsummarized in Table 1, and some X-ray diffraction d ata are shown in Fig. 2a-c. \nSample 345 was a perpendicular spin valve structure  with two Mn 3Ga layers, \nseparated by a 10 nm Cr spacer. The relatively high  magnetization and low spin \npolarization of the Mn 3Ga layer grown on Cr indicate that it has a high de gree of \ndisorder. \n  \nFig. 2. a) X-ray diffraction patterns of Mn 3Ga layers grown on Pt, Cr and MgO seed \nlayers. b) Substrate temperature ( Ts) dependence of Mn 3Ga layers grown on Pt (001); \nthe inset shows the order parameter S vs. Ts, at Ts = 375 °C the sample is not single \nphase. c) φ scans showing the four-fold symmetry of the films a nd the MgO (001) \nsubstrate. d) Atomic force microscope images of fil ms grown on MgO, e) Pt, f) Cr, all \ngrown at Ts = 350 °C. \n \n \n \nTable 1. Structural, magnetic and spin polarization  data on Mn 3Ga films. Films grown \non well oriented Pt and MgO exhibit lower magnetiza tion accompanied with a small \nin-plane canted moment and higher spin polarization , whereas the films grown on Cr \nhave higher moment and reduced polarization. Sample  345a and b represent a bilayer \nMn 3Ga structure; MgO/Pt/Mn 3Ga(60)/Cr(10)/Mn 3Ga(55). \n \nSample Buffer Ts t c M // M  ⊥ m µ0Hc P  (300K)  (300K)  (300K)  (2.2K)  \n#  °C nm  pm  MA m -1 MA m -1 µB fu -1 T % \n343  MgO/Pt  250  90  707  0.022  0.11  0.65  1.80  45  \n346  MgO/Pt  250  90  712  0.017  0.10  0.59  1.90  55  \n388  MgO  350  60  707  0.015  0.14  0.81  1.80  n.a.  \n408  MgO/Cr  350  40  696  0 0.14  0.81  1.68  n.a.  \n345a  MgO/Pt  250  60 708  n.a.  0.15  0.89  1.50  58  \n345b  Cr  250  55  700  n.a.  0.2 2 1.3 0 1.02  40  \n* 1 MA m -1 corresponds to 5.93 µB fu -1 \n \n \n \nThe composition of one of the films grown on MgO wa s determined by ICP \nmass spectroscopy; the Mn:Ga atomic ratio was 3.0:1 .0. The films were metallic, and \nthe resistivity of those grown on Pt was estimated as 1.6 ×10 -6 Ω m after correcting \nfor  the resistance of the underlying Pt. Resistivi ty of a film grown directly on MgO \ncould not be measured because of the discontinuous island growth mode in this case, \nillustrated in Fig. 2d.  \n \nAn order parameter S is defined by the square root of the intensity rat io of \n(101) to (204) reflections, divided by the theoreti cal ratio. The variation of S with \nsubstrate temperature is shown in the inset of Fig.  2b. The highest value of 0.8 is \nmeasured for the samples grown on Pt (001) at 350 °C. Platinum has the smallest \nlattice mismatch with tetragonal Mn 3Ga (0.4%) and almost strain-free epitaxial \ngrowth takes place easily. The surfaces of Mn 3Ga films exhibit variations depending on the substrate type and growth temperature as sho wn in Fig. 2d-f. The pinhole-free \nareas of films grown on Pt seed layers at 350 °C have an rms surface roughness of 0.8 \nnm, whereas the samples grown on thin Cr seed layer s were pinhole free, but much \nrougher (4-5 nm rms). In addition, there are large,  randomly distributed islands ~ 50 \nnm high on the surface (Fig. 2f).  Moreover, the Mn 3Ga films grown on Cr have a \nshorter c-axis, which is probably caused by the large lattic e mismatch with Cr (4.2 \n%). The D022  unit cell has to expand in the ab - plane for epitaxial growth. The films \non Pt and MgO grow in cube on cube mode (Fig. 2c), whereas the films on Cr are \nrotated in-plane by 45 degrees to facilitate epitax ial growth. The reason for the island \ngrowth mode on MgO is thought to be the large latti ce mismatch (7.7%), as islands \ncan relax the lattice strain. The top surface of th ese films is smooth, but continuous \nfilm growth was not possible for thicknesses up to 60-70 nm. The pinholes seen on \nthe surface of Mn 3Ga grown on Pt simply reflect pinholes formed on th e surface of \nthe thin Pt (001) seed layer, which arise because o f the lattice mismatch between Pt \nand MgO. Pinhole free surfaces can be obtained on t hick Pt (001) seedlayers (> 200 \nnm) grown on MgO (001) substrates, which  show feat ureless surfaces 21 .  \n \nThe relatively high surface roughness of epitaxial Mn 3Ga may limit its use to \ngiant magnetoresistance (GMR) spin valve devices. T his may not be a disadvantage. \nAs the dimensions of the memory bits decreases belo w 50 nm the large impedance of \ntunnel junctions can limit their use in high-speed circuits, whereas the impedance of \nGMR devices based on Mn 3Ga could be engineered to 50 Ω in these dimensions, \nwhich is compatible with high-speed operation. \n \nA typical room-temperature hysteresis loop is shown  in Fig. 3a. Data are corrected for substrate diamagnetism. The substrate  also exhibits weak Curie-law \nparamagnetism at low temperatures, which is attribu ted to 50 ppm of Fe 2+  impurities \npresent in the MgO. The broad hysteresis loops show  a spontaneous magnetization of \n130 ± 20 kA m -1 and coercivity of up to 1.9 T at 300 K. Epitaxial Mn 3Ga films grown \non MgO and Pt also exhibit a small canted magnetic moment which appears in the \nmagnetization measurements with field parallel to t he plane. There is considerable \nvariation in coercivity, but similar values of magn etization are found in all samples, \ncorresponding to a moment of 0.59 – 0.89 µB fu -1. Values reported in bulk samples \ndepend on the stoichiometry of the D022  compound, increasing from about 1.0 µB fu -1 \nfor a material with a 3:1 Mn:Ga ratio to 1.4 µB fu -1 for a 2:1 ratio 16-19 . The \nmagnetization in thin films is similar, and it has been shown to depend on the degree \nof atomic ordering 22, 23 . The magnetization is consistent with an essential ly \nantiparallel arrangement of moments on the 4 d and 2 b sites, determined by neutron \ndiffraction for a sample of composition Mn 2.85 Ga 1.15  to be 1.6 ± 0.2 and -2.8 ± 0.3 µB, \nrespectively 15 . A higher moment in the 2 b site is consistent with the calculations of \nKübler et al .24 . Electronic structure calculations  predict larger  moments on both sites, \nand indicate that the compound is nearly half-metal lic with a spin polarization of 88 \n% at the Fermi level 19 .  \n \nThe strength of the uniaxial anisotropy can best be  determined from the \nperpendicular and parallel magnetization curves. Th e anisotropy field determined \nfrom the extrapolation in Fig. 3a is 16 T, which co rresponds to a uniaxial anisotropy \nconstant K1 = 0.89 MJ m -3. A somewhat larger value was reported by Wu et al . in \nMn 2.5 Ga, which is mainly due to the higher magnetization  on account of non-\nstoichiometric composition 23 . The inverse correlation of coercivity and magneti zation indicated in Table 1 reflects the variation of the anisotropy field Ha = 2 K1 /Ms. The \nsmallest volume V for which the stability condition KuV/kBT  ≥  60 is satisfied at room \ntemperature is 280 nm 3. The corresponding nanopillar with height equal to  diameter \nhas dimensions 7 nm. \n \nIn uniformly magnetized, homogeneous uniaxial magne ts, the anisotropy field \nsets an upper limit on the coercivity as pointed ou t by Stoner  and Wohlfarth 25 . In \nreality, the coercivity is always much lower, a res ult known as Brown’s paradox 26 . \nThe explanation is that material is never perfectly  homogeneous; reversal begins in a \nlocalized nucleation volume of δB3, where δB = π √(A/Ku) is the Bloch wall width, and \nA is the micromagnetic exchange constant. Assuming A ≈ 10 pJ m -1, we find δB = 10 \nnm.  The expected activation volume is of order 100 0 nm 3. We anticipate that \nnanostructured Mn 3Ga elements will exhibit stable coherent reversal a t dimensions \nwhich are less than 10 nm. It therefore promises sc aling to these sizes.  \n \nMultilayer spin valve stacks were realized by growi ng the second layer of Mn 3Ga \non a Cr spacer. The magnetization measurements reve al a two step switching curve \nfor the trilayer structure as shown in Fig. 3b. The  magnetization reversal of the film \ngrown on Cr is sharper with a coercivity of 1T, whe reas the film grown on Pt switches \nat 1.5 T. The highest measured spin polarization of  the Mn 3Ga films is 58 %, which \ncompares with the calculated value of 88% 19 . The polarization decreases with order, \nstrain as well as Mn deficiencies, which increases magnetization.  \n  \n  \nFig.  3. (a) Magnetization curves for a 90 nm thick  film of tetragonal Mn 3Ga measured at room \ntemperature with the field applied perpendicular an d parallel to the plane of the film. Anisotropy fie ld \nat 300 K was determined by extrapolating in plane d ata through origin and is 16 T. (b) Magnetization \nof Mn 3Ga/Cr/Mn 3Ga bilayer structure. (c) Point-contact Andreev ref lection spectrum of a Mn 3Ga film \ngrown on Pt.  \n \n \nThe tetragonal D022 -Mn 3-xGa (x = 0 - 1) offers a wide variety of magnetic \nproperties that can be engineered to suit a specifi c application. In the case of \nstoichiometric Mn 3Ga, presented here, the magnetization is reduced du e a higher \ndegree of compensated spins and high anisotropy wou ld allow thermally-stable sub-\n10 nm spin torque devices. For magnetic random acce ss memories, a low \nmagnetization has the advantage that it requires a lower current to switch by STT. A \nparticular advantage is that the Mn 3Ga nanopillars would be switchable by STT but \nimmune to magnetic field of even NdFeB permanent ma gnets due to the high \ncoercivity of the films, and therefore would not re quire any magnetic shielding.   \nIn conclusion, tetragonal Mn 3Ga thin films look promising for nanoscale spin-\ntransfer torque memory and logic applications. Ther e is sufficient anisotropy to ensure \nthermal stability to sub-10 nm dimensions, and the small magnetization is \nadvantageous for spin torque switching. There is a degree of flexibility in the D022  \nstructure, in terms of composition and degree of at omic order, which should enable \nthe magnetic properties to be optimized for a speci fic magnetic application.  The immediate challenge now is to observe spin torque s witching a Mn 3Ga nanopillar. \n \n \n \n \nAcknowledgements \nThis work was supported by SFI as part of the MANSE  project 2005/IN/1850, and \nwas conducted under the framework of the INSPIRE pr ogramme, funded by the Irish \nGovernment's Programme for Research in Third Level Institutions, Cycle 4, National \nDevelopment Plan 2007-2013. \n \n \n  \n \n \n1 I. Galanakis, P . Mavropoulos, and P . H. Dederichs, Journal of Physics D: Applied Physics 39 , \n765 (2006). \n2 I. Galanakis, P . H. Dederichs, and N. Papanikolaou,  Physical Review B 66 , 174429 (2002). \n3 P . J. Webster and K. R. A. Ziebeck, in Landolt-Börnstein, New Series, Group III , edited by W. H. \nR. J. (Springer, Berlin, 1988), Vol. 19c, p. 77  \n4 H. C. Kandpal and et al., Journal of Physics D: App lied Physics 40 , 1507 (2007). \n5 B. Hillebrands and C. Felser, Journal of Physics D:  Applied Physics 39  (2006). \n6 R. A. de Groot, F. M. Mueller, P . G. v. Engen, and K. H. J. Buschow, Physical Review Letters 50 , \n2024 (1983). \n7 S. Wurmehl, H. C. Kandpal, G. H. Fecher, and C. Fel ser, Journal of Physics Condensed Matter \n18 , 6171 (2006). \n8 S. Chadov, X. Qi, J. Kübler, G. H. Fecher, C. Felse r, and S. C. Zhang, Nat Mater 9, 541 (2010). \n9 H. Lin, L. A. Wray, Y . Xia, S. Xu, S. Jia, R. J. Ca va, A. Bansil, and M. Z. Hasan, Nat Mater 9, 546 \n(2010). \n10 W. Wang, H. Sukegawa, R. Shan, S. Mitani, and K. In omata, Applied Physics Letters 95 , 182502 \n(2009). \n11 S. Tsunegi, Y . Sakuraba, M. Oogane, K. Takanashi, a nd Y . Ando, Applied Physics Letters 93 , \n112506 (2008). \n12 N. Tezuka, N. Ikeda, S. Sugimoto, and K. Inomata, A pplied Physics Letters 89 , 252508 (2006). \n13 Y . Sakuraba, M. Hattori, M. Oogane, Y . Ando, H. Kat o, A. Sakuma, T. Miyazaki, and H. Kubota, \nApplied Physics Letters 88 , 192508 (2006). \n14 T. Iwase, Y . Sakuraba, S. Bosu, K. Saito, S. Mitani , and K. Takanashi, Applied Physics Express 2, \n063003 (2009). \n15 E. Krén and G. Kádár, Solid State Communications 8, 1653 (1970). \n16 H. Niida, T. Hori, and Y . Nakagawa, Journal of the Physical Society of Japan 52 , 1512 (1983). \n17 H. Niida, T. Hori, H. Onodera, Y . Yamaguchi, and Y .  Nakagawa, Journal of Applied Physics 79 , \n5946 (1996). \n18 B. Balke, G. H. Fecher, J. Winterlik, and C. Felser , Applied Physics Letters 90 , 152504 (2007). 19 J. Winterlik, B. Balke, G. H. Fecher, C. Felser, M.  C. M. Alves, F. Bernardi, and J. Morais, \nPhysical Review B 77 , 054406 (2008). \n20 N. Yamada, Journal of the Physical Society of Japan  59 , 273 (1990). \n21 G. Cui, V. Buskirk, J. Zhang, C. P . Beetz, J. Stein beck, Z. L. Wang, and J. Bentley, in Materials \nResearch Society Symposium  Proceedings 310 , 1993), p. 345. \n22 F. Wu, E. P . Sajitha, S. Mizukami, D. Watanabe, T. Miyazaki, H. Naganuma, M. Oogane, and Y . \nAndo, Applied Physics Letters 96 , 042505 (2010). \n23 F. Wu, S. Mizukami, D. Watanabe, H. Naganuma, M. Oo gane, Y . Ando, and T. Miyazaki, Applied \nPhysics Letters 94 , 122503 (2009). \n24 J. Kübler, A. R. William, and C. B. Sommers, Physic al Review B 28 , 1745 (1983). \n25 E. C. Stoner and E. P . Wohlfarth, Philosophical Tra nsactions of the Royal Society of London. \nSeries A, Mathematical and Physical Sciences 240 , 599 (1948). \n26 W. F. Brown, Reviews of Modern Physics 17 , 15 (1945). \n \n "    },    {        "title": "1209.0004v1.Magnetic_symmetry_of_the_plain_domain_walls_in_the_plates_of_cubic_ferro__and_ferrimagnets.pdf",        "content": "1 \n \n MAGNETIC SYMMETRY OF THE PLAIN DOMAIN WALLS IN THE PLATES OF \nCUBIC FERRO- AND FERRIMAGNETS \nB. M. Tanygin 1, O. V. Tychko 2 \nKyiv Taras Shevchenko National University, Radiophy sics Faculty, Volodumiurska 64,  \nKyiv, MSP 01601, Ukraine \n1E-mail: b.m.tanygin@gmail.com, 2E-mail: a.tychko@gmail.com  \nAbstract.  Magnetic symmetry of possible plane domain walls i n arbitrary oriented plates of \nthe crystal of hexoctahedral crystallographic class  is considered.  The symmetry classification \nis applied for ferro- and ferrimagnets. \nPACS: 61.50 Ah, 75.60 Ch \n1.  Introduction \nFor sequential examination of static and dynamic pr operties of domain walls (DWs) \nin magnetically ordered media it is necessary to ta ke into account their magnetic symmetry \n[1,2]. The complete symmetry classification of plan e 180°-DWs in magnetically ordered \ncrystals [1], similar classification of these DWs w ith Bloch lines in ferromagnets and ferrites \n[2] and magnetic symmetry classification of plane n on 180 0-DWs (all possible DW types \nincluding 0 0-DWs [3]) in ferro- and ferrimagnets [4] were carri ed out earlier. These DW \nsymmetry classifications allows arbitrary crystallo graphic point symmetry group of the \ncrystal. The influence of the spatially restricted sample surfaces on the DW magnetic \nsymmetry wasn’t considered in works [1-4]. The real  magnetic sample restricts a spatial (3D) \nmagnetization distribution. Therefore, it modifies the DW symmetry in general case. This \npaper presents the investigation of the influence o f the restricted sample surfaces on the \nsymmetry of the all possible (0 0-, 60 0-, 70.5 0-, 90 0-, 109.5 0-, 120 0- and 180 0-DW [4,5]) plane \n(i.e. DW with 0r>> δ, where 0r is the curvature radius of the DW [1]) DWs in an a rbitrary \noriented plate of the cubic (crystallographic point  symmetry group m3m) ferro- and \nferrimagnets. \n 2 \n \n 2.  Domain wall symmetry in the restricted sample \nThe DW symmetry can be described by the magnetic sy mmetry classes (MSCs) kG \nwhere k is a MSC number [1]. The MSC kG of DW is the magnetic symmetry group \nincluding all symmetry transformations (all transla tions are considered as unit operations) that \ndo not change the spatial distribution of magnetic moments in the crystal with DW. The \nabove-mentioned group is a subgroup of the magnetic  (Shubnikov’s) symmetry group ∞\nPG of \nthe crystal paramagnetic phase [6]. Total number of  MSCs of arbitrary type DWs (i.e. DWs \nwith arbitrary α2 angle (0 0≤α2≤180 0) between the unit time-odd axial vectors 1m and 2m \ndirected  along magnetization vectors 1M and 2M in neighboring domains) in ferro- and \nferrimagnets is equal to 64. General enumeration of  MSCs contains 42 MSCs ( 42 1≤≤k) of \n180°-DWs [1], 10 MSCs ( 13 7≤≤k and 18 16 ≤≤k) of non 0°/180°- DWs and 42 MSCs ( \nk=2, 13 6≤≤k, 19 16 ≤≤k, k=22, 24, 26, 30, 32, 37, 39 and 64 43 ≤≤k) of 0°-DWs [4]. \nThe MSCs ( k=25, 28, 37-41, 52, 54, 61-63) with six-fold symmet ry axes (including inversion \naxes) do not realized in the cubic crystals [4]. \nThe unified co-ordinate system z y x O~~~ is chosen as [][ ]W12 zyx naaeee ,, ,, ~~~ −=  where \nWn is the unit polar time-even vector along the DW pl ane normal [4]. For the 180°-DWs the \nvectors 1a and 2a are given early [1] as vectors 1τ and 2τ respectively. For the case of \n°≠180 2α the unit vector 1a coincides with the direction of the vector ()mnnm Δ−ΔWW  (at \n0≠Δb and 0=Σb) or []Wna×2 (at 0=Δb or 0≠Σb) where 12mmm−=Δ , \n[]mnΔ ×=Δ Wb , []Σ Σ×=mnWb . Here the unit vector 2a coincides with the direction of \nvector ()Σ Σ−mnnmWW  (at 0≠Σb) or []1Wan× (at 0≠Δb and 0=Σb) or else with an \narbitrary direction in the DW plane ( Wna⊥2 at 0 ==ΔΣbb ) where 2 1mmm +=Σ . The \nmutual orientation of the vectors 1m, 2m and Wn is determined by the parameters \n()Σ Σ=mnWa , ()mnΔ=Δ Wa , ()CW Camn= ,  Σb and Δb, where []21mmm ×=C . The mutual 3 \n \n orientation of the vectors 1m, 2m, Wn and Sn is determined by parameters ()Sana11=, \n()S 2 2ana=, ()SWnann= , []1S 1b an×= , []2S 2b an×=  and []WSnb nn×= , where Sn \nis sample plane normal. \nThe MSC PG of restricted sample of crystal in paramagnetic ph ase could be defined \nas S P P GGG ∩=∞ where sample shape MSC SG is 1/mmm ′ ∞  for volumetric plate. MSCs of \nDWs in volumetric plate should satisfy the conditio n PkGG⊂. The MSCs of the all \npossible plane DWs in the arbitrary oriented plate of cubic crystals of hexoctahedral class \n(crystallographic point symmetry group m 3 m  in the p aramagnetic phase [6]) are presented \nin table. Here symmetry axes are collinear with vec tors 1a and 2a and reflection planes are \nperpendicular to them. For MSCs with k =24, k =26, k =27, 29≤k≤36, 42 ≤k≤51, k =53, \n55≤k≤60 and k=64 only generative symmetry elements are represent ed.  \nGeneral enumeration of MSCs of 0°-DWs contains MSCs  with k=2, 13 6≤≤k, \n19 16 ≤≤k, k=22, 24, 26, 30, 32, 51 43 ≤≤k, k=53, 60 55 ≤≤k and k=64. The 60 0- and \n120°-DWs are represented by MSCs with k=10, 16, 18 and k=11, 13, 16 respectively. The \nMSCs of the 70.5 0-, 90 0- (both for <100> and <110> like easy magnetization  axis [5]) and \n109.5 0-DWs are the MSCs with 7< k<13, 16< k<18.  The general list of 180°-DWs includes \nMSCs with 42 1≤≤k except for k=25, 28, 37-41. \n \n \n \n \n \n \n \n 4 \n \n 3.  Conclusions \nThe complete collection of ( nml )-plates with all possible orientations includes th e full \nlist of MSCs of α2 -DWs in cubic m 3 m  crystal. For separate ( nml )-plates with fixed \ncombination of Miller indexes this list is limited.  Such limitation depends on plate orientation. \nIt is minimal and maximal for the samples with high -symmetry (such as (100)-, (110)- or \n(111)-plates) and low-symmetry (the ( nml )-plates, where indexes are non-zero and have \ndifferent absolute values) developed surface respec tively. Maximal quantity of MSCs of α2-\nDWs is for (100)-plates. The MSC with k=16 is the MSC of all above-mentioned α2-DWs \nin arbitrary oriented plate of cubic m 3 mcrystal.  \n \nReferences \n[1] V. Baryakhtar, V. Lvov, D. Yablonsky, JETP  87 , 1863 (1984) \n[2] V. Baryakhtar, E. Krotenko, D. Yablonsky, JETP  91 , 921(1986) \n[3]    R. Vakhitov, A Yumaguzin, J. Magn. Magn. Mater.  215-216, 52 (2000) \n[4]    B. M. Tanygin, O. V. Tychko, Physica B: Condensed Matter.  Article in Press. \n[5] A  Hubert and R. Shafer, Magnetic Domains. The Analysis of Magnetic \nMicrostructures , Springer, Berlin 1998 \n[6]     L. Shuvalov, Modern Crystallography IV : Phys. Prop. Cryst.,  Springer, Berlin 1988 \n \n \n \n \n \n \n \n 5 \n \n Table.  MSCs of the plane α2-DWs in plates of the cubic m3m crystal. \n \nk  \n{nml }-sample  1m, 2m, Wn and \nSn: mutual \norientation**. 1m, 2m and Wn: \nmutual orientation  Symmetry \n     elements*** MSC \nsymbol \n1 {100}, {110}  0=nb or 1b= 0  0===ΔΣΣaba  ()()1 , 12 , 2 , 2 , 112×n mmm  \n2 {100}, {110}  0=nb or 1b= 0  0===ΣΔΔaba  or \n0===ΔΣΣaba  n2 , 2 , 2 , 12 1′′  mm′2′ \n3 {100}, {110}  0=nb or 1b= 0  0===ΔΣΣaba  n2 , 2 , 2 , 112  mm 2 \n4 {nml }*  0=1a or 0=1b 0===ΔΣΣaba  1, '1,1' 2,12 /m 2′ \n5 {nml }*  0=na or 0=nb 0===ΔΣΣaba  1, '1,n' 2 ,n2 /m 2′ \n6 {nml }* 0a2=or 0b2= 0===ΔΣ Caaa \n22 , 1 m \n7 {100}, {110}  0=nb or 1b= 0  0==ΔΣaa  1, 12′,22,n2′ 22′2′ \n8 {nml }*  0=na or 0=nb 0==ΔΣaa  1, n2′ 2′ \n9 {100}, {110}  0=nb or 1b= 0  0===ΣΔbaaC  1, 12′,22′,n2  mm′2′ \n10  {nml }*  0=1aor 0=1b 0=Δa 1, 12′ 2′ \n11 {nml }* 0=na or 0=nb 0===ΣΔbaaC  1, n2 m \n12  {nml }* 0=1aor 0=1b 0=Ca 1, 12′ m′ \n13  {nml }*  0a2=or 0b2= 0=Σa 1, 22 2 \n14 {nml }* 0a2=or 0b2= 0==ΣΣba  1, 1,22′,22′ m / 2′′ \n15 Arbitrary Arbitrary 0==ΣΣba  1, '1 '1 \n16  Arbitrary  Arbitrary  Arbitrary  1 1 \n17  {100}, {110}  0=nb or 1b= 0  0===ΔΣbaaC  1, 12′,22,n2′ m′m′2 \n18  {nml }*  0=na or 0=nb 0===ΔΣbaaC  1, n2′ m′ \n19  {nml }*  0=na or 0=nb 0==ΣΔbb  1, n2 2 \n20 {nml }*  0=na or 0=nb 0 ===ΔΣΣbba  1, '1,n2,n2′ 2/m  \n21  {100}, {110}  0=nb or 1b= 0  0===ΔΣΣbba  1, 12 ,22 ,n2 222  \n22  {100}, {110}  0=nb or 1b= 0  0==ΣΔbb  1, 12′,22′,n2 m′m′2 \n23  {100}, {110}  0=nb or 1b= 0  0===ΔΣΣbba  ( )()' 1 , 12 , 2 , 2 , 121×n mmm'′′ \n24  {111} 0=nb 0==ΣΔbb  n3 3 \n26  {111} 0=nb 0==ΣΔbb  12 , 3′n m 3′ \n27 {111} 0=nb 0===ΔΣΣbba  12 , 3n 32 \n29 {111} 0=nb 0===ΔΣΣbba  12 , ' 3′n m ' 3′ \n30  {100}  0=nb 0==ΣΔbb  n4 4 \n31  {100}  0=nb 0 ===ΔΣΣbba  nn' 2 , 4  m / 4′ \n32  {100} 0=nb 0==ΣΔbb  12 , 4′n mm4′′ \n33 {100} 0=nb 0===ΔΣΣbba  12 , 4n 422 \n34 {100} 0=nb 0===ΔΣΣbba  nn' 2 , 2 , 41′ mm/m'4 ′′\n35  {100}  0=nb 0===ΔΣΣbba  n' 4  '4 \n36 {100} 0=nb 0===ΔΣΣbba  12 , ' 4n m2'4′ \n42  {111}  0=nb 0 ===ΔΣΣbba  n' 3  ' 3 6 \n \n Table.  MSCs of the plane α2-DWs in plates of the cubic m3m crystal (continue).  \n \nk  \n{nml }-sample  1m, 2m, Wn and \nSn: mutual \norientation. 1m, 2m and Wn: \nmutual orientation  Symmetry \nelements MSC \nsymbol \n43 {100}, {110}  0=nb or 1b= 0  0===ΣΔΔaba  ()()1 , 12 , 2 , 2 , 12 1×′′n mmm′′ \n44  {100}, {110}  0=nb or 1b= 0  0===ΣΔΔaba  n2 , 2 , 2 , 12 1′′  mm′2′ \n45  {nml}* 0a2=or 02=b 0===ΣΔΔaba  1, 1,22,22 2/m \n46  {nml }*  0=na or 0=nb 0===ΣΔΔaba  1, 1,n2′,n2′ m / 2′′ \n47  {nml }*  0=1aor 0=1b 0==ΔΔba  1, 1,12′,12′ m / 2′′ \n48 Arbitrary Arbitrary 0==ΔΔba  1, 1 1 \n49 {nml }* 0=na or 0=nb 0===ΣΔΔbba  1, 1,n2,n2 2/m \n50 {100}, {110}  0=nb or 1b= 0  0===ΣΔΔbba  1, 12′,22′,n2 22′2′ \n51 {100}, {110}  0=nb or 1b= 0  0===ΣΔΔbba  ()()1 , 12 , 2 , 2 , 12 1×′ ′n mmm′′ \n53  {111} 0=nb 0===ΣΔΔbba  12 , 3′n 2 3′ \n55  {111} 0=nb 0===ΣΔΔbba  12 , 3′n m 3′ \n56  {100}  0=nb 0===ΣΔΔbba  nn2 , 4 4/m  \n57  {100}  0=nb 0===ΣΔΔbba  12 , 4′n 224′′ \n58  {100}  0=nb 0===ΣΔΔbba  nn2 , 2 , 41′ mm/m 4′′ \n59  {100}  0=nb 0===ΣΔΔbba  n4 4 \n60  {100}  0=nb 0===ΣΔΔbba  12 , 4′n m24′ ′ \n64  {111}  0=nb 0===ΣΔΔbba  n3 3 \n \n*   ( nml )-plates with arbitrary Miller indexes except non z ero values n≠m≠l≠n \n** At ()()02 1 = =ananW W   \n  *** The possible symmetry elements are rotations around two-fold symmetry axes n2 , n2′ or \n12, 12′ or else 22, 22′ that are collinear with the unit vectors Wn or 1a or else 2a, \nrespectively, reflections in planes n2, n2′ or 12 , 12′ or else 22 , 22′ that are normal to the \nabove mentioned vectors, respectively, rotations ar ound three-, four-fold symmetry axes n3 , \nn4  that are collinear with the vector Wn, rotations around three-, four-fold inversion \nsymmetry axes n3 ,n3′,n4 ,n4′ that are collinear with the vector Wn, inversion in the symmetry \ncenter 1, '1 and identity 1. Here an accent at symmetry element s means a simultaneous use \nof the time reversal operation [6]. \n "    },    {        "title": "1207.6039v2.High_cooperativity_in_coupled_microwave_resonator_ferrimagnetic_insulator_hybrids.pdf",        "content": "High cooperativity in coupled microwave resonator ferrimagnetic insulator hybrids\nHans Huebl,1,\u0003Christoph W. Zollitsch,1Johannes Lotze,1Fredrik Hocke,1Moritz\nGreifenstein,1Achim Marx,1Rudolf Gross,1, 2and Sebastian T. B. Goennenwein1\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany\n2Physik-Department, Technische Universit at M unchen, Garching, Germany\n(Dated: August 30, 2013)\nWe report the observation of strong coupling between the exchange-coupled spins in gallium-\ndoped yttrium iron garnet and a superconducting coplanar microwave resonator made from Nb. The\nmeasured coupling rate of 450 MHz is proportional to the square-root of the number of exchange-\ncoupled spins and well exceeds the loss rate of 50 MHz of the spin system. This demonstrates\nthat exchange coupled systems are suitable for cavity quantum electrodynamics experiments, while\nallowing high integration densities due to their spin densities of the order of one Bohr magneton per\natom. Our results furthermore show, that experiments with multiple exchange-coupled spin systems\ninteracting via a single resonator are within reach.\nPACS numbers: 85.25.-j, 85.70.Ge, 42.50.Pq, 76.30.+g\nKeywords: ferromagnetic resonance, ferrimagnetic resonance, strong coupling, cavity quantum electrody-\nnamics, YIG, yttrium iron garnet, low temperatures, microwave spectroscopy\nThe study of the interaction of matter and light on the\nquantum level is at the core of solid state quantum infor-\nmation systems. Strong [1, 2] and ultra-strong coupling\n[3] has been achieved, allowing for the coherent transfer\nof quantum information. For the practical implementa-\ntions of quantum information systems, the use of hybrid\nsystems has been suggested. In such hybrids, natural\nmicroscopic systems (atoms, molecules, electron spins,\nand nuclear spins) are coupled with arti\fcial meso-scale\nstructures such as superconducting quantum circuits by\nmeans of microwave photons [4{6]. Whereas the former\nhave long coherence times due to su\u000ecient decoupling\nfrom environmental noise, the latter allow for fast qubit\ngates due to strong coupling to electromagnetic \felds [7].\nEnsembles of electron spins as quantum memories [8, 9]\nseem promising and their coupling to superconducting\nresonators [10{16] and \rux qubits [17] has been studied\nrecently. Although the coupling strength gof an indi-\nvidual spin to the electromagnetic mode of a supercon-\nducting microwave resonator is small (typically 10 Hz),\nthe coupling of an ensemble of Nspins is enhanced by\na factor ofp\nN[18, 19]. In this way, strong coupling\nge\u000b=gp\nN\u001d\u0014;\rcan be realized, where \u0014and\rare the\nloss rates of the resonator and spin system, respectively.\nWith loss rates in the order of MHz, typically 1012spins\nare needed to reach the strong coupling regime. Until\ntoday, mostly paramagnetic systems consisting of ensem-\nbles of noninteracting spins have been studied. The co-\nherent coupling of microwave resonators to ferromagnetic\nsystems with strongly exchange coupled spins remains\nto be explored. Soykal and Flatt\u0013 e [20, 21] theoretically\ndiscussed the strong coupling of photonic and magnetic\nmodes in exchange locked ferromagnetic systems. Two\nparticular advantages of ferromagnetic systems are (i)\ntheir higher spin density, such that for the same number\nNof spins, their volume can be reduced considerablycompared to dilute paramagnetic systems, and (ii) the\nfact that below the magnetic ordering (Curie) tempera-\nture the system essentially is fully polarized, in contrast\nto the thermal polarization in uncoupled spin ensembles.\nThis should allow to couple multiple spin ensembles to\nthe same microwave resonator, e.g. for realizing an ad-\njustable coupling between the magnetic subsystems or for\nthe exchange of individual quanta between them.\nIn this letter, we investigate the coupling between the\nelectromagnetic modes of a superconducting coplanar\nwaveguide microwave resonator and the magnetic modes\nof the exchange-locked ferrimagnet yttrium iron garnet\n(Y3Fe5O12or YIG) doped with gallium (YIG:Ga). We\nmeasure a coupling rate of ge\u000b=2\u0019= 450 MHz exceed-\ning both the spin relaxation rate \r=2\u0019= 50 MHz and\nthe resonator decay rate \u0014=2\u0019= 3 MHz. That is, we\nobserve strong coupling. The measured e\u000bective cou-\npling strength follows ge\u000b=gp\nN, where the number\nof spins interacting with the resonator is estimated from\nthe sample geometry. Furthermore, the measured re-\nlaxation rate of the spin system is fully consistent with\nthe natural linewidth of YIG:Ga obtained from ferromag-\nnetic resonance (FMR) measurements [22]. Pure YIG is\none of the prime candidates for studying strong coupling\nbetween exchange locked spins and the electromagnetic\nmodes of a microwave resonator, in particular because\nof its very small FMR linewidth of \u001910\u0016T at 4 K and\n!=2\u0019= 9:3 GHz [23]. This narrow linewidth corresponds\nto aT2time in the order of microseconds [24]. Since\nhigh quality YIG thin \flms can be prepared on various\nsubstrates (gadolinium gallium garnet [25, 26], Si and\nGaAs [27]) and doped with rare earth elements in order\nto adjust the FMR linewidth in a controlled way, YIG\nseems an ideal material for ferromagnet based quantum\nhybrids.\nAs pointed out by Soykal and Flatt\u0013 e [20, 21], inarXiv:1207.6039v2  [quant-ph]  29 Aug 20132\nA\nYIG:GaB C\nT=50 mK/uni03BC0HT=0.7 KT=4.2 K\n3dB\n1dB3dB3dB\ncable cablecable cable3dBT=293 Kvector network analyzercable cable\nFIG. 1. Schematic of the experimental setup. The (purple)\ngallium doped YIG sample is cemented on top of one of the\nniobium microwave resonators which are arranged to allow\nfor multiplexing. Experiments are performed at millikelvin\ntemperatures in transmission by vector network analysis in a\nsuperconducting solenoid magnet.\na macrospin approximation the Hamiltonian for the\nferromagnet-resonator system can be expressed as\nH= \u0016h!raya+gs\u0016BBe\u000b\nzSz+ \u0016hg(aS++ayS\u0000):(1)\nHere,ayandaare the photon creation/annihilation oper-\nators,!rthe resonator frequency, gsthe electron g-factor,\n\u0016Bthe Bohr magneton, and Be\u000b\nzthe magnetic \feld [28].\nThe macrospin operator S= (S+^e\u0000\u0000S\u0000^e+)=p\n2 +Sz^z\nwith ^e\u0006=\u0007(^x\u0006{^y)=p\n2 is expressed in terms of the spin\nlowering and raising operators\nS\u0006\f\f\f\fN\n2;m\u001d\n=s\u0012N\n2\u0007m\u0013\u0012N\n2\u0006m+ 1\u0013\f\f\f\fN\n2;m\u00061\u001d\n:\n(2)\nHere,j`;miare the eigenstates of the macrospin and we\nhave assumed that the macrospin state is \fxed at its\nmaximal value `=N=2. In the Dicke model [29] of Nin-\ndependent paramagnetic spins this would correspond to\na fully excited spin system with no photons in the cavity.\nIn contrast to the Dicke model, for our macrospin model\nstates with ` < N= 2 are not accessible. Due to strong\nexchange coupling, states with ` < N= 2 are separated\nin energy and require the excitation of magnons. We\nnote that the coupling between the photonic and mag-\nnetic system is a magnetic dipole transition and that the\nHamiltonian (1) conserves the total excitation number\nZ=n+m, wherenis the photon number in the cavity\nandjmj\u0014`=N=2 the magnetic quantum number. As-\nsuming that Sis antiparallel to Bzandn= 0 initially,\nwe haveZ=N=2 and, hence, can index the basis states\njn;miof the resonator-spin systems either by the photon\nnumbern(jn;N\n2\u0000ni) or the magnetic quantum number\nm(jN\n2\u0000m;mi). Evidently, these basis states are similar\nto those of the Dicke model [29] for a paramagnetic en-semble ofNnoninteracting spins coupled to a resonator,\nwith`=N=2 taking the role of the cooperation number.\nDue to the analogy with the Dicke model, we expect\nthat the coupling strength of the ferromagnet-resonator\nsystem is given by the e\u000bective coupling strength ge\u000b=\ngp\nNof a paramagnet-resonator system, where g=\ngs\u0016B\n2\u0016hB1;0is the coupling rate of an individual spin with\nthe magnetic quantum number m= 1=2 to the resonator\n[29]. It is determined by the magnetic component of the\nrf vacuum \feld B1;0=p\n\u00160\u0016h!r=2Vmwhich depends on\nthe resonance frequency of the microwave resonator !r\nand its mode volume Vm[30]. Hence, for a given res-\nonance frequency, ge\u000b=gs\u0016B\n2\u0016hp\n\u00160\u001a\u0016h!rV=2Vmdepends\nonly on the spin density \u001a=N=V and the \flling factor\nV=V m, whereVdenotes the volume of the resonator \feld\nmode \flled with the spin system. Non-interacting spin\nensembles like paramagnetic centers in semiconductors or\ninsulators typically have a spin density \u001ain the order of\n1015\u0014\u001a\u00141018cm\u00003[10, 11, 13]. For !r=(2\u0019)\u00195 GHz,\nthis results in coupling rates in the order of 10 MHz as-\nsumingV=V m'1. In contrast, exchange coupled sys-\ntems naturally have a spin density in the order of one\nper atom (e.g. Fe, Ni, Co) or in the case of YIG 40 per\nunit cell (unit cell volume - 1 :8956 nm3), corresponding to\na spin density of 2 \u00021022cm\u00003[31]. Due to the increase\nof at least four orders of magnitude in the spin density\nwe expect a two orders larger coupling strength in ex-\nchange coupled systems as compared to non-interacting\nspins. Therefore, the exchange coupled system sample\nvolume can be reduced by a factor of 104while keeping\nthe coupling rate constant, enabling a higher integration\ndensity.\nIn our experiments, we study the coupling between the\nexchange locked system YIG:Ga and a superconducting\nNb resonator. The resonator structure is patterned into\na 100 nm thick Nb \flm deposited onto an intrinsic sili-\ncon substrate using optical lithography and reactive ion\netching [30]. Figure 1 shows the layout of the microwave\ncircuitry consisting of an input line, three resonators with\nresonance frequencies of fA= 5:90 GHz,fB= 5:53 GHz,\nandfC= 5:30 GHz, and an output line. This con\fgura-\ntion allows to compare loaded and unloaded microwave\nresonators on the same chip. A 2 \u00020:5\u00020:7mm3sized\n(length\u0002width\u0002thickness) commercial YIG:Ga crystal\nis cemented onto resonator Awith the highest microwave\nfrequencyfA= 5:90 GHz. The number of spins interact-\ning with the resonator is roughly estimated from the over-\nlap of the YIG:Ga crystal with the meandering coplanar\nwaveguide with a center conductor width of 6 \u0016m and a\ngap of 12 \u0016m. With the overlap length of 2 :5 mm and\nassuming that the vertical extension of the microwave\n\feld into the YIG crystal is about 30 \u0016m, the total num-\nber of spins coupled to the resonator is estimated to\nN\u00194:5\u00021016. To preserve the superconducting state\nof the microwave resonators, the surface of the chip is\ncarefully aligned in parallel to the applied magnetic \feld3\nA\nB\nC\nDFrequency (GHz)\nMagnetic Field B     (mT)\n|S21|2 (10-5)1.6\n1.4\n1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n06.5\n4.04.55.05.56.0\n0 200 100 300 0 200 100 300a b\next\nz\nFIG. 2. Transmission spectrum of the setup including the\nYIG-microwave resonator hybrid as a function of the applied\nmagnetic \feld Bext\nz, taken at T= 50 mK [33]. The copla-\nnar waveguide resonators ( A-C) show a slightly decreasing\nresonance frequency with increasing in-plane magnetic \feld.\nAdditionally, resonator Ashows a pronounced avoided cross-\ning at 170 mT, where the resonator frequency fAmatches the\nFMR frequency !FMR=(2\u0019). Resonance Dis a parasitic mode\npresent in the sample box. Panel a) shows the raw (uncali-\nbrated) transmission data as measured. Panel b) shows the\nsame data again superimposed with a \ft according to eq.(3)\nplotted as red line.\nBzgenerated by a superconducting solenoid. The mi-\ncrowave transmission experiments are performed at the\nbase temperature of a dilution refrigerator of 50 mK us-\ning a commercial vector network analyzer. To thermally\nanchor the center conductor of the microwave input and\noutput lines, attenuators are used at the 4 K, the still and\nthe mixing chamber stages (cf. Fig. 1). Considering only\nthe attenuators, we estimate a microwave \feld tempera-\nture of about 70 K (or 290 thermally excited photons on\naverage) in the resonator. [32]\nFigure 2 shows the microwave transmission jS21j2raw\ndata as a function of frequency and applied magnetic\n\feld [33]. In the spectrum at Bz= 0, four transmis-\nsion peaks are visible corresponding to the resonance fre-\nquencies of the coplanar microwave resonators A,B, and\nC. The broad feature labeled Dstems from a parasitic\nmode of the metallic microwave box in which the sam-\nple is mounted. As expected, the resonators BandC\nshow only a weak magnetic \feld dependence, because\nthey are not interacting with the YIG:Ga crystal due to\nthe absence of physical overlap (cf. Figs. 1 and 2). On\nthe contrary, resonator Aand the box mode Dcouple\nto the YIG:Ga. While mode Dallows us to probe the\nferromagnetic resonance independently of the strongly\ncoupled mode ( A) and to determine the FMR disper-\nsion relation, resonator Ashows a distinct anticrossing\natBext\nz(\u0001 = 0) = BFMR = 170 mT where the FMR\ndispersion relation \u0016 h!FMR =gs\u0016BBe\u000b\nz[34] is degeneratewith the resonator \u0016 h!r.\nTo derive the e\u000bective coupling rate ge\u000bfrom the mea-\nsured data, we simplify the discussion of (1), by modeling\nthe system as two coupled harmonic oscillators. The dis-\npersion of the resonance frequency is then given by [35]\n!=!r+\u0001\n2\u00061\n2q\n\u00012+ 4g2\ne\u000b: (3)\nHere, \u0001 = !FMR\u0000!r=gs\u0016B(Bext\nz\u0000BFMR)=\u0016his\nthe \feld dependent detuning between the resonator fre-\nquency!r=(2\u0019) =fAand the \feld dependent FMR fre-\nquency!FMR=(2\u0019). The experimental data agree very\nwell with this model prediction. Fitting the data yields\nge\u000b= 450\u000620 MHz and BFMR = 170\u00065 mT, and the\ng-factor of the ferrimagnetic resonance gs= 2:17\u00060:05\n(red line in Fig.2). Here, BFMR is reduced with respect\nto the bare electron spin resonance \feld of 194 mT due to\nthe presence of an anisotropy \feld Ba= 24 mT [28],[34].\nAdditionally, the experimentally observed gsis not ex-\nactly identical to the literature value for pure YIG at 2 K\nofgs;lit= 1:99 [36] or YIG:Ga at 5 K of gs;lit= 2:1[22].\nThe observed di\u000berence might be due to the higher Ga\nconcentration compared to Ref. [22] or the lower tempera-\nture. Note, that the g-factor and the magnetic anisotropy\nof YIG and doped YIG is not well established and re-\nquires further investigations. For our resonator we esti-\nmateg=(2\u0019)'5 Hz [30] and N'4\u00021016. With these\nnumbers we expect ge\u000b=(2\u0019) = (g=2\u0019)p\nN'1 GHz cor-\nroborating our experimental result within a factor of two\ndespite of the rough estimate for N.\nTo determine the relaxation rate \rof the spin system\nwe analyze the evolution of the linewidth of the resonator\nmodeAas a function of the magnetic \feld Bzby \ftting\na Lorentzian lineshape in the frequency domain for ev-\nery measured magnetic \feld magnitude [16]. Figure 3(a)\nshows the resonance frequency obtained from such a \ft\nas red crosses superimposed on the color-coded dataset.\nIn Fig. 3(b) the corresponding (FWHM) linewidth (red\ncrosses) is shown. At low magnetic \felds, the resonator\nmodeAis essentially decoupled from the spin system,\nsuch that the measured linewidth is given by \u0014. Closer to\nthe ferromagnetic resonance, the linewidth of the system\nis given by the combined relaxation rate of the spin sys-\ntem and the microwave resonator leading to an increase\nin the observed linewidth.\nTo quantify the coupling and loss rates in our system,\nwe use a standard input-output formalism [10, 16, 19,\n37]. Within this framework the transmission amplitude\nof microwaves from the input to the output port of the\nmicrowave resonator is given by\nS21=\u0014c\n{(!\u0000!r)\u0000(\u0014c+\u0014i) +jgeffj2\n{(!\u0000!FMR)\u0000\r=2:(4)\nHere,!=2\u0019is the frequency of the microwave probe tone,\n\u0014cis the external coupling rate between the microwave4\nMagnetic Field B     (mT)ext\nz250 200 150 100 50 298Frequency\n(GHz)Frequency\n(GHz)\n|S21|2 (10-6)2.5\n2.0\n1.5\n1.0\n0.5\n0Peak Width\n(MHz)6.2\n5.86.06.2  20  40  60  805.86.0a\ncb\nDataSimulation\nFIG. 3. Analysis of the resonance frequency and linewidth as\na function of the external magnetic \feld Bext\nz. Panel (a) shows\nthe resonator transmission jS21j2(same data as Fig. 2) as a\nfunction of frequency and applied magnetic \feld [33]. The red\ncrosses mark the resonance frequency determined by \ftting a\nLorentzian lineshape to the data at constant Bext\nz. The red\ncrosses in panel (b) show the extracted FWHM linewidths\ncorresponding to 2 \r=(2\u0019) and 2\u0014=(2\u0019). In addition, the blue\ncircles show the linewidths obtained from the numerical sim-\nulation of the transmission spectra plotted in panel (c). The\nsimulation is based on the input-output formalism resulting\nfrom eq.(4) [10, 16, 19, 37].\nresonator and the feed line, and \u0014isummarizes the in-\ntrinsic loss rate of the microwave resonator. In our case\nwe have\u0014i\u001d\u0014c, resulting in a total microwave res-\nonator relaxation rate \u0014'\u0014i(cf. [34]). Fig. 3(c) shows\nthe calculated transmission using ge\u000b=2\u0019= 450 MHz,\n\r=2\u0019= 50 MHz, and \u0014=2\u0019= 3 MHz. Evidently, all\nfeatures of the experimental data of Fig. 3(a) are nicely\nreproduced. Moreover, the two transmission peaks ex-\npected atBFMR = 170 mT cannot be resolved due to the\nlimited signal to noise ratio in the experimental data.\nAdditonally, we can analyze the simulation data shown\nin Fig. 3(c) in the same way as the experimental data\nin Fig. 3(a). The result is shown by the blue circles in\nFig. 3(b), where in contrast to the experimental data,\nthe modeled data is noise-free allowing to predict the\nlinewidth for all magnetic \feld values. The good agree-\nment between experimental and simulation data again\ndemonstrates that the parameters chosen in the simula-\ntion well reproduce the experimental situation. In sum-\nmary, our analysis shows that ge\u000b\u001d\u0014;\rwith a cooper-\nativityC=g2\ne\u000b=\u0014\r'1350. That is, the strong coupling\nregime has been reached for the ferrimagnet-resonator\nsystem [38].\nNext, we compare the experimentally determined re-\nlaxation rate \rwith the temperature dependence of\nthe FMR linewidth measured at 9 :43 GHz. Typically,\nYIG:Ga exhibits signi\fcantly larger damping (larger\nlinewidth) than pure YIG. Rachford et al. [22] report\nlinewidths for YIG:Ga that decrease from 1 mT at 4 :2 Kto 0:1 mT at room temperature, corresponding to 28 MHz\nand 2:8 MHz, respectively. Our sample has a higher\nGa-doping concentration [39], and thus larger linewidth,\nwhich coroborates the relaxation rate \r=2\u0019= 50 MHz\nwe measured at millikelvin temperatures. However, note\nthat little is known about the damping in YIG for T\u0014\n2 K. According to Sparks and Kittel [40] the limiting re-\nlaxation mechanism is spin-lattice coupling, expected to\nbe well below 1 MHz for YIG. This calls for further ex-\nperiments in this temperature regime to verify the pro-\nposed relaxation mechanism and to eludicate the maxi-\nmum spin coherence time achievable in YIG for T\u00142 K.\nNote also, that relaxation mechanisms in ferromagnets\nare fundamentally di\u000berent from relaxation mechanisms\nin diluted paramagnetic systems. In the latter, spin-spin\ninteractions can cause dephasing and decoherence [41].\nIn the former, it is possible to simultaneously have high\nspin density and low damping.\nFinally, we \fnd that ge\u000bis independent of the mi-\ncrowave power from 10 fW to 10 nW. This is expected,\nbecause here the number of excitations (photons in the\nmicrowave resonator) is much smaller than the number\nof spinsN\u00191016[14]. Only if the number of excita-\ntions (photons) becomes comparable to or exceeds N, a\nquenching of the observed anticrossing is expected.\nIn conclusion, we experimentally observed strong cou-\npling between a superconducting microwave resonator\nand a gallium doped yttrium iron garnet ferrimagnet.\nThe e\u000bective coupling rate of 450 MHz reaches 8% of the\nresonator frequency !r=(2\u0019) and by far exceeds the re-\nlaxation rate \r=(2\u0019) = 50 MHz of the spin system at\nabout 50 mK, which is rather large owing to the gallium\ndoping. Considering the much smaller linewidth in pure\nYIG, even higher cooperativities can be anticipated. Our\nresults establish that exchange coupled spin systems in-\ndeed can be used for cavity quantum electrodynamics.\nFurthermore, the large coupling rates achievable in ex-\nchange coupled systems allow to place more than one\nmagnetic system in the microwave resonator. This should\nallow to study e.g. the exchange of magnetic excitations\nvia a cavity bus similar to the approaches pursued in the\n\feld of cavity QED [7].\nWe acknowledge technical support by M. Opel. This\nwork is supported by the German Research Foundation\nthrough SFB 631 and the German Excellence Initiative\nvia the \\Nanosystems Initiative Munich\" (NIM).\n\u0003corresponding author huebl@wmi.badw.de\n[1] A. Wallra\u000b, D. I. Schuster, A. Blais, L. Frunzio, R.-S.\nHuang, J. Majer, S. Kumar, S. M. Girvin, and R. J.\nSchoelkopf, Nature 431, 162 (2004).\n[2] R. J. Schoelkopf and S. M. Girvin, Nature 451, 664\n(2008).\n[3] T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel,5\nF. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco,\nT. H ummer, E. Solano, A. Marx, and R. Gross, Nat.\nPhys. 6, 772 (2010).\n[4] A. Andr\u0013 e, D. DeMille, J. M. Doyle, M. D. Lukin, S. E.\nMaxwell, P. Rabl, R. J. Schoelkopf, and P. Zoller, Nat.\nPhys. 2, 636 (2006).\n[5] J. Verd\u0013 u, H. Zoubi, C. 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Lett. 71, 2617 (1997).\n[28] The magnetic induction Be\u000b\nz=Bext\nz+Bahas two contri-\nbutions: i) the externally applied magnetic \feld Bext\nzand\nii) internal \felds accounting for the magnetic anisotropy\nBa. (cf. [34]).\n[29] R. H. Dicke, Phys. Rev. 93, 99 (1954).\n[30] T. Niemczyk, F. Deppe, M. Mariantoni, E. P. Menzel,\nE. Ho\u000bmann, G. Wild, L. Eggenstein, A. Marx, and\nR. Gross, Supercond. Sci. Tech. 22, 034009 (2009).\n[31] M. Gilleo and S. Geller, Phys. Rev. 110, 73 (1958).\n[32] This accounts for the \fxed attenuators and does not in-\nclude the lossy lines. For further details refer to [34].\n[33] Since we have not calibrated our setup at millikelvin tem-\nperatures, we show raw, uncalibrated, as-measured S21\ntransmission data in Figs. 2 and 3. In our opinion, this is\nthe most honest way of presenting the data in lack of a\nproper full calibration. .\n[34] \\Sublemental material,\".\n[35] S. Haroche and J. M. Raimond,\nExploring the Quantum: Atoms, Cavities and Photons\n(Oxford University Press, Oxford, 2006).\n[36] K. P. Belov, L. A. Malevskaya, and V. I. Sokolov, Soviet\nPhysics JETP 12, 1074 (1960).\n[37] A. A. Clerk, S. M. Girvin, F. Marquardt, and R. J.\nSchoelkopf, Rev. Mod. Phys. 82, 1155 (2010).\n[38] Our claims are based on the analysis of the lineshape\nand resonance frequency dependence as function of the\napplied magnetic \feld and do not rely on the absolute\ntransmission intensity. We therefore have not calibrated\nthe setup with respect to the amplitude information.\n[39]YIG:Ga Datasheet - www.ferrisphere.com.\n[40] M. Sparks and C. Kittel, Phys. Rev. Lett. 4, 232 (1960).\n[41] A. M. Tyryshkin, S. Tojo, J. J. L. Morton, H. Riemann,\nN. V. Abrosimov, P. Becker, H.-J. Pohl, T. Schenkel,\nM. L. W. Thewalt, K. M. Itoh, and S. A. Lyon, Nature\nMaterials 11, 143 (2011)."    },    {        "title": "1804.08719v1.Unidirectional_Loop_Metamaterials__ULM__as_Magnetless_Artificial_Ferrimagnetic_Materials__Principles_and_Applications.pdf",        "content": "1\nUnidirectional Loop Metamaterials (ULM)\nas Magnetless Artificial Ferrimagnetic Materials:\nPrinciples and Applications\nToshiro Kodera, Senior Member, IEEE, and Christophe Caloz, Fellow, IEEE\nAbstract —This paper presents an overview of Unidirectional\nLoop Metamaterial (ULM) structures and applications. Mimick-\ning electron spin precession in ferrites using loops with unidi-\nrectional loads (typically transistors), the ULM exhibits all the\nfundamental properties of ferrite materials, and represents the\nonly existing magnetless ferrimagnetic medium . We present here\nan extended explanation of ULM physics and unified description\nof its component and system applications.\nIndex Terms —Unidirectional Loop Metamaterials (ULM),\nnonreciprocity, ferrimagnetic materials and ferrites, gyrotropy,\nFaraday rotation, metamaterials and metasurfaces, transistors,\nisolators, circulators, leaky-wave antennas.\nI. I NTRODUCTION\nOver the past decades, nonreciprocal components (isola-\ntors, circulators, nonreciprocal phase shifters, etc.) have been\nhave been almost exclusively implemented in ferrite tech-\nnology [1]–[8]. This has been the case in both microwaves\nand optics, despite distinct underlying physics, namely the\npurely magnetic effect (electron spin precession) in the former\ncase [3], [9], [10] and the magneto-optic effect (electron\ncyclotron orbiting) [11]–[13] in the latter case. However,\nferrite components suffer from the well-known issues high-\ncost, high-weight and incompatibility with integrated circuit\ntechnology, and magnetless nonreciprocity has therefore long\nbeen consider a holy grail in this area [14], [15].\nThere have been several attempts to develop magnetless\nnonreciprocal components, specifically 1) active circuits [16]–\n[21], and space-time [15] 2) modulated structures [22]–[27]\nand 3) switched structures [28] (both based on 1950ies para-\nmetric (e.g. [29], [30]) or commutated (e.g. [31]) microwave\nsystems). All have their specific features, as indicated in Tab. I.\nTABLE I\nCOMPARISON (TYPICAL AND RELATIVE TERMS )BETWEEN DIFFERENT\nMAGNETLESS NONRECIPROCITY TECHNOLOGIES PLUS FERRITE .\nmaterialPconsum. bias cost noise\nferrite yes zero magnet high N/A\nact. circ. no low DC low low\nswitched no med. RF high high\nmodulated no med. RF med. med.\nULM YES low DC low med\nWe introduced in 2011 [32] in a Unidirectional Loop\nMetamaterial (ULM) mimicking ferrites at microwaves and\nrepresenting the only artificial ferrite material, or metamaterial,\nexisting to date. This paper presents an overview of the ULM\nand its applications reported to date.\nT. Kodera is with the Department of Electrical Engineering, Meisei Univer-\nsity, Tokyo Japan (e-mail: toshiro.kodera@meisei-u.ac.jp). C. Caloz is with the\nDepartment of Electrical and Engineering, ´Ecole Polytechnique de Montr ´eal,\nMontr ´eal, QC, H2T 1J3 Canada.II. O PERATION PRINCIPLE\nA Unidirectional Loop Metamaterial (ULM) may be seen\nas a physicomimetic1artificial implementation of a ferrite in\nthe microwave regime . Its operation principle is thus based\nonmicroscopic unidirectionality , from which the macroscopic\ndescription is inferred upon averaging.\nA. Microscopic Description\nMicrowave magnetism in a ferrite is based on the precession\nof the magnetic dipole moments arising from unpaired electron\nspins about the axis of an externally applied static magnetic\nbias field, B0, as illustrated in Fig. 1(a), where B0k^ z. This is\na quantum-mechanical phenomenon, that is described by the\nLandau-Lifshitz-Gilbert equation [3], [10], [33]\ndm\ndt=\u0000\rm\u0002B0+\u000b\nMsm\u0002dm\ndt; (1)\nwhere mdenotes the magnetic dipole moment, \rthe gyromag-\nnetic ratio,Msthe saturation magnetization, and \u000bthe Gilbert\ndamping term. Equation (1) states that the time-variation rate\nofmdue to a transverse2RF magnetic field signal, HRF\nt\n(k^t;^t?^ z), is equal to the sum of the torque exerted by B0\nonm(directed along +^\u001e,^\u001e: azimuth angle), and a damping\nterm (directed along\u0000^\u0012,^\u0012: elevation angle) that reduces\nthe precession angle,  , to zero along a circular-spherical\ntrajectory (conserved jmj) when the RF signal is suppressed\n(relaxation).\nClassically, magnetic dipole moments can be associated\nwith current loop sources , according to Amp `ere law. Decom-\nposing a ferrite magnetic moment, m, into its longitudinal\ncomponent, mz, and transverse component, m\u001a, as shown in\nFig. 1(a), one may thus invoke the effective current loops Imz\neff\nandim\u001a\neffassource models for the corresponding moments.\nAmong these currents, only im\u001a\neffmatters in terms of mag-\nnetism, since Imz\neff, as the source associated with HRF\nz, does\nnot induce any precession (Footnote 2). im\u001a\neffis thus the current\none has to mimic to devise an “artificial ferrite.” This current,\nas seen Fig. 1, has the form of a loop tangentially rotating on\nan imaginary cylinder of axis z.\n1The adjective “physicomimetic” is meant here, from etymology, as “mim-\nicking physics.”\n2The longitudinal ( z) component does not contribute to precession, and\nhence to magnetism. Indeed, since B0k^ z, thez-component of mproduced by\nHRF\nzwould lead to mRF\nz\u0002(B0+\u00160HRF^ z) = [mRF\nz(B0+\u00160HRF)](^ z\u0002^ z) =\n0, the only torque being produced by the transverse component ( HRF\nt,^t2xy-\nplane), mRF\nt\u0002(B0+\u00160HRF^t) = (mRF\ntB0)(^t\u0002^ z)6= 0. In the rest of the\ntext, we shall drop the superscript “RF,” without risk of ambiguity since mt,\nis exclusively produced by the RF signal.arXiv:1804.08719v1  [physics.app-ph]  16 Apr 20182\nFig. 1. “Physicomimetic” construction of the Unidirectional Loop Metama-\nterial (ULM) “meta-molecule” or particle. (a) Magnetic dipole precession,\narising from electron spinning in a ferrite material about the axis (here z) of an\nexternally applied static magnetic bias field, B0, with effective unidirectional\ncurrent loops Imz\neffandim\u001a\neff, and transverse radial rotating magnetic dipole\nmoment m\u001aassociated with im\u001a\neff. (b) ULM particle [32], typically (but not\nexclusively [34]) consisting of a pair of broadside-coupled transistor-loaded\nrings supporting antisymmetric current and unidirectional current wave (shown\nhere with exaggeratedly small wavelength for the sake of visibility), with\nresulting radial rotating magnetic dipole moment emulating that in (a).\nGiven its complexity, the current im\u001a\neffmay a priori seem\nimpossible to emulate. However, what fundamentally matters\nfor magnetism is not this current itself, but the moment m\u001a,\nfrom which magnetization will arise at the macroscopic level\n(Sec. II-B). This moment may be fortunately also produced\nby a pair of antisymmetric \u001e-oriented currents, rotating on the\nsame cylinder, which can be produced by a pair of conducting\nrings operating in their odd mode [35], as shown in Fig. 1(b).\nIf this ring-pair structure is loaded by a transistor [32], as\ndepicted in the figure, or includes another unidirectionality\nmechanism such as the injection of an azimuthal modula-\ntion [34], m\u001awillunidirectionally rotate about zwhen excited\nby an RF signal, and hence mimic the magnetic behavior of the\nelectron in Fig. 1(b). The structure in Fig. 1 constitutes thus\ntheunit-cell particle of the ULM at the microscopic level .\nOne may argue there there is a fundamental difference\nbetween the physical system in Fig. 1(a) and its presumed\nartificial emulation in Fig. 1(b): the ferrite material also\nsupports the longitudinal moment mzwhereas the ULM does\nnot include anything alike. However, as we have just seen\nabove, particularly in Footnote 2, mzdoes not contribute to\nthe magnetic response. It is therefore inessential and does thus\nnot need to be emulated. So, the particle in Fig. 1(b), with its\nmoment m\u001ais all that is needed for artificial magnetism !\nDoes this mean that the ULM particle includes no counter-\npart to the static alignment of the dipoles due to B0(and\nproducing mz) in the ferrite medium? In fact, there isa\ncounterpart, although mz= 0 in the ULM. The fundamental\noutcome of the static alignment of the dipoles along zin\nthe ferrite is the alignment of the relevant magnetic dipoles\nm\u001ain the plane perpendicular to ^ z(or within the xy-plane)\nacross the medium, for otherwise the m\u001a’s of the different\ndomains [3], [10] would macroscopically cancel out. Such an\norientation of the m\u001a’s perpendicularly to ^ zis essential to\nemulate magnetism. How is this provided in the ULM? Simply\nbyfixing the rings in a mechanical support , such as a substrate,\nas will be seen later. So, the counterpart of the ferrite static\nalignment of dipoles is simply mechanical orientation in the\nULM.B. Macroscopic Description\nSince it mimics the relevant magnetic operation of a ferrite\nat the microscopic level, the unit-cell particle in Fig. 1(b) must\nlead to the same response as bulk ferrite at the macroscopic\nlevel when repeated according to a subwavelength 3D lattice\nstructure so as to form a metamaterial as shown in Fig. 2(a).\nThe ULM in Fig. 2(a), just as a ferrite3, forms a 3D\narray of magnetic dipole moments, mi, whose average over a\nsubwavelength volume V,\nM=1\nVX\ni=1mi= \n1\nVX\ni=1m\u001a;i!\n^\u001a=M\u001a^\u001a (2)\ncorresponds to the density of magnetic dipole moments, or\nmagnetization , as the fundamental macroscopic quantity de-\nscribing the metamaterial4.\n(a) (b)\nFig. 2. ULM structures obtained by periodically repeating the unit cell with\nthe particle in Fig. 1. (a) Metamaterial (3D), described by the Polder volume\npermeability (3a). (b) Metasurface (2D metamaterial), described by a surface\npermeability [36].\nFrom this point, one may follow the same procedure as in\nferrites [8], [37] to obtain the Polder ULM permeability tensor\n\u0016=2\n4\u0016 j\u0014 0\n\u0000j\u0014 \u0016 0\n0 0\u00160;3\n5; (3a)\nwith\u0016=\u00160\u0012\n1 +!0!m\n!2\n0\u0000!2\u0013\nand\u0014=\u00160!!m\n!2\n0\u0000!2;(3b)\nwhere!0and!mare the ULM resonance frequency (or\nLarmor frequency) and effective saturation magnetization fre-\nquency , respectively5, that will be derived in the next section.\nAs in ferrites, the effect of loss can be accounted for by\nthe substitution !0 !0+j\u000b!, where\u000ba damping factor\nin (1) [8].\nSo, a ULM may really be seen as a an artificial ferrite\nmaterial producing magnet-less artificial magnetism . However,\nitsnonreciprocity is achieved from breaking time-reversal\n(TR) symmetry by a TR-odd current bias , originating in the\ntransistor (DC) biasing, instead of a TR-odd external magnetic\nfield [14].\nULMs have been implemented only in a 2D format so far.\nThe corresponding structure is shown in Fig. 2(b), and may\n3The difference is essentially quantitative : while in the ferrite p=\u0015< 10\u00006\n(p: molecular lattice constant), in the ULM p=\u0015\u00191=10\u00001=5(p:\nmetamaterial lattice constant or period), but homogeneization works in both\ncases.\n4Whereas in a ferrite, we have M=Ms+MRF= (Ms+MRF\nz)^ z+MRF\nt\u0019\nMs^ z+MRF\n\u001a^\u001a, whereMsis the saturation magnetization of material, in the\nULMMs= 0. We shall subsequently drop the superscript “RF” also in M\u001a.\n5In a ferrite,!0=\rB0and!m=\r\u00160Ms.3\nbe referred to as a Unilateral Loop Metasurface (ULMS) .\nSection IV will present ULMS Faraday rotation and Sec. V-A\nwill discuss related applications.\nIII. ULM P ARTICLE AND DESIGN\nULMs may be implemented in different manners. Figure 3\nshows a ULM particle implemented in the form of a microstrip\ntransistor-loaded single ring placed on PEC plane. Assuming\na distance much smaller than the wavelength between the\nring and the PEC plane, the structure is equivalent, by the\nimage principle, to the antisymmetric double-ring structure in\nFig. 1(b) [35].\nFig. 3. ULM particle microstrip implementation in the form of a transistor-\nloaded single ring on a PEC plane, supporting the odd effective current\ndistribution and unidirectional current wave as in Fig. 1(b). The transistor\nbiasing circuit is not shown here.\nThe transistor-loaded ULM particle in Fig. 3, or Fig. 1(b),\nis essentially a ring resonator , whose total electrical size is\ngiven by [38]\n\fmsa(2\u0019\u0000\u000bTR) +'TR= 2\u0019; \f ms=k0p\u000fe=!\ncp\u000fe;(4)\nwhere\fmsis the microstrip line wavenumber ( \u000fe: effective\nrelative permittivity), ais the average radius of the ring, \u000bTR\nis the geometrical angle subtending the transistor chip, and\n'TRis the phase shift across it. Solving Eq. (4) for !provides\nthe resonance frequency of the resonator, and hence the ULM\nresonance frequency ,\n!0=\f\f\f\f(2\u0019\u0000'TR)c\nap\u000fe(2\u0019\u0000\u000bTR)\f\f\f\f; (5)\nin (3). The parameter !min the same relations follows from\nthe mechanical orientation of the moments, as explained in\nSec. II-B: although we do not have here a saturation magne-\ntizationMsleading to the frequency parameter !m=\r\u00160Mm\nin the ferrite, we have have an equivalent phenomenological\nparameter!massociated with the orientation of the rings,\nwhich may be found by extraction, as will be seen in Sec. IV.\nNote that ULMs may be designed for multi-band operation\nand enhanced-bandwidth operation. The former, in contrast\nto ferrites that are restricted to a single ferromagnetic reso-\nnance!0=\rB06, can in principle accommodate multiple\nresonances by simply incorporating rings of different sizes,\nas illustrated in Fig. 4. The latter, in contrast to ferrite whose\nbandwidth is inversely proportional to loss due to causality, can\nbe achieved by leveraging overlapping coupled resonances [2].\n6This restriction can be somewhat overcome in a structured ferromagnetic\nstructure, such as a ferromagnetic nanowire membrane supporting a remanent\nbistable population of up and down magnetic dipole moments with corre-\nsponding resonances !\"\n0=\r\u00160H\"\n0and!#\n0=\r\u00160H#\n0[39], [40].\n(a) (b)Fig. 4. Unique magnetic properties of the ULM attainable by using multiple\nrings of resonance frequencies !0n, here two rings with resonance frequencies\n!01and!02. (a) Multiband operation using independent rings with separate\nresonance frequencies. (b) Enhanced bandwidth using coupled-resonant rings\nwith overlapping resonance frequencies.\nThe single-ring-PEC ULM implementation of Fig. 3 is\nideal for microstrip components [38] and reflective metasur-\nfaces [32]. However, for 3D ULM [Fig. 2(a)] structures and\ntransmissive metasurfaces (Sec. IV), a transparent version of\nthat ULM is required. This could theoretically be realized with\na pair of rings, as in Fig. 1(b), but may be more conveniently\nimplemented in the form of circular slots in a Coplanar Wave-\nGuide (CPW) type technology, as reported in [41].\nIV. F ARADAY ROTATION\nFaraday rotation is one of the most fundamental and useful\nproperties of magnetic materials. Given their artificial ferrite\nnature (Sec. III), ULMs can readily support this effect. The\nFaraday angle is given by [3], [8]\n\u0012F(z) =\u0000\u0012\f+\u0000\f\u0000\n2z\u0013\n;with\f\u0006=!p\n\u000f(\u0016\u0006\u0014);(6)\nwhere\u0016and\u0014are the Polder tensor components in (3b),\nwith the resonance ( !0) given by (5) and the saturation\nmagnetization frequency ( !m) discussed in Sec. III. Inter-\nesting, the ULM allows the option to reverse the direction\nof Faraday rotation by simple voltage control (instead of\nmagnet mechanical flipping in a conventional ferrite) using\nan antiparallel transistor pair load, as demonstrated in [42].\nFigure 5 shows a reflective Faraday ULM metasurface\n(ULMS) structure, based on the particle in Fig. 3, and re-\nsponse, initially reported in [32]. The results confirm that the\nULM works exactly as a ferrite, whose equivalent parameters\nare given in the caption.\nFig. 5. Reflective Faraday rotating ULMS based on the particle in Fig. 3 [32].\n(a) Perspective representation of the metasurface with rotated plane of polar-\nization. (b) Theoretical [Eq. (6)] and experimental polarization rotation angle\nversus frequency. Here, \u000fr= 2:6,!0=2\u0019= 7:42GHz (Bequiv.\n0= 0:265 T),\n!m=2\u0019= 28 MHz (\u00160Mequiv.\ns = 1 mT) and\u000b= 1:9\u000210\u00003Np\n(\u0001H=\u000b!0=\r\u0016 0= 0:4mT.)\nULM Faraday rotation has also been reported in transmis-\nsion, using the circular-slot ULM structure mentioned at the4\nend of Sec. III. Using slots, and hence equivalent magnetic\ncurrents, instead of rings supporting electric currents, that\nstructure really operates as an artificial magneto-optic material,\nwith a permittivity tensor replacing the magnetic tensor in (3a).\nA similar Faraday rotation effect may also be achieved using\narrays of twisted dipoles loaded by transistors [43].\nV. A PPLICATIONS\nA. Metasurface Isolators\nThe transmissive ULMS in [41] can be straightforwardly\napplied to build a Faraday isolator [3], [4], [44], [45], as shown\nin Fig. 6. As the wave propagates from the left to the right, its\npolarization is rotated 45\u000eby the left ULMS in the rotation\ndirection imposed by the transistors (here, clock-wise). It thus\nreaches the polarizer with its electric field perpendicular to\nthe conducting strips and therefore unimpededly crosses it. It\nis finally rotated back to its initial (vertical) direction by the\nright ULMS, whose rotation direction is opposite to the left\none (here, counter-clockwise). In the opposite direction, the\nright ULMS rotates the wave polarization in such a manner\nthat its electric field is parallel to the conducting strips of the\npolarizer, so that the wave is completely reflected. It is then\nrotated again by the right polarizer and gets back to the right\ninput orthogonal to the original wave7.\nFig. 6. Isolator using two transmissive Faraday rotation ULMSs [41] and a\n45\u000epolarizer. The bottom-left inset shows the transmissive “magneto-optic”\nslot-ULM demonstrated in [41], that may be used for this application.\nFaraday rotation is not the only approach to realize spatial\nisolation , as in Fig. 6. Such isolation may be simply achieved,\nwithout any gyrotropy but still magnetlessly, with a metasur-\nface consisting of back-to-back antenna arrays interconnected\nby transistors [47]; this nonreciprocal metasurface may exhibit\nan ultra wideband response and provide transmission gain.\nB. Nonreciprocal Antenna Systems\nThe ULM structure may be used in various nonreciprocal\nradiating (antenna, reflector and metasurface) applications.\nFigure 7 shows a nonreciprocal antenna system and its re-\nsponse [48]. The structure [Fig. 7(a)] is a ULM magnet-\nless version of the nonreciprocal ferrite-loaded Composite\nRight/Left-Handed (CRLH) open-waveguide leaky-wave an-\ntenna introduced in [49], [50], with the ferrite material re-\nplaced by a 1D ULM structure. This structure may be used\nas a nonreciprocal full-space scanning antenna [Figs. 7(b)\nand (c)], whose unidirectionality provides protection against\ninterfering signals, or as an antenna diplexer system , where\nnonreciprocity effectively plays the role of a circulator with\nhighly isolated uplink ( 3!1) and downlink ( 2!3) paths.\n7Lossy polarizers can be added, if necessary (The final wave could be\nreflected back), for dissipative (rather than reflective) isolation [46].\nFig. 7. ULM CRLH leaky-wave isolated-antenna or antenna-duplexer sys-\ntem [48]. (a) Prototype with port definitions. (b) Measured radiation patterns.\n(c) Measured scattering parameters.\nC. Isolators and Circulators\nULM technology also enables various kinds of nonre-\nciprocal components. Figure 8(a) shows a ULM microstrip\nisolator [38]. The ULM structure below the microstrip line is\ncomposed of two rows of transistor-loaded rings with opposite\nbiasing, and hence opposite allowed wave rotation directions.\nAs the wave from the microstrip line reaches a ring pair,\nits mode is coupled into a stripline mode with strip pair\nconstituted by the longitudinal sections of the overlapping\nrings, and usual antisymmetric currents. In the propagation\ndirection where these currents are co-directional, with allowed\nrotation direction of the ULM, the stripline mode is allowed\nto propagate, whereas in the opposite propagation direction, it\nis inhibited and dissipates in matching resistors on the rings.\nFigure 8(b) shows a ULM microstrip circulator [38], which is\nbased on mode-split counter-rotating modes as all circulators.\nFig. 8. ULM components [38]. (a) Isolator [51]. (b) Circulator [38].\nVI. C ONCLUSION\nWe have presented an overview of ULM structures and\napplications. The ULM physics has been described in great\ndetails, revealing that the ULM really represents an artificial\nferrite medium. It is in fact the only existing medium of\nthe kind. It has been pointed out that the ULM may offer\nunique extra benefits compared to ferrites, such a multiband\noperation, ultra broadband and electronic Faraday rotation\ndirection switching.5\nREFERENCES\n[1] A. G. Fox, S. E. Miller, and M. T. Weiss, “Behavior and applications\nof ferrites in the microwave region,” The Bell System Technical Journal ,\nvol. 34, no. 1, pp. 5–103, Jan 1955.\n[2] G. L. Matthaei, E. M. T. Jones, and S. B. Cohn, “A nonreciprocal,\nTEM-mode structure for wide-band gyrator and isolator applications,”\nIRE Transactions on Microwave Theory and Techniques , vol. 7, no. 4,\npp. 453–460, October 1959.\n[3] B. Lax and K. J. Button, Microwave ferrites and ferrimagnetics .\nMcGraw-Hill, 1962.\n[4] L. J. Aplet and J. W. 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In recent \nyears, the spin-lattice coupling has been recognized  to be essential for ultrafast magnetization \ndynamics . Magnetoelectric multi ferroics of type II possess intrinsic correlation s among  \nmagnetic  sublattice s and electric polarization ( P) through spin -lattice coupling , enabling \nfundamentally  coupled  dynamics between spins and lattice . Here we report on ultrafast \nmagnetization dynamics in a room -temperature multiferroic hexaferrite possessing \nferrimagnetic and antiferromagnetic  sublattices, revealed by time -resolved resonant x -ray \ndiffraction . A femtosecond above -bandgap excitation tri ggers a coherent magnon in which \nthe two magnetic sublattices entangle and give rise to a transient modulation of P. A novel \nmicroscopic mechanism for triggering  the coherent magnon in this ferrimagnetic insulator \nbased on the  spin-lattice coupling is prop osed. Our finding opens up a novel but general \npathway for ultrafast control of magnetism .  \n* To whom correspondence should be addressed : hiroki.ueda@psi.ch  and urs.staub@psi.ch   \n    \n2 \n Magnetoelectric multiferroics of type II commonly exhibit magnetic order that \ninduces ferroelectricity, and hence, there is naturally a  strong coupling between magnetism \nand lattice, i.e., spin -lattice coupling . This class of materials has attracted enormous interest \nin the last two decades [ 1]. Interest in these multiferroics lies, for example, in electric \n(magnetic) manipulation of magnetism (polarization) [ 2-4] and in coupled spin-lattice \nexcitations called  electromagnons [ 5,6]. This inherent correlation between magnetism and \nferroelectricity enables simultaneous control of principal excitation characters, magnons and \nphonons, in a wide dynamic range. However , time-resolved studies on the coupling, e.g., \noptical excitation of electro magnons and attempts  to understand the microscopic origin of \ntheir creation, have been limited  so far  [7-10].  \nOptical manipulation of magnetic moments and identification of the underlying  \nexcitation mechanism are important objectives towards new -generation storage technologies \nand have been intensely debated  in the context of spin -lattice  coupling . These are, e.g., \nultrafast non -thermal magnetization switching/reorientation via a change in  \nmagnetocrystalline anisotropy [ 11-13], a coherent excitation of magnon s by driving an  \nelectromagnon  or optical phonon mode [ 8,14,15], and an ultrafast demagnetization that \ngenerates a transverse strain wave via the Einstein -de Haas effect [ 16]. The importance of \nspin-lattice coupling is well established in multiferroics with their non -trivial magnetic orders \nresult ing in remarkable magnetoelectric effects. Often , multiferroicity  emerge s due to \nmagnetic frustration resulting in a high  sensitivity  of the property  to bond angle s. Tuning a \nlattice parameter , e.g. , using strain , is indeed a standard approach to obtain multiferroicity \n[17]. The well -documented knowledge of spin -lattice coupling in multiferroics inspires us to \nexplore a new pathway to create magnons via ultrafast modulation in the lattice other than \nknown mechanisms, e.g., coherent displacive excitation [ 11] and the inverse Cotton -Mouton \neffect [ 18].  \nIn this paper, we study the ultrafast magnetization dynamic s in a room -temperature \nmultiferroic Z -type hexaferrite Sr 3Co2Fe24O41 with a  ferrimagnetic (FM)  and an \nantiferromagnetic (AFM) cycloidal sublattice  [19,20]. The dynamics of the individual \nsublattices upon a femtosecond  above -bandgap excitation were investigated by time -resolved \nresonant x -ray diffraction (tr -RXD). A coherent entangled magnon mode of the AFM and FM \nsublattices  is observed and is described by  the Landau -Lifshitz -Gilbert (LLG) equation  as an \nelectromagnon mode . Nonreciprocal directional dichroism  (NDD)  in microwave transmission  \nspectra further confirm s its origin as an electromagnon mode.  An alternative excitation  \nmechanism for the magnon creat ion based on the ultrafast Einstein -de Haas effect  is \nproposed . Such  a novel excitation mechanism for optical driving of magnetic modes  possibly \nneeds to be  more  general ly considered  in frustrated magnetic insulators .  \nZ-type hexaferrite Sr 3Co2Fe24O41 crystals are commonly  described in space group \nP63/mmc  with lattice constants of a ≈ 5.87 Å and c ≈ 52.07 Å [ 19]. The crystal structure,   \n3 \n displayed in Fig. 1a , comprises two basic magnetic blocks, S  block s and L block s, which \nalternately stack along [001]. Because of many magnetic Fe sites in the chemical unit cell, the \nmagnetic structure of hexaferrites is described by a collinear -FM structure in each magnetic \nblock. Arrows in Figs. 1b  and 1c represent the local structures: a red  (blue)  arrow represents \nthe magnetic moment of an L  (S) block  μL (μS). There is magnetic frustration at the interface \nbetween S and L blocks, which is tunable by chemical sub stitution that changes the specific \nFe-O-Fe bond  angle (see the green d otted box in Fig. 1a ) [21]. As a result, a non -collinear \nmagnetic structure shown in Fig. 1b  is stabilized below the multiferroic transition \ntemperature TMF ≈ 410 K [ 20,22], whereas a collinear -FM structure shown in Fig. 1c  appears \nabove TMF.  \n \nFigure 1 . Crystal structure and magnetic structure s of Z-type hexaferrite \nSr3Co2Fe24O41 and time -resolved resonant x -ray diffraction  setup.  (a) Crystal structure \nof the Z -type hexaferrite Sr 3Co2Fe24O41 drawn by VESTA [38]. Co2+ sits on some Fe3+ \nsites. A green dotted box shows an enlarged view of the local structure at the interface \nbetween adjacent magnetic blocks, where some atoms not relevant to the magnetic \nfrustration are omitted. ( b,c) Simplified magnetic structure s of Sr 3Co2Fe24O41 in (b) the \nconical phase and ( c) the collinear -ferrimagnetic phase. Red and blue arrows denote net \nmoments in magnetic L and S blocks ( μL and μS), respectively. In the  black dotted box , the \nprecession of μL in the magnetic excitation mode  is shown by black arrows . Here μS is \nomitted because of its small precession amplitude. ( d) Sketch of the experimental setup \nused for the time -resolved resonant diffraction experiments (details are described in \nMethod ). (e) Resonant diffraction profile fr om Sr 3Co2Fe24O41 along (00 L) in the conical \nphase.   \n \nThe non -collinear transverse -conical structure is composed of a collinear -FM \ncomponent with a propagation vector k = (0,  0, 0) and a non -collinear cycloidal component \nwith k = (0,  0, 1). Hereafter, we refer to the former component as the FM compone nt and to \nthe latter as the AFM component. Whereas the FM component hosts magnetization ( M), the \n  \n4 \n AFM component gives rise to electric polarization ( P) through spin -orbit  couplings \n[19,23,24]. A previous RXD  study on Sr3Co2Fe24O41 investigated the magnetic sublattices \nseparately using two reflections sensitive to the individual sublattices  [24] [see Fig 1e  for a \ntypical resonant diffraction profile along (00 L) in the conical phase] . It revealed a strong \ncoupling between the two sublattices in the adiabatic limit, essential for the magnetoele ctric \neffect.  \nTHz time -domain spectroscopy (THz-TDS) revealed the presence of an \nelectromagnon mode around 1 THz [ 25,26], indicative of the magnetoelectric coupling on \nultrafast timescales. This mode is the higher -frequency mode of the two possible \nelectromagnon modes with AFM  resonance obtained by solving the LLG equation. The \nlower -frequency  mode though has not been reachable in the THz-TDS experiment. The \nexpected d ynamics of the slower mode are visualized in the d otted box of Fig. 1b , which  \ninvolves anti -phase oscillation between adjacent two μL and also between two μS with two -\norder smaller motions for the latter . In addition, an even slower magnetic mode is observed \nby the microwave  spectroscopy technique at GHz frequenc ies [27,28], which might  be \nreferred to as the toroidal -magnon mode or Nambu -Goldstone mode [ 29], the lowest -energy \nmagnon mode in the transverse conical state .  \n \nRESULTS  \nWe collected intensities of the (003) space -group forbidden and of the (004) allowed \nBragg reflections sensitive to the AFM and the FM component, respectively, in addition to \nthe fluorescence signal. The optical pump beam wavelength was 400 nm, which resul ts in an \nabove -bandgap excitation. Figure 2  shows tr -RXD intensities of the (003) and (004) Bragg \nreflections measur ed with  polarization at 709.6 eV , normalized by unperturbed intensities \n(IOFF). Following the laser excitation, both reflections show ( O-1) a rapid decrease within \n~200 fs (see Figs. 2 a and 2b), (O-2) oscillations  (see Figs. 2 c and 2d), (O-3) a slow response \nthat is either a decrease [for (003) and (004) with low fluence] or an increase [for (004) with \nhigh fluence] (see Figs. 2 e and 2f), and ( O-4) a recovery following t he precedent transient \nchanges . The  time traces of the normalized intensities  (ION/IOFF) are fitted by these four \ncomponents defined as  \n𝐼ON\n𝐼OFF(𝑡)=−𝐴rap[1−e−(𝑡−𝑡0)𝜏rap⁄]+𝐴osce−(𝑡−𝑡0)𝜏osc⁄cos[2𝜋𝑓𝑡−𝜑]−𝐴slow[1−\ne−(𝑡−𝑡0)𝜏slow⁄]+𝐴rec[1−e−(𝑡−𝑡0)𝜏rec⁄]+1.  (1)  \nHere Arap (rap), Aosc (osc), Aslow (slow), and Arec (rec) are amplitudes (relaxation times ) of ( O-\n1) the rapid decrease, ( O-2) oscillation , (O-3) slow response , and ( O-4) recovery , \nrespectively , f is the oscillation frequency  of the mode , t0 is time zero,  and  is a phase shift. \nThe fit results in an oscillation frequency of f1 ≈ 42.0±1.0 GHz at room temperature ( shown   \n5 \n as translucent blue curves in Fig. 2 c,d). Best f its for the parameters are listed in \nSupplementary Information.   \nThe oscillations exhibit the same frequency for the two reflections, but they are \nopposite in phase. Because the (003) and (004) reflections sampl e the AFM and FM \ncomponents, respectively, the oscillations with the same frequency indicate a coherent \nexcitation  of a magnon mode entangl ing both components. Cooling to 84 K shows \nqualitatively similar results except for; (I) f1 shifts to a slightly higher frequency  [~46.3±0.3 \nGHz (see the black translucent curves in Figs. 2 c,d, representing  the fitted oscillation \ncomponent)] , (II) the oscillation lives more protracted , and (III) the slow response of (004) \nturns from an increasing to decreasing behavior despite the equal fluence used (compare \nblack and da rk green curves ).  \n \nFigure 2 . Time traces of (003) AFM and (004) FM reflections.  (a,c,e) (003) AFM and \n(b,d,f) (004) FM reflections measured with  polarization of incident x -ray beams  at 709.6 \neV; (a,b) short time trace around t0 (from 0.5 ps to 1.5 ps ), (c,d) medium time range ( 5 \nps to 40 ps) to improve visibility of the intensity oscillations , and (e,f) extended time trace \n(from 50 ps to 200 ps) . Each color corresponds to a different laser fluence  or temperature  \ndepicted with the label of  the same c olor, and solid curves are best fits. A translucent curve \nin (c) and ( d) shows the oscillation component in the best fit. All data were taken at room \ntemperature except for black curves  taken at 84 K. Vertical and horizontal dotted lines \nindicate t0 and the intensities before t0, respectively.  \n \nMagnetic c ircular dichroic signals of the (004) reflection can directly capture M [24] \nand its time evolution upon laser excitation. Figure 3  show s tr-RXD intensities taken with \ncircular polarization at room temperature and extracted circular dichroic signals . An \nadditional feature appears in the circular dichroic  signals, ( O-5) a significantly slower \noscillation  at f2 ≈ 5.7±0.1 GHz  (see black data in Fig. 3a ). The observation  of a rapid decrease \nwithin < 200 fs followed by  an increase of the overall signals  indicate s an ultrafast \ndemagnetization followed by a  delayed and  slower increase of the FM component. The \n  \n6 \n increase in the FM component goes hand in hand with the reduction of the AFM component \n(see Fig. 2 e), suggesting a transformation of AFM to FM moment  component s.  \n \nFigure 3 . Time traces of the circular dichroic (004) FM reflection . (a) Extended time \ntrace with high fluence ( 50 ps to 500 ps, 5.3 mJ/cm2) and ( b) short time range around t0 \n(2 ps to 27 ps, 5.3 mJ/cm2 and 0.6 mJ/cm2), measured at room temperature . Red  (blue) \nplots are normalized intensities taken with C+ (C), while black plots are the difference \nbetween them, i.e., IC+  IC. In ( b), data taken with C  are vertically shifted by 6 [arb. \nunits] . Inset shows d ata taken with low fluence (0.6 mJ/cm2) for better visibility of the \noscillation  with f1. Errors are shown  as translucent filling in ( b).  \n \nTo better capture the mode s observed in the time traces , we performed microwave \nspectroscopy measurements . Figure 4  shows the NDD signals of microwave transmission \nspectra in the relevant frequency range s taken at room temperature and in various magnetic \nfields up to ±0.35 T. Resonant feature s with clear NDD are noticeable in both frequency \nranges, consistent with f1 and f2. Furthermore, the NDD signals indicate both modes are \nelectric - and magnetic -dipole active [ 6], meaning  that these modes have a n electromagnon  \ncharacter . Note that the higher -frequency  f1 mode exhibits extraordinar y large NDD of up to \n3.5 dB, which translates  to a change in  transmitted power  > 100% and is much larger than the  \nmaximum change of  ~11% for the lower -frequency  f2 mode . This implies  a considerable  \nresonator strength coupled to the electric field of the microwave  and corresponding to  \nsignificant modulation s in M and P when the mode  is resonantly excited . While NDD is less \npronounced in the low -frequency range, microwave absorption is resonantly enhanced  (see \nSupplementary Information ), reminiscent of previous NDD/ microwave studies on a chiral \nmagnet  [30]. All modes harden for increasing magnetic fields; resonan ce frequenc ies in the \nlower -frequency regime  are almost linear in the amplitude of the static magnetic field \nconsistent with  the Zeeman effect, whereas that of the higher -frequency mode is strongly \nnon-linear.  Furthermore, the modes soften for elevat ed temperatures, as shown in Fig. S6  (see \nSupplementary Information).  These results indicate that the oscillation with f2 could  be \nassigned to the toroidal magnon or Nambu -Goldstone  mode , the lowest -energy magnon mode \n  \n7 \n showing NDD  [31], while the oscillation with f1 would be  assigned to a more complicated \nelectromagnon excitation . Together with no evident heat -driven lattice expansion  shown in \nSI, we conclude that the oscillations in tr -RXD are not caused by longitudinal strain waves \nbut are of magnetic origin.  \n \nFigure 4. Nonreciprocal directional dichroic signals in microwave transmission \nspectra.  Around  (a) low - (3-20 GHz , f2) or (b) high - (35-45 GHz , f1) frequency range in \nvarious magnetic fields  at room temperature  with a coplanar waveguide setup (see \nMethods) .  \n \nAs the slow non-oscillating response in tr-RXD intensities of the (004) reflection (O-\n3) qualitatively differs for lower and higher laser fluences, we varied the laser fluence on \n(003) and (004) reflections for a given time delay of 200 ps  (see Fig. 5). On the one hand, the \nintensity of the (004) reflection decreases for increasing fluence up to a threshold  laser \nfluence of approximately 0.7 mJ/cm2 at room temperature , above which the intensity starts to \nincrease and saturates at ~3.1 mJ/cm2 far above its initial value. In contrast, the (003) \nintensity decreases monotonically over the whole range. On the other hand , the tr -RXD \nintensities of both the reflections monotonically reduce for increasing laser fluence  at 84 K  \n(see blue data points i n Fig. 5).  \nThis behavior can be explained by the absorbed energy of the optical excitation . When \nexcited at room temperature , the temperature increase s above TMF after equilibration . Above \nTMF, the AFM component is absent, but the FM component is enhanced . Note that the in -\nplane M in the multiferroic phase does not strongly depend on temperature [ 22], indicating \nthat any thermal effect on the FM component becomes relevant only for temperature s above  \nTMF. This view is supported by the estimated lattice expansion shown in Fig. S1a , consistent \nwith a temperature of ~410 K  [32], which is precisely TMF. Note also that there is a depth \nprofile of the deposited energy ( effective temperature ), and the estimated temperature is the \nmean value within the probe  depth . Thus, the top layers have an effective temperature well \nbeyond TMF. This scenario explains both the suppression of the (003) reflection and the \nincrease of the (004) reflection.  \n  \n8 \n  \nFigure 5. Fluence dependence of (003) AFM and (004) FM reflections . (a) (003) AFM \nand ( b) (004) FM  measured , taken at room temperature (red) and 84 K (blue) , and at 200 \nps delay time . Dotted lines indicate normalization baselines.  \n \nUnderstanding the fast dynamics occurring within the first ~200 fs  after excitation , as \nshown in Figs. 2 a and 2b, is not straightforward as it is close to the time resolution of the \nbeamline [ 33]. Nevertheless, tr -RXD intensities  for both AFM and FM components  show a \nrapid decrease of magnetic signals within ~200 fs. Such a quick response cannot directly \nrelate to a lattice expansion but is possibly due to an electron ic change [ 34]. To gain more \ninformation, we collected time-resolved  x-ray absorption  spectra  (tr-XAS) in fluorescence \nmode . Figure 6c shows tr -XAS signals for  polarization below or above the prominent main \npeak of the Fe L3 edge 0.5 ps after the excitation (see Fig. 6a). At both  photon energies, \ntransient signals show a rapid variation within ~200 fs, the same time scale as found in the \nresponse of the  tr-RXD intensities.  \nAs the fast response with a tiny amplitude is also present in the time-resolved circular \ndichroic  signals of the (004) reflection sensitive to M (Fig. 3b ), the quick response is \nexpected to be relate d to the ultrafast demagnetization. Note that the rapid change in \nfluorescence signals results in an intermediate state with a long lifetime with a monotonic \namplitude increase for increasing laser fluence, as shown in Fig. 6b. This long -lived change \nin XAS remains up to  500 ps (see Fig. 6d), consistent with a  magnetic origin , whereas we \n  \n9 \n expect that an electronic change would be very short -lived  [35]. It is likely caused by  \nvariations in  x-ray magnetic linear dichroism due to demagnetization  (from the AFM and /or \nthe FM component) , though here the  changes present only the variation of the single  \npolarization channel (). The absence of a clear oscillation with the frequency of f2 implies \nthat this mode is unlikely  the simple Kittel mode.  However, a more detailed investigation \nwith improved statistic s would be  required to clarify what exactly causes the tr -XAS signals.  \n \nFigure 6. Time -resolved f luorescence signals.  (a) Fluorescence spectra taken with  normal \nincidence of  -polariz ed x-ray beams  for the excited (red, ON, 8.2 mJ/cm2, 0.5 ps after the \nexcitation) and unperturbed (blue, OFF) sample and the difference between them (black). \n(b) Fluence dependence o f normalized fluorescence signal s taken with the photon energy \nof 708.5 eV at a time delay of 0.5 ps. Time traces of normalized fluorescence signal s (c) \nwith a time range around t0 (1 ps to 1 ps) and ( d) up to 500 ps. Red, blue, and black \nsymbols were taken with the photon energy of 708.5 eV, 710.6 eV, and 708.9 eV, \nrespectively. The solid curves in ( c) and ( d) are represents the best fit to Eq. (1) excluding \nthe oscillation term. Note that the long -lived change indicates an additional recovery term \ntaking more than 500 ps. Horizontal and vertical dotted lines indicate a normalization \nbaseline and t0, respectively.  \n \nDISCUSSION  \nWe have found that there are; ( O-1) a rapid response within ~200 fs, observed in tr -\nRXD intensities of both the (003) and (004) reflections, a time-resolved circular dichroic  \n  \n10 \n contrast  in (004) , as well as in the tr-XAS, ( O-2) oscillations with the frequency of f1 (≈ \n42.0±1.0 GHz at room te mperature) in tr -RXD intensities of both the reflections that are of \nmagnetic origin as supported by the microwave spectra , (O-3) a slow response subsequently \nfollowing the rapid one, which we interpret as energy transfer from the electronic to the \nlattice  system, (O-4) a recovery that takes longer than 500 ps , and ( O-5) oscillations with the \nfrequency of f2 (≈5.7±0.1 GHz) in the time-resolved circular dichroic signals of the (004) \nreflection most likely representing the lowest -energy magnon  mode.  Here we discuss ( i) the \noscillation s with f1 represent ing a coherent magnon mode and ( ii) how an above -bandgap \nexcitation launches the magnon mode.  \nI. Assignment of the entangled magnon mode  \nWe assume  a simple phenomenological model to describe  the experimental ly \nobserved entangled magnon mode , i.e., the frequency  of f1 with opposite phase oscillation s \nfor the  two reflections . We consider two order parameters with temporal modulation s; an \nAFM order parameter 𝑆AFM(𝑡)=𝑆AFM[1+𝛿e𝑖(𝜔𝑡−𝛷)] and an FM order parameter 𝑆FM(𝑡)=\n𝑆FM[1−𝛿e𝑖(𝜔𝑡−𝜑)]. Here SAFM and SFM are static order parameters of the AFM component \nand the FM component, respectively, δ is the amplitude of temporal modulation, ω is an \nangular frequency, and Φ is an arbitrary phase. The (004) reflection at resonance involves \nscattering  proces ses of magnetic, charge, or orbital  (or electric quadrupole)  origin. We refer \nto a parameter describing the latter two scatterings as SCO.  \nThe form factor 𝐹̂ including the polarization dependence at resonance can be written \nas a tensor of the form  \n𝐹̂=(𝐹𝜎−𝜎′𝐹𝜋−𝜎′\n𝐹𝜎−𝜋′𝐹𝜋−𝜋′).  (2)  \nAs the magnetic field in the tr -RXD experiments lies in both the scattering and the basal \nplane s, the magnetic part of the form factor Fm at (004), i.e., the FM component, is in the \nscattering plane [ 𝐅𝑚𝐐=∑𝐦𝑗exp(𝑖𝐐∙𝐫𝑗) 𝑗 , where mj and rj are the magnetic moment and \npositional vector of a magnetic atom j, respectively, and Q is the scattering vector.],  whereas  \nFm at (003), i.e., the AFM component, is normal to M. For  polarized incoming x -rays, \nmagnetic scattering occurs in both the π -σ’ and π -π’ channels for the (003) reflection. As \nthere is no charge/orbital scattering  contribution  at (003), the (003) intensities are \nproportional to |𝐅𝑚(003)|2\n and thus |𝑆AFM|2 as  \n𝐼(003)≈|𝑆AFM|2.  (3)  \nOn the contrary, the charge/orbital scattering at (004) appear s in the π -π’ chan nel for  \npolarized incoming x -rays (see Supplementary Information ), whereas magnetic scattering \nappears in the π -σ’ channel  because Fm is in the scattering plane . Due to  the orthogonal   \n11 \n relation between the π-π’channel ( charge/orbital scattering ) and π-σ’ channel ( the magnetic \nscattering ), the (004) intensities are  \n𝐼(004)≈|𝑆FM|2+|𝑆CO|2=𝐼FM+𝐼CO.  (4)  \nTr-RXD intensities of the reflections are  \n𝐼(003)(𝑡)≈|𝑆AFM(𝑡)|2=𝑆AFM∗𝑆AFM[1+𝛿e−𝑖(𝜔𝑡−𝜑)][1+𝛿e𝑖(𝜔𝑡−𝜑)] \n≈𝐼(003)[1+2𝛿cos(𝜔𝑡−𝜑)]  (5)  \nand  \n𝐼(004)(𝑡)≈|𝑆FM(𝑡)|2+|𝑆CO|2=𝑆FM∗𝑆FM[1−𝛿e−𝑖(𝜔𝑡−𝜑)][1−𝛿e𝑖(𝜔𝑡−𝜑)]+|𝑆CO|2 \n≈𝐼FM[1−2𝛿cos(𝜔𝑡−𝜑)]+𝐼CO=𝐼(004)−2𝛿𝐼FMcos(𝜔𝑡−𝜑).  (6) \nEquation s (5) and (6) describe an entangled mode of the two sublattices and reproduce the \nexperimental observation that is of a single  frequency but opposite phase between the (003) \nreflection and (004) reflection. The initial phase of the oscillations shown in Fig. 2  result s in a \npositive sign of δ, representing an increase of the AFM component subsequent  to the \nexcitation. Although  chang es in magnetic exchange interactions  through lattice \nthermalization  can launch magnetic excitations, it should result in a negative sign of δ due to \nthe reduction of the AFM component at higher temperatures .  \nUsing an effecti ve magnetic Hamiltonian and solving the LLG equation for the \nhexaferrite, Chun and coworkers obtained two k = 0 magnon modes with different \nfrequencies [ 26]. They found that the two magnetic moments in adjacent L blocks or S bl ocks \nhave antiphase motions in the magnon modes. The parameters  that quantitatively reproduc e \nthe higher -frequency electromagnon mode observed by THz-TDS at low temperature (~1 \nTHz at 20 K), provide a slower electro magnon mode frequency of ~58 GHz. Th is frequency \nis close to f1 (≈46.3±0.3 GHz at 84 K) found  in the tr -RXD intensities . This mode softens \nwhen increasing temperature s, for example, ~42.0±1.0 GHz at room temperature , resembling \nthe softening of the higher -frequency electromagnon mode  by increa sing temperatures  [26]. \nThis indicates  that f1 obtained from  the tr -RXD intensities represents  the coherent excitation \nof the slower electro magnon mode  with two entangle d sublattices  as sketched in  Fig. 1b that \nis domina ted by the  μL motion. The antiphase motion of magnetic moments in adjacent L \nblocks or S blocks creates transient P along [001] through exchange striction, similar to the \nreported electromagnon mode [ 25,26]. Inducing transient P through exchange striction \nimplies  a more intense resonator strength for the electromagnon mode than for the toroidal \nmagnon mode, which modulates static P originated from spin -orbit couplings , consistent with \nthe observed strengths in the NDD signals (see Fig. 4 ).  \nII. Mechanism of magnon excitation  \nThe known mechanisms for optical magnon excitations are based on a change in \neffective magnetic -field direction due to; ( i) a photomagnetic effect changing magnetic \ncrystalline anisotropy or known as a coherent displacive excitation [ 11] and (ii) the inverse   \n12 \n Cotton -Mouton effect or known as an impulsive stimulated Raman scattering [ 18]. The \nantiphase spin motions require the opposite direction of transient effective magnetic  field \nbetween adjacent  two L blocks . Among the possible tensor components that can change the \neffective magnetic -field direction  in su ch a way , however , the symmetry of the hexaferrite \ndoes not support such a transient change through these photomagnetic effects (see \nSupplementary Information for details). Furthermore, photothermalization changing magnetic \nexchange interactions is ruled o ut, as mentioned above.  \nFollowing the intensive laser excitation, ultrafast demagnetization of a FM sublattice \noccurs as observed in a rapid drop of the transient (004) circular dichroic signal. It has been \nshown that m ost of the lost angular momentum can be transferred within 200 fs into  the \nangular momentum of lattice , creating a transverse force that triggers a strain wave through \nthe ultrafast Einstein -de Haas effect [ 16]. In ferromagnets with uniform M, the force only \nappears at the surface because a microscopic torque for spatially uniform M cancels in bulk. \nOn the other hand, in ferrimagnets, possessing two blocks with opposite M as drawn in Fig. \n7a, the force has the same direction at the interface and is therefore maximal there. This will \ninitiate shear waves (see Fig. 7b and Supplementary  Information ).  \nThere i s the magnetic frustration at the  interfaces of the two magnetic blocks, L and S, \nthat creates the  non-collinear magnetic structure in the hexaferrite family. Chemical \nsubstitution and thus angle change s of specific iron -oxygen bonds (see Fig. 1a ) alter the \nmagnetic exchange interaction that strongly affect s the frustration and correspondingly the \nmagnetic structure . As the launched shear wave is on a timescale much faster  (< 200 fs)  than \nthe period of the magnetic excitation , it can launch the ele ctromagnon excitation  owing to the \nsensitivity of the magnetic structure due to the tiny displacements of interfacial bonds .  \nWe can estimate the amplitude of the spin precession from the oscillation amplitude \nof the tr -RXD intensities of the (003) AFM ref lection. The fitting results in Fig. 2  yield the \noscillation  amplitude  in normalized intensities as ~3% at room temperature  with 0.5 mJ/cm2, \nand ~5% at 84 K  and 1.2 mJ/cm2. This roughly corresponds to a precession angle of μL as \n~1.1° and ~1.8° at room temperature and 84 K, respectively (see Supplementary  Information  \nfor details). These angle s are smaller than that in the Y-type hexaferrite  Ba0.5Sr1.5Zn2(Fe 1-\nxAlx)12O22 (x = 0.08)  manifesting a different excitation mechanism, namely  a coherent \ndisplacive excitation type, which is allowed by symmetry [ 10]. However , the amplitudes in \nangle displacement found in our experiment  for the Z -type hexaferrite  are comparable to the \nreported large -amplitude spin dynamics upon electromagnon resonance excitation in \nTbMnO 3 [8].    \n13 \n  \nFigure 7. Sketch es of the temporal r esidual strain at the interface of adjacent \nmagnetic blocks due to angular momentum transfer from  ultrafast demagnetization \nof the ferrimagnetic  blocks ( the ultrafast Einstein -de Haas effect ). Ultrafast \ndemagnetization of the ferrimagnetic component , red a nd blue arrows  in (a), resulting in  \nangular momentum transfer , yellow and black arrows in ( a) and black circle arrows  in (b), \nresulting in uncompensated residual strain at the interface of two magnetic blocks as \ndenoted by thick black arrows.  \n \nIn summary, we investigat ed the ultrafast magneti zation dy namics  of the room -\ntemperature multiferroic Z-type hexaferrite Sr 3Co2Fe24O41 that exhibit s two magnetic \nsublattices ( ferrimagnetic one and pure antiferromagnetic one ), utilizing time -resolved \nresonant x-ray diffraction upon a femtosecond above -bandgap excitation. Our observations \nshow a coherent entangled magnetic excitation of the two sublattices instantaneously \nfollowing the optical excitation , which  reflects a  simultaneous optical control of two \ncharacters in a multiferroic material, magnetization and electric polarization, i.e., an \nelectromagnon excitation. We propose that a direct tuning of magnetic frustration due to \nshear waves launched by ultrafast demagne tization through the Einstein -de Haas effect could \nbe the origin of the coherent excita tion of the electro magnon. This novel non-thermal  \nmechanism can trigger transient spin modulation in materials with magnetic frustration. The \noptical  manipulat ion of  lattice, based on ultrafast demagnetization and shear waves via the \nEinstein -de Haas effect , could be, in general , a more competing channel to explore \ncorrelation s among electronic degrees of freedom and the lattice on an ultrafast timescale .  \n \nMethods  \nTime -resolved resonant x -ray diffraction.  Tr-RXD experiments were performed at the \nRSXS end -station of the SSS beamline in the PAL -XFEL [ 33]. The photon energies of \nmonochromatic x -ray beams were in the vicinity of the Fe L3 edge (≈ 710 eV). The \npolarization of the beams was either linear (π) or circular (C+/C ) controlled by the insertion \nof a magnetized Fe-foil with in -plane M (±) at an angle of approximately 45º with respect to \n  \n14 \n the x -ray beam, acting as a circular polarizer [ 36]. The circular polarization degree estimated \nfrom the transmitted intensities is ~80%. Switching M of the Fe -foil with an electromagnet \nreverses the circularly polarized component.  The optical pump beams  (400 nm)  were p-\npolarization  and collinear to the probe beam.  A single -crystal of Sr 3Co2Fe24O41 was mounted \ntogether with a pair of permanent magnets (≈ 0.1 T along [100]  and parallel to the scattering \nplane ) on the diffractometer implemented in the end -station (see Fig. 1d ). Two avalanche \nphotodiodes collected simultaneous diffraction and fluorescence signals. A gas -monitor \ndetector collected incident x -ray pulse intensities and normalized the signals on a shot-to-shot \nbasis. The penetration dept hs of the beams are estimated to be ~34 nm and ~13 nm for the \npump (obtained from ellipsometry measurements) and for the probe (obtained from x -ray \nabsorption spectroscopy measurements [ 37]), respectively. The measurements were  \nperformed at either room temperature or base temperature of the LN 2-cooled cryostat ~84 K. \nThe repetition rates of x -ray probe and optical pump pulses were 60 Hz and 30 Hz , \nrespectively . The length of the x -ray pulses is ~80 fs and is used for convolution  of each time \ntrace with the corresponding Gaussian function.  \nMicrowave transmission  spectroscopy.  Room temperature  microwave transmission \nexperiments  were performed  at the Paul Scherrer Institute  and employed  a coplanar \nwaveguide setup . The s ame crystal as used for the x-ray experiments  was placed on top of the \ncoplanar waveguide, with the [001] direction along the microwave propagation direction.  \nBefore the experiments, the sample was processed with magnetic/electric fields, so -called \nmagneto electric poling procedures, to create a single -domain multiferroic state as \nexemplified  in Ref. [ 24]. An electromagnet provided a calibrated magnetic field up to 0.35 T \nalong the [100] direction, which realize d the Voigt geometry , typical for NDD studies with \ncoplanar waveguides [ 30,31]. In the frequency range below 20 GHz, a commercial vector \nnetwork analyzer recorded the transmitted microwave power . To reach frequencies up to 50 \nGHz, a custom microwave setup was used. The NDD  of transmitted microwaves wa s \nobtained by taking the ratio of transmission signals with  opposite magnetic field polarity.  In \nparticular, data recorded for an up -sweep in the ma gnetic field was divided by data recorded \nfor the corresponding down -sweep. The Supplementary Information contains further \ntechnical details on the microwave setup , transmission data, and NDD  at elevated \ntemperature s and a different magnetic field orientation.   \n \nData availability  \nExperimental and model data are accessible from the PSI Public Data Repository [ 39].  \n \nReferences    \n15 \n 1. Fiebig,  M., Lottermoser,  T., Meier,  D. & Trassin , M. The evolution of multiferroics. Nat. \nRev. 1, 16046 (2016).   \n2. Kimura,  T., Goto,  T., Shintani,  H., Ishizaka,  K., Arima,  T. & Tokura , Y. Magnetic control \nof ferroelectric polarization, Nature  426, 55-58 (2003) .  \n3. Tokunaga,  Y., Taguchi,  Y., Arima,  T. & Tokura,  Y. Electric -field-induced generation and \nreversal of ferromagnetic moment in ferrites. Nat. Phys.  8, 838 -844 (2012) .  \n4. 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Lett.  71, 1446 -1448 \n(1997) .    \n18 \n 37. Ueda , H., Tanaka , Y., Wakabayashi , Y., Tsurumi , J., Takeya , J. & Kimura , T. Multiple \nmagnetic order parameters coexisting in multiferroic hexaferrites resolved by soft x rays. \nJ. Appl. Phys.  128, 174101 (2020) .  \n38. Momma , K. &  Izumi , F. VESTA 3 for three -dimensional visualization of crystal, \nvolumetric and morphology data. J. Appl. Crystallogr.  44, 1272 -1276 (2011) .  \n39. DOI: https://doi.org/10.16907/  efb03af6 -f28b-414f-8eb1 -77b31a035fb7 .  \n \nAcknowledgements  \nWe thank M. Burian for his advice  in data analysis.  The time -resolved resonant x -ray \ndiffraction experiments were performed at the RSXS end -station of the SSS beamline in the \nPAL -XFEL under proposal No. 2020 -2nd-SSS-011. The static resonant x -ray diffraction \nexperiments were performed at the X11MA beamline  in the Swiss Light Source during in -\nhouse access and at the BL17SU in the SPring -8 under proposal No. 20180021. H.U. was \nsupported by the National Centers of Competence in Research in Molecular Ultrafast Science \nand Technology (NCCR MUST -No. 51NF40 -183615 ) from the Swiss National Science \nFoundation and from the European Union’s Horizon 2020 research and innovation \nprogramme under the Marie Skłodowska -Curie Grant Agreement No. 801459 – FP-\nRESOMUS. H.J. acknowledges the support by the National Research Found ation grant \nfunded by the Korea government (MSIT) (Grant No. 2019R1F1A1060295). S.H.C. was \nsupported by National Research Foundation of Korea (2019R1C1C1010034 and \n2019K1A3A7A09033399). N. O. H. was supported by the Swiss National Science \nFoundation ( No. 2 00021_169017 ). A.D. acknowledges funding from the PSI -internal CROSS \ninitiative.  T.K. and Y.T. were supported by JSPS KAKENHI Grant Number JP19H00661.  \n \nAuthor contributions  \nH.U. and U.S. conceived and designed the project . H.U. and T.K. prepared a single  \ncrystal of Sr 3Co2Fe24O41. H.U., N.O.H., Y.T., and U.S. performed synchrotron measurements \nto characterize the sample. S.F. fabricated the circular polarizer. H.U., H.J., S.H.C., H. -D.K., \nand U.S. performed the time -resolved resonant x -ray diffraction experiment, and H.U. \nanalyzed  the experimental data. S. -Y.P. established the data acquisition system for the \nbeamline experiment. M.K. prepared the optical laser pump setup. V.O. and M.S. measured \noptical ellipsometry of the sample. A.D. built up the microwave  spectroscopy setup and \nperformed the experiment. H.U., A.D., and U.S. interpreted the experimental results and \nwrote the manuscript. All authors contributed to its final version.  \n 1 \n Supplementary Information for  \nOptical excitation of electromagnons in hexaferrite   \nHiroki Ueda1,*, Hoyoung Jang2, Sae  Hwan Chun2, Hyeong -Do Kim2, Minseok Kim2, Sang -\nYoun Park2, Simone Finizio1, Nazaret Ortiz Hernandez1, Vladimir Ovuka3, Matteo Savoini3, \nTsuyoshi Kimura4, Yoshikazu Tanaka5, Andrin Doll1, and Urs Staub1,*  \n1Swiss Light Source, Paul Scherrer Institute, 5232 Villigen -PSI, Switzerland.  \n2 PAL-XFEL, Pohang Accelerator Laboratory, Pohang, Gyeongbuk 37673, South Korea.  \n3 Institute for Quantum Electronics, Physics Department, ETH Zurich, 8093 Zurich, \nSwitzerland.  \n4 Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277 -8561, \nJapan.  \n5 RIKEN SPring -8 Center, Sayo, Hyogo 679 -5148, Japan.  \n \n* To who m correspondence should be addressed: hiroki.ueda@psi.ch  and urs.staub@psi.ch   \n  2 \n Contributions of orbital scattering  \nIn general, scattering from an aspherical electron density created by the occupation  \nvalence electrons  in the orbitals (orbital scattering) can contribute to diffraction intensities at \nresonance. Here, w e perform a symmetry analysis to clarify if orbital scattering contributes to \nthe (003) and (004) reflec tions. Orbital scattering is described by the anisotropic x -ray \nsusceptibility tensor 𝑓̂. The tensor elements are restricted by  the local (site) symmetry of the \nresonant  atom, and the sum of the respective tensors with their phase factor gives a form \nfactor and intensity [ 1].  \nWe denote the symmetric tensor 𝑓̂as  \n𝑓̂=(𝑓𝑥𝑥𝑓𝑥𝑦𝑓𝑥𝑧\n𝑓𝑥𝑦𝑓𝑦𝑦𝑓𝑦𝑧\n𝑓𝑥𝑧𝑓𝑦𝑧𝑓𝑧𝑧)  (S1) \nin a Cartes ian coordinate system where x is along [100], y is along [120], and z is along [001].  \nThere are 10 Fe sites in a Z -type hexaferrite [ 2], and their local symmetries are summarized \nin Table S1. All Fe sites except the Fe4 and Fe8 sites have the three -fold rotational symmetry \nC3 along [001]. The local symmetry requires the relation 𝑓̂=𝐶3𝑓̂𝐶3−1, which results in fxx = \nfyy and fxy = fyz = fzx = 0. The symmetry -adopted 𝑓̂ is then  \n𝑓̂=(𝑓𝑥𝑥00\n0𝑓𝑥𝑥0\n00𝑓𝑧𝑧).  (S2) \nThe Fe4 and Fe8 sites contain  the mirror symmetry m that is  along  <100> . The symmetry -\nadopted tensor follows the relation 𝑓̂=𝑚𝑓̂𝑚−1 and is represented as  \n𝑓̂=(𝑓𝑥𝑥00\n0𝑓𝑦𝑦𝑓𝑦𝑧\n0𝑓𝑦𝑧𝑓𝑧𝑧).   (S3)  \nFor the sites lacking local C3 symmetry,  i.e., the Fe (4) and Fe (8), there are three atoms at the \nsame z coordinate due to the global C3 symmetry.  For (00L) reflections,  only the z coordinate \nis relevant , allowing to  define a n average  𝑓̂ for these three atoms that is equivalent to Eq. S2.  \nAs the residual  𝑓̂’s are isotropic in the basal plane, the form factor 𝐹̂ for the (003) \nreflection is zero while that for the (004) reflection is   \n𝐹̂=(𝐹𝑥𝑥00\n0𝐹𝑥𝑥0\n00𝐹𝑧𝑧).  (S4)  \nEquation S4 implies that orbital scattering appears only in the polarization -unrotated \nchannels, i.e., σ -σ’ and π -π’ and is independent of the azimuthal angle.   \n 3 \n Table S1. Ten Fe sites in a Z -type hexaferrite with their Wyckoff positions and local \nsymmetries, referring to Ref.  [2].  \nFe site  Wyckoff position  Local symmetry  \nFe(1) 2a 3̅m. \nFe(2) 4f 3m. \nFe(3) 4e 3m. \nFe(4) 12k .m. \nFe(5) 4e 3m. \nFe(6) 4f 3m. \nFe(7) 4f 3m. \nFe(8) 12k .m. \nFe(9) 4f 3m. \nFe(10) 2c 6̅m2 \n \n  4 \n Lattice expansion due to photothermalization  \nTo test if the Bragg peak position (2 θ) shifts due to heat -driven lattice expansion we \ncollected (00 L) diffraction profiles in the vicinity of the (004) reflection at several delay times \nfollowing laser excitation . Figure S1  shows peak shifts extracte d from Gaussian fits of the \n(004) reflection profiles reflecting possible changes in the lattice constant c. There is a tiny \nchange around ~4 ps after the laser excitation, and recovery sets in on a timescale of 500 ps \nsubsequently following the expansion.  These peak shifts are more than an order of magnitude \nsmaller than the peak width and therefore do not result in observable variation in the intensity  \nof the peak maxima used in the  traces. The peak width of the (004) reflection gets broader by \nthe laser excitation again at ~4 ps after the laser excitation (see  Fig. S1 ). However, these \nvariations are clearly too small to affect the observed oscillations in tr -RXD.   \n \nFigure S1. Fitted results of 2 θ/θ profiles around (004) measure d. (a) Peak center in 2 θ \nand ( b) full width at half maximum (FWHM) as a function of time delay. Red and blue \nplots indicate data for the excited (ON) and unperturbed (OFF) samples, re spectively, at \nroom temperature, whereas black plots indicate a change in the lattice parameter c by the \nexcitation. The inset shows 2 θ/θ profiles at 40 ps of time delay. Laser fluence is 5.7 \nmJ/cm2, and dotted lines indicate either a baseline or t0.  \n  \n5 \n Comparison with other known mechan isms of optically excited coherent magnons  \nHere we discuss the observed coherent magnon with the other known mechanism s, \ni.e., (i) a photomagnetic effect changing magnetic crystalline anisotropy or known as a \ncoherent displacive excitation [ 3] and (ii) the inverse Cotton -Mouton effect or known as an \nimpulsive stimulated Raman scattering [ 4]. These mechanisms are based on a change in \neffective magnetic -field direction since fo r an effective magnetic field Heff along the initial \nmagnetization M, the Landau -Lifshitz equation is  \nd𝐌\nd𝑡=−𝛾𝐌×𝐇eff=𝟎,  (S5)  \nwhere γ is the gyroscopic ratio. Therefore, we consider if the mechanisms can generate Heff \nthat changes the magnetic -field direction. Besides, t he magnon mode involves two -order \nlarger motions for μL than μS, and so we more specifically consider if the mechanisms  can \ngenerate Heff driving μL. To launch the antiphase motion of magnetic moments, Heff needs to \npoint in opposite directions between adjacent L blocks.  \n(i) The photo -induced anisotropy field δ Hi through a coherent displacive excitation is \nphenomenologically described as  \nδ𝐇𝑖(0)=𝜒̂𝑖𝑗𝑘𝑙𝐄𝑗(𝜔)𝐄𝑘∗(𝜔)𝐌𝑙(0),  (S6)  \nwhere 𝜒̂𝑖𝑗𝑘𝑙 is a fourth -rank polar tensor that is symmetric for j and k, E is the electric field of \nlight [ 3]. We use the Cartesian coordinate where x // [100] (= 1), y // [120] (= 2), and z // \n[001] (= 3). In our experimental setup, the scattering plane is spanned by the z-x plane, μL is \nin the x-y plane, and the pump beam is p-polarized. With this condition, p ossible components \nthat explain the antiphase spin motion are twelve components am ong 𝜒̂𝑖𝑗𝑘𝑙 (χ2111, χ2131, χ2331, \nχ3111, χ3131, χ3331, χ1112, χ1132, χ1332, χ3112, χ3132, and χ3332).  \nΤhe symmetry of the L block belongs to point group 6̅m2 and allows only χ3131 among \nthe possible components. However, the symmetry postulates the component invariant \nbetween two adjacent L blocks, connected by the two -fold symmetry along [100]. Therefore, \nthe component can give only an in -phase but not an antiphase spin motion between two \nadjacent L blocks.  \n(ii) The inverse Cotton -Mouton effect in a magnetic media is due to a photo -induced \nmodulation in dielectric permittivity that is proportional to M and is phenomenologically \ndescribed as  \n𝐇𝑖ICME(0)=𝑔̂𝑖𝑗𝑘𝑙𝐄𝑗(𝜔)𝐄𝑘∗(𝜔)𝐌𝑙(0).  (S7)  \nHere 𝑔̂𝑖𝑗𝑘𝑙 is a fourth -rank polar tensor that is symmetric for j and k and represents a photo -\ninduced modulation in the permittivity of the media, and HiICME is the magnetic field induced \nby the Cotton -Mouton effect [ 5]. As 𝑔̂𝑖𝑗𝑘𝑙 has the same intrinsic symmetry with 𝜒̂𝑖𝑗𝑘𝑙, the \ninverse Cotton -Mouton effect can also not explain the observed antiphase spin motion.   6 \n Estimation of strain at the interface of magnetic blocks  \n(i) Local symmet ry of magnetic blocks and elastic stiffness tensor  \nAt first, we discuss the local symmetry of the magnetic blocks to find the elastic \nstiffness tensor components that are allowed by symmetry. The crystal structure of a Z -type \nhexaferrite comprises two magnetic blocks, an S block and an L block, as shown in  Fig. 1(a) . \nAn S block locate s at z = 0 or  1/2 and is alternating with an L block located at z = 1/4 or 3/4. \nThe space group of the crystal structure is P63/mmc , and the point group is 6/ mmm . The point \ngroup of an S block is 3̅m, while that of an L block is 6̅m2. Note that these point groups are \nsubgroups of 6/ mmm . The symmetry -adapted elastic stiffness tensor Cij of the point groups \nare  \n𝐶𝑖𝑗=\n(   𝐶11\n𝐶12\n𝐶13\n𝐶14\n0\n0   𝐶12\n𝐶11\n𝐶13\n−𝐶14\n0\n0   𝐶13\n𝐶13\n𝐶33\n0\n0\n0   𝐶14\n−𝐶14\n0\n𝐶44\n0\n0   0\n0\n0\n0\n𝐶44\n𝐶14   0\n0\n0\n0\n𝐶14\n𝐶112⁄−𝐶122⁄)   \n  (S8)  \nfor 3̅m and  \n𝐶𝑖𝑗=\n(   𝐶11\n𝐶12\n𝐶13\n0\n0\n0    𝐶12\n𝐶11\n𝐶13\n0\n0\n0     𝐶13\n𝐶13\n𝐶33\n0\n0\n0     0\n0\n0\n𝐶44\n0\n0     0\n0\n0\n0\n𝐶44\n0   0\n0\n0\n0\n0\n𝐶112⁄−𝐶122⁄)   \n  (S9) \nfor 6̅m2 [6].  \n \n(ii) Strain through the ultrafast Einstein -de Haas effect  \nFollowing Ref. [ 7], we estimate the strain at the interface of two magnetic blocks. The \nultrafast demagnetization causes a transient volume torque density τ described as  \n𝛕=−1\n𝛾d𝐌\nd𝑡  (S10)  \n[8]. Based on a continuum model, τ contributes to the off -diagonal components of a \nmagnetization -dependent antisymmetric stress tensor 𝜎𝑀̂ as 𝜎12𝑀=−𝜎21𝑀=𝜏3, 𝜎23𝑀=−𝜎32𝑀=\n𝜏1, and 𝜎31𝑀=−𝜎13𝑀=𝜏2. The structural dynamics follows the equation of motion with a \ngiven stress tensor 𝜎̂  \n𝜌𝜕2𝑢𝑖\n𝜕𝑡2=∑𝜕𝜎𝑖𝑗\n𝜕𝑥𝑗𝑗 ,  (S11)  \nwhere ρ is the mass density of the material, xj (j = 1, 2, 3) spans the Cartesian coordinate, and \nui is the displacement along xi. Here 𝜎̂ contains three contributions written as  7 \n 𝜎𝑖𝑗=∑𝐶𝑖𝑗𝑘𝑙𝜂𝑘𝑙 𝑘𝑙 +𝜎𝑖𝑗𝑀+𝜎𝑖𝑗Ext.,  (S12)  \nwhere 𝜎𝑖𝑗Ext. is an external stress tensor component, and 𝜂𝑘𝑙 is the strain described as  \n𝜂𝑘𝑙=1\n2(𝜕𝑢𝑘\n𝜕𝑥𝑙+𝜕𝑢𝑙\n𝜕𝑥𝑘),   (S13)  \nwhere k, l = 1, 2, 3.  \nWe describe the tr -RXD setup as x1 // M and x3 // [001], namely τ = (τ 1, 0, 0). Then \nonly 𝜎23𝑀=−𝜎32𝑀=𝜏1 are non -zero in 𝜎𝑀̂. Here we ignore the diagonal components of 𝜎𝑀̂ \nsince such components do not contain a torque density. From Eqs. ( S10) and ( S11), we find, \nfor example,  \n𝜌𝜕2𝑢2\n𝜕𝑡2=𝜕𝜎23\n𝜕𝑥3=𝜕𝜏1\n𝜕𝑥3=−1\n𝛾𝜕\n𝜕𝑥3(d𝑀1\nd𝑡),  (S14)  \nindicating that the spatial derivative of the ultrafast demagnetization accelerates th e \ndisplacement. The displacements are allowed only along x2 // [120] when we assume the \npump area is significantly larger than the probe area. To simpl ify the discussion, we assume \nthat the magnetization and ultrafast demagnetization are both uniform in a magnetic block. \nWithin t his assumption , the strain in the interior of a magnetic block is uniform. Besides, the \nforces from the internal torques cancel ev erywhere except at the interfaces of the two \nmagnetic blocks, similar to the case of the uniformly magnetized Fe film [ 7]. The stress \nthrough the torque density in a magnetic block appears at both the interfaces with adjacent  \nmagnetic blocks, with the same amplitude but opposite sign. Assuming an interface of two \nmagnetic blocks is identical with the surface of the material, we have the same boundary \ncondition at the interfaces with the surface;  \n𝜎3𝑗=0.   (S15)  \nThis means at the interfaces  \n∑𝐶32𝑘𝑙𝜂𝑘𝑙 𝑘𝑙 =−𝜎32𝑀=−1\n𝛾d𝑀1\nd𝑡.  (S16)  \nThe elastic stiffness tensor is C32kl = C4j (j = k if k = l, otherwise j = 9kl). From Eqs. ( S9) \nand ( S16), we find only C44 = C3232 is allowed in the left -hand side of Eq. ( S16) for both S \nand L blocks. Therefore, the transverse strain at the two interfaces that each magnetic block \nhas is  \n𝜂32=−1\n2𝛾𝐶3232d𝑀1\nd𝑡.  (S17)  \nSince the circular dichroic  signals on the (004) FM reflection are proportional to \nmagnetization [ 9], the ultrafast demagnetization is quantitatively estimated from the rapid \ndrop of the signals displayed in Fig. S2 . The lost magnetization within 200 fs upon the optical \nexcitation with fluence of 5.3  mJ/cm2 is ~8% of the initial value, corresponding to 0.96 \nμB/f.u. as the magnetization at 1 kG is ~12 μB/f.u. (see Fig. S3 ) We s uppos e that the timescale \nand the ratio of angular momentum transfer to the lattice through the ultra fast Einstein -de 8 \n Haas effect are the same as those of the Fe film, shown in Ref. [ 7]; 80% of the lost angular \nmomentum transfers to the lattice within 200 fs. There are two interfaces between the \nmagnetic blocks for each formula unit. Thus, each interface acquires 0.38 μB/f.u. within 200 \nfs. Note that our discussion is based on the simpl e model with uniform magnetization  in each \nmagnetic block , meaning a local ferromagnetic structure but not a ferrimagnetic structure as \nessentially the hexaferrite is.  \nUsing the reported values of γ (= 2.76 π106 /s/T) and C3232 (= 3.761010 J/m3) of a \nsimilar hexaferrite [ 10,11], we obtain the strain η32 ≈ 3.510-2 and the estimated transverse \ndisplacements at the interfaces are ~2.2 pm , which is the comparable scale with observed \nlongitudinal phonon modes in, e.g., a Bi or Fe film  [12, 7].  \nFig. S2. Time traces of the (004) FM reflection with circular polarization of incident x -rays \naround t0 measured at room temperature . The fluence is 5.3 mJ/cm2. Red and blue plots are \nnormalized intensities taken with C+ and C  while black plots are the difference between \nthem. Data taken with C  are vertically shifted to match its baseline with data taken with C+ .  \n \nFig. S3. Magnetiza tion curve measured at room temperature. An applied magnetic field is \nparallel to the basal plane.   \n9 \n Estimation of spin -precession angle in the electromagnon  \nWe here quantify the spin -precession angle in the electromagnon mode upon the \nfemtosecond  laser excitation. Based on the model shown in Fig. 1(b) , the transient magnetic \nform factor for (003) Fm(003)(t) is  \n𝐅𝑚(003)(𝑡)=2(0\n−𝑖𝜇L(cosΔsin𝛽+sinΔcos𝛽cos𝜔𝑡)\n𝜇Ssin𝛼−𝑖𝜇LsinΔsin𝜔𝑡),  (S18)  \nwhere μL(S) and β(α) are the amplitude of the magnetic moment in an L (S) block and the half \nopening angle of the conical structure of an L (S) block, respectively  (see Fig S4). Δ stands \nfor the angle between the transient magnetic moment of an L block and its equilibrium \ndirection  (see Fig. 1b ), and ω is the angular frequency of the electromagnon mode. Here we \nuse the Cartesian coordinate  where x is parallel to M, z is parallel to the scattering vector, and \ny is perpendicular to both x and z (see Fig. 1d for the experimental setup) . The wave vectors \nof incoming x -ray beam ( q) and outgoing x -ray beam ( q’) are then  \n𝐪=(cos𝜃\n0\n−sin𝜃),𝐪′=(cos𝜃\n0\nsin𝜃), and 𝐪′×𝐪=(0\nsin2𝜃\n0),  (S19) \nwith θ being the Bragg angle of the (003) reflection. For incoming π x-ray polarization, \nmagnetic scattering occur s in the πσ’ ( Iπσ’) and the ππ’ channels ( Iππ’), which are described by  \n𝐼𝜋𝜎′=|𝐅𝑚(003)∙𝐪|2\n and 𝐼𝜋𝜋′=|𝐅𝑚(003)∙(𝐪′×𝐪)|2\n,  (S20) \nrespectively. Using Eqs. ( S18)-(S20), the tr -RXD intensities of (003) are obtained as  \n𝐼(003)(𝑡)≈4sin2𝜃(4𝜇L2cos2𝜃sin2𝛽+𝜇S2sin2𝛼)+2𝜇L2sin22𝜃sin2Δsin2𝛽cos𝜔𝑡,\n (S21)  \nwhich is qualitatively the same as Eq. 5. Thus, the normalized intensity ION/IOFF is  \n𝐼ON(𝑡)\n𝐼OFF=1+2𝜇L2cos2𝜃sin2Δsin2𝛽\n4𝜇L2cos2𝜃sin2𝛽+𝜇S2sin2𝛼cos𝜔𝑡,  (S22)  \nand Δ is directly defined by the oscillation amplitude extracted from the fits.  \nFig. S4. Definition of the angles in the magnetic structure ; (a) α [(b) β] is the half opening \nangle of an S [L] block cone.  \n  \n10 \n Fitting of tr -RXD intensities  \nTable S2 shows refined  parameters  using  Eq. 1  convoluted with the time resolution  to \ndescribe the  tr-RXD intensities of the (003) reflection and (004) reflection following the \noptical excitation with fluence of 0.5 mJ/cm2 at room temperature. The d escription of each \nparameter is given in the main text.  \n \nTable S2. Refined  parameters f rom the tr -RXD intensities following the optical excitation \nwith fluence of 0.5 mJ/cm2 at room temperature. Strong correlation s among parameters \nprevent obtaining unique parameters for ( O-3) and ( O-4), which are omitted here.  \nReflection  Arap  Aosc τrap [ps]  τosc [ps]  f [GHz ]  φ []  \n(003)  1.4±0.010-2 3.0±0.010-2 0.25±0.04  18±2  42.0±1.0 179±4  \n(004)  8±110-2 4±110-2 0.13±0.02  32±6  40.8±0.9 1±1 \n \n  11 \n Setup for microwave transmission measurements  \nThis section provides details on the experimental setup used for the microwave \ntransmission measurements. As stated in the main text, the sample was mounted on top of a \ncoplanar waveguide  (see also Fig. S8  below) . For optimum perf ormance up to high \nmicrowave frequencies, a commercial coplanar waveguide board was used (Model B4350 -\n30C-50, Southwest Microwave) with suitable 2.92 mm  coaxial  connectors (Model 1093 -01A-\n5, Southwest Microwave) . Experiments in the sub -20 GHz regime utilized  a commercial \nvector network analyze r (Model 8720ES, Agilent Technologies) with output power set to  1 \nmW. This vector network analyzer (VNA) directly yielded the microwave transmission \nparameter S21 as a function of microwave frequency  f. In the sub -20 GHz regime, primary \nexperimental data were therefore transmission parameters S21(f) from 1 to 20 GHz recorded at \ndifferent magnetic field strengths.  The magnetic field was stepped in 10 mT steps along a full \ncycle over its maximum range of ±0.35 T. Since the spectral  features in recorded data were \nquite broad, a  smoothing procedure with 240 MHz span was implemented in post -processing  \nto reduce noise contributions . \nTo reach frequencies up to 50 GHz, a custom setup that implements microwave \nfrequency multiplication was built. In particular, the output frequency of  a microwave source \n(Model EraSynth+, Era Instruments ; max. 15 GHz ) was multiplied by four. Two frequency \ndoublers (Model TB -973-CY244C +, Mini -Circuits) realized  the four -fold frequency \nmultiplication.  After passing the first frequency doubler, sufficient drive level and signal \npurity for the second frequency doubler w ere provided by a driving amplifier (Model EVAL -\nHMC383LC4 , Analog Devices ) followed by  a filter  (Model TB-883-1832C+ , Mini -Circuits) . \nMicrowave power levels were measured with a power detector (Model ZV47 -K44RMS+ , \nMini -Circuits). To keep the high -frequency signal path as short as possible, the second \nfrequency doubler and the power detector were directly connected to the coplanar waveguide \nprobe. To ease the voltage readout of the power detector, a modulation scheme with a lock -in \namplifier (Model elockin 203, Anaftec Instruments)  was implemented: The microwave \nexcitation power was modulated sinus oidally at 910 Hz and the power detector output was \ndemodulated accordingly. The filter time -constant upon demodulation was set to 0.1 s and a \nsettling time of 0.6 s preceded any power level readout.  Primary experimental data of this \nhigh frequency setup a re therefore the transmitted power Ptr at a particular frequency and a \nparticular magnetic field.  At each microwave frequency, the same magnetic field cycle as for \nthe sub -20 GHz experiments was used.  \nThe high -frequency setup had a lower -frequency cutoff around 30 GHz due to the \namplification and filtering stage in -between the two frequency doublers. The upper frequency \ncutoff was around 51 GHz due to high -frequency limitations of the utilized components.  In \norder to verify  the operation of the frequency m ultiplication chain  around 40 GHz, we have \ntemporarily added an extra bandpass filter in front of the coplanar waveguide probe (Model 12 \n FB-3700 , Marki Microwave) . This filter exclusively passed the frequency range from 33 – 41 \nGHz  and approved  the features observed in Fig. 4b within the filter passband.  In order to \naccess intermediate frequencies between 20 and 30 GHz, the setup was operated with only \none of the two frequency doublers.   \n  13 \n  \nMicrowave t ransmission data  \nIn this section, transmission data related  to the NDD data in the main text are shown.  \nAn important aspect for transmission data is the normalization, which allows compensating  \nfor instru mental features . For sub -20 GHz data, normalization was achieved by e ither data at \nthe highest field  for frequencies below 16.7 GHz  or data at  zero field elsewhere. The \nparticular choice of the frequency crossover at 16.7 GHz  was due to the resonance crossing \nthe instrumental baseline. Figure S5a shows the normalized transmission data obtained in this \nway. Corresponding NDD data is shown in Fig. 4a  in the main text. Except for clearly visible \nfringes in the transmission data, the absorption peaks show a similar  field-frequ ency trend as \nthe NDD data. By comparing the data at positive and negative fields, one can also infer the \nstronger attenuation for negative fields due to NDD.  \nTo facilitate peak extraction, an additional normalization was applied to the signal \nattenuation |S21|n – 1: For each frequency f > 5.75 GHz, the attenuation was renormalized to \nthe mean peak attenuation for f > 5.75 GHz , where peak attenuation refers to the maximum \nattenuation over the scanned magnetic field cycle . In this way, frequency -dependent fringes \nwere reduced  to a constant attenuation value, as is evident in Fig. S5 b. \nNote that  contrary to transmission data,  there is no need for any special normalization \nfor NDD extraction, since we extracted NDD by changing the direction of the magnetic field.  \nWe have obtained equivalent results by extracting NDD upon  changing the microwave \npropagation direction. However, ch anging the magnetic field direction yielded superior data \nquality, since the  underlying data originated  from exactly the same microwave transmission \npathway.  \nFig. S5. Normalized transmission data |S21|n (f) at different magnetic fields . (a) Data \nnormalized to instrumental background extracted  at highest field ( f < 16.7 GHz)  and zero -\nfield (f > 16.7 GHz). (b) Additional scaling of attenuation for fringe suppression in order to \nease peak extraction.   \n14 \n In the higher frequency range around 40 GHz, data normalization was less evident \nsince there was no clear absorption peak as in the sub-20 GHz  data.  Primary un -normalized \ndata are shown in Fig. S6 a, where the voltage reading of th e power detector was scaled by the \nnominal detector characteristics, i.e. a 34.4 dB/V slope  and an output voltage of 1.0 V for a -5 \ndBm input signal.  For frequencies around 39 GHz, a pronounced asymmetry along the field \naxis is already visible in this prim ary dataset.  Normalized data are shown in  Fig. S6b , where \nthe normalization is the maximum power level  over field  at each frequency.  For negative \nfields, the heat map resembles ordinary microwave transmission data with a visually \ntraceable field -frequency evolution of the absorption peak. For positive fields, however, the \nNDD -induced asymmetry renders the absorption peak difficult to trace. In this case, resonant \nNDD is thus more pronounced than resonant absorp tion. This is quite different to the sub -20 \nGHz data, where resonant absorption is predominant and resonant NDD is only visible as a \nsmall additional change ( Fig. S6a). Note that for the sub -20 GHz data  in Fig. S5a , a \nmaximum -based normalization  as applied  here around 40 GHz  yields a qualitatively \ncomparable  heat map .  \nIn analogy to the sub -20 GHz data, we have verified that we obtain the same NDD \ncharacteristics by reversing the microwave propagation direction. With the setup at 40 GHz, \nthis implied to flip the direction of the coplanar waveguide probe.  \nFig. S6. Transmission power  Ptr (f) around 40 GHz at different magnetic fields . (a) Primary \npower levels. (b) Data normalized to maximum power level  over field at each frequency.  \n \n \n15 \n Microwave data at elevated temperature  \nIn order to support the softening of the 40 GHz mode with temperature observed in \ntime-resolved data, additional microwave transmission spectra at elevated temperature were \nrecorded. For this purpose, a heat gun set to 80 °C was fixed in proximity of the s ample. A \nthermometer brought in contact with the sample displayed a temperature of 76 °C, which we \nconsider as an upper limit for the actual sample temperature .  \nIn the sub -20 GHz regime, softening with temperature was approved by following the \nevolution o f the absorption peak in microwave transmission ( Figs. S7a) at room (blue) and \nelevated temperature s (orange) . It is readily seen that for a given magnetic field, heating \nresult ed in mode softening. Moreover, the effect progresse d with magnetic field strength. The \ncorresponding NDD spectra at room and elevated temperature s are shown in Figs. S7b and \nS7c, respectively.  For comparison, the evolution in the absorption peak of the related \ntransmission spectra  is superimposed o nto the NDD spectra  as a black curve . Note that \nsudden jumps in the absorption peak evolution are due to the apparent dual -mode nature of \nthe recorded absorption and NDD spectra. Especially at the largest fields, the branch with \nslightly higher resonance frequency bec ame dominant (see  Fig. S5 ).  \nIn the 40 GHz regime, temperature effects were traced by following the field -\nfrequency evolution of the most intense  NDD  (Fig. S7 d) at room (blue) and elevated \ntemperature s (orange). Also in this case, the experimental data indicate field -progressive \nmode softening upon heating , even though less pronounced as in sub -20 GHz data . The \ncorresponding NDD spectra at room and elevated temperature are shown in Figs. S7e and \nS7f, respectively. When compared to the NDD spectrum in Fig 4b  in the main text, NDD was \nweaker  in these experiments.  For further reference, the evolution of the NDD peak of the \ndataset in the main text is shown in green in Fig. S7 . We tentatively attribute these \ndifferences in NDD to the fact that the sample was re -mounted onto the coplanar wav eguide \nprobe in -between these two experiments. We believe that the geometry, orientation and \ndegree of electric polarization of the sample face that is exposed to microwaves are critical \nparameters for the resultant microwave properties. Importantly, these  observations warrant \nfurther microwave studies of Sr3Co2Fe24O41 with a prospect of even larger NDD effects.  \nLikewise, variations in NDD strength were observed among different runs with re -positioned \nsamples  in the sub -20 GHz regime . However, NDD strengths stayed within ±0.3 dB with \nrespect to the data showed in Figs. 4 and S7. For sub -20 GHz data acquired at a different \nmagnetic field orientation, however, quite important changes in NDD were observed upon r e-\npositioning  (see next section).   16 \n  \nFig S7. Change in microwave resonances upon heating in the sub -20 GHz (a -c) and 40 GHz \n(d-f) frequency range. (a) Field -frequency evolution of absorption peak in transmission data \nat room temperature (blue) and elevated  temperature (orange). There are two curves at each \ntemperature , field up - and down -sweep , and peak extraction was according to Fig. S5b. (b,c) \nNDD data at room and elevated temperature s, respectively. The superimposed black curve s \nare the absorption peaks shown in panel (a). Data in panel (b) is the same as in Fig. 4a  in the \nmain text. (d) Field -frequency evolution of resonant NDD peak at room (blue) and elevated \ntemperature s (orange). The green curve depicts the NDD peak for the NDD spectrum shown \nin Fig. 4b  in the main text. (e,f) NDD data at room and elevated temperature s, respectively.  \n \n \n  \n17 \n Microwave data at different field orientation  \nIn previous sections, u ltrafast time -resolved X FEL and frequency -domain microwave \ntransmission  experiments were performed with the static magnetic field along the [100] \ndirection. Since the two techniques bear different excitation mechanisms, we also performed \nmicrowave experiments with the field along the [120] direction. The orientation of the sa mple \nwith respect to the field and coplanar waveguide is depicted in Figs. S8a and S8b. The \nphotograph in Fig. S8c shows the actual setup with field along the [100] direction  and the \nmicrowave propagation direction kμw as indicated. Rotation of the magnetic field around \n[001] was accomplished by fixation of the coplanar waveguide probe at a different angle. \nSuch a rotation does not only change the orientation of the  relevant  magnetic field in the \nbasal plane, but also changes the relative orientation between static and microwave fields.  \nFigure S8d  illustrates  schematically  the direction of the  electric and magnetic microwave \nfields due to the coplana r waveguide.  Changes observed upon sample rotation are thus a \ncombination of (i) anisotropy with respect to the external field within the basal plane  and (ii) \nanisotropy with respect to the microwave excitation fields.   \n \nFig S8. Sample and probe geometry for microwave transmission e xperiments. (a, b) \nSchematic layout of sample (black) placed on top of coplanar transmission waveguide cross \nsection (beige) with the external field along the [100] or [120] direction, respectively. (c) \nPhotograph of the sample m ounted on top of the coplanar waveguide. The  center trace has a \nwidth of 43 mils (1.1 mm).  (d) Indicative illustration of  typical microwave field distribution \nwithin the waveguide cross section.  The arrows represent field lines of constant intensity.  \n \nFigure S9  shows microwave tr ansmission (a,c) and NDD (b,d)  in the sub -20 GHz \nregime  with field along the [120] direction. Two different datasets are shown in order to point \nat the subtle influence of sample positioning and microwave properties. For upper dataset, \nremarkably large NDD was observed for frequencies around 17.5 GHz . For the lower dataset, \nthe NDD pattern is in qualitative agreement, but at much lower intensity. In accordance to \nour observations in the 40 GHz regime ( Fig. S7), we anticipate the possib ility to fine tune the \nNDD response  and reach even larger effects with Sr3Co2Fe24O41. In contrast to the 40 GHz \n18 \n data, however, sub-20 GHz data feature pronounced resonant microwave absorption that can \nbe compared against each other via  the corresponding transmission spectra in panels (a) and \n(c). Interestingly, these transmission data show quantitatively comparable absorption leve ls. \nThe apparent differences in the color scale are mainly caused by the different NDD response s \nthat superimpose on normalized transmission data.  \nAs is evident when comparing Fig. S9 and Fig. S5, there are important changes when \nflipping the magnetic field direction. To further ease the comparison, the field -frequency \nevolution of the principal absorption peak from Fig. S5b  was superimposed as black line onto \ntransmission spectra in Fig. S9. These superimposed lines follow the highest -frequency ridge \nin the transmission spectra. We thus tentatively conclude that there is a common mode that is \nexcited  with the field along either [100] or [120]. With the field along [100], this common \nmode is the lowest -frequency ridge in the transmission spectrum, while with the field along \n[120], the common mode becomes the highest -frequency ridge in the transmission spectrum. \nIt is also interesting to note that the with the field along [1 20], the lowe st-frequency ridge has \na resonance frequency of 7 GHz at 0.1 T, which is very close to the beating observed in laser -\nexcited time-domain  XMCD data  in the main text .  \nFor further insight into the origin of the observed anisotropy in microwave \ntransmission, additional experiments with the magnetic field at intermediate orientations \nbetween [100] and [120] have been performed (data not shown). These experiment s indicated \nthat field orientations along [100]  or [120] constitute extremum  constellatio ns, from which \nwe tentatively deduce a two -fold rotational symmet ry of the apparent  anisotropy. As \nmentioned above, the anisotropy can originate from  both anisotropy related to the sample or \nrelated to the microwave excitation geometry . Since the sample ha s hexagonal symmetry in \nthe basal plane, we consider anisotropy with respect to the microwave excitation fields to be \nthe more  likely cause for the two -fold rotational symmetry. Within this viewpoint , there is the \npossibility that laser -excited microwave mode  in time -domain expe riments with the field \nalong [100] corresponds to the lowest -frequency ridge of the microwave -excited modes with \nthe field along [120] . Given the anisotropy and multitude of microwave -excited modes in the \nsub-20 GHz regime, however,  the microscopic relationship between the microwave - and \nlaser -excited modes is beyond the scope of the current study and subject to further research.   \nAn anisotropy in microwave properties was also observed for the mode around 40 \nGHz. In this case, the NDD response with field along the [100] direction ( Fig. S10 a) changed  \nits characteristic field -frequency evolution when applying the field along the [120] direction  \n(Fig. S10 b). As is re adily observed,  the resonance was  almost field-independent in this \nconfiguration.  Note that the broad er range scans in Fig. S10  were performed with  a ten times \nlower resolution in the microwave frequency axis than in other illustrated NDD spectra at 40 \nGHz.  19 \n  \nFig S9. Microwave response in sub -20 GHz regime with magnetic field along [120], showing \nnormalized microwave transmission (a,c) and NDD (b,d) of the same sample mounted twice . \nFig S10. 40 GHz NDD with 1 GHz frequency steps and magnetic field along [100] (a) or \n[120] (b).  \n  \n20 \n References  \n1. Dmitrienko , V. E.  Forbidden reflections due to anisotropic x -ray susceptibility of crystals. \nActa Cryst.  A39, 29-35 (1983) .  \n2. Tachibana , T., Nakagawa , T., Takada , Y., Izumi , K., Yamamoto , T. A. , Shimada , T. & \nKawano , S. X-ray and neutron diffraction studies on iron -substituted Z -type hexagonal \nbarium ferrite: Ba 3Co2-xFe24+xO41 (x = 0-0.6). J. Magn. Magn. Mater.  262, 248 -257 \n(2003) .  \n3. Hansteen , F., Kimel , A., Kirilyuk , A. & Rasing , T. Femtosecond photomagnetic \nswitching of spins in ferrimagnetic garnet films. Phys. Rev. Lett.  95, 047402 (2005) .  \n4. Kalashnikova , A. M. , Kimel , A. V. , Pisarev , R. V. , Gridnev , V. N. , Usachev , P. A. , \nKirilyuk , A. & Rasing , Th. Impulsive excitation of coherent magnons and phonons by \nsubpicosecond laser pulses in the weak ferromagnet FeBO 3. Phys. Rev . B 78, 104301 \n(2008) .  \n5. Shen , L. Q. , Zhou , L. F. , Shi, J. Y., Tang , M., Zheng , Z., Wu , D., Zhou , S. M. , Chen , L. \nY. & Zhao , H. B.  Dominant role of inverse Cotton -Mouton effect in ultrafast stimulation \nof magnetization precession in undoped yttrium iron garnet films by 400 -nm laser pulses. \nPhys. Rev. B  97, 224430 (2018) .  \n6. Gallego , S. V. , Etxebaria , J., Elcoro , L., Tasci , E. S.  & Perez -Mato , J. M.  Automatic \ncalculation on symmetry -adapted tensors in magnetic and non -magnetic materials: new \ntool of the Bilbao Crystallographic Server.  Acta Cryst.  A75, 438 -447 (2019) .  \n7. Dornes , C., Acremann , A., Savoini , M., Kubli , M., Neugebauer , M. J. , Huber , L., Lantz , \nG., Vaz , C. A. F. , Lemke , H., Bothschafter , E. M. , Porer , M., Esposito , V., Rettig , L., \nBuzzi , M., Alberca , A., Windsor , Y. W. , Beaud , P., Staub , U., Zhu , D., Song , S., \nGlownia , J. M.  & Johnson , S. L.  The ultrafast Einstein -de Haas effect. Nature  565, 209 -\n212 (2019) .  \n8. Einstein , A. & de Haas , W. J.  Experimenteller Nachweis der Ampèreschen \nMolekularströme. Verhandl. Deut. Phys. Ges.  17, 152 -170 (1915).  \n9. Ueda , H., Tanaka , Y., Wakabayashi , Y. & Kimura , T. Insights into magnetoelectric \ncoupling mechanism of the room -temperature multiferroic Sr 3Co2Fe24O41 from domain \nobservation. Phys. Rev. B  100, 094444 (2019) .  \n10. Gonzalez , A. V., Lütgemeier , H. & Zinn , W. Field dependence of the NMR frequency in \nBaFe 12O19. Sol. Stat. Commun.  17, 1599 -1601 (1975) .  \n11. Gudkov , V. V. , Sarychev , M. N. , Zherlitsyn , S., Zhevstovskikh , I. V., Averkiev , N. S. , \nVinnik , D. A. , Gudkova , S. A. , Niewa , R., Dressel , M., Alyabyeva , L. N. , Gorshunov , B. \nP. & Bersuker , I. B. Sub-lattice of Jahn -Teller centers in hexaferrite crystal.  Sci. Rep.  10, \n7076 (2020) .  \n12. Fritz , D. M. , Reis , D. A. , Adams , B., Akre , R. A. , Arthur , J., Blome , C., Bucksbaum , P. \nH., Cavalieri , A. L. , Engemann , S., Fahy , S., Falcone , R. W. , Fuoss , P. H. , Gaffney , K. J., \nGeorge , M. J. , Hajdu , J., Hertlein , M. P. , Hillyard , P. B. , Horn -von Hoegen , M., 21 \n Kammler , M., Kaspar , J., Kienberger , R., Krejcik , P., Lee, S. H., Lindenberg , A. M. , \nMcFarland , B., Meyer , D., Montagne , T., Murray , E. D. , Nelson , A. J., Nicoul , M., Pahl , \nR., Rudati , J., Schlarb , H., Siddons , D. P. , Sokolowski -Tinten , K., Tschentscher , Th., von \nder Linde , D. & Hastings , J. B. Ultrafast bond softening in bismuth: mapping a solid’s \ninteratomic potential with x -rays. Science  315, 633 -636 (2007) .  \n \n "    },    {        "title": "1711.10790v1.Thermal_contribution_to_the_spin_orbit_torque_in_metallic_ferrimagnetic_systems.pdf",        "content": "1 \n Thermal contribution to the spin -orbit torque in metallic/ferrimagnetic systems  \nThai Ha Pham1, S.-G. Je1,2, P. Vallobra1, T. Fache1, D. Lacour1, G. Malinowski1, M. C. Cyrille3, G. Gaudin2, O. \nBoulle2, M. Hehn1, J.-C. Rojas -Sánchez1* and S. Mangin1 \n1 . Institut Jean Lamour, CNRS UMR 7198, Université de Lorraine, F-54011 Nancy, France  \n2 . CNRS, SPINTEC, F -38000 Grenoble \n3. Leti, technology research institute, CEA, F-38000 Grenoble  \n*juan -carlos.rojas -sanchez@univ -lorraine.fr  \n \nAbstract  \nWe report a systematic study of current -induced perpendicular magnetization switching in \nW/Co xTb1-x/Al thin films with strong perpendicular  magnetic anisotropy . Various Co xTb1-x \nferrimagnetic alloys with different magnetic compensation temperatures are presented. The \nsystem s are characterized using MOKE , SQUID and  anomalous Hall resistance at  different \ncryostat temperature ranging from 10K to 350 K. The current -switching experiments  are \nperformed in the spin–orbit torque geometry where the current pulses are  injected in plane and the \nmagnetization reversal is detected by measuring the Hall resistance. The full reversal magnetization has \nbeen observed in all samples . Some experimental results could only be explained by the strong  sample  \nheating effect during the current pulse s injection . We have found that, for a given composition  x \nand switching polarity , the devices always reach the same temperature Tswitch (x) before \nswitching independently of the cryostat  temperature. Tswitch  seems to scale with the Curie \ntemperature of the CoxTb1-x ferrimagnetic alloys. This explain s the evolution of the critical \ncurrent (and critical current density) as a function of the alloy  concentration.  Future application \ncould take advantages of this heatin g effect which allows reducing  the in -plane external field.  \nUnexpected double magnetization  switching has been  observed when the heat generated by \nthe current allow s cross es the compen sation temperature .  \n \nI. Introduction  \nSpin Orbit Torque switching with perpendicularly magnetized material in Hall bar based devices \noffers  a simple and powerful geometry to probe current induced magnetization reversal  and \nhad opened a new way to manipulating magnetization at the nanoscale . The underlying physics \nis quite rich and complex including origin of spin-orbit torque (SOT), interfacial effects and \nthermal contributions.  Magnetization switching by SOT was first observed  in heavy \nmetal/ferromagnetic, HM/FM, ultrathin films  [1–3]. The torque is mainly related to the spin Hall \neffect (SHE)  [4-10], where the charge current flowing in the heavy metal is converted into a 2 \n vertical spin current due to the large spin -orbit coupling. This spin current is then transferr ed to \nthe FM magnetization, which leads to a torque, namely the spin orbit torque.  It has been  \ndemonstrated for instance using  Pt [4–9], Ta [9–12], W [13–16] as HM and FM layers  with \nperpendicular magnetization like C oFeB [7,9–12,14,15] , Co [4–6], CoFeAl  [16] or  (Co/Ni)  [8] \nmultilayers. Interface effects can play a key role in the SOT i n particular, interfacial spin memory  \nloss  [17] and spin transparency  [18] which affects the transmitted spin . Furthermore, \nadditional charge to spin current conversion can also occur due to Edelstein effect   [19] in \nRashba  [20] and topological insulator interfaces  [19]. There is some attempts to unify a \nmodel  [21–23] including the aforementioned effects.  It might be also an interfacial DMI \n(Dzyaloshinskii -Moriya interaction ) which  favors  formation of  chiral Neel domain wall  [24]. It \nwas shown  in FM/ HM systems that the reversal of the magnetization occurs  first by a magnetic \ndomain nucleation followed by a domain wall prop agation thanks to SHE and iDMI  [8,25 –28]. \nThe thermal contribution  [6,29]  is usually neglected in those experiments.  \nFor possible application s the critical switching current need s to be reduced while maintaining a \nsufficient thermal stability. In the literature the critical  current density to reverse the \nmagnetization , Jcc, is typically of the order of ~1010 to 1012 A/m2 depending on the applied \ncurrent pulse duration and on the in -plane external magnetic field  [4,5,8,30] . Jcc is proportional \nto the m agnetization times the thickness of the FM layer ( Jcc  Mt F). Recently , transition metal -\nrare earth TM -RE ferrimagnetic materials start ed to attract large attention for spin -orbitronics \napplications  [31–36]. In these  ferrimagnetic alloys  the net magnetization  is given by the sum of \nthe magnetization of the two magnetic sub-lattices (r are earth and transition metal)  which are \nantiferromagnetically coupled . The most advantage of ferrimagnetic materials is that its net \nmagnetization M can be tuned by changing its composition or temperature  [37]. As a result , a \nmagnetic compensation point with zero magnetization can occur for a certain alloy \nconcentration, xMcomp , or temperature, TMcomp , where the magnetization of both sub -lattices \ncompensates . Moreover TM -RE thin f ilms are characterized by a large bulk magnetic anisotropy \nperpendicular to the film plane (PMA) which make easier to integrate TM -RE with different NM \nmaterials while keeping large thermal stability   [38]. Furthermore, the control of magnetization \nswitching using ultrafast femtosecond laser pulse has been demonstrated recently for various \nTM-RE materials  [39,40] . Those features are encouraging  to combine the control of \nmagnetization by both optical and electrical means. Concerning the Spin Orbit Torque  (SOT) \nswitching, reports on  experiments with TM-RE alloys claim  that the spin -orbit torque efficiency \nreaches a maximum at the magnetic compensation point  [31–36], however the critical current is \nnot minimum  at this point  [33,35,36] . In this study we address the SOT-switching experimen ts \non well characterized  //W/Co xTb1-x/Al systems  for various concentrations . We demonstrate that \nthermal effects are keys to explain the current induced magnetization reversal in this system . \nWhen the current is injected in the bilayer the Joule heating leads to a large increase of the \nsample temperature . Using systematic SOT measurement s at different temperatures and alloy 3 \n compositions, we establish that for each concentration x the current induced magnetization \nswitching occurs for a unique sample tempe rature Tswitch (x). Tswitch  scales with the Curie \ntemperature ( TC) of the alloy. Those new f indings  open new rooms to explore combination of \nSOT and thermal contribution towards reducing critical current density to reverse M and \nconsequently  low power consu mption  applications . In the specific case where Tswitch  is close to \nTMcomp an unexpected “double switching ” is observed.  \n \nII. Basic characterization  \nTo study SOT magnetization switching in RE -TM alloys, a model system composed of CoxTb1-X \nferrimagnetic alloy s deposited  on a tungsten heavy metal  with high charge to spin conversion \nefficiency  [13] was considered .  The samples were grown by dc magnetron sputtering on \nthermally oxide Si  substrates (Si -SiO 2). The full stacks of the samples are Si -SiO 2//W(3  \nnm)/Co xTb1-x(3.5 nm)/Al(3  nm) with  0.71  x  0.86 . The 3 nm thick Al (naturally oxidized and \npassivated  after the deposition ) is used to cap the ferrimagnetic layer. The W and CoTb lay ers \nhave amorphous structure.  As described in the introduction , ferrimagnetic alloys like CoTb can \nshow a compensation point at which the Co and Tb moments  cancel  each other, resulting in \nzero net magnetization . When the net magnetization of the alloy is parallel (resp. antiparallel ) to \nthe magnetization of the Terbium sub -lattice the alloy will be call Terbium rich  (resp. Cobalt \nrich) . The samples were characterized by a SQUID -VSM  magnetometer and Magneto -optic ally \nKerr effect (MOKE)  at room temperature . The SQUID measurements obtained at room \ntemperature are presented in Fig 1a . Magnetization compensation is observed  for a \nconcentration x Mcomp =0.77 where  the coercivity Hc diverge s and the net saturation \nmagneti zation M s tends to zero . This value is close to  the one report ed for bulk and thicker \nCoxTb1-x films  [37,41]  at room temperature . Additionally to the divergence of Hc, MOKE \nmeasurements show that  the Kerr angle rotation changes its sign between Co-rich and Tb-rich \nsamples , which can be explained by the fact that  Kerr rotation is mainly sensitive to the Cobalt \nsub-lattice  (see for instance fig . S1 in supplementary material  [42]).  Both SQUID and MOKE \nresults clearly show that all CoTb films studied have a strong out of plane magnetic  anisotropy.  \nTo study Spin Orbit Torque switching,  the stacks were patterned by standard UV lithography \ninto micro -sized Hall crosses with a channel of 2, 4, 10 and 20 m. The results  shown ar e \nobtained for  a width of 20 m unless other wise  specified. Ti(5)/Au(100) ohmic contacts were \ndefined  by evaporation deposition and lift-off method  on top of W layers . By measuring the \nanomalous Hall resistance RAHE of the hall crosses while sweeping the external perpendicular \nmagnetic field Hz at different temperatures , we could determin e the magnetic compensa tion \ntemperature of the samples . Fig. 1b shows  the temperature dependence of Hc for 78% of Co.  \nThe coercive  field  Hc diverge s around  280 K which determines TMcomp  for this composition. 4 \n Moreover, we can observe in the insets that the RAHE(Hz) cycle is reversed for Tb -rich (T<280 K) \nand Co -rich( T>280 K) phases , namely change of field switching polarity (Field -SP). The latter  is \ndue to the fact the Anomalous hall resistance is sensitive to the cobalt sub -lattice . Van der Pauw \nresistivity measurements leads to a resistivity of W in Si-SiO 2//W(3 nm)MgO(3 nm) of W = 162 \n.cm. Then  we could deduce the Co 0.72Tb0.28 resistivity CoTb = 200 .cm which  decrease s to \n135 .cm when the Cobalt concentration reaches Co0.86Tb0.14 in accord with previous \nresults  [43]. Despite this trend,  the a mplitude of  RAHE(Hz), RAHE, increases as a function of the \nCo concentration verifying that RAHE is mainly sensitive  to the Cobalt sub -lattice  (see also Fig \nS2  [42]). \n \nIII. Thermal ly assisted and spin -orbit torque switching  \nFig. 2a shows a scheme of a Hall bar along with the convention s used  for current injection , \nvoltage probe  and directions  axes .  Typical RAHE(Hz) cycles obtained at room temperature with a \nlow in -plane d c current of 400 A (charge current density of about 2.4 109A/m2 flowing in each \nlayer) for a Tb-rich ( resp. Co-rich)  sample  is shown in Fig. 2b ( resp. Fig. 2e) . As expected , a \nchange of  Field-SP is observed since the alloy net magnetization is parallel to  the magnetization \nof the Cobalt sub -lattice in one case and antiparallel  in the other . For the same samples the \ncurrent -induced switching cycles are shown in Fig. 2c -d (Tb -rich) and 2f -g (Co -rich) with a n in-\nplane bias field of Hx=100 mT  (Fig. 2c and 2f) and -100 mT  (Fig. 2d and 2g). The current injection  \nwas performed with pulse duration  of 100 s using a K6221 source coupled to a K2182  Keithley  \nnanovoltmeter. The Hall voltage is measured during the pulse. We have observed the current \ninduced magnetization switching in all the samples for 0.72  x  0.86. The Hall resistance \namplitudes  are the same  for the current -switching and the field -switching cycles indicating that \nthe reversal  of magnetization is fully achieved  in both cases . The da ta of the full series are \nshown in Fig. S3   [42]. Sharp current switching  are observed and  the critical current reduce s \nwhen Hx increases  following similar trends that for ferromagnetic materials  [30] as shown in \nFig. S4 . Remarkably, we observe a full magnetization reversal even for a n in-plane field  Hx as low \nas 2 mT. The role of the in -plane field can be understood as the field to balance the iDMI to \npropagate domain  walls which have in -plane magnetization after nucleation of magnetic \ndomains or the  field to break the symmetry and to allow for a deterministic switching  [8,30] . If \nthe SOT depends on the Co moment , the SOT acts as an effective field HSHE  m   [24,44]  \nwhere  m is the magnetic moment and  the spin polarization of the spin current Js injected from \nthe W layer into the CoTb layer.  is along the y direction in our measurement geometry (it \nchanges between +y and –y when the direction of the injected current is inverted). m changes \nits sign upon  the change of the in -plane field  direction . Then the sign of the Hall cycle vs current, \nRAHE(i), is reversed when Hx is reversed  as observed in Fig 2 c-f. Additionally in ferrimagnetic 5 \n alloys, the effective field HSHE can be reversed if a Co -rich s ample is replaced by a Tb-rich one (a \nschematic is shown in Fig. 2h for Hx>0 and i>0). The identical effect will be observed if the same \nsample is kept and the magnetic compensation temperature is crossed. The fact that the \nsample s which have been identified as Co-rich and Tb -rich at room temperature are showing \nthe same current -switching polarity (Fig 2c and 2f) can only be understood if the so call Tb -rich \nsample  has cross ed compensation to become Co -Rich. This compensation crossing is due to the \nJoule heating effect.  This assumpt ion was tested by measuring  RAHE(Hz) cycles for different \napplied current pulses on the “Tb-rich” sample shown in Figure 3. We observe that for applied \ncurrent s i< 19.5 mA the sign of the cycle demonstrate a “ Tb-rich” nature, however  for current \ni>19.5 mA the RAHE(Hz) cycles are reversed and demonstrate a  “Co-rich” nature . This is clear \nexperimental evidence that  for current close to 19.5 mA the device reach es the sample \ncompensation temperature  (TMcomp  ~320 K). This demonstrates  that the sample is strongly \nheated during the current -switching experiments. The temperature can be determined by the \nresistance value as explained in the next section. In Figure 3 the corresponding temperature is \nshown using color code.  We can determine  that a temperature of 460 K is reached  for 24mA \nwhich is the critical current to switch M (Fig. 2c) . Th is current -switching  of 24 mA is then \nobtained for a temperature above TMcomp which explain why the sign of the RAHE(i) cycle  is the \none expected for a Co -rich sample.  We have performed RAHE(Hz) cycles with intensity current \npulse as high as 34 mA  (~525 K)  where we can observe that the device shows a  ferromagnetic \nhysteresis loop and remain perpendicular ly magneti zed. TC is then higher than 525 K . \n \nIV. Characteristic temperatures of switching  \nSince we have addressed the reason of the observed switching polarity  several questions  are \narising :  i) how much are the devices heated when the switching  occurs ? ii)  Does the \ntemperature  at which  the switching occurs changes with the initial temperature (temperature \nat which the experiment is carried out, T cryostat ) ?  iii) How d oes the switching current and \nswitching temperature depend on composition? And iv ) What is the physical meaning of this \nswitching temperature: Angular compensation temperature T Acomp ? In order to address all those  \nquestions we have performed a series of temperature dependence experiments for various  \nsamples.  \nFig. 4 a shows the RAHE(ipulse) cycles for Hx>0 at different cryostat temperature for W/Co0.73Tb0.27 \n(Tb-rich at room temperature). We have observed the Down -Up current -switching polarity  at \n300 K . For 150 K Tcryostat  250 K a double  current switching  loop is observed. This type of \ndouble current switching can be explain ed when the switching temperat ure is close to TMcomp  \nand its origin will be discussed later in the paper (Fig. 4a show s only the case of 150 K for \nclarity ). For  10 K T 100 K we obser ve only Down -Up current -switching polarity  (we didn’t 6 \n increase too much the pulse current to avoid burn ing the device). One can calibrated the real \nsample temperature at different pulse -current performing the following protocol: i) measuring \nthe resistance of the current channel Rchannel (ipulse) as function of pulse current intensity as \nshown in Fig. 4b for  different cryostat temperatures, and ii) measuring the temperature \ndependence of the current channel Rchannel (T) as shown in F ig 4c (for which we use a very low dc \nbias current of only 400 A). Interesting ly, we observe that for Co -rich current -switching polarity \n(Down -Up) the device reaches the same resistance (1.373 k ) and consequently the same \nswitching temperature Tswitch  = 435 K  25 K  for this W/Co 0.73Tb0.27. We note that the resistance \ndecreases when T increases which is a feature and confirmation of amorphous materials  [45]. \nWe have performed the same protocol for various compositions and different devices. An \nexam ple for Co -rich sample at room temperature is shown in Fig. 4 d-f (W/Co 0.79Tb0.21) where \nwe also observe that the critical current heat s up the device to the same channel resistance  (Fig. \n4e), so the same T switch  (~485 K for this Co xTb1-x sample)  irrespective  of the initial temperature . \nMoreover , on this particular sample TMcomp  is about 500 K  and we observe no change of C urrent -\nSP even for T as low as 10 K  which is well below its TMcomp .  \nAdditionally to the characteristic Tswitch  we have just disc ussed , one can also investigate  the \ntemperature dependence of the critical current as shown in Fig . 5a for a Co0.78Tb0.22 sample ( Co-\nrich at room temperature ).  The extrapolation of the linear dependence to zero current is \ndefined as T*. In Fig. 5b is found out that T*~470 K for W/ Co 0.78Tb0.22.  \nFig. 6a show s RAHE(ipulse) for Co 0.72Tb0.28 performed for a cryostat temperature of 150 K to \nhighl ight the observation of the two switching. At lower current  (37.5 mA) , i.e. lower Joule \nheating effect,  we observed Up-Down current -switching polarity  while  the second one which \noccurs  at higher current  (43 mA)  is Down -Up. This can be  understood considering that to \nachieve the first switching the device reaches a temperature below its TMcomp  so the sample is  \nstill Tb -rich and the Up -Down current -switching polarity observed is as expected (i.e Fig 2h) . If \nwe continue  increas ing the intensity of the applied in -plane pulse current we overcome TMcomp  \nand the n the  perpendicular component of effective torque field HSHE now changes its sign as \ndiscussed previous ly. Consequently, the second observed switching agree well for Co -rich phase  \n(Down -Up). In Fig. 6b is shown the temperature dependence of both switching current s.  As \ndiscussed, the first  (second) switching  agrees with a Tb -rich (Co -rich) current -switching polarity \nand happens for T<TMcomp  (T>TMcomp ). The linear extrapolation of both switching currents  \nroughly tends to  350 K (Tb-rich switching ) and T*~455 K (Co-rich switching ). The value of T* \nseems to be slightly higher than Tswitch  (~435 K   20 K  as determined in Fig 4c).  \n \n 7 \n V. T-x switching  phase diagram  and conclusions  \nIn figure 7a the different characteristic temperature s of our W/CoTb systems can be plotted in  \nthe (T,xCo) phase diagram. The determine d TMcomp  decreases linearly with the Co  concentration \nas reported for bulk CoTb and thick CoGd films (300 nm)  [41,46] . However Tswitch  and T* \nincrease l inearly with the Co-concentration  and scale with the Curie temperature Tc thus  \ndepend ing on composition , and independent of initial temperature. It is remarkable that the \nTswitch  and T* are nearly the same, indicating that, to achieve the switching, one has to reach a \nspecific temperature. The three first questions in section IV  are answered . Now let’s discuss the  \nphysical meaning of these switching temperatures. It is clear that for Co -Current SP the \ntemperature of switching is above TMcomp  and below Tc. Fig. 7a also shows Tc in bulk CoTb after \nHans et al.   [41]. The angular composition temperature , TAcomp , scales with  TMco mp. Indeed, \ntypically TAcomp  ~(TMcomp +30 K) for Co0.775Gd 0.225 thick films  (300 nm)   [46]. This is explained by \nthe relationship between the angular moment  L, magnetic moment and the gyromagnetic ratio \n or Landé g -factor ( LTM, RE =MTM, RE /TM, RE  and =gB/hb). Therefore  in CoTb it is expected that the \ntrends of TAcomp  and TMcomp  are similar and they decreas e with increasing Co concentration . \nHowever Tswitch  increases with Co concentration which indicates  that Tswitch  is not scaling with \nTAcomp  or TMcomp. For sake of comparison we plotted the switching -current for experiment \nperformed at room temperature  together with the temperature increase T = TswitchTcryostat ,  \ndue to Joule heating effect, Fig. 7b . We can observe that the temperature is increased between \n100 K and 300 K. This variation will increase when we reduce Tcryostat .  Considering that resistivity \nof both, CoTb  and W layers, change similarly with temperature and using the resistivity \nmeasured at room temperature we can estimate the critical current density JCC flowing on each \nlayer as displayed in Fig. 7c. We observe that Jcc on W is reduce d by a factor of ~2 wh ile varying \nthe composition of CoTb . We observe the minimum of Jcc at the lower Co concentration \nmeasured. This c an be explained by the fact that TC and Tswitch  decrease with de creas ing Co-\nconcentration  but a relation ship between Tswitch  and TC will need additional studies . \nConclusions  \nIn conclusion w e have fully characterized the current -induced switching experiment in a series \nof //W(3nm)/Co xTb1-x(3.5nm)/AlO x(3nm) samples. In addition to the SOT effect we demonstrate \na strong thermal contribut ion to achieve the magnetization reversal.  For the Co -rich current -\nswitching polarity the device needs  to reach the same temperature  Tswitch  to achieve the \nswitching. This Tswitch  increase s with Co -concentration which then scale with Curie temperature  \nTc. It is then unlikely that Tswitch  corresponds to angular momentum compensation temperature \n(which scales with TMcomp  decreasing with Co -concentration) . Those results highl ight the \nimportance of considering thermal contributions in SOT switching experiment s and the fact that  \nthe spin Hall angle determination might be overestimated when thermal contribution is 8 \n neglected . The use of resistive W layer i ncrease the heating of the device, reducing strongly the \nexternal in -plane needed to assist the SOT . Those results are important for the full \nunderstanding of current -induced magne tization switching and may lead the way to  new \ntechnological applications  taking advantages of the rather strong heating  \nAcknowledgements  \nThis work was supported partly by the french PIA project “Lorraine Université d’Excellence”, \nreference ANR -15-IDEX -04-LUE. by the ANR -NSF Project, ANR -13-IS04 -0008 - 01, “COMAG” by \nthe ANR -Labcom Project LSTNM, Experiments were  performed using equipment fro m the \nTUBE —Daum funded by FEDER (EU), ANR, the Region Lorraine and Grand Nancy.  We thank  J. \nSampaio and S.  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Insets show Hall resistance cycle for a temperature below (left) and above \n(right) TMcomp . The change of field - switching polarity (Field -SP) evi dences that RAHE is mainly sensitive to \nthe magnetization of the Cobalt sub -lattice.  \n  \n13 \n  \n \nFigure 2. Anomalous Hall effect and Current -induced magnetization reversal at room temperature on Si -\nSiO 2//W(3)/Co xTb1-x (3.5)  /Al(3). a) Scheme of the Hall bar along with geometry used. The spin \npolarization  is along y axes. (b and g ) Sweeping perpendicular field (H ||z) with a low dc bias of 400 A. \n(c,d,e and f ) Sweeping in -plane current (i pulse ||x) with an in -plane field Hx=±100 mT .The width of the \nchannel current is 20 m. The current switching polarity (current -SP) for Hx>0 is Down -Up for Co -rich and \nTb-rich samples (c,f). The Current -SP in (c) is opposite than predicted for Tb -rich samples as shown in the \nschematic in (h). The p erpendicular effective torque field HSHE is proportional to m and should lead to \na change of the sign in the RAHE(i) cycle  when changing from Co -rich to Tb -rich phase.  \n  \n14 \n  \n \nFigure 3. R AHE(Hz, ipulse)  cycles on Si -SiO 2//W(3)/Co 0.76Tb0.24(3.5)/Al(3) Hall bar measured at room \ntemperature. The cycles are vertically offset for clarity. It is observed that for i <  19.5 mA the cycle has a \nsignature corresponding to Tb -rich phase according to our convention. However for i> 19.5 mA the cycle \nchanges their sign and now corresponds to Co -rich phase.  It is an evidence of the Joule heating effect \nwhen high pulse current is applied. 19.5 mA roughly corresponds to TMcomp . The critical current for this \ndevice is about 24 mA. Thus during the electrical switching T device>TMcomp >300 K for this Co 0.76 system.  \n  \n15 \n  \n \n \nFig4. a) RAHE(ipulse)  at different cryostat temperatures. Cycles for x=0.73 (Tb -rich at room temperature ). b) \nRChannel (ipulse)  and c) RChannel (T)  at different cryostat temperatures. The vertical dashed line points out the \ncritical current to reverse M. It is observed that independently of the initial temperature, the device \nalways reaches the same value of  longitudinal  resistance (1373 ) which means it reaches the same \ntemperature .  The linear extrapolation of RChannel (T)  allows us to know the temperature corresponding to \nthe current -induced magnetization reversal . Such a temperature is defined as Tswitch. RChannel (T)  is \nperformed with  a low bias current of 400 A. For Co 0.73 we found that Tswitch= 435 K  25 K.  d) RAHE(ipulse)  \ncycles for x=0.79 (Co -rich at room temperature)  at different cryostat temperatures. e) RChannel (ipulse)  and f) \nRChannel (T)  at different cryostat temperatures. It is also observed that independently of the initial \ntemperature, the device always reaches the same resistance , thus is the same temperature . In this case \nit corresponds to 1291  and Tswitch ~ 485 K.  \n  \n16 \n  \n \nFigure 5 . a) RAHE(ipulse)  at different cryostat temperatures. Cycles for x=0.78 (Co -rich at r oom \ntemperature ). b) The critical current to reverse M increase s linearly when T decrease and saturate for T< \n50 K. The extrapolation of th e linear behavior at higher temperature for zero current is defined as T*. \n  \n17 \n  \n \nFig6. a) RAHE(ipulse)  at 150K for x=0.73 (Tb -rich at room temperature ). There are two switching s: i) at lower \ncurrent it agrees with a Tb-rich switching polarity (Up-Down) . ii) The reversal with hi gher current agrees \nwith a Co -rich switching polarity (Down -Up).  b) Temperature dependence o f the critical current s for this \ncomposition. TMcomp  would be between 350 K and 455 K.  \n  \n18 \n  \n \nFigure 7 . a) Characteristic temperatures as function of Co concentration: T*, Tswitch and TMcomp  \nindependently measured in Hall bar patterned devices (lines are guides for the eyes). It is observed that \nTMcomp  follows the same behavior reported for bulk Co xTb1-x alloys. T* and Tswitch have the same trend \nthan that of the Curie temperature Tc. The gre en dashed line stands for Tc in bulk CoTb after Hans et \nal.  [41]. b) The total critical current injected to reverse M when the experiment is performed at room \ntemperature (= Tcryostat ), and the variation of temperature TswitchTcryostat  to reach the switching. c) The \ncritical current density, calculated from b, flowing in W and CoTb layers, respectively.  \n \n1 \n Supplementary Material  \nThermal contribution to the spin -orbit torque in metallic/ferrimagnetic systems  \nThai Ha Pham1, S.-G. Je1,2, P. Vallobra1, T. Fache1, D. Lacour1, G. Malinowski1, M. C. Cyrille3, G. Gaudin2, O. \nBoulle2, M. Hehn1, J.-C. Rojas -Sánchez1* and S. Mangin1 \n1 . Institut Jean Lamour, CNRS UMR 7198, Université de Lorraine, F-54011 Nancy, France  \n2 . CNRS, SPINTEC, F -38000 Grenoble \n3. Leti, technology research institute, CEA, F -38000 Grenoble  \n*juan -carlos.rojas -sanchez@univ -lorraine.fr  \n \nS1- Magneto -optically Kerr effect measurements at room temperature  \nFigure  S1 show s the coercitivity obtained by MOKE of W/Co xTb1-x thin film as a function of x (the Co-\nconcentration ). The  results show that coercive field Hc diverges about Co -concentration of 7 7 % in \nagreement with SQUID results (Fig. 1a). The insets highlight the  opposite sign of Kerr angle  rotation \nwhen the CoTb alloy change from Tb -rich to Co -rich phase.  \n \n \nFigure S1 . //W(3)/Co xTb1-x (3.5)  /Al(3) : Coercive field Hc obtained by MOKE measurements at room \ntemperature  as a function of the  Cobalt concentration . Insets show raw data of MOKE cycles for a Tb -\nrich (left) and Co -rich (right) samples.  \n \n \n2 \n S2- Hall resistance amplitude | RAHE|, channel resistance Rchannel , and Co xTb1-x \nresistivity CoxTb1 -x at room temperature  \nFigure S2 present the evolution of the Hall resistance amplitude |RAHE|, the channel resistance Rchannel , \nand the CoxTb1-x resistivity CoTb at room temperature  as a function of the Co -concentration . Despite the \ndecreas e of the channel resistance  with increasing Co -concentration, we observe that | RAHE| increase s \nwith Co-concentration . Thus corroborate  that the Hall resistance is mostly sensitive to Co magnetic \nmoment.  The W resistivity, W = 162 .cm, is of similar order than CoTb alloys.  \n \n \nFigure S2 . Si-SiO 2//W(3)/Co xTb1-x (3.5)  /Al(3) : The change of Hall resistance amplitude  |RAHE|, channel \nresistance  Rchannel , and Co xTb1-x resistivity as a function of the Cobalt concentration at room temperature . \nThe vertical blue dashed line points correspond to the magnetic compensation point at room \ntemperature. The horizontal red dashed line shows the value of W resistivity (ρ W).  \n \n \n \n3 \n S3- Current -switching in the full W/CoxTb1 -x series for different in -plane field at \nroom temperature  \nFigure S3 (S4) presents the c urrent -induced magnetization reversal performed at room temperature on \nSi-SiO 2//W(3  nm)/Co xTb1-x (3.5 nm) /AlOx(3 nm) with an in -plane field Hx=100 mT (5 mT).   \n \n \nFigure S3 . Current -induced magnetizati on reversal at room temperature : Hall resistance as a function of \nthe injected current measured for various  ipulse Si-SiO 2//W(3  nm)/Co xTb1-x (3.5 nm) /AlOx(3 nm), with an \nin-plane field of Hx= 100 mT.  \n4 \n  \nFigure S4 . Current -induced magnetization reversal at  room temperature : Hall resistance as a function of \nthe injected current measured for various  ipulse Si-SiO 2//W(3  nm)/Co xTb1-x (3.5 nm) /AlOx(3 nm), with an \nin-plane field of Hx= 5 mT. \n \n  \n5 \n S4- Power consumption  in the W/Co xTb1-x Hall bar with 20 m of width  at room \ntemperature  \nFigure S 5 shows the electrical power consumption to  revers e the magnetization i n Hall bar of length L= \n100 m and width  of w  = 20 m at room temperature on  various  Si-SiO 2//W(3  nm)/Co xTb1-x (3.5 nm) \n/AlOx (3nm) with an in -plane field  of 100 mT.  \n \n \nFigure S5 . Si-SiO 2//W(3)/Co xTb1-x (3.5)  /AlOx(3): The electrical power consumption to switch  the \nmagnetization at room temperature of a Si-SiO 2//W(3)/Co xTb1-x (3.5)  /AlOx(3) hall bar as a function of  \nCobalt concentration. The current channe l dimensions are length L= 100 m, and width of w = 20 m. \n  \n6 \n S5- Hx-I switching phase diagram in the W/Co xTb1-x  \nFigure S6 (S7) presents a  2D plot summarizing the c urrent –switching cycles performed under different \nexternal in -plane field Hx (phase di agram ) at room temperature  for a channel width of w= 20 m (10 \nm). Figure S6  are the results obtained for   Si-SiO 2//W(3  nm)/Co 0.86Tb14 (3.5 nm) /AlOx(3 nm) and fig. S7 \nfor Si-SiO 2//W(3  nm)/Co 0.84Tb0.16  (3.5 nm) /AlOx(3 nm). \n \n \n \nFigure S6 . 2D-plot of c urrent –switching cycles performed at room temperature on Si -SiO 2//W(3  \nnm)/Co 0.86Tb14 (3.5 nm) /AlOx(3 nm) for a channel width of w  = 20 m. The R(ipulse, H x) cycles were carried \nout with different  applied in plane  field between –3 kG and + 3 kG. The red (blue) color region stand for \nUp (Down) magnetic configuration according the schematic Hall bar shown in Fig. 2a.  \n \n7 \n  \nFigure S 7. 2D-plot of c urrent –switching cycles performed at room temperature on Si -SiO 2//W(3  \nnm)/Co 0.84Tb16 (3.5 nm) /AlOx(3 nm) for a channel width of w  = 10 m. The R(ipulse, H x) cycles were carried \nout with different  applied in plane  field between –6.5 kG and +6.5 kG. The red (blue) color region stand \nfor Up (Down) magnetic configuration according the schematic Hall bar shown in Fig. 2a. \n \n"    },    {        "title": "0811.2118v1.Selection_rules_for_Single_Chain_Magnet_behavior_in_non_collinear_Ising_systems.pdf",        "content": "arXiv:0811.2118v1  [cond-mat.mtrl-sci]  13 Nov 2008Selection rules for Single-Chain-Magnet behavior in non-c ollinear Ising systems\nAlessandro Vindigni1∗and Maria Gloria Pini2\n1Laboratorium f¨ ur Festk¨ orperphysik, Eidgen¨ ossische Te chnische Hochschule Z¨ urich, CH-8093 Z¨ urich, Switzerlan d\n2Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche,\nVia Madonna del Piano 10, I-50019 Sesto Fiorentino (FI), Ita ly\n(Dated: November 2, 2018)\nThe magnetic behavior of molecular Single-Chain Magnets is investigated in the framework of a\none-dimensional Ising model with single spin-flip Glauber d ynamics. Opportune modifications to\nthe original theory are required in order to account for reci procal non-collinearity of local anisotropy\naxes and the crystallographic (laboratory) frame. The exte nsion of Glauber’s theory to the case of\na collinear Ising ferrimagnetic chain is also discussed. Wi thin this formalism, both the dynamics\nof magnetization reversal in zero field and the response of th e system to a weak magnetic field,\noscillating in time, are studied. Depending on the geometry , selection rules are found for the\noccurrence of slow relaxation of the magnetization at low te mperatures, as well as for resonant\nbehavior of the a.c.susceptibility as a function of temperature at low frequenc ies. The present\ntheoryapplies successfully tosome real systems, namely Mn -, Dy-, andCo-based molecular magnetic\nchains, showing that Single-Chain-Magnet behavior is not o nly a feature of collinear ferro- and\nferrimagnetic, but also of canted antiferromagnetic chain s.\nPACS numbers: 75.10.-b, 75.10.Pq, 75.50.Xx, 75.60.Jk\nI. INTRODUCTION\nSlow dynamics of the magnetization reversal is a cru-\ncial requirement for potential applications of Single-\nChain Magnets (SCM’s)1,2,3, and nanowires in general,\nin magnetic-memory manufacture. For nanowires with\na biaxial anisotropy, provided that their length is much\ngreater than the cross section diameter but smaller than\nexchange length, this phenomenon is governed by ther-\nmal nucleation and propagation of soliton-antisoliton\npairs; the associated characteristic time is expected to\nfollow an Arrhenius law4,5. For genuine one dimensional\n(1D) Ising systems with single spin-flip stochastic dy-\nnamics, a slow relaxation of the magnetization was first\npredicted by Glauber6in 1963. Through Glauber’s ap-\nproach,manyphysicalsystemswereinvestigated,ranging\nfrom dielectrics7,8,9to polymers7,10,11. More fundamen-\ntally, this model has been employed to justify the use\noftheKohlrauch-Williams-Wattsfunction10,12(stretched\nexponential) to fit the relaxation of generalized 1D spin\nsystems. Also the universality issue of the dynamic crit-\nical exponent13,14,15,16,17of the Ising model18, as well as\nstrongly out-of-equilibrium processes (magnetization re-\nversal19, facilitated dynamics19, etc.) have been studied\nmoving from the basic ideas proposed by Glauber.\nIn this paper, single spin-flip Glauber dynamics is\nused to investigate theoretically the slow dynamics of\nthe magnetization reversal in molecular magnetic sys-\ntems. In particular, we extend Glauber’s theory6to\nthe Ising collinear ferrimagnetic chain, as well as to the\ncase of a chain in which reciprocal non-collinearity of lo-\ncal anisotropy axes and the crystallographic(laboratory)\nframe is encountered. Such extensions are motivated by\nthefact that(i) inmolecular-basedrealizationsofSCM’s,\nantiferromagnetic coupling typically has a larger inten-\nsity than the ferromagnetic one; in fact, the overlappingof magnetic orbitals, which implies antiferromagnetic ex-\nchange interaction between neighboring spins, can be\nmore easily obtained than the orthogonality condition,\nleadingtoferromagnetism20,21,22; (ii)non-collinearitybe-\ntweenlocalanisotropyaxesandthecrystallographic(lab-\noratory) frame takes place quite often in molecular spin\nchains. Besides magnetization reversal, the dynamic re-\nsponse of the system to a weak magnetic field, oscillating\nin time at frequency ω, is also studied. Depending on the\nspecific experimental geometry, selection rules are found\nfor the occurrence of resonant behavior of the a.c.sus-\nceptibility as a function of temperature (stochastic reso-\nnance) at low frequencies, as well as for slow relaxation\nof the magnetization in zero field at low temperatures.\nThe paper is organized as follows. In Sect. II we ex-\ntend Glauber’s theory6, originally formulated for a chain\nofidenticalandcollinearspins, tothemoregeneralmodel\nof a chain with non-collinear spins, possibly with Land´ e\nfactors that vary from site to site. In Sect. III we use\ntwo different theoretical methods (the Generating Func-\ntions approach and the Fourier Transform approach) to\ninvestigate the relaxation of the magnetization after re-\nmoval of an external static magnetic field, starting from\ntwo different initial conditions: fully saturated or par-\ntially saturated. In Sect. IV we calculate, in a linear ap-\nproximation, the magnetic response of the system to an\noscillatingmagnetic field. Fora chain of Nspins, the a.c.\nsusceptibility is expressed as the superposition of Ncon-\ntributions, eachcharacterizedby itstime scale; througha\nfew simple examples, we show that, depending on the ge-\nometryof the system ( i.e., the relativeorientationsofthe\nlocal easy anisotropy axes and of the applied field), dif-\nferent time scalescan be selected, possibly giving rise, for\nlow frequencies, to a resonant peak in the temperature-\ndependence of the complex magnetic susceptibility. In\nSect. V we show that the theory provides a satisfactory2\naccount for the SCM behavior experimentally observed\nin some magnetic molecular chain compounds, charac-\nterized by dominant antiferromagnetic exchange interac-\ntionsandnon-collinearitybetween spins. Finally, inSect.\nVI, the conclusions are drawn and possible forthcoming\napplications are also discussed.\nII. THE NON-COLLINEAR ISING-GLAUBER\nMODEL\nIn a celebrated paper6, Glauber introduced, in the\nusual 1D Ising model18, a stochastic dependence on the\ntime variable t:i.e., the state of a spin lying on the k-\nth lattice site was represented by a two-valued stochastic\nfunction σk(t)\nHI=−N/summationdisplay\nk=1/parenleftbig\nJIσkσk+1+gµBHe−iωtσk/parenrightbig\n, σk(t) =±1.\n(1)\nJIis the exchange coupling constant, that favors nearest\nneighboring spins to lie parallel ( JI>0, ferromagnetic\nexchange) or antiparallel ( JI<0, antiferromagnetic ex-\nchange); gis the Land´ e factor of each spin, and µBthe\nBohr magneton. In the original paper6a 1D lattice of\nequivalent and collinear spins was studied; there the re-\nsponse to a time-dependent magnetic field H( t), applied\nparallel to the axis of spin quantization and oscillating\nwith frequency ω, as in typical a.c.susceptibility exper-\niments, was also considered.\nIn order to investigate the phenomena of slow relax-\nation (for H=0) and resonant behavior of the a.c.sus-\nceptibility (for H ∝ne}ationslash= 0) in molecular SCM’s, we are going\nto generalize the Glauber model (1) accounting for non-\ncollinearity of local anisotropy axes and crystallographic\n(laboratory) frame. To this aim, we adopt the following\nmodel Hamiltonian\nH=−N/summationdisplay\nk=1/parenleftbig\nJIσkσk+1+GkµBHe−iωtσk/parenrightbig\n, σk(t) =±1.\n(2)\nJIis an effective Ising exchange coupling that can ap-\nproximately be related to the Hamiltonian parameters\nof a real SCM23,24: see later on the discussion in Sec-\ntion V. Like in the usual Ising-Glauber collinear model\n(1), the spins in Eq. (2) are described by classical, one-\ncomponent vectors that are allowed to take two integer\nvaluesσk(t) =±1, but now the magnetic moments may\nbe oriented along different directions, ˆzk, varying from\nsite to site. Within this scheme, the Land´ e tensor of a\nspin on the k-th lattice site has just a non-zero compo-\nnent,g/bardbl\nk, along the local easy anisotropy direction ˆzk.\nDenoting by ˆeHthe direction of the oscilla.ting magnetic\nfield,H(t)=He−iωtˆeH, we define the generalized Land´ e\nfactorGkappearing in Eq. (2) as\nGk=g/bardbl\nkˆzk·ˆeH (3)i.e., a scalar quantity that varies from site to site. Fol-\nlowing Glauber6, we define the single spin expectation\nvaluesk(t) =∝an}bracketle{tσk∝an}bracketri}htt, where brackets denote a proper en-\nsemble average, and the stochastic magnetization along\nthe direction of the applied field\n∝an}bracketle{tM∝an}bracketri}htt=µBN/summationdisplay\nk=1Gk∝an}bracketle{tσk∝an}bracketri}htt=µBN/summationdisplay\nk=1Gksk(t).(4)\nThe basic equation of motion of the Glauber model12,14\nreads\nd\ndtsk(t) =−2∝an}bracketle{tσkwσk→−σk∝an}bracketri}ht, (5)\nin which wσk→−σkrepresents the probability per unit\ntime to reverse the k-th spin, through the flip + σk→\n−σk. For a system of Ncoupled spins, this probability\nis affected by the interaction with the other spins, with\nthe thermal bath and, possibly, with an external mag-\nnetic field. Among all possible assumptions for the tran-\nsition probability wσk→−σkas a function of the N+ 1\nvariables11,19,25,26{σ1,...,σk,...,σN,t}, again following\nGlauber6we require wσk→−σkto be independent of time\nand to depend only on the configuration of the two near-\nest neighbors of the k-th spin. In zero field, these re-\nquirements are fulfilled by\nwH=0\nσk→−σk=1\n2α/bracketleftbigg\n1−1\n2γσk(σk−1+σk+1)/bracketrightbigg\n(6)\nwhile in the presence of an external field\nwH\nσk→−σk=wH=0\nσk→−σk(1−δkσk), (7)\nis usually chosen; the attempt frequency1\n2α(i.e., the\nprobability per unit time to reverse an isolated spin) re-\nmains an undetermined parameter of the model; γac-\ncounts for the effect of the nearest neighbors; the param-\netersδkhave the role of stabilizing the configuration in\nwhich the k-th spin is parallel to the field, and destabiliz-\ning the antiparallel configuration. Thanks to the partic-\nular choices (6) and (7) for the transition probability, by\nimposing the Detailed Balance conditions6it is possible\nto express γandδkas functions of the parameters in the\nspin Hamiltonian (2)\nγ= tanh(2 βJI), δk= tanh(βGkµBH).(8)\nwhereβ=1\nkBTis the inverse temperature in units of\nBoltzmann’s constant. Another advantage of Glauber’s\nchoices (6) and (7) is that the equation of motion (5)\ntakes a simple form. In particular, for H = 0, Eq. (5)\nwith the choice (6) becomes\nds(t)\ndt=−αAs(t), (9)\nwheres(t) denotes the vector of single-spin expectation\nvalues{s1(t), s2(t),···, sN(t)}andAis a square N×N3\nsymmetric matrix, whose non-zero elements are A k,k=1\nand A k,k−1=Ak,k+1=−γ\n2, with A 1,N=AN,1=−γ\n2if peri-\nodic boundary conditions are assumed for the N-spin\nchain. A closed solution of this set of first-order differen-\ntial equationscan be obtained expressingthe expectation\nvalue of each spin, sk(t), in terms of its spatial Fourier\nTransform (FT) /tildewidesq\nsk(t) =/summationdisplay\nq/tildewidesqeiqke−λqt. (10)\nSubstituting Eq. (10) into Eq. (9), one readily obtains\nthe dispersion relation\nλq=α(1−γcosq), q=2π\nNn (11)\nwithn= 0,1,..., N−127. For ferromagnetic cou-\npling (JI>0, hence γ >0) the smallest eigenvalue\nλq=0=α(1−γ) occurs for n=0, independently of the\nnumber of spins Nin the chain. For antiferromagnetic\ncoupling ( JI<0, hence γ <0) andNeven, the smallest\neigenvalue λq=π=α(1− |γ|) occurs for n=N\n2; while in\nthe case of Nodd, the smallest eigenvalue corresponds\ntoα/bracketleftbig\n1−|γ|cos/parenleftbigπ\nN/parenrightbig/bracketrightbig\n, thus depending on the number of\nspins in the antiferromagnetic chain28. The characteris-\ntic time scales of the system, τq, are given by\nτq=1\nλq=1\nα(1−γcosq). (12)\nAt finite temperatures the characteristic times τqare fi-\nnite because |γ|<1; forT→0 one has that 1 −|γ|van-\nishes irrespectively of the sign of JI, because γ→JI\n|JI|=\n±1. Thus, for H= 0, there is one diverging time scale\nin theT→0 limit:τq=0for ferromagnetic coupling and\nτq=πfor antiferromagneticcoupling (and even N). In the\npresence of a non-zero, oscillating field H( t) = He−iωt,\nthe equation of motion (5) with the choice (7) takes a\nform (see Eq. 24 in Section IV later on) which can still\nbe solved, though in an approximate way6, for a suffi-\nciently weak intensity of the applied magnetic field.\nIII. RELAXATION OF THE MAGNETIZATION\nIN ZERO FIELD\nThe originalGlauber model was formulated for a chain\nof collinear spins with the same Land´ e factors: i.e., in\nEq. (2) one has Gk=g,∀k= 1,···,N. Assuming\nthat the system has been fully magnetized by means of\na strong external field, one can study how the system\nevolves if the field is removed abruptly. This corresponds\nto take a fully saturated initial condition\nsk(0) = 1,∀k. (13)\nIn ferrimagnetic chains, on the other hand, a “par-\ntial” saturation can be reached, provided the antiferro-\nmagnetic coupling ( JI<0) between nearest neighbors is“strongenough”, in a sense that will be clarified later on.\nIn fact, if the Land´ efactorsfor odd and even sites arenot\nequal (go∝ne}ationslash=ge), through the application of an opportune\nfield the sample can be prepared in a configuration with\n/braceleftBigg\nsk(0) = +1 ,fork= 2r+1 (kodd)\nsk(0) =−1,fork= 2r(keven).(14)\nWith respect to the case considered by Glauber, it is\nconvenient to separate explicitly the expectation values\nof the odd sites, s2r+1(t), from those of the even sites,\ns2r(t). Thus, for H = 0, the set of Nequations of motion\n(9) can be rewritten as\n/braceleftBigg\nd\ndts2r=−α/bracketleftbig\ns2r+1\n2γ(s2r+1+s2r−1)/bracketrightbig\nd\ndts2r+1=−α/bracketleftbig\ns2r+1+1\n2γ(s2r+s2r−2)/bracketrightbig(15)\nIn the following, the solutions of (15) will be found using\ntwo different approaches that yield identical results.\nA. The Generating Function approach\nThe Generating Function approach, which closely fol-\nlows the original Glauber’s paper, is exposed in detail in\nAppendix. Here, in order to distinguish between the fer-\nromagnetic and ferrimagnetic relaxations, we specialize\nthe general solution, Eq. (A5) and Eq. (A6), to the two\ndifferentkindsofinitialconditions, Eq.(13)andEq.(14).\nIn both cases, we will assume that the exchange cou-\nplingJIis negative. The “partially saturated” configu-\nration, Eq. (14), reflects a typical experimental situation,\nin which the antiferromagnetic coupling is much bigger\n(JI≈100÷1000 K) than the Zeeman energy associated\nwith accessible magnetic fields. On the other hand, the\ninitial configurationwith allthe spins alignedin the same\ndirection, Eq. (13), clearly reflects the experimental situ-\nation of a fully saturated sample. This condition is easily\nobtained for ferromagnetic coupling ( JI>0), while it\nmay require very strong fields (eventually unaccessible)\nfor antiferromagnetic coupling ( JI<0).\nLet us startfrom the saturatedconfiguration, Eq.(13).\nSubstituting the initial condition sk(0) = 1 for all kin\nboth (A5) and (A6), we obtain\n\n\ns2r(t) =e−αt+∞/summationtext\nm=−∞/bracketleftbig\nI2(r−m)(γαt)+I2(r−m)−1(γαt)/bracketrightbig\ns2r+1(t) =e−αt+∞/summationtext\nm=−∞/bracketleftbig\nI2(r−m)(γαt)+I2(r−m)+1(γαt)/bracketrightbig\n.\nHence, exploiting the property (A4) of the Bessel func-\ntions (taking y= 1), and redefining the sums by a unique\nindexj, we get\n\n\ns2r(t) =e−αt+∞/summationtext\nj=−∞Ij(γαt) =e−α(1−γ)t\ns2r+1(t) =e−αt+∞/summationtext\nj=−∞Ij(γαt) =e−α(1−γ)t.(16)4\nThis meansthat, startingwith all the spinsalignedin the\nsamedirection,eachspinexpectationvalue(bothoneven\nand odd sites) decays obeying a mono-exponential law\nwith relaxation time τq=0= [α(1−γ)]−1, which is just\nthe characteristic time scale obtained as the inverse of\nthe dispersion relation λqwith zero wave number q= 0,\nseeEq. (11). Noticethat τq=0candivergefor T→0only\nin the case of ferromagnetic coupling, JI>0 (γ >0).\nLet us now consider the partially saturated configu-\nration, Eq. (14), in which sk(0) = 1 for kodd and\nsk(0) =−1 forkeven. Substituting these initial con-\nditions in both (A5) and (A6), we obtain\n\n\ns2r(t) =−e−αt+∞/summationtext\nm=−∞/bracketleftbig\nI2(r−m)(γαt)−I2(r−m)−1(γαt)/bracketrightbig\ns2r+1(t) =e−αt+∞/summationtext\nm=−∞/bracketleftbig\nI2(r−m)(γαt)−I2(r−m)+1(γαt)/bracketrightbig\nand, still exploiting the property (A4) (but now for y=\n−1), we get\n\n\ns2r(t) =−e−αt+∞/summationtext\nj=−∞(−1)jIj(γαt) =−e−α(1+γ)t\ns2r+1(t) =e−αt+∞/summationtext\nj=−∞(−1)jIj(γαt) =e−α(1+γ)t.\n(17)\nAlso in this case all the spins of the system relax with\na mono-exponential law, but now the relaxation time is\nτq=π= [α(1+γ)]−1, which corresponds to the inverse of\nthe eigenvalue λqwith wave number q=π, see Eq. (11).\nNotice that τq=πcan diverge for T→0 only in the case\nof antiferromagnetic coupling, JI<0 (γ <0).\nSummarizing, according to the sign of the exchange\nconstant, both time scales τq=0(forJI>0) andτq=π\n(forJI<0) diverge in the low temperature limit T→0,\nfollowing an Arrhenius law\nτ=1\n2αe4β|JI|(18)\nwith energy barrier 4 |JI|(slow relaxing mode). It is\nworth noting that the remaining relaxation times, given\nby the inverse of the eigenvalues in Eq. (11) with q∝ne}ationslash= 0\nandq∝ne}ationslash=π, always remain of the same order of mag-\nnitude as α−1(fast relaxing modes). This time scale is\ntypically very small ( ∼ps) in real systems, and negligible\nwith respect to the characteristic times involved in any\nexperimental measurement we refer to.\nB. The Fourier Transform approach\nThe solutions, (16) and (17), to the set of equations\n(15) can alternatively be deduced within the Fourier\nTransform (FT) formalism, which has already been ex-\nploited to obtain the dispersion relation (11). Recalling\nthe definition (10) of sk(t) and its spatial FT\n/tildewidesq=1\nN/summationdisplay\nksk(t)e−iqkeλqt, (19)we evaluate /tildewidesqat time t= 0,/tildewidesq=1\nN/summationtext\nksk(0)e−iqk,\nfor the two initial conditions of interest, (13) and (14).\nStarting from the all-spin-up configuration, Eq. (13), we\nhave\n/tildewidesq=1\nN/summationdisplay\nke−iqk=δq,0. (20)\nHence the solution for the expectation value of a spin\nlocalized on the klattice site at time tis\nsk(t) =/summationdisplay\nqδq,0eiqke−λqt=e−λq=0t,(21)\nwhich is identical to (16) since λq=0=α(1−γ).\nStarting from the partially saturated configuration,\nEq. (14), it is useful to rewrite it as sk(0) =−eiπk, so\nthat the FT at t= 0 is\n/tildewidesq=−1\nN/summationdisplay\nkeiπke−iqk=−δq,π.(22)\nHence the solution is readily obtained\nsk(t) =−/summationdisplay\nqδq,πeiqke−λqt=−eiπke−λq=πt,(23)\nwhich is identical to (17) since λq=π=α(1+γ).\nFinally, we observe that Eqs. (21) and (23) hold even\nfor a ring with a finitenumberNof spins, while Eqs.\n(16) and (17) were obtained in the infinite-chain limit.\nC. Slow versus fast relaxation of the spontaneous\nmagnetization\nThe expectation values, sk(t), of spins localized on\nthe even and odd sites of a linear lattice at time t,\ncomputed either with the Generating Functions or the\nFourier Transform approach, have been shown to dis-\nplay a mono-exponential relaxation, see Eqs. (16) and\n(17), with different time scales, τq=0= [α(1−γ)]−1and\nτq=π[α(1 +γ)]−1respectively, depending on the differ-\nent initial conditions, Eqs. (13) and (14). As a conse-\nquence, also the macroscopic magnetization, expressed\nby Eq. (4), displays the same mono-exponential relax-\nation as the single site quantities sk(t).\nA chain in which all the magnetic moments are equal\ncan be prepared only in the saturated initial configu-\nration, Eq. (13), with all the spin aligned in the same\ndirection, through the application of an external field.\nThus, when the field is abruptly removed, such a sys-\ntem will relax slowly at low temperature only if the ex-\nchange coupling is ferromagnetic ( JI>0). In contrast, if\nthe exchange coupling is antiferromagnetic ( JI<0) and\nthe chain is “forced” in the saturated state by a strong\napplied magnetic field, the system will relax very fast\n(in a typical time of the order of α−1) when the field\nis removed. Let us now discuss how these results, first\nobtained by Glauber6, are generalized to the case of a5\nchain in which the magnetic moments are collinear, but\nnotequal on each site.\nAs pointed out in the introduction, a model with an-\ntiferromagnetic coupling ( JI<0) but non-compensated\nmagnetic moments on the two sublattices is more akin\nto real SCM’s1,3,29. Yet it is very interesting since, de-\npending on the intensity of the applied magnetic field,\nthe system can be prepared either in the saturated ini-\ntial configuration, Eq. (13), where all spins are parallel\nto each other, or in the partially saturated one Eq. (14),\nwhere nearest neighbors are antiparallel. In the former\ncase, a very strong field is required in order to overcome\nthe antiferromagnetic coupling between nearest neigh-\nbors; once the field is removed, the relaxation of the\nmagnetizationis expected to be fast at lowtemperatures,\non the basis of the solution (16). In the latter case, the\npartiallysaturatedinitial configuration(14) can easilybe\nobtainedthroughtheapplicationofasmaller,experimen-\ntally accessible magnetic field; when the field is abruptly\nremoved, the relaxation is expected to be slow accord-\ning to the solution (17). The solution (17) justifies the\nobservation of SCM behavior in ferrimagnetic quasi-1D\ncompounds like CoPhOMe29(see Sect. V).\nSummarizing, we have found that when a collinear\nferrimagnetic chain is prepared in an initial state –\nfully or partially saturated depending on the intensity\nof the applied magnetic field – once the field is re-\nmoved abruptly, the spin system can show fast or slow\nrelaxation, respectively. Fast relaxation corresponds to\nstrongerfields; unfortunatelyforthe quasi-1Dchaincom-\npound CoPhOMe29,30, the antiferromagnetic exchange\nconstant is so large ( |JI| ∼100K) that the realization\nof a fully saturated initial configuration would require a\nvery high, almost unaccessible field ( ∼1000 kOe). Thus\nthis compound is not a good candidate for such a kind\nof experiments31.\nIV. MAGNETIC RESPONSE TO AN\nOSCILLATING MAGNETIC FIELD\nIn the presence of a magnetic field H, the transition\nprobability to be put in the equation of motion (5) is\nwH\nσk→−σk, defined in Eq. (7). One obtains\ndsk(t)\ndt=−α/braceleftBig\nsk(t)−γ\n2[sk+1(t)+sk−1(t)]\n+γδk\n2[∝an}bracketle{tσkσk+1∝an}bracketri}htt+∝an}bracketle{tσk−1σk∝an}bracketri}htt]−δk/bracerightBig\n(24)\nthat differs from Eq. (15), considered earlier for H= 0, in\nthe presence of both a non-homogeneous term, δk, and\nthe time-dependent pair-correlation functions ∝an}bracketle{tσkσk±1∝an}bracketri}htt.\nThe latter ones, assuming that the field is so weak to\ninduce just small departures from equilibrium, can be\napproximated by their time-independent counterparts32\n∝an}bracketle{tσkσk+1∝an}bracketri}htt=∝an}bracketle{tσk−1σk∝an}bracketri}htt≈tanh(βJI)≡η.(25)As it is usual in a.c.susceptibility measurements,\nwe also assume the time-dependent magnetic field\nH(t)=He−iωtˆeH, oscillating at frequency ω, to be weak\nso that the δkparameters can be linearized\nδk= tanh(βµBGkH(t))≈βµBGkH(t).(26)\nThe system of equations of motion (24) then takes the\nform\ndsk(t)\nd(αt)=−sk(t)+γ\n2[sk+1(t)+sk−1(t)]\n+βf(βJI)µBGkH(t), (27)\nwhere\nf(βJI) = 1−γη=1−η2\n1+η2(28)\nis a function of the reduced coupling constant βJIand\nwe have taken into account that γ= 2η/(1+η2). After\na brief transient period, the system will reach the sta-\ntionary condition in which the magnetic moment of each\nspin oscillates coherently with the forcing term at the\nfrequency ω. Expressing the expectation value of a spin\non thek-th lattice site, sk(t), through its spatial FT, /tildewidesq,\nthe trial solution is\nsk(t) =/summationdisplay\nq/tildewidesqeiqke−iωt. (29)\nSubstituting the latter in the system (27) we get\n/tildewidesq=βf(βJI)µBHα/tildewideGq\nα(1−γcosq)−iω,(30)\nwhere/tildewideGqis the FT of Gk:\n/tildewideGq=1\nNN/summationdisplay\nk=1e−iqkGk. (31)\nThe average of stochastic magnetization can readily be\nobtained from (4) as\n∝an}bracketle{tM∝an}bracketri}htt=µBe−iωtN/summationdisplay\nk=1/summationdisplay\nqq′/tildewideGq/tildewidesq′eiqkeiq′k,(32)\nwhich accounts for non-collinearity of local anisotropy\naxes with respect to the field direction. Performing the\nsum over all the lattice sites ( kindices) yields a factor\nNδq,−q′in Eq. (32); substituting the expression (30) for\n/tildewidesq, one obtains\n∝an}bracketle{tM∝an}bracketri}htt=Nµ2\nBβf(βJI)He−iωt/summationdisplay\nqq′α/tildewideGq/tildewideGq′δq,−q′\nα(1−γcosq)−iω.\n(33)\nThen, considering that /tildewideGq/tildewideG−q=|/tildewideGq|2, thea.c.suscep-\ntibility is finally obtained dividing (33) by H e−iωt\nχ(ω,T) =Nµ2\nBβf(βJI)/summationdisplay\nqα|/tildewideGq|2\nα(1−γcosq)−iω.(34)6\nFIG. 1: Temperature dependence of the imaginary part of the c omplex susceptibility, Eq. (38), of a collinear one-dimens ional\nIsing model with alternating spins. Resonant behavior in re sponse to an oscillating magnetic field is possible, at low fr equency,\nonly when magnetic moments are uncompensated (a,c,d), whil e a broad peak is found when the net magnetization is zero (b).\n(The curves refer to reduced frequency ω/α= 0.001)\nIn principle, the a.c.susceptibility of a chain with N\nspinsadmits Npoles, correspondingtothe Neigenvalues\nλqin Eq. (11). Each mode is related to a different time\nscaleτq= 1/λq. Inpractice, notallthetimescaleswillbe\ninvolved in the complex susceptibility χ(ω,T), butonly\nthe ones selected by /tildewideGq. A result similar, at first glance,\nto Eq. (34) was deduced by Suzuki and Kubo27, but in\ntheir case the relationship was between the time scale τq\nand the wave-vector-dependent susceptibility χ(q,ω). In\ncontrast, in an a.c.susceptibility experiment only the\nzero-wave-vector susceptibility χ(q= 0,ω) is accessible;\nthe peculiarity of Eq. (34) is that other time scales, dif-\nferent from τq=0, can be selected thanks to the depen-\ndence of the gyromagnetic factors Gkand of the local\nanisotropy axes on the site position k. This is the main\nresult of our study and will be clarified hereafter through\na few examples.\nA. The a.c.susceptibility of a collinear Ising\nferrimagnetic chain\nLet us start considering the case of a one-dimensional\nIsing model with two kinds of spins (aligned parallel on\nantiparallel to the chain axis) alternating on the odd\nand even magnetic sites of the lattice with Land´ e fac-torsG2r+1=goandG2r=ge(integerr), respectively.\nStrictly speaking, a collinear Ising ferrimagnet is char-\nacterized by an antiferromagnetic coupling JI<0, but\nalso the case JI>0 can be treated through Eq. (34).\nIn fact, since the local axis of anisotropy has the same\ndirection for all the spins, the FT of the site-dependent\nLand´ e factor is\n/tildewideGq=1\nNN/2/summationdisplay\nr=1[gee−iq2r+goe−iq(2r−1)]\n= (ge+eiqgo)1\nNN/2/summationdisplay\nr=1e−iq2r. (35)\nTaking into account that, in the presence of periodic\nboundary conditions, one has\nN/2/summationdisplay\nr=1e−iq2r=N\n2(δq,0+δq,π), (36)\nitfollowsthatthe onlynon-zerovaluesof /tildewideGqareforq= 0\nandq=π\n/tildewideG/bardbl\nq=1\n2[(ge+go)δq,0+(ge−go)δq,π].(37)\nThus, according to Eq. (34), the parallel a.c.suscepti-\nbility (∝bardbl=zz) of a collinear Ising chain with alternating7\nspins is\nχ/bardbl(ω,T) =Nµ2\nBβ f(βJI)\n×1\n4/bracketleftbigg(ge+go)2\n(1−γ)−i(ω\nα)+(ge−go)2\n(1+γ)−i(ω\nα)/bracketrightbigg\n(38)\nIt appears that both the relaxation times obtained by\nSuzuki and Kubo27for the ordinary and the staggered\nsusceptibility of the usual Ising model, namely τq=0=\n[α(1−γ)]−1andτq=π= [α(1 +γ)]−1respectively, do\ncoexist in the a.c.susceptibility (38). Notice that, in\ntheω→0 limit, the static susceptibility of the Ising\nferrimagnet in zero field17is recovered from Eq. (38),\nsince one hasf(βJI)\n1∓γ=1−η2\n1+η21\n1∓γ=e±2βJI.\nAs regards the dynamic response of the system to an\noscillating magnetic field applied along the chain axis,\ndepending on the sign of the effective exchange coupling\nconstant JI, the ferromagnetic ( ge+go) or the antifer-\nromagnetic ( ge−go) branch of the parallel susceptibility\n(38) are characterized by a diverging time scale at low\ntemperature. In particular, for a collinear Ising ferri-\nmagnet one has JI<0, so that τq=πis diverging, while\nτq=0is short (of the order of α−1, the attempt frequency\nof an isolated spin). Thus, for JI<0, a resonant be-\nhavior of the a.c.susceptibility versus temperature (at\nlow frequencies ω/α≪1) can only be observed in the\ncasege∝ne}ationslash=go(see Fig. 1d) when magnetic moments are\nuncompensated , while a broad peak is found in the case\nge=gowhen the net magnetization is zero (see Fig. 1b).\nClearly, for JI>0, a resonant peak is found in both\ncases (see Fig. 1a and 1c), because a net magnetization\nis always present in the system.\nSuch a resonant behavior of the a.c.susceptibility ver-\nsusT, in ferromagnetic33as well as in ferrimagnetic17\nIsing chains with single spin-flip Glauber dynamics, is a\nmanifestation of the stochastic resonancephenomenon34:\ni.e., the response of a set of coupled bistable systems to a\nperiodic drive is enhanced in the presence of a stochastic\nnoise when a matching occurs between the fluctuation-\ninduced switching rate of the system and the forcing fre-\nquency. In a magnetic chain, the role of stochastic noise\nis played by thermal fluctuations and the resonant peak\nin the temperature-dependence of the a.c.susceptibil-\nity occurs when the statistical time scale, associated to\nthe slow decay of the magnetization, matches with the\ndeterministic time scale of the applied magnetic field\nτq(Tpeak)≈1\nω. (39)\nB. The a.c.susceptibility of an n-fold helix\nNext, as an example of a non-collinear spin arrange-\nment, we consider a system of spins with the local axes\nof anisotropy arranged on an n-fold helix (see Fig. 2); θ\nis the angle that the local axes form with z, the unique\naxis of the helix ( i.e., the chain axis). In this case the\nLand´ e factors are equal on all lattice sites, but differentFIG. 2: Thick arrows denote the projections on the xyplane,\nperpendicular to the chain (helix) axis z, of magnetic mo-\nments in a one-dimensional Ising helimagnet, for different f old\nsymmetries ( n= 2,3,4,6). Dashed lines are the projections\nof the local axes of anisotropy, ˆzk.\nspins experience different fields because of the geometri-\ncal arrangement of magnetic moments. In the following\nwe will make the approximation that the Land´ e tensor\nof a spin on the k-th lattice site has just a non-zero com-\nponentgalong the easy anisotropy direction ˆzk, so that\nGk=gˆzk·ˆeH(see Eq. (3)). In the crystallographicframe\n(x,y,z), the directors ˆzkread (integer k)\nˆzk= sinθ/bracketleftbigg\ncos/parenleftbigg2πk\nn/parenrightbigg\nˆex+sin/parenleftbigg2πk\nn/parenrightbigg\nˆey/bracketrightbigg\n+cosθˆez.\n(40)\nLet us first consider the case of an oscillating magnetic\nfield H applied parallel to z, the helix axis. All the spins\nactually undergo the same field, and since Gk=gcosθ\nindependently of the lattice site k, the only peak in the\nFT/tildewideGqoccurs at q= 0\n/tildewideGz\nq=gcosθ δq,0(∀n). (41)\nFollowing the same procedure as in the previous para-\ngraph, the parallel a.c.susceptibility ( ∝bardbl=zz) takes an\nexpression (valid for any value of the fold index nof the\nhelix)\nχ/bardbl(ω,T) =Nµ2\nBβ f(βJI)g2cos2θ\n(1−γ)−i(ω\nα)(∀n) (42)\nthat differs from Glauber’s result for the collinear Ising\nchain6only by the geometrical factor cos2θ. For fer-\nromagnetic coupling, JI>0, the relaxation time τ0=\n[α(1−γ)]−1diverges as T→0, and a resonant behavior\nof thea.c.parallel susceptibility versus temperature is\nfound, at low frequency, when the oscillating field is ap-\nplied parallel to the helix axis, z, along which spins are\nuncompensated : see Fig. 3a, which refers to the case of a\ntwo-fold helix ( n= 2) .8\nFIG. 3: (color online) Temperature dependence of the imagin ary part of the parallel (42) and perpendicular (45) complex\nsusceptibility of an Ising chain with two-fold helical spin arrangement. The local axes ˆz1andˆz2were assumed to form an angle\nθ=π\n3withz, the chain axis (unique axis of the helix). Different curves r efer to different values of ω/α: 0.0001 (continuous, red\nline); 0.0002 (dashed, green line); 0.0005 (dashed single- dotted, blue line); 0.0010 (dashed double-dotted, violet l ine). Resonant\nbehavior in response to an oscillating magnetic field is poss ible, at low frequency, only for field applied in a direction w here\nmagnetic moments are uncompensated (a,c), while a broad pea k is found (b,d) when there is no net magnetization along the\nfield direction.\nLet us now consider the case of an oscillating magnetic\nfield H applied perpendicularly to the chain axis. In this\nconfiguration, it is useful to distinguish the case n= 2\nfrom the general case n >2.\n•n= 2\nIn this case, it is worth noticing that for H parallel\ntoy, one has identically Gr≡0 for any lattice\nsiter. Thus, /tildewideGy\nq≡0 and the corresponding a.c.\nsusceptibility is identically zero\nχyy(ω,T)≡0 (43)\n(not shown). In contrast, for H parallel to x, one\nhasGr=−gsinθon odd sites and Gr= +gsinθ\non even sites. The FT is\n/tildewideGx\nq=1\nNN/2/summationdisplay\nr=1gsinθ(−e−iq(2r−1)+e−iq2r)\n=gsinθ1\n2(δq,0+δq,π)(1−eiq)\n=gsinθ δq,π (44)\nwhere we have taken into account Eq. (36). Thus,\nfor ferromagnetic coupling, JI>0, the relaxationtimeτπ= [α(1+γ)]−1does not diverge as T→0,\nand the perpendicular a.c.susceptibility\nχxx(ω,T) =Nµ2\nBβf(βJI)g2sin2θ\n(1+γ)−i(ω\nα)(45)\ndoes not present a resonant behavior as a function\nof temperature; rather, it presents a broad max-\nimum (see Fig. 3b). Clearly, in the case of an-\ntiferromagnetic coupling, JI<0, the behavior of\nthe susceptibility components is reversed: a broad\nmaximum is found for the temperature dependence\noftheparallelsusceptibility χzz(ω,T)(seeFig. 3d),\nwhile a resonant behavior is found for the perpen-\ndicular susceptibility χxx(ω,T) (see Fig. 3c).\n•n >2\nIn this case, denoting by ˆex·ˆeHandˆey·ˆeHthe\ndirectors of the in-plane field, the FT’s of Gkare\ngiven by\n\n\n/tildewideGx\nq=1\nN(ˆex·ˆeH)gsinθ/summationtextN\nk=1cos/parenleftbig2πk\nn/parenrightbig\ne−iqk\n= (ˆex·ˆeH)gsinθ1\n2/parenleftBig\nδq,2π\nn+δq,−2π\nn/parenrightBig\n/tildewideGy\nq=1\nN(ˆey·ˆeH)gsinθ/summationtextN\nk=1sin/parenleftbig2πk\nn/parenrightbig\ne−iqk\n= (ˆey·ˆeH)gsinθ1\n2i/parenleftBig\nδq,2π\nn−δq,−2π\nn/parenrightBig.(46)9\nRemarkably, as just |/tildewideGq|2appears in Eq. (34), the\ngeneral result for the in-plane susceptibility turns\nout to be independent of the field direction. Thus,\nforn >2, the perpendicular ( ⊥)a.c.susceptibility\nof then-fold helix is given by\nχ⊥(ω,T) =Nµ2\nBβ f(βJI)\n×1\n2sin2θg2\n[1−γcos/parenleftbig2π\nn/parenrightbig\n]−i(ω\nα),(47)\nwhere we have exploited the fact that cos/parenleftbig\n−2π\nn/parenrightbig\n=\ncos/parenleftbig2π\nn/parenrightbig\nfor the term appearingat the denominator\nof Eq. (34).\nSummarizing, in the general case of an Ising chain\nwith an n-fold helical spin arrangement ( n≥2), we\nhave explicitly shown that a resonant behavior of the\na.c.susceptibility versus temperature, similar to the one\ndisplayed by ferromagnetic6,33and ferrimagnetic17Ising\nchains with collinear spins, is possible only for field ap-\nplied in a direction where magnetic moments are uncom-\npensated. In contrast, a broad peak is found when there\nis no net magnetization along the field direction.\nV. APPLICATION TO REAL SINGLE CHAIN\nMAGNETS\nIn this Section we will apply the developed formalism\nto some real compounds as representative realizations of\nSCM’s; for the three selected systems – we know this\nrestriction is far from being exhaustive1,3–a.c.suscep-\ntibility data on single crystal are available, which is a\nfundamentalrequirementforcheckingtheproposedselec-\ntion rules. The considered systems29,35,36are character-\nized by the alternation of two types of magnetic centers\nalong the chain axis, so that at least two spins per cell\nhave to be considered; moreover, the magnetic moments\nare not collinear, the dominant exchange interactions are\nantiferromagnetic and a strong single-ion anisotropy is\npresent, which favors magnetization alignment along cer-\ntain crystallographic directions ˆzk. The static properties\nof these compounds, like magnetization and static sus-\nceptibility, are generally well described using a classical\nHeisenberg model with an isotropic exchange coupling J\nand a single-ion anisotropy D. Thus, in order to describe\nthe dynamic behavior in response to a weak, oscillating\nmagnetic field by means of the previously developed the-\nory, it is necessary to relate the Hamiltonian parame-\nters of such a classical spin model to the exchange con-\nstantJIof the effective Ising model (2). In the following\nwe will show, through a few examples on real systems,\nthat indeed, depending on the geometry, selection rules\nare obeyed for the occurrence of slow relaxation of the\nmagnetization at low temperatures ( β|JI| ≫1), as well\nas for resonant behavior of the a.c.susceptibility as a\nfunction of temperature at low frequencies. As regards\nthe frequencies involved in an a.c.susceptibility exper-\niment on real SCM’s, generally1,3they lie in the range10−1÷104Hz, while the attempt frequency αis of the\norder of 1010÷1013Hz. Thus, for a typical experiment,\na resonant peak in the a.c.susceptibility can safely be\nobserved provided that at least one of the characteristic\ntime scales τqinvolved in (34) diverges at low T, in order\nfor the condition (39) to be satisfied.\nA. The MnIII-based Single Chain Magnet\nIn the one-dimensional molecular magnetic compound\nof formula [Mn(TPP)O 2PPhH]·H2O, obtained by re-\nacting Mn(III) acetate mesotetraphenylporphyrin with\nphenylphosphinic acid35, hereafter denoted by MnIII-\nbased SCM, the phenylphosphinate anion transmits a\nsizeable antiferromagnetic exchange interaction that,\ncombined with the easy axis magnetic anisotropy of the\nMnIIIsites, gives rise to a canted antiferromagnetic ar-\nrangement of the spins. The static single-crystal mag-\nnetic propertieswereanalyzedin the frameworkofaclas-\nsical spins Hamiltonian\nH=−N/2/summationdisplay\nr=1{JS2r−1·S2r+D/bracketleftbig\n(Sz1\n2r−1)2+(Sz2\n2r)2/bracketrightbig\n+e−iωtµBHαgαβ[Sβ\n2r−1+Sβ\n2r]} (48)\nwhereJ <0 is the antiferromagnetic nearest neighbor\nexchange interaction between S= 2 spins. D >0 is\nthe uniaxial anisotropy favoring two different local axes,\nalternating along odd and even sites respectively; both\naxes form an angle θ= 21.01owith the crystallographic\ncaxis, while they form opposite angles of modulus φ=\n56.55owith the aaxis (see Fig. 4). Thus we can write\nˆz2r−1= sinθcosφˆex−sinθsinφˆey+ cosθˆezandˆz2r=\nsinθcosφˆex+sinθsinφˆey+cosθˆez.\nA best fit of the static single-crystal magnetic suscep-\ntibilities, calculated via a Monte Carlo simulation35pro-\nvidesJ=−1.34 K and D= 4.7 K; the gyromagnetic\ntensorGαβis diagonal and isotropic with g/bardbl= 1.97.\nEquivalent results can be obtained calculating the static\npropertiesofmodel(48)viaatransfermatrixapproach37.\nSince the uniaxial anisotropy Dis rather strong with re-\nspect to the exchange coupling |J|, as a first approxima-\ntion one can assume the two sublattice magnetizations\nto be directed just along the two easy axes, ˆz2r−1and\nˆz2r, so that the chain system (48) can be described by a\nnon-collinear Ising model formally identical to Eq. (2),\nwith an effective38Ising exchange coupling JIand a gen-\neralized Land´ e factor Gkdefined as, respectively\nJI=JS(S+1)cos( ˆz2r−1·ˆz2r)\nGr=g/bardbl\nr/radicalbig\nS(S+1) (ˆzr·ˆeH). (49)\nDepending on the orientation of the oscillating mag-\nnetic field with respect to the crystallographic axes, the\nFT of the generalized Land´ e factor takes the following\nforms\n/tildewideGq=g/bardbl/radicalbig\nS(S+1)10\nFIG. 4: (color online) Disposition of local axes ( ˆz2r−1and\nˆz2r) and magnetic moments (red arrows) in the MnIII-based\nreal SCM, discussed in Sect. V.A, with antiferromagnetic\neffective Ising exchange coupling JI<0 . Right: Schematic\nview of the chain structure ( zis the chain axis) along the\ncrystallographic xaxis. Left: projections oflocal axes(dashed\nlines) and of magnetic moments (red arrows) in the xyplane,\nperpendicular to the chain axis.\n×1\nNN/2/summationdisplay\nr=1e−iq2r[eiq(ˆz2r−1·ˆeH)+(ˆz2r·ˆeH)]\n=g/bardbl/radicalbig\nS(S+1)\n\nsinθ1cosφ1δq,0,H∝bardblx\nsinθ1sinφ1δq,π,H∝bardbly\ncosθ1δq,0,H∝bardblz(chain axis)(50)\nThe corresponding a.c.susceptibility takes the expres-\nsion\nχ(ω,T) =Nµ2\nBβf(βJI)(g/bardbl)2[S(S+1)]\n×\n\nsin2θcos2φ1\n(1−γI)−i(ω\nα),H∝bardblx\nsin2θsin2φ1\n(1+γI)−i(ω\nα),H∝bardbly\ncos2θ1\n(1−γI)−i(ω\nα),H∝bardblz(chain axis)(51)\nTakingintoaccountthat, fortheMnIIISCMunderstudy,\nthe “true” exchange coupling, Jin Eq. (48), is antifer-\nromagnetic, and that the angle between the two easy\nanisotropy axes ˆz1andˆz2isδ= 34.6o<90o(see Fig.\n4, right), from Eq. (49) it follows that also the effective\nIsing exchange coupling is antiferromagnetic, JI<0. As\na consequence, in the low temperature limit β|JI| → ∞,\nthe relaxation time τq=πdiverges, while τq=0does not.\nThus, for low frequencies ω/α≪1, thea.c.susceptibil-\nity presents a resonant behavior only when the oscillat-\ning magnetic field is applied along the crystallographic yaxis,i.e.the direction, perpendicular to the chain axis,\nalong which the magnetizations of the two sublattices\nareuncompensated (see Fig. 4). In contrast, when H\nis applied parallel to z(the chain axis) or to x, namely\ntwo directions along which the magnetizations of the two\nsublattices are exactly compensated, no resonant behav-\nior is expected. These theoretical predictions turn out\nto be in excellent agreement with experimental a.c.sus-\nceptibility data35obtained in a single crystal sample of\n[Mn(TPP)O 2PPhH]·H2O, thus confirming that such a\nMnIII-based canted antiferromagnet is a bona fide SCM.\nB. The DyIII-based Single Chain Magnet\nThe molecular magnetic compound of formula\n[Dy(hfac) 3(NITPhOPh)], hereafter denoted by DyIII-\nbased SCM, belongs to a family of quasi one-dimensional\nmagnets in which rare earth ions (with spin S) and or-\nganic radical ions (with spin s= 1/2) alternate them-\nselvesalongthe chainaxis, z, which in this compound co-\nincides with the crystallographic baxis. Static measure-\nments in single crystal samples suggest36that there is an\nantiferromagnetic exchange interaction between neigh-\nboring DyIIIions, whose easy anisotropy axes are canted\nwithrespecttothechainaxisinsuchawaytogeneratean\nuncompensated moment along b, while the components\nin theacplane are compensated. Thus, as far as the\ndominant exchange interaction J <0 between DyIIIions\nis taken into account, the spin Hamiltonian of the sys-\ntem is quite similar to Eq.(48). However, with respect to\nthe MnIII-based chain, the crystal structure of the DyIII-\nbased SCM is more complicated, not only owing to the\npresence of two kinds of magnetic centers (the DyIIIions\nand the organic radical ions), but mainly because the\nsystem is formed by two different families of chains, with\ntwo almost orthogonal projections of the easy axes in\ntheacplane, perpendicular to the chain axis: this “ac-\ncidental” (in the sense that it is not imposed by symme-\ntry) orthogonality is the reason for the nearly isotropic\nmagnetic behavior displayed by the system within such\na plane36.\nWe adoptasimplified model formallyequivalentto Eq.\n(48). Taking into account only the dominant antiferro-\nmagnetic exchange interaction ( J <0) between neigh-\nboring DyIIIions (which indeed are next nearest neigh-\nbors in the real system) and their uniaxial anisotropy\n(D >0), the system can approximately be described by\nthe classical spins Hamiltonian (48), where now |Sk|= 1.\nBy means of a classical Transfer Matrix calculation, the\nstatic properties of the DyIII-based SCM turn out to be\nsatisfactorily fitted36byJ=−6 K,D= 40 K, g/bardbl= 10,\nwith the two easy anisotropy axes ˆz2r−1,ˆz2rforming\nequal angles θ≈75owith the chain axis z. (Notice that\nthe latter propertyholds true forboth families ofchains.)\nAlso in the case of the DyIII-based SCM, the uniaxial\nanisotropy Dturns out to be sufficiently strong with re-\nspect to the exchange coupling |J|in order to assume,11\nFIG. 5: (color online) Disposition of odd and even local axes\n(ˆz2r−1andˆz2r) and magnetic moments (thick arrows) in the\nDyIII-based real SCM, discussed in Sect. V.B, with ferro-\nmagnetic effective Ising exchange coupling JI>0 . Top:\nSchematic view of the chain structure ( zis the chain axis),\ndisplaying the two families of chains (A, with red magnetic\nmoments, and B, with green magnetic moments). Bottom:\nprojections of magnetic moments in the xyplane, perpendic-\nular to the chain axis.\nas a first approximation38, the two sublattice magneti-\nzations of DyIIIto be directed just along the two easy\naxes. Thus one can define an equivalent non-collinear\nIsing model (2), where the effective Ising exchange cou-\nplingJIand the generalized Land´ e factor Grare now\ndefined as\nJI=Jcos(ˆz2r−1·ˆz2r)\nGr=g/bardbl\nr(ˆzr·ˆeH). (52)\nDepending on the orientation of the oscillating magnetic\nfield with respect to the crystallographic axes, the FT of\nthe generalized Land´ e factor takes the form\n/tildewideGq=g/bardbl1\nNN/2/summationdisplay\nr=1e−iq2r[eiq(ˆz2r−1·ˆeH)+(ˆz2r·ˆeH)]\n∝g/bardbl/braceleftBigg\ncosθ δq,0,H∝bardblz(chain axis)\nsinθ δq,π,H⊥z.(53)\nIt is important to notice that this result holds true for\nbothfamilies(A,B)ofchains. Next, weobservethatsince\nin the DyIII-basedSCM, the spinsonopposite sublattices\narecoplanarwith the chain axis, the angle between ˆz2r−1\nandˆz2risjust2θ≈150o>90o. Takingintoaccountthat\nthe “true”exchangeconstantinEq. (48) isantiferromag-\nnetic,J <0, from Eq. (52) it follows that the effective\nIsingexchangecouplingisnowferromagnetic, JI>0(see\nFig. 5, top). As a consequence, in the low temperaturelimitβJI→ ∞, the relaxation time τq=0diverges, while\nτq=πdoes not. Thus, the a.c.susceptibility\nχ(ω,T)∝Nµ2\nBβf(βJI)(g/bardbl)2\n×/braceleftBigg\ncos2θ1\n(1−γI)−i(ω\nα),H∝bardblz(chain axis)\nsin2θ1\n(1+γI)−i(ω\nα),H⊥z(54)\nis expected to have a resonant behavior, for low frequen-\nciesω/α≪1, only when the oscillating magnetic field is\nappliedparalleltothechainaxis, z,alongwhichthemag-\nnetizations of the two sublattices are uncompensated (see\nFig. 5, top). Such a theoretical prediction turns out to\nbe in excellent agreementwith the experimental a.c.sus-\nceptibility data36obtained in a single crystal sample of\n[Dy(hfac) 3(NITPhOPh)] ∞, thus confirming that also the\nDyIII-based canted antiferromagnet is a bona fide SCM.\nTheonlyqualitativedifference, withrespecttotheMnIII-\nbased chain is that, due to the different geometry of the\nspin arrangement and of the local anisotropy axes with\nrespecttothechainaxis,theresonantbehaviorofthe a.c.\nsusceptibility is now observed for field applied parallel to\nthe chain axis, rather than perpendicular to it.\nC. The CoPhOMe (CoII-based) Single Chain\nMagnet\nIn the molecular magnetic compound of for-\nmula [Co(hfac) 2NITPhOMe], hereafter denoted by\nCoPhOMe29,30, the magnetic contribution is given by\nCobalt ions, with an Ising character and effective S=\n1/2, and by NITPhOMe organic radical ions, magnet-\nically isotropic and with s= 1/2. The spins are ar-\nranged on a helical structure, schematically depicted in\nFig. 6, right, whose projections in a plane perpendic-\nular to the helix axis z(coincident with the crystallo-\ngraphiccaxis), are represented in Fig. 6, left. The prim-\nitive magnetic cell is made up of three Cobalts (black\narrows) and three organic radicals (red arrows). Al-\nthough the effective spins of the two types of magnetic\ncenters have the same value, the gyromagnetic factors\nare different: gCo∝ne}ationslash=gR; thus, since the nearest neighbor\n(Cobalt-radical) exchange interaction is negative (and\nstrong,|J| ≈100 K)30, the sublattice magnetizations are\nnot compensated along z, whereas they are compensated\nwithin the xyplane perpendicular to the chain axis z.\nFor this compound, which was the first to display SCM\nbehavior29,30, static measurements on single-crystalsam-\nples has not been interpreted in terms of a simple model\nyet, due to the complexity of the system itself. Thus,\na relationship such as (49) and (52), which associate the\nIsing Hamiltonian (2) parameters( JIandGk) with those\nofa more realistic Hamiltonian, is still missing. However,\nthe dynamic behavior has been thoroughly investigated\ntreating - for the sake of simplicity - both the CoIIand\nthe organic radical spins as Ising variables, with σ=±1.12\nFIG. 6: (color online) Disposition of even and odd local axes\n(dashed lines) and magnetic moments (thick arrows) in the\nCoPhOMe real SCM, discussed in Sect. V.C, with antiferro-\nmagnetic effective Ising exchange coupling JI<0 . Right:\nSchematic view of the chain structure ( zis the chain axis)\nalong the crystallographic yaxis. Left: projections of local\naxes (dashed lines) and of magnetic moments (thick arrows)\nin thexyplane, perpendicular to the chain axis.\nThe effective Ising Hamiltonian reads\nH=−N/6/summationdisplay\nl=13/summationdisplay\nm=1{JIσl,2m[σl,2m−1+σl,2m+1]+e−iωtµBH\n[gRσl,2m−1(ˆz2m−1·ˆeH)+gCoσl,2m(ˆz2m·ˆeH)]}(55)\nwithlmagneticcellindexand msitelabelwithboundary\nconditions σl,7=σl+1,1. Since all the local axes ˆzk(k=\n1,···,6) form the same angle θ≈55owith the zaxis,\nwhen a magnetic field is applied along z, the FT of the\ngeneralized Land´ e factor is simply given by\n/tildewideG/bardbl\nq= cosθ1\nNN/2/summationdisplay\nr=1(gCoe−iq(2r−1)+gRe−iq2r)\n=cosθ\n2[(gCo+gR)δq,0+(gCo−gR)δq,π] (56)\nwhich, except for the prefactor cos θ, is quite similar to\nEq. (37) for the collinear Ising chain with alternating\nspins. Thus, the parallel a.c.susceptibility is\nχ/bardbl(ω,T) =Nµ2\nBβ f(βJI)cos2θ\n4\n×[(gCo+gR)2\n(1−γI)−i(ω\nα)+(gCo−gR)2\n(1+γI)−i(ω\nα)].(57)\nTaking into account that the effective exchange coupling\nof CoPhOMe is negative ( JI<0), the antiferromagnetic\nbranch of the parallel susceptibility is characterized by\na diverging time scale τq=π= [α(1+γI)]−1at low tem-\nperature, so that, for low frequencies ω/α≪1,χ/bardbl(ω,T)\ndisplays a resonant behavior.Let us now consider the case of a field applied in the\nplane perpendicular to the chain axis. For H ∝bardblx(see Fig.\n6, left) one has, letting k0=π\n3\nGx\n2r−1=gRsinθcos[k0(2r−1)]\nGx\n2r=gCosinθcos[k02r] (58)\nso that the FT takes the form\n/tildewideGx\nq= sinθ1\nNN/2/summationdisplay\nr=1e−iq2r/parenleftBig\ngCocos(k02r)\n+gReiq[cos(k0)cos(k02r)+sin(k0)sin(k02r)]/parenrightBig\n=1\n4sinθ[(gCo+gRei(q−k0))(δq,k0+δq,π+k0)\n+ (gCo+gRei(q+k0))(δq,−k0+δq,π−k0)] (59)\nwhere, as usual, we have exploited Eq. (36). Thus it\nfollows that\n/tildewideGq=±π\n3=sinθ\n4(gCo+gR)\n/tildewideGq=π±π\n3=sinθ\n4(gCo−gR).(60)\nThe corresponding relaxation times are τq=±π\n3=α\n1−1\n2γ\nandτq=±2π\n3=α\n1+1\n2γso that, summing the four contribu-\ntions we obtain the perpendicular a.c.susceptibility\nχ⊥(ω,T) =Nµ2\nBβ f(βJI)sin2θ\n8\n×/bracketleftbigg(gCo+gR)2\n(1−1\n2γ)−i(ω\nα)+(gCo−gR)2\n(1+1\n2γ)−i(ω\nα)/bracketrightbigg\n.(61)\nIn conclusion, for the six-fold helix model with alter-\nnating spins and Ising exchange coupling in Eq. (55),\nthe parallel and perpendicular components of the a.c.\nsusceptibility, χ/bardbl(ω,T) andχ⊥(ω,T), display a behavior\nsimilar to that of a ferrimagnetic chain with alternating\nspins (see (38)) and of an n-fold helical spin arrangement\nwith equivalent spins (see (47)), respectively. In spite of\nthe approximations involved in model (55) to describe\nthe real CoPhOMe molecular magnetic chain, the two\ncalculated susceptibilities (57) and (61), qualitatively re-\nproduce the dynamic behavior of this compound29,30. In\nfact, no out-of-phase a.c.susceptibility (imaginary part)\nis observed when the field is applied in the plane per-\npendicular to the chain axis, z, for the experimental fre-\nquencies (1 ÷105Hz)30. In contrast, when the oscillating\nfield is parallel to z, a resonant behavior is observed as\na function of temperature. Even though our theoretical\ntreatment holds only for small deviations from equilib-\nrium, it is worth mentioning that the absence of slow\nrelaxation for fields applied in the perpendicular plane is\nevidenced in the low temperature magnetization curve as\nwell: at low enough temperatures, a finite-area hysteresis\nloop is present only when a static field is applied parallel\nto the chain axis, while no hysteresis is observed in the\nin-plane magnetization curve29,30.13\nVI. CONCLUSIONS\nIn conclusion, in the framework of a one-dimensional\nIsing model with single spin-flip Glauber dynamics,\ntaking into account reciprocal non-collinearity of local\nanisotropy axes and the crystallographic (laboratory)\nframe, we have investigated: (i) the dynamics of mag-\nnetization reversal in zero field, and (ii) the response of\nthe system to a weak magnetic field, oscillating in time.\nWe have shown that SCM behavior is not only a feature\nof collinear ferro- and ferrimagnetic, but also of canted\nantiferromagnetic chains. In particular, we have found\nthat resonant behavior of the a.c.susceptibility versus\ntemperature in response to an oscillating magnetic field\nis possible, at low frequency, only for fields applied in a\ndirection where magnetic moments are uncompensated.\nIn contrast, a broad peak is expected when there is no\nnet magnetization along the field direction.\nThe role played by geometry in selecting the time\nscales involved in a process is an important and pecu-\nliarresult, typicalofmagneto-molecularapproachtolow-\ndimensional magnetism. In fact, magnetic centers with\nuniaxial anisotropyusually correspondto building blocks\nwith low symmetry39,40, which – in turn – often crystal-\nlize in more symmetric space groups, realizing a recip-\nrocal non-collinearity between local anisotropy axes as a\nnatural consequence1,2. Thus the family of real SCM’s,\nto which our model applies, does not restrict to ad-hoc\nsynthesized compounds but, instead, is expected to grow\nlarger in the future3. As a validity check of our selec-\ntion rules (as well as a tutorial exemplification), we have\nshown how our theory applies successfully to three differ-\nent molecular-based spin chains; when possible, we have\nput the parameters of our model Hamiltonian (2) in re-\nlationship with those of more general models, typically\nused tofit the staticpropertiesofthe correspondingcom-\npounds. Needless to say that the possibility of schema-\ntizing the chosen three compounds with Hamiltonian (2)\nrelies on the fact that at low enough temperatures they\nbehave as chains consisting of two-level units coupled by\na fully anisotropic exchange interaction. The latter as-\nsumption is expected to hold also for spin larger than\none-half in the presence of strong single ion anisotropy,\nprovided that domain walls still remain sharp24,38. In\nthis case each single magnetic center follows a thermally\nactivated dynamics, with an energy barrier ∆ 0, and well\nestablished heuristic arguments41suggest to replace the\nattempt frequency by α=α0e−β∆0.\nA naive application of our 3-fold-helix results (42)\nand (47) to the recently synthesized non-collinear Dy 3\ncluster would prevent the observation of slowrelaxation,\nwhile Single-Molecule-Magnet dynamics is indeed there\nobserved even in the presence of compensated magnetic\nmoments42. However, such a behavior in the classical\nregime,i.e.far from level crossings where underbarrier\nprocesses of quantum origin are important, is observed\nfor Dy 3in non-zero field and the resonant behavior is\ndue to a change of the relative population between thelowest and the first excited Kramers doublets of each Dy\nion: For sure this mechanism cannot be accounted for\nwhen dealing with two-level elementary variables, like σk\nin Hamiltonian (2). An extension of our model to mul-\ntivalued σk’s definitely deserve to be considered in the\nnext future.\nBeyondmolecularspinchains, ourapproachmightalso\nbe used to model monatomic nanowires showing slow re-\nlaxation ofthe magnetization at low temperatures43and,\npossibly, one-dimensional spin glasses44(provided that\nquenched disorder is somehow taken into account). In\nthis regard, the question of distinguishing between SCM\nand spin-glass behavior in quasi-1D systems is still a hot\ntopic of discussion45,46,47,48.\nAfter the successful organization of Single-Molecule\nMagnets onto surfaces49,50,51, the grafting of properly\nfunctionalized SCM’s on substrates represents a foresee-\nable goal as well as a fundamental step for their possi-\nble use as magnetic-memory units3. Technologies em-\nploying more traditional materials but based on alter-\nnative geometrical arrangement of magnetic anisotropy\naxes with respect to the switching field, such as in per-\npendicular recording52or processional switching53, are\nalready at the stage of forthcoming implementation in\ndevices. Were SCM’s to be considered as a possible route\nto tackle the main issues of high-density magnetic stor-\nage –i.e.optimization of the signal-noise ratio, thermal\nstability and writability52– the proposed selection rules\nfor slow relaxation, and related bistability, might find an\napplication in magnetic-memory manufacture as well.\nAcknowledgments\nWe wish to thank R. Sessoli and A. Rettori for stim-\nulating discussions, and J. Villain for interest and fruit-\nful suggestions in the early stages of this research work.\nFinancial support from ETHZ, the Swiss National Foun-\ndation, and Italian National Research Council is also ac-\nknowledged.\nAPPENDIX A: THE GENERATING FUNCTIONS\nAPPROACH\nAssuming periodic boundary conditions and defining\nthe two generating functions\nL(y,t) =+∞/summationdisplay\nr=−∞y2r+1s2r+1(t)\nG(y,t) =+∞/summationdisplay\nr=−∞y2rs2r(t), (A1)\nEqs.(15) taketheformoftwodifferentialequations(with\nthe dot indicating the first derivative with respect to the14\nadimensional variable αt)\n/braceleftBigg˙L(y,t) =−L(y,t)+1\n2γ(y+y−1)G(y,t)\n˙G(y,t) =−G(y,t)+1\n2γ(y+y−1)L(y,t).(A2)\nThe system (A2) can be decoupled through the substi-\ntution\nU(y,t) =L(y,t)+G(y,t)\nW(y,t) =L(y,t)−G(y,t), (A3)\nfrom which we directly get\n/braceleftBigg˙U(y,t) =−(1−ν)U(y,t)\n˙W(y,t) =−(1+ν)W(y,t),\nwithν=1\n2γ(y+y−1). The solutions of these two\nequations are U(y,t) =U(y,0)e−(1−ν)αtandW(y,t) =\nW(y,0)e−(1+ν)αtthat, exploiting the property\nexp/bracketleftbigg1\n2(y+y−1)x/bracketrightbigg\n=+∞/summationdisplay\nk=−∞ykIk(x) (A4)\nof the Bessel functions of imaginary argument Ik(x), can\nbe rewritten as\n\n\nU(y,t) =U(y,0)e−αt+∞/summationtext\nk=−∞ykIk(γαt)\nW(y,t) =W(y,0)e−αt+∞/summationtext\nk=−∞(−1)kykIk(γαt)\nPerforming the inverse transformation of (A3), the solu-\ntions to the system (A2) can be obtained\nL(y,t) =1\n2e−αt×\n+∞/summationdisplay\nk=−∞yk/bracketleftbig\nU(y,0)Ik(γαt)+(−1)kW(y,0)Ik(γαt)/bracketrightbig\nG(y,t) =1\n2e−αt×\n+∞/summationdisplay\nk=−∞yk/bracketleftbig\nU(y,0)Ik(γαt)−(−1)kW(y,0)Ik(γαt)/bracketrightbig\n.\nThen, separatingthe k-oddfromthe k-eventermsinboth\nsums, we get\nL(y,t) =e−αt+∞/summationdisplay\nr=−∞[y2rL(y,0)I2r(γαt)+y2r+1G(y,0)I2r+1(γαt)]\nG(y,t) =e−αt+∞/summationdisplay\nr=−∞[y2rG(y,0)I2r(γαt)\n+y2r−1L(y,0)I2r−1(γαt)].\nAccording to (A1), now L(y,0) andG(y,0) can be ex-\npressed again in terms of the initial single spin expecta-\ntion values s2r(0) ands2r+1(0) respectively\nL(y,t) =e−αt+∞/summationdisplay\nr=−∞\n×[y2r+∞/summationdisplay\nm=−∞y2m+1s2m+1(0)I2r(γαt)\n+y2r+1+∞/summationdisplay\nm=−∞y2ms2m(0)I2r+1(γαt)]\nPuttingk′=k+mwe have\nL(y,t) =e−αt+∞/summationdisplay\nk′=−∞y2k′+1+∞/summationdisplay\nm=−∞\n[s2m+1(0)I2(k′−m)(γαt)+s2m(0)I2(k′−m)+1(γαt)].\nComparing this latter result with the definition of L(y,t)\n(A1) and requiring for the terms corresponding to the\nsame power of yto be equal, an explicit function for the\nodd spin expectation values is readily obtained\ns2r+1(t) =e−αt+∞/summationdisplay\nm=−∞[s2m+1(0)I2(r−m)(γαt)\n+s2m(0)I2(r−m)+1(γαt)]. (A5)\nSubstituting L(y,0) eG(y,0) in the solution found for\nG(y,t) and performing similar passages, we obtain the\nexpectation value for even sites\ns2r(t) =e−αt+∞/summationdisplay\nm=−∞[s2m(0)I2(r−m)(γαt)\n+s2m+1(0)I2(r−m)−1(γαt)]. (A6)\n∗Electronic address: vindigni@phys.ethz.ch\n1C. Coulon, H.Miyasaka, and R.Cl´ erac, Struct. Bond. 122,\n163 (2006), and references therein.\n2H. MiyasakaandM. Yamashita, DaltonTrans., 399 (2007),\nand references therein.3L. Bogani et al., J. Mater. Chem. 18, 4750 (2008), and\nreferences therein.\n4H. B. Braun, Phys. Rev. Lett. 71, 3557 (1993).\n5H.B. Braun, J.Appl.Phys. 85, 6172(1999), andreferences\ntherein.15\n6R. J. Glauber, J. Math. Phys. 4, 294 (1963).\n7J. E. Anderson, J. Chem. Phys. 52, 2021 (1969).\n8S. Bozdemir, Phys. Status Solidi B 103, 459 (1981).\n9S. Bozdemir, Phys. Status Solidi B 104, 37 (1980).\n10J. L. Skinner, J. Chem. Phys. 79, 1955 (1983).\n11G. O.Berim andE.Ruckenstein, J. Chem.Phys. 119, 9640\n(2003).\n12J. J. Brey and A. 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Back et al., Science 285, 864 (1999)."    },    {        "title": "1304.2826v1.Magnetization_and_spin_dynamics_of_the_spin_S_1_2_hourglass_nanomagnet_Cu5_OH_2_NIPA_4_10H2O.pdf",        "content": "Magnetization and spin dynamics of the spin S=1\n2hourglass nanomagnet\nCu5(OH) 2(NIPA) 4\u000110H 2O\nR. Nath,1A. A. Tsirlin,2, 3,\u0003P. Khuntia,2O. Janson,2, 3T. F orster,2, 4M. Padmanabhan,5\nJ. Li,6Yu. Skourski,4M. Baenitz,2H. Rosner,2and I. Rousochatzakis7,y\n1School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695016, India\n2Max Planck Institut f ur Chemische Physik fester Sto\u000be, N othnitzer Str. 40, 01187 Dresden, Germany\n3National Institute of Chemical Physics and Biophysics, 12618 Tallinn, Estonia\n4Dresden High Magnetic Field Laboratory, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany\n5School of Chemistry, Indian Institute of Science Education and Research, Thiruvananthapuram-695016, India\n6Department of Chemistry &Chemical Biology, Rutgers University, Piscataway, NJ 08854, USA\n7Institute for Theoretical Solid State Physics, IFW Dresden, 01171 Dresden, Germany\nWe report a combined experimental and theoretical study of the spin S=1\n2nanomagnet\nCu5(OH) 2(NIPA) 4\u000110H 2O (Cu 5-NIPA). Using thermodynamic, electron spin resonance and1H nu-\nclear magnetic resonance measurements on one hand, and ab initio density-functional band-structure\ncalculations, exact diagonalizations and a strong coupling theory on the other, we derive a micro-\nscopic magnetic model of Cu 5-NIPA and characterize the spin dynamics of this system. The ele-\nmentary \fve-fold Cu2+unit features an hourglass structure of two corner-sharing scalene triangles\nrelated by inversion symmetry. Our microscopic Heisenberg model comprises one ferromagnetic and\ntwo antiferromagnetic exchange couplings in each triangle, stabilizing a single spin S=1\n2doublet\nground state (GS), with an exactly vanishing zero-\feld splitting (by Kramer's theorem), and a very\nlarge excitation gap of \u0001 '68 K. Thus, Cu 5-NIPA is a good candidate for achieving long electronic\nspin relaxation ( T1) and coherence ( T2) times at low temperatures, in analogy to other nanomag-\nnets with low-spin GS's. Of particular interest is the strongly inhomogeneous distribution of the\nGS magnetic moment over the \fve Cu2+spins. This is a purely quantum-mechanical e\u000bect since,\ndespite the non-frustrated nature of the magnetic couplings, the GS is far from the classical collinear\nferrimagnetic con\fguration. Finally, Cu 5-NIPA is a rare example of a S=1\n2nanomagnet showing\nan enhancement in the nuclear spin-lattice relaxation rate 1 =T1at intermediate temperatures.\nPACS numbers: 75.50.Xx, 75.10.Jm, 75.30.Et, 76.60.Jx\nI. INTRODUCTION\nThe \feld of molecular nanomagnets has enjoyed an\nenormous experimental and theoretical activity over\nthe last few decades.1Owing to the nanoscopic size\nof their elementary magnetic units, these compounds\nprovide experimental access to a plethora of quantum\nmechanical (QM) e\u000bects, including quantum tunneling\nof the magnetization2,3or the N\u0013 eel vector,4quantum\nphase interference,5level-crossings and magnetization\nplateaux.6They also allow to probe on the macroscopic\nscale the crossover from quantum to classical physics.7\nFinally, molecular magnets are promising materials for\nspintronic applications8and quantum computing.9\nHere we report on the magnetic behavior, mi-\ncroscopic magnetic model, and spin dynamics of\nCu5(OH) 2(NIPA) 4\u000110H 2O, hereinafter referred to as\nCu5-NIPA, where the acronym NIPA stands for the\n5-nitro-isophtalic acid ligand. It is an hourglass-shaped\nmolecular magnet comprising \fve Cu2+spin-1\n2ions. The\nground state (GS) of this magnet has a low spin value\nS=1\n2with a very large spin gap of \u0001 '68 K. Therefore,\nCu5-NIPA behaves as a rigid spin S=1\n2entity in a wide\ntemperature range, and resembles other spin-1\n2molecu-\nlar magnets, such as V 6.10,11Given that both compounds\ncomprises=1\n2spins, we expect similarly long electron\nspin-phonon relaxation times T1, which allow for the ob-servation of rich hysteresis e\u000bects in pulsed \felds.10,12\nHowever, in contrast to V 6, here we do not expect abrupt\nsteps in the magnetization curve,10because the present\ncompound features an odd number of half-integer spins,\nthus the zero-\feld splitting vanishes exactly by Kramer's\ntheorem.\nIn analogy with other molecular magnets with low-\nspin GS's, such as iron-sulfur clusters,13{15heterometallic\nrings,16iron trimers,17V15,18and single-molecule mag-\nnets (SMM),19we expect that Cu 5-NIPA manifests also\nlong coherence times T2, which is a crucial step towards\nimplementations in quantum computing.20\nAnother attractive feature of Cu 5-NIPA is the pres-\nence of two corner-sharing scalene triangles, related\nby inversion symmetry.21,22The spin triangle is the\nmost elementary unit for highly frustrated magnetism,23\nwhile its chirality may induce a \fnite magnetoelec-\ntric coupling,24,25and may also be used as a qubit.26\nIn molecular magnetism, the spin triangle is com-\nmon in many compounds,1such as V 15,12,27Cu3\nclusters,28the chiral Dy 3cluster,29the cuboctahedron\nCu12La8,30as well as the giant icosidodecahedral ke-\nplerates Mo 72Fe30,31W72Fe30,32Mo72Cr30,33Mo72V3034\nand W 72V30,35which host highly frustrating kagome-like\nphysics.36\nThe distinct feature of the present compound is the\nstrong distortion of its regular spin triangles, featuringarXiv:1304.2826v1  [cond-mat.mtrl-sci]  10 Apr 20132\nb\nCu3\n217 K\n217 KCu1\nCu1\n62□K62□K\n/c4562 K\n/c4562 KCu2\nCu2a\nc\n32\n21\n1\nCC\nOO\nO/c109\n3-OH/c1093-OH\nO\nFIG. 1. (Color online) Left panel: crystal structure of Cu 5-NIPA showing the stacking of the Cu 5units (one unit is highlighted\nby gray shading) through the NIPA ligand molecules, into a rigid three-dimensional metal-organic framework, with the shortest\ndistance of 8.6 \u0017A between the magnetic units. Middle and right panels: structure of the Cu 5magnetic unit and relevant\nmagnetic interactions according to the \ft of the experimental susceptibility data (Fig. 13). The numbers 1 \u00003 denote the\ncrystallographic positions Cu1{Cu3, with Cu3 residing at the inversion center. Arrows show the classical ferrimagnetic ground\nstate.\nthree nonequivalent Cu positions, denoted by Cu1, Cu2,\nand Cu3 (see Fig. 1). Each spin triangle is centered by the\n\u00163-OH group37providing Cu{O{Cu superexchange path-\nways, while the carboxyl (COO\u0000) group of the NIPA lig-\nand creates an additional Cu1{Cu3 pathway (see Fig. 1,\nmiddle). This topology leads to three drastically di\u000berent\ninteractions, one of which is ferromagnetic (FM). Despite\nthe fact that these couplings do not compete with each\nother classically, the GS has strong QM character, which\nis partly re\rected in a strongly inhomogeneous distribu-\ntion of the magnetic moment over the \fve Cu2+spins.\nOf particular interest is our experimental \fnding of\nan enhancement of the1H nuclear spin-lattice relaxation\nrate 1=T1at a characteristic temperature slightly below\nthe spin gap ( T'40 K). While such an enhancement\nhas been repeatedly found in numerous antiferromag-\nnetic (AFM) homometallic38and heterometallic39rings\nof spinss >1\n2, it is very rare for s=1\n2. We argue that\nthe origin of the peak is the same in both cases, namely\nthe slowing down of the phonon-driven magnetization\n\ructuations.38,40{42However, there are several qualita-\ntive di\u000berences with the case of homometallic rings, re-\nlated to the sparse excitation spectrum of Cu 5-NIPA and\nthe presence of inequivalent Cu2+sites.\nThe organization of this article is the following. We\nbegin in Sec. II with details on the sample prepara-\ntion and methods. Next, we discuss bulk speci\fc heat\nand magnetization (Sec. III A), as well as local electron\nspin resonance (Sec. III B) and1H nuclear magnetic res-\nonance data (Sec. III C). The microscopic description of\nCu5-NIPA in terms of the isotropic Heisenberg model\nis based on ab initio density-functional band-structure\ncalculations (Sec. IV A) and facilitates the evaluation of\nmagnetic properties (local magnetizations, nature of the\nGS, etc.) using exact diagonalizations (Sec. IV B) and a\nstrong coupling theory (Sec. IV C). Finally, we conclude\nwith a discussion and some interesting perspectives of\nthis study in Sec. V.II. METHODS\nA powder sample of Cu 5-NIPA was prepared according\nto the procedure described in Ref. 21. Sample purity was\nchecked by powder x-ray di\u000braction (Huber G670 Guinier\nCamera, CuK \u000b1radiation, 2 \u0012= 3\u0000100\u000eangular range).\nThe magnetic susceptibility ( \u001f) was measured in the\ntemperature range 1 :8 K\u0014T\u0014400 K in an applied \feld\nof\u00160H= 1 T using the commercial Quantum Design\nMPMS SQUID. The magnetization isotherm Mvs.H\nwas measured at T= 2 K in \felds up to 14 T using the\nvibrating sample magnetometer (VSM) option of Quan-\ntum Design PPMS. Additionally, pulsed-\feld measure-\nments in \felds up to 60 T were performed at 1.4 K in the\nDresden High Magnetic Field Laboratory. Details of the\nmeasurement procedure are described in Ref. 43.\nThe heat capacity Cp(T) was measured on a\nsmall piece of pellet over the temperature range\n2 K\u0014T\u0014300 K in zero \feld using the Quantum De-\nsign PPMS.\nThe electron spin resonance (ESR) measurements were\nperformed at Q-band frequencies ( f= 34 GHz) using a\nstandard spectrometer together with a He-\row cryostat\nthat allows us to vary the temperature from 1.6 to 300 K.\nESR probes the absorbed power Pof a transversal mag-\nnetic microwave \feld as a function of a static and exter-\nnal magnetic \feld B. To improve the signal-to-noise ra-\ntio, we used a lock-in technique by modulating the static\n\feld, which yields the derivative of the resonance signal\ndP=dB .\nThe nuclear magnetic resonance (NMR) measurements\nwere carried out using pulsed NMR technique on1H nu-\nclei (nuclear spin I=1\n2and gyromagnetic ratio \rn=2\u0019=\n42:576 MHz/T) in the 2 \u0000230 K temperature range. The\nNMR measurements were performed at two di\u000berent ra-\ndio frequencies of 70 MHz and 38.5 MHz, which corre-\nspond to an applied \feld of about 1 :6608 T and 0 :9135 T,3\n600\n0.6\n0.4\n0.2\n0.00.81.0 dehydrated\npristine1.21.4\n/c99*T(emu□K/mol□Cu) 400\n200\n0\n0 100 100 10 2 200 300 400\nTemperature□(K) Temperature□(K)80010001200\ndehydrated□Cu -NIPA\n=□1.75 , =□9□K5\neff B/c109 /c113 /c109hydrated□Cu -NIPA5\neff B/c109 /c109 /c113=□1.75 , =□34□K\nground□state\n1/ (emu/mol□Cu)/c99/c451\n1\n2\n/c109 /c109 /c113eff B=□0.9 , =□0\nground□statehigh- paramagnet T\n1\n2\nFIG. 2. (Color online) Left panel: Inverse magnetic susceptibility of Cu 5-NIPA measured upon heating from 2 K (circles,\nhydrated form) and upon cooling from 400 K (triangles, dehydrated form). Lines show Curie-Weiss \fts to Eq. (1), as described\nin the text. Right panel: \u001f\u0003Tplot showing the formation of the\f\f1\n2istate in the pristine sample ( \u001f\u0003T=1\n4) and the high-\ntemperature paramagnetic regime of the dehydrated sample ( \u001f\u0003T=5\n4). Here,\u001f\u0003=\u001f\u0010\nNg2\u00162\nB\nkB\u0011\u00001\n,N=NA=5, andg= 2:2\naccording to ESR (Sec. III B). For the \u001f(T) plot, see Fig. 13.\nrespectively. Spectra were obtained by Fourier transform\nof the NMR echo signal. The1H spin-lattice relaxation\nrate, 1=T1, was measured by using the conventional sat-\nuration pulse sequence.\nThe electronic structure of Cu 5-NIPA was calculated in\nthe framework of density functional theory (DFT) using\ntheVASP code44for crystal structure optimization and\nFPLO45for the evaluation of magnetic parameters. The\ngeneralized gradient approximation (GGA)46exchange-\ncorrelation potential was augmented with the mean-\feld\nDFT+Ucorrection for Coulomb correlations in the Cu\n3dshell. The DFT+ Uparameters were chosen as Ud=\n9:5 eV (on-site Coulomb repulsion), Jd= 1 eV (on-site\nHund's exchange), and fully-localized-limit (FLL) \ra-\nvor of the double-counting correction, following earlier\nstudies of Cu2+-based magnets.47Reference calculations\nwith other exchange-correlation potentials and double-\ncounting corrections arrived at qualitatively similar re-\nsults. The reciprocal space was sampled by a kmesh\nwith 64 points in the \frst Brillouin zone. The conver-\ngence with respect to the kmesh was carefully checked.\nDetails of the computational procedure are reported in\nSec. IV A.\nThermodynamic and GS properties of the Cu 5\nmolecule were evaluated by full numerical diagonaliza-\ntions.\nIII. EXPERIMENTAL RESULTS\nA. Thermodynamic properties\n1. Magnetization\nThe magnetic susceptibility of Cu 5-NIPA was mea-\nsured for the as-prepared sample under \feld-cooling (FC)\ncondition upon heating from 1.8 K to 400 K. At 400 K,the sample was kept inside the SQUID magnetometer for\nabout 20 minutes, and \u001f(T) was measured again upon\ncooling from 400 K to 1.8 K. The drastic di\u000berence be-\ntween the heating and cooling curves (Fig. 2) indicates\nthe decomposition of Cu 5-NIPA shortly above room tem-\nperature. Indeed, the sample color changed from blue to\ngreen, and the weight loss of 13 % was detected. This\nweight loss is in good agreement with the preceding ther-\nmogravimetric data of Ref. 22 that reported the weight\nloss of 13.6 % at 380 \u0000400 K. Although the decomposi-\ntion process is tentatively ascribed to the release of water\nmolecules (expected weight loss of 13.15 %)22, no detailed\ninformation on the decomposed sample is available in the\nliterature.\nThe pristine sample shows a sharp decrease in the mag-\nnetic susceptibility (increase in 1 =\u001f) upon heating and a\nbend around 30 K followed by the nearly linear regime of\n1=\u001fabove 200\u0000250 K. In contrast, the decomposed sam-\nple remains paramagnetic down to at least 10 K. Below\n320 K, where the decomposition process is \fnished, the\ninverse susceptibility of dehydrated Cu 5-NIPA follows a\nstraight line that can be \ftted with the Curie-Weiss law\nin the 20\u0000320 K temperature range,48\n\u001f=C\nT+\u0012: (1)\nThe resulting Curie constant C'0:383 emu K/(mol Cu)\nleads to an e\u000bective moment \u0016e\u000b'1:75\u0016B, which is\nclose to 1.73 \u0016Bexpected for spin-1\n2. The Weiss temper-\nature is\u0012'9 K.\nThe inverse susceptibility of the pristine Cu 5-NIPA\nsample does not have a well-de\fned paramagnetic re-\ngion, because the data above 320 \u0000350 K are a\u000bected\nby the decomposition. A tentative Curie-Weiss \ft in\nthe 220\u0000320 K range yields an e\u000bective moment of\n\u0016e\u000b'1:75\u0016B(C= 0:383 emu K/(mol Cu)), which is\nsame as in the dehydrated sample. However, the Weiss4\n0.8\n0.4\n0.0\n5\nFiel d□(T)0 10Experiment\nBrillouin□function\n( =□2.38)g\nT=□1.8□K\n151.2\nMs=□1.19 /f.u. /c109B\nMagnetization□( /f.u.)/c109B\nFIG. 3. (Color online) Magnetization curve of Cu 5-NIPA\nmeasured at 1.8 K up to 14 T. The dashed line denotes the\nsaturation of the\f\f1\n2istate atMs= 1:19\u0016B/f.u. The solid\nline is the Brillouin function calculated at 1.8 K with g= 2:38.\ntemperature of \u0012'34 K in the pristine sample is notably\nlarger than \u0012'9 K observed in the dehydrated sample.\nThe conspicuous di\u000berence between the hydrated and\ndehydrated samples suggests a huge e\u000bect of dehydra-\ntion on the magnetism of the Cu 5molecule. The dehy-\ndration proceeds in a single step and results in the loss\nof all 10 water molecules per formula unit. As 4 out\nof these 10 molecules enter the \frst coordination sphere\nof Cu (Fig. 1, left), a large e\u000bect on the local environ-\nment of Cu sites and, thus, on the magnetism should\nbe expected.49Unfortunately, no structural data for the\ndehydrated version of Cu 5-NIPA are presently available.\nTherefore, we restrict ourselves to a detailed study of the\nhydrated compound.\nAt low temperatures, the inverse susceptibility of Cu 5-\nNIPA approaches another linear regime, with the Curie\nconstantC= 0:101 emu K/(mol Cu) ( \u0016e\u000b'0:90\u0016B)\nand vanishingly small \u0012'0:5 K. This low-temperature\nparamagnetic state can be also seen in the \u001f\u0003Tplot\n(\u001f\u0003=\u001f\u0000\nNg2\u00162\nB=kB\u0001\u00001is the reduced susceptibility),\nwhere\u001f\u0003Tapproaches the value of1\n4expected for the\ntotal spinS=1\n2per molecule (Fig. 2, right). Likewise,\nthe magnetization curve of Cu 5-NIPA (Fig. 3) saturates\natMs= 1:19\u0016B/molecule, which is, however, higher\nthan the free-electron value of 1 \u0016B/f.u. This di\u000berence\ncan be well accounted for by the large g-value of 2.38 ac-\ncording toMs=gS\u0016B. The same g-value explains the\nlow-temperature e\u000bective moment: C=Ng2\u00162\nB=3kB=\n0:106 emu K/(mol Cu), where we use N=NA=5 for\nthe magnetic moment of1\n2per Cu 5molecule. As we ex-\nplain later, this change in the g-value (Sec. IV B) can be\ntraced back to the presence of three non-equivalent Cu\npositions with di\u000berent g-tensor anisotropies. At room\ntemperature, the Cu spins are nearly independent, and\ntheirg-values average with same weights for all Cu posi-\ntions. At low temperatures, the powder averaging of the\ng-values is determined by the distribution of the magne-\ntization in the S=1\n2GS.\n4\n2\n0\n10 100 26\nC Tp/ (J□mol K )/c451/c452\nTemperature□(K)100 200 00400800\nCp(J□mol K )/c451/c45/c492 4 60.51.0\nC Tp/FIG. 4. (Color online) Speci\fc heat of Cu 5-NIPA divided\nby temperature ( Cp=T). The inset in the upper left corner\nmagni\fes the data at low temperatures. The inset in the\nbottom right corner shows the speci\fc heat ( Cp).\nOur results suggest that at low temperatures Cu 5-\nNIPA is in the paramagnetic state with the total moment\nofS=1\n2per Cu 5molecule. This state is further denoted\nas\f\f1\n2i. The experimental magnetization curve follows\nthe Brillouin function\nB(H) =g\u0016BS\u0002th\u0012g\u0016BSH\nkBT\u0013\n(2)\nwithS=1\n2,g= 2:38, andT= 1:8 K (Fig. 3). A marginal\ndeparture of the experimental curve toward higher \felds\nmay be due to anisotropies and/or intermolecular cou-\nplings. The maximum deviation of about 0.3 T puts an\nupper limit of about 0.5 K on the total energy of such\ncouplings. This low energy scale is in agreement with the\nlarge spatial separation between the molecules (Fig. 1,\nright).\nBelow 4 K, the susceptibility of Cu 5-NIPA departs\nfrom\u001f\u0003T=1\n4(Fig. 2, right). This decrease in \u001f\u0003Tmay\nindicate an evolution of the system toward a long-range\nmagnetic order between the Cu 5molecules. However, we\nwere unable to see any clear signatures of a magnetic\ntransition in the susceptibility data measured down to\n2 K.\n2. Speci\fc heat\nThe speci\fc heat ( Cp) of Cu 5-NIPA is smooth down\nto 2 K (Fig. 4). It steeply increases up to 60 \u000070 K\nand shows a slower increase at higher temperatures, as\nshown by a broad maximum in the temperature depen-\ndence ofCp=T. The signal is dominated by the phonon\ncontribution, whereas the magnetic part is quite small.\nThe total magnetic entropy of Cu 5-NIPA is 5 Rln 2'\n28:8 J mol\u00001K\u00001, which is only 2 % of the total entropy\nof about 1100 J mol\u00001K\u00001released up to 200 K (the\nlatter is obtained by integrating the temperature depen-\ndence ofCp=T).5\nAt 200 K, the heat capacity of Cu 5-NIPA is still very\nfar from its maximal Dulong-Petit value of Cp= 3RN'\n2767 J mol\u00001K\u00001, because Cu 5-NIPA features a large\nnumber of high-energy phonon modes related to the O{H,\nC{C, C{N, and C{O vibrations. The entropy associated\nwith these modes can be released at very high temper-\natures, only. The complexity of the Cu 5-NIPA struc-\nture and the presence of multiple phonon modes of di\u000ber-\nent nature hinder a quantitative analysis of the speci\fc\nheat data. A non-magnetic reference compound would\nbe ideal to extract the magnetic contribution and ana-\nlyze it in more detail. Unfortunately, such a reference\ncompound is presently not available.\nAt low temperatures, the heat capacity of Cu 5-NIPA\ndoes not fall smoothly to zero. The temperature depen-\ndence ofCp=Tshows an increase below 3 K (see the up-\nper left inset of Fig. 2). The origin of this behavior is\npresently unclear. The increase in Cp=Tcould signify a\nSchottky anomaly or a proximity to the magnetic order-\ning transition. However, our susceptibility data, as well\nas NMR (Sec. III C), rule out any magnetic transition\ndown to 2 K. The heat capacity data are consistent with\nthese observations.\nB. ESR\nThe ESR spectrum of Cu 5-NIPA could be detected\nfrom the lowest measured temperature (5 K) up to 135 K.\nAbove 135 K, the ESR line broadens and eventually be-\ncomes invisible. Between 70 K and 135 K, the spec-\ntra contain only one line that can be \ftted with a sin-\ngle Lorentzian function (1L-Fit) providing the ESR pa-\nrameters, the linewidth \u0001 Bandg-factor,g=h\u0017\n\u0016BBres,\nwhereBresis the resonance \feld. Below 70 K, the sin-\ngle ESR line splits, so that at 40 \u000060 K the \ft is only\npossible with two Lorentzian lines (2L-Fit), whereas at\neven lower temperatures a plethora of narrow lines ap-\npear (Fig. 5). At 5 K, more than 20 lines are visible\nin the spectrum. This complex structure is typical in\nlow-temperature ESR spectra of molecular magnets and\nis related to the hyper\fne coupling.50,51The complexity\nof the signal and the presence of multiple Cu sites and\nanisotropy parameters prevent us from a detailed anal-\nysis of the low-temperature powder spectra. One would\nneed much higher frequencies and, preferably, single crys-\ntals of Cu 5-NIPA, in order to discriminate the multiple\nresonances that are visible below 40 K. So we restrict\nourselves to the analysis of the data above 40 K.\nFig. 6 shows the temperature evolution of the ESR g-\nfactor, linewidth, and line intensities. Above 70 K, the\nobserved powder-averaged values of g'2:2 are typical\nfor Cu2+in the planar oxygen environment.28,50,51Our\nmicroscopic insight into the energy spectrum of the Cu 5\nmolecule (Sec. IV B) suggests that the formation of two\nlines below 70 K is related to two lowest S=1\n2energy\nlevels, which are separated by \u0001 '68 K. Indeed, lines\n1 and 2 show di\u000berent temperature evolution. While the\ndP dB/ (arb.□units)\n1.0 1.1 1.2 1.365□K5□K\nx20\nFiel d□(T)FIG. 5. (Color online) Typical ESR signal of Cu 5-NIPA at\n5 K (solid line) and 65 K (dashed line). For better visibility,\nthe intensity of the high-temperature signal was increased by\na factor of 20.\n2.2\n2.1\n300\n200\n100\n0\n0\n40 60 80 100\nTemperature□(K)120 14012\nIntensity□(arb.□units)Linewidth□(mT)2.31L-fit 2L-fit,□line□1 2L-fit,□line□2\nESR -factorg\nFIG. 6. (Color online) Temperature dependence of the ESR\nsignals:g-factor (top), linewidth (middle), intensity (bot-\ntom). The data at higher temperatures (diamonds) were ob-\ntained by \ftting the spectra with one Lorentzian line (1L-Fit).\nOpen circles and triangles show the data from \ftting the spec-\ntra with two Lorentzian lines (2L-Fit). Representative error\nbars are shown for several data points, only. All intensities\nare scaled to the intensity of line 2 at 65 K.\nintensity of line 2 is reduced upon cooling, the intensity\nof line 1 is growing. In Sec. IV B below, we provide the\nexplicit dependence of the e\u000bective g-tensors for the GS\nand the \frst excited doublet (in terms of the individual\ng-tensors of the three inequivalent Cu sites) in order to6\n0 50 100 150 2000.5\n0.0\n80120160\nFWHM□(kHz)69.5 69.6 69.7/c1090H=□1.6608□T\n/c1090H=□1.6608□T/c110(MHz)69.8 69.94.2□K\n9.9□K\n20□K\n69.8□K\n140□K\n189.8□K\n230.5□K\n70.01.0\nIntensity, /I Imax\n250\nTemperature□(K)\nFIG. 7. (Color online) Top panel: Fourier-transform1H\nNMR spectra measured at di\u000berent temperatures. Arrows\nshow the shifts of the left and right shoulders upon cooling.\nNote that while the right shoulder is moving toward higher\nfrequencies, the left shoulder shows a non-monotonic behav-\nior. Bottom panel: Full width at half-maximum (FWHM)\nof1H NMR spectra plotted as a function of temperature ( T)\nmeasured in an external \feld \u00160H= 1:6608 T.\ndemonstrate the origin of their di\u000berence.\nC.1H NMR\n1. Linewidth\nConsidering the nuclear spin I=1\n2of the1H nu-\ncleus, one expects a single spectral line for each of the\n17 nonequivalent proton sites (see Table III of App. B).\nHowever, these lines merge into a single broad line with\na nearly Gaussian line shape over the whole measured\ntemperature range (Fig. 7, top). The linewidth at 215 K\nis 95 kHz and almost comparable with the linewidth re-\nported for other molecular magnets (see, e.g., Ref. 52).\nThe line position shifts weakly with temperature. The\nbottom panel of Fig. 7 shows the temperature evolu-\ntion of the full width at half-maximum (FWHM) of the\n1H NMR line measured in an applied \feld of \u00160H=\n1:6608 T. It is almost temperature-independent at high\ntemperatures and increases progressively upon cooling\nbelow about 30 K.\nThe shape and width of the1H NMR spectra are gov-\nerned by two main interactions: the nuclear-nuclear dipo-\nlar interactions and the hyper\fne couplings between the\nproton and unpaired electrons at the Cu sites. Therefore,we can write FWHM as:53{55\nFWHM/p\nh\u0001\u00172id+h\u0001\u00172im; (3)\nwhere the broadening due to the nuclear-nuclear dipolar\ninteractions (h\u0001\u00172id) is temperature-independent, while\nthe broadening due to the hyper\fne couplings ( h\u0001\u00172im)\nscales with the local susceptibilities. For each given pro-\nton sitep,\nq\nh\u0001\u00172i(p)\nm\nH'X\njA(p)\nj\u001fj; (4)\nwhereA(p)\njis the dipolar coupling constant between the\nproton and the Cu2+ions at site j, and\u001fjis the lo-\ncal susceptibility. We should note here that, in general,\nthe temperature dependence of \u001fjis di\u000berent from that\nof the bulk susceptibility \u001f(Fig. 2), especially at higher\ntemperatures, and they are also di\u000berent from each other\n(see Fig. 15). The sharp increase in FWHM at low\ntemperatures can be ascribed to the corresponding low-\ntemperature increase in \u001fj.\nThe shallow minimum in FWHM observed around\n70 K is paralleled by the peculiar evolution of the spec-\ntral line (Fig. 7, top). On cooling down, the right side of\nthe line shifts weakly but monotonously to the right, fol-\nlowing the central position of the line. The left side, on\nthe other hand, shows a non-monotonic behavior, shifting\nbackwards for the data at 20 K and below. The origin of\nthis feature (and the minimum in FWHM) can in princi-\nple be traced back to the temperature dependence of the\nlocal Cu2+moments, and indeed such a non-monotonic\nbehavior is shown by the moments on the Cu2 sites. As\nwe discuss in detail below (Sec. IV B and Fig. 15), at\nT\u001c\u0001'68 K, the moments of the two Cu2 sites are\nantiparallel to the \feld, owing to the large negative ex-\nchange \feld exerted by the neighboring Cu1 and Cu3\nsites. These exchange \felds are balanced by entropy at\nsome characteristic temperature T\u0003, at which the Cu2\nmoments turn positive. Our calculations based on the\nactual exchange couplings give T\u0003'38 K (see Fig. 15\nbelow). At even higher temperatures, the Cu2 moments\nattain their paramagnetic Curie-like behavior of isolated\nspins. The contribution to the second moment from the\nCu2 sites is then expected to decrease down to zero and\nthen increase again as we cool down, starting from the\nhigh-temperature paramagnetic to the low-temperature\n\\ferrimagnetic\" S=1\n2state of Cu 5-NIPA.\n2. Wipe-out e\u000bect\nThe wipe-out e\u000bect is common in molecular\nnanomagnets,39,56{58and refers to the gradual loss\nin the NMR signal intensity below a characteristic\ntemperature. Here, we have measured the temperature\ndependence of Mxy(0)T, whereMxy(0) is the transverse7\n10 100/c1090H=□0.9135□T0.6\n0.4\n0.2\n0.00.81.01.2\nIT I T/max max\nTemperature□(K)\nFIG. 8. (Color online) Normalized integrated intensity\n(IT=I maxTmax) measured at \u00160H= 0:9135 T. The strong\ndrop in the intensity below 100 K is the wipe-out e\u000bect. The\nline shows the maximum intensity ( ImaxTmax) attained at high\ntemperatures.\nmagnetization at time t= 0, obtained by the extrapola-\ntion of theMxy(t) recovery curve to t=0. The integrated\nintensityMxy(0)Tis proportional to the total number\nof protons resonating at the irradiated frequency. The\nnormalized integrated intensity of the NMR signal in\nCu5-NIPA as a function of temperature is shown in\nFig. 8. We \fnd that the loss of the NMR signal begins\nbelow 150 K and becomes more pronounced at low\ntemperatures. The onset of the NMR signal loss around\n150 K coincides with the regime where 1 =T1starts\nincreasing towards the maximum (see Fig. 9 below).\nThe loss in the NMR signal is related to the slowing\ndown of the electron spin dynamics which, as we explain\nbelow, is also responsible for the enhancement in 1 =T1.\nOn decreasing T, a progressively larger fraction of pro-\ntons close to the magnetic Cu2+ions attain spin-spin\nrelaxation times ( T2) shorter than the dead time ( \u001cd) of\nthe spectrometer, hence the signal from these protons\ncannot be detected. At low temperatures, only the pro-\ntons that are very far from the magnetic ions contribute\nto the signal intensity. For further details, see Refs. 56\nand 57.\n3. Nuclear spin-lattice relaxation rate 1=T1\nThe spin-lattice relaxation rate 1 =T1for Cu 5-NIPA was\nmeasured at two di\u000berent applied \felds. The recovery of\nthe longitudinal nuclear magnetization after a saturation\npulse was \ftted well by the stretched exponential func-\ntion (top panel of Fig. 9, inset)\n1\u0000M(t)\nM0'A0e\u0000(t=T1)\f; (5)\nwhereM(t) is the nuclear magnetization at a time tafter\nthe saturation pulse, and M0is the equilibrium magne-\ntization. The value of the exponent \ffor both \felds was\n0.4\n0.0\n0\n0 50 100 150 200 250\nTemperature□(K)2460.81.21.6\n1/ (ms )T1/c451\n1/( (emu□ms□K/mol)/c99TT1/c451\n)10 20 30 t(ms)0.010.114.3□K 22□K 43□K\n160□K\n1 ( )//c45M t M0\n100 50 0 150/c1090H=□0.9135□T\n200Orbach□process\n250\nTemperature□(K)1\n0/c119(10 rad/sec)920.9135□T\n1.6608□TFIG. 9. (Color online) Top panel: Spin-lattice relaxation\nrate 1=T1vs.Tmeasured at two di\u000berent values of external\n\feld, 0.9135 T and 1.6608 T. The inset shows the longitudi-\nnal recovery curves at four representative temperatures; solid\nlines are the \fts using Eq. (5). Bottom panel: 1 =(\u001fT1T)\nvs.Tfor two di\u000berent magnetic \felds. The inset shows the\ntemperature-dependent !c(decay rate of the total moment\nSz) extracted from the 1 =T1data; the solid line is the \ft\nusing Eq. (7).\nfound to decrease slowly from 0.95 to 0.7 upon cooling.\nThe deviation of \ffrom unity re\rects the distribution of\nrelaxation rates according to di\u000berent hyper\fne couplings\nbetween the1H nuclei and Cu2+ions.57,59,60Accordingly,\nthe spin-lattice relaxation rate 1 =T1obtained by \ftting\nthe recovery with Eq. (5) provides a value averaged over\nthe whole distribution.\nTheTdependence of 1 =T1is presented in the top panel\nof Fig. 9 for two values of the external \feld. At high tem-\nperatures ( T>\u0018150 K), 1=T1is almostT-independent.\nThis behavior is typical for uncorrelated paramagnetic\nmoments \ructuating fast and at random.61,62At lower\ntemperatures, 1 =T1increases and passes through a max-\nimum atT'40 K. Such a characteristic enhancement\nhas been found in numerous AFM homometallic38and\nheterometallic39rings built of spins s>1\n2, but in spin-1\n2\nsystems it is very rare. The only example known to us\nis the Cu 6magnet with a high-spin S= 3 GS.63,64Sys-\ntems with predominantly AFM couplings and low-spin\nGS (e.g., V 12having anS= 0 GS, Ref. 65) do not show\nsuch a feature in 1 =T1.\nIn homometallic rings, the maximum in 1 =T1essen-\ntially signals the slowing down of phonon-driven spin\n\ructuations.38,40{42,57The equivalence between the spins\n(by virtue of the nearly perfect rotation symmetry of the\nring) allows to express 1 =T1in terms of the spectral den-\nsity of the total magnetic moment Szof the molecule.40\nAs shown numerically40and by a microscopic theory,41,428\nthe phonon-driven decay of Szproceeds independently\nfrom the remaining observables of the problem, which in\nturn leads to a single Lorentzian form for 1 =T1:\n1\nT1'A\u001fT!c(T)\n!2c(T) +!2\nL; (6)\nwhereAis the average square of the transverse hyper\fne\n\feld,!Lis the nuclear Larmor frequency, and !c=1\n\u001c(T)\nis the decay rate of the total moment Sz. The enhance-\nment in the spin-lattice relaxation rate 1 =T1takes place\nwhen!capproaches the order of magnitude of !L.\nThere are two qualitative di\u000berences between Cu 5-\nNIPA and homometallic rings that should be emphasized,\nthough. First, Cu 5-NIPA features a very sparse excita-\ntion spectrum (Sec. IV B), which means that there are\nvery few spin-phonon channels available for relaxation.\nAccording to Ref. 42, this also means that here !cis\nnot expected to show the strong power-law \u0018T3orT4\nbehavior. Instead, one expects a much weaker Tdepen-\ndence, with longer relaxational times in a wide tempera-\nture range.\nThe second di\u000berence is the fact that the Cu 5molecule\ncomprises three inequivalent Cu sites with di\u000berent local\nmagnetizations, and so the above single-Lorentzian for-\nmula for 1=T1is not valid any longer, but instead a multi-\nLorentzian form should be expected on general grounds,\nas in the case of the heterometallic Cr 7Ni ring.66\nGiven the large spin gap, \u0001 '68 K, one could still\nargue in favor of the single-Lorentzian formula of Eq. (6)\nover a wide low- Trange, since the Wigner-Eckart theo-\nrem allows to replace individual spin operators with the\ntotal spin operators times a constant. However, such a\ntreatment would only capture the resonant spin-phonon\ntransitions between the two Zeeman-split levels of the\nGS, but these transitions are prohibited by Kramer's the-\norem. Instead, for temperatures well below the spin gap\n\u0001'68 K, the relaxation dynamics of the spins must\nbe controlled by inter-multiplet Orbach transitions67be-\ntween the GS doublet and the lowest magnetic excitation\nwith the energy \u0001 (see below). Again, this is quite sim-\nilar to Cr 7Ni, with the only di\u000berence that the spin gap\nis much smaller there ( '14 K).66\nWith the above remarks in mind, we may still use the\nabove single-Lorentzian formula for 1 =T1to extract the\ntemperature dependence of !c, but the latter is now a\nrepresentative measure of the relaxation dynamics (as\nprobed by NMR), and not necessarily the relaxation rate\nof the total moment. For simplicity, we have plotted\n1=\u001fT 1Tvs.Tin the bottom panel of Fig. 9 that clearly\nshows a broad maximum re\recting the slowing down of\n\ructuating moments. Following Eq. (6) and the pro-\ncedure outlined in Ref. 38, we extracted !cfrom the\n1=\u001fT 1Tdata at\u00160H= 0:9135 T. The resulting Tde-\npendence of !c(shown in the lower inset of Fig. 9) con-\n\frms that it is much weaker compared to the strong\npower-law dependence found in homometallic rings.38In\nparticular, there is a wide low- Trange over which !c\nfollows the inter-multiplet Orbach relaxation processes\n300\n200\n100\n0\n0\nEnergy□(eV)4CuTotal\n/c454 /c458400500\nDOS□(eV )/c451100200\nTotal\nCu\nO\n0\n0 0.2 /c450.2 /c450.4FIG. 10. (Color online) LDA density of states for Cu 5-NIPA.\nThe shading shows the contribution of Cu orbitals. The inset\nmagni\fes \fve bands at the Fermi level ( E= 0), with the\natomic contributions denoted by the dashed (Cu) and solid\n(O) lines.\nmentioned above. Following Ref. 67,\n!c'3\u00152\u00013\n2\u0019\u0016h4\u001amv5\u00021\ne\u0001=T\u00001; (7)\nwhere\u001am= 2043 kg/m3is the mass density,22vis the\nsound velocity in Cu 5-NIPA (typical value c\u00181500\nm/sec in nanomagnets68), and\u0015stands for the spin-\nphonon coupling energy parameter, which is related e.g.\nto the \ructuating portion of the Dzyalozinskii-Moriya in-\nteractions present in this system. Note that we neglect\nthe Zeeman splitting contribution to the resonance en-\nergy, because it is negligible compared to \u0001. By \ftting\nthe!cdata with Eq. (7), we obtain \u0015=kB'3:9(v=c)5\n2K.\nA more complete quantitative understanding of 1 =T1\n(e.g. the high-temperature behavior and \feld depen-\ndence) must take into account the multi-exponential be-\nhavior of the relaxation, discussed above, but also the\npresence of the wipe-out e\u000bect discussed above, see e.g.\nRef. 66.\nIV. THEORY\nA. Microscopic magnetic model\nTo determine individual exchange parameters in the\nCu5molecule, we calculate the electronic structure of\nthe Cu 5-NIPA compound. This procedure requires reli-\nable crystallographic information, including precise posi-\ntions of all atoms in the unit cell. However, the available\nstructural data21,22are obtained from x-ray di\u000braction\nthat has only limited sensitivity to the positions of hy-\ndrogen atoms. Therefore, we used the literature data as\na starting model and optimized the hydrogen positions,\nwhereas all other atoms were kept \fxed. The equilibrium\npositions of hydrogen are listed in Table III of App. B.\nThe relaxed structure is 18.6 eV/f.u. lower in energy9\n/c450.2\n/c450.40.0\nX M Y Z T R A à Ã0.2\nEnergy□(eV)\nFIG. 11. (Color online) LDA band structure of Cu 5-\nNIPA (thin light lines) and the \ft with the tight-binding\nmodel (thick dark lines). The Fermi level is at zero en-\nergy (white dashed line). The notation of kpoints is as fol-\nlows: \u0000(0;0;0),X(1\n2;0;0),M(1\n2;1\n2;0),Y(0;1\n2;0),Z(0;0;1\n2),\nT(1\n2;0;1\n2),R(1\n2;1\n2;1\n2), andA(0;1\n2;1\n2), where the coordinates\nare given in units of the reciprocal lattice parameters.\nTABLE I. Interatomic distances d(in\u0017A), Cu{O{Cu bridging\nangles'(in deg), hopping parameters ti(in meV), and ex-\nchange integrals Ji(in K) in Cu 5-NIPA. The Jivalues are\nobtained from DFT+ Ucalculations. The AFM contribu-\ntionsJAFM\ni are evaluated as 4 t2\ni=Ue\u000b, whereUe\u000bis the ef-\nfective on-site Coulomb repulsion. The FM contributions are\nJFM\ni=Ji\u0000JAFM\ni.\ndCu{Cu'Cu{O{Cu tiJAFM\niJFM\niJi\nJ13 3.20 107.9 \u00000:054 34\u000089\u000055\nJ12 3.33 114.5 \u00000:128 191\u0000134 57\nJ23 3.51 125.7 \u00000:161 302\u000034 268\nthan the starting model taken from the literature. This\nenergy reduction should be ascribed to the elongation of\nO{H distances that are unrealistically short (about 0.8 \u0017A)\nin the experimental structural data.22\nThe LDA energy spectrum of Cu 5-NIPA (Fig. 10) com-\nprises narrow lines that represent molecular orbitals of\nthe NIPA molecules and the Cu 5(OH) 2unit. The states\nat the Fermi level belong to the narrow Cu 3 dbands with\na sizable admixture of O 2 p. The local environment of Cu\natoms resembles the conventional CuO 4plaquette units\n(four-fold coordination, see Fig. 1, left). Therefore, the\nhighest-lying Cu 3 dbands have predominantly x2\u0000y2\norigin, following the crystal-\feld levels of Cu2+withx\nandyaxes lying in the plane of the CuO 4plaquette.\nThe \fve Cu atoms of the Cu 5molecule (one molecule\nper unit cell) give rise to \fve bands, with the middle\nband crossing the Fermi level (Figs. 10 and 11). This\nspurious metallicity is due to the strong underestimate\nof electronic correlations in LDA. DFT+ Ucalculations\nreveal the robust insulating behavior with a band gap of\nabout 2.2 eV in reasonable agreement with the light-blue\ncolor of the sample.\nHOO\nO O\nOOH\nCu1/c1093-OH\nHCCFIG. 12. (Color online) Cu dx2\u0000y2-based Wannier function\nfor the Cu1 site in Cu 5-NIPA. Note that the ligands (COO\u0000,\n\u00163-OH\u0000, H2O) a\u000bect the positions of oxygen atoms and their\norbitals. However, only the orbitals of four oxygen atoms\ncontribute to the Wannier function.\nTo evaluate the magnetic couplings, we \ft the \fve Cu\nbands with a tight-binding model (Fig. 11) and extract\nthe relevant hopping parameters tiusing Wannier func-\ntions (WFs) based on the Cu dx2\u0000y2orbital character.69\nTheti's are further introduced into an e\u000bective Hub-\nbard model with the on-site Coulomb repulsion Ue\u000b. As\nthe conditions of the half-\flling and strong correlations\n(ti\u001cUe\u000b) are ful\flled, the lowest-lying excitations can\nbe described by a Heisenberg model with the AFM ex-\nchangeJAFM\ni = 4t2\ni=Ue\u000b. The values of tiandJAFM\ni\nobtained from the LDA band structure are listed in Ta-\nble I and provide an overview of the magnetic couplings\nin Cu 5-NIPA. Considering only nearest-neighbor interac-\ntions, we \fnd a strong coupling between Cu2 and Cu3, a\nsomewhat weaker coupling between Cu1 and Cu2, and a\nrelatively weak coupling between Cu1 and Cu3. Long-\nrange couplings are several times smaller than JAFM\n13.\nThe long-range exchanges in the Cu 5molecule are within\n10 K, whereas the interactions between the molecules are\nbelow 3 K (note the experimental estimate of 0.5 K in\nSec. III A). As the long-range couplings are much weaker\nthanJ12,J13, andJ23, we further restrict ourselves to\nthe minimum microscopic model that comprises three\nnearest-neighbor interactions, only.\nPossible FM contributions to the short-range cou-\nplings require that the evaluation of JAFM\ni is supplied\nby an independent estimate of total exchange couplings\nJi. In a so-called supercell approach, total energies of\ncollinear spin con\fgurations are mapped onto the Heisen-\nberg model to yield the Jivalues, as listed in Table I.\nAll nearest-neighbor interactions have sizable FM compo-\nnentsJFM\ni=Ji\u0000JAFM\ni that reduce J12andJ23, whereas\nJ13eventually becomes ferromagnetic. This way, the tri-\nangular units in the Cu 5molecule feature a combination\nof FM interaction J13and AFM interactions J12andJ23\n(Fig. 1, left).\nThe microscopic origin of the magnetic couplings can\nbe understood from the analysis of Cu-based Wannier\nfunctions (Fig. 12). Each WF features the Cu dx2\u0000y210\n1234\n0\n0 100 200 300 400\nTemperature□(K)/c99(10 emu/mol□Cu)/c452\n400 300 200Experiment□(1□T)\nFit: = =62□K,\n/ =3.5J12/c45J\nJ J13\n23 12\n100 08001200\n400\n0/c99/c451(emu/mol□Cu)/c451\nFIG. 13. (Color online) Fit of the magnetic suscepti-\nbility (\u001f) and inverse magnetic susceptibility ( \u001f\u00001, inset)\nwith the exact-diagonalization result for the Cu 5pentamer\n(J12=\u0000J13= 65 K and J23=J12= 3:5).\norbitals together with the 2 porbitals of the surround-\ning oxygen atoms. These oxygen atoms have di\u000berent\nchemical environment and belong to one of the ligands:\n\u00163-OH\u0000, COO\u0000, and H 2O. While the ligands a\u000bect the\npositions of oxygen atoms (note the slight downward dis-\nplacement of the O atoms in COO\u0000groups) and modify\nthe \\shape\" of the porbitals (note the oxygen atom be-\nlonging to the H 2O molecule), they do not provide any\nsizable long-range contributions to the WFs. The short-\nrange nature of the WFs underlies the diminutively small\nlong-range exchange in Cu 5-NIPA. Further microscopic\naspects of the magnetic interactions in Cu 5-NIPA are dis-\ncussed in Sec. V.\nOur microscopic model can be directly compared to the\nexperimental data. Using full diagonalization for the Cu 5\nmolecule, we calculate thermodynamic properties and re-\n\fne theJiparameters as to match the experiment. This\nway, we \ft the experimental magnetic susceptibility curve\nwithJ12=\u0000J13= 62 K and J23= 217 K (so that\nJ12=J23'2\n7), as well as g= 2:22 and the temperature-\nindependent contribution \u001f0=\u00001:7\u000210\u00004emu/mol Cu\n(Fig. 13). The experimental estimates of Jiare in excel-\nlent agreement with the DFT results (Table I), whereas\ntheg-value matches the high-temperature g'2:2 from\nESR (Sec. III B). Regarding the sizable \u001f0, it may origi-\nnate from the diamagnetic gelatin capsule that was used\nin the susceptibility measurement.\nThe combination of the FM coupling J13and AFM\ncouplingsJ12andJ23implies that the spin triangles in\nCu5are non-frustrated. In a classical picture, all three\ncouplings are satis\fed in a con\fguration with Cu3 and\nCu1 spins pointing up and Cu2 spins pointing down, thus\nleading to the total moment of1\n2per molecule. Indeed,\nat low temperatures Cu 5-NIPA shows paramagnetic be-\nhavior with the magnetic moment of1\n2per molecule (\f\f1\n2i\nstate). In high magnetic \felds, the\f\f3\n2iand, eventually,\f\f5\n2istates will be stabilized. The formation of states with\nthe higher magnetic moment can be seen from the sim-\n2\n1\n0\n0\n0 20 40 60\nFiel d□(T)243\nstatic□fiel d□( =□1.8□K) T\nsimulation□( =□1.8□K) Tpulsed□fiel d□( =□1.4□K) T\nMagnetization(arb.□units)\n( /f.u.)/c109B\n1\n23\n2FIG. 14. (Color online) Upper panel: magnetization of\nCu5-NIPA measured in pulsed magnetic \feld (in arb. units).\nBottom panel: magnetization measured in static \feld up to\n14 T and the simulated curve for J12=\u0000J13= 65 K and\nJ23=J12= 3:5 (both in units of \u0016B/f.u.) The arrow denotes\nthe bend in the pulsed-\feld curve that matches the transition\nbetween the\f\f1\n2iand\f\f3\n2istates of the Cu 5molecule.\nulated magnetization curve of Cu 5-NIPA. The\f\f1\n2istate\npersists up to about 50 T, where the magnetization in-\ncreases abruptly until the\f\f3\n2istate is reached.\nWe endeavored to verify this prediction with magneti-\nzation measurements in pulsed \felds. Unfortunately, the\nresults are strongly a\u000bected by dynamic e\u000bects. The typ-\nical duration of the pulse (about 20 \u0016s) is insu\u000ecient to\nchange the magnetic moment of the molecule. Therefore,\nall features are blurred and, moreover, the curve shows\na non-zero slope above 5 T (Fig. 14, top), in contrast to\nthe \rat plateau that is observed in static \feld (Fig. 14,\nbottom). The apparent mismatch between the pulsed-\n\feld and static-\feld data also prevents us from scaling\nthe high-\feld curve. Nevertheless, the bend observed at\n50\u000055 T is likely the signature of the\f\f1\n2i\u0000!\f\f3\n2itran-\nsition and con\frms our expectations.\nB. Exact Diagonalization\nWe are now ready to evaluate the magnetic energy\nspectrum of each Cu 5molecule and deduce its basic prop-\nerties, according to the DFT values of the exchange cou-\nplingsJi. Using a site labeling convention in accord with\nthe site symmetries, the spin Hamiltonian (disregarding\nanisotropies) reads:\nH=J23S3\u0001S220+J12(S1\u0001S2+S10\u0001S20\u0000S3\u0001S110);(8)\nwhere S220\u0011S2+S20,S110\u0011S1+S10, and we have used\nthe relation J13=\u0000J12. In addition to the full SU(2)\nspin rotation group, this Hamiltonian is also invariant\nunder the inversion operation (in real space) through the\ncentral site S3, which maps S3$S3,S1$S10, and11\nS2$S20. So all eigenstates can be characterized by the\ntotal spinS, its projection Malong some quantization\naxisz, and the parity p(even or odd) under inversion.\nGlobal spectral structure. A straightforward numerical\ndiagonalization of Hyields the energy spectrum given in\nTable II. For each level we also provide the good quantum\nnumbersSandp, as well as the expectation values of the\nsquare of the composite spins S220,S3220\u0011S3+S220, and\nS110. The essential \fnding is that the GS is a total spin\nS=1\n2doublet and the lowest excitation (also a S=1\n2\ndoublet) has a very high energy gap \u0001=0 :313J23'68 K.\nThus the Cu 5molecule behaves as a rigid S=1\n2entity\nforT\u001c\u0001, con\frming the previous experimental picture\nfrom magnetization measurements (Sec. III A 1).\nLooking at Table II, we also \fnd that the spectrum is\norganized in three compact groups of states, which are\nseparated by horizontal lines. All states belonging to a\ngiven group show a very similar value of hS2\n3220i, suggest-\ning that the trimer of S3,S2, and S20plays a special\nrole in the physics of Cu 5-NIPA. Moreover, the expecta-\ntion values of S2\n220,S2\n3220, and S2\n110are for all states very\nclose to the formula S(S+ 1) for some integer or half-\nintegerS, suggesting that these composite spins are al-\nmost conserved quantities. Below, we shall demonstrate\nthat all these spectral features as well as the GS proper-\nties can be physically understood quite naturally on the\nbasis of a strong coupling expansion around the limit of\nJ12=jJ13j=0.\nFrom the spectrum, we can also obtain the level-\ncrossing \felds at which the GS of the molecule changes\nfromS=1\n2toS=3\n2and fromS=3\n2toS=5\n2. We \fnd:\ng\u0016BHc1=kB'0:40J23; g\u0016BHc2=kB'1:43J23:(9)\nWithJ23= 217 K and the average g'2:38 (see pre-\nviously), these numbers yield Hc1'54 T (compare to\nFig. 14) and Hc2'194 T.\nGS properties. To probe the nature of the GS, we\ncalculate several experimentally relevant GS expectation\nvalues. We \frst consider how the total S=1\n2moment of\nthe GS (at low T, this moment can be attained by a small\napplied \feld) is distributed among the \fve Cu sites. At\nT\u001c\u0001, we \fnd\nhSz\nii=hSz\nii0\u0002tanh\u0012g\u0016BH\n2kBT\u0013\n; (10)\nwherehSz\nii0are the GS expectation values (in an in-\n\fnitesimal \feld):\nhSz\n3i0=0:1416;hSz\n2i0=hSz\n20i0=\u00000:1415;\nhSz\n1i0=hSz\n10i0=0:3207:(11)\nThe signs of the above numbers are in agreement with the\nclassical ferrimagnetic GS discussed above (Fig. 1, right),\nwhereby the spins S3,S1, and S10are aligned parallel\nto each other, but antiparallel to S2andS20. However\nthe strong deviation (especially for S3,S2andS20) from\ntheir maximum (classical) value of1\n2indicates that thisTABLE II. The exact spectrum of the spin Hamiltonian H\nof Eq. (8) at y=J12=J23=2\n7. For each state we also provide\nits total spin S, its parity punder the real-space inversion\nthrough the central site S3, as well as the expectation values\nofS2\n220,S2\n220, and S2\n220. The horizontal lines di\u000berentiate the\nthree unperturbed manifolds of the strong coupling limit y=0\n(see text and Fig. 16).\ntotal S parity ptotalE=J 23hS2\n220i hS2\n3220i hS2\n110i\n1\n2even\u00001:3289 1.969 0.795 1.969\n1\n2odd\u00001:0157 1.976 0.75 0.024\n3\n2even\u00000:9286 2 0.893 2\n3\n2odd\u00000:1732 0.086 0.879 1.914\n1\n2even\u00000:0395 0.222 1.011 0.222\n1\n2odd 0.3014 0.024 0.75 1.976\n5\n2even 0.5 2 3.75 2\n3\n2odd 0.5303 1.914 3.621 0.086\n3\n2even 0.5714 2 3.607 2\n1\n2even 0.5827 1.809 3.444 1.8087\n0 10 20 30 40 50−0.100.10.20.30.4\nT(K)  \n0 50 100 150 200−0.0100.010.02\nT (K)  \nCu1\nCu2Cu3Cu3total \nSz\nCu2total \nSz\nT*=38 KCu1\nFIG. 15. (Color online) Temperature dependence of the local\nmoments of the three inequivalent Cu sites in Cu 5-NIPA as\nobtained by exact diagonalization using our ab initio exchange\ncoupling parameters and a \feld of \u00160H= 1:6608 T. The\ninset shows the same results in a di\u000berent scale, so that the\ndeviation from the low- Tbehavior of Eq. (10), shown by the\ndashed lines, is better visible. Also, T\u0003'38 K denotes the\ncharacteristic temperature where the Cu2 moments change\nsign.\nstate has strong QM corrections, as can be directly seen\nfrom the explicit form of the wavefunction (not shown).\nThis feature will be explained on the basis of the strong\ncoupling description developed below.\nThe deviation of the local moments from Eq. (10) at\nhigh temperatures is shown in Fig. 15 at a \feld of 1 :6608\nT. We note in particular the non-monotonic temperature\ndependence of the magnitude of the Cu2 moments, which\nwas discussed in Sec. III C 1 above in relation to the NMR\nlineshape.\nTo further probe the nature of the GS, we consider the\nstrength of various spin-spin correlations eij\u0011hSi\u0001Sji0.12\nWe \fnde31=e310'+0:2122,e32=e320' \u00000:4810,\ne12=e1020' \u0000 0:4299,e120=e102' \u0000 0:2858, and\ne110=e220'+0:2345. These values corroborate the\nabove QM ferrimagnetic picture for the GS, and in addi-\ntion highlight that the bonds with the largest exchange\ncouplingJ23(i.e., the bonds 3-2 and 3-20) exhibit also\nthe strongest correlations. Moreover, using the above\ncorrelations and the values of the exchange couplings,\none \fnds that the J23-bonds contribute about 72% of\nthe total GS energy E0'\u00001:3289J23.\nWe should remark here that the above result e110=\ne220does not arise from the symmetry of the Hamilto-\nnian, but it is an exact property of the particular GS\nonly (see explanation below).\nE\u000bective g-tensors. To make contact with our ESR\n\fndings in Sec. III B we calculate the e\u000bective g-tensors\nof the GS and the \frst excited state in terms of the in-\ndividual g-tensors of the three inequivalent Cu sites. To\nthis end, one needs the equivalent spin operators within\neach multiplet in order to rewrite the Zeeman Hamilto-\nnian as\nX\niB\u0001gi\u0001Si7!B\u0001ge\u000b\u0001S: (12)\nFor the GS, the equivalent operators can be extracted\nfrom Eq. 11 above, which yields\nge\u000b, GS = +0:0708g3\u00000:1415g2+ 0:3207g1:(13)\nSo the deviation from the isotropic g= 2 value is coming\nmainly from that of g1.\nFor the \frst excited state, our exact wavefunctions give\nthe equivalent operators\nSz\n3!\u00001\n3S; Sz\n2;Sz\n20!+0:1647S;\nSz\n1;Sz\n10!+0:002S:(14)\nleading to\nge\u000b, 1st exc. =\u00001\n3g3+ 0:3294g2+ 0:004g1:(15)\nSo here the deviation from the isotropic g= 2 value is\nmainly governed by that of g2andg3. The contribution\nfromg1is extremely small because in the \frst excited\ndoublet the pair of S1andS10forms almost a singlet\n(see Table II and Sec. IV C below).\nThese expressions demonstrate why the e\u000bective g-\ntensors of the two ESR lines appearing below 70 K are\ndi\u000berent.\nC. Strong coupling description\nWe are now going to show that almost all properties\nof the model at y\u0011J12=\u0000J13=2\n7can be adiabatically\ntraced back { even on a quantitative level { to the strong\ncoupling limit y= 0.Global spectral structure. Aty= 0, the spins S1and\nS10are isolated and are, thus, free to point up or down.\nOn the other hand, the spins S3,S2, and S20form an\nAFM spin trimer with the Hamiltonian\nH0=J23=S3\u0001(S2+S20) =1\n2\u0000\nS2\n3220\u0000S2\n220\u0000S2\n3\u0001\n:(16)\nHence the spectrum of H0can be derived analytically in\nterms of the good quantum numbers S220andS3220:\nS220; S3220;parity;deg; E(0)\n11\n2even 8\u0000J23\n01\n2odd 8 0\n13\n2even 16 + J23=2(17)\nwhere we have also indicated the parity and the degen-\neracy (deg) of each unperturbed level. The latter takes\ninto account the four possible states of the space of S1\nandS10, and the Zeeman degeneracy. So we \fnd three\nwell separated unperturbed manifolds. The remaining\ncouplingsV\u0011H\u0000H 0split these manifolds but, as we\nshow below, most of these splittings are fairly weak, thus\nexplaining the overall spectral structure at y=2\n7(Ta-\nble II).\nSince each unperturbed manifold has a well-de\fned\nS3220, we can \fnd the \frst-order splitting with the use\nof the \\equivalent operators\":\nS3!\u00153S3220;S2!\u00152S3220;S20!\u001520S3220;(18)\nwhere the values of \u00151;2;20can be derived from the cou-\npling (Clebsh-Gordan) coe\u000ecients of the respective an-\ngular momenta of each manifold (see below). Using\n\u00152=\u001520, which holds in all manifolds, and replacing (18)\ninV, we obtain\nV!Je\u000bS3220\u0001S110=1\n2Je\u000b\u0000\nS2\u0000S2\n3220\u0000S2\n110\u0001\n;(19)\nwhereJe\u000b= (\u00152\u0000\u00153)y. Thus, to lowest order, the per-\nturbation gives rise to an e\u000bective exchange coupling be-\ntween S110andS3220.\nFor the lowest eight-dimensional manifold, we couple\nthe spinS3=1\n2to the spin S220= 1 to get an S3220=1\n2\nobject, which gives (see Appendix A): \u00153=\u00001\n3,\u00152=\n\u001520= +2\n3, and thusJe\u000b=y>0. The resulting correction\nto the lowest unperturbed manifold reads:\nS220; S3220; S110; S; parityp;deg;\u0001E(1)\n11\n211\n2even 2\u0000y\n11\n201\n2odd 2 0\n11\n213\n2even 4 + y=2(20)\nSimilarly, the second unperturbed manifold has S220=\n0 and thus\u00152=\u001520=0 and\u00153=1, givingJe\u000b=\u0000y, which\nis now FM. So the \frst-order splitting of this manifold has\nthe opposite structure from that of the lowest manifold,13\n/g4/g4/g2/g5 /g4/g2/g6 /g4/g2/g7 /g1/g5/g1/g4/g2/g9 /g4/g4/g2/g9 \n/g20/g10/g12 /g5/g6 /g10/g1/g12/g5/g7 /g11/g16/g14/g18/g15/g20/g3/g12 /g6/g7 \n/g1/g1\n/g4/g4/g2/g5 /g4/g2/g6 /g4/g2/g7 /g1/g4/g2/g6 /g1/g4/g2/g5 /g4/g4/g2/g5 /g4/g2/g6 /g4/g2/g7 /g4/g2/g8 /g4/g2/g9 \n/g20/g10/g12 /g5/g6 /g10/g1/g12/g5/g7 /g4/g4/g2/g5 /g4/g2/g6 /g4/g2/g7 /g1/g4/g2/g9 /g1/g4/g2/g6/g9 /g4/g4/g2/g6/g9 \n/g20/g10/g12 /g5/g6 /g10/g1/g12/g5/g7 /g2/g13/g6/g21/g1/g2/g13/g5/g21/g1/g2/g13/g19/g17/g19/g21/g1\n/g2/g13/g7/g21/g1/g14/g7/g5 \n/g14/g6/g5/g22\n/g14/g5/g6 /g14/g7/g6 /g14/g5/g5/g22(a) Energy spectrum (b) local moments (c) spin-spin correlations \nS=1/2, odd \nS=1/2, even S=3/2, even S=3/2, odd S=5/2, even \nFIG. 16. (Color online) Accuracy of the strong-coupling description. Evolution of the energy spectrum (a), the local moments\n(b), and the spin-spin correlations eij=hSi\u0001Sji(c) from the strong coupling limit y=0 up toy=2\n7. The energies are given in\nunits ofJ23.\nnamely:\nS220; S3220; S110; S; parityp;deg;\u0001E(1)\n01\n213\n2odd 4\u0000y=2\n01\n201\n2even 2 0\n01\n211\n2odd 2 + y(21)\nThe third manifold has S3220=3\n2and thus\u00153=\u00152=\n\u001520=1\n3, which in turn gives Je\u000b= 0. Thus, the third\nunperturbed manifold remains intact in lowest order, i.e.,\nall 16 states have \u0001 E(1)=0:\nS220; S3220; S110; S; parityp;deg;\u0001E(1)\n13\n211\n2even 2 0\n13\n213\n2even 2 0\n13\n215\n2even 4 0\n13\n203\n2odd 4 0(22)\nIn particular, the energy of the state with the maximum\nspin (S=5\n2) remains equal to J23=2 in all orders of\nperturbation theory, as can be easily checked, e.g., for its\nmaximum polarized M=5\n2portion, which is a trivial\neigenstate ofH.\nUp to this lowest order, the level-crossing \felds Hc1\nandHc2are given by\ng\u0016BHc1=kB'3y=2; g\u0016BHc2=kB'(3J23\u0000y)=2:(23)\nWithJ23= 217 K,y= 2J23=7, and the average g'2:2,\nthese numbers give: Hc1'54 T, andHc2'170 T, which\nare close to the exact numbers at y=2\n7given above.\nFigure 16 shows the evolution of the spectrum from\ny= 0 up to y=2\n7, and demonstrates that the above\n\frst-order spectral structure reproduces quantitatively\nthe spectrum at y=2\n7.\nWe should add here that S220,S3220, andS110do not\nremain good quantum numbers in higher orders of theperturbative expansion, which is expected, since the cor-\nresponding angular momenta are not conserved quanti-\nties under the full Hamiltonian. Still, their expectation\nvalues reported in Table II are very close to the above\nstrong-coupling values.\nAs a \fnal remark, we note that one may use the above\nbasis provided by the lowest-order theory and the good\nquantum numbers S,M, andpto \fnd the invariant\nsubspaces ofH. Most of the subspaces turn out to be\ntwo-dimensional, except for the space with S=M=1\n2\nandp= even (in which the GS belongs), which is three-\ndimensional. Here, it may be readily checked that each\nof the three states has S110=S220, thus explaining the\nequalitye110=e220discussed above. The excited states\ndo not share this property, so at higher temperatures the\ntwo correlations deviate from each other.\nGS properties. Let us now return to the nature of the\nGS. According to the above lowest order result, the GS is\na totalS=1\n2doublet with even parity and total energy\nE(1)=\u0000J23\u0000J12. To understand the nature of this state,\nwe note that it arises from the coupling of a spin s=1\n2\n(here S3220) and a spin s= 1 (here S110) to a total S=1\n2\nstate. So we may again use the \\equivalent operators\"\nS3220!\u00001\n3S;S1;S10!+2\n3S; (24)\nwhich in conjunction with the ones relating S1,S3, and\nS5toS3220(see Eq. (18) above) gives the central rela-\ntions:\nS3!+1\n9S;\nS2;S20!\u00002\n9S;S1;S10!+2\n3S:(25)\nThis equation provides a very clear explanation for\nthe peculiar local moment distribution discussed above,\nsince, when the total S=1\n2moment is saturated (by14\napplying a small \feld), this equation gives:\nhSz\n3i= +1\n18;\nhSz\n2i=hSz\n20i=\u00001\n9;hSz\n1i=hSz\n10i= +1\n3:(26)\nThe same numbers can be extracted directly from the\nexplicit form of the GS wavefunction, which for the M=\n+1\n2portion reads:\nj*i=\u00002\n3j\"i3jt1i110jt\u00001i220+1\n3j\"i3jt0i110jt0i220\n+p\n2\n3j#i3\u0000\njt1i110jt0i220\u0000jt0i110jt1i220\u0001\n; (27)\nwherejt1iij=j\"i\"ji,jt\u00001iij=j#i#ji, andjt0iij=\n(j\"i#ji+j#i\"ji)=p\n2. The leading term of Eq. (27)) is\nthe classical ferrimagnetic con\fguration. However, the\noverall form of the wavefunction demonstrates explicitly\nthe presence of strong QM corrections to the GS.\nThe strong coupling limit reproduces the ferrimag-\nnetic arrangement of the local moments and even their\nstrengths, except for S3which has almost tripled its\nmoment at y=2\n7. This is further demonstrated in\nFig. 16(b), which shows the evolution of the local mo-\nments from y=0 up toy=2\n7. All moments, except hSz\n3i,\nexhibit a small slope, i.e., higher-order corrections are\nweak.\nIn a similar fashion, the strong coupling prediction\nfor the spin-spin correlations are: e31=e310=1\n6,e32=\ne320=\u00001\n2,e12=e1020=\u00001\n3,e120=e102=\u00001\n3, and\ne110=e220=1\n4, which are in close agreement with the\nexact values for y=2\n7given earlier. This is again demon-\nstrated in Fig. 16(c) that shows the evolution of the spin-\nspin correlation strengths from y= 0 up toy=2\n7. All\ncorrelations exhibit weak corrections, except e12which\nshows a somewhat larger slope in y.\nV. DISCUSSION AND SUMMARY\nThe Cu 5-NIPA molecule comprises two non-frustrated\nspin triangles with two AFM couplings and one FM\ncoupling each (Fig. 1, left). This coupling regime is\nrather unusual and deserves a further analysis with re-\nspect to the underlying crystal structure and interacting\norbitals. Three leading couplings { J12;J13, andJ23{\nrepresent the Cu{O{Cu superexchange running via the\n\u00163-O atom in the middle of the Cu 3triangle. According\nto Goodenough-Kanamori rules, high values of the bridg-\ning angle should lead to an AFM interaction, whereas low\nbridging angles close to 90\u000efavor FM couplings. Our re-\nsults for Cu 5-NIPA (Table I) follow this general trend.\nHowever, a closer examination pinpoints two peculiari-\nties of this compound.\nFirst, the coupling remains FM for the bridging angle\nof 107:9\u000e, although the standard and commonly accepted\nthreshold value of the FM-AFM crossover is slightly be-\nlow 100\u000e.70Second,JFM\n12is much larger than JFM\n13, eventhough the respective bridging angle is also larger and\nshould lead to a smaller FM contribution. Both pecu-\nliarities should be traced back to the twisted con\fgu-\nration of the interacting CuO 4plaquettes (Fig. 1, left).\nThe systematic work on the angular dependence of the\nexchange coupling is usually restricted to systems with\ntwo plaquettes lying in the same plane or only weakly\ntwisted (the dihedral angle between the planes is close to\n180 deg). The Cu 5molecule represents an opposite limit\nof strongly twisted CuO 4plaquettes, with the dihedral\nangles of 91.8 deg ( J12), 125.2 deg ( J13), and 99.9 deg\n(J23) between the CuO 4planes.71\nThe large FM contribution to J12(JFM\n12=\u0000134 K)\ncan be ascribed to the nearly orthogonal con\fguration\nof the interacting CuO 4plaquettes. However, the mech-\nanism of this FM interaction is yet to be determined.\nThe large FM coupling for the bridging angle of 90 deg\nis generally understood as the Hund's coupling on the\noxygen site72or the direct FM exchange between Cu and\nO.73Both mechanisms depend solely on the Cu{O{Cu\nangle and should be rather insensitive to the mutual ori-\nentation of the CuO 4plaquettes. Therefore, other ef-\nfects, such as the direct Cu{Cu exchange and the Cu{\nO{Cu interaction involving fully \flled Cu 3 dorbitals,\nmay be operative. Experimental information on the mag-\nnetic exchange between the strongly twisted CuO 4pla-\nquettes remains scarce and somewhat unsystematic. The\nCu{O{Cu angle of 104.5 deg combined with the sizable\ntwisting give rise to a FM nearest-neighbor interaction\nin the kagome material kapellasite Cu 3Zn(OH) 6Cl2,74\nwhereas a similar geometry with the Cu{O{Cu angle of\n107.6 deg results in an overall AFM coupling in dioptase,\nCu6Si6O18\u00016H2O.75\nWe also note that Cu 5-NIPA brings a fresh perspec-\ntive on magnetostructural correlations in Cu 3triangular\nmolecules, where the Cu{X{Cu bridging angle was previ-\nously considered the key geometrical parameter. As long\nas the ligand atom X lies in the Cu 3plane, the bridg-\ning angles remain close to 120 deg and should generally\nlead to the AFM exchange. The FM exchange would\nonly be possible when the ligand atoms are shifted out of\nthe plane, as in the Cl- and Br-containing Cu 3molecules\nwhere the Cu{X{Cu angles are below 90 deg.76The twist-\ning of the CuX 4plaquettes provides another opportunity\nfor creating the FM exchange and, moreover, for intro-\nducing it selectively. Surprisingly, organic ligands have\nonly a weak e\u000bect on the magnetic interactions. Al-\nthough carboxyl groups (COO\u0000) provide an additional\nsuperexchange pathway between Cu1 and Cu3 (Fig. 1,\nleft), their molecular orbitals do not in\ruence the Cu\ndx2\u0000y2Wannier functions (Fig. 12). For example, the\nFM contribution to J12exceeds that to J13. This demon-\nstrates that the FM exchange is basically unrelated to the\norganic ligands. It should be understood as a joint e\u000bect\nof the the low Cu{O{Cu angle and twisting.\nThe combination of FM and AFM couplings also has\nan important e\u000bect on the magnetic GS of Cu 5-NIPA.\nA regular spin triangle entails the four-fold degenerate15\nGS that further splits into two close-lying doublets by\nvirtue of anisotropy,27residual interactions between the\ntriangles,10or a marginal distortion of the triangle.28In\nCu5-NIPA, there is only one doublet state, which is sep-\narated by about 68 K from the \frst excited state (see\nTable II). According to Kramer's theorem, in zero \feld\nthe degeneracy of this state can not be lifted, hence the\ntunnel splitting is exactly zero, and no magnetization\nsteps (Landau-Zener-St uckelberg transitions) should oc-\ncur, in contrast to, e.g., the V 6molecule.10Therefore,\nbroad butter\ry hysteresis e\u000bects but without tunneling\nare expected in the magnetization process of Cu 5-NIPA.\nOf particular interest is our experimental \fnding of the\nenhanced1H nuclear spin-lattice relaxation rate 1 =T1at\na characteristic temperature slightly below the spin gap\n(T'40 K). While such an enhancement has been re-\npeatedly found in numerous AFM homometallic38and\nheterometallic39rings of spins S >1\n2, it is very rare for\nS=1\n2and has been observed only in molecules with a\nhigh-spin GS.63,64The origin of the peak is likely the\nsame in both cases, namely, the slowing down of the\nphonon-driven magnetization \ructuations.38,40{42How-\never, the sparse excitation spectrum and the presence\nof nonequivalent Cu sites with di\u000berent local magnetiza-\ntion render Cu 5-NIPA dissimilar to typical homometallic\nrings. We argue that in Cu 5-NIPA inter-multiplet Or-\nbach processes make the dominant contribution to the\nspin-lattice relaxation process, as in the heterometallic\nring Cr 7Ni at very low temperatures.66\nThe use of molecular magnets in quantum computing\nis severely restricted by their short coherence time. Co-\nherence times on the order of 100 \u0016s could be achieved\nin, e.g., V 15by arranging magnetic molecules in a self-\nassembled layer formed by an organic surfactant.18This\nmethod is fundamentally similar to the formation of the\nmetal-organic framework in Cu 5-NIPA. Therefore, it may\nbe interesting to study the decoherence process in this\nmolecular magnet and, more generally, explore the role\nof organic bridges in the decoherence process.\nFinally, we would like to note that only a few examples\nof magnetic pentamers based on spin-1\n2ions have been re-\nported so far. The [Cu( \u0016-L)3]2Cu3(\u00163-OH)(PF 6)3\u00015H2O\n(L\u0000= 3;5-bis(2-pyridil)pyrazolate) shows a somewhat\nsimilar phenomenology with the spin S=1\n2GS and the\nrespective magnetization plateau ranging up to 30 T.77\nHowever, the geometrical structure of this compound is\na trigonal bipyramid that is notably di\u000berent from the\nhourglass shape of the magnetic cluster in Cu 5-NIPA.\nThe magnetic pentamer based on Cu 5(OH) 4(H2O)2(A-\n\u000b-SiW 9O33)2] is more similar to our case and also reveals\nthe spinS=1\n2GS.51,78Nevertheless, the equivalence of\nJ23andJ13, as well as sizable long-range couplings, dis-\ntinguish its spectrum and magnetic properties from that\nof Cu 5-NIPA.\nIn summary, we have studied thermodynamic proper-\nties, spin dynamics, microscopic magnetic model, and en-\nergy spectrum of the spin-1\n2Cu5pentamer in Cu 5-NIPA.\nThis magnetic molecule has an hourglass shape with twoAFM couplings and one FM coupling on each of the two\ntriangles that form the non-frustrated pentamer. In zero\n\feld, the ground state of Cu 5-NIPA is a doublet with the\ntotal spin of S=1\n2. This ground state is separated from\nthe \frst excited state by an energy gap of \u0001 '68 K. Our\nresults evidence a highly inhomogeneous distribution of\nmagnetization over rhe nonequivalent Cu sites accord-\ning to the quantum nature of the magnetic ground state.\nThe maximum in the spin-lattice relaxation rate 1 =T1is\nvery rare among molecular magnets with spin-1\n2ions and\na low-spin ground state.\nACKNOWLEDGMENTS\nRN was funded by MPG-DST (Max Planck\nGesellschaft, Germany and Department of Science\nand Technology, India) fellowship. AT was supported\nby the Alexander von Humboldt Foundation and the\nMobilitas program of the ESF (grant MTT77). OJ\nacknowledges partial support of the Mobilitas grant\nMJD447. IR was funded by the Deutsche Forschungsge-\nmeinschaft (DFG) under the Emmy-Noether program.\nThe high-\feld magnetization measurements were sup-\nported by EuroMagNET II under the EC contract\n228043. Finally, we would like to thank M. Belesi for\nfruitful discussions.\nAppendix A: Coupling of a spin 1 to a spin 1/2\nobject and \\equivalent operators\"\nHere we work out the coupling of two spins, one with\nspinSa= 1=2 and the other one with Sbc= 1. For the\nlatter, we shall imagine that there are two spins Sb=\nSc= 1=2 forming a triplet, namely jt1i=j\"\"i,jt\u00001i=\nj##i, andjt0i=1p\n2(j\"#i+j#\"i). The states of the system\ncan be labeled by jS;Mi, whereSis the total spin (here\nS=1\n2or3\n2), andMis the projection along some axis z.\nThe states with S=3\n2are given by:\nj3\n2;3\n2i=j\"ia\njt1ibc;\nj3\n2;1\n2i=1p\n3\u0010\nj#ia\njt1ibc+p\n2j\"ia\njt0ibc\u0011\n;\nj3\n2;\u00001\n2i=1p\n3\u0010\nj\"ia\njt\u00001ibc+p\n2j#ia\njt0ibc\u0011\n;\nj3\n2;\u00003\n2i=j#ia\njt\u00001ibc:\nOn the other hand, the two states of the S=1\n2doublet\nread:\nj1\n2;1\n2i=1p\n3\u0010\nj\"ia\njt0ibc\u0000p\n2j#ia\njt1ibc\u0011\n;\nj1\n2;\u00001\n2i=1p\n3\u0010\n\u0000j#ia\njt0ibc+p\n2j\"ia\njt\u00001ibc\u0011\n:\nUsing the above explicit relations, it is straightforward to\n\fnd out the equivalent operators within the 2 \u00022 manifold16\nof theS=1\n2doublet. We \fnd:\nSa!\u00001\n3S;Sb!+2\n3S;Sc!+2\n3S: (A1)\nAppendix B: Structural data\nTable III shows the equilibrium positions of the 17 hy-\ndrogen nuclei in the structure.\nTABLE III. Relaxed hydrogen positions in Cu 5-NIPA. Last\ncolumn lists atoms bonded to the given hydrogen atom.\nThe notation of atoms follows Ref. 22, a= 10:8303 \u0017A,\nb= 11:4692 \u0017A, andc= 11:5697 \u0017A.\nx=a y=b z=c\nH2 0.0341 0.9481 0.3828 (C2)\nH4 0.4295 0.6383 0.4380 (C4)\nH6 0.4442 0.0066 0.2381 (C6)\nH10 0.4563 0.4564 0.2062 (C10)\nH12 0.1278 0.8023 0.0923 (C12)\nH13 0.0814 0.3482 0.1695 (O13)\nH14 0.5705 0.7834 0.9955 (C14)\nH14A 0.1105 0.1813 0.9704 (O14)\nH14B 0.0012 0.2993 0.9133 (O14)\nH15A 0.6634 0.2163 0.2633 (O15)\nH15B 0.8008 0.0961 0.2947 (O15)\nH16A 0.9178 0.6522 0.4705 (O16)\nH16B 0.8144 0.5794 0.5639 (O16)\nH17A 0.3996 0.2783 0.2640 (O17)\nH17B 0.4504 0.1667 0.3606 (O17)\nH18A 0.0954 0.1403 0.6165 (O18)\nH18B 0.1939 0.1190 0.6921 (O18)\n\u0003altsirlin@gmail.com\nyi.rousochatzakis@ifw-dresden.de\n1O. Kahn, Molecular Magnetism (UCH, Berlin, 1990);\nD. Gatteschi, R. Sessoli, and J. Villain, Molecular Nano-\nmagnets (Oxford University Press, New York, 2006).\n2L. Gunther and B. 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In this work, we investigate the spin S= 1=2 Cairo\npentagonal lattice with respect to selective exchange coupling (which e\u000bectively corresponds to a\nvirtual doping of x= 0;1=6;1=3), in a nearest-neighbour antiferromagnetic Ising model. We also\ndevelop a simple method to quantify geometric frustration in terms of a frustration index \u001e(\f;T),\nwhere\f=J=~J, the ratio of the two exchange couplings required by the symmetry of the Cairo lattice.\nAtT= 0, the undoped Cairo pentagonal lattice shows antiferromagnetic ordering for \f\u0014\fcrit= 2,\nbut undergoes a \frst-order transition to a ferrimagnetic phase for \f > \f crit. The results show that\n\u001e(\f;T= 0) tracks the transition in the form of a cusp maximum at \fcrit. While both phases show\nfrustration, the obtained magnetic structures reveal that the frustration originates in di\u000berent bonds\nfor the two phases. The frustration and ferrimagnetic order get quenched by selective exchange\ncoupling, and lead to robust antiferromagnetic ordering for x= 1/6 and 1/3. From mean-\feld\ncalculations, we determine the temperature-dependent sub-lattice magnetizations for x= 0;1=6\nand 1=3. The calculated results are discussed in relation to known experimental results for trivalent\nBi2Fe4O9and mixed valent BiFe 2O4:63. The study identi\fes the role of frustration e\u000bects, the ratio\n\fand selective exchange coupling for stabilizing ferrimagnetic versus anti-ferromagnetic order in\nthe Cairo pentagonal lattice.\nPACS numbers: 81, 75.10.-b, 61.14.-x\nI. INTRODUCTION\nGeometric frustration on a lattice is the inability to\nconsistently satisfy all pair-wise spin interactions be-\ntween lattice sites as de\fned by a Hamiltonian. The most\ncommon example of geometric frustration is the clas-\nsical nearest-neighbour antiferromagnetic (NNAF) Ising\nmodel on a triangular lattice in two dimensions.1This\nseemingly simple case does not show magnetic order\ndown to the lowest temperature, but instead exhibits a\n\fnite entropy at T= 0. Interestingly, the more general\nor complex case, with the Ising-spins replaced by spin\n1/2 Heisenberg-spins has been shown to exhibit an or-\ndered ground state, which corresponds to the quantum\nanalogue of the classical Neel ground state.2This may be\ncompared with another case of frustration-induced zero-\npoint entropy and absence of ordering, namely, the quan-\ntum spin-liquid known for the S= 1=2 kagome lattice\nHeisenberg antiferromagnet.3These contradictory results\ni.e. presence or absence of ordering, typify the uncer-\ntain role of frustration in two dimensional systems with\ntriangular plaquettes. Extensive studies on a variety of\nlattice spin models in two and three dimensions have in-\ndeed revealed and established a plethora of exotic phases\nand properties due to frustration.4{6. Well-known exam-\nples include resonating valence bond superconductivity7,\n'order by disorder'8, spin-ice9, spin liquid with charge\nfractionalization10, magnetic monopoles11, and so on.\nAlthough the triangular plaquette is the smallest unit\nwith intrinsic frustration, any larger plaquette with anodd number of edges (or vertices) would also naturally\ngive rise to frustration. In particular, a pentagonal pla-\nquette also leads to frustration. While a regular pen-\ntagonal plaquette cannot form a Bravais lattice, there\nare 14 known pentagonal tesselations based on irregular\npentagons. Early studies12{14discussed frustration ef-\nfects for the two-dimensional pentagonal lattice obtained\nfrom the hexagonal lattice by cutting each hexagon into\nhalves with parallel lines(Fig. 1 of ref. 12). In the follow-\ning, we call this pentagonal lattice the P1 lattice. Using\na transfer matrix approach, Waldor et al. showed that\nthe NNAF Ising model for the P1 lattice had a \fnite\nground state entropy due to frustration.12For the NNAF\nHeisenberg model on the P1 lattice, Bhaumik and Bose\nidenti\fed the possibility of a collinear Neel type ground\nstate order.13Moessner and Sondhi investigated the P1\nlattice in terms of a NNAF Ising model with a trans-\nverse \feld, and they identi\fed a novel sawtooth state\nconsisting of zigzag antiferromagnetic stripes coexisting\nalternately with frustrated ferromagnetic stripes.14Uru-\nmov, on the other hand, investigated15a di\u000berent tesel-\nlation, the so-called two-dimensional Cairo pentagonal\nlattice (See Fig. 1). Urumov presented an exact solu-\ntion of the nearest-neighbour ferromagnetic Ising model\nfor the two dimensional Cairo pentagonal lattice by map-\nping it onto a Union Jack lattice with nearest and second\nnearest-neighbour non-crossing interactions.15\nHowever, more recently, since the discovery of an ap-\nproximate experimental realization of the Cairo pen-\ntagonal lattice in Bi 2Fe4O9, which exhibits magneto-arXiv:1412.6944v1  [cond-mat.mtrl-sci]  22 Dec 20142\nelectric coupling16and frustration induced non-collinear\nmagnetism,17there has been a signi\fcant resurgence of\ninterest in the Cairo pentagonal lattice.18{24Ralko dis-\ncussed the phase diagram of the XXZ spinS= 1=2\nsystem under an applied magnetic \feld and showed that\nfrustration leads to unconventional phases such as a ferri-\nmagnetic super\ruid.18Rojas et al. employed a direct dec-\noration transformation approach in order to investigate\nantiferromagnetic as well as ferromagnetic coupling for\nan exact solution of the Cairo pentaonal lattice.19They\ncould thereby show that the phase diagram includes a so\ncalled disordered/frustrated state, in addition to ferro-\nmagnetic and ferrimagnetic phases. In an extensive study\nof the NNAF Heisenberg model on the Cairo pentagonal\nlattice, Rousochatzakis et al. revealed the role of an or-\nder by disorder mechanism and a possible spin-nematic\nphase with d-wave symmetry.20The authors addressed\nthe evolution of the phase diagram from the classical to\nthe quantum limit as a function of \f=J=~J, whereJ\nand ~J(J43andJ33, respectively, in ref. 20 ) are the two\ntypes of exchange couplings which originate in the two\ntypes of lattice sites. Jis the exchange coupling between\na 4-co-ordinated and a 3-co-ordinated site, while ~Jis the\nexchange coupling between two 3-co-ordinated sites, re-\nspectively, of the Cairo pentagonal lattice(see Fig. 1(a)\nand its caption).\nOn the experimental front, Retuerto et al. addressed21\nthe role of oxygen non-soichiometry in BiFe 2O5\u0000\u000eand\ncon\frmed that the mixed valence of iron, with a nominal\nvalency of Fe3:2+in BiFe 2O4:63, leads to an antiferro-\nmagnetic ground state with TN= 250K. In the related\nsystem Bi 4Fe5O13F, Abakunov et al. showed22that the\npresence of frustrated exchange couplings lead to a se-\nquence of magnetic transitions at T1= 62 K,T2= 71 K\nandTN= 178 K. Using neutron di\u000braction and thermo-\ndynamic measurements, the authors could show the for-\nmation of a non-collinear antiferromagnet below T1= 62\nK, while the structure between T1andTNwas partially\ndisordered. Pchelkina and Streltsov carried out ab-initio\nband structure calculations23for Bi 2Fe4O9and showed\nthat a complete description required going beyond the\ntwo-dimensional Cairo pentagonal lattice. However, con-\nsidering only the two largest in-plane exchange coupling\nparameters, the obtained ground state was found to be\nconsistent with experiment. And very recently, Nakano\net al. addressed24the magnetization process in the two\ndimensional spin S= 1=2 Heisenberg model on the Cairo\npentagonal lattice. Using a numerical diagonalization\nmethod, they discussed the role of the ratio of the two\ntypes of exchange couplings in driving a quantum phase\ntransition, and very interestingly, found a 1/3 magneti-\nzation plateau usually associated with triangular plaque-\nttes.\nThe results described above clearly show that frustra-\ntion e\u000bects play an important role in determining the\nmagnetic ordering in the Cairo pentagonal lattice. How-\never, the quanti\fcation of frustration and the role of spe-\nci\fc or selective exchange couplings associated with the\nFIG. 1. (Color online) (a) The unit cell of the two-dimensional\nCairo pentagonal lattice(thick blue lines) used in this paper.\nThe undoped ( x= 0) case corresponds to ~J1=~J2=~J, and\nJ1=J0\n1=J2=J0\n2=J, and the magnetic structure depends\non the ratio \f=J/~J. The two magnetic structures obtained\nforx= 0, with \fnite but di\u000berent values of frustration index\n\u001e(T= 0), are shown : (b) for \f= 1, it is antiferromagnetic,\nand (c) for \f= 3, it is ferrimagnetic. For x= 1/6, the ob-\ntained 'star' antiferromagnetic structure is the same as shown\nin (b), while (d) shows the 'boat' antiferromagnetic structure\nobtained for x= 1/3.\ntwo types of sites in the lattice has not been addressed\nyet. Further, while experimental results for the hole-\ndoped system BiFe 2O5\u0000\u000ehave been reported, it is im-\nportant to theoretically investigate the role of doping for\nmagnetic ordering in the Cairo pentagonal lattice. In\nthe present study, we address these issues for the spin\nS= 1=2 Cairo pentagonal lattice in a NNAF Ising model.\nWe \frst develop a method to quantify geometric frustra-\ntion in terms of a frustration index \u001e(\f;T). We \fnd\nthat, for the undoped case (x = 0), \u001e(\f;T= 0) exhibits\na cusp maximum as a function of \f=J=~J. Further,\nin the presence of \fnite frustration, we identify antiferro-\nmagnetic ordering at low \f, which undergoes a \frst-order\ntransition to a ferrimagnetic phase for \f >\fcrit= 2. The\nfrustration and ferrimagnetic order get suppressed by se-\nlective exchange coupling, and lead to antiferromagnetic\nordering for x = 1/6 and 1/3. Mean-\feld calculations\nare carried out to determine the temperature-dependent\nmagnetization for x= 0, 1/6 and 1/3. The results are\ndiscussed in relation to known experimental results for\ntrivalent Bi 2Fe4O9and mixed valent BiFe 2O4:63. The\nstudy identi\fes the role of frustration e\u000bects, the ratio\n\fand selective exchange coupling in relation to ferri-\nmagnetic versus anti-ferromagnetic ordering in the Cairo\npentagonal lattice.3\nII. CALCULATIONAL DETAILS\nThe unit cell is as shown in Fig. 1 (thick blue lines)\nand consists of six Ising spins: four 3-co-ordinated spins\ns1;s2;s3;s4, a 4-co-ordinated spin \u001b0at the center, and\nanother 4-co-ordinated spin \u001b1at the bottom-left corner.\nEach of these can take values of S=\u00061=2. For the most\ngeneral case, the couplings (that are symmetric in the\nindices) are denoted as follows:\nJs1;s3=~J1; Js2;s4=~J2;\nJs1;\u001b0=Js3;\u001b0=J0\n1;\nJs2;\u001b0=Js4;\u001b0=J0\n2;\nJs1;\u001b=Js3;\u001b=J1;\nJs2;\u001b=Js4;\u001b=J2: (1)\nWhile the above detailed notations for the couplings\nare necessary for describing selective exchange couplings,\nwhich e\u000bectively represent virtual doping content ( x), the\nundoped case is obtained by setting ~J1=~J2=~J, and\nJ1=J0\n1=J2=J0\n2=J. As detailed in the following,\nappropriate limits of these parameters allow us to calcu-\nlate physical quantities for x= 1/6 and 1/3. The Ising\nHamiltonian is\nH=X\niHi; (2)\nwhere\nHi=X\n\u000bK(i)\n\u000bs(i)\n\u000b (3)\ncorresponds to the ithunit cell of the two-dimensional\nCairo lattice. Here the symbols K(i)\n\u000bare de\fned by\nK(i)\n1=J1\u001b(i)+J0\n1\u001b0(i)+~J1s(i\u0000y)\n3;\nK(i)\n2=J2\u001b(i+x)+J0\n2\u001b0(i)+~J2s(i+x)\n4;\nK(i)\n3=J1\u001b(i+x+y)+J0\n1\u001b0(i)+~J1s(i+y)\n1;\nK(i)\n4=J2\u001b(i+y)+J0\n2\u001b0(i)+~J2s(i\u0000x)\n2; (4)\nin which the superscripts i\u0006x; i\u0006y; i +x+yare\nindices of unit cells neighboring i. We perform a mean-\n\feld decoupling of the Hamiltonian Eq.(2) according to\nSiSj'hSiiSj+hSjiSi\u0000hSiihSji (5)\nfor a product of any two spins SiandSj, wherehSi=\nTr(Se\u0000HMF=kBT)=Tr(e\u0000HMF=kBT) for anyS. This ap-\nproximation linearizes the unit cell Hamiltonian Eq.(3)\nin the spin variables:\nHMF=c0+4X\n\u000b=1c\u000bs\u000b+c5\u001b0+c6\u001b: (6)\nFIG. 2. (Color online) Plots of (a) frustration index \u001e(\f;0)\n(Eq. (14)), (b) zero-temperature free energy f(\f;0) (Eq.\n(10)), and (c) Emin(\f) (Eq. (13)) as a function of \f=J=~Jfor\nthe undoped case, x= 0 ;\u001e(\f;0) exhibits a non-monotonic\nbehaviour and a cusp at \f= 2.\nWe have suppressed the unit cell index i. The coe\u000ecients\nare\nc0=\u00002~J1m1m3cos(ky)\u0000J1m3m6cos(kx+ky)\n\u00002~J2m2m4cos(kx)\u0000J1m1m6\n\u0000J2m2m6cos(kx)\u0000J2m4m6cos(ky)\n\u0000(J0\n1m1+J0\n1m3+J0\n2m2+J0\n2m4)m5;\nc1= 2~J1m3cos(ky) +J1m6+J0\n1m5;\nc2= (2 ~J2m4+J2m6) cos(kx) +J0\n2m5;\nc3= 2~J1m1cos(ky) +J1m6cos(kx+ky) +J0\n1m5;\nc4= 2~J2m2cos(kx) +J2m6cos(ky) +J0\n2m5;\nc5=J0\n1m1+J0\n1m3+J0\n2m2+J0\n2m4;\nc6=J1m1+J2m2cos(kx) +J1m3cos(kx+ky)\n+J2m4cos(ky): (7)\nFor the sub-lattice magnetizations, we have introduced\nthe notation m\u000b=hs\u000bi(for\u000b= 1;2;3;4),m5=\nh\u001b0i; m 6=h\u001bi. Since the single unit-cell Hamiltonian\n(Eq.(3)) involves couplings with spins in the neighboring\nunit cellsi\u0006x\u0006y, we have introduced a spin density wave\n(SDW) vector kwhose components kx= 2\u0019=\u0015 1; ky=\n2\u0019=\u0015 2appear in the expressions for the coe\u000ecients. The\nsix magnetizations m\u000b(\u000b= 1;\u0001\u0001\u0001;6), and the two wave\nlengths\u00151;\u00152together form an eight-component mean-\n\feld order parameter vector \u0016:\n\u0016= (m1; m 2; m 3; m 4; m 5; m 6; \u00151; \u00152):(8)\nThe thermodynamics of the model is determined by the\nunit cell free energy f(T):\ne\u0000f(T)=kBT=Tr(e\u0000HMF=kBT); (9)4\nFIG. 3. (Color online) Components of the order parameter\n\u0016at zero temperature plotted as a function of \fatx= 0.\n(a) Plots of the sub-lattice magnetizations m1tom6. The\njumps seen for m2,m4andm5indicate a change in the mag-\nnetic order from an antiferromagnetic phase for \f\u00142 to a\nferrimagnetic phase for \f > 2. (b) The SDW lengths \u00151;\u00152\nplotted as a function of \findicate a doubling of the magnetic\nlattice along x and y axes for the antiferromagnetic phase\nfor\f\u00142(panel c) compared to the ferrimagnetic phase for\n\f >2(panel d).\nwhere the trace is performed over the six spins of the unit\ncell, each of which takes the values \u00061=2. Therefore\nf(T) =c0\u00006kBTln(2)\u0000kBT6X\n\u000b=1ln\u0014\ncosh\u0012c\u000b\n2kBT\u0013\u0015\n:\n(10)\nThe magnetizations are determined using the self-\nconsistency equations\n2m\u000b+ tanh\u0012c\u000b\n2kBT\u0013\n= 0; \u000b = 1;\u0001\u0001\u0001;6; (11)\nFIG. 4. Sub-lattice magnetizations mi;i= 1\u00006, plotted as a\nfunction of temperature for the undoped case ( x= 0), for (a)\n\f= 1 (antiferromagnetic phase) and (b) \f= 3 (ferrimagnetic\nphase).\nwhile the SDW lengths are determined by free energy\nminimization @f=@\u0015k= 0; k= 1;2:\n@c0\n@\u0015k+6X\n\u000b=1m\u000b@c\u000b\n@\u0015k= 0; k = 1;2: (12)\nWe solve the eight equations in (11) and (12) numerically\nto obtain the order parameter vector \u0016(fJijg;T).\nWe de\fne a frustration index \u001e(\f;T) in the follow-\ning manner. Consider a reference state for which each\nbond between neighboring spins were to be fully satis-\n\fed. Then, Eq.(3) would give a zero temperature energy\nof\nEmin=\u00001\n2(J1+J0\n1+J2+J0\n2+~J1+~J2): (13)\nThe zero temperature free energy f(\f;T = 0) deviates\nfromEmin(\f) because of frustration, so if we de\fne\n\u001e(\f;T) = 1\u0000f(\f;T)\nEmin(\f); (14)5\nFIG. 5. Sub-lattice magnetizations mi;i= 1\u00006, plotted as\na function of temperature for the antiferromagnetic phases of\nx= 1=6) for (a)\f= 1 and (b) \f= 3.\nthen\u001e(\f;T = 0) is a convenient measure of frustration.\nIt provides a quantitative measure of frustration as a\nfunction of \fand selective exchange coupling.\nIII. RESULTS AND DISCUSSIONS\nWe \frst compute the order parameter vector \u0016by nu-\nmerically solving the eight equations in (11) and (12)\nfor the undoped case with ~J1=~J2=~J, andJ1=\nJ0\n1=J2=J0\n2=J. In Fig. 2, we plot the frustration\nmeasure\u001e(\f;T = 0) (Eq. (14)), the zero-temperature\nfree energy f(\f;T = 0) (Eq. (10)), and Emin(\f) (Eq.\n(13)) as a function of \f=J=~J. The frustration measure\n\u001e(\f;T = 0) exhibits a clear cusp maximum at \f= 2.\nThe cusp originates in the zero-temperature free energy\nf(\f;T= 0) which exhibits a derivative discontinuity at \f\n= 2. This indicates a \frst-order transition as a function\nof\f. As shown in Fig. 3, the order parameter plots as\na function of \findicate an antiferromagnetic phase for\n\f\u00142, which transforms into a ferrimagnetic phase for\n\f >2.\nFIG. 6. Sub-lattice magnetizations mi;i= 1\u00006, plotted as\na function of temperature for the antiferromagnetic phases of\nx= 1=3) for (a)\f= 1 and (b) \f= 3.\nWe see discontinuities in the sub-lattice magnetizations\nm2,m4andm5at\f= 2: while m2andm4\rip from\n1=2 to\u00001=2,m5responds by changing from 0 to 1 =2 to\nminimize the free energy (Fig. 3(a)). The SDW lengths\n\u00151;\u00152also have discontinuities (Fig. 3(b)): \u00151;\u00152= 2\nfor\f\u00142, and\u00151;\u00152= 1 for\f >2. Thus, the magnetic\nunit cell of the antiferromagnetic phase is doubled along\nthe x and y axes, compared to the ferrimagnetic unit cell\nas shown in Fig. 3(c) and (d), respectively. Interestingly,\nwhile the system remains frustrated in both these phases,\nthe frustration index \u001e(T= 0)!0 as\f!0, but goes to\na \fnite value at large \fasymptotically. Thus, in the limit\n\f= 0, we have isolated dimers on the 3-co-ordinated sites\nwhile for\f=1, we retain the ferrimagnetism.\nThe ferrimagnetic phase obtained for \f >2 with a to-\ntal magnetization M(T= 0) = 1=3, is identical to earlier\nstudies from (i) exact calculations for the Ising15,19, (ii)\nhard-core bosons18and (iii) the Heisenberg20models. It\nis noted that, for the antiferromagnetic phase, the sub-\nlattice magnetization m5=h\u001b0i= 0 for\f\u00142, (see Fig.\n3(a)) due to the fact that out of its 4 nearest-neighbour\nsites, two sites have up-spins and two have down-spins.6\nThis leads to an e\u000bective cancellation of the magnetiza-\ntion contribution from the \u001b0site. In earlier work, Rojas\net al.19reported a frustrated phase of Ising spins from ex-\nact calculations, corresponding to the antiferromagnetic\nphase obtained from our mean-\feld calculations, while\nRousochatzakis et al.20obtained a so called orthogonal\nphase for the Heisenberg case.\nWe now turn to investigating the role of selective ex-\nchange coupling at T= 0. We carry out this exercise\nmainly to identify the role of speci\fc couplings of the 3-\nand 4-co-ordinated sites in the Cairo lattice, and to e\u000bec-\ntively simulate virtual dopings of x = 1/6 and 1/3. From\nthe way we have de\fned our couplings (see Eq. (1)), it\ncan be observed that if we set J0\n1=J0\n2= 0, the central\nspin\u001b0gets disconnected from the lattice. Consequently,\nthe central spin \u001b0does not participate in the magnetic\nordering and the system e\u000bectively represents a virtual\nhole doping content of x= 1=6. In the same way, it can\nbe observed that when J2=~J2=J0\n2= 0, the spins s2\nands4get disconnected, corresponding to a virtual hole\ndoping ofx= 1=3. Our results indicate that for both\nthese cases, the system is antiferromagnetic for all \f, i.e.\n\u00151;\u00152= 2. Forx= 1=6, the sub-lattice magnetizations\nare exactly the same as those for x= 0;\f\u00142 (Fig. 3),\nand we therefore refrain from showing these plots in a\n\fgure. For x= 1=3, the actual magnetic structure gets\nmodi\fed as discussed below. Another important di\u000ber-\nence from the x= 0 case is that \u001e(\f;0) = 0 for both\nvalues ofx= 1=6 and 1=3 and is independent of \f, indi-\ncating the absence of frustration.\nThe absence of a \f-driven transition for the cases\nx= 1=6;1=3 is easy to understand. In the x= 1=6\ncase, the central spin \u001b0does not participate in the mag-\nnetic ordering. The remaining spins can be looked upon\nas forming a \\star\" con\fguration with 12 bonds on its\nperiphery, all of which can be satis\fed in the antifero-\nmagnetic phase, thus removing frustration fully. An in-\ncrease of\fonly strengthens the antiferromagnetic order,\nincreasingTN(see Fig. 7 and associated description). In\nthex= 1=3 case, the spins \u001b2;\u001b4do not participate in the\nmagnetic ordering. The remaining spins can be looked\nupon as forming a \\boat\" con\fguration with 10 bonds\non its periphery, all of which can be satis\fed in the antif-\neromagnetic phase, again fully removing frustration(Fig.\n1(d)). An increase of \fin this case also, merely strength-\nens the antiferromagnetic order by increasing TN. While\nboth thex= 1=6 andx= 1=3 cases exhibit antifer-\nromagnetic order, the actual spin ordering is found to\nbe di\u000berent: the former corresponds to a repetition of\n\\star\"(Fig. 1(b)) unit cells, and the latter, a repetition\nof \\boat\"(Fig. 1(d)) unit cells.\nIn Figs. 4(a) and 4(b), we plot the sub-lattice magne-\ntizationsmi;i= 1;\u0001\u0001\u0001;6 for the undoped case ( x= 0),\nas a function of temperature in units of ~Jfor\f= 1\nand\f= 3, i.e. in the antiferromagnetic and ferrimag-\nnetic phases, respectively. We \fnd that the transition\ntemperature TCandTN(the ferrimagnetic and antifer-\nromagnetic transition temperatures, respectively) for de-\nFIG. 7. (a) Plots of the transition temperature Tc;TNas a\nfunction of \ffor x = 0, 1/6 and 1/3. A jump in the ordering\ntemperature is seen only for x = 0 at \f= 2. In (b) we present\na magni\fed view of the region near Tc.\nstruction of magnetic order primarily depends on \f. This\nis further borne out by the temperature-dependence plots\nforx= 1=6 presented in Figs. 5(a) and 5(b) for \f= 1 and\n\f= 3, respectively. The temperature-dependence plots\nforx= 1=3, presented in Figs. 6(a) and 6(b) for \f= 1\nand\f= 3, respectively, also re\rect this. In particular,\nTNincreases on increasing \fforx= 1=6 and 1=3. In\nfact, the ordering temperatures increase smoothly with\n\fas can be seen in Fig. 7(a), where we plot TCand\nTNas a function of \ffor the three cases x= 0;1=6;1=3.\nThe discontinuity at the \frst-order transition for x= 0\nat\f= 2 is quite clear and remarkable. In Fig. 7(b), we\npresent a magni\fed view of the plot around \f= 2.\nWe can also see in Fig. 7(a) that the dependence of\nTNonxis negligible for all \f\u00142. But for \f > 2,\nwhile thex= 1=6;1=3 plots follow the same course as\n\f\u00142, thex= 0 curve follows a completely di\u000berent\ncourse because of the discontinuity. The larger TNof7\nthe antiferromagnetic phase compared to the TCof the\nferrimagnetic phase, just above \f= 2, suggests a higher\nstability of the antiferromagnetic phase. However, for\n\fvalues greater than \u00182:5, the ferrimagnetic TCfor\nx= 0 becomes larger than the antiferromagnetic TNfor\nx= 1=6 and 1=3. The results also show that TNgoes to\na \fnite value as \f!0, but increases linearly for high \f.\nWe now describe the physical picture of the vari-\nous phases obtained by varying the model parameters.\nFirstly, the crucial role that frustration plays in the \f-\ndriven \frst-order transition is seen from the results of the\nx= 0 case. In the x= 0 antiferromagnet region obtained\nfor\f\u00142, the frustration is con\fned to the core of the\nunit cell, i.e. two out of the four J0-bonds (either the\nJ0\n1or theJ0\n2bonds) connecting to the central spin \u001b0are\nfrustrated; in this situation, the frustration measure \u001e(0)\nis an increasing function of \f(see Fig. 2, \f\u0014\fcrit= 2 ).\nWith further increase in \fbeyond\fcrit= 2, it becomes\nenergetically favorable to \\eject\" the frustration from the\ncore to the ~Jbonds lying on the periphery of the unit\ncell i.e. the the four ~J-bonds connecting 3-co-ordinated\nsites become frustrated. Thus, the \f-driven transition is\nattended by a change in the location of the frustrated\nbonds in the unit cell. Secondly, it is surprising to \fnd\nthat the obtained TNvalues do not depend on xbut only\non\f. However, experimental results have also indicated\nthatTNdoes not depend signi\fcantly on doping content.\nFor example, studies on Bi 2Fe4O9(\u0011BiFe 2O4:5; nominal\nvalency of Fe3:0+), the material recently recognized as\na realization of the Cairo pentagonal lattice, reported a\nTN= 238 K for single crystals17,25, while for polycrys-\ntals,TNwas reported to be \u0018260 K.16,26,27For mixed\nvalent polycrystalline BiFe 2O4:63with a nominal valency\nof Fe3:2+, which corresponds to a hole doping of \u001820%,\nRetuerto et al. reported a value of TN= 250 K.21Fur-\nther, forx= 1=6 and 1=3, since the magnetic structures\nhave no frustration, the zero temperature free energy\nf(T= 0) =Emin. However, the two magnetic structures\nhave di\u000berent values of Emin. From Eq.(3), we obtain\nEmin=\u0000(J+~J) forx= 1=6, andEmin=\u0000(J+~J=2 for\nx = 1/3. Thus, even with di\u000berent values of the ground\nstate energies for x= 1=6 and 1=3, our results suggest\nthatTNdepends only on \f=J=~J.\nFinally, the present results also show that the ab-\nsolute values of the sub-lattice magnetizations at high\ntemperatures(\u00180:25\u00000:5TN/TCtoTN/TC)) depend on\nthe co-ordination of the sites and the value of \f. For ex-\nample, as can be seen in Fig. 4(a) for x= 0 and\f= 1,\nthe absolute values of sub-lattice magnetizations of the\n3-co-ordinated sites mi;i= 1;\u0001\u0001\u0001;4 are exactly the same,\nbut for\u00180:5TNtoTN, they are slightly lower than m6,\nwhich is a 4-co-ordinated site. Similarly, for the ferrimag-\nnetic phase with \f= 3, Fig. 4(b) shows that absolutevalues ofmi;i= 1;\u0001\u0001\u0001;4(3-co-ordinated sites) are exactly\nthe same, but for \u00180:25TCtoTC, they are lower than\nm5andm6which are 4-co-ordinated sites. This was also\npointed out by Urumov for the ferrimagnetic phase us-\ning an exact calculation, although the changes were very\nsmall.15Forx= 1=6, the changes are the same as for\nx= 0 and\f= 1, but the di\u000berence between 3 and 4-co-\nordinated sites get enhanced on increasing \f= 3. Very\ninterestingly, we see the opposite behaviour for x= 1=3\ncompared to x= 1=6. For\f= 1(Fig. 6(a)), due to selec-\ntive exchange coupling, the structurally 4-co-ordinated\nsites become magnetically 2-co-ordinated sites(see Fig.\n1(d)), and consequently, the absolute values of m5and\nm6get suppressed compared to m1andm3(structurally\nand magnetically 3-cordinated sites) for temperatures be-\ntween\u00180:25TNtoTN. In contrast, for \f= 3, the dif-\nference between absolute values of m5,m6compared to\nm1,m3become smaller as the magnetization pro\fles get\ndominated by the larger value of Jcompared to ~J.\nIV. CONCLUSIONS\nIn conclusion, we have investigated the spin S= 1=2\nCairo pentagonal lattice with respect to selective ex-\nchange coupling, in a nearest-neighbour antiferromag-\nnetic Ising model. We have developed a simple method\nto quantify geometric frustration in terms of a frustration\nindex\u001e(\f;T), where\f=J=~J, the ratio of the two ex-\nchange couplings required by the symmetry of the Cairo\nlattice. At T= 0, the undoped Cairo pentagonal lattice\nshows a \frst order phase transition with antiferromag-\nnetic order for \f\u0014\fcrit= 2, which transforms to a\nferrimagnet for \f > \fcrit.\u001e(\f;T = 0) exhibits a cusp\nmaximum at \fcrit. The obtained magnetic structures\nreveal that the frustration originates in di\u000berent bonds\nfor the two phases. The frustration and ferrimagnetic\norder get suppressed by selective exchange coupling, and\nthe system shows antiferromagnetic ordering for a virtual\ndoping ofx= 1/6 and 1/3. 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Dahlem Center for Complex Quantum Systems and Fachbereich Physik . \n*,+: Corresponding author s: victor@usal.es , edumartinez@usal.es   \n \nAbstract  \nBoth helicity -independent and helicity -dependent all -optical  switching  processes  driven \nby single ultrashort laser pulse ha ve been experimentally  demonstrated in ferrimagnetic \nalloys  as GdFeCo. Although the switching has been previously reproduced by atomistic \nsimulations, the lack of a robust micromagnetic  framework  for ferrimagnets  limits the \npredictions to small nano -systems, whereas the experiments are usually performed  with \nlasers and samples of tens  of micrometers. Here we develop a micromagnetic model based \non the extended Landau- Lifshitz -Bloch equation, which is  firstly validated by directly \nreproducing atomistic results for small samples  and uniform laser heating. After that, the \nmodel is used to study ultrafast single  shot all -optical switching in ferrimagnetic alloys  \nunder realistic conditions. We find that the helicity -independent switching under a \nlinearly polarized laser pulse is a pure thermal phenomenon, in which the size  of inverted \narea directly correlates with the maximum electron  temperature  in the sample . On the \nother hand, the  analysis of the helicity -dependent processes  under circular polarized \npulses in ferrimagnetic alloys with different composition indicates qualitative differences \nbetween the results predicted by  the magnetic circular dichroism and the ones from \ninverse Faraday effect . Based on these predictions, we propose experiments that would \nallow to resolve the controversy over the physical phenomenon that underlies these \nhelicity -dependent all optical processes . \n  2 \n I. INTRODUCTION \nAll optical switching (AOS) refers to  the manipulation of the magnetic state of a \nsample through the application of short laser pulses. The discovery of subpicosecond \ndemagnetization of a nickel sample   [1] upon application of  a short laser pulse, ranging \nfrom tens of femtosecond to several picoseconds , opened the path for other experiments \nto manipulate the magnetization using ultrashort laser pulses in ferromagnetic  [2,3] , \nsynthetic antiferromagnetic   [4–6] and ferrimagnetic materials  [7 –9]. While the AOS in \nferromagnetic materials is usually described by the Magnetic Circular Dichrois m \n(MCD)  [10,11]  or the Inverse Faraday Effect (IFE)  [12–14] , and it requires multiple \nshots of circularly polarized laser pulses  [15] , the inversion of the magnetization of \nferrimagnetic materials can be achieved by single- shot pulse  [16] , even with linear \npolarization. In these Helicity -Independent AOS  (HI-AOS)  processes, the reversal takes \nplace as the two antiferromagnetically coupled sublattices demagnetize at different rates \nwhen submitted to a laser pulse of adequate duration and energy. Since exchange \nprocesses conserve total angular momentum, the system transits through a ferromagnetic -\nlike state despite being ferrimagnetic at the ground state   [17] . The switching of the \nmagnetization is completed when the sublattices relax back to their thermodynamic \nequilibrium  [18] . On the other hand, several experimental studies  [7,9]  have also \nobserved that the magnetic state of ferrimagnetic alloys can be also reversed under \ncircular polarized laser pulses within a narrow range of laser  energies, resulting in a \nHelicity -Dependent AOS (HD -AOS ) which could be useful to develop ultrafast magnetic \nrecording devices purely controlled by optical means. While the single -shot HI -AOS can \nbe caused by the strong non- equilibrium due to the heating induced by the laser pulse, the \nphysical mechanisms behind the HD -AOS are still not completely understood, and \nseveral works participated by  the same authors ascribe it either to the IFE  [9]  or the \nMCD  [8]. Although several attempts have been performed to explain such AOS \nprocesses, a realistic numerical description of experimental observations is still missing . \nIndeed, some  theoretical studies usually adopt an atomistic description which is limited \nto small samples   [16,19] , with dimensions at the nanoscale, well below the size of the \nexperimentally studie d samples, with lateral sizes of several hundreds of microns . Such \natomistic approach cannot describe the non -uniform heating caused by laser beams of \nseveral microns, so it does not predict some multidomain patterns typically observed in \nthe experiments   [9]. On the other hand, other numerical attempts  have been carried out 3 \n by describing the ferrimagnetic alloy as an effective ferromagnetic sample, without \nconsidering  the individual nature of the two sublattices forming the ferrimagnet  [9] . \nAlthough these micromagnetic studies predict some features of the AOS processes, so  \nfar, the structure of the ferrimagnetic alloys has not been taken into account to investigate \nthe reversal of magnetic samples of micrometer size under realistic excitation conditions . \nAs the switching happens due to angular momentum transfer between subl attices , \nsomething impossible to account for within an effective ferromagnetic description, it is needed  to develop studies considering the two sublattice nature of the such alloys to \nnaturally evaluate their role on the reversal processes.  \nHere we present  a micromagnetic framework that is able to reproduce accurately \nthe atomistic results  of the laser -induced switching  by the extension of the conventional \nLandau -Lifshitz -Bloch (LLB) model  for ferrimagnets   [20,21] . Note that the  conventional \nLLB model does not allow to accurately describe AOS as indicated in  [21] , and here we \nextend it to solve this limitation . However, and differently from  atomistic simulations, \nwhich are limited to small samples at the nanoscale submitted to uniform laser heating, \nour micromagnetic formalism allows us to realistically describe AOS experimental \nobservations  by directly evaluating  extended samples at the microscale and non- uniform \nenergy absorption from the laser  pulse . The procedures here developed are essential to \nunderstand the physical aspects underlying these experiments , and will be useful for the \nfuture development of novel ultra -fast devices based on  these AOS processes.  After \npresenting and validating both the atomistic and the extended micromagnetic models for \nsample size of ten s of nanometers, the upper size limit of the atomistic spin models , we \ndescribe the results for HI -AOS processes in realisti c samples at the microscale for a \ntypically ferrimagnetic alloy  (GdFeCo) . Later on, we focus our attention to the description \nof the HD -AOS processes by exploring the role of the IFE and MCD separately for two \ndifferent ferrimagnetic alloys where the relative composition is slightly varied. Our \nresults allow us to suggest future  experiments which could be useful to infer the \ndominance of the IFE or the MCD  in single -shot HD-AOS  in ferrimagnetic alloys . \n \nII. ATOMISTIC AND MICROMAGNETIC MODELS  \nTypical ferrimagnetic (FiM) samples formed by a Transition Metal (TM:Co, CoFe) \nand a Rare Earth (RE:Gd) are considered here. Square samples in the 𝑥𝑥𝑥𝑥  plane with side 4 \n length ℓ and with thickness  𝑡𝑡𝐹𝐹𝐹𝐹𝐹𝐹 =5.6 nm are studied. At atomistic level the FiM sample \nis formed by a set of coupled spins, and the magnetization dynamics is described the \nLangevin -Landau -Lifshitz -Gilbert eq uation \n𝜕𝜕𝑆𝑆⃗𝐹𝐹\n𝜕𝜕𝑡𝑡=−𝛾𝛾0\n(1+𝜆𝜆2)�𝑆𝑆⃗𝐹𝐹×�𝐻𝐻��⃗𝐹𝐹+𝐻𝐻��⃗𝑡𝑡ℎ,𝐹𝐹 �+𝜆𝜆𝑆𝑆⃗𝐹𝐹×�𝑆𝑆⃗𝐹𝐹×�𝐻𝐻��⃗𝐹𝐹+𝐻𝐻��⃗𝑡𝑡ℎ,𝐹𝐹 ��� (1) \nwhere  𝑆𝑆⃗𝐹𝐹 is the localized magnetic moment and 𝐻𝐻��⃗𝐹𝐹 is the local effective field including \nintra- and inter -lattice exchange and anisotropy contributions . 𝐻𝐻��⃗𝑡𝑡ℎ,𝐹𝐹 is the local stochastic \nthermal field .  𝛾𝛾0 and 𝜆𝜆 are the gyromagnetic ratio and the damping parameter \nrespectively   [22] . Except the contrary is indicated, t ypical parameters  of Gdx(CoFe )1-x \nwith relative composition 𝑥𝑥 =0.25 were considered  [22] . See Supplemental Material \nNote  SN1  [23]  for further details , including material and numerical parameters . \nStarting from an initial uniform state of the FiM  with the spins  of the two sublattices  \nantiparallelly aligned each other along the easy axis 𝑧𝑧, a laser pulse is applied , and the \nirradiated sample absorbs energy from the laser pulse. The laser spot is assumed to have \na spat ial Gaussian profile ( 𝜂𝜂(𝑟𝑟)), with 𝑟𝑟0 being the radius  spot (𝑑𝑑0=2𝑟𝑟0 is the full width \nat half maximun, FWHM). Its temporal profile ( 𝜉𝜉(𝑡𝑡)) is also Gaussian, with 𝜏𝜏𝐿𝐿 \nrepresenting the pulse duration (FWHM). The  absorbed power density can be expressed \nas 𝑃𝑃(𝑟𝑟,𝑡𝑡)=𝑄𝑄𝜂𝜂(𝑟𝑟)𝜉𝜉(𝑡𝑡) where 𝜂𝜂(𝑟𝑟)=exp[−4ln(2)𝑟𝑟2/ (2𝑟𝑟0)2 ]  is the spatial profile \nwith 𝑟𝑟=�𝑥𝑥2+𝑥𝑥2 being the distance from the center of the laser spot, and 𝜉𝜉 (𝑡𝑡)=\nexp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/ 𝜏𝜏𝐿𝐿2] the temporal profile. 𝑄𝑄 is the maximum value of the \nabsorbed power density reached at 𝑡𝑡=𝑡𝑡0 just below  the center of the laser spot. \nLaser pulse heats the FiM  sample, and consequently, it is transiently dragged into a \nnon-equilibrium thermodynamic state, where its magnetization changes according to the \ntemperature dynamics. The temperature evolution is described by the Two Temperatures Model (TTM)  [9,24]  in terms of two subsystems: the electron ( 𝑇𝑇\n𝑒𝑒=𝑇𝑇𝑒𝑒(𝑟𝑟⃗,𝑡𝑡)) and the \nlattice (𝑇𝑇𝑙𝑙=𝑇𝑇𝑙𝑙(𝑟𝑟⃗,𝑡𝑡)), \n𝐶𝐶𝑒𝑒𝜕𝜕𝑇𝑇𝑒𝑒\n𝜕𝜕𝑡𝑡=−𝑘𝑘𝑒𝑒∇2𝑇𝑇𝑒𝑒−𝑔𝑔𝑒𝑒𝑙𝑙(𝑇𝑇𝑒𝑒−𝑇𝑇𝑙𝑙)+𝑃𝑃(𝑟𝑟,𝑡𝑡)−𝐶𝐶𝑒𝑒(𝑇𝑇𝑒𝑒−𝑇𝑇𝑅𝑅)\n𝜏𝜏𝐷𝐷 (2) \n𝐶𝐶𝑙𝑙𝜕𝜕𝑇𝑇𝑙𝑙\n𝜕𝜕𝑡𝑡=−𝑔𝑔𝑒𝑒𝑙𝑙(𝑇𝑇𝑙𝑙−𝑇𝑇𝑒𝑒) (3) \nwhere 𝐶𝐶𝑒𝑒 and 𝐶𝐶𝑙𝑙 denote the specific heat of electrons and  lattice subsystems, respectively. \n𝑘𝑘𝑒𝑒 is the electronic thermal conductivit y. 𝑔𝑔𝑒𝑒𝑙𝑙 is a coupling parameter between the electron 5 \n and lattice subsystems, and 𝜏𝜏𝐷𝐷 is the characteristic heat diffusion time to the substrate and \nthe surrounding medi a  [25] . Conventional values were adopted  (see  [1,16,26] and \nSupplemental Material Note SN1  [23]). \nThe approach that consists on solving Eq. (1 ) coupled to Eqs. ( 2)-(3) is named as \nAtomistic Spin Dynamics  (ASD), and due to computational restrictions,  its numerical \nsolution is limited to small samples at the nanoscale ( ℓ ≲ 100  nm, see Supplemental \nMaterial Note  SN2  [23]) . While A SD predicts the single -shot switching in small FiM \nnano- samples  [16,22,27] , the lack of a realistic micromagnetic framework for micro -size \nsamples and non- unifom laser spot limits the description of many experimental \nworks  [9] . In particular, the appearance of central regions with a multi -domain \ndemagnetized pattern s  [28] , or the observation of rings  of switched magnetization  under \nirradiation with lasers of tens of micrometers   [29] cannot be reproduced by AS D due to \nsuch computing limitations. In order to overcome the ASD limitations , here we develop \nan extended continuous micromagnetic model that describes the temporal evolution of \nthe reduced local magnetization  𝑚𝑚��⃗𝐹𝐹(𝑟𝑟 ⃗,𝑡𝑡) of each sublattice 𝑖𝑖 :RE,TM based on the \nconventional ferrimagnetic Landau -Lifshitz -Bloch (LLB) Eq  [21,30], \n𝜕𝜕𝑚𝑚��⃗𝐹𝐹\n𝜕𝜕𝑡𝑡=−𝛾𝛾0𝐹𝐹′�𝑚𝑚��⃗𝐹𝐹×𝐻𝐻��⃗𝐹𝐹�+ \n−𝛾𝛾0𝐹𝐹′𝛼𝛼𝐹𝐹⊥\n𝑚𝑚𝐹𝐹2𝑚𝑚��⃗𝐹𝐹×�𝑚𝑚��⃗𝐹𝐹×�𝐻𝐻��⃗𝐹𝐹+𝜉𝜉⃗𝐹𝐹⊥��+ \n+𝛾𝛾0𝐹𝐹′𝛼𝛼𝐹𝐹∥\n𝑚𝑚𝐹𝐹2(𝑚𝑚��⃗𝐹𝐹·𝐻𝐻��⃗𝐹𝐹)𝑚𝑚��⃗𝐹𝐹+𝜉𝜉⃗𝐹𝐹∥ (4) \nwhere 𝐻𝐻��⃗𝐹𝐹=𝐻𝐻��⃗𝐹𝐹(𝑟𝑟 ⃗,𝑡𝑡) is the local effective field on sublattice magnetic moment 𝑖𝑖 at location \n𝑟𝑟 ⃗ of the FiM sample , 𝛼𝛼𝐹𝐹∥ and 𝛼𝛼𝐹𝐹⊥ are the longitudinal and perpendicular damping \nparameters, and 𝜉𝜉⃗𝐹𝐹∥ and 𝜉𝜉⃗𝐹𝐹⊥ are the longitudinal and perpendicular stochastic thermal fields. \nDetails of the LLB model can be found in [21,30] and in Supplemental Material Note  \nSN2  [23]. C ontrary to the ASD, where the spatial discretization is imposed by the \natomistic scale ( 𝑎𝑎= 0. 35 nm ), within the micromagnetic model the sample is discretized \nin elementary cells with dimensions of  Δ𝑥𝑥=Δ𝑥𝑥~1nm a\nnd Δ𝑧𝑧=𝑡𝑡𝐹𝐹𝐹𝐹𝐹𝐹. Therefore, it is \npossible to numerically  evaluate, with manageable  computing effort, extended samples at \nthe microscale  (ℓ~100  μm), three order s of magnitude larger than the  ones which can be \ndealt with  the ASD model. 6 \n Numerically solving Eq. (4) coupled to Eqs . (2)- (3) under ultra -short laser pulses \nprovides a micromagnetic description of several AOS processes in ferromagnetic \nsystems  [13] . However, when dealing with ferr imagnetic  samples we checked that some \ndisagreement with the predictions of the ASD mode l were observed (see Supplemental \nMaterial Note  SN3  [23] ), which are related to the lack of a proper description of the \nangular moment exchange between sublattices during the non- equilibrium transient state \npromoted by the laser pulse. Indeed, magnetization dynamics in FiMs  is driven by \ndissipative processes of relativistic and exchange nature. The relativistic ones allow \nexchange of angular momentum between the magnetization and the lattice degree of \nfreedom due to the spin- orbit coupling between them, and are phenomenologically \ndescribed by the usual damping terms in the LLB Eq. (4). Additiona lly, in multisublattice \nmagnets as FiMs, another different pathway opens local exchange of angular momentum \nbetween both sublattices of the FiM, and to account for  it, the LLB Eq . (4) has to be \nenhanced by an additional exchange relaxation torque  [18,31–34] . The simplest model \nto describe the sublattice -specific magnetization dynamics in FiMs, was derived from \nOnsager’s relations  [31]  within a macrospin approach based on a microscopic spin \nmodel. In this simplified description, the magnetization dynamics of sublattice 𝑖𝑖  can be \nexpressed as 1\n𝛾𝛾0𝑖𝑖𝑑𝑑𝑚𝑚𝑖𝑖\n𝑑𝑑𝑡𝑡=𝛼𝛼𝐹𝐹𝐻𝐻𝐹𝐹+𝛼𝛼𝑒𝑒𝑒𝑒�𝜇𝜇𝑖𝑖\n𝜇𝜇𝑗𝑗𝐻𝐻𝐹𝐹−𝐻𝐻𝑗𝑗�, where 𝑖𝑖,𝑗𝑗:RE,TM, 𝜇𝜇𝐹𝐹 and 𝐻𝐻𝐹𝐹 are the \nmagnitude of the magnetic moment and the e ffective field acting on macrospin of \nsublattice 𝑖𝑖 respectively . The relativistic relaxation parameter in this model, 𝛼𝛼𝐹𝐹, \ncorresponds  to the longitudinal damping parameter in the LLB E q. and it depends on the \ntemperature of the thermal bath  to which ang ular momentum and energy is dissipated. \nDifferently, it is assumed that the exchange relaxation parameter 𝛼𝛼 𝑒𝑒𝑒𝑒, only depends on \nthe non- equilibrium sublattice magnetizations, 𝛼𝛼𝑒𝑒𝑒𝑒=𝛼𝛼𝑒𝑒𝑒𝑒(𝑚𝑚𝐹𝐹,𝑚𝑚𝑗𝑗). Considering that \nexchange relation rate should be symmetric with respect to the sublattice index, \n𝛼𝛼𝑒𝑒𝑒𝑒(𝑚𝑚𝐹𝐹,𝑚𝑚𝑗𝑗)=𝛼𝛼𝑒𝑒𝑒𝑒(𝑚𝑚𝑗𝑗,𝑚𝑚𝐹𝐹), a simple functional fulfilling these heuristic conditions \nyields to 𝛼𝛼𝑒𝑒𝑒𝑒(𝑚𝑚𝐹𝐹,𝑚𝑚𝑗𝑗)=𝜆𝜆𝑒𝑒𝑒𝑒𝑚𝑚𝑖𝑖+(𝑒𝑒𝑗𝑗𝜇𝜇𝑗𝑗/𝑒𝑒𝑖𝑖𝜇𝜇𝑖𝑖 )𝑚𝑚𝑗𝑗\n𝑚𝑚𝑖𝑖𝑚𝑚𝑗𝑗 where 𝜆𝜆𝑒𝑒𝑒𝑒 is a phenomenological parameter  \nrepresenting the exchange relaxation rate and 𝑥𝑥𝐹𝐹 the concentration of each specimen 𝑖𝑖. \nInspired by this two sublattice phenome nological model based on Onsager’s relations, \nhere we add an additional torque 𝜏𝜏⃗𝐹𝐹𝑁𝑁𝑁𝑁 to the micromagnetic LLB Eq. (4) that accounts for \nnon-equilibrium magnetic moment exchange  between sublattices , and becomes crucial to \ndescribe AOS ultra -fast switching in FiMs  under realistic conditions . The torque reads as  7 \n 𝜏𝜏⃗𝐹𝐹𝑁𝑁𝑁𝑁=𝛾𝛾0𝐹𝐹′𝜆𝜆𝑒𝑒𝑒𝑒𝛼𝛼𝐹𝐹∥𝑥𝑥𝐹𝐹𝜇𝜇𝐹𝐹𝑚𝑚𝐹𝐹+𝑥𝑥𝑗𝑗𝜇𝜇𝑗𝑗𝑚𝑚𝑗𝑗\n𝜇𝜇𝐹𝐹𝑚𝑚𝐹𝐹𝜇𝜇𝑗𝑗𝑚𝑚𝑗𝑗�𝜇𝜇𝐹𝐹𝐻𝐻��⃗𝐹𝐹∥−𝜇𝜇𝑗𝑗𝐻𝐻��⃗𝑗𝑗∥� (5) \nwhere and 𝐻𝐻��⃗𝐹𝐹∥ and 𝐻𝐻��⃗𝑗𝑗∥ are the longitudinal effective field s for each lattice 𝑖𝑖:RE,TM  [21] , \n𝑥𝑥𝐹𝐹≡𝑥𝑥 and 𝑥𝑥𝑗𝑗=1−𝑥𝑥𝐹𝐹=1−𝑥𝑥 are the concentration s of each specimen, and 𝜆𝜆𝑒𝑒𝑒𝑒 is a \nparameter representing the exchange relaxation rate  [18] . By including Eq. (5) in the \nRHS of Eq. (4), and numerically solving it coupled to TTM Eqs. (2) -(3), w e can provide \na realistic description of the magnetization dynamics in FiM systems  under ultra -short \nlaser pulses. In what follows, we refer to this formalism as the extended micromagnetic \nLLB model  (eLLB) , to distinguish it from the conventional  LLB model  (LLB) when \n𝜏𝜏⃗𝐹𝐹𝑁𝑁𝑁𝑁=0. See Supplemental Material Note  SN2  [23]  for the rest of details.  \n \nIII. RESULTS  AND DISCUSSION  \nBefore presenting the predictions of the extended micromagnetic model for realistic \nFiM samples and laser  beam s at the microscale, here we firstly compare the results \nobtained from the extended LLB model ( eLLB) to the ones resulting from the atomistic \nspin dynamics simulations (ASD) for a small FiM dot at the nano -scale (ℓ≈25 nm ). As \ntypical laser spots have radius  of 𝑟𝑟0~1 μm−10 μm or even larger , we assume here that \nthe power absorbed by the FiM dot the from the laser pulse is uniform, that is, 𝜂𝜂(𝑟𝑟)=1. \nThe pulse duration is 𝜏𝜏 𝐿𝐿=50 fs. Typical results showing  the temporal evolution of the \nout-of-plane averaged mag netization ( 𝑚𝑚𝑧𝑧𝐹𝐹) for each sublattice ( 𝑖𝑖:TM (red),  RE (blue)) are \nshown in FIG.  1 for two different values of 𝑄𝑄. 8 \n  \nFIG. 1 . Comparison between atomistic simulation (ASD, solid lines) and micromagnetic model \n(eLLB , dashed lines) results for the temporal evolution of the out -of-plane magnetization for the \ntransition metal (TM, red) and the rare earth (RE, blue) under two different laser power densities: \n(a) 𝑄𝑄=6×1021 W/m3 and (b) 𝑄𝑄=12×1021 W/m3. Four consecutive laser pulses with 𝜏𝜏𝐿𝐿=\n50 fs are applied every 20  ps. (c) shows a detailed view of first pulse switching event as in ( a), \nwhile (d) shows temporal evolution of the electron ( 𝑇𝑇𝑒𝑒) and lattice ( 𝑇𝑇𝑙𝑙) temperatures for 𝑄𝑄=\n6×1021 W/m3. (e) and  (f) corresponds to 𝑄𝑄=12×1021 W/m3. Shaded interval  in (e)  shows \nthe transient ferromagnetic state. The pulse length  is 𝜏𝜏𝐿𝐿=50 fs. The eLLB results were obtained \nwith 𝜆𝜆𝑒𝑒𝑒𝑒=0.013. \n \nA remarkable agreement between both ASD and e LLB models with similar \ndynamics  for both sublattices  is observed in FIG. 1 (a) and (b) . For low 𝑄𝑄 values ( FIG. \n1(a) and (c) ) there is no switching, but when 𝑄𝑄 increases above a threshold, which \ndepends on the pulse length ( 𝜏𝜏𝐿𝐿), the deterministic AOS  is predicted by both ASD and \neLLB models ( FIG.1(b)  and (e) ). It is important to note that similar switching was  also \nobtained within the deterministic e LLB framework, that is, in the absence of thermal \n9 \n fluctuations  (𝜉𝜉⃗𝐹𝐹⊥=𝜉𝜉⃗𝐹𝐹∥=0 in Eq. (4) , see FIG. S4(b) in Supplemental Material Note  \nSN3  [23] ). On the contrary, the conventional  LLB model ( LLB, 𝜏𝜏⃗𝐹𝐹𝑁𝑁𝑁𝑁=0) fails to \nreproduce the switching of FIG. 1(b)  (see FIG. S4(c) -(d) in Supplemental Note \nSN3  [23] ). FIG. 1(c)  show s the details of the temporal evolution 𝑚𝑚𝑧𝑧𝐹𝐹 for 𝑄𝑄=\n6×1021 W/m3 during the first laser pulse , while the corresponding evolutions  of 𝑇𝑇𝑒𝑒 and \n𝑇𝑇𝑙𝑙 are depicted in FIG. 1(d) , which also shows the laser pulse . Corresponding results for \n𝑄𝑄=12×1021 W/m3 are shown in FIG.  1(e) and (f) respectively . For 𝑄𝑄=\n6×1021 W/m3, the electron temperature reaches a peak maximum value of  𝑇𝑇𝑒𝑒≈850  K \nat the end of the laser pulse, but this is not enough to achieve the switching . For 𝑄𝑄=\n12×1021 W/m3, 𝑇𝑇𝑒𝑒 reaches  a peak  of 𝑇𝑇𝑒𝑒≈1150  K and switching takes place . Not ice \nthat this value is well above the Curie temperature ( 𝑇𝑇𝐶𝐶≈600  K), and therefore the system \nneeds to be significantly heated above the Curie threshold to achieve the deterministic \nAOS  in FiM . These processes  are explained by the different demagnetization rates of the \nRE and TM sublattices, that lead to a transient ferromagnetic aligment. Such transient \nferromagnetic state is observed during a  short transient  (see shaded interval in FIG. 1(e)), \nand it is only present when the system is far away from the thermodynamic equilibrium, \nas caused by the ultrafast laser heating.  Except the contrary is indicated, all eLLB results \nwere obtained wi th 𝜆𝜆𝑒𝑒𝑒𝑒=0.013, (s ee FIG. S 5 and its corresponding discussion in \nSupplemental Material Note SN3 (c)  [23]  for results with other values of 𝜆𝜆𝑒𝑒𝑒𝑒). \n \nHelicity -Independent A ll Optical Switching (H I-AOS) . Once validated the e LLB \nformalism by reproducing the ASD results for small nano- sample s under uniform \nlinearly -polarized laser pulses, we can now use it to explore the influence of the laser \nduration ( 𝜏𝜏𝐿𝐿) and maximum absorbed power  density  (𝑄𝑄) in realistic extended samples at \nthe microscale (ℓ ~10 μ m). This is illustrated  in the phase diagram of FIG.  2(a), which \nshows the final state under a single  linearly -polarized laser  pulse starting from a uniform \nstate of the FiM . White color indicates the combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) where the sample \nreturns to the original state after the pulse  (no-switching) . The blue region corresponds to \ncombinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) presenting  deterministic HI-AOS  after each pulse, and red \ncorresponds to combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) resulting in a final demagnetized multidomain \nconfiguration. It is noted that there is a correlation b etween the final state and the \nmaximum electron temperature reached in the sample, which is shown by the overlapping 10 \n solid black lines in FIG.  2(a). As it is clearly observed, s olid lines coincide with \nboundaries between the three possible behaviors  already discussed. Indeed, the transition \nbetween no -switching (white) to the deterministic switching range (red) is limited by the \n~1000  K curve , whereas the transition to the thermal demagnetization (blue) occurs when \n𝑇𝑇𝑒𝑒≳1400  K, as shown in FIG.  2(a). Instead of 𝑄𝑄 , the information collected in the phase \ndiagram of FIG. 2(a)  could be also presented in terms  of the laser fluence ( 𝐹𝐹≡𝑄𝑄 𝜏𝜏𝐿𝐿 𝑡𝑡𝐹𝐹𝐹𝐹𝐹𝐹), \nas it is done in FIG.  S6 of Supplemental Material Note SN 4  [23] . Note, that such phase \ndiagram is also in good qualitative agreement with recent experimental \nobservations  [35] .  \n \nFIG. 2 . (a) Phase diagram of the final state as a function of 𝑄𝑄  and 𝜏𝜏𝐿𝐿 for a small nano -sample ℓ=\n25 nm under uniform laser heating ( 𝜂𝜂(𝑟𝑟)=1). White, red and blue colors represent no -\nswitching, deterministic switching and thermal demagnetization behaviors respectively. Solid \nlines are isothermal curves showing the maximum electron temperature ( 𝑇𝑇𝑒𝑒) reached due to the \npulse. (b) Typical mi cromagnetic snapshots of the initial and final magnetization of RE and TM \nfor three combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿). I: (8×1021 W/m3,20 fs), II: (15×1021 W/m3,30 fs) and \nIII: (20×1021 W/m3,60 fs). Here extended samples ( ℓ=20 μm) with a laser spot of 𝑑𝑑0=ℓ/2 \nwere considered. Dashed purple lines in the images of the initial state indicate the FWHM of  the \nlaser spot . The results of the phase diagram  (a) coincide with (b) for magnetization at the center \nof the laser spot . \n \nThe main advantage of the extended e LLB model over ASD simulations is that it  \nallows us  to explore  realistic samples and laser beams with dimensions that are not \naccessible with ASD models. e LLB model ( Eqs. (4) and (2) -(3)) has been used to simulate \n11 \n samples with lateral size of ℓ=20 μ m. From no w on , the spatial Gaussian dependence \nof the laser beam is considered  (𝜂𝜂(𝑟𝑟)=exp[−4ln(2)𝑟𝑟2/ (2𝑟𝑟0)2 ]), with a laser spot \ndiameter  of 𝑑𝑑0=2𝑟𝑟0=ℓ/2. Typical initial and final states corresponding to three  \nrepresentative combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) are shown in FIG. 2(b). Our micromagnetic \nsimulations point out again that the three types of behaviors observed experimentally (see  \nfor example Fig. 4(a) in   [9] or  [6,28] ) are also achieved under these realistic conditions, \nwith samples and laser spots at the microscale. Not ice that now the final magnetic state \ndepends on the local position because the power absorption from laser pulse does . The \nfinal states depict a radial symmetry  around the center of the laser spot, which coincides \nwith the center of the  FiM  sample at (𝑥𝑥𝑐𝑐,𝑥𝑥𝑐𝑐)=(0,0). \nIn order to further describe such spatial dependence, FIG.  3 plots the final state of \n𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹 as a function of 𝑥𝑥 along the central line of the FiM sample ( 𝑥𝑥=0) for the same \nthree representative combinations  of (𝑄𝑄,𝜏𝜏𝐿𝐿) as in FIG.  2(b). The maximum electron \ntemperature 𝑇𝑇𝑒𝑒=𝑇𝑇𝑒𝑒(𝑥𝑥) is also plotted in top graphs by blue curves. Bottom graphs in \nFIG. 3 show the final state over the sample plane (𝑥𝑥,𝑥𝑥). These graphs clearly correlate \nthe local final magnetic state ( 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹(𝑥𝑥,𝑥𝑥)) with the maximum electron temperature \n𝑇𝑇𝑒𝑒(𝑥𝑥,𝑥𝑥). In the no- switching regime (I), the electron temperature does not reach 1000  K \nat any point . For combinations (𝑄𝑄,𝜏𝜏𝐿𝐿) as II, 𝑇𝑇𝑒𝑒(𝑥𝑥)≳1000  K is only reached in the cent ral \nregion , whose dimensions  fit the local part of the sample that switches its magnetization. \nNote that 𝑇𝑇 𝑒𝑒 remains below 𝑇𝑇 𝑒𝑒(𝑥𝑥)≲1400  K. Finally, the demagnetiz ed case (multi -\ndomain pattern, III) occurs in the part of the sample where 𝑇𝑇𝑒𝑒≳1400  K, but deterministic \nswitching is s till obtained in the ring region, where 1000  K≲𝑇𝑇𝑒𝑒(𝑥𝑥)≲1400  K. \nMicromagnetic images  (bottom graphs in FIG. 3 ) are in good agreement with typical \nexperimental HI -AOS observations   [6,28] . 12 \n  \nFIG. 3 . Final out -of-plane magnetization ( 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹(𝑥𝑥)) and maximum electron temperature ( 𝑇𝑇𝑒𝑒(𝑥𝑥)) \nas function of 𝑥𝑥  for 𝑥𝑥=0 and for the t hree representative combinations of (𝑄𝑄,𝜏𝜏𝐿𝐿) as in FIG. 2(b)  \n(top graphs). The corresponding final states over the sample plane (𝑥𝑥,𝑥𝑥) are shown in bottom \ngraphs.  \n \nThe inferred  correlation between the maximum electron temperature and the final \nmagnetic state allows us to predict the size of the inverted region by studying the \nmaximum electron temperature  reached in the sample by using the TTM (Eqs. (2)-(3)) in \ncombination with t he switching diagram of FIG. 2(a) . The radius of the switched area \n(𝑅𝑅𝑠𝑠) calculated from micromagnetic simulations (dots, e LLB), and the one predicted by \nthe TTM (lines, TTM) is shown in FIG. 4  as function of 𝜏𝜏 𝐿𝐿 for two different values of 𝑄𝑄 \nwithin the deterministic switching range  (1000  K≲𝑇𝑇𝑒𝑒(𝑥𝑥)≲1400  K). Again, good \nagreement is obtained , a fact that points out that the origin of these HI -AOS processes \nunder linearly polarized laser pulse is a purely thermal  phenomenon. Indeed, as the local \nmaximum electron temperature  reached in the sample only depends on the absorbed \npower from the laser ( 𝑄𝑄) and the laser pulse length ( 𝜏𝜏𝐿𝐿), the size of the inverted region \ncan be directly obtained from the TTM by the condition of 𝑇𝑇 𝑒𝑒≳1000  K. \n13 \n  \nFIG. 4 . Radius of the switched area ( 𝑅𝑅𝑠𝑠) as a function of laser duration ( 𝜏𝜏𝐿𝐿) for two values of the \nmaximum absorbed power ( 𝑄𝑄). Dots are micromagnetic results from the e xtended e LLB model . \nLines are predictions from the TTM, where 𝑅𝑅𝑠𝑠 is inferred from the condition that the local \nmaximum electron temperature reaches 𝑇𝑇𝑒𝑒≳1000  K. \n \nHelicity -Dependent All Optical Switching (H D-AOS) . Previous results were \ncarried out by applying laser pulses with linear polarization  (𝜎𝜎=0), and show that the \nHI-AOS can be achieved in a controlled manner with an adequate election of the laser \npower (𝑄𝑄) and duration ( 𝜏𝜏𝐿𝐿): the magnetization switches its direction in the p icoseconds  \nrange independently on the initial state. While this is interesting for toogle memory \napplications, the procedure to store and record a bit  using linearly  polarized  laser pulses \nwould still require two steps: ( 𝑖𝑖) a pre liminary  reading operation of the magnetic state, \nand after that, ( 𝑖𝑖𝑖𝑖) deciding or not to apply the laser pulse depending on the  preceeding \nstate. This two -step procedure can be avoided by using circularly polarized laser pulses, \nresulting in  Helicity -Dependent AOS  processes  (HD- AOS) . However, as it was already \ncommented the physics behind these HD -AOS observations still remains un clear, and \nboth the Magnetic Circular Dichroism (MCD)  [8,10]  and the Inverse Faraday Effect \n(IFE)  [12,14]  have been suggested as responsible of the experimental observations. In \nwhat follows, we explore both mechanisms in a separated manner by including them in \nthe extended micromagnetic model.  \nLet firstly consider the Magnetic Circular Dichroism . It has been suggested to be \nplay a dominant  a role on these HD- AOS  processes  in GdFeCo ferrimagnetic samples, \nwhich are known for its strong magneto- optical effect   [8]. According to the MDC \nformalism,  right -handed (𝜎𝜎+) and  left-handed (𝜎𝜎−) circular ly polarized laser  pulses \n14 \n experience different refractive indices, and consequently a differen ce in energy \nabsorption of the FiM sample for 𝜎𝜎+ and 𝜎𝜎− pulses is expected. The MCD coefficient \ncan be calculated from the total absorption for each polarization, resulting in MCD≡𝑘𝑘=\n(𝐴𝐴−−𝐴𝐴+)/(1\n2(𝐴𝐴++𝐴𝐴−)), where 𝐴𝐴± represent the total absorption  for each polarization, \n(±≡𝜎𝜎±). Indeed, the MCD makes the power absorbed by the sample ( 𝑃𝑃(𝑟𝑟,𝑡𝑡)) to depend \non the laser helicity ( 𝜎𝜎±=±1 for right -handed and left -handed circular helicities ) and \non the initial net magnetic state ( 𝑚𝑚𝑁𝑁(0)=𝑀𝑀𝑆𝑆𝑇𝑇𝐹𝐹𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹(0)+𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁(0)), up (↑: \n𝑚𝑚𝑁𝑁(0)>0) or down (↓:𝑚𝑚𝑁𝑁(0)<0). Note that 𝑚𝑚 𝑧𝑧𝑇𝑇𝐹𝐹(0)=±1 and 𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁(0)=∓1, \nwhereas 𝑀𝑀𝑆𝑆𝑇𝑇𝐹𝐹 and 𝑀𝑀𝑆𝑆𝑅𝑅𝑁𝑁 are both positive. Under a right -handed laser pulse  (𝜎𝜎+), an \ninitially up (down)  magnetic state is expected to absorb more (less)  energy than the \ninitially down  (up) state . Therefore, 𝑃𝑃(𝑟𝑟,𝑡𝑡) in Eq. (2) is replaced by 𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)𝑃𝑃(𝑟𝑟,𝑡𝑡) \nwith 𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)=�1+12𝑘𝑘𝜎𝜎±sign (𝑚𝑚𝑁𝑁)� describing  the different absorption power  for \nup and down magnetization states as depending on the laser helicity.  See further details \non the implementation of the MCD in Supplemental Material Note SN 5  [23] . \nWe have evaluated the role of the MCD in the eLLB model with several values of \nthe MCD coefficient  (𝑘𝑘). The isothermal curve delimiting the border between the no-\nswitching and switching  regimes  now depends on the combination of helicity and initial \nnet magnetic state (see such isothermal threshold curves for different values of the MCD \ncoefficient in of SN 5). Considering  a realistic value of  MCD≡𝑘𝑘~2%, as estimated \nin  [8], the electron temperature variation is quite small, typically  a few  units of K, and \ntherefore small variations in the phase diagram are obtained with respect to the one for \nlinearly -polarized pulses  FIG. 2(a) (see also FIG. S 7 in Supplemental Material Note \nSN5  [23] ). However, when exciting with circular polarized pulses close to the no-\nswitching /switching  boundary, the FiM  switches or not depending on the helicity and \ninitial net state, only within a narrow  interval  of 𝑄𝑄. This is represented in FIG.  5(a), where \nthe H D-AOS is shown for 𝜏𝜏 𝐿𝐿=50 fs pulses with different 𝑄𝑄 in a sample with  ℓ=20 μm. \nNote that these results correspond to a F iM alloy Gdx(FeCo) 1-x with 𝑥𝑥=0.25, and that for \nthis relative composition the RE is the dominant sublattice at room temperature, 𝑀𝑀 𝑠𝑠𝑅𝑅𝑁𝑁>\n𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹 at 𝑇𝑇=300  K (see inset of FIG. S3 of Supplemental Material  [23] or FIG. 6(a) ). No \nswitching is achieved for low energy values (see left colum in FIG. 5(a) for 𝑄𝑄=\n5.7×1021 W/m3). However, if 𝑄𝑄 increases to 𝑄𝑄 =5.8×1021 W/m3, the system shows  \nthe so -called H D-AOS: if 𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁 is initially down (up), the reversal is only achieved for  15 \n left-handed helicity, 𝜎𝜎−=−1 (right -handed helicity, 𝜎𝜎+=+1). Consequently, the final \nstate can be selected by chosing the laser helicity, which is relevant  for ultrafast memory \napplications. It is important to note that this H elicity Dependent AOS is only obtained in \na very narrow range of 𝑄𝑄 around the H elicity Independent AOS boundary . Indeed, a small \nincrease of the absorbed power results again in H I-AOS as the linear polarized case (see \n3rd and 4th columns in FIG. 5(a)  for 𝑄𝑄 =5.9×1021 W/m3 and 𝑄𝑄=9.0×1021 W/m3). \nFor high values of 𝑄𝑄, the final state depicts a ring around a central demagnetized state, \nsimilar to HI -AOS case (see right column in FIG. 5(a)  for 𝑄𝑄=18×1021 W/m3). In this \ncase, the maximum electron temperature overcomes the  𝑇𝑇𝑒𝑒≃1400  K threshold in the \ncentral region below the laser spot , resulting in a central  demagnetized or multidomain \nstate. However, the maximum 𝑇𝑇𝑒𝑒 remains with in the range of HD -AOS deterministic \nswitching ( 1000  K≲𝑇𝑇𝑒𝑒(𝑥𝑥)≲1400  K) in the ring  around the central part . These \nmicromagnetic predictions , including the narrow range of HD -AOS,  are in good \nagreement with several experimental observations  (see, for instance,  Fig. 4 and 7(a) \nin  [9] , Fig. 1(b) in  [4] , or Fig. 3 in  [36] ).  \n \n16 \n FIG. 5. Helicity -Dependent AOS  predicted by the MCD . (a) Snapshots of the final RE  magnetic \nstate (𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁) after a laser pulse of 𝜏𝜏𝐿𝐿=50 fs for five  different values of the absorbed power density \n(𝑄𝑄). Results are shown for four combinations of the initial state ( ↑,↓) and helicities ( 𝜎𝜎±) as \nindicated at the left side.  The HD- AOS is shown in panel corresponding to 𝑄𝑄=\n5.8×1021 W/m3. (b) RE magnetic state after every pulse for 𝑄𝑄=5.9×1021 W/m3, showing \nthe appearance of a ring due to the MCD and the switching of the central  part. Here pulses with \nleft-handed chirality are applied ( 𝜎𝜎+). The sample side is ℓ=20 μm and the laser spot diameter \nis 𝑑𝑑0=ℓ/2. \n \nMoreover, the inclusion of the MCD in our e LLB  model allows us to explain the \nexperimental observation of rings  [28,36,37]  which appear after the application of a \nsecond laser pulse. This is illustrated  in FIG. 5(b)  for pulses with 𝑄𝑄=5.9×1021 W/m3 \nand 𝜏𝜏𝐿𝐿=50 fs. The central part of the sample reaches temperatures that lead to H I-AOS, \nand ther efore, its magnetization reverses after each pulse. On the contrary, the ring around \nof the inverted region is within the H D-AOS  regime , and therefore, its local magnetic \nstate (going from up to down) is only reversed by the first pulse. For the second and \nsubsequent  pulses , the ring maintains its down state while the inner part changes again to \nup (white). This is repeated every pulse, with the inversion of the central part and the \nmaintenance of bla ck ring in the external shell, as it is clearly seen in even pulses (see \nsnapshots after pulses #2 and #4 in FIG. 5(b) ). Note that this ring structure differs from \nthe ones shown in FIG. 2(b)  and FIG.3 , as they were caused by the inversion of the \nmagnetiza tion around the central demagnetized part under high -power linear pulses  (𝜎𝜎=\n0). For circularly polarized pulses  (𝜎𝜎±=±1) the images correspond to alternative \nswitching and the H D-AOS  without the central demagnetized (multidomain) state . Again, \nthese results are in good agreement with recent experimental observations (see figures \nin  [28,36,37] ). \n \nInstead  of the MCD, several other works claim that the observations of the HD -\nAOS can be ascribed to the Inverse Faraday Effect (IFE)  [9]. Within this formalism, the \nlaser pulse generates an effective out -of-plane magneto -optical field which direction \ndepends on the laser pulse helicity , 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)=𝜎𝜎±𝐵𝐵𝐹𝐹𝑀𝑀𝜂𝜂(𝑟𝑟)𝜖𝜖(𝑡𝑡)𝑢𝑢�⃗𝑧𝑧, where 𝜂𝜂(𝑟𝑟)=\nexp[−4ln(2)𝑟𝑟2/ (2𝑟𝑟0)2 ]  is the spatial field profile, and 𝜖𝜖(𝑡𝑡) is its temporal profile. \nNote that the spatial dependence of 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) is the same as the one of the  absorbed power \ndensity. However, a ccording to the literature   [9], the so- called magneto -optical field \n𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) has some temporal persistence with respect to the laser pulse, and therefore  its 17 \n temporal profile is different for 𝑡𝑡 <𝑡𝑡0 and 𝑡𝑡 >𝑡𝑡0: 𝜖𝜖(𝑡𝑡<𝑡𝑡0)=exp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/\n 𝜏𝜏𝐿𝐿2], and 𝜖𝜖 (𝑡𝑡≥𝑡𝑡0)=exp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/(𝜏𝜏𝐿𝐿+𝜏𝜏𝐷𝐷)2], where 𝜏𝜏 𝐷𝐷 is the delay time of \nthe 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) with respect to the laser pulse. We have evaluated this IFE scenario by \nincluding this field 𝐵𝐵 �⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) in the effective field of Eq. (4). The  results  for the same \nFiM alloy considered up to here ( Gdx(FeCo) 1-x, with x=0.25, see SN 5), are similar to the \nones already presented in FIG. 5  for the MCD considering a maximum magneto -optical \nfield of 𝐵𝐵𝐹𝐹𝑀𝑀=20 T  with a delay time of 𝜏𝜏𝐷𝐷=𝜏𝜏𝐿𝐿. These IFE results can be seen in FIG. \nS8 in Supplemental Material Note  SN6  [23] . Therefore, we could conclude from this \nanalysis that, from the  micromagnetic  modeling point of view, both the MCD and the IFE \nare compatible with experimental observations of the HD -AOS . At this point, it is worth \nto mention here that in real experiments there is not a clear distinction between MCD and \nIFE phenomena. Indeed, the modeling of the IFE for absorbing materials can account for \nabsorption phenomena as MCD (see for instance  [38,39] ). These works suggested that \nin micromagnetic simulations the IFE could induce  a change of the magnetic moment \n(Δ𝑚𝑚��⃗𝐹𝐹) modifying the initial magnetic moments in the two sublattices of the FiM  when \nsubmitted to circular polarized laser pulses. We have evaluated in our modeling this alternative manner of studying the role of the IFE by adding such an induced magnetic \nmoment in the eLLB Eq. (4), and compared the results to the case where the IFE is \nsimulated by the magneto -optical field 𝐵𝐵\n�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) as discussed above. As presented and \ndiscussed in Supplemental Material Note S N7  [23] , both alternatives ( 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) or Δ𝑚𝑚��⃗𝐹𝐹) \nare equivalent from the simulation point of view. Therefore, in what follows we will \nsimulate the IFE as an effective out -of-plane magneto- optical field.  \nIn order to get a further understanding on the physics of these two mechanisms, \neither the MCD or th e IFE, we have explored the switching probability as a function of \n𝑄𝑄 for laser pulses with  fixed duration ( 𝜏𝜏𝐿𝐿=50 fs ) in two FiM alloys with slightly \ndifferent composition: Gd x(FeCo) 1-x with x=0.25 and x=0.24 respectively . The \ncorresponding parameters to numerically evaluate these two alloys are given in SN8, and  \nthe temperature dependence of the saturation magnetization of each sublattice (RE: Gd; \nTM: CoFe) are shown in FIG. 6  (a) and (c) respectively . Note that magnetization \ncompensation temperature at which the net magnetization of the sample vanishes ( 𝑇𝑇𝐹𝐹) is \nabove and below room temperature for x=0.25 and x=0.24 respectively. In other words, \nthe FiM sample is dominated by the RE (TM) at 𝑇𝑇 =300  K for x=0.25 (x=0.24) \ncompositions . The MCD and IFE parameters remain fixed as indicated above (MCD: 18 \n 𝑘𝑘~2%,; IFE: 𝐵𝐵 𝐹𝐹𝑀𝑀=20 T, 𝜏𝜏𝐷𝐷=𝜏𝜏𝐿𝐿). The two  possible initial states prior the laser pulse  \n(𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹:(↑,↓)), and the three laser polarizations (linear: 𝜎𝜎 =0, and circular 𝜎𝜎±=±1) were \nevaluated. The switching probability was computed by evaluating ten different stochastic \nrealization s for each for each 𝑄𝑄, and t he results  are presented in FIG. 6(b) and  (d) for \nx=0.25 and x =0.24 respectively . As in experimental observations   [9], the HD -AOS takes \nplace o nly in a narrow range of 𝑄𝑄 around the threshold value of 𝑄𝑄  at which the switching \nprobability abruptly changes from 0 to 1 under linear polarized pulses ( black dots in FIG. \n6(b) and (d)) . \n \n \nFIG. 6. Temperature dependence of the spontaneous magnetization of each sublattice (RE:Gd; \nTM:CoFe) of the FiM alloy (Gd x(CoFe) 1-x) for two different compositions: (a) x=0.25 and (c) \nx=0.24. The vertical grey line indicates the initial room temperature prior the laser pulse ( 𝑇𝑇=\n300  K). Probability of switching as a function of the absorbed power density ( 𝑄𝑄) for a laser pulse \nof 𝜏𝜏𝐿𝐿=50 fs for different combinations of the initial state  (𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹:(↑,↓)) and the polarization  \n(linear: 𝜎𝜎=0 (black dots), and circ ular 𝜎𝜎±=±1) of the laser pulse as indicated  in the legend \nand in the main text: (b) corresponds to 𝑥𝑥 =0.25 and (d) to x =0.24.  MCD results are shown by \nsolid dots, whereas IFE results are presented by open symbols. Lines are guide to the eyes.  \n \nFor 𝑥𝑥=0.25, as all result s presented up to here, the FiM is dominated by the RE:Gd \nat room temperature: 𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹<𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁 at 𝑇𝑇=300K , see FIG. 6(a) . In this case, the switching \nrequires  less 𝑄𝑄 with circular polarization ( 𝜎𝜎±) with respect to the linearly polarized case \n19 \n (𝜎𝜎=0) for two different combinations of the circular laser polarization and the initial \nstate of the FiM: ( 𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↑) and (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↓). This happens for both MCD (solid \nsymbols) and IFE scenarios (open symbols) as it is shown in FIG. 6(b) . Note that the \ninitial state in the TM is the opposite to the RE: 𝑚𝑚 𝑧𝑧𝑇𝑇𝐹𝐹↑ (up) corresponds to 𝑚𝑚 𝑧𝑧𝑅𝑅𝑁𝑁↓ (down) \nand vice, and the AOS is independent on the initial state for linear polarization (HI -AOS), \nwhereas under circular polarized laser pulses the switching is helicity -dependent (HD -\nAOS). For the rest of combinations, either (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↑) or (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↓), a higher 𝑄𝑄 is \nneed ed to achieve 100% of switching probability with respect to the linear polarized laser \npulse, and again for x =0.25 both MCD and IFE scenarios result in similar behavior of the \nswitching probability ( FIG. 6(b) ).  \nRemarkably, the MCD and IFE r esults are qualitatively different w hen the \ncomposition is slightly modified to x =0.24, where the FiM becomes dominat ed by the \nTM:CoFe at room temperature: 𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹>𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁 at 𝑇𝑇=300K , see FIG. 6(c) . In this case  \n(x=0.24) , the results in the IFE scenario  are qualitatively  similar to the ones already \nobtained for 𝑥𝑥=0.25: the HD -AOS occurs with small 𝑄𝑄  with respect to the linearly \npolarization case for (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↑) and (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↓) (open blue symbols in FIG. 6(d) ), and \nit requires high  𝑄𝑄 for the two other combinations ( (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↑), (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹↓), open red \nsymbols in FIG. 6(d) ). However, for this concentration ( x=0.24), the results in the MCD \nscenario (see solid symbols in FIG. 6(d) ) are opposite as for x =0.25, and also opposite to \nthe ones obtained in the IFE scenario.  \nThese results can be understood as follows. In the MCD scenario, the HD -AOS \ndepends on the net initial magnetization at room temperature ( 𝑚𝑚𝑁𝑁=𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹+\n𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁, with 𝑚𝑚 𝑧𝑧𝑇𝑇𝐹𝐹=±1 and 𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁=∓1) and on the laser helicity ( 𝜎𝜎±=±1): if \ninitially 𝑚𝑚𝑁𝑁>0, a laser pulse with 𝜎𝜎+=+1 promotes the reversal, and this happens \neither for 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹=−1 (𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁=+1) when  x=0.25 because 𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁>𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹 at 𝑇𝑇=300K , or \nfor 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹=+1 (𝑚𝑚𝑧𝑧𝑅𝑅𝑁𝑁=−1) if x=0.24 because  now 𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹>𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁 at 𝑇𝑇=300K . On the \nother hand, the HD -AOS within the IFE scenario is essentially determined by the \ndominant sublattice magnetization just below the Curie threshold  (𝑇𝑇≲𝑇𝑇𝐶𝐶), due to the \npersistence of the magneto -optical field 𝐵𝐵 �⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) when the laser puls e has already \nfinished. Note that for both concentrations 𝑀𝑀𝑠𝑠𝑇𝑇𝐹𝐹>𝑀𝑀𝑠𝑠𝑅𝑅𝑁𝑁 for 𝑇𝑇≲𝑇𝑇𝐶𝐶. Indeed, during the \ncooling down after the pulse, the magneto- optical field 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)∝𝜎𝜎±𝑢𝑢�⃗𝑧𝑧 promotes up \nor down magnetic state for the TM for 𝜎𝜎+ and 𝜎𝜎− respectively, and therefore, if 𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹 is 20 \n initially up (𝑚𝑚𝑧𝑧𝑇𝑇𝐹𝐹=+1), 𝐵𝐵�⃗𝐹𝐹𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) promotes the reversal for 𝜎𝜎− and vice. Our analysis \nsuggests a set of experiments which could help to elucidate the  physical mechanism \nbehind these HD -AOS , just by evalu ating the switching probability as function of the \ninitial state and o f the laser pulse helicity for two different concentrations x , one resulting \nin a TM -dominated and other in a RE -dominated FiM alloy at room temperature.  Another \nalternative could be to use a single FiM with a given composition, and working with a \ncryostat to fix different initial temperatures below and above the magnetization \ncompensation temperature. Similar results to ones obtained by changing the composition \nx for a fixed temperature of the thermal bath are also  predicted by our simulations  when \nit is the temperature of the thermal bath what is varied for a given composition  (see Fig. \nS10 in Supplemental Material Note SN 9  [23] ): both IFE and MCD scenarios give similar \nresults below compensation and opposite above it . These theoretical predictions on the \nHD-AOS could be checked by experiments, which all together would allow us to shed \nlight on the real scope of these two mechanisms . \n \nIV. CONCLUSIONS  \nAs summary, the extension of the two sublattice Landau- Lifshitz -Bloch equation \nwith the angular momentum non- equilibrium exchange is proven to be a powerful tool to \nstudy ultrafast AOS switching in ferrimagnetic alloys . The formalism here developed \nreproduces  the atomistic spin dynamics  results for small samples at the nanoscale , while \nit opens the possibility to  numerically study realistic extended micro -size systems, with \ndimensions  comparable to the experiment al one s. The deterministic single -shot switching  \nand the demagnetization at high power regime are found to be in very good agreement with the experimental observations of Helicity -Independent AOS  under linearly polarized \nlaser pulses . The phase diagram combined with  the thermal analysis allow ed us to \ndetermine and predict the size of the inverted regions as depending the absorbed power \nand duration of the laser pulse . Moreover, w e have also explored and reproduced \nexperimental observations for the Helicity -Dependent AOS within the two physical \nmechanism s suggested in the literature: M agnetic Circular Dichroism  and Inverse \nFaraday Effect. According to the  Magnetic Circular Dichroism, the absorbed power by \nthe FiM depends on the laser helicity under  circularly polarized pulses , and our model \nalso predicts the main features of the Helicity -Dependent AOS measurements. Indeed, 21 \n both the Helicity -Dependent AOS  and the appearance of rings around the circularly \npolarized laser beam appear naturally  in our simulations . Additionally, similar r esults of \nthe HD -AOS switching were also obtained in Inverse Faraday Effect scenario, where the \ncircular polarization has been suggested to generate a persistent magneto -optical field \npromoting the switching for proper combinations of initial magnetic stat e and laser pulse \nhelicity . By exploring FiM samples with different compositions resulting in TM -\ndominated or in a RE -dominated FiM alloy at room temperature, we have found a \ndifference between the predictions of the IFE and the MCD scenarios. These result s could \nbe tested by performing the corresponding experiments, and consequently helping \ntogether to elucidate the  true basis of such  HD-AOS processes . Therefore, our methods \nwill be useful to understand recent and future experiments on AOS , and a lso to the \ndevelop novel recording devices where the information can be manipulated by optical \nmeans in an ultra -fast fashion. \n \nACKNOWLEDGMENTS  \nThis work was supported by projects MAT2017- 87072- C4-1-P funded by \nMinisterio de Educacion y Ciencia and PID2020117024GB -C41 funded by Ministerio de \nCiencia e Innovacion , both from the Spanish government , projects No. SA299P18 and \nSA114P20 from Consejer ia de Educaci on of Junta de Castilla y León, and project \nMagnEFi, Grant Agreement 860060, (H2020- MSCA -ITN-2019) funded by the European \nCommission.  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Li, All -Optical \nHelicity- Dependent Magnetic Switching by First -Order Azimuthally \nPolarized Vortex Beams , Applied Physics Letters 113, 171108 (2018).  \n[37] C. Banerjee, K. Rode, G. Atcheson, S. Lenne, P. Stamenov, J. M. D. Coey, and J. Besbas, Ultrafast Double Pulse All -Optical Reswitching of a \nFerrimagnet , Physical Review Letters 126 , 177202 (2021).  \n[38] M. Battiato, G. Barbalinardo, and P. M. Oppeneer, Quantum Theory of the Inverse F araday Effect , Physical Review B 89, 014413 (2014).  \n[39] M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer, Ab Initio Theory of \nCoherent Laser -Induced Magnetization in Metals , Physical Review Letters \n117, 137203 (2016).  \n \n 1 \n Supplementa l Material: Realistic micromagnetic description of all -optical ultrafast \nswitching  processes in ferrimagnetic alloys  \n \nV. Raposo1,*, F. García- Sánchez1, U. Atxitia2, and E. Martínez1,+  \n \n1. Applied Physics Department, University of Salamanca. \n2. Dahlem Center for Complex Quantum Systems and Fachbereich Physik . \n*,+ Corresponding authors : victor@usal.es , edumartinez@usal.es   \n \n \nCONTENT:  \n \n Supplemental Note SN1. Atomistic Spin Dynamics (ASD) model  \n Supplemental Note SN2. Micromagnetic LLB model s: conventional (LLB) and \nextended (eLLB) cases  \n Supplemental Note SN3. Comparison between atomistic, conventional -LLB \nand extended- LLB models  \n Supplemental Note SN4. Phase diagram in terms of the fluence and the pulse \nduration  \n Supplemental Note SN5. Helicity -Dependent AO S (HD-AOS) and M agnetic \nCircular Dichroism (MCD)  \n Supplemental Note SN 6. Helicity -Dependent AO S (HD-AOS) and Inverse \nFaraday Effect ( IFE) \n Supplemental Note SN7. I nverse Faraday Effect: magneto-optical field or \ninduced  magnetic moment  \n Supplemental note SN 8. Material inputs for two different compositions  \n Supplemental note SN 9: Helicity -Dependent All Optical Switching: MCD & \nIFE for different compositions and initial temperatures  \n \n \n  2 \n SN1. Atomistic S pin Dynamics  (ASD) model  \nWith the growing power of modern computers, numerical modeling has become an \nessential tool for scientific research, especially when handling systems as complex as \nmagnetic devices. For magnetic modeling, there are several  methodological choices in \nterms o f dimensions and time scales. In this section  we review the bases of the Atomistic \nSpin Dynamic  (ASD) model used to study the magnetization dynamics  in small nano -size \nferrimagnetic FiM samples submitted to  ultra -short of laser pulses. The physical basis o f \nthe ASD model is the  localization of unpaired electrons to atomic sites, leading  to an \neffective local atomistic magnetic moment at site 𝑖𝑖 , (𝜇𝜇⃗𝑖𝑖), which is  treated as a classical \nspin of fi xed length. The FiM alloys are composed of  two sublattices: rare -earth (RE) and \ntransition -metal (TM). Typical examples of these FiM alloys are GdCo, GdFe , GdFe Co, \nTbFeCo , etc. We consider  that the ordered TM alloy is represented by the fcc -type lattice , \nand t o simulate the amorphous character of t he TM -RE alloy, 𝑥𝑥𝑅𝑅𝑅𝑅⋅100% lattice sites are \nsubstituted randomly with RE  magnetic moments.  An example of a typical atomistic \narrangement is shown in Fig. S1  for a Gd x(FeCo) 1-x with 𝑥𝑥=𝑥𝑥𝑅𝑅𝑅𝑅=0.25 being the \nrelative composition of the sublattices  (𝑥𝑥=𝑥𝑥𝑅𝑅𝑅𝑅, and 𝑥𝑥 𝑇𝑇𝑇𝑇=1−𝑥𝑥𝑅𝑅𝑅𝑅). Each atom, either \nRE (Gd) or TM (FeCo), has its local atomistic magnetic moment 𝜇𝜇⃗𝑖𝑖 at site 𝑖𝑖 . \n \nFIG. S1. Atomistic scheme showing a FiM Gd x(FeCo) 1-x alloy consisting on two sublattices of \nRE (Gd, red) and TM (CoFe , grey) magnetic moments, with 𝑥𝑥 representing the relative \ncomposition of each sublattice. The sample dimensions along the Cartesian coordinate’s  \ndirections  are (ℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇) with 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇=5.6 nm. Atomistic simulations are restricted to samples \nat the na noscale (ℓ≈25 nm). The dimensions of an elementary computational cell ( Δ𝑥𝑥=Δ𝑦𝑦=\n3 nm) within the micromagnetic approach in shown at the bottom, along with the indication of \nthe lattice constant ( 𝑎𝑎=0.32 nm) for comparison.  \n \n3 \n The basis of ASD model  (see for instance  [1,2]  and references  therein ) is a classical \nspin Hamilt onian based on the Heisenberg  exchange formalism. The spin Hamiltonian ℋ \ntypically  has the form:  \n ℋ=ℋ𝑒𝑒𝑒𝑒𝑒𝑒+ℋ𝑎𝑎𝑎𝑎𝑖𝑖+ℋ𝑑𝑑𝑖𝑖𝑑𝑑+ℋ𝑎𝑎𝑑𝑑𝑑𝑑 (eS1) \nwith the terms on the RHS representing respectively exchange, anisotropy, dipolar and \nZeeman terms. The exchange energy for a system of interacting atomic moments is given by the expression  \n ℋ𝑒𝑒𝑒𝑒𝑒𝑒=−�𝐽𝐽𝑖𝑖𝑖𝑖𝑆𝑆⃗𝑖𝑖·𝑆𝑆⃗𝑖𝑖\n𝑖𝑖≠𝑖𝑖 (eS2)  \nwhere 𝐽𝐽𝑖𝑖𝑖𝑖 is the exchange interaction betwe en atomic sites 𝑖𝑖 and 𝑗𝑗, 𝑆𝑆⃗𝑖𝑖 is a unit vector \ndenoting the local spin moment direction ( 𝑆𝑆⃗𝑖𝑖=𝜇𝜇⃗𝑖𝑖/𝜇𝜇𝑠𝑠𝑖𝑖 with 𝜇𝜇𝑠𝑠𝑖𝑖=|𝜇𝜇⃗𝑠𝑠𝑖𝑖|) and 𝑆𝑆⃗𝑖𝑖 is the spin \nmoment direction of neighboring atoms. The sum in Eq. (eS2) is truncated to nearest neighbors only. As the FiM alloy is formed by two sublattices of magnetic moments, we \ncan split the exchange interaction ( 𝐽𝐽\n𝑖𝑖𝑖𝑖) between ferromagnetic intra- lattice  (𝐽𝐽𝑖𝑖𝑖𝑖>0) and \nantiferromagnetic inter -lattice (𝐽𝐽𝑖𝑖𝑖𝑖<0) exchange interactions. In what follows, we adopt \nthe notation of 𝐽𝐽 𝑇𝑇𝑇𝑇−𝑇𝑇𝑇𝑇>0, 𝐽𝐽𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅>0 for intra -lattice interactions, and 𝐽𝐽𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅<0 for \nthe inter -lattice interactions.  \nThe second term in Eq. (eS2) is the magnetic anisotropy. Here, a standard uniaxial \nanisotropy along the easy -axis (𝑧𝑧, out-of-plane  direction ) is considered ,  \n ℋ𝑎𝑎𝑎𝑎𝑖𝑖=−𝑑𝑑𝑢𝑢��𝑆𝑆�⃗𝑖𝑖·𝑢𝑢�⃗𝐾𝐾�2\n𝑖𝑖 (eS3) \nwhere 𝑑𝑑𝑢𝑢 is the anisotropy energy per atom and 𝑢𝑢�⃗𝐾𝐾=𝑢𝑢�⃗𝑧𝑧 is the  unit vector denoting the \npreferred direction . The last two terms in Eq. ( eS1) account for the dipolar  (ℋ𝑑𝑑𝑖𝑖𝑑𝑑) and \nexternal applied field (ℋ𝑎𝑎𝑑𝑑𝑑𝑑) contributions . Since the demagnetizing field is usually \nrelatively small in FiMs samples , this term is generally ignored  here. No external field is \napplied in the present work. \nThe equilibrium state of the FiM  sample can be obtained by minimiz ing of the total \nenergy. Here, it consists o f the two antiferromagnetic coupled sublattices aligned along \nthe easy -axis in anti -parallelly . Its dynamics response is governed by the  Langevin -\nLaudau -Lifshitz -Gilbert equation  for each atomistic moment ( 𝑆𝑆⃗𝑖𝑖)  [1,2] : \n𝜕𝜕𝑆𝑆⃗𝑖𝑖\n𝜕𝜕𝑡𝑡=−𝛾𝛾0\n(1+𝜆𝜆2)�𝑆𝑆⃗𝑖𝑖×�𝐻𝐻��⃗𝑖𝑖+𝐻𝐻��⃗𝑡𝑡ℎ,𝑖𝑖 �+𝜆𝜆𝑆𝑆⃗𝑖𝑖×�𝑆𝑆⃗𝑖𝑖×�𝐻𝐻��⃗𝑖𝑖+𝐻𝐻��⃗𝑡𝑡ℎ,𝑖𝑖 ��� (eS4) \nwhere 𝜆𝜆=0.02 is the Gilbert damping parameter  and 𝛾𝛾0=2.21×105 m/(A⋅s) the \ngyromagnetic ratio . 𝐻𝐻��⃗𝑖𝑖 is the local  effective magnetic field obtained from the spin \nHamiltonian  as \n𝜇𝜇0𝐻𝐻��⃗𝑖𝑖=−1\n𝜇𝜇𝑠𝑠𝑖𝑖𝜕𝜕ℋ\n𝜕𝜕𝑆𝑆⃗𝑖𝑖 (eS5) 4 \n and 𝐻𝐻��⃗𝑡𝑡ℎ,𝑖𝑖 is the stochastic thermal field is given by:  \n𝜇𝜇0𝐻𝐻��⃗𝑡𝑡ℎ,𝑖𝑖 =Γ⃗𝑖𝑖(𝑡𝑡)�2𝜆𝜆𝑘𝑘𝐵𝐵𝑇𝑇\n𝛾𝛾0𝜇𝜇𝑠𝑠𝑖𝑖Δ𝑡𝑡 (eS6) \nwith 𝑘𝑘𝐵𝐵 is the Boltzmann constant, 𝑇𝑇  the temperature, Δ 𝑡𝑡 the integration time step . Γ⃗𝑖𝑖(𝑡𝑡) \nis a local vector wh ose components a re Gaussian -distributed white -noise random \nnumbers  with zero mean value . \nUnder a laser pulse, the FiM  sample absorbs energy and its temperature changes in time. \nThe temperature 𝑇𝑇 that enters in Eq. (eS6) is the temperature of the electronic subsystem \n(𝑇𝑇≡𝑇𝑇𝑒𝑒), which is coupled to the lattice subsystem temperature ( 𝑇𝑇𝑙𝑙) and to the laser pulse \nas given b y the Two Temperature Model ( TTM ). Details of the power absorbed by sample \nas due to the laser pulse and the Eqs. (2)-(3) of the TTM were already discussed in the \nmain text. Conventional TTM values were adopted  [3 –5]: 𝐶𝐶𝑒𝑒=1.8×105 J/(m3K) at \n𝑇𝑇𝑅𝑅= 300  K, 𝐶𝐶𝑙𝑙=3.8×106 J/(m3K), 𝑘𝑘𝑒𝑒=91 W/(m K), 𝑔𝑔𝑒𝑒𝑙𝑙=7×1017 W/(m3K) \nand 𝜏𝜏𝐷𝐷=10 ps. \nEq. (eS4)  [or Eq. (1) in the main text] is numerically solved coupled to the T TM \nEqs. ((2) -(3) in the main text) with a home -made solver using a 4th-order Runge -Kutta  \nscheme with Δ𝑡𝑡=0.1 fs. Typical Gd 0.25(CoFe )0.75 parameters were considered   [2]: intra-\nlattice exchange energies:  intra-lattice exchange energies 𝐽𝐽 𝑇𝑇𝑇𝑇−𝑇𝑇𝑇𝑇=3.58×10−21 J, \n𝐽𝐽𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅=1.44×10−21 J, inter -lattice energy 𝐽𝐽𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅=−1.25×10−21 J, magnitude of \nthe local magnetic moments: 𝜇𝜇𝑇𝑇𝑇𝑇=1.92 𝜇𝜇𝐵𝐵, 𝜇𝜇𝑅𝑅𝑅𝑅=7.63 𝜇𝜇𝐵𝐵, and anisotropy energy of \n𝑑𝑑𝑢𝑢=8.07×10−24 J. The FiM sample dimensions for atomistic simulations are \nℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 (see FIG. S1) with ℓ ≃25 nm  and 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 =5.6 nm, and magnetic moment  \nof the two sublattices are randomly generated with 25% and 75% of RE:Gd and \nTM:(FeCo) respectively (FIG. S1). The lattice constant is 𝑎𝑎 =0.35 nm . TTM parameter \nThe power absorbed by the laser pulse is assumed to be uniform over the nano- size FiM  \nsample considered in the main text for atomistic simulations  (ASD) .  \n \nSN2. Micromagnetic LLB models: conventional (LLB) and extended (eLLB) cases  \nAs it was already stated, due to memory and computation limitations , atomistic \nsimulations are restricted to small samples at the nano -scale (ℓ≲100  nm ). However, \nexperimental studies on All Optical Switching are typical carried out in extended samples \nwith several micro meters  in length ( ℓ≃100  μm ). Also,  the size of conventional laser \nspot (FWHM) are usually at the micro -scale (𝑑𝑑0~1−10 μm). It is worthy  to provide an \nestimation  of the maximum size that c ould be numerically manage d in atomistic \nsimulations  (ASD) . For instance, in the present work we used one of the most power ful \nGraphics Processing Units (GPUs) , model RTX3090 (24GB) , and performing atomistic \nsimulations of FiM samples of ℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 with ℓ≃150  nm and 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇=5.6 nm \n(𝑎𝑎~0.35 nm ) will be already out of the computational power of such GPU. Therefore, in \norder to explain experimental observations another coarse- grained formalism is needed. \nHere, we adopt the mesoscopic description ( micromagnetic model)  which assumes that \nthe magnetization of each sublattice is a continuous vector field: 𝑚𝑚��⃗𝑖𝑖=𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡), where 5 \n here the sub- index 𝑖𝑖 refers to the local magnetic magnetization of each sublattice, \n𝑖𝑖:RE,TM. The FiM sample, with dimension ℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇, is discretized using a 2D finite \ndifferences scheme, using computational cells of volume Δ 𝑉𝑉=Δ𝑥𝑥2 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇. The continuous \ndescription ( Δ𝑥𝑥≫𝑎𝑎) is justified by the short- range of the exchange interaction, which \npromote s the ferromagnetic and the antiferromagnetic orders for intra - and inter -lattices \ninteractions respectively. At the same time, in ord er to resolve magnetic patterns such as \ndomain walls, the cell size Δ𝑥𝑥 must be smaller than the characteristic length scale along \nwhich the magnetization varies significantly (for instance, the so -called domain wall \nwidth , 𝛿𝛿≫Δ𝑥𝑥). Therefore, within the m icromagnetic approach, at each cell location ( 𝑟𝑟⃗) \nthere are two magnetic spin s representing the magnetization of the two sublattices of the \nFiM, 𝑚𝑚��⃗𝑖𝑖=𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)=𝑀𝑀��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)/𝑀𝑀𝑠𝑠𝑖𝑖(𝑇𝑇) where 𝑀𝑀��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡) is the local magnetization of \nsublattice 𝑖𝑖  in units of A/m, and 𝑀𝑀 𝑠𝑠𝑖𝑖(𝑇𝑇) is the corresponding spontaneous magnetization  \nat temperature 𝑇𝑇. FIG. S2 shows these ideas, along with different spatial scales of the \nASD and micromagnetic model.  \n \n \nFIG. S2 . (a) Arrange of atoms for atomistic simulations , limited to nanoscale samples, ℓ≈\n25 nm. (b) Micromagnetic discretization scheme for extended samples at the microscale, ℓ ≈\n20 μm. The FiM sample is discretized using a 2D finite differences scheme using computational  \ncells of volume 𝑉𝑉 =Δ𝑥𝑥2 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇. Each cell contains two micro -magnetic moments, one for each \nsublattice , and its size would include thousands of atomistic magnetic moments  (𝑆𝑆⃗𝑖𝑖). The \nmagnetization of  each sublattice i s a continuous  vector function (𝑚𝑚��⃗𝑖𝑖=𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)) over the FiM \nsample.  \n \nIn order to overcome the ASD limitations here we  adopt  an extended continuous \nmicromagnetic model that describes the temporal evolution of the reduced local \nmagnetization  𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡) of each sublattice 𝑖𝑖 :RE,TM based on the stochastic  Landau -\nLifshitz -Bloch (LLB) equation   [1,6]  \n \n6 \n 𝜕𝜕𝑚𝑚��⃗𝑖𝑖\n𝜕𝜕𝑡𝑡=−𝛾𝛾0𝑖𝑖′�𝑚𝑚��⃗𝑖𝑖×𝐻𝐻��⃗𝑖𝑖�+ \n−𝛾𝛾0𝑖𝑖′𝛼𝛼𝑖𝑖⊥\n𝑚𝑚𝑖𝑖2𝑚𝑚��⃗𝑖𝑖×�𝑚𝑚��⃗𝑖𝑖×�𝐻𝐻��⃗𝑖𝑖+𝜉𝜉⃗𝑖𝑖⊥��− \n+𝛾𝛾0𝑖𝑖′𝛼𝛼𝑖𝑖∥\n𝑚𝑚𝑖𝑖2(𝑚𝑚��⃗𝑖𝑖·𝐻𝐻��⃗𝑖𝑖)𝑚𝑚��⃗𝑖𝑖+𝜉𝜉⃗𝑖𝑖∥ (eS7) \nwhere 𝛾𝛾0𝑖𝑖′=𝛾𝛾0𝑖𝑖/(1 +𝜆𝜆𝑖𝑖2) is the reduced gyromagnetic ratio, which is defined via the \ncoupling parameter 𝜆𝜆𝑖𝑖 of sublattice 𝑖𝑖  to the heat bath . 𝛼𝛼𝑖𝑖∥ and 𝛼𝛼𝑖𝑖⊥ are the longitudinal and \nperpendicular damping parameters . 𝐻𝐻��⃗𝑖𝑖=𝐻𝐻��⃗𝑖𝑖(𝑟𝑟 ⃗,𝑡𝑡) is the local effective field at location 𝑟𝑟 ⃗ \nacting  on sublattice magnetic moment 𝑖𝑖, and 𝜉𝜉⃗𝑖𝑖∥ and 𝜉𝜉⃗𝑖𝑖⊥ are the longitudinal and \nperpendicular stochastic thermal fields.  \nFor 𝑇𝑇<𝑇𝑇𝐶𝐶, the damping constants for sublattice 𝑖𝑖 are \n𝛼𝛼𝑖𝑖∥= 2𝜆𝜆𝑖𝑖𝑘𝑘𝐵𝐵𝑇𝑇𝑚𝑚𝑒𝑒,𝑖𝑖\n𝐽𝐽0,𝑖𝑖𝑖𝑖𝑚𝑚𝑒𝑒,𝑖𝑖+�𝐽𝐽0,𝑖𝑖𝑖𝑖�𝑚𝑚𝑒𝑒,𝑖𝑖 (eS8) \n𝛼𝛼𝑖𝑖⊥=𝜆𝜆𝑖𝑖�1−𝑘𝑘𝐵𝐵𝑇𝑇𝑚𝑚𝑒𝑒,𝑖𝑖\n𝐽𝐽0,𝑖𝑖𝑖𝑖𝑚𝑚𝑒𝑒,𝑖𝑖+�𝐽𝐽0,𝑖𝑖𝑖𝑖�𝑚𝑚𝑒𝑒,𝑖𝑖� (eS9) \nwhere 𝑚𝑚𝑒𝑒,𝑖𝑖 and 𝑚𝑚𝑒𝑒,𝑖𝑖 are the equilibrium magnetization in sublattices 𝑇𝑇 𝑀𝑀 a nd 𝑅𝑅𝑅𝑅 \nrespectively . These values are obtained from the mean field approximation  (see \nRef.  [1,6]) . The exchange energies became 𝐽𝐽0,𝑖𝑖𝑖𝑖=𝑧𝑧𝑥𝑥𝑖𝑖𝐽𝐽𝑖𝑖−𝑖𝑖, 𝐽𝐽0,𝑖𝑖𝑖𝑖=𝑧𝑧𝑥𝑥𝑖𝑖𝐽𝐽𝑖𝑖−𝑖𝑖, with 𝑧𝑧  the \nnumber of nearest neighbors and 𝐽𝐽𝑖𝑖−𝑖𝑖 and 𝐽𝐽𝑖𝑖−𝑖𝑖 the atomistic exchange energy between \nspecies. 𝑥𝑥𝑖𝑖 is the concentration  of sublattice 𝑖𝑖 . Above the Curie temperature,  𝑇𝑇>𝑇𝑇𝐶𝐶, the \ndamping parameters are  \n𝛼𝛼𝑖𝑖∥=𝛼𝛼𝑖𝑖⊥=2𝜆𝜆𝑖𝑖𝑇𝑇\n3𝑇𝑇𝐶𝐶 (eS10 ) \nThe effective field consists on several contributions: \n𝐻𝐻��⃗𝑖𝑖=𝐻𝐻��⃗𝑎𝑎𝑎𝑎𝑖𝑖,𝑖𝑖+𝐻𝐻��⃗𝑒𝑒𝑒𝑒,𝑖𝑖+𝐻𝐻��⃗𝑖𝑖∥ (eS11 ) \nwhere 𝐻𝐻��⃗𝑎𝑎𝑎𝑎𝑖𝑖,𝑖𝑖 and 𝐻𝐻��⃗𝑒𝑒𝑒𝑒,𝑖𝑖 are the anisotropy and exchange fields given by  \n𝐻𝐻��⃗𝑎𝑎𝑎𝑎𝑖𝑖,𝑖𝑖=2𝐾𝐾𝑖𝑖\n𝜇𝜇0𝜇𝜇𝑖𝑖𝑚𝑚��⃗𝑖𝑖·𝑢𝑢�⃗𝐾𝐾 (eS12 ) \n𝐻𝐻��⃗𝑒𝑒𝑒𝑒,𝑖𝑖=2𝐴𝐴𝑒𝑒𝑒𝑒,𝑖𝑖\n𝜇𝜇0𝑀𝑀𝑠𝑠𝑖𝑖(∇2𝑚𝑚��⃗𝑖𝑖)−𝐽𝐽0,𝑖𝑖𝑖𝑖\n𝜇𝜇0𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖2�𝑚𝑚��⃗𝑖𝑖× (𝑚𝑚��⃗𝑖𝑖×𝑚𝑚��⃗𝑖𝑖)� (eS13 ) \nwhere the first term in 𝐻𝐻��⃗𝑒𝑒𝑒𝑒,𝑖𝑖 is the continuous exchange between neighbors’ cells, and the \nsecond one accounts for the inter -lattice contribution. Finally, the term 𝐻𝐻��⃗𝑖𝑖∥ is a f ield \ncomputed differently below 𝑇𝑇 𝐶𝐶, \n𝐻𝐻��⃗𝑖𝑖∥(𝑇𝑇<𝑇𝑇𝐶𝐶) =−1\n𝜇𝜇0�1\n𝜒𝜒 �𝑖𝑖∥+�𝐽𝐽0,𝑖𝑖𝑖𝑖�\n𝜇𝜇𝑖𝑖𝜒𝜒 �𝑖𝑖∥\n𝜒𝜒 �𝑖𝑖∥�𝛿𝛿𝑚𝑚𝑖𝑖\n𝑚𝑚𝑒𝑒,𝑖𝑖 𝑚𝑚��⃗𝑖𝑖+1\n𝜇𝜇0�𝐽𝐽0,𝑖𝑖𝑖𝑖�\n𝜇𝜇𝑖𝑖𝛿𝛿𝜏𝜏𝐵𝐵,𝑖𝑖\n𝑚𝑚𝑒𝑒,𝑖𝑖𝑚𝑚��⃗𝑖𝑖 (eS14 ) 7 \n and above 𝑇𝑇𝐶𝐶 \n𝐻𝐻��⃗𝑖𝑖∥(𝑇𝑇 ≥ 𝑇𝑇𝐶𝐶) =−1\n𝜇𝜇0�1\n𝜒𝜒 �𝑖𝑖∥+�𝐽𝐽0,𝑖𝑖𝑖𝑖�\n𝜇𝜇𝑖𝑖𝜒𝜒�𝑖𝑖∥\n𝜒𝜒 �𝑖𝑖∥�𝑚𝑚��⃗𝑖𝑖+1\n𝜇𝜇0�𝐽𝐽0,𝑖𝑖𝑖𝑖�\n𝜇𝜇𝑖𝑖𝜏𝜏𝐵𝐵,𝑖𝑖\n𝑚𝑚𝑖𝑖𝑚𝑚��⃗𝑖𝑖 (eS15 ) \nwith 𝜏𝜏𝐵𝐵,𝑖𝑖=�𝑚𝑚��⃗𝑖𝑖·𝑚𝑚��⃗𝑖𝑖�/𝑚𝑚𝑖𝑖, 𝛿𝛿𝑚𝑚𝑖𝑖=𝑚𝑚𝑖𝑖−𝑚𝑚𝑒𝑒,𝑖𝑖 and 𝛿𝛿𝜏𝜏𝐵𝐵,𝑖𝑖=𝜏𝜏𝐵𝐵,𝑖𝑖−𝜏𝜏𝑒𝑒,𝐵𝐵,𝑖𝑖. The longitudinal \nsusceptibilities  𝜒𝜒 �𝑖𝑖∥ are calculated from mean field approach  [1] . Below the Curie \ntemperature ( 𝑇𝑇<𝑇𝑇𝐶𝐶): \n𝜒𝜒 �𝑖𝑖∥(𝑇𝑇<𝑇𝑇𝐶𝐶)\n=𝜇𝜇𝑖𝑖ℒ𝑖𝑖′(𝜁𝜁𝑖𝑖)�𝐽𝐽0,𝑖𝑖𝑖𝑖�ℒ𝑖𝑖′�𝜁𝜁𝑖𝑖�+𝜇𝜇𝑖𝑖ℒ𝑖𝑖′(𝜁𝜁𝑖𝑖)�𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖ℒ𝑖𝑖′�𝜁𝜁𝑖𝑖��\n�𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖ℒ𝑖𝑖′(𝜁𝜁𝑖𝑖)��𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖ℒ𝑖𝑖′�𝜁𝜁𝑖𝑖��−�𝐽𝐽0,𝑖𝑖𝑖𝑖�ℒ𝑖𝑖′(𝜁𝜁𝑖𝑖)�𝐽𝐽0,𝑖𝑖𝑖𝑖�ℒ𝑖𝑖′�𝜁𝜁𝑖𝑖� (eS16 ) \nwhere ℒ𝑖𝑖 is the Langevin function with argument 𝜁𝜁𝑖𝑖=𝐽𝐽0,𝑖𝑖𝑖𝑖𝑚𝑚𝑖𝑖+�𝐽𝐽0,𝑖𝑖𝑖𝑖�𝑚𝑚𝑖𝑖\n𝑘𝑘𝐵𝐵𝑇𝑇 and ℒ𝑖𝑖′ is the \nderivative respect 𝜁𝜁𝑖𝑖. Above the Curie temperature ( 𝑇𝑇>𝑇𝑇𝐶𝐶):  [7] \n𝜒𝜒 �𝑖𝑖∥(𝑇𝑇 ≥ 𝑇𝑇𝐶𝐶) =𝜇𝜇𝑖𝑖�𝐽𝐽0,𝑖𝑖𝑖𝑖�+𝜇𝜇𝑖𝑖�3𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖�\n��3𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖��3𝑘𝑘𝐵𝐵𝑇𝑇−𝐽𝐽0,𝑖𝑖𝑖𝑖�−�𝐽𝐽0,𝑖𝑖𝑖𝑖��𝐽𝐽0,𝑖𝑖𝑖𝑖�� (eS17) \n \nStochastic fluctuations due to temperature are introduced through thermal fields 𝜉𝜉 ⃗𝑖𝑖∥ \nand 𝜉𝜉⃗𝑖𝑖⊥. These fields  are time and space uncorrelated, and they are  generated from white \nnoise random numbers with zero mean field and variance given by  [6]: \n〈𝜉𝜉𝑖𝑖,𝛼𝛼𝜂𝜂(𝑟𝑟⃗,𝑡𝑡)𝜉𝜉𝑖𝑖,𝛽𝛽𝜂𝜂(𝑟𝑟⃗′,𝑡𝑡′)〉=2𝐷𝐷𝑖𝑖𝜂𝜂𝛿𝛿𝛼𝛼𝛽𝛽𝛿𝛿(𝑟𝑟⃗−𝑟𝑟⃗′)𝛿𝛿(𝑡𝑡−𝑡𝑡′) (eS18) \nwhere 𝛼𝛼,𝛽𝛽 are the cartesian components of the stochastic thermal fields , 𝑖𝑖 denotes \nsublattice 𝑇𝑇\n𝑀𝑀 o\nr 𝑅𝑅𝑅𝑅 and 𝜂𝜂:∥,⊥ represents parallel or perpendicular field components. \nThe diffusions constants are given by: \n𝐷𝐷𝑖𝑖⊥=�𝛼𝛼𝑖𝑖⊥−𝛼𝛼𝑖𝑖∥�𝑎𝑎3𝑘𝑘𝐵𝐵𝑇𝑇\n(𝛼𝛼𝑖𝑖⊥)2𝛾𝛾0𝑖𝑖′𝜇𝜇0𝑛𝑛𝑎𝑎𝑡𝑡𝑥𝑥𝑖𝑖𝜇𝜇𝑖𝑖Δ𝑉𝑉 (eS19) \n𝐷𝐷𝑖𝑖∥=𝛼𝛼𝑖𝑖∥𝛾𝛾0𝑖𝑖′𝑎𝑎3𝑘𝑘𝐵𝐵𝑇𝑇\n𝑛𝑛𝑎𝑎𝑡𝑡𝑥𝑥𝑖𝑖𝜇𝜇0𝜇𝜇𝑖𝑖Δ𝑉𝑉 (eS20) \nwith Δ𝑉𝑉=Δ𝑥𝑥2𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 the discretization volume, 𝑎𝑎 is the lattice constant and 𝑛𝑛𝑎𝑎𝑡𝑡 is the \nnumber of atoms per unit cell.  \n \nA\ns it will be shown in Supplemental Note N 3(b), we have confirmed that the \nconventional LLB Eq. (eS7) (or Eq. (4) in the main text)  [1,6,8] is not able to fully \nreproduce the atomistic model predictions of some all optical switching processes in FiM  \nsystems. Indeed, atomistic simulations show that the sudden change in the temperature \ndue to the laser pulse results in transient non-equilibrium states  where  a transfer of angular \nmomentum between sublattices  takes place. This transference is not fully reproduced by \nthe conventional LLB Eq. (eS7). In order to overcome this limitation, in the present work 8 \n we exten d the LLB E q. (eS7) by add ing an additional term  to the RHS  that account s of \nthe angular momentum transfer between lattices in non -equilibrium states . This non-\nequilibrium torque is  given  \n𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅=𝛾𝛾0𝑖𝑖′𝜆𝜆𝑒𝑒𝑒𝑒𝛼𝛼𝑖𝑖∥𝑥𝑥𝑖𝑖𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖+𝑥𝑥𝑖𝑖𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖\n𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖𝜇𝜇𝑖𝑖𝑚𝑚𝑖𝑖�𝜇𝜇𝑖𝑖𝐻𝐻��⃗𝑖𝑖∥−𝜇𝜇𝑖𝑖𝐻𝐻��⃗𝑖𝑖∥� (eS21) \nwhere and 𝐻𝐻��⃗𝑖𝑖∥ and 𝐻𝐻��⃗𝑖𝑖∥ are the longitudinal effective fields for each lattice 𝑖𝑖 :RE,TM  [6], \n𝑥𝑥𝑖𝑖≡𝑥𝑥 and 𝑥𝑥𝑖𝑖=1−𝑥𝑥𝑖𝑖≡1−𝑥𝑥 are the concentrations of each specimen, and  𝜆𝜆𝑒𝑒𝑒𝑒 is a \nparameter representing the exchange relaxation rate  [9]. By including Eq. (eS21) in the \nRHS of Eq. (eS7), and numerically solving it coupled to TTM Eqs. (2) -(3), we can provide \na realistic description of the magnetization dynamics in FiM systems  under ultra -short \nlaser pulses. In what follows, and in order to distinguish it from the conventional  LLB \nmodel  (LLB, 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅=0), we refer to this formalism as the extende d micromagnetic LLB \nmodel  (eLLB, 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅≠0), Note that the  additional torque has not been previously included \nin the LLB model, and therefore, here we name it  here as the extended LLB model  \n(eLLB) . A comparison between atomistic (ASD) predictions and the results without  \n(conventional LLB)  and with (extended LLB, eLLB)  the additional t orque 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅 is shown \nin FIG. S4, discussed in the next section.  \nIn the present work , FiM samples of ℓ×ℓ×𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 are discretized using a 2D finite \ndifferences scheme, using computational cells of volume Δ 𝑉𝑉=Δ𝑥𝑥2 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇. Typical \nmaterial parameters for Gd 0.25(FeCo) 0.75 were adopted:  𝐽𝐽0,𝑇𝑇𝑇𝑇−𝑇𝑇𝑇𝑇=2.59×10−21 J, \n 𝐽𝐽0,𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅=1.35×10−21 J, 𝐽𝐽0,𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅=−1.13×10−21 J. 𝜆𝜆𝑇𝑇𝑇𝑇=𝜆𝜆𝑅𝑅𝑅𝑅=0.02. \nMicromagnetic parameters needed to solve the LLB Eq.  can be obtained from atomistic \ninputs as follows   [2]: the spontaneous magnetization at zero temperature of each \nsublattice is 𝑀𝑀 𝑠𝑠𝑖𝑖(0)=𝜇𝜇𝑖𝑖𝑥𝑥𝑖𝑖/(𝑝𝑝𝑓𝑓⋅𝑎𝑎3), where 𝑝𝑝𝑓𝑓 is a packing factor which depends on the \ncrystalline structure ( 𝑝𝑝𝑓𝑓=0.74 for fcc,  [2]): 𝑀𝑀𝑆𝑆,𝑇𝑇𝑇𝑇(0)=0.41 MA /m, 𝑀𝑀𝑆𝑆,𝑅𝑅𝑅𝑅(0)=\n0.55 MA /m. The continuous exchange stiffness constant   [10]  is 𝐴𝐴𝑒𝑒𝑒𝑒,𝑖𝑖=𝑛𝑛𝑛𝑛𝑥𝑥𝑖𝑖2𝐽𝐽𝑖𝑖/(2𝑎𝑎)  \nwith 𝑛𝑛=2 being the number of atoms pe r unit cell for fcc, 𝑛𝑛=0.79 the spin wave mean \nfield correction value also for FCC, and 𝑥𝑥𝑖𝑖 the concentration of sublattice 𝑖𝑖  (𝑥𝑥𝑖𝑖=1−𝑥𝑥𝑖𝑖): \n𝐴𝐴𝑒𝑒𝑒𝑒,𝑇𝑇𝑇𝑇=3.2 pJ/m, 𝐴𝐴𝑒𝑒𝑒𝑒,𝑅𝑅𝑅𝑅=0.19 pJ /m. The perpendicular anisotropy parameter is \n𝐾𝐾𝑖𝑖=𝑑𝑑𝑢𝑢𝑥𝑥𝑖𝑖/(𝑝𝑝𝑓𝑓⋅𝑎𝑎3), see  [2] , so 𝐾𝐾𝑢𝑢,𝑇𝑇𝑇𝑇=1.87 MJ /m3, 𝐾𝐾𝑢𝑢,𝑅𝑅𝑅𝑅=0.62 MJ /m3. For the \npresented results we take 𝜆𝜆𝑒𝑒𝑒𝑒=0.013 for the exchange relaxation rate. Within the \nmicromagnetic  model, the  laser -induced magnetization dynamics is evaluated by \nnumerically solving Eq. (eS7) coupled to the TTM Eqs. (2) -(3) using the Heun algorithm \nwith Δ𝑡𝑡=1 fs with cell sizes of Δ𝑥𝑥=3 nm. This was done  by implementing the models \nin a home -made CUDA -based code, which was run a RTX3090 GPU . It was also checked \nin several tests that decreasing the cell size to Δ 𝑥𝑥=1 nm does not change the results. \n \nSN3. Comparison between atomistic, conventional LLB and extended LLB models  \n(a) In order to validate the extended LLB model  (eLLB, 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅≠0), we have firstly \ncompared the predictions of the conventional  LLB model (LLB , 𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅=0) with the \natomistic results  (ASD)  by computing for the equilibrium magnetization of each 9 \n sublattice ( 𝑖𝑖:RE,TM ) as a function of the temperature of the thermal bath (w hich coincides \nwith the electronic temperature). This study was carried out in a small FiM  sample  with \nℓ≈25 nm, and the results are shown in F IG. S3. Both ASD and eLLB models lead to \nsimilar results . \n \nFIG. S3. Temperature dependence of the reduced equilibrium magnetization  (𝑚𝑚) of the two \nsublattices . Solid (dashed ) lines corresponds to ASD ( eLLB) results. Blue and red color \ncorrespond to the RE and TM sublattices respectively.  The inset shows the corresponding \ndependence of the spontaneous magnetiz ation value of each sublattice ( 𝑀𝑀𝑠𝑠𝑖𝑖(𝑇𝑇) vs 𝑇𝑇). \n \n(b) In the main text and also in S N2, we claimed that the conventional LLB mode l \n(LLB , 𝜏𝜏⃗𝑁𝑁𝑅𝑅,𝑖𝑖=0) is not able to fully reproduce the atomistic results  (ASD)  for some all -\noptical switching processes in FiM samples. On the other hand, when such term is \nconsidered ( 𝜏𝜏⃗𝑁𝑁𝑅𝑅,𝑖𝑖≠0), the extended LLB model (eLLB) naturally reproduces the \natomistic temporal variation of the magnetization in small FiM samples under ultra -short \nlaser pulses. This is shown in F IG. S4 for the same study as in Fig. 1(b) of the main text.  \nContrary to the ASD and the eLLB models, the conventional LLB (𝜏𝜏⃗𝑁𝑁𝑅𝑅,𝑖𝑖=0) does not \npredicts switching for 𝑄𝑄=12×1021 W/m3 pulses ( Fig. S4(c) -(d)). Moreover, as shown \nin FIG. S4(a) -(b), t he extended LLB model  (eLLB) , which includes 𝜏𝜏⃗𝑁𝑁𝑅𝑅,𝑖𝑖≠0, repro duces \nsimilar results  even in the absence of therma l fluctuations ( 𝜉𝜉⃗𝑖𝑖∥=𝜉𝜉⃗𝑖𝑖⊥=0). \n10 \n  \nFIG. S4. Temporal evolution  of the out -of-plane component  (𝑚𝑚𝑧𝑧 vs 𝑡𝑡) calculated by three different \nmodels : ASD, conventional LLB and extended eLLB models for the case studied in Fig. 1(b) of \nthe main text. Solid lines in all  graphs correspond to the ASD results. Red and blue c urves \ncorrespond to the TM and RE respectively . In top graphs, dashed lines correspo nd to extended \neLLB results  (𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅≠0): (a) including thermal fluctuations 𝜉𝜉⃗𝑖𝑖∥,𝜉𝜉⃗𝑖𝑖⊥≠0, and (b) without thermal \nfluctuations ( 𝜉𝜉⃗𝑖𝑖∥=𝜉𝜉⃗𝑖𝑖⊥=0, deterministic case) . In bottom  graphs, dashed lines correspond to LLB \nresults (𝜏𝜏⃗𝑖𝑖𝑁𝑁𝑅𝑅=0): (a) including thermal fluctuations 𝜉𝜉⃗𝑖𝑖∥,𝜉𝜉⃗𝑖𝑖⊥≠0, and (b) without thermal \nfluctuations ( 𝜉𝜉⃗𝑖𝑖∥=𝜉𝜉⃗𝑖𝑖⊥=0, deterministic case) . \n \n(c) Except the contrary is indicated, all presented results with the extended eLLB \nmodel were computed with 𝜆𝜆𝑒𝑒𝑒𝑒=0.013. FIG. S5 show the comparison of ASD results \nto eLLB  results for three different values of 𝜆𝜆𝑒𝑒𝑒𝑒. A good quanti tative agreement  in the \ntime traces of the out -of-plane components of each sublattice  predicted by the ASD model \nis achieved time when 𝜆𝜆𝑒𝑒𝑒𝑒=0.013 is considered  (central graph in FIG. S5) . For a laser \npulse of 𝜏𝜏 𝐿𝐿=50 fs, and using 𝜆𝜆 𝑒𝑒𝑒𝑒=0.013, the threshold laser power density to achieve \nthe switching is 𝑄𝑄=7.9×1021 W/m3 within the ASD, whereas the threshold is 𝑄𝑄=\n9.9×1021 W/m3 with the eLLB model. However, if 𝜆𝜆𝑒𝑒𝑒𝑒=0.024 both models give the \nsame threshold  (𝑄𝑄=9.9×1021 W/m3). \n \n11 \n  \nFIG. S5 . Temporal evolution of the out -of-plane component ( 𝑚𝑚𝑧𝑧 vs 𝑡𝑡) calculated with  three \ndifferent  values of 𝜆𝜆𝑒𝑒𝑒𝑒 as indicated within each graph.  Here 𝑄𝑄 =10×10−21 W/m3. The rest of \ninputs are the same as in  FIG. S4 .  \n \nSN4. Phase diagram in terms of the f luence and the pulse duration  \nFIG. 2(a) of the main text presents the phase diagram of the three possible final \nstates as function of the pulse length ( 𝜏𝜏𝐿𝐿) and the absorbed energy from the laser pulse \n(𝑄𝑄). Same results can be also presented in terms of the absorbed fluence, which is given \nby 𝐹𝐹=𝑄𝑄𝜏𝜏𝐿𝐿𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇, where 𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 is the thickness of the ferrimagnetic alloy. This is shown in \nFIG. S 6 for an extended range of laser pulse  durations  (𝜏𝜏𝐿𝐿). These results are also in \nagreement with recent experimental studies   [11]  \n \n12 \n  \nFIG. S 6. Same r esults as FIG. 2(a) of the main text showing the three possible final states as a \nfunction of the fluence, 𝐹𝐹=𝑄𝑄𝜏𝜏𝐿𝐿𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 for an extended range of pulse durations ( 𝜏𝜏𝐿𝐿). White, red \nand blue colors represent no -switching, deterministic switching  (HI-AOS)  and thermal \ndemagn etization behaviors respectively . \n \nSN5. Helicity -Dependent AOS (HD -AOS) and Magnetic Circular Dichroism (MCD)  \nConsidering the Magnetic Circular Dichroism (MCD)  scenario , the absorbed power \nby the FiM  under circular polarized laser pulses , 𝑃𝑃(𝑟𝑟,𝑡𝑡), also depends on the magnetic \nstate of the system  and the helicity of the pulse . Therefore , 𝑃𝑃(𝑟𝑟,𝑡𝑡) in Eq. (2) of the TTM \nis replaced by  𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)𝑃𝑃(𝑟𝑟,𝑡𝑡), where 𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁) describes  the different absorption  \npower for up (𝑚𝑚𝑁𝑁> 0,↑) and down (𝑚𝑚𝑁𝑁< 0,↓) magnetization states as depending on \nthe laser helicity  (𝜎𝜎±) of the laser pulse, \n𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)=�1 +1\n2𝑘𝑘𝜎𝜎±sign(𝑚𝑚𝑁𝑁)� (eS22) \nwhere 𝑚𝑚𝑁𝑁 is the local net out of plane magnetization, ( 𝑚𝑚𝑁𝑁=𝑀𝑀𝑠𝑠𝑇𝑇𝑇𝑇𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇+𝑀𝑀𝑆𝑆𝑅𝑅𝑅𝑅𝑚𝑚𝑧𝑧𝑅𝑅𝑅𝑅), \n𝜎𝜎±= ±1  is the helicity  of the laser pulse ( 𝜎𝜎+= +1  for right -handed helicity and 𝜎𝜎−=\n−1 left-handed helicity) , and  sign (𝑚𝑚𝑁𝑁) is the sign of the initial net magnetization.  Note that \n𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇= ±1  corresponds to 𝑚𝑚𝑧𝑧𝑅𝑅𝑅𝑅=∓1, but 𝑀𝑀𝑠𝑠𝑇𝑇𝑇𝑇 and 𝑀𝑀 𝑠𝑠𝑅𝑅𝑅𝑅 are both positive . 𝑘𝑘 is a f actor  \nthat determines the difference in the power absorption with respect to the linearly \npolarized case ( 𝑘𝑘= 0). According to this criterium, a state with local net magnetization \n𝑚𝑚𝑁𝑁> 0 (↑: up) absorb s more energy for a right -handed laser 𝜎𝜎+ and less for left -handed \nhelicity (𝜎𝜎−) than in the linearly polarization case  (𝜎𝜎= 0). The opposite happens starting \nfrom  a state with local down net magnetization 𝑚𝑚𝑁𝑁< 0 (↓: down).  Considering \n𝜓𝜓(𝜎𝜎±,𝑚𝑚𝑁𝑁)𝑃𝑃(𝑟𝑟,𝑡𝑡) in the TTM Eqs (2) -(3) of the main text, we obtain isothermal curves  \nof 𝑇𝑇=𝑇𝑇𝑒𝑒=1000  K showing the transition from no-s witching to helicity -dependent \nswitching as a f unction of 𝑄𝑄  and 𝜏𝜏𝐿𝐿 for different combinations of the laser helicity and \nthe initial magnetic state under uniform illumination.  \nThe results are shown  in FIG. S 7(a) and (b) for 𝑘𝑘= 0.1  and 𝑘𝑘= 0. 02 respectively , \nwhich correspond to differences in the power absorption of 10% and 2% between up  and \ndown states . Solid black line corresponds to the  case of linear polarization ( 𝜎𝜎= 0) already \n13 \n plotted in Fig. 2(a) of the main text. Cases with (𝜎𝜎+,↑) and (𝜎𝜎−,↓) are represented by the \nsolid -blue curve, whereas solid -red curve corresponds to cases with (𝜎𝜎+,↓) and (𝜎𝜎−,↑). \nAs it can be observed  for 𝑘𝑘=0.1 (FIG. S7(a)), the isothermal curves  𝑇𝑇=𝑇𝑇𝑒𝑒=1000  K \nfor combinations (𝜎𝜎+,↑) and (𝜎𝜎−,↓) slightly reduce the values of 𝑄𝑄  and 𝜏𝜏𝐿𝐿 needed to \nachieve switching with respect to the linear polarization case ( 𝜎𝜎=0). On the contrary, \nthe isothermal curve for combinations (𝜎𝜎+,↓) and (𝜎𝜎−,↑) slightly moves towards high \nvalues of 𝑄𝑄  and 𝜏𝜏𝐿𝐿 with respect to linear polarization case ( 𝜎𝜎=0). If 𝑘𝑘 =0.02 (FIG. \nS7(b)), the differences are even smaller, and the three curves almost overlap . This fact  \nagrees  with experimental observation s, where the Helicity -Dependent AOS is only \nachieved  in a narrow range of absorbed power ( 𝑄𝑄) for a fixed pulse length ( 𝜏𝜏𝐿𝐿). Dashed \nlines in FIG. S 7(a)-(b) correspond to the isothermal curves of 𝑇𝑇 =𝑇𝑇𝑒𝑒=1400  K, and \nindicate the transition between s witching  and demagnetized m ultidomain behaviors . \n \nFIG. S 7. Phase diagram in the MCD scenario for different values of the  MCD  coefficient 𝑘𝑘. (a) \nand ( b) show the isothermal curves of 𝑇𝑇=𝑇𝑇𝑒𝑒=1000  K (solid lines) as a  function of 𝑄𝑄 and 𝜏𝜏𝐿𝐿 \nindicating  the transition between no- switching to switching , as computed from the TTM  Eq. (2) -\n(3) for 𝑘𝑘 =0.1, and 𝑘𝑘=0.02 respectively . Dashed curves are isothermal curves indicating the \ntransition between the switching  to m ultidom ain regimes ( 𝑇𝑇𝑒𝑒=1400  K). Black curve  \ncorresponds to the case of linear  polarization  (𝜎𝜎=0) already plotted in Fig. 2(a) of the main text. \nCases with (𝜎𝜎+,↑) and (𝜎𝜎−,↓) are represented by the blue curve, whereas red curve corresponds \nto cases with (𝜎𝜎+,↓) and (𝜎𝜎−,↑). Here ↑ and ↓ represent the initial  net magnetic state (𝑚𝑚𝑁𝑁), either \nup or down  respectively.   \n \nSN6. Helicity -Dependent AOS (HD -AOS) and Inverse Fadaray Effect  (IFE)  \nAs mentioned in the main text, several works claim that the observations of the HD -\nAOS can be ascribed to the Inverse Faraday Effect (IFE)  [12] . Within this formalism, \nthe laser pulse generates an effective out -of-plane magneto -optical field whose  direct ion \ndepends on the laser pulse helicity  as 𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)=𝜎𝜎±𝐵𝐵𝑇𝑇𝑀𝑀𝜂𝜂(𝑟𝑟)𝑛𝑛(𝑡𝑡)𝑢𝑢�⃗𝑧𝑧, where 𝜂𝜂(𝑟𝑟)=\nexp[−4ln(2)𝑟𝑟2/ (2𝑟𝑟0)2 ]  is the spatial field profile, and 𝑛𝑛(𝑡𝑡) is its temporal profile. \nNote that there the spatial dependence of 𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) is the same as the one of the absorbed \npower density. A ccording to the literature  [12] , the so- called magneto -optical field \n𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) has some persistence with respect to the laser pulse, and therefore its temporal \nprofile is different for 𝑡𝑡<𝑡𝑡0 and 𝑡𝑡>𝑡𝑡0: 𝑛𝑛(𝑡𝑡<𝑡𝑡0)=exp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/ 𝜏𝜏𝐿𝐿2], and \n14 \n 𝑛𝑛(𝑡𝑡≥𝑡𝑡0)=exp[−4ln(2)(𝑡𝑡−𝑡𝑡0)2/(𝜏𝜏𝐿𝐿+𝜏𝜏𝐷𝐷)2], where 𝜏𝜏𝐷𝐷 is the delay time of the \n𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) with respect to the laser pulse. We have evaluated this scenario by including \nthis field 𝐵𝐵 �⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡) in the effective field of Eq. (4) . The results for the same FiM alloy \nconsidered up to here ( Gdx(FeCo) 1-x, with x =0.25), (see Supplemental Note SN5 ), are \nshown in FIG. S 8 by considering a maximum magneto -optical field of 𝐵𝐵𝑇𝑇𝑀𝑀=20 T  with \na delay time of 𝜏𝜏𝐷𝐷=𝜏𝜏𝐿𝐿. \n \n \nFIG. S8. Micromagnetic results of the HD -AOS computed within the IFE scenario.  (a) Snapshots \nof the final RE magnetic state ( 𝑚𝑚𝑧𝑧𝑅𝑅𝑅𝑅) after a laser pulse of 𝜏𝜏𝐿𝐿=50 fs for four different values of \nthe absorbed power density ( 𝑄𝑄). Results are shown for four combinations of the initial state ( ↑,↓) \nand helicities ( 𝜎𝜎±) as indicated at the left side. (b) RE magnetic state after every pulse with 𝜎𝜎+ \nfor 𝑄𝑄=5.9×1021 W/m3, showing the appearance of a ring around the central part. The sample \nside is ℓ=20 μm and the laser spot diameter is 𝑑𝑑0=ℓ/2. The IFE was evaluated by considering \na maximum magneto -optical field of 𝐵𝐵 𝑇𝑇𝑀𝑀=20 T with a delay time of 𝜏𝜏𝐷𝐷=𝜏𝜏𝐿𝐿. The material \nparameters correspond to a FiM alloy Gd x(CoFe) 1-x with x =0.25 . \n \nSN7. Inverse Faraday Effect: magneto- optical  field or induced  magnetic moment  \nIn previous section SN6 and in the main text, the Inverse Faraday Effect (IFE) was \nconsidered  by adding an effective out -of-plane magneto -optical field  (𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)) whose \ndirection depends on the laser pulse helicity . Other authors   [13,14]  have alternatively \npointed out  that under circularly polarized laser pulses , the IFE c an be taken into account \nin micromagnetic simulations by adding a helicity -dependent  induced magnetic moment \n15 \n on each sublattice ( Δ𝑚𝑚��⃗𝑖𝑖). We have also explored this alternative by adding, instead of the \n𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡), an additional term  to the right -hand side of the eLLB Eq. ( eS7). This term can \nbe expressed as 𝛾𝛾0′ Δ𝑚𝑚��⃗𝑖𝑖, where Δ𝑚𝑚��⃗𝑖𝑖 represents the laser induced magnetization for \nsublattice 𝑖𝑖 :RE,TM, and i t is given by  \nΔ𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)=(𝜎𝜎±)𝐾𝐾𝐼𝐼𝐹𝐹𝑅𝑅,𝑖𝑖𝐼𝐼\n𝑐𝑐𝑢𝑢�⃗𝑧𝑧 (eS23)  \nwhere 𝐾𝐾𝐼𝐼𝐹𝐹𝑅𝑅,𝑖𝑖 is the IFE constant  for sublattice 𝑖𝑖 :RE,TM, 𝐼𝐼=𝑃𝑃(𝑟𝑟,𝑡𝑡)𝑡𝑡𝐹𝐹𝑖𝑖𝑇𝑇 is the laser \nintensity  with 𝑃𝑃(𝑟𝑟,𝑡𝑡)=𝑄𝑄𝜂𝜂(𝑟𝑟)𝜉𝜉(𝑡𝑡) (see main text) , and 𝑐𝑐 the speed of light. In FIG. S9 \nwe compare both options to account for the IFE: either by a magneto- optical field \n(𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)≠0 with Δ𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)=0, solid lines  in FIG. S9)  as described in SN6, or by an \ninduced magnetic moment ( 𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)=0 with Δ𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)≠0, dashed lines in F IG. S9). \nAs it can be observed, both alternatives provide very similar  results, so we conclude , as \nin  [14] , that both alternatives are essentially equivalent within the scope of our numerical \nstudy . Moreover, the  adopted value  of 𝜎𝜎±𝐾𝐾𝐼𝐼𝐹𝐹𝑅𝑅,𝑖𝑖=±0.0014  T−1 (see FIG. S9) is also in \ngood quantitative agreement with typical values deduced  in  [14]  from an ab initio  \nformalism. \n \nFIG. S9 . Comparison of two micromagnetic implementations of the IFE  for both laser helicities: \n(a) 𝜎𝜎+=+1 and (b) 𝜎𝜎−=−1. Solid lines correspond to results obtained assuming that the IFE \ngenerates a magneto -optical field ( 𝐵𝐵�⃗𝑇𝑇𝑀𝑀(𝑟𝑟⃗,𝑡𝑡)≠0), whereas dashed lines are for the induced \nmagnetic moment alternative ( Δ𝑚𝑚��⃗𝑖𝑖(𝑟𝑟⃗,𝑡𝑡)≠0). The values of the maximum magneto -optical field \nis 𝐵𝐵𝑇𝑇𝑀𝑀=20 T, wher eas the IFE constant  is 𝜎𝜎±𝐾𝐾𝐼𝐼𝐹𝐹𝑅𝑅 ,𝑖𝑖=±0.0014  T−1. The  rest of inputs are the \nsame as in Fig. S8.  \n \nSN8. Material inputs for two different compositions  \nIn the las t part of the manuscript we studied the probability of switching ( see FIG. \n6) for two FiMs ( Gdx(CoFe) 1-x:RE x(TM) 1-x) with two different compositions : x=0.25 and \nx=0.24. The inputs used in these simulations are collected in the following Table S 1. \n \n x=0.25: Gd x(CoFe) 1-x x=0.24: Gd x(CoFe) 1-x \n𝑀𝑀𝑠𝑠,𝑇𝑇𝑇𝑇  / 𝑀𝑀𝑠𝑠,𝑅𝑅𝑅𝑅 (MA/m) 0.412 / 0.546  0.412 / 0.52  \n𝐴𝐴𝑒𝑒𝑒𝑒,𝑇𝑇𝑇𝑇  / 𝐴𝐴𝑒𝑒𝑒𝑒,𝑅𝑅𝑅𝑅 (pJ/m) 3.27 / 0.189  3.35 / 0 .174 \n𝐽𝐽𝑇𝑇𝑇𝑇−𝑇𝑇𝑇𝑇  / 𝐽𝐽𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅 (×10−21J) 2.59 / 1.35  2.59 / 1.35  \n16 \n 𝐽𝐽𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅  (×1021J)  -1.12  \n𝐾𝐾𝑢𝑢,𝑇𝑇𝑇𝑇  / 𝐾𝐾𝑢𝑢,𝑅𝑅𝑅𝑅 (MJ/m3) 1.87 / 0.62  1.89 / 0.59  \n𝜇𝜇𝑇𝑇𝑇𝑇  / 𝜇𝜇𝑅𝑅𝑅𝑅 (𝜇𝜇𝐵𝐵) 1.92 / 7.63  1.92 / 7.63  \n𝛼𝛼𝑇𝑇𝑇𝑇  / 𝛼𝛼𝑅𝑅𝑅𝑅 (  ) 0.02 / 0.02  0.02 / 0.02  \n𝛾𝛾𝑇𝑇𝑇𝑇  / 𝛾𝛾𝑅𝑅𝑅𝑅 (1011×(T⋅s)−1) 1.847 / 1.759  1.847 / 1.759  \n𝑔𝑔𝑇𝑇𝑇𝑇  / 𝑔𝑔𝑅𝑅𝑅𝑅 (  ) 2.1 / 2.0  2.1 / 2.0  \n𝑎𝑎 (nm)   0.352  \n   \n𝑘𝑘𝑒𝑒 (W/ (K⋅m))   91 \n𝐶𝐶𝑒𝑒(300  K) (×105J/ (K⋅m3))  1.8 \n𝐶𝐶𝑙𝑙 (×106J/ (K⋅m3))  3.8 \n𝑔𝑔𝑒𝑒𝑙𝑙(300  K) (×1017 W/m3)  7 \n \nTABLE  S1. Material inputs adopted to explore the switching probability for two different FiM \nalloys, with different composition s x. These inputs were used to get the results of FIG. 6 in the \nmain text.  \n \nSN9. Helicity -Dependent All Optical Switching : MCD & IFE for different \ncompos itions and initial temperatures  \nIn FIG. 6 of the main text we explore the switching probability predicted by both \nthe MCD and IFE mechanism s for two different compositions of the FiM and considering \nthat the initial temperature of the thermal bath was room temperature. Similar results can \nbe also obtained by fixing the compos ition of the FiM alloy and chan ging the temperature \nof the thermal bath with a cryostat. These results are shown in FIG. S10 for with x=0.25 \n(left column) and with x =0.24 (right column) compositions and three different \ntemperatures of the thermal bath: 𝑇𝑇 =260  K, 𝑇𝑇=300  K, and 𝑇𝑇 =340  K. For x=0.25, \nboth the IFE and MCD predict similar behavior for 𝑇𝑇 =260  K (FIG. S10(b)) and 𝑇𝑇 =\n300  K (FIG. S10(c) ): respect to the linear  polarized laser pulse, the 100% switching \nprobability occurs with smaller  Q for circular polarization when (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇↑) and \n(𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇↓), and with larger Q  when (𝜎𝜎+,𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇↑) and (𝜎𝜎−,𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇↓). However, at 𝑇𝑇=\n340  K (FIG. S10(d)), the IFE scenario results in similar behavior but the MCD results are \nthe opposite. This can be easily understood as explained in the main, because the FiM \nalloy with 𝑥𝑥=0.25 is RE -dominated for 𝑇𝑇=260  K and 𝑇𝑇=300  K, whereas becomes \nTM-dominated for 𝑇𝑇=340 K. (Fig. S8(a)). Note that the magnetization compensation \ntemperature for 𝑥𝑥=0.25 is 𝑇𝑇 𝑇𝑇≈320  K. \nSimilar results are also achieved for 𝑥𝑥=0.24 (right column in F IG. S10), but now \nboth IFE and MCD only give similar results for 𝑇𝑇=260  K (FIG. S10(f)), whereas they \npredict opposite behavior for 𝑇𝑇=300  K (FIG. S10(g)) and 𝑇𝑇 =300  K (FIG. S10(h)). \nNote that for 𝑥𝑥=0.24, the FiM is only RE -dominated for temperature below the 𝑇𝑇𝑇𝑇, \nwhich now is 𝑇𝑇 𝑇𝑇≈280  K. \n 17 \n  \nFIG. S 10. Temperature dependence of the spontaneous magnetization of each sublattice (RE:Gd; \nTM:CoFe) of the FiM alloy (Gd x(CoFe) 1-x) for two different compositions: (a) x=0.25 and ( e) \nx=0.24. Probability of switching as a function of the absorbed power density ( 𝑄𝑄) for a laser pulse \nof 𝜏𝜏𝐿𝐿=50 fs for different combinations of the initial state ( 𝑚𝑚𝑧𝑧𝑇𝑇𝑇𝑇:(↑,↓)) and the polarization \n(linear: 𝜎𝜎=0 (black dots), and circular 𝜎𝜎±=±1) of the laser pulse as indicated in the legend \nand in the main text: (b) , (c) and (d) corresponds for 𝑇𝑇=260  K, 𝑇𝑇=300  K, and 𝑇𝑇=340  K \nfor 𝑥𝑥=0.25, whereas  (f), (g) and (h)  to x=0.24. MCD results are shown by solid dots, whereas IFE \nresults are presented by open symbols. Lines are guide to the eyes.  \n \n \n18 \n REFERENCES  \n[1] U. Atxitia, P. Nieves, and O. 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Oppeneer, Quantum Theory of the \nInverse Faraday Effect , Physical Review B.  89, 014413 (2014).  \n[14] M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer, Ab Initio Theory of \nCoherent Laser -Induced Magnetization in Metals , Physical Review Letters 117, \n137203 (2016).  \n  "    },    {        "title": "1310.1782v1.Ferrimagnetic_Slater_Insulator_Phase_of_the_Sn_Ge_111__Surface.pdf",        "content": "arXiv:1310.1782v1  [cond-mat.mes-hall]  7 Oct 2013FerrimagneticSlater InsulatorPhaseoftheSn/Ge(111)Sur face\nJun-Ho Lee, Hyun-Jung Kim, and Jun-Hyung Cho∗\nDepartment of Physics and Research Institute for Natural Sc iences,\nHanyang University, 17 Haengdang-Dong, Seongdong-Ku, Seo ul 133-791, Korea\n(Dated: June 7, 2021)\nWe have performed the semilocal and hybrid density-functio nal theory (DFT) studies of the Sn/Ge(111)\nsurface to identify the origin of the observed insulating√\n3×√\n3 phase below ∼30 K. Contrasting with the\nsemilocal DFT calculation predicting a metallic 3 ×3 ground state, the hybrid DFT calculation including van\nder Waals interactions shows that the insulating ferrimagn etic structure with√\n3×√\n3 structural symmetry is\nenergetically favored over the metallic 3 ×3 structure. It is revealed that the correction of self-inte raction error\nwith a hybrid exchange-correlation functional gives rise t o a band-gap opening induced by a ferrimagnetic\norder. The results manifest that the observed insulatingph ase is attributedtothe Slatermechanism via itinerant\nmagnetic order ratherthan the hithertoaccepted Mott-Hubb ard mechanism via electron correlations.\nPACS numbers: 71.30.+h, 73.20.At, 75.50.Gg\nTwo-dimensional (2D) electronic systems formed at crys-\ntal surfaceshave attractedmuch attentionbecause of their in-\ntriguing physical phenomena such as charge density waves\n(CDW), magnetic order, Mott insulators, and 2D supercon-\nductivity [1–3]. One of the most popular quasi-2D systems\nis the√\n3×√\n3 phase formed by the 1/3-monolayer adsorp-\ntion of group IV metal atoms, Sn or Pb, on the Si(111) or\nGe(111) surface [4–12]. Here, adatoms locating at T4sites\n[Fig. 1(a)] saturate all the dangling bonds (DBs) of Si or\nGe surface atoms, leaving a single DB for each adatom [13].\nSincetheseDBelectronsareseparatedasfaras ∼7˚A,there-\nsulting narrow half-filled DB band is likely to invoke variou s\ninstabilitiesandstrongelectroncorrelations.\nWe here focus on a prototypical example of quasi-2D sys-\ntems,Sn/Ge(111). Below ∼220K,thissystemundergoesare-\nversible phase transition from a√\n3×√\n3R30◦(hereafter des-\nignated as√\n3×√\n3) structure to a 3 ×3 structure [14]. This\nphase transition was initially interpreted in terms of a sur -\nface CDW formation stabilized by electron correlation ef-\nfects in the√\n3×√\n3 structure [14, 15]. However, the high-\ntemperature√\n3×√\n3 phase was later explained as a dynam-\nical effect in which inequivalent Sn atoms interchange thei r\nverticalpositions[16–18]: i.e.,thethreeSnatoms[up(U) and\ndown (D) atoms in Fig. 1(b)] of two different heights within\nthe3×3structurefluctuatebetweentwopositionsastempera-\nture increases, apparentlyshowinga√\n3×√\n3 structural sym-\nmetry [18]. Nevertheless, the ground state of Sn/Ge(111)\nis still subject to much debate because, as the temperature\nfurther decreases to ∼30 K, various experimental techniques\nsuch as scanning tunneling microscopy (STM), low-energy\nelectron diffraction, and angle-resolved photoemission s pec-\ntroscopy (ARPES) explored a phase transition from the 3 ×3\nstructureto a new√\n3×√\n3 structure wherethree Sn atomsin\n3×3 unit cell have an equivalent height [19]. This structural\nphase transition was observed to accompany simultaneously\na metal-insulator transition (MIT) [19]. However, previou s\ndensity-functional theory (DFT) studies [20–25] with the l o-\ncaldensityapproximation(LDA)andgeneralizedgradienta p-\nproximation(GGA) predicted that the√\n3×√\n3 structure wasnot only metallic but also less stable than the 3 ×3 structure,\nthereby failing to describe the electronic and energetic pr op-\nerties of the observed insulating√\n3×√\n3 phase. To resolve\nthis problem of LDA, Profeta and Tosatti [4] took into ac-\ncountCoulombinteractions(Hubbard U)using the LDA+ U\nscheme. They found that electron correlations can stabiliz e\na magnetic insulator with√\n3×√\n3 structural symmetry,indi-\ncating a Mott-Hubbard insulating ground state. On the other\nhand,Flores et al.pointedoutthattheeffectofelectroncorre-\nlations cannotinduce a transition to a Mott insulating grou nd\nstate [22]. Thus, it remains elusive whether the formation o f\nthe insulating phase of Sn/Ge(111) is attributed to electro n\ncorrelations[26].\n(a) (b)\nD D\n(c)U\nFIG. 1: (Color on line) (a) Top and (b) side views of the metall ic\n3×3 structure of Sn/Ge(111). The side view of the insulating fe rri-\nmagneticstructureisgivenin(c). Thedarkandgraycircles represent\nSn and Ge atoms, respectively. U and D in (b) represent the up a nd\ndownSnatoms,respectively. TheSnatomsarelocatedatthe T4site,\ni.e.,asingle threefoldhollowsiteabove asecondlayerGea tom. For\ndistinction, Ge atoms in the subsurface layers are drawn wit h small\ncircles. In (a), the 3 ×3 and√\n3×√\n3 unit cells are indicated by the\nsolidand dashedlines, respectively. The xandzdirections in(b) are\n[112]and [111], respectively.\nIn the present work, we propose a new origin of the ob-\nserved insulating phase in Sn/Ge(111) due to itinerant mag-\nnetism. This magneticallydriveninsulatingphase via an it in-\nerant single-electron approach is characterized as a Slate r in-\nTypesetbyREVT EX2\nsulator[27]. ItisfoundthattheDBstatesofSnatomsexhibi t\nan itinerant character because of their significant hybridi za-\ntion with the Ge surface states, differing from the case of th e\npreviously proposed Mott-Hubbard insulator [4] where each\nDBelectronwastreatedtobelocalizedatSnadatomsite. We\nnotethat theLDAandGGA tendto stabilizeartificially delo-\ncalized electronic states due to their inherent self-inter action\nerror (SIE), because delocalization reduces the spurious s elf-\nrepulsion of electron [28, 29]. In this regard, previous LDA\nand GGA calculations for Sn/Ge(111) may overestimate the\nstabilityofthemetallic3 ×3structure[20–25]. Itis,therefore,\nvery interesting to examine if the observed insulating phas e\ncan be predicted by the correction of SIE with an exchange-\ncorrelationfunctionalbeyondtheLDAorGGA.\nIn this Letter, we present a new theoretical study for\nSn/Ge(111) based on the hybrid DFT scheme including\nvan der Waals (vdW) [30] interactions (termed DFT+vdW\nscheme). We find that the correction of SIE with the hybrid\nexchange-correlation functional of Heyd-Scuseria-Ernze rhof\n(HSE) [31] stabilizes the insulating ferrimagnetic (FI) st ruc-\nture where the band-gap opening occurs by a FI spin order-\ning of three Sn atoms within the 3 ×3 unit cell. Here, the\nbuckling of three Sn atoms is suppressed to show apparently\na√\n3×√\n3 structural symmetry, and the stability of the FI\nstructure is further enhanced by the inclusion of vdW inter-\nactions. The calculated magnetic moment of the FI structure\nis well distributed over Sn adatoms as well as Ge substrate\natoms, giving rise to a magnitude of 1 µBper 3×3 unit cell.\nIt is thus demonstrated that the observed insulating phase i n\nSn/Ge(111) can be represented as a Slater insulator through\nitinerantmagnetism,notasthepreviously[4]proposedMot t-\nHubbardinsulatorbyCoulombinteractions U.\nThe present semilocal and hybrid DFT calculations were\nperformed using the FHI-aims [32] code for an accurate,\nall-electron description based on numeric atom-centered o r-\nbitals, with “tight” computationalsettings. For the excha nge-\ncorrelation energy, we employed the GGA functional of\nPerdew-Burke-Ernzerhof (PBE) [33] as well as the hybrid\nfunctional of HSE [31]. The k-space integration was done\nwith the 15 ×15 and 9×9 uniformmeshesin the surface Bril-\nlouin zones of the√\n3×√\n3 and 3×3 unit cells, respectively.\nThe Ge(111) substrate was modeled by a 6-layer slab with\n∼34˚Aofvacuuminbetweentheslabs[34]. Here,weusedthe\noptimizedGelatticeconstants a0=5.783,5.718,and5.667 ˚A\nfor the PBE, HSE, and HSE+vdW calculations, respectively.\nThe HSE+vdWlattice constant [35, 36] agreesmost with the\nexperimentalvalueof5.658 ˚A[37]. EachGe atominthebot-\ntom layer was passivated by one H atom. All atoms except\nthe bottom layer were allowed to relax along the calculated\nforces until all the residual force components were less tha n\n0.02 eV/ ˚A. The employed HSE+vdW scheme was success-\nfully applied to determine the energy stability of the metal lic\nandinsulatingphasesinindiumnanowiresonSi(111)[38].\nWe begin to optimize the nonmagnetic (NM)√\n3×√\n3 and\n3×3 structures using the PBE functional. The optimized top\nand side views of the 3 ×3 structure are displayed in Fig.TABLEI:Calculatedtotalenergies(inmeVper√\n3×√\n3unitcell)of\nthe 1U2D, FM, and FI structures relative to the NM√\n3×√\n3 struc-\nture. For comparison, the previous LDA and GGA results are al so\ngiven.\n1U2D FM FI\nPBE −10.1 − −\nHSE+vdW −31.6 −18.7 −37.7\nLDA (Ref. [21]) −5 − −\nLDA (Ref. [24]) −7.5 − −\nLDA (Ref. [4]) −9 − −\nGGA (Ref. [17]) −5 − −\n-1-0.5 0 0.5 1\nΓ KM ΓEnergy (eV)\n-1-0.5 0 0.5 1\nΓ KM ΓEnergy (eV)\nS1S1S2ΓΜ\nΚ\nS3\n→S2S3\n→\nDUDDUD\n(b)(a)→\nFIG.2: (Coloronline)(a)Surfacebandstructureofthe1U2D struc-\nture computed using the PBE functional. The spin-polarized surface\nband structure of the FI structure computed using the HSE+vd W\nscheme is given in (b). The band dispersions are plotted alon g the\nsymmetrylinesoftheBrillouinzone ofthe3 ×3unitcell[seethein-\nset in(a)]. The Γ-M line corresponds to [11 2] direction. The charge\ncharacters of the DB states at the Γpoint are also displayed with an\nisosurfaceof0.006e/ ˚A3. TheenergyzerorepresentstheFermilevel.\nThe majority and minoritybands in (b) are drawn with the soli d and\ndashed lines,respectively.\n1(a) and 1(b), respectively. It is seen that the U atom posi-\ntions higher than the two D atoms by 0.35 ˚A, in good agree-\nment with those (ranging from 0.26 to 0.36 ˚A) of previous\nDFT studies [4, 20–25, 39]. This so-called 1U2D structure\nis more stable than the NM√\n3×√\n3 structure by 10.1 meV\nper√\n3×√\n3 unit cell, which is well comparable with previ-\nous DFT results (see Table I). As shown in Fig. 1S of the\nSupplemental Material [40] [Fig. 2(a)], the calculated ban d\nstructure for the NM√\n3×√\n3 (1U2D) structure exhibits the3\npresenceofoccupiedDBstate(s)attheFermilevel,indicat ing\na metallic feature. It is revealed that the charge character s of\nthe DB states in the 1U2D structure [see Fig. 2(a)] represent\na chargetransferfromtheD to the U atoms[24]. Thischarge\ntransfergivesrisetoareducedCoulombrepulsionbetweenS n\nDB electrons, resulting in a more stabilization over the NM√\n3×√\n3 structure. Ourspin-polarizedPBE calculationswere\nnotabletofindanyspinorderingwithinthe√\n3×√\n3and3×3\nunit cells. Thus, we can say that the semilocal DFT scheme\nwiththePBEfunctionalcannotpredicttheinsulating√\n3×√\n3\nphaseobservedbelow ∼30K [19].\nItisnoteworthythatarecenthigh-resolutionphotoemissi on\nstudy [5] for the Sn 4 dcore level of the metallic 3 ×3 struc-\nture resolved three components,which were assigned to each\nof the three Sn atoms within the 3 ×3 unit cell. On the basis\nofthisphotoemissiondatatogetherwith STMimages,Tejeda\net al.[5] concluded that the two D atoms position at slightly\ndifferentheights, formingan inequivalent-down-atoms(I DA)\nstructure. Our PBE calculation shows that the IDA struc-\nture with a height difference of ∼0.03˚A between the two D\natoms is almost degenerate in energy (less than 0.1 meV per\nSn atom)with the 1U2D structure,consistent with a previous\nLDA calculation [39]. For the 1U2D and IDA structures, we\ncalculate the Sn 4 dcore-levelshifts using initial-state theory,\nwhere the shift is defined by the difference of the eigenval-\nues of the Sn 4 dcore level at different sites. The results are\ndisplayedinFig. 3andcomparedtothephotoemissionexper-\niment[5]. WefindthateachSn4 dcorelevelissplitintothree\nsublevels, i.e., the degenerate Cd1(Cd2) sublevel arising from\nthedxyanddx2−y2(dyzanddxz) orbitals and the Cd3sublevel\nfrom the dz2orbital. Note that such a crystal field splitting\nis conspicuous for the two D atoms but negligible for the U\natom. On the basis of the calculated initial-state core leve ls,\nthe observed C1,C2, andC3components (see Fig. 3) can be\nassociated with the three sublevels of the U atom, Cd1of the\ntwo D atoms, and Cd2andCd3of the two D atoms, respec-\ntively. Thus, we can say that the observed two components\nof higher binding energy are attributed to the effect of crys -\ntal field splitting [41] on the Sn 4 dcore levels of the two D\natoms,ratherthanto differentcorelevelsforthetwo inequ iv-\nalentDatomsintheIDAstructure[5]. Theinitial-statethe ory\nfor the 1U2D (IDA) structure shows that the Cd1,Cd2, and\nCd3sublevels for the two D atoms shift to higher binding en-\nergy by 155 (157 ±1), 237 (238 ±2), and 256 (257 ±3) meV,\nrespectively, relative to the average value of three sublev els\nfor the up atom. These shifts are smaller than those (230 ±40\nand 390±40 meV) of C2andC3relative to C1, which were\nresolved from the high-resolution photoemission spectra [ 5].\nThis differenceof Sn 4 dcore-level shifts between the initial-\nstatetheoryandthephotoemissionexperimentmayreflectth e\nfinal-statescreeningeffects[42].\nIn order to provide an explanation for the observed insu-\nlating√\n3×√\n3 phase, Profeta and Tosatti [4] performed the\nLDA +Ucalculation to propose the Mott-Hubbardinsulator,\nwhere the inclusion of electron correlations strongly modi -\nfies the ground state of Sn/Ge(111) from a NM 3 ×3 metal0.40.30.20.1 00.40.30.20.1 0Experiment\nC3 C2 C1\nTheory-1U2Dz z\nd2z\ndC3dC2dC1 U\nD1\nD2\n0.40.30.20.1 0Theory-IDA\nBinding energy (eV) dC3dC2dC1\ndyz\nd2x2-y dxzdxy\nx y x y\nFIG. 3: (Color on line) Calculated Sn 4 dsurface core-level shifts of\nthe1U2DandIDAstructures,incomparisonwiththehigh-res olution\nphotoemission experiment [5]. The shifts for the two D atoms (D1\nand D2) are given with respect to the average of the three subl evels\nfor the U atom. The positive sign indicates a shift to higher b inding\nenergy. The five dorbitals of the D1 atom are displayed with an\nisosurface of ±0.2(e/˚A3)1/2.\ntoamagneticinsulatorwith√\n3×√\n3structuralsymmetry[4].\nHere, the FI spin order in the Mott-Hubbard insulating state\nwasstabilizedwithina localizedorHeisenbergpicturewhe re\nanunpairedelectronwaslocalizedateachSnadatomsitewit h\na spin moment of 1 µB. This localized picture for magnetism\ncontrastswiththepresentitinerantorSlatermagnetismwh ere\nthe magnetic moment is well delocalized over Sn adatoms\nas well as Ge substrate atoms, as discussed below. Since\nthe Sn DB state is largely delocalized through hybridizatio n\nwith the Ge surface states [see Fig. 2(a)], the SIE inherent\nto the PBE functional may cause the incorrect prediction of\nthe metallic 3 ×3 structure as a ground state. To circumvent\nsuch an over-delocalization of Sn DB electrons, we use the\nHSE+vdW scheme to optimize the NM and magnetic struc-\ntures of Sn/Ge(111). The calculated total energies of the\n1U2D, FI, and ferromagnetic (FM) structures relative to the\nNM-√\n3×√\n3 structure are given in Table I. We find that the\nFI structure is energetically favored over the 1U2D and FM\nonesby6.1and19.0meVper√\n3×√\n3unitcell,respectively.\nThe optimized geometry of the FI structure is shown in Fig.\n1(b), where the buckling of three Sn atoms within the 3 ×3\nunit cell is suppressed to become flat, leading to a√\n3×√\n3\nstructural symmetry [43]. In Fig. 2(b), the calculated band\nstructure of the FI structure shows a band-gap opening of 71\nmeV, in goodagreementwith the ARPES measurementof60\nmeV [19]. Therefore, the present HSE+vdW calculation pre-4\ndicts an insulating FI ground state with√\n3×√\n3 structural\nsymmetry, consistent with the observed insulating√\n3×√\n3\nphase[19].\nSince the total energyobtained from the HSE+vdW calcu-\nlation is composed of the HSE energy ( EHSE) and the vdW\nenergy(EvdW), the total energydifferencebetween the 1U2D\nand FI structures can be divided into the two components,\nΔEHSEandΔEvdW. For this decomposition,we obtain ΔEHSE\n=4.9meV[44],whichislargerthan ΔEvdW=1.2meV.Thus,\nwecansaythatthecorrectionofSIEwiththeHSEfunctional\ngivesa more dominantcontributionto the stabilization of t he\ninsulatingFIstructureoverthemetallic1U2Dstructure,c om-\nparedtothat fromvdWinteractions.\nFigure2(b)alsoshowsthechargecharactersofthespin-up\n(denotedas S1↑andS3↑)andspin-down( S2↓)DBstatesinthe\nFIstructure. Itisrevealedthat S1↑andS3↑representsomehy-\nbridization of two DB electrons. All of the three DB states\nstronglyhybridizewith the pzorbitalsofthesurfaceandsub-\nsurface Ge atoms. This strong hybridization gives rise to a\nlarge delocalization of spin moments up to deeper Ge atomic\nlayers (see Fig. 4). The sum ( m) of the spin moments of\nSn atoms or Ge atoms in each layer is also given in Fig. 4.\nHere, the spin moment of each atom is calculated by Mul-\nliken analysis. We find that the Sn layer has m= 0.22µB,\nwhile the first and third Ge layers have m= 0.40 and 0.22\nµB, respectively, which are significantly larger than those ob -\ntained from other Ge layers. Note that the total spin moment\nis1µBper3×3unitcell. Theresultofalargespindelocaliza-\ntion over Sn atoms and Ge substrate atoms contrasts with the\ncase of the previously proposed Mott-Hubbard insulator [4]\nwhereeachSn atom hasa localizedspinmomentof1 µBasa\nconsequenceofelectroncorrelations. Itisremarkabletha tthe\npresent FI order is determined by an itinerant single-elect ron\napproach with the correction of SIE, thereby representing a\nSlaterinsulatordrivenbyitinerantmagnetism.\n(µ  )Bm\nSn layer 0.22\n1st Ge layer 0.40\n2nd Ge layer 0.03\n3rd Ge layer 0.22\n4th Ge layer 0.04\n5th Ge layer 0.05\n6th Ge layer 0.03\nFIG.4: (Coloronline)SpindensityoftheFIstructure. Them ajority\n(minority)spindensityisdisplayedindark(bright)color withaniso-\nsurfaceof0.01( −0.01)e/˚A3. Thesum( m)ofthespinmomentsofSn\natoms or Ge atoms in each layer is also given. For the spin mome nt\nof eachGe atom, see Table ISof the Supplemental Material[40 ].\nWe note that the total energy difference ΔEbetween the\nmetallic 1U2D structure and the insulating FI structure is\n6.1 meV per Sn atom. Although the MIT temperature can\nbe predicted by the precise entropy-related free energy dif -\nference (TΔS) between the 1U2D and the FI structures, we\nroughly estimate it by considering only the electronic con-tribution to the entropy, as done by Profeta and Tosatti [4].\nAssuming that the FI structure has a lack of spin entropy\nand a charge gap, TΔSwas approximated to γT2whereγis\nthe electronicspecific heat coefficient(roughlyof order ∼0.1\nmeV/site K2) [4, 45]. We thus estimate the MIT temperature\nas∼8K,whichissomewhatbelowtheobservedMITtemper-\nature of∼30 K [19]. This deviation of the MIT temperature\nmayreflect that the 1U2Dand FI structureshavedifferentvi-\nbrationalcontributionstotheentropy,whicharenottaken into\naccountin TΔS.\nIn conclusion, our semilocal and hybrid DFT calculations\nshowed the different predictions for the ground state of the\nSn/Ge(111) surface. Contrasting with the PBE functional\npredicting a metallic 3 ×3 ground state, the HSE functional\nshowed that the correction of SIE cures the delocalization\nerror to predict an insulating FI ground state with√\n3×√\n3\nstructural symmetry. We found that the magnetic moment of\nthe FI structure is well distributed over Sn adatoms as well\nas Ge substrate atoms. It is thus demonstrated that the ob-\nserved insulating phase in Sn/Ge(111) can be represented as\na Slater insulator through itinerant magnetism rather than a\nMott-Hubbardinsulator driven by Coulomb interactions. We\nnotice that the Sn/Si(111) surface has also been much stud-\nied to determine its exact crystallographicarrangement,e lec-\ntronic structure, and ground state [9, 10]. 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Status Solidi B\n248, 2142 (2011).\n[40] See Supplemental Material at\nhttp://link.aps.org/supplemental/xxxx for the band stru c-\nture of the√\n3×√\n3 structure computed using the PBE\nfunctional and the spinmoments of Snand Geatoms.\n[41] F.A.Cotton, Chemical Applicationsof GroupTheory (Wiley,\nNew York, 1990).\n[42] E.Pehlke andM. Scheffler,Phys. Rev. Lett. 71, 2338 (1993).\n[43] Since the HSE calculation without vdW interactions pre dicts\nthe insulating FI ground state with the disappearance of Sn\nbuckling, we can say that the flat structure of Sn atoms is not\ndue to vdW interactions but tothe FIspin ordering between Sn\natoms. It is found that the Sn-Ge bond lengths and the height\ndifference between the up and down Sn atoms (in the 1U2D\nstructure) obtained usingHSE+vdWdecrease bylessthan0.0 2\n˚A, compared tothose obtained using HSE.\n[44] It is notable that ΔEHSEobtained using the HSE calcula-\ntion without vdW interactions is 2.0 meV. This HSE value\nis somewhat different from that (4.9 meV) obtained from the\nHSE+vdW calculation, possibly due to the use of two differen t\nGe lattice constants optimized usingHSEand HSE+vdW.\n[45] K. Kanoda, J.Phys.Soc. Jpn. 75, 051007 (2006)."    },    {        "title": "1907.00647v1.Robust_Formation_of_Ultrasmall_Room_Temperature_Neél_Skyrmions_in_Amorphous_Ferrimagnets_from_Atomistic_Simulations.pdf",        "content": "1 \n Robust  Formation of Ultrasmall  Room -Temperature  Neél \nSkyrmions in Amorphous Ferrimagnets  from Atomistic \nSimulations  \nChung Ting Ma1, Yunkun Xie2, Howard Sheng3, Avik W. Ghosh1,2, and  S. Joseph Poon1* \n1Department of Physics , University of Virginia, Charlottesville, Virginia 22904 USA  \n2Department of Electrical and Computer Engineering , University of Virginia, Charlottesville, \nVirginia 22904 USA  \n3Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030 USA  \n* sjp9x@virginia.edu  \n \nNeél skyrmions originate from interfacial Dzyaloshinskii Moriya interacti on (DMI) . Recent studies \nhave explored us ing thin -film ferromagnets and ferrimag nets to host Ne él skyrmions f or \nspintronic applications. However, it is unclear if ultrasmall (10 nm or less) skyrmions can ever be \nstabilized at room temperature for practical use in high density parallel racetrack memories. \nWhile  thicker films can improve stability, DMI decay s rapid ly away from the interface.  As such, \nspins far away from the interface would experience near -zero DMI, raising question on whether \nor not unrealistically large DMI is needed to stabilize skyrmions, and whether skyrmions will also \ncollapse  away from the in terface. To address these questions, we have employed atomistic \nstochastic Landau -Lifshitz -Gilbert simu lations to investigate skyrmion s in amorphous \nferrimagnetic GdCo . It is revealed that a significant reduction in DMI below that of Pt is sufficient \nto st abilize ultrasmall skyrmions even in films as thick as 15 nm . Moreover, skyrmions are found \nto retain a uniform columnar shape across the film thickness due to the long ferrimagnetic  \nexchange len gth despite the decaying DMI . Our r esults show that increasing thickness and \nreducing DMI in GdCo can further reduce the size of skyrmions at room temperature, which is \ncrucial to improve the density and energy efficiency in skyrmion based devices.  \nIntroduction  \nMagnetic skyrmions have  topologically protected spin textures. Their potential in advancing  \nmemo ry density and efficiency  has drawn extensive  investigation in recent years1-25. In magnetic \nmaterials, skyrmions are stabilized th rough  the Dzyaloshi nskii Moriya interaction (DMI)26-27,  \ngenerated by  either  inherent chiral  asymmetries or by interfacial symmetry breaking . Intrinsic  \nDMI arises in non-centrosymmetric crystal s such as B20 alloys , where  Bloch skyrmions have been  \nfound in MnSi and FeGe at low temperature12-13. On the other hand , interfacial DMI originates  \nfrom inversion symmetry breaking by an interfacial layer with strong spin-orbit coupling . \nMultilayer stacks, such as  Pt/Co/Os/Pt,  Ir/Fe/Co/Pt  and Pt/Co/Ta , have been  found to host  40 nm \nto over 1 μm Ne él skyrmions at room temperature14-16. Several challenges remain in developing \nskyrmion based memory  and logic  devices  - for instance , skyrmion Hall effect can pr esent  a \nsignificant challenge in  guid ing skyrmion s linearly along racetracks  19-22. More critically , aggressive  \nreduction in skyrmion sizes is needed to optimize skyrmion based devices , whereupon their room \ntemperature  stability becomes a problem . Thicker magnetic layers are required  in most  cases to \nincrease stability17-18. However , for ferromagnet  (FM) /heavy meta l multilayer stacks, increase in \nthickness of the magnetic layer can lead to a loss of interfacial anisotropy  and the reduction of \nthe strength of average DMI28-31, both of which are critical for skyrmion formation. To overcome \nthese challenges, we need to consider a suite of  materials  and understand their underlying physics , \nespecially with varying film thickness.  2 \n Amorphous rare -earth-transitional -metal ( RE-TM) ferrimagnet s (FiM) are potential candidates  to \novercome  these challenges.  Several properties of RE -TM alloys provide a favorable environment  \nto host small skyrmion s at room temperature. Their isotropic structure helps with avoiding defect \npinning18, while  their intrinsic  perpendicular magnetic anisotropy (PMA) 32-35 helps stabil ize small \nskyrmion s by allowing the use of thicker films ( > 5 nm) . However, the effectiveness of interfacial \nDMI decreases significantly away from the interface28-31, which is the focus of our present \ninvestigation.  Besides PMA, the magnetization of RE -TM alloys vanishes at the compensation \ntemperature36. With near zero magnetization  and near compensated angular momentum37, the \nskyrmion Hall effect is vastly  reduced18-20,23, and the skyrmion velocity is predicted to  be maximum \nnear the compensation point of angular moment um61. Recently,  all-optical helicity  dependent \nultrafast switching  has been demonstrated in RE-TM alloys  using a circularly polarized  laser38-45. \nThis gives an additional tool to control spins in device  structures . Indeed, RE-TM alloys have begun \nto draw  interest in the field of skyrmion  research . Large skyrmions of ~150 nm have  been \nobserved in Pt/Gd FeCo/MgO23, and skyrmion bound pairs have been  found in Gd/ Fe multilayers24. \nRecently , small skyrmions approaching  10 nm were  found in Pt/GdCo/TaO x films25. Understanding \nthese results will enable f urther tuning to reduce the size of the skyrmion s. To guide experiments , \nnumerical simulation  has served  as an important tool , especially for co mplex systems such as RE-\nTM alloys40,46-50. Several  methods, such as atomistic Landau -Lifshitz -Gilbert (LLG) algorithm40,46-49 \nand micromagnetic Landau -Lifshitz -Bloch (LLB) algorithm49 have been employed  to provide \ndeeper understanding of  the magnetic properties in RE-TM alloys . \nIn this study , an atomistic LLG algorithm40,46-48 is employed to investigate  the properties  of \nskyrmions  in Gd Co with interfacial DMI . Although the sign of DM I at FM/heavy metal interface  is \nwell studied51-57, the sign of DMI involv ing a  FiM remains complex  and is rarely discussed . Here,  \nwe consider two scenario s for the DMI between Gd and Co (dGd-Co). First,  DMI  between the \nantiferromagnetic  (AFM) pair is set to the same sign as DMI  between ferromagnetic  pair, i.e. dGd-\nCo > 0. Second , the case of d Gd-Co < 0 is considered. The latter appears to be f avored by the sign of \nAFM inter action27. Moreover, t o incorporate  DMI being an interfacial effect, an exponential ly \ndecay ing DMI is utilized. Simulation results find that with a switched DMI sign, near 10 -nm \nskyrmions remain robust in Gd Co films as thick as 15 nm at room temperature .  Through numerical \ntomography  maps , we find  that skyrmions at room temperature are distributed as a  near uniform  \ncolumn in thicker samples , despite  a spatially  decaying DMI.   \nResults and Discussion  \nWe will now begin to investigate if ultrasmall skyrmions in GdCo can survive an exponential  DMI \nreduction over thick sample sizes. To incorporate the amorphous nature of  GdCo, we employ an \namorphous structure of RE 25TM 75 from ab initio molecular dynamics calculations , as shown in Fig. \n1. As shown in Fig. 2 , at 300 K , the magnetization of amorphous Gd 25Co75 is 5 x 104 A/m , and it has \na compensation  temperature near 250 K. We begin with a n exponential  DMI  decay  away from the \ninterface, as shown in Fig. 3. The DMI value discussed herein is the interfacial DMI D 0. The decay \nlength of DMI is based on both previous simulations and experiments.  DMI calculation in Co/Pt \ninterface finds a significant decrease in DMI beyond the second layer of Co from the Pt interface30. \nExperimental  results also find similar decay in Co -Alloy/Pt interface52-54. In amorphous GdCo, we \nadapted the “second layer” as decay length for direct comparison wi th these  findings.  \nA range of interfacial DMI values, from dCo-Co = 0.1 x 10-22 J to dCo-Co = 2.0 x 10-22 J (D = 0.12 to 2. 38 \nmJ/m2), and three thicknesses, 5 nm, 10 nm, and 15 nm, are considered. Only those show \nskyrmions are shown herein. To shorten computational time, thicker samples of 10 nm and 15 nm 3 \n are simulated using  a 5 nm  thick  sample  by conserving DMI energy density  across the  film. To \nconserve the total DMI energy, a faster decay is employed in a 5 nm sample to simulate 10 nm \nand 15 nm thick samples to keep the sum of DMI energy to be the same. To check the validity of \nthis simplification, we have compared the results of 10 nm  thick samples and 5 nm thick samples  \nwith faster DMI decay to  verif y that the two sets of samples produce identical results.  First, we \nconsider two scenario s for the sign of d Gd-Co, as both + and - signs have been reported in \nantiferromagnetically coupled  systems59,60. Fig. 4 shows the color maps of equilibrium spin \nconfigurations at 300 K for both d Gd-Co > 0 and d Gd-Co < 0 . For the case of d Gd-Gd, dCo-Co > 0 and dGd-\nCo > 0, the simulation with  dCo-Co = 0.25 x 10-22 J, dGd-Gd = 2.96 x 10-22 J and dGd-Co = 0.86 x 10-22 J \ncorresponds to an average DMI of D = 0. 21 mJ/m2. The value of dGd-Gd and dGd-Co is calculated from \ndCo-Co by multiplying the ratio of Gd moment μGd over Co moment μCo. Further discussion in the \nsupplementary  material shows that for a given average DMI, the energy minimum and thus \nskrymion size  is independent of how each DMI term varies . Eq. 3 shows the formula used for \nconverting  atomistic DMI to average DMI for Gd xCo1-x. \nD=2\nπ1\n𝑛̅[(1−𝑥)(𝑛𝐶𝑜−𝐶𝑜 ̅̅̅̅̅̅̅̅̅𝑑𝐶𝑜−𝐶𝑜\n𝑟𝐶𝑜−𝐶𝑜 ̅̅̅̅̅̅̅̅2+𝑛𝐶𝑜−𝐺𝑑 ̅̅̅̅̅̅̅̅̅|𝑑𝐺𝑑−𝐶𝑜|\n𝑟𝐶𝑜−𝐺𝑑 ̅̅̅̅̅̅̅̅2)+x(𝑛𝐺𝑑−𝐺𝑑 ̅̅̅̅̅̅̅̅̅𝑑𝐺𝑑−𝐺𝑑\n𝑟𝐺𝑑−𝐺𝑑 ̅̅̅̅̅̅̅̅̅2+𝑛𝐺𝑑−𝐶𝑜 ̅̅̅̅̅̅̅̅̅|𝑑𝐺𝑑−𝐶𝑜|\n𝑟𝐺𝑑−𝐶𝑜 ̅̅̅̅̅̅̅̅2)](1) \nWhere 𝑛̅ is the average number of nearest neighbor s around all atoms, 𝑛𝐴−𝐵̅̅̅̅̅̅̅ is the average \nnumber of atom s A that are nearest neighbor s to atom B,  𝑟𝐴−𝐵̅̅̅̅̅̅ is average distance between \natom s A and nearest neighbor ing atom B. The 2\nπ factor comes from averaging of the cross product \n𝒔𝒊×𝒔𝒋 in DMI energy.  \nFor 5 nm GdCo , with d Gd-Co < 0, dCo-Co < 0.25 x 10-22 J, only ferrimagnetic states are observed. At dCo-\nCo > 1.0 x 10-22 J, skyrmions are elongated  due to boundary effect in the simulation or stripes states \nare observed. The range of DMI, where skyrmions are found, is smaller compared to calculation \nby Cort et al.21. This is due to a reduction  in anisotropy an d exchange stiffness in GdCo.  With less \nDMI energy required to create skyrmions, smaller DMI value is needed to create skyrmions and \nstripes in FiM. Furthermore, experiment results have measured DMI value  greater than 1 mJ/m2 \nonly at ordered FM/heavy metal  interface50-56. The DMI value at amorphous  FiM/heavy metal \nremains unknown . Due to disorder nature of  amorphous materials,  the DMI value in amorphous \nFiM can be much smaller than the DMI value observed in ordered FM.  \nAs shown in Fig. 4, with ferromagnetic DMI (d Gd-Gd and d Co-Co) that are  positive, two scenarios  of \nAFM  DMI (d Gd-Co) are considered.  At 300 K, in all thicknesses, larger DMI is needed to form \nskyrmions with positive  dGd-Co than with negative dGd-Co. In 5 nm sample, D = 0.5 5 mJ/m2 is needed \nto stabilize skyrmions  with dGd-Co > 0. In comparison, with  dGd-Co < 0, a smaller DMI  of D = 0.21 \nmJ/m2 is needed to stabilize skyrmions. Similar behaviors are also found in 10 nm and 15 nm \nsamples. With dGd-Co > 0, the smallest skyrmions are found at D = 1.26 m J/m2 in 10 nm sample and \nD = 2.31 m J/m2 in 15 nm sample. On the other hand, with dGd-Co < 0, the smallest skyrmions are \nfound at D = 0.84 m J/m2 in 10 nm sample and D = 1.68 m J/m2 in 15 nm sample.   \nTo understand  such  intriguing behavior in a FiM, the in-plane spin configurations and the chirality \nof the skyrmion wall are investigated. Fig. 5 summarizes the chirality of the skyrmion wall s in the \nCo sublattice . Using  dGd-Gd, dCo-Co > 0 and dGd-Co < 0, in the  Co sublattice , the spins are turning in \ncounter -clockwise direction  across the skyrmion wall . For Gd sublattice, the spins in the skyrmion \nwall are also turning counter -clockwise. This can be explained by the DMI in the system . AFM  \ncoupling s between Gd and Co align the spins of Gd and Co in nearly antiparallel direction s, except \na small canting due to the presence of DMI. With  dGd-Gd and dCo-Co > 0, turning counter -clockwise  4 \n is energetically favorable . However , with dGd-Gd, dCo-Co > 0 and dGd-Co > 0, the chirality of the \nsimulated skyrmion wall  is found to be opposite. The DMI torque between  the AFM pairs  now \noppose s the DMI torques with in each sublattice . In the presence of a strong er inter -sublattice \nDMI torque , the spins  in each sublattice  now turn clockwise across the skyrmion wall.  \nTo better illustrate  the change in ch irality, the total DMI energies  between Co -Co, Gd -Gd and Gd -\nCo are computed using the equilibrium configurations at 0 K. Table 1 summarizes the sign of the \ntotal DMI energies for different nearest neighbor pair s. With dGd-Gd, dCo-Co > 0 and dGd-Co> 0, spins \nare turning  counter -clockwise . With this configuration, the total DMI energy between Gd -Gd pair \nEDMI(Gd-Gd) and Co-Co pair E DMI(Co-Co) are negative, and the total DMI energy between Gd and \nCo pair E DMI(Gd-Co) is also negative . This means that with dGd-Gd, dCo-Co > 0, it is energetically \nfavorable for spins to turn counterclockwise.  On the other hand,  with dGd-Gd, dCo-Co > 0 and dGd-Co > \n0, spins are revea led to turn clockwise from  the simula ted configurations. As a result of the sign \nchange in chirality,  EDMI(Gd-Gd) and E DMI(Co-Co) become  positive . On the other hand, EDMI(Gd-Co) \nremains  negative , bec ause both chirality and d Gd-Co change s sign. This implies  that it is \nenergetically favorable for Gd -Co pair to turn clockwise across, but it is energetically unfavorable \nfor Gd -Gd and Co-Co pairs to do so. In other word , AFM DMI d Gd-Co is able to overcome \nferromagnetic DMI d Gd-Gd and dCo-Co, resulting in energy favorable configuration s for Gd -Co pairs.  \nTo summarize , in a FiM, if the DMI of ferromagnetic pair and AFM  pair have the same sign, a \ncancellation of DMI occurs because it is preferable for a ferromagnetic pair to turn in the opposite \ndirection of an AFM  pair. No cancellation occurs if the DMI of ferromagnetic pair and AFM  pair \nhave the opposite  sign. These  also explain the differences in size of skyrmion  between d Gd-Co < 0 \nand d Gd-Co > 0. The dGd-Co < 0 scenario has larger  skyrmions because both ferromagnetic and AFM  \npairs are contributing to the formation of a skyrmion, which means the DMI effect is stronger  \noverall.  \nTo investigate  the minimal size  of room temperature skyrmions in GdCo, D -K phase diagrams with \nexponential ly decaying DMI at 300 K are simulated for 5, 10 and 15 nm GdCo films. In this section, \nwe focus  on the d Gd-Co < 0 scenario . Since energy barrier is a function  of exchange stiffness and \nthickness18, the minimal skyrmions size found in d Gd-Co < 0 scenario can also apply to d Gd-Co > 0 \nscenario, except a larger DMI is required . For each thi ckness, anisotropy ranges from 0.05 x 105 \nJ/m3 to 4 x 105 J/m3 are investigated.  Experimentally, GdCo h as anisotropy in the order of 104 \nJ/m3.25,36 For DMI, larger  interfacial DMI is explored in thicker samples, because a s thickness \nincreases, the average  DMI decreases,  and larger interfacial DMI is needed  to stabilize skyrmions.  \nIn 5 nm samples, interfacial DMI of 0 to 2 mJ/m2, which corresponds to dCo-Co of 0 to 2.38 x 1022 J, \nare investigated . Fig. 6 (a) shows the D -K phase diagram of 5 nm GdCo at 300 K. In 5 nm GdCo, \nskyrmions range from 12 nm to 40 nm are stabilized in the simulated range of interfacial DMI and \nanisotropy. Lines of 15 to 30 nm indicate the size of skyrmions at various DMI and anisotropy. As \nDMI decreases or anisotropy increase s, skyrmions become smaller and eventually collapse into \nFiM states. At the opposite side of D -K diagram, with large DMI and  small anisotropy , skyrmions \nlarger than 40 nm becomes elongated or collapsed due to the boundary of the simula tion space \n(50.7 nm x 50.7 nm).  This elongation of skyrmions was also seen earlier in Fig. 4 at large DMI \nvalues.  Overall, for a given anisotropy, as interfacial D MI increases from 0 to 2.0 mJ/m2, the \nequilibrium configuration goes from FiM to skyrmions, then to stripes. For a fixed DMI, as \nanisotropy increases , size of skyrmions decreases, and finally, skyrmions collapse into FiM states.  \nThese  behavior of skyrmion s in FiM GdCo as a function of DMI and anisotropy  is the same as  what \nhas been observed in a ferromagnet17,18.  5 \n For 10 nm and 15 nm GdCo, DMI of 0 to 3 mJ/m2 (dCo-Co of 0 to 3.57  x 1022 J) and 0 to 4 mJ/m2 (dCo-\nCo of 0 to 4.76 x 1022 J) are explored respectively.  The overall trend  of skyrmions as a function of \nDMI and anisotropy in 10 nm and 15 nm GdCo are identical to that of 5 nm GdCo , where  increase \nin DMI leads to larger skyrmions, and increase in anisotropy  results in smaller skyrmions. However, \none difference in thicker sample s from 5 nm sample is that ultrasmall skyrmions as small as 7 nm \nare stable in room temperature. For both 10 nm and 15 nm GdCo, there is a region of DMI and \nanisotropy where ultrasmall  skyrmions are stablized . In 10 nm GdCo,  ultrasmall skyrmions are \nfound in the region of DMI ranges from 0.8 to 1.0 mJ/m2 and anisotropy ranges from (0.1 to 0.8) \nx 105 J/m3. For 15 nm GdCo, this region lays within DMI ranges from 1.5 to 1.8 mJ/m2 and \nanisotropy ranges from (0.1 to 1.0) x 105 J/m3. For both 10 nm and 15 nm GdCo, the anisotropy \nfalls within the same range as what has being measured experimentally in GdCo25,36, which is in \nthe order of 104 J/m3. However, the interfacial DMI is  less than what has been observed at a Pt \ninterface. Ab-inito calculation has found Interfacial DMI of up to 12 mJ/m2 is reported at a n ideal  \nPt/Co interface30. On the other hand,  the interfacial  DMI measured in Co/Pt and other Co -alloy/Pt \nfilms are around 1.2 to 1.5 mJ/m2. 52-54 Thus, some reduction s of DMI from that of Pt are needed \nto experimentally obtain ultrasmall  skyrmion in  5 and 10 nm  GdCo films. Reduction of DMI can be \nobtained by sandwiching GdCo between two Pt layers with one Pt layer being diluted by other \nelements. Since GdCo is amorphous, we have more flexibility of tuning the underlayer and the \ncapping layer of a multilayer sandwich. With its intrinsic anisotropy and flexibility , GdCo films are \npromising materials to obtain ultrasmall skyrmions at room temperature  through DMI tuning .  \nFor device applications, especially in thicker films, we will also need to consider the growth of \nskyrmions away from the interface. With decaying DMI away from the interface, spins at the top \nof a thicker  sample experience effectively zero DMI. Without DM I, one might expect spins near \nthe top to align parallel for FM neighbors and antiparallel for AFM neighbors, and skyrmions to \ndisappear far away from the interface. If skyrmions collapse far away from the interface, the \nreliability of such memory devices would be vastly reduced. To investigate whether skyrmions \nremain robust in thicker samples, a numerical tomography is employed to image simulated \nultrasmall  skyrmions at 300 K. Fig. 7 shows the numerical tomography plot of a ultrasmall \nskyrmion  in 10 nm Gd Co. This skyrmion corresponds to D = 0.84 mJ/m2 and K = 0.3 x 105 J/m3. The \nsame skyrmion was shown  in Fig. 4(b) and as the smallest skyrmions (Star Symbol) at K = 0.3 x 105 \nJ/m3 in Fig. 6(b). In the 3D plots at the center of Fig. 7, color s are made to be somewhat \ntransparent to reveal the skyrmions structure near the center. For Co sublattice, red to orange \ncolor shows that most of the spins are pointing down . A region of green and blue  that appears \nnear the center corresponds to the simula ted skyrmion  at 300 K. As evidenced by the columnar \ndistribution of blue color, the skyrmion retain s a uniform columnar growth from the bottom to \nthe top. Columnar distribution of skyrmion is also found in Gd sublattice , where a column of red \nis distribute d uniformly from the bottom to the top . This feature can be understood in terms of \nthe large magnetic exchange length >20 nm  due to the low magnetization  in the ferrimagnet . \nTo further demonstrate  the uniform  columnar distribution  of skyrmion, in -plane and out -of-plane \ncross sections of the skyrmion are also plotted in Fig. 7. On the left of Fig. 7, in-plane cross section \nof spin configuration within 0.5 nm of the interface  and 0.5 nm of the top are mapped. The \nskyrmion s at the interface and near the top have  identical  size an d shape.  Compare to the \nmapping of spin configuration s in Fig. 4(b), size of the skyrmion remain the same. This shows that \nthe size of skyrmions rema in the uniform throughout a sample.  On the right side of Fig. 7, out -of-\nplane cross section s are shown for Gd and Co sublattice s. The blue color in Co sublattice and the \nred color in Gd sublattice correspond  to the center of the skyrmion. For both sublattice, out -of-\nplane cross sections show a columnar distribution of skyrmion from the bottom in terface to the 6 \n top. These results provide important evidences that skyrmion remain robust through a thicker  \nsample, and further support of using thicker GdCo samples  to increase skyrmion stability at room \ntemperature . \nConclusions  \nUsing atomistic stochastic  LLG simulations, ultrasmall skyrmions  are shown  to be stable at room \ntemperature  in ferri magnetic GdCo. Despite the rapid decay of Dzyaloshinskii Moriya interaction \n(DMI)  away from the interface, a rea listic range of DMI values  is seen to stabilize skyrmions in \nGdCo films as thick as 15 nm  irrespective of the sign of DMI between antiferromagnetic coupled \nGd and Co,  Furthermore, the low DMI values needed to form ultrasmall skyrmion in GdCo  indicate \nopportunity for design ing magnet ic materials to host ultrasmall Neel skyrmions . Through \ntomography of an ultrasmall skyrmion in 10 -nm thick GdCo  film, it is discovered that the skyrmion  \nassumes a columnar configuration that extends uniformly across the film thickness  despite  having \nnear zero DMI far away from the interface. These findings argue for using thicker magnetic films \nto host  ultrasmall skyrmions, providing an  important strategy for developing high density and high \nefficiency  skyrmion based devices.    \nMethods   \nThe classical atom istic Hamiltonian H in Eq. (1) is employed to investigate magnetic textures in \namorphous FiMs.  \n𝐻=−1\n2∑ 𝐽𝑖𝑗𝒔𝒊∙𝒔𝒋\n<𝑖,𝑗>−1\n2∑ 𝐷𝑖𝑗∙(𝒔𝒊×𝒔𝒋)\n<𝑖,𝑗>−𝐾𝑖(𝒔𝒊∙𝑲𝒊̂)2 \n−𝜇0𝜇𝑖𝑯𝒆𝒙𝒕∙𝒔𝒊−𝜇0𝜇𝑖𝑯𝒅𝒆𝒎𝒂𝒈 ∙𝒔𝒊  (2) \nwhere 𝒔𝒊,𝒔𝒋 are the normalized spins and  𝜇𝑖,𝜇𝑗 are the atomic moments at sites i, j respectively. \nThe atomic moment is absorbed into the exchange constant, 𝐽𝑖𝑗=𝜇𝑖𝜇𝑗𝑗𝑖𝑗, the DMI interaction \n𝑫𝒊𝒋=𝜇𝑖𝜇𝑗𝒅𝒊𝒋, which is proportional to ri x rj, the po sitional vector between the atoms i, j and the \ninterface , and the effective anisotropy  𝐾𝑖=𝜇𝑖𝑘𝑖. 𝑯𝒆𝒙𝒕 and 𝑯𝒅𝒆𝒎𝒂𝒈  are the external field and \ndemagnetization field respectively.  \nOnly nearest neighbor interactions are considered in the exchange and DMI interactions. Periodic \nboundary conditions  are enforced in the x and y direction s.  \nTo find the ground state, the spins are evolved under the following stochastic Landau -Lifshitz -\nGilbert (LLG) equation,  \n𝑑𝑴\n𝑑𝑡=−𝛾\n1+𝛼2𝑴×(𝑯𝒆𝒇𝒇+𝝃)−𝛾𝛼\n(1+𝛼2)𝑀𝑠𝑴×[𝑴×(𝑯𝒆𝒇𝒇+𝝃)] (3) \nwhere 𝛾 is the gyromagnetic ratio, 𝛼 is the Gilbert damping constant,  𝑯𝒆𝒇𝒇 is the effective field, \n𝝃 is the Gaussian white noise term for thermal fluctuations and 𝑀𝑠 is the saturation magnetization.  \nThe parameters used in our simulation are listed in Table 2. Exchange couplings 𝐽𝑖𝑗are calibrated \nbased on Ostler et al.49 to maintain the same Curie temperature and compensation temperature \nfor a given compensation. At 300 K, the magnetization of Gd 25Co75 is 5 x 104 A/m. Anisotropy \nenergy is determined based on Hansen et al.36 7 \n To incorporate the amorphous short range order, an amorphous structure of a 1.6 nm x 1.6 nm x \n1.6 nm box containing  250 atoms is generated from ab initio  molecular dynamics calculations by \nSheng et al .58. The composition used in the simulation is Gd 25Co75. Fig. 1 shows a plot of RE and \nTM atoms in the amorphous structure. For a 4.8 nm thick sample, replicas of th is box (32 x 32 x 3) \nare placed next to each  other to expand the simulated sample  to 50.7 nm x 50.7 nm x 4.8 nm and \n768000 atoms. On average, we find that each Co atom has 6.8 Co neighbor s and 4.1 Gd neighbor s, \nwhile  each Gd atom has 11.7 Co neighbors  and 3.5 Gd neighbor s. We have also employed a FC C \nlattice to study skyrmions in GdCo. We found that with the same compensation temperature and \nmagnetization, a larger DMI is needed to stabilize skyrmion in a FCC lattice structures  than \namorphous structure. This is because the overall effectiveness of DM I is affected by the structure. \nOnly results using  the amorphous structure are shown herein.  \nIn the simulations, the initial states are skyrmion of 20 nm based on the 2 -pi model18. Various \ninitial states, includes random initial states and 10 -30 nm skyrmions, have  been tested and found \nto produce  the same final states.  The size of skyrmions are defined as the diameter for which M z \n= 0. Since skyrmions are not perfectly symmetric, size of skyrmion is the avera ge diameter .  \nData Availability  \nThe datase ts generated during and/or analyzed  during the current study are available from the \ncorresponding author on reasonable request.  \nReferences:  \n1. Rößler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in \nmagnetic metals. Nature  442, 797–801 (2006).  \n2. Yu, X. Z. et al . Real -space observation of a two -dimensional skyrmion crystal. Nature  465, \n901–904 (2010).  \n3. 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The content of the information does not necessarily reflect the position or the \npolicy of the Government, and no official endorsement should be inferred. Approved for public \nrelease; distribution is unlimited . This work was  partially supporte d by NSF -SHF-1514219.  \nAdditional Information  \nAuthor Contributions  \nS.J.P. conceived the project and supervised the simulation, C.T.M.  performed the simulation, \nY.X. assisted in the simulation and provided comments, A .W.G. discussed skyrmion stability and \nsuggested improve ment to the manuscript , H.S. provided the amorphous structure  from ab \ninitio molecular dynamics calculations.  \nCompeting Interests  \nThe authors declare no competing interests.  \n \n \n \n \n 11 \n   \nFigure 1 Amorphous st ructure of RE 25TM 75 from ab initio molecular dynamics calculations . Red \natoms are rare -earth, and blue atoms are transition -metal.  \n \nFigure 2 Simulated saturation magnetization vs. temperature of amorphous Gd 25Co75. The \ncompensation temperature of amorphous Gd 25Co75 is near 250 K , and the magnetization is small \nat room temperature.  \n12 \n  \nFigure 3 Exponential decay DMI in 5 nm GdCo as function of distance from bottom interface (z) . \nIn this model, DMI remains constant  (D0) within 0.35 nm of the bottom int erface, as indicated by \nthe red line. Away from the interface, the strength of DMI decays exponentially as shown.  \n13 \n  \nFigure 4 Color mapping of equilibrium spin configurations  for various DMI values (exponentia lly \ndecay ing DMI) at 300K for dGd-Co > 0 and dGd-Co < 0 in (a ) 5 nm, (b) 10 nm, and (c) 15 nm GdCo . \nOut-of-plane component s of reduced magnetizations  (mz) are mapped  in the x -y plane  using  the \ncolor bar shown in (c). Ultrasmall skyrmions are revealed in 10 nm and 15 nm GdCo samples.  \n14 \n  \nFigure 5 Simulated s kyrmion configurations  of Co sublattice for dGd-Co < 0 and dGd-Co > 0 with metal \ninterface  at the bottom . (Top) A n overhead view of simulated skyrmion configurations for dGd-\nCo < 0 and dGd-Co > 0. (Bottom) For dGd-Co < 0, the skyrmion wall is turning counter -clockwise. The  \nd(FM) vector is pointing in the opposite direction of  Si x Sj. EDMI (FM)= dij · (Si x Sj ) is negative, \nwhich is favorable.  For dGd-Co > 0, the skyr mion wall is turning clockwise. The d(FM) vector  and Si x \nSj are pointing in the same direction, resulting in positive  EDMI (FM) . Identical signs of the DMI \nenergy are also found in the Gd sublattice.  \nScenario  EDMI(Gd-Gd)   EDMI(Co-Co)  EDMI(Gd-Co)  \ndGd-Gd, dCo-Co > 0, dGd-Co < 0 - - - \ndGd-Gd, dCo-Co > 0, dGd-Co > 0 + + - \nTable.  1 Sign of total DMI energy E DMI computed from equilibrium spin configurations at 0 K.   \n15 \n  \nFigure 6 D-K phase diagram of (a) 5 nm, (b) 10 nm and (c) 15 nm GdCo at 300 K with dGd-Co < 0. \nStar corresponds to smallest skyrmions simulated at K = 0 .3 x 105 J/m3. Ultrasmall skyrmions are \nrevealed in 10 nm and 15 nm GdCo. Due to limits of simulation space (50.7 nm x 50.7 nm), with \nperiodic boundary conditions in x -y direction, large skyrmions (> 40 nm) become either elongated \nor collapsed.  \n \n16 \n  \n \nFigure 7 Tomograph  of a  simulated  ultrasmall  skyrmion in 10 nm GdCo  at 300 K . It reveals \ncolumnar skyrmion distribution throughout the 10 nm GdCo sample.  The figure  shows Co -\nsublattice spins (top box), Gd -sublattice spins (bottom box), in -plane cross sections of near the \ntop and bottom interface (left), and out -of-plane cross sections (right).    \nParameter  Value  \nGyromagnetic ratio (ϒ) 2.0023193  \nGilbert Damping (α) 0.05 \nGd moment ( μGd) 7.63 μB \nCo moment ( μCo) 1.72 μB \nGd-Gd exchange constant (J Gd-Gd)  1.26 x 10-21 J \nCo-Co exchange constant (J Co-Co) 3.82 x 10-21 J \nGd-Co exchange constant (J Gd-Co) -1.09 x 10-21 J \nMagnetic Field (H)  0.01 T  \nTable. 2 Values of parameters used in the simulation.  \n \n"    },    {        "title": "2309.01965v1.Strong_and_nearly_100_____spin_polarized_second_harmonic_generation_from_ferrimagnet_Mn___2__RuGa.pdf",        "content": "arXiv:2309.01965v1  [cond-mat.mtrl-sci]  5 Sep 2023Strong and nearly 100 %spin-polarized second-harmonic generation from ferrimag net\nMn2RuGa\nY. Q. Liu,1M. S. Si∗,1and G. P. Zhang†2\n1School of Materials and Energy, Lanzhou University, Lanzho u 730000, China\n2Department of Physics, Indiana State University, Terre Hau te, IN 47809, USA\n(Dated: September 6, 2023)\nSecond-harmonic generation (SHG) has emerged as a promisin g tool for detecting electronic and\nmagnetic structures in noncentrosymmetric materials, but 100% spin-polarized SHG has not been\nreported. In this work, we demonstrate nearly 100% spin-pol arized SHG from half-metallic ferri-\nmagnet Mn 2RuGa. A band gap in the spin-down channel suppresses SHG, so t he spin-up channel\ncontributes nearly all the signal, as large as 3614 pm/V abou t 10 times larger than that of GaAs.\nIn the spin-up channel, χ(2)\nxyzis dominated by the large intraband current in three highly d ispersed\nbands near the Fermi level. With the spin-orbit coupling (SO C), the reduced magnetic point group\nallows additional SHG components, where the interband cont ribution is enhanced. Our finding is\nimportant as it predicts a large and complete spin-polarize d SHG in a all-optical spin switching\nferrimagnet. This opens the door for future applications.\nPACS numbers:\nI. INTRODUCTION\nThe interaction between an intense optical field and a\nmaterial is always fascinating. This gave birth to non-\nlinear optics [1, 2]. Second-harmonic generation (SHG),\na special case of sum frequency generation, has received\nenormous attention worldwide. SHG only exists in non-\ncentrosymmetricmaterialswith brokeninversionsymme-\ntryI[3], while it is absent in centrosymmetric systems.\nIn general, impurities and surfaces introduced in a mate-\nrial can break I. For instance, Iof the NV center is bro-\nken by introducing nitrogen-vacancies in diamond [4–7].\nMoreover, a few layers of crystal, created by mechanical\nexfoliation, exhibit different symmetry properties. Odd\nlayers of MoS 2and h-BN belong to the noncentrosym-\nmetric space group, different from their bulk, can also\ngenerate SHG [8–10]. So far, most of the materials stud-\nied are nonmagnetic. For magnetic materials, magnetic\norder can break time reversal symmetry T. The sizable\nSHG appears in the antiferromagnetic (AFM) CrI 3and\nthe even septuple layers of MnBi 2Te4[11–13], where the\nAFM orderingbreaks I. In these two cases, SOC destroys\nthe symmetry of band structure thereby enhancing SHG.\nAlthough nearly 100% spin polarization at the Fermi\nlevel is observed in materials such as half-metal Cr 2O3\n[14], they possess I, where only the odd-order harmon-\nics are observed. Until now, little is known about 100%\nspin-polarized SHG in a half-metallic ferrimagnet.\nIn this work, we predict a strong SHG signal from the\nhalf-metallic Heusler Mn 2RuGa. We show that a single\nspin channel mainly contributes to SHG in Mn 2RuGa.\nThe band gap in the spin-down channel is open and lim-\nits SHG, resulting in SHG mainly from the spin-up chan-\nnel. Surprisingly, χ(2)\nxyzreaches as large as 3614 pm/V for\nthe spin-upchannel, atleastanorderofmagnitudelarger\nthan that ofGaAs. It is found that the intrabandcurrent\ndominates this large χ(2)\nxyz, which originates from threehighly dispersive bands near the Fermi level. To con-\nfirm our conclusion, we remove the Ru atoms to obtain\nMn2Ga, where both the spin-up and spin-down channels\nare metallic. It is found that a highly dispersive band\nnear the Fermi level appears in the spin-down channel,\nwhich does enhance the SHG spectrum χ(2)\nxyz. With SOC,\nthe spin-up and spin-down channels arecoupled. As a re-\nsult, the restricted transitions in the spin-polarized case\nare now SOC-allowed between the flat valence and con-\nduction bands. The underlying physics stems from the\nreduced magnetic point group induced by SOC, where\nthe magnetization field is applied along the z-axis. This\ndirectly leads to the appearance of additional SHG com-\nponents such as χ(2)\nxxz, where the allowed interband tran-\nsitions play a role. Our study demonstrates that nearly\n100% spin-polarized SHG can detect the half-metallicity\nin Heusler alloy Mn 2RuGa.\nThe rest of the paper is arrangedas follows. In Sec. II,\nwe show our theoretical methods. Then, the results and\ndiscussions are given in Sec. III. Finally, we conclude our\nwork in Sec. IV.\nII. COMPUTATIONAL METHODS\nA. First-principle electronic structure calculations\nThe electronic structures of Mn 2RuGa are calculated\nwithin the first-principle density functional theory using\ntheprojector-augmentedwave(PAW)[15,16]method, as\nimplemented in the Vienna Ab intio Simulation Package\n(VASP) [17–20]. The generalized gradient approxima-\ntion (GGA) [21] is employed within the Perdew-Burke-\nErnzerhof (PBE) scheme as the exchange-correlation\nfunctional. We self-consistently solve the Kohn-Sham2\nequation\n/bracketleftbigg\n−¯h2\n2me∇2+Vne(/vector r)+e2\n4πǫ0/integraldisplayn(/vector r)\n|/vector r−/vector r′|d3/vector r′+Vxc(/vector r)/bracketrightbigg\n×ψn/vectork(/vector r) =εn/vectorkψn/vectork(/vector r).\n(1)\nThe first term is the kinetic energy and next three terms\nare the potential energy, the Coulomb, and the exchange\ninteractions, respectively. meis the electron mass and\nn(/vector r) is the electron density. ψn/vectork(/vector r) denotes the Bloch\nwave function of band nat crystal momentum /vectork, and\nεn/vectorkis the band energy. The cutoff energy is set to 500\neV. The structural optimizations and self-consistent are\ncarried out by Γ-centered k-point mesh of 15 ×15×15.\nThe density of states (DOS) is calculated using a denser\nk-point mesh of 21 ×21×21. In order to obtain accurate\nresults, we set the energy convergence less than 10−6eV.\nB. First-principle nonlinear optical response\ncalculations\nWe use the length gauge to compute SHG [22–\n24]. The nonlinear polarization is given by Pa(2ω) =\nχabc(2ω;ω,ω)Eb(ω)Ec(ω), whereχabcdenotes the SHG\nsusceptibility, and Eb(ω) is thebcomponent of the op-\ntical electric field at frequency ω. In general, χabccon-\ntains three major contributions: the interband transi-\ntionsχabc\ninter, the intraband transitions χabc\nintra, and the\nmodulation of interband terms by intraband terms χabc\nmod,\nand can be expressed as\nχabc(2ω;ω,ω) =χabc\ninter(ω)+χabc\nintra(ω)+χabc\nmod(ω),\n=χabc\n2ph,inter(ω)+χabc\n1ph,inter(ω)\n+χabc\n2ph,intra(ω)+χabc\n1ph,intra(ω)\n+χabc\nmod(ω),(2)\nwherethe subscripts2 phand1phrepresenttwo-andone-\nphoton transitions, respectively. The interband and in-\ntraband transitions are\nχabc\ninter(ω) =χabc\n2ph,inter(ω)+χabc\n1ph,inter(ω),\nχabc\nintra(ω) =χabc\n2ph,intra(ω)+χabc\n1ph,intra(ω).(3)\nThedetailedexpressions[25]ofthesefourtermsaregiven\nby\nχabc\n2ph,inter(ω) =e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlra\nnm(k)/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\nωln(k)−ωml(k)\n×2fnm\nωmn(k)−2ω−2iη,\n(4)\nχabc\n1ph,inter(ω) =e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlra\nnm(k)/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\nωln(k)−ωml(k)\n×/braceleftbiggfml\nωml(k)−ω−iη+fln\nωln(k)−ω−iη/bracerightbigg\n,\n(5)χabc\n2ph,intra(ω) =e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmra\nnm(k)/braceleftbig\n∆b\nmn(k)rc\nmn(k)/bracerightbig\nω2mn(k)\n×−8ifnm\nωmn(k)−2ω−2iη\n−e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlra\nnm(k)/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\nω2mn(k)\n×2fnm(ωln(k)−ωml(k))\nωmn(k)−2ω−2iη,\n(6)\nχabc\n1ph,intra(ω) =e3\n¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlra\nnm(k)/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\n×ωmn(k)/braceleftbiggfnl\nω2\nln(k)(ωln(k)−ω−iη)\n−flm\nω2\nml(k)(ωml(k)−ω−iη)/bracerightbigg\n.\n(7)\nWe know that χabc\nmodin Eq. (2) contributes little to SHG\nin comparison with χabc\ninterandχabc\nintra, and it has the form\nas\ni\n2ωχabc\nmod(2ω;ω,ω) =\nie3\n2¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlωnl(k)ra\nlm(k)/braceleftbig\nrb\nmn(k)rc\nnl(k)/bracerightbig\nfnm\nω2mn(k)(ωmn(k)−ω−iη)\n−ie3\n2¯h2/integraldisplaydk\n4π3/summationdisplay\nnmlωlm(k)ra\nnl(k)/braceleftbig\nrb\nlm(k)rc\nmn(k)/bracerightbig\nfnm\nω2mn(k)(ωmn(k)−ω−iη)\n+ie3\n2¯h2/integraldisplaydk\n4π3/summationdisplay\nnmfnm∆a\nnm(k)/braceleftbig\nrb\nmn(k)rc\nnm(k)/bracerightbig\nω2mn(k)(ωmn(k)−ω−iη).\n(8)\nIn Eqs. (4)-(8), ωmn(k) =ωm(k)−ωn(k), and the\nenergy of band nis ¯hωn. ∆a\nmn(k) =υa\nmm(k)−\nυa\nnn(k), whereυa\nnm(k) is theacomponent of the ve-\nlocity matrix elements, and fmn=f(¯hωm)−f(¯hωn),\nwheref(¯hωn) is the Fermi Dirac function. The ma-\ntrix elements of position operator rmn(k) is given by\nra\nmn(k) =−iυa\nnm(k)/ωmn(k), and/braceleftbig\nrb\nml(k)rc\nln(k)/bracerightbig\n=\n1\n2/bracketleftbig\nrb\nml(k)rc\nln(k)+rc\nml(k)rb\nln(k)/bracketrightbig\n. The matrix elements of\nposition operator rmn(k) are directly computed by the\nfirst-principle calculations, which accounts for the effect\nof non-local potentials [26]. The damping parameter η\nis set to 0.003 Hartree. In the realistic calculations, the\nnumber ofkpoints and energy bands affect the accuracy\nofχabc. A large number of kpoints are required to ob-\ntain an accurate NLO response, so a very dense k-point\nmesh of32 ×32×32is used. The number of energybands\nis set to 32 to converge the spin-polarized SHG spectra,\nand 64 for the SHG spectra with SOC.3\nIII. RESULTS AND DISCUSSIONS\nA. Crystal, electronic structures and\nspin-polarized SHG of Mn 2RuGa\n(a)\nHalf-metallic ferrimagnet0.0 0.5 1.0 1.5 2.0 2.5 3.030006000900012000\n0Spin-up Spin-dn(b)\n|\n\u0001xyz| (pm/V)(2)\n0 1 2 3 4 5 6100200300\n0(c)\nE (eV) GaAsMn2RuGa2 \u0002\n7 8zLaser\nDetector\nEF\nSpin-up Spin-dngap2\u0000 2\u0003\n\u0004\u0005\n\u0006400\n|\n\u0007xyz| (pm/V)(2)\nFIG. 1: (a) A sketch of the nearly 100% spin-polarized SHG\nin half-metallic ferrimagnets, where the contribution fro m the\nspin-up channel is significant, while that from the spin-dow n\nchannel is largely suppressed. (b) The absolute value of the\nSHG susceptibility χ(2)\nxyzfor Mn 2RuGa without SOC. The red\nsolid andblack dashed lines denote the spin-upand spin-dow n\nchannels, respectively. The damping parameter η= 0.003\nHartree is taken. (c) χ(2)\nxyzof GaAs is also given for compari-\nson.\nSHG appears in nonmagnetic materials with broken I,\nbut it does not distinguish spin. For ferrimagnetic ma-\nterials, both spin-up and spin-down channels contribute\nto SHG. However, SHG from two spin channels is differ-\nent, which is closely related to the spin-polarized band\nstructures near the Fermi level. For a half-metallic ferri-\nmagnet with nearly 100% spin polarization at the Fermi\nlevel, SHG mainly comes from one spin channel show-\ning metallicity, as schematically displayed in Fig. 1(a).\nMn2RuGa is such a half-metallic ferrimagnet with many\ndifferent structures [27–29]. We choose the most stable\nXAHeusler structure, which belongs to the space group\nF-43m [30], as shown in Appendix A. This structure is\nmore consistent with the experimental results [31]. The\nexperimental lattice parameters are a=b=c= 5.97\n˚A [32]. In Mn 2RuGa, there are two Mn atoms, Mn 1\nand Mn 2, with different Wyckoff positions 4a(0, 0, 0)\nand 4c(1/4, 1/4, 1/4). The positions of Ru and Ga are\n4d(3/4, 3/4, 3/4) and 4b(1/2, 1/2, 1/2), respectively.\nThe magnetic moments of two Mn atoms are M4a= 3.13\nµB, andM4c=−2.29µB, consistent with the prior\nstudy [30], and they are antiferromagnetically coupled.\nHere, the magnetic moment direction is along the zdi-\nrection. It is found that the total magnetic moment is\nMtot= 1.03µB, which satisfies the Slater-Paulingrule as\nreported in the literature [33]. This rule provides a sim-\nple relationshipbetween the totalmagneticmoment Mtot\nand the valence electron Z. For Mn 2RuGa, they satisfy\nMtot=Z−24, and the number of valence electron is 25,TABLE I: The SHG susceptibilities for different materials.\n|χ(2)|and|χ(2)|SPrepresent the absolute values of SHG sus-\nceptibilities without and with spin polarization, respect ively,\nin units of pm/V. The unit of the photon energy is eV.\ncMQWs is the abbreviation of the coupled metallic quantum\nwells.\nMaterial |χ(2)| |χ(2)|SPPhoton energy Reference\nMn2RuGa 3614 0.38 This work\nGaAs 358 1.80 This work\nGaAs 350 1.53 Ref.[35]\nTaAs 3600 1.55 Ref.[36]\nCo3Sn2S21050.05 Ref.[37]\ncMQWs 1500 1.35 Ref.[38]\nBiFeO 315-19 0.80 Ref.[39]\nCaCoSO 6.9 1.17 Ref.[40]\nsoMtotis about 1µB[34].\nIn orderto understand the nonlinear optical properties\nof Mn 2RuGa, we have calculated the second-order non-\nlinear optical susceptibilities. Nonmagnetic Mn 2RuGa\nbelongs to the point group T d. There are six equiva-\nlent nonvanishing SHG susceptibilities χ(2)\nxyz=χ(2)\nxzy=\nχ(2)\nyxz=χ(2)\nyzx=χ(2)\nzxy=χ(2)\nzyx. When the antiferromag-\nnetic coupling appears along the zdirection, the symme-\ntry is reduced from 24 to 8, belonging to the magnetic\npoint group -42m. This changes SHG. The system con-\ntains three independent nonvanishing elements, namely,\nχ(2)\nxyz=χ(2)\nyxz,χ(2)\nxzy=χ(2)\nyzx, andχ(2)\nzxy=χ(2)\nzyx. The SHG\nsusceptibility satisfies the intrinsic permutation symme-\ntry, that is χ(2)\nabc=χ(2)\nacb. Thus, there are only two inde-\npendent nonvanishing elements, χ(2)\nxyz=χ(2)\nxzy=χ(2)\nyxz=\nχ(2)\nyzx, andχ(2)\nzxy=χ(2)\nzyx.\nFor the SHG susceptibility χ(2)\nxyz, the absolute value of\nthe spin-up channel is given by the red solid line in Fig.\n1(b). It is shown that |χ(2)\nxyz|is at the maximum value\nof 11670.76 pm/V when the photon energy approaches 0\neV. Due to the metallic nature for the spin-up channel\nin Mn 2RuGa, the intensity of |χ(2)\nxyz|at 0 eV is not accu-\nrate, which also depends on the damping parameter [see\nAppendix A for more details]. When the photon energy\nis between 0 and 0.22 eV, |χ(2)\nxyz|decreasesmonotonically.\nAt 0.22 eV, the value of |χ(2)\nxyz|is close to zero. As the\nphoton energyincreases, a dramaticpeak appearsat 0.38\neV, as shown in Table I. The intensity of this peak is as\nlarge as 3614.37 pm/V. We also note that this peak is in-\nsensitive to the damping parameter, as discussed in Ap-\npendix A. As the energy further increases, the spectrum\noscillates and gradually decreases, and the final intensity\nis close to zero. By contrast, the spin-down |χ(2)\nxyz|[see\nthe black dashed line in Fig. 1(b)] is much smaller, with\na maximum of about 652.46 pm/V in the entire energy\nrange. This is consistent with the experimental finding\n[41], where only the majority spin channel is optically4\nexcited highly. When the energy is less than 0.2 eV, the\nintensity difference of |χ(2)\nxyz|between the spin-down and\nspin-up channels is largest. In the energy range of 0.2\nto 1.6 eV, the spin-down |χ(2)\nxyz|change is relatively sta-\nble, but the spin-up decreases, so the difference between\nthe two spins decreases. When the energy is larger than\n1.6 eV, the spin-down |χ(2)\nxyz|is very close to the spin-up,\nand the intensity is almost zero. Therefore, we can con-\nclude that SHG in Mn 2RuGa is mainly contributed by\nthe spin-up channel [41], while the contribution of spin-\ndown channel is negligible. As a result, we obtain nearly\n100% spin-polarized SHG in Mn 2RuGa. We also notice\nthat GaAs and Mn 2RuGa share the same point group.\nHowever, as shown in Fig. 1(c), the maximum intensity\nof|χ(2)\nxyz|in GaAs is only about 358.19 pm/V at 1.8 eV,\nwhich is about 10 times smaller than the spin-polarized\nSHG of Mn 2RuGa. The SHG susceptibilities for other\nmaterials are also presented in Table I. We believe that\nsuch a large spin-polarized SHG in Mn 2RuGa has poten-\ntial applications in spin-filter devices [42].\nB. Intraband and interband contributions\nTo reveal insights into this nearly 100% spin-polarized\nSHG, we resort to the band structure of Mn 2RuGa, as\nshown in Fig. 2(a). The red line indicates the spin-up\nchannel, and the black line indicates the spin-down chan-\nnel. The band of spin-up channel crosses the Fermi level\nat multiple kpoints, indicating a metallic state. This re-\nsultcanbeverifiedfromthetotaldensityofstates(DOS),\nas shown in Fig. 2(b). Bands in the energy range from\n−0.08 to 0.12 eV are mainly occupied by dorbitals, in\nwhich Mn-3 dorbitals are dominant, Ru-4 dorbitals con-\ntribute less, and the contribution from Ga-3 dorbitals\nis negligible. However, the spin-down band only touches\nthe Fermi level near the Γ point. The total DOS near the\nFermi level is close to 0. Therefore, Mn 2RuGa is a half\nmetal, which is consistent with previous report [29, 30].\nThe band gap disappears in the spin-up channel, but ap-\npears in the spin-down channel. Therefore, the appear-\nance of band gap hinders the transition of electrons from\nthe valence band to the conduction band.\nNext, we further analyze the difference between the\nspin-up and the spin-down χ(2)\nxyzof Mn 2RuGa by exam-\nining the real and imaginary parts separately. For the\nspin-up channel, the real and imaginary parts of χ(2)\nxyzare\ngiven in red solid and black dashed lines, respectively, as\nshown in Fig. 2(c). The real part of χ(2)\nxyz(Re(χ(2)\nxyz))\ndecreases monotonously as the photon energy increases\nfrom 0 to 0.16 eV. When the photon energy reaches 0.16\neV,itsvaluereachesaminimum of-1099.71pm/V.When\nthe energy is larger than 0.16 eV, the spectrum oscillates\nand finally approaches zero. For the imaginary part of\nχ(2)\nxyz, that is Im( χ(2)\nxyz), a Lorentzian-like resonance ap-\npears in the energy range between 0 and 0.4 eV. Com-\npared with the real part, its first negative peak shiftsΓ L K0.01.0\n-0.5\n-1.0W \b X0.5\n0.03.06.09.0\n-3.0\n-6.0\n-9.0\n0.0 0.5 1.0 1.5 2.0 -0.5-1.0-1.5-2.0DOS (States/eV)\nE (eV) E (eV)(a) (b)Total Mn-3d Ru-4d\nSpin-up Spin-dnW1\nW2\nW3\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(c)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) \n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (d)\n(e) (f)Spin-up Spin-up\nSpin-dn Spin-dnInter (\t) Intra (\n)\nInter (2\u000b) Intra (2\f)\nInter (\r) Intra (\u000e)\nInter (2\u000f) Intra (2\u0010)\n05001000\n-500\n-10001500\n-150004000800012000\n-4000\n-8000\n-12000\nIm(χxyz) (pm/V)(2)Im(χxyz) (pm/V)(2)χxyz (pm/V)(2)χxyz (pm/V)(2)04000800012000\n-4000\n-8000\n-12000\n05001000\n-500\n-10001500\n-1500\nFIG.2: (a)BandstructurewithoutSOCforMn 2RuGa, where\nred line represents the spin-up channel, and black line repr e-\nsents the spin-down channel. The blue circles represent the\ndxy,dyzanddxzorbitals of Mn atoms for the bands W 1, W2\nand W 3. The dashed line denotes the Fermi level. (b) To-\ntal DOS of Mn 2RuGa, where the partial DOS of Mn-3 dand\nRu-4dstates are also given and represented by red and green\nlines, respectively. The upward arrow indicates the spin-u p\nchannel, and the downward arrow indicates the spin-down\nchannel. The vertical dashed line denotes the Fermi level. ( c)\nReal and imaginary parts of the SHG susceptibility χ(2)\nxyzfrom\nthe spin-up channel in Mn 2RuGa without SOC. (d) Calcu-\nlated Im( χ(2)\nxyz) from inter( ω)/(2ω) (black solid, black dashed\ncurve)andintra( ω)/(2ω)(red solid, reddashed-dottedcurve)\nparts. (e) and (f) are similar to (c) and (d), but from the spin -\ndown channel.\nto the lower energy by about 0.06 eV, and the intensity\nis decreased to 7768.47 pm/V. When the photon energy\nfurther increases, the spectrum oscillates.\nIn general, the imaginary part of χ(2)\nxyzreflects the op-\ntical absorption in NLO experiments. From the band\nstructure, the optical absorption involves in the intra-\nand interband currents. Thus, we decompose Im( χ(2)\nxyz)\ninto the inter- and intraband parts for the spin-up chan-\nnel, asshowninFig. 2(d). It clearlyshowsthattheintra-\nband contribution dominates the spectrum in the lower\nphoton energy window from 0 to 0.4 eV. By contrast,\nthe interband contribution from both single- and two-\nphoton resonances is much smaller. In the energy range\nof around 0.4 −2.0 eV, these four spectra are comparable\nand have opposite signs for the single- and two-photon\nresonances, leading to the oscillation of Im( χ(2)\nxyz) in this5\nenergy range. Thus, the negative characteristic peak of\nIm(χ(2)\nxyz) in the lower energy is determined by the in-\ntraband current. In other words, the interband current\ncontributes little to the negative characteristic peak in\nthe lower energy. We know that the negative character-\nistic peak locates in the energy range of 0 −0.4 eV. The\ntwo-photon resonance would correspond to the energy\nrange of around 0 −0.8 eV. In this perspective, the bands\nrelated to the intraband current would locate in the en-\nergy range from −0.4 to 0.4 eV near the Fermi level. As\nshown in Fig. 2(a), we can see that only three bands W 1,\nW2and W 3appear in this energy range. The bands W 1\nand W 2are degenerate along the L-Γ direction, while\nthe bands W 2and W 3are degenerate along the Γ- Xdi-\nrection. More importantly, these three bands disperse\nquadratically along the L-Γ-Xdirection. This means\nthat they highly disperse along this high-symmetry line,\ncontributingalargenormalvelocitytothe intrabandcur-\nrent [43]. This is the reason why the intraband current\ndominates the negative characteristic peak in the lower\nenergy range.\nWealsonoticethatthereexisttworegionsfortheinter-\nbandtransitionsamongthesethreebands. Oneislocated\nnear theLpoint along the L-Γ direction, where the dou-\nble degenerate bands of W 1and W 2form the conduction\nbands while the band W 3is the valence band. The other\ninterband transition appears near the Xpoint along the\nΓ-Xdirection,wherethebandW 1istheconductionband\nwhile the two-fold degenerate bands of W 2and W 3are\nthe valence bands. According to the selection rules, the\ninterband transitions from these two regions are largely\nlimited. This is because these three bands are mainly\nformed by the dxy,dyzanddxzorbitals of Mn atoms\n[see Fig. 2(a) for more details]. The interband transi-\ntions between the same dorbitals are not allowed. As\na result, the interband current contributes little to the\nnegative characteristic peak in the lower energy range.\nThis is generic for metallic ferro- or ferrimagnets as the\nspin-splitting states near the Fermi level are dominated\nby thedorbitals.\nIn the case of spin-down channel, the real and imag-\ninary parts of χ(2)\nxyzare comparable, as shown in Fig.\n2(e). Both oscillate around zero as the photon energy\nincreases. However, they are largely suppressed in com-\nparison with those of spin-up channel. Similarly, we also\ndecompose Im( χ(2)\nxyz) into the inter- and intraband con-\ntributions, as shown in Fig. 2(f). It is found that both\nthe inter- and intraband currents are comparable and os-\ncillate around zero, which are much smaller than those\nof the spin-up channel. This is because only few elec-\ntrons are allowed to transit from the valence band to the\nconduction band near the Γ point. However, the large\nband gap of the spin-down channel limits both the inter-\nand intraband currents. This explains why SHG of the\nspin-down channel is much smaller.C. SHG in metallic Mn 2Ga\nTo further confirm the contribution of SHG from the\nspin-up quadratic bands in Mn 2RuGa, we artificially re-\nmove the Ru atoms in Mn 2RuGa and obtain the crystal\nstructure of Mn 2Ga [30]. In Mn 2Ga, two Mn atoms are\nantiferromagnetic coupled and their magnetic moments\nare close to 3 µB, which can compensate each other. The\nmagnetic moments of Ga atoms are very small, so the to-\ntal magnetic moment of the unit cell is nearly zero. This\ncoincides with the previous report [34]. However, the\nantiferromagnetic coupling in Mn 2Ga has a huge effect\non the spin-polarized band structures, as shown in Fig.\n3(a). It is found that both the spin-up and spin-down\nbands cross the Fermi level, indicating that Mn 2Ga is a\nmetal. This is also confirmed from the PDOS, as shown\nin Fig. 3(b). We can see that obvious DOS exits both\nfor the spin-up and spin-down channels near the Fermi\nlevel, where the Mn-3 dorbitals dominates.\nΓ L K0.02.0\n-1.00.5\n-2.0\nW W\u0011-1.5-0.51.01.5\nSpin-up Spin-dnE (eV)(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.050001000015000\n0\nE (eV) (c)0.03.0\n6 \u0012 \u00139 \u0014 \u0015-3.0- \u0016 \u0017 \u0018\u0019 \u001a \u001b \u001c\n0.0 0.5 1.0 1.5 2.0 -0.5-1.0-1.5-2.0DOS (States/eV)\nE (eV) (b)\n(d)Total Mn-3d\nSpin-dn\nInter (\u001d) Intra (\u001e)\nInter (2\u001f) Intra (2 )\nx ! \" # $ % & ' ( ) * + , . /\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) 03000\n-30000 1 2 3 45 7 8 : ;Im(\n<xyz) (pm/V)(2)|= abc\n>?@ABCD(2)Spin-up\nSpin-dn\n-12000\nFIG. 3: (a) Band structures for Mn 2Ga, where the red and\nblack lines represent the spin-up and spin-down channels, r e-\nspectively. The dashed line denotes the Fermi level. (b) Tot al\nDOS of Mn 2Ga, where the partial DOS of Mn-3 dorbitals is\nalso given and represented by red line. The upward arrow\nindicates the spin-up channel, and the downward arrow indi-\ncates the spin-down channel. The vertical dashed line denot es\nthe Fermi level. (c) The absolute value of the SHG suscep-\ntibilityχ(2)\nabcfor Mn 2Ga, where abcrefers to xyz,xzy,yxz,\nandyzx. The red solid and black dashed lines denote the\nspin-up and spin-down channels, respectively. (d) Calcula ted\nIm(χ(2)\nxyz) from inter( ω)/(2ω)(black solid, black dashed curve)\nand intra ( ω)/(2ω) (red solid, red dashed-dotted curve) parts\nfor the spin-down channel.\nFor Mn 2Ga, SHG has two independent nonvanishing\nelements, namely, χ(2)\nxyz=χ(2)\nxzy=χ(2)\nyxz=χ(2)\nyzx, and\nχ(2)\nzxy=χ(2)\nzyx. The absolute value of χ(2)\nxyzis displayed\nin Fig. 3(c). For the spin-up channel, the first peak\nappears at the photon energy 0.14 eV and its intensity\nis 5602.48 pm/V. Then, the intensity sharply decreases\nand finally approaches zero as the photon energy further6\nincreases. In the case of the spin-down channel, χ(2)\nxyz\nis similar to that of the spin-up channel, but the inten-\nsity is much larger. However, this is contrast to that of\nthe spin-down channel in Mn 2RuGa, where χ(2)\nxyzis much\nsmaller. This is because no band gapappears in the spin-\ndown channel of Mn 2Ga and three metallic bands cross\nthe Fermi level. More importantly, one of them disperses\nquadratically with knear the Γ point, which largely con-\ntributes to the intraband part of χ(2)\nxyz. This implies that\nthe presence of band gap in the spin-down channel of\nMn2RuGa limits χ(2)\nxyz, while the absence of band gap or\nthe quadratic band near the Fermi level enhances χ(2)\nxyz.\nTo this end, we decompose Im( χ(2)\nxyz) of the spin-down\nchannel into the inter- and intraband contributions, as\nshown in Fig. 3(d). It clearly shows that the intraband\ncurrent dominates, while the interband contribution is\nnearly neglected. In addition, the two-photon resonance\nis obviously larger than that of one-photon resonance.\nThis would be easily detected by SHG in experiment.\nD. Role of SOC in SHG in Mn 2RuGa and the\ngroup symmetry\nWith SOC, the spin-up and the spin-down channels\nmix together. The band structure of Mn 2RuGa is shown\nin Fig. 4(a), which almost coincides with the spin-\npolarized band structures. Figure 4(b) shows that DOS\nhas a peak between −0.1 and 0.1 eV, and mainly comes\nfrom the Mn-3 dorbitals, which is the same as the spin-\nup channel. This is because the spin-down DOS is close\nto zero in this energy range. However, in the energy\nrange below −0.1 eV and above 0.1 eV, DOS under SOC\nchanges compared to the spin-up channel, which is due\nto the contribution of the spin-down channel. This will\naffect SHG.\nIn fact, if we only include SOC, but ignore the mag-\nnetic field direction, the symmetries of the system re-\nmain unchanged. In real calculations, SOC is considered\nthrough a tiny magnetic field applied along the z-axis.\nThe symmetry is reduced and belongs to the magnetic\npoint group of -4. The remaining four symmetry op-\nerations are the identity operation E, the twofold rota-\ntional symmetry C2zwith the binary axis as the zaxis,\nand two combination operations IC4zandIC−1\n4z.IC4z\ndenotes the rotation of π/2 around the z-axis, followed\nby a mirror symmetry σxy.IC−1\n4zis similar to IC4z,\nbut with a rotation −π/2 around the z-axis. As a re-\nsult, there are six independent nonvanishing elements,\nχ(2)\nxyz=χ(2)\nyxz,χ(2)\nxzy=χ(2)\nyzx,χ(2)\nzxy=χ(2)\nzyx,χ(2)\nxxz=−χ(2)\nyyz,\nχ(2)\nxzx=−χ(2)\nyzy, andχ(2)\nzxx=−χ(2)\nzyy. Based on the in-\ntrinsic permutation symmetry, only four components in-\ndeed appear, χ(2)\nxyz=χ(2)\nxzy=χ(2)\nyxz=χ(2)\nyzx,χ(2)\nzxy=χ(2)\nzyx,\nχ(2)\nxxz=χ(2)\nxzx=−χ(2)\nyyz=−χ(2)\nyzy, andχ(2)\nzxx=−χ(2)\nzyy. It\nshould be noted that χ(2)\nxxzandχ(2)\nzxxare induced from the\nreduced magnetic point group.Γ L K0.01.0\n-0.5\n-1.0\nW W X0.5E (eV)(a)\n0.03.0\nE F GH I J0.0 0.5 1.0 1.5 2.0 -0.5-1.0-1.5-2.0DOS (States/eV)\nE (eV) (b)\nTotal Mn-3d Ru-4d\n0.0 0.5 1.0 1.5 2.0 2.5 3.03000\nK L M NO P Q R0\nE (eV) (c)S T U V Y Z [ \\ ] ^ _ ` a b c\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (d)d e f g h i j k l m n o p q r s tz u v w y { } ~       \n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(e)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (f)\nInter ( ) Intra ( )\nInter (2 ) Intra (2 )020004000\n   \n-2000\n-4000    020004000\n   \n-2000\n-4000    \n abc\n¡¢£¤(2)¥¦ abc\n§¨©ª«¬(2) Im(\n®xxz) (pm/V)(2)¯xxz (pm/V)(2)3000\n° ± ² ³´ µ ¶ ·0\nFIG. 4: (a) Band structure for Mn 2RuGa under SOC. The\ndashed line denotes the Fermi level. The rectangles denote\nthe flat valence and conduction bands and the arrow labels\nthe transitions between those flat bands. (b) Total DOS of\nMn2RuGa, where the partial DOS of Mn-3 dand Ru-4 dstates\nare also given and represented by red and green lines, re-\nspectively. The vertical dashed line denotes the Fermi leve l.\n(c) and (d) The absolute values of the SHG susceptibility\nχ(2)\nabcfor Mn 2RuGa under SOC. (e) Real and imaginary parts\nofχ(2)\nxxz. (f) Calculated Im( χ(2)\nxxz) from inter( ω)/(2ω) (black\nsolid, black dashed curve) and intra ( ω)/(2ω) (red solid, red\ndashed-dotted curve) parts.\nFigure 4(c) shows the absolute values of SHG suscep-\ntibilitiesχ(2)\nxyzandχ(2)\nzxy. It clearly shows that those two\nSHG spectra are nearly the same as that of the spin-\nup channel [see Fig. 1(d)]. This is because the contri-\nbution from the spin-down channel is negligible. Thus,\nSOC has little effect on these six SHG spectra, which\nalso appear in the spin-polarized case. It is also found\nthat the vanishing SHG spectra χ(2)\nxxzandχ(2)\nzxxin the\nspin-polarized case are recovered now, as shown in Fig.\n4(d). The intensity of χ(2)\nzxxis obviously larger than that\nofχ(2)\nxxz. The appearance ofthem directly comes from the\nreduced magnetic point group. In the following, we use\nthe four remaining symmetries to understand the nonva-\nnishing SHG spectrum χ(2)\nxxz. The matrix representations\nofEandC2zare diag{1,1,1}and diag {−1,−1,1}. The7\nother twoIC4zandIC−1\n4zare\nIC4z=\n0 1 0\n−1 0 0\n0 0 −1\n,IC−1\n4z=\n0−1 0\n1 0 0\n0 0 −1\n.\n(9)\nThe transformations of position operator under these\nfour symmetry operations are as follows. E: (x,y,z)→\n(x,y,z),C2z: (x,y,z)→(−x,−y,z),IC4z: (x,y,z)→\n(y,−x,−z), andIC−1\n4z: (x,y,z)→(−y,x,−z). As a\nresult, we can get:\nE:χ(2)\nxxz→χ(2)\nxxz,\nC2z:χ(2)\nxxz→χ(2)\n(−x)(−x)z=χ(2)\nxxz,\nIC4z:χ(2)\nxxz→χ(2)\nyy(−z)=−χ(2)\nyyz,\nIC−1\n4z:χ(2)\nxxz→χ(2)\n(−y)(−y)(−z)=−χ(2)\nyyz.(10)\nIt clearly shows that these four SHG susceptibilities do\nnot cancel out each other. We can obtain these induced\nSHG spectra via the above symmetry analysis, where\nχ(2)\nxxzandχ(2)\n−yyzare protected by C2zandIC4z, respec-\ntively.\nThe reduced magnetic point group is closely related\nto SOC and the applied magnetic filed direction. This\nmeans that the induced SHG spectra must have a deep\nrelation to them. Here, we take χ(2)\nxxzto reveal the un-\nderlying physics. The decomposed real and imaginary\nparts ofχ(2)\nxxzare displayed in Fig. 4(e). We can see\nthat both have a similar manner. Sizable intensities are\nmainly located in the photon energy range of 0 −0.75 eV.\nTo understand the contributions of inter- and intraband\ncurrents, we decompose Im( χ(2)\nxxz) into the inter- and in-\ntraband parts, as shown in Fig. 4(f). There also exists a\nlarge contribution from the intraband current, where the\nhighly dispersed bands near the Fermi level play a role.\nIt should be noted that the interband contributions are\nlargely enhanced, which are nearly limited in the spin-\npolarized case. We find that many transitions are now\nallowed between the flat valence and conduction bands,\nas shown in rectangles of Fig. 4(a). However, those tran-\nsitions are not allowed in the spin-polarized case. This is\nbecause the flat valence bands are originally spin-up po-\nlarized, while those flat conduction bands are spin-down\npolarized. The direct transition from the spin-up band\nto the spin-down band is forbidden. But, this does occur\nunder SOC as the conservation of spin is not needed. To\ncheck it, we calculate some matrix elements of the posi-\ntionoperatorbetweenthoseflatbandsatseveral kpoints,\nas shown in Table II. We can see that the y-componentof\nmatrix elements are small but not zero. However, the x-\nandz-components are much larger. The maximum abso-\nlute value reaches as large as 8.02 a 0with a 0being the\nBohrradius. Thesenonzeromatrixelementsconfirmthat\nthe transitions are not allowed in the spin-polarized case,\nbut did occur with SOC. This tells us that the reduced\nmagnetic point group or the remaining four symmetriesprotect the induced SHG spectra, which is similar to our\nprevious study [44].\nTABLE II: The matrix elements of position operator between\nthose flat bands near the Fermi level under SOC in Mn 2RuGa\nfor three kpoints, where the atomic unit is used.\nkpointx y z\nRe Im Re Im Re Im\n(0,0.139,0) -0.83 0.32 0.05 0.12 -0.68 -1.74\n(0,0.209,0) -1.07 -2.34 0.01 -0.01 -2.49 1.13\n(0,0.278,0) 1.93 2.45 -0.09 0.07 6.29 -4.96\nIV. CONCLUSIONS\nIn conclusion, we have demonstrated that large nearly\n100% spin-polarized SHG carries rich information about\nthe electronic structures of the half-metal Mn 2RuGa.\nThe band gap in the spin-down channel limits SHG.\nIn contrast, the spin-up channel is metallic and gives\nrise toχ(2)\nxyzas large as 3614 pm/V, which is about 10\ntimes largerthan that of typical nonlinear materials such\nas GaAs. For the spin-up channel, the intraband cur-\nrent mainly contributes to χ(2)\nxyz, which stems from three\nhighly dispersed bands near the Fermi level. In addition,\nSOC under the z-axis magnetization field induces addi-\ntionalSHG susceptibilities suchas χ(2)\nxxzfrom the reduced\nmagnetic point group, where the interband transitions\ndominate. Our study would provide a good guide in fu-\nture application of largespin-polarized SHG in spin-filter\ndevices.\nACKNOWLEDGMENTS\nThis work was supported by the National Science\nFoundationofChinaunderGrantNo. 11874189. Wealso\nacknowledge the Fermi cluster at Lanzhou University for\nproviding computational resources. GPZ was supported\nby the U.S. Department of Energy under Contract No.\nDE-FG02-06ER46304. The research used resources of\nthe National Energy Research Scientific Computing Cen-\nter,whichissupportedbytheOfficeofScienceoftheU.S.\nDepartment of Energy under Contract No. DE-AC02-\n05CH11231.\n∗sims@lzu.edu.cn\n†guo-ping.zhang@outlook.com\nAPPENDIX A: The crystal structure of Mn 2RuGa\nand the effect of damping parameters on SHG\nHeusler alloyshavethree different structures belonging\nto different space groups [45]. The normal full-Heusler\nX2YZalloys belong to group symmetry L2 1(No. 225).8\n0.0 0.5 1.0 1.5 2.0 2.5 3.050001000015000\n0\nE (eV) Spin-up(b)\n|χxyz| (pm/V)(2)η¸ ¹ º » ¼ ½ ¾¿ À Á  à Ä\nηÅ Æ Ç È É Ê ËÌ Í Î Ï Ð Ñ\nη Ò Ó Ô Õ Ö × ØÙ Ú Û Ü Ý Þ200002500030000\nz\nxyMn1\nMn2Ruß à(a)\nFIG. 5: (a) Crystal structure of Mn 2RuGa. Purple, or-\nange, grey, and green spheres represent Mn 1, Mn2, Ru, and\nGa atoms, respectively. The arrows marked on the Mn atoms\nindicate the local magnetic moments, forming the antiferro -\nmagnetic configuration. (b) The absolute value of the spin-u p\nχ(2)\nxyzof Mn 2RuGa without SOC for different damping param-\netersη.\nThe Half-Heusler XYZcompounds have group symme-\ntryC1b(No. 216), and the inverse-Heusler X2YZalloys\nwith group symmetry XA(No. 216) [28]. We select\na stable one, that is XAstructure, as our example, as\nshown in Fig. 5(a), where the sublattice Mn 2RuGa are\nferrimagnetic ordering. The magnetic moments of these\nsublattice Mn atoms are 3.13 and -2.29 µB, respectively.\nAs aresult, the net magneticmoment ofunit cellisabout\n1µB, which agrees well with the Slater-Pauling rule.\nDue to the metallic nature for the spin-up channel in\nMn2RuGa, the nonlinear optical response at zero fre-\nquency must be estimated, because the energy difference\n¯hωnm=En−Emis zero for those metallic states near\nthe Fermi level. As a result, ωnm−ωorωnm−2ωin\nthe denominators of Eqs. (4)-(7) in the main text di-\nverges. To avoid this, the damping parameter ηis in-\ntroduced to estimate |χ(2)|at zero frequency. However,\nthe obtained results are still not accurate. As shown in\nFig. 5(b), we can see that the intensity of the spin-up\n|χ(2)\nxyz|at 0 eV largely depends on η. Whenη= 0.004\nHartree, the intensity of spin-up |χ(2)\nxyz|at 0 eV is about\n6150.37 pm/V. When ηis decreased to 0.003 Hartree,\nthe value is about twice larger than that for η= 0.004\nHartree. When ηis further decreased to 0.002 Hartree,\nthe intensity of spin-up |χ(2)\nxyz|is dramatically increased.\nThis means that |χ(2)|usually diverges at zero frequency.\nIn other words, it is a challenge to accurately compute\n|χ(2)|at zero frequency. This is also the case of linear\nresponse in Hall effect. According to the Drude model,\nthe frequency-dependent conductivity σ(ω) is given by,\nσ(ω) =σ0\n1−iωτ, (11)\nwhereσ0is the DC Drude conductivity without the ex-\nternal magnetic field and τis the relaxation time. At\nzero frequency, we can see σ(ω) reduces to σ0=ne2τ/m\nwithn,e, andmbeing the electron density, the electron\ncharge, and the electron mass, respectively. This shows\nthe failure of zero-frequency response already exists in\nthe linear response, which would be our future researchfocus of SHG.\nWhen the photon energy is further increased to about\n0.38 eV, a stable peak of spin-up |χ(2)\nxyz|with respect to\nηappears. The intensity of this peak increases as η\ndecreases. But the change is small. This implies that\nthe intensity of this second peak is insensitive to η. We\nalso note that the intensity of this peak corresponding to\nη= 0.003Hartree is as large as3614.37pm/V, which can\nbe compared with SHG spectra of other materials such\nas GaAs, TaAs, and CaCoSO [see Table I].\nAPPENDIX B: The SHG susceptibility for\nMn2RuGa without SOC\n0.0 0.5 1.0 1.5 2.0 2.5 3.030006000900012000\n0\nE (eV)Spin-up\nSpin-dn\ná â ã ä å æ çèχabc\néêëìíîï(2)\nFIG. 6: The absolute value of the SHG susceptibilities χ(2)\nabc\nfor Mn 2RuGa without SOC. The red solid and black dashed\nlines denote the spin-up and spin-down channels, respectiv ely.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) \n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (b)\n(c) (d)Spin-up Spin-up\nSpin-dn Spin-dnInter (ω) Intra (ω)\nInter (2ω) Intra (2ω)\nInter (ω) Intra (ω)\nInter (2ω) Intra (2ω)\n05001000\n-500\n-10001500\n-150004000800012000\n-4000\n-8000\n-12000\nIm(χzxy) (pm/V)(2)Im(χzxy) (pm/V)(2)χzxy (pm/V)(2)χzxy (pm/V)(2)04000800012000\n-4000\n-8000\n-12000\n05001000\n-500\n-10001500\n-1500\nFIG. 7: (a) The real and imaginary parts of the SHG sus-\nceptibility χ(2)\nzxyfrom the spin-up channel in Mn 2RuGa with-\nout SOC. (b) Calculated Im( χ(2)\nzxy) from inter( ω)/(2ω) (black\nsolid, black dashed curve) and intra ( ω)/(2ω) (red solid, red\ndashed-dotted curve) parts. (c) and (d) are similar to (a) an d\n(b), but from the spin-down channel.\nNext, theabsolutevalueofthe SHGsusceptibility χ(2)\nzxy\nfor Mn 2RuGa without SOC is given in Fig. 6. For the\nspin-upchannel,theintensityatzerofrequencyis9936.41\npm/V, which is smallerthan that of χ(2)\nxyz. The secondary9\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(c)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (d)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(e)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (f)0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) Re\nIm(a)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nE (eV) (b)Inter (ω) Intra (ω)\nInter (2ω) Intra (2ω)\nInter (ω) Intra (ω)\nInter (2ω) Intra (2ω)\nInter (ω) Intra (ω)\nInter (2ω) Intra (2ω)4000800012000\n0\n-8000\n-12000-4000χxyz (pm/V)(2)χzxy (pm/V)(2)χzxx (pm/V)(2)\nIm(χxyz) (pm/V)(2)Im(χzxy) (pm/V)(2)Im(χzxx) (pm/V)(2)4000800012000\n0\n-8000\n-12000-4000\n4000800012000\n0\n-8000\n-12000-4000\n4000800012000\n0\n-8000\n-12000-40004000800012000\n0\n-8000\n-12000-4000\n4000800012000\n0\n-8000\n-12000-4000\nFIG. 8: (a) The real and imaginary parts of χ(2)\nxyz. (b) Calcu-\nlated Im( χ(2)\nxyz) from inter( ω)/(2ω) (black solid, black dashed\ncurve)andintra( ω)/(2ω)(redsolid, reddashed-dottedcurve)\nparts. (c) and (d) are similar to (a) and (b), but for χ(2)\nzxy. (e)\nand (f) are similar to (a) and (b), but for χ(2)\nzxx.\npeak appears at 0.41 eV with an intensity of 3452.36\npm/V. When the photon energy is larger than 0.59 eV,the spectrum oscillates and then approaches to 0. In\ncontrast,the intensityofthespin-down χ(2)\nzxyisverysmall\nin the energy range from 0 to 3 eV. Therefore, in the low\nenergy region, the intensity of the spin-up χ(2)\nzxyis much\nstronger than that of the spin-down channel, which is\nsimilar toχ(2)\nxyz. This is because the spin-up channel has\nno band gap, while a gap in the spin-down channel limits\nSHG.\nThe real and imaginary parts of the spin-up channel\nforχ(2)\nzxyis shown in Fig. 7(a). We decompose Im χ(2)\nxyz\nof the spin-up channel into the inter- and intraband con-\ntributions, as shown in Fig. 7(b). It is found that the\nintraband contributions are dominant in the low energy\nregion. For the spin-down channel, the real and imag-\ninary parts are very small, as shown in Fig. 7(c). It\nmainly comes from the intraband and interband tran-\nsitions, as shown in Fig. 7(d). These results of both\nspin-up and spin-down channels are similar to χ(2)\nxyz.\nAPPENDIX C: The SHG susceptibility for\nMn2RuGa with SOC\nWith SOC, χ(2)\nxyzandχ(2)\nzxyare similar to the spin-up\nχ(2)\nxyzandχ(2)\nzxy, respectively, as shown in Figs. 8(a)-8(d).\nIt shows that SOC has little effect on these two compo-\nnents. On the contrary, the reduced symmetries induce\nχ(2)\nxxzandχ(2)\nzxx, showing different characteristics. The\nreal and imaginary parts of χ(2)\nzxxare shown in the red\nand black lines in Fig. 8(e). It is contributed by inter-\nand intraband transitions, as shown in Fig. 8(f). Com-\npared with the spin-polarized χ(2)\nxyzandχ(2)\nzxy, the inter-\nband contribution increases, which is similar to χ(2)\nxxz.\n[1] Y. R. Shen, The Principles of Nonlinear Optics (Wiley,\nNew York, 1984).\n[2] R. W. 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Phys. 113, 17101 (2013)."    },    {        "title": "1109.2705v2.Interplay_between_non_equilibrium_and_equilibrium_spin_torque_using_synthetic_ferrimagnets.pdf",        "content": "arXiv:1109.2705v2  [cond-mat.mes-hall]  10 Apr 2012Interplay between non equilibrium and equilibrium spin tor que\nusing synthetic ferrimagnets\nChristian Klein\nSPSMS, UMR-E 9001 CEA / UJF-Grenoble 1, INAC, Grenoble, F-38 054, France,\nPhysikalisches Institut, Universit¨ at Karlsruhe (TH), Wo lfgang-Gaede Strasse 1, 76131 Karlsruhe, Germany\nCyril Petitjean and Xavier Waintal\nSPSMS, UMR-E 9001 CEA / UJF-Grenoble 1, INAC, Grenoble, F-38 054, France\n(Dated: November 27, 2018)\nWe discuss the current induced magnetization dynamics of sp in valves F 0|N|SyF where the free\nlayer is a synthetic ferrimagnet SyF made of two ferromagnet ic layers F 1and F 2coupled by RKKY\nexchange coupling. In the interesting situation where the m agnetic moment of the outer layer\nF2dominates the magnetization of the ferrimagnet, we find that the sign of the effective spin\ntorque exerted on the free middle layer F 1is controlled by the strength of the RKKY coupling:\nfor weak coupling one recovers the usual situation where spi n torque tends to, say, anti-align the\nmagnetization of F 1with respect to the pinned layer F 0. However for large coupling the situation\nis reversed and the spin torque tends to align F 1with respect to F 0. Careful numerical simulations\nin the intermediate coupling regime reveal that the competi tion between these two incompatible\nlimits leads generically to spin torque oscillator (STO) be havior. The STO is found in the absence\nof magnetic field, with very significant amplitude of oscilla tions and frequencies up to 50 GHz or\nhigher.\nPACS numbers: 72.25.Ba, 75.47.-m, 75.70.Cn, 85.75.-d\nSince the first successful experimental manipulation of\nmagnetic configurations by spin-polarized currents [1, 2],\nthe interest in spintronics devices entirely controlled by\nelectrical currents has rapidly increased [3–10, 12, 13].\nThe key concept behind theses experiments is the no-\ntion of spin transfer torque (STT) predicted by Slon-\nczewski[14] andBerger[15]: acurrentgetsspinpolarized\nbyafirst(pinned)magneticlayerandthentransferssome\nof its polarization to a second free layer, hence exerting\na torque on its magnetization. As a result the magneti-\nzation can switch to a different static configuration [2, 3]\nor even to a dynamical stationary regime where the mag-\nnetization of the free layers shows a sustained preces-\nsion [4–7]. This dynamical regime, known as the spin\ntorque oscillation (STO), allows to convert a dc voltage\ninto a microwaves signal at several GHz and has a real\npotential for applications (nanoscale microwave source\nor detector, high frequencies possibly above CMOS tech-\nnology, narrow band). There are still technological is-\nsues with STO based devices, however, such as very lim-\nited output power and the need of rather strong external\nfields. Indeed, the main mechanism for producing STO\nis based on a detailed tuning of the angular dependance\nof the spin torque [5] and requires the use of an exter-\nnal magnetic field. Recently, different mechanisms have\nemerged to enable STO behavior in the absence of exter-\nnal field: the first approach [9] relies on the precession\nof a non uniform magnetic structure, a magnetic vortex,\nthat can be excited at sub GHZ frequencies; a second\napproach [8, 16] uses a strongly asymmetric spin valve to\nincrease the anharmonicity of the angular dependance of\nthe spin torque, leading to the so-called ”wavy” behaviorof the STT [16–19]. Other approaches use an orthogonal\npolarizer [10] or a perpendicular free layer [11, 12].\nIn this letter we propose an entire new mechanism\nwhich naturally induces strong STO behavior, even in\nthe absence of magnetic field, using a synthetic ferrimag-\nnet (SyF) as the free layer of the spin valve. The SyF is\nmade of two ferromagnetic layers coupled through anti-\nferromagneticRKKYexchangeinteraction[20–22]. Here,\nwe predict that by simply tuning the effective strength\nof the RKKY exchange coupling (through the different\nthicknesses of the layers for instance), the signof the\neffective STT can be changed. At intermediate effective\ncoupling, the STT generically induces strong STO be-\nhavior irrespectively of the presence of applied magnetic\nfield.\nIn the following we first provide a simple physical pic-\nture that captures the essential features of our setup.\nOur main prediction uses very general conservation ar-\nguments and should therefore be quite robust and inde-\npendent of the details of the model. In a second step,\nwe provide more quantitative arguments to describe the\nstrong coupling regime. Last, we present a detailed nu-\nmerical study of the phase diagram of our model. We\nfind strong STO behavior at rather large frequencies in a\nwideportionofthephasediagramhintingthatSyFbased\nSTOs could be good candidates for application purposes.\nPhysical mechanism. We consider a nanopillar spin\nvalveF 0|N0|SyF made of apinned ferromagneticlayerF 0\nand a free synthetic ferrimagnetic layer SyF = F 1|N1|F2\n(where N 0and N 1are normal spacers), see Fig. 1a for a\ncartoon of the full stack. Upon injecting a current den-\nsityjthrough the nanopillar, a spin torque τττis exerted2\nFigure 1: Panel (a) is a cartoon of the magnetic multilayer\nstack, in which Miis the the magnetization of the layer F i.\nThe angle between M1andM0(M2) is respectively θ(α).\nPanels (b,c,d) show the stationary magnetization mz\n1along\nM0as a function of the current density jin A.cm−2, for a\nstack with d1= 2.nm,d2= 4d1and three different coupling\nstrength J; (b) weak coupling limit ( J= 0), (c) Intermediate\ncoupling limit ( J=−10−3J.m−2) (d) Strong coupling limit\n(J=−6.10−3J.m−2) Upper (Lower) inset : Sketch of the\ntorqueτττ1actionon M1(MSyF)fortheweak(strong)coupling\nregime.\non the magnetic moments of the layers [23]. This torque\nhas been extensively discussed in the literature and can\nbe strong enough to destabilize the initial magnetic con-\nfiguration. On the other hand, the magnetization M1\nandM2of F1and F 2are coupled by an oscillatory ex-\nchange (RKKY) interaction Jwhich is nothing but the\nspin torque present in the pillar at equilibrium [24, 25].\nThe coupling Jis reminiscent of Friedel oscillations and\ntypically behaves as J∼(cos2kFdN)/dNwherekFis\nthe Fermi momentum and dNthe width of the spacer\nN1.Jcan be tuned from negative (antiferromagnetic\ncoupling) to positive (ferromagnetic coupling), hence the\nname ”oscillatory”. Here we focus on negative values of\nJwhichstabilizean anti-alignedconfigurationof M1and\nM2. The goal of this letter is to discuss new phenomena\nthat arise from the competition between the equilibrium\nand the non equilibrium torques.\nIn order to gain a qualitative understanding of the un-\nderlying physics, let us discuss two extreme limits. We\nstart from a configuration where M1is aligned with M0whileM2is anti-aligned, and inject electrons from the\nright. Upon crossing the pinned layer F 0, the electrons\nget a polarization anti-parallel to M0. Denoting Jithe\nspincurrent(perunit surface)flowingin the normallayer\nNi, we have J0=/planckover2pi1pjm0/(2e) withm0=M0/|M0|and\npthe polarization of the current (0 ≤p≤1). Let us fo-\ncus on the weak coupling limit where J→0. When there\nis a finite angle θbetween M0andM1, the spin current\non the left J0and right J1of F1become different and\na torque τττ1=J0−J1is exerted on the magnetization\nof F1: conservation of angular momentum implies that\nwhatever spin lost by the conducting electrons is gained\nby magnetic degrees of freedom [14]. This conservation\nof angular momentum reads,\n∂\n∂t/bracketleftbiggd1M1\nγ1/bracketrightbigg\n=τττ1+... (1)\nwhered1is the width of F 1,γ1its gyromagnetic ratio\nand the ...indicate other contribution to the dynam-\nics (magnetic anisotropy, damping) to be discussed later\n(upper inset of Fig. 1). Ignoring (momentarily [26]) the\nrole of F 2, we arrive at ∂θ/∂t=γ1/planckover2pi1pjsinθ/(2ed1|M1|)\nand recover the usual phenomenology of spin torque: the\ntorque tends to stabilize the configuration where M1is\nanti-parallel to M0(or destabilize it if one reverses the\nsign of the current).\nLet us now discuss how this picture is modified when\none considers the opposite strong coupling J→ −∞\nlimit. Upon switching on J, the exchange energy (per\nunit surface of the pillar) EJ=−J(m1·m2) induces\na field like term in the RHS of Eq.(1) of the form\n+Jm2×m1. One needs also to consider the correspond-\ning equation for the dynamics of M2(see below Eq.(3)\nfor the full model) and one obtains a potentially rich cou-\npled dynamics for M1andM2. The situation simplifies\nin the limit where the exchange energy J→ −∞domi-\nnates: the relative configuration of M1andM2becomes\nfrozen in the anti-aligned position. Conservation of an-\ngular momentum now leads to a spin torque on the total\nmagnetic moment:\n∂\n∂t/bracketleftbiggd1M1\nγ1+d2M2\nγ2/bracketrightbigg\n=τττ1+... (2)\nwhere we have used the fact that τττ2=J1−J2vanishes\nwhenM1andM2are collinear. Note that the exchange\nfield, which conserves the total angular momentum, re-\ndistributes the torque τττ1onM1andM2so that they\nremain anti-aligned.\nEq.(2) allows two very different regimes: if\nd1|M1|/γ1> d2|M2|/γ2then the effective magnetiza-\ntion direction MSyFof the SyF layer points along M1\nand the torque is very similar to the weak coupling\nregime. This case has attracted some attention recently\nboth experimentally by Smith et al. [27] and theoreti-\ncally by Balaz et al. [28]. Here, we focus on the other3\nlimitd1|M1|/γ1< d2|M2|/γ2whereMSyFpoints in the\nopposite direction as M1(lower inset of Fig. 1). In\nthis limit, the torque tends to stabilize the configura-\ntion where MSyFis anti-aligned with M0hence where\nM0andM1are aligned: this is the opposite situation\nfrom the weak coupling J→0 limit.\nTo summarize, when going from J→0 toJ→ −∞,\nwithoutchangingthesignofthecurrent,thetorquetends\nto favor two different stationary situations. In the rest\nof this letter, we study this crossover in more details. In\nparticular, we find that the intermediate coupling regime\nreveals interesting, STO like, dynamical behavior.\nModel.We now turn to the modelisation of our sys-\ntem. The full stack we are considering has the form (see\nFig. 1a): AF |F0|N0|F1|N1|F2|N2, where AF is an antifer-\nromagnetic layer (typically IrMn) that pins the magne-\ntization of F 0. We set our reference spin axis ezparallel\ntoM0. Magnetization dynamics is described by two cou-\npled Landau Lifshiz Gilbert (LLG) equations that read\n(in SI units),\n∂m1\n∂t=γ1B1×m1+α1m1×∂m1\n∂t+γ1\n|M1|d1τττ1(3a)\n∂m2\n∂t=γ2B2×m2+α2m2×∂m2\n∂t+γ2\n|M2|d2τττ2,(3b)\nwheremi=Mi/|Mi|is the unit vector describing the\nmagnetization orientation of the layer Fiandαiis the\ndamping factor of the corresponding magnetic material.\nThe effective field Biseen by the magnetization in the\nlayer (i= (1,2)) is\nBi=Ki\nu\n|Mi|[mi·ez]ez+J\n|Mi|dim¯i,(4)\nwhere¯i= 3−icorresponds to the index of the other\nlayer. The effective field includes an uniaxial anisotropy\nKi\nufield and the RKKY exchange field. In the physical\nregimes studied below, RRKY coupling is the dominat-\ning energy so that our results are rather insensitive to\nthe presence of less important terms (for instance dipo-\nlar coupling between layers that we do not take into ac-\ncount). The spin torque τττiis calculated in the framework\nof CRMT (Continuous Random Matrix Theory) [16, 29].\nCRMT is a semiclassicalapproachthat generalizesValet-\nFert theory to non collinear situations. For discrete sys-\ntems, It is formally equivalent to the generalized circuit\ntheory [30] and is well adapted to the treatment of metal-\nlic magnetic multilayers.\nStrong coupling limit. When the RKKY coupling is\nlarge, one can derive an effective LLG equation for the\ndynamics of the SyF. It reads,\n∂m1\n∂t=γeffBeff×m1+αeffm1×∂m1\n∂t+γeff\n|M1|d1τττ1,(5)\nwhere the various effective parameters get renormalized\nas follows: γeff=γ1/(1−δ),αeff= (α1+δα2)/(1−δ),Keff\nu=K1\nu+K2\nud2/d1,Beff= (Keff\nu/|M1|)[m1·ez]ezand\nthe renormalization parameter δhas been defined as,\nδ=γ1|M2|d2\nγ2|M1|d1(6)\n(Eq.(5) is obtained by calculating ∂/∂t[m1+δm2] which\ncancels the RKKY contribution, and then setting m2=\n−m1andτττ2=0). From this equation we can clearly\nsee the inversion of the sign of the effective torque γeffτττ1\ndiscussed previously when δ >1.\nFigure 2: Upper: phase diagram of the system as a func-\ntion of the current density jand the thickness d1for a fixed\nδ= 4 and J=−5.10−3J.m−2. The various symbols indi-\ncate the observed stationary states in the corresponding re -\ngion (defined by different colors): ↑(↓) stands for mz\n1= 1\n(mz\n1=−1), while STOcorresponds to the presence a pre-\ncessional state. In the 2 STOregion, two different STO\nstates can be observed depending on the hysteresis history.\nLower: Resistance Ras function of time tfor a pillar surface\nof 1000 nm2andd1= 3nm. Upper (lower) plots correspond\nto initial conditions in the up (down) configuration. Left pa n-\nels:j= 2.8 107A.cm−2. Right panels: j= 5.8 107A.cm−2.\nThe amplitude of the oscillation with the up initial conditi on\nis small and invisible on the scale of the graphics\nNumerical simulations. Let us now go back to the full\nmodel Eq.(3) and study it numerically. We focus on a\nstack defined as IrMn 5|Py5|Cu10|Cod1|Ru0.3|Cod2|Cu10\nwhere the index indicate the widths of the layers in\nnm. We have used KCo\nu= 8.104J.m−3,MCo=\n1.42 106A.m−1and the transport parameters (spin re-\nsolved bulk resistivities, interface resistivities, spin flip\nlengths) have been taken from the database established\nin MSU and CNRS/Thales laboratory[31]. The coupling\nconstant Jcan be tuned by changing the Ruthickness\nwith typical values JRu≈ −5.10−3J.m−2[21]. The\ntorqueand resistanceshavebeen calculatedusing CRMT\nand the corresponding parametrization (as a function of\nthe angles θandα, see Fig.1a) incorporated into our nu-\nmerical LLG integrator.\nA first set of curve is presented in Fig. 1 b,c and d\nwhere the stationary magnetization mz\n1along the zaxis4\nis plotted against the current density jfor three values\nof the coupling constant: small (b), intermediate (c) and\nlarge coupling (d). For small coupling (b) , we recover\nthe usual behavior of current induced magnetic reversal\nin spin valves: at zero current the system has two sta-\nble configurations where m1is aligned ( mz\n1= 1) or anti\naligned ( mz\n1=−1) withm0. Upon injecting a strong\npositive current, one ”pushes” toward the anti-aligned\nconfiguration which becomes stable above a critical value\nof the current density j(0.5 107A.cm−2in this example).\nAt large coupling (d), this hysteretic curve is reversed, in\nagreement with the arguments developed above. A very\ninteresting regime is found at intermediate couplings (c)\nwhere one observes a rather large window where the sta-\ntionary value of mz\n1corresponds to a finite angle between\nm0andm1, i.e. to a dynamical state where a sustained\nprecession of m1is present (with frequencies around 15\nGHz in this example).\nThe coupling Jcan be varied by tuning the width dN\nof the normal (Ru) spacer. However, this is not very\neasy to control in practice, as the oscillatory character\nof the RKKY interaction make this variation non mono-\ntonic. Alternatively, one can explore the phase diagram\nof our stack by varying the thickness d1for a fixed ra-\ntioδ=d2/d1, hence varying the relative importance of\nbulk versus surface terms in the LLG equation. Fig. 2,\nshows the resulting phase diagram in the ( j,d1) plane for\nδ= 4. The different symbols indicate the possible sta-\ntionary states in the various regionsof the phase diagram\n(defined by the various colors). The presence of two dif-\nferent symbols (most regions)indicate that depending on\nthe hysteresis history, one or the other state is observed.\nIn particular, in the 2 STO region, two different STO\nstatescanbeobserveddepending, forinstance, ontheini-\ntial condition mz\n1(t= 0) =±1. These two different STO\nstates merge upon entering the 1 STO region. We have\ninvestigated other stacks with different material parame-\nters (not shown) and found very similar phase diagrams,\nincluding the presence of high frequency STO phases, as\nlong as one remains in the δ >1 regime. We observe two\nkind of STO behaviors, as evident from the Resistance\nversus time R(t) traces plotted in Fig. 2. When the am-\nplitude of the oscillating signal is small (lower left of Fig.\n2), we observe some beating behaviors with one well de-\nfined frequency and a second much smaller frequency less\nwell defined (corresponding to the precession of the two\nlayers respectively). When these two frequencies become\ncloser, the time dependent signal gets bigger and R(t)\nbecomes much more sinusoidal indicating phase locking\nbetween the precessional dynamics of the two magneti-\nzationsm1andm2.\nIn the last Fig.3, we focus on the region of the phase\ndiagram (lower right corner of Fig.2) where the ampli-\ntude of the STO signal is the highest. The upper curves\nshow the amplitude of the time dependent signal while\nthe lower curves show the corresponding frequency. Wefind that this large angle STO phase, which is stable in\nthe absence of magnetic field, leads to very high frequen-\ncies up to several tens of GHz while sustaining high pre-\ncession angles (of the order of π/6 ). The observed high\nfrequencies is tightly linked with the high value of the\nRRKY coupling which can be up to two orders of magni-\ntude larger than typical anisotropy or Zeeman energies.\nEveniftherelativeanglebetweenthetwomagnetizations\nof the synthetic ferrimagnet remain close to π, this gives\nrise to frequencies of several tens of GHz.\nToconclude, wehaveshownthatthe interplaybetween\nequilibriumandnonequilibriumtorqueinsyntheticferri-\nmagnets based spin valves can lead to interesting physics\nand potentially a new route to high frequency STOs.\nFigure 3: Upper plot: amplitude (in /permil) of the time depen-\ndent signal of the resistance as a function of d1for fixed\nj= 6.107A.cm−2. Lower plot: idem for the main oscillating\nfrequency of the resistance. Insets: corresponding colorp lots\nin the (j,d1) plane. δ= 4 and J=−5.10−3J.m−2.\nWe thank T. Rasing, W. Wulfhekel for very useful dis-\ncussions. This work was supported by , CEA NanoSim\nprogram, CEA Eurotalent and EC Contract Macalo.\n[1] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang,\nM. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80,\n4281 (1998).\n[2] E. Myers, D. Ralph, J. Katine, R. Louie, and\nR. Buhrman, Science 285, 867 (1999).\n[3] J. Katine, F. Albert, R. Buhrman, E. Myers, and\nD. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[4] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang,\nV. Tsoi, and P. Wyder, Nature 406, 46 (2000).\n[5] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).5\n[6] W. Rippard, M. Pufall, S. Kaka, S. Russek, and T. Silva,\nPhys. Rev. Lett 92, 027201 (2004).\n[7] I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kise-\nlev, D. C. Ralph, and R. A. Buhrman, Science 307, 228\n(2005).\n[8] O.Boulle, V.Cros, J.Grollier, L.G.Pereira, C.Deranlo t,\nF. Petroff, G. Faini, J. Barna, and A. 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Waintal, Phys. Rev. Lett. 103, 066602 (2009).\n[17] J. Manschot, A. Brataas, and G. Bauer, Phys. Rev. B\n69, 092407 (2004).\n[18] J. Barna´ s, A. Fert, M. Gmitra, I. Weymann, and\nV. Dugaev, Phys. Rev. B 72, 024426 (2005).\n[19] M. Gmitra and J. Barnas, Phys. Rev. Lett. 96, 207205\n(2006).\n[20] S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev.\nLett.64, 2304 (1990).\n[21] S. Parkin, Phys. Rev. Lett. 67, 3598 (1991).\n[22] P. Bruno and C. Chappert, Phys. Rev. B 46, 261 (1992).\n[23]j >0correspondstoinjectinganelectrons from theright.\n[24] J. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n[25] X. Waintal and P. W. Brouwer, Phys. Rev. B 65\n054407(2002).\n[26] In the case considered here, the effect of τττ2will only\nslightly increase the predicted effect.\n[27] N. Smith, S. Maat, M. J. Carey, and J. R. Childress,\nPhys. Rev. Lett. 101, 247205 (2008).\n[28] P. Bal´ aˇ z and J. Barna´ s, Phys. Rev. B 83, 104422 (2011).\n[29] S. Borlenghi, V. S. Rychkov, C. Petitjean, and X. Wain-\ntal, Phys. Rev. B 84, 035412 (2011).\n[30] G. E. W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando,\nand A. Brataas, Phys. Rev. B 67, 094421 (2003).\n[31] H. Jaffres private communication."    },    {        "title": "2101.10240v1.Raman_Spectroscopy_and_Aging_of_the_Low_Loss_Ferrimagnet_Vanadium_Tetracyanoethylene.pdf",        "content": "Raman Spectroscopy and Aging of the Low-Loss Ferrimagnet Vanadium\nTetracyanoethylene\nH. F. H. Cheung,1M. Chilcote,1H. Yusuf,2D. S. Cormode,2Y.\nShi,3M. E. Flatt\u0013 e,3E. Johnston-Halperin,2and G. D. Fuchs1, 4, ∗\n1School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA\n2Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\n3Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA\n4Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA\nVanadium tetracyanoethylene (V[TCNE] x,x\u00192) is an organic-based ferrimagnet with a high\nmagnetic ordering temperature T C>600 K, low magnetic damping, and growth compatibility with\na wide variety of substrates. However, similar to other organic-based materials, it is sensitive to air.\nAlthough encapsulation of V[TCNE] xwith glass and epoxy extends the \flm lifetime from an hour\nto a few weeks, what is limiting its lifetime remains poorly understood. Here we characterize encap-\nsulated V[TCNE] x\flms using confocal microscopy, Raman spectroscopy, ferromagnetic resonance\nand SQUID magnetometry. We identify the relevant features in the Raman spectra in agreement\nwith ab initio theory, reproducing C = C ;C\u0011N vibrational modes. We correlate changes in the\ne\u000bective dynamic magnetization with changes in Raman intensity and in photoluminescence. Based\non changes in Raman spectra, we hypothesize possible structural changes and aging mechanisms in\nV[TCNE] x. These \fndings enable a local optical probe of V[TCNE] x\flm quality, which is invaluable\nin experiments where assessing \flm quality with local magnetic characterization is not possible.\nI. INTRODUCTION\nIn the \feld of coherent magnonics, where function is\nderived through the creation and manipulation of long-\nlived spin wave modes, yttrium iron garnet (YIG) is the\nprototype material due to its exceptionally low damping\nat room temperature. However, it is di\u000ecult to integrate\nwith other materials, requiring deposition on a lattice\nmatched substrate, e.g. gadolinium gallium garnet, and\na synthesis temperature above 800\u000eC [1{3]. In compari-\nson, vanadium tetracyanoethylene (V[TCNE] x,x\u00192) is\na low loss organic-based material with intrinsic damping\ncomparable to YIG, and it can be grown as a \flm with\nchemical vapor deposition at a mild temperature (50\u000eC)\nand on nearly arbitrary substrates [4, 5]. Furthermore,\nV[TCNE] xcan be patterned using electron-beam lithog-\nraphy lift-o\u000b techniques while retaining its low magnetic\nloss [6]. Hence it is an attractive alternative for magnon-\nics. In particular, it is a candidate for coupling spin wave\nmodes to isolated defect centers [7{10]. The ability to\nshape magnon modes with small volumes allows the re-\nalization of strong defect spin-magnon coupling [9].\nA practical challenge in working with V[TCNE] xis\nthat, like many organic-based and monolayer materials, it\nis air sensitive. The recent development of encapsulation\n∗Corresponding author: gdf9@cornell.edutechniques, similar to those used to protect organic light\nemitting diodes (OLEDs) [11{14], can extend its lifetime\nat room temperature and in ambient atmosphere from\nan hour to a few weeks [15]. However, the limitations in\nthe lifetime of V[TCNE] xand its associated aging mech-\nanisms are not well understood. For example, one aging\nmechanism is the chemical reaction with oxygen and wa-\nter, which turns a V[TCNE] x\flm from blue-green opaque\nto transparent [15]. This is not the only mechanism, as\nevidenced by aging observed in samples stored in an argon\nglove box and the slow change in the magnetic properties\nof encapsulated samples that remain opaque. Another\npiece of evidence is the slowing down of aging when the\nsample is stored at low temperature (-30\u000eC) in argon,\nsuggesting an internal change being the next dominant\naging mechanism.\nFirst we study accelerated aging through laser heating,\nwhich provides a clean condition where reactions with\noxygen and water are absent. Studying sample response\nto high intensity laser illumination is also of particular\nrelevance to proposed quantum interfaces between spin\nwaves and isolated defects, which are probed with a fo-\ncused laser beam. We observe a nonlinear dependence of\nlaser damage on optical power, which is consistent with a\nheating-based laser damage mechanism. Instead of being\nmerely a nuisance, this local laser damage opens up a new\navenue of V[TCNE] xpatterning, which is an alternative\nto electron-beam patterning [6]. We show preliminary\nresults on laser patterning of V[TCNE] xand discuss itsarXiv:2101.10240v1  [cond-mat.mtrl-sci]  25 Jan 2021resolution limit.\nTo better deploy V[TCNE] xoutside of a pristine envi-\nronment, e.g. inert atmosphere, we need a better under-\nstanding of its aging mechanisms. V[TCNE] xis amor-\nphous with a local structural order [16], making Ra-\nman spectroscopy an e\u000bective method for studying its\nstructural properties locally, at the micron scale. In\naddition to this optical probe, we concurrently mea-\nsure magnetic properties using ferromagnetic resonance\n(FMR) and superconducting quantum interference device\n(SQUID) magnetometry to correlate changes in chemical\nproperties with changes in magnetic properties.\nThe paper is structured as follows. We \frst describe\nsample fabrication and then explain the measurement\nprocedures for confocal microscopy, micro-focused Ra-\nman spectroscopy, FMR, and SQUID magnetometry.\nNext, we characterize V[TCNE] xusing the above meth-\nods. In particular, we show Raman spectra and explain\nthe associated vibrational modes and their relation to\nchemical structure. Next we study laser damage in detail,\nmonitoring how photoluminescence and Raman spectra\nchange with laser exposure. We show proof-of-concept\npatterning examples using laser damage to selectively re-\nmove magnetism. After measuring the optical and mag-\nnetic properties of pristine V[TCNE] x, we next study its\naging by monitoring how the above properties change in\ntime. In particular, we observe aged samples are more\nsusceptible to laser damage. We also identify Raman\npeaks observed in pristine samples which are absent in\naged or laser damaged samples.\nII. METHODS\nA. Sample growth\nV[TCNE] x\flms are grown by chemical vapor deposi-\ntion, similar to previous reports [4]. We examine 4 sam-\nples in the main text. The \frst sample (sample 1) is a\nuniform 400 nm V[TCNE] x\flm with 73 nm of aluminum\ndeposited with an evaporator, encapsulated with epoxy\nand a glass cover slip. V[TCNE] xthickness is estimated\nfrom growth time. The aluminum layer is designed to\nprevent the laser light from exciting epoxy, ensuring we\nare only probing V[TCNE] xRaman spectra and photo-\nluminescence.\nSample 2 is a 1.6 \u0016m thick sample. Sample 3 and 4 are\n400 nm thick samples grown in the same batch. All three\nsamples are encapsulated with glass and epoxy only.B.Ab initio calculations\nThe electronic structure and phonon modes of\nV[TCNE] xare calculated using the Vienna ab initio Sim-\nulation Package (VASP) (version 5.4.4) with a plane\nwave basis and projector-augmented-wave pseudopoten-\ntials [17{20]. These pseudopotentials use the generalized\ngradient approximation (GGA) of Perdew, Burke, and\nErnzerhof (PBE) [21]. A Hubbard constant U=4.19, de-\ntermined via a linear response method [22], was used\nin the phonon calculations. The hybrid functional\nHeyd{Scuseria{Ernzerhof (HSE) [23] with the standard\nrange separation parameter !=0.2 was also tested for this\nsystem and showed consistent results with the PBE+U\napproach.\nC. Optical and Magnetic Measurements\nOptical measurements including micro-focused Raman\nspectroscopy and photoluminescence are performed in a\nhomebuilt confocal microscope with a 532 nm continuous\nwave laser. Integrated light collected from the sample is\ndetected using a single photon counting module, whereas\nRaman spectra are recorded using a Princeton Instru-\nments spectrometer and low dark count camera.\nWe measure angle-resolved, \feld modulated FMR in\na homebuilt spectrometer. The setup consists of a mi-\ncrowave signal generator, an electromagnet, a pair of\nmodulation coils, a microwave diode detector and a lock\nin ampli\fer. Film magnetization is measured using a\nQuantum Design MPMS 3 SQUID magnetometer. Ap-\nplied magnetic \feld is in plane.\nIII. RAMAN SPECTROSCOPY\nWe measure Raman spectra to characterize V[TCNE] x\nchemical bonds and structure. To better understand\nthe Raman spectrum, we also perform density func-\ntional theory (DFT) calculations using VASP code. We\nshow the calculated V[TCNE] xstructure in \fgure 1(a),\nwhere the geometry agrees with previous DFT studies\n[24{26]. In \fgure 2, we plot the experimental Raman\nspectrum with DFT phonon density of states. Based\non comparison with DFT calculations, we assign 1300-\n1500 cm\u00001Raman peaks to C = C stretching modes, and\n2200 cm\u00001Raman peaks to C \u0011N stretching modes.\nExample vibrational modes are shown in \fgure 1(b),\ndisplaying modes with dominant C \u0011N (2352 cm\u00001) or\nC = C (1451 cm\u00001) bond stretching.\n2(b)\nV\nN\nC(a)FIG. 1. V[TCNE] xstructure and vibrational modes based on\ndensity functional theory (DFT) calculations. (a) V[TCNE] x\nstructure, showing 2 nonequivalent TCNE molecules. The\nbottom TCNE is bonded with 4 vanadium atoms ( \u00164 bonded)\nwhile the top TCNE is bonded with 2 vanadium atoms (trans-\n\u00162 bonded) (b) C \u0011N (2352 cm\u00001) and C = C (1451 cm\u00001)\nvibrational modes. Shown is the bottom ( \u00164 bonded) TCNE.\nHigh wavenumber vibrational modes ( >1000cm\u00001) are local-\nized on one or the other TCNE, with non-degenerate vibra-\ntional frequencies.\nNext, we explain the \fne Raman features and com-\npare them with previous studies. Fine features near 2202,\n2225 cm\u00001are in agreement with previous IR studies\n2194, 2214 cm\u00001[15, 27]. We observe 2121 cm\u00001in Ra-\nman spectroscopy, while previous IR spectroscopy only\nobserved a peak at 2155 cm\u00001[15], which could be ex-\nplained by di\u000berent IR and Raman activities of these\nvibrational modes. The Raman peaks at 1308, 1411,\n1530 cm\u00001are absent in IR spectrum, suggesting vana-\ndium atoms are more symmetrically bonded to the cen-\nter C = C bond, leading to low IR activity [27]. We at-\ntribute low wavenumber peaks at 336, 457, 543 cm\u00001to\nlow energy TCNE vibration modes and V \u0000N stretching\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000035/uni00000044/uni00000050/uni00000044/uni00000051/uni00000003/uni0000004c/uni00000051/uni00000057/uni00000048/uni00000051/uni00000056/uni0000004c/uni00000057/uni0000005c\n/uni0000000b/uni00000046/uni00000053/uni00000056/uni00000012/uni00000046/uni000000501/uni0000000c\n /uni0000000b/uni00000044/uni0000000c\n/uni00000013 /uni00000014/uni00000013/uni00000013/uni00000013 /uni00000015/uni00000013/uni00000013/uni00000013\n/uni0000003a/uni00000044/uni00000059/uni00000048/uni00000051/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni0000000b/uni00000046/uni000000501/uni0000000c\n/uni00000033/uni0000004b/uni00000052/uni00000051/uni00000052/uni00000051/uni00000003/uni00000027/uni00000032/uni00000036/uni0000000b/uni00000045/uni0000000cFIG. 2. V[TCNE] xRaman spectrum and density of states\n(DOS). (a) Experimental spectrum of pristine encapsulated\nV[TCNE] x. A broad baseline is subtracted from the raw Ra-\nman spectrum (see SI). (b) Ab initio V[TCNE] xphonon DOS.\nIn the plot, each mode is broadened as a Lorentzian with a\nFWHM of 20 cm\u00001.\nmodes.\nHere we have characterized Raman spectra of pristine\nencapsulated V[TCNE] xand assigned Raman peaks to\nparticular vibrational modes. Low wavenumber modes\nare particularly relevant for magnetism as they are in-\n\ruenced by V{N bonds. In subsequent sections, we mea-\nsure how these Raman peaks evolve under aging and laser\nheating induced damage, and show how Raman spectra\ncould be used to assess \flm quality.\nIV. PHOTOLUMINESCENCE AND LASER\nDAMAGE SUSCEPTIBILITY\nIn the following section, we study laser damage in a\npristine encapsulated V[TCNE] x\flm (sample 2). Using\nlaser-induced damage allows us to study aging in the ab-\nsence of oxygen and water. There is also independent\ninterest in studying V[TCNE] xunder focused laser in-\ntensity, in particular in relation to experiments coupling\nV[TCNE] xwith defect centers [9]. The goal here is to\ngain insights on the nature of aging and a quantitative\nmeasure of laser damage susceptibility. We note that the\nquantitative laser damage susceptibility is sample and\nencapsulation dependent and we focus on reproducible\ntrends across samples and relative changes.\nBecause the laser light is tightly focused, local temper-\nature can be large enough to cause degradation. Indeed,\n3(a)\n(c)(b)\n(d)FIG. 3. (a) Laser damage susceptibility \u001fPLshowing a nonlinear dependence on optical power. Also shown is a guide to the\neye showing an exponential dependence with power. Inset - photoluminescence time trace at an optical power 0.23 mW. (b)\nRaw Raman spectra taken at 0.10 mW after exposing the laser of di\u000berent powers for 4 minutes. (c)-(d) Baseline subtracted\nintegrated Raman intensity around 1400 cm\u00001. (d) Integrated Raman intensity around 2200 cm\u00001.\nphotoluminescence increases under continuous illumina-\ntion, which is accompanied by the V[TCNE] x\flm turning\ntransparent. Here we characterize laser damage suscepti-\nbility\u001fPLas the fractional rate of change in photolumi-\nnescence per unit optical power, namely\n\u001fPL=1\nPopt1\nPL(t= 0)d\ndtPL(t) (1)\nwherePLis the photoluminescence, Poptis the incident\nlaser power. We linear \ft the initial photoluminescence\nincrease (Fig. 3(a) inset) and extract \u001fPL.\nLaser damage susceptibility increases nonlinearly in op-\ntical power, with a threshold near 0.25 mW (Fig. 3(a)).\nA nonlinear laser damage susceptibility is consistent witha laser heating damage mechanism. While the temper-\nature rise is proportional to laser power, chemical reac-\ntion rates increase nonlinearly with temperature, causing\nmore rapid aging at a high optical power.\nNext we quantify how laser damage alters the Ra-\nman spectra. All measurements are done in a local area\n20\u0016m\u000220\u0016m, where the \flm is uniform. We suc-\ncessively illuminate di\u000berent spots with di\u000berent optical\npower for 4 minutes, and then measure Raman spectra\nwith a low laser power (93 \u0016W) for 4 minutes, with a\npeak laser intensity 5 \u0002104W=cm2. As the spot is ex-\nposed to higher laser power, the Raman intensity near\n2121 cm\u00001decreases with a corresponding increase in the\n2202, 2225 cm\u00001features (Fig. 3(b)). The side peaks at\n1308, 1530 cm\u00001of C=C bond increase in intensity and\n4linewidth. The lower wavenumber peaks <600 cm\u00001dis-\nappear. In addition, the total \ruorescence background\nincreases with increasing laser power. The quantitative\nchanges in Raman intensity are shown in \fgure 3(c,d).\nFocusing on qualitative features, the disappearance of\nlow wavenumber ( <600 cm\u00001) and the 2121 cm\u00001Ra-\nman peaks are clear, qualitative signatures of \flm aging.\nIn particular, low wavenumber vibrational modes have\nsigni\fcant V{N stretching components, therefore the dis-\nappearance of those peaks suggest a reduction in vana-\ndium bonding to TCNE groups. Likewise, the 2121 cm\u00001\nRaman peak corresponds to a C \u0011N vibrational modes,\nwhich is sensitive to vanadium nitrogen bonding and is\nrelevant for magnetic ordering in V[TCNE] x.\n10 μm\n(a)\n(b)\nFIG. 4. Laser patterning of a 400 nm thick V[TCNE] xsam-\nple (sample 4). (a) Photoluminescence map. High count rate\nregions are laser damaged, low count rate regions are the re-\nmaining undamaged material. (b) Grayscale optical micro-\ngraph of the same area. Laser damaged areas are more trans-\nparent and appear brighter in the image. This is patterned\nwith 1.7 mW laser power, corresponding to an optical inten-\nsity of 1:3\u0002106W=cm2.\nInstead of merely being a nuisance, laser damage could\nalso be used for patterning with micron-scale spatial ex-\ntent. We next explore using laser damage to pattern a\nV[TCNE] x\flm. If thermal degradation is nonlinear with\nlocal temperature rise, the attainable feature size could\nbe much smaller than the di\u000braction limit [28].\nWe show a proof-of-concept demonstration by laser\npatterning the authors' a\u000eliations on a V[TCNE] x\flm\n(Fig. 4(a)), where laser written area is damaged, produc-\ning a much higher photoluminescence rate. Based on the\nphotoluminescence map and the optical image, the laser\npatterned feature size is on the order of 1 \u0016m. Note the\nphotoluminescence map only measures optical properties,\nand the magnetic properties of laser patterned samples,\ne.g. magnetization pro\fle, supported spin wave modes,\nFIG. 5. Angle-resolved FMR measured at 3 GHz microwave\nfrequency. Fitted Heff= 68Oe. (Inset) Field sweep at 50\u000e.\nFitted FWHM linewidth is 1.1 Oe.\nwill be the topic of a future study.\nV. FERROMAGNETIC RESONANCE\nWe next measure the e\u000bective magnetization by angle-\nresolved FMR. The resonance frequency !with an ex-\nternal \feld much greater than the e\u000bective \feld of the\nmaterial (H\u001dHeff) is [4]\n!=\rq\n(H\u0000Heffcos2\u0012)(H\u0000Heffcos 2\u0012) (2)\nwhere\ris the gyromagnetic ratio, His the external mag-\nnetic \feld,\u0012is the angle of the external \feld with respect\nto the \flm normal, Heff= 4\u0019Meff= 4\u0019Ms\u0000Hk, which\nis a combination of the saturation magnetization and the\nanisotropy \feld. One possible origin of the anisotropy\n\feld is from strain due to di\u000berential thermal expansion\nof V[TCNE] xand the underlying substrate [29, 30]. In\n\fgure 5, we show a \feld-modulated FMR signal measured\nat 3 GHz. Fitting angle-resolved FMR, we extract an ef-\nfective \feld 4 \u0019Meff= 68 Oe. The FWHM linewidth is\n1.1 Oe, which is comparable to previously reported values\n[6].\nVI. AGING\nHaving characterized pristine encapsulated V[TCNE] x,\nwe next monitor its properties as it ages at room temper-\nature in ambient atmosphere. First, we note a visual\nchange in V[TCNE] x, beginning at the sample edge and\npropagating under the glass coverslip. This material is\ndark grey-black as deposited. As it ages, a transparency\n5Day 4\nDay 15 (a)\n100 μm\n(b)\n(c)FIG. 6. (a) Visual indication of V[TCNE] xaging, showing the\naging front advancing from day 4 to day 15. The V[TCNE] x\n\flm turns transparent and reveals the underlying re\rective\naluminum layer. (b) Raw Raman spectrum on day 1 (red),\nday 32 (blue) and day 89 (green) near sample center. The\noverall \ruorescence/Raman background \roor increases and a\nplateau arises near 1400 cm\u00001. Over long time (89 days), low\nwavenumber peaks (300-600 cm\u00001) and the peak near 2120\ncm\u00001vanishes. (c) Reduction in 4 \u0019Ms\u0000Hkand increase\nin laser damage susceptibility. Laser damage susceptibility\nmeasured at the center of the sample, using a laser power of\n100 - 200\u0016W. Fitted lines decay rate is 0.72 Oe/day and laser\ndamage susceptibility increases with a 1 =etime constant of 6.4\ndays.front propagates from the sample edges (Fig. 6(a)). We\nattribute this chemical change to reaction with di\u000bused\noxygen and water across the epoxy encapsulation barrier\nat sample edges.\nIn contrast, V[TCNE] xfar from the edges does not\nshow a strong color change with time, However, the lack\nof an apparent color change does not mean no changes\noccur. We next characterize sample photoluminescence\nand Raman spectra at the center of the sample and show\nhow they change in time. Raman spectra show an in-\ncrease in the \ruorescence background and an increase in\nthe Raman intensity near 1300 - 1500 cm\u00001as a function\nof time (Figure 6(b)). In addition, peaks near 1300, 1500\ncm\u00001increase in Raman intensity compared to the cen-\nter 1400 cm\u00001peak. Looking at an even longer timescale\n(89 days), the low wavenumber peaks (336, 457, 543\ncm\u00001) vanish, and the 2120 cm\u00001peak further dimin-\nishes. These features are qualitatively similar to those\nobserved in laser damaged samples, with the strongest\nchanges appearing in V{N and C \u0011N bonding. This sug-\ngests that the slow processes present in room temperature\naging are similar to the chemical reactions accelerated by\nlaser damage.\nA more drastic signature of aging is an increase in\nlaser damage susceptibility \u001fPL, which increases expo-\nnentially in time as the sample ages (Fig. 6(c)). Concur-\nrently, the e\u000bective magnetization Heff= 4\u0019Meff, as\nmeasured from angle-resolved FMR, decreases over time\n(Fig. 6(c)). This establishes a link between optical prop-\nerties and magnetic properties, indicating optical mea-\nsurements could be a local probe of V[TCNE] x\flm qual-\nity.\nOne limitation of the angle-resolved FMR is that it\nonly measures an intensive quantity Meff, which is a\nsum of shape anisotropy and other anisotropy, e.g. strain\ninduced anisotropy [30]. Moreover, it is not sensitive to\nthe total magnetic moment of the sample. To disentangle\nthese contributions and understand the total reduction of\nmagnetic material, we monitor aging of another 400 nm\nthick V[TCNE] x\flm (sample 3) with SQUID magnetom-\netry and FMR.\nHere we compare 3 quantities, total magnetic mo-\nmentmtotmeasured by SQUID magnetometry, e\u000bective\nmagnetization (4 \u0019Meff) and weighted total moment mw\nmeasured by angle-resolved FMR. Since mtotandmware\nextensive quantities while 4 \u0019Meffis an intensive quan-\ntity, we explain below in detail the measurement proce-\ndure for each quantity and how we compare them.\nWe compute saturation magnetization from SQUID\nmagnetometry data by normalizing the total moment to\nthe V[TCNE] x\flm volume. The volume has an uncer-\ntainty of up to 20%, which is dominated by the thickness\n6uncertainty. This study lacks a direct measure of the\nsample thickness for each sample, and there is growth-to-\ngrowth variation in the nominal deposition rate as well as\nvariation of growth rate across di\u000berent positions in the\ngrowth chamber. Nonetheless, we are most interested in\nhowMschanges in the aging process, which is less de-\npendent on initial sample volume. We also note that the\nvolume of ferrimagnetic V[TCNE] xis not constant as the\nsample ages due to the oxidation front that propagates\nfrom sample edge to center (see Fig. 6(a)), resulting in a\n33% reduction in opaque area over 32 days. In \fgure 7(b),\nwe plot the moment normalized to the initial V[TCNE] x\nvolume. We denote this quantity as 4 \u0019Msand we inter-\npret it as a scaled total ferrimagnetic moment.\nNext we discuss FMR measurements. We note the sam-\nple has a low loss fraction throughout the course of the ex-\nperiment, with a Gilbert damping varies from 1 :5\u000210\u00004\nto 2:5\u000210\u00004from day 1 to day 20 (Fig. 7(a)). We can\nunambiguously detect a resonance line shape and the typ-\nical uncertainty in 4 \u0019Meffis near 1 Oe. Because we\nmeasure FMR with \feld modulation, we are insensitive\nto high damping magnetic material, which will appear as\na broad background.\nThis weighted moment mwis computed based on\nFMR signal strength. With a microwave drive below\nsaturation and a low modulation \feld, the double in-\ntegrated signal is proportional to total magnetic mo-\nment [31, 32]. For an absorption derivative line shape\nL0\nabs(H) =a\u0001H3(H\u0000H0)\n(\u0001H2+4(H\u0000H0)2)2, this double integrated sig-\nnal is proportional to the product of signal amplitude\nand FWHM a\u0001H. We normalize a\u0001Hto microwave\npower and modulation amplitude (see SI) to compute\nmw, which is proportional to the magnetic moment. Be-\ncause the sample (2450 \u0016m) is several times wider than\nthe microwave waveguide (430 \u0016m), FMR sensitivity is\nnon-uniform across the sample. Hence we note mwis\na measure of weighted magnetic moment and is more\nsensitive to the sample portion closer to the microwave\nwaveguide. Moreover, the sample has been remounted\nmultiple times as we alternate between the SQUID and\nFMR setup, so the coupling between the V[TCNE] x\flm\nand the microwave waveguide varies, which contributes\nan uncertainty of nearly a factor of 2. With the above\nsubtleties pointed out, we use FMR intensity mwto esti-\nmate weighted magnetic moment with low damping.\nOver the course of the study (38 days), 4 \u0019Meffand\nmwreduces by nearly 2 orders of magnitude (Fig. 7(b)).\nIn comparison, 4 \u0019Meffonly changes from 95 \u00062 Oe to\n45\u00061 Oe. These \fndings suggest an aging process where\nthe magnetic portion shrinks while remaining low damp-\ning. This is consistent with an aging front propagatingfrom sample edge to center, where the center is relatively\npristine. These measurements suggest intrinsic aging due\nto internal chemical reaction at room temperature does\nnot increase damping signi\fcantly over 20 days.\nTo summarize the results of this section over several\nobservations, we \fnd a qualitative correlation between\nmagnetic properties and optical properties of V[TCNE] x.\nNamely, a decrease in 4 \u0019Meffcorrelates with the dis-\nappearance of Raman features (300-600 cm\u00001) involving\nV{N bonds and an increase in laser damage susceptibil-\nity. DC magnetometry, in combination with FMR, re-\nveals non-uniform aging resulting in a large decrease in\ntotal magnetic moment. The combination of optical and\nmagnetic measurements allow a comprehensive study of\nV[TCNE] xproperties, relating change in chemical prop-\nerties and magnetic properties.\nVII. CONCLUSION\nWe characterize pristine encapsulated V[TCNE] xthin\n\flms using confocal microscopy, micro-focused Raman\nspectroscopy, FMR and SQUID magnetometry. Through\ncomparison with ab initio calculations, we associate the\nexperimentally observed Raman peaks with particular\nC\u0011N;C = C stretching modes. We measure how the\nsample photoluminescence depends on laser power and\nobserve that laser damage susceptibility has a nonlinear\ndependence on laser power, which is consistent with a\nheating based laser damage mechanism. We further ex-\nplore laser damage as a means of patterning. Studying\nthe spatial pro\fle of laser damaged V[TCNE] xand their\nspin waves mode is a subject for future studies.\nWe identify changes in Raman features and laser dam-\nage susceptibility as sample ages. Low wavenumber (300-\n600 Cm\u00001) Raman features associated with V{N bonds\nvanish as sample ages, suggesting changes in bonding be-\ntween vanadium and TCNE. These \fndings show that\noptical measurement is a local probe of V[TCNE] x\flm\nmagnetic quality and could assess magnetic microstruc-\nture quality.\nThe existence of a narrow FMR response in encapsu-\nlated V[TCNE] xover 20 days under ambient conditions\nsuggests that intrinsic aging (e.g. not oxidation) does not\nincrease damping over this time interval. This is promis-\ning for coherent magnonics using V[TCNE] xmicrostruc-\ntures that are positioned far from encapsulation edges.\nFor quantum applications that use V[TCNE] xat cryo-\ngenic temperatures, one expects that the damping prop-\nerties will be preserved over an even longer timescale.\n7(a)\n(b)FIG. 7. Magnetic properties of an encapsulated V[TCNE] x\n\flm as it ages under ambient conditions. (a) The Gilbert\ndamping parameter \u000bextracted from FMR measurements in\nthe range 2-11 GHz with an out of plane DC magnetic \feld.\n(Inset) Gilbert damping \ft on day 1. \u000b= (1:8\u00060:1)\u000210\u00004\n(b) Magnetic measurements vs. sample aging time. SQUID\nmagnetometry measurements and FMR intensity consistently\nshow a large decrease in magnetic moment. In comparison,\n4\u0019Meff= 4\u0019Ms\u0000Hkshows a much smaller change over the\nsame period of time.\nVIII. ACKNOWLEDGEMENTS\nOptical Raman experiments, optical writing experi-\nments, waveguide FMR studies of aging, \frst principles\ntheory, and sample growth of aging study samples were\nsupported by the US Department of Energy O\u000ece of\nScience, O\u000ece of Basic Energy Sciences under Award\nDE-SC0019250. Control sample growth and cavity FMR\ncharacterization, as well as growth of 2{4 year old sam-\nples were supported by NSF DMR-1808704 and DMR-\n1507775. We acknowledge use of the facilities of the Cor-nell Center for Materials Research, which is supported\nthrough the NSF MRSEC program (DMR-1719875), and\nthe Cornell NanoScale Facility, which is a member of\nthe National Nanotechnology Coordinated Infrastructure\nand supported the NSF (NNCI-2025233). We acknowl-\nedge helpful discussions with Brendan McCullian. We\nthank Hong Tang and Na Zhu for providing 2{4 year old\nV[TCNE] xreference samples.\nREFERENCES\n[1] Christoph Hauser, Tim Richter, Nico Homonnay, Chris-\ntian Eisenschmidt, Mohammad Qaid, Hakan Deniz, Di-\netrich Hesse, Maciej Sawicki, Stefan G. Ebbinghaus, and\nGeorg Schmidt. Yttrium iron garnet thin \flms with very\nlow damping obtained by recrystallization of amorphous\nmaterial. Scienti\fc Reports , 6(1):20827, Feb 2016.\n[2] Yiyan Sun, Young-Yeal Song, Houchen Chang, Michael\nKabatek, Michael Jantz, William Schneider, Mingzhong\nWu, Helmut Schultheiss, and Axel Ho\u000bmann. 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Rapid measurement of thermal conductiv-\nity of polymer \flms. Review of Scienti\fc Instruments ,\n53(6):884{887, 1982.\n[43] Shouhang Li, Xiaoxiang Yu, Hua Bao, and Nuo Yang.\nHigh thermal conductivity of bulk epoxy resin by bottom-\nup parallel-linking and strain: A molecular dynam-\nics study. The Journal of Physical Chemistry C ,\n122(24):13140{13147, 2018.\n[44] Quantum Design. Correcting for the Absolute Field Error\nusing the Pd Standard , 3 2018. Rev. A0.IX. SUPPLEMENTAL INFORMATION\nA. Confocal measurement\nV[TCNE] xsamples are measured in a homebuilt con-\nfocal microscope with a 532 nm continuous wave laser \fl-\ntered with a bandpass \flter (Iridian 532 BPF, ZX000163).\nIt is focused with a 50x, 0.7 NA objective (Olympus\nLCPLFLN50xLCD) on the sample. Re\rection and \ru-\norescence from the sample is collected and then \fl-\ntered by a long pass \flter (Iridian 532 LPF nano cuto\u000b,\nZX000850). This light is split between a silicon avalanche\nphotodiode (Excelitas SPCM-AQRH-13-FC) and a spec-\ntrometer (Princeton Acton SP-2500, focal length 500 nm\nwith a 300 g/mm grating) with a low dark count camera\n(PyLoN 100 BR).\nAn incident laser beam of 5 mm 1 =e2diameter illumi-\nnates the objective back aperture (5 mm diameter). At\nthis truncation ratio, the di\u000braction limited 1 =e2beam\ndiameter is 700 nm [33]. At a laser power 500 \u0016W, peak\nlaser intensity is 2 :6\u0002105W=cm2.\nB. Raman spectroscopy\nThe spectrometer is calibrated with IntelliCal wave-\nlength source and intensity source [34]. Absolute Ra-\nman spectra wavelength accuracy is limited by excita-\ntion laser wavelength accuracy, which limits the Raman\nspectroscopy accuracy to 4-8 cm\u00001.\nFor samples that are more susceptible to laser damage,\nwe reduce laser power and raster scan across \u0018100\u0016m2\nto reduce accumulated laser damage.\nWe remove cosmic rays from Raman spectra by re-\nmoving data points that have counts higher than its\nneighbor above a threshold. The Raman baseline is \ftt\nwith asymmetrically reweighted penalized least squares\nsmoothing [35] with the implementation in the Python\nmodule Rampy [36]. Processed Raman spectra are \ftted\nwith Lorentzian peaks.\nC.Ab initio calculations\nThe electronic structure and phonon modes of\nV[TCNE] xare calculated using the Vienna ab initio Sim-\nulation Package (VASP) (version 5.4.4) , which uses a\nplane-wave basis and pseudopotentials [17{20]. The pseu-\ndopotentials we used are default options from VASP's of-\n\fcial PAW potential set, with 5 valance electrons per V,\n104 valance electrons per C, and 5 valance electrons per N\natom [37, 38].\nThese pseudopotentials use the generalized gradient\napproximation (GGA) of Perdew, Burke, and Ernzerhof\n(PBE) [21]. A PBE+U approach with U=4.19 was used\nin the phonon calculation; with U determined via a linear\nresponse method [22]. This approach is chosen as simple\nGGA calculations fail to capture the d-orbital behavior\nof the V atom in our electronic structure calculations.\nHybrid functional Heyd{Scuseria{Ernzerhof (HSE) [23]\nwith standard range separation parameter !=0.2 was also\ntested for this system and showed consistent results with\nthis PBE+U approach. For the rest of the calculation,\nwe used 400 eV for the energy cuto\u000b, and a \u0000-centered\n2\u00022\u00022 k-mesh sampling.\nThe geometry of V[TCNE] xis consistent with the work\nof De Fusco et al. [24]. The unit cell has a triclinic\nstructure and consists of 1 V atom, 12 C atoms and 8 N\natoms. The structure used in the phonon calculation is\nour own relaxed system under the same set of parameters.\nD. Ferromagnetic resonance\nSample 1 is placed on a 50 \u0016m inner diameter loop an-\ntenna. A microwave signal is applied to the antenna with\na signal generator (Anristu MG3692C). The re\rected sig-\nnal is routed by a circulator (Narda-MITEQ Model 4923,\n2 - 4 GHz) to a diode detector (Herotek DHM185AB).\nWe sweep the magnetic \feld at a \fxed microwave fre-\nquency (3 GHz typical) and modulate an external \feld\nat a modulation frequency of 587 Hz. Detected power is\nfed into a lock in ampli\fer (Signal Recovery 7265) and\ndetected at the modulation frequency.\nSample 3 is placed on a microstrip test board (South-\nwest Microwave B4003-8M-50). We launch microwave\nsignal at one end and do lock in detection of the trans-\nmitted microwave power.\nAs the sample ages, we increase modulation \feld am-\nplitude and microwave power to maintain a large enough\nsignal to noise.\nE. FMR data analysis\nThe spectra are \ftted with sums of Lorentzian deriva-\ntives, each having an absorptive and a dispersive part\n[39].L0\nabs(H) =a\u0001H3(H\u0000H0)\n(\u0001H2+ 4(H\u0000H0)2)2(3)\nL0\ndisp(H) =d\u0001H2(\u0001H2\u00004(H\u0000H0)2)\n(\u0001H2+ 4(H\u0000H0)2)2(4)\nwhere \u0001His the full-width at half-maximum of the\nLorentzian.\nIn angle-resolved FMR, we \ft with the following func-\ntion, using \r;H eff;\u00120set as free parameters.\n!=\rq\n(H\u0000Heffcos2(\u0012\u0000\u00120))(H\u0000Heffcos 2(\u0012\u0000\u00120))\n(5)\nwhere\ris the gyromagnetic ratio, Heff= 4\u0019Meffis\nthe e\u000bective magnetization, and \u00120is a constant o\u000bset\nbetween the real \flm normal with respect to the nominal\n\flm normal.\nNext we explain the procedure of extracting the total\nmagnetic moment from the FMR signal strength. The\nchange in waveguide transmission parameter on magnetic\nresonance is [40, 41]\n\u0001S21(Hres)/\r\u00160Msldm\n8Z0\u000bw(6)\nwherelis the \flm length along the transmission line, dm\nis the \flm thickness, Z0is the characteristic impedance\nof the transmission line, and \u000bis the Gilbert damping.\nAt a \fxed resonance \feld, the absorption curve FWHM is\n\u0001Hfwhm = 2\u000bHres. Hence the integrated area under the\nabsorption curve with respect to external H\feld is pro-\nportional to a weighted magnetic moment, independent\nof damping.\nFrom the absorption derivative term L0\nabs(H) =\na\u0001H3(H\u0000H0)\n(\u0001H2+4(H\u0000H0)2)2, we extract an equivalent area aH0as\nan estimate of the total magnetic moment.\nWe adjust for microwave power by normalizing the de-\ntected signal to microwave power ( P0) and the power-to-\nvoltage conversion factor ( K(V/mW)) of the diode de-\ntector. Namely, the normalized signal is\nVsig\nP0K(7)\nIn modulation detection, when modulation \feld is\nmuch smaller than linewidth, the lock in detected signal\nis proportional to a derivative of a Lorentzian absorp-\ntion. When modulation \feld is comparable to linewidth,\ndetected signal is no longer a simple derivative and we\n11correct for that by modeling the line shape.\nNote the FMR line shape has both an absorptive\nderivative and a dispersive derivative. Here we only use\nthe absorptive derivative to extract the total magnetic\nmoment.\nF. Laser heating\nIn this section, we estimate local temperature rise due\nto laser heating of a V[TCNE] x\flm grown on a glass\nsubstrate. We approximate the glass substrate thermal\nconductivity to be \u00141= 1:4Wm\u00001K\u00001. We approx-\nimate V[TCNE] xand epoxy to have a similar thermal\nconductivity \u00142= 0:3Wm\u00001K\u00001[42, 43].\nBecause the laser spot size (350 nm) is much smaller\nthan the epoxy thickness (5-10 \u0016m), we model both the\nsubstrate and V[TCNE] xplus epoxy as semi-in\fnite. The\ntemperature rise pro\fle for a point heat source at the\ninterface is\nT(r) =1\n4\u0019\u0016\u0014_Q\nr(8)\nwhere \u0016\u0014= (\u00141+\u00142)=2, assuming temperature is \fxed at\n0 at in\fnity.\nWe model the actual heat source as a surface Gaussian\nheat source\n2P0\n\u0019w2exp\u0012\n\u00002r2\nw2\u0013\n(9)\nwherewis the 1=e2beam radius.\nThe maximum temperature rise occurs at the interface\nalong the beam center,\nTmax=1\n2p\n2\u0019\u0016\u0014P0\nw(10)\nAt 0.7 NA, 1 =e2beam radius w\u0019350 nm. 100 \u0016W of\npower causes a local temperature rise of 70 K.\nA more realistic estimate takes into account of the \fnite\nabsorption depth of V[TCNE] x, which is of the order 100-\n400 nm. As this is comparable to the beam size, the peak\ntemperature rise is of the same order of magnitude.\nG. Long term aging\nVisual inspection provides preliminary information on\nV[TCNE] x\flm quality, but it is neither a quantitative\nnor a de\fnitive measure. While discolored V[TCNE] xis almost certainly aged, even fully opaque \flm can be\nmagnetically inactive[15].\nTo better understand long term degradation, we ex-\namine 2 other V[TCNE] xsamples. One is a 4 year old\nuniform \flm, the other is a 2 year old, patterned 100\n\u0016mwide bars. Both of them are nominal 1 \u0016m thick,\nestimated from growth time. Both of them are visually\nopaque and yet are magnetically inactive.\nThe samples show qualitatively similar peaks (Fig. 8),\nwith 3 peaks near 1300-1500 cm\u00001and a single peak\nnear 2200 cm\u00001. Under strong laser illumination, 3 peaks\n1300-1500 cm\u00001merge into 2 peaks, and become brighter\nthan the 2200 cm\u00001peak. Crucially, the spectra di\u000ber\nsigni\fcantly from aged V[TCNE] x. Degraded samples\ndon't have low wavenumber peaks (300-600 cm\u00001) and\nthe small 2120 cm\u00001peak as seen in pristine samples,\nfurther supporting a link between these \fne features with\ngood magnetic properties.\nH. SQUID magnetometry\nWe measure a 400 nm thick V[TCNE] x\flm (sample 3)\nin a Quantum Design MPMS 3 SQUID magnetometer in\nthe vibrating sample magnetometer (VSM) mode. The\nsample is mounted to a quartz holder with a small dab\nof GE varnish. We measure moment vs. \feld at 300 K\nin a four-quadrant sweep, from {20,000 Oe to +20,000\nOe back to {20,000 Oe. We use a palladium reference to\ncorrect for the absolute \feld error [44]. The applied \feld\nis in the plane of the V[TCNE] x\flm.\nWe normalize the measured magnetic moment to the\ninitial \flm volume, which is estimated from the opaque\n\flm area (7.27 mm2) and the nominal \flm thickness 400\nnm.\nThe magnetization vs. \feld data has a large diamag-\nnetic background (Fig. 9(c)), which we subtracted away\nby \ftting to a linear background at \felds jHj>2000\nOe. We also measure the magnetic response GE varnish,\nwhich has a small diamagnetic and ferromagnetic compo-\nnent (Fig. 9(a,b)). The magnetic moment of GE varnish\nis normalized to the same initial V[TCNE] xvolume. Note\nthat this is an equivalent V[TCNE] xmagnetization error\ndue to GE varnish background but not the physical GE\nvarnish magnetization. As the amount of GE varnish ap-\nplied in di\u000berent runs varies, one cannot simply subtract\na \fxed background from the data. We note GE varnish\nhas a gradual moment vs. \feld dependence over {1000\nOe to +1000 Oe (Fig. 9(b)). Because the applied \feld is\nin plane, V[TCNE] xshould be close to fully magnetized\nat 200 Oe. Therefore, we estimate V[TCNE] x4\u0019Msto\nbe the magnetization at 200 Oe, where background mag-\n12(a)\n(d)\n(g)(b)\n(e)\n(h)(c)\n(f)\n(i)\nFIG. 8. Long term study of encapsulated V[TCNE] xRaman spectra. The Raman spectra are baseline subtracted. We probe\nthe center opaque areas away from encapsulation edges in all 3 samples. (a) 4 year old, 1 \u0016m thick uniform \flm. (b) 2 year old,\n1\u0016m thick, patterned 100 \u0016m wide bar sample. (c) (Sample 1 in the main text) 89 days old, 400 nm thick \flm with aluminum\nencapsulation. Note that the V[TCNE] x\flm has been remounted on the antenna and their relative position di\u000ber from that\nin the main text. (d)-(f) The laser is raster scanned across near a 50 \u0016m2area to reduce laser damage. Laser power is (d) 190\n\u0016W, (e) 190 \u0016W, (f) 51\u0016W . (g)-(i) The laser power is 1.9 mW with the laser focused at one spot to evaluate the e\u000bect of laser\ndamage.\nnetization is small. We estimate the uncertainty as the di\u000berence between magnetization at 200 Oe and 500 Oe.\n13(a) (b)\n(c) (d)FIG. 9. Magnetization 4 \u0019Mvs. \feld for GE varnish and sample 3. (a)-(b) Quartz sample holder and GE varnish response.\nMeasured moment is normalized to initial V[TCNE] xvolume. (a) Total magnetization (b) Linear background subtracted\nmagnetization. (c)-(d) V[TCNE] xsample on day 3. (c) Total magnetization (d) Linear background subtracted magnetization.\n14"    },    {        "title": "1706.06379v1.Magnetic_signatures_of_quantum_critical_points_of_the_ferrimagnetic_mixed_spin__1_2__S__Heisenberg_chains_at_finite_temperatures.pdf",        "content": "arXiv:1706.06379v1  [cond-mat.stat-mech]  20 Jun 2017Journal of Low Temperature Physics manuscript No.\n(will be inserted by the editor)\nMagnetic signatures of quantum critical points of\nthe ferrimagnetic mixed spin-(1/2, S) Heisenberg\nchains at finite temperatures\nJozef Streˇ cka ·Taras Verkholyak\nReceived: date / Accepted: date\nAbstract Magneticpropertiesoftheferrimagneticmixedspin-(1/2, S)Heisen-\nberg chains are examined using quantum Monte Carlo simulations for t wo dif-\nferent quantum spin numbers S= 1 and 3/2. The calculated magnetization\ncurves at finite temperatures are confronted with zero-temper ature magne-\ntization data obtained within density-matrix renormalization group m ethod,\nwhich imply an existence of two quantum critical points determining a b reak-\ndown of the gapped Lieb-Mattis ferrimagnetic phase and Tomonaga -Luttinger\nspin-liquid phase, respectively. While a square-root behavior of the magneti-\nzation accompanying each quantum critical point is gradually smooth ed upon\nrising temperature, the susceptibility and isothermal entropy cha nge data pro-\nvide a stronger evidence of the quantum critical points at finite tem peratures\nthrough marked local maxima and minima, respectively.\nKeywords ferrimagnetic Heisenberg chains ·quantum critical point ·\nquantum Monte Carlo\nPACS75.10.Pq ·75.10.Kt ·75.30.Kz ·75.40.Cx ·75.60.Ej\nThis work was financially supported by the grant of The Minist ry of Education, Science,\nResearch and Sport of the Slovak Republic under the contract No. VEGA 1/0043/16, as\nwell as, by grants of the Slovak Research and Development Age ncy provided under Contract\nNos. APVV-0097-12 and APVV-14-0073.\nJozef Streˇ cka\nInstitute of Physics, Faculty of Science of P. J. ˇSaf´ arik University,\nPark Angelinum 9, 040 01 Koˇ sice, Slovak Republic\nE-mail: jozef.strecka@upjs.sk\nTaras Verkholyak\nInstitute for Condensed Matter Physics, NASU,\n1 Svientsitskii Street, 790 11 L’viv-11, Ukraine\nE-mail: werch@icmp.lviv.ua2 Jozef Streˇ cka, Taras Verkholyak\n1 Introduction\nQuantum phase transitions traditionally attract a great deal of at tention, be-\ncause they are often accompanied with several remarkable signat ures exper-\nimentally accessible at nonzero temperatures [1]. One-dimensional q uantum\nspin chains provide notable examples of condensed matter systems , which\nbring a deeper understanding into exotic forms of magnetism nearb y quan-\ntum critical points based on rigorous calculations [2]. Apart from con ven-\ntional solid-state compounds affording experimental realizations o f the quan-\ntum spin chains [3], one may simulate a quantum phase transition in spin\nchains through ultracold atoms confined in an optical lattice [4]. Frac tional\nmagnetizationplateaux[5],magnetizationcuspsingularities[6]andsp in-liquid\nground states [5,6,7,8] can be regarded as the most profound m anifestations\nof zero-temperature magnetization curves of the quantum spin c hains.\nThe ferrimagnetic mixed spin-( s,S) Heisenberg chains [9,10,11,12,13,14,\n15,16,17,18] with regularly alternating spins s= 1/2 andS >1/2 display\nthe intermediate magnetization plateau inherent to the gapped Lieb -Mattis\nferrimagnetic ground state, as well as, the gapless Tomonaga-Lu ttinger spin-\nliquid phase. It has been argued [18] on the grounds of Lieb-Mattis t heorem\n[19] and Oshikawa-Yamanaka-Affleck rule [20] that the ferrimagnet ic mixed\nspin-(1/2, S)Heisenberg chains should exhibit just one plateau at the following\nvalue of the total magnetization m/ms= (2S−1)/(2S+1) normalized with\nrespect to its saturation. The gapped Lieb-Mattis ferrimagnetic g round state\nbreaks down at a quantum phase transition towards the gapless To monaga-\nLuttinger spin-liquid phase, which is accompanied with a singular squar e-root\nbehaviorof the magnetization. The same type of the magnetization singularity\nalsoappearsjustbelowthesaturationfield,whichdeterminesaqua ntumphase\ntransition between the Tomonaga-Luttinger spin liquid and the fully p olarized\nphase.Inthepresentworkwewillexaminemagneticsignaturesoft hequantum\ncritical points of the ferrimagnetic mixed spin-(1/2, S) Heisenberg chains at\nfinite temperatures by making use of quantum Monte Carlo simulation s.\n2 Model and method\nLetusconsidertheferrimagneticmixedspin-(1/2, S)Heisenbergchainsdefined\nthrough the Hamiltonian\nˆH=JL/summationdisplay\nj=1ˆSj·(ˆsj+ˆsj+1)−hL/summationdisplay\nj=1(ˆSz\nj+ ˆsz\nj), (1)\nwhereˆsj≡(ˆsx\nj,ˆsy\nj,ˆsz\nj) andˆSj≡(ˆSx\nj,ˆSy\nj,ˆSz\nj) denote the standard spin-1/2\nand spin- Soperators, respectively. The first term in the Hamiltonian (1) take s\ninto account the antiferromagnetic Heisenberg interaction J >0 between the\nnearest-neighborspins and the second term h=gµBHincorporatingthe equalMagnetic signatures of quantum critical points of the Heise nberg chains 3\nLand´ e g-factors gs=gS=gand Bohr magneton µBaccounts for the Zee-\nmann’s energy of individual magnetic moments in an external magnet ic field.\nSince the elementary unit contains two spins an overall chain length is 2L,\nwhereas a translational invariance is ensured by the choice of perio dic bound-\nary conditions ˆsL+1≡ˆs1.\nTo explore magnetic properties of the ferrimagnetic mixed spin-(1/ 2,S)\nHeisenberg chains at nonzero temperatures, we have implemented a directed\nloopalgorithminthestochasticseriesexpansionrepresentationof thequantum\nMonte Carlo (QMC) method [21] from Algorithms and Libraries for Phy sics\nSimulations (ALPS) project [22]. The QMC method allows a straightfor ward\ncalculation of the magnetization data at finite temperatures, which will be\nalso confronted with recent zero-temperature magnetization da ta calculated\nwithin the density-matrix renormalization group (DMRG) method [18] serv-\ning as a useful benchmark at low enough temperatures. The susce ptibility of\nthe ferrimagnetic mixed spin-(1/2, S) Heisenberg chains can also be directly\ncalculated from a directed loop algorithm of QMC method, while the isot her-\nmal entropy change can be obtained from the relevant magnetizat ion data\nusing the Maxwell’s relation\n∆ST=/integraldisplayh\n0/parenleftbigg∂m\n∂T/parenrightbigg\nhdh. (2)\nTo avoid an extrapolation due to finite-size effects, we have perfor med QMC\nsimulations for a sufficiently large system size with up to L= 128 units (256\nspins), whereas adequate numerical accuracy was achieved thro ugh 750 000\nMonte Carlo steps used in addition to 150 000 steps for thermalizatio n.\n3 Results and discussion\nLet us proceed to a discussion of the most interesting results for t he magneti-\nzation and susceptibility data of the ferrimagnetic mixed spin-(1/2, S) Heisen-\nberg chains. Fig. 1(a) shows a three-dimensional (3D) plot of the m agneti-\nzation of the ferrimagnetic mixed spin-(1/2,1) Heisenberg chain aga inst the\nmagnetic field and temperature. As one can see, the one-third plat eau due to\nthe gapped Lieb-Mattis ferrimagnetic ground state diminishes upon increas-\ning of temperature until it becomes completely indiscernible above ce rtain\ntemperature kBT/J≈0.5. To explore the temperature effect in more detail,\nFig. 1(b) compares a zero-temperature magnetization curve obt ained within\nDMRG method [18] with low-temperature magnetization data stemmin g from\nQMC simulations. It is quite evident that the square-root singularity of the\nmagnetization, which emerges at both quantum critical points dete rmining an\nupper edge of the one-third plateau and saturation field, is gradua lly rounded\nupon raising temperature.\nA stronger evidence of two quantum phase transitions of the ferr imagnetic\nmixed spin-(1/2,1)Heisenberg chain thus provides the susceptibility , which is\nplotted in Fig. 2. As a matter of fact, the susceptibility still displays a t low4 Jozef Streˇ cka, Taras Verkholyak\n/s48/s49/s50/s51/s52\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s107\n/s66/s32\n/s84\n/s32/s47/s32\n/s74/s109/s32/s47/s32/s109/s115\n/s104/s32/s47/s32/s74/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s109 /s32/s47/s32 /s109\n/s115\n/s40/s97/s41\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s98/s41/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s48/s32/s40/s68/s77/s82/s71/s41\n/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s49/s32/s32/s40/s81/s77/s67/s41\n/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s50/s32/s32/s40/s81/s77/s67/s41\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s51/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s49/s46/s55/s54/s49/s47/s51 /s112/s108/s97/s116/s101/s97/s117/s49/s47/s50 /s49\n/s32/s32/s109/s32/s47/s32/s109\n/s115\n/s104 /s32/s47/s32 /s74\nFig. 1 The total magnetization of the ferrimagnetic mixed spin-(1 /2,1) Heisenberg chain\nnormalized with respect to its saturation value: (a) 3D surf ace plot as a function of temper-\nature and magnetic field constructed from QMC data; (b) Zero- temperature DMRG data\nversus QMC magnetization curves at low enough temperatures .\n/s48\n/s49\n/s50\n/s51\n/s52/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s40/s97/s41\n/s32/s74\n/s107/s66/s32/s84/s32/s47/s32/s74\n/s104\n/s32/s47/s32\n/s74/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s74\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s40/s98/s41/s107\n/s66/s32/s84 /s32/s47/s32 /s74/s32\n/s32/s48/s46/s48/s32/s40/s68/s77/s82/s71/s41\n/s32/s48/s46/s48/s53/s32/s40/s81/s77/s67/s41\n/s32/s48/s46/s49/s48/s32/s40/s81/s77/s67/s41\n/s32/s48/s46/s50/s48/s32/s40/s81/s77/s67/s41\n/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s51/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s49/s46/s55/s54/s115/s112/s105/s110/s32/s108/s105/s113/s117/s105/s100/s49/s47/s50 /s49\n/s32/s32/s74\n/s104 /s32/s47/s32 /s74\nFig. 2The susceptibility of the ferrimagnetic mixed spin-(1/2,1 ) Heisenberg chain per unit\ncell: (a) 3D surface plot as a function of temperature and mag netic field constructed from\nQMCdata; (b) Zero-temperature DMRGdata versus QMCsimulat ions at lowtemperatures.\nenough temperatures two sharp peaks in a vicinity of the magnetic fi elds driv-\ning the investigated quantum spin system towards the quantum crit ical points\nin addition to a pronounced divergence observable at zero magnetic field. Of\ncourse, these local maxima are gradually suppressed and merge to gether upon\nincreasing of temperature.\nTo illustrate a general validity of the aforedescribed conclusions, F igs. 3\nand 4 show similar plots for the magnetization and susceptibility of ano ther\nferrimagnetic mixed spin-(1/2,3/2) Heisenberg chain. It is quite app arent that\nthe same general trends can be reached as far as the temperatu re effect is con-\ncerned. There are only two quantitative differences. The first diffe rence refers\nto a height of the intermediate Lieb-Mattis plateau, which is in the pre sent\ncase at one-half of the saturation magnetization. The second diffe rence lies\nin absolute values of two field-driven quantum critical points, which d eter-\nmine the end of the intermediate one-half plateau and the beginning o f the\nsaturation magnetization, respectively.Magnetic signatures of quantum critical points of the Heise nberg chains 5\n/s48/s49/s50/s51/s52/s53\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s107\n/s66/s32\n/s84\n/s32/s47/s32\n/s74/s109/s32/s47/s32/s109/s115\n/s104/s32/s47/s32/s74/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s109 /s32/s47/s32 /s109\n/s115\n/s40/s97/s41\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s98/s41/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s48/s32/s40/s68/s77/s82/s71/s41\n/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s49/s32/s32/s40/s81/s77/s67/s41\n/s32/s107\n/s66/s84/s32 /s47/s32 /s74 /s32/s61/s32/s48/s46/s50/s32/s32/s40/s81/s77/s67/s41\n/s49/s47/s50 /s112/s108/s97/s116/s101/s97/s117\n/s49/s47/s50 /s51/s47/s50\n/s32/s32/s109/s32/s47/s32/s109\n/s115\n/s104 /s32/s47/s32 /s74/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s52/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s50/s46/s56/s52\nFig. 3The total magnetization of the ferrimagnetic mixed spin-(1 /2,3/2) Heisenberg chain\nnormalized with respect to its saturation value: (a) 3D surf ace plot as a function of temper-\nature and magnetic field constructed from QMC data; (b) Zero- temperature DMRG data\nversus QMC magnetization curves at low enough temperatures .\n/s48\n/s49\n/s50\n/s51\n/s52\n/s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s40/s97/s41\n/s32/s74\n/s107/s66/s32/s84/s32/s47/s32/s74\n/s104\n/s32/s47/s32\n/s74/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s52/s48/s46/s53/s74\n/s48/s46/s48 /s48/s46/s53 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s40/s98/s41/s107\n/s66/s32/s84 /s32/s47/s32 /s74/s32\n/s32/s48/s46/s48/s32/s40/s68/s77/s82/s71/s41\n/s32/s48/s46/s48/s53/s32/s40/s81/s77/s67/s41\n/s32/s48/s46/s49/s48/s32/s40/s81/s77/s67/s41\n/s32/s48/s46/s50/s48/s32/s40/s81/s77/s67/s41\n/s115/s112/s105/s110/s32/s108/s105/s113/s117/s105/s100/s49/s47/s50 /s51/s47/s50\n/s32/s32/s74\n/s104 /s32/s47/s32 /s74/s104\n/s115/s32/s47/s32 /s74/s32 /s61/s32/s52/s46/s48/s48/s104\n/s99/s32/s47/s32 /s74/s32 /s61/s32/s50/s46/s56/s52\nFig. 4 The susceptibility of the ferrimagnetic mixed spin-(1/2,3 /2) Heisenberg chain per\nunit cell: (a) 3D surface plot as a function of temperature an d magnetic field constructed\nfrom QMC data; (b) Zero-temperature DMRG data versus QMC sim ulations at low enough\ntemperatures.\nLast but not least, let us examine the magnetic-field variations of th e\nisothermal entropy change, which represents one of two basic ma gnetocaloric\npotentials. For this purpose, Fig. 5 illustrates typical dependence s of the\nisothermal entropy change of two considered ferrimagnetic mixed spin-(1/2, S)\nHeisenberg chains as a function of the magnetic field at a few differen t temper-\natures. It is quite clear that the isothermal entropy change −∆STexhibits a\nsteep increase when the magnetic field is lifted from zero, which can b e related\nto an abrupt field-induced increase of the magnetization from zero towardsthe\nferrimagnetic Lieb-Mattis plateau m/ms= (2S−1)/(2S+1). The isothermal\nentropy change −∆STthen reaches a constant value within the field range,\nwhich is nearly equal to a width of the intermediate magnetization plat eau\nat a given temperature. This interval of the magnetic fields determ ines range\nof applicability of the ferrimagnetic mixed spin-(1/2, S) Heisenberg chains for\ncooling purposes. Namely, the isothermal entropy change −∆STconsecutively6 Jozef Streˇ cka, Taras Verkholyak\n/s48 /s49 /s50 /s51 /s52/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48\n/s48/s46/s50\n/s48/s46/s49\n/s107\n/s66/s32/s84\n/s32/s47/s32 /s74/s32 /s61/s32/s48/s46/s48/s53/s49/s47/s50 /s49\n/s32/s32/s83\n/s84/s32/s47/s32/s107\n/s66\n/s40/s97/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s104 /s32/s47/s32 /s74/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48\n/s48/s46/s50\n/s48/s46/s49\n/s107\n/s66/s32/s84\n/s32/s47/s32 /s74/s32 /s61/s32/s48/s46/s48/s53/s49/s47/s50 /s51/s47/s50\n/s32/s32/s83\n/s84/s32/s47/s32/s107\n/s66\n/s40/s98/s41 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s104 /s32/s47/s32 /s74\nFig. 5 The isothermal change of the entropy versus the magnetic fiel d at three different\ntemperatures for the ferrimagnetic mixed spin-(1/2, S) Heisenberg chain: (a) S= 1; (b)\nS= 3/2.\nacquiresrelativelydeep minima near both field-driven quantum critica lpoints,\nwhich are gradually lifted and smoothed upon increasing of temperat ure. The\nminima in the isothermal entropy change −∆STcan be thus regarded as an-\nother faithful indicators of the quantum critical points of the fer rimagnetic\nmixed spin-(1/2, S) Heisenberg chains.\n4 Conclusion\nIn the present work we have examined magnetic and thermodynamic proper-\ntiesoftheferrimagneticmixedspin-(1/2, S)Heisenbergchainsatfinitetemper-\natures using QMC simulations. In particular, we have focused our at tention to\na detailed examination of the temperature effect upon the magnetiz ation pro-\ncess in a close vicinity of the field-driven quantum critical points. It h as been\nevidenced that a singularsquare-rootbehaviorof the magnetizat ion, which ac-\ncompanies both quantum phase transitions connected with a break downof the\ngapped Lieb-Mattis ferrimagnetic phase and the gapless Tomonaga -Luttinger\nspin-liquid phase, undergoes a gradual rounding upon increasing of tempera-\nture.Themagnetizationcurveatnon-zerotemperaturesisthus almostwithout\nany clear signature of the quantum critical points unlike other ther modynamic\nresponse functions.\nThe susceptibility and isothermal entropy changes contrarily displa y close\nto quantum phase transitions relatively sharp maxima and minima, res pec-\ntively, which are gradually suppressed and smoothed upon increasin g of tem-\nperature. The respective maximum in the susceptibility and minimum in t he\nisothermal entropy change can be accordingly regarded as a faith ful indicator\nof the field-driven quantum critical point of the ferrimagnetic mixed spin-\n(1/2,S) Heisenberg chains. Although the present study was restricted j ust to\nthe mixed-spin Heisenberg chains with the quantum spin numbers S= 1 and\n3/2, the foregoing studies [9,10,11,12,13,14,15,16,17,18] h ave already proven\nlargely universal behavior of the ferrimagnetic mixed spin-(1/2, S) HeisenbergMagnetic signatures of quantum critical points of the Heise nberg chains 7\nchains also for other spin values for which qualitatively the same beha vior\nshould be expected.\nIt has been also demonstrated that the most efficient cooling throu gh the\nadiabatic demagnetization of the ferrimagnetic mixed spin-(1/2, S) Heisenberg\nchains can be achieved in a range of the magnetic fields from zero up t o nearly\na half of the intermediate magnetization plateau, while the isotherma l entropy\nchange implies for larger magnetic fields a less efficient cooling as it rapid ly\ndrops down near both quantum critical points. It is our hope that t heoretical\nresults of the present work are ofobvious relevance for a series o f bimetallic co-\nordinationcompoundsMM’(pba)(H 2O)3·2H2O [23] andMM’(EDTA) ·6H2O\n[24] (M,M’ = Cu, Ni, Co, Mn), which represent experimental realizatio ns of\nthe ferrimagnetic mixed-spin Heisenberg chains. High-field magnetiz ation and\nmagnetocaloric measurements on these or related series of bimeta llic com-\nplexes are however needed for experimental testing of the prese nt theoretical\npredictions.\nReferences\n1. S. Sachdev, Quantum Phase Transitions . Cambridge University Press, Cambridge\n(1999).\n2. D.C. Mattis, The Many-Body Problem: An Encyclopedia of Exactly Solved Mo dels in\nOne Dimension . World Scientific, Singapore (1993).\n3. J.S. Miller, M. Drillon, Magnetism: Molecules to Materials I . Wiley-VCH Verlag, Wein-\nheim (2001).\n4. J. Simon, W.S. Bakr, R. Ma, M.E.Tai, P.M.Preiss,M. Greine r,Nature472,307 (2011).\n5. A. Honecker, J. Schulenburg, J. Richter, J. Phys.: Condens. Matter 16, S749 (2004).\n6. K. Okunishi, Prog. Theor. Phys. Suppl. 145, 119 (2002).\n7. G. Misguich, Quantum Spin Liquids , inExact Methods in Low-Dimensional Statistical\nPhysics and Quantum Computing , eds. J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban,\nL. F. Cugliandolo. Oxford University Press, Oxford (2008).\n8. Y. Zhou, K. Kanoda, T.-K. Ng, preprint arxiv: 1607.03228\n9. T. Kuramoto, J. Phys. Soc. Jpn. ,67, 1762 (1998).\n10. S. Yamamoto, T. Sakai, J. Phys.: Condens. Matter ,11, 5175 (1999).\n11. T. Sakai, S. Yamamoto, Phys. Rev. B ,60, 4053 (1999).\n12. N.B. Ivanov, Phys. Rev. B ,62, 3271 (2000).\n13. A. Honecker, F. Mila, M. Troyer, Eur. Phys. J. B ,15, 227 (2000).\n14. S. Yamamoto, T. Sakai, Phys. Rev. B ,62, 3795 (2000).\n15. T. Sakai, S. Yamamoto, Phys. Rev. B ,65, 214403 (2002).\n16. A.S.F. Ten´ orio, R.R. Montenegro-Filho, M.D. Coutinho -Filho,J. Phys.: Condens. Mat-\nter,23, 506003 (2011).\n17. N.B. Ivanov, S.I. Petrova, J. Schnack, Eur. Phys. J. B ,89, 121 (2016).\n18. J. Streˇ cka, preprint arxiv: 1607.03617.\n19. E. Lieb, D. Mattis, J. Math. Phys. ,3, 749 (1962).\n20. M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. ,78, 1984 (1997).\n21. A.W. Sandvik, Phys. Rev. B 59, 14157 (1999).\n22. B. Bauer, L.D. Carr, H.G. Evertz, A. Feiguin, J. Freire, S . Fuchs, L. Gamper, J. Gukel-\nberger, E. Gull, S. Guertler, A. Hehn, R. Igarashi, S.V. Isak ov, D. Koop, P.N. Ma, P.\nMates, H. Matsuo, O. Parcollet, G. Pawlowski, J.D. Picon, L. Pollet, E. Santos, V.W.\nScarola, U. Schollw¨ ock, C. Silva, B. Surer, S. Todo, S. Treb st, M. Troyer, M.L. Wall, P.\nWerner, S. Wessel, J. Stat. Mech.: Theor. Exp. ,2011, P05001 (2011).\n23. O. Kahn, Struct. Bonding (Berlin) ,68, 89 (1987).\n24. M. Drillon, E. Coronado, D. Beltran, R. Georges, J. Appl. Phys. ,57, 3353 (1985)."    },    {        "title": "2105.10795v1.Ferromagnetic_resonance_modes_in_the_exchange_dominated_limit_in_cylinders_of_finite_length.pdf",        "content": " 1 Ferromagnetic resonance modes in the exchange dominated limit  \nin cylinders of finite length  \n \nJinho Lim1, Anupam Garg1 and J. B. Ketterson1,2 \n \n1. Department of Physics and Astronomy, Northwestern University, Evanston, IL, \n60208.  \n2. Department of Electrical and C omputer Engineering Northwestern University, \nEvanston, IL, 60208.  \nAbstract  \nWe analyze the magnetic mode structure of axially -magnetized, finite -length, nanos copic \ncylinders in a regime where the exchange interaction dominates, along with simulations of the  \nmode frequencies of the ferrimagnet yttrium iron garnet. For the bulk modes we find that the \nfrequencies can be represented by an expression given by Herring and Kittel by using \nwavevector components obtained by fitting the mode patterns emerging from the se simulations. \nIn addition to the axial, radial, and azimuthal modes that are present in an infinite cylinder, we \nfind localized “cap modes” that are “trapped” at the top and bottom cylinder faces by the \ninhomogeneous dipole field emerging from the ends. Semi -quantitative explanations are given \nfor some of the modes in terms of a one -dimensional Schrodinger equation which is valid in the \nexchange dominant case. The assignment of the azimuthal mode number is carefully discussed \nand the frequency splitting o f a few pairs of nearly degenerate modes is determined through the \nbeat pattern emerging from them.  \n \n1. Introduction  \n Measurements of the resonant microwave response of the spins of unpaired electrons in \nferromagnetic materials have a long history, beginni ng with its initial observation by Griffiths1. \nThe vast majority of the subsequent experimental studies have involved “macroscopic” samples \nwhere the spins excited by an external microwave field precess in the combined field arising \nfrom an externally appl ied constant field together with the field produced by the sample’s own \nmagnetization (the so -called demagnetization field) under conditions where both are nearly \nuniform. Under such conditions the strength of the exchange interaction, though itself \nrespon sible for the materials magnetization, does not affect the resonance frequency.  \n In the decade after Griffiths’ work, it was found that attempts to drive the resonance to \nlarger amplitudes did not work, and that it saturated at values of the microwave pow er far below \nthat expected from the observed resonance linewidth. The explanation is that small \ninhomogeneities in the magnetization of the sample are amplified by the inhomogeneities \ngenerated in turn in the dipolar or demagnetizing field, and leads to wh at are now called the Suhl \ninstabilities2,3. In addition to studies of the uniformly precessing mode, spatially -nonuniform  2 modes have also been intensively studied, both experimentally and theoretically, as we shall \ndescribe below.  \n In this paper, we repor t on an exhaustive numerical study, using the OOMMF \nmicromagnetic simulation code4, of the resonance modes of Yttrium Iron Garnet (YIG)  \ncylinders, primarily of diameter d = 75nm and height h = 300 nm, although some aspects have \nbeen studied for other value s of h (7.5 –1200 nm). Our methodology shares many features with \nthe work of McMichael and Stiles5 on two -dimensional elliptical disks and three dimensional \nthin cylindrical discs. Our three -dimensional  geometry displays a much richer mode structure, \nhoweve r, requiring a more detailed theoretical framework. In addition, we have also developed \ntechniques to resolve modes that are nearly degenerate in frequency.  \n The motivation for our work is as follows. Firstly, in recent years many authors have \nexamined whe ther the magnetization of small particles (in the size range of order 102 nm) can be \nreversed by applying microwave fields (so -called microwave assisted switching) with a view to \napplications in magnetic storage6,7,8,9. The reasoning is that at small sizes , the exchange \ninteraction favors parallel alignment of the spins and suppresses the dipolar instabilities. In \nprevious work, we have used OOMMF to simulate YIG cylinders with (d, h) = (25, 50)nm, \n(25,100)nm and (75,150)nm, where we have attempted to rever se the magnetic moment of the \ncylinder by applying pi -pulses10,11, as in NMR12. We found that by chirping the frequency of the \npulse, we could achieve reversal in the 25 nm diameter samples, but not in the larger ones. If pi -\npulses or other reversal protocols  are to be successfully implemented in more realistic larger \nsamples, then an understanding of the normal modes in restricted geometries is a necessary prior \nstep. The problem is also of intrinsic interest, and we have found novel features not seen in \nellipsoidal geometries (including degnerate forms thereof) that have been the subject of almost \nall other work to date.  \n Secondly, with recent advances in sub -micron patterning techniques, the study of arrays \nof objects (so as to have large signals) for which the largest dimension is few hundred \nnanometers or less is attracting increasing attention. With advanced techniques it is even possible \nto probe the magnetic properties of individual sub -micron particles13,14,15. Measurements on such \nsamples can even be perf ormed in the absence of an external field, i.e., solely in the presence of \nthe internal demagnetization field (for shapes where such a field exists), provided the sample is \nsmall enough to be in a single domain state16. Modes with an odd number of maxima an d minima  3 can be excited directly with a uniform microwave field; coupling to modes with higher wave \nnumbers will be more challenging17.  \n Here we will primarily be concerned with size-quantization effects  arising from finite \nsample dimensions. In particular  we will examine the mode spectrum in samples having a \ncylindrical shape with radius a (and corresponding diameter d = 2a) and height h, both \nanalytically and numerically. Due to the ease of preparation of some materials as wires, such \nsamples are widely s tudied experimentally, e.g., in permalloy Py18 and  in Ni19,20. Cylinders of \nfinite length with h/d ratios of order unity and larger can be readily patterned using optical and e -\nbeam lithography by creating hole arrays in a resist followed by deposition and l iftoff21.  \n1.1. Theoretical background    \n Free spins in a magnetic field H precess at the Larmor frequency, \nH  =  , where \ng | e | /2mc=\n with g, e, and m being the electron g -factor, charge and mass. As noted, in \nmaterials having an internal magnetization additional fields are present which can alter the \nprecession frequency. To describe this and related effects, Landau and Lifshitz22 (LL) introduced \nthe following equation of motion  \n \n= −  −  \n0d()dt MMM H M M H ;      (1.1)  \nhere \nH  is the total field at a given position within the sample arising from the external field as \nwell as that produced by the magnetization itself and an effective field arising from quantum \nmechanical exchange; it can also includ e crystalline anisotropy, but this is suppressed in what \nfollows. The second term on the right -hand  side of Eq. (1) is incorporated to phenomenologically \naccount for damping, which will largely be neglected in what follows. In addition to satisfying \nEq. (1 ) \nand  MH  must satisfy appropriate boundary conditions at the surface of the body.  \n For ellipsoidal samples (including degenerate forms thereof), and in the presence of a \nhomogeneous external field \nH , the magn etization \nM  is nominally homogeneous as is the \nresulting demagnetization field; one can then observe sharp absorption lines in ferrommagnetic \nresonance (FMR) experiments (in the absence of strong damping), all spins then seeing th e same \nlocal field. The resonance frequency of this uniformly precessing mode in a spheroidal sample \n(where two of the principal axes of the ellipsoid are identical) with the external field \n0H  along \nthe axis of rotation is given by  what is commonly called the Kittel formula,23  4  \n( ) 00H 4 (N N )M⊥  =  +  −\n ,       (1.2)  \nwhere \nN  and N⊥\n  are coefficients accounting for the effect of demagnetization perpendicular \nand parallel to the rotation axes (with \n2N N 1⊥+=\n ), and \n0M  is the internal magnetization, \ntaken as a constant; note \n  may differ from the free -space value due to atomic and solid -state \neffects.  \n In addition to the uniformly precessing mode ther e exist non -uniform modes24 which we \ncan characterize by some effective wavelength, \n . At shorter (nanometer) scale wavelengths, the \nexchange interaction dominates, and the associated modes are termed exchange modes, first \nintroduc ed by Bloch25. The importance of modes with longer wavelengths (in suitably large \nsamples), was emphasized by Clogston, Suhl, Walker and Anderson26,27. They arise from a \nsolution of Eq. (1) together with \n0 =B  and the Maxwell boundary co nditions; they are \ncommonly referred to as magnetistatic modes . Modes in the region where both exchange and \nmagnetostic effects compete are called dipole/exchange modes.  \n For the case of a sphere some of the low -lying magnetostaic modes were first examined  \nby Mercerau and Feynman28. They were later studied in much greater detail for spheroidal \nsamples by Walker29.  \n The limiting case of a finite thickness, infinite -area, slab (\nz N 0, N 4⊥= =  ) with both \nthe wave vector \nk  (\n| k | 2 /=   ) and the external magnetic field \n0H  lying in plane was treated \nby Damon and Eshbach30; the case with \n0H  perpendicular to the film was examined by Damon \nand Van De Vaart31. For an in-plane field Damon and Eshbach identified two classes of \nmagnetostatic modes: a surface wave and a family of bulk waves. The first of these, now \ndesignated as the DE mode, decays exponentially within the interior. The DE mode is most \ncommonly studied for  wave vectors \n0ˆ () k H z\n  where \nˆz  is the plane normal; it has a positive \ngroup velocity but also has the unusual property that it propagates only on one side  of the slab \nfor a given field direction, switching to th e opposite side on reversing the field. The second class \nis a family of bulk modes which are quantized in the film thickness direction; they are typically \nstudied with \n0 kH\n . They propagate in both directions and the lowest lying mode  has the \nunusual property that its group velocity is negative for small k and therefore it is called a \nbackward volume mode . The frequencies of all the volume modes asymptotically approach the  5 free spin precession frequency, \n0H  =  , as \nk→  (in the absence of exchange); they approach \nthe in -plane Kittel frequency, \n0 0 0(H (H 4 M ) +  , as \nk0→ .  \n For \n0H  perpendicular to the film we again have modes quantized a long the film \nthickness that now propagate isotropically in plane. The lowest has a positive group velocity for \nsmall in -plane wave vector k; it is therefore referred to as a forward volume mode . All these \nmodes approach the perpendicular Kittel frequency \n0(H 4 M) −   as \nk0→  and \n0 0 0(H (H 4 M ) − \n as \nk→ . Exchange effects have been considered by De Wames and \nWolfram32 and more completely by Arias33. \n For the case of an infinitely long cyl inder (\nz N 2 ;N 0⊥=  = ) with \n0H  parallel to the \nrotational axis, which is relevant to the work presented here, the mode structure was first studied \nby Joseph and Schlomann34. Here we encounter families of purely azimutha l modes and \nanalogous to the backward volume modes we have radially quantized modes propagating up and \ndown the cylinder axis, which also approach \n0H  =   at large k (in the absence of exchange). \nRecently this problem was reexamined by Arias and Mills35 who also considered the effects of \nexchange via perturbation theory.  \n At shorter wavelengths the effects of exchange contribute. In this regime the frequency of \na mode with wave vector \nk 2 /=    for a spheroidal sample wi th the external field \n0H  aligned \nalong the rotational axis can be described by the Herring -Kittel formula,36 which we discuss in \nAppendix A  \n \n2 2 20 0 ex 0 0 ex 0(H 4 N M D k )(H 4 N M D k 4 M sin )  =  −  + −  + +  \n ;  (1.3)  \nhere \nexD  is a parameter measuri ng the strength of the exchange (see below), \n2 2 2z k k k⊥=+ , \nzk  and k⊥\n are the components of the wavevector parallel and perpendicular to the spheroid axis, \nand \n1z tan (k / k )−⊥ =  is the angle between the spin wave p ropagation direction and the \nspheroid axes. Note that at k = 0  the factor involving \nN⊥ that appears in Eq. (1.2) is absent  from \nEq. (1.3), since it is assumed that the transverse demagnetization field is “screened out” at short \nwavelengths. Indeed \n  is ill defined at precisely k = 0 since \n  is ambiguous. This shortcoming \nalso signals the importance of the magnetostatic modes at intermediate k values; i.e., as the \nsample size is red uced there is a crossover between dipole dominated and exchange dominated  6 modes. Modes with k values straddling these regimes are the dipole -exchange modes mentioned \nabove. At short wavelengths, which will be the case in sufficiently small samples, Eq. (1. 3) \nshould provide a representation of the mode structure in rotationally symmetric samples, \nprovided quantized values of \nz k  and  k⊥  satisfying the boundary conditions are available; we \nwill utilize Eq. (1.3) to represent some of our finite -size sample simulations in what follows.  \n More generally and in the absence of exchange effects, magnetostatic effects would \ndominate the mode frequencies, which for a spheroid would lie in the range  \n \n0 0 0 0 0(H 4 N M 2 M ) (H 4 N M ) −  +      − \n ;     (1.4)  \nnote the numbe r of modes in this interval is bounded only by the number of spins; i.e., the mode \ndensity is very high, making the resolution of the individual modes extremely difficult at shorter \nwavelengths (where they pile up). When exchange is present the mode freque ncies are spread \nover a much wider interval.  \n In an inhomogeneous external field, or for samples with an arbitrary shape, one might \ninitially expect to observe a line -broadening (as happens in most nuclear magnetic resonance \nexperiments). However,  in the p resence of exchange this is not the case and well-defined  modes \nemerge as will be discussed further below. Exchange modes are not restricted to magnetic \nmaterials; they have been observed in experiments on Fermi liquids at low temperatures, \nexamples being certain metals37 and the normal state of liquid 3He 38; the theory for the latter was \nfirst given by Silin39 for homogeneous fields and later extended by Leggett40 to describe the \ninhomogeneous case.  \n1.2. Plan of the paper  \n We develop the theoretical framewor k for our problem in Sec. 2, beginning with a \ndiscussion of cylindrical symmetry and the resulting angular momentum quantum number (or \nazimuthal mode number) in Sec. 2.1. In the magnetostatic limit it is convenient to take this as the \ntotal angular momentu m, m, as done by Walker, and by Joseph and Schlomann34. In the \nexchange dominated limit, it becomes more important to understand the separation of the angular \nmomentum into its orbital and spin parts. The major component of a mode has spin \nsm1=  and \norbital angular momentum \nmpl= , and there is a small admixture of \nsm1=−  and \nm p 2l=+ . \nAccordingly, we find it better to label the modes by the orbital angular momentum p of the major  7 component. This is especially so when examining the computer -generated  mode patterns since \nthe orbital behavior of any component of \nM  is immediately apparent.  \n That the two components of a mode are so unequal goes hand in ha nd with the fact that \nmodes with m l = p and −p are nearly degenerate, as we discuss in Sec. 2.2. A clear understanding \nof this issue is important, as this near degeneracy can lead to some confusion when looking at the \nmode patterns. In one case, we have re solved this degeneracy (see Sec. 3. 4) by exciting and \nexamining the beat pattern between the ±p modes.  \n In Sec. 2.3, we show that exchange dominated modes in long cylinders are approximately \ndescribed via a Schrodinger -like equation for a particle in a cylindrical box with a modified \nbounda ry condition, such that axial and radial dependence of the mode function factorizes, and \nthe resulting quantization gives rise to axial and radial mode numbers. In an infinite cylinder this \nseparation is exact, which is exploited to good effect in the anal yses of Joseph and Schlomann34 \nand of Arias and Mills35. In a finite cylinder, the separation is approximate since the \ndemagnetizing field is non -uniform, and flares away from the axis near the perimeter of caps at z \n= 0 and z = h. We give a semi -quantitat ive argument in Sec. 2.4 that the Schrodinger equation \npossesses bound state solutions near these caps, corresponding to “cap modes” which we see \nvery clearly in our simulations. For any p, there are two such modes (one for each cap), whose \nfrequencies lie  below those of the bulk modes with the same p. This means that the uniform \nFMR or Kittel mode, which is the lowest bulk mode with p = 0, is not the lowest frequency mode \nof the body. For this case, we present numerical results for the solution of the Schr odinger \nequation in Sec. 2.5, and find good agreement with the simulations.  \n The cap modes are a novel and unexpected feature of our study, as they do not exist in an \ninfinite cylinder or a finite sized ellipsoid of revolution. Similar “end modes” were fou nd by \nMcMichael and Stiles5, who did not however investigate their origin. We expect that such \nlocalized modes will exist near the surfaces of other sample shapes as well whenever the \ndemagnetizing field departs significantly from uniformity.  \n Our simulati onal approach is described in Sec. 3. It is based on the OOMMF code \ndeveloped at the National Institute of Standards and Technology.  After finding the static \nequilibrium magnetization, \neq()Mr   (Sec. 3.1), we can excite the system by a pplying pulses that \nare localized in either space, time or both41. The spatial center and the width and frequency \nbandwidth are varied depending on which mode(s) we wish to excite. The resulting time  8 development of \n( ,t)Mr  is Fourier tra nsformed, and the point -wise power spectrum is added over \nall the cells. The resulting sum displays peaks at many mode frequencies, and by honing in on \nindividual peaks, we can identify the magnetization patterns for each mode as explained in Sec. \n3.3. \n Once a particular mode pattern is obtained in a simulation it can be used as is or altered in \nsome way, say by combining it with some other mode, to study the subsequent development in \ntime. This is a potentially promising way to study mode -mode coupling or large amplitude \nresponses which we hope to pursue in the future. As an application of this idea, and as noted \nabove, ±p modes are sometimes nearly degenerate, as are the even and odd super -positions of the \ncap modes. In Sec. 3.4 we show that by starting th e simulation in a suitable real -space pattern we \ncan find a beat pattern in the time development of the magnetization from which we can obtain \nthe frequency splitting of the modes. We have performed this exercise for only a few cases as it \nis computational ly intensive, and the physical principles are the same for the other cases.  \n In Sec. 4 we tabulate the frequencies of all the modes we have found (approximately 90) \nand discuss the assignment of mode numbers further. The assignment of the longitudinal \nquantum number \nzn  on the basis of the one -dimensional Schrodinger equation is particularly \ntricky as the existence of the cap modes forces nodes in the bulk mode functions near the caps, \nand prevents accurate fitting of the lowest few  bulk modes to a sinusoidal form \nz sin(k z)  with \nzk  \nstrictly equal to π/h times an integer. Nevertheless, an unambiguous labeling of the modes is \npossible.  \n In Sec. 5 we show that the mode frequencies that we obtain agree surprisingly well with \nthe Herring -Kittel expression (1.3) provided we identify \nk⊥  and \nzk  in this formula correctly. \nWe give reasons why this agreement might be so good, explain how the wavevector components \nare found, and how this allows us to organize the normal mode spectrum into familie s of modes \nlabeled by p.  \n Spatial patterns for a variety of modes are given in Sec’s. 6 and 7 (in Figs. 6.1 –6.8 and \nFig. 7.2). These patterns are the centerpiece of our paper, and show beautiful regularity and \nsymmetry. In Sec. 6 we consider only the d = 7 5 nm, h = 300 nm sample, while in Sec. 7 we \nconsider the lowest three p = 0 modes as a function of h. We find that at small h/d (disc -like \nsample), the symmetric cap mode (which has the lowest frequency of all three) is in fact the \nmode that one would rega rd as the uniform FMR or Kittel mode, and its frequency is well fit by  9 the Kittel formula with an appropriate choice of demagnetization coefficients. For large h/d \nhowever, it is the lowest bulk mode (whose frequency lies above the two cap modes) that shou ld \nbe identified with the Kittel mode. For intermediate values of h/d ≃ 6–8, the Kittel formula does \nnot actually describe any of the modes. To our knowledge this point has not been appreciated \nbefore. Once again it illustrates the richness of the normal m ode spectrum in non-ellipsoidal  \nsamples.  \n Finally, Sec. 8 summarizes our conclusions. Here we take the opportunity to  emphasize \nthe importance that  simulations of the small amplitude mode structure  in nano -structures  have \nfor present and possible future ap plications, some of which are currently speculative in character.  \n2.  Modes in the exchange dominated limit  \n In the presence of an isotropic exchange interaction, and neglecting the effects of damping, \nEq. (1.1) takes the form42 \n \n\n= −  −  \n= −  − ex 2\n2\n0\nex 2\n0d 2A\ndt M\nD ,MMM H M\nM H M          (2.1)  \nwhere \nexA  is a parameter fixing the strength of the exchange interaction and \nex ex 0D 2A / M  \nHere \nH  is the applied magnetic field, \n0ˆHz , plus the dipolar o r demagnetizing field generated by \nM\n.  In a cylinder of finite height, the dipolar field is not uniform, especially near the caps, and \nso the static equilibrium field, \neq()Mr  is not everywhere parallel to \nˆz . A linearized normal \nmode analysis should therefore consider deviations \neq ( ,t)⊥M r M , which do not lie in the x -y \nplane. If exchange is strong, however, the non -uniformity in \neqM  is very small (this is true for \nall the simulations we have performed), and we may then take \nz0 =M . This assumption makes \nit much easier to discuss the physics, and relaxing it only obscures the key ideas without adding \nsubstance. We stress that it  is not essential to our argument, especially with respect to the \nsymmetries and the azimuthal quantum number. With this assumption, we may write  \n \n( )2 1/20 ˆ ( ,t) M (1 m ) ( ,t) = − + M r z m r ,        (2.2)  \nwhere \nm  has only x and y components and is dimensionless since we have scaled out \n0M . \n For small deviations, \n| | 1m , the linearized LL equation can be cast as   10  \n( ) =   − − 2\nz 0 d exd ( ,t)ˆH ( ,z) ( ,t) M ( ,t) D ( ,t)dtmrz r m r h r m r .   (2.3)  \nHere \nzH ( ,z)r  consists of the app lied field, \n0ˆHz , together with the position dependent \nlongitudinal demagnetization field arising from the static magnetization, and \ndh  is the (small) \ndemagnetization field induced by \nm . (We use cylindrical coordinates \n=(r, ,z)r  here and \nbelow.)  \n2.1 Assignment of the angular momentum quantum number  \n Equation (2.3) defines an eigenvalue problem with cylindrical symmetry, so there must \nexist solutions with definite a zimuthal mode \nnumber. In the zero -exchange or magnetostatic limit, \nthe analysis is best done in terms of a scalar  \nmagnetic potential ψ, which varies as \nime  in the \neigenmodes; the integer m (which we must be \ncareful to distinguish fr om the scalar value of \nm ) is \nthen naturally interpreted as the angular momentum \nquantum number. In the strong exchange limit, the \nproblem is better formulated in terms of \nm  directly, \nwhich as a  vector  field transforms differently under rotations than a scalar field43 (such as \n()r  \nin the Schrodinger equation).  \n Let us examine the effect of a rotation on the \nvector \nm  at a point \n(x,y,z)  by an angle \n  about the z axis to a vector \nm  at the point  \n(x ,y ,z) . \nWe need only carry this analysis out to leading order in \n . The components of the rotated vect or \nm\n are then (see Fig. 2.1.)  \n \n   = − x x ym (x ,y ,z) m (x,y,z) m (x,y,z) ,        (2.4a)  \n \n   =  +y x ym (x ,y ,z) m (x,y,z) m (x,y,z) .       (2.4b)  \nThe coordinates \n(x ,y ,z)  themselves are related to \n(x,y,z)  as \n \n=  +  = − x xcos ysin x y ,          (2.5a)  \n \n=  +  = + y ycos xsin y x .         (2.5b)  \nIf we expand the left side of (2.4a,b) to first order in \n , note that to zeroth order \nFig. 2. 1. Transf ormation of a vector field.   11 \n   = (x ,y ,z) (x,y,z)mm, and recall the definition of the (di mensionless) orbital angular \nmomentum operator \nzl  in quantum mechanics as \n \n  = − − = −  z i x y iyxl ,         (2.6)  \nwe can write \nm  in terms of \nm  as \n \n = −  −  −x z x ym (x,y,z) (1 i )m (x,y,z) i [ im (x,y,z)] l ,     (2.7a)  \n \n = −  − y z y xm (x,y,z) (1 i )m (x,y,z) i [im (x,y,z)] l .      (2.7b)  \nEquation (2.7)  can be rewritten in the form    \n \n −       −  = −          −        x x x z\ny y y zm m m 1 i 0 0iim m m 0 1 i i0l\nl .     (2.8)  \n For a scalar  field, \n()r , we would simply have \n = −  z ( ) (1 i ) ( )rr l , but the presenc e of  \nthe last term  in (2.8)  mixes  the two components of the vector  field \nm . This can be interpreted as \narising from an  “internal” or “spin” angular momentum of \n=sm1  that is added to or subtracted \nfrom the o rbital angular momentum \nml  associated with our vector field \nm  (a tensor of rank 1). \nA similar separation exists in the description of light fields44.  \n Consider the case of a vector field of the form  \n  \n=ip\nxm ( ) a(r,z)er ,   \n=ip\nym ( ) b(r,z)er      (a, b arbitrary).    (2.9)    \nThis field has orbital angular momentum \nmpl , but has no definite spin angular momentum. \nFor it to have a definite spin a and b must be proportion al according to     \n \n   =   x\nsym 1m1m i          or  \n   = −  − x\nsym 1m1m i .          (2.10a,  b) \nWriting \n=xy m m im  it then follows that   \n \n\n+− =   =     = +ip\ns totm 1yields m 0, m e and m p 1       (2.11a)  \nand \n \n\n+− = −     =   = −ip\ns totm 1yields m e , m 0 and m p 1       (2.11b)  \nwhere we wrote \n=+tot sm m ml . It follows that an eigen mode with total angular momentum \np1+\n must be of the form   12   \n− + −\n−+    =+     −     x i(p t) i((p 2) t)\nym ( , t) 11F (r,z) e F (r,z) em ( , t) iir\nr     (2.12)  \nwhere we have adopted an \n−ite  time dependenc e following standard practice. The physical \nsolution is obtained by taking the real part of this complex -valued solution. We shall see below \nthat for positive frequency solutions, \n−+FF   in the strong exchange limit, and it is often \nconvenient to neglect \n+F  entirely. It is then more useful to label the modes by p, the orbital \nangular momentum of the dominant component, \n−F . This is especially so when looking at mode \npatterns generated by  OOMMF, since we can read off p by seeing how many times \nm  turns as \nwe go around a circle in the x -y plane. For example, in a \np = 0  mode (see Figs. 6.1a and 6.1b) \nm\n appears uniform, while in p = −1 (Fig. 6.3) and p = 1 (Fig. 6.4) modes \nm  winds by 2π and \n−2π, respectively as we go anti -clockwise around a circle.  \n In the magnetostatic limit by contrast, \n+−  F / F O(1) , and the m labeling is better.  Thus, \nfor the sphere, while we would describe the uniform or Kittel mode as having p = 0, Walker29 \nassigns m = 1 to it (see his Fig. 3 where the mode is labeled (110)).  \n2.2. Near degeneracy of p and − p modes  \n When exchange dominates over dipole -dipole i nteractions, we may as a first approximation \nneglect \ndh  in Eq. (2.3). In component form the equation then reads  \n \n( )x y 2ex\ny xm ( ,t) m ( ,t) dH(r,z) Dm ( ,t) m ( ,t) dt−  =  −    r r\nr r       (2.13)  \nor \n \n( )2exdm ( ,t) H(r,z) D m ( ,t)dt =  − rr        (2.14)  \nwith \n \nxy m ( ,t) m ( ,t) im ( ,t) =r r r .         (2.15)  \nWe seek solutions of the form  \n \nitm ( ,t) m ( )e− =rr            (2.16)  \nand demand that \n0 , with the understanding that the physical solution will be given  \nby the real part. These solutions then obey   13  \n( )2z exH (r,z) D m ( ) m ( )   −  = \n rr .       (2.17)  \nThis equation is like a one -particle Schrodinger equation, and since γ > 0 in our convention, the \noperator on the left is a positive operator which cannot have negative eigenvalues. Since we also \ndemand that ω > 0, we must  choose \nm0+= . Finally, since our finite cylinder retains full \nazimuthal symmetry, the solution for \nm−  takes the form  \n \nipm (r, ,z) F (r,z)e−−= ,          (2.18)  \nwhere \nF (r,z)−  can be chos en to be real, and ω has the same (positive) value for either sign of p.  \nIn terms of the general form (2.12), this solution corresponds to putting \nF0+= , and a physical \nsolution  \n \nx i(p t)\nym ( ,t) 1 cos(p t)F (r,z) e c.c. 2F (r,z)m ( ,t) i sin(p t)−−− −      = + =      −      r\nr .   (2.19)  \n If we now inclu de the dipolar field \ndh  as a perturbation, we can expect that \nF+  will \nbecome nonzero, with \n20 ex F / F 4 M / D k+− \n , where \n2 / k  is the typical length scale on \nwhich the solution va ries. \n The source of the degeneracy with respect to ±p is that Eq. (2.13) is invariant under \nreflection in the yz plane provided we do not also reflect the vector \nm .  Hence the operation  \n \nxxm (x,y,z) m ( x,y,z) →−           (2.20 a) \nand \n \nyym (x,y,z) m ( x,y,z) →−            (2.20b)  \nalso produces a solution. This operation is equivalent to φ → − φ, or alternatively to p → − p. \nInclusion of the dipole -dipole interaction destroys this invariance: the field \ndh  produced by the \noperation is not the same field as before.  \n Strictly speaking therefore, modes differing only in the sign of p are not degenerate, \nalthough the non -degeneracy may be small.  Indeed, as explained below, we have spent \nsignificant effort to n umerically resolve the splitting and have not always succeeded. The \nphysical origin of this non -degeneracy is just that the applied external field breaks time reversal \nand parity symmetries. In the magnetostatic limit, this point emerges directly from the solution in \nterms of the scalar potential. Joseph and Schlomann45 find that \n|m| |m|−    for the volume  14 modes (where we have here used m instead of p to label the modes), and that only m > 0 \nsolutions exist for the surface modes. This is an e xtreme form of the nondegeneracy, and is the \ncylindrical analog of Damon and Eshbach’s discovery of one -sided surface modes in the slab \ngeometry. Joseph and Schlomann34 also find that the ±p splitting becomes smaller with \nincreasing \nzk  or increasing radial mode number (see their Fig. 5). The same behavior is found \nfor the general spheroid by Walker,46. Arias and Mills35 on the other hand, appear to us to be \nfinding that modes with the opposite sign of the angular momentum are degener ate; we are \nunable to pinpoint why.  \n The near degeneracy of ±p modes also underlies whether one sees azimuthal standing or \nrunning wave patterns in the OOMMF simulations. We discuss this issue in Sec. 3.3 below.   \n2.3. The long cylinder in the exchange domi nated approximation  \n  For an infinite cylinder the variables in the Schrodinger equation separate and we can \nwrite  \n \n( ) ( ) 0 p z z F (r,z) m J (k r) Acos k (z h / 2) Bsin k (z h / 2)−⊥ = − + −     (2.21)  \nyielding  \n \n( ) ( )x\n0 p z z\nym ( ,t) cos(p t)m J (k r) Acos k (z h / 2) Bsin k (z h / 2)m ( ,t) sin(p t)⊥ −   = − + −    −  −  r\nr\n .           (2.22)  \nHere \npJ  is the Bessel  function of order p. For a long but finite cylinder Eq. (2.21) should be a \ngood approximation except for the cap modes.  \n If we take the modes \np+  and \np−  as degenerate we can superimpose them and form \nstand ing waves in \n , an operation we carry out in the next section.  Inserting any of these forms \ninto (2.8) yields the frequencies  \n \n( )22\n0 ex zH D (k k )⊥ =  + + .        (2.23)  \nIf we adopt the boundary condition (discussed below)  \n \nd0d=nmr            (2.24)  \nwhere \nn  is a vector normal to the surface, the values of \nk⊥  will be fixed by the condition   15  \npdJ (k a)0dk⊥\n⊥=           (2.25) \nwhere a is the cylinder radiu s. We write the solutions of Eq. (2.25) as \nrp,nk  where \nrn  denotes  \nthe number of additional zeros of \npJ  (other than those for \np0J  at r = 0) within the cylinder o f \nradius a. We will find that (2.25) agrees quite well with the simulations.   \n For the finite cylinder we present the argument in two stages. In the first stage we assume \nthat the inhomogenity in the static demagnetization field can be ignored and take \n00 H(r,z) H 4 N M = − \n with \nN\n  being the longitudinal demagnetization coefficient. The \nsolution (2.21) continues to hold but the mode frequencies are given by  \n \n( )22z 0 0 ex z(k ,k ) H 4 N M D (k k ) ⊥ ⊥ =  −  + +\n .     (2.26)  \nThe allowed values of \nk⊥  are given by \nrp,nk as discussed above, but the quantization of \nzk  is \nless simple. If the end caps are taken to be at \nz 0 and z h=   = , then to have a definite parity under \nreflec tion in the mid plane at \nz h / 2= , the mode function must depend on z as either \n( ) z cos k (z h / 2) −\n (even parity) or \n( ) z sin k (z h / 2) −  (odd parity), but the association between \nzk\n and the parity  depends on the boundary condition applied at the caps.  \n If the boundary condition is taken as \n()0=nm , then the allowed \nzk  values are  \n \nz z zk , 0, 1, 2h=   =  .        (2.27)  \nEven parity is associated with e ven \nz  and odd parity with odd \nz . \n If instead the boundary condition is taken as \n0=m , the allowed values of \nzk  are \n \nz z zk , 1, 2h=   =          (2.28 ) \nNow even parity is associated with odd \nz  and odd parity with even \nz . \n The boundary condition obeyed by OOMMF mode functions is closer to \n()0=nm  \nthan to \n0=m . In addition,  they do have definite parity. Except for the two lowest frequency \nmodes, which we call “cap modes” and which require a separate discussion,  they are well fit by  \nthe \n( ) z cos k (z h / 2) −  and \n( ) z sin k (z h / 2) −  forms. However, i t is advantageous to allow for a \nshift and write   16  \nzzkh =           (2.29)  \nwhere  \n \nz z z =  +   \nz( 2)         (2.30)  \nWe can refer to \nz  as an “end defect” analogous to the conc ept of a quantum defect in atomic \nspectroscopy47. With this correction Eq. (2.26) continues to be a good approximation to the mode \nfrequencies.  \n We comment further on the boundary condition (2.24) that the normal derivative vanishes \nat the surface. The iso tropic continuum exchange field \n2exDM−  arises from a microscopic \nijSS\n Heisenberg interaction, which has the property that for any pair of spins, the torque on \niS  \ndue to \njS  cancels that on \njS  due to \niS . Thus,  the total exchange torque on the body vanishes and \nEq. (2.24) is the continuum expression of this fact. This argument dates back to Ament and \nRado48 and has been  used by many authors since.  Aharoni49 offers a different derivation. Thus,  \nit would appear to be very general, and valid for any \nexD , however small. For the magnetostatic \nlimit, \nexD0= , there is however no such  condition on \nM . Turning on \nexD  perturbatively would \nthen appear to lead to a contradiction. This is not so for the following reason.  \n In the boundary value problem for the spatial form of the eigenmodes, \nexD  multiplies the \nhighest derivative, and is thus a singular perturbation from the mathematical point of view. Such \nperturbations are known to lead to thin boundary layers where the solution changes character \nrapidly50. Thus, while t he normal derivative at the surface may formally be zero, there could be \nlarge curvature in the boundary layer, and the derivative of \nm  as we approach this layer need not \nbe small. This is especially relevant for our OOMMF simulat ions, where the discretization into \ncells may: (a) be too coarse to reveal any boundary layer behavior, and (b) fundamentally \npreclude measurements of this derivative by fitting to the mode functions. In this case adoption \nof an end defect \nz  is an effective practical procedure.  \n2.4. A variational solution for a cylinder of finite length  \n In the second stage of our argument, we attempt to include the inhomogeneity in the static \ndemagnetizing field by adopting the trial form  \n \n± p m ( ) = Z(z) J (k r)cos(p ) ⊥  r .         (2.31)   17 If we substitute this form in Eq. (2.17), together with some radially averaged z -dependent \nmagnetic field \nH(z) , we obtain the following one -dimensional eigenvalue problem  \n \n22ex2dZ(z) H(z) D k Z(z) 0\ndx⊥ \n −  − − =            (2.32)   \nwith the boundary conditions  \n \nZ (0) = Z (h) = 0 .           (2.33)  \nIn the spirit of this variational approach we could obtain\nH(z) ) by averaging with respect to \n2pJ (k r) ⊥\n, which would lead  to slight differences between modes with differing \nk⊥ .  \nAlternatively,  we can use the analytic expression for the dipole field along the cylinder axis \nzH (z,r 0) =\n that arises from spins which are fully aligned (as expected for the case where the \nexchange is totally dominant)51. Assuming a cylinder of radius a and height h, and setting z = 0 \nand z = h at the caps, the resulting demagnetization field along z -axis is  \n \n( )\n( )demag 02 2 2 2h z zH r 0,z 2 M 2\nza h z a−= = −  − − ++ −+  .   (2.34)  \n In the limit of \na / h 0→ , \n( ) demagH r 0,z 0 ==  (corresponding to an infinite rod) and in \nthe limit of \na / h→ , \n( ) demag 0H r 0,z 4 M = = −   (corresponding to a thin disk). Fig. 2.2 shows \nthe resulting magnetic field for YIG cylinders having a diameter of 75nm and lengths of 75, 150, \n300, 600, and 1200nm as calculated from Eq. (2.34) (dashed lines) and along the r = 0 axis by \nOOMMF. The close correspondence arises from the dominance of exchange in these small \ndiameter samples.  \n2.5. Zeros of mode functions and mode labels  \n The demagnetizing field plays the role of an external potential in the Schrodinger equation \n(2.32), and the strong decrease in this field near the end caps leads to surface bound states or cap \nstates whose wav e functions die off exponentially away from the caps. In principle there could \nbe many bound states, but for our parameters we find only one state at each cap. All higher \nenergy states are extended along the z direction, and since the demagnetization field  is \nessentially uniform in the bulk of the cylinder, their wavefunctions behave approximately as \nsinusoidal standing waves.   18 \n \n \n Let us now recall that for a one -dimensional Schrodinger equation with a reflection sym - \nmetric potential, the states with succ essively higher energy alternate in parity and have \nsuccessively increasing number of zeros, with the lowest energy state having no nodes and even \nparity. Further their wave functions must be mutually orthogonal. For our problem, these \ntheorems are satisfi ed as follows. The two cap states are nearly degenerate but they are admixed \nby tunneling to form even and odd parity states with zero and one node respectively. (See, \nhowever, the discussion in Sec. 7 on how the inclusion of dipole -dipole interactions mod ifies the \nenergy ordering.) The first extended state must then have even parity and two nodes. To be \napproximated by \n( ) z cos k (z h/2) −  and to be orthogonal to the cap states, we must have \nzk 2 / h\n corresponding to \nz2 =  and a negative end defect \nz . Higher extended states must \nhave higher values of \nz . In this way we see the need for the restriction \nz2   in Eq. (2.30) and \nfor the end d efect at the same time.  \n  For each value of p and \nrn , we could label the differently quantized modes along z by the \nnumber of their zeros. The lowest extended or bulk mode in any family with given p and \nrn  Fig. 2.2. The dashed line shows the analytic demagnetization field calculated from Eq. \n(2.32) using \n=0 4 M 1750Oe  for YIG cylinders with a diameter of 75nm and five different \nlengths of 75, 150, 300, 600, 1200nm i n a field of 2000Oe. For comparison the solid line \nshows the field computed by OOMMF along the line r = 0.  Note how for long cylinders the \nfield profile near one cap is insensitive to the presence of the other cap.   19 would then have the label \nz2 = , while the cap mode would be labeled \nz0 = . This is \nunaestehetic and also does not differentiate between the physically different character of the cap \nmodes vis a vis the bulk mode s. We therefore label the bulk modes by an index \nzn , with  \n \nzzn2=  −             (2.35)  \nFor the cap modes we replace the number \nzn  by the letters ‘g’ (gerade, even parity) and ‘u’ \n(ungerade , odd parity). For reference we summarize the correspondence between the number of \nzeros and the mode labels as follows:   \n  No. of zeros   Mode label  \n         0         g \n         1         u \n        \nz  \nzzn2=  −  \nThe modes are labeled by the scheme \nrz(p n n )   with the letters ‘g’ or ‘u’ for cap modes in lieu \nof \nzn . In particular the mode nominally identified as the uniform FMR mode has the label (000) \n(but see the discus sion in Sec. 7).  \n2.6. Numerical results for the variational approximation and comparison with simulations  \n We now describe some results from the numerical integration of Eq. (2.32) together with \nthe position dependence of \n( ) demagH r 0,z =  given by Eq. (2.34). Imposing the boundary \ncondition (2.33) at the faces then yields \nZ(z)  together with the eigenvalues \n( ) zrn (p,n )  =  , \nwhere for the general case \nzrn (p,n )  denotes the eigenvalue for given va lues of the azimuthal \nand radial mode numbers, p and \nrn . Given that we have neglected the transverse dipolar field in \nobtaining Eq. (2.32) we expect the resulting eigenvalues to be most accurate in the limit of large \nzn\n mode numbers, and particularly when both \nr p 0 and n 0=   =  (which corresponds to \n0=  in \nEq. (1.3)).  \n The dashed lines in figure 2.3 show the resulting form of  Z(z) for the lowest lying cap \nmode with p = 0 an d no radial nodes for cylinders with a diameter of 75nm and heights of 75, \n150, 300, 600, and 1200 nm in a field of 2000G. Note the approximately exponential decay of \nthe amplitude as we proceed deep into the interior for the longer samples confirming thei r \nsurface like character.  Also shown are the OOMMF simulations obtained using procedures to be \noutlined below (the fact that their amplitudes do not go strictly to zero in longer samples arises  20 from a contamination from other modes). Accompanying anti symmetric modes (not shown) are \nalso highly localized while in addition having a node at the cylinder midpoint . \n As is evident, the semi -analytic results for Z(z) are surprisingly good for \nh 300nm . The \nfrequencies however are not. These c ould be improved by including the transverse dipolar field \nusing perturbation theory, which will raise the frequency. We have not attempted this exercise \nsince our approximate treatment is quite rough in the first place and it would not add to our \nqualitat ive understanding.  \n \n \nFig. 2.3.  The dashed lines show the behavior of the mode function Z(z) vs. z obtained from integrating Eq. \n(2.32) for the (00g) cap mode for cylinders with a diameter of 75nm and heights of 75, 150, 300, 600, and \n1200nm. The solid li nes show the OOMMF simulation results for the same parameters.  \n \n As a crude estimate of the cap mode frequency we can compare it with the frequency of a \nhat box (disc) with a radius equal to its height. Reported demagnetization coefficients52 for this \naspec t ratio are \nN 0.4745=\n  and \nN 0.2628⊥= . For a field \n0H 2.000 kOe=  and \n0 4 M 1.750 kOe =  \n Eq. (1.2) yields f = 4.568  GHz. For our exchange dominated sample it is \nreasonable to add a correction of order  \n2\nexD / a 1.0 kOe =   which raises the frequency to 5.6  \nGHz which is to be compared with the OOMMF value of 6.64  GHz.   21 \n \n Fig. 2.4. The dashed line shows the mode function Z(z) vs. z obtained from \nintegrating Eq. (2.35) for the (00g), (000), (002), a nd (004) modes. The solid line \nshows the corresponding forms arising from OOMMF. Note the behavior at the \ncylinder faces closely conforms with the boundary condition (2.24).  \n  \n    Table I. End defect for (00n) modes.  \nMode Label  z hkz,fit/ z \n000 2 1.62 −0.38 \n001 3 2.50 −0.50  \n002 4 3.68 −0.32 \n003 5 4.78 −0.22  \n004 6 5.84 −0.16 \n005 7 6.96 −0.04  \n006 8 7.90 −0.10 \n007 9 8.92 −0.08  \n008 10 10.01  +0.01  \n009 11 10.91  −0.09  \n00,11  13 12.96  −0.04  \n00,13  15 14.98  −0.02  \n00,15  17 16.92 −0.08  \n Calculations for the extended states with mode number \nzn 0,1,=     were also performed.  \nHere we encounter progressively higher mode frequencies scaling approximately as \n2\nzn .    Figure \n2.4 below shows the result of such calculations for the (00g), (000), (002), and (004) modes.  \nTable I lists values of \nzk h/π , \nz and \nz  for these and neighboring modes.  Note that \nz0  →   22 with increasing \nz . We will explain why modes (00,10), (00,12), and (00,14) are not in this table  \nat the end of Sec. 3.2.   \n 3. Computational Approaches  \n The material studied here is YIG which was chosen for its long mode lifetim es. Whether \nthese long lifetimes survive in submicron structures is an open question. The majority of the \nstudies were for a sample  with h = 4d = 300  nm. The material parameters used  are typical for \nYIG 53:  γ = 2π × 2.8 GHz/kOe, saturation magnetization M s = 139emu/cm3, damping constant \n–5 =5  10\n, and exchange constant \n–7exA  = 3.5 10  erg/cm. The applied field was 2 kOe \nalong z direction.  Damping was turned on to relax the system to its initial state, and turned  off \nafter the sy stem was excited for most simulations. In the few that it was not, it was too small to \nhave any significant effect.  \n As noted above our simulations were carried out with the OOMMF code developed by the \nU. S. National Institute for Standards and Technology.  This program divides a chosen sample \ninto cells on a rectilinear grid and numerically integrates the LL equation in time for their \nmagnetizations, \ni(t)M , as they evolve under the influence of the torques acting on them arising \nfrom a n external field, the nearest neighbor exchange interaction, and the dipolar fields of the \nremaining cells (anisotropy fields can be included but will be ignored in what follows). Each \nmagnetic moment is located at the center of the cell.  The number of cel ls scales with the cube of \na characteristic sample dimension but was nominally fixed at 1/54  cells/(nm)3 corresponding to a \ncell size of \n3 nm 3 nm 6 nm   for the \nd 75 nm, h 300 nm=  = sample.  There are 50 cells in the z \ndirection, and 489 c ells in the xy plane (489/625 = 0.7824 vs. /4 = 0.7854).  In Sec. 7, we \nsimulate samples with other values of h.  As described there,  we then use cells with the same x \nand y dimensions (3 nm × 3 nm), but depending on the value of h, the dimension z is adjusted \nappropriately.  \n3.1. Static equilibrium  \n Prior to exciting the system,  the spins were initially aligned along the cylinder axis \n(parallel to the external field) after which the system was evolved in time (with damping) until it \nstabilizes in an equilibrium configuration. Various tests can be ap plied to determine that it is a \nglobal equilibrium state. This part of our simulations yields the static magnetic field distribution \nwhich could also be used for the calculations in Section 2.   23 3.2. Exciting the system  \n Several different excitation schemes  were utilized. In the simplest of these, all spins were \ntipped by a small fixed angle relative to their equilibrium orientations in a plane containing the z \naxis as an initial condition. This favors the excitation of uniformly precessing modes. To drive a  \nparticular  non-uniform mode the spins were tipped from their equilibrium positions in a manner \nthat mimics the mode  (such as that obtained as the mode pattern in a prior simulation)54. To drive \na broader spectrum of modes  that is localized around a time \n0t  and some position \n0r  we tip the \nspins in some direction according to the function41  \n \ny0 0 x 0 x 0\n0 0 0 0sin[ k (y y )] sin[ (t t )] sin[ k (x x )] sin[ k (z z )]F(t, ) A(t t ) (x x ) (y y ) (z z )−  −  −  −=− − − −r  (3.1)  \nwhere \nx y x, k , k  and k     control the extent to which the excitation is local ized in time and \nspace. Here x, y, z denote cell coordinates. Such pulses can also be introduced at multiple times \nand positions to favor the excitation of modes with differing spatial and temporal properties. In \nparticular inclusion of only the last facto r induces modes propagating along z. Forms can be \nconstructed that favor the excitation of radial or azimuthal modes. Finally,  some simulations \nwere performed in which individual spins were tipped in random directions within some \nspecified average angular range. This excites a very broad range of modes and if the tipping \nangles are large (e.g., approaching 180o) generates a “hot” system, from which it is difficult to \nextract clear modes.  Altogether, we tried more than ten different excitation pulses in an e ffort to \nidentify modes with different symmetry and numbers of nodes. Despite this, our mode table (see \nSec. 4) has gaps. In some cases, modes are nearly degenerate [for example modes (003), ( -10g), \nand ( -10u)] and cannot be easily resolved. In others, the y were too high in frequency to be seen \nwith the particular excitation pulse employed. We are confident that these modes exist and that \nour mode classification is complete.  \n3.3. Identifying modes  \n As the system state simulated by OOMMF evolves in time  from  some chosen initial \nconfiguration, the magnetization vectors \ni( ,t)mr  at the (discrete) cell sites \nir  are recorded at \nregular time intervals. From this data set we can perform a cell by cell fast -Fourier transfo rm \n(FFT) within some chosen time interval available from the simulation to obtain the complex \nquantities \ni( , ) mr . We stress that the OOMMF simulation does not assume that \neq()Mr  is  24 along \nˆz  or that the deviations \ni( ,t)mr  are in the x -y plane, although the most useful information \nis contained in these components for low amplitude mode studies. From the FFT, we follow \nMcMichael and Stiles5 and construct the cell -wise power s pectra,  \n \n2x i x iS ( , ) | m ( , ) |  = rr ,        (3.2)  \ntogether with their sum over the entire sample,  \n \nx x i\niS ( ) S ( , ) =  r ,         (3.3)  \nand likewise for \nyiS ( , ) r  and \nyS ( ) . As noted by them, this definition of a power spectrum is \nvery different from the power spectrum of the integrated magnetization (total magnetic moment \nof the sample), which is what makes them so useful in mode identification; in particular, the \nfrequencies where these total sample power sp ectra have sharp maxima are identified as possible \nmode frequencies of the system. As an example, the power spectra in Eq. (3.3) are given in Fig. \n3.1 which shows various modes including a very high frequency mode that is aliased to less than \n5 GHz due to a Nyquist critical frequency of 50 GHz . \n Suppose a mode has been identified at a frequency \na . With the sign conventions used in \nour numerical FFT program, the mode pattern associated with this frequency is given by  \n \n( )a (a) i( t )ii( ,t) ( , )e +=   m r m r .       (3.4)  \nThe phase  \n  is arbitrary and amounts to a choice of the zero of time. To avoid unnecessary  \nminus signs, we choose \n3 / 2 =   which gives  \n \n(a)i i a( ,0) ( , ) =   m r m r , \n(a)i i a( ,T / 4) ( , ) =   m r m r ,    (3.5)  \nwith \na T 2 /=    being the time period of the mode. Hence, by examining the imaginary and real \nparts of the vector \nia( , ) mr  we can, respectively, obtain the real -space vector magnetization at \nsome ti me and a quarter cycle later for the spatial pattern associated with some nominal mode at \nthe specific frequency. By plotting these vector fields,  we can get a highly visual depiction of the \nmode, permitting easy mode assignment and further analysis.  \n  25 \n \nFig. 3.1. An example of an FFT spectrum. The left and right y axes show \n()2\nx sample2SωN  and \n()2\nsampleSω2Nz\n. See Eq. (3.3).  A broad sinc pulse  as described in Eq. (3.1) was used to excite this \nspectrum . \n \nIn constructing the power spec trum, modes with higher frequencies than the inverse of the \nchosen integration time step, which violate the Nyquist sampling criterion and are then aliased to \nlower frequencies, must be identified and rejected. In the exchange dominated samples \nconsidered here some of them can be identified as spurious peaks with frequencies lower than \nthe known uniform modes, but in dipole dominated larger samples genuine modes below the \nuniform modes are expected. More generally they must be identified by altering the tim e interval \nover which the transform is performed to determine if some mode moves its position. Most \nsimulations were done with a time step of 10 ps and for a duration of 10.24 ns.  \n A curious spatial aliasing was also observed (as evidenced by a rapid spati al variation of \nthe mode intensity on the scale of the cell period) in some patterns; it is thought to be associated \nwith a spatial FFT that is performed to calculate the dipole field in the underlying program. Such \nmodes must also be rejected.   \n3.4. Impli cations of the p degeneracy for OOMMF patterns  \n Using the procedures described we can construct mode maps in chosen planes by plotting \nthe complex cell amplitudes, \nxy ˆ ˆ ( , ) m ( , ) m ( , ) =  +  m r r x r y , at frequencies where the power  26 spectrum shows maxima. On the b asis of these patterns we are typically able to assign \napproximate mode numbers p, n r and n z, and designate them as \nr z r z(pn n ) (pn n )( , ) mr  for that \nfrequency, \nrz(pn n ) =  . The p mode number requires special attention as we now discuss.  In \nwhat immediately follows we will drop the mode designation, regarding it as being understood.  \n Writing the complex function \n( , ) mr  in component form as  \n \nx x x\ny y ym ( , ) m ( , ) im ( , )\n( , )m ( , ) m ( , ) im ( , )     +      = =         +    r r r\nmrr r r ,      (3.6)  \nthe corresponding behavior in the tim e domain follows as    \n \nx x x it\ny y ym ( ,t) m ( , ) im ( , )\n( ,t) em ( ,t) m ( , ) im ( , )−   +     = =       +    r r r\nmrr r r  \n \nxx\nyym ( , )cos t m ( , )sin t\nm ( , )cos t m ( , )sin t    +   =    +  rr\nrr .     (3.7)  \nIf the modes of the system have a pure p character, as in Eq. (2.11), we can write the above \ncomponents as  \n  \nxm ( , ) m(r,z,p, )cos(p ) =  r , \nxm ( , ) m(r,z,p, )sin(p ) =  r ,           (3.8a,  b) \n \nym ( , ) m(r,z,p, )sin(p ) = −  r ,  \nym ( , ) m(r,z,p, )cos(p ) =  r .                  (3.8c,  d) \nHence,  we can write \n( ,t)mr  as \n  \nx\nym (r,z,p,t) cos(p t)( ,t) m(r,z,p, )m (r,z,p,t) sin(p t)  −  = =  −  −  mr  .        (3.9)  \nHere the magnetization vector rotates as \n  changes with a radially symmetric  amplitude. In our \napproximation, where the variables r and z separate, we would write the solutions that have even \nparity as  \n \n( )x\n0 z pym ( , t) cos(p t)m cos k (z h / 2) J (k r)m ( , t) sin(p t)⊥  −  =−  −  −  r\nr .   (3.10)  \n If the modes of the system have a pure p character and in addition p and \np−  are \ndegenerate we can form symmetric and anti symmetric standing wave super -positions of the two \nforms of Eq. (3.10) to obtain   27  \nxx\nyxm (r,z,p, t) m (r,z, p, t)\n( , t)m (r,z,p, t) m (r,z, p, t)− =−mr  \n \ncos(p t) cos( p t)           m(r,z,p, )sin(p t) sin( p t) −   −  −  = −  −  −  − \n     (3.11)  \nwhich results in the following two forms  \n \ncos( t)m(r,z,p, )cos(p )sin( t)           (3.12a)   \n \nsin( t)m(r,z,p, )sin(p )cos( t)  − ,              (3.12b)  \nor if our model product form is assumed,  \n \n( ) 0 z pcos( t)m cos k (z h / 2) J (k r)cos(p )sin( t)⊥ −   ,     (3.13a)  \n \n( ) 0 z psin( t)m cos k (z h / 2) J (k r)sin(p )cos( t)⊥ − − .     (3.13b)  \nHere again the spin direction rotates (in both senses) but now the amplitude is modulated in \n . \nA similar discussion applies to the solutions that are odd parity.  \n In Appendix B we discuss how a pure p mode pattern can be e xtracted from a \nsuperposition of +p and –p OOMMF patterns.   \n3.5.  Separating nearly degenerate modes  \n As noted above, in the absence of the dipole interaction modes with +p and –p are \ndegenerate.  When this interaction is present such modes are split; i.e ., according to the \ndiscussion of section 3. 4 our eigen modes will be running waves in \n . But the splitting rapidly \ndecreases for larger mode numbers, and for running times less than \n1t−  , where \n  is the \nsplitting, the power spectrum displays a single (slightly broadened) peak at the mode frequency. \nIn order to resolve the splitting in a power spectrum the OOMMF run times must be increased.  \n When the splitting is not resolved in the power spectrum the resulting mode patterns \ndisplay a standing wave character. For a few of these we used the standing wave mode pattern as \nan initial configuration and ran the program long enough to display a beat pattern in time from \nwhich the splitting c ould be accurately determined. An example of this technique is shown in \nFig. 3.2 for the cylinder with d = 75  nm and h = 300 nm. Fig. 3.2a shows the case of the \n( 105)   28 mode s with an average frequency of 17.29  GHz. Note a beat waist occ urs at t = 32.68  ns from \nwhich we calculate the mode splitting as 30. 60 MHz . \n \n \nFig. 3.2.  Beat  pattern s emerging from the time evolution arising from the splitting \nof (a) the \n( 105)  modes with an average frequency 17.29 GHz and a spli tting of \n30.60 MHz , (b) the \n,r p 0 n 0== , g and u cap modes with  an average frequency \n6.54 GHz and a splitting of 14.4MHz.  \n   29  A very small splitting also occurs between the symmetric (or gerade, denoted g) and anti-\nsymmetric  (ungerade, denot ed u) combinations of the cap modes. In the Schrodinger equation \nlanguage of Sec. 2, this is a tunnel splitting between the surface bound states. This splitting is \nintrinsically small and hard to resolve in long cylinders, although we have resolved it for the \nr p 0, n 0==\n, g and u cap modes for the cylinder with h = 300  nm. Now  \nf 6.54 GHz=  and \nf 14.4 MHz=\n. The corresponding beat pattern is shown in Fig. 3.2b.   \n4. Frequencies of low -lying modes  \n Most computation s were carried out on a YIG sample with \nh = 4d =8a = 300 nm  in a \nstatic field of \n0H 2000 Oe= . To test the behavior at small and large \nzk  some calculations were \ncarried out for samples with h = 7.5, 37.5, 75,  150, 600 and 1200  nm. The material parameters \nused are typical for YIG, as given earlier in section 3.  \n Table II lists the frequencies \nr z r zpn n pn nf / 2 =    of low lying modes as obtained from \nthe peaks in the power spectrum; all entries are for \nh = 4d =8a = 300 nm  and in a static field of \n0H 2000 Oe=\n. The mode numbers come from a comparison of the accompanying mode pattern \nwith the forms discussed in Sec. 2 with special attention to the number of radial and longitudinal \nzeros  and how \nm  winds around the z axis. By fitting to the form (2.21) we can assign discrete \nwavevectors, \nr,nk⊥  and \nznk ; these values are used for a comparison with the HK formula as we \ndescribe in the next section. All modes with \np0  have a node at \nr = 0 ; for larger values of \nk⊥ , \nadditional radial nodes can be present and their mode number is denoted as \nrn0 . Modes listed \nas \np  are nearly degenerate in the sense discussed above, and the mode patterns display \nstanding waves in \n  as described by Eq. (3.9a,b).  \nTable II.  The mode frequencies for a YIG  cylinder with a height of 300nm and a diameter of \n75nm in a static magnetic field \n0H = 2000  Oe, organized into families with given mode \nnumbers p and n r.  Multiply listed modes (e.g., ( –115) and (\n 115)) we re observed with both \npure and with mixed +p and –p character, depending on the methodology used  to extract \nthem (e.g. FFT vs. beat pattern) .  \n \n  30  \n \n Table II  \nMode Frequencies vs. p, \n  rzn , and n  \np nr nz f (GHz)  p nr nz f (GHz)  p nr nz f (GHz ) \n0 0 Even cap (g)  6.543  1 0 g 10.35  ±1 0 5 17.29  \n0 0 Odd cap (u)  6.543  1 0 u 10.35  ±1 0 7 21.78  \n0 0 0 7.813  1 0 1 11.91  ±1 0 11 33.98  \n0 0 1 8.105  1 0 3 13.96  ±1 1 g 35.16  \n0 0 2 8.887  1 0 5 17.29  ±1 1 u 35.16  \n0 0 3 10.06  1 0 7 21.78  ±1 1 1 36.91  \n0 0 4 11.52  1 0 9 27.44  ±1 1 3 39.26  \n0 0 5 13.38  1 0 11 33.98  ±1 1 5 42.68  \n0 0 6 15.53  1 0 13 41.5 ±1 1 9 52.83  \n0 0 7 17.97  0 1 0 22.66  ±2 0 g 15.82  \n0 0 8 20.7 0 1 1 23.24  ±2 0 u 15.82  \n0 0 9 23.63  0 1 2 24.22  ±2 0 1 18.07  \n0 0 11 30.27  0 1 3 25.49  ±2 0 2 18.36  \n0 0 13 37.79  0 1 4 27.05  ±2 0 3 19.63  \n0 0 15 46.09  0 1 5 28.91  ±2 0 5 22.95  \n−1 0 g 10.06  0 1 6 30.96  ±2 0 9 33.11  \n−1 0 u 10.06  0 1 7 33.4 ±2 0 11 39.65  \n−1 0 1 11.72  0 1 8 36.04  ±2 0 15 55.37  \n−1 0 2 12.7 0 1 9 38.96  ±3 0 g 24.02  \n−1 0 3 13.87  0 1 10 42.19  ±3 0 u 24.02  \n−1 0 4 15.43  0 1 11 45.51  ±3 0 1 25.59  \n−1 0 5 17.19  0 1 12 49.12  ±3 0 3 27.83  \n−1 0 6 19.34  −1 1 g 35.16  ±3 0 5 31.25  \n−1 0 7 21.78  −1 1 u 35.16  ±3 0 7 35.84  \n−1 0 8 24.41  −1 1 3 39.26  ±3 0 11 47.95  \n−1 0 9 27.34  −1 1 5 42.68      \n−1 0 10 30.57  −1 1 6 44.82      \n−1 0 11 33.98  0 2 g 55.57      \n−1 0 12 37.6 0 2 0 56.64     \n−1 0 13 41.5 0 2 2 58.3     \n−1 0 14 45.51  0 2 4 61.23      \n−1 0 15 49.71  0 2 6 65.53      \n    0 2 8 70.41       31 \n \n \n \n \nFig. 5.1. a) Typical fit of the radial OOMMF amplitude to the function \n1J (k r) ⊥  for \nthe ( –105) mode.  b) T ypical fit of the axial OOMMF amplitude to the function \n()z cosk z h / 2 −\n for the (006) mode.  \n \n  32 5. Comparison of simulated frequencies with the Herring -Kittel expression  \n It is interesting to examine the extent to which the OOMMF mode frequencies  in Table II \ncan be represented by the Herring -Kittel frequencies as given by Eq, (1.3). To do this we need \nvalues of \nzk  and \nrp,nk  for the extended modes. A preliminary value of \nzk  follows from \ncounting the number of nodes along z. Better values emerge55 from fitting \nrm (z,r 0, ) =  to \n( ) z cos k (z h / 2) −\n or \n( ) z sin k (z h / 2) − . Values for \np,nk  were obtained by fitting \nrm (z,r, )   to \np p,nJ (k r)\n at some z with \np,nk  as an adjustable parameter. Fig. 5.1a,  b shows examples of such \nfits. Note that although we do not employ it to find \nzk  and k⊥ , the boundary condition at the \nfaces for this mode closely approximate s maximum amplitude as opposed to maximum \nderivative.  \n We now show some plots of the frequencies, \nr z r z(pn n ) (pn n )f / 2 =    inferred from the \nsimulations, for various modes \nrz(pn n )  versus \nzk  at fixed \np,nk  with these latter values \nobtained by the above procedures.  Also shown are the frequencies predicted by the H - K \nexpression, Eq. (1.3), for the same wave -vector components and a demagnetization  coefficient of \nN 0.098=\n.  \n  The \n  symbols in Fig. 5.2 show the results of the OOMMF simulation for the \nz (00n )\n modes, as a function of \nzk  in units of \n/h , for which the lowest frequency is \n7.81GHz. Not included are the accompanying cap -mode frequencies which are both 6.41GHz \nwithin the resolution. The continuous curve shows the frequencies predicted by the Herring -\nKittel (H - K) formula. The ag reement is surprisingly good, especially at small \nzk  where H - K \nis expected to break down. To some extent this may arise from the fact that the internal magnetic \nfield is position dependent, being lower at the caps, and thereby pr oducing an effect partially \ncompensating that of \nN⊥ . The \n  symbols  show the predictions of the one -dimensional \nSchrodinger  equation as described in section 2.2.  \n \n \n \n  33  \n \nFig. 5.2. The frequencies of the \nz (00 n )  modes vs. \nzk  as obtained from OOMMF \nsimulations (  symbols), Herring -Kittel formula (continuous line) and the \nSchrodinger equation  (\n ), as discussed in section 2.2. For the Schrodi nger \nequation data \nzk h /   is a proxy for \nzn2+ . \n \n \nFig. 5.3. The  symbols  show the OOMMF simulations for the mode frequencies versus the \ndimensionless wave -vector \nzk  for p = 0 and \nr    n 0, 1 and 2= ; the continuous curves show \nthe predictions of the H – K expression.   34  \n The symbols  in Fig. 5.3 show the OOMMF simulations of p = 0 mode frequencies as a function \nof the dimensionless \nzk  for \nrn 0, 1 and 2=  while the continuous curves show the predictions for the \ncorresponding mode numbers of the H -  K formula. Again,  the agreement is excellent. Readers may \nnotice that some modes are missing in these plots. This is because they were not excited with t he \nprotocols used, but we are confident they exist and that their mode patterns conform with the general \nframework presented here.  \n \nFig. 5.4. The OOMMF mode frequencies vs. \nzk  for \nrn0=  and \n   −   p 0, 1, 2, 3=\n. The curves show the predictions of the H – K equation.  \n \n \n Lastly, Fig. 5.4 shows the OOMMF simulations for mode frequencies vs. \nzk  for \nrn0=  \nand \np 0, 1, 2, 3 = −    . The H - K formula again  gives an excellent overall representation.  \n Analogous to our approximating the position dependence of the bulk states with a form \nz cos(k z)\n we can use \nze−  and \n(z h)e−  to qualitatively describe  the amplitude in the vicinity of \nthe cylinder faces for the cap mode; i.e., we take k as imaginary by writing \nki= , in which case \n2k\n is replaced by \n2−  in a Herring -Kittel like expres sion, which pushes the frequency below  \nthat of the first extended mode.  \n5.1. Why the H - K formula works so well   35  The H - K relation is sometimes referred to as a spin -wave dispersion relation, and indeed \nit is tantamount to saying that the normal modes o f the body can be described by a continuous (or \nquasicontinuous) variable k. As explained in Appendix A, this assertion is grossly incorrect \nwhen the spatial variation of m is on a scale comparable to the dimensions of the body56; the \nmode functions must th en take account of the shape  of the body, and be described by discrete \nsets of appropriate mode numbers, which may or may not be wavevector -like. \n Nevertheless, if it should turn out that some discrete modes with slow spatial variation are \nwell described b y wavevector -like variables, then for those modes the H - K expression can be \nexpected to give the frequency to good approximation, especially if one is in the exchange \ndominated limit. This is the situation in the present investigation. We have seen that for every \nbulk mode, we can identify a reasonably well -defined \nzk  and an orbital angular momentum \nquantum number p. We also see a radial dependence in \n(r)m  that pretty well matches a Bessel \nfunction \npJ (k r) ⊥  from which can obtain a \nk⊥ . \n That the mode functions should look this way is not an accident. We have some support for \nthis functional behavior from the theory of the infinite cylinder in the exchange dominated l imit. \nWe intend to publish the details of this theory separately, and here we only summarize the key \nresults. In first order perturbation in the small parameter  \n \n0\n2exM=  \nDk             (5.1)  \nwe find that the mode function is given by  \n \n( )zx i(p t) i((p 2) t) ik zp\nym 11 1J (k r) f (r) e f (r)e em ii 2− − + + − ⊥     = + +      −       .  \n               (5.2)  \nThe corrections \nf  are both \nO( ) . We add here that we discovered the properties of the mode \nfunctions by experimenting with OOMMF first, and developed the theoret ical framework later.  \n6. Examples of mode patterns  \nTable II in section 4 lists approximately 90 modes which we have identified. We will  \nnow present some accompanying mode patterns which display various behaviors. All figures  in \nthis section pertain to YIG  cylinders with a diameter d = 2a = 75 nm and a height  h = 300 nm in \na magnetic field H 0 = 2000  Oe.  36 There is a wealth of information in these figures as we now explain.  They all depict \nvarious aspects of the frequency space Fourier amplitude m(r, ), which is  a complex vector, \ni.e., its x, y, and z components are all complex numbers. The real parts  make a vector, and so do \nthe imaginary parts, which we can call the real and imaginary parts  of the complex vector, \n( , )mr\n and \n( , )mr . We discard the z component leaving the  xy projection m⊥(r, ). \nCircular panels such as Figs. 6.1a and 6.1b show xy cross sections  of the cylinder, while \nrectangular panels such as Fig. 6.1c show yz cross sections through a  diameter  of the cylinder. In \nthe circular panels, the arrows show the directions and relative  magnitudes of either \n( , )⊥mr  \nor \n( , )⊥mr , while the thin black lines show contour  levels of the magnitudes of these same \nvectors, i .e., either \n( , )⊥mr  or \n( , )⊥mr . These contour levels are also color coded \naccording to the scale on the right. As explained  in Sec. 3.3 — see Eqs. (3.4) and (3.5) — the \n  \npanel shows the spi ns a quarter -cycle after  the \n  panel. That is, time proceeds from \n  to \n , \nwhich is reflected in the sequence of  panels a) and b) when both \n  and \n  parts are shown. In \nthe rectangular panels we show  contours of \nxm ( , ) r ; again, the contours are color coded. \nBecause the circular panels  show the magnitude of the vector in the xy plane, while the \nrectangular panels  show only  the x component, the contour levels in the two types of panels \ncannot be directly compared.  Furthermore, depending on just how a particular mode is excited, \nthe spins can have larger  projections along the x or y directions at the particular time  captured in \nthe xy cross  sections, and this can further affect the values of the contour levels in the rectangular \npanels  vis-a-vis the circular ones. The most salient feature is the variation or the relative Fourier  \namplitude within a panel. We add here that our simulations are done with 50 vertical layers of \ncells of height 6 nm each. There is a layer extending from z = 144 to 150 nm, and another from z \n= 150 to 156 nm. Hence, circular panels such as in Figs. 6.1a and 6.1b that are labeled z = 147  \nnm cor respond to the midpoint of the cell layer just below the midplane of the  cylinder;  panels \nlabeled z = 3 nm such as in Fig. 6.2a show the lowest layer; panels labeled  z = 39 nm such as in \nFigs. 6.5a and 6.5b show the 7th layer from the bottom. However, in  the text and figure captions \nwe have described the panels at z = 3 nm and z = 147 nm as  lying at z = 0 and z = h/2, as this is \nmore natural and intuitively easier to understand.  \nIn the interest of clarity, we shall repeat these points as necessary, and add further  \ninformation about the patterns as we discuss them one by one.   37 We start with the lowest lying modes: the nominal bulk uniform precession or cylindrical  \nKittel (0, 0, 0) mode with f = 7.813  GHz, which is concentrated within the body of the  cylinder, \naway from the caps, together with the even (0, 0, g) and odd (0, 0, u) cap modes  concentrated on \nthe top and bottom cylinder faces with a mean frequency of 6.543  GHz and  a splitting that is too \nsmall to be resolved.  \nFig. 6.1a shows \n()000(x,y,z h 2, )⊥ = m  while Fig. 6.1b shows \n()000(x,y,z h 2, )⊥ = m\n for the (0, 0, 0) bulk mode with f = 7.813 GHz; here \n( , , 2, )x y z h⊥ = m\n and \n( , , 2, )x y z h⊥ = m  denote the normalized vector fields of the \nFourier amplitude given by following prescript ions:  \ni\n12\n2 1\ncell i\ni( , )K\nN ( , )⊥\n−\n\n \nmr\nmr\n and \ni\n12\n2 1\ncell i\ni( , )K\nN ( , )⊥\n−\n\n \nmr\nmr  (6.1)  \nwhere K is a global constant scale factor whose value is chosen as 1000 for convenience  for all \nmode patterns (this and subsequent ones). The arrows indicate the xy projection  of the \nmagnetization. The lines and color coding depict the contours of constant amplitude,  either \n()i,⊥mr\n or \n()i,⊥mr . Ideally these would be concentric circles but there is  always \ncontamination at some level from other mode s. There may also be numerical errors  associated \nwith the discretization. Time proceeds from \n  (panel (a)) to \n  (panel (b)), a  quarter cycle later.  \nFig. 6.1c shows \n()000\nxm (x 0,y,z, ) =  , again with contour lines together with color  \ncoding. Note the contour lines are quite parallel to the faces, a behavior that arises from the  \nstrong influence of exchange in these small samples and validates the factorized form  Eq. (2.21) \nfor \n(), F r z− . \nFig. 6.2a shows \n()00g(x,y,z 0, )⊥ = m  for the symmetric (g) cap mode with f = 6.543  \nGHz. Figs. 6.2b and 6.2c show \n()00g\nxm (x 0,y,z, ) =   and \n()00u\nxm (x 0,y,z, ) =   for the even \n(g) and odd (u) cap modes respectively. We see that the mod e intensity is strongly  concentrated  38 near the cylinder faces, dropping off rapidly as one proceeds to the interior.  Note the anti -\nsymmetric character of the u mode is clearly apparent as seen from the node  at z = h/2.  \nFigs. 6.3a,  b show \n( )10u(x,y,z 0, )−\n⊥ = m  and \n( )10u(x,y,z 0, )−\n⊥ = m  of the (−1, 0, u)  \nantisymmetric 10.06  GHz cap mode which has a node at r = 0. Note how the spins wind  through \nan angle of 2  as we proceed counter -clockwise around the line r = 0. Fig. 6.3c  shows \n( )10u\nxm (x 0,y,z, )− = \n where the antisymmetric behavior in z is evident.  \nFigs. 6.4a,  b show \n()10u(x,y,z 0, )⊥ = m  and \n()10u(x,y,z 0, )⊥ = m  for the \nneighboring  p = +1 mode with a frequency of 10.35  GHz. Here one encounters the “retrograde” \nmotion  associated with the oppositely winding sense of m with the azimuthal angle . \nWe next consider a mode with multiple nodes along z. As remarked earlier the splitting \nbetween ±p modes diminishes as the overall mode numbers increase, so we designate them with \nboth signs since our mode projection method generally yields a superpositi on. Here  we consider \nthe (±1, 0, 5) mode(s) with f = 17.29  GHz. Figs. 6.5a,  b show \n( )105(x,y,z 39nm, )\n⊥ = m  and \n( )105(x,y,z 39nm, )\n⊥ = m\n while Fig. 6.5c shows  \n( )105(x 0,y,z, )\n⊥ = m . (We recall that z = \n39 nm is the midplane of the 7th layer of cells from the bottom of the cylinder.) Note the mode \npatterns in the xy plane now display nodes  since the modes with p = +1 and p = −1 interfere to \nform a partial standing wave. Our  plot in the yz plane contains the two end nodes arising from \northogonality to the cap modes  discussed above (the patterns for which we do not show) as well \nas the five interior nodes.  If we use the projecti on technique described in Appendix B, we can \nagain separate the two  modes. This is shown in Figs. 6.6a and 6.6b where we plot \n( )105(x,y,z 39nm, )+\n⊥ = m\n and \n( )105(x,y,z 39nm, )−\n⊥ = m . These are also the modes for \nwhich we resolved the splitting  via the beat pattern in Fig. 3.2a.  \nAs an example of a mode with a larger azimuthal mode number and mixed p character,  \nFig. 6.7 shows a plot of \n( )305(x,y,z 39nm, )\n⊥ = m , which has a frequency of 31.25  GHz.   39 Finally, we present a mode with additional radial nodes.  Such a mode will have a high  frequency \nconsidering the relatively small diameter of our sample. Figs. 6.8a and 6.8b show  \n()020(x,y,z h 2, )⊥ = m\n and \n()020\nxm (x 0,y,z, ) =   for the (020) mode with a frequency  of f \n= 56.64  GHz.  \n \n  40  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 6.1. Mode pattern for the lowest (Kittel -like) bulk (0 0 0) mode, with a frequency of \n7.813 GHz . Parts a) and b) show an x - y cross section of th e imaginary and real \nparts of the Fourier transform amplitude through the cylinder mid point while part \nc) is the real y - z cross section cont aining the cylinder axis. The arrows show the \ndirection of the spins. The spin orientations for the real part correspond to a time 1/4 \ncycle later than that  for the imaginary part. The lines show contours of constant \n| ( , ) |  mr\n in (a) and (b) and of constant \n( , )x mr  in (c) ; these values are also color \ncoded according to th e scale given to the right.  \n 41  Fig. 6.2.  The p = 0, n r = 0 cap \nmodes. Part a) shows the real x - y \ncross section at z = 0 of the  \n(00g)\nsymmetric cap mode; this \nmode has the globally lowest \nfrequency of 6.543 GHz . Parts b) \nand c) show the real y - z cross \nsections of the g and u modes \ncontaining the cylinder axis of the \nfrom which the surface \nconfinement is apparent . \n \n 42  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 6.3. Parts  a) and b) show the  imaginary and real x  - y cross \nsections at z = 0 of the ( –1 0 u) antisymmetric cap mode with a \nfrequency of 10.06  GHz. Part c) is the real y - z cross section \ncontaining the cylinder axis from which the surface \nconfinement is apparent.  \n \n 43  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 6.4. Parts a) and b) show th e imaginary and real x - y cross sections at z = 0 of the \n(10u) antisymmetric cap mode with a frequency of 10.35  GHz. Important to note here is \nthe retrograde character of the spin winding.   44  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 6.5. Parts a) and b) show  the imaginary and real  x - y cross sections at z = 39 nm  of the \n( 105)\n modes with a frequency of 17.29 GHz. Note the standing wave behavior of these \ncross -sections arising from the superposition of azimuthally counter propagating mode s.  \nHowever, at a fixed point in space the spins still precess counterclockwise. Part c) is the real y \n- z cross section containing the cylinder axis.  \n \n 45  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 6.6. Here the projection technique described in Appendix B has been used to separate the modes \nwith p = +1 and p = –1 from the same data used to construct figure 6.5.  Note the azimuthal intensity is \nnow constant as appropriate for running waves.  \n \n \nFig. 6.7.   The real x - y cross section of \nthe \n( 305)  bulk mode with a \nfrequency o f 31.25  GHz showing a \nmultiplicity of azimuthal nodes. Note \nalso the deep central node due to the \n3r\n behavior of \n3J (k r) ⊥ . \n  46  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 7.1. The depen dence of the frequency on h for the  lowest \nzk  bulk mode  (000),  symmetric cap \nmode  (00g), and  antisymmetric cap mode  (00u) of a YIG cylinder with d =75nm. Also shown is the \nfrequency predicted by the Kittel expression. The inset sh ows the region where the symmetric cap \nmode crosses over the  antisymmetric cap mode for small h. In the region below about 1 00nm the \n(00g) mode replaces the (000) mode as the quasi uniform mode we associate with the Kittel formula.  \nSee discussion.  \n Fig. 6.8. The real x -y and y -z cross -sections of the (02 0) mode with a frequency \nof 56.64 GHz exhibiting multiple radial nodes.  \n \n 47 7. Evol ution of the low -lying mode behavior with cylinder height  \n Although the majority of our simulations for mode patterns were carried out for a YIG \ncylinder with d = 2a = 75  nm and h = 300  nm, the behavior of the three lowest lying p  = 0 \nmodes, (000), (00g), and (00u), was studied over the much wider range of h extending from \n7.5nm to 1200  nm57. Fig. 7.1 shows the frequencies emerging from the OOMMF simulations \ntogether with the predictions based on the Kittel expression, Eq. (1.2), according to the \ndemagnetiza tion coefficients found by Joseph and Schl omann58.  Importantly we see that the \n(00g) “cap mode” evolves into the dominant mode in the thin -disc limit , while the (000) bulk \nmode becomes dominant (although lying slightly higher in frequency) in the long cyli nder limit. \nThe Kittel formula, which is a single equation, actually describes two different modes in these \nlimits and does not apply well to any mode for  150 nm \n  h \n 300 nm . The level crossing of the \n(00g) and (00u) modes does not violate the Wigner -von Neumann anti -crossing theorem due to \ntheir even and odd character. Furthermore,  the apparent violation of the ordering of energy levels \nof 1d Schrodinger equations is resolved by noting that the dipole -dipole interaction is a non-local  \nperturbation which puts the eigenvalue problem outside the Sturm -Liouville class.  \n \nFig. 7.2. The contours of constant intensity for three modes of a YIG cylinder with d = 75  nm and h = 37.5  \nnm. a) The 4.59  GHz symmetric (00 g) cap mode, corresponding to the Kittel mode in the thin -disc limit; note \nit has no nodes. b) The 13.57  GHz antisymmetric (00u) cap mode.  c) The 42.97  GHz (000) “bulk” mode.  \n \n It is clear from this discussion that the true behavior of the dominant FMR re sponse \ncannot be described by a simple Kittel -like expression; a characterization in terms of  48 demagnetization coefficients is often inadequate and glosses over the spatial complexity of the \nmode with the greatest spectral weight in a uniformly excited samp le. \n Some mode patterns for a YIG cylinder with d = 75  nm and h = 37.5  nm are shown in \nfigure 7.2. The cell size for these simulations is 3 x 3 x 1.5 nm^3 . Fig7.2a shows the shows the  \n4.59 GHz symmetric (00g) “cap” mode which for these dimensions has becom e the dominant \nmode. It is a cap mode in name only since it spans the entire sample. Rather than the planar \nconstant amplitude contours encountered in the longer cylinders, this mode now has \napproximately cylindrical ones. Figure 7.2b shows the antisymmetric  cap mode for a cylinder \nwith the same dimensions which now has a significantly higher frequency of 13.57  GHz; here \nthe contours of constant amplitude run approximately parallel to the faces indicating that the odd \nmode has substantially changed charact er from the even one. Lastly,  we show the bulk mode in \nFig. 7.2c. Consistent with the results in Fig. 7.1 this mode has the highest frequency, 42.97  GHz. \nThe contours of constant amplitude run approximately parallel to the faces and we obtain a quite \nregul ar sine wave with one complete wavelength along z.  \n \n8. Conclusions  and possible applications  \n We have explored the mode structure of nanoscopic cylinders of yttrium iron garnet, both \nanalytically and through many -spin simulations, in a regime where the ef fects of exchange \ndominate the response.  In addition to the extended (bulk) modes, which can be classified in \nterms of the azimuthal, radial, and axial mode numbers designated as \nrzp,n ,n , we find \nsymmetric and antisymmetric combination s of cap modes, that are localized at each of the \ncylinder faces. In all cases they lie lower in frequency than the accompanying family of modes \nzn\n for given azimuthal and radial mode numbers \nr p and n . When exami ning the height \ndependence,  we find that the dominant FMR response  cannot be precisely described by a simple \nKittel -like formula. How this picture would change in passing to the dipole dominated limit is \ndeserving of additional study.  \n By way of applicatio ns, there is a growing effort directed at using magnetic bits for \ncomputation as well as data storage. In particular it has been demonstrated that the required logic \noperations can be accomplished with lines and arrays of dipole -coupled, single -domain, bar  \nmagnets with dimensions of a few hundred nanometers59. While promising, this approach is still  49 restricted to what can be done using binary macro -spin flips (and cascades thereof) of the \nindividual magnets.  \n Looking farther ahead it is natural to ask if lo gic functions can be performed by \nexploiting the internal  dynamics of nano particles . For the case of wave -guide based operations \nthis is already a world -wide activity60. But here we envision exciting (and mixing) large \namplitude resonant modes within a sin gle nanoparticl e. This can involve a  single or multiple \ninputs applied simultaneously or sequentially, with different microwave frequencies and/or \npolarizations . When the particles are small the various  modes are well separated and can be \naddressed individ ually and rapidly. By exploiting intrinsic nonlinearities and optimizing the \nsample dimensions (to tune the mode frequencies), different pump frequencies can be efficiently \nmixed, a topic we are exploring independently. Progress in this direction requires an \nunderstanding of the low amplitude mode structure of the particles  involved.  \n Apart from the cap modes, the modes studied here are all extended in character and in \nthat sense are the standing wave counter part of the plane wave modes  that were the start ing \npoint of the non -linear analysis  carried out by  Suhl discussed in the introduction .  However \nongoing simulations of nano -scale cylinders and elliptical discs at large precession amplitudes \ndisplay instabilities involving the edge -nucleation of dynamic vortex and antivortex modes. \nPossibly related instabiliti es occur a t domain walls in stripes61,62. The connection between such \nstates and low -lying extended mode s in understanding large amplitude precession  dynamics in \nnanomagnets is currently unclear.  \n  \nAcknowledgement  \n This research was carried out under the support of U.S. Department of Energy through \ngrant DE -SC0014424. This research was supported in part through the computational resources \nand staff contributions provided for the Quest high performance computing facility at \nNorthwestern University which is jointly supported by the Office of the Provost, the Office for \nResearch, and Northwestern University Information Technology.  \n \nAppendix A: The Herring - Kittel equation  \n The Herring - Kittel (H - K) exp ression, given earlier as Eq. (1.3), is   50  \n2 2 2 2 2in,z ex in,z 0 ex(H D k )(H 4 M sin D k )  =  + +   +      (A.1)  \nwhere we wrote \nin,z 0 0H H 4 N M  − \n  with \nN\n  an axial demagnetization coefficient, and  \n \nexex\n02ADM= ,            (A.2)  \nwhich fixes  the exchange energy density used in the OOMMF simulation,  \n \njj exex2iii,j0MM AExx M= .          (A.3)  \nTo understand the remarkable agreement between our OOMMF results and this formula,  \nlet us recall how it is derived.  \n The linearized Landau -Lifshitz eq uation is  \n \nex 2in,z 0\n0d 2AˆHMdt M=   − + mz m h m        (A.4)  \nwhere h is the dynamic demagnetization (or dipolar) field induced by m. Fourier transforming \nwith respect to space and assuming a time dependence,  \nite−  we obtain  \n \nex 2k in,z k 0 k k\n02Aˆ i H M kM−  =   − −m z m h m .      (A.5)  \nThe field \nh  is governed by the Maxwell equations,  \n \n = 4π ,       = 0  −  h m h   ,         (A.6)  \ntogether with the requirements that the normal component of \n4 = + b h m  and the tangential \ncompon ent of h be continuous at the boundary of the particle. If we now Fourier transform Eq.’s \n(A.6), and simply  ignore  the effects of the boundary conditions we obtain  \n \nkk24\nk= − kmhk .            (A.7)  \nInserting (A.7) into Eq. (A.5), requiring th e equations for the resulting two components  to be \ncompatible, and taking \nin,zH  to be homogeneous, leads immediately to the HK formula, Eq. \n(A.1).  \n From the above derivation we see that, qualitatively, the HK approximation accounts f or \nthe axial (static) demagnetization but neglects some part of the transverse contribution.  Since the \ndipole -dipole interaction is long ranged, this neglect is qualitatively profound, and quantitatively  51 valid only for short wavelengths, when \nka  1 . When this condition is satisfied we can argue \nthat the magnetic charges induced on the “lateral” surface by m change sign rapidly on a length \nscale \n1k− , so the field produced by them dies off on the same length scal e and may be ignored in \nthe bulk of the particle. To further clarify this behavior,  we will derive the HK formula in a \nsecond way.  \n As is known from magnetostatics, the (normalized) magnetic field, \n()hr , can be regarded \nas arising f rom a magnetic charge density \n()mr  according to  \n \n3\n3 V()( ) 4 ( )d r\n||+−   = −  \n−rrh r m r\nrr .        (A.8)  \nThe integral here is taken to extend infinitesimally beyond the particle volume, as indicated by \nthe superscript in \nV+ . In this way both volume and surface charges are included. If we Fourier \ntransform this equation, we obtain  \n \n3\n24 d k   \nk− = −  \nkk k kkmhk          (A.9)   \nwhere \nk  is the Fourier transform of unity over the particle volume:  \n \ni3\n3V1e d r\n(2 )+−=\nkrk  .          (A.10)  \nWe can write (A.9) more compactly as  \n \nk24\nk= −  kkkmhk           (A.11)    \nwhere ∗ denotes a convolution. For a cylinder of height h and radius a,  \n \n2z1\n3za h sin k h 2J (k a)k h k a (2 )⊥\n⊥=\nk .         (A.12)  \nWhen \nzk h 1\n  and  \nka⊥ \n  in the convolution in Eq. (A.11), \nk  can be approximat ed by a \ndelta -function, \n()k , and we recover Eq. (A.7). For smaller k this approximation is invalid.  \n  \nAppendix B.  Relation between standing and running waves in \n  \nThe magnetization fields corresponding to running waves associated with orbital angular \nmomentum p and –p (with p > 0) have the form,   52  \n()() ( ) ( ) p ˆ ˆ ,t F r,z cos p t sin p t=  −  −  −  m r x y ,    (B.1a)  \n \n()() ( ) ( ) p ˆ ˆ ,t F r,z cos p t sin p t− = −  −  − −  −  m r x y ,    (B.1b)  \nwhere\n F(r,z)  is an unspecified function. By superposing these fiel ds, we obtain a standing wave \npattern,  \n \n() ()() () ()s\np p p ˆ ˆ ,t F r,z cos p cos t sin t− = + =   +  m r m m x y .   (B.2)  \nWe wish to recover the running wave patterns from a knowledge of the standing wave pattern. \nTo do this, we first transform the latter by rotating the amplitude by p, and the ve ctor by \ndirection by . The transformed field is,   \n \n() ( )( ) ( )\n()() () ()Ts\npp,t r,z, 2p ,t 2\nˆ ˆ F r,z sin p sin t cos t .=  −  +  \n=  −  +  m r m\nxy     (B.3)  \nIt is now easy to see that the difference of the transformed and the original stationary wave \npattern gives us \npm : \n \n() () ()sT\np p p,t ,t ,t 2 =−m r m r m r .       (B.4a)  \nLikewise, the sum gives \np−m : \n \n() () ()sT\np p p,t ,t ,t 2−=+m r m r m r .      (B.4b)  \n What this means is the following. Supposing OOMMF has produced a pattern which has \np nodal lines in  at some fixed time. We denote this patte rn by \ns\npm  as above. We then consider \nthe vector fields  \n \n( )\n( )( )\n( )( )\n( )ss\np,x p,y p,x\nssp,yp,y p,xm r,z, m r,z, 2p m r,z, 11\n22 m r,z, m r,z, m r,z, 2p\n     −  −      =         −      \n .   (B.5)  \nIf these combinations are used as initial conditions in OOMMF, they should evolve into \np+  and \np−\n running waves as indicated. 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Mills, Theory of spin excitations and the microwave response of cylindrical \nferromagnetic nanowires, Phys. Rev. B 63, 134439 (2001).  \n36 C. Herring and C. Ki ttel, On the theory of spin waves in ferromagnetic media, Phys. Rev. 81, \n869 (1951) .  55  \n37 S. Shultz and G. Dunifer, Observation of spin waves in sodium and potassium, Phys. Rev. \nLett. 18, 283 (1967).  \n38 N. Masuhara, D. Candela, D. O. Edwards, R. F. Hoyt, H. N.  Scholz, D. S. Sherrill  and R. \nCombescot, Collisionless Spin Waves in Liquid 3He, Phys. Rev. Lett. 53 (12), 1168 (1984).  \n39 V. P. Silin, Oscillations of a Fermi -liquid in a magnetic field, Zh. Eksp. Teor. Fiz. 33, 1227 \n(1957) [Sov. Phys. JETP 6, 945 (1958)] . \n40 A. J. Leggett, Spin diffusion and spin echoes in liquid 3He at low temperature, J. Phys. C : \nSolid State Phys.  3, 448 (1970) . \n41 G. Venkat, D.Kumar, M. Franchin, O. Dmytriiev, M. Mruczkiewicz, H. Fangohr, A.Barman, \nM. Krawczyk, and A. Prabhakar, Proposal  for a Standard Micromagnetic Problem: Spin Wave \nDispersion in a Magnonic Waveguide IEEE Trans. Magn . 49, 524 –529 (2013).  \n42 L. D. Landau and E. M. Lifshitz, Statistical Physics Part II, Pergamon Press, Section 69; the \nform written here follows Herring and Kittel.  \n43 This argument is standard, and can be found in many books. See, e.g., R. Shankar, Principles \nof Quantum Mechanics, Plenum, New York, 1980, Exercise 12.5.1 and Fig. 12.1.  \n44 See, e.g., Anupam Garg, Classical Electromagnetism in a Nutshell, Princeto n University Press, \nPrinceton, N. J., 2012, pp. 174 –177, and 239 –240. \n45 See Eqs. (19), (23), and (24) in Ref. 34. \n46 See the unnumbered equation two equations above Eq. (22) in Ref. 29.  \n47 H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Elec tron Atoms, \nSpringer, Berlin (1957), Sections 29 and 30.  \n48 W. S. Ament and G. T. Rado, Electromagnetic Effects of Spin Wave Resonance in Ferro -\nmagnetic Metals, Phys. Rev. 97, 1558 (1955)  \n49 Amikam Aharoni, Introduction to the Theory of Ferromagnetism, Oxfor d, 1996, p. 178.  \n50 See, e.g., C. M. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and \nEngineers, McGraw -Hill, 1978, Chapter 9.  \n51 This has the virtue of replacing the discrete OOMMF field profile with a continuous one; it \nalso gave some what better agreement with the simulated OOMMF mode profiles.  \n52 M. Sato and Y. Ishii, Simple and approximate expressions of demagnetizing factors of \nuniformly magnetized rectangular rod and cylinder, J. Appl. Phys. 66 (2), 983 (1989). DOI: \n10.1063/1.343481 . \n53 A. G. Gurevich and G. A. Melkov, Magnetization oscillations and waves, CRC Press, Boca \nRaton, 1996.  \n54 It can also drive additional modes having some overlap on the chosen mode at some initial \ninstant in time.  \n55 Alternatively, we could Fourier transform  the simulated form of \nrm (z,r, )   along z, where the \nbar indicates a radial average, and identify the peak in the spectrum. Since our description is \nonly semi quantitative this procedure was not attempted.  \n56 In particular Eq. (A.11) cannot  be replaced by Eq. (A.7) in Appendix A. In this regard, we find \nthe discussion by L. R. Walker, Resonant Modes of Ferromagnetic Spheroids, J. Appl. Phys.  \n29, 318 -324 (1958)  and his Fig. 6 to be particularly germane.  \n57 The simulations were done with cell d imensions of 3 nm × 3nm in the x and y directions, and \nthe dimension z along the z direction was adjusted so as to give varying numbers of layers \ndepending on the height h. The number of layers varied from 10 for h = 7.5 nm, to 25 for h= \n16.8 to 145 nm, a nd either 25 or 50 for h ≥ 150nm. The choice of 25 layers is enough to  56  \ncapture the spatial variation of the three modes we are seeking in this section. It is more \nimportant to increase the run time (by as much as a factor of 8) in order to resolve the mode s in \nfrequency.  \n58 R. I. Joseph and E. Schlomann, Demagnetizing field in non -ellipsoidal bodies, J. Appl. Phys. \n36, 1579 (1965).  \n59 For a review see: M. T. Niemier, G. H. Bernstein, G. Csaba, A. Dingler, X. S. Hu, S. Kurtz, S.  \nLiu, J. Nahas, W. Porod, M. Siddiq and E. Varga, Nanomagnet logic: progress toward system -\nlevel integration,  J. Phys. Condens. Matter  23, 493202  (2011) . \n60 Abdulqader Mahmoud, Florin Ciubotaru, Frederic Vanderveken, Andrii V. Chumak, Said  \nHamdioui, Christoph Adelmann, and Sorin Cotofana, Introduction to spin wave computing, J.  \nAppl. Phys. 128, 161101 (2020); https://doi.org/10.1063/5.0019328  \n61 D. J. Clarke, O. A. Tretiakov, G. -W. Chern, Ya. B. Bazaliy, and O. Tchernyshyov, Dynamics  \nof a vortex domain wall in a magne tic nanostrip: Application of the collective -coordinate  \napproach, Phys. Rev. B  78, 134412 (2008).  \n62 M.  Klaui,  Topical Review: Head -to-head domain walls in magnetic nanostructures,  J. Phys.  \nCondens. Matter 20, 313001 (2008).  "    },    {        "title": "2102.02116v2.Infinite_Series_of_Ferrimagnetic_Phases_Emergent_from_the_Gapless_Spin_Liquid_Phase_of_Mixed_Diamond_Chains.pdf",        "content": "arXiv:2102.02116v2  [cond-mat.str-el]  6 Apr 2021Journal of the Physical Society of Japan FULL PAPERS\nInfinite Series of Ferrimagnetic Phases Emergent from the Ga pless Spin\nLiquid Phase of Mixed Diamond Chains\nKazuo Hida∗\nProfessor Emeritus, Division of Material Science, Graduat e School of Science and Engineering,\nSaitama University, Saitama, Saitama, 338-8570\n(Received )\nThe ground-state phases of mixed diamond chains with ( S,τ(1),τ(2)) = (1/2,1/2,1), where Sis the\nmagnitude of vertex spins, and τ(1)andτ(2)are those of apical spins, are investigated. The apical spin s\nτ(1)andτ(2)are connected with each other by an exchange coupling λ. Other exchange couplings are set\nequal to unity. This model has an infinite number of local cons ervation laws. For large λ, the ground state\nis equivalent to that of the uniform spin 1 /2 chain. Hence, the ground state is a gapless spin liquid. For\nλ≤0, the ground state is a Lieb-Mattis ferrimagnetic phase wit h spontaneous magnetization msp= 1\nper unit cell. For intermediate λ, we find a series of ferrimagnetic phases with msp= 1/pwhereptakes\npositive integer values. The phases with p≥2 are accompanied by the spontaneous breakdown of the\np-fold translational symmetry. It is suggested that the phas e with arbitrarily large p, namely infinitesimal\nspontaneous magnetization, is allowed as λapproaches the transition point to the gapless spin liquid p hase.\n1. Introduction\nThe quantum effects in low-dimensional frustrated\nmagnets have been extensively studied in recent con-\ndensed matter physics.1,2)Various exotic quantum\nphases emerge from the interplay of quantum fluctuation\nand frustration. Among them, the diamond chain,3–9)\nwhose lattice structure is shown in Fig. 1, is known as a\nmodel with an infinite number of local conservationlaws.\nThe ground states can be classified by the correspond-\ning quantum numbers. If the two apical spins have equal\nmagnitudes, which is the case widely investigated, each\npair of apical spins can form a singlet dimer. It cuts the\ncorrelation between both sides and the ground state is a\ndirect product of the cluster ground states separated by\ndimers.\nThe ground states of spin-1/2 diamond chains have\nbeen investigated in Ref. 3. In addition to the spin clus-\nter ground states, the ferrimagnetic state with sponta-\nneous magnetization msp= 1/2 per unit cell is found.\nIn the latter phase, the apical spins form triplet dimers\nand all the spins collectively form a long-range ordered\nferrimagnetic state. Extensive experimental studies have\nbeen also carried out on the magnetic properties of nat-\nural mineral azurite that is regarded as an example of\ndistorted spin-1/2 diamond chains.8,9)\nThe ground states of spin-1 diamond chains have been\nalso investigated in Refs. 3 and 4. In addition to the spin\ncluster ground states, the nonmagnetic Haldane state\nand the ferrimagnetic states with spontaneous magne-\ntizationmsp= 1 and 1/2 are found. It should be noted\nthat the latter ferrimagnetic state is accompanied by a\nspontaneous translational symmetry breakdown.\n∗E-mail address: hida@mail.saitama-u.ac.jpOn the other hand, if the magnitudes of the two apical\nspins are not equal to each other, they cannot form a sin-\nglet dimer. Hence, all spins in the chain inevitably form a\nmany-body correlatedstate. In manycases, (quasi-)long-\nrange order evolves including the vertex spins. As a sim-\nple example of such cases, we investigate the mixed di-\namond chain with apical spins of magnitude 1 and 1/2,\nand vertex spins, 1/2 in the present work. Remarkably,\nwe find an infinite series of ferrimagnetic phases.\nThispaperisorganizedasfollows.InSect.2,themodel\nHamiltonian is presented. In Sect. 3, the ground-state\nphase diagram is determined numerically. The behavior\nof the spontaneous magnetization in each phase is pre-\nsented and analyzed. The last section is devoted to a\nsummary and discussion.\n2. Hamiltonian\nWe consider the Hamiltonian\nH=L/summationdisplay\nl=1/bracketleftBig\nSl(τ(1)\nl+τ(2)\nl)\n+(τ(1)\nl+τ(2)\nl)Sl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,(1)\nwhereSl,τ(1)\nlandτ(2)\nlare spin operators with magni-\ntudesSl=τ(1)\nl= 1/2 andτ(2)\nl= 1. The number of the\nunit cells is denoted by L, and the total number of sites\nis 3L. Here, the parameter λcontrols the frustration as\ndepicted in Fig. 1.\nThe Hamiltonian (1) has a series of local conservation\nlaws. To see it, we rewrite Eq. (1) in the form,\nH=L/summationdisplay\nl=1/bracketleftbigg\nSlTl+TlSl+1+λ\n2/parenleftbigg\nT2\nl−11\n4/parenrightbigg/bracketrightbigg\n,(2)\n1J. Phys. Soc. Jpn. FULL PAPERS\nSlλ\n11τl(1)\nSl+1\nτl(2)11 S=τ(1)=1/2\nτ(2)=1\nFig. 1. Structure of the diamond chain investigated in this work\nS=τ(1)= 1/2 andτ(2)= 1.\nwhere the composite spin operators Tlare defined as\nTl≡τ(1)\nl+τ(2)\nl(l= 1,2,···,L).(3)\nThen, it is evident that\n[T2\nl,H] = 0 (l= 1,2,···,L). (4)\nThus, we have Lconserved quantities T2\nlfor alll. By\ndefining the magnitude Tlof the composite spin Tlby\nT2\nl=Tl(Tl+1), we have a set of good quantum numbers\n{Tl;l= 1,2,...L}whereTl= 1/2 and 3/2. The total\nHilbertspaceoftheHamiltonian(2)consistsofseparated\nsubspaces, each of which is specified by a definite set of\n{Tl},i.e.,asequenceof1/2and3/2.Apairofapicalspins\nwithTl= 1/2 is called a doublet (hereafter abbreviated\nas d) and that with Tl= 3/2 a quartet (abbreviated as\nq).\n3. Ground-State Phase Diagram\n3.1 Ground states for λ≫1\nForλ≫1,∀l Tl= 1/2. Hence, this model is equiva-\nlent to the spin-1/2 antiferromagnetic Heisenberg chain\nwhose ground state is a gapless spin liquid.\n3.2 Ground states for λ≪1\nForλ≪1,∀l Tl= 3/2. Hence, this model is equiv-\nalent to the spin-1/2-3/2 alternating antiferromagnetic\nHeisenberg chain whose ground state is a ferrimagnetic\nstate with spontaneous magnetization msp= 1 per unit\ncell according to the Lieb-Mattis theorem.10)Here,msp\nis defined by\nmsp=1\nLL/summationdisplay\nl=1(/an}bracketle{tSz\nl/an}bracketri}ht+/an}bracketle{tTz\nl/an}bracketri}ht) (5)\nwhere/an}bracketle{t/an}bracketri}htdenotes the expectation value in the ground\nstate with an infinitesimal symmetry breaking magnetic\nfield inz-direction.\n3.3 Intermediate λ\nIn the absence of spontaneous translational symmetry\nbreakdown, only above two phases are allowed. To pur-\nsue the possibility of other phases, we employ the finite\nsize DMRG method with the geometry of Fig. 2. TheSlλ\n11τl(1)\nSl+1\nτl(2)11\nFig. 2. Lattice structure used for the finite size DMRG calcula-\ntion\ncorresponding Hamiltonian is given by\nH=L/summationdisplay\nl=1SlTl+L−1/summationdisplay\nl=1TlSl+1+L/summationdisplay\nl=1λ\n2/parenleftbigg\nT2\nl−11\n4/parenrightbigg\n.(6)\nThis geometry is chosen to allow for the nonmagnetic\nground state for λ≫1. Here and in what follows, the\nnumber of states χkept in each subsystem in the DMRG\ncalculation ranged from 240 to 360. We find that the\nresults with χ= 240 are accurate enough in the present\nwork.\nThe ground-state energies of the Hamiltonian (6) up\ntoL= 16 for all possible configurations {Tl}are calcu-\nlated. The configurations that give the lowest energy are\nidentified for each λ. The spontaneous magnetizationper\nunit cell is given by\nmsp=1\nL/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleL/summationdisplay\nl=1(Tl−Sl)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(7)\nfromtheLieb-Mattistheorem10)fortheHamiltonian(6).\nFigure 3 shows our numerical result for the λ-\ndependence of msp. In addition to the two phases de-\nscribed in Sect. 3.1 and Sect 3.2, a quantized ferrimag-\nnetic ground state with msp= 1/2 is clearly observed.\nThis state has the configuration with Tl= q for odd\nlandTl= d for even l. We denote this configuration\nas (qd)L/2. In the thermodynamic limit, this configura-\ntion tends to the (qd)∞configuration.This configuration\nis equivalent to the (dq)∞configuration in the thermo-\ndynamic limit. Hence, this phase is doubly degenerate\nand is accompanied by the spontaneous breakdown of\ntwofold translational symmetry. In what follows, similar\nnotations are employed for other configurations. Within\nthe finite-size DMRG calculation, the intermediate fer-\nrimagnetic phases are observed between this phase and\nthe nonmagnetic phase. Although two intermediate fer-\nrimagnetic phases are also observed near msp= 1, these\nareartifactsofthefinite sizeeffect, sincethey correspond\ntomsp= 1−1/Land 1−2/L.\nFor finite chains, however, the values of mspare con-\nstrainedbythe systemsize.Hence, itis notclearwhether\nmspis quantized or continuously varying with λfor\n0< msp<1/2 in the thermodynamic limit.\nHence, we resort to the infinite-size DMRG calcula-\ntion. However, it is not possible to carry out the calcu-\nlation for all possible configurations of {Tl}for infinite\n2J. Phys. Soc. Jpn. FULL PAPERS\n0.4 0.6 0.800.51\nλmsp\nL=8\nL=10\nL=12\nL=14L=14\nL=16(qd)∞\nFig. 3. λ-dependence of mspcalculated by the finite size DMRG\nmethod.\nS S S\nsingle q and (p−1) d’s\np S’s (S=1/2)1 segment = p unit cells\n=  qdp−1q d d q\nFig. 4. (qdp−1)∞configuration.\nchains. Hence, we analyze this regime in the following\nway: We start with plausible candidates of periodic con-\nfigurations of Tl’s and find the configuration that gives\nthe lowest energy ground state among them. This leads\nto a stepwise λ-dependence of msp. Then, we check the\nstability of these steps against the formation of defects.\nAs plausible candidates of the ground states, we consider\nthe configurations(qdp−1)∞withmsp= 1/pthat consist\nof an infinite array of segments qdp−1with length of p\nunit cells as depicted in Fig. 4. These configurations can\nbe obtained from the configuration d∞corresponding to\nthe nonmagnetic ground state by inserting a q every p\nunit cells periodically. It should be remarked that these\nstates are accompanied by the p-fold spontaneous trans-\nlational symmetry breakdown. Hereafter, this series of\nconfigurations is called the main series configurations.\nThe case p= 1 corresponds to the q∞configuration that\ncorresponds to the Lieb-Mattis type ferrimagnetic phase\nwithmsp= 1.\nTheλ-dependence of mspin the main series is shown\nin Fig. 5. The spontaneous magnetization msprises con-\ntinuously from msp= 0 at the critical value of λgiven0.6 0.7 0.800.51\nλmsp\np≤38λc(∞)−~0.807p=2p=1\np=3\np=4\nFig. 5. λ-dependence of mspfor the main series configurations\ncalculated by the infinite size DMRG method.\n0 0.05 0.1 0.150.780.790.80.81λ\n1/p\nFig. 6. Extrapolation scheme of λc(p,p−1) andλc(p+1,p) to\np→ ∞. The filled and open circles correspond to the fits by Eq.\n(9) and Eq. (10), respectively. The filled square is the extra polated\nvalueλc(∞).\nby\nλc(∞)≡lim\np→∞λc(p,p−1)≃0.807 (8)\nwhere the boundary between the (qdp−1)∞phase with\nmsp= 1/pand (qdp−2)∞phase with msp= 1/(p−1) is\ndenoted by λc(p,p−1). The extrapolation to p→ ∞is\ncarried out using the data for 38 ≥p≥11 assuming the\nfollowing two asymptotic forms\nλc(p,p−1) =λc(∞)+C1\np+C2\np2, (9)\n3J. Phys. Soc. Jpn. FULL PAPERS\n0.8 0.802 0.804 0.806 0.80800.020.04\nλmsp\np≤38λc(∞)−~0.807\nEqs. (9),(10)\nFig. 7. λ-dependence of mspfor the main series configurations\nnearλ=λc(∞). The open circles are the results of the infinite\nsize DMRG calculation. The small filled circles and solid lin es are\nextrapolation by Eqs. (9) and (10).\nqdp−1qdp−1qdp−2qdp−1qdp−2\nn segments=qdp−2+(n−1)×qdp−1(a)\nqdp−1qdp−1qdpqdp−1qdp\nn segments=qdp+(n−1)×qdp−1(b)\nFig. 8. (a) (qdp−2(qdp−1)n−1)∞configuration and (b)\nqdp((qdp−1)n−1)∞configuration.\nλc(p+1,p) =λc(∞)+C′\n1\np+C′\n2\np2.(10)\nThe extrapolation procedure is plotted in Fig. 6. The\nboth extrapolations give the same value for λc(∞) up\nto the above digit. It should be noted that Eq. (9) and\nEq. (10) correspond to the extrapolation of left and right\nends of the steps, respectively. Using Eqs. (9) and (10),\ntheλ-dependence of mspis plotted down to msp= 0 in\nFig. 7.\nThe remaining question is whether the intermediate\nconfigurations with 1 /p < m sp<1/(p−1) can be a\nground state. To answer this question, it is necessary to\ncalculatethe ground-stateenergiesforall possibleconfig-\nurations of {Tl}, which is impossible. Hence, we confine\nourselves to the following configurations that are plausi-\nble to compete with the (qdp−1)∞configurations.\n(1) (qdp−2(qdp−1)n−1)∞configuration( msp=n/(p(n−1) + (p−1))) depicted in Fig. 8(a): A qdp−1seg-\nment is replaced by a qdp−2segment per every n\nsegments in the (qdp−1)∞configuration. For n= 1,\nthis configuration reduces to the (qdp−2)∞configu-\nration with msp= 1/(p−1) that corresponds to the\nneighboring step of the main series with higher msp.\nForn= 2, this configuration reduces to that with\nalternating qdp−1and qdp−2segments.\n(2) (qdp(qdp−1)n−1)∞configuration ( msp=\nn/(p(n−1)+(p+1))) depicted in Fig. 8(b):\nA qdp−1segment is replaced by a qdpsegment per\neverynsegments in the (qdp−1)∞configuration.\nForn= 1, this configuration reduces to the\n(qdp)∞configuration with msp= 1/(p+1) that\ncorresponds to the neighboring step of the main\nseries with lower msp. Forn= 2, this configuration\nreduces to that with alternating qdp−1and qdp\nsegments.\n(3) The configurations with spatial periodicity less than\n12 and spontaneous magnetization msp≤1/4 have\nbeen thoroughly checked.\nFor configurations (1) and (2), the computational cost\nincreases with an increase of pandn. We have numeri-\ncally confirmed that these states with 2 ≤n≤4 are not\nthe ground state for 2 ≤p≤11. It is further confirmed\nthat those with 2 ≤n≤6 are not the ground state for\n2≤p≤8. Within our numerical accuracy, we find no\nconfigurations that give lower energy than the main se-\nries configurations. Within the available numerical data,\nthe configurations with higher nare even less favorable.\nHence, it is highly plausible that the state intermedi-\natemspis not a ground state at least for p≤11. Also,\nconfigurations (3) do not give the ground state except\nfor the main series configurations. Thus, we expect that\nonly the configurations in the main series are realized in\nthe ground state.\n4. Summary and Discussion\nThe ground-state phases of diamond chains (1) with\n(S,τ(1),τ(2)) = (1/2,1/2,1) are investigated. Between\nthe gapless spin liquid phase for large λand Lieb-Mattis\nferrimagnetic phase with msp= 1 forλ≤0, we find a\nseries of quantized ferrimagnetic phases with msp= 1/p\nwhereptakes all positive integer values.\nTheλ-dependence of the spontaneous magnetization\nmspis very different from other diamond chains with fer-\nrimagnetic ground states. Although the quantized ferri-\nmagneticphasesarepresenteveninundistorteddiamond\nchains, the allowed values of spontaneous magnetization\nare limited to several simple rational values.3,4)\nThere are some examples of ferrimagnetic ground\nstates of diamond chains that are induced by the lat-\ntice distortion.5,6)In these cases, the ground state of\nthe undistorted chain is a paramagnetic state consist-\ning of clusters with finite magnetic moments. The sizes\n4J. Phys. Soc. Jpn. FULL PAPERS\nof the clusters are limited even in the absence of dis-\ntortion. The distortions induce ferromagnetic interac-\ntions between the cluster spins leading to the quantized\nferrimagnetic phases. The quantum fluctuations of the\nlengths of clusters are also induced by distortion leading\nto the partial ferrimagnetic phases.5)\nIn the present case, the ferrimagnetic phases are\npresent even in the absence of distortion. In contrast to\nthe cases of Refs. 3–6, arbitrarily large segments are al-\nlowed leading to the infinitesimally small steps around\nλ=λc(∞). However, no fluctuations of the lengths of\nthe segments are allowed in the present model, since the\nmagnitudes of the composite spins Tlremain good quan-\ntum numbers. Hence, partial ferrimagnetic phases are\nabsent in contrast to the cases of Refs. 5 and 6.\nIn the spin-1 alternating bond diamond chain with\nbond alternation δ, the nonmagnetic phase is equivalent\ntothegroundstateofthespin-1alternatingbondHeisen-\nberg chain with bond alternation δ.7)In this model, an\nintermediate ferrimagnetic phase is observed in the tiny\nregion close neighborhood of the point ( λ,δ) = (λc,δc)≃\n(1.0832,0.2598) that corresponds to the endpoint of the\nHaldane-dimer critical line.11–14)In Ref. 7, it has been\nspeculated that this region is the partial ferrimagnetic\nphase. However, considering the similarity of the gapless\nspin-liquid phase of the present model and the Haldane-\ndimercriticallineofthespin-1alternatingbonddiamond\nchain, it would be more reasonable to speculate that the\ninfinite series of quantized ferrimagnetic phases similar\nto those discussed in the present work is realized also in\nthis case. Unfortunately, the numerical confirmation is\ndifficult due to the smallness of the width of this region.\nAs mentioned above, in some examples of the quan-\ntized ferrimagnetic phases in undistorted diamond\nchains, the allowed values of spontaneous magnetization\nare limited to several rational values.3,4)In these exam-\nples, the nonmagnetic phases neighboring the ferrimag-\nnetic phases are spin-gap phases. On the other hand, the\nnonmagnetic phase neighboring the ferrimagnetic phase\nin the present model with infinitesimal step is the gap-\nless spin liquid phase. This seems to suggest that the\ninfinitesimal energy scale of the gapless spin liquid phase\nhelps the emergence of the exotic ferrimagnetic phase\nwith infinitesimal spontaneous magnetization. A further\nanalytical approach would be required to get insight into\nthe physical implication of the present phenomenon.So far, the infinite series of ferrimagnetic phases pro-\nposed in this work have not been found in real mate-\nrials. However, since the gapless spin liquid phases are\ngeneric critical states in quantum spin chains, it would\nbe possible that these series of phases are realized in\nthe presence of appropriate frustrating exchange inter-\nactions. Nevertheless, in more realistic cases, the pertur-\nbation that does not preserve the conservation laws (4)\nis inevitable. In such cases, the infinitesimal structure\nof spontaneous magnetization might be smeared. In this\ncontext, it would be an interesting problem to investi-\ngate the effect of lattice distortion in the present model.\nThese studies are left for future investigation.\nA part of the numerical computation in this work has\nbeencarriedoutusingthe facilitiesofthe Supercomputer\nCenter, Institute for Solid State Physics, University of\nTokyo,andYukawaInstituteComputerFacilityatKyoto\nUniversity.\n1)Introduction to Frustrated Magnetism: Materials, Experi-\nments, Theory , ed. C. Lacroix, P. Mendels, and F. Mila\n(Springer Series in Solid-State Sciences, Springer, Heide lberg,\n2011).\n2)Frustrated Spin Systems , ed. H. T. Diep, (World Scientific,\nSingapore, 2013) 2nd ed.\n3) K. Takano, K. Kubo, and H. Sakamoto, J. Phys.: Condens.\nMatter8, 6405 (1996).\n4) K. Hida and K. Takano, J. Phys. Soc. Jpn. 86, 033707 (2017).\n5) K. Hida, K. Takano, and H. Suzuki, J. Phys. Soc. Jpn. 79,\n114703 (2010).\n6) K. Hida, J. Phys. Soc. Jpn. 88, 074705 (2019).\n7) K. Hida, J. Phys. Soc. Jpn. 89, 024709 (2020).\n8) H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T.\nTonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. Ohta,\nPhys. Rev. Lett. 94, 227201 (2005).\n9) H.Kikuchi,Y.Fujii,M.Chiba,S.Mitsudo,T.Idehara,T.T one-\ngawa, K. Okamoto, T. Sakai, T. Kuwai T, K. Kindo, A. Mat-\nsuo, W.Higemoto, K.Nishiyama, M.Horovi´ c, and C.Bertheir ,\nProg. Theor. Phys. Suppl. 159, 1 (2005).\n10) E. Lieb and D. Mattis, J. Math. Phys. 3, 749 (1962).\n11) Y. Kato and A. Tanaka, J. Phys. Soc. Jpn. 63, 1277 (1994).\n12) S. Yamamoto, J. Phys. Soc. Jpn. 63, 4327 (1994); Phys. Rev.\nB51, 16128 (1995).\n13) K.Totsuka, Y.Nishiyama, N.Hatano, and M.Suzuki, J.Phy s.:\nCondens. Matter 7, 4895 (1995).\n14) A. Kitazawa and K. Nomura, J. Phys. Soc. Jpn. 66, 3944\n(1997).\n5"    },    {        "title": "2212.04423v1.Strong_photon_magnon_coupling_using_a_lithographically_defined_organic_ferrimagnet.pdf",        "content": "Strong photon-magnon coupling using a lithographically de\fned\norganic ferrimagnet\nQin Xu,1Hil Fung Harry Cheung,1Donley S. Cormode,2Tharnier O. Puel,3Huma Yusuf,2\nMichael Chilcote,4Michael E. Flatt\u0013 e,3Ezekiel Johnston-Halperin,2and Gregory D. Fuchs4\n1Department of Physics, Cornell University, Ithaca NY 14853\n2Department of Physics, The Ohio State University, Columbus, OH 43210\n3Department of Physics and Astronomy,\nUniversity of Iowa, Iowa City, IA 52242\n4School of Applied and Engineering Physics,\nCornell University, Ithaca NY 14853\n(Dated: December 9, 2022)\nAbstract\nWe demonstrate a hybrid quantum system composed of superconducting resonator photons and\nmagnons hosted by the organic-based ferrimagnet vanadium tetracyanoethylene (V[TCNE] x). Our\nwork is motivated by the challenge of scalably integrating an arbitrarily-shaped, low-damping\nmagnetic system with planar superconducting circuits, thus enabling a host of quantum magnonic\ncircuit designs that were previously inaccessible. For example, by leveraging the inherent properties\nof magnons, one can enable nonreciprocal magnon-mediated quantum devices that use magnon\npropagation rather than electrical current. We take advantage of the properties of V[TCNE] x,\nwhich has ultra-low intrinsic damping, can be grown at low processing temperatures on arbitrary\nsubstrates, and can be patterned via electron beam lithography. We demonstrate the scalable,\nlithographically integrated fabrication of hybrid quantum magnonic devices consisting of a thin-\n\flm superconducting resonator coupled to a low-damping, thin-\flm V[TCNE] xmicrostructure.\nOur devices operate in the strong coupling regime, with a cooperativity as high as 1181(44) at\nT\u00180.4 K, suitable for scalable quantum circuit integration. This work paves the way for the\nexploration of high-cooperativity hybrid magnonic quantum devices in which magnonic circuits\ncan be designed and fabricated as easily as electrical wires.\n1arXiv:2212.04423v1  [quant-ph]  8 Dec 2022Hybrid quantum systems are attractive for emerging quantum technologies because they\ntake advantage of the distinct properties of the constituent excitations [1, 2]. This is impor-\ntant because no single quantum system is ideal for every task, e.g. scalable quantum infor-\nmation processing, quantum sensing, long-lived quantum memory, and long-range quantum\ncommunication all have di\u000berent requirements. Some hybrid systems that have been ex-\nplored are microwave photons hybridized with spins [3{15], optical photons hybridized with\natomic degrees of freedom [16{19], and superconducting qubits hybridized with phonons [20{\n23]. In creating hybrid systems, it is advantageous to operate in the strong-coupling, low-loss\nregime, where the relaxation rates of the two distinct quantum systems are exceeded by the\ncoupling rate between them. This allows the hybrid system to operate as a quantum in-\nterconnect, wherein quantum information can be passed from one excitation to another [2].\nThus, a central challenge is to couple distinct quantum systems strongly, with all elements\nmaintaining long coherence times. An equally critical challenge is to fabricate the hybrid\nquantum devices using scalable and integrable approaches so that their engineered properties\ncan be used in applications.\nCombining magnons (quantized spin waves) and microwave photons into a hybrid quan-\ntum system harnesses the unique properties of magnetic materials [24{27]. For example,\nmagnetic materials break time-reversal symmetry, providing a natural opportunity to create\nnonreciprocal devices. Circulators | classical but nonreciprocal circuit components based\non ferromagnetic resonance | are indispensable for superconducting qubits. Magnon-cavity\ninteractions provide new opportunities for nonreciprocal behavior [28, 29], which is useful\nalso for quantum devices [30, 31]. Additionally, magnetic materials are promising for cou-\npling to long-lived quantum spin systems [32{35], potentially enabling quantum intercon-\nnects between important quantum technologies based on microwave photons and on color\ncenter spins [36].\nThe challenge of integrating a microwave resonator with a low-damping magnetic ma-\nterial that can be lithographically patterned has limited the scalability of hybrid quantum\nmagnonic systems. Landmark initial demonstrations used large (millimeter-scale) crystals\nof the ferrimagnetic insulator yttrium iron garnet (YIG) because of its record-low damping,\neven at temperatures below 1 K for bulk crystals [37, 38]. These e\u000borts used either copper\nthree-dimensional microwave cavities [38{41] or a planar waveguide cavity [37]. With focused\nion beam milling and nanomanipulators, a micron-scale YIG crystal has been integrated with\n2a superconducting resonator [42]. Unfortunately, direct lithographic integration of thin-\flm\nYIG with superconducting planar circuits is an outstanding challenge, likely because lattice-\nmatched substrates such as gadolinium gallium garnet are strongly paramagnetic and thus\na lossy microwave substrate [43, 44]. Another approach was the integration of permalloy\n(Ni0:81Fe0:19), a ferromagnetic alloy that can be directly grown and patterned on a planar\ncavity without the need for high-temperature processing or lattice matching, which substan-\ntially enhances the scalability of this hybrid system [45, 46]. Unfortunately, permalloy has\norders-of-magnitude larger intrinsic magnetic damping as compared with high-quality bulk\nYIG crystals.\nIn this work, we demonstrate an alternate pathway to a strongly-coupled hybrid magnonic\nsystem in which a low-damping magnetic \flm is patterned directly on a superconducting\nresonator under gentle growth conditions. We develop a process to integrate lithographically-\npatterned \flms of the organic-based ferrimagnet vanadium tetracyanoethylene (V[TCNE] x)\nwith a planar superconducting microwave resonator. V[TCNE] xhas a typical Gilbert damp-\ning coe\u000ecient in the range \u000b= (4\u000020)\u000210\u00005[47{49], which is comparable to YIG bulk\ncrystals and high-quality YIG \flms grown on lattice matched substrates [50, 51]. We demon-\nstrate strong coupling between V[TCNE] xmagnons and resonator photons in two devices\n(3.6 GHz and 9.2 GHz) with a cooperativity as high as \u0018103. This is critically enabling\nfor integration and scaling, permitting future designs in which magnonic waveguides can be\ntailored as couplers or can mediate interactions between di\u000berent quantum excitations. Fo-\ncusing on the 3.6 GHz device, we present a detailed microwave transmission spectrum that\nreveals not only the expected avoided level crossing of the resonator mode and the uniform\nmagnon mode (the Kittel mode, or simply magnon mode unless otherwise stated) but also\nthe resonator mode hybridized with a discrete spectrum of excited magnon modes. Finally,\nwe study the relaxation rate of the hybrid system as a function of the system detuning\nin the frequency domain and the time domain. This work represents a paradigmatic shift\nin hybrid magnonic quantum systems by establishing an integrated and scalable platform\nenabling arbitrary design of the magnonic elements.\nThe Hamiltonian describing a magnon mode coupled to a resonator mode is [24, 38, 39,\n45, 52{54]:\nH0=~=!r\u0012\n^ay^a+1\n2\u0013\n+!m(B0)^by^b+g\u0010\n^by^a+^b^ay\u0011\n(1)\nwhere the \frst two terms describe, respectively, the energy of resonator photons and magnons\n3at a static \feld B0, while the third term describes the photon-magnon coupling. We have\nused ^aand ^ayto describe the creation and annihilation of resonator photons, and ^band^by\nto describe the creation and annihilation of magnons | these are derived from the Holstein-\nPrimako\u000b transformation on spin operators [55]. The collective coupling rate between the\nresonator photons and magnons is estimated as [38] g=gsp\nN, whereNis the number of\nspins within the magnetic material coupled to the resonator. One can estimate the single\nspin coupling rate from the geometry as [45, 56] gs=ge\u0016Bbrf!r=p8~Zr, wherebrf=\u00160=2w\nis the magnitude of the magnetic \feld experienced by V[TCNE] xspins per unit current in\nan inductor wire of width wwhen the spins are in close contact. We have also used the\nelectron Land\u0013 e gfactorge, the Bohr magneton \u0016B, and the characteristic impedance of the\nresonatorZr=p\nL=C.\nTo design a hybrid system that is useful for quantum circuit integration, we desire a sys-\ntem operating in the strong-coupling, low-loss regime in which both the resonator damping\nrate\u0014rand the magnon damping rate \u0014mare smaller than g. It is also useful to parameter-\nize the system in terms of its cooperativity, C= 4g2=\u0014r\u0014m, which exceeds 1 when the two\nsystems are strongly coupled [38]. In that case, when we tune the system into resonance,\nthe excitations are best described as hybrids of resonator photons and magnons.\nFirst we discuss the requirements of the magnetic material. Low intrinsic (Gilbert) damp-\ning is critical for minimizing \u0014m. Additionally, it is a major materials challenge to precisely\npattern and integrate low-damping magnetic material with the superconducting resonator.\nFor this purpose V[TCNE] xis advantageous because it can be grown on nearly any sub-\nstrate without the need for high temperature processing [57{60], giving it wide substrate\ncompatibility. Moreover, it can be patterned using electron beam lithography and lift-o\u000b\ntechniques without compromising the ultra-low damping [47, 61], which overcomes the chal-\nlenges of working with bulk-grown crystals and thus has considerable advantages for scaling\nand integration. One of the unusual properties of V[TCNE] xas a magnetic material is that\nit has a relatively low value of the saturation magnetization \u00160Ms\u001810 mT [47, 62, 63]. On\none hand, this could be a disadvantage in reaching a largep\nNto enable strong coupling.\nOn the other hand, it is advantageous from a device design point of view because it allows\none to work at comparatively small applied magnetic \feld, which avoids superconducting\nvortex formation.\nFor the microwave resonator we select a lumped-element design consisting of a planar in-\n4terdigitated capacitor that is shorted by a narrow inductor wire. This device is structurally\nsimilar to a transmon qubit that has a capacitor and a narrow inductor, except our structure\ndoes not include a Josephson junction as a part of the inductor. Instead, our narrow induc-\ntor wire e\u000eciently couples resonator excitations to the magnon mode when the magnetic\nmaterial is patterned directly on the wire surface. The resonator mode concentrates the\noscillating current through the wire, generating an Oersted magnetic \feld that excites the\nspins [14, 42, 45]. Using our device's characteristic impedance Zr=17.0(4.5) \n and inductor\nwidthw= 10\u0016m, we estimate gs= 36(5) Hz. Using N= 2:195\u00021012(from the magnetic\nvolume described below, and Ms), the total coupling rate is estimated to be g= 54(8) MHz.\nThe resonator is capacitively coupled to a coplanar microwave feedline that we use to excite\nand detect the coupled resonator-magnon system. Microwave electromagnetic simulations\nof the bare resonator design are available in the supplementary materials.\nThe fabrication procedure must integrate the growth and patterning of inorganic super-\nconducting \flms with organic-based V[TCNE] x[48, 64]. We begin by sputtering 60 nm of\nNb on a sapphire wafer and use photolithography and dry etching to form the microwave\nresonator. After wafer dicing, we spin a poly-methyl methacrylate (PMMA) bilayer on an\nindividual die, which we expose using ebeam lithography. Once developed and passivated,\nthe resulting structure is a lift-o\u000b template for CVD-deposited V[TCNE] x[47]. Inside an in-\nert gas glovebox, we deposit the V[TCNE] xat\u001860\u000eC [57, 58, 65] and then perform lift-o\u000b.\nThe V[TCNE] xis then encapsulated by epoxy and a glass coverslip, stabilizing the material\nfor weeks under ambient conditions [48, 64]. Figure 1(a) shows an optical micrograph of\nan encapsulated device with a resonator frequency !r=2\u0019\u00183.6 GHz. A zoom into the\nnarrow inductor shows the bright 10- \u0016m-wide Nb wire with a dark gray strip of patterned\nV[TCNE] xrunning down its center. The V[TCNE] xstrip is 300 nm thick, 6 \u0016m wide and\n600\u0016m long.\nWe start the electrical characterization of the device at 0.0809 T, a magnetic \feld at which\nthe resonator and magnons are strongly detuned. We measure the microwave transmission\nthrough the feedline using a vector network analyzer (VNA) at sample temperature T=\n0.43 K. The result is shown in Fig. 1(b). The transmission coe\u000ecient of a microwave\nresonator coupled to a transmission line can be modeled as [66]\njS21(!)j=\f\f\f\fa\u0002\u0012\n1\u0000(Ql=jQcj)ei\u001e\n1 + 2iQl(!=!res\u00001)\u0013\f\f\f\f; (2)\n5FIG. 1. (a) Microscope image of the 3.6 GHz resonator after V[TCNE] xwas deposited on its\ninductor wire. The V[TCNE] xcan be seen as a dark gray strip on the bright Nb wire at the\ncenter of the structure. The curved line is the border of the encapsulating epoxy. (b) Transmission\nspectrum (blue) and \ft (black dashed curve) at 0.0809 T when the device is at 0.43 K.\nwhereais the attenuation coe\u000ecient, !res\u0019!ris the mode resonance frequency and Qlis\nthe loaded quality-factor that is used to calculate the total damping rate via \u0014l=!res=Ql.\njQcjis the magnitude of the coupling quality-factor that parameterizes the loss of photons\nfrom the resonator to the waveguide and \u001eis the phase of Qc.\nFitting to Eqn. 2 reveals Ql= 4302,jQcj= 11200, and !r=2\u0019= 3.604 GHz. Since these\nmeasurements are far detuned from the magnon resonance, the measured damping rate is\napproximately the resonator damping \u0014r=!r=Ql= 2\u0019\u00020:8377 MHz. As discussed below,\nwe \fnd that !rand\u0014rhave weak but non-zero magnetic \feld dependence. We attribute\nthis behavior to vortices in the superconducting \flm that can increase \u0014r[67, 68]. We model\nthese contributions phenomenologically, assuming that they vary linearly with magnetic\n\feld. As discussed below, we estimate \u0014r=2\u0019to be 0:902(32) MHz at the resonance \feld\nBreswhere!m(Bres) =!r.\nNext we tune the external magnetic \feld closer to the avoided level crossing between the\nmicrowave resonator mode and the magnon mode to study the coupling between the two\nsystems. The coupled eigenfrequencies are [37]\n!\u0006=!r+ \u0001=2\u0006p\n\u00012+ 4g2=2; (3)\nwhere \u0001 = !m(B0)\u0000!ris the system detuning, and the magnon frequency is !m(B0) =\n\rp\nB0(B0+\u00160Me\u000b) for the static magnetic \feld B0applied parallel to the long axis of the\n6V[TCNE] xstrip. Here Me\u000b=Ms\u0000Hkdescribes the di\u000berence between Msand the uniaxial\nanisotropy \feld Hkof V[TCNE] x. The transmission spectrum through the feedline will be\nmodi\fed by the resonator and its interactions with the magnons [25]. The total transmission\ndue to line attenuation and magnon coupling is [37]\nS21(!;B 0) =S21;0(!)\u0012\n1 +(\u0014ext=2)e\u0000i\u001e\ni(!\u0000!r)\u0000\u0014r=2 +g2fi[!\u0000!m(B0)]\u0000\u0014m=2g\u00001\u0013\n;(4)\nwhere\u0014extdescribes the coupling rate between the feedline and the resonator, and S21;0(!)\nis the background transmission.\nFigure 2(a) shows the corresponding experimental data \u0001 jS21j= 20log 10(jS21j=jS21;0j),\nmeasured at T= 0.43 K as a function of magnetic \feld and frequency. It shows a strong\navoided level crossing | direct evidence of strong coupling between resonator photons and\nmagnons. These data are acquired with a VNA using \u000075 dBm of microwave power at the\nsample, which is far below the resonator's nonlinear power level, corresponding to about\n3:9\u0002106resonator photons for large \u0001 [69] (see supplementary materials for details). At\neach \feld, we extract the resonance frequencies !\u0006and resonance linewidths \u0014\u0006of the upper\nand lower branch by \ftting to Eqn. 2.\nTo resolve the features at small \u0001 and to determine gand\u0014maccurately, we measure\nS21with \fner magnetic \feld steps. The data are presented in Fig. 2(b). Qualitatively, the\nstrongest features are the two hybrid photon-magnon branches dispersing as !\u0006given by\nEqn. 3. Additionally, we observe fainter modes that are linearly dispersing with a slope of\n28 GHz/T. In contrast to the uniform magnon mode, we attribute those to magnon modes\nwith a \fnite wavevector k[70{72], which we discuss in further detail below.\nFigure 2(c) shows a line cut of \u0001 jS21jas a function of frequency acquired at B0=Bres\n(e.g. \u0001 = 0), with corresponding \fts to Eqn. 2 for each branch. The two resonances are the\n!\u0006as de\fned by Eqn. 3. We note that although !\u0000has a symmetric resonance lineshape, !+\nhas a Fano-like lineshape. Nevertheless, we can use the splitting between these resonances\nto \fndg= [!+(Bres)\u0000!\u0000(Bres)]=2 = 2\u0019\u0002[90:31(8)] MHz. The damping rate of the two\nbranches,\u0014\u0006are related to resonator and magnon damping rates via [25]\n\u0014\u0006= [\u0014r=2 +\u0014m=2\u0007Imp\n(\u0000!r+i\u0014r=2 +!m\u0000i\u0014m=2)2+ 4g2]: (5)\nFrom the linewidths in Fig. 2(c) we \fnd \u0014\u0000(Bres)=2\u0019= 16:37(29) MHz and \u0014+(Bres)=2\u0019=\n15:15(17) MHz. In the strong coupling regime ( g\u001d\u0014r;\u0014m),\u0014+=\u0014\u0000= (\u0014r+\u0014m)=2.\n7FIG. 2. Microwave measurement of V[TCNE] xcoupled to a 3.6 GHz resonator at 0.43 K. (a)\n\u0001jS21jplotted as a function of magnetic \feld and frequency. The white dashed lines mark the\nresonator and the magnon mode frequencies in the case where they are not coupled. (b) \u0001 jS21j\nacquired in \fner \feld steps near resonance \feld. We attribute the faint diagonal lines with slope\n28 GHz/T to be k6= 0 magnon modes. (c) \u0001 jS21jline cut atB0=Bres(e.g. \u0001 = 0) showing the\ntwo branch resonances at !+and!\u0000. The dashed lines are \fts as discussed in the text.\nTherefore we take the average of experimental \u0014+;\u0014\u0000to estimate ( \u0014r+\u0014m)=2. Together\nwith\u0014r(Bres)=2\u0019= 0.902(32) MHz, we estimate \u0014m=2\u0019= 30:62(34) MHz. We \fnd that this\ndevice operates with C= 1181(44) and ful\flls g>\u0014r;\u0014m.\nHaving probed the system in the frequency domain, we now turn to measurements of\nsystem relaxation in the time domain. For this, we apply pulsed microwave excitation to the\nfeedline and detect the ring-up/-down response in the time domain using a homodyne circuit.\nThe full circuit and measurement protocol is discussed in the supplementary materials. The\nmicrowave pulses are long enough to excite the system into driven steady-state oscillations.\nWhen the microwave drive turns o\u000b suddenly, the system will continue to oscillate, however,\nit will do so at its natural frequency. Its relaxation will include both intrinsic relaxation\n8FIG. 3. Ring-down experiment with homodyne circuit for the 3.6 GHz sample at 0.43 K, \u000065 dBm\nexcitation power. (a) Schematic plot of how the homodyne detected signal (red) changes with\nrespect to the microwave excitation power at the sample (green). When the microwave excitation\npower turns on (o\u000b), system will ring up (down) to steady-state. (b) Amplitude vs time plot of ring-\ndown at 0.101 T for on resonance excitation. The \ftted voltage ring-down time is 170.0(2.6) ns.\n(c) Amplitude vs time plot with 5 MHz detuning. (d) Total system damping calculated from\nring-down experiments (orange) and VNA experiments (blue).\ninto the environment and relaxation through radiation into the feedline that we detect. We\namplify and mix this signal with the reference tone, and digitize the result. Figure 3(a)\nshows the schematic plot of microwave power at the sample and the oscilloscope detected\nvoltage vs time.\nFigure 3(b) shows a log-linear plot of amplitude relaxation after driving the system\nwith\u000065 dBm of microwave power at the resonance frequency !+atT= 0.43 K and\nB0= 0:101 T. Under these conditions, the hybrid state has more resonator character than\nmagnon character, however, !+\u0000!r= 2\u0019\u000243.9 MHz\u0018g=2, indicating substantial magnon\n9participation. We observe exponential relaxation with a voltage ring-down time constant\nof 170.0(2.6) ns. Therefore, the energy relaxation time constant is \u001c= 85:0(1:3) ns, corre-\nsponding to a decay rate of \u0014+=2\u0019= 1:872(29) MHz. We can also detune the microwave\nexcitation with respect to the hybrid mode. In Fig. 3(c) we plot ring-down data acquired\nat the same \feld, except using a driving frequency that is detuned by 5 MHz from !+=2\u0019.\nIn this case, the homodyne signal beats with respect to the reference oscillator, giving rise\nto a decaying sinusoidal response. We recover a decay rate of 1.924(28) MHz on top of the\n5 MHz beat, which is consistent with the on-resonance driving result.\nTo explore the relaxation rates at di\u000berent detunings \u0001, we perform on-resonance ring-\ndown measurements as a function of B0. We plot the resulting damping rates along with\nthe damping rates extracted from VNA full-width at half-maximum (FWHM) linewidth\nmeasurements vs B0in Fig. 3(d). We see excellent agreement between the two methods,\nindicating that our VNA measurements are acquired in the linear response regime. Fig-\nure 3(d) underscores that the magnon damping rate is one order of magnitude larger than\nthe bare resonator damping rate. The damping rate of the upper (lower) branch is increasing\nwhen the \feld is tuned closer to Bresfrom below (above), since the hybrid state has a larger\nparticipation from the magnon mode as \u0001 approaches 0. The upper branch damping rate\nat 0.0809 T and the lower branch damping rate at 0.1281 T are assumed to be the resonator\ndamping rates with no magnon contribution since these \felds are far away from Bres. By\nlinear interpolation, we estimate \u0014r=2\u0019to be 0:902(32) MHz at Bres, which we used above\nto calculateC.\nWe now turn our attention to the additional magnon modes that are evident in Fig. 2(b),\ni.e., the lines indicated by arrows within the avoided level crossing region. For reference, we\nplot a dashed line showing the uniform magnon mode position if it were not coupled to the\nresonator. It separates the additional modes into higher frequency (red arrows) and lower\n(white arrows) frequency modes, which likely have distinct origins. Here, we treat all magnon\nmodes as spinwave modes characterized by a wavevector kand quantized by the boundary\nconditions at the surfaces of the V[TCNE] x. The case k= 0 refers to the uniform magnon\nmode, while the additional modes are k6= 0 magnon modes. The width and length of the\nV[TCNE] xstrip are long enough that quantization constrained in those directions cannot be\nresolved; only modes that are quantized in the thickness direction ( jkj\u0011kperpendicular to\nthe magnetic \feld) are resolvable. We assign the features at frequencies above the dashed\n10FIG. 4. (a) Energy spectrum of Hj ii=~!ij iicolored by the coe\u000ecient j ii=\u0010\nc(i)\nr;:::\u0011\nrelated\nto the resonator. This simulation used realistic values as follows: L= 300 nm,\r=2\u0019= 28 GHz/T,\n\u00160Me\u000b= 53:7 mT,\u00152\nex= 0:25\u000210\u000016m2,!r=2\u0019= 3:593 GHz,g=2\u0019= 90 MHz, and gn=g=(n+1)\nforn= 1;:::; 4. (b) The results labeled by `magnons' repeat the plot (a), while the red lines include\nonly the coupling between the uniform magnon mode (Kittel mode) and the resonator mode (both\nindependently identi\fed by the dashed lines).\nline to be thickness quantized k6= 0 magnon modes. We do not have a de\fnitive assignment\nfor the modes lying at frequencies below the dashed line. A more complete discussion can\nbe found in the supplemental material.\nTo better understand the transmission spectrum, we now discuss a model for multiple\nmagnon modes coupled to the resonator. We extend the Hamiltonian H0given in Eqn. 1 by\nadding a termHsthat describes the k6= 0 magnon modes as\nHs=~=X\nn=1!n^sy\nn^sn+ X\nn=1gn(^sy\nn^a+ ^sn^ay)!\n: (6)\nWe introduce creation and annihilation operators ^ snand ^sy\nnfor thek6= 0 magnons, and\ntheir direct coupling to the resonator has strength gn.\nWe consider magnon modes described by dipole-exchange interactions [73]; their fre-\nquency dispersion is given in Ref. [74],\n!n=\rp\n(B0+\u00160Me\u000b\u00152\nexk2\nn) (B0+\u00160Me\u000b+\u00160Me\u000b\u00152\nexk2\nn): (7)\nThe quantization index n= 1;2;:::is along the thickness direction, where the wavevectors\nare constrained by knL=n\u0019. Settingn= 0 recovers the frequency of the uniform magnon\n11mode (or Kittel mode), while the extra terms are due to the presence of exchange interac-\ntions with amplitude described by the exchange-length constant \u0015ex. The expression above\nassumesB0is oriented in the plane of V[TCNE]x.\nMagnon modes with k6= 0 can be directly excited by the magnetic \feld generated by the\ninductor only if their spatially-averaged amplitude does not vanish; therefore, their coupling\nto the resonator is highly dependent on the spin-pinning boundary conditions [47, 75]. For\ninstance, there is no direct coupling for totally unpinned boundaries, while for complete\npinning only even n-index modes couple. Here, we consider an intermediate situation where\nallgnare allowed to exist, which can happen for partial pinning [76]. To demonstrate the\nessential features of this interaction, we chose gn=g=(n+1) as a qualitative description of the\nfeatures observed, which is likely an overestimate. We solve the equation Hj ii=~!ij ii\nas a function of B0and plot the results in Fig. 4(a). The two branches !\u0006represent the\nhybridized modes due to the strong uniform magnon-resonator coupling. The k6= 0 modes\nhave weak \u0001jS21j(Fig. 2) because the j iihave small resonator amplitudes, however, they\nbecome stronger as their frequencies approach the !+branch (Fig. 4(a)). We truncate at\nn= 4 capturing only spin states that lie within the avoided-crossing gap, however, we note\nthat the spacing between the k6= 0 magnon modes is sensitive to the parameter \u0015exthrough\na linear dependence on \u00152\nex. The coupling of the k6= 0 magnons with the resonator increases\nthe total gap between the two branches !\u0006, as shown in Fig. 4(b).\nThese advances in our understanding of the quantum magnonic properties of V[TCNE] x\nhighlight its potential for applications in quantum information devices and the substan-\ntial potential for further optimization. In this study, the uniform mode magnon damp-\ning rate\u0014matT= 0:43 K corresponds to a FWHM ferromagnetic resonance (FMR)\nlinewidth of 1.09 mT, which is comparable to the room temperature 0.57 mT linewidth\nof the \\witness\" sample that was deposited at the same time as the V[TCNE] xused in\nthis device and measured at \u00189.8 GHz. This points to the opportunity for improvement\nthrough V[TCNE] xgrowth and patterning considering that room temperature thin \flm\nV[TCNE] xFMR linewidths as narrow as 0.094 mT [62], and ebeam patterned V[TCNE] x\nFMR linewidths as narrow as 0.12 mT, at 9.8 GHz have been demonstrated [47]. More-\nover, prior measurements of the FMR linewidth at temperatures down to 5 K revealed a\nstrong temperature-dependent strain e\u000bect, highlighting one avenue for improvement [63].\nV[TCNE] xis a promising material for quantum magnonic applications with properties that\n12rival and exceed that of YIG, is compatible with a broad range of materials, and can be\nlithographically integrated with planar superconducting circuits.\nIn conclusion, we have demonstrated a hybrid quantum system composed of supercon-\nducting resonator photons and magnons in the strong coupling regime with C= 1181. A\nkey advance of this work is the integration of lithographically patterned and low-damping\nmagnetic material with a superconducting circuit platform, enabling new quantum tech-\nnologies in which the electrical and the magnonic structures are designed and fabricated\non an equal basis. This capability can lead to nonreciprocal quantum circuit elements,\nnew forms of hybrid-system couplers, new opportunities for tunability, and the creation\nof transmon qubits with integrated magnonic properties. Beyond quantum technologies,\nthis hybrid system can o\u000ber sensitive readout of coherent magnonic excitations. The broad\ndesign space o\u000bered by lithographic patterning allows experiments that selectively probe dif-\nferent magnon wavevectors and { in combination with superconducting qubits { enables the\ncreation, manipulation, and detection of single magnons in a scalable, integrated platform.\nACKNOWLEDGEMENTS\nWe thank Brendan McCullian for useful conversations. The resonator design and fabri-\ncation, V[TCNE] xgrowth and growth optimization, and theory of uniform magnon mode\ncoupling were supported through the Center for Molecular Quantum Transduction (CMQT),\nan Energy Frontier Research Center supported by the Department of Energy O\u000ece of Sci-\nence, Basic Energy Sciences (DE-SC0021314). The development and design of resonator-\nV[TCNE] xintegration and lithography, measurement techniques, and theoretical analysis\nofk6= 0 magnon modes were supported by the Department of Energy O\u000ece of Science,\nBasic Energy Sciences Quantum Information Sciences program (DE-SC0019250). All mea-\nsurements were done using the CMQT low temperature facility at Cornell. This work also\nmade use of facilities at the Cornell NanoScale Facility, an NNCI member supported by\nthe NSF (NNCI-2025233) and the Cornell Center for Materials Research Shared Facilities\nwhich are supported through the NSF MRSEC program (DMR-1719875). 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DEVICE SIMULATIONS\nWe simulate the resonator with Keysight PathWave Advanced Design System (ADS)\nsoftware. Fig S1(a) shows the simulated transmission coe\u000ecient \u0001 jS21jvs frequency for the\n3.6 GHz resonator. The simulated resonance frequency is 3.933 GHz. We then simulate the\ndevice being driven at the resonance frequency and plot the time averaged magnitude of\ncurrent density in the superconducting \flm. The result is shown in \fg S1(b), in which red\nindicates larger current density and blue indicates smaller current density. This resonance\nmode has a large current density in the inductor wire for e\u000ecient coupling to magnon modes\nof a magnetic material deposited on the wire. After fabrication, we measure the resonator's\ntransmission spectrum using a VNA at 0 \feld before V[TCNE] xdeposition. The result is\nshown in \fg S1(c), where the resonance is at 3.804 GHz with Q-factor ( Ql) of 4922. This\nresonance frequency is lower than the simulation because of the \fnite kinetic inductance of\nNb [67, 68], which is not included in the simulation. This kinetic inductance adds to the\ngeometric inductance of the LC resonator and decreases the resonance frequency. Fig S1(d)\nshows the transmission spectrum at 0 \feld after V[TCNE] xdeposition and encapsulation.\nThe encapsulation epoxy and cover glass increase the e\u000bective dielectric constant of the\nresonator environment, which increases the capacitance C, resulting in a lower resonance\nfrequency. We speculate that the decrease of Q (2546) with epoxy encapsulation is caused by\nthe loss tangent of the epoxy and glass, and loss from the non-uniform V[TCNE] xmagnetiza-\ntion at 0 \feld. Then we increase the \feld to 0.0944 T to saturate V[TCNE] xmagnetization.\nFig 1(b) in the main text shows the resulting transmission spectrum. We attribute the\nincrease in Q (4302) to the increase of the uniformity of the V[TCNE] xmagnetization.\nII. DEVICE FABRICATION\nWe deposit Nb as the superconducting material for our device on a sapphire wafer by\nsputtering at 600 Celsius. The resulting Nb is 60 nm thick and has a critical temperature\n(Tc) of 7.3 K. The high T cof Nb and the low loss tangent of sapphire help limit the damping\nrate of the superconducting device.\nWe pattern the Nb \flm using photolithography followed by dry etching. We \frst spin-\n22FIG. S1. (a) Microwave transmission \u0001 jS21jfrom the ADS simulation of the 3.6 GHz resonator.\nWe \fnd a resonance frequency of 3.93 GHz. (b) The time averaged magnitude of current density\nplot of the resonator at the resonance frequency shown in (a). The zoom shows a large current\ndensity in the inductor wire. (c) Experimental measurement of \u0001 jS21jatB0= 0 for a resonator\nat 0.43 K without V[TCNE] x. Fitting with Eqn. 2 gives !r=2\u0019= 3:804 GHz and Ql= 4922. The\nresonance frequency is lower than the simulation because of the \fnite kinetic inductance of Nb in\nthe real device, which is not included in the simulation. (d) Experimental measurement of \u0001 jS21j\natB0= 0 for the V[TCNE] x-resonator device discussed in the main text, at 0.43 K. Fitting with\nEqn. 2 gives !r=2\u0019= 3:600 GHz and Ql= 2546.\ncoat the wafer with photoresist, expose using a 5x stepper, and then develop. Then we etch\nthe Nb using ion milling before stripping the photoresist.\nWe use an electron beam lithography lift-o\u000b process to pattern the V[TCNE] xas discussed\nin the main text. First we spin-coat individual resonator chips with a bilayer of PMMA\nresist and expose the resist layer. We then develop the PMMA layers, leaving an opening\n23for V[TCNE] xdeposition directly on the Nb. The V[TCNE] xis deposited using CVD as\ndiscussed in the main text. The PMMA is then dissolved using dichloromethane, which\nleaves the V[TCNE] xin the patterned regions untouched. The \fnal step is to encapsulate\nthe resonator and patterned V[TCNE] xusing epoxy and a coverslip. The \fnal dimensions\nof the V[TCNE] xis 600\u0016m\u00026\u0016m\u0002300 nm.\nIII. RESONATOR AND MAGNON MODE SATURATION POWER\nUsing power-dependent S21measurements at B0= 0:0994 T, we determine the saturation\npower (of the kinetic inductance non-linearity) for the 3.6 GHz resonator to be larger than\n\u000065 dBm applied to the feedline port. To make sure we are not saturating the magnetic\nresonance, we also make power-dependent S21measurements near Bres. No power depen-\ndence is observed in the range of \u000065 dBm to\u000085 dBm. The data in Fig. 2 are acquired\nwith\u000075 dBm of microwave power at the sample, which is well below saturation for both\nthe resonator and the magnet.\nWe now theoretically estimate the magnon precession cone angle. When we drive on\nresonance with the upper or lower branch !res, the average number of excitations in the\nresonator-magnet system is [69]\nhni=4Pin\n~!2\nresQ2\nl\njQcj;\nwherePin=\u000075 dBm = 3 :16\u000210\u000011W is the excitation power at the sample, Qlis the\nloaded Q-factor and Qcis the coupling Q-factor. For the upper branch at B0=Bres, we \fnd\nQl= 242:1,jQcj= 23000 and the resonance frequency !res=2\u0019= 3:669 GHz from \ftting\nto Eqn. 2. Under these conditions, we estimate hni= 5800 and the average number of\nmagnonshnmi=hni=2 = 2900 at B0=Bres.\nThe cone angle \u0012of the uniform magnon mode satis\fes\n1\u0000cos\u0012\u00191\n2\u00122=hnmi~!res\n1\n2N~!res;\nwhereN\u00192:195\u00021012is the estimated number of V[TCNE] xspins in the sample. Therefore,\nwe estimate \u0012\u00192p\nhnmi=N= 7:2\u000210\u00005rad = 0:0041\u000e.\nSimilarly, in Fig. 1(b) with !res=2\u0019= 3:604 GHz,Ql= 4302 andjQcj= 11200, we get\nthe average number of resonator photons hnri'hni= 3:9\u0002106.\n24IV. BACKGROUND TRANSMISSION\nThe background transmission S21;0(!) is assumed to be independent of B0and is measured\nat some (B0's,!'s) that are far away from the upper or lower branch (see Fig. 2(a), 2(b)).\nIn Fig. 2(a) for the 3.6 GHz device, S21;0(!) is measured from 3.48 GHz to 3.595 GHz at\n0.1024 T, and from 3.595 GHz to 3.71 GHz at 0.1064 T. In Fig. 2(b), S21;0(!) is measured\nfrom 3.395 GHz to 3.495 GHz and from 3.595 GHz to 3.795 GHz at 0.1071 T, and 3.495 GHz\nto 3.595 GHz at 0.1017 T.\nSimilarly, in Fig. 5(b) for the 9.2 GHz device, S21;0(!) is measured from 8.9 GHz to\n9.25 GHz at 0.2913 T, and from 9.25 GHz to 9.6 GHz at 0.3021 T. In Fig. 5(d), S21;0(!)\nis measured from 8.63 GHz to 9.23 GHz and from 9.47 GHz to 9.83 GHz at 0.2886 T, and\n9.23 GHz to 9.47 GHz at 0.3021 T.\nV. ERROR ANALYSIS\nUncertainty is calculated via standard error analysis from least-squared \ftting. We use\nthe python package \\scipy.optimize.curve \ft\" [77]. The output includes the optimized values\nof all the \ftting parameters and a 2-D array \\pcov\" which gives the covarience matrix. We\nreport the square root of the diagonal entries of the covariance matrix as the standard error.\nVI. EXTRACTED MAGNETIC PARAMETERS\nUsing the data shown in Fig. 2(a), we \ft !\u0006by \ftting to Eqn. 2. These results are then\n\ft to Eqn. 3 where !mis given by the Kittel formula with \r=2\u0019= 28 GHz/T. Also, we\nassume!r=!r0+\rrB0and treat!r0and\rrtogether with Me\u000bandgas free parameters.\nWe have used Me\u000b=Ms\u0000Hk, whereHkis the uniaxial anisotropy \feld. We obtain \u00160Me\u000b\n= 53.614(63) mT. Such a large Me\u000bis likely caused by the large strain applied to the\nV[TCNE] xdue to di\u000berential thermal expansion [63], which can induce a value of Hk>Ms.\nNext we determine the value of Bresusing the data shown in Fig. 2(b), again extracting\n!\u0006by \ftting to Eqn. 2. Figure S2 shows the extracted splitting !+\u0000!\u0000as a function of B0.\nFrom Eqn. 3 we know that !+\u0000!\u0000=p\n\u00012+ 4g2, where \u0001 = !m\u0000!r=\rrm\u0002(B0\u0000Bres),\nand\rrmis the change of !m\u0000!rper unit increase of B0. Treating g,Bresand\rrmas\nfree parameters, we get g=2\u0019= 90:43(8) MHz, Bres= 0:103429(18) T and \rrm=2\u0019=\n25FIG. S2. Upper and lower branch frequency di\u000berence vs \feld and the \ftting to extract gand\nBres. At the resonance \feld, the frequency di\u000berence is the smallest and is equal to 2 g.\n52:1(2:6) GHz/T. The best-\ft curve is shown as the black dashed line in Fig. S2. First, we\n\fnd a value of gthat is consistent with the value we extracted from the line cut shown in\nFig. 2(c). However, the \ftted \rrm=2\u0019is larger than the expected value of 28 GHz/T. Possible\nreasons for the discrepancy include (1) that interactions between the spin wave modes and\nthe upper (lower) branch distorts the shape of !+(B0>Bres) and!\u0000(B0<Bres) from the\nmodel prediction of Eqn. 3 (see Fig 4(b)); (2) Although the \ft requires a \rrm, the resonances\nthat we use to extract it are strongest for larger values of \u0001, where its in\ruence is smallest.\nThus, we are not as sensitive to \rrmthan we are to other parameters, and small covariances\nand distortions can lead to an unphysical value. Nevertheless, the most relevant parameters\nfor this analysis, gandBresare insensitive to the exact value of \rrm.\nVII. MAGNON MODES BELOW THE UNIFORM MAGNON MODE\nFREQUENCY\nIn general, modes that appear at frequencies below the uncoupled uniform magnon mode\nfrequency have been attributed to backwards volume modes [70, 71]. Forward volume modes\nand surface spin waves have higher frequencies than the uniform magnon mode [72], so we\ndo not consider them here as potential sources for these magnon modes below the uniform\n26mode frequency. Our experimental geometry makes the excitation and detection of quantized\nbackwards volume modes unlikely because the magnetic \feld is applied parallel to the long\naxis of the magnetic material, with an estimated misalignment of less than 3\u000e. Because\nbackwards volume modes require a wavevector parallel to the magnetic \feld, this would\nrequire quantization along the (600 \u0016m) long axis and cannot explain the magnon modes\nbelow the uniform mode frequency because this scenario cannot impose visible quantization.\nWe also note that the cross section of the sample is not rectangular at the edges due to the\ngrowth process [47], which in some situations could give rise to a nonuniform demagnetization\n\feld, however this mechanism would also require a substantial magnetic \feld perpendicular\nto the long axis of the magnet, and is ruled out by our misalignment estimate above. Finally,\nwe have also considered the possibility of a nonlinear process in which a resonator photon\ncouples to the uniform mode magnon and induces a transition from a uniform mode magnon\ninto ak6= 0 magnon. This scenario is consistent with the appearance of modes at frequencies\nbelow the uncoupled uniform magnon mode. Our estimate of the resonator photon and\nmagnon populations suggests that such a nonlinear process would be orders of magnitude\nsmaller than observed. Due to these inconsistencies with respect to typical explanations for\nmagnon modes below the uniform magnon mode, we cannot make a de\fnitive assignment\nof the lower frequency quantized modes.\nVIII. RING-DOWN EXPERIMENTS\nThe setup for the ring-down experiments is shown in Fig S3. From the top-left of this\ndiagram: the microwave (MW) signal generator generates a sinusoidal voltage at a frequency\nrelevant to !+or!\u0000, which is then split using a directional coupler. The top branch, which\nwe label the local oscillator, carries 11 dBm of MW power. The lower branch, which we\nlabel the signal carrier, carries \u00009 dBm of MW power. In the lower branch, a function\ngenerator outputs a square wave that controls a microwave switch that chops the lower\nbranch microwave power in a square wave. After the switch, the attenuation from the\ncoaxial cable, a 20 dB attenuator outside the cryostat, and a 30 dB attenuator inside the\ncryostat attenuate the MW signal to \u000065 dBm before it reaches the sample. The signal from\nthe sample is ampli\fed outside the cryostat before it's mixed with the local oscillator signal,\nwith a power limiter before the mixer to protect it from overloading. We adjust the phase\n27FIG. S3. Circuit diagram for the ring-down experiment.\nshifter in the local oscillator branch so that the DC component of the mixed ring-down signal\nis the highest. At the end, a low pass \flter rejects components at 2 !and we are left with\na DC component of the mixed signal that we digitize using a sampling oscilloscope. The\nresulting voltage is proportional to the amplitude of magnon-resonator ring-down, which we\n\ft as discussed in the main text.\nIX. TEMPERATURE DEPENDENCE\nUsing the techniques discussed in the main text, we also investigate properties of our\nhybrid resonator-magnon system at higher temperatures. We plot \u0014mandgin Fig. S4(a) and\nS4(b) respectively. We see that gdecreases and \u0014mincreases with increasing temperature.\nX. SECOND DEVICE RESULTS\nOther than the 3.6 GHz resonator as described in the main text, we also study another\nresonator-magnon device that is designed with a larger resonance frequency. It has the\nsame inductor wire dimensions, but a smaller capacitance. The device resonance frequency\nis around 9.2 GHz, measured at zero \feld after V[TCNE] xdeposition and encapsulation.\nA microscope image of the device after V[TCNE] xdeposition is shown in \fg S5(a). In\nthis device there is incomplete lift-o\u000b. A large area of V[TCNE] xremains on the device,\nwhich is connected to the V[TCNE] xstrip on the inductor wire. This may result in the\nV[TCNE] xstrip coupled to a continuum of magnon modes.\n28FIG. S4. MW measurement of V[TCNE] xcoupled to the 3.6 GHz resonator at di\u000berent tem-\nperatures. (a) Temperature dependence of gextracted from the S21measurements at Bres, (b)\nTemperature dependence of \u0014mextracted from S21linewidth measurements at Bres. The error\nbars for\u0014mare too small to be seen on this scale.\nFigure S5(b) shows \u0001 jS21jas a function of magnetic \feld and frequency acquired at\n0.43 K and\u000065 dBm of microwave power. As with the device discussed in the main text,\nwe verify that we operate the device in the linear response regime; here between \u000055 dBm to\n\u000065 dBm. We again see a strong avoided level crossing, indicating that this device operates\nin the strong coupling regime. Both \u0014rand\u0014mare larger as compared to the 3.6 GHz device,\nand the upper and lower branch resonance signals are weaker at B0=Bres. We \ft the data\nusing Eqn. 4 of the main text to extract !+,\u0014+from 0.249 T to 0.291 T, and !\u0000,\u0014\u0000from\n0.299 T to 0.344 T. As in the main text, we then substitute !\u0006into Eqn. 2 of the main\ntext and treat g,!r0,\rrandMe\u000bas free parameters. We obtain g=2\u0019= 147:21(29) MHz\nandMe\u000b= 72:40(11) mT, !r0=2\u0019= 9:2529(15) GHz and \rr=2\u0019=\u000071:7(5:1) MHz/T. Also,\nassuming linear dependence of \u0014ronB0, we evaluate \u0014+(B0= 0:249 T)\u0019\u0014r(B0= 0:249 T)\nand evaluate \u0014\u0000(B0= 0:344 T)\u0019\u0014r(B0= 0:344 T). We \fnd \u0014r(Bres)=2\u0019= 7:917(15) MHz.\nTo extract \u0014m, we use [25]\n~!\u0006= ~!r+~\u0001=2\u0006q\n~\u00012+ 4g2=2; (8)\nwhere ~!=!\u0000i\u0014/2 and ~\u0001 = ~!m(B0)\u0000~!r. The real part of this equation gives Eqn. 2\nwhen!r;m\u001d\u0014r;m, and the imaginary part gives\n\u0014\u0006= [\u0014r=2 +\u0014m=2\u0007Imp\n(\u0000!r+i\u0014r=2 +!m\u0000i\u0014m=2)2+ 4g2]: (9)\n29FIG. S5. MW measurement of V[TCNE] xcoupled to the 9.2 GHz resonator at 0.43 K. (a) Micro-\nscope image of V[TCNE] xon the 9.2 GHz resonator. (b) S21measured as a function of B0and\n!. (c) Damping rate of the upper branch vs B0from 0.249 T to 0.291 T, and that of the lower\nbranch from 0.299 T to 0.344 T. The \ftted magnon damping rate is \u0014m= 117:7(5:3) MHz. (d)\nS21measurements with \fner steps in B0showing spin wave modes around 150 MHz higher than\nthe lower branch frequency.\nUsing this expression of \u0014\u0006, and treating \u0014m,Me\u000bandgas free parameters, we obtain\n\u0014m=2\u0019= 115 MHz. The data and the \ft are shown in Fig. S5(c). Therefore, this\n9.2 GHz resonator-magnon hybrid system also operates in the strong coupling regime with\nC= 93:0(4:2). For the same system at 3.0 K, we extract Me\u000b= 72:94(10) mT, \u0014r(Bres)=2\u0019=\n10:456(31) MHz, \u0014m=2\u0019= 112:9(4:8) MHz,g=2\u0019= 146:20(26) MHz. Therefore the cooper-\nativity isC= 72:4(3:2).\nAs in the main text, we measure \u0001 jS21jwith higher resolution around the avoided level\ncrossing at 0.43 K [Fig. S5(d)]. We also see faint spin wave modes within the avoided-\n30FIG. S6. Experimental data for the static \feld at the sample B0vs the magnet coil current (blue\ndots) and the linear \ft (black dashed line) to extract the calibration factor of 60.64(10) mT/A.\nThe black dashed line extrapolates to the origin as expected.\ncrossing gap between !\u0006.\nXI. MAGNETIC FIELD CALIBRATION\nTo calibrate the magnetic \feld of the cryostat electromagnet at the sample location, we\nmeasure absorption electron spin resonance (ESR) of 2,2-diphenyl-1-picrylhydrazyl (DPPH)\nin contact with the central conductor of a broadband coplanar waveguide. DPPH is a stable\nradical with a well-studied gyromagnetic ratio across a broad range of temperatures. The\nESR is measured at 4.7 K, where the gfactor is 2.083 [78]. We measure ESR from 4.5 GHz\nto 8.5 GHz in steps of 0.5 GHz. At each frequency we sweep the magnet current as we\nacquire microwave transmission with a vector network analyzer. We extract the magnet\ncurrent associated with the spin resonance at each frequency. We \fnd that the line formed\nby plotting the resonant currents as a function of frequency extrapolates to the origin as\nexpected. Using these data we calculate a calibration factor of 60.64(10) mT/A with an\nuncertainty in absolute magnetic \feld of 0.4 mT. Fig. S6 shows a plot of the static \feld at\nthe sample B0vs the magnet coil current.\n31"    },    {        "title": "2312.07678v2.Dzyaloshinskii_Moriya_interaction_inducing_weak_ferromagnetism_in_centrosymmetric_altermagnets_and_weak_ferrimagnetism_in_noncentrosymmetric_altermagnets.pdf",        "content": "Dzyaloshinskii-Moriya interaction inducing weak ferromagnetism in centrosymmetric\naltermagnets and weak ferrimagnetism in noncentrosymmetric altermagnets\nCarmine AutieriID,1,∗Raghottam M SattigeriID,1,†Giuseppe CuonoID,1,‡and Amar Fakhredine2\n1International Research Centre Magtop, Institute of Physics,\nPolish Academy of Sciences, Aleja Lotnik´ ow 32/46, 02668 Warsaw, Poland\n2Institute of Physics, Polish Academy of Sciences, Aleja Lotnik´ ow 32/46, 02668 Warsaw, Poland\nThe Dzyaloshinskii-Moriya interaction (DMI) has explained successfully the weak ferromagnetism\ninsome centrosymmetric antiferromagnets. However, in the last years, it was generally claimed that\nthe DMI is not effective in centrosymmetric systems. We reconciled these views by separating the\nconventional antiferromagnets and altermagnets. Altermagnets represent collinear antiferromagnetic\ncompounds with spin-up and spin-down sublattices connected only by mirror and roto-translational\nsymmetries. Consequently, the system shows even-parity wave spin order in the k-space lifting the\nKramer’s degeneracy in the non-relativistic band structure. We emphasize that the DMI can create\nweak ferromagnetism in centrosymmetric altermagnets while it is not effective in centrosymmetric\nconventional antiferromagnets. Additionally, DMI can create weak ferromagnetism or weak ferri-\nmagnetism in noncentrosymmetric altermagnets. Once the spin-orbit coupling is included in an\naltermagnetic system without time-reversal symmetry, the components of spin moments of the two\nsublattices along the N´ eel vector are antiparallel but the other two spin components orthogonal\nto the N´ eel vector can be either parallel or antiparallel for centrosymetric systems. For noncen-\ntrosymmetric systems, we can have different bands showing parallel or antiparallel spin components\nresulting in weak ferrimagnetism. We can divide the altermagnetic compounds into classes based on\nthe weak ferromagnetism or weak ferrimagnetism properties. In the case of weak ferromagnetism,\nthe Hall vector is orthogonal to the N´ eel vector while in the case of weak ferrimagnetism the Hall\nvector has a component parallel to the N´ eel vector too. Therefore, an electric field breaking the in-\nversion symmetry could generate a magnetization parallel to the N´ eel vector. We find a sign change\nof the magnetization, and possibly of the anomalous Hall effect, as a function of the band filling and\nN´ eel vector. We describe the dependence of the weak ferromagnetism from the charge doping and\nelectric field. The weak ferromagnetism and weak ferrimagnetism induced by DMI is a property\nexclusively of the altermagnets, not present in either ferromagnets or conventional antiferromagnets\nso we propose that altermagnets should be classified based on this property.\nI. INTRODUCTION\nIt was discovered that non-relativistic spin-splitting\nin electronic bands can be a feature in antiferromag-\nnetic materials having two equivalent sublattices with\nmagnetic moments related by crystal-symmetry1,2. This\nmagnetic phase has been named altermagnetism (AM)3–7\nor antiferromagnetism with non-interconvertible spin-\nstructure motif pair8. The existence of AM necessi-\ntates that the electronic charge of spin-up (down) atoms\nshould be mapped into spin-down (up) atoms not ex-\nclusively via translations, inversions, or their combina-\ntions but through the application of roto-translations or\nmirrors6,7. As a result, AM can be observed in specific\nspace groups, and more precisely, in the magnetic space\ngroups of type-I and type-III9. Numerous transition\nmetal oxides9and rare-earth compounds have been docu-\nmented to present altermagnetism10. Altermagnets have\na wide range of potential applications, they are promis-\ning to enable highly efficient spin-current generation11,\nas well as contributing to the development of giant and\ntunneling magnetoresistance12. It was shown that spin-\nindependent conductance in altermagnets can be effi-\nciently exploited in spintronics13–15. Furthermore, alter-\nmagnets can play a significant role in spincaloritronics16\nand can be practical use in Josephson junctions17.The Dzyaloshinskii-Moriya interaction (DMI) is a\nrelativistic magnetic interaction that has successfully\nexplained the weak ferromagnetism described for the\nfirst time in centrosymmetric antiferromagnets18–20with\nspace group 167. This weak ferromagnetism has a rel-\nativistic origin and is the only kind of weak ferromag-\nnetism discussed in this paper. Some compounds of\nthe centrosymmetric space group 62 exhibit weak ferro-\nmagnetism and lead to a significant ferromagnetic be-\nhavior, which was later explained by the mechanism\nof the Dzyaloshinskii Moriya exchange based on torque\nmeasurements21,22. The spin canting has been attributed\nto the DMI in most cases, although other effects share\nthe same requirements to be imposed on the crystal\nsymmetry23. The weak ferromagnetism can also be\npresent in complex systems like solids with molecules\nas cations24and biphasic systems25. In other litera-\nture, the weak ferromagnetism from DMI in bulk systems\nwas associated with noncentrosymmetric systems26. Fur-\nthermore, in ferroelectric BiFeO 3, a subtle DMI which\nconstitutes about 1% of the antiferromagnetic exchange\nstrength causes a tilting of the two spin sublattices, con-\nsequently giving rise to ”weak ferromagnetism”20,27. The\nDMI as explained by definition, appears as an effect\nof antisymmetric interactions, which necessitates a lack\nof inversion symmetry between two spins. Neverthe-\nless, a structure can possess an overall centrosymmet-\nTypeset by REVT EXarXiv:2312.07678v2  [cond-mat.mtrl-sci]  26 Dec 20232\nEFEF\nEFEF\nEFK1K2\nK1K2Sz\nSz0\n0Г\nГMz = 0 ; σxy = 0\nMz ≠ 0 ; σxy ≠ 0a) b)\nc)\nd) e)f)\nFIG. 1. Origin of weak ferromagnetism in altermagnets. (a)\nSchematic band structure of an altermagnet with SOC, N´ eel\nvector in the xy plane and Fermi level in the gap. The red and\nblue colors represent the negative and positive S zcomponent\nof the total system. (b) Local S zcomponent for the magnetic\natom as a function of the energy for the energy spectrum (a),\nin the centrosymmetric case S zfor the two atoms are equal.\n(c) Magnetic configuration without M zand AHE ρxy. (d)\nSame as (a) with metallicity. (e) S zcomponent as a function\nof the energy for the energy spectrum (d). (f) Magnetic con-\nfiguration with non-zero M zcomponents and non-zero AHE\nρxy.\nric crystal symmetry while also accommodating a DMI\ninteraction28. DMI plays an important role in forming\nskyrmions with a fixed chirality and vorticity in the case\nof noncentrosymmetric magnets since it allows the tilting\nof adjacent spins under special rules related to crystal\nsymmetry. In contrast, in centrosymmetric structures,\nother mechanisms are believed to be responsible for com-\nplex magnetism29.\nThe non-relativistic spin-splitting is proportional to\nthe hopping parameters which tend to increase with the\natomic number. Large spin-splittings have been found in\ncompounds with heavy elements as MnTe, RuO 2, CrSb,\netc ... which are also the most representative and most\nstudied compounds among the altermagnetic systems.\nBeyond the large spin-splitting, these heavy atoms own\na huge spin-orbit coupling (SOC). Therefore, the study\nof the spin-orbital effects is extremely relevant, especially\nfor the most representative altermagnets. Beyond that,\nthe anomalous Hall effect (AHE) strongly depends on\nthe size of the SOC. In the case of a metallic system, the\nDMI produces weak ferromagnetism in AM compounds\nthat display spontaneous AHE3,30which can be enhanced\nby Weyl points31. The orientation of the Hall vector\nstrongly depends on the N´ eel vector, which is defined as\nthe difference between the magnetization vectors of the\ntwo distinct magnetic sublattices15,32,33. While the direc-\ntion of the Hall vector has been explored using symmetry\nconsiderations, there is a notable absence of information\nin the literature regarding how to establish this orienta-tion based on fundamental electronic properties through\ndensity functional theory calculations.\nIn this work, our objective is to have a general under-\nstanding of the appearance of weak ferromagnetism in\ncentrosymmetric systems, while we found weak ferromag-\nnetism and weak ferrimagnetism in noncentrosymmetric\nsystems. We discuss why weak ferromagnetism and weak\nferrimagnetism are relativistic effects that can appear\nonly in altermagnets. Once the time-reversal symmetry\nis broken and the Kramers degeneracy is lifted, the spin\ncomponents of the two sublattices orthogonal to the N´ eel\nvector are related by symmetry in centrosymmetric sys-\ntems and therefore must be parallel or antiparallel. The\nscenario changes in noncentrosymmetric systems where\ndifferent sets of bands can generate at the same time\nparallel or antiparallel spin components in which we es-\ntablish that the direction of the Hall vector is orthogonal\nto the N´ eel vector. A schematic representation of our\nmain results is presented in Fig. 1. We propose that the\nband structure reported in Fig. 1(a) can be present in\nall altermagnets for a suitable N´ eel vector, as a conse-\nquence of this band structure, the local density of states\n(DOS) for the component of the spin orthogonal to the\nN´ eel vector have a relativistic spin-splitting as shown in\n1(b). If the Fermi level is in the gap, we still observe a\nnet zero magnetic configuration M zand zero anomalous\nHall effect in altermagnets owing to a complete antiparal-\nlel spin configuration in two sublattices. However, when\nbands cross the Fermi level (as in Fig. 1(d)) which is\nalso evident from the increase in the contribution from\nthe S zcomponent (as in Fig. 1(e)), the SOC creates\nan effective non-zero magnetic configuration giving rise\nto ferromagnetic spins aligned perpendicular to the N´ eel\nvector (as in Fig. 1(f)) leading to a non-zero anoma-\nlous Hall effect in altermagnetic compounds. Integrating\nin energy the spin component orthogonal to the N´ eel in\na realistic multiband system we will obtain the magne-\ntization as a function of the energy, this magnetization\ncan be zero at the same energies by compensation of the\nspins, however, the compensation of the spins does not\nmean that we have a compensation of the Berry curva-\nture. Therefore, a null magnetization is not associated\nwith a null anomalous Hall effect34. The weak ferromag-\nnetism is difficult to detect especially on the nanoscale\nlevel, so our work provides a guide to theoretically pre-\ndict the direction of the weak ferromagnetism to support\nthe experimental investigation. Using density functional\ntheory calculations, the results of the self-consistent cal-\nculation will report parallel or antiparallel components\ndepending on which of them minimizes the total energy.\nWe calculate the spin components of the two sublattices\nas a function of the N´ eel vector for the centrosymmetric\nd-wave RuO 2, the centrosymmetric g-wave CrSb and the\nnoncentrosymmetric g-wave MnSe. The paper is orga-\nnized as follows: the second Section describes the com-\nputational framework while the third Section is dedicated\nto the results for the three chosen compounds. Finally,\nin the fourth Section, the authors draw their conclusions.3\n-0.3 0 0.3\n S1  Γ  S2 Energy (eV) (a)Sx\nn || x\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n S1  Γ  S2 Energy (eV) (b)Sy\nn || x\nMY≠0\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n S1  Γ  S2 Energy (eV) (c)Sz\nn || z\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. 2. Band structure of d-wave RuO 2along the S 1-Γ-S 2low-symmetry k-path. (a) The positive and negative S xcomponents\nare plotted in red and blue, respectively, while the N´ eel vector is along the x-axis. (b) The positive and negative S ycomponents\nare plotted in red and blue, respectively, while the N´ eel vector is along the x-axis. (c) The positive and negative S zcomponents\nare plotted in red and blue, respectively, while the N´ eel vector is along the z-axis. S x, Syand S zrepresent the spin components\nfor the total system. The Fermi level is set to zero. The coordinates of the k-points are S 1=(0.5,0.5,0) and S 2=(-0.5,0.5,0) in\ndirect coordinates.\n-0.3 0 0.3\n R1  Γ  R2 Energy (eV) (a)Sy\nn || x\nMY≠0\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n R1  Γ  R2 Energy (eV) (b)Sy\nn || z\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. 3. Band structure of d-wave RuO 2along the R 1-Γ-R 2\nlow-symmetry k-path with the N´ eel vector along (a) the x-\nand (b) z-axis. The positive and negative S yand S xcompo-\nnents are plotted in red and blue, respectively. S x, Syand\nSzrepresent the spin components for the total system. The\nFermi level is set to zero. The coordinates of the k-points are\nR1=(0.5,0,0.5) and R 2=(-0.5,0,0.5) in direct coordinates.II. COMPUTATIONAL DETAILS\nThe electronic properties were computed within the\nframework of density functional theory calculations based\non plane wave basis set and projector augmented wave\nmethod using VASP35–37package. A plane-wave energy\ncut-off of 400 eV was chosen and the generalized gra-\ndient approximation of Perdrew, Burke, and Ernzerhof\nhas been used38. The Hubbard U effects for the 4d-\norbitals of Ru4+, the 3d-orbitals of Cr and 3d-orbitals\nof Mn2+have been adopted39. We have used the value\nof U = 2 eV40,41keeping J H= 0.15U. The value of U\nstrengthens the gap in MnSe and establishes a robust\nmagnetic moment in RuO 2, however, all qualitative re-\nsults are independent of the value of U. We have per-\nformed the calculations using 20 ×20×28 and 24 ×24×18\nk-points centered in Γ for RuO 2and CrSb, respectively.\nA denser k-grid of 28 ×28×18 was used for wurtzite MnSe,\nwhich is necessary to have an accurate description of the\nDOS, especially in the presence of relatively small spin-\norbit coupling. The lattice constants are a=4.4825 ˚A\nand c=3.1113 ˚A for the RuO 2, a=4.103 ˚A and c=5.463\n˚A42for the CrSb and a=4.12 ˚A with c=6.72 ˚A for the\nMnSe43. The experimental N´ eel vectors are along the\nz-axis for RuO 2and CrSb44. All calculations were per-\nformed in the presence of SOC. The information on the\nindividual spinor components is available for the site’s\nprojected density of states in the VASP code. The site-\nprojected orbital- and energy-resolved spin density S x, Sy\nand S zare available. We sum over all orbitals to obtain\nthe total S x, Syand S zcomponents. The smearing value\nfor the DOS was 50 meV. Regarding the magnetic space\ngroup (MSG), the RuO 2has MSGs 58.398 and 136.499\nfor the N´ eel vector along the x-axis and z-axis, respec-\ntively. CrSb has MSGs 63.457, 63.462 and 194.268 for\nthe N´ eel vector along the x-axis, y-axis, and z-axis, re-\nspectively. Wurtzite MnSe has the magnetic space group\n194.268 when the N´ eel vector is along the z-axis. The4\n-6-5-4-3-2-1 0 1 2 3 4 5 6\n-0.3  0  0.3DOS (10-1states/eV)\nEnergy (eV)(a)\nn || xSx Ru1\nSx Ru2\nTotal DOS\n-5-4-3-2-1 0 1 2 3 4 5\n-0.3  0  0.3DOS (10-2states/eV)\nEnergy (eV)(b)\nn || xSy Ru1\nSy Ru2\nTotal DOS\nEnergy (eV)-6-5-4-3-2-1 0 1 2 3 4 5 6\n-0.3  0  0.3DOS (10-1states/eV)(c)\nn || zSz Ru1\nSz Ru2\nTotal DOS\nFIG. 4. DOS for the spin-components as a function of the energy for the d-wave RuO 2. (a,b) DOS of the S xand S ycomponents\nwith the N´ eel vector along the x-axis, respectively. (c) DOS of the S zcomponent with N´ eel vector along the z-axis. The other\ncomponents are antiparallel. The Fermi level is set to zero. The total DOS is plotted in green and renormalized to fit within\nthe plot.\nmagnetic point group without spin-orbit coupling32,45of\nthe wurtzite MnSe is 6′m′m, different from that of the\nhexagonal CrSb in the NiAs structure which is 6′/m′m′m.\nHowever, both groups belong to the same magnetic Laue\nclass (6′/m′m′m) proving that the spin splitting of elec-\ntron bands without SOC is qualitatively the same in both\nsystems. This could be established using the formalism\nof spin groups6,7.\nIII. RESULTS\nIn a centrosymmetric conventional antiferromagnet, we\nhave Kramers degeneracy therefore the DOS of the spin-\nup sublattice and the spin-down sublattice are always\nequal for all components. The two sublattices have al-\nways antiparallel spin components and no weak ferromag-\nnetism is possible. In a noncentrosymmetric conventional\nantiferromagnet with SOC, there is an additional Rashba\nspin-splitting where the spin components belonging to\ntwo sublattices are antiparallel.46Therefore, in noncen-\ntrosymmetric conventional antiferromagnets, there is no\nweak ferromagnetism as well.\nWe move to the study of the electronic properties re-\nsponsible for weak ferromagnetism in centrosymmetric\nand noncentrosymmetric altermagnets. We study the\nelectronic properties of d-wave centrosymmetric alter-\nmagnet RuO 2in the first subsection. In the second and\nthird subsections, we compare the g-wave altermagnets\nCrSb and wurzite MnSe, where the former is centrosym-\nmetric and the latter is noncentrosymmetric. We will see\nhow SOC acts differently on centrosymmetric and non-\ncentrosymmetric space groups. All selected compounds\nhave two atoms per unit cell which we will call M 1and\nM2(M= Ru, Cr, Mn) to represent the properties of the\ntwo sublattices with opposite spins.A. Weak ferromagnetism in centrosymmetric\ntetragonal altermagnet RuO 2\nIn this subsection, we describe the metallic centrosym-\nmetric d-wave RuO 2. Several authors described the al-\ntermagnetic properties of RuO 23,47, however, we will fo-\ncus on the band structure and the DOS for the spin-\ncomponents which were never investigated before. The\nappearance of the weak ferromagnetism derives from the\nbreaking of the time-reversal symmetry, therefore, there\nis the chance for the spins to be not antiparallel when\nSOC is included. The self-consistent DFT calculations\nprovide the energetic ground state for the parallel or an-\ntiparallel configuration of the spin components orthogo-\nnal to the N´ eel vector. Since RuO 2is metallic, we expect\nthat this weak ferromagnetism could be observed in the\nDFT calculations. Indeed, the magnetic moment orthog-\nonal to the N´ eel vector is 0.022 µBper Ru atom along\nthe y-axis when the N´ eel vector is along the x-axis. In\nFigs. 2 and 3, we report the band structures of RuO 2pro-\njected on different spin components for the two inequiva-\nlent directions of the N´ eel vector (x-axis and z-axis) along\nthe S 1-Γ-S 2and R 1-Γ-R 2low-symmetry k-paths, respec-\ntively. When the N´ eel vector is along the x-axis, the S x\nspin component of the total system for every energy has\na value along S 1-Γ and an equal but opposite value along\nΓ-S2as reported in Fig. 2(a). As a result, the magne-\ntization component M xof the entire system that is the\nintegral over the energy up to the Fermi level of S xis\nalways zero. On the contrary, the spin components S yof\nthe total system show a band structure where the value\nof Syalong S 1-Γ is equal to the value along Γ-S 2. For\nevery value of the energy, the value of the magnetization\nMyas integral of S ywill be in general non-zero resulting\nin a weak magnetization in the y-direction as we can see\nfrom Figs. 2(b) and 3(a). The origin of the canted spin\nmoment derives exclusively from the spin-orbit coupling\nbut this picture is equivalent to the DMI picture. In\nthe case of RuO 2with spins along the z-axis, we have a\npure antiparallel order without weak ferromagnetism, as\nshown in Figs. 2(c) and 3(b). The other results were not5\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(a)Sx\nn || x\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(b)Sy\nn || x\n-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(d)Sx\nn || y\n-0.6-0.4-0.2 0 0.2 0.4 0.6\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(e)Sy\nn || y\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(g)Sx\nn || z\nMX≠0\n-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(h)Sy\nn || z\nMY≠0\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(c)Sz\nn || x\nMZ≠0\n-0.6-0.4-0.2 0 0.2 0.4 0.6\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(f)Sz\nn || y\nZ≠\n-0.6-0.4-0.2 0 0.2 0.4 0.6\n0 M\n-0.3 0 0.3\n L1  Γ  L2 Energy (eV)(i)Sz\nn || z\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. 5. Band structure of centrosymmetric CrSb with SOC along the L 1-Γ-L 2low-symmetry k-path. With the N´ eel vector\nalong the x-axis, we report the projection of the spin-component (a) S x, (b) S yand (c) S z. With the N´ eel vector along the\ny-axis, we report the projection of the spin-component (d) S x, (e) S yand (f) S z. With the N´ eel vector along the z-axis, we\nreport the projection of the spin-component (g) S x, (h) S yand (i) S z. Sx, Syand S zrepresent the spin components for the\ntotal system. The Fermi level is set to zero. The coordinates of the k-points are L 1=(0.5,0,0.5) and L 2=(-0.5,0,0.5) in direct\ncoordinates.\nshown because they are either equivalent to the shown\nresults or with null spin components. The band degen-\neracy is broken at Γ for the N´ eel vector along the x-axis,\nbut not for the N´ eel vector along to z-axis. Therefore,\nwe have lifting of the degeneracy at Γ when the weak\nferromagnetism rises.\nTo further extend our consideration to the entire Bril-\nlouin zone, we also show the density of states for the\nnon-zero and inequivalent spin components in Fig. 4 for\nthe N´ eel vector along the x- and z-axis. When the N´ eel\nvector is along the x-axis, the local DOSs of the S xcom-\nponents for the two sublattices are equal and opposite as\ndescribed by the local DOS related to atoms Ru 1and Ru 2\nin Fig. 4(a), therefore, the S xcomponents are equal andopposite for the two sublattices. From the DOS of the S y\ncomponent reported in 4(b), it is further clear that there\nis a magnetic moment along the y-axis on the Ru 1and\nRu2atoms that are equal and in the same direction re-\nsulting in a weak ferromagnetism. The DOSs of the spin\ncomponent S yfor the atoms Ru 1and Ru 2are perfectly\nequal, therefore, the size of the local magnetic moments\nof Ru 1and R 2are equal as expected since the two atoms\nare connected by roto-translation symmetries. We can\nsee that we have a sign change in M yas a function of\nthe energy, this could be accompanied by a sign change\nof the AHE. Comparing the DOS of the spin-component\nSxwith the total DOS, we can see a strong reduction\nof Sxaround -0.4 eV together with the maximum of S y,6\n-0.4-0.2 0 0.2 0.4\n-0.3  0  0.3DOS (states/eV)\nEnergy (eV)(a)\nn || xSx Cr1\nSx Cr2\nTotal DOS\n-0.3-0.2-0.1 0 0.1 0.2 0.3\n-0.3  0  0.3DOS (states/eV)\nEnergy (eV)(b)\nn || zSz Cr1\nSz Cr2\nTotal DOS\n-2-1.5-1-0.5 0\n-0.3  0  0.3DOS (10-3states/eV)\nEnergy (eV)(c)\nn || x Sz Cr1\nSz Cr2\nTotal DOS-0.2 0 0.2 0.4 0.6 0.8 1 1.2\n-0.3  0  0.3DOS (10-3states/eV)\nEnergy (eV)(d)\nn || ySz Cr1\nSz Cr2\nTotal DOS\n 0 1\n-0.3  0  0.3DOS (10-3states/eV)\nEnergy (eV)(e)\nn || zSx Cr1\nSx Cr2\nTotal DOS\n 0 0.2 0.4\n-0.3  0  0.3DOS (10-3states/eV)\nEnergy (eV)(f)\nn || zSy Cr1\nSy Cr2\nTotal DOS\nFIG. 6. DOS for the spin-components S x, Syand S zas a\nfunction of the energy for the centrosymmetric CrSb. (a,b)\nDOS for the components S xand S zwith the N´ eel vector along\nthe x-axis and along the z-axis, respectively. (c,d) DOS for\nthe spin component S zwith the N´ eel vector along the x-axis\nand along the y-axis, respectively. Both of them show weak\nferromagnetism. (e,f) DOS for the spin component S xand\nSywith the N´ eel vector along the z-axis. Both of them show\nweak ferromagnetism. The Fermi level is set to zero. The\ntotal DOS is renormalized to fit within the plot.\ntherefore, there is a strong rotation of the S xcomponent\ninto the S ycomponent at -0.4 eV. In the case of the N´ eel\nvector along the z-axis, the DOS of all spin-components\nrelative to Ru 1and Ru 2are equal and opposite, there-\nfore, the Ru spins are perfectly compensating each other\nas it is shown in Fig. 4(c), consequently, there is no weak\nferromagnetism. From the DOS we can also understand\nthat the system presents ferromagnetic or antiferromag-\nnetic features depending on the orientation of the N´ eel\nvector, and this property in the k-space is the same in\nthe whole Brillouin zone and for all bands.\nB. Weak ferromagnetism in centrosymmetric\nhexagonal altermagnet CrSb\nWe repeat the investigation for the g-wave centrosym-\nmetric altermagnet CrSb. In Figs. 5, we report the band\nstructures of CrSb projected on different spin compo-\nnents (S x, Syand S z) for the three different directions of\nthe N´ eel vector along the L 1-Γ-L 2focusing on the region\nnear Γ point. The ground state is altermagnetic phase\nwith the N´ eel vector along the z-axis, we have an energydifference of 0.2 meV per formula unit with respect to\nthe in-plane N´ eel vector. When the N´ eel vector is along\nthe x-axis as in Fig. 5(a,b,c), we have weak ferromag-\nnetism only for S zresulting in a non-zero value of M z.\nWe have qualitatively similar results when the N´ eel vec-\ntor is along the y-axis as in Fig. 5(d,e,f). We have weak\nferromagnetism only for S zresulting in a non-zero value\nof M z. When we have the N´ eel vector along the z-axis,\nwe have weak ferromagnetism for the S xand S ycompo-\nnents resulting in a non-zero M xand M yas shown in\n5(g,h,i). Overall, we find a weak ferromagnetism out-of-\nplane when the N´ eel vector is in-plane and a weak ferro-\nmagnetism in-plane when the N´ eel vector is out-of-plane.\nAdditionally, we find a strong change in the relativistic\nband structure as a function of the N´ eel vector, especially\nbetween in-plane and out-of-plane directions of the N´ eel\nvector. Qualitatively similar results are observed for the\nhigh-symmetry k-path K 1-Γ-K 2(not reported) with the\ndifference that S zis always zero when the N´ eel vector is\nalong the z-axis since the non-relativistic spin splitting is\nabsent. We check the electronic properties in the DOS\nas well where we have information along the entire Bril-\nlouin zone and not for a limited number of k-points. The\nDOS for S xand S zwhen the N´ eel vector is along the x-\nand z-axis are reported in 6(a) and (b), respectively. We\ncompare the DOS of the spin components with the total\nDOS illustrated in green. At energy 0.2 eV above the\nFermi level, the spin-component resolved DOS with spin\nalong the N´ eel vectors has less weight with respect to the\ntotal DOS. This could mean that 0.2 eV above the Fermi\nlevel there is DOS from non-magnetic atoms or there is a\nrotation of the spins with the weight of the DOS moving\ntowards the spin components orthogonal to the N´ eel vec-\ntor. When the N´ eel vector is along the x- and y-axis, we\nhave non-zero magnetization along the z-axis. The values\nof DOS for S zare reported in 6(c) and (d) for the N´ eel\nvector along the x- and y-axis. The case with N´ eel vec-\ntor along the x-axis gives the larger weak ferromagnetism\nfor CrSb with a value of -0.002 µBper formula unit. We\ncan observe a value of the total DOS above 0.2 eV larger\nthan the DOS of the spin-component, therefore, the spins\nin this region are rotated from the main component by\nspin-orbit. We can see that the DOS of S zis positive or\nnegative depending on whether the N´ eel is along the y-\nor x-axis. This means that by rotating the N´ eel vector\nin the xy plane, we will have a sign change of M z. This\nsign change of M zcould also produce a sign change of the\nanomalous Hall effect. Recent studies on the sign change\nof the AHE in ferromagnets have explained this effect\nwhich is relatively rare in experimental conditions48,49.\nWhen the N´ eel vector is along the z-axis, we observe\nweak ferromagnetism for both spin components for S x\nand S yreported in 6(e) and (f).7\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (g)Sx\nn || z\nMX≠0\n-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (d)Sx\nn || y\n-0.3-0.2-0.1 0 0.1 0.2 0.3-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (a)Sx\nn || x\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (b)Sy\nn || x\n-0.3-0.2-0.1 0 0.1 0.2 0.3\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (e)Sy\nn || y\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (c)Sz\nn || x\nMZ≠0\n-0.3-0.2-0.1 0 0.1 0.2 0.3\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (f)Sz\nn || y\nMZ≠0\n-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (h)Sy\nn || z\nMY≠0\n-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4\n-0.3-0.2-0.1 0\n L1  Γ  L2 Energy (eV) (i)Sz\nn || z\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\nFIG. 7. Band structure of noncentrosymmetric MnSe with SOC along the L 1-Γ-L 2low-symmetry k-path. With the N´ eel vector\nalong the x-axis, we report the projection of the spin-component (a) S x, (b) S yand (c) S z. With the N´ eel vector along the\ny-axis, we report the projection of the spin-component (d) S x, (e) S yand (f) S z. With the N´ eel vector along the z-axis, we\nreport the projection of the spin-component (g) S x, (h) S yand (i) S z. Sx, Syand S zrepresent the spin components for the\ntotal system. S x, Syand S xrepresent the spin components for the total system. The Fermi level is set to zero. The coordinates\nof the k-points are L 1=(0.5,0.0.5) and L 2=(-0.5,0,0.5) in direct coordinates.\nC. Weak FM and Weak ferrimagnetism in\nnoncentrosymmetric hexagonal altermagnet MnSe\nTo understand the effect of the breaking of the in-\nversion symmetry, we analyze MnSe which is a hexag-\nonal g-wave altermagnet like CrSb. Indeed, MnSe with a\nwurtzite crystal structure represents the case of an alter-\nmagnet without inversion symmetry with a small number\nof atoms in the unit cell. The structural and electronic\nproperties without SOC for wurtzite MnSe are reported\nin the literature43. The non-relativistic spin-splitting in\nwurzite MnSe can reach a maximum of 200-300 meV in\nthe Mn-bands, the SOC of Mn is of the order of 30 meV\nwhile the SOC of Se is 140 meV. Therefore, the SOC\nwould be a perturbation at least in the region where the\nspin-splitting is maximum. For MnSe, the top of the va-\nlence band at Γ and A are mainly p x/pywith a sizeablecontribution from Mn orbitals. When the N´ eel vector is\nmoved from in-plane to out-of-plane, the top of the va-\nlence band rises. We have a reduction of the band gap of\n80 meV (from 1.72 to 1.64 eV) when we move from AFM\nwith the N´ eel vector in the plane to AFM with the N´ eel\nvector out-of-plane. The same sensitivity of the band gap\nfrom the N´ eel vector was found for the centrosymmetric\ng-wave altermagnet MnTe.50\nWe calculate the band structure for the three different\nspin components along the low symmetry k-path L 1-Γ-\nL2where we have the non-relativistic spin-splitting. The\ntop of the valence band at the Γ point without SOC is\nfour-fold degenerate. Starting from the low-symmetry k-\npath L 1-Γ-L 2reported in Fig. 7, we find a relativistic\nspin-splitting in all cases. When the N´ eel vector is in\nthe plane, the eigenvalues at the Γ point split into two\neigenvalues twofold degenerate with zero magnetization8\nEFEF\nL1L2 0 ГMy ≠ 0 ; σxz ≠ 0\nMy = 0 ; σxz = 0a)\nb)\nc)B1B1\nEFEF\nL1L2 Sy0 ГMn1\nMn2\nB2\nB2Mn1\nMn2\nEFEF\nL1L2 Sy0 ГB1\nB2 Mn1\nMn2B2\nB1\nMy ≠ 0 ; σxz ≠ 0Sy\nFIG. 8. Schematic band structure of noncentrosymmetric al-\ntermagnets with N´ eel vector along the z-axis. The red and\nblue colors represent the negative and positive S ycomponents\nof the total system in the band structure. DOS of the spin\ncomponent S yand magnetic configuration for three selected\ncases. a) Weak ferromagnetism in some bands (B1) of non-\ncentrosymmetric altermagnets. b) Weak antiferromagnetism\nin some bands (B2) of noncentrosymmetric altermagnets. c)\nWeak ferrimagnetism as the superposition of weak FM and\nweak AFM from bands B1 and B2.\nin the plane (see Figs. 7(a,b,d,e)) and non-zero magneti-\nzation along the z-axis (see Figs. 7(c,f)). When the N´ eel\nvector is along the z-axis, there is a strong splitting with\nthe two highest eigenvalues single-degenerate and the one\neigenvalue double-degenerate at the Γ point. The single-\ndegenerate highest eigenvalues produce non-zero M xand\nMyas shown in Fig. 7(g) and 7(h), respectively. How-\never, the double degenerate bands show antiparallel spin\ncomponents. Without SOC, there is a fourfold degener-\nacy in the top of the valence band43. When the system\nis noncentrosymmetric as in the wurzite structure, we\ncan have a combination of parallel and antiparallel spin\norientation despite both sets of bands coming from the\nsame fourfold degenerate point without SOC. As a con-\nsequence in the DOS, we will see that we do not have\na completely parallel or antiparallel DOS weight. Fi-\nnally, 7(i) shows the sign and weight of S zon the two\nsublattices that are equal and opposite when the N´ eel\nvector is along the z-axis. Qualitatively similar results\nare observed for the K 1-Γ-K 2path (not reported) with\nthe difference that S zis always zero when the N´ eel vector\nis along the z-axis since the non-relativistic spin splitting\nis absent. We find weak ferromagnetism for g-wave com-\npounds along the same direction with or without inver-\nsion symmetry. The main difference is the coexistence of\n-0.1 0 0.1\n-0.3 -0.2 -0.1  0DOS (states/eV)\nEnergy (eV)(a)\nn || xSx Mn1\nSx Mn2\nTotal DOS\n-0.1 0 0.1\n-0.3 -0.2 -0.1  0DOS (states/eV)\nEnergy (eV)(b)\nn || zSz Mn1\nSz Mn2\nTotal DOS\n-0.1 0 0.1\n-0.3 -0.2 -0.1  0DOS (10-4states/eV)\nEnergy (eV)(c)\nn || xSz Mn1\nSz Mn2\nTotal DOS\n-0.2 0 0.2 0.4\n-0.3 -0.2 -0.1  0DOS (10-4states/eV)\nEnergy (eV)(d)\nn || ySz Mn1\nSz Mn2\nTotal DOS\n-0.1 0\n-0.3 -0.2 -0.1  0DOS (10-4states/eV)\nEnergy (eV)(e)\nn || zSx Mn1\nSx Mn2\nTotal DOS Total DOS\n-0.04-0.02 0 0.02 0.04\n-0.3 -0.2 -0.1  0DOS (10-4states/eV)\nEnergy (eV)(f)\nn || zSy Mn1\nSy Mn2FIG. 9. DOS for the spin-components S x, Syand S zas a\nfunction of the energy for the wurtzite MnSe. (a,b) DOS for\nthe spin components S xand S zwith the N´ eel vector along\nthe x-axis and along the z-axis, respectively. (c,d) DOS for\nthe spin component S zwith the N´ eel vector along the x-axis\nand along the y-axis, respectively. (e,f) DOS for the spin\ncomponent S xand S ywith the N´ eel vector along the z-axis,\nrespectively. The solid and dashed lines represent the local\nspin components S x, Syand S zfor the two sublattices repre-\nsented by Mn1 and Mn2. The Fermi level is set to zero. The\ntotal DOS is renormalized to fit within the plot.\nboth parallel and antiparallel magnetic configurations for\nthe different bands of the in-plane spin-components when\nthe N´ eel is along the z-axis. Basically, we have the super-\nposition of weak ferromagnetism [see Fig. 8](a)] and weak\nantiferromagnetism [see Fig. 8](b)]. The superposition\nof both kinds of magnetism as reported schematically in\nFig. 8(c) ends up forming weak ferrimagnetism (weak\nFiM). In this case, there is no exact compensation of the\nmagnetization along the N´ eel vector and the Hall vector\nhas a component along the N´ eel vector too.\nTo further check the results from the band structure,\nwe report in Fig. 9 the resolved DOS for the spin-\nsublattice and spin components S x, Syand S zfor dif-\nferent directions of the N´ eel vector, namely along the x-,\ny- and z-axis. The DOSs for the spin components along\nthe N´ eel vector are reported in Fig. 9(a,b). The two\nsublattices, represented by the Mn 1and Mn 2local DOS,\nhave equal and opposite DOS of the spin components\nsince the spins are antiparallel. Summing the DOSs of\nthe components along the N´ eel vector for the two sub-\nlattices, we would obtain zero. Now, we focus on the\ncomponents orthogonal to the N´ eel vector. The size of9\nthese DOSs is 3-4 orders of magnitude smaller than the\nsize of the DOSs for the spin-components along the N´ eel\nvector. When the N´ eel vector is along the x-axis, we ob-\nserve from Fig. 9(a,c) that the in-plane S xcomponents of\nthe two sublattices are antiparallel, while the spin com-\nponents S zare parallel. Since the filling was completed\nand the MnSe is an insulator, the weak ferromagnetism\nwas not obtained in the computational details but in the\ncase of p-doping, the weak ferromagnetism would be ob-\nservable. Assuming a p-doping, the magnetization M z\nobtained integrating the spin density up to the Fermi\nlevel will be non-zero and the system will host a weak\nferromagnetism along the z-direction. The same happens\nfor the N´ eel vector along the y-axis, the in-plane compo-\nnents are antiparallel while we have weak ferromagnetism\nalong S zas shown in Fig. 9(d). When the N´ eel vector\nis along the z-axis, we have both M xand M ycompo-\nnents different from zero, as shown in Fig. 9(e,f). When\nthe N´ eel vector is in the xy plane, the weak ferromag-\nnetism is along the z-axis. When the N´ eel vector is along\nthe z-axis, the weak ferromagnetism is in the xy plane.\nThis weak ferromagnetism generates AHE with the Hall\nvector along the direction of the weak ferromagnetism.\nWe observe that the Hall vector is always orthogonal to\nthe N´ eel vector. Even if some band structures resemble\nthe Rahsba-like dispersions, the spin textures of the al-\ntermagnets are different and more complicated than the\nRashba effect9. Of course, a Rashba spin texture rises on\ntop of the altermagnetic spin texture when the system\nlacks inversion symmetry like in the wurtzite structure.\nFor the wurtzite with N´ eel vector along the z-axis, we\nhave a coexistence of bands with parallel and antipar-\nellal components regarding S xand S yas shown in Fig.\n7(g) and (h), respectively.\nIV. DISCUSSION AND CONCLUSIONS\nIn this paper, we propose that all compounds host-\ning weak ferromagnetism induced by the relativistic\nDzyaloshinskii-Moriya interaction18,19belong to the fam-\nily of the altermagnets. While Dzyaloshinskii and\nMoriya focused on the interplay between magnetic in-\nteractions and symmetries, the recent studies on alter-\nmagnetism provided a deeper knowledge of the crystal\nsymmetries, extended the investigation on the magnetic\nproperties and discovered new effects on the electronic\nproperties1,4–7,33. The connection between these two\nfields emphasized in this paper paves the way to simplify\nthe physical scenario and merge the two fields giving the\nchance to connect the literature developed from two dif-\nferent perspectives which until now had a minor overlap.\nIn Table I, we report several altermagnetic compounds\nwith different space groups where the non-relativistic\nspin-splitting and the weak ferromagnetism were theoret-\nically predicted or experimentally detected. All the com-\npounds described in the seminal papers of Dzyaloshin-\nskii and Moryia are altermagnets present in this table19.Compound NO SOC Weak FM\n(No. space group) splitting from DMI\n(Sr,Ca) 3Ru2O7(36) - Exp.51\nCa3Mn2O7(36) Th.9Exp.52\nMn(N(CN) 2)2(58) Th.9Exp.53\nCa2RuO 4(61) Th.54-\nYVO 3(62) Th.54Exp.55\nLaMnO 3(62) Th.56Exp.57\nCaMnO 3(62) Th.9Exp.58\nYCrO 3(62) Th.9Exp.59\nCaCrO 3(62) Th.60Th.60\nBa2CoGe 2O7(113) Th.9Exp.61\nRuO 2(136) Th.6Th.*, Exp.47\nLa2NiO 4(138) Th.9Exp.62\nLiFeP 2O7(140) Th.9Exp.63\nKMnF 3(140) Th.9Exp.64\nMnTiO 3(161) Th.9Exp.65\nα-Fe2O3(167) Th.9Th.19, Exp.66\nMnCO 3(167) Th.9Th.19, Exp.67\nCoCO 3(167) Th.9Th.19, Exp.68\nCrF 3(167) Th.9Th.19, Exp.69\nCoF 3(167) Th.9Exp.70\nFeF 3(167) Th.9Exp.70\nFeBO 3(167) Th.9Exp.71\nNi1/3NbS 2(182) Th.72-\nNi1/3TaS 2(182) Th.72-\nMnSe (186) Th.43Th.*\nMnTe (194) Th.6, Exp.73Th.74, Exp.74\nCrSb (194) Th.6, Exp.42Th.*, Exp.75\n* Present work.\nTABLE I. List of selected altermagnetic compounds where\nthe non-relativistic spin-splitting appears with weak ferro-\nmagnetism induced by DMI. The words ”Th.” and ”Exp.”\nare reported for the theoretical and experimental verification\nof the non-relativistic spin splitting or weak ferromagnetism.\nThe space groups 36, 182 and 186 lack inversion symmetry.\nDue to the lifting of the Kramers degeneracy, all alter-\nmagnetic systems could potentially show weak ferromag-\nnetism, however, for particular N´ eel vector orientations\nthe DMI is zero as in the case of the RuO 2with N´ eel vec-\ntor along the z-axis. Once separating the centrosymmet-\nric and noncentrosymmetric compounds, we propose to\ndivide the centrosymmetric altermagnets into two classes\nas shown in Fig. 10. The g-wave CrSb falls into the first\ncategory which exhibits weak ferromagnetism for every\nN´ eel vector. In the second category, are some altermag-\nnets that do not display weak ferromagnetism for a given\nN´ eel vector, such as the d-wave CaCrO 360and RuO 2\nwhere we have shown that the N´ eel vector is oriented\nalong the z-axis. In the DMI picture, the symmetries\ndo not allow the presence of the DMI vector for partic-\nular N´ eel vector orientation. The compounds belonging\nto this second class are probably a limited number and\nseem to have higher symmetry with respect to the first10\nclass. As a consequence of this proposed classification,\nthe family of altermagnets is wider than the family of\nweak ferromagnets since there are altermagnets that do\nnot exhibit weak ferromagnetism. For the noncentrosym-\nmetric systems, we have found that weak FM or weak\nFiM are present depending on the N´ eel vector orienta-\ntion. To extend this classification, further studies are\nneeded especially for the noncentrosymmetric systems.\nIn this prospect, interesting compounds are the metal-\nlic d-wave Ca 3Ru2O7and the metallic g-wave Ni 1/3MS2\n(M=Nb, Ta). These results will be presented elsewhere.\nBeyond the crystal symmetries and the N´ eel vector\norientation, the only other requirement for weak FM or\nweak FiM is to have a non-compensated magnetization\nas happens in metallic systems or doped semiconduc-\ntors. As shown in Fig. 1, when the system is insulat-\ning no weak ferromagnetism is observable since there is\na spin compensation, therefore, a small doping is neces-\nsary as in the case of MnTe74and other semiconductive\naltermagnets. From the DOS of the spin-components,\nwe can understand that doping can change the Fermi\nlevel, strongly modify the weak ferromagnetism, and can\neven reverse the sign of the DMI. This explains the de-\npendence of the weak ferromagnetism from electron- or\nhole-doping concentration in several cases76,77. What\nwas defined as ”weak altermagnetism” is the band struc-\nture effect that gives rise to the DMI in centrosymmet-\nric altermagnets.73The electric field increases or acti-\nvates the non-relativistic spin-splitting depending on the\nfield orientation56,78, therefore, the electric field can in-\ncrease and manipulate the DMI in altermagnets79. We\ncan make another observation regarding centrosymmet-\nric altermagnetic systems that can host an active DMI.\nIn the case of magnetic systems with frustrated mag-\nnetic interactions, the presence of an active DMI in al-\ntermagnetic crystal structure can further push the sys-\ntem towards helical magnetic structures, spin spiral and\ncomplex magnetism as observed for instance in transi-\ntion metal monopicnitides80,81. The component of the\nweak ferromagnetism is proportional to the spin-orbit,\ntherefore, to look for a large canting angle we need to\nsearch among systems with a strong spin-orbit and a\nsmall magnetic moment. In the weak ferromagnets, an\nextremely relevant role is given by the magnetocrystalline\nanisotropy (MCA) that decides a key ingredient as the\ndirection of the N´ eel vector. Additionally, the size of the\nMCA determines how easy it is for the spin to rotate,\ntherefore a low MCA is fundamental to favor the weak\nferromagnetism. Even if there are altermagnets where\nthe weak ferromagnetism is zero in the bulk as the com-\npounds belonging to the second class, the weak ferromag-\nnetism can be obtained by breaking crystal symmetries\non the surfaces. Further studies are necessary for the\nevolution of the DMI and weak ferromagnetism on the\nsurfaces of the altermagnets.56,82\nIn conclusion, we propose that weak ferromagnetism\ninduced by DMI can appear in centrosymmetric and non-\ncentrosymmetric altermagnetic compounds but not in\nAltermagnetic system\nCentrosymmetric Non -centrosymmetric\nMnSe\nCCaCrO3CrSbRRuO2Weak \nferromagnetism \npresent for all \ndirectionsWeak \nferromagnetism  \npresent for suitable \ndirections of the \nNéel vector Weak ferromagnetism \nor weak \nferrimagnetism \ndepending on the N éel \nvectorFIG. 10. Altermagnets classified into centrosymmetric and\nnoncentrosymmetric. We further classify the centrosymmet-\nric systems into two classes; the first contains compounds in\nwhich the weak FM is always present, and the second contains\ncompounds in which the weak FM is present for suitable direc-\ntions of the N´ eel vector. For the noncentrosymmetric MnSe,\nwe have weak ferromagnetism with the N´ eel vector in the\nxy plane and weak ferrimagnetism with the N´ eel vector out-\nof-plane. Likely, this classification would be improved and\nextended in coming years.\ncentrosymmetric and noncentrosymmetric conventional\nantiferromagnets. Weak FiM can appear in noncen-\ntrosymmetric altermagnets. This weak ferromagnetism\noriginates exclusively from the spin-orbit coupling in al-\ntermagnets and this picture is equivalent to the DMI pic-\nture. The weak ferromagnetism and the AHE are always\northogonal to the N´ eel vector for centrosymmetric sys-\ntems. The weak ferromagnetism depends on the filling,\ncan be tuned by an electric field and can change its sign\nby rotating the N´ eel vector in hexagonal systems. From\nour results, the weak ferromagnetism is absent for insula-\ntors and in lightly doped semiconductive systems would\nbe proportional to the carriers. We provide a recipe to\nestablish the presence of weak ferromagnetism and the\ncomponents of the Hall vector from a simple relativistic\ndensity of state and band structure calculations without\ncalculating the DMI vector. There is a correlation be-\ntween the rise of weak FM and weak FiM with the lifting\nof the electronic degeneracy at the Γ point. We have the\ncentrosymmetric altermagnetic and the noncentrosym-\nmetric altermagnets. We further divided the centrosym-\nmetric altermagnets into two classes, the first class always\nhosts weak ferromagnetism for every direction of the Hall\nvector while the other class hosts weak ferromagnetism\nfor suitable directions of the N´ eel vector. For noncen-\ntrosymmetric systems, we have weak ferromagnetism in\nthe x-y plane while parallel and antiparallel spin compo-\nnents for different bands can coexist at the same energy\nresulting in weak ferrimagnetism and a component of the\nHall vector along the N´ eel vector. Therefore, an exter-11\nnal electric field breaking the inversion symmetry could\ngenerate a weak ferromagnetic component and an AHE\nalong the N´ eel vector. We have predicted weak ferro-\nmagnetism in CrSb and weak ferrimagnetism in wurztite\nMnSe. The size of weak ferromagnetism, weak ferrimag-\nnetism and AHE strongly depends on the spin-orbit cou-\npling, therefore, further research directions should focus\non heavy elements. In summary, the family of the alter-\nmagnets is mainly composed of weak ferromagnets with\na minor number of compounds that are weak ferrimag-\nnets (MnSe with N´ eel vector along the z-axis) and zero\nmagnetization compounds (RuO 2with N´ eel vector along\nthe z-axis). It was shown that altermagnetism displays\nproperties that either belong to ferromagnets or conven-\ntional antiferromagnets1, the weak FM and weak FiM\ninduced by DMI are properties belonging exclusively to\naltermagnets not present neither in ferromagnets or in\nconventional antiferromagnets. If we want to classify al-\ntermagnets based on their macroscopic physical proper-\nties, we can define altermagnets as magnetic systems with\nspin-up and spin-down sublattices connected by symme-\ntry where the DMI allows the rise of weak ferromagnetism\nfor a suitable N´ eel vector orientation.ACKNOWLEDGMENTS\nC. A., R. M. S. and G. C. contributed equally to\nthis work. We acknowledge M. Cuoco, K. V´ yborn´ y,\nM. Gryglas-Borysiewicz and M. Grzybowski for useful\ndiscussions. The work is supported by the Founda-\ntion for Polish Science through the International Re-\nsearch Agendas program co-financed by the European\nUnion within the Smart Growth Operational Programme\n(Grant No. MAB/2017/1). A.F. was supported by\nthe Polish National Science Centre under project no.\n2020/37/B/ST5/02299. We acknowledge the access to\nthe computing facilities of the Interdisciplinary Center\nof Modeling at the University of Warsaw, Grant g91-\n1418, g91-1419 and g91-1426 for the availability of high-\nperformance computing resources and support. We ac-\nknowledge the CINECA award under the ISCRA initia-\ntive IsC99 ”SILENTS”, IsC105 ”SILENTSG” and IsB26\n”SHINY” grants for the availability of high-performance\ncomputing resources and support. 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Ekiz2\n1Institut f¨ ur Theoretische Physik, RWTH Aachen, 52056 Aachen, Germany\n2Department of Physics, Adnan Menderes University, 09010 Aydin, Turkey\nAbstract. We study Ising ferrimagnets on square lattices with antiferromagn etic\nexchange couplings between spins of values S=1/2 and S=1 on neighb ouring sites,\ncouplings between S=1 spins at next–nearest neighbour sites of th e lattice, and\na single–site anisotropy term for the S=1 spins. Using mainly ground s tate\nconsiderationsandextensiveMonteCarlosimulations, weinvestigat evariousaspectsof\nthephasediagram,includingcompensationpoints, criticalpropert ies,andtemperature\ndependent anomalies. In contrast to previous belief, the next–ne arest neighbour\ncouplings, when being of antiferromagnetic type, may lead to compe nsation points.\nPACS numbers: 75.10.-b, 75.10.Hk, 75.40.Mg, 75.50.Gg\nSubmitted to: Institute of Physics Publishing\nJ. Phys.: Condens. Matter\n1. Introduction\nMixed spin ferrimagnetic Ising models have been studied for some time , with a renewed\ninterest, especially, in connection with ’compensation points’, see, f or instance, [1-\n11]. These points occur at temperatures, below the critical tempe rature, at which the\nsublattice magnetizations cancel exactly, giving zero total momen t. As the temperature\nis tuned through such a point the total magnetization changes sign , which may be used\nin technological applications, most notably in magnetic recording.\nOne of the simplest such models consists of classical Ising spins S=1/ 2 and S=1\non a square or simple cubic lattice with the spins of different type being located on\nneighbouring sites of the lattice. Its Hamiltonian may be written in the form\nH=J1/summationdisplay\n/angbracketlefti,j/angbracketrightσiSj+D/summationdisplay\nj∈BSj2(1)\nwhereJ1>0 denotes the antiferromagnetic coupling between spins σi=±1/2 on\nthe sites of sublattice ’A’, and neighbouring spins Sj= 1,0,−1 on sites forming the\nsublattice ’B’. Then, spins on each sublattice tend to order ferroma gnetically, withMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 2\nopposite sign for the two types of spins. Dis the strength of a single–site anisotropy\n(or crystal–field) term acting only on the S=1 spins of sublattice B.\nOne may also choose σi=±1 rather than ±1/2. The change has to be taken\ninto account when calculating sublattice magnetizations and defining the compensation\npoint. Otherwise, themodifiedconvention simplyamountstoaresca lingoftheexchange\ncoupling [5, 10].\nThe mixed spin Ising model, eq. (1), is known, see, e.g., [3, 5, 10], to le ad to\ncompensation points for simple cubic lattices, in accordance with mea n–field theory [2],\nbut not for square lattices, in contrast to mean–field theory. As h as been observed\nalready some years ago by Buendia and Novotny [5, 12], compensat ion points may\noccur on square lattices, when adding a (ferromagnetic) coupling b etween next–nearest\nneighbouring (nnn) spins on the A sublattice. On the other hand, th e authors did not\nfind any evidence for compensation points, when considering nnn co uplings for B spins,\ninstead of the ones for A spins.\nIn the following article we shall challenge the latter suggestion, which seems to have\nbeen taken for granted by others, see, e.g., [7]. Indeed, based on extensive Monte Carlo\n(MC) simulations, we shall present clear evidence for compensation points due to nnn\nantiferromagnetic interactions between B, or S=1, spins on the sq uare lattice. In fact,\nthe effect is not easy to identify.\nThe outline ofthe article is as follows. Insection 2, to set the scene f or the following\nparts, we define the model with nnn couplings between S=1 spins, dis cuss its ground\nstates, and describe details of the simulations. The following section deals with the\ncompensation points. In section 4, MC results on critical propertie s and temperature\ndependent anomalies of the model will be presented. A brief summar y is given in the\nfinal section.\n2. Model, ground states, and simulations\nWe shall study the following mixed spin Ising model on a square lattice\nH=J1/summationdisplay\n/angbracketlefti,j/angbracketrightσiSj+D/summationdisplay\nj∈BSj2−J2/summationdisplay\n/angbracketlefti,j/angbracketright∈BSiSj (2)\nwith antiferromagnetic nearest neighbour couplings, J1>0 between A spins, σi=±1,\nand B spins, Sj= 0,±1, a crystal–field term of strength Dacting on the B spins, and\nnnn couplings, J2, between B spins. Note that we set A spins equal to ±1, in accordance\nwithprevious work [5, 10]. To identify possible compensation points, c are isthenneeded\nin defining the magnetization of the sublattice A, see above and below .\nTo determine the ground states of the model in the ( D/J1,J2/J1) plane, we first\nselect, like before [5, 12, 13], the structures which are stable amon g the ones described\nby 2×2 cells of spins on the square lattice. One readily observes the possib ility of\ndegenerate ground states, eventually with indefinitely large unit ce lls. Finally, we check\nthe tentative ground state structures by monitoring spin configu rations at very lowMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 3\ntemperatures obtained from careful cooling MC runs for lattices o f various sizes. The\nresulting ground state phase diagram is depicted in figure 1.\nThephasediagramatvanishingtemperature, T= 0,comprisesfourstructures. The\nantiferromagnetic(AF) structure isstableforferromagneticor weakly antiferromagnetic\nnnn couplings, J2, and negative or sufficiently small positive values of the crystal field ,\nD. The spins on each of the two sublattices order ferromagnetically, ±1, with opposite\nsign on the sublattices A and B.\nIn two of the remaining other three ground states, S=1 spins may b e in the state 0,\ndue to the single–site anisotropy term, D. Actually, for sufficiently large values of D, all\nspins of the sublattice B are in state 0. We abbreviate the resulting s tructure as ’0II’.\nThe structure is highly degenerate, with each A spin being either −1 or 1, leading to a\n2NA–fold degeneracy for NAsites on the sublattice A. In contrast, in the ’0I’ structure,\nsee figure 1, only half of the S=1 spins are in the state 0. They form, on the sublattice\nB, in the standard Wood’s notation for overlayers [14], a periodic c( 2×2) superlattice.\nThe other half of the spins on the B sites are ferromagnetically orde red,±1. The spins\non the sublattice A are ferromagnetically ordered as well, with oppos ite sign, compared\nto that of the B spins.\n-2-1012 3 4 56\nD/J1-3-2.5-2-1.5-1-0.500.51J2/J1AF\n0I 0II\nB-AF\nFigure 1. Ground state phase diagram of the mixed spin model, eq. (2), on a sq uare\nlattice, with solid lines separating the different phases. The dashed lin e in the 0I phase\nrefers to equation (7), see text.\nIn the fourth ground state structure, all spins on the sublattice B are\nantiferromagnetically aligned, due to the dominant antiferromagne tic nnn coupling, J 2.\nThe structure may beabbreviated asB–AF. Eachspin onthesublat tice Amay beeither\n−1 or 1, leading, as for 0II, to a large degeneracy.\nNote that there may be additional degeneracies at borderlines bet ween different\nground state structures [10].\nTo determine thermal properties of the model, (2), we mainly perfo rm MC\nsimulations, using the Metropolisalgorithmwithsingle–spin flips, provid ing, indeed, theMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 4\nrequired accuracy, so thatthereisno needtoapplyother techniq ues like cluster–updates\nor the Wang–Landau approach [15]. We consider lattices with LxLsites, employing full\nperiodic boundary conditions. Lranges from 4 to 120, to study finite–size effects.\nTypically, we do runs of 5 ×106to 107Monte Carlo steps per spin, where averages and\nerror bars may be obtained from evaluating a few of such runs, usin g different random\nnumbers. These quite long runs lead to reliable data, as before [10]. T he estimated\nerrors are usually smaller than the sizes of the symbols in the figures , and they are\nshown there only in a few cases.\nWe record the energy per site, E, the specific heat, C, both from the energy\nfluctuationsandfromdifferentiating Ewithrespecttothetemperature, andtheabsolute\nvalues of the sublattice magnetizations of the two sublattices\n|mA|=<|/summationdisplay\nAσi|> /(2(L2/2)) (3)\nand\n|mB|=<|/summationdisplay\nBSj|> /(L2/2) (4)\nas well as the absolute value of the staggered magnetization of sub lattice B, describing\nthe ordering for the B–AF structure,\n|mst\nB|=<|/summationdisplay\nB+Si−/summationdisplay\nB−Sj|> /(L2/2) (5)\nwith B+,−denoting an obvious bipartition of sublattice B. The brackets <>denote\nthe thermal average. Note the factor of 1/2 in the definition of |mA|, taking into\naccount the correct length of the S=1/2 spins, so that |mA(T= 0)|= 1/2 for the\nferromagnetic ground state of sublattice A. In addition, the corr esponding sublattice\n(staggered) susceptibilities, χA,χB, andχst\nB, have been computed from the fluctuations\nofthe(staggered)sublattice magnetizations. Wealsoanalyse the fourth–ordercumulant\nof various (sublattice) order parameters, the Binder cumulant [16 ], defined by\nU= 1−< m4> /(3< m2>2) (6)\nwith< m2>and< m4>being the second and fourth moment of (staggered)\nmagnetizationsofsublatticeAorB.Finally, wemonitortypicalequilibr iumMonteCarlo\nconfigurations, illustrating the microscopic behaviour of the syste m and providing, e.g.,\nevidence for the ground states structures, as mentioned above .\nTo test the accuracy of the simulations, we compared our MC data t o those of\nprevious accurate numerical work and to exact results by enumer ating all possible\nconfigurations for small lattices with L= 4 [5, 10].Mixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 5\n3. Compensation points\nCompensationpointsoccurattemperatures Tcomp,wherebothsublatticemagnetizations\ncancel each other, with a vanishing total magnetization. In finite s ystems, as one is\nstudying in MC simulations, a convenient and efficient way [5] to locate such points is to\nuse the crossing condition |mA|(Tcomp) =|mB|(Tcomp). At low temperatures sufficiently\nfar below the phase transition, finite–size effects are expected to be very weak, because\nthere the compensation effect is not related to critical phenomena .\n0 0.2 0.4 0.60.81 1.2 1.4 1.6\nkBT/J10.10.20.30.40.50.60.70.80.91|mA|, |mB|\nFigure 2. Sublattice magnetizations |mA|(open symbols, dotted lines) and |mB|(full\nsymbols,solidlines)atfixed D/J= 3.0andvarying J2/J1=0.25(circles), 0.0(triangles\ndown),−0.1 (squares), −0.24 (diamonds), and −0.26 (triangles left). Lattices of size\n602are simulated.\nIn the AF ground state, one has |mB|(T= 0)=1, while |mA|(T= 0)= 1/2. Now,\nat non–zero temperatures, |mB|(T) may fall off quite rapidly due to antiferromagnetic\ncouplings J2and due to a relatively large single–site anisotropy term, D, which favours\nflips of S=1 spins to state 0. Furthermore, at zero temperature, |mB|(T= 0) drops from\n1 to 1/2, when passing through the borderline between the AF and 0 I ground states.\nAccordingly, one might speculate that compensation points, at low t emperatures in the\nAF phase, may show up close to the AF–0I borderline at T=0. In fact, our preliminary\nmean–field calculations seem to suggest that lines of compensation p oints may spring\nfrom this AF–0I borderline.\nHowever, in our MC simulations, investigating carefully several case s, we find no\nevidence forsuch compensationpointsintheAFphase. Anexample is depicted infigure\n2. There, D/J1is fixed at 3.0, and J2/J1is varied, from +0.25 to −0.26, so that the\nAF–0I border at J2/J1=−0.25 is approached and crossed. One observes, that |mB|is\nalways larger than |mA|, albeit the difference between the two sublattice magnetizations\nmay get quite small when decreasing J2. MC data for lattices of fixed size, L= 60, areMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 6\ndisplayed. The observation on the absence of compensation points in the AF phase also\nholds for other lattice sizes as well as for other values of J2/J1andD/J1.\n0.10.2 0.3 0.4 0.50.60.7 0.8 0.9 1\nkBT/J10.10.20.30.40.50.60.70.80.91|mA|, |mB|\nTcomp\nFigure 3. Sublattice magnetizations |mA|(open symbols, dotted lines) and |mB|\n(full symbols, solid lines) at fixed J2/J1=−0.2, and varying D/J1= 3.0 (circles), 3.4\n(squares), 3.57 (diamonds), and 3.7 (triangles up), simulating lattic es of size L= 60.\nAtD/J1=3.57, a compensation point shows up, Tcomp.\nOn the other hand, compensation points will be argued below to occu r in the 0I\nphase, for J2/J1>−1.0, arising at vanishing temperature from the line\nJ2/J1=−2+D/(2J1) (7)\nwhich is depicted as the dashed line in the ground state phase diagram , figure 1.\nLet us first present numerical support for this claim, followed then by low\ntemperature energy considerations backing it up. A numerical exa mple is shown in\nfigure 3, where, for lattices with 602sites, the temperature dependences of the sublattice\nmagnetizations, |mA|and|mB|, are shown at fixed nnn coupling, J2/J1=−0.2, and\nvarying the strength of the single-site anisotropy term, D/J1, from 3.0 to 3.7. Note\nthat for this ratio of couplings, at T= 0, the AF–0I border is at 3.2, and the origin,\nat vanishing temperature, of the line of compensation points would b e, according to\nequation (7), at D/J= 3.6. As discussed in the context of figure 2, one observes\nthat|mB|(T) is always larger than |mA|(T) when being in the AF phase. However,\nin the 0I phase, here, at D/J1= 3.57, there seems to be a compensation point, with\nkBTcomp/J1≈0.26, clearly below the phase transition: At lower temperatures, |mB|is\nstill larger than |mA|, with reverse ordering at T > T comp. By further increase of D,\nD/J1= 3.7,|mA|(T) is larger than |mB|(T), at all temperatures up to Tc.\nTo establish numerically the presence of compensation points, care ful finite–size\nanalyses are required, as illustrated in figure 4. Here, MC data for v arious lattices sizes,\nwithLranging from 20 to 120, aredisplayed, at J2/J1=−0.2 andD/J1= 3.59, i.e. veryMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 7\n0.05 0.1 0.15 0.2 0.25 0.3\nkBT/J10.360.380.40.420.440.460.480.5|mA|, |mB|\nTcomp\nFigure 4. Sublattice magnetizations |mA|(open symbols, dotted lines) and |mB|\n(full symbols, solid lines) at J2/J1=−0.2 andD/J1= 3.59, with lattice sizes L= 20\n(squares), 40 (diamonds), 60 (triangles left), 80 (triangles down ), and 120 (circles).\nThe compensation point is located at kBT/J1≈0.16.\nclose to the point, from which, according to (7), the line of compens ation points may\narise. In fact, a compensation point may be located at kBTcomp/J1≈0.16. Below that\ntemperature, |mB|(T)supercedes |mA|(T), withfinite–size dependences being extremely\nsmall. Then, at Tcomp, a crossing of the two sublattice magnetizations occurs. The finite\nsize effects are still very weak up to kBT/J1≈0.25, providing clear evidence on the\nexistence of the compensation point in the thermodynamic limit.\nIndeed, similarfinite-sizeanalysesallowonetolocatetheratherste eplyrisinglineof\ncompensationpoints, atfixednnncouplings J2/J1=−0.2. Results onthecompensation\npoints,Tcomp, are summarized in figure 5, together with estimates for the trans ition line,\nTc, to the paramagnetic phase.\nThe critical line, Tc, has been determined by monitoring the size dependent\npositions of maxima in the specific heat C, in the susceptibility of the sublattice A,\nand in the intersection points of the Binder cumulant, especially, of s ublattice A. The\nintersections of the Binder cumulant U(L,T) for successive lattice sizes [16] turn out\nto have the smallest finite–size effects, thereby being most efficient in estimating the\ncritical temperature.\nThe line of compensation points seems to start at zero temperatur e atD/J1= 3.6\nforJ2/J1=−0.2, in agreement with equation (7). Furthermore, it extends only ov er\nquite a small region, with the compensation point coinciding with the cr itical point at\naboutD/J1= 3.52±0.02.\nLet us now turn to the low temperature considerations leading to (7 ). Specifically,\nwe consider, for the 0I structure, the one–spin flip excitations on sublattice B. For\nconcreteness, we analyse ground states with spins being in state −1 or 0 on sublatticeMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 8\n33.13.2 3.3 3.4 3.53.63.7 3.8 3.9 4\nD/J100.10.20.30.40.50.60.70.80.911.1kBT/J1\nAF 0ITc\nTcompparamagnetic\nFigure 5. Phase boundary to the paramagnetic phase, Tc(solid line), and\ncompensation points, Tcomp(dashed line), at J2/J1=−0.2, varying D/J1.\n22.12.2 2.3 2.4 2.52.62.7\nD/J100.10.20.30.40.50.60.70.80.9kBT/J1\n0Iparamagnetic\nTcompTc\nFigure 6. Phase boundary to the paramagnetic phase, Tc(solid line), and\ncompensation points, Tcomp(dashed line), at J2/J1=−0.8, varying D/J1.\nB, and in state 1 on sublattice A. Then there are four possible flips of S=1 spins,\nnamely flipping a spin from its state 0 to the state (i) −1 or (ii) 1, or flipping a spin\nfrom its state −1 to state (iii) 0 or (iv) +1. Obviously, only flip (i) will increase the\nsublattice magnetization |mB|above its value at zero temperature, |mB|(T= 0) = 1/2,\nwhile otherwise |mB|(T) will have a negative slope at low temperatures. Now, one\nmay readily calculate the energies of all four elementary flips. One ob tains that flip\n(i) costs the lowest energy either followed by flip (iii), when D/J1<2J2/J1+ 4,\nor followed by flip (ii), when J2/J1>−1.0. Equation (7) is then obtained underMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 9\nthe assumption, that an initial increase of |mB(T)|is needed to have a compensation\npoint, with |mB|>|mA|at sufficiently low temperatures. Otherwise, in the 0I phase,\n|mA|>|mB|at all temperatures 0 < T < T c, excluding a compensation point.\nWe further checked numerically the hypothesis, (7), in determining lines of\ncompensation points by fixing the nnn coupling J2/J1not only at −0.2, but also at\n−0.4 and−0.8, varying the strength of the single–site anisotropy term, D/J1. Indeed,\nwe observe compensation points in the 0I structure, which seem to arise, at zero\ntemperature, from D/J1= 3.2 at J2/J1=−0.4 and from D/J1= 2.4 at J2/J1=−0.8,\nin agreement with (7). The latter case is displayed in figure 6.\nNote that the range of values of D/J1, in which there is a line of compensation\npoints, shrinks appreciably as one decreases J2/J1. This may be seen by comparing\nfigures 5 and 6. Thence, it becomes increasingly difficult to locate com pensation points\nfor lower values of J2/J1.\n4. Phase transitions and anomalies\nWhile mixed spin Ising models are of much interest because of the pote ntial presence of\ncompensations points, they may exhibit other intriguing thermal pr operties as well.\nWe shall focus on two aspects, the characterization of the phase transitions\nassociatedwiththevariousgroundstates, andanomaliesinthetem peraturedependence,\nespecially, of the sublattice magnetization and the specifc heat C.\nOne expects that there is no phase transition associated with the h ighly degenerate\n0II ground state for the following reasons: At zero temperature , all spins of sublattice\nB are in state 0, while each spin of sublattice A may be either −1 or 1. The single–site\nanisotropy term, favouring the spin state 0 on sublattice B, does n ot support long–range\norder, in close analogy to the situation in the Blume–Capel model [17]. The spins on\nsublattice A act merely like a fluctuating random field, and may not lead to a phase\ntransition neither. In fact, our MC data, for various quantities, lik e the specific heat C\nor the probability to encounter a spin in state 0, give no indication for singular thermal\nbehaviour.\nIn case of the other three ground states, AF, 0I, and B–AF, we d etermine the\nuniversality class [18, 19] by analysing specific heat and susceptibilitie s. In particular,\nwemonitorthesize, L,dependenceofthatmaximumofthespecificheat, Cmax(L), which\ngoesover into a singularity inthe thermodynamic limit, L→ ∞(note that there may be\nother maxima as will be discussed below in the context of anomalies). F urthermore, we\nstudy the size dependence of the maxima in (sublattice) susceptibilit ies. For the AF and\n0I structures, we focus on χA,max(L). For the B–AF structure, we record the staggered\nsusceptibility of the antiferromagnetically ordered sublattice B, χst\nB,max(L). In all three\ncases, the critical behaviour is found to be consistent with having p hase transitions in\nthe universality class of the standard two–dimensional Ising model.\nTypical MC results on the size, L, dependence of maxima in the specific heat,\nCmax, are depicted in figure 7. For all three types of ground states, AF , 0I as wellMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 10\n2.2 2.4 2.62.8 3 3.2 3.4 3.63.8 4 4.2 4.4\nln(L)0.80.911.11.21.31.41.51.61.71.81.92Cmax\nFigure 7. Height of the critical maximum of the specific heat, Cmaxversus logarithm\nof the lattice size, ln L, forL= 10, 20, 40, 60, and 80 at (a) (circles) J2/J1=−0.3,\nD/J1=1.5,(b)(squares) J2/J1=−1.0,D/J1=1.5,and(c)(diamonds) J2/J1=−1.5,\nD/J1= 2.5. The three cases refer to the (a) AF, (b) 0I, and (c) B–AF st ructures.\nas B–AF, one approaches to a good degree, already for lattices of moderate sizes, the\nformCmax∝lnL, being characteristic for the two–dimensional Ising universality cla ss.\nNote that the prefactor in front of the logarithmic term depends s trongly on J2/J1and\nD/J1. Certainly, such Ising–like critical behaviour may be expected in the AF case. In\ncase of the B-AF phase, the spins on sublattice B order antiferrom agnetically, with the\nA spins providing effectively a randomly fluctuating field, which may be a rgued to be\nirrelevant for the universality class. Finally, in the 0I case, the Ising –like criticality may\nbe understood by the ferromagnetic order of the spins on sublatt ice A.\nThe analysis of the susceptibilies confirms the findings on the specific heat. Typical\nexamples, for the same model parameters as in figure 7, are shown in figure 8. The size\ndependent maxima in the susceptibilities are observed to follow the fo rmχmax∝L7/4,\nwith rather small corrections, for all three cases, AF, 0I, and B– AF. This behaviour\nmay be quantified by calculating the slope between successive points in the doubly\nlogarithmic plot of the MC data. The resulting effective local critical e xponent of the\nsusceptibility is near 7/4, even for moderate lattice sizes. Thence, in all three cases,\ncriticality seems to belong to the two–dimensional Ising universality c lass.\nCare is needed when attempting to determine the universality class f rom the value\nof the critical Binder cumulant U∗=U(Tc,L−→ ∞), because that value is known to\ndepend on boundary conditions, shape, and anisotropy of the cor relations [16, 20, 21].\nNotethatinthecases westudied, itsvalueseems tobeclose tothat ofthestandardtwo–\ndimensional Ising model with periodic boundary conditions for lattice s of square shape,\nU∗≈0.6107 [22]. However, the dependence, especially on anisotropy, may be very weak\n[20, 21], and a reliable analysis may require an exact or, at least, extr emely accurateMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 11\n2.2 2.4 2.62.8 3 3.2 3.4 3.63.8 4 4.2 4.4\nln(L)0123456ln(χA(B),max(st))\nFigure 8. Log–log plot for maxima of (staggered) sublattice susceptibilities χmax\nversus lattice size Lfor the AF, 0I, and B–AF cases, with the same parameters and\nnotation as in figure 7. In case of AF and 0I structures, χA,max, in case of the B–AF\nstructure, χst\nB,maxis recorded. For comparison, the solid line shows χmax∝L7/4.\ndetermination of the critical point. We therefore refrained from s uch an analysis here.\n0 0.2 0.4 0.60.81 1.2 1.4 1.6 1.8 2 2.2\nkBT/J100.10.20.30.40.5 C\nFigure 9. Specific heat Cas a function of temperature, kBT/J1, atJ2/J1=−0.2 and\nD/J= 3.4, simulating lattices of sizes L= 20 (circles), 40 (squares), 60 (diamonds),\nand 80 (triangles left).\nLet us now turn to discussing anomalies, exhibiting intriguing non–mon otonic\ntemperature dependences, which are not related to phase trans itions. Especially, we\nmonitor the magnetizations of sublattice B, |mB|, the specific heat C, and the thermallyMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 12\naveraged occupancy of B sites with spins in the state 0, pB(0).\nAn example for such anomalies is the overshooting of |mB|(T) at low temperatures\nin the vicinity of the 0I ground state, illustrated in figures 2 and 3. As discussed above,\nthe overshooting is related to decreasing the number of spins in sta te 0, by flipping such\nspins to, say, −1, costing minimal energy. Indeed, the anomaly in |mB|is monitored to\nbe accompanied by a pronounced decrease in pB(0).\nAnomalies may also occur in the specific heat, being caused either by a rapid\nincrease or decrease in the number of S=1 spins in state 0. For insta nce, such an\nanomaly in Chas been observed before [10] in the AF phase at J2/J1= 0, close to the\n0II structure. There, monitoring C(T), a three–peak–structure of the specific heat has\nbeen found, with a non-critical maximum at low temperatures, due t o easy flips of spins\nin the state 0, followed, at higher temperature, by a strongly size d ependent critical\npeak, and, at even larger temperature, another non–critical ma ximum, due to flipping\nof single spins for sublattices with rather large clusters of ’+’ or ’ −’ spins.\nWe observe a similar thermal behaviour of the specific heat in the 0I p hase, when\nturningonthennncoupling, J2<0. Anexample isdepicted infigure9, at J2/J1=−0.2\nandD/J1= 3.4. However, here the peak at lowest temperature, Tl, is due to a rather\ndrastic decrease of B spins in state 0, which has been discussed abo ve. Actually, by\nenhancing the strength of the single–site anisotropy term, D/J1= 3.6, the position of\nthat peak, at Tl, shifts first to somewhat higher temperatures. It still correspo nds to a\nlowering, with temperature, of the average number of 0’s for the s ublattice B, pB(0)(T).\nWhen moving even closer to the 0IIstructure, D/J1= 3.8 and 3.9, Tltends to be shifted\ntowards lower temperatures with increasing D. The peak position seems to follow the\nsame dependence as for vanishing J2[10], namely kBTl/J1∝4−D/J1. As may be seen\nfrom monitoring pB(0), the peak is then, as in the limit J2= 0, due to an increase in\nthe number of B spins in state 0 at low temperatures.\n5. Summary\nWehave studied a mixed spin Ising model with antiferromagneticcoup lings,J1, between\nspinsS=1/2andS=1onneighbouring sitesofasquarelattice, augme ntedbycouplings,\nJ2, between spins S= 1 on next–nearest neighbouring sites of the latt ice. An additional\nquadratic single–site anisotropy term, D, acts upon the S=1 spins. Based mainly on\ngroundstateconsiderationsandonextensive standardMonteCa rlosimulations, wehave\ndetermined the ground state phase diagram in the ( D/J1,J2/J1) plane and identified\ncompensation points, types of phase transitions corresponding t o different ground state\nstructures as well as anomalies for various physical quantities.\nIn particular, compensation points are found to exist, in contrast to previous belief.\nThey exist for antiferromagnetic nnn couplings, J2>−1.0, in the 0Iphase, springing, at\nzero temperature, from the line J2/J1=−2+D/(2J1). Across that line, different types\nof one–spin flips on the S=1 sublattice cost lowest energy, accompa nied by a change in\nthesignoftheslopeofthemagnetizationofthatsublatticeatlowte mperatures. AtfixedMixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 13\nvalue ofJ2/J1, the range, in D/J1, of compensation points is quite narrow, becoming\nextremely small for decreasing nnn couplings, as we have shown by d ecreasing J2/J1\nfrom−0.2 to−0.8.\nThe transition to the paramagnetic phase appears to be always in th e universality\nclass of the two–dimensional Ising model, with the critical maxima of t he specific\nheat growing with lattice size, L, in a logarithmic fashion, and with the maxima\nof the appropriate (staggered) sublattice susceptibilities growing with lattice size\nproportionally to L7/4. These results hold in case of AF, 0I, and B–AF ground states of\nthe model. In case of the 0II ground state, there is no phase tran sition.\nThe model is observed to exhibit interesting anomalies by showing non –monotonic\ntemperature dependences for various quantities, including sublat tice magnetization and\nspecific heat. The anomalies are typically caused by energetically fav oured flips of S=\n1 spins between the state 0 and a non–zero state.\nIn conclusion, our study provides clear evidence for non–expecte d compensation\npoints in a rather simple mixed Ising model with antiferromagnetic nex t–nearest\nneighbour couplings between S=1 spins on a square lattice. Extensio ns to three\ndimensional lattices are well beyond the present study, and may be investigated in\nthe future.\n6. Acknowledgements\nC E thanks the Department of Physics at the RWTH Aachen for the k ind hospitality\nduring his stay there. We thank Prof. Mark Novotny for a very use ful discusssion as\nwell as Dr. Mukul Laad for interesting conversations.\nReferences\n[1] N´ eel L 1948 Ann. Phys. (Paris) 3137\n[2] Kaneyoshi T and Chen J C 1991 J. Magn. Magn. Mater. 98201\n[3] Zhang G-M and Yang C-Z 1993 Phys. Rev. B489452\n[4] Bob´ ak A and Jascur M 1995 Phys. Rev. B5111533\n[5] Buendia G M and Novotny M 1997 J. Phys.: Condens. Matter 95951\n[6] Oitmaa J and Zheng W-H 2003 PhysicaA328185\n[7] Godoy M, Leite V S, and Figueiredo W 2004 Phys. Rev. B69054428\n[8] Jascur M and Strecka J 2005 Condens. Matter Phys. 8869\n[9] Oitmaa J and Enting I G 2006 J. Phys.: Condens. Matter 1810931\n[10] Selke W and Oitmaa J 2010 J. Phys.: Condens. Matter 22076004\n[11] Zukovic M and Bob´ ak A 2010 PhysicaA3895402\n[12] Buendia G M and Novotny M A 1997 J. Phys. IV (C1) France 7175\n[13] Buendia G M and Liendo J A 1997 J. Phys.: Condens. Matter 95439\n[14] Oura K, Lifshits V G, Saranin A A, Zotov A V and Katayama M 2003 Surface Science: An\nIntroduction (Berlin: Springer)\n[15] Landau D P and Binder K 2005 A Guide to Monte Carlo Simulations in Statistical Physics\n(Cambridge: Cambridge University Press)Mixed Ising ferrimagnets with next-nearest neighbour coup lings on square lattices 14\n[16] Binder K 1981 Z. Physik B43119\n[17] Blume M 1966 Phys. Rev. 141517; Capel H W 1966 Physica (Utr.) 32966\n[18] Fisher M E 1974 Rev. Mod. Phys. 46597\n[19] Pelissetto A and Vicari E 2002 Phys. Rep. 368549\n[20] Chen X S and Dohm V 2004 Phys. Rev. E70056136; Dohm V 2008 Phys. Rev. E77061128\n[21] Selke W and Shchur L N 2005 J. Phys. A: Math. Gen. 38L739; Selke W 2006 Eur. Phys. J. B\n51223\n[22] Kamieniarz G and Bl¨ ote H W J 1993 J. Phys. A: Math. Gen 26201"    },    {        "title": "1702.03609v4.Lieb_and_hole_doped_ferrimagnetism__spiral__resonating_valence_bond_states__and_phase_separation_in_large_U__AB__2___Hubbard_chains.pdf",        "content": "Lieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separation\nin large-U AB 2Hubbard chains\nV . M. Martinez Alvarez1and M. D. Coutinho-Filho1\n1Departamento de F ´ısica, Laborat ´orio de F ´ısica Te ´orica e Computacional,\nUniversidade Federal de Pernambuco, Recife 50670-901, Pernambuco, Brazil\nThe ground state (GS) properties of the quasi-one-dimensional AB2Hubbard model are investigated taking\nthe effects of charge and spin quantum fluctuations on equal footing. In the strong-coupling regime, we derive\na low-energy Lagrangian suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-\ndoped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice\nmagnetizations, and Lieb total spin per unit cell of the effective quantum Heisenberg model, in very good\nagreement with previous results. In the challenging hole doping regime away from half filling, we derive the\ncorresponding t-JHamiltonian. Under the assumption that charge and spin quantum correlations are decoupled,\nthe evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially\nmodulated spin structures and the emergence of phase separation in the presence of resonating-valence-bond\nstates. We also calculate the doping-dependent GS energy and total spin per unit cell, in which case it is shown\nthat the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in\nthe doped regime are in very good agreement with density matrix renormalization group studies, where our\nassumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in\nanti-phase with the modulation of the magnetic structure.\nI. INTRODUCTION\nMuch attention has been given to quantum phase transi-\ntions [1, 2], which are phenomena characterized by the change\nof the nature of the ground state (GS) driven by a non-thermal\nparameter: pressure, magnetic field, doping, Coulomb repul-\nsion, or competitive interactions. In this context, the study\nof quasi-one-dimensional (quasi-1D) compounds with ferri-\nmagnetic properties [3, 4] has attracted considerable theoreti-\ncal and experimental interest because of their unique physical\nproperties and very rich phase diagrams. In particular, the GS\nof quasi-1D quantum ferrimagnets with AB2orABB0unit\ncell topologies (diamond or trimer chains) described by the\nHeisenberg or Hubbard models [5] exhibit unsaturated spon-\ntaneous magnetization, ferromagnetic and antiferromagnetic\nspin-wave modes, effect of quantum fluctuations, and field-\ndependent magnetization plateaus, among several other fea-\ntures of interest.\nOf special interest is the topological origin of GS mag-\nnetic long-range order associated with the unit cell structure\nof the lattice [5–12]. These studies have been motivated\nand supported by exact solutions and rigorous results [13–\n18]; in particular, at half filling, the total spin per unit cell\nobeys Lieb-Mattis [13] (Heisenberg model) or Lieb’s theo-\nrem [15] (Hubbard model). On the other hand, it has been\nverified that the ferrimagnetic GS of spin- 1=2Heisenberg and\nHubbard=t-J AB 2chains, under the effect of frustration [19–\n23] or doping [7, 11, 24, 25], are strongly affected by quan-\ntum fluctuations that might cause its destruction and the occur-\nrence of new exotic phases: spiral incommensurate (IC) spin\nstructures, Nagaoka ( U!1 ) and resonating-valence-bond\n(RVB) states, phase separation (PS), and Luttinger-liquid be-\nhavior. These features can enhance the phenomenology in\ncomparison with a linear chain, which is dominated by the\nnontrivial Luttinger-liquid behavior that exhibits fractional\nexcitations [26, 27], emergent fractionalized particles [28],and fractional-exclusion statistic properties [29] in the spin-\nincoherent regime [30]. In addition, investigations of transport\nproperties in AB2chains, and related structures, have also un-\nveiled very interesting features [31].\nOn the experimental side, studies [32–34] of the magnetic\nproperties of homometallic phosphate compounds of the fam-\nilyA3Cu3(PO 4)4(A = Ca ,Sr,Pb) suggest that in these\nmaterials the line of trimers formed by spin- 1=2 Cu+2ions\nantiferromagnetically coupled do exhibit ferrimagnetism of\ntopological origin. Further, compounds Ca3M3(PO 4)4(M =\nNi;Co) with a wave-like layer structure built by zigzag M-\nchains exhibit antiferromagnetic ordering ( M = Ni ) or param-\nagnetic behavior ( M = Co ) [35]. On the other hand, bimetal-\nlic compounds, such as CuMn (S 2C2O2)2\u00017:5H2O[36], can\nbe modeled [36–38] by alternate spin- 1=2- spin- 5=2chains\nand support interesting field-induced quantum critical points\nand Luttinger-liquid phase [37]. In addition, frustrated dia-\nmond (AB2topology) chains can properly model the com-\npound azurite, Cu3(CO 3)2(OH) 2, in which case the occur-\nrence of the 1=3magnetization plateau is verified at high\nfields [39] in agreement with topological arguments [40] akin\nto those invoked in the quantum Hall effect. The spin-1/2\ntrimer chain compound Cu3(P2O6OH) 2, with antiferromag-\nnetic interactions only, also display the 1/3 magnetization\nplateau [41]. Interestingly, it has been established that in azu-\nrite the magnetization plateau is a dimer-monomer state [42],\ni.e., the chain is formed by pairs of S= 1=2monomers and\nS= 0dimers, with a small local polarization of the diamond\nspins [43], in agreement with density functional theory [44].\nThese dimer-monomer states have been found previously in\nthe context of modeling frustrated AB2chains [45–47], and\nconfirmed through a modeling using quantum rotors [48]. In\ncontrast to azurite, whose dimers appear perpendicular to the\nchain direction, in the spin-1/2 inequilateral diamond-chain\ncompounds [49] A3Cu3AlO 2(SO 4)4(A = K;Rb;Cs), the\nmagnetic exchange interactions force the dimers to lie along\nthe sides of the diamond cells and the monomers form a 1DarXiv:1702.03609v4  [cond-mat.str-el]  13 Dec 20182\nHeisenberg chain. In fact, the low-energy excitations of these\nnew compounds have been probed and a Tomonaga-Luttinger\nspin liquid behavior identified [50]. It is worth mentioning\nthat strongly frustrated AB2chains can exhibit ladder-chain\ndecoupling [20], in which case the ladder is formed via the\ncoupling between dimer spins in neighboring AB2unit cells.\nOn the other hand, besides the above-mentioned quasi-\n1D compounds and related magnetic properties, consider-\nable efforts have been devoted to the study of supercon-\nductivity and intriguing magnetic/charge ordered phases in\ndoped materials [51, 52], in particular the formation of spin-\ngapped states in compounds such as the family of doped\n(La;Sr;Ca)14Cu24O41. This compound is formed by one-\ndimensional CuO 2diamond chains, (Sr;Ca)layers, and two-\nlegCu2O3ladders [53]. These results certainly stimulate\nexperimental and theoretical investigations of quasi-1D com-\npounds in the hole-doped regime, which is the main focus of\nour work, as described in the following.\nIn this work, we shall employ an analytical approach suit-\nable to describe the strongly coupled Hubbard model on doped\nAB2chains, which were the object of recent numerical studies\nthrough density matrix renormalization group (DMRG) tech-\nniques [25]. Our functional integral approach, combined with\na perturbative expansion in the strong-coupling regime, was\noriginally proposed to study the doped Hubbard chain [54],\nand later adapted to describe various doped-induced phase\ntransitions in the U=1AB2Hubbard chain [55]. In ad-\ndition, this approach was used to describe the doped strongly\ncoupled Hubbard model on the honeycomb lattice [56], whose\nresults are very rewarding, particularly those for the GS en-\nergy and magnetization in the doped regime, which com-\npare very well with Grassmann tensor product numerical stud-\nies [57].\nThe paper is organized as follows: in Sec. II we review the\nfunctional integral representation of the Hubbard Hamiltonian\nin terms of Grassmann fields (charge degrees of freedom) and\nspinSU(2)gauge fields (spin degrees of freedom). In Sec. III\nwe diagonalize the Hamiltonian associated with the charge de-\ngree of freedom and obtain a perturbative low-energy theory\nsuitable to describe the ferrimagnetic phase at half filling and\nthe phases in the hole-doped regimes. In Sec. IV, we show\nthat the resultant Hamiltonian at half filling and large-U maps\nonto the spin- 1=2quantum Heisenberg model. In this regime,\na perturbative series expansions in powers of 1=Sof the spin-\nwave modes is presented, which allows us to calculate the\nGS energy, sublattice magnetizations, and Lieb GS total spin\nper unit cell in very good agreement with previous estimates.\nIn Sec. V, we derive the low-energy effective t-JHamilto-\nnian, which accounts for both charge and spin quantum fluc-\ntuations. We also present the evolution of the second-order\nspin-wave modes, GS energy and total spin per unit cell un-\nder hole doping, thus identifying the occurrence of spatially\nmodulated spin structures, with non-zero and zero GS total\nspin, and phase separation involving the later spin structure\nand RVB states at hole concentration 1=3. Remarkably, these\npredictions are in very good agreement with the DMRG data\nreported in Ref. [25]. Lastly, in Sec. VI, we present a sum-\nmary and concluding remarks concerning the reported results.II. FUNCTIONAL-INTEGRAL REPRESENTATION\nThe Hamiltonian of the one-band Hubbard model on chains\nwithAB2unit cell topology is given by [7, 8, 10]:\nH=\u0000X\nhi\u000b;j\fi\u001bft\u000b\f\nij^cy\ni\u000b\u001b^cj\f\u001b+H.c.g+UX\ni\u000b^ni\u000b\"^ni\u000b#;(1)\nwherei= 1;:::;Nc(=N=3)is the specific position of the\nunit cell, whose length is set to unity, Nc(N) is the num-\nber of cells (sites), \u000b;\f=A; B 1; B2denote the type of site\nwithin the unit cell, ^cy\ni\u000b\u001b(^ci\u000b\u001b) is the creation (annihilation)\noperator of electrons with spin \u001b(=\";#) at site\u000bof celli,\nand^ni\u000b\u001b= ^cy\ni\u000b\u001b^ci\u000b\u001bis the occupancy number operator. The\nfirst term in Eq. (1) describes electron hopping, with energy\nt\u000b\f\nij\u0011t, allowed only between nearest neighbors A-B1and\nA-B2linked sites of sublattices AandB(bipartite lattice),\nand the second one is the on-site Coulombian repulsive inter-\nactionU > 0, which contributes only in the case of double\noccupancy of the site i\u000b.\nAt this point, it is instructive to digress on some fundamen-\ntal aspects of the formalism used in our work [54–56]. With\nregard to the large-U doped Hubbard chain [54], U=1AB2\nHubbard chain [55] and the Hubbard model on the honeycomb\nlattice [56], it has been shown that the particle density product\nin Eq. (1) can be treated through the use of a decomposition\nprocedure, which consists in expressing ^ni\u000b\"^ni\u000b#in terms of\ncharge and spin operators:\n^ni\u000b\"^ni\u000b#=1\n2^\u001ai\u000b\u00002(^Si\u000b\u0001ni\u000b)2; (2)\nwhere\n^Si\u000b= 1=2X\n\u001b\u001b0^cy\ni\u000b\u001b0\u001b\u001b0\u001b^ci\u000b\u001b; (3)\nand\n^\u001ai\u000b= ^ni\u000b\"+ ^ni\u000b#; (4)\nare the spin-1/2 and charge-density operators, respectively,\n\u001b\u001b0\u001bdenotes the Pauli matrix elements (~\u00111), andni\u000bis\nan arbitrary unit vector. In fact, Eq. (2) follows from the iden-\ntity:1\n2^\u001ai\u000b\u0000^ni\u000b\"^ni\u000b#= 2( ^Sx;y;z\ni\u000b)2= 2( ^Si\u000b\u0001ni\u000b)2. The\nconvenience of using the decomposition defined in Eq. (2),\nwith explicit spin-rotational invariance for the large-U Hub-\nbard model, was discussed at length in Refs. [54–56].\nWe start by using the Trotter-Suzuki formula [58, 59],\nwhich allows us to write the partition function,\nZ=Tr[exp(\u0000\fH)], at a temperature kBT\u00111=\f, as\nZ=Trf^TQM\nr=1exp[\u0000\u000e\u001cH(\u001cr)]g, where ^Tdenotes the\ntime-ordering operator, the total imaginary time interval\nis formally sliced into Mdiscrete intervals of equal size\n\u000e\u001c=\u001cr\u0000\u001cr\u00001,r= 1;2;:::;M; with\u001c0= 0;and\n\u001cM=\f=M\u000e\u001c , under the limits M!1 and\u000e\u001c!0. We\nshall now introduce, between each discrete time interval, an\novercomplete basis of fermionic coherent states [58, 59], 1 =\u0001Q\ni\u000b\u001bdcy\ni\u000b\u001bdci\u000b\u001bexp\u0010\n\u0000P\ni\u000b\u001bcy\ni\u000b\u001bci\u000b\u001b\u0011\njfci\u000b\u001bgihfci\u000b\u001bgj,3\nwherefcy\ni\u000b\u001b;ci\u000b\u001bgdenotes a set of Grassmann fields satisfy-\ning anti-periodic boundary conditions: cy\ni\u000b\u001b(0) =\u0000cy\ni\u000b\u001b(\f)\nandci\u000b\u001b(0) =\u0000ci\u000b\u001b(\f); while the set of unit vectors\ndefines the vector field fni\u000bg, satisfying periodic ones:\nni\u000b(0) = ni\u000b(\f), under a weight functional (see below).\nThereby, following standard procedure [58, 59], the partition\nfunction reads:\nZ=\u0002Y\ni\u000b\u001bDcy\ni\u000b\u001bDci\u000b\u001bY\ni\u000bD2ni\u000bW(fni\u000bg)e\u0000\u0001\f\n0L(\u001c)d\u001c;\n(5)\nwhere the pertinent measures are defined by\nDcy\ni\u000b\u001bDci\u000b\u001b\u0011 lim\nM!1;\u000e\u001c!0M\u00001Y\nr=1dcy\ni\u000b\u001b(\u001cr)dci\u000b\u001b(\u001cr);(6)\nD2ni\u000b\u0011 lim\nM!1;\u000e\u001c!0M\u00001Y\nr=1d2ni\u000b(\u001cr); (7)\nthe weight functional, W(fni\u000bg), satisfies a normalization\ncondition at each discrete imaginary time \u001cr:\n\u0002Y\ni\u000bd2ni\u000bW(fni\u000b(\u001cr)g) = 1; (8)\nand the Lagrangian density L(\u001c)is written in the form:\nL(\u001c) =X\ni\u000b\u001bcy\ni\u000b\u001b@\u001cci\u000b\u001b\u0000X\nij\u000b\f\u001b(t\u000b\f\nijcy\ni\u000b\u001bcj\f\u001b+H.c.)\n+UX\ni\u000b[\u001ai\u000b\n2\u00002(Si\u000b\u0001ni\u000b)2]: (9)\nIn order to fix W(fni\u000bg)one should notice that, in the op-\nerator formalism: ^\u001a2\ni\u000b= ^\u001ai\u000b+ 2^ni\u000b\"^ni\u000b#. Therefore, using\nEq. (2), the following identity holds [54]:\n2(^Si\u000b\u0001ni\u000b)2=^\u001ai\u000b(2\u0000^\u001ai\u000b)\n2; (10)\nwhich means that the square of the spin component opera-\ntor along the ni\u000bdirection has zero eigenvalues if the site\nis vacant or doubly occupied, and a nonzero value only for\nsingly occupied sites, i.e., (^Si\u000b\u0001ni\u000b)2= 1=4. Now, taking\nadvantage of the choice of ni\u000b, the local spin-polarization and\nspin-quantization axes are both chosen along the ni\u000bdirec-\ntion. Therefore, for singly occupied sites, we find Si\u000b\u0001ni\u000b=\npi\u000b=2, withpi\u000b=\u00061, corresponding to the two possible\nspin-1/2 states. Further, by incorporating vacancy and double\noccupancy possibilities, corresponding to the four possible lo-\ncal states of the Hubbard model, one can write [54]\npi\u000b^Si\u000b\u0001ni\u000b=^\u001ai\u000b(2\u0000^\u001ai\u000b)\n2; (11)\nwithp2\ni\u000b= (\u00061)2. We stress that, due to fermion operator\nproperties, the square of Eq. (11) reproduces Eq. (10), and\na comparison between them implies, at arbitrary doping andU value, the formal equivalence between 2(^Si\u000b\u0001ni\u000b)2and\npi\u000b(^Si\u000b\u0001ni\u000b). In this context, we remark that the origi-\nnal Coulomb repulsion term of the Hubbard Hamiltonian in\nEq. (1) is formally and energetically (eigenvalues) equivalent\nto both that in Eq. (9) or in its linear version through the fol-\nlowing replacement: 2(^Si\u000b\u0001ni\u000b)2!pi\u000b(^Si\u000b\u0001ni\u000b). In-\ndeed, using the constraint in Eq. (11) we find, UP\ni\u000b[\u001ai\u000b\n2\u0000\npi\u000b(Si\u000b\u0001ni\u000b)] =UP\ni\u000b[\u001ai\u000b\n2\u00001\n2\u001ai\u000b(2\u0000\u001ai\u000b)], which is zero\nfor\u001ai\u000b= 0;1; whereas, as expected, for double occupied\nsites,\u001ai\u000b= 2, the local energy is U. Therefore, Eq. (11) in its\nGrassmann version, can be enforced by a proper choice of the\nnormalized weight functional:\nW(fni\u000bg) = lim\nM!1;\u000e\u001c!0MY\nr=1W(fni\u000b(\u001cr)g)\n=Cexpn\n\u0000\u0002\f\n0d\u001c\rX\ni\u000b[pi\u000bSi\u000b\u0001ni\u000b\u0000\u001ai\u000b\n2(2\u0000\u001ai\u000b)]2o\n;\n(12)\nwhere\r!1 in the continuum limit ( M!1 ,\u000e\u001c!0),\nwith delta-function peaks at the four local states of the Hub-\nbard model, andCis a normalization factor such that Eq. (8)\nholds. In fact, the product of W(fni\u000b(\u001cr)g)in Eq. (12) gener-\nates a sum in rin the exponential of the suitable chosen Gaus-\nsian function, i.e., W(fni\u000bg)is such that in the continuum\nlimit,M!1;\u000e\u001c!0, Eq. (12) obtains with a diverging \r,\nas pointed out in Ref. [54]. In this way, using Eq. (12) for the\nweight functional in Eq. (5) for the partition function Z, and\nintegrating overfni\u000bg, the Lagrangian density L(\u001c)in Eq.\n(9) can thus be written in the following linearized form [54]:\nL(\u001c) =X\ni\u000b\u001bcy\ni\u000b\u001b@\u001cci\u000b\u001b\u0000X\nij\u000b\f\u001b(t\u000b\f\nijcy\ni\u000b\u001bcj\f\u001b+H.c.)\n+UX\ni\u000b[\u001ai\u000b\n2\u0000pi\u000b(Si\u000b\u0001ni\u000b)]; (13)\nwhere the constraint in Eq. (11) was explicitly used.\nNow, since we are interested in studying the GS properties\nof theAB2Hubbard chains, we choose the staggered factor\npi\u000b= +1 (\u00001)at sites\u000b=B1; B2(A), consistent with the\nlong-range ferrimagnetic GS predicted by Lieb’s theorem at\nhalf filling and for any Uvalue [7, 8, 15], in which case we\nassume broken rotational symmetry along the z-axis. In this\ncontext, by considering the symmetry exhibited by the ferri-\nmagnetic order, let us define the SU(2)=U(1)unitary rotation\nmatrix [60]\nUi\u000b=2\n4cos\u0010\u0012i\u000b2\u0011\n\u0000sin\u0010\u0012i\u000b2\u0011\ne\u0000i\u001ei\u000b\nsin\u0010\u0012i\u000b2\u0011\nei\u001ei\u000b cos\u0010\u0012i\u000b2\u00113\n5;(14)\nwhere\u0012i\u000bis the polar angle between the z-axis and the unit lo-\ncal vector ni\u000band\u001ei\u000b2[0;2\u0019)is an arbitrary azimuth angle\ndue to theU(1)gauge freedom of choice for Ui\u000b. Moreover,\na new set of Grassmann fields, fay\ni\u000b\u001b;ai\u000b\u001bgcan be obtained,\naccording to the transformation:\nci\u000b\u001b=X\n\u001b0(Ui\u000b)\u001b\u001b0ai\u000b\u001b0; (15)4\nthat locally rotates each unit vector ni\u000bto thez-direction. On\nthe other hand, if we express the product \u001b\u0001ni\u000bin matrix\nform:\n\u001b\u0001ni\u000b=\u0014\ncos (\u0012i\u000b) sin (\u0012i\u000b)e\u0000i\u001ei\u000b\nsin (\u0012i\u000b)ei\u001ei\u000b\u0000cos (\u0012i\u000b)\u0015\n; (16)\nwe obtain, after using Eq. (14),\nUy\ni\u000b(\u001b\u0001ni\u000b)Ui\u000b=\u001bz; (17)\nwhich explicitly manifest the broken rotational symmetry\nalong thez-axis. In this way, by substituting Eqs. (14) and\n(15) into Eq. (3), and using the above result, we find\nSi\u000b\u0001ni\u000b=1\n2X\n\u001b\u001b0ay\ni\u000b\u001b[Uy\ni\u000b(\u001b\u0001ni\u000b)Ui\u000b]\u001b\u001b0ai\u000b\u001b0\n=1\n2X\n\u001b\u001b0ay\ni\u000b\u001b(\u001bz)\u001b\u001b0ai\u000b\u001b0\u0011Sz\ni\u000b; (18)\nthereby, the constraint in Eq. (11) can be written in the form\nSi\u000b\u0001ni\u000b=pi\u000b\u001ai\u000b(2\u0000\u001ai\u000b)\n2=1\n2(ay\ni\u000b\"ai\u000b\"\u0000ay\ni\u000b#ai\u000b#);\n(19)\nwherepi\u000b= +1 (\u00001)at sites\u000b=B1; B2(A). The choice\nofpi\u000babove implies Lieb’s ferrimagnetic ordering with the set\nf\u0012iA=\u0012iB1=\u0012iB2= 0g, for alli, at half filling. However,\nin the hole doped regime away from half filling, the \u0012i\u000b’s can\nbe nonzero (e.g., \u0012i\u000b=\u0019for a spin flip, leading to a change\nin the sign of Sz\ni\u000b); further,Sz\ni\u000bcan be zero either by the pres-\nence of holes or doubly occupied sites ( ay\ni\u000b\"ai\u000b\"=ay\ni\u000b#ai\u000b#).\nLastly, using Eqs. (15) and (19) into the Lagrangian, Eq. (13),\nwe find, after suitable rearrangement of terms,\nL(\u001c) =L0(\u001c) +Ln(\u001c); (20)\nwhere both Lagrangians are quadratic in the Grassmann fields:\nL0(\u001c) =X\ni\u000b\u001bay\ni\u000b\u001b@\u001cai\u000b\u001b\u0000X\ni\u000bj\f\u001b(t\u000b\f\nijay\ni\u000b\u001baj\f\u001b+H.c.)\n+U\n2X\ni\u000b\u001b(1\u0000pi\u000b\u001b)ay\ni\u000b\u001bai\u000b\u001b; (21)\nand\nLn(\u001c) =X\ni\u000b\u001b\u001b0ay\ni\u000b\u001b0(Uy\ni\u000b@\u001cUi\u000b)\u001b0\u001bai\u000b\u001b\n\u0000X\ni\u000bj\f\u001b\u001b0t\u000b\f\nij[ay\ni\u000b\u001b0(Uy\ni\u000bUj\f\u00001)\u001b0\u001baj\f\u001b+H.c.];(22)\nwith the first term in both Eqs. (21) and (22) being originated\nfrom the first term in Eq. (13), the second ones come from\nthe hopping term in Eq. (13), after a rearrangement of terms,\nwhile the last one in Eq. (21) (proportional to U) is obtained\nby using Eq. (19) in the last term of Eq. (13). It is worth\nmentioning that only charge degrees of freedom (Grassmann\nfields) appear inL0(\u001c), and spin degrees of freedom under theconstraint in Eq. (19) [ SU(2)gauge fieldsfUy\ni\u000b;Ui\u000bg, which\ncarry all the information on the vector field fni\u000bg] are now\nrestricted toLn(\u001c), which includes both spin and charge de-\ngrees of freedom.\nIn the large-U regime, double occupancy is energetically\nunfavorable and the factor 2\u0000\u001ai\u000bis no longer needed in\nEq. (19), i.e., Si\u000b\u0001ni\u000b=pi\u000b\u001ai\u000b\n2, with\u001ai\u000b= 0 or1. In\nthis case, a proper perturbative analysis will allow us to study\nhole doping effects in Sec. V in a macroscopic fashion, so we\ndefine\n\u000e= 1\u00001\nNX\ni\u000bh\u001ai\u000bi; (23)\nwhich measures the thermodynamic average of hole doping\naway from half filling. In this context (strong-coupling limit),\nwe take advantage of results derived from L0(\u001c)(charge ef-\nfects in Sec. III), and at half filling (Sec. IV), in which case\ncharge degrees of freedom are frozen.\nIII. CHARGE DEGREES OF FREEDOM AND THE\nSTRONG-COUPLING LIMIT\nIn this section, we shall first diagonalize the Hamiltonian\nassociated with the Lagrangian L0(\u001c)through the use of a\nspecial symmetry property of the AB2chains and a canonical\ntransformation in reciprocal space. Then, by introducing a\nperturbative expansion in the strong-coupling regime, a low-\nenergy effective Lagrangian for the AB2Hubbard chains at\nhalf filling and in the doped regime will be obtained.\nA. Charge degrees of freedom\nWe begin our discussion by considering the Lagrangian L0\nin Eq. (21), and its corresponding Hamiltonian H0, free of\ntheSU(2)gauge fields. By performing the Legendre trans-\nformation:H0=\u0000P\ni\u000b\u001b@L0\n@(@\u001cai\u000b\u001b)@\u001cai\u000b\u001b+L0, where\n@L0\n@(@\u001cai\u000b\u001b)=ay\ni\u000b\u001b, the resultingH0is given by\nH0=\u0000X\nhi\u000b;j\fi\u001b(t\u000b\f\nijay\ni\u000b\u001baj\f\u001b+H.c.)\n+U\n2X\ni\u000b\u001b(1\u0000pi\u000b\u001b)ay\ni\u000b\u001bai\u000b\u001b: (24)\nFurther, sinceH0(L0) is quadratic in the Grassmann fields,\nthe solution for the energy of the system is given by H0in its\ndiagonalized form [59].\nTheAB2unit cell topology exhibits a symmetry [9, 11, 24,\n25, 55] under the exchange of the labels of the Bsites in a\ngiven unit cell. Thus, we can construct a new set of Grass-\nmann fields possessing this symmetry, i.e., either symmet-\nric or antisymmetric with respect to the exchange operation\nB1$B2:\n(di\u001b;ei\u001b) =1p\n2(aiB1\u001b\u0006aiB2\u001b); bi\u001b=aiA\u001b: (25)5\nIn addition, as a signature of the quasi-1D structure of the\nAB2chains, we notice that the B1andB2sites are located at\na distance 1=2(in units of length) ahead of the Asite. There-\nfore, after Fourier transforming the above Grassmann fields,\ni.e.,fdi;\u001b;ei;\u001b;bi;\u001bg=1pNcP\nkeikxifdk;\u001b;ek;\u001b;bk;\u001bg, it is\nconvenient to introduce a phase factor eik\n2through the follow-\ning transformation [55]: (Ak\u001b;Bk\u001b) =1p\n2(dk\u001b\u0006eik\n2bk\u001b), so\nthatH0in Eq. (24) thus becomes\nH0=X\nk\u001b\"k[Ay\nk\u001bAk\u001b\u0000By\nk\u001bBk\u001b] +U\n2X\nk\u001b(1\u0000\u001b)ey\nk\u001bek\u001b\n+U\n2X\nk\u001b[Ay\nk\u001bAk\u001b+By\nk\u001bBk\u001b\u0000\u001b(Ay\nk\u001bBk\u001b+By\nk\u001bAk\u001b)];\n(26)\nwhere\n\"k=\u00002p\n2tcos(k=2); (27)\nwithk= 2\u0019j(3\nN)\u0000\u0019, andj= 1;:::;N= 3. We can now ex-\nactly diagonalizeH0through the following Bogoliubov trans-\nformation:\nAk\u001b=uk\u000bk\u001b\u0000\u001bvk\fk\u001b; Bk\u001b=\u001bvk\u000bk\u001b+uk\fk\u001b;(28)\nwithukandvksatisfying the canonical constraint: (uk)2+\n(vk)2= 1 , to maintain the anticommutation relations of\nthe Grassmann fields. Due to the ferrimagnetic order of the\nGS, the above transformation is subject to a 4\u0019periodicity\nof the Bogoliubov functions fuk;vkgand Grassmann fields\nf\u000bk\u001b;\fk\u001bg. The diagonalized H0thus reads:\nH0=\u0000X\nk\u001b(Ek\u0000U\n2)\u000by\nk\u001b\u000bk\u001b+X\nk\u001b(Ek+U\n2)\fy\nk\u001b\fk\u001b\n+U\n2X\nk\u001b(1\u0000\u001b)ey\nk\u001bek\u001b; (29)\nwhere\n(uk;vk) =1p\n2\u0012\n1\u0006j\"kj\nEk\u00131=2\n; (30)\nand\nEk=q\n\"2\nk+U2=4: (31)\nAs one can see from Eq. (29), the non-interacting tight bind-\ning (U= 0) spectrum ofH0present three electronic bands:\na nondispersive flat band (related to the Grassmann fields\nfey\nk\u001b;ek\u001bg, macroscopically degenerate), and two dispersive\nones. InAB2chains, flat bands are closely associated with\nferrimagnetism (unsaturated ferromagnetism) [5, 7, 8] at half\nfilling, in agreement with Lieb’s theorem [15, 16], or fully\npolarized ferromagnetism [17] associated with the flat lowest\nband. We also stress that even at this level of approximation\nand in the weak coupling regime ( U= 2t), it was shown [7]\nthat hole doping [parametrized by \u000edefined in Eq. (23)] candestroy the ferrimagnetic order and/or induce phase separa-\ntion inAB2chains. As depicted in Fig. 1(a), the U= 0 spin\ndegeneracy of the flat bands is removed by the Coulombian\nrepulsive interaction, in which case a gap Uopens between\ntheek\u001bmodes:ek\"= 0, where spins at sites B1andB2are\nup, andek#=U, where these spins are down. On the other\nhand, the two dispersive bands are spin degenerated, and also\ndisplay a Hubbard gap Useparating the low (\u000bk\u001b)-energy and\nhigh(\fk\u001b)-energy modes [55].\nB. Strong-coupling limit\nIn this subsection, we shall introduce a perturbative expan-\nsion in the strong-coupling regime ( U\u001dt) in order to obtain\na low-energy effective Lagrangian for the AB2Hubbard chain\nat half filling and in the doped regime. First, we resume the re-\nsults of the previous section by writing the Grassmann fields\ndi\u001bandbi\u001bin terms of the Grassmann (Bogoliubov) fields\n\u000bk\u001band\fk\u001b:\n(di\u001b;bi\u001b) =1p2NcX\nk(eikxi;eik(xi\u00001\n2))\n\u0002[(uk\u0006\u001bvk)\u000bk\u001b\u0006(uk\u0007\u001bvk)\fk\u001b];(32)\nwhere the phase factor e\u0000ik\n2signalizes the quasi-1D AB2\nstructure, and the antisymmetric Grassmann field ei;\u001bremains\nas defined in Eq. (25). In the strong-coupling limit, however,\nit will prove useful to define a set of auxiliary spinless Grass-\nmann fields [54, 55] in direct space associated with di\u001band\nbi\u001b:\n(\u000bi;\fi) =r\n1\nNcX\nk;\u001b\u0012(\u0006\u001b)eikxi(\u000bk\u001b;\fk\u001b); (33)\nand a similar equation for (\u000b1\n2\ni;\f1\n2\ni)$(\u000bk\u001b;\fk\u001b)is obtained\nby the replacements: \u0012(\u0006\u001b)!\u0012(\u0007\u001b)andxi!xi\u00001=2,\nwhere\u0012(\u001b)is the Heaviside function, while for the antisym-\nmetric component, one has\nei;\u001b=r\n1\nNcX\nkeikxiek;\u001b: (34)\nNow, by expanding (uk;vk)in Eq. (30) in powers of t=U:\n(uk;vk)\u00191p\n2\u0014\n1\u0006j\"kj\nU+O\u0012t2\nU2\u0013\u0015\n; (35)\nsubstituting these results into the Eq. (32), and using the in-\nverse transformation of Eq. (33), we can derive a perturbative\nexpansion in powers of t=U for the Grassmann fields di\u001band\nbi\u001bin terms of the spinless Grassmann fields as follows:\ndi\u001b=\u0012(\u001b)\u000bi+\u0012(\u0000\u001b)\fi+p\n2t\nU\u0012(\u0000\u001b)(\u000b1\n2\ni+\u000b1\n2\ni+1)\n+t\nU\u0012(\u001b)[p\n2(\f1\n2\ni+\f1\n2\ni+1)\u0000t\nU(2\u000bi+\u000bi+1+\u000bi\u00001)]\n+O(t2=U2); (36)6\nbi\u001b=\u0012(\u0000\u001b)\u000b1\n2\ni\u0000\u0012(\u001b)\f1\n2\ni+p\n2t\nU\u0012(\u001b)(\u000bi+\u000bi\u00001)\n\u0000t\nU\u0012(\u0000\u001b)[p\n2(\fi+\fi\u00001) +t\nU(2\u000b1\n2\ni+\u000b1\n2\ni+1+\u000b1\n2\ni\u00001)]\n+O(t2=U2): (37)\nIn the above derivation, we have used that \u0012(\u001b)\u0012(\u001b0) =\n\u0012(\u001b)\u000e\u001b;\u001b0. Notice that, sincet\nU\u001c1, in Eqs. (36) and (37)\nwe can identify the fields \u000b1\n2\ni\u0019aiA#and\u000bi\u0019(aiB1\"+\naiB2\")=p\n2, a result fully consistent with the low-energy spin\nconfiguration of the ferrimagnetic state discussed previously.\nAnalogously, for the high-energy bands, the opposite spin\nconfiguration is observed, with spin up (down) present at sites\nA(B1,B2).\nIntroducing Eqs. (36) and (37) into Eq. (24), with the aid\nof Eq. (25), we obtain a perturbative expression for H0(low-\nenergy sector) in terms of the spinless Grassmann fields up to\norderJ= 4t2=U:\nH0=\u0000JX\ni[\u000by\ni\u000bi+\u000b(1\n2)y\ni\u000b1\n2\ni\u0000\fy\ni\fi\u0000\f(1\n2)y\ni\f1\n2\ni]\n\u0000J\n2X\ni[\u000by\ni\u000bi+1+\u000b(1\n2)y\ni\u000b1\n2\ni+1\u0000\fy\ni\fi+1\u0000\f(1\n2)y\ni\f1\n2\ni+1+H.c:]\n+UX\ni[\fy\ni\fi+\f(1\n2)y\ni\f1\n2\ni+ey\ni#ei#]:(38)\nBy applying Fourier transform to the above expression and\nrearranging the terms, we obtain\nH0=\u0000X\nk2Jcos2(k=2)(\u000by\nk\u000bk+\u000b(1\n2)y\nk\u000b1\n2\nk)\n+X\nk[2Jcos2(k=2) +U](\fy\nk\fk+\f(1\n2)y\nk\f1\n2\nk)\n+U\n2X\nk\u001b(1\u0000\u001b)ey\nk\u001bek\u001b: (39)\nIn Fig. 1 we plot the electronic spectrum of the Hamilto-\nnianH0, both in the weak and strong-coupling regime: (a)\nEq. (29) for U= 2tand (b) Eqs. (29) and (39) for U= 12t\n(J= 4t2=U= 1=3), respectively, with t\u00111. We can notice\nthe presence of the shrinking phenomenon [7] as Uincreases\nfrom 2tto12t(strong-coupling regime) and that, for U= 12t,\nEq. (39) is a very good approximation to Eq. (29). Notice-\nably, thet\u001cUexpansion of the fields allow us to identify\n\u000b1\n2\nk\u0019akA#,\u000bk\u0019(akB1\"+akB2\")=p\n2(triplet state) and\nek\"\u0019(akB1\"\u0000akB2\")=p\n2(singlet state), as the low-energy\nspin configuration of the ferrimagnetic state with single occu-\npancy, where spins at sites A(B1;B2) are down (up), in agree-\nment with Lieb’s theorem [7, 8, 15].\nIn order to describe the most relevant low-energy pro-\ncesses that take place in this regime, one has to additionally\nproject out the high-energy bands from H0, that is, terms con-\ntaining only fields related to the high-energy bands are ex-\ncluded. Therefore, after the Legendre transformation, H0=\n\u0000P\ni;\u0011i@L0\n@(@\u001c\u0011i)@\u001c\u0011i+L0, where\u0011i=\u000bi;\u000b1\n2\ni, andei\"(fields\n-1 0 1\nk/π-2024 E(a)\nβkσ\nek↓\nek↑\nαkσU= 2t\n-1 0 1\nk/π-2061214(b)\nU= 12tβkβ1\n2\nk\nek↓\nek↑\nαkα1\n2\nkFigure 1. (Color online) Electronic spectrum of the Hamiltonian H0:\n(a) Eq. (29) for U= 2tand (b) Eqs. (29) and (39) for U= 12t(J=\n4t2=U= 1=3), witht\u00111. Notice the band shrinking phenomenon\nasUincreases from 2tto12t(strong-coupling regime). The t\u001c\nUexpansion of the fields identifies \u000b1\n2\nk\u0019akA#,\u000bk\u0019(akB1\"+\nakB2\")=p\n2andek\"\u0019(akB1\"\u0000akB2\")=p\n2, where spins at sites\nA(B1;B2) are down (up), in agreement with Lieb’s theorem [15].\nrelated to the low-energy bands), with@L0\n@(@\u001c\u0011i)=\u0011y\ni, the La-\ngrangian associated with H0(up to order J) is given by\nL0=X\ni;\u0011i\u0011y\ni@\u001c\u0011i\u0000JX\ni(\u000by\ni\u000bi+\u000b(1\n2)y\ni\u000b1\n2\ni)\n\u0000J\n2X\ni(\u000by\ni\u000bi+1+\u000b(1\n2)y\ni\u000b1\n2\ni+1+H.c:):(40)\nWe shall now focus on the U\u001dtperturbative expansion of\nLn, Eq. (22), which amounts to consider the most significant\nlow-energy processes, after the use of Eqs. (36) and (37) for\ndi\u001bandbi\u001bin terms of the spinless Grassmann fields. How-\never, terms allowing interband transitions between low- and\nhigh-energy bands do exist in Ln. In this context, we apply a\nsuitable second-order Rayleigh-Schr ¨odinger perturbation the-\nory [54, 55], consistent with the strong-coupling expansion,\nso that the modes associated with the high-energy bands are\neliminated. Lastly, by adding L0to the perturbative expansion\nofLn, which leads to the cancellation of the exchange terms\nin Eq. (40), the effective low-energy Lagrangian density of the\nAB2Hubbard model in the strong-coupling limit (up to order\nJ) reads:\nLeff(\u001c) =L(I)+L(II)+L(III)+L(IV); (41)\nwhere\nL(I)=X\ni\u000by\ni@\u001c\u000bi+X\ni\u000b(1\n2)y\ni@\u001c\u000b(1\n2)\ni+X\niey\ni\"@\u001cei\";(42a)7\nL(II)=X\ni\u001bn\n\u0012(\u0000\u001b)(U(b)y\ni@\u001cU(b)\ni)\u001b;\u001b\u000b(1\n2)y\ni\u000b(1\n2)\ni\n+\u0012(\u001b)1\n2[(U(d)y\ni@\u001cU(d)\ni)\u001b;\u001b+ (U(e)y\ni@\u001cU(e)\ni)\u001b;\u001b]\n\u0002(\u000by\ni\u000bi+ey\ni\"ei\") +\u0014\n\u0012(\u001b)1\n2[(U(d)y\ni@\u001cU(e)\ni)\u001b;\u001b\n+(U(e)y\ni@\u001cU(d)\ni)\u001b;\u001b]\u000by\niei\"+H.c.io\n; (42b)\nL(III)=\u0000tX\ni\u001bn\n\u0012(\u0000\u001b)(U(b)y\niU(d)\ni)\u001b;\u0000\u001b\u000b(1\n2)y\ni\u000bi\n+\u0012(\u001b)(U(d)y\niU(b)\ni+1)\u001b;\u0000\u001b\u000by\ni\u000b(1\n2)\ni+1\n+\u0012(\u0000\u001b)(U(b)y\niU(e)\ni)\u001b;\u0000\u001b\u000b(1\n2)y\niei\"\n+\u0012(\u001b)\u0010\nU(e)y\niU(b)\ni+1\u0011\n\u001b;\u0000\u001bey\ni\"\u000b(1\n2)\ni+1+H.c.\u001b\n;(42c)\nL(IV)=\u0000J\n4X\ni;i0=i;i+1;\u001b\u0012(\u001b)j(U(d)y\niU(b)\ni0)\u001b;\u001bj2\u000by\ni\u000bi\n\u0000J\n4X\ni;i0=i;i+1;\u001b\u0012(\u001b)j(U(e)y\niU(b)\ni0)\u001b;\u001bj2ey\ni\"ei\"\n\u0000J\n4X\ni;i0=i;i\u00001;\u001b\u0012(\u0000\u001b)[j(U(b)y\niU(d)\ni0)\u001b;\u001bj2\n+ (U(b)y\niU(e)\ni0)\u001b;\u001bj2]\u000b(1\n2)y\ni\u000b(1\n2)\ni; (42d)\nwhere\nU(b)\ni=UiA; U(d;e)\ni =1p\n2(UiB1\u0006UiB2); (43)\nin which case we took advantage of the symmetry of the AB2\nchain under the exchange operation B1$B2, in correspon-\ndence with Eq. (25). From the above equations, we see that the\nkinetic term is represented by L(I)and is related to the charge\ndegrees of freedom only, whereas L(II)describes the dynam-\nics of the spin degrees of freedom coupled to the charge fields.\nOn the other hand, L(III)exhibit first-neighbor hopping con-\ntributions between charge degrees of freedom in the presence\nofSU(2)gauge fields, while L(IV)is the spin exchange term\nin the presence of the charge Grassmann fields.\nIV . HALF-FILLING REGIME\nLet us now discuss some basic aspects of the localized mag-\nnetic properties related to the spin degrees of freedom. At half\nfilling, i.e.,\u000e= 0, we haveh\u000by\ni\u000bii= 1,h\u000b(1=2)y\ni\u000b(1=2)\nii= 1,\nhey\ni\"ei\"i= 1, andh\u000by\niei\"i= 0(no band hybridization) as the\nelectrons tend to fill up the lower-energy bands, whereas the\nhigher-energy ones remain empty. As a consequence, a fer-\nrimagnetic configuration of localized spins emerges, i.e., the\ncharge degrees of freedom are completely frozen, such that\nh\u000by\ni@\u001c\u000bii=h\u000b(1=2)y\ni@\u001c\u000b(1=2)\nii=hey\ni\"@\u001cei\"i= 0, with for-\nbidden hopping. Therefore, only terms from LIIandLIVinEqs. (42b) and (42d), respectively, give nonzero contributions\nand the resulting effective strong-coupling Lagrangian at half\nfilling, defined in Eq. (41), reads:\nLJ\neff=X\ni\u000b\u001b\u0012(pi\u000b\u001b)(Uy\ni\u000b@\u001cUi\u000b)\u001b;\u001b\n\u0000J\n4X\nhi\u000b;j\fi\u001b\u0012(pi\u000b\u001b)\f\f\f(Uy\ni\u000bUj\f)\u001b;\u001b\f\f\f2\n;(44)\nwhere the staggered factor pi\u000bwas defined in Eq. (11), and use\nwas made of the matrix transformations defined in Eq. (43) in\norder to sum up the squares of the SU(2)gauge field prod-\nucts in the exchange contribution from LIVin Eq. (42d).\nNow, using the following Legendre transform: HJ\neff =\n\u0000P\ni\u000b\u001b@LJ\neff\n@(@\u001cUi\u000b)\u001b;\u001b(@\u001cUi\u000b)\u001b;\u001b+LJ\neff;where@LJ\neff\n@(@\u001cUi\u000b)\u001b;\u001b=\n\u0012(pi\u000b\u001b)(Uy\ni\u000b)\u001b;\u001b;we get the respective quantum Heisenberg\nHamiltonian written in terms of the SU(2)gauge fields at half\nfilling as\nHJ\neff=\u0000J\n4X\nhi\u000b;j\fi\u001b\u0012(pi\u000b\u001b)\f\f\f(Uy\ni\u000bUj\f)\u001b;\u001b\f\f\f2\n: (45)\nFurther, using the definition of the SU(2)=U(1)unitary\nrotation matrix Eq. (14), it is possible to write [54–56]\f\f\f(Uy\ni\u000bUj\f)\u001b;\u001b\f\f\f2\n=1\n2(1 + ni\u000b\u0001nj\f);where ni\u000b=\nsin(\u0012i\u000b) [cos(\u001ei\u000b)^x+ sin(\u001ei\u000b)^y] + cos(\u0012i\u000b)^zis the unit vec-\ntor pointing along the local spin direction. Lastly, by using the\nconstraint as given in Eq. (19), we can identify the spin field\nfSi\u000bgat the single occupied sites:\nSi\u000b=pi\u000bni\u000b=2; (46)\nwherepi\u000b= +1 (\u00001)at sites\u000b=B1; B2(A), in order to\nobtain\nHJ\neff=JX\nih\n(SB1\ni+SB2\ni)\u0001(SA\ni+SA\ni+1)i\n\u0000JNc:(47)\nThe above expression is indeed that of the quantum antifer-\nromagnetic Heisenberg spin-1/2 model on the AB2chain in\nzero-field, which takes into account the effects of zero-point\nquantum spin fluctuations. In fact, to achieve this goal, we an-\nalyze the Hamiltonian, Eq. (47), by means of the spin-wave\ntheory, which has proved very successful in describing the\nproperties of the GS and low-lying excited states of spin mod-\nels. The predicted results provide a check of the consistency\nof our approach and will be fully used in our description of\nthe doped regime.\nWe shall first introduce boson creation and annihilation op-\nerators via the Holstein-Primakoff [58] transformation:\nSA;z\ni=\u0000S+ay\niai;\nSA;+\ni= (SA;\u0000\ni)y=p\n2Say\nifA(S);(48)\nfor a down-spin on the Asite, and\nSBl;z\ni=S\u0000by\nlibli;\nSBl;+\ni = (SBl;\u0000\ni)y=p\n2SfB(S)bli;(49)8\nfor an up-spin on the Blsite, withl= 1;2, and\nfr(S) =\u0010\n1\u0000nr\n2S\u00111=2\n= 1\u00001\n2nr\n2S+:::; (50)\nwhereSis the spin magnitude, and nr=ay\niaiorby\nlibli. The\noperatorsay\niandai(orby\nli,bli) satisfy the boson commutation\nrules. Under the above transformation, the spin Hamiltonian,\nEq. (47) is mapped onto the boson Hamiltonian:\nHJ\neff=E0\u0000JNc+H1+H2+O(S\u00001); (51)\nwhere\nE0=\u00004S2JNc; (52)\nis the classical GS energy and H1andH2are the quadratic\nand quartic (interacting) terms of the boson Hamiltonian, suit-\nable to describe the quantum AB2Heisenberg model via a\nperturbative series expansion in powers of 1=S. By Fourier\ntransforming the boson operators, we find\nH1= 2JSX\nk(2ay\nkak+X\nlby\nlkblk)\n+X\nk;l=1;22JS\rk(ay\nkby\nlk+akblk); (53)\nwhere we have defined the lattice structure factor as\n\rk=1\nzX\n\u001aeik\u001a= cos\u0012k\n2\u0013\n; (54)\nwithzdenoting the coordination number ( z= 4for theAB2\nchain), while \u001a=\u00061=2connects the nearest neighbors A-B1\nandA-B2linked sites of sublattices AandB, and\nH2=\u00003J\n2NX\n1234;l=1;2\u000e12;34n\n4\r1\u00004ay\n1a4by\nl3bl2\n+(\r1ay\n1by\nl4by\nl3bl2+\r1+2\u00003ay\n1ay\n2a3by\nl4+H.c.)o\n:(55)\nFor simplicity, we use the convention 1fork1,2fork2, and so\non. Also, the \u000e12;34=\u000e(k1+k2\u0000k3\u0000k4)is the Kronecker\n\u000efunction, and expresses the conservation of momentum to\nwithin a reciprocal-lattice vector G.\nWe shall consider H1first, which is the term leading to\nlinear spin-wave theory (LSWT). In fact, H1is diagonalized\nusing the following Bogoliubov transformation:\nak=uk\fk\u0000vk\u000by\nk;\nblk=1p\n2[uk\u000bk\u0000vk\fy\nk+ (\u00001)l\u0018k];withl= 1;2;(56)\n(uk;vk) =(3 +p\n9\u00008\r2\nk;2p\n2\rk)q\n(3 +p\n9\u00008\r2\nk)2\u00008\r2\nk; (57)\nwhereukandvksatisfy the constraint u2\nk\u0000v2\nk= 1. Thus,\nH1=E1+X\nk(\u000f0(\u000b)\nk\u000by\nk\u000bk+\u000f0(\f)\nk\fy\nk\fk+\u000f0(\u0018)\nk\u0018y\nk\u0018k);(58)E1=JSX\nk(q\n9\u00008\r2\nk\u00003); (59)\n\u000f0(\u000b;\f)\nk=JS(q\n9\u00008\r2\nk\u00071); \u000f0(\u0018)\nk= 2JS; (60)\nwhereE1is theO(S1)quantum correction to the GS energy,\nand\u000f0(\u000b;\f)\nk,\u000f0(\u0018)\nkare the three spin-wave branches provided\nby LSWT, both in agreement with previous results [19, 61].\nIn fact, it is well known that systems with a ferrimagnetic GS\nnaturally have ferromagnetic and antiferromagnetic spin-wave\nmodes as their elementary magnetic excitations (magnons).\nFor theAB2chain, there are three spin-wave branches: an\nantiferromagnetic mode ( \u000f0(\f)\nk) and two ferromagnetic ones\n(\u000f0(\u000b)\nkand\u000f0(\u0018)\nk). The mode \u000f0(\u000b)\nkis gapless at k= 0, i.e., the\nGoldstone mode, with a quadratic (ferromagnetic) dispersion\nrelation\u000f0(\u000b)\nk\u0018k2. The other two modes are gapped. No-\ntice that the gapped ferromagnetic mode \u000f0(\u0018)\nkis flat, and is\nclosely associated with ferrimagnetic properties at half fill-\ning [7, 17]. Since the dispersive modes preserve the local\ntriplet bond, they are identical to those found in the spin- 1=2\n- spin-1 chains [62–65]. These chains also exhibit interesting\nfield-induced Luttinger liquid behavior [66].\nNow, our aim is to obtain the leading corrections to LSWT,\ni.e., second-order spin-wave theory to the GS energy, sublat-\ntice magnetizations and Lieb GS total spin per unit cell. In do-\ning so, we develop a perturbative scheme for the description\nof this quartic term. First, we decompose the two-body terms\nby means of the Wick theorem, via normal-ordering protocol\nfor boson operators. Conservation of momentum to within a\nreciprocal-lattice vector, implies: k1=k+q,k2=p\u0000q,\nk3=kandk4=p. Then, we need to look at the possible\npairings of the 4 operators, as for example, in the first term of\nEq. (55):\nay\nk+qapby\nl;kbl;p\u0000q;ay\nk+qapby\nl;kbl;p\u0000q;ay\nk+qapby\nl;kbl;p\u0000q:\nUnder this procedure, and by substituting the Bogoliubov\ntransformation, Eqs. (56)-(57), into Eq. (55), we find\nH2=E2+X\nk(\u000e\u000f(\u000b)\nk\u000by\nk\u000bk+\u000e\u000f(\f)\nk\fy\nk\fk+\u000e\u000f(\u0018)\nk\u0018y\nk\u0018k);(61)\nwhere\nE2=Nc=\u00002J(q2\n1+q2\n2\u00003p\n2q1q2); (62)\nand the corresponding corrections for the spin-wave disper-\nsion relations read:\n\u000e\u000f(\u000b)\nk=J[u2\nk(p\n2q2\u00002q1) + 2v2\nk(p\n2q2\u0000q1)]\n+ 4J\rkukvk\u00143\n2p\n2q1\u0000q2\u0015\n+O(S\u00001);(63)\n\u000e\u000f(\f)\nkis obtained from \u000e\u000f(\u000b)\nkthrough the exchange of uk$\nvk, and\n\u000e\u000f(\u0018)\nk=J(p\n2q2\u00002q1) +O(S\u00001): (64)9\nIn Eqs. (62)-(64) above, the quantities q1andq2are defined\nby (thermodynamic limit)\nq1=1\n2\u0019\u0002\u0019\n\u0000\u0019dk(v2\nk); q 2=1\n2\u0019\u0002\u0019\n\u0000\u0019dk(\rkukvk):(65)\nWe remark that in deriving Eqs. (62)-(64), we have neglected\nterms containing anomalous products, such as, \u000by\nk\fy\nkand ver-\ntex corrections.\nLastly, the above results of our perturbative 1=Sseries ex-\npansion lead to the effective Hamiltonian:\nHJ\neff=EJ\nGS\u0000JNc+X\nk(\u000f\u000b\nk\u000by\nk\u000bk+\u000f(\f)\nk\fy\nk\fk+\u000f(\u0018)\nk\u0018y\nk\u0018k);\n(66)\nwhere\nEJ\nGS=E0+E1+E2; (67)\nwhich can be read from Eqs. (52), (59), and (62), respectively,\nis the second-order result up to O(1=S)for the GS energy,\nand\n\u000f(s)\nk=\u000f0(s)\nk+\u000e\u000f(s)\nk;withs=\u000b; \f; \u0018; (68)\nare the corresponding second-order spin-wave modes, where\nthe linear and the second-order correction terms are given by\nEq. (60) and Eqs. (63)-(65), respectively.\nA. Second-order spin-wave analysis\nOur perturbative 1=Sseries expansion approach is able to\nimprove the LSWT result for the gap \u0001 =Jof the antiferro-\nmagnetic mode, which should be compared with the second-\norder result derived from \u000f(\f)\nk, Eqs. (60), (63) and (68), at\nk= 0:\u0001 = (1 +p\n2q2)J'1:676J, in full agreement\nwith similar spin-wave calculations for AB2[19] and spin-\n1=2-spin- 1[64, 65] chains, and in agreement with numeri-\ncal estimates using exact diagonalization, \u0001 = 1:759J, for\nbothAB2[5] and spin- 1=2-spin- 1[63] chains. On the other\nhand, the LSWT predicts a gap \u0001flat=Jfor the flat fer-\nromagnetic mode ( \u000f(\u0018)\nk) inAB2chain, whereas our second-\norder spin-wave theory finds, using Eqs. (60), (64) and (68):\n\u0001flat= (1\u00002q1+p\n2q2)J'1:066J, in full agreement\nwith a similar spin-wave procedure [19]. Surprisingly, the es-\ntimated value from Exact Diagonalization (ED) [5]: \u0001flat=\n1:0004J, lies between these two theoretical values. In fact,\nanalytical approaches are still unable to reproduce the ob-\nserved level crossing found in numerical calculations [5, 19]\nfor the two ferromagnetic modes. This is probably due to the\nfact that the different symmetries exhibited by the localized\nexcitation (flat mode) and the ferromagnetic dispersive mode\nare not explicitly manifested in the analytical approaches, so\nthe levels avoid the crossing.B. Ground state energy\nIn the thermodynamic limit, the second-order result for the\nGS energy of the AB2chain per unit cell reads:\nEJ\nGS\nNc=\u00004JS2+JS\n2\u0019\u0002\u0019\n\u0000\u0019dk\u0012q\n9\u00008\r2\nk\u00003\u0013\n\u00002J(q2\n1+q2\n2\u00003p\n2q1q2): (69)\nWe remark that, at half filling, we shall not consider the con-\nstant term\u0000JNcin Eq. (51), with the purpose of compari-\nson with preceding results. Performing the integration over\nthe first BZ and taking S= 1=2, we obtain that the GS en-\nergy per site at zero-field is given by \u00000:4869J. This result\nagrees very well with values obtained using exact diagonal-\nization [45] (\u00000:485J) and DMRG [67] ( \u00000:4847J) tech-\nniques. For the spin- 1=2- spin-1 chain, the value obtained\nusing DMRG [62] is \u00000:72704J. To compare it with our find-\ning, we need to multiply this value by 2=3(ratio between the\nnumber of sites of the two chains), yielding \u00000:48469J.\nC. Sublattice magnetizations and Lieb GS total spin per unit\ncell\nIn order to derive results beyond LSWT, we intro-\nduce staggered magnetic fields coupled to spins SA;z\ni\nandSBl;z\ni, withl= 1;2, through the Zeeman terms:\n\u0000hAP\niSA;z\niand\u0000hBlP\niSBl;z\ni, which are added to HJ\neff\nin Eq. (47). Thus, hSA;ziandhSBl;zicorresponding\nto sublattices AandBlare obtained from hSA;zi=\n\u0000(1=Nc)P\ni=1;2[@Ei(hA)=@hA]jhA=0, and an analogous\nequation forhSBl;ziusing Eqs. (59) and (62):\n(hSA;zi;hSBl;zi) =\u0007S\u0006\u00121\n2;1\n4\u00131\n\u0019\u0002\u0019\n\u0000\u0019dkv2\nk\n\u0007\u00121\n2;1\n4\u0013q1\n\u0019S\u0002\u0019\n\u0000\u0019dk\r2\nk\n(9\u00008\r2\nk)3=2+O(1\nS2):(70)\nCarrying out the above integration, we obtain hSA;zi=\n\u00000:316343 andhSBl;zi= 0:408172 . These results are in\ngood agreement with those obtained using DMRG [11] and\nED [5] techniques: hSA;zi=\u00000:2925 andhSBl;zi=\n0:3962 , respectively, and with values for hSA;ziand\n2hSBl;zifor the spin- 1=2- spin-1 chain [62–65]. Although\nat zero temperature, the sublattice magnetizations are strongly\nreduced by quantum fluctuations, as compared with their clas-\nsical values, the unit cell magnetization remains SL\u00111=2,\nwhereSLis the Lieb GS total spin per unit cell, in full agree-\nment with Lieb’s theorem [5, 15] for bipartite lattices:\nSL=1\n2kNA\u0000NBk; (71)\nwithNA(NB)denoting the total number of spins in sublattice\nA(B)per unit cell.10\nV .t-JHAMILTONIAN: DOPING-INDUCED PHASES,\nGROUND STATE ENERGY AND TOTAL SPIN\nIn this section, we shall derive the corresponding t-J\nHamiltonian suitable to describe the strongly correlated AB2\nHubbard chain in the doped regime, in which case both charge\n(Grassmann fields) and spin [ SU(2)gauge fields] quantum\nfluctuations are considered on an equal footing. Indeed, the\nt-JHamiltonian can be derived by means of the following\nLegendre transformation to Eq. (41):\nHt-J\neff=\u0000X\ni;\u0016=b;d;e@Leff\n@(@\u001cU(\u0016)\ni)\u001b;\u001b(@\u001cU(\u0016)\ni)\u001b;\u001b\n\u0000X\ni;\u0017i@Leff\n@(@\u001c\u0017i)@\u001c\u0017i+Leff; (72)\nwhere@Leff\n@(@\u001c\u0017i)=\u0017y\niwith\u0017i=\u000bi;\u000b1\n2\ni;ei\";@Leff\n@(@\u001cU(b)\ni)\u001b;\u001b=\n\u0012(\u0000\u001b)(U(b)y\ni)\u001b;\u001b\u000b(1\n2)y\ni\u000b(1\n2)\ni, and@Leff\n@(@\u001cU(d;e)\ni)\u001b;\u001b=\n\u0012(\u001b)1\n2[(U(d;e)y\ni )\u001b;\u001b(\u000by\ni\u000bi+ey\ni\"ei\") + (U(e;d)y\ni )\u001b;\u001b(\u000by\niei\"+\ney\ni\"\u000bi)], from which we can write the effective t-JHamilto-\nnian as\nHt-J\neff=Ht+HJ; (73)\nwhere\nHt=\u0000tX\ni\u001bf\u0012(\u0000\u001b)(U(b)y\niU(d)\ni)\u001b;\u0000\u001b\u000b(1=2)y\ni\u000bi\n+\u0012(\u001b)(U(d)y\niU(b)\ni+1)\u001b;\u0000\u001b\u000by\ni\u000b(1=2)\ni+1\n+\u0012(\u0000\u001b)(U(b)y\niU(e)\ni)\u001b;\u0000\u001b\u000b(1=2)y\niei\"\n+\u0012(\u001b)(U(e)y\niU(b)\ni+1)\u001b;\u0000\u001bey\ni\"\u000b(1=2)\ni+1+H.c.g; (74)\nand\nHJ=\u0000J\n4X\ni;i0=i;i+1;\u001b\u0012(\u001b)j(U(d)y\niU(b)\ni0)\u001b;\u001bj2\u000by\ni\u000bi\n\u0000J\n4X\ni;i0=i;i+1;\u001b\u0012(\u001b)j(U(e)y\niU(b)\ni0)\u001b;\u001bj2ey\ni\"ei\"\n\u0000J\n4X\ni;i0=i;i\u00001;\u001b\u0012(\u0000\u001b)[j(U(b)y\niU(d)\ni0)\u001b;\u001bj2\n+j(U(b)y\niU(e)\ni0)\u001b;\u001bj2]\u000b(1\n2)y\ni\u000b(1\n2)\ni: (75)\nNotice that Eqs. (74) and (75) are identical to Eqs. (42c) and\n(42d), since Eqs. (42a) and (42b) were eliminated through the\nLegendre transformation.\nSome digression on Ht-J\neffis in order. One of the key prop-\nerties of quasi-1D interacting quantum systems is the phe-\nnomenon of spin-charge separation, leading to the formation\nof spin and charge-density waves, which move independently\nand with different velocities. It has been demonstrated [24]\nthat for\u000e > 2=3the low-energy physics of the doped AB2\nHubbard chain in the U=1coupling limit is described interms of the Luttinger-liquid model, with the spin and charge\ndegrees of freedom decoupled. Most importantly, it has been\nshown that for the AB2t-JHubbard chains [25] charge and\nspin quantum fluctuations are practically decoupled, as sug-\ngested by the emergence of charge-density waves in anti-\nphase with the modulation of the ferrimagnetic order. One\ncan make use of this feature to formally split each term of the\nt-JHamiltonian, Eq. (73)-(75) , into a product of two inde-\npendent terms acting on different Hilbert spaces, i.e., we can\nenforce spin-charge separation and calculate the charge and\nspin correlation functions in a decoupled fashion.\nTherefore, from the above discussion, we shall consider\nthat the charge correlation functions are well described by an\neffective spinless tight-binding model [24, 55, 68], since the\nhole (charge) density waves develop along the x-axis and in\nanti-phase with the modulation of the ferrimagnetic structure,\nas numerically observed in Fig. 2(b) of Ref. [25]. So, using\nEqs. (33), with a=2!a(effective lattice spacing of the lin-\near chain: distance between AandBsites, see Fig. 2(a) of\nRef. [25]), we find\nh\u000b(1=2)y\ni\u000bii=1\nNcX\nkk0e\u0000ik(xi\u00001)eik0xih\t0j\u000by\nk\u000bk0j\t0i\n=1\n\u0019\u0002kF(\u000e)\n\u0000kF(\u000e)eikdk=2\n\u0019sin[kF(\u000e)]; (76)\nwithj\t0ibeing the hole-doped ferrimagnetic GS,\nwherekF(\u000e) =\u0019Nh\nN\u0011\u0019\u000e is the Fermi wave vec-\ntor of the spinless tight-binding holes. In the\nsame fashion: h\u000by\ni\u000b(1=2)\ni+1i=2\n\u0019sin[kF(\u000e)] and\nh\u000b(1=2)y\niei\"i=hey\ni\"\u000b(1=2)\ni+1i= 0; whileh\u000by\ni\u000bii =\nh\u000b(1=2)y\ni\u000b(1=2)\nii=hey\ni\"ei\"i= (1\u00001\n2\u0019\u0001kF(\u000e)\n\u0000kF(\u000e)dk) =\n(1\u0000\u000e). Here, we remark that the itinerant holes away from\nhalf filling are associated with the lower-energy dispersive \u000bk\nand\u000b(1=2)\nkbands [see Fig. 1(b) in Sec. (II)], thus contributing\nto the kinetic Hamiltonian in Eq. (74). On the other hand,\nthe local correlations related to the lower-energy bands\n\u000bk,\u000b(1=2)\nk, andek\", contribute equally to the exchange\nHamiltonian in Eq. (75). Thereby, using the above tight-\nbinding results for the charge correlation functions, Ht-J\neff\nin Eqs. (73)-(75) gives rise to the \u000e-dependent Hamiltonian,\nHt-J\neff(\u000e) =Ht\neff(\u000e) +HJ\neff(\u000e), written below:\nHt-J\neff(\u000e) =\u0000t2\n\u0019sin[kF(\u000e)]X\ni[(U(b)y\niU(d)\ni)#\"\n+ (U(d)y\niU(b)\ni+1)\"#+H.c.]\n\u0000J(1\u0000\u000e)\n4X\nhi\u000b;j\fi\u001b\u0012(pi\u000b\u001b)j(Uy\ni\u000bUj\f)\u001b;\u001bj2;(77)\nwhere the sum over \u001bwas evaluated in Eq. (74) and the square\nof theSU(2)gauge field products in the exchange contribu-\ntion have been summed up in Eq. (75), so that this contribu-\ntion is just (1\u0000\u000e)timesHJ\neffat half filling, Eq. (45), or\nalternatively, in terms of spin fields, Eq. (47), or spin-waves,\nEqs. (66)-(68). On the other hand, the SU(2)gauge fields ma-11\ntrix elements:\u0010\nU(b)y\niU(d)\ni\u0011\n#\"and\u0010\nU(d)y\niU(b)\ni+1\u0011\n\"#, that ap- pear in the kinetic contribution of Eq. (77), can be written in\nterms of the spin fields [55, 56] as\n(U(b)y\niU(d)\ni)#\"+H.c.=X\nl1p\n2\u0012q\n1\u00002SBl;z\ni\u00002SA;z\ni+ 4SA;z\niSBl;z\ni+q\n1 + 2SA;z\ni+ 2SBl;z\ni+ 4SA;z\niSBl;z\ni\u0013\n;(78)\nand\u0010\nU(d)y\niU(b)\ni+1\u0011\n\"#is obtained from Eq. (78) through the replacement SA;z\ni!SA;z\ni+1, in which case we took advantage of the\nU(1)gauge freedom and Eq. (46). Notice that these square-root matrix elements depend on z-spin components only.\nAt this stage, it will prove useful, in the calculation of the GS total spin in the doped regime, to consider Ht-J\neff(\u000e;h)which\ndescribes the system in the presence of a homogeneous magnetic field h=h^ z= (\u0000hA+hB1+hB2)^ z, where the staggered fields\npoint along the local corresponding magnetizations in the ferrimagnetic phase have the same magnitude h. The magnetic field\ncouple with the spin fields through the Zeeman term (see Sec. IV C) andwith the charge degrees of freedom through the magnetic\norbital coupling in the Landau gauge: A=hx^ y. Since our aim is to study doping effect on the magnetization, we shall assume\nvanishingly small magnetic field in the context of linear response theory and perturbative expansion in the strong-coupling\nregime. Additionally, the magnetic orbital coupling can be considered through the so-called Peierls substitution [27, 69]: t!\ntei\u0001j\f\ni\u000bA\u0001dl, wherei\u000bandj\fare first-neighbor sites, and the flux quantum \u001e0=hc=e\u00111. If one consider that the carrier is\nat the siteiA, we have four hopping possibilities: iA!iB1;2andiA!(i+ 1)B1;2, so the total phase \u001eacquired by the\ncarrier in this prescription satisfies Stokes’ theorem: \u001e=\u0017\nunit cellA\u0001dl=s\nSh\u0001dS=ha2(a\u00111). We also remark that,\nin order to obtain real values for the zero-field staggered magnetizations, we have considered, for convenience, an imaginary\ngauge transformation [56, 70]: A!iA. Therefore, by placing Eq. (78) and the similar matrix element into the kinetic term in\nEq. (77), making the above Peierls substitution, and using the Holstein-Primakoff and Bogoliubov transformations introduced in\nEqs. (48)-(50) and Eqs. (56)-(57), respectively, up to order O(S\u00001), we arrive at the following diagonalized kinetic Hamiltonian\nHt\neff(\u000e;h):\nHt\neff(\u000e;h) =\u00004p\n2\n\u0019te\u0000(\u0000hA+hB1+hB2)sin[kF(\u000e)]X\nk[4S\u00003v2\nk\u0000(u2\nk+ 2v2\nk)\u000by\nk\u000bk\u0000(2u2\nk+v2\nk)\fy\nk\fk\u0000\u0018y\nk\u0018k];(79)\nwhere the doped-induced contributions for the spin dispersion relations are evidenced in the last three terms. On the other hand,\nby adding the Zeeman terms (see Sec. IV C) to the exchange contribution HJ\neff(\u000e), given in Eq. (77), we obtain HJ\neff(\u000e;h).\nLastly, by adding the kinetic and the exchange contributions, we arrive at the effective t-JHamiltonian in the presence of a\nmagnetic field:\nHt-J\neff(\u000e;h) =\u00004p\n2\n\u0019te\u0000(\u0000hA+hB1+hB2)sin[kF(\u000e)]X\nk(4S\u00003v2\nk) +J(1\u0000\u000e)(EJ\nGS\u0000JNc)\n+X\nk[\u000f(\u000b)\nk(\u000e)\u000by\nk\u000bk+\u000f(\f)\nk(\u000e)\fy\nk\fk+\u000f(\u0018)\nk(\u000e)\u0018y\nk\u0018k]\u0000hAX\niSA;z\ni\u0000hB1X\niSB1;z\ni\u0000hB2X\niSB2;z\ni;(80)\nwhereEJ\nGSis given by Eq. (67) and (69), and the corresponding spin-wave modes [see Eqs. (79), (60), (63)-(65), and (68)] of\nthe dopedAB2t-Jchain read:\n\u000f(\u000b)\nk(\u000e) =4p\n2\n\u0019tsin(\u0019\u000e)[u2\nk+ 2v2\nk] + (1\u0000\u000e)(\u000f0(\u000b)\nk+\u000e\u000f(\u000b)\nk); (81)\n\u000f(\f)\nk(\u000e)is obtained from \u000f(\u000b)\nk(\u000e)through the exchange uk$vkand the replacement \u000b!\f, while\n\u000f(\u0018)\nk(\u000e) =4p\n2\n\u0019tsin(\u0019\u000e) + (1\u0000\u000e)(\u000f0(\u0018)\nk+\u000e\u000f(\u0018)\nk): (82)\nWe find it instructive to comment on the analytical structure of the above equations. Firstly, we mention the presence of the\nBogoliubov parameters [see Eqs. (57)] in a symmetric form in the kinetic terms of Eq. (81) and its analogous for \u000f(\f)\nk(\u000e); besides,\nalthough the flat mode is strongly affected by the presence of holes, it remains dispersionless. In addition, using Eqs. (80) and\n(65), the total GS energy (no spin-wave excitations) per unit cell in the thermodynamic limit is readily obtained:\nEt-J\nGS(\u000e;h)=Nc=\u00004p\n2\n\u0019te\u0000(\u0000hA+hB1+hB2)sin(\u0019\u000e)(4S\u00003q1)\n+ (1\u0000\u000e)(EJ\nGS=Nc\u0000J)\u0000hSA;zihA\u0000X\nl=1;2hSBl;zihBl; (83)12\nwherehSA;ziandhSBl;ziare the calculated sublattice magnetizations, at half filling and zero-field, given by Eqs. (70).\nIn subsections V A, V B, and V C, we will show that the\nunderlying competing physical mechanisms: the magnetic\norbital response and the Zeeman contribution embedded in\nEqs. (80)-(83) will dramatically affect the behavior of the sys-\ntem under hole doping and, in particular, will lead to spiral\nIC spin structures, the breakdown of the spiral ferrimagnetic\nGS at a critical value of the hole doping, a region of phase\nseparation, and RVB states at \u000e\u00191=3.\nA. Doped regime: Spin-wave modes\nBefore we go one step further to discuss relevant macro-\nscopic quantities, i.e., the GS energy and total spin in the\ndoped regime, we shall first undertake a detailed study, at a\nmicroscopic level, of the hole-doping effect on the calculated\nspin-wave branches given by Eqs. (81)-(82).\nFig. 2 depicts the second-order spin-wave dispersion rela-\ntions atJ=t= 0:3and for the indicated values of \u000e. Without\nloss of generality, we set t= 1in our numerical computations.\nAt half filling, the antiferromagnetic mode \u000f(\f)\nk, together with\nthe two ferromagnetic modes: the dispersive \u000f(\u000b)\nkand the flat\none\u000f(\u0018)\nk, are shown in Fig. 2(a), which are defined in Eq. (68),\nand can be plotted using Eqs. (60) and (63)-(65).\nAs the hole doping increases slightly, the abrupt decrease\nof the peaks at k= 0 andk=\u0019of the numerical DRMG\nstructure factor (see Fig. 3 of Ref. [25]), associated with the\nferrimagnetic order, manifests itself here through the opening\nof a gap in the ferromagnetic Goldstone mode \u000f(\u000b)\nk, as seen in\nFig. 2(b), thus indicating that the system loses its long-range\norder. Note that the antiferromagnetic mode \u000f(\f)\nkis also sim-\nilarly shifted. On the other hand, although the dispersion re-\nlation is modified for small values of the wave vector k, the\nminimum value of \u000f(\u000b)\nkstill remains at k= 0 up to the onset\nof the formation of spiral IC spin structures at \u000ec(IC) = 0:043\n(a value that should be compared with the numerical DMRG\nestimate of\u000e\u00190:055\u00060:012), characterized by the flatten-\ning of the dispersive spin-wave branches around zero. Upon\nfurther increase of \u000e, two minima form (around k= 0) and\nmove away from each other as one enhances the hole doping.\nThis behavior is the signature of the occurrence of spiral IC\nspin structures (see Fig. 3 of Ref. [25]).\nFig. 2(c) shows the onset of phase separation (PS) at\n\u000e(PS) = 0:165forJ= 0:3, which is characterized by the\noverlap of the two ferromagnetic modes at k= 0. The sig-\nnature of this regime is the spatial coexistence of two phases:\nspiral IC spin structures at \u000e(PS) = 0:165and RVB states\nat\u000e\u00191=3, in very good agreement with the numerical es-\ntimate of\u000eIC\u0000PS\u00190:16[25]. At\u000e\u00191=3, the flat mode\nhas the lowest energy, as illustrated in Fig. 2(d). This be-\nhavior indicates that the RVB state is the stable phase at\n\u000e\u00191=3andJ= 0:3[25], and also in agreement with the\nnumerical DMRG studies [11, 24] and analytical prediction at\nU=1[55].\nOnset of IC\nOnset of PS\n(b) (a)\n(c) (d)Figure 2. (Color online) Evolution of the zero-field second-order\nspin-wave dispersion relations of the AB2t-Jchain as a function of\nhole doping ( \u000e): dispersive ferromagnetic \u000f(\u000b)\nkand antiferromagnetic\n\u000f(\f)\nkmodes and the flat ferromagnetic one \u000f(\u0018)\nk, at (a) half filling; (b)\nthe onset of the spiral IC spin structures at \u000ec(IC) = 0:043, in which\ncase the flattening of the gap of \u000f(\u000b)\nkaroundk= 0 is observed; (c)\nthe onset of PS at \u000e(PS) = 0:165, characterized by the overlap of\nthe two ferromagnetic modes at k= 0and by the spatial coexistence\nof two phases: spiral IC spin structures, with modulation fixed at\n\u000e(PS) , and RVB states at \u000e\u00191=3. (d) At\u000e= 1=3the flat mode\npresents the lowest energy, thus indicating that the short-range RVB\nstate is the stable phase.\nIn order to better understand the rich variety of doping-\ninduced phases in the system, in Fig. 3 we plot the evolution of\nthe wave vector kmincorresponding to the local minimum of\n\u000f(\u000b)\nk(\u000e), upon increasing the hole doping \u000efrom 0to1=3. The\nwave vector kminremains zero until it hits the onset doping\nvalue\u000ec(IC) = 0:043, beyond which a square-root growth be-\nhavior takes place [71]: [\u000e\u0000\u000ec(IC)]1=2(blue line), for \u000eclose\nto\u000ec(IC). The square-root growth behavior is the signature\nof the occurrence of a second-order quantum phase transition\nfrom the doped ferrimagnetic phase to the IC spiral ferrimag-\nnetic state with a non-zero value of the total GS spin, SGS.\nThis result is supported by the behavior of \u0001k\u0011kmax\u0000\u0019\nat which the local maximum of the numeric DMRG structure\nfactorS(k)neark=\u0019is observed, as shown in the inset of\nFig. 3 (taken from the inset of Fig. 3(b) of Ref. [25]). For\nfurther increase of hole doping our result deviates from the\nsquare-root growth behavior and some very interesting fea-\ntures are to be noticed. The value of \u000ec= 0:08indicates\nthe breakdown of the total SGSin the IC phase, as will be\nconfirmed by the explicit calculation of SGS, a macroscopic\nquantity, in Section V C. Thus, for 0:08<\u000e < 0:165the sys-\ntem displays an IC phase with zero SGS, in agreement with\nthe DMRG data (see Fig. 1(c) of Ref. [25]). At \u000e(PS) = 0:16513\n0 0.043 0.08 0.165 1/300.10.20.3kmin/π\nDoped FerriIC\nSGS/negationslash= 0IC\nSGS= 0PS\nδc(IC)δcδδ(PS)∼\nRVB\nAnalytic\n[δ−δc(IC)]1/2\n0 0.05 0.1δ00.10.20.30.4\n∆k/π\nFigure 3. (Color online) Evolution of kmin(value ofkat the local\nminimum of \u000f(\u000b)\nk(\u000e)neark= 0) as a function of \u000e: doped ferri-\nmagnetism for 0< \u000e < \u000ec(IC)\u00190:043; spiral IC spin structures\nwith non-zero (zero) SGSfor\u000ec(IC)< \u000e < \u000ec\u00190:08(\u000ec< \u000e <\n\u000e(PS)\u00190:165), with a second-order quantum phase transition at\n\u000ec(IC) characterized by a square-root behavior [\u000e\u0000\u000ec(IC)]1=2(blue\nline), and a first-order transition at \u000e(PS) involving the IC spin struc-\nture, with modulation fixed at \u000e(PS) , and short-range RVB states at\nhole concentration 1/3. The inset shows DMRG data from Ref. [25]\nfor\u0001k\u0011kmax\u0000\u0019as a function of \u000e, wherekmaxis the value of\nkat the local maximum of the structure factor S(k)neark=\u0019, in\nqualitative agreement with the second-order transition at \u000ec(IC) .\nthe system exhibits a first-order transition accompanied by the\nspatial phase separation regime: the IC phase with zero SGS\nand modulation fixed by \u000e(PS) in coexistence with the short-\nrange RVB states at \u000e\u00191=3, also consistent with the DMRG\ndata plotted in Fig. 4 of Ref. [25].\nLastly, we emphasize that, despite the occurrence of several\ndoping-induced phases in the DMRG studies [25]: Lieb fer-\nrimagnetism, spiral IC spin structures, RVB states with finite\nspin gap, phase separation, and Luttinger-liquid behavior, it\nis surprising and very interesting that the second-order spin-\nwave modes remain stable up to \u000e\u00191=3, with predictions\nin very good agreement with the DMRG studies [25]. In this\ncontext, it is worth mentioning the long time studied case of\nrare earth metals [72], where an external magnetic field can in-\nduce non-trivial phase transitions involving spiral spin struc-\ntures, well described by spin-wave theory.\nB. Doped regime: Ground state energy\nPerforming the integration over the first BZ in Eqs. (65) and\n(69) and setting S= 1=2in Eq. (83), we find that the AB2t-J\nground state energy per unit cell as a function of hole doping\nin zero-field reads:\nEt-J\nGS(\u000e)=JNc=\u00001:9543t\nJsin (\u0019\u000e)\u00002:4608 (1\u0000\u000e):(84)\nWe shall now examine the case of small hole doping away\nfrom half filling, i.e., with hole concentration ranging from\n\u000e= 0 up to\u000e= 0:2for two values of J:0:1and0:3. In\nFig. 4, we show the evolution of the GS energy per unit cell\n0 0.05 0.1 0.15 0.2\nδ0\n-5\n-10\n-15Et-J\nGS(δ)/JNc\nAnalyticJ= 0.3\nNumericJ= 0.3\nAnalyticJ= 0.1\nNumericJ= 0.1Figure 4. (Color online) Analytical prediction for the GS energy per\nunit cell of the AB2t-Jchain as a function of doping, and compar-\nison with numerical data from DMRG technique for J=t= 0:1and\nJ=t= 0:3[25]. At half filling ( \u000e= 0), both results meet at the\nexpected prediction [25]: \u0019\u00002:4678 . Note that we have added the\nterm\u0000JNcwith the intention of comparison with numerical calcu-\nlation.\nof theAB2t-Jmodel as a function of hole doping for both\nmentioned values of J, and the comparison was made with the\nnumerical DMRG data [25]. From the two results at J= 0:3,\nthe only quantitative difference induced by the increase of the\nhole concentration is a crossing feature around \u000e\u00190:1, where\nour analytical result slightly change its behavior by lowering\nthe energy with respect to the numerical data [25]. In fact, be-\ncause our model assumes a ferrimagnetic state as the starting\npoint, this change of behavior suggests that we have entered in\na region of strong magnetic instabilities, and possibly indicat-\ning a smooth transition to an incommensurate phase with zero\nGS total spin beyond \u000e\u00190:1, as confirmed by the numerical\ndata in Ref. [25] and illustrated in Fig. 3. On the other hand,\natJ= 0:1, although our results reproduce the numerical data\nwith an acceptable agreement, we observe a discrepancy that\nincreases with \u000e. The cause of such discrepancy will be dis-\ncussed in the next subsection.\nWith the purpose of determining the interplay between the\ncontribution of magnetic exchange and the itinerant kinetic\nenergy to the zero-field GS energy Eq. (83), we take J= 0:3\nand show its evolution with doping in Fig. 5. We can see\nin the insets, Fig. 5(a) and Fig. 5(b), the competitive behav-\nior of the two energetic contributions, i.e., the contribution of\nthe exchange energy increases linearly with \u000e, while a practi-\ncally linear decrease of the hopping term is observed as one\nenhances the hole doping. This competition indicates that a\nphase transition to a paramagnetic phase should occur at some\ncritical concentration value.\nC. Doped regime: Ground state total spin\nThe existence of a transition from an IC spiral ferrimag-\nnetic phase to an IC paramagnetic one is a most interest-14\n0 0.05 0.1 0.15 0.2\nδ0\n-2\n-4\n-6Et-J\nGS(δ)/JNc\nJ= 0.3\nExchange\nHopping0 0.05 0.1 0.15 0.2\nδ-2\n-2.2\n-2.4EJ(δ)/JNc\n0 0.05 0.1 0.15 0.2\nδ0\n-1\n-2\n-3Et(δ)/JNc\nFigure 5. (Color online) Ground-state energy per unit cell for the\nAB2t-Jchain as a function of \u000eforJ= 0:3. In the insets, we\nillustrate the two energetic contribution due to (a) exchange and (b)\nhopping terms.\ning feature observed numerically in doped AB2t-JHubbard\nchains [25]. In order to firmly corroborate the mentioned\ntransition, we have calculated the GS total spin per unit cell\nas a function of hole doping, SGS(\u000e) =P\n\u000bhS\u000b;zi(\u000e),\nwith\u000b=A;B 1;B2, by means of the zero-field derivative\nof Eq. (83):\nhS\u000b;zi(\u000e) =\u0000(1=Nc)[@Et-J\nGS(\u000e;h)=@h\u000b]jh\u000b=0:(85)\nWe thus find\nhS\u000b;zi(\u000e) =hS\u000b;zi\u00064p\n2\n\u0019tsin(\u0019\u000e)(4S\u00003q1);(86)\nwhere +(\u0000) corresponds to sublattice \u000b=A(\u000b=B1;2),\nandhSA;ziandhSBl;ziare given by Eqs. (70). Therefore,\nby performing the integration over the first BZ of the three\ncontributions in Eq. (86), we finally obtain:\nSGS(\u000e)\nSL= 1\u00003:9086 sin(\u0019\u000e); (87)\nwhereSL=P\n\u000bhS\u000b;zi= 1=2is Lieb’s reference value for\nthe GS total spin per unit cell at half filling and zero-field (see\nSection IV C).\nIn Fig. 6 we plot the evolution of SGS, normalized by SL,\nas a function of \u000e, and compare it with the numerical data\nfrom DMRG and Lanczos techniques [25], for J= 0:3(red\nsquares) and J= 0:1(blue circles). In the latter (former) case,\nthe system undergoes a transition from the modulated itin-\nerant ferrimagnetic phase to an incommensurate phase with\nzero (nonzero) SGS. Notice that, in both cases, the transition\nis characterized by a decrease of SGSfromSLto0or to a\nresidual value, regardless of the value that SGStakes after the\ntransition. Indeed, at J= 0:1and\u000e > 0:1, the formation\nof magnetic polarons (onset of the Nagaoka phenomena that\nsets in asU!1 ) with charge-density waves in phase with\nthe modulation of the ferrimagnetic structure, as indicated by\nFigure 6. (Color online) Ground-state total spin SGSper unit cell\n(solid magenta line), normalized by its value in the undoped regime:\nSL=1\n2, as a function of hole doping \u000efor the indicated values of\nJ. In the figure, \u000ec\u00190:08indicates the critical value of doping\nat which the magnetic order is suppressed and a second-order phase\ntransition takes place.\nthe DMRG data [25], leads to an incommensurate phase with\nnonzeroSGS.\nMost importantly, we can observe in Fig. 6 that the value of\nSGSdecreases practically linearly with \u000euntil the magnetic\norder is completely suppressed at \u000ec\u00190:08. This behavior is\nsupported by numerical results [25], particularly in the regime\nwhere the Nagaoka phenomenon is not manifested, that is, at\nJ= 0:3, as indicated in Fig. 3. In this regime, spin and charge\nquantum fluctuations destabilize the ferrimagnetic structure\nand trigger a transition to an incommensurate paramagnetic\nphase at\u000ec, withSGS\u0018(\u000e\u0000\u000ec)!0.\nVI. CONCLUSIONS\nIn summary, we have presented a detailed analytical study\nof the large-U Hubbard model on the quasi-one-dimensional\nAB2chain. We used a functional integral approach combined\nwith a perturbative expansion in the strong-coupling regime\nthat allowed us to properly analyze the referred system at and\naway from half filling.\nAt half filling, our model was mapped onto the quantum\nHeisenberg model, and analyzed through a spin-wave pertur-\nbative series expansion in powers of 1=S. We have demon-\nstrated that the GS energy, spin-wave modes, and sublat-\ntice magnetizations are in very good agreement with previ-\nous results. In the challenging hole doping regime away\nfrom half filling, the corresponding t-J(= 4t2=U)Hamilto-\nnian was derived. Further, under the assumption that charge\nand spin quantum correlations are decoupled, the evolution\nof the second-order spin-wave modes in the doped regime\nhas unveiled the occurrence of spatially modulated spin struc-\ntures and the emergence of phase separation (first-order tran-\nsition) in the presence of resonating-valence-bond states. The\ndoping-dependent GS energy and total spin per unit cell are15\nalso calculated, in which case the collapse of the spiral mag-\nnetic order at a critical hole concentration was observed.\nRemarkably, our above-mentioned analytical results in the\ndoped regime are in very good agreement with density matrix\nrenormalization group studies, where our assumption of spin-\ncharge decoupling is numerically supported by the formation\nof charge-density waves in anti-phase with the modulation of\nthe ferrimagnetic structure.\nFinally, we stress that our reported results evidenced that\nthe present approach, also used in a study on the compatibil-\nity between numerical and analytical outcomes of the large-U\nHubbard model on the honeycomb lattice, was proved suitablefor theAB2chain (a quasi-1D system), where the impact of\ncharge and spin quantum fluctuations are expected to manifest\nin a stronger way. 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Elliott\n(Plenum Press, 1972), Chapter 1, p. 187."    },    {        "title": "1901.03072v2.Ultrafast_magnetization_dynamics_in_uniaxial_ferrimagnets_with_compensation_point__GdFeCo.pdf",        "content": "Ultrafast magnetization dynamics in uniaxial ferrimagnets with compensation point.\nGdFeCo\nM. D. Davydova,1, 2,\u0003K. A. Zvezdin,1, 2A. V. Kimel,3, 4and A. K. Zvezdin1, 2,y\n1Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia\n2Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia\n3Moscow Technological University (MIREA), 119454 Moscow, Russia\n4Institute for Molecules and Materials, Radboud University, Nijmegen 6525 AJ, The Netherlands\nWe derive an e\u000bective Lagrangian in the quasi-antiferromagnetic approximation that allows to\ndescribe the magnetization dynamics for uniaxial f-d(rare-earth - transition metal) ferrimagnet\nnear the magnetization compensation point in the presence of external magnetic \feld. We perform\ncalculations for the parameters of GdFeCo, a metallic ferrimagnet with compensation point that is\none of the most promising materials in ultrafast magnetism. Using the developed approach, we \fnd\nthe torque that acts on the magnetization due to ultrafast demagnetization pulse that can be caused\neither by ultrashort laser or electrical current pulse. We show that the torque is non-zero only in\nthe non-collinear magnetic phase that can be acquired by applying external magnetic \feld to the\nmaterial. The coherent response of magnetization dynamics amplitude and its timescale exhibits\ncritical behavior near certain values of the magnetic \feld corresponding to a spin-\rop like phase\ntransition. Understanding the underlying mechanisms for these e\u000bects opens the way to e\u000ecient\ncontrol of the amplitude and the timescales of the spin dynamics, which is one of the central problems\nin the \feld of ultrafast magnetism.\nPACS numbers: 79.20.Ds, 75.50.Gg, 75.30.Kz, 75.78.-n\nINTRODUCTION\nMost of the prominent advances in the \feld of ultrafast magnetism have been achieved by using thermal mechanism\nof magnetization control [1{7]. These studies rooted from the pioneering work by Beaurepaire et al. [8] on ultrafast\nlaser-induced demagnetization of Ni. In this experiment, partial destruction of magnetic order was found at much\nfaster rates that were believed to be possible prior to that publication. Since then, the \feld of ultrafast magnetism has\nbeen rapidly growing and the possible channels of ultrafast angular momentum transfer have been studied extensively\n[9]. Ultrafast demagnetization can be achieved by applying ultrashort laser pulses [1{7], or, alternatively, by using\nshort pulses of electric currents [10, 11].\nIn the last decades, GdFeCo and other rare-earth - transition metal compounds (RE-TM) have been in the center\nof attention in this regard [12]. For example, all-optical switching has been demonstrated for the \frst time in GdFeCo\n[13]. It was found that the switching is possible due to di\u000berent rates of sublattice demagnetization, which enables\nultrafast magnetization reversal to occur because of the angular momentum conservation [1, 2].\nIn many RE-TM compounds, GdFeCo and TbFeCo being part of them, realization of the magnetization compen-\nsation point is possible. At this point, the magnetizations of the two antiferromagnetically coupled sublattices with\ndi\u000berent dependencies on temperature become equal and the total magnetization of the material turns to zero. In\nthe presence of the external magnetic \feld, a record-breaking fast subpicosecond magnetization switching was found\nin GdFeCo across the compensation point [13]. In addition, a number of anomalies in the magnetic response was\nobserved near this point [14{16], which has never been explained theoretically. All said above illustrates the impor-\ntance of understanding the role of the compensation point in the dynamics and working out an appropriate tool for\nits description.\nE\u000ecient control of the amplitude and the timescales of the response of the magnetic system to an ultrafast de-\nmagnetizing impact on a medium is one of the most important issues in the area of ultrafast magnetism nowadays\n[9, 14, 15]. Understanding of the mechanisms and of the exhaustive description of the subsequent spin dynamics is\nalso a long-standing goal that will help to promote the achievements of this area towards practical applications in\nmagnetic recording [3, 17], magnonics [18] and spintronics [19]. In this work, we expand the understanding of response\nof magnetic system of a uniaxial f-dferrimagnet near the compensation point in the external magnetic \feld to an\nultrafast demagnetizing pulse, which can be induced either by a femtosecond laser or an electric current pulse. We\npresent a theoretical model and calculations, which allow to describe the ultrafast response of the system that resides\nin an angular phase before the impact. We show that in this magnetic phase the coherent precessional response is\npossible and the subsequent magnetization dynamics may become greatly nonlinear and is governed by large inter-\nsublattice exchange \feld [20]. We derive the e\u000bective Lagrangian that governs the dynamics of the system near thearXiv:1901.03072v2  [cond-mat.mtrl-sci]  28 Jan 20192\ncompensation point and obtain the torque acting on the magnetizations of the two sublattices due to demagnetiza-\ntion. In ref. [15], the critical response of the amplitude and the time of the signal rise have been found in GdFeCo\nin external magnetic \feld along the easy axis. At given laser pump \ruences, the response was found to be negligible\nin collinear phase, but it was dramatically large in angular one. We elaborate on this example and show that the\ncritical behavior of the response is the consequence of the second-order magnetic phase transition from collinear to an\nangular in the external magnetic \feld. We \fnd that these e\u000bects are pronounced in the vicinity of the compensation\npoint, where the phase transitions cross each other[21{23]. Thus, the proposed model explains a range of important\nexperimental observations as well as allows for developments of methods and tools of magnetization control by setting\nthe temperature near the compensation point and applying magnetic \feld. Moreover, by changing the composition\nof the alloy, the [24], the position of the magnetization compensation point can be tuned arbitrary close to the room\ntemperature. Our results might open new ways for technologies for ultrafast optical magnetic memory.\nEFFECTIVE LAGRANGIAN AND RAYLEIGH DISSPATION FUNCTION\nOur approach is based on Landau-Lifshitz-Gilbert equations for a two-sublattice (RE-TM) ferrimagnet. These\nequations are equivalent to the following e\u000bective Lagrangian and Rayleigh dissipation functions:\nL=Mf\n\r(1\u0000cos\u0012f)@'f\n@t+Md\n\r(1\u0000cos\u0012d)@'d\n@t\u0000\b(Mf;Md;H); (1)\nR=Rf+Rd;Rf;d=\u000bMf;d\n2\r\u0010\n_\u00122\nf;d+ sin2\u0012f;d_'2\nf;d\u0011\n(2)\nwhere\ris the gyromagnetic ratio, MdandMfare the magnetizations, \u0012d(TM) and\u0012f(RE) are the polar, 'dand\n'fare the azimuthal angles of d- andf- sublattices correspondingly in the spherical system of coordinates with z-axis\naligned along the external magnetic \feld H. \b(Mf;Md;H) is the thermodynamic potential for the system that we\ntake in the following form:\n\b =\u0000MdH+\u0015MdMf\u0000MfH\u0000Kf(Mfn)2\nM2\nf\u0000Kd(Mdn)2\nM2\nd; (3)\nwhere\u0015is the intersublattice exchange constant, nis the direction of the easy axis and Kf;dare the anisotropy\nconstants for f- andd- sublattices, respectively.\nNext, we transfer to description in terms of the antiferromagnetic L=MR\u0000Mdand the total magnetization\nM=MR+Mdvectors. In the vicinity of the compensation point the di\u000berence between the sublattice magnetizations\njMR\u0000Mdj\u001cLis small. The two vectors are parametrized using the sets of angles \u0012;\"and';\f, which are de\fned\nas:\n\u0012f=\u0012\u0000\"; \u0012d=\u0019\u0000\u0012\u0000\";\n'f='+\f; 'd=\u0019+'\u0000\f:(4)\nIn this case the antiferromagnetic vector is naturally de\fned as L= (Lsin\u0012cos';Lsin\u0012sin';Lcos\u0012).\nFollowing the work [25] we use the quasi-antiferromagnetic approximation to describe the dynamics near the mag-\nnetization compensation point. F In this approximation the canting angles are small \"\u001c1,\f\u001c1, and we can\nexpand the Lagrangian (1) and the corresponding thermodynamic potential up to quadratic terms in small variables:\nL=\u0000m\n\r_'cos\u0012\u0000M\n\rsin\u0012\u0010\n_'\"\u0000\f_\u0012\u0011\n\u0000\b;\n\b =\u0000K(l;n)2\u0000Hmcos\u0012\u0000\"MH sin\u0012+\u000e\n2\u0000\n\"2+ sin2\u0012\f2\u0001\n:(5)\nHerem=MR\u0000Md,M=MR+Md,K=KR+Kdis the e\u000bective uniaxial anisotropy constant, l=L=Lis\nthe unit antiferromagnetic vector \u000e= 2\u0015MdMRand we assume the anisotropy to be weak K\u001c\u0015M. For GdFeCo\nwith 24% Gd and compensation point near 283 K, we assume the following values of parameters: M\u0019800 emu/cc,\nK= 1:5\u0002105erg/cc,\u0015= 18:5 T/\u0016B,\u000e\u0019109erg/cc and mchanges in the range between 150 emu/cc and \u000050\nemu/cc at \felds H\u0019H\u0003\u00194 T. The characteristic values of small angles \u000fand\fare of the order of 10\u00002.3\nNext, we exclude the variables \"and\fby solving the Euler-Lagrange equations. Substituting them into the\nLagrangian (5), we obtain the e\u000bective Lagrangian, which describes the dynamics of a uniaxial ferrimagnet in the\nvicinity of the compensation point:\nLeff=\u001f?\n2 _\u0012\n\r!2\n+mcos\u0012\u0012\nH\u0000_'\n\r\u0013\n+\u001f?\n2sin2\u0012\u0012\nH\u0000_'\n\r\u00132\n+K(l;n)2; (6)\n\beff(H) =\u0000mHcos\u0012\u0000\u001f?\n2H2sin2\u0012\u0000K(l;n)2; (7)\nReff=\u000bM\n2\r\u0010\n_\u00122+ sin2\u0012_'2\u0011\n(8)\nwhere\u001f=2M2\n\u000e. In GdFeCo \u001f\u00191:6\u000210\u00003and\u000b\u00190:05. In the derivation above we assumed the gyrotropic factor\n\rand Gilbert damping constant \u000bto be the equal for both sublattices. Taking into account the di\u000berence between\nthese values for di\u000berent sublattices will lead to the angular momentum compensation e\u000bect at certain temperature.\nThe Lagrangian, Rayleigh function and equations of motion preserve the same form in this case if we substitute the\nparameters \rand\u000bwith temperature-dependent factors e\rande\u000bde\fned as:\n1\ne\r=1\n\u0016\r\u0012\n1 +M\nm\rf\u0000\rd\n\rf+\rd\u0013\n=Md\n\rd\u0000Mf\n\rf\n(Md\u0000Mf);1\n\u0016\r=1\n2\u00121\n\rd+1\n\rf\u0013\n;e\u000b=(\u000bd\rf+\u000bf\rd)\n(\rf+\rd)1\n1 +M\nm\rf\u0000\rd\n\rf+\rd(9)\nThis allows to reproduce the angular moment compensation phenomenon, which was studied experimentally in ref.\n[14].\nEXCITATION OF THE SPIN DYNAMICS\nThe proposed approach presents a powerful tool allowing analyzing coherent magnetization dynamics in ferrimagnets\nthat occurs under a broad range of conditions. Let us consider the following example that poses an important problem\nin the \feld of ultrafast magnetism. An femtosecond laser pulse strikes the uniaxial ferrimagnet (for instance, of\nGdFeCo, TbFeCo type) in the presence of external static magnetic \feld. The impact of the laser pulse leads to the\ndemagnetization of one or both of the sublattices. What coherent magnetization dynamics will occur as a consequence\nof this impact? The proposed model can be further developed in order to answer to this question and is applicable\nfor small values of demagnetization \u000eM.\nIn our framework the spin dynamics in ferrimagnet is described by Euler-Lagrange equations of the formd\ndt@L\n@_q\u0000@L\n@q=\n\u0000@R\n@_q, whereq=\u0012; ' are the polar and azimuthal angles describing the orientation of the antiferromagnetic vector\nL, correspondingly. Let us consider a particular case when the easy magnetization axis is aligned with the external\nmagnetic \feld, which leads to the presence of azimuthal symmetry in the system. In this case n= (0;0;1). In this\nparticular case the Euler-Lagrange equations can be rewritten as:\n\u001f?\n\r2\u0012=@Leff\n@\u0012\u0000@Reff\n@_\u0012;d\ndt@Leff\n@_'=\u0000@Reff\n@_'(10)\nThe nonlinear equations that are similar to Eqs. (10) and describe the spin dynamics of two-sublattice ferrimagnets\nwere obtained in the work [26] under the conditions H= 0 andReff= 0. Over the short time of demagnetization the\nsecond equation can be approximately treated as a conservation law and the conserving quantity (angular momentum\nof magnetization precession J) stays approximately constant as @L=@'= 0 due to the Noether theorem:\nJ=@Leff\n@_'=\u00001\n\r\u0014\nmcos\u0012+\u001f?sin2\u0012\u0012\nH\u0000_'\n\r\u0013\u0015\n=const (11)\nLet the moment of time t= 0\u0000denote the moment before the laser pulse impact and system initially is in the\nground state de\fned by the ground state angles \u0012(0\u0000) =\u00120,'(0\u0000) ='0, and their derivatives _ '(0\u0000) = 0, _\u0012(0\u0000) = 0.4\nDepending on the external parameters and preparation of the sample, the system might reside in one of the two\npossible antiferromagnetic collinear phases or in angular phase, which are separated by the magnetic phase transition\nlines [27]. If the demagnetization due to the laser pulse action is small, it produces the changes in the values of Mf,Md\nandMof the order of percent or less, whereas the change of m(which is approximately equal to total magnetization\nnear the compensation point) may be of several orders of magnitude, as its value is almost compensated. In what\nfollows, we assume that the demagnetization is associated only with change of m, namelym=m0+ \u0001m(t). As we\nwill see below, the change in this quantity already leads to several drastic e\u000bect in dynamics.\nTherefore, the conservation law (11) leads to the emergence of azimuthal dynamics _ '(t) at the demagnetization\ntimescales (\u0001 t) due to demagnetization pulse \u0001 m(t):\n_'(t) =\r\u0001m(t)\n\u001f?cos\u00120\nsin2\u00120(12)\nWe see that the torque is non-zero only in the angular phase, where 0 <\u00120<\u0019. Emergence of the azimuthal spin\nprecession as a result of demagnetization of the medium is similar to the well-known Einstein-de-Haas e\u000bect, where\nthe demagnetization leads to azimuthal precession of the body. Subsequently, this azimuthal spin dynamics leads to\nthe emergence of polar dynamics \u0012(t), which is most commonly measured in pump-probe experiments of ultrafast\nmagnetism, by acting as an e\u000bective \feld Heff=H\u0000_'\n\rin the Lagrangian (5). We can then view the Lagrangian\nas depending only on variable \u0012and the e\u000bective \feld Heff. At demagnetization \u000em\u00180:01Min GdFeCo the value\nof _'can reach up to 1 THz, and the corresponding e\u000bective magnetic \feld is of the order of 10 T. Note that initial\nstate of the system corresponds to the condition@\b\n@\u00120(Heff=H) = 0. We can rewrite the Euler-Lagrange equation\nfrom eq. (10) for polar angle as follows:\n\u001f?\n\r2\u0012+@\b(Heff)\n@\u0012=\u0000\u000bM\n\r_\u0012: (13)\nOr, alternatively:\n\u001f?\n\r2\u0012+msin\u0012Heff\u0000\u001f?sin\u0012cos\u0012\u0012\nH2\neff\u00002K\n\u001f?\u0013\n=\u0000\u000bM\n\r_\u0012 (14)\nBy integrating this equation over the short demagnetization pulse duration \u0001 twe obtain the state of system after\nthe laser pulse impact at t= 0+, which is characterized by the initial conditions\n\u0012(0+) =\u00120;_'(0+); '(0+) =Z\u0001t\n0_'(t)dt;_\u0012(0+) =Z\u0001t\n0\u0012(t)dt\nThe value \u0001 tis of the order of the optical pulse length. It may also include the time of restoration of the magnetization\nlength (or the value of m). After the moment of time 0+ free magnetization precession occurs in the model. Analysis\nof the spin dynamics under laser pump excitation will lead to emergence of critical dynamics near the second-order\nphase transitions to the collinear phases where \u0012= 0;\u0019, as is already seen from (12). We will discuss this behavior\nbelow.\nCRITICAL DYNAMICS\nIn a simple case of a quick decay of demagnetization (at the exciton relaxation timescales) with \u0001 m(t) = \u0001m,\n0<t< \u0001t; \u0001m(t) = 0,t>\u0001t, we obtain the initial condition from (14):\n_\u0012(+0)\u0019\u0014\n\u0000\u0012\n2cos2\u00120\nsin\u00120+ sin\u00120\u0013\nH+m0\n\u001f?cos\u00120\nsin\u00120+\u0001m\n\u001f?cos\u00120\nsin3\u00120\u0015\r2\n\u001f?\u0001m\u0001t=B(\u00120)\u0001m+O(\u0001m2): (15)\nThis quantity de\fnes the initial angular momentum of the polar spin precession that is induced in the system due to\nthe optical spin torque created by the femtosecond laser pulse. The amplitude of oscillations is proportional to the\ninitial condition (15). Its dependence on the external magnetic \feld is illustrated in Fig. 1 for di\u000berent temperatures\nfor magnetic parameters of GdFeCo uniaxial ferrimagnet. At low values of external magnetic \felds there is only\ncollinear ground state in the ferrimagnet and above certain \feld Hsfthe transition to an angular state occurs [21].\nThe schematic of the magnetic phase diagram for GdFeCo is shown in insertions in Fig. 1. At T= 275 K and T= 2885\nFIG. 1. The amplitude of the magnetization precessional response after the demagnetization due to the femtosecond laser\npulse action in GdFeCo ferrimagnet near the compensation point at di\u000berent temperatures. Insertions: the schematic of the\nmagnetic phase diagram. There are two antiferromagnetic collinear phases with Mddirected along(opposite) to the external\nmagnetic \feld above(below) the compensation temperature TM. They are separated by the \frst-order phase transition line\n(blue). Above them, an area where the angular phase exists, which is \flled with gray color. The black solid lines are the\nsecond-order phase transition lines. The dashed lines corresponds to the \fxed temperature in the plot. The red dot is the point\nof phase transition for this temperature.\nK the phase transitions are of the second order, which corresponds to a smooth transition from angle \u00120= 0 to\u00120>0,\nand the divergence of the response occurs at Hsf. Immediately above the compensation temperature the transition is\nof the \frst order and the behavior of the response above the is more complex; however, there is no critical divergence.\nThe critical behavior of the signal amplitude was observed experimentally for GdFeCo in ref. [15].\nAnother feature in the dynamics described by the proposed model is the critical behavior of the characteristic\ntimescales that occurs in the vicinity of the second-order phase transitions. To demonstrate this e\u000bect analytically,\nwe assume small deviations of \u0012during oscillations: \u0012(t) =\u00120+\u000e\u0012(t). We obtain:\n\u000e\u0012+!2\nr(\u00120)\u000e\u0012=\u0000\u000b!ex\u000e_\u0012; (16)\nwhere!2\nr(\u00120) =\r2h\nm\n\u001f?Hcos\u00120+\u0010\n2K\n\u001f?\u0000H2\u0011\ncos 2\u00120i\n,!ex=\rM\n\u001f?. The initial conditions are \u0012(0) =\u00120and eq.\n(15). In the limit of small oscillations and !r<\u000b!ex=2 (is ful\flled near the second-order transition) the solution has\nthe form\u000e\u0012(t) =Ae\u0000\ftsinh!t, where\f=\u000b!ex=2,!2=\f2\u0000!2\nr,A=B(\u00120)=!. The rise time can be estimated from\nthe condition _\u0012(\u001crise) = 0:\n\u001crise\u0019atanh!\n\f\n!=atanhp\n\f2\u0000!2r\n\fp\n\f2\u0000!2r(17)\nThe time of the oscillations decay (relaxation time) is proportional to the imaginary part of eigenfrequency and can\nbe estimated by the following expression:\n\u001crelax\u00194\u0019\f\n!2r: (18)\nNear second-order phase transition the mode softening occurs and the eigenfrequency turns to zero: !r!0,\nand we observe growth of the both timescales. The critical behavior of the rise time has been observed in GdFeCo\nexperimentally [15] and the typical values of \u001crisewere of the order of 10 ps.6\nCONCLUSIONS\nTo sum up, the developed theoretical model based on quasi-antiferromagnetic Lagrangian formalism proved to\nbe suitable for description of the coherent ultrafast response of RE-TM ferrimagnets near the compensation point\ndue to an ultrashort pulse of demagnetization in the presence of external magnetic \feld. We have found that the\ntorque acting on magnetizations is non-zero in the noncollinear phase only. We have explained the experimentally\nobserved critical behavior of the response amplitude and characteristic timescales as the consequence of the second-\norder magnetic phase transition from collinear to an angular in the external magnetic \feld and the mode softening\nnear it. These e\u000bects are vivid in the vicinity of the compensation point in external magnetic \feld. Understanding\nthe ultrafast response to demagnetizing optical or electrical pulses and subsequent spin dynamics can facilitate future\ndevelopments in the \felds of ultrafast energy-e\u000ecient magnetic recording, magnonics and spintronics.\nACKNOWLEDGMENTS\nThis research has been supported by RSF grant No. 17-12-01333.\n\u0003davydova@phystech.edu\nyzvezdin@gmail.com\n[1] T. Ostler, J. Barker, R. Evans, R. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader, E. Men-\ngotti, L. 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Phys. 82, 2731 (2010).\n[10] R. B. Wilson, J. Gorchon, Y. Yang, C.-H. Lambert, S. Salahuddin, and J. Bokor, Phys. Rev. B 95, 180409 (2017).\n[11] Y. Yang, R. B. Wilson, J. Gorchon, C.-H. Lambert, S. Salahuddin, and J. Bokor, Science Advances 3(2017), 10.1126/sci-\nadv.1603117, http://advances.sciencemag.org/content/3/11/e1603117.full.pdf.\n[12] R. Chimata, L. Isaeva, K. K\u0013 adas, A. Bergman, B. Sanyal, J. H. Mentink, M. I. Katsnelson, T. Rasing, A. Kirilyuk,\nA. Kimel, O. Eriksson, and M. Pereiro, Phys. Rev. B 92, 094411 (2015).\n[13] C. D. Stanciu, A. Tsukamoto, A. V. Kimel, F. Hansteen, A. Kirilyuk, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 217204\n(2007).\n[14] C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, Phys. Rev. B 73, 220402\n(2006).\n[15] J. Becker, A. Tsukamoto, A. Kirilyuk, J. C. Maan, T. Rasing, P. C. M. Christianen, and A. V. Kimel, Phys. Rev. Lett.\n118, 117203 (2017).\n[16] Z. Chen, R. Gao, Z. Wang, C. Xu, D. Chen, and T. Lai, Journal of Applied Physics 108, 023902 (2010),\nhttps://doi.org/10.1063/1.3462429.\n[17] A. Moser, K. Takano, D. T. Margulies, M. Albrecht, Y. Sonobe, Y. Ikeda, S. Sun, and E. E. Fullerton, Journal of Physics\nD: Applied Physics 35, R157 (2002).\n[18] B. Lenk, H. Ulrichs, F. Garbs, and M. Mnzenberg, Physics Reports 507, 107 (2011).\n[19] J. Walowski and M. Mnzenberg, Journal of Applied Physics 120, 140901 (2016), https://doi.org/10.1063/1.4958846.\n[20] M. Liebs, K. Hummler, and M. F ahnle, Phys. Rev. B 46, 11201 (1992).\n[21] B. Goransky and A. Zvezdin, JETP Lett. 10, 196 (1969).\n[22] A. Zvezdin and V. Matveev, JETP 35, 140 (1972).\n[23] C. K. Sabdenov, M. Davydova, K. Zvezdin, D. Gorbunov, I. Tereshina, A. Andreev, and A. Zvezdin, J. Low. Temp. Phys\n43, 551 (2017).\n[24] M. Ding and S. J. Poon, Journal of Magnetism and Magnetic Materials 339, 51 (2013).\n[25] A. Zvezdin, JETP Lett. 28, 553 (1979); arXiv preprint arXiv:1703.01502 (2017).\n[26] B. Ivanov and A. Sukstanskii, Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 84, 370 (1983).7\n[27] A. Zvezdin, Handbook of Magnetic Materials 9, 405 (1995)."    },    {        "title": "2205.13802v3.Magnonic_Casimir_Effect_in_Ferrimagnets.pdf",        "content": "Magnonic Casimir Effect in Ferrimagnets\nKouki Nakata1,\u0003and Kei Suzuki1,y\n1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n(Dated: March 2, 2023)\nQuantum fluctuations are the key concepts of quantum mechanics. Quantum fluctuations of\nquantum fields induce a zero-point energy shift under spatial boundary conditions. This quantum\nphenomenon, called the Casimir effect, has been attracting much attention beyond the hierarchy\nof energy scales, ranging from elementary particle physics to condensed matter physics together\nwith photonics. However, the application of the Casimir effect to spintronics has not yet been in-\nvestigated enough, particularly to ferrimagnetic thin films, although yttrium iron garnet (YIG) is\none of the best platforms for spintronics. Here we fill this gap. Using the lattice field theory, we\ninvestigate the Casimir effect induced by quantum fields for magnons in insulating magnets and find\nthat the magnonic Casimir effect can arise not only in antiferromagnets but also in ferrimagnets\nincluding YIG thin films. Our result suggests that YIG, the key ingredient of magnon-based spin-\ntronics, can serve also as a promising platform for manipulating and utilizing Casimir effects, called\nCasimir engineering. Microfabrication technology can control the thickness of thin films and realize\nthe manipulation of the magnonic Casimir effect. Thus, we pave the way for magnonic Casimir\nengineering.\nIntroduction. —Toward efficient transmission of infor-\nmation that goes beyond what is offered by conven-\ntionalelectronics, thelasttwodecadeshaveseenasignifi-\ncant development of magnon-based spintronics [1], called\nmagnonics. The main aim of this research field is to use\nthe quantized spin waves, magnons, as a carrier of in-\nformation in units of the reduced Planck constant ~. A\npromising strategy for this holy grail is to explore insu-\nlating magnets. Thanks to the complete absence of any\nconducting metallic elements, insulating magnets are free\nfrom drawbacks of conventional electronics, such as sub-\nstantial energy loss due to Joule heating. This is the\nadvantage of insulating magnets. Thus, exploring quan-\ntum functionalities of magnons in insulating magnets is\na central task in the field of magnonics.\nQuantum fluctuations of photon fields induce a zero-\npoint energy shift, called the Casimir energy, under spa-\ntial boundary conditions. This Casimir effect is a funda-\nmentalphenomenonofquantumphysics, andtheoriginal\nplatform for the Casimir effect was the photon field [2–\n4], which is described by quantum electrodynamics. The\nconcept can be extended to various fields such as scalar,\nvector, tensor, and spinor fields. Nowadays, the Casimir\neffect has been attracting much attention beyond the hi-\nerarchyofenergyscales, rangingfromelementaryparticle\nphysics to condensed matter physics [5, 6]. As an exam-\nple, see Refs. [7–14] for Casimir effects in magnets [15].\nHowever, the application of the Casimir effect to spin-\ntronics has not yet been studied enough, particularly to\nferrimagnetic thin films (see Fig. 1), although yttrium\niron garnet (YIG) [16] has been playing a central role in\nspintronics.\nHere we fill this gap. In terms of the lattice field the-\nory, we investigate the Casimir effect induced by quan-\n\u0003nakata.koki@jaea.go.jp; equal contribution.\nyk.suzuki.2010@th.phys.titech.ac.jp; equal contribution.\nFIG. 1. Schematic of the ferrimagnetic thin film for the\nmagnonic Casimir effect. Two kinds of magnons (circles) arise\nfrom the alternating structure of up and down spins (arrowed\nlines). Wavy lines represent quantum-mechanical behaviors\nof magnons in the discrete energy.\ntum fields for magnons in insulating magnets and refer to\nit as the magnonic Casimir effect. We study the behav-\nior of the magnonic Casimir effect with a particular fo-\ncus on the thickness dependence of thin films, which can\nbeexperimentallycontrolledbymicrofabricationtechnol-\nogy [17, 18]. Then, we show that the magnonic Casimir\neffect can arise not only in antiferromagnets (AFMs) but\nalso in ferrimagnets with realistic model parameters for\nYIG thin films. Our study indicates that YIG, an ideal\nplatform for magnonics, can serve also as a key ingredi-\nent of Casimir engineering [19] which aims at exploring\nquantum-mechanical functionalities of nanoscale devices.\nAntiferromagnets. —We consider insulating AFMs de-\nscribed by the quantum Heisenberg Hamiltonian which\nhas U (1)symmetry about the quantization axis andarXiv:2205.13802v3  [quant-ph]  1 Mar 20232\nstudy the behavior of the magnonic Casimir effect with\na focus on the thickness dependence. The AFM is a\ntwo-sublattice system, and the ground state has the Néel\nmagnetic order [20]. From the spin-wave theory with the\nBogoliubov transformation, elementary excitations are\ntwo kinds of magnons designated by the index \u001b=\u0006\nhaving the spin angular momentum \u001b~. Owing to the\nU(1)symmetry, the Hamiltonian can be recast into the\ndiagonal form with the magnon energy dispersion for the\nwave number k= (kx;ky;kz)as\u000f\u001b;kwhere the total spin\nangular momentum of magnons is conserved. Two kinds\nof magnons ( \u001b=\u0006) are in degenerate states. Hence, we\nstudy the low-energy magnon dynamics of the insulat-\ning AFM by using the quantum field theory of complex\nscalar fields, i.e., the complex Klein-Gordon field the-\nory [21, 22]. Then, we can see that there exists a zero-\npoint energy [23]. This is the origin of the Casimir effect.\nNote that throughout this study, we focus on clean insu-\nlating magnets and work under the assumption that the\ntotal spin along the quantization direction is conserved\nand thus remains a good quantum number.\nThrough the lattice regularization, the Casimir energy\nECasis defined as the difference between the zero-point\nenergyEint\n0for the continuous energy \u000f\u001b;kand the one\nEsum\n0for the discrete energy \u000f\u001b;k;nwithn2Z. In two-\nsublattice systems, such as AFMs and ferrimagnets (see\nFig. 1), the wave numbers on the lattice are replaced by\n(kja)2!2[1\u0000cos(kja)]in thejdirection for j=x;y;z,\nwhereais the length of a magnetic unit cell. Here, by\ntaking the Brillouin zone (BZ) into account, we set the\nboundary condition for the zaxis in wave number space\nk= (kx;ky;kz)askz!\u0019n=Lz, i.e.,kza!\u0019n=Nz,\nwhereLz:=aNzis the thickness of magnets, Nj2N\nis the number of magnetic unit cells in the jdirection\nforj=x;y;z, andn= 1;2;:::;2NforN2N. The\nnumberofunitcellsonthe xyplaneis 4NxNy,andthatof\nmagneticunitcellsis NxNy. Thus, themagnonicCasimir\nenergy per the number of magnetic unit cells NxNyon\nthe surface for Nz=Nis described as [24–28]\nECas:=Esum\n0(N)\u0000Eint\n0(N); (1a)\nEsum\n0(N) =X\n\u001b=\u0006Z\nBZd2(k?a)\n(2\u0019)2\"\n1\n2\u00101\n22NX\nn=1j\u000f\u001b;k;nj\u0011#\n;(1b)\nEint\n0(N) =X\n\u001b=\u0006Z\nBZd2(k?a)\n(2\u0019)2\"\n1\n2NZ\nBZd(kza)\n2\u0019j\u000f\u001b;kj#\n;\n(1c)\nwherek?:=q\nk2x+k2y,d2(k?a) =d(kxa)d(kya), the\nintegral is over the first BZ, and the factor 1=2arises\nfrom the zero-point energy for the scalar field. Since\nthe constant terms which are independent of the wave\nnumber do not contribute to the Casimir energy, we drop\nthemthroughoutthisstudy. Toseethedependenceofthe\nCasimir energy ECason the thickness of magnets Lz:=\naNz, it is convenient to introduce the rescaled Casimir\n−1−0.8−0.6−0.4−0.2 0\n 0  2  4  6  8 10  12  14  16  18  20Antiferromagnetic thin film\nNzCasimir energy ECas [meV]\nNz: Thickness of magnet in units of aδ=2.0×10−3\nδ=1.0×10−3\nδ=0 −1−0.8−0.6−0.4−0.2 0\n 0  5  10  15  20CCas[3]=Nz3ECas [meV]FIG. 2. Plots of the magnonic Casimir energy ECasand its\ncoefficient (inset) C[3]\nCas=ECas\u0002N3\nzin the AFM [see Eq. (3a)]\nas a function of Nzfor the thickness of magnets Lz=aNz.\nenergyC[b]\nCasin terms of Nb\nzforb2Ras\nC[b]\nCas:=ECas\u0002Nb\nz: (2)\nThen, we refer to C[b]\nCasas the magnonic Casimir coeffi-\ncient in the sense that ECas=C[b]\nCasN\u0000b\nz.\nHere, we consider magnons in AFMs on a cubic lattice\nwith the energy dispersion \u000f\u001b;k=\u000fAFM\n\u001b;k[29]:\n\u000fAFM\n\u001b;k=~!0r\n\u000e+\u0010ka\n2\u00112\n; (3a)\n~!0:=2p\n3JS; (3b)\n\u000e:=3h\u0010K\n6J\u00112\n+ 2\u0010K\n6J\u0011i\n; (3c)\nwherek:=jkj,J >0parametrizes the exchange interac-\ntion between the nearest-neighbor spins of the spin quan-\ntum number S, andK\u00150is the easy-axis anisotropy,\nwhich provides the magnon energy gap and ensures the\nNéel magnetic order. Two kinds of magnons ( \u001b=\u0006)\nare in degenerate states. In the absence of the spin\nanisotropy, K= 0, the energy gap vanishes \u000e= 0, and\nthegaplessmagnonmodebehaveslikearelativisticparti-\nclewiththelinearenergydispersion. Fromtheresultsob-\ntainedinRefs.[30–32],weroughlyestimatethemodelpa-\nrameter values for Cr 2O3, as an example, as follows [29]:\nJ= 15meV,S= 3=2,K= 0:03meV, anda= 0:496 07\nnm. These parameter values provide ~!0\u001877:94meV\nand\u000e\u00182\u000210\u00003.\nFigure 2 shows that the magnonic Casimir effect arises\nin the thin film of the AFM. The magnonic Casimir en-\nergyECasof the magnitude O(10\u00002)meV is induced\nforNz\u00152. Even in the presence of the magnon en-\nergy gap\u000e6= 0, the absolute value amounts to O(10\u00002)\nmeV and decreases as the magnon energy gap increases.\nThus, the magnonic Casimir energy takes a maximum3\nabsolute value in the gapless mode \u000e= 0, where the\nmagnon behaves like a relativistic particle with the lin-\near energy dispersion. We remark that in the case of\nthe gapped magnon modes, the absolute value of the\nmagnonicCasimircoefficient C[3]\nCas=ECas\u0002N3\nzdecreases\nas the thickness of the thin film increases. This behav-\nior is similar to the Casimir effect known for massive\ndegrees of freedom [33, 34]. In the case of the gapless\nmode, the magnonic Casimir coefficient C[3]\nCasapproaches\nasymptotically to a constant value, and the magnonic\nCasimir energy exhibits the behavior of ECas/1=N3\nzas\nthe thickness increases. The asymptotic value of C[3]\nCas\nfor the gapless magnon mode \u000e= 0given in the nu-\nmerical result (see Fig. 2) is estimated approximately as\n(\u0000\u00192=720)\u0002(~!0=2)\u0018\u00000:5341meV from an analyt-\nical calculation. The factor of \u0000\u00192=720is well known\nas the analytic solution for the conventional Casimir\neffect of a massless complex scalar field in continuous\nspace [34]. Thus, although the magnonic Casimir effect\nis realized on the lattice, it is qualitatively and quanti-\ntatively analogous to the continuous counterpart, except\nfora-dependent lattice effects.\nFerrimagnets. —We develop the study of AFMs into\nferrimagnets where the ground state has an alternating\nstructure of up and down spins on a cubic lattice (see\nFig. 1). In contrast to AFMs, the spin quantum num-\nber on the two-sublattice is different from each other in\nferrimagnets. Hence, the degeneracy for two kinds of\nmagnons (\u001b=\u0006) is intrinsically lifted. In ferrimagnetic\nthinfilms, dipolarinteractionsduetothenonzeromagne-\ntization play a key role. Still, at low temperatures where\nthe magnon-magnon interaction of the fourth order in\nmagnon operators is negligibly small [35], the number of\nmagnons and the total spin angular momentum are con-\nserved, and the Hamiltonian for the ferrimagnetic thin\nfilm can be diagonalized with the magnon energy disper-\nsion\u000f\u001b;k=\u000fferri\n\u001b;k.\nHere, we consider magnons in the thin film of clean\ninsulating ferrimagnets on a cubic lattice subjected to\nin-plane magnetic fields at such low temperatures. Still,\ndue to the competition between dipolar and exchange\ninteractions, the minimum energy point shifts from the\nzero mode of magnons, k= 0, to a finite wave number\nmode which is characterized by the thickness of the thin\nfilmLz=aNz(see Fig. 1).\nThe magnon energy dispersion along the in-plane di-\nrection is provided in Refs. [36, 37], whereas the disper-\nsion along the zaxis in the thin film has not yet been\nestablished [38]. Hence, taking into account the compe-\ntition between dipolar and exchange interactions in the\nthin film, we phenomenologically assume the behavior\nthat the power of kz,l2R, approaches asymptotically\ntol= 2in the bulk limit, whereas it slightly differs from\nl= 2as long as we consider the thin film (see Fig. 1).\nUsing this assumption and Refs. [36, 37], the magnon\n−120−100−80−60−40−20 0 20\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnetic thin film\nNzCasimir energy ECas [µeV]\nNz: Thickness of magnet in units of al=2.1\nl=2.0l=1.99\nl=1.9\n−100−50 0 50 100\n 0  5  10  15  20CCas[l]=NzlECas [µeV]FIG. 3. Plots of the magnonic Casimir energy ECasand its\ncoefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforl= 2:1,l= 2:0,l=\n1:99, and l= 1:9in the ferrimagnetic thin film [see Eq. (4a)]\nas a function of Nzfor the thickness of magnets Lz=aNz\nwith the model parameter values for YIG of Dz=D.\nenergy dispersions are\n\u000fferri\n\u001b;k=r\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l(4a)\n\u0002r\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l+\u001b~!MFk;\nFk:=Pk?(1\u0000Pk?)\u001b~!M\n\u001bH0+ \u0001\u001b+D\na2(k?a)2+Dz\na2(jkzja)l\u0010kx\nk?\u00112\n+1\u0000Pk?\u0010ky\nk?\u00112\n; (4b)\nPk?:=1\u00001\u0000e\u0000k?Lz\nk?Lz; (4c)\nwhere the external magnetic field is applied along the y\naxis (see Fig. 1) and \u001bH0represents the resulting Zee-\nman energy, \u0001\u001b\u00150is the (intrinsic) magnon energy\ngap in ferrimagnets, D(z)>0is the spin stiffness con-\nstant,k?:=q\nk2x+k2y,~!M:= 4\u0019\rMswith the sat-\nurated magnetization density Msand the gyromagnetic\nratio\r, and the termFkis responsible for the shift of\nthe minimum energy point from the zero mode to a finite\nwave number mode due to the competition between dipo-\nlar and exchange interactions in the ferrimagnetic thin\nfilm: The first term of Fk[see Eq. (4b)] reproduces the\nDamon-Eshbach magnetostatic surface mode [39], and\nthe last term reproduces the backward volume magneto-\nstatic mode [39].\nFrom the results obtained in Refs. [40–42], the model\nparameter values for YIG thin films are estimated as fol-\nlows:D=a2\u00183:376 45meV [35] with a= 1:2376nm [43],\nH0\u00188:103 73\u0016eV [35], and ~!M\u001820:3369\u0016eV [35].\nThen, we estimate the magnon energy gap \u0001\u001bas [44]\n\u0001\u001b=\u0000\u0000\u0001\u001b=+\u001839:848 81meV with \u0001\u001b=+\u00182:131 91\nmeV and \u0001\u001b=\u0000\u001841:980 72meV, which satisfy the con-4\ndition \u0001\u001b\u001d~!M. In this condition, we study the low-\nenergy magnon dynamics of the ferrimagnetic thin film\nby using the quantum field theory of real scalar fields,\ni.e., the real Klein-Gordon field theory [21, 22]. Then,\nwe can see that there exists a zero-point energy [35].\nThe magnonic Casimir energy through the lattice reg-\nularization is given as Eq. (1a). We remark that the\nzero-point energy arises from quantum fluctuations and\ndoes exist even at zero temperature. The zero-point en-\nergy defined at zero temperature does not depend on the\nBose-distribution function [Eqs. (1b) and (1c)]. Hence,\nnot only the low-energy mode ( \u001b= +) but also the high-\nenergy mode ( \u001b=\u0000) contribute [45] to the magnonic\nCasimir energy.\nUnderthephenomenologicalassumptionthatthevalue\nofl[see Eq. (4a)] approaches asymptotically to l= 2in\nthe bulk limit, in this work focusing on the thin film,\nwe study the behavior of the magnonic Casimir effect\nby changing the value slightly from l= 2. As exam-\nples, we consider the cases of l= 2:1,l= 1:99, and\nl= 1:9. Figure 3 shows that the magnonic Casimir ef-\nfect arises in the ferrimagnetic thin film. There is the\nmagnonic Casimir energy ECasof the magnitude O(1),\nO(10), andO(10)\u0016eV forl= 1:99,l= 1:9, andl= 2:1,\nrespectively, in Nz\u00152. As the thickness increases, the\nmagnonic Casimir coefficient C[l]\nCasapproaches asymptot-\nically to a constant value, and the magnonic Casimir en-\nergy exhibits the behavior of ECas/1=Nl\nz. We also\nfind from Fig. 3 that the sign of the magnonic Casimir\ncoefficient and energy for l= 2:1is positive C[2:1]\nCas =\nECas\u0002N2:1\nz>0inNz\u00152, whereas that for l= 1:9is\nnegativeC[1:9]\nCas=ECas\u0002N1:9\nz<0. This means that the\nCasimir force works in the opposite direction.\nNote that even if l= 2, the magnonic Casimir effect\narises in the ferrimagnetic thin film. We have numer-\nically confirmed that although the value is small, there\ndoes exist the magnonic Casimir energy of the magnitude\njECasj\u0014O(0:1)neV forl= 2. This strong suppression of\nthe Casimir energy is a general property of the Casimir\neffect for quadratic dispersions on the lattice [28], and\nthe survival values originate from the dipolar interaction\nin the ferrimagnetic thin film.\nWe remark that as long as we consider thin films,\nthe value of Dzcan differ from D. Even in that case,\nthe magnonic Casimir effect arises. When the value\nofDzchanges from Dto0:8Das an example, the\nmagnonic Casimir energy ECasincreases approximately\nby0:8times. For more details about its dependence on\nthe parameters Dzandl, see the Supplemental Material.\nProposal for experimental observation. —Themagnonic\nCasimir energy of the ferrimagnetic thin film depends\nstronglyonexternalmagneticfieldsthroughZeemancou-\npling as in Eq. (4a) and contributes to magnetization of\nmagnets, whereas the photon and phonon Casimir ef-\nfects [46] do not usually. On the other hand, in the\npresenceofmagnetostriction[47–50], thephononCasimir\neffect is influenced by magnetostriction, and its correc-\ntion for the phonon Casimir energy depends on magneticfields and contributes to magnetization. However, such a\ncontribution to magnetization from the phonon Casimir\neffect should be negligibly small by the factor of 10\u00006\ncompared with that from the magnonic Casimir effect\nof ferrimagnets because the magnetostriction constant\n(i.e., the correction for the lattice constant) is known to\nbe10\u00006for YIG [47, 48]. Hence, even in the presence\nof magnetostriction, the magnonic Casimir effect can be\ndistinguished from the others. Thus, we expect that our\ntheoreticalprediction, themagnonicCasimireffectinfer-\nrimagnets, can be experimentally observed through mea-\nsurement of magnetization and its film thickness depen-\ndence. For more details, see the Supplemental Material.\nForobservation, afewcommentsareinorder. First, we\nremark on edge/surface magnon modes. The magnonic\nCasimir effect in our setup (see the thin film of Fig. 1)\nis induced by magnon fields with wave numbers kzdis-\ncretized by small Nz, and its necessary condition is a\nkz-dependent dispersion relation under the discretization\nofkz. Throughout this study, we consider thin films of\nNz\u001cNx;Ny. Even if additional edge/surface magnon\nmodesexistaswellastheDamon-Eshbachmagnetostatic\nsurface mode and the backward volume magnetostatic\nmode [see Eq. (4b)], they are confined only on the x-\nyplane. Then, their wave number in the zdirection\nis always zero, i.e., kz= 0, and its energy dispersion\nrelation is independent of kz. Since a kz-independent\ndispersion relation cannot induce the Casimir effect, the\nedge/surface modes cannot contribute to the magnonic\nCasimir effect. In this sense, our magnonic Casimir effect\nis not affected by the existence of edge/surface magnon\nmodes.\nNote that details of the edge condition, such as the\npresence or absence of disorder, may change the bound-\nary condition for the wave function of magnons, but the\nexistence of the magnonic Casimir effect remains un-\nchanged. Even if there is a change in the spectrum near\nthe edge, the magnonic Casimir effect is little influenced\nas long as one does not assume an ultrathin film such\nasNz= 1;2;3. In this sense, we expect that the fol-\nlowing size of thin films is appropriate for observation of\nour prediction: Nz\u001810, i.e., the film thickness of YIG\nisLz=aNz\u001812:376nm. Note that microfabrication\ntechnology [17, 18] can control the thickness of thin films\nand realize the manipulation of the magnonic Casimir\neffect.\nNext, we remark on the magnon band structure.\nSince our magnonic Casimir effect is induced by the kz-\ndependent dispersion, its Casimir energy of ferrimagnets\nis mainly characterized by the Dz-term in Eq. (4a), i.e.,\nDz(jkzja)l. Hence, we have investigated its dependence\non bothlandDz(see Fig. 3 and the Supplemental Mate-\nrial). Even if the magnon band structure is affected due\nto some reasons, the magnonic Casimir effect of ferrimag-\nnets is little influenced by other details of the magnon\nband structure except for landDz.\nLastly, we remark on thermal effects. At nonzero tem-\nperature, thermal contributions to the Helmholtz free en-5\nergy arise. However, at low temperatures compared to\n\u000f\u001b;k[51], the thermal contribution is exponentially sup-\npressed due to the Boltzmann factor and becomes negli-\ngibly small. Hence, at such low temperatures, the contri-\nbution of the magnonic Casimir energy given as Eq. (1a)\nis dominant.\nConclusion. —We have shown that the magnonic\nCasimir effect can arise not only in antiferromagnets but\nalso in ferrimagnets with realistic model parameters for\nYIG. Since the lifetime of magnons in YIG thin films is\nthelongestamongknownmaterials, andmagnonsexhibit\nlong-distance transport over centimeter distances [52],\nYIG is the key ingredient of magnonics [1, 16], which\nhas already realized the magnon transistor [53]. Ourresult suggests that YIG can serve also as a promis-\ning platform for Casimir engineering [19]: Because the\nmagnonic Casimir effect contributes to the internal pres-\nsure of thin films, it will provide the new principles of\nnanoscale devices such as highly sensitive pressure sen-\nsors and magnon transistors. 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Volovik, arXiv:1003.4889 .7\nSupplemental Material\nIn this Supplemental Material, first, we provide some\ndetails about the dependence of the magnonic Casimir\neffect on the parameters landDzin ferrimagnetic thin\nfilms. Next, we remark on its film thickness dependence.\nThen, we provide another point of view for its robust-\nness against disorder effects. Lastly, we comment on the\ndistinction between the Casimir effect and the thermal\nCasimir effect.\nI. THE PARAMETER l- AND Dz-DEPENDENCE\nIn the main text, we have studied the magnonic\nCasimirenergy ECasandthecoefficient C[l]\nCas=ECas\u0002Nl\nz\nforl= 2:1,l= 2:0,l= 1:99, andl= 1:9in the ferrimag-\nnetic thin film by using the model parameter values for\nYIG with fixed Dz=D. Here, we provide more details\nabout its dependence on the parameters landDz.\nFirst, we consider the cases of l= 1:5andl= 1:0with\nsettingDz=D. Figure S1 shows that the magnonic\nCasimir effect still arises in the ferrimagnetic thin film.\nThere is the magnonic Casimir energy ECasof the mag-\nnitudeO(10\u00002)meV,O(10\u00001)meV, andO(10\u00001)meV\nforl= 1:9,l= 1:5, andl= 1:0, respectively, in Nz\u00152.\nAs the value of ldecreases from l= 2and approaches\ntol= 1, the magnitude of the magnonic Casimir energy\nincreases. Note that it amounts to O(10\u00001)meV even in\nNz=O(10)forl= 1:0. As the thickness increases, the\nmagnonic Casimir coefficient C[l]\nCasapproaches asymptot-\nically to a constant value and the magnonic Casimir en-\nergy exhibits the behavior of ECas/1=Nl\nz.\nThen, we consider the cases of Dz=D= 0:3,Dz=D=\n0:5, andDz=D= 0:8by fixingl= 1:99. Figure S2 shows\nthat the magnonic Casimir effect still arises in the ferri-\nmagnetic thin film. When the value of Dzchanges from\nDto0:8Das an example, the magnonic Casimir energy\nECasincreases approximately by 0:8times. Thus, the\nvalue ofECasis approximately proportional to Dz.\nII. REMARKS ON THE THICKNESS\nDEPENDENCE OF MAGNETIZATION\nIn the main text, we have remarked that our predic-\ntion, the magnonic Casimir effect in ferrimagnets, can\nbe observed through measurement of magnetization and\nits film thickness dependence. Here, we add an expla-\nnation about it. At zero temperature, the Helmholtz\nfree energy of magnon fields in thin films (i.e., the\nsum over discrete kz) isEsum\n0(Nz)NxNy, and that un-\nder the bulk approximation (i.e., the integral with re-\nspect to continuous kz) isEint\n0(Nz)NxNy[see Eqs. (1a)-\n(1c)]. The difference between them is characterized by\nthe magnonic Casimir energy ECasasEsum\n0(Nz)NxNy=\nEint\n0(Nz)NxNy+ECasNxNy, where the magnon energy\n−1−0.8−0.6−0.4−0.2 0\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnet\nNzCasimir energy ECas [meV]\nNz: Thickness of magnet in units of al=2.0\nl=1.9l=1.5\nl=1.0\n−1−0.8−0.6−0.4−0.2 0\n 0  5  10  15  20CCas[l]=NzlECas  [meV]FIG. S1. Plots of the magnonic Casimir energy ECasand\nthe coefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforl= 2:0,l= 1:9,\nl= 1:5, and l= 1:0intheferrimagneticthinfilmasafunction\nofNzfor the thickness of magnets Lz=aNzunder the model\nparameter values for YIG with fixed Dz=D.\n−14−12−10−8−6−4−2 0\n 0  2  4  6  8 10  12  14  16  18  20Ferrimagnet\n(l=1.99)\nNzCasimir energy ECas [µeV]\nNz: Thickness of magnet in units of aDz=0.3D\nDz=0.5DDz=0.8D\nDz=D\n−14−12−10−8−6−4−2 0\n 0  5  10  15  20CCas[l]=NzlECas [µeV]\nFIG. S2. Plots of the magnonic Casimir energy ECasand the\ncoefficient (inset) C[l]\nCas=ECas\u0002Nl\nzforDz=D= 0:3,Dz=D=\n0:5,Dz=D= 0:8, and Dz=D= 1:0in the ferrimagnetic thin\nfilmasafunctionof Nzforthethicknessofmagnets Lz=aNz\nunder the model parameter values for YIG with l= 1:99.\ndispersion of ferrimagnets (i.e., magnets including dipo-\nlar interactions) is Eq. (4a). Note that the magnetic-\nfield derivative (i.e., H0-derivative) of the Helmholtz free\nenergy is magnetization. Then, magnetization of thin\nfilms consists of two parts: The bulk component and the\nmagnonic Casimir energy. Since Eint\n0(Nz)/Nz, whereas\nECas/1=(Nz)l, magnetization of thin films exhibits a\ndifferentNz-dependence from the bulk component, and\nits difference is characterized by the magnonic Casimir\nenergy. In other words, magnetization of thin films ex-\nhibits a different film thickness dependence from the bulk\ncomponent due to the magnonic Casimir effect. Hence,\nour prediction, the magnonic Casimir effect in ferrimag-8\nnetic thin films (i.e., magnetic thin films including dipo-\nlar interactions), can be observed through measurement\nof magnetization and its film thickness dependence.\nNote that if dipolar interactions are relevant also in\nantiferromagnets, its low-energy magnon dynamics is es-\nsentially described by Eq. (4a) given for ferrimagnets.\nThe only difference is that the spin quantum number for\neachsublatticeisidenticalinantiferromagnets, wherethe\n(intrinsic) magnon energy gap for each mode \u001b=\u0006can\nbe identical \u0001\u001b=+= \u0001\u001b=\u0000[see Eq. (4a)]. In this sense,\nitsCasimireffectexhibitsqualitativelythesamebehavior\nas Fig. 3.\nIII. REMARKS ON DISORDER EFFECTS\nIn the main text, we have remarked that details of the\nedge condition, such as the presence or absence of dis-\norder, may change the boundary condition for the wave\nfunction of magnons, but the existence of the magnonic\nCasimir effect remains unchanged. Here, we add a com-\nment on disorder effects. Since the magnonic Casimir\nenergy does not depend on the Bose-distribution func-\ntion [see Eqs. (1a)-(1c)], not only the low-energy magnon\nmode (\u001b= +) but also its high-energy mode ( \u001b=\u0000) in\nferrimagnets contributes to the magnonic Casimir effect.\nTherefore, it can be expected that as long as disorder\neffects on the bulk are weak enough that the high-energy\nmode is little influenced, the existence of the magnonic\nCasimir effect in ferrimagnets remains unchanged.\nIV. THERMAL CASIMIR EFFECT\nIn the main text, we have explained that thermal con-\ntributions to the Helmholtz free energy arise at nonzero\ntemperature. Here, we add a remark on it. Although its\nthermal contribution is called the “thermal Casimir en-\nergy”, there is a crucial distinction between the Casimir\neffect and the thermal Casimir effect: The zero-point en-\nergy, which is the key concept of quantum mechanics and\nplays a crucial role in the Casimir effect, is absent in the\nthermal Casimir effect. It should be noted that the zero-\npoint energy is one of the most striking phenomenon of\nquantum mechanics in the sense that there are no clas-\nsical analogs. The Casimir effect arises from the zero-\npoint energy due to quantum fluctuations and is not af-\nfected by temperatures, whereas the thermal Casimir ef-\nfect arises from thermal fluctuations and is exponentially\nsuppressed at low temperatures. The thermal Casimir\neffect vanishes at zero temperature, whereas the Casimir\neffect does exist even at zero temperature. Thus, there is\na significant distinction between the Casimir effect and\nthe thermal Casimir effect."    },    {        "title": "1510.05087v1.Ferrimagnetic_nanostructures_for_magnetic_memory_bits.pdf",        "content": "1 \n Ferrimagnetic nanostructures for magnetic memory bits  \nAUTHOR NAMES  \nA. A. Ünal ,1 S. Valencia,1 D. Marchenko,1 K. J. Merazzo,2 F. Radu,1 M. Vázquez,2 and J. \nSánchez -Barriga1,* \nAUTHOR ADDRESS  \n1Helmholtz -Zentrum Berlin für Materiali en und Energie, Albert -Einstein -Str. 15, 12489 \nBerlin, Germany  \n2Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Madrid, Spain \nKEYWORDS  Heat -assisted magnetic recording, bit- patterned media, magnetic data storage, \nferrimagnetic antidots  \n \n \n     2 \n ABSTRACT   \nIncreasing the magnetic data recording density requires reducing the size of the individual \nmemory elements of a recording layer as well as employing magn etic materials with \ntemperature- dependent functionalities. Therefore, it is predicted that t he near future of \nmagnetic data storage technology involves a combination of energy -assisted recording on \nnanometer -scale magnetic media. We present the potential of heat -assisted magnetic \nrecording on a patterned sample; a ferrimagnetic alloy composed of a rare earth and a transition metal, DyCo\n5, which is grown on a hexagonal -ordered nanohole array membrane. \nThe magnetization of the antidot array sample is out- of-plane oriented at room temperature \nand rotates towards in- plane upon heating above its spin- reorientation temp erature (TR) of \n~350 K, just above  room temperature.  Upon cooling back to room temperature (below T R), we \nobserve a well -defined and unexpected in- plane  magnetic domain configuration modulating \nwith ~45 nm. We discuss the underlying mechanisms giving rise to this behavior by \ncomparing  the magnetic properties of the patterned sample with the ones of its extended thin \nfilm counterpart .  Our results pave  the way for  novel applications of ferrimagnetic  antidot \narrays  of superior functionality  in magnetic nano -devices near room temperature.   \nINTRODUCTION  \nThe research on artificially engineered spintronic materials in the for m of layered thin films \nand low -dimensional structures has experienced a tremendous boost due to the versatility of \ntheir applications in computing technology. This is reflected by the development of magnetic \ntunneling based read- out technologies [ 1], magnetic sensors [ 2,3], and non- volatile \nmagnetoresistive random -access memory devices [ 4]. Because conventional magnetic data \nstorage relies on two -dimensional arrays of magnetic bits, developing higher density, cheaper \nand faster devices relies on reducing the lateral size of individual memory elements or data 3 \n storage bits [ 5]. Ultimately controlling the magnetic anisotropy and  shape of nanostructures, \nas well as understanding their magnetic properties in reduced dimensions are key aspects that \nhave received a sustained interest over the past decades [ 6]. For instance, in antiferromagnetic \n(AF) materials, which have crucial impo rtance in exchange- bias systems, finite -size effects \nlead to a scaling of the magnetic properties such as the ordering temperature or the magnetic anisotropy [ 7], while in ferromagnetic (FM) materials control of the shape anisotropy can be \nachieved dependi ng on the geometry of the objects [ 8]. \nNanostructured magnetic materials often exhibit novel  properties over their bulk \ncounterparts as a consequence of reduced atomic coordinates and modified density of states. Magnetic systems of low dimensionality like artificially grown or self -organized FM \nnanostructures [ 9,10], arrays of nanostructures [ 11,12], nanowires [ 13,14,15], and antidot \narrays [ 16] have been intensively investigated, mainly triggered by the discovery of \nspontaneous self -organization of magnetic FePt  nanoparticles on a surface [ 11]. In all these \nfields, there is a rapid decrease of the relevant length scales of the magnetic structures down to the sub- 100 nm range. To illustrate this , a storage density of 1 Tbi ts/in\n2 would require  a bit \nlength of around 25 nm. The underlying idea is to replace the comparatively large  randomly -\noriented magnetic grains in conventional media with a nanoscopic region of a single magnetic \ndomain [17]. As the magnetic stability of individual particles scales with the  material  \nanisotropy constant and the particle volume , the use of FM FePt alloyed nanoparticles with \nvery high magnetic anisotropy (only in their chemically ordered face -centered tetragonal L1 0 \nphase) has been  considered  one of the promising routes towards  future ultrahigh- density \nrecording -media applications [ 18,19]. In this case, the suppression of superparamagnetic  \neffects at very small volumes  might prove crucial to achieve  stable and enhanced magnetic \nproperties at the smallest length scales.  On the other hand, the use of ferrimagnetic  \nnanoparticles with controlled coercivity near room temperature might be  a more desirable  \ncompromise  to additionally  achieve full  control ove r the magnetic properties.  4 \n Recently, ferrimagnetic (FI) alloy systems  with relatively low coercive fields , such as  the \ncobalt (Co) -  dysprosium (Dy) alloy  DyCo 5, have been proposed as ideal candidates for  \nmagnetic data storage and spintronic  applications  [20] owing to their flexibility in controlling \na tunable perpendicular exchange -bias effect at room temperature [ 21]. Here the Dy-Co alloy \nacts as a hard FI material with  relatively low coercive field and well-defined perpendicular \nmagnetic anisotropy at room temperature . The system  exhibits a spin -reorientation \ntemperature (TR) of ~350 K above which the uniaxial magnetic anisotropy rotates from out -\nof-plane to in -plane . Such  a remarkable change of the anisotropy axis within a narrow \ntemperature window nea r room temperature  renders  this type of F I alloys  one of the  \ntechnologically relevant and  promising  materials  for future spintronic  applications. Therefore , \nparticularly taking into account that the magnetic properties of DyCo 5 nanostructures has \nremained unexplored, patterning  of this type of  FI alloys  down to the nanoscale  as well as \nunderstanding their magnetic properties  at ultimate length scales  are both of scientific and \ntechnological interest.  \nAmong the different growth methods of  magnetic nanostructure s, the anodization of \nalumina templates and the subsequent deposition of  magnetic antidot array s have  certain \nadvantages that overcome  the unwanted effects of  superparamagnetism.  This technique \nemploys ultrahigh -density substrates th at are patterned down to the nanoscale  as temp lates to \ngrow thin films on top  [16,22]. In the case of nanoporous alumina membranes, nanoholes are \narranged in a fully -controllable hexagonal  symmetry [ 23,24], and growing a thin film on top \nresults in an array of antidots  [25]. The antidots act as well -ordered nonmagnetic inclusions \nwithin the magnetic thin film, promoting the creation of single nano- domain structures whose \nsizes can be manipulated depending on the anodization conditions. This is a  much  less \nexpensive technique compared to lithographic patterning methods. These nanostructures can \nbe promising candidates for a new generation of ultrahig h-density magnetic -storage media 5 \n mainly due to the absence of a superparamagnetic limit, as there are  no isolated small \nmagnetic entities . This is due to the fact that the nanoholes introduce a locally distributed \nshape anisotropy [ 26], act themselves as pi nning centers for magnetic  wall displacements, and \ntheir periodic distribution determines the whole magnetization process as well as  the \nmagnetoresistance behavior [ 16]. Investigations of FI antidot arrays with tunable and \ncontrollable magnetic anisotropies  might in fact  provide superior functionalities that have not \nbeen addressed  so far .  \nIn this work, we present a nanoscale -patterned, hexagonally -ordered FI DyCo 5 antidot array \nwhere the local nanoscopic distribution of the magnetization  at room temperature  can be \ncontrolled via heati ng the sample slightly above  its spin -reorientation temperature . We report \non controlling the magnetic anisotropy as a function of temperature and we compare the nano-\npatterned sample with its plain film counterpart. U sing x -ray magnetic circular dichroism \n(XMCD) in combination with photoelec tron emission microscopy (PEEM), we reveal that the \ntemperature- dependent changes of the magnetic anisotropy in the presence of antidots results \nin well- defined and stable magnetic nano -domain configurations modulating within ~45 nm at \nroom temperature . We discuss the underlying mechanisms giving rise to this behavior . Our \nresults pave the way for future applications of FI nanostructures in heat-assisted magnetic \nnano- devices . \nMETHODS   \n   We prepared  antidot arrays of DyCo 5 by magnetron sputtering  on top of  hexagonally -\nordered alumina me mbranes , which were f abricated u sing a two -step anodization process  [20, \n25]. For the present experiments, we produced  membranes containing nanopores  with a \ntypical diameter of ~68 nm and  center -to-center  inter-pore distances of ~105 nm  [see Fig. \n1(a)] . A DyCo 5 extended film was  grown  on top of an un- patterned alumina substrate for 6 \n comparison purposes . The growth of the antidot arrays  and the extended film  was perform ed \nsimultaneously within the sputtering chamber, and their thickness was ~25 nm. The samples \nwere  obtained by  co-deposition from Co and Dy targets keeping the substrate temperature at \n~200°C. The co-deposition was done with a relative composition of 1:5 to ensure  the spin-\nreorientation transition to be closest to room temperature. The samples were capped with a 2  \nnm thick Al layer to avoid oxidation upon transport in air. The magnetic properties of the  \nextended films , their compensation and spin reorientati on temperatures, coercive fields, and \nmagnetic anisotropy axes were determined by element specific hysteresis loops at the Dy and \nCo absorption [ 21] edges u sing XMCD in transmission and scattering geometries  [27]. \n   High -resolution magnetic domain configurations around the nanoholes and across the spin-\nreorientation transition were  obtained by XMCD -PEEM imaging [ 28]. Experiments were \nperformed  at the UE49 -PGM a beamline of the synchrotron BESSY II . XMCD element -\nspecific images  were obtained by tuning the synchrotron photon energy  to the L 3 and M 5 \nresonances of  Co (779.2 eV) and Dy  (1293.9 eV), respectively .  Each of the XMCD images \nwas calculated from a sequence of images taken with circular polarization (90% of circular \nphoton polarization) and alternating the light helicity. After normalization to a bright- field \nimage, the sequence was  drift-corrected, and frames recorded at the same photon energy and \npolarization were averaged. T he difference of the two resulting  images  with opposite helicity, \ndivided by their sum , showed Co and Dy magnetic -domain contrasts, which represent the \nmagnetization vector pointing along the incidence direction of the x -ray be am. Due to the 16º  \nshallow incidence angle of the x -rays on the sample, our XMCD- PEEM measurements are  \nmainly  sensitive to the in -plane magnetization.  The samples were priorly  magnetized  by \napplying a magnetic field of ~0.5 T perpendicular to the surface, so that the XMCD images in the initial remanent  magnetic state at room temperature exhibited nearly -zero magnetic \ncontrast.  7 \n RESULTS AND DISCUSSION  \nWe investigated the formation of magn etic domains in  FI DyCo 5 grown on extended \nsubstrates and on nanoporous membranes, controlling the sample tempe rature above and \nbelow the spin- reorientation transition temperature of ~ 350 K. Scanning electron microscopy \n(SEM) [Fig. 1(a)] and XPEEM images [ Fig. 1(b)] from the nanohole array show the \nhexagonal  arrangement of  antidots. Circles and lines on the SEM image mark several \npunctual defects and structural domains that occur during the self -assembling anodization \nprocess. The distance between  the borde rs of two nanoholes, where  the metallic alloy is \nsitting, is around 37 nm. The XPEEM image shows  photoelectrons emitted from the material \ndeposited on the surroundings of the nanoholes as the bright intensity contrast, and dark \ncontrast corresponds to the nanohole locations. By varying the x -ray photon energy and \nsimultaneously rec ording XPEEM images, we obtain x -ray absorption spectra  (XAS)  for Co \n[Fig. 1(c)] and Dy  [Fig. 1(d)], showing L 2,3 and M 4,5 absorption edges, respectively.  \nAt a temperature of  385 K, just above T R = 350 K, the XMCD -PEEM magnetic contrast  \nshown in Fig. 2 reveals  clear  in-plane  ferrimagnetic ordering of Co and Dy elements, both \nbeing anti -ferromagnetically aligned with respect to each other in the imaged x -y surface \nplane.  Very c learly, t he red (blue) regions of the Co XMCD images in Fig. 2 correspond to \nblue (red) regions of the Dy XMCD  signal . Figure s 2(a -b) show  the magnetic configuration \nfrom the antidot array sample and Fig s. 2(c -d) from its  extended thin film counterpa rt. Re d \nand blue -color contrast correspond to magnetic domains pointing in the surface plane and \nprojected along the incident x -ray beam direction, with a maximum of ~10 % XMCD \nasymmetry. White (or faint) color contrast indicates eith er out -of-plane magnetization , \nmagnetization perpendicular to the incident beam direction or the zero net magnetization as  it \nwould be expected from the nanohole positions. Comparing  the XMCD -PEEM images of the \nantidot array to  the ones of the extended film sample reveals clear  differ ences in the size of  8 \n the magnetic domains; the  antidot sample exhibits nanometer -sized domains separated by \nwhite color contrast, whereas the extended thin film sample exhibits domains in the order of \nseveral micrometers. This observation suggests that the  antidots act as p inning centers for the \nmagnetization  stabilizing  magnetic nano -domains that are  naturally separat ed from each other. \nBecau se the extended thin film lacks of such pinning centers, small nucleation domains  which \ninitiate  the magneti zation reversal have  collapsed to  form larger ones . \nIn the following, we present Co XMCD -PEEM  images for three selected temperatures \nfrom a heating -and-cooling cycle; initial state at 300 K, high -temperature state at 470 K, and \nthe final state  reached  after cooling back to 300 K again, for both the extended thin film and \nantidot array samples. Figures 3(a) and 3(d) show the initial states for both samples, where nearly -zero local magnetization  signal  along the in- plane orientation is observed. The reason \nfor the vanishing magnetic contrast  is the out -of-plane spin confi guration  below  T\nR. Upon \nheating  to 470 K, which is well above the transition temperature, we clearly observe the \nappearance of  in-plane oriented magnetic domains , as shown by the  XMCD images in Fig. \n3(b) and 3(e). When  we cool the samples  back to room temperature  and re -examine the \nresulting magnetic  properties , the extended thin film sample re -establishes its original out-of-\nplane  spin configuration,  as expected [Fig. 3(c)] ; however, the antidot array exhibits  a non-\nzero local magnetization as if the magnetic domains are mostly pinned to the in-plane  spin \norientation even in the absence of any  magneti c fields upon cooling [Fig. 3(f)]. To take a \ncloser look at the remanent local  magnetization of the antidot array, we extract line profiles \nalong red and blue bits of magnetic domains as well as the real str ucture as seen by SEM and \nXPEEM. The results are summarized in Fig s. 3(g) and 3(h), where we show l ine profiles  \nextracted  along magnetic im ages for opposite magnetization directions  (blue and red lines) . \nWe clearly observe magnetic information repeating every 45- 50 nm (between M ↑ and M=0 \nstates, and between M ↓ and M=0 states) , which correlates well with the difference between 9 \n the center -to-center inter -pore distance and the nanohole diameter . One bit of magnetic \ninformation being ~ 45 nm, one can estimate a data storage density of around 75 Gbits/inch2, \nwhich is in the same order of magnitude with the stora ge density of conventiona l hard -disk \ndrives.  \nFigure 4 (a) summarizes  key experimental results together with a schematic representation \nof the magnetic anisotropy Ku of the samples as a function of temperature. It shows various \nchannels of writing and storing data using  a medium of  DyCo 5 antidot array s. The easy axis \nof the magnetization rotates from out -of-plane to in- plane as the samples are heated above T R, \nand spins of the two sublattices of Co and Dy align in the sample plane. The change of the \neasy axis  is outlined by the  Co X MCD images from an antidot hexagonal unit cell and fr om \nthe extended thin film sample  on the right -hand side of  the schematic representation depicting \nthe in -plane anisotropy (K||) at high t emperatures.  C ooling the extended thin film sample back \nto room t emperature leads to  either one of the out -of-plane anisotropy configurations in the \nabsence of an external field , as seen in Fig. 3(c) . However, the antidot array sample has more \noptions upon cooling: it can decay into two in- plane and two  out-of-plane ani sotropy \nconfigurations , as shown by the Co XMCD -PEEM images on the left -hand side  of Fig. 4(a) . \nRegions within  the antidot sample with out -of-plane magnetization show a faint color \ncontrast;  therefore not much is seen in these images. Two regions of the sa mple with two \nhexagonal unit cells of antidots showing in- plane magnetization are also shown. We attribute \nthe occurrence of  this magnetic multi- domain state  to the presence of nanoholes , which act as \npinning centers for  the magnetic anisotropy of the samp le. To illustrate this in more detail, in \nFigs. 4(b -d) we show the results of micromagnetic simulations within about one hexagonal \nunit cell obtained using the OOMMF package [ 29]. In Fig. 4(b) we show a SEM image from \nthe antidot array superimposed with blue circles around the nanoholes. The circles are used as \na mask to construct the input for the calculations. For simplicity, we simulate a Co layer with 10 \n the magnetic anisotropy oriented along the + y direction and an initial out- of-plane \nmagnetization.  In Figs. 4(c) and 4(d) we show the calculated magnetization state for the \nantidot array and the extended film in the fully -relaxed state, respectively. Very clearly, the \nantidot ar ray in Fig. 4(c) exhibits a magnetic multi- domain configuration, while the extended \nfilm remains in a single -domain state in qualitative agreement with the experiments. The \ncalculations clearly reveal that the nanoholes pin the magnetization so that it follows  circular \ncontours around them , leading to local changes in the magnetic anisotropy.  These changes are \ncaused by the fact that every nanohole gives rise to six constrictions with its first neighbors, so that magnetic domains and domain walls are pinned or trapped by these constrictions. In \nthis way, every domain wall experiences a spatially -dependent dipolar interaction originating \nfrom a landscape of pinning potentials within the whole structure, resulting in local rotations of the magnetization and thus in magnetic multi- domain configurations .  \nHaving observed these variety of magnetic domains  in the experiment , one can consider \ncontrolling their occurrence by applying either in -plane or out -of-plane magnetic field pulses \n(H\n‖ or H┴) to selectively force the system to choose one of the four possibilities  mentioned \nabove . The experimental results  summarized in Fig. 4 (a) are therefore quite analogous to the \ncase where one applies a local magnetic field pulse during the cooling process to manipulate the mag netism of the nano- domains . It would be equally possible to employ this method for \nboth longitudinal and perpendicular recording as they follow the same basic principle. When \nthe energized write head passes by the medium, it leaves behind a magnetization p attern. For \nperpendicular recording, this magnetization pattern is ‘up’ and ‘down’ rather than ‘left’ and ‘right’ for longitudinal recording.  \nContrary to the high anisotropy HAMR materials (among others: FePt, CoPt, SmCo\n5), FI \nantidots such as the ones st udied here  would be a promising candidate that facilitates writin g \ndata by means of manipulating the magnetic easy and hard axes as a function of mild-11 \n temperature treatment. T he critical temperatu re of  our magnetic  system is not the Curie \ntemperature TC but the  spin reorientation temperature  TR, which is very close to room  \ntemperature.  Note that  differently from this, high anisotropy HAMR materials need to be \nheated to temperatures close to their T C, which are around 750 -1000 K [ 30], to sufficiently \nreduce their high coercivity so that a moderate write field can induce a new magnetic state. In \nthis respect, there have been also attempts to reduce the high T C of the recording layers with \ndoping. In one example, Thiele et al.  used Ni doping to reduce the T C of FePt; however, at the \nsame time the anisotropy decreases [ 31]. Assuming that in the near future  plasmonics  and \nnear-field optics  will focus and transmit optical energy to spot sizes of around (25- 50 nm)2 \nand demagnetize the sample [ 19,30 ], lower  phase -transition  temperatures  would still be more \nfeasible considering  issues su ch as power consumption, excess -heat dissipation and risk of \ndamaging medium materials.  \nSummarizing , we fabricated ferrimagnetic  DyCo 5 antidot arrays for heat -assisted and bit -\npatterned magnetic recording. Our work de monstrates a novel functionality  of ferrimagnetic  \nbit-patterned materials that can be used to write and store  magnetic data  near room \ntemperature. The size of the nanoholes used in this work was around ~ 68 nm and their \nhexagonal lattice constant  ~105 nm. W e have shown that the magnetization of the antidot \narray sample can be tuned from an out -of-plane to an in -plane spin configurat ion upon heating \nabove its spin- reorientation temperature of ~ 350 K. We have compared the magnetic \nproperties of the patterned s ample with the ones of its extended thin film counterpart and \nobserved remarkable differences in the size of the magnetic domains.  Our results reveal  the \nformation of small magnetic nano -domains in between antidots with well -defined and stable \nmagnetic dom ain configurations. Magnetic modulations of the nano- domains contrast are \nobserved to occur within ~45 nm for opposite magnetizations . We have attributed this \nbehavior to the change in the magnetic anisotropy caused by the nonmagnetic inclusions, 12 \n which at the same time act as pinning centers upon formation of  multiple  magnetic domains  \nin agreement with our micromagnetic simulations . Finally, w e have discussed the \nimplications of our findings for heat -assisted  magnetic recording using local magnetic probes.  \nOur work demonstrates that ferrimagnetic antidot arrays exhibit tunable and controllable \nmagnetic traits with the potential to provide superior functionality for energy -assisted bit -\npatterned media near room temperature.  \n \n   \n  13 \n FIGURES  \n \n \n \n \nFigure 1.  (a) SEM and (b) XPEEM images of the hexagonal lattice antidot array of 68 nm pore size \nand 105 nm separation between pore centers . XAS spectra of (c) Co and (d) Dy elements at their L 2,3 \nand M 4,5 edges, as obtained from XPEEM energy scan s using linear horizont al polarized x -rays from \nthe field of view seen in (b).  \n  \n14 \n  \n \n \n \nFigure 2.  XMCD -PEEM images measured at 385 K at the Co L 3 and Dy M 5 edges from the antidot \narray (a- b) and the extended thin film sample (c -d) showing the ferrimagnetic ordering of the Co and \nDy magnetic moments via the corresponding blue -red XMCD contrast. Antidots stabilize nanometer -\nsized domains  separate d from each other.  \n \n   \n15 \n  \n \nFigure 3.  XMCD -PEEM images measured at Co L 3 edge from the extended thin film sample (a -c) \nand from the antidot array sample (d -f) at the labeled temperatures. (g) Line profile along the structure \nof five neighbor ing antidots, XPEEM (black line) and SEM (dashed line). (h) Line profiles along \nXPEEM magnetic images for opposite magnetizations (blue and red lines) show  magnetic information \nrepeating every 45 -50 nm (between M ↑ and M=0 states, and between M ↓ and M=0 states).  \n16 \n  \n \nFigure 4. (a) S chematic representation of the heat -assisted magnetic recording process and the spin \nreorientation transition as observed by XPEEM. A selection of images at the Co  L3 edge summarizes  \nthe experimental results.  On the left, the room -temperature results correspond to different XMCD \nimages of the antidot array.  (b-d) Results of micromagnetic simulations. In (b ), a SEM image from the \nantidot array superimposed with blue circles around the nanoholes is shown. The circles are used as a \nmask to construct the input for the calculations. The magnetic anisotropy is oriented along the + y \ndirection and the initial magnetization is out -of-plane. (c ) Calculate d multi -domain configuration of  \nthe antidot array in the relaxed state. Small arrows depict the magnetization directions, blue and red \ncolors the projection of the magnetization along the y  direction. (d) Same calculations as in (c ) for the  \nextended film, revealing a single- domain configuration.  \n17 \n AUTHOR INFORMATION \nCorresponding Author  \n*E-mail: jaime.sanchez -barriga@helmholtz -berlin.de  \nACKNOWLEDGMENT  \nK. J. M and M. V gratefully acknowledge the support from MINECO under project \nMAT2013- 48054- C2-1-R. \n \nREFERENCES  \n                                                           \n[1]  S. S. P. Parkin , C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S.- H. \nYang,  Nature Mater.  3, 862 ( 2004).  \n[2]  M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne, G. \nCreuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).  \n[3]  G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989).  \n[4]  S. S. P. Parkin, K. P. Roche, M. G. Samant, P. M. 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Appl. \nPhys. 91, 6595 (2002).  "    },    {        "title": "1705.01335v1.Synthetic_ferrimagnet_spin_transfer_torque_oscillator__model_and_non_linear_properties.pdf",        "content": "Synthetic ferrimagnet spin transfer torque oscillator: model and non-linear properties\nB. Lacoste,1M. Romera,2,∗U. Ebels,2and L. D. Buda-Prejbeanu2\n1International Iberian Nanotechnology Laboratory, Braga, Portugal\n2Univ. Grenoble Alpes, CEA, INAC-SPINTEC, CNRS, SPINTEC F-38000 Grenoble, France\n(Dated: Date of submission October 12, 2018)\nThe non-linear parameters of spin-torque oscillators based on a synthetic ferrimagnet free layer\n(two coupled layers) are computed. The analytical expressions are compared to macrospin simula-\ntions in the case of a synthetic ferrimagnet excited by a current spin-polarized by an external fixed\nlayer. It is shown that, of the two linear modes, acoustic and optical, only one is excited at a time,\nand therefore the self-sustained oscillations are similar to the dynamics of a single layer. However,\nthe non-linear parameters values can be controlled by the parameters of the synthetic ferrimagnet.\nWith a strong coupling between the two layers and asymmetric layers (different thicknesses), it is\ndemonstrated that the non-linear frequency shift can be reduced, which results in the reduction of\nthe linewidth of the power spectral density. For a particular applied field, the non-linear parameter\ncan even vanish; this corresponds to a transition between a red-shift and a blue-shift frequency\ndependence on the current and a linewidth reduction to the linearlinewidth value.\nKeywords: spin transfer torque, synthetic ferrimagnet, non-linear auto-oscillator\nSpin transfer torque oscillators (STOs) have promis-\ning applications as high frequency microwave generators.\nA typical STO nano-pillar is composed of two magnetic\nlayers separated by a metallic spacer or an isolating bar-\nrier. The magnetization of the first magnetic layer re-\nmains fixed in-plane or out-of-plane. It acts as a spin\npolarizer for the current flowing through the nano-pillar.\nThe magnetization of the second layer can be driven into\nself-sustained oscillations by an applied DC current due\nto spin transfer torque (STT)1–3. The oscillation of the\nfree layer magnetization gives rise to a variation of the\nresistance of the pillar, so that an alternative voltage ap-\npears at its boundaries. For a single domain free layer,\nthe generated microwave signal is typically in the GHz\nrange. However, the large linewidth, in the order of tens\nof MHz, is an obstacle for functional devices.\nIn order to improve the STO characteristics, an accu-\nrate and simple model describing the dynamics is funda-\nmental. For a single-layer (SL) STO, the general frame-\nwork of non-linear auto-oscillators (NLAO), proved to\nbe a particularly well adapted model4–7. Indeed, most\nof the features exhibited by experimental devices could\nbe explained within this framework, such as the field\nand current dependence of the frequency, the broadened\nlinewidth8, but also synchronization to an external signal\nor to other STOs9. More importantly, this model defines\na key parameter for understanding the STO behavior:\nthe non-linear amplitude-phase coupling parameter. By\nevaluating this non-linear parameter from the magnetic\nproperties of the layer, it was found that the linewidth of\nthe STO was reduced when applying a transverse field,\nfor instance5. However this model is confined to a SL free\nlayer and some recent works studied STO devices where\nthe free layer is composed of two coupled layers consti-\ntuting a synthetic ferromagnet (SyF)10,11. Allegedly, the\nadditional coupling energy would increase the magnetic\nstiffness and reduce the fluctuations. However, coupled\nsystems are also more complicated to understand and ageneral analytical model is necessary to explain and de-\nfinetheimportantparametersofitsdynamics. Typically,\nit would be useful to be able to calculate the non-linear\namplitude-phase coupling parameter of a SyF-STO.\nToanswerthisquestion, weproposetoextendthisframe-\nwork, the NLAO model, to describe the dynamics of two\ncoupled layers subjected to spin transfer torque. In order\nto treat the most general case, different coupling are in-\ncluded : the Ruderman-Kittel-Kasuya-Yosida (RKKY)\ninteraction, the dipolar coupling and the mutual STT.\nThe non-adiabatic STT (or field-like torque) is also in-\ncluded, although its effect was found to be negligible in\ntheparticularconfigurationsexaminedinthispaper. Itis\nfundamental in the dynamics of self-polarized STO12,13,\nthough.\nThe NLAO theory is based on a change of coordinates\nto complex variables to represent the magnetization dy-\nnamics of the layers. The phase and amplitude of the\ncomplexvariablesdescribethenon-lineardynamicsofthe\nauto-oscillator. The validity of this approach is limited\nto quasi-conservative trajectories, for which the energy is\nalmost constant, and to small oscillation amplitudes. Us-\ning common diagonalization techniques, the conservative\npart of the magnetization equation of motion is simpli-\nfied to two terms : a linear and a non-linear contribution.\nThe dissipative part, which is supposed to be small com-\npared to the conservative part, defines the equilibrium\nenergy of the auto-oscillator by balancing the negative\nGilbert damping and the positive STT.\nThe first and second parts of this work describe the steps\nto extract the auto-oscillator equation for two coupled\nSyF layers in the macrospin approximation. In the third\nand fourth part, we describe the dynamics of the SyF-\nSTO defined by two coupled equations, so the STO can\nbe described by two modes. However, only one of them\nis usually excited into steady-state at a time, so the SyF-\nSTO is equivalent to a single-layer (SL) STO. This is an\nimportant result of this paper. The parameters of thearXiv:1705.01335v1  [cond-mat.other]  3 May 20172\nsingle-mode SyF-STO are computed, especially the non-\nlinear parameter, which is responsible for the frequency\ntunability, the large linewidth and the synchronization\nbandwidth. Another important result of this paper is\nthe link between the vanishing of the non-linear parame-\nter and the transition between a redshift and a blueshift\nregime. Finally in the fifth part, we study how to de-\ncrease the linewidth of a SyF-STO by changing the cou-\npling strength and the thickness of the layers.\nI. DESCRIPTION OF THE SYSTEM\nA. Landau-Lifshitz-Gilbert-Slonczewski equation\nWe consider the system in Figure 1 of two magnetic\nlayers, labeled 1 and 2 constituting a synthetic ferromag-\nnet (SyF). The total free energy Eholds the demagnetiz-\ning energy, the uniaxial anisotropy energy and the Zee-\nman energy (including an exchange energy) of both lay-\ners, plus a conservative coupling term between the two\nlayers, consisting of an RKKY interaction coupling and\nthe dipolar coupling :\nE=µ0\n2V1M1Hd1(m1·uz)2−µ0\n2V1M1Hk1(m1·ux)2\n+µ0\n2V2M2Hd2(m2·uz)2−µ0\n2V2M2Hk2(m2·ux)2\n−˜Dxm1xm2x−˜Dym1ym2y−˜Dzm1zm2z\n−µ0V1M1(Hx+Hex 1)(m1·ux)\n−µ0V2M2(Hx+Hex 2)(m2·ux) (1)\nHereµ0is the permeability of free space. V1=t1S\nandV2=t2Sare the volumes of the layers, with\nthicknesses t1andt2and surface S.M1andM2are\nthe saturation magnetizations of the layers, Hd1,Hd2\ntheir demagnetizing fields (supposed positive), Hk1,Hk2\nthe uniaxial anisotropy fields. For each layer labeled\nbyi= (1,2), we define the demagnetizing coefficients\n(Ni\nxx,Ni\nyy,Ni\nzz), and the interface anisotropy constant\nKSi, so thatHdi= (Ni\nzz−Ni\nyy)Mi−2KSi/(µ0Miti)\nandHki= (Ni\nyy−Ni\nxx)Mi.Hex1,Hex2are the exchange\nfields acting on each layer (for instance from a coupling\nwith a fixed anti-ferromagnet), and Hxthe applied field\nalong the easy axis.\nThe coefficients ˜Dx,˜Dyand ˜Dzaccount for the (conser-\nvative) coupling between the two layers. They include\nRKKY interaction term and the dipolar coupling, such\nthat, fori= (x,y,z ),˜Di=SJRKKY +Di, where\ntheDiare the dipolar coupling energy coefficients in\nmacrospin. Note that a negative JRKKYcorresponds to\nan anti-ferromagnetic coupling between the layers. Two\nadjacent layers give rise to negative DxandDy, and\npositiveDz.\nThe layers are also subject to a spin transfer torque\n(STT) due to a current flowing perpendicular to the lay-\ners. A positive current corresponds to electrons flowing\nFigure 1. Schematics of the synthetic ferrimagnet (SyF) as a\nfree layer with a fixed in-plane magnetized reference layer.\nfrom layer 2 towards layer 1, and then to the reference\nlayer. Thus, layer 1 is subjected to the STT from the\nreference layer and to the STT from layer 2 (with a neg-\native factor because layer 1 receives reflected electrons\nfrom layer 2). Layer 2 is subjected to the STT from the\nreferencelayerandfromlayer1(becauseofelectronsthat\nwere spin polarized after passing through layer 1). These\nspin torques acting on the two layers are modeled by two\nspin torque potentials, for layer 1 and 2 respectively, P1\nandP214:\nP1=−~\n2|e|Iη1m1·ux+~\n2|e|Iη21m1·m2\nP2=−~\n2|e|Iη2m2·ux−~\n2|e|Iη12m1·m2\nIis the current flowing through the layers. η1(resp.η2)\nis the effective spin-polarization of the current in layer\n1 (2) due to the fixed in-plane polarizer positioned be-\nfore layer 1 according to the direction of the current. η12\n(resp.η21) is the effective spin-polarization of the current\nin layer 2 (1) due to layer 1 (2).\nMoreover, the two layers are subjected to perpendicu-\nlar (or field-like) spin transfer torque (pSTT), from the\nreference layer and from the other layer. The pSTT is\nmodeled by two potentials, similar to the spin torque po-\ntentials defined above :\n˜P1=−~\n2|e|Iβ1m1·ux+~\n2|e|Iβ21m1·m2\n˜P2=−~\n2|e|Iβ2m2·ux−~\n2|e|Iβ12m1·m2\nThe equation of motion is given by the Landau-\nLifshitz-Gilbert-Slonczewski (LLGS) equation. In this\nform, the damping is defined with respect to the time-\nderivative of the magnetization vector; after moving all\nthe time-derivatives on the left-hand-side, the LLGS3\nwrites :\nµ0V1M1dm1\ndt=γ0m1×∂E\n∂m1+γ0m1×∂˜P1\n∂m1\n+γ0m1×/parenleftbigg\nm1×∂\n∂m1(P1−α1E)/parenrightbigg\nµ0V2M2dm2\ndt=γ0m2×∂E\n∂m2+γ0m2×∂˜P2\n∂m2\n+γ0m2×/parenleftbigg\nm2×∂\n∂m2(P2−α2E)/parenrightbigg\nThe Gilbert damping coefficients of the two layers are\ngiven byα1andα2. The correction to the gyromagnetic\nratio due to the damping coefficient has been neglected,\nsoγ0=µ0γwhereγis the gyro-magnetic ratio.\nAccording to the form of the LLGS equation used in\nthis paper, the coefficients βj(j= (1,2,12,21)) of the\nfield-like torques can contain a term proportional to the\ncoefficients ηjfrom the damping-like STT and to the\nGilbert damping constants of the two layers, that we call\npseudo-field-like torque. Namely β1=α1η1,β2=α2η2,\nβ12=α2η12andβ21=α1η21. Such additional terms\nwould be coming from the transformation of the STT\nfrom the Gilbert-form of the LLGS equation to the\nLandau-form.\nBy writing the LLGS equation in this form, the\nfree energy part, which is common to both layers, is\nseparated from the rest. This will allow to use a similar\nformalism as for the description of a single layer in\nprevious publications4,5.\nIn order to simplify the notations, we introduce the\nlayer asymmetry β, the geometrical mean magnetic vol-\numeMand the following normalized hamiltonian and\npotentials :\nβ=/radicalbigg\nM2t2\nM1t1M=µ0S/radicalbig\nM1t1M2t2\nH=γ0E\n2M∆1=γ0˜P1\n2M∆2=γ0˜P2\n2M\nΓ1=α1H−γ0P1\n2MΓ2=α2H−γ0P2\n2M\nIn the following, dotted variables represent their time\nderivative. Therefore the LLGS equation rewrites :\n1\n2β˙m1=m1×∂H\n∂m1+m1×∂∆1\n∂m1\n+m1×/parenleftbigg\nm1×∂Γ1\n∂m1/parenrightbigg\n(2)\nβ\n2˙m2=m2×∂H\n∂m2+m2×∂∆2\n∂m2\n+m2×/parenleftbigg\nm2×∂Γ2\n∂m2/parenrightbiggWe decompose the right-hand-side of the LLGS equation\nin two parts that will be treated separately: (i) the con-\nservative hamiltonian terms (simply called conservative\nin the following) that are composed of the first terms\non the right-hand-side and depend only on H. (ii) the\nconservative non-hamiltonian terms and the dissipative\nterms (simply called dissipative in the following because\nthe dissipative terms play a more important role) that\nare composed of the other two terms (respectively) on\nthe right-hand-side.\nIn general, the damping constants α1,α2are considered\nto be small ( <0.1) and the applied current is reasonably\nsmall, so the conservative part is larger than the dissipa-\ntive part. The two different orders of magnitude further\nsupport the distinction made between the two parts.\nB. Numerical parameters\nThe results from the extended NLAO model will be\ncompared to macrospin LLGS simulations. The case of\nan asymmetric SyF shows interesting properties, espe-\ncially in terms of linewidth reduction. As all cases can-\nnot be reproduced here, we focus on a SyF with thickness\nasymmetry between the two layers. However, asymme-\ntry can also be introduced by submitting one layer to an\nexchange field or by reducing the effective demagnetiz-\ning field of one of the layers with perpendicular interface\nanisotropy.\nFor the inter-layer coupling, two regimes are considered,\nsmall coupling JRKKY =−2×10−4J/m2and large cou-\nplingJRKKY =−5×10−4J/m2. The dipolar coupling\nis neglected in the macrospin simulations. This is sup-\nported by the fact that, in the macrospin approximation\nandinnano-pillarswithcircularcross-section,thedipolar\ncoupling is an antiferromagnetic coupling in the in-plane\ndirections ( xandydirections in our convention) and a\nferromagnetic coupling in the normal direction ( zdirec-\ntion). Because of the high demagnetizing field in thin\nlayers, the trajectories have a small out-of-plane compo-\nnent, so contribution from the dipolar coupling is com-\nparable to a low RKKY antiferromagnetic coupling. For\nthe layer thicknesses considered, the dipolar field is lower\nthan the RKKY coupling field, so the dipolar coupling is\nsimply neglected.\nThe rest of the parameters are defined in Table I.\nAccording to the value of the area Sof the pillars, cur-\nrents expressed in mA correspond to current densities of\n1011A/m2.\nThe current is considered to be unpolarized after going\nthrough the first layer, so η2= 0. However the same\nqualitative results were obtained15if we suppose that\nη2=±η1.\nThe conservative part is the most important to de-\nscribe the self-sustained oscillations because the trajec-\ntories of the self-sustained oscillations are close to the\nconstant energy trajectories. For this reason, a change of\nvariables that describes accurately the conservative part4\nIdentical properties Value\nMs1,Ms2 1×106A/m\nHd1,Hd2 0.9×106A/m\nHk1,Hk2 10×103A/m\nHex1,Hex2 0\nα1,α2 0.02\nS 10−14m2\nη21,η12 0\nβ1,β2,β12,β21 0\nDifferent properties Values\nt1andt21.8 and 2.2 nm\nη1andη2 0.5 and 0\nTable I. Properties of the magnetic layers.\nand treats the dissipative part as a small perturbation is\nadapted to describe the dynamics of the STO. This will\nbe developed in the next part.\nII. TRANSFORMATION TO COMPLEX\nVARIABLES\nA. Complex variables: conservative part\nWe intend to rewrite the LLGS equation in complex\nform representing the evolution of two modes a1anda2.\nLeta= (a1,a2)be a 2-dimensional complex vector. The\ngoal is to write the conservative part of the LLGS equa-\ntion in the form :\n˙a=−i∂H\n∂a†(3)\nThe elements of the basis, a1anda2, represent uni-\nform modes around the equilibrium position, with com-\nplex conjugates a†= (a†\n1,a†\n2). In the following, we focus\nonly on the modes around the parallel equilibrium state\n(or antiparallel depending on the sign of the RKKY cou-\npling constant and the dipolar coupling), i.e. the syn-\nthetic ferrimagnet (SyF) is in the plateau region. The\nequilibrium position is represented by :\nmeq\n1x=mux meq\n2x=mnux\nHeren= sign( ˜Dx)reflects the ferromagnetic or anti-\nferromagnetic type of coupling between the two lay-\ners:n= +1ferromagnetic coupling, n=−1anti-ferromagnetic coupling. The direction of layer 1 rela-\ntively to the fixed reference layer is given by m:m= +1\nfor a parallel (P) orientation, m=−1for an antiparallel\n(AP) orientation. The initial state is then defined by a P\nor AP configuration (with respect to the reference layer)\nand a ferromagnetic or anti-ferromagnetic coupling be-\ntween layer 1 and layer 2.\nWe proceed to a change of coordinate system so that\nthe equilibrium magnetizations have the same definitions\nfor all the layers. They are defined by meq\ni=ui\nζfor\ni= (1,2):\nu1\nζ=mux u2\nζ=mnux\nu1\nξ=muy u2\nξ=mnuy\nu1\nη=uz u2\nη=uz\nThe expressions of a1anda2with respect to the local\nmagnetization coordinates have to be chosen adequately\nso that the conservative part of LLGS in this new system\nof coordinates take the hamiltonian form of Eq. (3). For\nthat we set :\na1=1√βm1ξ−im1η/radicalbig\n2(1 +m1ζ)(4)\na2=/radicalbig\nβm2ξ−im2η/radicalbig\n2(1 +m2ζ)(5)\nNotice that there are other choices of (H,a1,a2)that al-\nlows to rewrite the LLGS equation in the hamiltonian\nform of Eq. (3), notably by multiplying a1anda2by the\nsame constant term CandHbyC2. The quadratic part\n(as it will be defined later) of the Hamiltonian would re-\nmain unchanged by changing this factor, but the quartic\n(and the other orders) part would be affected. Hence, it\nis not possible to compare coefficients of quartic or higher\norder for different geometries, as their definition depends\non the arbitrary choice of the constant C. Instead, nor-\nmalized coefficients should be compared.\nThe expression of Hwith respect to the new variables\n(a1,a2)and their complex conjugates (a†\n1,a†\n2)can be di-\nvided asH=H2+H4by dropping the constant term\nand neglecting higher order hamiltonian terms. In terms\nof the complex variables, H2is the quadratic part and\nH4is the quartic part.\nH2=A1a1a†\n1+A2a2a†\n2+1\n2/parenleftbig\nB1a2\n1+B2a2\n2+c.c./parenrightbig\n+/parenleftbig\nC12a1a2+D12a1a†\n2+c.c./parenrightbig\nH4=U1a2\n1a†2\n1+U2a2\n2a†2\n2+W12a1a2a†\n1a†\n2\n+/parenleftbig\nV1a3\n1a†\n1+V2a3\n2a†\n2+c.c./parenrightbig\n+/parenleftbig\nY12a2\n1a†\n1a2+Y21a1a2\n2a†\n2+c.c./parenrightbig\n+/parenleftbig\nZ12a1a†2\n1a2+Z21a†\n1a2\n2a†\n2+c.c./parenrightbig\nWe introduce new parameters that correspond to the5\ncharacteristic frequencies :\nω1\nk=γ0Hk1,ω2\nk=γ0Hk2,ω1\nd=γ0Hd1,ω2\nd=γ0Hd2,\nω1\na=γ0m(Hx+Hex1),ω2\na=γ0nm(Hx+Hex2),\nω0\nc=γ0\nMn˜Dx,ω−\nc=γ0\nMn˜Dy−˜Dz\n2,ω+\nc=γ0\nMn˜Dy+˜Dz\n2.\nUsing these notations, the coefficients of the hamiltonian\nare given by :\nA1=ω1\nk+ω1\nd\n2+ω1\na+βω0\nc,A2=ω2\nk+ω2\nd\n2+ω2\na+ω0\nc\nβ,\nB1=−ω1\nd\n2,B2=−ω2\nd\n2,C12=−ω−\nc,D12=−ω+\nc,\nU1=−βω1\nk−β\n2ω1\nd,U2=−ω2\nk\nβ−ω2\nd\n2β,\nW12=−2ω0\nc,V1=β\n4ω1\nd,V2=ω2\nd\n4β,\nY12=β\n2ω−\nc,Y21=ω−\nc\n2β,Z12=β\n2ω+\nc,Z21=ω+\nc\n2β.\nIn matrix form,H2rewrites :\nH2=1\n2/parenleftbig\na†\n1a†\n2a1a2/parenrightbig\nA1D12B1C12\nD12A2C12B2\nB1C12A1D12\nC12B2D12A2\n\na1\na2\na†\n1\na†\n2\n\nThe notation xis used for the complex conjugate of\nx, to distinguish scalar coefficients from the magnetiza-\ntion complex variables (a1,a2)with complex conjugates\n(a†\n1,a†\n2). As for a SL oscillator, it is possible to diago-\nnalize the quadratic part H2of the hamiltonian16. In\nfact it is possible to do so for any number of layers, al-\nthough it becomes difficult to find analytical expressions\nformorethantwolayers. Thenewcomplexbasisiscalled\n(bop,bac), with :\n\na1\na2\na†\n1\na†\n2\n=Tab\nbop\nbac\nb†\nop\nb†\nac\n(6)\nTab=\nuop\n1uac\n1vop\n1vac\n1\nuop\n2uac\n2vop\n2vac\n2\nvop\n1vac\n1uop\n1uac\n1\nvop\n2vac\n2uop\n2uac\n2\n\nWe note/hatwideIthe 4x4 block matrix /hatwideI=/parenleftbigg\nI20\n0−I2/parenrightbigg\nwithI2\nthe2×2unity matrix. T†\nabis the transpose conjugate of\nTab. It verifies :\nT−1\nab=/hatwideIT†\nab/hatwideIIn the new basis, H2takes the simple form :\nH2=ωopbopb†\nop+ωacbacb†\nac\nThe complex variables (bop,bac)are eigenvectors of the\nlinear hamiltonian. They correspond to the two linear\nmodes of the SyF-STO: optical and acoustic. This base\nof eigenvectors is then used to express the non-linear part\nof the hamiltonian.\nThe expressions of ωop,ωacand of the coefficients of\nthe matrix Tabcome from diagonalizing the matrix ˜H2:\n˜H2=\nA1D12B1C12\nD12A2C12B2\n−B1−C12−A1−D12\n−C12−B2−D12−A2\n\nThe expression of ˜H2is for the general case, for any di-\nrection of the equilibrium magnetizations. In the con-\nfiguration studied here, with equilibrium configurations\nand applied fields along the easy axis, all the coefficients\nare real. We take this assumption in the following.\nFrom computing the eigenvalues of ˜H2, the following val-\nues are obtained for ωop/ac:\nω2\nav= (A2\n1+A2\n2)/2.−(B2\n1+B2\n2)/2.+D2\n12−C2\n12\n∆ =/parenleftBig\nA2\n1−A2\n2−B2\n1+B2\n2/parenrightBig2\n+ 4/parenleftBig\nC12(B1+B2)−D 12(A1+A2)/parenrightBig2\n−4/parenleftBig\nC12(A1−A 2)−D 12(B1−B2)/parenrightBig2\nω2\nop/ac =ω2\nav±√\n∆\n2(7)\nThefrequencies ωopandωaccorrespondtothetwomodes\noptical and acoustic that are observed in ferromagnetic\nresonance (FMR) experiments with SyFs. By definition,\nthe optical mode corresponds to the mode with the high-\nest frequency. The expressions of the two mode frequen-\ncies are in agreement with the expressions found in the\nliterature14,17.\nThe eigenvectors of ˜H2, which correspond to the\ncolumns of the matrix Tab, have complicated expressions.\nHowever, due to normalization conditions, they can be\nexpressed by 6 angles. For the two labels j= (op,ac),\nthe elements of the matrix Tabare given by :\nuj\n1= coshθjcosφj\nuj\n2=−coshθjsinφj\nvj\n1=−sinhθjcosψj\nvj\n2= sinhθjsinψj\nThe details about the coefficients are given in Ap-\npendix A.\nThe angles φjandψjare related to the coupling6\nbetween the two layers. In fact, if the coupling vanishes\n(C12=D12= 0), these angles vanish for one mode,\nsay the acoustic mode, φac=ψac= 0, whereas for the\nother mode, φop=ψop=π/2. So the optical mode bop\ndepends only on the layer 2 complex variable a2and the\nacoustic mode bacon the layer 1 and a1.\nThe angles θjcorrespond to the mixing between the\ndiagonal termsA1,A2and the off-diagonal terms B1,\nB2, by analogy to the transformation coefficients for a\nsingle layer.\nHowever, it is not possible to obtain an exact diago-\nnalization of the quartic part H4of the hamiltonian but\nnon-canonical transformations provide good approxima-\ntions. We distinguish the resonant terms, for which the\noverall phase vanishes, like bopb†\nop, from the non-resonant\n(or off-diagonal) terms, for which the overall phase varies\nwith time, like bopb†\nac.\nBecause in this configuration, all along the easy axis,\nthere is no cubic term in the Hamiltonian, the (non-\ncanonical) transformation to remove the conservative\nnon-resonant terms18does not affect the value of the di-\nagonal quartic terms. Equivalently, we then assume that\nthe off-diagonal terms of the quartic term are negligi-\nble. However this assumption is valid only if the mode\nfrequencies ωopandωacare large compared to the off-\ndiagonal terms. Concretely, when applying an external\nfield that is comparable to the spin-flop field, the acous-\ntic mode frequency almost vanishes and the previous as-\nsumption is no longer valid. In this case the dynamics is\nmore complicated because the conservative non-resonant\nterms become important. We therefore limit the major\ndiscussion to the field range below the spin-flop field.\nNeglecting the non-resonant terms, the quartic part has\nthe simple expression :\nH4=Nop\n2b2\nopb†\nop2+Nac\n2b2\nacb†\nac2+Tbopb†\nopbacb†\nac\nWhereNac(resp.Nop) is the acoustic (optical) non-\nlinearfrequencyshiftcoefficientand Tisthemixed-mode\nnon-linear frequency shift coefficient. They are all real.\nAll these coefficients come from the conservative part of\nthe LLGS equation, they depend on the demagnetizing\nfields of the layers, applied field and coupling energy.\nHowever, they are independent of the damping coeffi-\ncients of the layers and of the applied current.\nB. Complex variables: dissipative part\nWenowfocusonthedissipativepartoftheLLGSequa-\ntion. After the transformation to the complex variables\na1,a2, the LLGS equation writes :\n˙a=−i∂H\n∂a†−Fa (8)\nWhere Fa= (Fa1,Fa2)is a vector with two complex\ncomponents. The two dissipative complex componentsFa1,Fa2are truncated to contain only linear and cubic\nterms ina1,a2,a†\n1anda†\n2. The polynomial coefficients\nare noted with 4 indices (k,l,m,n ), so that :\nFai=/summationdisplay\nk,l,m,nfk,l,m,n\naia1ka2la†\n1ma†\n2nfori= 1,2\nTheexpressionsofthecoefficientsofthedissipativeterms\nare given in the Appendix B. Using the linear transform\nwiththematrix Tab,similarcoefficientsforthe b-variables\nare obtained, with b= (bop,bac):\n˙b=−i∂H\n∂b†−Fb\nFb= (Fbop,Fbac)and fori=op, ac :\nFbi=/summationdisplay\nk,l,m,nfk,l,m,n\nbibopkbaclbop†mbac†n\nSo that :\n/parenleftbiggFb\nF†\nb/parenrightbigg\n=T−1\nab·/parenleftbiggFa\nF†\na/parenrightbigg\nWhere FaandF†\naare expressed in terms of b-variables\nusing the transform of equation (6).\nAll these fk,l,m,n\nbicoefficients in the b-coordinates\nare complex in general. However, if the coefficients\nof the conservative terms are real ( B1,B2,C12,D12,\nV1,V2, etc...), only the field-like torque contributes to\nthe imaginary part. For the magnetic configuration\nstudied in this paper, the applied field is aligned with\nthe magnetization, so the conservative coefficients are\nreal. The field-like torque is also set to zero, so the\ndissipative coefficients are real. In any case, the real part\nis the really important part, as it defines the power as it\nwill be shown in the next section. The imaginary part\nonly gives a contribution to the phase equation and it is\nnegligible compared to the contribution from the con-\nservative part in the configuration studied in this paper,\nwith an external polarizer. Without external polarizer,\nbut taking into account the mutual spin-torque in a\nself-polarizer structure, the contribution of the field-like\ntorque is non-negligible as shown in reference 19.\nOf all the dissipative terms, the most important are\nthe resonant terms, i.e. the terms that are similar to the\nresonant terms from the conservative part. Taking into\naccount only these resonant terms, the dissipative part\nreduces to :\nFbop=bop/parenleftBig\nγop+Qopbopb†\nop+Ropbacb†\nac/parenrightBig\nFbac=bac/parenleftBig\nγac+Qacbacb†\nac+Racbopb†\nop/parenrightBig\n(9)\nγac(resp.γop) is the acoustic (optical) linear relaxation\nrate.Qac(Qop) is the acoustic (optical) non-linear\nrelaxation rate coefficient. Rac(Rop) is the coefficient of\nthe acoustic (optical) non-linear mode mixing relaxation7\nrate.\nBecause of the linear dependence of the STT amplitude\nwith respect to the applied current hypothesized in this\npaper, these coefficients depend linearly on the applied\ncurrent. The linear coefficients, γopandγac, are positive\nfor zero current, in agreement with the fact that the\nGilbert damping is a relaxation to the minimum energy\nconfiguration. They decrease with the current if the\ncurrent is applied in the direction that destabilizes the\nmagnetization. The dissipative coefficients also depend\non the demagnetizing fields and coupling energy, like the\nconservative coefficients.\nThe analytical expressions of the coefficients are very\nlengthy and therefore they are not presented here in de-\ntail. Instead, for each value of field and current, the co-\nefficients are calculated numerically through the various\ntransformations, using the materials parameters given in\nsection IB. The variation of the different coefficients with\nfield and current are given in section III, where the cou-\npled complex equations are solved.\nIII. DYNAMICS WITH RESONANT TERMS\nONLY\nIn order to illustrate some of the basic features of the\ncoupled system, in a first approximation only the reso-\nnant terms, i.e.H2,H4and Eq. 9 are considered for the\ntime evolution of bopandbac:\n˙bop=−ibop(ωop+Noppop+Tpac)\n−bop(γop+Qoppop+Roppac)\n˙bac=−ibac(ωac+Nacpac+Tpop)\n−bac(γac+Qacpac+Racpop)(10)\nWherepop=bopb†\nopandpac=bacb†\nacare the powers of\nthe two modes. All the coefficients are supposed to be\nreal.\nWe notice that the dynamics of the coupled system does\nnot reduce to two independent oscillator equations. Even\nif the two modes are decoupled in the linear regime\n(pop,pac/lessmuch1), the acoustic and optical modes are cou-\npled through the non-linear coefficients.\nIntroducing the phases φop,φacof the two modes, let’s\ndefine :\nbop=√pope−iφop\nbac=√pace−iφac\nUsing the definitions of bopandbac, one can derive\nseparate equations for the power and the phase. These\nwill be discussed in the next sections. It is reminded\nthat for a single layer the equivalent analytical equations\nyield as a stationary solution a constant oscillation\npower (cancellation of the dissipative part). In the next\nsection it is shown that the coupled Eq. 10 can reduce\nto a single mode equation under specific conditions.\nFigure 2. Linear and non-linear dissipative coefficients versus\napplied current IforHx=−40kA/m and JRKKY =−5×\n10−4J/m2. (a) Optical coefficients, (b) acoustic coefficients.\nAll values are divided by 2πto be in units of Hz and not in\nrad/s.\nFor this we start discussing the solutions to the power\nequations.\nA. Power equations\nTheequationsoftimeevolutionofthepowerandphase\nare derived from the complex equations 10. The equa-\ntions of evolution of the powers of both modes are given\nby the generalized Lotka-Volterra (LV) equations20:\n˙pop=−2pop(γop+Qoppop+Roppac)\n˙pac=−2pac(γac+Qacpac+Racpop)(11)\nLotka-Volterra systems are well known for modeling the\nevolution of predator-prey populations. We define the\nsingle-mode equilibrium powers ¯popand¯pacas :\n¯pop=−γop\nQop¯pac=−γac\nQac(12)\nTheeffective linear coefficients are defined by :\ndop=γop+ ¯pacRopdac=γac+ ¯popRac\nAnd the inter-mode mixing coefficient ∆is defined by :\n∆ = 1−RopRac\nQopQac\nThe convergence to equilibrium for the LV system is\ndescribed in reference 21 and references 22, 23 provide a\nclassification with state diagrams. The two-modes sys-\ntem has four equilibriums, their conditions for existence\nand stability are defined by :8\n•P0= (0,0)ifγop>0andγac>0:\nNo mode is excited, this is the subcritical regime\nwith only damped modes.\n•Pop= (¯pop,0)ifγop<0,Qop>0anddac>0:\nOnly the optical mode is excited and the acoustic\nmode vanishes.\n•Pac= (0,¯pac)ifγac<0,Qac>0anddop>0:\nOnly the acoustic mode is excited and the optical\nmode vanishes.\n•P∗= (p∗\nop,p∗\nac)ifdopQac<0,dacQop<0,\nQopQac∆>0and(dop+dac)/∆<0:\nThe system converges to a mixed-mode equilibrium\nwhere both modes have a finite power given by:\np∗\nop=−dop\nQop∆p∗\nac=−dac\nQac∆(13)\nNotice thatP0andP∗are compatible, they can be\nstable local equilibriums at the same time, but they are\nincompatible with PopandPac. And reciprocally, Pop\nandPaccan be stable at the same time, but not at the\nsame time asP0andP∗.\nGiven the specific conditions are fulfilled, each equilib-\nrium is defined and locally stable. However, the global\nconvergence to this equilibrium depends on the initial\nconditions, if they are in the basin of convergence of this\nequilibrium. For instance, PopandPaccan be stable\nat the same time, it depends on the initial conditions if\nthe system converges to one or the other equilibrium,\nor even if it diverges (which corresponds to a switching\nof one or both layers). See Fig. 4 in reference 20 for a\nphase portrait of pacversuspop— notedn1andn2.\nThe coefficients of Eq. (11) are plotted in figure 2\nversus applied current Ifor the macrospin parame-\nters defined previously and for Hx=−40kA/m and\nJRKKY =−5×10−4J/m2. Two threshold currents\nfor the modes excitations, Iac\ncandIop\nc, are defined by\nthe vanishing of the linear coefficients γopandγac,\nrespectively. For this particular set of parameters, the\nacoustic threshold current Iac\ncis lower than the optical\nthreshold current Iop\nc. Therefore the critical current Ic\ncorresponds to the acoustic threshold current, which is\nIc= 3.4mA in this particular case. Above the critical\ncurrentIc, the acoustic mode is excited, and because Qac\nis positive (not shown in Figure 2 above 2mAQacin-\ncreases linearly), the power converges to the equilibrium\nacoustic power; the optical mode remains zero. Above\nthe optical threshold current, the equilibrium acoustic\npower still exists and it is stable, because dop>0(not\nshown on the figures). However, Qopis negative, so\nno equilibrium optical power is defined and the optical\nmode may diverge. Therefore, the final state depends\non the initial conditions : if the acoustic power is close\nto the equilibrium ¯pacand the optical power is close to\n0, the system converges to the powers {0; ¯pac}; if theoptical mode diverges faster than the acoustic mode\nconverges to its equilibrium value, the whole system\nwill diverge, which corresponds to a reversal of the layers.\nHaving defined the equilibrium powers, the oscillation\nfrequency is given by the phase equations that will be\nanalyzed in the next section.\nB. Phase equations\nThe corresponding phase equations of Eq. 10 including\nonly resonant terms are :\n˙φop=ωop+Noppop+Tpac\n˙φac=ωac+Nacpac+Tpop (14)\nWe notice that the phase velocities ˙φopand ˙φacof the\ntwo modes are constant if the powers are at equilibrium\n(˙pop= ˙pac= 0). Moreover, the phase of each mode\ndepends not only on its own power, but also on the power\nof the other mode through the non-linear phase mixing\nT. But both phases are independent of each other : each\nmode oscillates at its own constant frequency. Note that\nthis is true only if the non-resonant terms are excluded,\nas shown in section IV below.\nLet’s consider the case of a single-mode excitation of\nthe acoustic mode, as it is observed in the simulations\nshown in this paper. In this case, pop= 0andpac=\n¯pac=−γac\nQac>0. Therefore the magnetization oscillates\nat the frequency fof the excited acoustic mode, which\nis given by :\n2πf=ωstt= Ωac=ωac+Nac¯pac (15)\nThis equation is equivalent to the phase equation\nof an STO composed of a single-layer (SL) free layer\nas described in previous work5. The power increases\nwith the applied current, and the frequency decreases\nor increases depending on the sign of Nac. Figure 3\nshows the transition between the two regimes, red-shift\n(frequency decrease with the current) and blue-shift\n(frequency increase) with JRKKY =−5×10−4J/m2, by\nchanging the applied field. The frequency is computed\nfrom the extended NLAO model and compared to\nthe frequency obtained from macrospin simulations,\nboth show a transition between red-shift and blue-shift\nat around−75kA/m. The change of regime with\napplied field in an asymmetric SyF was already observed\nnumerically24and experimentally10.\nAs stated, this transition corresponds to Nacchanging\nsign. The value of the non-linear coefficients of Eq. (14)\nis plotted versus applied field Hxin Figure 4 (a). Nac\nchanges signs at around Hx=−75kA/m, which, indeed,\ncorresponds to the red-shift/blue-shift transition. The\nself-sustained oscillations frequency f=ωstt/(2π)versus9\nFigure 3. Self-sustained oscillations frequency versus applied\ncurrentIwithJRKKY =−5×10−4J/m2and for different\napplied fields, from top to bottom : −40kA/m to −90kA/m.\nThe frequency scale is identical in all the panels, from 0 to\n4 GHz. Solid red line : computed from the extended NLAO\nmodel. Dashed blue line : extracted from LLGS simulations.\nBeyond the spin-flop transition, for Hx=−90kA/m, the\nextended NLAO model is not applicable.\nfieldHxatI= 4mA is reported in Figure 4 (b)\nand compared to the acoustic FMR frequency ωac\nand the frequency obtained from the simulations. We\ndifferentiate four regions, from low to high fields : (i)\nbelow the spin-flop field, at −90kA/m, the extended\nNLAO model is not valid. (ii) for higher fields but below\n−75kA/m, the acoustic mode is excited in the blue-shift\nregime, so ωstt> ωac. The discrepancy between the\nfrequency obtained from the extended NLAO model and\nthe simulation is high, as expected because the model is\nnot valid anymore if ωacis small. (iii) above −75kA/m,\nthe acoustic mode is excited, in the red-shift regime,\nωstt< ωac. The frequency computed from the model\nagrees with the simulations. (iv) above −20kA/m,\nthe applied current is too low to excite a mode, the\noscillator is in sub-critical mode. Notice that in the\nvicinity of the field value at which Nacvanishes, the\noscillator frequency does not change much with the\napplied field, in agreement with the simulations. At this\nfunctioning point, the oscillator frequency is not very\nsensitive neither to the applied field, nor to the applied\ncurrent.\nWe showed that the frequency of the self-sustained os-\ncillationscanbepredictedbytheextendedNLAOmodel,\nin the next section the model will be compared to numer-\nical simulations to define its validity range.\nFigure4. (a)Non-linearand(b)linearfrequencytermsversus\napplied field HxforI= 4mA andJRKKY =−5×10−4J/m2.\n(a) Non-linear coefficients: optical Nop(green), acoustic Nac\n(red),inter-mode T(blue). (b)Linearcoefficients: dottedma-\ngenta line, linear ωac; red solid line, self-sustained oscillations\nfrequency from the model ωstt= Ω ac=ωac+pacNac; dashed\nblue line, self-sustained oscillations frequency from simula-\ntions. The field range is divided in four regions, from low\nto high fields : model non-applicable (NA), blue-shift regime\n(BS), red-shift regime (RS) and no excitation (NE). All values\nare divided by 2πto be in Hz units and not in rad/s.\nC. Single-mode description of the SyF-STO\nTwo sets of simulations are presented, showing the self-\nsustained oscillations frequency versus applied current\nand field for two coupling strengths: (i) Figure 5 in the\nsmallcouplingregime JRKKY =−2×10−4J/m2, (ii)Fig-\nure6the largecouplingregime JRKKY =−5×10−4J/m2.\nIn both figures, the frequency of the m1ycomponent of\nthe magnetization of layer 1 from macrospin simulations\nis plotted in the top panels (a). The frequency computed\nfrom the extended NLAO model is plotted in the bottom\npanels (b). State diagrams for these values of JRKKYare\ndisplayed in reference 25.\nWe observe a qualitative agreement between the model\nand the simulations, especially in the region close to\nthe critical current. First, above the acoustic critical\ncurrentIac\nc(region on the right of the red solid line), the\nmodel predicts self-sustained acoustic-like oscillations,\njust like the simulations (and other publications25).\nJust above the optical critical current Iop\nc(region on\nthe right of the green solid line and on the left of\nthe red solid line), there is no oscillation and the two\nlayersswitch, aspredictedbytheequationsofthepowers.\nThere are also several discrepancies, that will be dis-\ncussed in the following.10\nFigure 5. Frequency of the self-sustained oscillations versus\napplied current and field (a) from macrospin numerical sim-\nulations and (b) from the formulas for the power and phase\nfrom Eq. (15). The RKKY coupling is of −2×10−4J/m2.\nRed (green) solid lines represent Ic(Hx)the vanishing of the\nacoustic (optical) linear dissipative coefficient γac(op). Dotted\nlines correspond to the vanishing of the quadratic dissipative\ncoefficientQac(op).\nFirst, the out-of-plane precession (OPP) region is not\npredicted by the model. OPP are oscillations around\nthe energy maximum, which are not considered in this\nmodel. To describe the OPP, the projection base for the\ncomplexa-coordinates should be changed to the out-of-\nplane axes, instead of the equilibrium in-plane axes, and\nall the coefficients should be computed again.\nSecond, according to the simulations, self-sustained os-\ncillations are expected when the field is larger than the\nspin-flop field. However the extended NLAO model is\nnot valid in the spin-flop region. In fact it is not valid in\nthe vicinity of the spin-flop field either, as it was already\nmentioned. That is why for JRKKY =−2×10−4J/m2,\nFigure 5, the red-shift/blue-shift transition at around\nHx=−45kA/m, is not predicted by the extended\nNLAO model : it is too close to the spin-flop field\nvalue of−50kA/m. On the contrary, for JRKKY =\n−5×10−4J/m2, Figure 6, the red-shift/blue-shift tran-\nsition at around Hx=−70kA/m, with a spin-flop field\nat−90kA/m, is well predicted by the extended NLAO\nmodel.\nLast, the model predicts a much larger region of oscil-\nlations than the simulations. In the region on the right of\nthe optical critical current Iop\nc(green solid line), the dif-\nFigure 6. Same as Figure 5 with an RKKY coupling of −5×\n10−4J/m2.\nference between the model and the simulations becomes\nreally important. This was also shown in Figure 3. In\nthis region, the power is large, which is a known limit\nfor the validity of the NLAO model. But there could be\nanother explanation, because in this region, the model\npredicts a single-mode excitation with pop= 0, whereas\nthe simulations show that popdoes not vanish (not shown\nin the figures).\nTo explain the failure of the model in this region, we\npropose to study the influence of other terms that we\nfirstdiscardedinthemodel, namelythelinearcoefficients\nfrom the dissipative part that are non-resonant. The lin-\near terms are important corrections as they depend lin-\nearly in the powers, contrary to higher order terms. Also\nthey can be easily computed, which is not the case of\nhigher order terms.\nIV. CORRECTION DUE TO NON-RESONANT\nTERMS\nAs was shown in Section IIIC, Eq. 10 cannot capture\nall the features of the dynamics, in particular the fre-\nquency versus current. Therefore, in order to obtain a\nbetter description of the phase, we also include in Eq. 10\nnon-resonant, off-diagonal terms. This leads to the fol-11\nlowing equation:\n˙bop=−(iΩop+ Γop)bop−˜γopb†\nop−˜ϑopbac−ϑopb†\nac\n˙bac=−(iΩac+ Γac)bac−˜γacb†\nac−˜ϑacbop−ϑacb†\nop(16)\nHere Γop=γop+Qoppop+Roppacis the optical dissi-\npative part with only resonant terms from equation (9).\nIdentically, Γac=γac+Qacpac+Racpopis the acoustic\nresonant dissipative part. For the conservative part,\nΩop=ωop+Noppop+TpacandΩac=ωac+Nacpac+Tpop.\nThe coefficients of the non-resonant terms\n(˜γop,˜γac,ϑop,ϑac,˜ϑop,˜ϑac) are independent of the\npowers; for simplicity, we take the coefficients to be real,\nbut taking into account the imaginary part does not\nchange the general conclusions.\nThe equations for the amplitude and phase rewrite as :\n˙pop=−2(Γop+ ˜γopcos(2φop))pop\n−2√poppac/parenleftbig˜ϑopcos(φop−φac) +ϑopcos(φop+φac)/parenrightbig\n˙pac=−2(Γac+ ˜γaccos(2φac))pac\n−2√poppac/parenleftbig˜ϑaccos(φop−φac) +ϑaccos(φop+φac)/parenrightbig\n(17)\n˙φop= Ωop+ ˜γopsin(2φop)\n+/radicalbiggpac\npop/parenleftbig˜ϑopsin(φop−φac) +ϑopsin(φop+φac)/parenrightbig\n˙φac= Ωac+ ˜γacsin(2φac)\n+/radicalbiggpop\npac/parenleftbig˜ϑacsin(φac−φop) +ϑacsin(φop+φac)/parenrightbig\n(18)\nThe equations including the non-resonant terms are\nmore complicated, therefore each term will be treated\nseparately.\nWe first present a qualitative interpretation of each term\nand then evaluate its effect on the dynamics in Figure 7 :\nLLGS equation (Eq. (2)), extended NLAO model with\nonly resonant terms (Eq. (10)), with the addition of the\nlinear dissipative terms (Eq. (16)) and with all the terms.\nA. Inter-mode phase locking\nAn important disagreement between the LLGS sim-\nulation (Eq. (2)) and equation (10) is the phase of the\nnon-excited mode, as can be seen from the comparison\nof Fig. 7 (a) and 7 (c). In section IIIB, it was shown\nthat without the non-resonant terms the two modes have\ndifferent frequencies. However, in the LLGS simulations,\nFig. 7 (a), the two modes are locked, they have the same\nfrequency (although they can have an opposite sign19).\nThis discrepancy can be corrected by including the terms\nwiththe ˜ϑacand˜ϑopcoefficients, asisshowninFig.7(d).\nLet’s suppose that only an acoustic-like mode is ex-\ncited, buttheopticalmodedoesnotvanishtotally( pop≈0andpac= ¯pac). The powers are considered to be con-\nstant.\nThe differential equation for the phases of the two modes\nare :\n˙φop= Ωop+/radicalbiggpac\npop˜ϑopsin(φop−φac)(19)\n˙φac= Ωac+/radicalbiggpop\npac˜ϑacsin(φac−φop)(20)\nIn the acoustic phase equation (20), the second term\nis negligible compared to the constant frequency Ωac\nbecause of the powers ratio, so in the first order, the\nacoustic mode has a constant frequency ˙φac= Ωac, so\nφac= Ωact. However, in the optical phase equation (19),\nthe second term on the right-hand-side is dominant, also\nwith respect to the left-hand-side. This leads to the rela-\ntionsin(φop−φac) =/radicalbiggpop\npac/parenleftbigg˙φop−Ωop\n˜ϑop/parenrightbigg\n≈0, so in the\nfirst order, φop≈Ωact, orφop≈π+ Ωact. This means\nthat the frequency of the non-excited mode is locked to\nthe frequency of the excited mode in the supercritical\nregime.\nAt the second order, the phase difference is given approx-\nimately by :\nφop−φac=/radicalbiggpop\npac/parenleftbiggΩac−Ωop\n˜ϑop/parenrightbigg\n+kπwithk∈Z\n(21)\nSimilarly, the terms with the ϑacandϑopcoefficients\nare responsible for a locking with opposite frequency\n(same absolute frequency, but opposite phase sign), of\nthe form :φop+φac≈0, withφac(t) = Ωact.\nIf both ˜ϑopandϑopare included simultaneously, there\nis a competition between the two terms for the locking\nof the non-excited mode, to the same or the opposite\nfrequency as the excited mode. The resulting relation\nbetween the two phases is more complicated then. How-\never, regarding the time-average of the frequency, the\nnon-excited mode is locked to the frequency of the ex-\ncited mode if|˜ϑop|>|ϑop|, and to the opposite frequency\nif|˜ϑop|<|ϑop|. In other words, the coefficient with the\nhighest value (in norm) determines the type of locking,\ndirect or opposite. An example for opposite frequency\nlockingistheself-polarizedconfigurationdiscussedinref-\nerence 19.\nB. Power oscillations and second harmonics\nSecond, let’s focus on the term with the ˜γaccoefficient\n(itwillbesimilarforthetermin ˜γop). Weconsidera pure\nsingle-mode excitation of the acoustic mode, so pop= 0.\nNote that this analysis is valid for any single-mode non-\nlinear oscillator equation, including the SL case.12\nWithout the other non-resonant terms, the power and\nphase equations of the acoustic mode write :\n˙pac=−2Γacpac−2˜γaccos(2φac)pac\n˙φac= Ωac+ ˜γacsin(2φac)\nIn the assumption that the perturbation due to the ˜γac\nterm is small, one can use Lindstedt’s series to solve this\nsystem of equations26. If/epsilon1=˜γac\nΩacis small, then the\npowerpacand phaseφaccan be written as power series\nof/epsilon1:pac=p0+/epsilon1p1andφac=φ0+/epsilon1φ1. In the zeroth\norder,p0= ¯pacandφ0=¯Ωact, with ¯Ωac=ωac+Nac¯pac.\nIn the first order, the equation for the power deviation\np1and phase deviation φ1are :\n˙p1=−2¯pacQacp1−2¯pac¯Ωaccos(2 ¯Ωact)\n˙φ1=Nacp1+¯Ωacsin(2 ¯Ωact)\nWe use the fact that ¯pacQac=−γac/lessmuch¯Ωac, so the first\nterm on the right-hand side of the power equation is ne-\nglected. Therefore, in the first order and in the perma-\nnent regime, the power pacwrites :\npac(t) = ¯pac/parenleftbigg\n1−˜γac\nΩacsin(2 ¯Ωact)/parenrightbigg\nUp to the first order, the phase φacis given by :\nφac(t) =¯Ωact−˜γacωac\n2¯Ω2accos(2 ¯Ωact)\nTherefore the term ˜γacgives rise to oscillations of the\npower but also a second harmonics in the frequency spec-\ntrum. As a consequence, it also contributes to the STO\nsynchronization by an AC current on the second harmon-\nics. Notice that this term is also present in STO based on\na SL free layer but was omitted in previous descriptions4.\nC. Simulations and trajectories\nThe effect of the non-resonant terms on the dynamics\nis best seen by simulating the different equations.\nOn Fig. 7, we compare the simulations of different\nequations and performed in different coordinate sys-\ntems, and projected afterwards in the (pop,pac,φop,φac)-\ncoordinates for comparison. In Fig. 7 (a), the simulation\nis performed in the (m1x,m1y,m1z,m2x,m2y,m2z)-\ncoordinates, like the usual LLGS simulations, according\nto Eq. (2).\nIn Fig. 7 (b), the simulation is performed in the com-\nplexa-coordinates, using equation (8). The trajectory is\nvery similar to the LLGS trajectory. That is because the\nterms of order superior to 3 in (a1,a2)were dropped after\nthe canonical transformation (m1,m2)−→(a1,a2)and\nwith powers of the order of 10−2, this approximation isperfectly valid.\nIn Fig. 7 (c-e), the simulations are performed in the com-\nplexb-coordinates, from Equation (16). In Fig. 7 (c), all\nthe off-diagonal terms are omitted (which corresponds to\nEq.(10)). Thetrajectoryexhibitsaconstantfiniteacous-\ntic power, a vanishing optical power, and instantaneous\nfrequency for the acoustic and optical mode being con-\nstant but with different values. The constant power and\nfrequency of the acoustic mode are close to the averaged\nvalues computed from the LLGS equation.\nInFig.7(d), ϑop,ϑac,˜ϑopand˜ϑacaretakenintoaccount.\nThe powers are very similar to the powers obtained in\nFig. 7 (c), which justifies the approximation of constant\npowers used in the previous section. The frequency of\nthe non-excited mode, the optical mode, is locked to the\nacoustic frequency. The optical frequency is not constant\nthough, this is because of the competition between the\ntwo types of locking, direct and opposite. But its average\nvalue is close to the value of the acoustic frequency.\nIn Fig. 7 (e), ˜γopand˜γacare also included, so the sim-\nulated equation is exactly Eq. (16). The powers are not\nconstant anymore, but oscillate around the average value\ninstead. Although the average acoustic power is over-\nestimatedcomparedto theLLGSequation (0.043instead\nof 0.036, 20%over-estimated), the average frequencies\nmatch more accurately (-3.40 GHz instead of 3.46 GHz,\n2%under-estimated).\nIn conclusion, Eq. (10) with only resonant terms\npredicts accurately the excitation of the acoustic mode\nfor this set of parameters and it gives a good estimation\nfor the average values of the power and frequency of\nthe excited mode. In order to account for second order\nfeatures, like phase locking of the non-excited mode to\nthe excited mode and first harmonic oscillation, the\ncorrected equation (16) should be used. However, this\ncorrected model is not enough to explain the discrepancy\nwith the LLGS equation in the average power. When\nthe field becomes closer to the spin-flop field, this error\nbecomes so large that extended NLAO model is not\nvalid anymore. As stated in section IIA, the error is\nprobably due to higher order terms but this is out of\nthe scope of this paper. Similarly, the extended NLAO\nmodel fails at large applied currents and this cannot be\nexplained by the correction terms. It is also probably\ndue to higher order terms.\nWith the restrictions of the model of Eq. (10) in mind,\nin the next section we make predictions on how to reduce\nthe generation linewidth of the SyF-STO, which is a very\nimportant parameter for application. The value of the\nlinewidth given by the model were compared to LLGS\nsimulations.13\nFigure 7. Simulations for Hx=−40kA/m,I= 4mA andJRKKY =−5×10−4J/m2performed in the (a) m-variables, (b)\na-variables, (c) b-variables, only with resonant terms from Eq. (10), (d) b-variables with non-resonant terms from Eq. (16) but\n˜γop= ˜γac= 0and (e)b-variables with all non-resonant terms from Eq. (16). The results of the simulations are transformed to\ntheb-coordinates to compare them easily. Insets in (a) and (e) : zoom between 30 and 31 ns. Top panel figures : powers pac\n(red) andpop(green). Bottom panel figures : phase velocity or instantaneous frequency in GHz,∂φac\n∂t(red) and∂φop\n∂t(green).\nV. APPLICATION: REDUCE THE STO\nLINEWIDTH\nA. Thermal noise\nSo far, the system was supposed to be at zero tem-\nperature, however stochastic fluctuations arise at non-\nzero temperature. The effect of these fluctuations can\nbe estimated in regions where the single-mode approx-\nimation is valid. We consider single-mode acoustic-like\nself-sustained oscillations, but the same reasoning apply\nto any single-mode non-linear oscillator.\nWith finite temperature, the power and phase of the os-\ncillator are given by :\n˙pac=−2pac(γac+Qacpac) +/radicalbig\n4pacDacηp(22)\n˙φac=ωac+Nacpac+/radicalBigg\nDac\npacηφ (23)\nWhereηpandηφrepresent white Gaussian noise with\nnormalized variance and the diffusion coefficient Dacis\ndefined by :\nDac= Γ+\nacωT\nΩacwithωT=γ0kBT\n2Mwith Γ+\nacthe positive damping (without the contribution\nfrom the STT) computed at ¯pacandΩac=ωac+Nac¯pac.\nBecause of the thermal noise, the auto-oscillator\nexhibits a finite generation linewidth ∆ω, typical of a\nnon-linear single-mode oscillator6,7. The spectral density\ncan be Lorentzian or Gaussian depending on the value\nof the damping rate of the power fluctuations (or power\nrelaxation rate) Γp= ¯pacQac. The characterization of\na non-linear single-mode oscillator in the presence of\nthermal noise is detailed in Appendix C.\nIf the correlation time of the power fluctuations ( 1/Γp)\nissmallcomparedtothecharacteristicphasedecoherence\ntime(theinverseofthegenerationlinewidthbeingagood\nestimation), ∆ω/lessmuchΓp, the spectral density is Lorentzian\nand the full width at half-maximum (FWHM) ∆ωLis\ngiven by :\n∆ωL= ∆ω0/parenleftbig\n1 +ν2\nac/parenrightbig\n(24)\nwithνac=Nac/Qacand ∆ω0= Γ+\nacωT\n¯pacΩac(25)\nWhere ∆ω0is thelineargeneration linewidth and νacis\nthe normalized non-linear frequency shift coefficient.14\nOn the other hand, if the correlation time of the power\nfluctuations is much larger than the decoherence time,\n∆ω/greatermuchΓp, the spectral density is Gaussian with standard\ndeviation ∆ωGgiven by :\n∆ωG=|νac|/radicalbig\n∆ω0Γp (26)\nThe FWHM is given by√\n8 ln 2 ∆ωG.\nB. Key parameters to the linewidth\nThe expressions of Eq. 24 and 25 identify three pa-\nrameters that can be changed to reduce the value of\nthe linewidth to make functional devices : (i) increase\nthe power relaxation rate Γp, (ii) decrease the linear\nlinewidth ∆ω0and (iii) decrease the normalized non-\nlinear parameter νac.\n•The power relaxation rate Γp= ¯pacQac=|γac|, is\nproportional to the difference between the applied\ncurrent and the critical current Ic. An analytical\nexpression of γacis given in reference 14. In order\nto increase Γpwithout increasing Ic, the absolute\nvalue of the slope of |γac|versusIshould be in-\ncreased without increasing |γac|atI= 0.\n•The linewidth is proportional to the square of the\nnormalized non-linear parameter νac(if the nor-\nmalized non-linear parameter is large, which is the\ncase for STOs). Therefore, one way of reducing the\nlinewidth would be to reduce the non-linear param-\neterNacto zero. For the SyF structure discussed\nhere, this is the case at the transition from the red-\nshift to the blueshift regime. At the transition, the\nlinewidthisequaltothelinearlinewidthvalue ∆ω0.\nIn SL-STO, the vanishing of the non-linear param-\neter can be achieved by changing the equilibrium\nmagnetic state from in-plane along the easy axis to\nin-plane along the hard axis or out-of-plane4. This\nusually requires an external field. In SyF-STO, the\nvanishing of νaccan be achieved by applying an\nin-plane magnetic field along the easy axis. Such\na magnetic field can be generated by the dipolar\nfield from another magnetic layer with the same\neasy axis direction. Notice that a vanishing non-\nlinear parameter Nacmeans that the frequency of\nthe STO becomes independent of its power, and\nthen of the applied current; this loss of tunability\ncan be detrimental for applications. The synchro-\nnization bandwidth with an external signal is also\nproportional to the normalized non-linear parame-\nterνac5, so it should not be too small.\n•The linear linewidth is inversely proportional to\nthe geometrical mean magnetic volume M(see\nEq. (25)). With a SL, the critical current is pro-\nportional to the magnetic volume, so it is counter-\nproductive to increase it. For a SyF however, one\nFigure 8. Power relaxation rate Γpat constant super-\ncriticalityζ= 1andHx= 0versus RKKY coupling energy\nby area, plotted for different layer thicknesses (in nm). The\nother layer properties are the same as in Table I without ap-\nplied field,Hx= 0.Γpis divided by 2πto be expressed in Hz\ninstead of rad/s.\ncan think of a thin layer subjected to the spin-\ntransfer torque from the reference layer, coupled to\na thick layer not subjected to spin transfer torque.\nThus the critical current remains low, whereas the\nmean magnetic volume is increased.\nIn the next sections, we give some ideas about improving\nthese three parameters using a SyF-STO.\nC. Dependence of Γpon the coupling strength\nFirst, we study the variation of the power relaxation\nrate Γpwith some parameters of the SyF. However, be-\ncause Γpis related to the critical current Ic, we need\nto somehow normalize its value. To start, the super-\ncriticalityζis used instead of the current :\nζ=I−Ic\nIc\nUsing this normalized quantity, one can compare the\nvalues of Γpat twice the critical current value, which\ncorresponds to ζ= 1.\nThe applied field dependence of Γpis non-trivial but\nits value at zero field, Hx= 0, is interesting for ap-\nplications. The value of Γpat zero field, for the same\nsuper-criticality ζ= 1, is plotted in Figure 8 for different\nthicknesses of the two layers. It shows that Γpincreases\nwith the RKKY coupling strength, although it remains\nin the same order of magnitude as with a single layer\n(asymptotic value for JRKKY→0).\nD. Vanishing of the non-linear parameter Nac\nBecause of the quadratic dependence of the linewidth\non the normalized non-linear parameter νac, the most\neffective action to reduce the oscillator linewidth is to15\nFigure 9. Linewidth of m1y(yellow diamonds) from LLGS\nsimulations at 300 K, compared to the linewidth (solid red\nline), linearlinewidth (dotted red line) and Γp(dashed blue\nline) computed from the extended NLAO model, versus ap-\nplied field for a current of I= 4mA andJRKKY =−5×\n10−4J/m2.\ndecreaseNacby applying an in-plane field so the oscil-\nlator is excited close to the transition between red-shift\nand blue-shift.\nFigure 9 shows a comparison of the linewidth from LLGS\nsimulations at 300 K and from the extended NLAO\nmodel. The linewidth is plotted versus applied field,\natI= 4mA andJRKKY =−5×10−4J/m2. For\nthe simulations, the linewidth is calculated from a fit\nto a Lorentzian function. We observe a decrease of the\nlinewidth of almost two orders of magnitude between\nHx= 0andHx=−70kA/m. The linewidth decrease is\nassociated to the vanishing of the non-linear parameter\nNac. For small fields, |Hx|<50kA/m, the linewidth is\nmuch larger than the power relaxation rate, which cor-\nresponds to a Gaussian spectrum. On the other hand,\naroundHx=−70kA/m, the spectrum has a Lorentzian\nprofile. In the simulations, the spectrum appears to be\nindeed Lorentzian around Hx=−70kA/m. It is difficult\nto conclude about the line shape at lower absolute field\nvalue, though, because the noise is too large and both\nprofiles interpolate well the simulated spectrum.\nFigure 10 shows the linewidth versus field for a low cou-\npling,JRKKY =−2×10−4J/m2, and forI= 3mA.\nAs was shown above, the model does not predict a van-\nishing ofNac, therefore the predicted linewidth remains\nlarge in the whole field range. However, the macrospin\nsimulations show a redshift/blueshift transition at Hx=\n−45kA/m and a decrease of the linewidth to its linear\nvalueatthisfield. Infact, atthisfield, thefrequencydoes\nnot change with the applied current. In a single-mode\nmodel, it means that the phase does not depend on the\npower, so the linewidth is given by the linear linewidth\nalone. Therefore, in the low coupling regime, the os-\ncillation looks like it is single-mode, according to the\nmacrospin simulations, but the extended NLAO model\nis not sufficient to estimate the characteristic parameters\nof the oscillator.\nFigure 10. Same as Figure 9 with JRKKY =−2×10−4J/m2\nandI= 3mA.\nFigure11. Linewidthof m1yversusfield, comparisonbetween\na 2 nm thick single layer (blue) and a 2 nm layer coupled with\na 20 nm thick layer (red-orange), separated by (a) 1 nm and\n(b) 20 nm spacer. Symbols : LLGS simulations, solid lines :\nextended NLAO model, dotted lines : linearlinewidth from\nthe extended NLAO model.\nE. Coupling to a thick layer\nFinally, the last parameter that can be tuned to\nreduce the linewidth is the linear linewidth ∆ω0. The\nlinear linewidth does not depend much on the coupling\nstrength, but more on the magnetic volume, as stated\nbefore. In order to increase the total magnetic volume\nand keep a reasonable critical current, we can imagine a\nthinlayerof2nmcoupledtoathicklayerof20nm. With\nthis geometry, where the layers are very asymmetric,\nthe coupling strength plays an important role. Contrary16\nto the asymmetric case studied previously where the\nnon-linear parameter vanishes with the combination\nof asymmetric layers and strong coupling25, in the\nfollowing example, the non-linear parameter is reduced,\nso the linewidth is decreased, although not to the level of\nthelinearlinewidth. However, the linewidth reduction\nhappens at lower fields, more suitable for application,\nthan in the previous case.\nThe SyF of this example is compared to a nano-pillar\nbased on a single free layer. The SL-STO is composed\nof three layer : (1) a reference layer with in-plane\nfixed magnetization, a spin polarization of 0.3and\ncompensated dipolar fields (the total stray field is zero),\n(2) a tunnel barrier, and (3) a 2 nm thick free layer, with\nsaturation magnetization of 1×106A/m and damping\nconstant of 0.02. The nano-pillar has an elongated shape\nof150×100nm, giving a shape anisotropy to the free\nlayer along the x-axis. The SyF-STO of study comprises\nthe same SL nano-pillar, plus two additional layers : (4)\na spacer of variable thickness, 1 nm or 20 nm, and (5)\na 20 nm thick free layer with saturation magnetization\nof1×106A/m and damping constant of 0.02. The\nmagnetizations of the thick and the thin layers are\ncoupled through dipolar field, whose strength is lower or\nhigher depending on the thickness of the spacer. For the\ntwo cases, strong and weak coupling, the coefficients of\nequation (1) take the values :\nStrong coupling ( tMgO= 1nm) :\n(˜Dx,˜Dy,˜Dz)/S= (−1.9,−2.9,4.8)×10−4J/m2\nWeak coupling ( tMgO= 20nm) :\n(˜Dx,˜Dy,˜Dz)/S= (−0.9,−1.4,2.3)×10−4J/m2\nDue to the shape anisotropy, the magnetization of the\nthick layer is more stable than that of the thin layer, but\nit is still free to move. Like for the other stacks stud-\nied in this paper, the current is spin polarized between\nthe reference layer and the 2 nm thin layer, but it is\nconsidered unpolarized at any other point, including be-\ntween the thin and thick layers. The simulations were\nperformed at 300 K and the linewidth is computed do-\ning a Lorentzian fit of the power spectral density of m1y,\nthe magnetization of the thin layer along the y-axis. The\nlinewidth computed from the extended NLAO model and\nextracted from the simulation are showed in Figure 11.\nWhen the thin and thick layers are separated by 20 nm\nso they are weakly coupled, Figure 11 (a), the linewidth\nis of the same order of magnitude with or without the\nthick layer, in the hundreds of MHz range. Around\n10kA/m, which is the coupling field (at which the thick\nand thin layers have the same FMR frequency), the ex-\ntendedNLAOmodelpredictsanincreaseofthelinewidth\nabove the SL value, that is not observed in the simula-\ntions. On the contrary, the simulations show a decreaseof the linewidth around 10kA/m that we cannot explain.\nOverall, the value of the SyF linewidth is essentially com-\nparable to the value of the SL linewidth.\nIn the strongly coupled case, with a 1 nm spacer, Fig-\nure 11 (b), below the coupling field (around -10 kA/m),\nthe model predicts a reduction of the linewidth of one or-\nderofmagnitudebetweentheSyFandtheSLcase; above\nthe coupling field, an increase of the linewidth is pre-\ndicted. The simulations show a decrease of the linewidth\nofalmostoneorderofmagnitudefortheSyFcomparedto\nthe SL case for fields smaller than the coupling field, with\na minimum of 5 MHz at -25 kA/m, in agreement with\nthe model. Notice that the decreased linewidth is still\none order of magnitude larger than the linearlinewidth.\nAround the coupling field, the SyF and SL linewidths are\nequivalent, around 100 MHz. Above the coupling field,\nthe linewidth of the SyF is half the linewidth of the SL,\nin disagreement with the model.\nIn conclusion, we observe a reduction of the linewidth\nwhen a thin layer is strongly coupled to a thick layer.\nThe linewidth reduction occurs for all the fields except\nfor the coupling field, at which the linewidth value is as\nhigh as for a single layer.\nVI. CONCLUSION\nWe presented an extension of the NLAO model to de-\nscribe the self-sustained oscillations of a SyF composed\nof two layers coupled with RKKY coupling, dipolar cou-\npling and mutual STT. The analysis was restricted to\nthe plateau region of the SyF, where the two layers are\naligned along the same direction at equilibrium, paral-\nlel or anti-parallel. However, nothing prevents one from\napplying the same analysis to arbitrary initial configura-\ntions (and an arbitrary number of layers), by taking into\naccount a transverse field for instance, although the di-\nagonalization of the hamiltonian matrix would be more\ncomplicated and only numerical solution would be avail-\nable.\nIn the extended model, the SyF dynamics is described\nby two coupled complex non-linear equations, which cor-\nrespond, in the linear regime, to the acoustic and optical\nmode. Inthispaper, wefocusedonSyFswithfixedexter-\nnal polarizer and, for the set of parameters that we chose,\nonly one mode is excited at a time, the acoustic-like self-\noscillation. Therefore, the dynamics can be described by\na single mode power and a phase equation, as in the case\nof a single layer. It means that the self-sustained oscil-\nlations are defined by a constant power, resulting from\nthe balance between natural damping and STT. The fre-\nquency consists of a linear part and a non-linear part,\nproportionaltothepowerandtothenon-linearfrequency\nshiftNac. Identically, the linewidth of the power spectral\ndensity consists of a linear part and a non-linear part.\nIt was found that with a strong coupling and if the\ntwo layers are asymmetric, for instance if they have dif-\nferent thicknesses, the non-linear frequency shift Naccan17\nbe reduced strongly, so the linewidth is also strongly re-\nduced of one order of magnitude. In particular cases, Nac\ncan even vanish at a given field, which corresponds to a\ntransition between a red-shift and a blue-shift frequency\nversus current dependency. At this field, the linewidth is\nreduced to its linearlinewidth value, which is a reduction\nof almost two orders of magnitude. The power relaxation\nrateΓpwas not found to change much compared to the\nvalues found for a single layer STO.\nThis work confirms the robustness of the NLAO model\ntodescribesmalloscillationsofthemagnetizationaround\nthe equilibrium and it shows that it can be extended to\nseverallayers. Italsopresentedarelativelysimplesystem\nto study the interaction between oscillating modes and\nwe hope it can be extended to more general cases.\nACKNOWLEDGMENTS\nThisworkwassupportedbytheEuropeanCommission\nundertheFP7programNo.316657SpinIcurandtheFP7\nprogram No. 317950 MOSAIC.18\nAppendix A: Hamiltonian diagonalization :\ntransformation a-b\nThe expression of the coefficients of the transformation\nmatrixTabare given by the 6 angles : φj,ψjandθjfor\ni= (op,ac).\nFirst the angles φj(forj= (op,ac)) are computed :\nR+\nj= (A1+ωj)(A2+ωj)−D2\n12\nusj=A1−ωj+D12+C12D12−B1(A2+ωj)\nR+\nj(B1+C12)\n+B1D12−C12(A1+ωj)\nR+\nj(B2+C12)\nucj=A2−ωj+D12+B2D12−C12(A2+ωj)\nR+\nj(B1+C12)\n+C12D12−B2(A1+ωj)\nR+\nj(B2+C12)\nunj=/radicalBig\nus2\nj+ uc2\nj\nsinφj=usj\nunjcosφj=ucj\nunj\nNext the angles ψj(forj= (op,ac)) :\nR−\nj= (A1−ωj)(A2−ωj)−D2\n12\nvsj=A1+ωj+D12+C12D12−B1(A2−ωj)\nR−\nj(B1+C12)\n+B1D12−C12(A1−ωj)\nR−\nj(B2+C12)\nvcj=A2+ωj+D12+B2D12−C12(A2−ωj)\nR−\nj(B1+C12)\n+C12D12−B2(A1−ωj)\nR−\nj(B2+C12)\nvnj=/radicalBig\nvs2\nj+ vc2\nj\nsinψj=vsj\nvnjcosψj=vcj\nvnj\nAnd finally, the angles θj(forj= (op,ac)) are com-\nputed :\nF1=A1−B1−C12F2=A2−B2−C12tanhθj=−cosφj(F1−ωj)−sinφj(F2−ωj)\ncosψj(F1+ωj)−sinψj(F2+ωj)\nAppendix B: Coefficients of the dissipative part\nThe dissipative part is expressed as a power series in\nthea-coordinates, truncated after the cubic term :\nFai=/summationdisplay\np,q,r,sfp,q,r,s\naia1pa2qa†\n1ra†\n2sfori= 1,2\nWe use the following notations :\nν1=−mγ0\n2M~\n2|e|Iη1ν2=−mnγ0\n2M~\n2|e|Iη2\nν21= +γ0\n2M~\n2|e|Iη21ν12=−γ0\n2M~\n2|e|Iη12\nκ1=−mγ0\n2M~\n2|e|Iβ1κ2=−mnγ0\n2M~\n2|e|Iβ2\nκ21= +γ0\n2M~\n2|e|Iβ21κ12=−γ0\n2M~\n2|e|Iβ12\nHence the non-vanishing coefficients of Fa1andFa2with\nindices (p,q,r,s )are given by ( iis the imaginary unit,\ni2=−1) :\nFa1:\n(1,0,0,0) :α1A1+ 2nβν 21+ 2βν1−2inβκ 21−2iβκ1\n(0,1,0,0) :α1D12−(1 +n)ν21+i(1 +n)κ21\n(0,0,1,0) :α1B1\n(0,0,0,1) :α1C12+ (1−n)ν21−i(1−n)κ21\n(2,0,1,0) :−α1βA1+ 2α1U1−2nβ2ν21−2β2ν1\n(1,1,0,1) :α1W12−4nν21+ 4inκ21\n(0,2,0,1) :α1Z21+1 +n\n2βν21−i1 +n\n2βκ21\n(0,1,0,2) :α1Y21−1−n\n2βν21+i1−n\n2βκ21\n(1,1,1,0) : 2α1Z12+ (1 +n)βν21−iβ(1 +n)κ21\n(1,0,1,1) : 2α1Y12−(1−n)βν21+iβ(1−n)κ21\n(2,0,0,1) : 3α1Z12+ 31 +n\n2βν21−i1 +n\n2βκ21\n(2,1,0,0) : 3α1Y12−31−n\n2βν21+i1−n\n2βκ21\n(1,0,2,0) : 3α1V1\n(3,0,0,0) : 3α1V119\nFa2:\n(1,0,0,0) :α2D12−(1 +n)ν12+i(1 +n)κ12\n(0,1,0,0) :α2A2+2n\nβν12+2\nβν2−i2n\nβκ12−i2\nβκ2\n(0,0,1,0) :α2C12+ (1−n)ν12−i(1−n)κ12\n(0,0,0,1) :α2B2\n(0,2,0,1) :−α2\nβA2+ 2α2U2−2n\nβ2ν12−2\nβ2ν2\n(1,1,1,0) :α2W12−4nν12+ 4inκ12\n(2,0,1,0) :α2Z12+1 +n\n2βν12−i1 +n\n2βκ12\n(1,0,2,0) :α2Y12−1−n\n2βν12+i1−n\n2βκ12\n(1,1,0,1) : 2α2Z21+1 +n\nβν12−i1 +n\nβκ12\n(0,1,1,1) : 2α2Y21−1−n\nβν12+i1−n\nβκ12\n(0,2,1,0) : 3α2Z21+ 31 +n\n2βν12−i1 +n\n2βκ12\n(1,2,0,0) : 3α2Y21−31−n\n2βν12+i1−n\n2βκ12\n(0,3,0,0) : 3α2V2\n(0,1,0,2) : 3α2V2\nAppendix C: Thermal noise and Fokker-Planck\nequation\nThermal noise is introduced in Eq. (10) in the form :\n˙bop+bop(iΩop+ Γop) =/radicalbig\n2Dopηop\n˙bac+bac(iΩac+ Γac) =/radicalbig\n2Dacηac(C1)\nThe noise amplitudes DopandDac, also called diffusion\ncoefficients, are not constant and depend on the mode\npowers :Dop(bop,bac)andDac(bop,bac), but this depen-\ndence is omitted for clarity. They will be determined\nlater. Ωop,Ωac,ΓopandΓacare the conservative (for op-\ntical and acoustic modes) and the dissipative determin-\nistic coefficients. They also depend on the mode powers.\nηopandηacare two independent white noise sources with\nzero mean and correlators given by :\n/angbracketleftηi(t)/angbracketright= 0, for i∈(op, ac)\n/angbracketleftηi(t)ηj(t/prime)/angbracketright= 0, for i,j∈(op, ac)2\n/angbracketleftηi(t)¯ηj(t/prime)/angbracketright=δijδ(t−t/prime), fori,j∈(op, ac)2\nThe expressions of the diffusion coefficients are de-\ntermined by insuring that the equilibrium probability\ndensity function (PDF) for the powers and phase re-\nduces to the Boltzmann distribution without applied cur-\nrent5. Considering the Stratonovich stochastic differen-\ntial equation (SDE) (C1), the time evolution of the PDFP(pop,pac,φop,φac,t)is given by the following Fokker-\nPlanck (FP) equation :\n∂P\n∂t−∂\n∂pop(2popΓopP)−∂\n∂pac(2pacΓacP)\n+∂\n∂φop(ΩopP) +∂\n∂φac(ΩacP)\n=∂\n∂pop/parenleftbigg\n2popDop∂P\n∂pop/parenrightbigg\n+∂\n∂pop/parenleftbigg\nP∂\n∂pop(popDop)/parenrightbigg\n+∂\n∂pac/parenleftbigg\n2pacDac∂P\n∂pac/parenrightbigg\n+∂\n∂pac/parenleftbigg\nP∂\n∂pac(pacDac)/parenrightbigg\n+Dop\n2pop∂2P\n∂φ2op+Dac\n2pac∂2P\n∂φ2ac\nHere, we considered that the diffusion coefficients depend\nonlyonthemodepowers. Thetermsintheleft-hand-side\ncome from the deterministic equation, or drift, whereas\nthe terms in the right-hand-side represent the thermal\ndiffusion. At equilibrium/parenleftbigg∂P\n∂t= 0/parenrightbigg\n, the PDFP0is a\nuniform distribution for the phases, so we can remove the\nlast two drift terms of the left-hand-side. Moreover, the\nsecond and fourth diffusion terms of the right-hand side\nshould be compensated by two terms of drift that are\nusually neglected. They arise from the renormalization\nof the multiplicative noise terms27(see reference28where\nthese extra drift terms are included for a SL free layer).\nThe extra drift terms can be incorporated in (C1) to give\nthe correct equation in Stratonovich form :\n˙bop+bop(iΩop+ Γop) +fopbop=/radicalbig\n2Dopηop\n˙bac+bac(iΩac+ Γac) +facbac=/radicalbig\n2Dacηac\n(C2)\nWith :\nfop=−1\n2pop∂(popDop)\n∂pop\nfac=−1\n2pac∂(pacDac)\n∂pac\nInterestingly, these extra drift terms contribute only to\nthe power equations. In particular, they are responsible\nfor the non-zero average power below threshold (when\nsolving ˙p= 0,p= 0is not a solution anymore).\nAfter eliminating the extra drift terms, the FP equa-\ntion at equilibrium reduces to :\n0 =∂\n∂pop/parenleftbigg\n2popΓ+\nopP0+ 2popDop∂P0\n∂pop/parenrightbigg\n+∂\n∂pac/parenleftbigg\n2pacΓ+\nacP0+ 2pacDac∂P0\n∂pac/parenrightbigg\nWhere Γ+\nopandΓ+\nacare the dissipative terms at zero ap-\nplied current, i.e. the natural damping.20\nA solutionP0(pop,pac)of the former equation is :\nP0=Z−1exp/parenleftbigg\n−/integraldisplaypop\n0Γ+\nop\nDopdpop−/integraldisplaypac\n0Γ+\nac\nDacdpac/parenrightbigg\nWhereZis a normalization constant. The equilibrium\nPDF should correspond to the Boltzmann distribution,\nwhichisequalto Z/prime−1exp/parenleftbigg\n−E\nkBT/parenrightbigg\n, whereZ/primeisanother\nnormalization constant, Eis the energy of the system as\ndefined in Eq. (1) and Tis the temperature. Then the\ndiffusion coefficients are given by :\nDop= Γ+\nopkBT/parenleftbigg∂E\n∂pop/parenrightbigg−1\n= Γ+\nopωT\nΩop(C3)\nDac= Γ+\nackBT/parenleftbigg∂E\n∂pac/parenrightbigg−1\n= Γ+\nacωT\nΩac(C4)\nWe now consider the self-oscillation regime with a\nsingle-mode excitation of the acoustic mode, with ther-\nmal noise. The stochastic differential equation of the\npower and phase is expressed in the It¯ o form, which is\npreferred when solving analytically stochastic equations\nbecause the solutions are martingales. For clarity, the ac\nindex is dropped on the power pand phaseφ:\n˙p=−2p/parenleftBig\nγac+Qacp+˜fac(p)/parenrightBig\n+/radicalbig\n4pDacηp(C5)\n˙φ=ωac+Nacp+/radicalBigg\nDac\npηφ (C6)\nWhereηp=Re(√\n2ηaceiφac)andηφ=Im(√\n2ηaceiφac)\nare real stochastic variables with zero average and\n/angbracketleftηi(t)ηj(t/prime)/angbracketright=δijδ(t−t/prime), fori,j∈(p,φ).˜fac= 2fac\nis the extra drift term in the It¯ o form, computed from its\nStratonovich form and the diffusion coefficients.\nDue to the extra ˜facterm, the stationary power is dif-\nferent from the power p0without temperature. How-\never, above the threshold, we suppose that the stationary\npower ˜p0is close to the zero-temperature value:\n˜p0=p0(1 +δp0) withδp0/lessmuch1\nIt can be shown that δp0is given by:\nδp0=−˜fac(p0)\np0Qac=∆ω0\nΓp−ν∆ω0\nΩ0Γ+(p∞)\nΓ+(p0)Where ∆ω0=Dac(p0)\np0is thelineargeneration\nlinewidth, Γp=p0Qacis the power relaxation\nrate, Ω0=ωac+p0Nacis the stationary frequency,\nν=Nac/Qacis the normalized non-linear frequency\nshift coefficient and p∞=−ωac\nNac, with positive or\nnegative value. If Nacis negative, p∞corresponds to the\nmaximum oscillation power, for which ωac+Nacp∞= 0.\nAs long as ∆ω0/lessmuchΓpand the oscillation frequency Ω0is\nhigh enough ( Ω0/greatermuchν∆ω0), the effect of the extra drift\nterm can be neglected and ˜p0≈p0.\nThen, weconsiderfluctuationsofthepoweraroundthe\nequilibrium power p0and of the phase around φ0(t) =\nΩ0t:δp=p−p0(withδp/lessmuchp0) andδφ=φ−φ0:\n˙δp=−2p0˜Qδp+/radicalbig\n4p0Dacηp (C7)\n˙δφ=Nacδp+/radicalBigg\nDac\np0ηφ (C8)\nWhere the effective non-linear relaxation rate coefficient\nis˜Q=Qac+∂˜fac\n∂pac/vextendsingle/vextendsingle/vextendsingle/vextendsingle\np=p0.\nThe correction due to the temperature-dependent term\non the non-linear relaxation rate writes as :\n˜Q\nQac−1 =∆ω0\nΓp+ν∆ω0/parenleftbigg2ωac−Ω0\nΩ2\n0/parenrightbiggΓ+(p∞)\nΓ+(p0)\nThe same conditions that assured that δp0/lessmuch1lead to\n˜Q≈Qac.\nBecause the stochastic equations are linear, the power\nand phase fluctuations are Gaussian processes with zero\nmean. There are contributions to the linewidth from the\nphase noise ( ηφ) and from the amplitude noise ( Nacδp).\nNotethattheothermode, theopticalmode, isconsidered\nto be subcritical, so its power is almost zero, and in any\ncase much smaller than the power of the acoustic mode.\nTherefore its contribution to the power spectral density\nis neglected.\nThe power is a weakly stationary process but the phase\nis a non-stationary Gaussian random walk. We obtain\nthe expression of the power variance ∆p2=/angbracketleftδp2/angbracketrightand\nthe phase variance ∆φ2=/angbracketleftδφ2/angbracketright7:\n∆p2=p2\n0∆ω0\nΓp\n∆φ2= ∆ω0/bracketleftbigg\n(1 +ν2)|t|−ν21−e−2Γp|t|\n2Γp/bracketrightbigg\n∗M. Romera is now working at Unité Mixte de Physique,\nCNRS, Thales, Univ. Paris-Sud, Université Paris-Saclay,\n91767 Palaiseau, France1S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. 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This process adds two extra drift terms\nthat do not balance each other.\n28T. Taniguchi, Applied Physics Express 7, 053004\n(2014), ISSN 1882-0778, URL http://stacks.\niop.org/1882-0786/7/i=5/a=053004?key=crossref.\n0d01d0a8539d0d8a726bd42873186e86 ."    },    {        "title": "2401.13617v1.Current_Driven_Domain_Wall_Motion_in_Curved_Ferrimagnetic_Strips_Above_and_Below_the_Angular_Momentum_Compensation.pdf",        "content": "Current-Driven Domain Wall Motion in Curved Ferrimagnetic Strips Above and\nBelow the Angular Momentum Compensation\nD. Osuna Ruiz *1,∗O. Alejos2, V. Raposo1, and E. Mart´ ınez1\n1Department of Applied Physics, University of Salamanca, Salamanca 37008, Spain and\n2Department of Electricity and Electronics, University of Valladolid, Valladolid, Spain.\n(Dated: January 25, 2024)\nCurrent driven domain wall motion in curved Heavy Metal/Ferrimagnetic/Oxide multilayer strips\nis investigated using systematic micromagnetic simulations which account for spin-orbit coupling\nphenomena. Domain wall velocity and characteristic relaxation times are studied as functions of\nthe geometry, curvature and width of the strip, at and out of the angular momentum compensation.\nResults show that domain walls can propagate faster and without a significant distortion in such\nstrips in contrast to their ferromagnetic counterparts. Using an artificial system based on a straight\nstrip with an equivalent current density distribution, we can discern its influence on the wall terminal\nvelocity, as part of a more general geometrical influence due to the curved shape. Curved and narrow\nferrimagnetic strips are promising candidates for designing high speed and fast response spintronic\ncircuitry based on current-driven domain wall motion.\nI. INTRODUCTION\nA magnetic domain wall (DW) is the transition re-\ngion that separates two uniformly magnetized domains\n[1]. These magnetic configurations are interesting due to\nfundamental physics, but also due to potential technolog-\nical applications [2, 3]. In fact, during the last decades\nDWs have been at the core of theoretical and experimen-\ntal studies which have provided with a deep understand-\ning of different spin-orbit coupling phenomena [4–8]. For\ninstance, straight stacks where an ultra-thin ferromag-\nnetic (FM) layer is sandwiched between a heavy metal\n(HM) and an oxide (Ox), present perpendicular mag-\nnetic anisotropy (PMA) and therefore, the domains are\nmagnetized along the out-of-plane direction of the stacks:\nup(+⃗ uz) or down (−⃗ uz). DWs in these HM/FM/Ox\nstacks adopt an homochiral configuration due to the\nDzyaloshinskii-Moriya interaction (DMI) [5, 7, 9]. Adja-\ncent DWs have internal magnetic moments along the lon-\ngitudinal direction ( ⃗ mDW=±⃗ ux), and the sense is im-\nposed by the sign of the DMI, which in turns depends on\nthe HM [5]. For left-handed stacks such as Pt/Co/AlO,\nup-down (UD) and down-up (DU) DWs have internal mo-\nments with ⃗ mDW=−⃗ uxand⃗ mDW= +⃗ ux, respectively\n[5]. These DWs are driven with high efficiency by inject-\ning electrical currents along the longitudinal direction of\nthe HM/FM/Ox stack [4]. Due to the spin-Hall effect [5],\nthe electrical current in the HM generates a spin polar-\nized current which exerts spin-orbit torques (SOTs) on\nthe magnetization of the FM layer, and drives series of\nhomochiral DWs which are displaced along the longitudi-\nnal direction ( x-axis). DW velocities of VDW∼500 m/s\nhave been reported upon injection of current densities of\nJHM∼1 TA/m2in Pt/Co/AlO [4]. Consequently, these\nstacks have been proposed to develop highly-packed mag-\nnetic recording devices, where the information coded in\n∗osunaruiz.david@usal.esthe domains between DWs can be efficiently driven by\npure electrical means. Both UD and DU DWs move\nwith the same velocity along straight stacks, but some\nimplementations of these memory or logic devices would\nrequire to design 2D circuits, where straight parts of\nHM/FM/Ox stack are connected each other with curved\nor semi-rings sections. However, recent experimental ob-\nservations [10] and theoretical studies [11] have pointed\nout that adjacent UD and DU DWs move with differ-\nent velocity along curved HM/FM/Ox stacks, which is\ndetrimental for applications because the size of the do-\nmain between adjacent DWs changes during the motion,\nwith the perturbation of the information coded therein.\nTherefore, other systems must be proposed in order to\ndesign reliable 2D circuits for DW-based memory and\nlogic devices.\nOther stacks with materials and/or layers with anti-\nferromagnetic coupling, such as synthetic antiferromag-\nnets (SAF) and ferrimagnetic (FiM), have proven to out-\nperform FM in terms of current-driven DW dynamics\n[11–15]. Ultrafast magnetization dynamics in the THz\nregime, marginal stray field effects and insensitivity to\nexternal magnetic fields are other significant advantages\nof materials with antiferromagnetic coupling with respect\nto their FM counterparts. As conventional antiferromag-\nnets (AFs), FiM alloys are also constituted by two spec-\nimens, typically a rare earth (RE) and transition metal\n(TM), that form two ferromagnetic sublattices antiferro-\nmagnetically coupled to each other. GdFeCo, GdFe or\nTbCo are archetypal FiM alloys, with the RE being Gd\nor Tb and the TM being FeCo or Co. In contrast to AFs\nwith zero net magnetization, the magnetic properties of\nFiMs, such as magnetization and coercivity, are largely\ninfluenced by the relative RE and TM composition (or\nequivalently, temperature). This fact offers additional\ndegrees of freedom to control the current-driven DW ve-\nlocity. The spontaneous magnetization of each sublattice\nMS,ican be tuned by changing the composition of the\nFiM and/or the temperature of the ambient ( T) [13, 14].\nFor a given composition of the FiM (RE xTM 1−x), therearXiv:2401.13617v1  [cond-mat.mtrl-sci]  24 Jan 20242\nare two relevant temperatures below the Curie thresh-\nold. One is the magnetization compensation temperature\n(TM) at which the saturation magnetization of the two\nsublattices are equal ( Ms1(TM) =Ms2(TM)), so the FiM\nbehaves as a perfect antiferromagnetic material, with\nzero net magnetization and diverging coercive field. The\nother is the temperature at which the angular momentum\ncompensates, TA, at which Ms1(TA)/γ1=Ms2(TA)/γ2,\nwhere γiis the gyromagnetic ratio of each sublattice\n(i:1,2 for 1:TM and 2:RE). As the gyromagnetic ratio\ndepends on the Land´ e factors ( gi) which are different for\neach sublattice, the angular compensation temperature\nTAis in general different from the magnetization compen-\nsation temperature ( TM). Consequently, the FiM have\na net magnetization at TA, so conventional techniques\nused for FMs can be also adopted to detect the magnetic\nstate of FiM samples [16]. Moreover, recent experimen-\ntal observations have evidenced that the current-driven\nDW velocity along straight HM/FiM stacks can be sig-\nnificantly optimized at the angular momentum compen-\nsation temperature ( T=TA), with velocities reaching\nVDW∼2000 m/s for typical injected density current of\nJHM∼1 TA/m2along the HM underneath [14]. The\nDW velocity drops either below ( T < T A) and above\n(T > T A) angular momentum compensation. Note that\nalternatively to tuning the temperature for a fixed com-\nposition xof the FiM alloy RE xTM 1−x, even working at\nroom temperature ( T= 300 K) the DW velocity peaks\nat a given composition where angular momentum com-\npensates [13]. Therefore, both studies, either fixing the\ncomposition ( x) and changing temperature of the ambi-\nent (T), or fixing the ambient temperature and modifying\nthe FiM composition are equivalent for our purposes of\nDW dynamic. Although the current-driven DW motion\n(CDDWM) along HM/FiM stacks suggests their poten-\ntial for memory and logic applications, previous studies\nhave been mainly focused on straight FiM strips [13–15].\nThe further develop of novel DW-based devices also re-\nquires to analyze the dynamics of DWs along HM/FiM\nwith curved parts which would connect straight paths to\ndesign any 2D circuit. Such investigation of the dynam-\nics along curved is still missing, and it is the aim of the\npresent study.\nHere we theoretically explore the CDDWM along\ncurved HM/FiM stacks by means of micromagnetic ( µm)\nsimulations. Our modeling allows us to account for the\nmagnetization dynamics in the two sublattices indepen-\ndently. We explore the CDDWM below, at and above\nthe angular momentum compensation (AMC) for differ-\nent curved samples, with different widths and curvatures,\nand considering the realistic spatial distribution of the\ninjected current along the HM. In particular, we will in-\nfer and isolate the relevance of different aspects govern-\ning such dynamics, as the role of the non-uniform cur-\nrent and other purely geometrical aspects of the curved\nshape. This work completes previous studies on straight\nsamples [11, 13–15], and will be practical for designing\nmore compact and efficient DW-based devices. The restof the paper is organized as follows. In Sec. II we de-\nscribe the numerical details of the micromagnetic model\nalong with the material parameters and the geometri-\ncal details of the evaluated samples. Sec. III presents\nthe micromagnetic results of the CDDWM in different\nscenarios. Firstly, exploring the role of the FiM sam-\nple width ( w) for a fixed the curvature ( ρ, given by the\ninverse of the average radius, ρ= 1/re), and secondly\nfixing the width and varying the curvature. After that,\nwe present results which allow us to infer the role of non-\nuniform current and geometrical aspect ( w, ρ) comparing\ncurved and straight samples. The main conclusions are\nsummarized in Sec. IV.\nII. MATERIALS AND METHODS\nCDDWM is numerically studied here along curved\nHM/FiM stacks as schematically shown in Fig. 1, where\nriandroare the inner and outer radius rorespectively,\nandre= (ro+ri)/2 is the mean effective radius. wand\ntFiMare the width and the thickness of the FiM respec-\ntively. The relaxed magnetization configuration of the\nsublattice i= 1, shown in Fig. 1 (opposite configuration\nin sublattice i= 2), serves as the initial state to study\nthe CDDWM upon of current injection along the HM un-\nderneath. The temporal evolution of the magnetization\nof each sublattice is given by the Landau-Lifshitz-Gilbert\nequation (LLG) [17],\nd⃗ mi(t)\ndt=−γ0,i⃗ mi(t)×⃗Heff,i+αi⃗ mi×d⃗ mi(t)\ndt+⃗ τSOT,i ,\n(1)\nwhere here the sub-index istands for i: 1 and 2 sub-\nlattices respectively. γi=giµB/ℏandαiare the gy-\nromagnetic ratios and the Gilbert damping constants,\nrespectively. giis the Land´ e factor of each layer, and\n⃗ mi(⃗ r, t) =⃗Mi/Ms,iis the normalized local magnetization\nto its saturation value ( Ms,i), defined differently for each\nsublattice: Ms,i(i: 1,2). In our micromagnetic model the\nFiM strip is formed by computational elementary cells,\nand within each cell we have two magnetic moments, one\nfor each component of the FiM. The respective effective\nfield ( ⃗Heff,i) acts on the local magnetization of each sub-\nlattice ( ⃗ mi(⃗ r, t)), and it is the sum of the magnetostatic,\nthe anisotropy (PMA), the DMI and the exchange fields\n[11, 15]. The magnetostatic field on each local moment\nin the sublattice is numerically computed from the av-\nerage magnetization of each elementary cell using simi-\nlar numerical techniques as for the single FM case (see\n[11, 15]). We checked that the demagnetising field has a\nmarginal influence in the simulation results compared to\nother contributions to the effective field. For the PMA\nfield, the easy axis is along the out-of-plane direction ( z-\naxis), and the anisotropy constants for each sublattice are\nKu,i(PMA constant). Diis the DMI parameter for each\nsublattice i: 1,2 [11, 15]. The exchange field of each sub-\nlattice includes the interaction with itself (intra-lattice3\nexchange interaction, ⃗Hexch,i ) and with the other sublat-\ntice (inter-lattice exchange interaction, ⃗Hexch, 12). The\ninter-lattice exchange effective field is computed as for\na single FM sample, Hexch,i =2Ai\nµ0Ms,i∇2⃗ mi, where Aiis\nthe intralattice exchange parameter. The inter-lattice ex-\nchange contribution ⃗Hexch, 12to the effective field ⃗Heff,i,\nacting on each sublattice is computed from the corre-\nsponding energy density, ωexch,i =−Bij⃗ mi·⃗ mj, where\nBij(in [J m−3]) is a parameter describing the inter-lattice\nexchange coupling between sublattices (here, we used the\nnotation i: 1 and j: 2).\nIn Eq.(1), ⃗ τSOT,i are the SOTs acting on each sub-\nlattice, which are related to the electrical current along\nthe HM ( ⃗JHM). Based on preliminary studies [18], here\nwe assume that ⃗ τSOT,i is dominated by the spin Hall ef-\nfect (SHE), so ⃗ τSOT,i =−γ0HSL⃗ mi×(⃗ mi×⃗ σ) where\nHSL=ℏθSH,iJHM\n2|e|µ0MstFiM[19]. ℏis the Planck constant, and\nθSH,i is the spin Hall angle, which determines the ra-\ntio between the electric current and the spin current\n(Js=θSHJHM) for each sublattice. ⃗ σ=⃗ uJ×⃗ uzis\nthe unit vector along the polarization direction of the\nspin current generated by the SHE in the HM, being\northogonal to both the direction of the electric current\n⃗ uJand the vector ⃗ uzstanding for the normal to the\nHM/FiM interface. For a longitudinal current ( ⃗ uJ=⃗ ux),\nthe spin current is polarized along the transverse di-\nrection, ⃗ σ=−⃗ uy. For curved samples where the cur-\nrent density ⃗JHM=JHM(r)⃗ uJhas azimuthal direction\n(⃗ uJ=−⃗ uϕ), the direction of the polarization is radial,\n⃗ σ=⃗ uJ×⃗ uz=⃗ ur, as shown in Fig 1. A potential dif-\nference is applied between the ends of the curved track\nto inject current in the right circulation. Therefore, a\ngap of 25 nm is also modelled, leading to a split ring\nshape for the strip (see inset in Fig. 1). The spatial\ndistribution of current as a function of the radial coor-\ndinate ( ri< r < r o) is taken from [11, 20], and it de-\npends on the width ( w) and the radial distance ( r) as\nJHM(r) =J0w/(rlog (1 + w/ri)), where J0is the nom-\ninal, uniform current density, in an equivalent straight\nstrip of same cross-section ( w×tHM, where tHMis the\nthickness of the HM strip).\nIn order to illustrate the current-driven DW dynam-\nics along curved HM/FiM stacks we fix tFiM = 6 nm,\nand samples with different widths ( w) and radii ( re)\nwere evaluated. The following common material pa-\nrameters were adopted for the two sublattices i: 1,2:\nAi= 70 pJ/m, Ku,i= 1.4×106J/m3,αi= 0.02,\nDi= 0.12 J/m2,θSH,i = 0.155. The strength of the\nantiferromagnetic coupling between the sublattices was\nfixed to Bij≡B12=−0.9×107J/m3. The gyro-\nmagnetic ratios ( γi=giµB/ℏ) are different due to the\ndifferent Land´ e factor: g1= 2.05 and g2= 2.0. The\nsaturation magnetization of each sublattice Ms,ican be\ntuned with the composition of the FiM and/or with the\ntemperature of the ambient ( T). Here, we assume the\nfollowing temperature dependences for each sublattice:Ms,i(T) =Ms,i(0)\u0010\n1−T\nTC\u0011ai\n, where TC= 450 K is the\nCurie temperature of the FiM, Ms,1(0) = 1 .4×106A/m\nandMs,2(0) = 1 .71×106A/m are the saturation magne-\ntization at zero temperature, and a1= 0.5 and a2= 0.76\nare the exponents describing the temperature depen-\ndence of the saturation magnetization of each sublattice.\nThe temperature at which the net saturation magneti-\nzation vanishes ( Ms,1(TM) =Ms,2(TM)) is TM= 241 .5\nK, and the angular momentum compensation temper-\nature corresponding to Ms,1(TA)/g1=Ms,2(TA)/g2, is\nTA= 260 K. We evaluate the CDDWM below, at and\nabove the angular momentum compensation adopting\nthree representative temperatures: T= 220 K < T A,\nT= 260 K = TAandT= 300 K > TA. Samples were\ndiscretized using a 2D finite difference scheme using com-\nputational cells with ∆ x= ∆y= 0.2 nm and ∆ z=tFiM.\nSeveral tests were carried to certify that the presented re-\nsults are free of discretization errors.\nFIG. 1. Scheme showing the relaxed states of spins in sub-\nlattice i= 1, in the positive z-direction (white domain), in\nthe negative z-direction (black domain) and in the plane of\nthe strip for an ‘Up to Down’ (UD) domain wall (purple)\naccording to the current direction, for an exemplary curved\nstrip. The direction of the applied electric current (red arrow)\nin the Heavy Metal beneath the magnetic strip, generated\nfrom a potential difference ∆ V(see inset), is shown as well\nas the geometrical parameters of the strip.\nIII. MICROMAGNETIC RESULTS\nDue to the several combination of parameters to con-\nsider in our study, we divided this section in three sub-\nsections: (A) The study on the influence of the strip\nwidth ( w); (B) the study on curvature ( ρ=r−1\ne); and (C)\nsame studies for a straight strip with identical material\nparameters, to explore by comparison the effects of curva-\nture on the DW dynamics. In parts (A) and (B), scenar-\nios for three different temperatures, T1= 220 K , T2= 2604\nK, T3= 300 K are considered, to study the DW motion\nbelow the AMC ( T1), at the AMC ( T2) and above the\nAMC ( T3). We also define and refer to T3= 300 K as for\n‘room temperature’ in our study. Note that a change in\ntemperature only affects MSin our model, therefore it\nhas equivalent effects to changing material composition\n[13]. In addition to DW velocity, we also characterize the\ninertial motion of the DW as a function of current den-\nsity. As an example, Fig. 2 shows typical results of the\nDW position and its velocity in a ring-like strip ( w= 256\nnm and re= 384 nm) under a density current JHM= 2\nTA/m2and at T= 260 K. Qualitatively similar results\nare obtained at T= 220 K and T= 300 K (not shown).\nInsets show the (clockwise) DW displacement as a func-\ntion of time for one sublattice ( i= 1).\nA. Influence of width for a fixed curvature\nIn this study, the curvature is fixed ( re= 384 nm) and\nwidth ( w) is varied from 56 nm to 296 nm in steps of 40\nnm. Fig. 3 shows the results for the terminal DW veloc-\nity (|VDW,i|) as a function of the nominal density current\nJHM=j0, equivalent to the homogeneous density cur-\nrent in a straight strip with the same cross-section. In the\nnext sections, we use the notation JHM= J for simplic-\nity. Fig. 3 shows that temperature has a noticeable effect\non the terminal velocity on the DW type equally, Up to\nDown domain (UD) or Down to Up domain (DU). As it\nwas expected from previous work on straight FiM strips\n[11, 13], at TAthe DW velocities are greater. Also, the\nDW velocities increase for narrower strips (red symbols).\nIn fact, the observed trend is very similar to straight\nstrips: the terminal velocity is maximum at the temper-\nature of AMC, TA= 260 K and significantly increased,\nexceeding 2000 m/s for the narrowest strip as compared\nto∼1800 m/s for the widest. These results also suggest\nthat the DWs velocities are equal for both types of DWs\n(UD and DU), which would lead to no distortion of the\nsize of a domain between two adjacent DWs travelling\nalong the curved strip. This result is significantly differ-\nent from FM systems [11].\nAtT̸=TA, the DW velocity is reduced either increas-\ning or reducing temperature with respect to TA, leading\nto velocities around 1100 m/s, generally regardless of the\nwidth and DW type. For a given Jvalue, as the strip\ngets wider, however, the velocity is slightly smaller but\nthese differences are negligible (see Fig. 3(a), (c), (d),\n(f)). This result contrasts with that of a FM strip, where\na greater difference of velocities between a DU and a UD\nalong a curved strip was shown [11].\nTo characterize the inertial motion of the DW, we eval-\nuate the temporal evolution of the DW velocity ( V(t),\ncomputed from the spatial averaging of m1,z(t)) as a\nfunction of time (or instant velocity) under a current\nsquare pulse of duration 0.1 ns and start at t= 0. The\nµm results can be fitted to the following exponentials:\nV∞(1−e−t/τr) during the duration of the pulse ( t≤0.1\nxf\n0.00 ns\n0.50 ns\n0.13 ns\n0.25 ns\n0.38 ns\n(a)\n(b)\n(c)(d)FIG. 2. Micromagnetic results of applying a uniform JHM=\nJ0= 2 TA/m2(a) showing the relative and final position xf\n(b) and velocity VDW(c), as a function of time t, of an UD\nDW in a curved strip ( w= 256 nm and re= 384 nm), at\nT= 260 K. Insets (d) are snapshots of the magnetic config-\nuration in sublattice i= 1 at different times (highlighted by\nthe vertical dotted lines in (a-c)). Green solid lines are for\nguiding the eye and are co-parallel with the strip radius.\nns), and V∞e−t/τf, after the pulse ends ( t > 0.1 ns),\nwhere V∞is the DW terminal velocity (see Fig. 2(a)-(f)).\nThe characteristic relaxation times τr(or rising time) and\nτf(or fall time) represent the duration of such transients\nand characterize the inertial motion of the DW. These\nparameters can be extracted by fitting the µm results to\nthe exponentials (see solid curves in Fig. 4(a)).\nAlthough simulations were performed for both types of\nDWs, note that we only present here results for the DU\ntype wall, for sake of simplicity. Identical results (not\nshown) were obtained for the UD DW. Fig. 4(a) show\nthe ‘instantaneous’ DW velocity V(t) and the relaxation\ntimes τfor three selected values of J (see solid symbols) at\nT=TAfor the widest strip ( w= 296 nm). Solid lines are5\n𝑤\n260 K 260 K220 K 220 K\n300 K 300 K(a)\n(b)\n(c)(d)\n(e)\n(f)\nT=220 K\nT=260 K\nT=300 K\nDUUD\n(g)\nw=296 nm\nww (nm)\n56 \n176 \n296 \nw=56 nmw (nm)\n56 \n176 \n296 UD\nDU\nT=220 K\nT=260 K\nT=300 K\nFIG. 3. Terminal velocities as a function of J for a UD (a-c)\nand a DU (d-f) DW obtained for sub-lattice i= 1 and for\nthree different temperatures: below, above and at the AMC\ntemperature (220 K, 300 K and 260 K, respectively). Strips\nfor the two limiting cases are shown in the red and blue con-\ntour insets at the top. (g) Terminal velocities as a function\nofwfor a UD (full symbols) and a DU (open symbols) type\nwall for J = 2 .35 TA/m2and the three chosen temperatures.\nInset in (g) shows J(r) for two values of w. Red dashed line\nindicates J = 2 .35 TA/m2.\nthe exponential curves to which the obtained simulated\ndata is fit. For each current, the minimal τis expected\nforT=TA= 260 K. Fig. 4(b) shows that τrandτffor\nthe two limiting cases ( w= 56 nm and w= 296 nm) are\nquantitatively similar, in the order of 0.02 ns, since they\nfall within the 95% confidence interval, set by the largest\nerror bars obtained for τfrom the fitted results, among\nall J. Also, all values are similar in order to the step-size\nJ = 2.35 TA/m2\nJ = 1.60 TA/m2\nJ = 0.85 TA/m2T = 260 K\n(a) (b)FIG. 4. Instantaneous DW velocities V(t) for three values of\nJ and for a curved strip of re= 384 nm and w= 296 nm at\nT=TA(a). Symbols are the µm data, from which τ(rise and\nfall times) are extracted for the narrowest and widest strips\n(b). Dashed lines in (b) are the upper and lower bounds of a\n95% confidence interval.\nused in simulations, 0.01 ns (see Fig. 4(d)).\nSimilar values of τwere obtained for strips of other\nwidths. Relaxation times are not noticeably influenced\nby temperature, and they generally remain within the\nrange of 0.01 ∼0.03 ns for T= 200 K and T= 300 K.\nThis is more than one order of magnitude smaller than\nin FM strips, the latter being about ∼1 ns according to\nRef.[21]. Besides, the relaxation times of current-driven\nDWs in curved strips found here are in good agreement\nwith those from field-driven or thermally driven DWs in\nantiferromagnetic straight strips, in the order of picosec-\nonds [22, 23].\nB. Influence of curvature for a fixed width\nIn this study, the strip width is fixed to w= 256 nm\nand the curvature parameter ρis varied. In other words,\nthe equivalent radii re(re=ρ−1) is varied from 134\nnm to 534 nm in steps of 50 nm. Fig. 5 shows the re-\nsults for the terminal velocity ( |VDW, 1|) of DU and UD\nDWs, for several values of rein nanometers, where the\nred (blue) curve corresponds to the smaller (greater) val-\nues, for three different temperatures.\nFig. 5(a)-(f) shows that, at T=TAand for a given\ncurvature, the DW velocities of DU and UD types are\nvery similar for the whole range of currents explored.\nDW velocity reduces as the curvature increases (see Fig.\n5(g)). It is worth noting that the latter cannot be a con-\nsequence of only a nonuniform J(r) as defined in [11]. In\nfact, for curved-most strips ( re= 134 nm), the spatial-\ndependent density current J(r) varies with rsimilarly as\nit does for changing w(see inset in Fig. 5(g) and in Fig.\n3(g)), which would suggest similar variations to DW ve-\nlocities as those found in Fig. 3(g). In other words, the\nimpact of the non-uniform J(r) is not so relevant to be\nthe only source of the big differences between the DW\nvelocities for large and small curvatures (orange symbols\nin Fig. 5(g) for re= 134 nm and re= 484 nm, respec-\ntively). Fig. 5(g) also shows that as the strip curvature6\n260 K 260 K220 K 220 K\n300 K 300 K(a)\n(b)\n(c)(d)\n(e)\n(f)\n𝑟𝑒\n(g)re (nm)\n134 \n384 \n584 \nre (nm)\n134 \n384 \n584 UD\nDU\nT=220 K\nT=260 K\nT=300 K\nDUUD\nT=220 K\nT=260 K\nT=300 Kre = 134 nm\nre = 534 nm\nFIG. 5. Terminal velocities as a function of J for a UD (a-c)\nand a DU (d-f) DW in the curved strip obtained for sub-\nlattice i= 1 and for three different temperatures: below,\nabove and at the AMC temperature (220 K, 300 K and 260\nK, respectively). Strips for the two limiting cases are shown\nin the red and blue contour insets at the top. (g) Terminal\nvelocities as a function of refor a UD (full symbols) and a DU\n(open symbols) type wall for J = 2 .35 TA/m2and the three\nchosen temperatures. Inset in (g) shows J(r) for two values\nofre. Red dashed line indicates J = 2 .35 TA/m2.\nis reduced, DW velocity converges to the straight strip\ncase ( re→ ∞ ).\nForT̸=TA, the dependence of the DW velocity with\ntemperature is minimal regardless of the DW type. As\nthe strip curvature increases there is a prominent change\nin the maximal terminal DW velocity for both DW types.\nHowever, the relative difference of velocities is almost\nJ = 2.35 TA/m2\nJ = 1.60 TA/m2\nJ = 0.85 TA/m2\n(a) (b)FIG. 6. Instantaneous DW velocities V(t) for three values of\nJ and for a curved strip of re= 584 nm and w= 256 nm at\nT=TA(a). Symbols are the µm data, from which τ(rise and\nfall times) are extracted for the narrowest and widest strips\n(b). Dashed lines in (b) are the upper and lower bounds of a\n95% confidence interval.\nnegligible. Therefore, results suggest that the strip cur-\nvature affects in a similar way to width, and equally, to\nboth DWs. In other words, the terminal velocity is signif-\nicantly reduced as curvature (or width) increases, while\nthe differences between DWs remain negligible (see Fig.\n5(g)). This behavior is even more pronounced at T=TA.\nAs discussed in section III.A, the latter would imply that\nthe robustness of a transmitted bit, encoded in a domain\nbetween two DWs, can be optimised in such curved-most\nstrips and reaches larger velocities in the strip.\nFig. 6(a) show the DW velocity as a function of time\nand for three selected values of J for an effective radius of\nre= 534 nm (least curved strip) and intermediate width\nw= 256 nm, at T=TA. Results look quantitatively sim-\nilar to those shown in Fig. 4(a), where rewas fixed to an\nintermediate value of 384 nm. As in Fig. 4(b), Fig. 6(b)\nshows that τrandτffor the two limiting cases ( re= 134\nnm and re= 534 nm) are quantitatively similar, in the\norder of 0.02 ns. For all the FiM curved strips explored\nat, above and below AMC, τrandτfremain within the\nrange of 0.01 ∼0.03 ns, approximately one order of mag-\nnitude less than their FM counterparts. This is in good\nagreement with results presented in the previous section\nand other work in straight strips [8], which further sup-\nports the negligible inertia of DWs in such FiM systems.\nC. Discussion on the effective influence of a curved\nshape on the wall velocity\nA non-uniform current distribution is expected to in-\nfluence the terminal velocity of the DW for a given cur-\nvature, specially for wide curved strips [11]. In this sec-\ntion, to explore further the degree of influence of the\nnon-uniform current, equivalent studies on wandρon\na straight strip with an artificially implemented non-\nuniform J( r=y) atT=TAare performed. A straight\nstrip is a bounding case for a curved strip that shows no\neffective curvature ( re→ ∞ , ρ→0) and an homogeneous\ndensity current J=J0. Therefore, we explore whether7\n(c)(a)\n(d)w = 256 nm re = 384 nm(b)\nre = 384 nm w = 256 nm\nJ = Jx(y, re)\nJx = J 0J = Jx(y, w)\nJx = J 0\nJ = Jφ(r, re)J = Jφ(r, w)\nJφ = J 0\nFIG. 7. DW terminal velocities in a straight strip for an in-\nhomogeneous J(y, w, r e) (blue circles) and for a uniform J0\n(black crosses) as a function of radius re(a) and width w(b)\nat same temperature ( T= 260 K). Insets in (a) show the mag-\nnetic configuration of sublattice i= 1 at t= 0 and schematics\nof the current spatial distribution in the strips as examples.\n(b) DW terminal velocities in a curved strips with the same\nparameters as a function of radius re(c) and width w(d). As\nan example, inset in (c) shows the radial dependence of an in-\nhomogenous current distribution in such a strip. Dotted lines\nhighlight the cases where the two geometrical parameters ( w\nandre) are coincident among all the studies.\n‘curvature ( ρ) effects’ are mainly dominated by the in-\ntrinsic inhomogenous current, or whether they can also\narise from the curved shape itself [24]. We aim to discern\nthe actual influence of an inhomogeneous current, as part\nof an more global effect due to the curved shape. For the\nfollowing study, and since the straight shape must be re-\ntained, re(or equivalently ρ) is artificially modified in the\nnon-uniform current distribution expression: ⃗J(y, w, r e)\n[11] in the x-direction, as if the strip was curved. Note\nthat ρrepresents the inverse of the averaged or effective\ncurvature radius of the strip ( ρ=r−1\ne) and not the cylin-\ndrical radial coordinate ( r) in the system.\nFig. 7(a-b) show the velocity of an UD wall in the\nstraight strip for J=J0= 2.35 TA/m2(black crosses)\nand for an inhomogeneous J(y) (blue circles), varying re\ninJ(y, re) (see insets) for a fixed width w= 256 nm, and\nvarying width ( w) for a fixed re,J(y, re= 384). While it\nis expected that an inhomogeneous J(y, w, r e) will influ-\nence the DW velocity (especially for curved-most strips,\nseere= 134 nm in (a)), the differences with the case of\nJ=J0are almost negligible. Fig. 7(c-d) show resultsfor curved strips. For these cases, wandreare naturally\nvaried in J(r, w, r e) by modifying the shape itself. From\nthe standpoint of the applied current, this is expected\nto be equivalent to doing it by changing the shape itself.\nResults from an inhomogeneous current (blue circles, re-\nproduced from Fig. 3(g) and 5(g)) consistently tend to\nconverge to the straight strip as ρis reduced. The strip\nis straight when ρ= 0 ( re→ ∞ ).\nWhen the strip is either straight (a-b) or curved (c-\nd), for both studies (fixing wand varying reor vice-\nversa), the DW velocity is found to be the same when\nthe geometrical parameters are coincident, i.e., w= 256\nnm and re= 384 nm, as expected (see horizontal dot-\nted lines in (a-b) or (c-d)). However, when the shape\nis different, even in the cases when wandreare coinci-\ndent (and therefore, J(w, re) is expected to also be the\nsame), different DW velocities are obtained (see vertical\ndotted lines in (a-c) and (b-d)) below and slightly above\n2000 m/s, respectively. Moreover, by modifying either re\norwin the curved strip by directly changing its shape\n(see (c-d) and previous sections), there is a clear larger\nimpact on DW velocity, than by artificially (but equiva-\nlently) modifying reorwin the straight strip (see (a-c)).\nIn the curved strips, the trend when varying reorwis\nqualitatively similar regardless to the homogeneity of the\napplied J(see black crosses and blue circles in either (c)\nor (d)). Since J( r) is modeled in an equivalent way in all\nstudies by simply changing J( r) = J( y) for the straight\nstrip, marked differences between the wall velocity in (a-\nb) (straight strip) and (c-d) (curved strip), specially for\nvery curved strips, suggest that not only the non-uniform\nJ(r) is influencing the UD DW motion.\nOur results suggest that the curved shape may have\nan intrinsic influence on the wall velocity, manifested as\na more marked dependence with wandre, regardless of\nthe inhomogeneity of the current density (see Fig.(c-d)),\nas the shape becomes more curved. An influence due to\nthe inhomogeneous current, still appears naturally in the\ncurved strip, but may have a lesser impact compared to\nother geometrical factors (see differences between black\ncrosses and blue circles).\nIV. DISCUSSION AND SUMMARY\nWe have provided a study on DW motion in curved\nFiM strips, particularised to one of the two strongly cou-\npled sub-lattices, for three different temperatures, and as\na function of geometrical parameters for a HM/FiM/Ox\nmultilayer structure. We observe an absence of tilting of\nthe DW and domain distortion at different temperatures,\n40 K above and below the angular momentum compen-\nsation temperature.\nWidth and curvature effects on the DW velocity are\ndiscussed. Besides contributions from a non-uniform\nJ(r), there is an overall significant influence from the\nshape of the strip itself on DW velocity. This implies\nthat, for a fixed temperature (or composition), DW ve-8\nlocity can be optimised by optimising the geometrical\nparameters of the curved strip. The relative differences\nbetween a DU and a UD walls are marginal in general.\nIn other words, geometrical factors affect them almost\nequally, which is positive for a robust transmission of a\nbit encoded in an Up or Down domain between two adja-\ncent DWs. With reducing current, differences in veloci-\nties between curved and straight strips are still minimised\nat the expense of slower DWs. Similar effect is observed\nas width or curvature is increased. This is beneficial for\ndesigning intricate 2D circuit tracks combining curved\nand straight sections, while preserving DW velocities still\nlarger than those found in their FM counterparts. Also,\nDWs in a curved FiM strip show a negligible inertia in\ncontrast to their FM counterparts ( τFiM << τ FM) for\nall the explored scenarios. The DWs start to move and\nstop almost immediately ( τFiM∼0.02 ns) after the ap-\nplication or removal of current.\nConsidering the obtained results altogether and assum-\ningT̸=TA, which will be most of the experimental cases\nat room temperature ( T= 300 K), our study allows us\nto conclude that narrow enough FiM strips are ideal can-\ndidates for designing curved tracks for 2D spintronic cir-\ncuits of an arbitrary shape based on CDDWM, where bits\nare encoded in domains separated by walls. This is dueto very fast rise and fall times ( τr∼τf<<0.1 ns), high\nvelocities ( VDW>1000 m/s) and negligible distortion of\nthe two types of DWs (UD and DU) in all the scenarios\nexplored in this work. Greater DW terminal velocities\nand smaller time responses in curved FiM strips than\nthose in their FM counterparts are obtained. These re-\nsults can help in the further research, development and\nimprovement of FiM-based spintronic circuitry that may\nrequire compactness and high-speed functionality with\nhigh robustness to DW (and/or domain) distortion.\nV. ACKNOWLEDGEMENTS\nThis work was supported by project SA114P20 from\nJunta de Castilla y Leon (JCyL), and partially supported\nby projects SA299P18 from JCyL, MAT2017-87072-C4-\n1-P and PID2020-117024GB-C41 from the Ministry of\nEconomy, Spanish government, and MAGNEFI, from\nthe European Commission (European Union). All data\ncreated during this research are openly available from\nthe University of Salamanca’s institutional repository at\nhttps://gredos.usal.es/handle/10366/138189\n[1] A. Hubert and R. Sch¨ afer, in Magnetic Domains: The\nAnalysis of Magnetic Microstructures (Springer-Verlag\nBerlin Heidelberg, 1998).\n[2] S. S. P. Parkin, “Shiftable magnetic shift register and\nmethod of using the same,” (US6834005B1, 2004).\n[3] S. Parkin and S.-H. 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Yang, East Asian Journal on Applied Mathematics 7,\n837–851 (2017)."    },    {        "title": "2201.09067v1.Universal_criteria_for_single_femtosecond_pulse_ultrafast_magnetization_switching_in_ferrimagnets.pdf",        "content": "Universal criteria for single femtosecond pulse ultrafast magnetization switching in\nferrimagnets\nF. Jakobs and U. Atxitia\u0003\nDahlem Center for Complex Quantum Systems and Fachbereich Physik,\nFreie Universität Berlin, 14195 Berlin, Germany\nSingle-pulse switching has been experimentally demonstrated in ferrimagnetic GdFeCo and\nMn2RuxGa alloys. Complete understanding of single-pulse switching is missing due to the lack\nof an established theory accurately describing the transition to the non-equilibrium reversal path\ninduced by femtosecond laser photo-excitation. In this work we present general macroscopic theory\nfor the magnetization dynamics of ferrimagnetic materials upon femtosecond laser excitation. Our\ntheory reproduces quantitatively all stages of the switching process observed in experiments. We\ndirectlycompareourtheorytocomputersimulationsusingatomisticspindynamicsmethodsforboth\nGdFeCo and Mn 2RuxGa alloys. We provide explicit expressions for the magnetization relaxation\nrates in terms of microscopic parameters which allows us to propose universal criteria for switching\nin ferrimagnets.\nUltrafast magnetization switching induced by a single\nfemtosecond laser pulse has attracted a lot of attention\nas a promising solution for low energy, faster memory\napplications. [1–15]. Until recently only GdFeCo\nalloys and synthetic ferrimagnets [8], presented the\nability to switch under either optical femtosecond laser-\n[3, 4, 8] or electric picosecond current-pulses [16,\n17]. Although several micro- and macroscopic models\nhave reproduced single-pulse switching in GdFeCo\nferrimagnets [18–33], complete understanding of the\nrole of electrons, lattice and spin sublattices and their\nmutual interactions remains a challenge [34]. The\nexisting criteria for switching rely on the existence of\ntwo antiferromagnetically coupled magnetic sublattices\nshowingdistinctdynamicalresponsetofemtosecondlaser\nphoto-excitation. While in single species ferromagnets\nsuch as 3d transition metals, relaxation of angular\nmomentum occurs via dissipation into other degrees\nof freedom – relativistic relaxation – in two-sublattice\nmagnets, additionally, relaxation can occur via angular\nmomentum exchange between sublattices – exchange\nrelaxation. By driving the spin system into a non-\nequilibrium state where exchange relaxation dominates,\na non-equilibrium ultrafast reversal path opens. One\nof the most outstanding, open questions is about the\nconditions or criteria for the onset of the exchange\ndominated relaxation regime. Crucially, it is unclear\nhow previous understanding gained from observation\nin GdFeCo translates into the recent discovery of\nsingle-pulse switching in the ferrimagnetic Heusler alloy\nMn2RuxGa [35], where the two antiferromagnetically\ncoupled Mn atoms are of the same kind in comparison\nto Gd and FeCo.\nIn this work we present a general theoretical\nframework for the description of single pulse switching\nof ferrimagnets. We provide explicit expressions for\nthe relativistic and exchange relaxation parameters as a\nfunction of microscopic material parameters, including\ntheir dependence on temperature and non-equilibriumsublattice magnetization. This allows us to uncover\nthe criteria for the onset of the exchange-dominated\nrelaxationregimeandswitching. Weverifythevalidityof\nthe model by direct comparison to atomistic spin model\nsimulations of both GdFeCo and Mn 2RuxGa alloys.\nWe shall describe each magnetic atom at site ias a\nclassical spin vector sof a unit length. The magnetic and\nmechanical moments of each atom/element are given by\n\u0016=\u0016ssandS=\u0016ss=\rwhere\u0016sisthemagneticmoment\nand\ris the gyromagnetic ratio. We consider a classical\nHeisenberg spin Hamiltonian[36]:\nH=\u0000X\ni6=jhijiJijsi\u0001sj\u0000X\nidz\ni(sz\ni)2:(1)\nTo model a ferrimagnet, one needs to consider two\nsublattices with different and antiparallel magnetic\nmoments\u0016aand\u0016b, with three different exchange\ncouplingconstants. Twoferromagneticcouplingsforeach\nsublattice coupling with itself ( JaandJb>0) and a\nthirdfortheantiferromagneticinteractionbetweenthem,\nJab<0[37]. The second term in Eq. (1) describes the\ncontribution to the energy of on-site uniaxial anisotropy\nwith easy-axis in z-direction and anisotropy constant, dz\ni.\nThe macroscopic model we propose is derived from a\nmicroscopic spin model, where the equation of motion for\nthe spin dynamics of each atomic spin is the stochastic\nLandau-Lifshitz-Gilbert (LLG) equation [36]:\n@si\n@t=\u0000j\rj\n(1 +\u00152)\u0016i[(si\u0002Hi)\u0000\u0015(si\u0002(si\u0002Hi))]:\n(2)\n\u0015is the local intrinsic atomic damping, the effective field\nHi=\u0010i\u0000@H\n@si, wherethermalfluctuationsarerepresented\nby the stochastic field \u0010i.\nThe non-equilibrium macroscopic magnetization dy-\nnamics of the element-specific angular momentum Sa=\n\u0016ahsai=\r, wherema=hsaiis the first moment of the\nnon-equilibrium distribution function, can be describedarXiv:2201.09067v1  [cond-mat.mtrl-sci]  22 Jan 20222\nelectrons \nlattice \nFIG. 1. Degrees of freedom involved in single-pulse switching;\nelectrons, lattice and element-specific spin systems. Electrons\nact as heat-bath to the spin system with an element-specific\ncoupling strength given by \u000ba(b). Exchange relaxation rate\n\u000bex\u0018\u000ba=ma+\u000bb=mb, wherema(b)are the normalized\nsublattice magnetization magnitude.\nby [19]:\ndSa\ndt=\u000ba\u0016aHa+\u000bex(\u0016aHa\u0000\u0016bHb)(3)\nwherea6=b. The macroscopic relativistic damping\nparameter in Eq. (3) reads [38]\n\u000ba= 2\u0015aL(\u0018a)\n\u0018a: (4)\nHere,L(\u0018)stands for the Langevin function. The\nrelaxation parameter strongly depends on temperature\nand non-equilibrium magnetization state through the\nthermal field \u0018a=\f\u0016aHMFA\na. In the exchange\napproximation, the MFA field acting on sublattice ais:\n\u0016aHMFA\na =zaJaama+zabJabmb: (5)\nHere,zais the number of nearest neighbours (n.n.) of\nspins of type a, andzabis the amount of n.n. of type\nb. The macroscopic damping increases with temperature\nup to a value \u000ba= 2=3at the critical temperature [38].\nThe exchange relaxation parameter \u000bexis given by\n\u000bex=1\n2\u0012\u000ba\nzabma+\u000bb\nzabmb\u0013\n: (6)\nThisexpressionistheextensionofthenon-localexchange\nrelaxation in ferromagnets to local exchange relaxation\nin ferrimagnets. The role of \u000bex,\u000baand\u000bbas the\ncoupling between the sublattices and heat baths is\nvisualized in Fig. 1. In single species ferromagnets,\n0 0.2 0.4 0.6 0.8 1ma00.20.40.60.81mb0.1110\nFIG. 2. Normalized exchange relaxation parameter \u000bex=\u0015\nas function of the sublattice magnetization maandmb, at\nT= 600K. System parameters correspond to GdFeCo.\n\u0015a=\u0015b=\u0015= 0:01. The dotted black line corresponds\nto\u000bex=\u000ba. The white dot represents the starting\nsublattice magnetization ( ma;mb). The closed dashed orange\nline describes a trajectory meeting a no-switching criteria.\nThe open solid white line describes a trajectory meeting a\nswitching criteria.\nsublattices aandbrepresent the same spin lattice,\nhence\u000ba=\u000bb. Therefore, \u000bex=\u000ba=(zma), and\n\u0016aHa\u0000\u0016bHb=\u0016aHexa2\n0\u0001ma, witha0representing\nthe lattice constant. Hence, \u0000non\u0000loc:\nex =\u000bex(\u0016aHa\u0000\n\u0016bHb) =\u000ba(A=M a(T))\u0001ma, whereAis the so-\ncalled micromagnetic exchange stiffness [39]. Ma(T) =\n(\u0016a=\u001da)mais the magnetization density at temperature\nT, where\u001dais the unit cell volume. Non-local exchange\nrelaxation plays a minimal role in the field of ultrafast\nmagnetization dynamics since \u0001ma\u001c1.\nThe non-equilibrium fields ( \u0016aHa= 0at equilibrium)\nare given by\n\u0016aHa=(ma\u0000L(\u0018a))\n\fL0(\u0018a): (7)\nwhere,L0(\u0018) =dL=d\u0018. Notethattheyaredifferenttothe\nMFA fields in Eq. (5), and have been derived previously\n[38, 40]. As the magnetic system approaches thermal\nequilibrium, the non-equilibrium fields can be cast into\nLandau-like expressions [19, 38].\nEquation (3) has been proposed before based on\nsymmetry arguments [19, 23] as a direct generalization\nof the Landau-Lifshitz equation with longitudinal\nrelaxation terms. These models have introduced the\nrelaxation parameters at a purely phenomenological level\nand to some extent their values are arbitrary. Moreover,\nsince they were taken as constant values, most of the\nnon-equilibrium spin physics was not taken into account.\nOur model overcomes these assumptions by providing\nexpressions for the relativistic and exchange relaxation\nparameters as a function of the sublattice specific3\natomic relaxation parameter, \u0015a(b), and normalized\nmagnetization ma(b).\nThis insight is paramount to find the criteria for\nthe onset of a exchange relaxation dominated state.\nIn Fig. 2 we present a diagram of the normalized\nexchange relaxation parameter \u000bex=\u0015for GdFeCo alloy\nparameters as a function of maandmbat a fixed\ntemperature, T= 600K, which corresponds to the\nCurie temperature of the alloy. We observe that the\nexchange relaxation parameter strongly depends on the\nmagnitude of sub-lattice magnetization and its value\nranges over three orders of magnitude, \u000bex=\u0015= 0:1\u0000\n10. Importantly, only when magnetic states reduce\nsignificantly, does the exchange relaxation dominates\nover relativistic relaxation.\nSo far only two classes of ferrimagnets have\nshown single-pulse switching, Gd x(FeCo) 1\u0000xalloys and\nMn2RuxGa. Switching in GdFeCo has been thoroughly\nstudied both experimentally [1–5] and theoretically [3,\n11, 20–23, 31, 32], whereas in Mn 2RuxGa has been only\nrecently demonstrated for a range of Ru concentrations\n(x\u00150:9) [35]. In GdFeCo alloy, switching is\ncharacterised by a fast response of the Fe sublattice and\nslower response of the Gd sublattice to femtosecond laser\nphoto-excitation. This difference roots to their distinct\nmagnetic moment, \u0016Gd= 7:6\u0016Band\u0016Fe= 1:92\u0016B.\nDifferently to GdFeCo, antiferromagnetically coupled\nMn spins in Mn 2RuxGa have similar atomic magnetic\nmoments. We demonstrate the universality of our theory\nby direct comparison of the photo-induced magnetization\nswitching to computer simulations based on atomistic\nspin dynamics (Eq. (2)) for GdFeCo (disordered spin\nstructure) and Mn 2RuxGa (ordered spin structure).\nTheir magnetic properties such as magnetic moments\nand exchange interactions differ substantially. We\nconsider two typical compositions, that show switching,\nGd25(FeCo) 75alloysandMn 2Ru0:86Ga. Itisnoteworthy;\nthat while for GdFeCo atomistic spin models have been\nused thoroughly, in Mn 2Ru0:86Ga similar simulations are\nmissing. We derive the necessary material parameters\nbased on experimental measurements [41–45]. We find\nthat for Mn 2Ru0:86Ga the atomic magnetic moments,\n\u00164a= 2.88\u0016Band\u00164c=4.05\u0016B, and the exchange\nparameters, Ja= 1:28\u00021021J,Jb= 4:0\u00021022J\nandJab=\u00004:85\u00021022J describe the equilibrium\nmagnetization well as function of temperature. Using\nthese parameters the Curie temperature becomes Tc=\n600K and the compensation temperature TM\u0019300K.\nWe note that in the experimental samples, those\ntemperatures are sensitive to the growth conditions as\nwell as material composition. However our temperatures\nare within the reported range of experimentally found\ntemperatures [41–45]. We use the so-called two-\ntemperature model (TTM) to describe the dynamics\nof the electron and phonon systems[46, 47]. For both\nmaterials we use the same parameters in the TTM andthus the electron and phonon temperatures are the same\nfor the same fluence in both materials [45]. The heat-\nbath to which the spins are coupled is represented by the\nelectron system.\nFigures 3 (a) and (b) show excellent agreement\nbetween our macroscopic model and atomistic spin\ndynamics simulations for both alloys for all stages of\nthe magnetization dynamics leading to switching, from\nfast demagnetization, transient ferromagnetic-like state,\nto magnetization recovery. Interestingly, the switching\ndynamics in Mn 2Ru0:86Ga (Fig. 3 (a)) differs to GdFeCo\n(Fig. 3 (b)), both Mn sublattices demagnetize at\nsimilar rate and switch almost simultaneously. Although\ndemagnetization timescales are similar in Mn 2Ru0:86Ga\nand GdFeCo, the recovery of the magnetization in\nMn2Ru0:86Ga is significantly slower. The relaxation of\nthesublatticemagnetization(Eq. 3)canbesplitintotwo\ncontributions, the relativistic relaxation: \u0000r\na=\u000ba\u0016aHa,\nand exchange relaxation \u0000ex=\u000bex(\u0016aHa\u0000\u0016bHb)(see\nFig. 3(c) and (d)). For both ferrimagnetic alloys, \u0000r\na\ndrivesthedynamicsinthefirsthundredsoffemtoseconds,\nuntil sublattice magnetization reduces sufficiently to\nenter the exchange relaxation dominated region \u0000ex>\u0000r\na\n(Fig. 2). In this regime, the exchange relaxation steers\nthe systems towards an intermediate metastable state\ndefined by the condition \u0016aHa=\u0016bHb. Under some\nconditionsthisintermediatestateprecedeswitching. Itis\ncumbersome however to directly analyse Eqs. (7) due to\nitshighlynon-linearcharacter. Thereforeinthefollowing\nwe investigate some limiting scenarios.\nIn the high temperature limit, \u0018a=\f\u0016aHMFA\na!\n0,\u0000r\na= 2\u0015akBTma, which is the so-called thermal\nfluctuation field. The corresponding relaxation time,\n\u001ca=\u0016a=(\r2\u0015akBT), associated to relativistic relaxation,\nhas been discussed before [19, 48]. Similarly, we can\nestimate the high-temperature limit of the exchange\nrelaxation rate\n\u0000ex\n1(ma;mb) =\u0015kBT\nz(ma+mb)(mz\na\u0000mz\nb)\nmamb:(8)\nHigh temperature limits are valid when the temperature\nis larger than the exchange energy acting on the spins.\nOtherwise, intermediate-to-high temperature limit are\nnecessary. This limit adds corrections to previous\nestimations. For instance, \u0016aHa\u0000\u0016bHb= 3kB((T\u0000\nTa\nc)ma+ (T\u0000Tb\nc)mb), whereJ0;a+J0;ab= 3kBTa\nc. The\nexchange relaxation rate is\n\u0000ex\nT(ma;mb) = \u0000ex\n1\u0012\n1\u00001\nTTa\ncma+Tb\ncmb\nma+mb\u0013\n(9)\nFor element-specific critical temperature Ta\nc\u0019Tb\nc, the\ncorrection can be cast as \u0000ex\nT= \u0000ex\n1(T\u0000Ta\nc)=T.Since\nthe relativistic relaxation rate is also modified as \u0000r\na=\n\u0000r\na;1(T\u0000Ta\nc)=T, we can fairly investigate the crossover\nfrom relativistic to exchange dominated regime by4\n0123456FeGd\ntime [ps]−ΓrFeΓrGdΓexexchange relaxationexchange relaxationrelativistic relativistic relativistic −1−0.500.51\n0123\n0123456magnetizationMn4aMn4cΓ[µB/ps]time [ps]−Γr4aΓr4cΓex\n(a)(c)(d)(b)\nFIG. 3. Element-specific magnetization dynamics of single-pulse switching in ferrimagnets using an atomistic spin dynamics\nmodel (symbols) and a macroscopic model (solid lines) for Mn 2Ru0:86Ga (a) and Gd 25FeCo (b). The red area corresponds to\nthe critical values ( mc\na;mc\nb) to enter exchange relaxation regime. The blue area corresponds to the time lapse where relativistic\nrelaxation dominates. Different contributions to the element-specific magnetization relaxation for the single-pulse switching\ndynamics in Mn 2Ru0:86Ga (c) and Gd 25FeCo (d). \u0000r\na(b)stands for the relativistic relaxation rate of sublattice a= 4a(c)\nanda=Fe (d), and b= 4c(c) andb=Gd (d). \u0000exis the exchange relaxation rate, which is the same for both sublattices.\n\u0015a=\u0015b=\u0015= 0:01.\ncomparing \u0000ex\n1and\u0000r\na;1. Two cases of interest exist,\ni) one sublattice is faster than the other, and ii) both\nsublattices demagnetize at the same rate.\nIn the first case, \u001ca\u001c\u001cb, sublattice ademagnetizes\nfaster than sublattice b. This corresponds to GdFeCo\n(Fig. (3)(b)). Soon after the application of a fs\nlaser pulse, ma\u001cmb, and consequently, \u0000ex\u0019\n\u0015kBTmb=(zma)(cf. Eq.(8)). We estimate the conditions\nforthetransitionfromrelativistictoexchange-dominated\nregimes (see Fig. (3)(d)): From \u0000ex>\u0000r\na,ma<p\nmb=2zab\u00190:288pmb, and from \u0000ex>\u0000r\nbandma\u0014\n1=2z= 0:0833(red colored area in Fig. 3 (b)). A second\ncase,\u001ca\u0019\u001cb, might also arise when demagnetization\ntimes of both sublattices are similar. This is the case\nof Mn 2Ru0:86Ga alloys (Fig. (3)(c)). One can estimate\nthe conditions for the transition \u0000ex= \u0000r\na. This\nhappens for both sublattices when ma;b= (ma(0) +\nmb(0))2=(ma(0)mb(0))=2zab. Assuming a realistic value\nofmb(0)=ma(0) = 0:9, the exchange-dominate regime is\nreached when ma;b= 0:334(red colored area in Fig. 3\n(c)). Notably, this condition only depends on the initial\nvalues of the magnetization, ma(b)(0). These results are\nsimple and general, and one of the main result of the\npresent work. One can interpret the transition from\nrelativistic to exchange-dominated regime as follows:\nas the magnetization of one sublattice decreases, the\nphase space of available states for the spins of the other\nsublattice to switch by exchange of angular momentum\ndramaticallyincreases. Whileswitchingspinviacouplingto an external bath becomes increasingly difficult as the\nmagnetization reduces.\nOnce the system has entered the exchange dominated\nregime the dynamics follows a path where total angular\nmomentum is conserved towards a magnetic state where\n\u0016aHa=\u0016bHb. In the high temperature limit, this\ncondition reduces to mz\na=mz\nb, meaning that the\nexchange relaxation drives the magnetization of both\nsublattices into the same polarity, i.e. a ferromagnetic-\nlike state [2]. From this condition arises, that in order to\nhave a final mz\na<0the following condition is necessary:\nSex\na+Sex\nb<0. HereSex\na(b)standsfortheangularmagnetic\nmoment when the system enters the exchange dominated\nregime. For example, in Mn 2RuxGa, since Mn spins are\nassumed to demagnetize at the same rate, this condition\nreduces to Sex\na+Sex\nb\u0019(Sa(0) +Sb(0)) exp(\u0000t=\u001ca)<\n0. Namely, only for a starting temperature below the\ncompensation temperature, Sa(0)+Sb(0)<0, conditions\nfor switching are fulfilled, in complete agreement to\nexperimental observations [35]. Yet, the exchange\nrelaxation regime needs to be active for a significant time\nin order for the magnetization to switch. We compare\nthe time scales associated to both the relativistic and\nexchange relaxation. Relativistic relaxation rate follows\n\u0000r\na= 2\u0015akB(T\u0000Ta\nc)mz\na, which is strongly reduced by\nthe ultrafast dynamics of mz\na, in only a few hundred of\nfemtoseconds \u0000r\na!0. Whereas \u0000exrather follows the\ndynamics of the temperature T, and therefore decays\nslower than \u0000r\na(Fig. (3)(c) and (d)).5\nThe characteristic time scale of the electron and lattice\ntemperature is described by the TTM and for common\nparameter values in the range of 2 \u00003 ps. As the\ntemperature reduces, the exchange relaxation drives the\nsystem towards (T\u0000Ta\nc)mz\na= (T\u0000Tb\nc)mz\nb. For\nTb\nc< T < Ta\nc, the exchange relaxation drives the\nsystem back into an antiferromagnetic, but switched\nsate. As the magnetization builds up in the opposite\ndirection, \u0000ex, decreases and the relativistic relaxation\ntakes over. Interestingly, our theory predicts that for\nsystems where Ta\nc\u0019Tb\ncswitching would be unlikely,\nwhich has been recently demonstrated in rare-earth free\nsynthetic ferrimagnets [49]. Further, one can accelerate\nthe transition from exchange relaxation dominated to\nthe relativistic regime, and speed up complete switching\nby increasing difference between Ta\ncandTb\nc, an effect\nwhich has been observed by the substitution of Fe by\nCo, namely, GdFe by GdCo [50]. While GdFe recovers\nin tens of picoseconds, GdCo alloys only need of a couple\nof picoseconds.\nTo summarize, in this work we have proposed a general\nmacroscopic theory for the magnetization dynamics\nof ferrimagnetic materials driven by femtosecond laser\nphoto-excitation. Our theory reproduces quantitatively\nall stages of the switching process observed in ex-\nperiments. Notably, we have directly compared our\ntheory to computer simulations using atomistic spin\ndynamics methods for both GdFeCo and Mn 2RuxGa\nalloys. The magnetization dynamics transits from a\nrelativistic relaxation path to an exchange dominated\nregime due to the strong enhancement of the exchange\nrelaxation. 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Petrillo,3Bert Koopmans,3Reinoud\nLavrijsen,3Andreas Kehlberger,2and Mathias Kläui1\n1)Institute of Physics, Johannes Gutenberg University Mainz, Staudingerweg 7, 55128 Mainz,\nGermany\n2)Sensitec GmbH, Walter-Hallstein-Straße 24, 55130 Mainz, Germany\n3)Department of Applied Physics, Eindhoven University of Technology, Eindhoven, 5612 AZ,\nNetherlands\n(Dated: 3 April 2023)\nSynthetic ferrimagnets are an attractive materials class for spintronics as they provide access to all-optical switching\nof magnetization and, at the same time, allow for ultrafast domain wall motion at angular momentum compensation.\nIn this work, we systematically study the effects of strain on the perpendicular magnetic anisotropy and magnetization\ncompensation of Co/Gd and Co/Gd/Co/Gd synthetic ferrimagnets. Firstly, the spin reorientation transition of a bilayer\nsystem is investigated in wedge type samples, where we report an increase in the perpendicular magnetic anisotropy\nin the presence of in-plane strain. Using a model for magnetostatics and spin reorientation transition in this type of\nsystem, we confirm that the observed changes in anisotropy field are mainly due to the Co magnetoelastic anisotropy.\nSecondly, the magnetization compensation of a quadlayer is studied. We find that magnetization compensation of this\nsynthetic ferrimagnetic system is not altered by external strain. This confirms the resilience of this material system\nagainst strain that may be induced during the integration process, making Co/Gd ferrimagnets suitable candidates for\nspintronics applications.\nI. INTRODUCTION\nRecent advances in spintronics have opened new possi-\nbilities for electronic applications beyond the CMOS stan-\ndard. New concepts of high density and ultrafast non-volatile\ndata storage have been proposed in magnetic memories1,2.\nThroughout the years, magnetic memories have evolved3,4\nexploiting different geometries5and new material platforms\nsuch as ferrimagnets6have been used to improve storage\ndensity7, reading and writing speed8and energy efficiency9,10.\nAt the same time, single-pulse optical-switching (AOS) of\nmagnetization has reduced the switching speed of the mag-\nnetization below ps timescale11–14. This bears promise for a\nnew generation of ultrafast data buffering, in a single chip that\nintegrates photonics with spintronics15–19.\nFerrimagnets are a class of magnets with unbalanced\nantiparallel-aligned sublattice moments. The compensation\nof the two inequivalent sublattices, combines the advantages\nof both antiferromagnets (antiparallel alignment of magnetic\nmoments) and ferromagnets (finite Zeeman coupling and spin\npolarization)16,20. Moreover, the drastic contrast between the\ntwo sublattices in non-adiabatic dynamics, could potentially\naccommodate AOS by a femtosecond laser pulse12,16. Single-\npulse AOS is typically observed in rare earth–transition metal\n(RE–TM) ferrimagnetic alloys like GdFeCo20or in multilayer\nsynthetic ferrimagnet, such as Co/Gd and [Co/Tb] n21,22. In\nparticular, the one based on multilayer of Co/Gd is a good can-\ndidate for integrated opto-spintronics devices as it shows AOS\n- without the constrains on the composition as imposed by al-\nloy system23,24- and at the same time exhibits magnetic and\nangular momentum compensation, allowing ultrafast domain\na)Electronic mail: gmascioc@uni-mainz.dewall motion25,26. For instance, the integration of Co/Gd syn-\nthetic ferrimagnets in an optically switchable magnetic tunnel\njunction has been recently reported27.\nWhen it comes to technological implementation, strain\ninduced effects must be considered, which could be in-\ncurred from processing steps such as packaging and layer\ndeposition28. Intrinsic stresses and strain could affect the\nmagnetic anisotropy via changes to the spin-orbit coupling\n(SOC)29or to the magnetization compensation of ferrimag-\nnets especially in RE–TM alloys30,31. However, in spite of\nbeing omnipresent in applications32–34, the effect of strain has\nnot yet been explored in these materials. In this work, we\npresent a systematic study of the effects of strain on Co/Gd\nsynthetic ferrimagnets. By the application of external strain,\nusing substrate bending, we investigate the impact of strain on\nthe perpendicular magnetic anisotropy (PMA) and the mag-\nnetization compensation of [Co/Gd] and [Co/Gd] 2multilay-\ners, respectively. Using wedge samples in a bilayer system of\nCo/Gd and polar magneto-optic Kerr effect (pMOKE) mea-\nsurements, we confirm that the PMA is increased by in-plane\ntensile strain and a negative magnetostriction is reported. By\nincluding the contribution of the strain-anisotropy for this sys-\ntem in a model for the magnetostatics, we show that the ef-\nfects of strain on the magnetization are mainly due to the\nmodification of the spin-orbit coupling within the magnetic\nlayer and at the the Pt/Co interface that increases the mag-\nnetic anisotropy via magnetoelastic coupling. Additionally,\nwe find that the magnetization compensation point is not af-\nfected significantly by strain, as the magnetoelastic coupling\naffects the anisotropy rather than the magnetization of the two\nsublattices. Our study explores the mechanisms that underlie\nthe influence of strain on the magnetic anisotropy of Co/Gd\nferrimagnets and contributes to a better understanding of the\nmagnetoelastic effects of ferrimagnetic multilayers. These re-\nsults could be employed for the optimization and developmentarXiv:2303.15191v4  [cond-mat.mtrl-sci]  31 Mar 20232\nof spintronics devices, as well as for potential applications in\nfields such as magnetic memory and sensing.\nII. METHODS AND SAMPLE FABRICATION\nThe samples were grown on a 1.5 µm thick, thermally oxi-\ndized SiOx on top of a 625 µm thick Si substrate by DC mag-\nnetron sputtering in a chamber with a typical base pressure\nof 5×10−9mBar. To obtain a variable thickness (wedge)\nalong the sample surface, a shutter in the close proximity of\nthe sample is gradually closed during deposition. This al-\nlows to study the compensation and spin reorientation tran-\nsition (SRT) within a single sample. Two types of sam-\nples are realized. Firstly, a bilayer of Ta(4 nm)/Pt(4)/ Co(0-\n2)/Gd(t Gd)/TaN(4) with a constant Gd layer on top of a Co\nwedge is considered to study the SRT. In addition, a quadlayer\nof Ta(4)/Pt(4)/Co(0.6)/Gd(0-2)/Co(0.6)/Gd(1.5)/TaN(4), this\ntime with a Gd wedge, is grown to study the magnetization\ncompensation.\nThe magnetic properties of these wedge samples were in-\nvestigated by pMOKE, where we are only sensitive to the\nout-of-plane (OOP) component of the Co magnetization at a\nwavelength of 658 nm. According to Fig. 1 (a), the surface\nof the sample is scanned along the y-direction using a focused\nlaser spot with a spot-size of /similarequal250µm diameter. Accord-\ningly, the local magnetic properties and hysteresis loops can\nbe measured as a function of layer thickness, with a negligi-\nble thickness gradient <0.025 nm within the used laser spot.\nAll the measurements are performed at room temperature. To\napply in-plane tensile strain to our multilayer, the substrate\nis mechanically bent using a three-point method35. A square\nsample of 1 by 1 cm is vertically constrained on two sides\nand pushed uniformly from below by a cylinder that has off-\ncentered rotation axis. The device generates a tensile strain in\nthe plane of the sample when the cylinder is rotated. As pre-\nviously reported, the tensile strain is uniaxial along xand uni-\nform in the measured area of the sample. The in-plane strain\nmagnitude is 0 .1% and has been measured with a strain gauge\n(RS PRO). More details about the strain generating device can\nbe found in section S2 of the supplementary information.\nIII. RESULTS AND DISCUSSION\nA. Spin reorientation transition in Co/Gd bilayer\nThe use of magnetic materials for high density data storage\nrequires magnetic systems that are OOP magnetized36,37. In\nthin films, an OOP magnetic easy axis can be obtained by\nmagnetocrystalline anisotropy induced at the interface with\nheavy metal38,39. In addition to that, strain has been shown\nto affect the magnetic easy axis direction in systems with\nPMA40. To understand the effect of external strain on Co/Gd\nsystems with PMA, we investigate bilayer samples consist-\ning of Ta(4 nm)/Pt(4)/ Co(0-2)/Gd( tGd)/TaN(4). Specifically,\nthe Co thickness is varied between 0 and 2 nm over a few\nmm along the ydirection, whereas tGdis constant (as in Fig.1 (a)). In this system, the balance between the interfacial\nanisotropy energy (magnetocrystalline anisotropy energy at\nthe Pt/Co interface) and demagnetization energy determines\nthe effective magnetic anisotropy. In such system, the demag-\nnetization energy increases with the thickness of the Co mag-\nnetic layer, and consequently, the magnetization will go from\nout-of-plane (OOP) to in-plane (IP). To probe the magnetiza-\ntion of our wedge sample, we record hysteresis loops from\nthe pMOKE signal. We repeat the measurement moving the\nlaser spot along the wedge in the y direction. Firstly, a sample\nwhere t Gd=0 is considered. This measurement can be seen in\nFigs. 1 (b) and (c). Fig. 1 (b) reports the magnetic response of\nthe Ta(4 nm)/Pt(4)/Co(0-2)/TaN(4) sample to an OOP mag-\nnetic film for different t Co. The effective anisotropy Ke f fwas\nestimated38recording hysteresis loops with magnetic field ap-\nplied OOP and IP and the corresponding anisotropy energy\nper unit area is Ks=1.7 mJ/m2. For tCo=1.35 nm the square-\nshaped loop indicates PMA with Ke f f= 1.5(2)×105J/m3. A\nvalue of MCo=1.3 MA/m was used in the calculation. As the\nthickness of Co is increased (moving the laser spot along the\nwedge direction - y) the remanence and squareness of the hys-\nteresis loop decreases together with the PMA of the system.\nFortCo=2.00 nm, the sample is IP magnetized and Ke f f=-\n0.8(2)×105J/m3is negative. The OOP to IP transition occurs\nattCo=1.85(2)nm in this system.\nTo investigate the effects of externally applied in-plane\nstrain, we repeat the measurement while the sample is me-\nchanically bent. The magnetization is coupled to the ex-\nternal strain and can be described by the expression for the\nanisotropy energy35:\nKME=−3\n2λsYε, (1)\nwhere λsis the saturation magnetostriction, Yis the\nYoung’s modulus and εis the strain. If the strain in the film\nis non-zero, the magneto-elastic coupling of Co contributes\nin principle to the effective anisotropy. Accordingly, the total\nanisotropy Ke f fof the magnetic stack is expected to change in\nthe presence of external strain. Fig. 1 (c) shows the OOP hys-\nteresis loops of Ta(4 nm)/Pt(4)/Co(1.85)/TaN(4) sample be-\nfore (blue) and after (red) the application of εxx=0.1%. We\nobserve that the anisotropy field is decreased after the appli-\ncation of in-plane strain. This happens because, in this sys-\ntem, the strain-induced magnetoelastic anisotropy KME=0.02\nmJ/m2is positive, as we expect from a material with negative\nmagnetostriction like Co40,41. More details about the calcu-\nlations of magnetoelastic anisotropy can be found in section\nS2 of the supplementary information. Accordingly, the PMA\nis increased by the applied strain, i.e. the system is expected\nto be OOP magnetized for thicker Co if compared to samples\nwithout strain.\nAfter this preliminary study on Pt/Co systems, we focused\nour attention on the magnetostriction of Co/Gd multilayers.\nIn Co-Gd alloys the magnetostriction has been reported to be\nstrongly dependent on the composition29,42due to the struc-\ntural modification occurring with different atomic content. In\ncontrast to this case, the effects of magnetostriction of a mul-3\nFIG. 1: (a) Sample sketch, red arrow indicates the direction of the applied strain. (b) Out of plane hysteresis loops of a Pt/Co/TaN stack for different Co\nthicknesses. (c) OOP hysteresis loops of Pt/Co(1.85 nm)/TaN before (blue) and after (red) application of 0 .1% in-plane strain. (d) MOKE intensity scan at\nremanence (no applied field) of Pt/Co/Gd/TaN films along the Co wedge.\ntilayer, are expected to be dependent on the magnetoelastic\ncoupling of the individual layers43.\nTo study the magnetostriction of a Co/Gd multilayer, a con-\nstant layer of Gd on top of the Co wedge is added. To perform\nthickness dependent studies, a thickness tGd=1 nm and 3 nm\nis considered. In the bilayer system, the magnetization in the\nGd layers is mainly induced at the interface with the Co layer,\nand couples anti-parallel the Co magnetization21. Accord-\ningly, tCorequired to reach SRT is expected to change with in-\ncreasing tGd44. To compare the SRT of Ta(4 nm)/Pt(4)/Co(0-\n2)/Gd( tGd)/TaN(4) samples with different tGdwe performed\nremanent intensity scan along our Co wedge, in addition\nto hysteresis loop measurements. After the sample is sat-\nurated with an OOP magnetic field of 1T, we determine\nthe thickness-dependent remanence from the pMOKE signal\nwithout external magnetic field. The remanent intensity scans\nare reported in Fig. 1 (d). As the pMOKE signal is mainly\nsensitive to the OOP component of Co magnetization, the nor-\nmalized remanent intensity will drop to zero at the SRT, when\nthe magnetization rotates IP. The SRT can be observed in Fig.\n1 (d) in samples with different thicknesses of Gd before and\nafter the application of strain. As previously reported44the\ncritical thickness tCo=tcat which SRT occurs, changes sig-\nnificantly in the presence of a Gd layer. For all the considered\nsamples, the in-plane strain shifts the OOP to IP transition\ntowards larger Co thickness. This suggests that the effective\nmagnetostriction of the Co/Gd bilayer is negative and its value\nλs=−10(5)×10−6is not significantly altered by the pres-ence of the Gd layer.\nTo obtain a quantitative understanding of the shape of the\nspin reorientation boundary, we employ an analytical model44\ndescribing the magnetostatic free energy of the anisotropy,\nwhich is zero at the SRT boundary. The first constituent ener-\ngies of the model are the demagnetization energies of the Co\nlayer\nEd,Co=1\n2µ0/integraldisplayy\n0M2\nCodq=1\n2µ0M2\nCoy (2)\nand of the Gd layer\nEd,Gd=1\n2µ0/integraldisplayx\n0M2\nGdexp(−2q/λGd)dq=\n1\n4µ0M2\nGdλGd/parenleftbigg\n1−exp/parenleftbigg−2x\nλGd/parenrightbigg/parenrightbigg (3)\nwhere λGdis the characteristic decay length of the Gd mag-\nnetization, which is induced at the Co/Gd interface, MCois the\nmagnetization of the Co layer, MGdis the effective Gd mag-\nnetization at the interface between Co and Gd and xandy\nare, respectively, the Gd and Co thicknesses in the diagram of\nFig.2 (a). The plot axes in Fig.2 (a) have been inverted for a\nbetter comparison with the other figures. The magnetocrys-\ntalline anisotropy is included with the term4\nEK=Ks−∆K/parenleftbigg\n1−exp/parenleftbigg−2x\nλK/parenrightbigg/parenrightbigg\n, (4)\nand it is also considered to decay with a characteristic de-\ncay length λKand magnitude ∆K. The second term in Eq.\n4 phenomenologically addressed the experimentally observed\ndecay in the effective anisotropy, which may be caused by\nsputter induced disordering of the Co45. Using a numeri-\ncal fit to the experimentally determined SRT, the parame-\ntersλK,λGdand∆Kfor our Co/Gd bilayer are determined.\nAll the other parameters were either experimentally measured\nor taken from literature and are reported in Table S.1 , sec-\ntion S1 of the supplementary information. In addition to the\nanisotropy term, and additional energy term Emixis included\nin the model. Emixtakes into account the mixing at the mag-\nnetic layer interfaces where the local net magnetization is\nzero. More details about the expression for this term and the\ndetermination of the fitting parameters can be found in the\nsupplementary information and in the work of Kools et al.44.\nIn this model, the expression of the total free energy density\nper unit area is, considering all the terms mentioned so far:\nEtot=−EK−Emix+Ed,Co+Ed,Gd. (5)\nThe magnetocrystalline anisotropy energy per unit area Ks,\ndue to the Pt/Co interface is assumed constant.\nEq. 5, describing the total energy of a\nTa(4nm)/Pt(4)/Co( tCo)/Gd( tGd)/TaN(4) sample, can be\nsolved for y (t Co) by imposing Etot=0 (spin reorientation\ntransition). The solution for the SRT obtained with the model\ndescribed above is reported in Fig. 2 (a) with a blue solid line\nin a phase diagram where tGd(x) and tCo(y) are continuously\nvaried from 0 to 3 nm and from 0 to 2 nm, respectively.\nTogether with the calculations, the SRT measured experimen-\ntally without externally applied strain is reported with blue\ndiamonds in Fig. 2 (a). The experimental data, follow well\nthe general trend of the calculations. Discrepancies between\nmodel and experimental values for tGd=0, might be due to\nadditional mixing between the layers.\nTo include the effects of strain, a magnetoelastic anisotropy\nKMEis added to Eq. 5 that becomes\nEtot=−EK−Emix−KME+Ed,Co+Ed,Gd. (6)\nIn our case KME=0.02 mJ/m2corresponds to the value of\nmagnetoelastic anisotropy induced with 0 .1% externally ap-\nplied in-plane strain in our experiments. As showed in Fig.\n1 (d), we do not observe significant changes to KMEwith in-\ncreasing tGd. Again considering the SRT-boundary to be at\nEtot=0, the solution of Eq. 6 (that includes the magnetoe-\nlastic term) is reported in Fig. 2 (a) with an orange solid line.\nAs expected from a material with negative magnetostriction,\nKMEsums to Ksand the PMA is enhanced by in-plane strain.\nThe SRT calculated including KMEto Eq. 6 is consequently\nshifted to larger values of tCo. This trend is in agreement with\nFIG. 2: (a) 2D phase diagram of the SRT of the a\nTa(4nm)/Pt(4)/Co( tCo)/Gd(t Gd)/TaN(4) stack as a function of tGd(x) and tCo\n(y). The axes have been inverted for a better comparison with other figures.\nBlue diamonds and red squares correspond to the experimental data,\nreported without and with strain applied, respectively. The solid lines\nindicate the calculated values using the model for the magnetostatics and Eq.\n6. A magnetoelastic anisotropy KME=0 and 0.02 mJ/m2is considered,\nrespectively, for the blue and orange curve. (b) Spin reorientation transition\nof a Ta(4)/Pt(4)/Co( tCo)/Gd(t Gd)/TaN(4) sample calculated for values of\ntGd=0, 1 and 3 nm and plotted as a function of tCo. The SRT is represented\nhere by a step function. Solid and dashed lines consider KME=0 and 0.02\nmJ/m2, respectively.\nthe experimentally determined SRT when and external strain\nεxx=0.1% is applied (orange squares in Fig.2 (a)).\nAnother way to visualize the SRT is solving Eq. 6 for fixed\nvalues of tGdand obtaining the critical thickness of tCosuch\nthatEtot=0. Then, the SRT can be represented as a step\nfunction in the diagram of Fig. 2 (b), analogue to the MOKE\nremanence scan shown in Fig. 1 (d). The values of Gd thick-\nnesses considered are tGd=0, 1 and 3 nm and are plotted in\nFig. 2 (b) with solid lines in black, blue and orange, in order.\nSolid lines consider KME=0 mJ/m2. Dashed lines consider\ninstead KME=0.02 mJ/m2in Fig. 2 (b). The information\ncontained here can be correlated to the experimental rema-\nnent intensity scan in Fig. 1 (d). Comparing Fig. 2 (b) with5\nFIG. 3: (a) Layerstack consisting of a Co/Gd quadlayer used to obtain magnetization compensation. (b) Coercivity and (c) remanent pMOKE intensity scan as\na function of tGd. Measurements before (blue) and after (orange) application of in-plane strain are reported. (d) Hysteresis loops in the Co dominated and (e)\nGd dominated state. Both curves with (orange) and without (blue) in-plane strain applied are shown.\nFig. 1 (d), a similar behavior can be observed. Firstly we can\nnote that the model predicts the SRT to shift when the thick-\nness of the Gd layer is tGd>0. Secondly, we observe a similar\nshift of the SRT point in Fig. 2 (b) and Fig. 1 (d) due to the\neffect of magnetoelastic anisotropy and of the external strain,\nrespectively. As we expect from a material with negative mag-\nnetostriction, Ksadds to KME, therefore the PMA is increased\nand the Co/Gd bilayer stays OOP magnetized for thicker Co\n(corresponding to larger Ed,Co). We confirm that the major ef-\nfect of strain on the Ta(4 nm)/Pt(4)/ Co(0-2)/Gd(t Gd)/TaN(4)\nsample is the alteration of the PMA. Moreover, the estimated\neffective magnetostriction of the stack - λs=−10(5)×10−6\n- is not significantly altered by the presence of the Gd layer in\nthe thickness range considered.\nIn this section, we examined the impact of in-plane strain\non the effective PMA of a Co/Gd ferrimagnetic bilayer. Our\nresults suggest negative magnetostriction of the stack for the\ninvestigated thickness values. We employ a recent model for\nthe magnetostatics of these type of systems, where we include\nthe effects of strain purely as magnetoelastic anisotropy. Our\nexperimental findings are in good agreement with the predic-\ntions made by this model, providing deeper understanding of\nthe response of this material platform to external strain.\nB. Magnetization compensation in quadlayer systems\nIn ferrimagnets, magnetization compensation can be\nachieved. This occurs when the net magnetization /vectorMtot=\n/vectorMGd+/vectorMCovanishes because the magnetization, coming from\nthe two sub-lattices, is equal in magnitude and opposite in\nsign.\nIn recent studies, changes to the saturation magnetization\nin the presence of strain were reported in epitaxial films31\nand rare earth free ferrimagnets30. To study the effectsof strain on magnetization compensation of synthetic fer-\nrimagnets, we consider a quadlayer sample44consisting of\nTa(4 nm)/Pt(4)/Co(0.6)/Gd(0-2)/Co(0.6)/Gd(1.5)/TaN(4) as\nschematically drawn in Fig. 3 (a). In this case, the thickness of\nthe bottom Gd layer is varied between 0 and 2 nm over a few\nmm, whereas all the other layers have constant thickness. The\nreason for this choice is that compared to the Co/Gd bilayer,\nthe magnetic volume of the Co is doubled while the number\nof Co/Gd interfaces where magnetization is induced in the Gd\nthrough direct exchange with the Co, is tripled. In this way\nmagnetization compensation can be more readily achieved.\nThe growing thickness of Gd, increases the contribution of\n/vectorMGdto/vectorMtot. For this reason, some areas of the wedge sam-\nple will be Co-dominated (for tGd<tcomp) and other will be\nGd-dominated (for tGd>tcomp) with /vectorMtot=0 attGd=tcomp.\nHere, tcomp is the thickness where magnetization compensa-\ntion is obtained. At magnetization compensation two effects\nare expected: a divergence of the coercivity and a sign change\nin the remanent pMOKE signal (Kerr rotation, normalized to\nits value in absence of Gd). The measurements for coercivity\nand intensity are reported in Figs. 3 (b) and (c), respectively.\nThe coercivity data were extracted from hysteresis loops mea-\nsured across the wedge direction (along y). The reason for\nthe sign change in the pMOKE signal, is the alignment of the\nGd magnetization along the field direction, in the Gd domi-\nnated regime. We report magnetization compensation in this\nquad-layer for tGd=1.25 nm.\nIn a similar fashion to what we have done investigating the\nPMA in the bilayer system, we repeat the experiment in the\npresence of εxx=0.1% in-plane strain. The results are reported\nin orange in Fig. 3 (b) and (c). Remarkably, the compensation\npoint of the Co/Gd quadlayer is unchanged by the application\nof this externally applied strain.\nFigs. 3 (d) and (e) contain OOP hysteresis loops\nof Ta(4 nm)/Pt(4)/Co(0.6)/Gd( tGd)/Co(0.6)/Gd(1.5)/TaN(4)6\nsamples for tGd=1.15 nm and tGd=1.35 nm, respectively,\nand further show the effects of magnetization compensation.\nThe sample is in this case OOP magnetized. As the thickness\nof Gd is increased, the magnetization of the sample goes from\nCo dominated (Fig. 3 (d)) to Gd dominated (Fig. 3 (e)). The\ninversion of hysteresis loops happens because for tGd>1.25\nnmthe Co-magnetization aligns antiparallel to the field, lead-\ning to the change in sign of the pMOKE signal. When the\nmeasurement is repeated in the presence of εxx=0.1% strain\n(orange line), no significant changes to the remanent intensity\nor coercivity are reported, if compared to the unstrained case\n(blue line). This suggests that magnetization compensation\ncan be achieved in these multilayer systems in the presence of\nexternal strain and, most importantly, that the magnetization\ncompensation point is unaffected.\nTo explain this, we can consider earlier studies about\nmagnetostatics of these types of systems. As previously\nreported25,44, magnetization compensation is due to the bal-\nance in Co magnetization and the Gd magnetization, induced\nin the Gd at the Co/Gd interfaces. In-plane strain in multilayer\nsamples with PMA modifies spin orbit coupling within one\nlayer46, thus altering the magnetocrystalline anisotropy en-\nergy of the system47. On the other hand the total magnetic mo-\nment per unit area /vectorMtotin synthetic ferrimagnets is obtained\nby integrating the magnetization of the Co and Gd sublattices\nover the respective layer thicknesses. Accordingly, in a mul-\ntilayer in-plane strain is not affecting the induced magnetic\nmoment from the Co onto the Gd, thus not altering magneti-\nzation compensation.\nIV. CONCLUSIONS\nThis work reveals the effect that external strain has on PMA\nand magnetization compensation of Co/Gd systems at room\ntemperature. Growing wedge samples, where the thickness of\none of the magnetic layers was varied, has allowed us to deter-\nmine thickness dependent transition in the magnetostatics of\nthis multilayer system. Deliberate in-plane strain was applied\nto the sample. In a bilayer Pt/Co/Gd system, we experimen-\ntally show that a sizable magnetoelastic coupling changes the\nSRT in the presence of strain. The contribution of the strain-\nanisotropy for this system has been included in a model for the\nmagnetostatics, describing the experimental observations well\nif an effective negative magnetostriction is considered. In a\nPt/Co/Gd/Co/Gd quadlayer we obtain magnetization compen-\nsation of the two sub-lattices by varying the thickness of the\nbottom Gd layer. Here, we find that the application of in-plane\nstrain does not affect the magnetization compensation. The\ninduced magnetic moment from the Co onto the Gd, being an\ninterface effect in a multilayer system, is not altered by such\nmechanical deformation. To conclude, this work provides a\nbroad understanding of the magnetoelastic properties of these\nmultilayer systems. As PMA and magnetic compensation are\nmaintained in the presence of externally applied strain, this\nmaterial system is a good candidate for technological imple-\nmentation of ferrimagnets.SUPPLEMENTARY MATERIAL\nSee supplementary material for magnetostatics model for\nthe spin reorientation transition and for more details about the\nsetup used for application of strain.\nACKNOWLEDGMENTS\nThis project has received funding from the European\nUnion’s Horizon 2020 research and innovation program un-\nder the Marie Skłodowska-Curie grant agreement No 860060\n“Magnetism and the effect of Electric Field” (MagnEFi), the\nDeutsche Forschungsgemeinschaft (DFG, German Research\nFoundation) - TRR 173 - 268565370 (project A01 and B02)\nand the Austrian Research Promotion Agency (FFG). 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Shull, “Strain-assisted magnetization reversal in Co/Ni multi-\nlayers with perpendicular magnetic anisotropy,” Scientific reports 6, 1–8\n(2016).Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nSupplementary material - Strain effects on magnetic compensation and spin\nreorientation transition of Co/Gd synthetic ferrimagnets\n(Dated: 3 April 2023)\n1arXiv:2303.15191v4  [cond-mat.mtrl-sci]  31 Mar 2023Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nS1 - Magnetostatics model for the Spin Reorientation Transition\nIn the expression of the free energy, a term describing the effect of intermixing at the multilayer\nsystem interfaces is added. The expression is\nEmix=1\n2µ0/integraltexta0x\n0M2\nCo+ (MGdexp(−q/λGd))2dq=\n1\n2µ0a0M2\nCox+1\n4µ0λGdM2\nGd/parenleftBig\n1−exp/parenleftBig\n−2a0x\nλGd/parenrightBig/parenrightBig\n.(S.1)\nTherefore, the total energy Etot=−EK−Emix−KME+Ed,Co+Ed,Gd, including all the terms\nwill be:\nEtot=−Ks+∆K/parenleftBig\n1−exp/parenleftBig\n−2x\nλK/parenrightBig/parenrightBig\n−KME−1\n2µ0a0M2\nCox\n−1\n4µ0λGdM2\nGd/parenleftBig\n1−exp/parenleftBig\n−2a0x\nλGd/parenrightBig/parenrightBig\n+1\n2µ0M2\nCoy\n+1\n4µ0M2\nGdλGd/parenleftBig\n1−exp/parenleftBig\n−2x\nλGd/parenrightBig/parenrightBig\n.(S.2)\nHere xandyare, respectively, the Gd and Co thicknesses in the phase diagram of Fig. S1. The\nvalue of the parameters used in our model are listed in Table S I .\nParameter Value Description\nKs 1.7 mJ/m2Interfacial anisotropy (from exp.)\nKME 0.02 mJ/m2Magnetoelastic anisotropy (from exp.)\nMCo 1.3 MA/m Cobalt magnetization (from exp.)\nMGd 1.4 MA/m Gadolinium magnetization at Co/Gd interface (from Ref.1)\na0 0.13 (-) Growth parameter of intermixing region (from exp.)\nλK 0.51(15) nm Change of PMA energy characteristic decay length (Fit parameter)\nλGd 0.59(22) nm Gd magnetization decay characteristic decay length (Fit parameter)\n∆K 3.96(41) ×10−4J/m2Change of PMA energy (Fit parameter)\nTABLE S I: Parameters used in the model for the magnetostatics of uncompensated Co/Gd synthetic ferrimagnets\nused for the calculations of the SRT. The term KMEis considered zero for when external strain is not applied to the\nsample.\nThe values of λK,λGdand∆Kare instead determined using a numerical fit and are reported\nin Table S I. To fit this equation to the phase diagram obtained experimentally, it is convenient to\n2Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nfind the Co-thickness (y) where the anisotropy energy ( Etot) is equal to zero (spin reorientation\ntransition, SRT). Solving Eq. S.2 for y gives:\ny0(x) =2\nM2\nCoµ0/parenleftBig\n−/parenleftBig\nKs−∆K/parenleftBig\n1−exp/parenleftBig\n−x\nλK/parenrightBig/parenrightBig/parenrightBig/parenrightBig\n−1\n2µ0a0M2\nCox\n−1\n4µ0λGdM2\nGd/parenleftBig\n1−exp/parenleftBig\n−2a0x\nλGd/parenrightBig/parenrightBig\n+1\n4µ0M2\nGdλGd/parenleftBig\n1−exp/parenleftBig\n−2x\nλGd/parenrightBig/parenrightBig\n.(S.3)\nNote that for determining the fit parameters, the measurements were taken without externally\napplied strain. Accordingly, the magnetoelastic energy term KMEis set to zero in Eq. S.3. The\nexperimental data used for the numerical fit are reported in Fig. S1. The sample used for the\nnumerical fit, explores a wide thickness range (Gd from 0 to 6 nm) in a double wedge fashion\nto improve accuracy. Consequently, the dimensions of this sample exceed the 1x1 cm size of\nthe bending device. For this reason, single wedge samples have been deposited for the strain-\ndependent study.\nFIG. S 1: Values for the SRT obtained experimentally on a Ta(4nm)/Pt(4)/Co(tCo)/Gd(tGd)/TaN (4)sample and\nused to extract the fitting parameters in Eq. S.3.\nThe feature around tGd=2 nm in Fig. S1 is not captured by out toy model, and might be due to\nthe additional intermixing caused during sputtering, not included in Eq. S.2.\n3Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nS2 -Application of strain\nTo obtain information about the magnetoelastic properties of the material, the substrate was\nbent mechanically with a 3 point bending sample holder, as shown schematically in Fig. S2 (a).\nA square sample of 1 by 1 cm is vertically constrained on two sides and pushed uniformly from\nbelow by a cylinder that has an off-centered rotation axis. The device generates a tensile strain\nin the plane of the sample up to 0 .1 % when the cylinder is rotated by 90◦. The strain is mostly\nuniaxial and has been measured with a strain gauge on the substrate surface.\nFIG. S 2: (a) schematic of the three point bending method used to externally strain the sample. The strain is mostly\nuniaxial along the xdirection. (b) hysteresis loops measured before (blue) and during (red) application of in-plane\nstrain for a sample of Pt/Co(1.85 nm)/Ta. The area highlighted in red corresponds to the magnetoelastic energy in the\nstrained system. The magnetic field was applied along the OOP direction ( z).\nMagnetic hysteresis loops are recorded before and after the application of the tensile strain\nand are used to estimate the magnetoelastic anisotropy. As previously reported2,3the magnetic\nanisotropy Ke f fis linked to the energy stored in the magnetization curves. For example the PMA\nenergy is given by the area enclosed between the magnetic loops measured with field along IP and\nOOP direction. If then the strain in the film is non-zero, the magneto-elastic coupling contributes\nin principle to the effective anisotropy. Two hysteresis loops measurements, before and after the\napplication of strain, are sufficient to estimate KME. Indeed the total anisotropy of the system\nisKe f f=KsandKe f f=Ks+KMEbefore and after the application of strain, respectively. The\nmagnetoelastic anisotropy KME=−3\n2λsYεis linked to reversible part of the hysteresis loops (close\nto the saturation) according to\n4Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nKME=Ms∆E=−3\n2λsYε (S.4)\nwhere ∆Eis the anisotropy energy measured by the difference in area below the strained and\nunstrained curves, εis the strain λsis the magnetostriction and Y is the Young’s mudulus of the\nmaterial. In our case ε=0.1% and Y=200 GPa. ∆Ecorresponds to the reversible part, i.e. the red\nmarked area in Fig. S2 (b). The value of magnetoelastic anisotropy was calculated using the value\nof saturation magnetization ( Ms) of the stack taken from literature and reported in Table S I.\n5Suppl. material - Strain effects on magnetic compensation and spin reorientation transition of ...\nREFERENCES\n1Kools, T. J., van Gurp, M. C., Koopmans, B., and Lavrijsen, R. (2022). Magnetostatics of room\ntemperature compensated Co/Gd/Co/Gd-based synthetic ferrimagnets. Applied Physics Letters,\n121(24), 242405.\n2Johnson, M. T., Bloemen, P. J. H., Den Broeder, F. J. A., and De Vries, J. J. (1996). Magnetic\nanisotropy in metallic multilayers. Reports on Progress in Physics, 59(11), 1409.\n3Baril, L., Gurney, B., Wilhoit, D., Speriosu, V . (1999). Magnetostriction in spin valves. Journal\nof Applied Physics, 85(8), 5139-5141.\n6"    },    {        "title": "1607.02358v3.Control_of_magnon_photon_coupling_strength_in_a_planar_resonator_YIG_thin_film_configuration.pdf",        "content": "arXiv:1607.02358v3  [cond-mat.mtrl-sci]  24 Nov 2016Control of magnon-photon coupling strength in a planar reso nator/YIG thin film\nconfiguration\nV. Castel,1A. Manchec,2and J. Ben Youssef3\n1)T´ el´ ecom Bretagne, Technopole Iroise-Brest, CS83818, 29 200 Brest,\nFrance.\n2)Elliptika (GTID), 29200 Brest, France.\n3)Universit´ e de Bretagne occidentale, Laboratoire de Magn´ etisme de Bretagne CNRS,\n29200 Brest, France\n(Dated: 26 July 2021)\nA systematic study of the coupling at room temperature between f erromagnetic res-\nonance (FMR) and a planar resonator is presented. The chosen ma gnetic material is\na ferrimagnetic insulator (Yttrium Iron Garnet: YIG) which is positio ned on top of a\nstop band (notch) filter based on a stub line capacitively coupled to a 50 Ω microstrip\nline resonating at 4.731 GHz. Control of the magnon-photon couplin g strength is dis-\ncussed interms ofthe microwave excitation configurationandtheY IG thickness from\n0.2 to 41 µm. From the latter dependence, we extract a single spin-photon co upling\nof g0/2π=162±6 mHz and a maximum of an effective coupling of 290 MHz.\nKeywords: Cavity spintronic, quantum detector, Yttrium Iron Ga rnet, notch filter,\nmagnon-photon coupling\n1I. INTRODUCTION\nA recent field, known as cavity spintronics1,2, is emerging from the progress of spintronics\ncombined with the advancement in Cavity Quantum Electrodynamics ( QED) and Cavity\nPolaritons3,4. Cavity QED allows the use of coherent quantum effects for quantu m informa-\ntion processing and offers original possibilities for studying the stro ng interaction between\nlight and matter in a variety of solid-state systems5–7. A superconducting two-level system\nis quantum coherently coupled to a single microwave photon and an an alogy to spintronics\n(spin two-level system) has been made. The high spin density of the ferromagnet used in\nRef.8,9hasmade it possible to create a strongly coupled magnonmode. Magn on-photoncou-\npling has been investigated in several experiments at room tempera ture where a microwave\nresonator (three-dimensional cavity10–18and planar configuration19–21) was loaded with a\nferrimagnetic insulator such as the Yttrium Iron Garnet (YIG, thin film and bulk). A study\non a transition metal like Py (structured thin film) coupled with a Split R ing Resonator\n(SRR) was done by Gregory et al.22in order to demonstrate the possibility to achieve YIG-\ntype functionalities and to overtake the working frequency limitatio n of YIG. More recently,\nL. Bai et al.15have developed an electrical method to detect magnons coupled wit h photons.\nThis method has been established by placing a hybrid YIG/Pt system in a microwave cavity\nshowing distinct features not seen in any previous spin pumping expe riments but already\npredicted by Cao et al.23.\nII. COMPACT DESIGN DESCRIPTION\nThe main objective of the present paper is to demonstrate the con trol of a magnon-\nphoton coupling regime at room temperature in a compact design bas ed on a stub line\ncoupled with a microstrip with YIG thin film. Control of a magnon-phot on coupling regime\nin such configuration offers manifold opportunities in the developmen t of integrated spin-\nbased microwave applications, such as a sensitive reconfigurable st op-band filtering function.\nInsteadofusingaSplitRingResonator(SRR)configuration,thech oicewasmadetostudy\ntheYIGthickness dependence ofthecoupling regime withastub lineg eometry forwhich the\nmicrowaveexcitationofthemagneticmediumissimplified. Figure1(a)r epresentsthesketch\nof the experimental setup based on the stub/YIG film system excit ed by a microwave signal\n2/s76\n/s115/s119\n/s115/s49/s103\n/s119\n/s115/s50\n/s119/s40/s99/s41/s40/s98/s41\n/s83/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s115/s32/s91/s100/s66/s93/s32/s83\n/s49/s49/s32/s40/s77/s41\n/s32/s83\n/s50/s49/s32/s40/s77/s41\n/s32/s83\n/s49/s49/s32/s40/s83/s41\n/s32/s83\n/s50/s49/s32/s40/s83/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s91/s71/s72/s122/s93/s40/s97/s41\n/s89/s73/s71/s47/s71/s71/s71\n/s72/s48\n/s53\n/s49/s53/s49/s48\nFIG. 1. (a) Experimental setup: A Vector Network Analyzer (V NA) is connected to a 50 Ω\nmicrostrip line which is capacitively coupled to the resona tor. Numbers from 0 to 15 are referred\nto the YIG sample position (an example is given for which the c enter of the sample is placed\nat x=10 mm=0.25 λ). (b) Dimension of the microwave stop band resonator configu ration. (c)\nFrequency dependence of S parameters measured (M) and simul ated (S) by CST simulation of the\nempty resonator (without YIG sample).\nunder anin-plane static magnetic field, H, at θ=0◦.θis defined by the angle formed between\nH and the stub line. P1 and P2 correspond to the 2 ports of the VNA f or which a TSOM\ncalibration were realized (including cables). The frequency range is fi xed from 3 to 6 GHz at\nan microwave power of P=-10 dBm. The circuit is fabricated on a pre- metallized (double-\nsided 25 µm copper coating) ROGERS substrate (3003) presenting a relative permittivity\nofεr=3 and losses tan δ=2×10−3(dimension are shown in Fig. 1 (b)). The narrow stop-\nband (notch) resonator configuration is based on a main 50 Ω micros trip line (W=1.23 mm)\ncoupled by a gap of g=150 µm to an open circuited half wavelength stub (L s=15.52 mm,\nWs1=280µm and W s2=1.0 mm). It has been designed with an attenuation of 13 dB at 4.75\nGHz and 80 MHz of bandwidth. The frequency dependence of the S1 1 and S21 parameters\n(measured andsimulated) oftheempty resonatorareshown inFig. 1(c). TheS21resonance\npeak has a half width at half maximum (HWHM) ∆F HWHMof 32 MHz indicating that the\ndampingoftheresonator(workingatthefrequency F0)isβ=∆FHWHM/F0=1/2Q=6.8 ×10−3.\nThis leads to a quality factor Q of 74. The latter definitions of the Q fa ctor and β(used\n3in the following discussion) are based on the expression extracted f rom recent studies in the\nfield of magnon-photon coupling (2D20,21and 3D cavities13–18). Nevertheless, this definition\ndoes not reflect properly the electrical performances of our not ch filter which are defined by\nQ0=F0//bracketleftbig\n∆FS21\n−3dB(1−S11F0)/bracketrightbig\n=153.\nSingle-crystal Y 3Fe5O12(YIG) samples from 0.2 to 41 µm were elaborated by Liquid\nPhase Epitaxy (LPE) on top of a 500 µm thick GGG substrate in the (111) orientation.\nYIG samples have been cut in rectangular shape (4 mm ×7 mm) and placed on the stub line\nas shown in Fig. 1 (a) with the crystallographic axis [1,1, ¯2] parallel to the planar microwave\nfield generated by the stub line. The magnetic losses (Gilbert damping parameter, α) of the\nsetofYIGsamplewereinvestigatedbyFMRmeasurementsusingahig hlysensitivewideband\nresonance spectrometer within a range of 4 to 20 GHz. Measureme nts were carried out at\nroom temperature with a static magnetic field applied in the plane of YI G samples. These\ncharacterizations have given rise to a parameter α≤2×10−4for the set of samples which is\nin agreement with previous studies24.\nIII. RESULTS AND DISCUSSION\nWe first studied the magnitude of the coupling strength as a functio n of the position\nof a 9µm YIG sample on top of the planar resonator, as shown in Fig. 1 (a). F igures\n2 (a) to (c) illustrate the dependence of the resonator features (at H=0 Oe), such as the\nresonant frequency F0, linewidth ∆F HWHM, damping of the resonator β(Q factor), and\nthe dependence of S11 and S21 at the resonant frequency F0.F0can be tuned from 4.35\nto 4.715 GHz (tuning of 8.4 %) and presents a maximum at x=0.25 λwhich is closed to\nthe resonant frequency of the empty resonator (represented by horizontal dash dot lines).\nNote that the YIG position corresponds to the center of the samp le as illustrated in Fig.\n1 (a). The wavelength is defined by λ=λ0√εeff, whereεeffcorresponds to the effective\npermittivity. The configuration of the notch filter (open circuit (OC ) at x=0.5 λ) induced\nnecessarily the definition of short circuit (SC) at x=0.25 λwhich explains the limited impact\nof YIG (at this position) on the resonator features. Introductio n of a YIG layer ( εr=15\nand losses tan δ=2×10−4) on CST simulation make it possible to correctly reproduce this\ndependence at zero field (solid black line) which is attributed to the mo dification of the\neffective permittivity. Here, only the electrical properties of the Y IG sample were taken into\n4FIG. 2. (a) to (c): YIG sample position dependence (at H=0 Oe) of (a)F0and ∆F HWHM,\n(b) Q factor and losses, (c) S11 and S21 at the resonant freque ncyF0. Horizontal dash dot\nlines correspond to parameters extracted from the empty res onator whereas the vertical dash dot\nline illustrates the position of YIG (respect to the center) at x=0.25 λas shown in Fig. 1 (a).\n(d) and (e): Signature of the coupling. Frequency dependenc e of the magnitude in dB of S21\n(d) and the associated phase φS21(e) for a YIG position at x=0.25 λ. Solid blue and red line\ncorrespond respectively to the response at H=0 Oe and at H=H RES.F1andF2(represented by\nvertical red dash lines) correspond to hybridized mode freq uencies whereas g eff/2πcorresponds\nto the coupling strength parameter. (f) Dependence of g eff/2π(measured, black triangles) and\nmicrowave field amplitude (CST simulation, solid red line) a s function of the YIG sample position.\nAll measurements have been carried on at room temperature on a YIG sample which presents\na thickness of (9 µm). The inset shows the spatial distribution of the microwav e magnetic field\nsimulated at F0.\naccount. Contrary to a ferromagnetic conductor, such as an ex tended thin film of Permalloy\n(Py, NiFe)22, no eddy current shielding effect of YIG on the stub line was observe d. YIG is\na ferrimagnetic insulator with a band gap of 2.85 eV and the high quality YIG samples used\n5in this study allow the reduction of negative impacts on the planar res onator. As shown\nin Fig. 2 (a), ∆F HWHMreaches a maximum of 34 MHz at x=0.3 λ, which represents an\nenhancement of only 2 MHz with respect to the empty resonator (a reduction of 5 MHz\nbeing achievable at x=0.45 λ). Slightly changes in the Q factor ( β) from 70 to 77 (6.4 to\n7.2×10−3) were obtained. In the meantime, attenuation represented by th e S11 parameter\natF0are closed to the value extracted from the empty resonator from x=0.4 to 0.3 λ(-2.35\ndB).\nFor each position of the YIG sample, measurement at room tempera ture of the frequency\ndependence of S parameters (magnitude and phase) at P=-10 dBm was done with respect\nto the applied magnetic field. Figure 2 (d) and (e) represent, respe ctively, the frequency\ndependence of S21 and Φ S21of the notch/YIG system at x=0.25 λ(as shown in Fig. 1\n(a)). Solid (dash) blue and red lines are associated with the experime ntal (analytic solution\nfrom Eq. (3) from Ref.18) response under an applied magnetic field of H=0 Oe and H=H RES,\nrespectively. TheFMRandthenotchfilterinteractbymutualmicro wavefields, generatedby\nthe oscillating currents in the stub and the FMR magnetization prece ssion which led to the\nfollowing features observed in Fig. 2 (d) and (e): (i) Hybridization of resonances (magnitude\nand phase25), (ii) Annihilation of the resonance at F0, and (iii) Generation of two resonances\natF1andF2. At the resonant condition H=H RES, the frequency gap, Fgap, between F1and\nF2is directly linked to the coupling strength of the system ( Fgap/2=geff/2π). Several models\ncan be used to analyze the hybridized mode frequency F1andF2in the system. Recently,\nHarder et al.18have examined the accuracy to describe the microwave transmissio n line\nshape of a cavity/YIG system through three different models: cou pled harmonic oscillators,\ndynamic phase correlation, and microscopy theory. Here, the ana lysis has been focussed on\nthe harmonic coupling model for which we can define theupper ( F2) andlower ( F1) branches\nby:\nF1,2=1\n2/bracketleftbigg\n(F0+Fr)±/radicalBig\n(F0−Fr)2+k4F2\n0/bracketrightbigg\n(1)\nTheFMRfrequency, Fr, ismodelledbytheKittelequation, Fr=γ\n2πµ0/radicalbig\nH(H+Ms), which\ndescribes the precession frequency of the uniform mode (without taking into account spin\nwave distribution) in an in-plane magnetized ferromagnetic film. The p arameter kused in\nEq. 1 corresponds to the coupling strength which is linked to the exp erimental data g eff/2π\n6by the following equation18:Fgap=F2−F1=k2F0. As shown in Fig. 2 (f), sensitive\ncontrol of g eff/2π(and thus k) can be achieved by adjusting the YIG sample position on\nthe resonator from 54 MHz ( k=0.1573) at x=0.45 λto 127 MHz ( k=0.2315) at x=0.25 λ.\nIn order to understand the dependence of g eff/2πon the YIG position, CST simulations\nwere carried out in order to determine the microwave field ( hMW) generated at each position\n(represented in solid red line). It ends up that hMWfollows exactly the same trend of the\ncoupling factor, in agreement with the fact that the effective coup ling strength depends on\nthe mutual microwave field interaction between the FMR and the stu b line. The latter\ndependence is defined by the following equation9,12:\ngeff\n2π=η\n4πγe/radicalbigg\n/planckover2pi1ω0µ0\nVc√\nN, (2)\nwhereγeistheelectrongyromagneticratioof2 π×28.04GHz/T, µ0isthepermeability ofthe\nvacuum, Vccorresponds to the volume of the cavity, and Nis the total number of spins. The\ncoefficient η≤1 describes the spatial overlap and polarization matching conditions between\nthe microwave field and the magnon mode. In agreement with Zhang e t al.12(Appendix\nA), we demonstrated the dependence of g eff/2πas function of the spatial distribution of the\nmicrowave magnetic field along the stub line which is maximum at x=0.25 λ(short circuit).\nTABLE I. Notch/YIG configuration versus SRR & cavity/YIG sys tems\nRef. β[10−3]geff/2π[MHz]F0[GHz] k\n151.8 80 10.506 0.1234\n160.708 31.8 9.650.0812\n172.3 65 10.847 0.1095\n180.3 31.5 10.556 0.0773\n13,141.92 130 3.5350.2712\n209.85 270 3.20.4108\n215.04 63 4.960.1594\nThis work[1]6.89 127 4.7160.2315\nThis work[2]6.89 290 4.7190.3508\nTable I gives a picture of recent work on the determination and cont rol of magnon-\nphoton coupling regimes in SRR20,21and cavity13–18/YIG systems. Ref.15–17correspond to\n7the research field associated with the electrical detection of magn ons coupled with photons\nvia combined phenomena in a hybrid YIG/Pt system. Despite the fact that these later\nstudies have been realized in a cavity, insertion of a hybrid stack inclu ding a highly electrical\nconductor induced an enhancement of the intrinsic loss rate β(factor of 515to 1216). The\nvalueofkobtainedattheoptimizedpositionatx=0.25 λissignificantlyhigherthanRef.15–18\nand comparable to Ref.13,14,21but still much smaller than the value obtained by Bhoi et al.20.\nIt should be noted that the normalization of kby the intrinsic loss rate βchanges the latter\ncomparison drastically.\nNext, the dependence of the coupling strength between the FMR a nd the notch filter\nwas investigated with respect to the YIG thickness from 0.2 to 41 µm. YIG samples were\nplaced at the optimized position which has been determined previously (x=0.25 λ). This\nparticular position gives an access to the highest coupling (determin ed at P=-10 dBm) and\npresents the best compromise in terms of the electrical performa nce of the notch filter.\nSample position was adjusted by tracking F0at H=0 Oe ( F0=4.715±0.002 GHz). It should\nbe noted that no dependence of the insertion rate of the resonat or (β=6.89±0.01 10−3) and\nattenuation (S11 F0=-2.45±0.04 dB) have been observed with respect to the YIG thickness.\nAs shown in Fig. 3 (a), we demonstrated a strong coupling regime via t he anti-crossing\nfingerprint. A good agreement of F1,2based on Eq. 1 (solid lines) with experimental data\nis obtained for the various YIG thicknesses. The color plot in Fig. 3 (a ) is associated with\nthe S21 parameter for which the dark area corresponds to a magn itude of -10 dB. This\nrepresentation underlines the complexity of the response by incre asing the YIG thickness\nfrom 0.2 to 41 m, well illustrated by the additional anti-crossing signa ture between 0.75\nand 0.90 kOe (upper resonance). In the following discussion, the ex traction of the coupling\nfactor is only based on the uniform mode without taking into account the dispersion relation\nof spin waves. Figure 3 (b) represents the frequency dependenc e of the transmission spectra\nfor the notch/YIG system at the resonant condition for which the effective coupling was\nextracted. Control of the frequency gap can be achieved from 5 9 to 581 MHz through an\nenhancement of the YIG thickness from 0.2 to 41 µm, respectively. Parameters associated\nwith the thicker YIG are summarized in Tab I (last row).\nThe originality of this study is described in Fig. 3 (c) which represents the dependence of\nthe effective coupling g eff/2πas a function of the square root of the YIG volume interacted\nwith the 1 mm width microwave resonator (V=4 mm ×1 mm×YIGthickµm). Cao et al.23\n8FIG. 3. Control of the coupling strength as function of the YI G thickness. (a) Magnetic field\ndependence of the frequency: Observation of the strong coup ling regime via the anti-crossing\nfingerprint. The color map is associated to the response of th e thicker YIG sample (41 µm). (b)\nFrequency dependence of S21 at the resonant condition for va rious YIG thickness (0.2, 9, and 41\nµm). (c) Coupling strength of the Kittel mode to the microwave resonator mode as a function of\nthe square root of the YIG volume. Colored triangles corresp ond to the dispersion of g eff/2πfrom\nFig. 2 (f). The inset represents the YIG thickness dependenc e ofk. All measurements were done\nat room temperature and at x=8 mm as shown in Fig. 1 (a).\nshows that the filling factor of magnetic medium in a cavity can be used as a measure of\nthe total number of spins, N. The large effective coupling strength is due to the large\nspin density of YIG, ρs=2.1×1022µBcm−3(µB; Bohr magneton). The linear fit of the\ndependence presented in Fig. 3 (c) gives rise to a slope of 742 ±29 MHz mm3/2which\n9makes it possible to extract the single spin-photon coupling g 0/2π=162±6mHz based on the\nfollowing equation9,16geff=g0√\nN. Tabuchi et al.9demonstrate a good agreement between\nthe evaluation of the single-spin coupling strength from the fitting ( 39 mHz) and theory\nderived from the quantum optics community (38 mHz). The latter va lue is calculated from\nEq.2 by taking the coefficient η= 1. The higher value of g 0/2πobtain in the present work\nis mainly due to the compactness of our resonator. A rough estimat ion of the volume of our\nnotch filter working at F0=4.750 GHz can be done by using a one dimensional transmission-\nline cavity26which defines the volume as Vc=πr2λ/2. By assuming r=0.5 mm (distance\nbetween the feed line and the ground), λ=c/(√εeffF0), andη= 1, we evaluate g 0/2π=177\nmHzwhenthecavityiscompletely filledbyair( εeff=εr=1)which isclosed totheextracted\nvalue of g 0/2πfrom the fitting. Nevertheless, this value does not reflect the fac t that the\ncavity is nonuniformly filled with other dielectric materials such as the s ubstrate, GGG, and\nYIG. An enhancement of g 0/2πfrom 177 to 219 mHz can be achieved by taking into account\nan effective permittivity of εeff=2.449. The latter value is determined27for the notch filter\nloaded with a YIG film at x=0.25 λwhich induced a diminution of F0from 4.750 to 4.715\nGHz.\nWe have demonstrated the presence of a strong coupling regime via the anti-crossing\nfingerprint of the FMR from an magnetic insulator and a planar reson ator. Control of\nthe coupling with respect to the YIG thickness from 0.2 to 41 µm makes it possible the\ndetermination of the single spin photon coupling of our system at roo m temperature. We\nhave found that g 0/2πin a thin film configuration is equal to 162 ±6 mHz. In the meantime,\nwe demonstrate an effective coupling strength of 290 MHz for the t hicker YIG. Improvement\non insertion losses of the planar resonator can be achieved in order to be more competitive\nregarding the 3D cavity system by changing the design of the reson ator (SRR, array of\nSRR, enhancement of the capacitive coupling) and/or by using a low lo ss substrate ( <\n10−2). 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Allende, 84081 Baronissi (SA), Italy \n \n \nWe have performed point contact spectroscopy  measurements on a sample constituted by a \nmetallic ferromagnetic oxide (SrRuO 3) bottom electrode and a tunnel ferrimagnetic (CoFe 2O4) \nbarrier. Andreev reflection is observed across the tunnel barrier. From the comparison of Andreev \nreflection in SrRuO 3 and across the CoFe 2O4 barrier we infer that th e ferrimagnetic barrier has a \nspin filter efficiency not larger than +13%. The observation of a moderate and positive spin filtering \nis discussed in the context of the microstructure of the barriers and symmetry-related spin filtering \neffects. I. INTRODUCTION \n \nThe spin polarization is a key quantity for the spin tronics. In the last few years, spin filters, \nconstituted by a ferro(ferri)magne tic tunnel juncti on, have emerged as promising alternative to \ncreate artificial spin polarized current sources. Ferromagnetic spin filters rely on the spin-dependent \ntransmittance of ferromagnetic tunnel barriers due to the existence of an exchange-split band gap. \nFollowing the pioneering work of Moodera et al.1 who showed spin-filtering at low temperature \nusing EuS ferromagnetic tunnel barriers, spin-filte ring has been demonstrated using perovskite \noxides2 (BiMnO 3) and more recently, spinel oxides: NiFe 2O4 (NFO)3,4 and CoFe 2O4 (CFO)5  \nbarriers. As the Curie temperature of ferrites is  well above room temperature, efficient room-\ntemperature spin-polarized sources could be obta ined using these oxides. The most simple spin-\nfilter structure is formed by two metallic electrodes: a non spin-polarized current source (M) and a ferromagnetic layer (FM), acting as a spin analyzer , with a ferro(ferri)ma gnetic tunnel barrier (FI) \nbetween them. Determination of the spin-filtering efficiency (P\nFI) of the FI barrier has been \nachieved by measuring the magnetoresistance of the tunnel junction and using the Jullière model to \nderive P FI. Using this approach low-temperature values  of +22% and -25% have been reported for \nNFO3,4 and CFO5 respectively. However, by using the Meservey-Tedrow technique, positive P CFO \nvalues (ranging from +6% to 26% depending on film growth condi tions) have been obtained in \nCFO-based spin filters6. Theoretical calculations of spin-depe ndent electronic structure of spinels \nindicate that the lowest ener gy conduction band is spin-down7, thus predicting a P FI < 0. It thus \nfollows that, in spite of its relevance for further progress in spintronics, no definitive determination \nof P FI is yet available for spinel-based spin filters.  \nThe Point Contact Andreev Reflection (PCAR) t echnique has been introduced as a tool to \nmeasure the spin polarization of carriers in ferromagnetic materials8. In the case of a normal \nmetal/superconductor junction, an incoming electron from the normal metal with energy smaller \nthan the superconducting gap cannot enter into the superconducting (S C) electrode and is reflected \nas a hole in the normal metal, simultaneously adding two electrons (a Cooper pair) to the condensate in the superconductor. This process,  known as Andreev reflection (AR), causes an \nincrease of the conductance around zero bias G(V≈0) compared to the conductance at voltages \nG(V) well above the superconducting gap ( ∆/e) by a factor of two. Since the reflected hole is \ncreated in the density of states with opposite sp in than the incoming electron, the AR process is \npartially suppressed when the respective densities of  states are not equal, as in the case of a ferromagnetic metal/superconductor (FM/ SC) interface. In particular, in  fully spin-polarized metals, \nall carriers have the same spin  orientation and the AR should be  totally suppressed because a hole \ncannot be created in the opposite de nsity of states. Thus, the absolute value ( but not the sign) of \ntransport spin polarization │P│ of a ferromagnetic material can be inferred from the grade by \nwhich AR is suppressed in a measured conductance spectra8,9. It has been recently predicted10 that \nthe insertion of a spin-filtering barrier to form a SC/FI/M structure should lead to the subsequent \nmodification of the AR process by spin-selective tunnelling across the ferromagnetic insulator.  \nIn this paper we report PCAR spectrosc opy experiments realized by pressing a \nsuperconducting Nb tip on a spin filter constituted by a CoFe 2O4/SrRuO 3 (CFO/SRO) thin film \nbilayer. We show that AR occurs across the ferrimagnetic CFO tunnel barrier of the \nNb/CoFe 2O4/SrRuO 3 (SC/FI/FM) structure demonstrating spin-preserved tunnelling through the \nCFO barrier. Analysis of the experimental co nductance data by means of the modified BTK \nmodel11 allows determining the polarization P of th e CFO/SRO bilayer. PCAR experiments have \nalso been conducted on the bare SRO layers, and we obtained P ≈ 42(1)%. These results indicate a \nvery modest filtering efficiency of the CFO barriers (< +13%). Implications of these findings are discussed. \n \n  \nII. SAMPLE PREPARATION AND PRE-CHARACTERIZATION \n \nCFO/SRO bilayers have been deposited by RF magnetron sputtering from stoichiometric \ntargets, on a single crystalline (111)SrTiO 3 substrate. The bottom electrode is a 25 nm thick \nferromagnetic (T Curie ≈ 120 K) and metallic ( ρ ≈ 200 μΩ •cm at 10K) SRO epitaxial film, deposited \nat ≈0.5 nm/min in a mixed atmosphere of argon and oxygen (ratio 3:2), with a total pressure of 100  \nmTorr; the substrate temperature was 725 ºC. The top layer is a 3 nm thick epitaxial CFO film, \ngrown at ≈ 0.1nm/min with an Ar/O 2 ratio of 10:1, and a total pre ssure of 250 mTorr at 500 °C. The \nroughness of the SRO and the CFO layer are lower than 0.3 nm. Th e saturation magnetization of a \nbare 3 nm thick CFO film12, measured by using a Quantum Design’s SQUID, was ≈524 emu/cm3. \nThis value is somewhat larger than the bulk valu e. Similar enhancement ha d been reported earlier \nfor ultrathin films of other spinel oxides13 and attributed to partial ca tion inversion of the spinel \nstructure and subsequent electronic and magnetic ordering modifications.  The bilayer structure was characterized by Conductive-Atomic Force Microscope (C-AFM), \nwith a Nanotec Cervantes AFM, using Nanosensors  tips (CDT-NCHR). Electrical measurements \nwere performed in 2-point electr ical configuration: the C-AFM probe was grounded, while the SRO \nbottom electrode was positively biased. C-AFM m easurements were conducted in dry nitrogen \natmosphere, using a feedback-force of ≈850 nN and an applied voltage of 800 mV. \nIn Fig. 1(a) we show a resistance map measured  by C-AFM: at each point of the surface the \nsystem records the current between tip and sample under constant bias voltage. This image indicates \na very high electrical hom ogeneity in a large area (3 μm x 3 μ m). The simultaneously recorded \ntopographic maps (not shown) confirmed the absence of particular defects and an extremely smooth \nsurface (rms < 0.3 nm). The histogram of the resist ance values, shown the Fig. 1(a)(inset) indicates \na narrow distribution of log R (half-width at half-maximum ≈ 3.5%), centred around ≈8.25 \n(≈178 MΩ). As the tunnelling current depends exponentia lly on the barrier thic kness, the data in \nFig. 1(a) signals also an extremely sm all variation of the barrier thickness14. Subsequent \nmeasurements were repeated in  the same area; neither the re sistance map nor the corresponding \ntopographic image evidenced significant changes in the surface properties other than a minor \nreduction of the overall resistance likely associat ed to residual surface contamination removal. No \ntraces of indentation or  scratching could be detected. Tests re peated at different locations on the \nsample surface yield very similar results thus co nfirming the homogeneity of the surface properties. \nThe Current-Voltage ( I-V) curves measured at different points on the surface (one example \nis shown in Fig. 1b) show clear characteristics of tunnelling behaviour. The asymmetry visible in \nI-V curves is due to the asymmetric contact conf iguration for forward and reverse biasing of the \ntunnel structure and it is of no re levance for PCAR experiments, performed in the subgap region, to \nbe described in the following. \n \n \nIII. POINT CONTACT SPECTROSCOPY \n \nPoint contact junctions were formed by pushing a soft Niobium tip, obtained by mechanical \ncutting and chemical etching a Nb wire (diameter ≈ 0.2 mm), onto the CFO surface of the \nCFO/SRO bilayer. The experimental setup utilized for the meas urements has been described \nelsewhere15. Due to the softness of Nb and to the hard ness of the CFO surface, we do not expect \nsignificant scratching or penetration of the CFO film by the tip. The e ffective electric contact radius d can be estimated by using the approximation16 R = 4ρ l / 3πd2 + ρ / 2d and employing the values of \nthe resistance R measured at high bias ( ≈200-300 Ohm), the resistivity of equivalent  SRO films \n(ρ ≈ 200  μΩcm) at 4 K and a mean free path l of about 15 Å, estimated from ρ17. It turns out that \nthe estimated contact size d is of about 45-60 Å, implying that although the contacts are larger than \nthe mean free path they are stil l in an intermediate regime d ≈ l. \nWe have recorded conductance curves at low temperature (T = 4.2 K) at different positions \non the film and with different pressures between tip  and sample. In Fig. 2 we show representative \nconductance spectra of two different types of junc tions we measured: Junctions 1 and 2. Data have \nbeen normalized by using the background conductance estimated at large voltage (V >> ∆/e) \nregions, where Δ is the superconducting gap of Nb ( Δ ≈ 1.5 meV). Both spectra display a clear \nincrease of the conductance around zero bias suggesting that the tr ansport is mainly due to the \nAndreev reflection process. More over, Andreev reflection, presumab ly spin-filter weighted, takes \nplace across the ferrimagnetic barrier as predicted in Re f. [10]. From data in Fig. 2 it is clear that the \nnormalized conductance G(≈0 V) < 2 thus indicating some suppr ession of AR as expected for a \nspin-polarized electron tunnelling.  \nIt is noteworthy in Fig. 2 that the features in the conductanc e spectrum appear at energies \nthat are sensibly higher than what is expected for Nb. This observation is  commonly attributed to \nthe presence of a spread resistance R S arising from the resistance of the sample between the junction \nand one of the measuring contacts18. The effect of R S is to shift the coherence peaks from V ≈ Δ to \nlarger voltages and subsequently changing the G (0) values. Following Woods et al.18 the spread \nresistance is included in our modelling of the PCAR curves by considering two contributions to the \nmeasured voltage19: \n () ()()I V I V I VS PC + =  (1)\nand so the measured conductance G(V)  will be fitted by using  \n \n()1−\n⎟\n⎠⎞⎜\n⎝⎛+ = =dIdV\ndIdV\ndVdIV GS PC (2)\n V PC and VS are the voltage drops at the FM/FI/SC junction and at R S, respectively. To \ncalculate VPC(I) we used the modified BTK model11 considering an effective P spin-polarized \ncurrent coming out from the spin-filter. For simplicity, a ballistic regime ( l >> d ) of transport across \nthe junction will be assumed. Its application to the present case, where l ≈ d, at first sight may seem \nproblematic. However, as shown in Ref. [18], th e potential errors introdu ced by applying ballistic formulas to diffusive contacts, as far as Z is not too small, have a negligible impact on the extracted \nP values and the uncertainty is translated into the barrier transparency Z. The other relevant fitting \nparameters are: the superconducting gap Δ and a smearing coefficient ( Γ) that allows modelling of \nthe broadening of the gap edges and in elastic scattering at the interfaces20. The local temperature \nhas been assumed to be the same like measured at the sample holder, which is immersed in a liquid \nHe bath. \nThe spread resistance R S was determined beforehand by setting a gap value Δ equal to the \nBCS value for Nb, thus eliminating a possible degeneracy in Δ and R S. Consequently, Δ cannot be \nconsidered a completely free parameter, but it is set close to the BCS value. In the subsequent \nfitting step, R S was kept fixed to the determined value and P, Z, Δ and Γ  were allowed to vary. The \nsolid line in Fig. 2a is the result of the optimal fit of G(V)  using P  ≈ 39%, Z ≈  0.13, Γ  ≈ 0.22 meV, \nΔ ≈ 1.50 meV and R S/RPC ≈ 0.5. The low interface barrier transparency Z used to fit our \nexperimental data indicates that our measurem ents are not significantly affected by a possible \ndependence of the values of P on Z21-23. The quality of the fit and its robustness on variations of P \ncan be better appreciated in Fig. 2c where we s how the low-voltage zoom of  data in Fig. 2a. In \nFig. 2c we also include fits obtained by fixing Δ (1.5 meV) and P  to be larger (45%) or smaller \n(35%) than P from optimal fit of Fig. 2a and allowing Z, Γ to vary. We notice in Fig. 2c that fixing \nlarger (smaller) P values lead to smaller (larger) Γ values as predicted in Ref. [22], just illustrating \nthat moderate inelasti c interface scattering ( Γ) decreases the AR probability. Thus the Γ and P \nparameters in the AR spectra mix together a nd distinction among both e ffects is challenging20. In \nspite of this, it is clear from these data that P J1 should be in the 45-35 % range for Junction 1. \nTo treat a possible degeneracy of fits  to PCAR spectra, it has been proposed25 to perform fits \nof (Δ, Z, Γ ) for different fixed P trial, and check the resulting sum of the squared deviations χ2 of the \nfits as function of P trial. In Fig. 3, results of such analysis are shown for the data  of Junction 1. A \nclearly defined minimum of χ2 at about 39(1)% is found, which falls well within the previously \ndetermined range of values, thus we conclude that P J1 ≈ 39(1)%. However, for the data of Junction \n2 (Fig. 2b, which will be discussed below), such pr ocedure proved to be ill-d efined when using the \nproximity reduced gap, due to the larger number of fit parameters. \nIt has been demonstrated25 that an error in the normalization of the conductance spectra can \nlead to false minima in χ2 vs. P trial. Therefore the χ2 analysis was repeated using a deliberately \nwrong normalization of the data in Fig 2a (divid ed by an additional factor n = 1.001). Still, χ2 \nreproduces the minimum but values of χ2 are considerably enlarged. For normalization just slightly smaller than 1 (factor 0.999) reas onable fits to the data are no longer possible and values of χ2 \nincrease drastically, demonstrating the validity of the original normalization. \nIn Fig. 2b we show an example of a G(V) curve measured in some different contacts \n(Junction 2). Data in Fig. 2b displa y characteristic features of AR reflection but also the occurrence \nof proximity effects and subsequent smaller gap formation ( ΔP)11,24. As CFO is a ferromagnetic \ninsulator it may be supposed that a gap reduction may occur in some part of th e tip, likely due to the \nstray field of the ferrimagnetic barrier. The mo dified BTK model has b een worked out including \nexplicitly two gaps ( Δ and ΔP)11. We used the corresponding expressi ons to fit the da ta in Fig. 2b. \nThe solid line is the result of the optimal fit of G(V) using, Δ ≈ 1.48 meV, ΔP ≈ 0.99 meV, \nΓ ≈ 0.0 meV and R S/RPC ≈ 0.36. The values Z ( ≈0.20) and P  (31%) determined for this contact are \nquite similar to those extracted from data of Junc tion 1 (Fig. 2a). In the zoom of the low-voltage \nregion (Fig. 2d) we include the f its obtained by fixing P to larger (35%) or smaller (25%) values \nwhile keeping Δ and Δ P constant. The small Γ values obtained from the best fits ( Γ ≈ 0.1), most \nprobably result from the fact that the broadening of the spectrum is captured during the fit by the \npresence of the two gaps required to fit the di ps. Although the available data do not allow to \ndisentangle both contributions, it is  clear from these data that P ≈ 31(3)% is a robust result for \nJunction 2. At this point it is wort h to recall that dips in PCAR c onductance spectra, as they appear \nin the data of Junction 2 have also been explai ned as arising from the superconductor reaching its \ncritical current and thus adding some normal conducting state finite resistance26. Although the \norigin of the dips in  the conductance spectr a cannot be decided with certainty, a smaller \nsuperconducting energy gap is in princi ple, consistent with the observation of critical current effects \nin Junction 2.  \nTherefore, it follows that Andreev reflecti on has been observed across a ferrimagnetic tunnel \nbarrier (Fig. 4). From the analysis of the PCAR we infer that representative values of the effective \npolarization of our CFO/SRO spin f ilter (3 nm thick) are of about P ≈ 31(3)% and 39(1)% as \ndetermined for Junctions 1 and 2 respectively. To  progress further and to extract the spin-filter \nefficiency of the CFO barrier re quires the knowledge of the spin-p olarization of electrons emitted \nfrom the SrRuO 3 electrode (P SRO).  \nSignificantly different P SRO results have been reported: Worledge and Geballe27 reported \nMeservey-Tedrow type measurements of tunnel j unctions having SRO as electrode, and determined \nPSRO = -9%; Nadgorny et al.23, using PCAR with Sn tips, inferred a much larger value \n(│PSRO│≈ 53%). Although this strong discrepancy is not fully understood23,27, consensus exist that PSRO < 0. Negative spin-polarization arises from the difference of Fermi velocities for spin-up and \nspin-down electrons emerging from SRO rather than the density of states at the Fermi level (which \nis practically identical)27. \nTo obtain a more reliable basis for assessing the PCAR results from the CFO/SRO structure, \nadditional experiments were perfor med on a SRO film equivalent to  the bottom electrodes used in \nthe bilayers. A representative PCAR spectrum of the SRO film is shown in Fig. 5. The polarization \nof SRO was again determined by fitting (solid lin e) to the modified BTK model and resulted in \nPSRO = 42%, somewhat lower but still in agreement w ith previously reported PCAR results within \ntheir variations23. In the inset of Fig. 5, the resulting χ2 for fits with various P trial to the SRO data \nwith a clear minimum at P SRO = 41.5(1)% is shown. \n \n \nIV. DISCUSSION \n \nWe are now in position to compare the effective polarization measured in  the spin filter and \nthe bare SRO electrode. One first notes that the values of P extracted from both junctions are \nsmaller than that of the bare SRO electrode. Due to the fact that P SRO is recognized to be negative \n(PSRO < 0), this observation implies that the spin pol arization of the CFO barrier must be positive \n(PCFO > 0). Notice that although PCAR experiments of a single ferromagnetic layer do not allow to \nextract the sign of the spin polar ization, this is possible in th e present structure, where two \nferromagnetic layers are involved, if the sign of the sp in polarizations of one of the layers (SRO in \nthis case) is known.  \nIt has been theoretically predic ted that the lowest energy barrie r in the exchange-split gap of \nCFO is the spin-down7; in such circumstances it could be expected that the spin-down channel is \ndominating (P FI < 0) the AR (Fig. 4), and thus one could anticipate an effective P in CFO/SRO \nbilayers larger than in the PCAR of the bare SRO electrode. Experimentally this is not the case as \nPJ1 and P J2 are found to be smaller than P SRO. We will discuss below on this discrepancy. \n We next consider the values of the spin filtering efficiency of the CFO layers. The two \nextracted values of P J1, J2 (≈ 39(1)% and ≈  31(3)%) represent two distinct situations. Whereas the \nsecond value (Junction 2) would indicat e a substantial spin filtering of CFO, this is not so for data \nfrom Junction 1, where a rather small different  with the bare SRO electrode is observed.  \nWe can define the spin-filter efficiency of the CFO barrier P CFO by28:  \nCFO SROCFO SRO\nP P 1P PP++=  (3)\nwhere P is the effective spin polarization measured in the PCAR experiment. Using P SRO = -\n42%, the two solutions  of Eq. (3) are P CFO ≈ +4% and +73% for P J1 and P CFO ≈ +13% and +67% for \nPJ2. Among these two sets of possible values, which phys ically arise due to th e fact that PCAR can \nnot determine the sign of the meas ured spin polarization, the larges t pair of values (73% and 67%) \nare most likely inappropriate as they could lead to  spin filtering efficiencies for CFO much larger \nthan reported values5,6. On the other hand, we notice that P CFO ≈ +13%, as determined for Junction \n2, is within the range of Meservey-Tedrow results (+6%  ≤  PCFO ≤ +26%, depending on the growth \nconditions)6 obtained on CFO barriers of nominally equal thickness (d ≈ 3 nm). Notably, the sign of \nthe effect agrees with the Meservey-Tedrow experi ments, which is, however, different from the one \ndetermined in tunnel magneto resistance measurements using Al 2O3 tunnel barriers. On the other \nhand data for Junction 1 appears to  indicate a very marginal spin filtering effect. This could be \nrelated to the fact that the CFO barrier is locally suppressed either by a mechanical effect associated \nto the Nb-tip pressure (although, as mentioned a bove, scratching effects have not been observed \nwhen using the harder C-AFM tip) or a locally poorer homogeneity of the insulating barrier. \nAlthough the CFO layers appears homogenous and r obust in CAFM measurem ents, the possibility \nthat the Nb tip penetrates the CFO layer either through pinholes or completely when crushed onto \nthe sample cannot be excluded. Under such circumstances, it turns out that P J2 and thus P CFO ≈ \n+13% constitutes the most represen tative value of the spin filtering efficiency of the present CFO \nbarriers. It could be argued that a similar result would be obtained if the CFO barrier does not spin-\nfilter at all but only contributes to depolarize the electron current from SRO. However, the experimental observation of spin-filt ering in CFO in tunnel structures\n5,6 does not support this view.  \nBefore concluding we would like to comment on the positive sign observed for P CFO. As \nmentioned, this observation is oppos ite to theoretical predictions7, which are based on the electronic \nconfiguration of an ideally inverse spinel structure of CFO where all Co2+ occupy the octahedral \nsites of the unit cell and the Fe3+ equally populate the octahedral and tetrahedral sites. However, \nmagnetization data of nanometric thin films13,29 of various spinels, including CoFe 2O4 and NiFe 2O4 \nhave provided conclusive evidence that the ca tionic distribution in nanometric thin films may \nlargely differ from their bulk c ounterparts. As shown by the calc ulations of Szotek et al.7, the \ninsulating gap of CFO closes by some 75% when th e cationic distribution is not that of the ideal \ninverse spinel structure but a nor mal one. It thus follows that th e tunnel transport may be overcome by other non-spin preserving transp ort channels and therefore a redu ced spin filtering efficiency \ncould be anticipated for cationically -disordered films. To what extent  the change of sign of the spin \nfiltering efficiency is related to the same effect or  not is not definitely settled. We also notice that \nthe spin filtering efficiency should not be simply related to the ex change splitting of the insulator \nbut the symmetry of the relevant wave  functions may also play a role. \n \n \nIV. SUMMARY \n \nIn summary we have shown evidence that Andreev reflection occurs at ferro(ferri)magnetic \ntunnel barriers. Data collected us ing point contact spect roscopy allowed to estimate the effective \nspin-polarization of a current th rough the interface of a spin-fil ter and a superconducting tip. It \nturned out that a possible spin filtering effect of the spinel oxide CoFe 2O4 tunnel barrier is limited to \nabout +13% with the accuracy of  these measurements. Observati on of a positive spin filtering \nefficiency is unexpected and may suggest the rele vance of spin-dependent orbital symmetry effects \non the tunnel probability in spin filters. To th e best of our knowledge this issue has not been \ntheoretically addressed yet.  \n \n \nACKNOWLEDGEMENTS \n \nFinancial support from the Ministerio de Ci encia e Innovación of th e Spanish Government \nProjects (MAT2008-06761-C03 and NANOSELECT CSD2007-00041) and from the European \nUnion [ProjectMaCoMuFi (FP6-03321) and FEDER  and Marie Curie IEF Project SemiSpinNano] \nis acknowledged. S.P. thanks B.L. Gallagher and C. J. Mellor for hosting the final stage of this research. \n REFERENCES \n1 J. S. Moodera, X. Hao, G. A. Gibson, and R. Meservey, Phys. Rev. Lett. 61, 637 (1988). \n2 M. Gajek, M. Bibes, A. Barthélémy, K. 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Clowes, W. R. Branford, M. Lake, I. Brown, A. D. Caplin, \nand L. F. Cohen, Phys. Rev. B 71, 104523 (2005). \n26 G. Sheet, S. Mukhopadhyay, and P. Raychaudhuri, Phys. Rev. B 69, 134507 (2004). \n27 D. C. Worledge and T. H. Geballe, Phys. Rev. Lett. 85, 5182 (2000). 28 This equation is derived by considering diffe rent transmission coefficient for the two \nindependent spin channels across the spin-filte ring CFO barrier and a spin polarized current \ncoming from the SRO. \n29 F. Rigato, M. Foerster, J. Fontcuberta, unpublished Figure Captions \n \nFig. 1: (Color online) C-AFM analysis of a CF O (3 nm)/ SRO (25 nm) bilayer on (111)STO: (a) \nresistance map in logarithmic color scale; inset: hi stogram of the resistance values over the surface. \n(b) I-V characteristic, with a schematic of the measurement circuit (inset). \n \nFig. 2: (Color online) Measured conductance spectra: (a) Junction 1; (b) Junction 2; the \ncontinuous line represents the best fits. (c), (d) Zoom around low bias; for Junction 1 and 2 \nrespectively: Comparison of the best fits with simulations achieved forcing P to slightly different \nvalues, demonstrating the accuracy  of the obtained polarizations. \n \nFig. 3 Characteristics of fits obt ained for data from Junction 1. χ2 as function of P trial for \nstandard normalization (n = 1) and varied normalization (n = 1.001).   \n \nFig. 4: (Color online) Schematic  of the density of states vs. energy of a SC-FI-FM (Nb-CFO-\nSRO) structure. The Andreev refl ection (AR) process is illustrate d (solid arrow). An incoming \nelectron from the SRO side is reflected as a hole in the spin-reversed density of states, while a \nCooper pair is added to the supe rconducting condensate. Parallel to AR, electron tunnelling into \nthermally excited states may occur (dashed arrow). For simplicity, only spin-down electrons are \ndepicted and low-lying spin-dow n barrier has been assumed.  \n \nFig. 5 (Color online) Measured co nductance spectra of a SRO thin film, equivalent to the base \nelectrodes used in CFO/SRO bilayers; the con tinuous line represents th e best fit. Inset: χ2 as \nfunction of P trial for fits to the data in the main panel. \n \n \n \n \n \n \n Figure 1 \n(b)\n-1,0 -0,5 0,0 0,5 1,0-1001020\nSTO (111)SROCFO\nSTO (111)SROCFO\nSTO (111)SROCFO\n Current (nA)\nVoltage (V)\n(a) 10 \n7 Figure 2 \n \n(c) (a) \n(d) (b) Junction 1 Junction 2 \n-20 -10 0 10 200,951,001,051,101,15\nΔNb=1.48 meV\nΔproximity =0.99 meV\nT=4.2 K\nRseries /RPC=0.36\n   Z=0.21\nP=0.305\nΓ=0  meV\nV(mV)dI/dVnormalized\n- 6 - 4 - 2 02460,951,001,051,101,151,20dI/dVnormalized\n   \n P=0.305\nΓ=0.0 meV\nZ=0.21\nP=0.35\nΓ=0.1 meV\nZ=0.1 exp. data\n P=0.25\n P=0.305\n P=0.35\nV(mV)P=0.25\nΓ=0.18 meV\nZ=0.16-20 -10 0 10 200,951,001,051,101,15\nΔNb=1.48 meV\nΔproximity =0.99 meV\nT=4.2 K\nRseries /RPC=0.36\n   Z=0.21\nP=0.305\nΓ=0  meV\nV(mV)dI/dVnormalized\n-20 -10 0 10 200,951,001,051,101,15\nΔNb=1.5 meV            \nT=4.2 K\nRseries /RPC=0.5dI/dVnormalized\n   Z=0.135P=0.39\nΓ=0.22 meV\nV(mV)\n- 6 - 4 - 2 02460,951,001,051,101,151,20dI/dVnormalized\n   P=0.305\nΓ=0.0 meV\nZ=0.21\nP=0.35\nΓ=0.1 meV\nZ=0.1 exp. data\n P=0.25\n P=0.305\n P=0.35\nV(mV)P=0.25\nΓ=0.18 meV\nZ=0.16\n-10 -5 0 5 101,001,051,101,15\nP=0.39\nΓ=0.22 meV\nZ=0.135dI/dVnormalized\n   P=0.45\nΓ=0.0 meV\nZ=0.0P=0.35\nΓ=0.52 meV\nZ=0.19 exp. data\n P=0.35\n P=0.39\n P=0.45\nV(mV)Figure 3 \n0,25 0,30 0,35 0,400,01,0x10-52,0x10-53,0x10-5\n  \n n=1\n n=1.001χ2\nPtrial\n \n Figure 4 \n \n \n \n \n Figure 5 \n-60 -40 -20 0 20 40 600,981,001,021,041,061,08\n0,36 0,40 0,449,0x10-61,0x10-51,1x10-51,2x10-5\n dI/dVnormalized\nV(mV)Δ = 1.45 meV\nZ = 0.185\nΓ = 0.27 meV\nP = 0.415\nRS/RPC = 1\nχ2\nPtrial\n \n "    },    {        "title": "2212.11887v2.Spin_wave_dispersion_of_ultra_low_damping_hematite___α_text__Fe__2_text_O__3___at_GHz_frequencies.pdf",        "content": "Spin wave dispersion of ultra-low damping hematite ( \u000b-Fe 2O3) at GHz frequencies\nMohammad Hamdi,1,\u0003Ferdinand Posva,1and Dirk Grundler1, 2,y\n1\u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), Institute of Materials,\nLaboratory of Nanoscale Magnetic Materials and Magnonics, CH-1015 Lausanne, Switzerland\n2\u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL),\nInstitute of Electrical and Micro Engineering, CH-1015 Lausanne, Switzerland\n(Dated: December 26, 2022)\nLow magnetic damping and high group velocity of spin waves (SWs) or magnons are two crucial\nparameters for functional magnonic devices. Magnonics research on signal processing and wave-\nbased computation at GHz frequencies focussed on the arti\fcial ferrimagnetic garnet Y 3Fe5O12\n(YIG) so far. We report on spin-wave spectroscopy studies performed on the natural mineral\nhematite (\u000b-Fe2O3) which is a canted antiferromagnet. By means of broadband GHz spectroscopy\nand inelastic light scattering, we determine a damping coe\u000ecient of 1 :1\u000210\u00005and magnon group\nvelocities of a few 10 km/s, respectively, at room temperature. Covering a large regime of wave\nvectors up to k\u001924 rad=\u0016m, we \fnd the exchange sti\u000bness length to be relatively short and only\nabout 1 \u0017A. In a small magnetic \feld of 30 mT, the decay length of SWs is estimated to be 1.1\ncm similar to the best YIG. Still, inelastic light scattering provides surprisingly broad and partly\nasymmetric resonance peaks. Their characteristic shape is induced by the large group velocities, low\ndamping and distribution of incident angles inside the laser beam. Our results promote hematite as\nan alternative and sustainable basis for magnonic devices with fast speeds and low losses based on\na stable natural mineral.\nIntroduction. | Spin waves (magnons) are collective\nspin excitations in magnetically ordered materials.\nThey exhibit promising functionalities for information\ntransmission and processing at GHz frequencies [1{3].\nTo realize energy e\u000ecient magnonic circuits [1, 2, 4{6]\nisotropic spin wave (SW) dispersion relations, high group\nvelocities, and low magnetic damping are essential. Until\ntoday, the arti\fcial garnet Y 3Fe5O12(YIG) [7] played a\nkey role for the exploration of magnonics functionalities\n[8]. Already in 1961, M. Sparks et al. coined the term\nthat YIG was to ferromagnetic resonance research what\nthe fruit \ry was to genetics research [9]. This was\nparticularly true for high-quality YIG grown by liquid\nphase epitaxy on the wafer scale [8, 10]. However,\nin a ferrimagnetic material like YIG, magnon bands\nin the regime of small wave vectors, k, and low GHz\nfrequencies are inherently anisotropic due to the dipolar\ninteraction between spins. To overcome this, a lot of\ne\u000bort has been put into the development of microwave-\nto-magnon transducers which allow for the excitation of\nexchange dominated SWs with isotropic properties at\nhigh frequencies [11{13]. The transducers su\u000ber however\nfrom typically a narrow bandwidth or require an applied\nmagnetic \feld in contrast to conventional transmission\nlines and coplanar waveguides (CPWs).\nIn antiferromagnetic (AFM) materials, exchange\ninteraction dominates the dispersion relation already\nat small wave vectors k. The dipolar interactions are\nvirtually absent due to net zero magnetization. Still,\nSWs can propagate with high group velocities. Values\nsimilar to thick YIG [8, 14] and as high as 30 km/s\nhave been reported [15{18]. However, the challenge\nwith most AFMs is their net zero magnetization and\nsub-THz frequencies which make on-chip integrationhard due to lack of e\u000ecient CPWs and THz sources\n(THz gap) [19, 20]. Recently, the natural mineral and\ncanted antiferromagnet hematite ( \u000b-Fe2O3) [21] gained\nparticular attention for magnonics [22, 23] after the\nobservation of long distance spin transport [24, 25]\nand enhanced spin pumping [26]. It is known that,\ndue to extremely low anisotropy in the basal plane\n[21, 27, 28], in the canted phase [Fig. 1(a) and (b)]\none branch of the magnon modes resides at around 10\nGHz at small k. Depending on the purity of hematite\ncrystals, a damping coe\u000ecient as low as 7 :8\u000210\u00006was\nreported for the magnetic resonance [29]. The hematite's\n\fnite net magnetization and strikingly small damping\nof below 10\u00005make it hence suitable for magnonic\napplications. They allow for inductive coupling to\nCPWs and long-distance SW transport, respectively.\nHowever, there is no experiment reporting a measured\nSW dispersion for kvalues accessible by CPWs with\nintegrated microwave-to-magnon transducers [11{13].\nThe dispersion measured over a large wave vector regime\nis of fundamental importance as it allows one to quantify\nthe exchange sti\u000bness length lewith large precision. le\nis the key parameter to estimate the maximum possible\nspin-wave velocity of hematite in the GHz frequency\nregime.\nHere, we study the magnon band structure of bulk\nhematite at di\u000berent wave vectors kby means of broad-\nband microwave spectroscopy and k-resolved inelastic\nBrillouin light scattering (BLS) (Fig. 1). Using a\nCPW in \rip-chip con\fguration we extract a magnetic\ndamping parameter of 1 :12\u000210\u00005which is similar to\nthe best YIG reported in Ref. 30. Still, the measured\nspectra with BLS show broad linewidths. Our modelling\nsubstantiates that the linewidth is explained by thearXiv:2212.11887v2  [cond-mat.mtrl-sci]  23 Dec 20222\n(b)\n(a)\n(c)\n(d)\n(e)\nFIG. 1. Hexagonal unit cell of the crystal structure of\nhematite from (a) the side and (b) top view. The cyan spheres\nand red arrows indicate the Fe atoms and the spins associated\nwith them, respectively (oxygen atoms are not shown). We\ndepict the canted antiferromagnetic state above the Morin\ntemperature for which the sublattice spins lie in the c-plane\nalong thea-axis with a small canting. They give rise to sub-\nlattice magnetization vectors M1andM2. (c) Sketch of the\nlow frequency quasi-ferromagnetic mode where a small mag-\nnetization (green arrow), m, precesses elliptically around the\napplied \feld (black arrow), H. (d) Schematics of the \rip-\nchip VNA measurement. The sample (gray disk) is placed\non a CPW. The static magnetic \feld, H, is applied in the\na-plane and with an angle, \u0012H, to the normal of the c-axis\nof the crystal. The rf magnetic \feld (orange double-headed\narrow), hrf, of the CPW is parallel to the c-axis. (e) Sketch\nof the BLS con\fguration. The magnetic \feld, H, is applied\nperpendicular to c-axis in the a-plane. The laser light (green)\nforms a Gaussian beam and is focused on the surface of the\nsample (gray). The cone angle of the objective lens is \u0012. The\nincident laser light with an incidence angle, 'is scattered by\nmagnons. We measure in back-scattering geometry (dashed\ngreen arrow).\nSW dispersion relation, the large SW velocity and the\nGaussian pro\fle of the laser used for inelastic light\nscattering. The data substantiate a high group velocity\nof 10 km/s for k= 2:5 rad/\u0016m which are excited\neasily by a micron-sized CPW [23, 31]. In an applied\n\feld of 90 mT, the velocity increases to 16 km/s near\nk= 5 rad/\u0016m and levels o\u000b to 23.3 km/s for k\u001525rad/\u0016m. The latter value has routinely been realized\nby transducers. Our \fndings substantiate hematite as\na very promising candidate for a sustainable future of\nmagnonics as its growth avoids the lead-based synthesis\nroute used for high-quality YIG [7, 30].\nProperties of hematite. | We \frst brie\ry review the\nrelevant magnetic properties of \u000b-Fe2O3. Hematite is\nthe stable end product of oxidation of magnetite [32] and\nknown for its great abundance as well as stability in an\naqueous environment [33]. It is an insulating antiferro-\nmagnet (AFM) with a corundum crystal structure. The\narrangement of the magnetic atoms of Fe in the crystal\nis shown in Fig. 1(a) [34]. At room temperature and\nabove the Morin transition temperature of TM= 262 K\nthe Fe+3magnetic moments lie in the c-plane due to an\neasy plane anisotropy, HA, and stack antiferromagneti-\ncally along the c-axis [Fig. 1(a)] [21, 28]. Within the\nc-plane, there is a weak 6-fold anisotropy around the c-\naxis,Ha, which favors the magnetic moments to align\nwith thea-axes [Fig. 1(b)]. The magnetic moments of\nthe two AFM sublattices are slightly canted away from\nthea-axis by the Dzyaloshinskii-Moriya (DM) interac-\ntion (Fig. 1(b)), resulting in a weak magnetic moment,\nm, perpendicular to both a- andc- axes at equilibrium\n[21, 28]. The magnetization amounts to about 2 kA/m\n[32]. The magnetization dynamics of hematite in this\nweak ferromagnetic state o\u000bers two modes namely the\nquasi-ferromagnetic mode (qFM or low frequency mode)\nand quasi-antiferromagnetic mode (qAFM or high fre-\nquency mode) [21, 27, 28].\nHere, we explore the qFM mode schematically depicted\nin Fig. 1 (c). The canted AFM sublattice magnetization\nvectors precess elliptically around their equilibrium direc-\ntion. This results in an elliptical precession of the weak\nmagnetic moment, m(green arrow in Fig. 1 (c)), around\nthe applied \feld, H[21, 27, 28]. The frequency of the\nqFM mode was derived by Pincus [21, 27] according to\nfr= (1)\nj\rj\u00160\n2\u0019p\nHsin\u0018(Hsin\u0018+HD) + 2HE(Ha+HME);\nwhere,\r,\u00160,HE,HDandHMEis the electron gyro-\nmagnetic ratio, vacuum permeability, exchange, DM and\nspontaneous magnetoelastic e\u000bective \feld, respectively.\n\u0018=\u0019=2\u0000\u0012His the polar angle between Hand thec-\naxis (z-direction). We de\fne \u0012Hin Fig. 1(d). Fink [35]\nderived the dynamic susceptibility \u001fzz(f;fr) for the qFM\nmode. The real and imaginary parts read\nRe [\u001fzz(f;fr)] =\u0000\nf2\nr\u0000f2\u0001\nf2\nr\n(f2r\u0000f2)2+ \u0001f2f2and (2)\nIm [\u001fzz(f;fr)] =ff2\nr\u0001f\n(f2r\u0000f2)2+ \u0001f2f2; (3)3\nrespectively. Here, fis the frequency of the radiofre-\nquency (rf) magnetic \feld hrf(Fig. 1 (d)). The frequency\nlinewidth, \u0001 f, is related to the magnetic damping pa-\nrameter,\u000b, by\n\u0001f\u00192\u000bHE(\r=2\u0019); (4)\nforH\u001cHE. Unlike ferromagnets and uniaxial antifer-\nromagnets, the resonance line width of a canted antifer-\nromagnet does not depend on frand is governed by the\n\feld-independent exchange frequency. Following Turov\n[15, 16], the SW dispersion of the qFM mode for hematite\nneark= 0 is given by\nfm(k) = (5)\nj\rj\u00160\n2\u0019p\nH(H+HD) + 2HE(Ha+HME+Ak2);\nwhereA=HEl2\neis the dispersion coe\u000ecient and leis the\ne\u000bective magnetic lattice parameter or exchange sti\u000bness\nlength. For Eq. 5, we considered the experimental\ngeometry for k-resolved BLS measurements with an\nangle\u0018=\u0019=2 as described below.\nExperimental techniques. | The broadband microwave\nspectroscopy was conducted at room temperature above\nTM(Fig. 1(d)). The disk-shaped a-plane\u000b-Fe2O3crystal\nhad a diameter of 2.3 mm and thickness of 0.5 mm. Mea-\nsurements were done in \rip-chip con\fguration for which\nthe sample was placed on a CPW with signal (ground)\nline width of 165 \u0016m (295\u0016m). Thea-axis of the crys-\ntal was perpendicular to the disk plane. The c-axis was\nin the plane and perpendicular to the CPW axis. In-\njecting a radio-frequency (rf) current, Iin\nrf, into the CPW\nby port 1 of a vector network analyzer (VNA) induced\nthe dynamic magnetic \feld hrf(orange double-head ar-\nrow). The rf current was collected on the other end of\nthe CPW by port 2 of the VNA ( Iout\nrf). The static mag-\nnetic \feld, H, was applied in the a-plane of the crystal\nin all VNA measurements ( yz-plane in Fig. 1 (d)). For\nthe angle dependent measurements an external \feld of\n90 mT was applied and the angle, \u0012H, was varied in steps\nof \u0001\u0012H= 2\u000e. In case of \feld sweep measurements, the\napplied \feld angle was perpendicular to the c-axis and in\nthea-plane of the crystal ( \u0012H= 0 deg). The \feld ampli-\ntude was varied in steps of \u0001 H= 0:5 mT.\nWave vector-resolved BLS measurements were done for\n\u0012H= 0 deg on a piece of the same crystal in back-\nscattering geometry (Fig. 1 (e)) using a green laser with\nwavelength, \u0015= 532 nm and wave vector k0= 2\u0019=\u0015=\n11:81 rad/\u0016m. The external \feld was applied in the a-\nplane and perpendicular to c-axis. The sample for BLS\nwas irregularly shaped. We ensured that it was tilted\nwith respect to the incident laser beam along the y-axis in\nsuch a way that the laser beam remained in the xz-plane\nformed by the a- andc-axis. Since the penetration depth\nof the green laser in hematite is on the order of 75 nm[36], linear momentum conservation holds only for the in-\nplane component of the transferred wave vectors. There-\nfore, we de\fne the transferred momentum from the light\nto magnons along the c-axis askz= 2(k0sin'+k0\nxcos'),\nwhere'is the angle between the incident beam and the\nnormal to the plane of the sample ( a-plane) [37]. We\nFIG. 2. (a) Color-coded \feld dependent Im( U) parame-\nter measured on the sample as indicated in Fig. 1(d) with\n\u0012H= 0 deg. Solid red circles and wine triangles are the fre-\nquencies extracted by \ftting Eq. 7 on the data. Blue solid\nline is obtained by \ftting Eq. 1 with extracted f1values. (b)\nImaginary and (c) real part of Uexp(red symbols) and U\ft\n(black curves) for 70 mT, respectively. (d) Imaginary and (e)\nreal parts of \u001f1(red curve), \u001f2(blue curve) and their sum\n(gray curve) for 70 mT, respectively.\nassume a Gaussian beam pro\fle giving rise to a Fourier\ntransform of the beam intensity Ias\nI(k0\nx) =ek02\nxw2\n0=2(6)\nwithw0=2\nk0NA.NAis the numerical aperture of the\nlens [38, 39]. The momentum k0\nxhas a projection on\nz-direction due to the focusing of the beam.\nBroadband microwave spectroscopy data. | Field-\ndependent VNA spectra are shown in Fig. 2 (a). We\ndepict the imaginary part of the quantity U(f;H) =\niln [S21(f;H)=S21(f;H = 0)] in a color-coded plot,\nwhereS21is the complex scattering parameter measured4\nby the VNA at a given \feld, H. We identify two branches\nwhich we label f1andf2for positive \felds. For a detailed\nanalysis, we consider that the parameter Ucontains the\nsusceptibility \u001fof the sample [40, 41] and the electro-\nmagnetic response of the rf circuit used in the \rip-chip\nmethod. To account for the di\u000berent contributions, we\nfollow Refs. [40, 41] and \ft the measured Uwith\nU\ft= (7)\nC[1 +\u001f0+\u001f(f;f1;\u0001f1)ei\u001e1+\u001f(f;f2;\u0001f2)ei\u001e2]\nas shown in Fig. 2(b) and (c) (black lines). Cis a\nreal-numbered scaling parameter, \u001f0is a complex-\nnumbered o\u000bset parameter, and \u001eiare phase shift\nadjustments ( i= 1;2). Using Eq. 7, we extract the\nresonance frequencies and linewidths \u0001 fifor the two\nmodes labelled by f1andf2. In Fig. 2 (b) and (c),\nwe display the measured imaginary and real parts of\nUwith red symbols. In Fig. 2 (d) and (e), we show\nthe extracted real and imaginary parts together with\nthe total susceptibilities (gray lines) from which the\ntwo resonant modes are identi\fed. Their frequencies\nextracted for di\u000berent Hare depicted in Fig. 2 (a) with\nsolid red circles and magenta triangles. We focus on\n\felds larger than 50 mT to ensure a saturated state.\nThe blue line in Fig. 2(a) results from \ftting\nEq. 1 to branch f1with\u0018=\u0019=2. From the \ft,\nwe obtain \u00160HE= 1003:61 T,\u00160HD= 2:34 T and\n\u00160(Ha+HME) = 88:64\u0016T. Using Eq. 4 and the\nexperimental value of \u0001 f1= 0:63 GHz for branch f1, we\nextract a damping parameter of \u000b= 1:12\u000210\u00005. This\nvalue is similar to the best value reported for YIG in\nRef. [30]. The branch f2will be discussed later.\nAngle-dependent VNA spectra are shown in Fig.\n3. To enhance the signal to noise ratio, we depict\n0 90 180 270 3603691215182124Fitted Curve\n-3E-3-2E-3-1E-30E-31E-32E-33E-3\nFIG. 3. Color-coded neighbor subtracted angle dependent\nVNA-FMR spectra measured on the sample as indicated in\nFig. 1(d) with \u00160H= 90 mT. Orange curve depicts Eq. 1 by\nextracted material parameters.\n\u0001jS12j=jS12(H+ \u0001H)\u0000S12(H)j. For such data, a\nzero-crossing (highlighted by the red curve) representsthe resonance frequency. The spectra show a 2-fold\nsymmetry as expected for the c-axis being in the plane\nof the applied magnetic \felds. Introducing the extracted\nparameters discussed above, Eq. 1 models well the\nresonance frequency of branch f1as a function of\nangle\u0012H(red line). The angular dependency is hence\nconsistent with the e\u000bective anisotropy \feld extracted\nfrom the \feld-dependent data.\nBLS data. | The BLS spectra for di\u000berent values of\nincident angle, ', are shown in Fig. 4 (a). Black sym-\nbols in Fig. 4 (b) depict the frequency position of the\nmaximum BLS peak as a function of wave vector calcu-\nlated from the incidence angles shown in the legend of\nFig. 4 (a). The red curve in Fig. 4 (b) is obtained by\nconsidering \frst the material parameters obtained from\nVNA measurements and then \ftting Eq. 5 to the black\nsymbols in the same graph. We obtain a dispersion co-\ne\u000ecient of\u00160A= 9:153\u000210\u00006T.\u0016m2which leads to an\nexchange sti\u000bness length of le= 0:955\u0017A. Using Eq. 5\nand the obtained parameters we calculate the SW group\nvelocity,vg, for\u00160H= 90 mT [blue line in Fig. 4 (c)].\nThe velocity vgincreases signi\fcantly with kand lev-\nels o\u000b at 23.3 km/s. Such a high group velocity is the\ndirect result of the strong exchange interaction and at\nthe same time vanishingly small net magnetization. Mi-\ncrostructured CPWs used in magnonics o\u000ber spin-wave\nwave vectors around k= 2:5 rad/\u0016m [23, 31]. For such a\nvaluek, the qFM spin wave in hematite exhibits a group\nvelocity ofvg= 10 km/s similar to SWs in thick YIG and\nabout a factor of 10 larger compared to ultra-thin YIG\n[42]. The decay length of the SWs is given by ld=vg\u001c,\nwhere\u001c= (1=2\u0019\u000bfm(k)) is the relaxation time of the\nSW. For 30 mT applied \feld and k= 2:5 rad/\u0016m we\ncalculatevg= 13:4 km/s and \u001c= 840:6 ns which leads\ntold= 11:3 mm, again similar to thick YIG. This is due\nto low damping and high group velocity of the qFM spin\nwaves in hematite.\nDespite the small damping of the hematite sample\nmeasured by the VNA, the resonance peaks of the qFM\nmode in the BLS spectra show a large width of 10 to 20\nGHz. Furthermore, the BLS peak taken at nearly nor-\nmal incidence of the laser beam (e.g. at '= 0 or 5 deg)\nshows a strong asymmetric shape. As 'increases, the\nintensity of the prominent peak reduces and it becomes\nmore and more symmetric. To understand these observa-\ntions, we calculated the partial density of states (pDOS)\nby considering the magnon dispersion relation and the\nGaussian beam pro\fle according to\npDOS (f;') = (8)\n\u00001\n\u0019Z+1\n\u00001dk0\nxcos'Im\"\nI(k0\nx)\nf\u0000fm(k) +i\u0001f#\n:\nAssuming only fully back re\rected light we have\nk= 2(k0sin'+k0\nxcos'). \u0001fis the resonance broaden-5\n255075100200400600800100012001400160018002000\n 0 deg \n5 deg \n10 deg \n15 deg \n20 deg \n25 deg \n30 deg \n35 deg 40 deg \n45 deg \n50 deg \n55 deg \n60 deg \n65 deg \n70 deg(a)( b)(\nc)(\nd)05101520250510152025\n153045607590 \n \nFIG. 4. (a) BLS spectra for di\u000berent incident angles, ', mea-\nsured as indicated in Fig. 1(e) with \u00160H= 90 mT. (b) Ex-\ntracted peak maxima frequency (black squares) as a function\nof corresponding transferred wave vector, kand \ftted disper-\nsion relation (red curve) by Eq. 5. (c) Group velocity (d)\npartial density of states and obtained from \ftted magnon dis-\npersion at\u00160H= 90 mT. Inset in (d) depicts pDOS for '= 0\ndegree.\ning due to the damping given by Eq. 4. The numerical\naperture of the objective lens was NA = 0:18. The\ncalculated pDOS for di\u000berent incident angles is plotted\nin Fig. 4 (d). Comparing Figs. 4(a) and (c), the\nexperimental BLS peaks are broad because they contain\na certain frequency regime of the band structure which\nis determined by the Gaussian wave vector distribu-\ntion ofk= 2k0sin'+ \u0001k. The central wave vector\niskc= 2k0sin'and the distribution function for\n\u0001k= 2k0\nxcos'is given by I(k0\nx) (Eq. 6). We attribute\nthe broad BLS peaks hence to the high group velocity\nof magnons which leads to a peak width proportional to\nvg\u0002\u0001k. The asymmetry of BLS peaks for small 'can\nbe understood in terms of the Van Hove singularity of\nthe pDOS near k= 0. As'increases, the part of the\nband structure which is relevant for the collected light\nshifts away from k= 0. Consequently, the peak gets\nmore symmetric with '. The peak intensity reduces\nwith'due to the factor cos 'in Eq. 8. We note that\nEq. 8 does not include all possible scattering processes.Still it provides a good qualitative understanding about\nthe contributions giving rise to the characteristics shape\nand signal strength of BLS peaks as a function of '.\nDiscussion. | The branch of mode f2in Fig. 2(a)\nis higher by about \u000ef= 1 GHz than f1at the same\n\feldH. A second branch was reported also in Ref. 43.\nHere the authors studied the qAFM mode at zero \feld\nand assumed a distribution of magnetic domains. We\napplied a large enough magnetic \feld to avoid domains.\nA domain formation can not explain our second branch.\nAnother possibility for f2is a standing wave along the\nthickness of the sample or a SW excited with a discrete\nwave vector coming from the CPW. However, for these\ncases a quantitative estimate based on the magnon\ndispersion obtained with BLS (Eq. 5) led to a frequency\nseparation of much smaller than 1 GHz.\nThe remaining explanation for f2is a nonuniform\nmagnetoelastic \feld in the sample induced when \fxing\nthe sample on the CPW. We attribute the observed\nfrequency splitting to di\u000berent strains in di\u000berent parts\nof the sample that is either in contact with the CPW\nconductors or \roating on the CPW gaps. A di\u000berence of\n\u000eHME= 24\u0016T would account for \u000ef=f2\u0000f1= 1 GHz.\nThe e\u000bect of magnetoelastic interaction by unidirec-\ntional compression, p, in the basal plane of the crystal,\non the qFM mode is given by replacing HMEwith\nH0\nME=HME\u0000Rpcos 2 [44, 45]. Here, R= 287\n\u0016T/bar [45] is a coe\u000ecient determined by the elastic and\nmagnetoelastic parameters of the crystal and  =\u0019=2\nis the angle between pandH. Using\u000eHME=Rp\nwe obtainp= 0:084 bar. This value corresponds to a\nforce of 14.6 mN on the parts of the crystal that are in\ncontact with the CPW conductors. The evaluated force\nis one order of magnitude smaller than the weight of our\nsample and indicates the known sensitivity of hematite\ntowards magnetoelastic e\u000bects.\nFrom our experiments, we do not determine Haand\nHMEseparately as they enter Eq. 1 in the same way.\nTo separate the small HafromHMEan angle dependent\nVNA measurement with the magnetic \feld applied\nin thec-plane of the crystal would be required. The\nexisting sample was not suitable for that.\nOur BLS data were acquired on a seven times larger\nwave vector regime compared to the data used in Ref. 23\nwhere, the authors extracted an exchange sti\u000bness length\nleof 1:2\u0017A. From our extended dataset, we evaluate\nthe valuele= 0:96\u0017A. The precise determination of le\nis of key importance as it determines the SW group\nvelocity. Contrary to the ferrimagnetic YIG, dipolar\ne\u000bects are not expected to play an important role in SW\ndispersion of hematite. Hence, hematite thin \flms can\nprovide similarly large spin-wave velocities as reported\nhere for the bulk crystal. As a consequence, they\npotentially outperform YIG thin \flms concerning speed\nand decay lengths at wave vectors which are realized by6\nthe state-of-the-art transducers. Furthermore, hematite\nis based on earth abundant elements and as an end\nproduct of oxidation of magnetite a stable natural\nmineral suggesting sustainable synthesis routes.\nConclusion. | We have measured the magnon dis-\npersion relation of hematite for wave vectors kwhich\nare relevant for timely experiments in magnonics. The\ndamping coe\u000ecient of the studied natural crystal was\n1:1\u000210\u00005at room temperature. This value is only 40 %\nlarger than the best value reported for pure hematite\nand is already as good as the best YIG. The estimated\nspin-wave decay length for k= 2:5 rad/\u0016m is larger than\n1 cm in a small magnetic \feld. In optimized thin \flms,\ncurrent microwave-to-magnon transducers are expected\nto achieve larger group velocities in hematite than in\nYIG. The reported properties suggest that hematite can\nbecome the fruit \ry of sustainable modern magnonics.\nNote added. | While completing the manuscript\nabout our experiments on hematite [22], we became\naware of Ref. 23. The authors measured group velocities\nand the spin-wave dispersion via integrated CPWs for\n1 rad=\u0016m\u0014k\u00143:5 rad=\u0016m. Our BLS measurements\ncover a seven times larger regime of kvalues enabling an\nimproved evaluation of the parameter le. This parameter\nis decisive to estimate the saturation velocity of GHz\nSWs in hematite.\nThe scienti\fc colour maps developed by Crameri et.\nal. 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Whereas a behavior of the magn etization curve in higher\nmagnetic fields can be understood within a process for the sep arate ferrimagnetic\nchain, an appearance of the singlet plateau at lower fields is an example of non-Lieb-\nMattis typeferrimagnetism. By usingthe exact diagonaliza tion technique for a finite\nclusters of sizes 4 ×8 and 4×10 we show that the interchain frustration coupling\nplays an essential role in stabilization of the singlet phas e. These results are comple-\nmented by an analysis of four cylindrically coupled ferrima gnetic (1,1/2) chains via\nan abelian bosonization technique and an effective theory bas ed on the XXZ spin-\n1/2 Heisenberg model when the interchain interactions are s ufficiently weak/strong,\nrespectively.\nI. INTRODUCTION\nDuring the last fifteen years, two-dimensional (2D) quantum spin s ystems have attracted\na lot of attention both from theoretical and experimental physicis ts. A competition between\nconventional classically ordered phases and more exotic quantum o rdered phases lies in\nthe focus of the investigations. Magnetic systems with a finite corr elation length at zero\ntemperatureandafinitespingapabovethesingletgroundstate, s pinliquids, realizeHaldane2\nprediction at the level of two space dimensions1.\nTo date, one can distinguish two main routes in studies of 2D spin gap c ompounds. A\nformation of spin gap in spin dimer systems, for example SrCu 2(BO3)22and CaV 4O93, is\nexplained by a modified exchange topology similar to Shastry-Suther land lattice4. Another\nway to increase quantum fluctuations and stabilize a spin liquid ground state is realized\nin kagome antiferromagnet5. Experimental candidates for 2D kagome antiferromagnets are\ncurrently available: herbertsmithite6–8and volborthite9,10. Both these strategies deal with\nantiferromagnetic compounds. In view of this, an observation of a singlet ground state with\na pronounced spin gap in 2D ferrimagnetic material BIPNNBNO seems exotic12.\nThe crystal structure of BIPNNBNO is shown in Fig. 1. A magnetic un it of the spin\nsystem presents organic triradical BIPNNBNO. Each of the molecu les includes three s=1/2\na\nb\nFIG. 1: Magnetic model of BIPNNBNO crystal. The black (white ) circles denote spins S=1\n(s=1/2).\nspins (see Fig. 2) with intramolecular ferromagnetic JFand antiferromagnetic JAFinterac-\ntions. The magnitude of |JF| ∼300 K is very large, and two spins coupled ferromagnetically\nbehave as a S - 1 moiety. Ferrimagnetic chains are stretched along t he b-axis. There are two\nkinds of antiferromagnetic interchain interactions along the a-axis . One is between the s-1/2\nspins, which connects the nearest neighboring chains. The other is between S-1 species,\nwhich connects the next nearest neighboring chains and introduce s spin frustration.3\nS=1S=1/2 S=1/2\nJAFJAFJAF\nJFN+\nO- NO\nN N\nO OJAFJAF\nJF\nFIG. 2: Molecular structure of BIPNNBNO and the elementary m agnetic cell at JF≫ |JAF|.\nThe puzzle is following. It iswell known that a low-energy physics of an isolated ferrimag-\nnetic (S,s) chain corresponds to a gapless (S-s) ferromagnet13. Quite predictably one might\nexpect an appearance of an ordered state with fluctuations in the form of spin waves near\nthe classical state. However, measurements of magnetization sh ows that an opening of a gap\nby analogy with the Haldane chain is likely scenario. Such a behavior is a m anifestation of\nnon-Lieb-Mattis type ferrimagnetism11. Namely, the magnetization measured at 400 mK is\nnearly zero below 4.5 T, increases rapidly above 4.5 T and exhibits a bro ad 1/3 plateau and\na narrow 2/3 one at 7-23 T and around 26 T, respectively. Above 29 T, the magnetization\nis completely saturated12.\nThepurposeofthepaperistoinvestigatethemagnetizationproce ss. Theproblemiscom-\nplicated by a lack of reliable information about intra- and interchain ex change interactions.\nSo, before studying of the 2D ferrimagnetic system BIPNNBNO we d evelop a simple quan-\ntum mechanical approach that models a magnetization process of t he ferrimagnetic chain\n(1,1/2). The treatment agrees qualitatively with a predictions of th e theory for quantum\nspin chains14and provides reasonable estimations of the exchange intrachain couplings. In\naddition, it captures a peculiarity of a magnetization process in the p rototype 2D material,\ni.e. an appearance of the intermediate 2/3 plateau. Given these est imations we examine the\nmagnetization process in BIPNNBNO by analyzing exact diagonalizatio n (ED) calculations\nfor a finite clusters of size N= 32 andN= 40. A main conclusion to be drawn from these\ncalculations that an emergence of the anomalous singlet plateau is a c onsequence of the\nfrustrating interchain interaction.\nTwo different mechanisms of formation of the plateau may be likely can didates: a gener-4\nalization of Haldanes conjecture to the weakly coupled ferrimagnet ic chains, and a valence-\nbond (dimerized) type ground state in the strong-coupling limit. To d etermine what of these\nscenarios is relevant we develop low-energy effective theories for t he 4-legs spin tube, which\nforms a minimal setup including the interchain couplings. In the regime of weakly coupled\nspin tube legs we apply abelian bosonization technique. The opposite lim it of a strong-ring\ninteraction is analyzed in terms of an effective Heisenberg XXZ model where the intrachain\ncoupling is perturbatively taken into account. Our analytical treat ment shows that only the\nfirst approach confirms an important role of frustration in stabiliza tion of the singlet phase.\nNotethatastudyofspintubesisofinterest byitselfbecauseboth offrustrationandquan-\ntum fluctuation arestrong15. Our model isdirectly related with thecompound BIPNNBNO,\nbut the main results are expected to apply to other frustrated sp in tubes as well. Re-\ncently, it has been reported that the experimental candidate for the four-leg spin tube,\nCu2Cl4·D8C4SO2, is available16.\nThe paper is organized as follows: In Sec. II we consider a magnetiza tion process in the\nferrimagnetic chain (1 ,1/2). In Sec. III we discuss results of magnetization process in two-\ndimensional ferrimagnetic system obtained via the exact diagonaliza tion method on a finite\ncluster. In Sec. IV we derive effective low-energy spin-1/2 Hamilton ian. A bosonisation\nstudy of the spin tube is carried out in Sec. V. A discussion of these r esults is relegated to\nthe Conclusion part.\nII. MAGNETIZATION OF AN ISOLATED FERRIMAGNETIC CHAIN\nThe issue that we address below is whether a calculation for an isolate d ferrimagnetic\nchain partially reproduces features of the magnetization curve ob served in the BIPNNBNO\ncrystal. We demonstrate that both the 1/3 plateau and the 2/3 pla teau can be recovered\nwithin a simple quantum mechanical analysis of a magnetization proces s of an isolated\nquantum (1 ,1/2) ferrimagnetic chain under an applied magnetic field. This enables to\nestimate the intrachain exchange parameters JAFandJ1.5\nS=N/2\nS=N/2+1\nS=N\nS=N-1\nS=N+1\nS=3N/2-1\nS=3N/2\nFIG. 3: Schematic picture of states of the ferrimagnetic cha in used in construction of a magneti-\nzation curve. Excited blocks are marked by the gray shadow.\nWe start from the values of the critical fields derived from the ED da ta17\n\n\nB1=∂ε\n∂m/vextendsingle/vextendsingle\nm→1/3+0≈E(N/2+1)−E(N/2),\nB2=∂ε\n∂m/vextendsingle/vextendsingle\nm→2/3−0≈E(N)−E(N−1),\nB3=∂ε\n∂m/vextendsingle/vextendsingle\nm→2/3+0≈E(N+1)−E(N),\nBsat=∂ε\n∂m/vextendsingle/vextendsingle\nm→1−0≈E(3N/2)−E(3N/2−1),(1)\nwhereε,mare the energy and the magnetization per an elementary cell of the (1,1/2)\nferrimagnetic chain, Nis a number of the elementary cells.\nTo estimate the energies E(S) in the right-hand side of Eqs.(1), where Sis a total spin\nof the chain, we use the Hamiltonian of the one-dimensional quantum (1,1/2) ferrimagnet\nˆHc=JAFN/summationdisplay\ni=1/vectorS1i/vector s2i+J1N/summationdisplay\ni=1/vector s2i/vectorS1i+1(S1= 1, s2= 1/2), (2)6\nand construct the required quantum states |SM/an}bracketri}htwith the given quantum numbers of the\ntotal spinSand itsz-projection M.\nWe suppose that the 1 /3 magnetization plateau corresponds to the ground state of the\n(1,1/2) ferrimagnetic chain with S=N/2. The wave function of the polarized state is given\nby\n|N/2,N/2/an}bracketri}ht=N/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\ni, (3)\nand presents a direct product of the spin states of the magnetic e lementary cells (1,1/2) [see\nFig. 3]. The corresponding energy eigenvalue equals to\nE(N/2) =−JAFN−1\n9J1N. (4)\nWith an increasing of a magnetic field the ground state (3) is destroy ing and the state\nwithS=N/2 + 1 stabilizes. A low-lying excitation may be qualitatively considered as a\nforming of one triplet bond. The trial new wave function is\n|N/2+1,N/2+1/an}bracketri}ht=1√\nNN/summationdisplay\nk=1\nN/productdisplay\ni(/negationslash=k)=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\ni\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\nk=N/summationdisplay\nk=1αkΨk.(5)\nIt is composed from all arrangements of the excited block within the chain taken with equal\nweightsαk.\nBy introducing the state and calculating the matrix element\n/an}bracketle{tΨk|ˆHc|Ψk′/an}bracketri}ht=/bracketleftbigg\nE(N/2)+3\n2JAF+7\n18J1/bracketrightbigg\nδkk′−1\n3J1δk,k′±1 (6)\none obtains the relationship for the coefficients αk\n−1\n3J1αk−1+/bracketleftbigg\nE(N/2)+3\n2JAF+7\n18J1−E(N/2+1)/bracketrightbigg\nαk−1\n3J1αk+1= 0,(7)\nwhich is tantamount to\nE(N/2+1) =E(N/2)+3\n2JAF+7\n18J1−1\n3J1/parenleftbiggαk−1\nαk+αk+1\nαk/parenrightbigg\n. (8)\nThis expression includes two independent variational parameters αk−1/αkandαk+1/αk. The\nminimal value\nEmin(N/2+1) =E(N/2)+3\n2JAF−5\n18J1 (9)7\nis reached provided αk−1/αk=αk+1/αk= 1. This yields the critical magnetic field B1\ndestroying the 1 /3 plateau\nB1=3\n2JAF−5\n18J1. (10)\nTo find the critical fields B2andB3of the beginning and the end of the 2/3 plateau,\nrespectively, we construct the trial states\n|N,N/an}bracketri}ht=N/2/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i, (11)\n|N−1,N−1/an}bracketri}ht=1√\nNN/2/summationdisplay\nk=1/bracketleftBiggk−1/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i/bracketrightBigg\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2k−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2k\nN/2/productdisplay\ni=k+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i\n,(12)\n|N+1,N+1/an}bracketri}ht=1√\nNN/2/summationdisplay\nk=1/bracketleftBiggk−1/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i/bracketrightBigg\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2k−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2k\nN/2/productdisplay\ni=k+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\n2i−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\n2i\n,(13)\nwhich are schematically shown in Fig. 3.\nBy the same manner we obtain\nB2=3\n2JAF+7\n18J1, B3=3\n2JAF+5\n6J1. (14)\nThe saturation field Bsatis determined with an aid of the trial wave functions\n|3N/2,3N/2/an}bracketri}ht=N/productdisplay\ni=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\ni, (15)\n|3N/2−1,3N/2−1/an}bracketri}ht=1√\nNN/summationdisplay\nk=1\nN/productdisplay\ni(/negationslash=k)=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg3\n23\n2/angbracketrightbigg\ni\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n11\n2/parenrightbigg1\n21\n2/angbracketrightbigg\nk(16)\nthat results in\nBsat=3\n2JAF+3\n2J1. (17)\nGiven the experimental estimations for the 2D BIPNNBNO system, B1∼31K,B2≈\nB3∼35K, andBsat= 39K, we obtain from Eqs.(10,17) the values of the intrachain\nexchange couplings, JAF≈21KandJ1≈3.5K. By substituting them into Eq.(14) we get8\nthe critical fields of the 2/3 plateau, B2≈32.8K(24.4T) andB3≈34.4K(25.6T). A\nqualitativebehavioroftheferrimagneticchainmagnetizationcurve builtfromthesereference\npoints is depicted in Fig. 4. We emphasize especially that an emergence of the intermediate\n2/3 plateau is not related with interchain frustration effects.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48\n/s49/s47/s51/s50/s47/s51/s77/s47/s77\n/s115/s97/s116\n/s66/s40/s84/s41\nFIG. 4: Qualitative magnetization curve of the ferrimagnet ic (1,1/2) chain.\nIII. MAGNETIZATION: EXACT DIAGONALIZATION\nnnJ\nnnnJAFJ 1J\nFIG. 5: Cluster of N= 32 sites used in the exact diagonalization. The fixed exchan ge couplings\nareJAF= 21K,J1= 3.5K,Jnn= 0.5J1= 1.75K.\nIn order to understand a role of the interchain couplings the magne tization process of\nthe BIPNNBNO ferrimagnet was examined by the variant of a numeric al diagonalization\nmethod with conservation of a total cluster spin18,19.9\nThe model Hamiltonian is given by\nˆHclust=JAF/summationdisplay\nij/vectorSi/vector sj+J1/summationdisplay\nij/vectorSi/vector sj+Jnn/summationdisplay\nij/vector si/vector sj+Jnnn/summationdisplay\nij/vectorSi/vectorSj, (18)\nwhere/vectorSi(/vector si) denotes spin-1 (spin-1/2) operator at site i. The sublattices and the network\nof the antiferromagnetic interactions, JAF,J1,JnnandJnnn, are shown in Fig. 5. We\nperform calculation of the N-step magnetization curve for the N=3 2 cluster depicted in the\nsame Figure. The intrachain parameters, JAFandJ1, have been estimated in the previous\nSection whereas the interchain ones, JnnandJnnn, are assumed to be less than J1. The open\nboundary conditions are used for the numerical calculations.\nThe magnetization process is compared with the results of the mode l of non-interacting\n(1,1/2) ferrimagnetic chains. A standard way to build magnetization curve atT= 0\nis to define the lowest energy E(N,M) of the Hamiltonian (2) in the subspace where\n/summationtextN\nj=1/parenleftbig\nSz\nj+sz\nj/parenrightbig\n=Mfor a finite system of Nelementary ( S,s) blocks. Applying a magnetic\nfieldBleads to a Zeeman splitting of the energy levels, and therefore level crossing occurs on\nincreasing the field. These level crossing correspond to jumps in th e magnetization until the\nfully polarized state is reached at a certain value of the magnetic field . The magnetization\nof four independent chains is then derived from\nm= 4M/N, M = max[M|E(N,M+1)−E(N,M)>B], (19)\nwhich gives a step curve.\nAn importance of the frustrating coupling is seen from comparison o f two magnetization\ncurves displayed in Figs. 6, 7. They correspond to no frustration c ase and a pronounced\nfrustrating coupling, respectively. The magnetization curves exh ibit several interesting fea-\ntures. For instance, the magnetization behavior in higher magnetic fields (B > B 1) is well\nreproduced within the model of non-interacting ferrimagnetic cha ins. Another remarkable\nfeature revealed by Figs. 6, 7 that the singlet ground state platea u emerges at non-zero\nfrustration interaction whereas the narrow 2 /3 plateau appears regardless of the frustration.\nWe numerically found that the width ∆ Sof the singlet plateau scales almost linearly with\naJnnnvalue (Fig. 8). To check into the case of the dependence we repeat cacluations on\na cluster of larger size, N= 40, with the same set of parameters that support the finding.\nThe observation points out that the zero magnetization plateau ha s a quantum origin with\na crucial role of frustration which destroys a long-range order an d drives the system into the10\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48\n/s50/s47/s51\n/s49/s47/s51\n/s66/s40/s84/s41/s77/s47/s77\n/s115/s97/s116\nFIG. 6: Magnetization curve for the 32-site cluster. The exc hange couplings are taken as JAF=\n21K,J1= 3.5K,Jnn= 0.5J1, andJnnn= 0 (no frustration). The dotted line marks a calculation\nvia the model of non-interacting (1 ,1/2) chains.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48/s77/s47/s77\n/s115/s97/s116\n/s66/s40/s84/s41/s49/s47/s51/s50/s47/s51\nFIG. 7: Magnetization curve for the 32-site cluster. The exc hange couplings are taken as JAF=\n21K,J1= 3.5K,Jnn= 0.5J1,Jnnn= 0.075J1. The dotted line marks a calculation via the model\nof non-interacting (1 ,1/2) chains.11\n0□00. 0□02 . 0□04 . 0□06 . 0□08 . 0□10 . 0□12 . 0□14 .0□00.0□02.0□04.0□06.0□08. /c68S/JAF\nJnnn/JAF\nFIG. 8: Value of the singlet plateau ∆ Sas a function of the frustrating coupling Jnnnobtained on\na cluster of size N= 32 (black circles) and N= 40 (white circles).\nsinglet phase. Below, we address analytically the issue in the regimes o f strong and weak\ninterchain couplings.\nIV. A FORMATION OF THE SINGLET PLATEAU\nIn low-dimensional Heisenberg systems frustrating couplings can d rive transitions to gap-\nfull quantum states, where local singlets forma groundstate. Th ese quantum gappedphases\nmay have long-rangedsinglet order (valence bondstate), or realiz ea resonating valence band\nspin liquid. In last case, a ground state is a coherent superposition o f all lattice-coverings\nby local singlets21.\nTorecognizefeaturesofthesephasesintheEDresultsweundert akeanalytical treatments\nof the four-legs spin tube shown in Fig. 9. The new system is infinite alo ng theb-axis,\nand periodic with the 4-site period along the a-axis. The tube forms a minimal setup\nincluding the interchain nearest- and next-to-nearest neighbor c ouplings and contains the\nsame number of ferrimagnetic chains parallel to the b-axis as the clu sters in the ED study.\nAs we demonstrate below, the simplified model elucidates an importan t role of frustration\nin stabilization of the singlet phase.12\n1J\nnnJ\nnnnJ\nJ\nJJ’1\n2\n3412\n34\nFIG. 9: 4-leg spin tube structure used in the limit of strong r ing coupling (up) and in the case\nof weakly interacting chains (below). The black (white) cir cles mean spin-1 (spin-1/2) sites. The\ngray circles denote renormalized spin-1/2 blocks.\nThe singlet phase may arise in the limit of strong ring coupling Jnn,Jnnn≫J1. In this\ncase, the problem can be analyzed in terms of Heisenberg XXZ model similar to ladders in a\nmagnetic field20. The opposite limit ( J1≫Jnn,Jnnn) results in a scenario of weakly interact-\ning chains. Based on a block renormalization procedure the original s ystem is then mapped\nonto the model of a spin tube with four ferromagnetic spin-1/2 legs . We mention that the\nground state properties of two-leg spin ladders with ferromagnet ic intrachain coupling and\nantiferromagnetic interchain couplings have been discussed in Refs .30,31in an absence of an\nexternal field. A magnetization process of those spin ladders with a n even number of legs\n(2 and 4) has been studied in Ref.32in the regime of weak ferromagnetic coupling along the\nlegs and strong antiferromagnetic coupling along the rungs.\nAn appearance of the singlet phase in the frustrated spin tube with four weakly coupled\nferromagnetic spin-1/2 legs can be studied through the bosonizat ion technique which proves\neffectiveness for quasi-one-dimensional spin-one-half systems. To the best of our knowledge,\nthe system has never been previously reported, however our fur ther analysis follows closely\nto that of given in Ref.24, where 4-legs spin tube with antiferromagnetic chains and a specific13\nform of diagonal rung interactions (but with no frustration) has b een treated. Note as well\nthat spin ladders with ferromagnetic and ferrimagnetic legs are muc h less studied26,27by the\nbosonization approach in comparison with ladder systems with antife rromagnetic legs. The\nmain problem arising here is that the formalism is well defined only if ther e is an easy-plane\nexchange anisotropy. In this regard, we note that measurement s of the angular dependence\nof the ESR linewidth for the BIPNNBNO system showed that the large st linewidth was\nobserved for the field direction perpendicular to the ab plane28. Due to the theoretical\nconsideration by Oshikawa and Affleck29a critical regime of XY-anisotropy is expected in\nthe compound. In addition we point out that the ED algorithm invoked in the previous\nSection enables to treat clusters of sufficiently large sizes due to us e of the rotational SU(2)\nsymmetry. The latter is broken by the anisotropy whose role in a sing let gap formation is a\nsubject of future ED studies.\nA. Spin tube: weakly interacting rings and a model of a single XXZ chain\nWe study the Hamiltonian of the spin tube (see Fig. 9)\nH=N/summationdisplay\nn=1Hring\nn+J1N/summationdisplay\nn=1(Sn,1sn+1,1+sn,2Sn+1,2+Sn,3sn+1,3+sn,4Sn+1,4)−BN/summationdisplay\nn=14/summationdisplay\ni=1/parenleftbig\nSz\nn,i+sz\nn,i/parenrightbig\n,\n(20)\nwhere the Hamiltonian of the separate ring is\nHring\nn=Jnn(sn,1sn,2+sn,2sn,3+sn,3sn,4+sn,4sn,1)+JnnnN/summationdisplay\nn=1(Sn,1Sn,3+Sn,2Sn,4).\nHere,S= 1 ands= 1/2,nis the index of the ring, Nis the total number of rings, and the\nindeximarks the (1,1/2) blocks inside the rings. Periodic boundary conditio ns along the\ntube direction are imposed. In our model it is suggested that Jnn,Jnnn≫J1.\nIn the limit J1= 0 the system decouples into a collection of nonineracting rings. At z ero\nmagnetic field the singlet and triplet states\n|ψ0/an}bracketri}ht=−√\n3\n2|00;00/an}bracketri}ht+1\n2|11;00/an}bracketri}ht,\n|ψ1/an}bracketri}ht=1√\n2|01;11/an}bracketri}ht+1√\n2|10;11/an}bracketri}ht (21)\nhave the lowest energies E0=−2Jnn/9+8Jnnn/9 andE1=−Jnn/9+8Jnnn/9, respectively.\nThestatesofthering |S12S34;SM/an}bracketri}htareobtainedviathecommonruleofadditionofmoments,14\nwhereS12(S34) is spin of dimer composed of the spins of the 1 and 2 (3 and 4) blocks. The\nsinglet and triplet states of the ring that enter into (21) are given in Appendix A.\nUpon increasing the magnetic field a transition between the singlet an d triplet states\noccurs atB=Jnn/9 and the the total magnetization jumps abruptly from zero to M=N.\nAtnon-zeroringcouplingthesharptransitionisbroadenedandsta rtsfromacriticalvalue\nB0. To find the field we derive the XXZ spin chain Hamiltonian by we using the standard\napproach that is analogous to study of spin-1/2 ladder with strong rung exchange20.\nThe Hamiltonian (20) is splitted into two parts\nH=H0+H1.\nH0=N/summationdisplay\nn=1Hring\nn−Bc4/summationdisplay\ni=1N/summationdisplay\nn=1(Sz\nin+sz\nin),\nH1=J1N/summationdisplay\nn=1[Sn,1sn+1,1+sn,2Sn+1,2+Sn,3sn+1,3+sn,4Sn+1,4]−(B−Bc)4/summationdisplay\ni=1N/summationdisplay\nn=1(Sz\nin+sz\nin),\nwhereBc=E1−E0. TheH1lifts the 2N-fold degeneracy of the ground state of the\nHamiltonian H0. The later can be either in the state |ψ0/an}bracketri}htor|ψ1/an}bracketri}ht. By using the standard\nmany body perturbation theory22the effective Hamiltonian can be derived\nHeff=Jeff\nxyN/summationdisplay\nn=1/parenleftBig\n˜Sx\nn˜Sx\nn+1+˜Sy\nn˜Sy\nn+1/parenrightBig\n+Jeff\nzN/summationdisplay\nn=1˜Sz\nn˜Sz\nn+1−Heff\nzN/summationdisplay\nn=1˜Sz\nn, (22)\nwhereJeff\nxy=−16J1/27,Jeff\nz=−J1/9 andBeff\nz=J1/9+B−Bc.\nTo get the expression the pseudo-spin ˜Si= 1/2 operators that act on the states |ψ0/an}bracketri}htand\n|ψ1/an}bracketri}htare introduced\n˜Sz\nn|ψ0/an}bracketri}htn=−1\n2|ψ0/an}bracketri}htn,˜Sz\nn|ψ1/an}bracketri}htn=1\n2|ψ1/an}bracketri}htn,\n˜S+\nn|ψ0/an}bracketri}htn=|ψ1/an}bracketri}htn,˜S+\nn|ψ1/an}bracketri}htn= 0,\n˜S−\nn|ψ0/an}bracketri}htn= 0,˜S−\nn|ψ1/an}bracketri}htn=|ψ0/an}bracketri}htn. (23)\nThe starting spin-1 operators and the pseudo-spin operators in t he restricted space are\nrelated by\nSz\nin=1\n6+1\n3˜Sz\nn,\nS+\nin= (−1)i−14\n3√\n3˜S+\nn, S−\nin= (−1)i−14\n3√\n3˜S−\nn. (24)15\nThe corresponding map for the spin-1/2 operators is\nsz\nin=−1\n24−1\n12˜Sz\nn,\ns+\nin=(−1)i\n3√\n3˜S+\nn, s−\nin=(−1)i\n3√\n3˜S−\nn. (25)\nThe Jordan-Wigner transformation maps the Hamiltonian (22) onto a system of inter-\nacting spinless fermions\nHsf=tN/summationdisplay\nn=1/parenleftbig\nc+\nici+1+c+\ni+1ci/parenrightbig\n+VN/summationdisplay\nn=1nini+1−µN/summationdisplay\nn=1ni, (26)\nwheret=Jeff\nxy/2,V=Jeff\nzandµ=Jeff\nz+Beff\nz.\nThe lowest critical field B0corresponds to that value of thechemical potential µfor which\nthe band of spinless fermions starts to fill up. This yields the conditio nµ=−2tand leads\nto the result\nB0=1\n9Jnn+16\n27J1. (27)\nThe critical value involves no frustration parameter Jnnnthat is clearly contrary to the ED\nresults.\nA similar analysis can be carried out for the saturation field. The deta ils of the calcula-\ntions are relegated to Appendix B.\nB. Spin tube: a model of weakly interacting ferromagnetic legs and abelian\nbosonization\nTo apply the bosonization we should map the initial system, consisting of two sorts of\nspins, spin-1/2 and spin-1, to the spin-1/2 system by using the qua ntum renormalization\ngroup (QRG) in real space based on the block renormalization proce dure23. To exploit\nthe real-space QRG technique one divide the spin lattice into small bloc ks, namely, the\nintrachain dimers (1,1/2), and obtains the lowest energy states {|α/an}bracketri}ht}of each isolated block.\nThe effect of inter-block interactions is then taken into account by constructing an effective\nHamiltonian Heffwhich now acts on a smaller Hilbert space embedded in the original one.\nIn this new Hilbert space each of the former blocks is treated as a sin gle site. The effective\nHamiltonians Heff=Q†HQis constructed via the projection operator Q=N/producttext\ni=1Qiwith16\nQi=m/summationtext\nα=1|α/an}bracketri}ht/an}bracketle{tα|of eachi-th block where mis the number of low energy states that are kept\nandNis a number of lattice cells.\nWe hold the lowest doublet S−1/2 to find an effective low-energy Hamiltonian. The\nhigher energy S−3/2 states are neglected. One can check that the reduced matrix ele ments\nare (S1= 1,s2= 1/2)\n/an}bracketle{t11/2;1/2||S1||11/2;1/2/an}bracketri}ht= 2/radicalbigg\n2\n3,\n/an}bracketle{t11/2;1/2||s2||11/2;1/2>=−1√\n6.\nTherefore the effective spin-1/2 operators of the renormalized c hain are\nQ†\ni/vectorS1iQi=4\n3/vectorSi, Q†\ni/vector s2iQi=−1\n3/vectorSi(S= 1/2). (28)\nThe renormalized Hamiltonian of the intrachain interactions corresp onds to the ferromag-\nnetic Heisenberg spin-1/2 model with the exchange coupling J=−4J1/9 (Fig. 9). The\ninterchain interactions between the nearest neighbors, spins -1/ 2, and next to the nearest\nneighbors, spins -1, are renormalized as J⊥=Jnn/9 andJ′\n⊥= 16Jnnn/9, respectively.\nConsider a four-legs spin tube consisting of spin-1 /2 chains. The Hamiltonian of the\nsystem is\nˆHtube=4/summationdisplay\nλ=1ˆHλ+ˆH⊥\n12+ˆH⊥\n23+ˆH⊥\n34+ˆH⊥\n14+ˆH′⊥\n13+ˆH′⊥\n24. (29)\nThe spins along the chains are coupled ferromagnetically, the Hamilto nian for the separate\nλ-th chain is\nˆHλ=−JxyN/summationdisplay\ni=1/parenleftbig\nSx\nλ,jSx\nλ,j+1+Sy\nλ,jSy\nλ,j+1/parenrightbig\n−JzN/summationdisplay\ni=1Sz\nλ,jSz\nλ,j+1,\nwhereSx,y,z\nλ,jare the spin S=1/2 operators at the jth site, the intraleg coupling is ferromag-\nnetic,J >0.\nThe interaction parts are given by\nˆH⊥\nλλ′=Jxy\n⊥,λλ′N/summationdisplay\nj=1/parenleftbig\nSx\nλ,jSx\nλ′,j+Sy\nλ,jSy\nλ′,j/parenrightbig\n+Jz\n⊥,λλ′N/summationdisplay\ni=1Sz\nλ,jSz\nλ,j, (30)\nˆH′⊥\nλλ′=J′xy\n⊥,λλ′N/summationdisplay\nj=1/parenleftbig\nSx\nλ,jSx\nλ′,j+Sy\nλ,jSy\nλ′,j/parenrightbig\n+J′z\n⊥,λλ′N/summationdisplay\ni=1Sz\nλ,jSz\nλ,j (31)17\nandincludes thenearest, J⊥>0, andthenext-to-nearest, J′\n⊥>0, antiferromagneticinterleg\ncouplings.\nThe unitary transformation keeping spin commutation relations\nSx,y\nλ,j→(−1)jSx,y\nλ,j, Sz\nλ,j→Sz\nλ,j\nmaps the Hamiltonian (29) to the Hamiltonian with antiferromagnetic le gs. It changes\nJxy→ −JxyandJz→Jz, and the ferromagnetic isotropic point is ∆ = Jz/Jxy=−1 in\nthe Hamiltonian with the antiferromagnetic legs.\nFollowing the general procedure of transforming a spin model to an effective model of\ncontinuum field, we convert the spin Hamiltonian of the spin tube with antiferromagnetic\nlegs to a Hamiltonian of spinless fermions using Jordan-Wigner transf ormation, then map\nit to a modified Luttinger model. The bosonic expressions for spin ope rators are\nS+\nλ(x) =S+\njλ\na=e−i√πΘλ\n√\n2πa/bracketleftBig\ne−i(πx/a)+cos/parenleftBig√\n4πΦλ/parenrightBig/bracketrightBig\n,\nSz\nλ(x) =Sz\njλ\na=1√π∂xΦλ+1\nπaei(πx/a)sin/parenleftBig√\n4πΦλ/parenrightBig\n, (32)\nwhere Φ and Θ are the bosonic dual fields, and xis defined on the lattice, xj=ja,ais a\nshort-distance cutoff.\nThe bosonized form of the Hamiltonian of the non-interacting chains is\nHλ=u\n2/integraldisplay\ndx/bracketleftbigg\nKΠ2\nλ+1\nK(∂xΦλ)2/bracketrightbigg\n, (33)\nwhere Π λ(x) =∂xΘλis canonically conjugate momentum to Φ λ. The Luttinger liquid\nparameters are fixed from the Bethe ansatz solution33\nK=π\n2(π−arccos∆), u=Jxyπ√\n1−∆2\n2arccos∆. (34)\nThe velocity uvanishes and Kdiverges for ∆ = −1. This corresponds to the ferromagnetic\ninstability point of a single chain.\nThe interchain interactions (30) between the nearest neighbor ch ains reads as\nH⊥\nλλ′=Jz\n⊥,λλ′/integraldisplaydx\nπ(∂xΦλ)(∂xΦλ′)+g1/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4π(Φλ+Φλ′)/parenrightBig\n+g2/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4π(Φλ−Φλ′)/parenrightBig\n+g3/integraldisplaydx\n(2πa)2cos/parenleftbig√π(Θλ−Θλ′)/parenrightbig18\n+g4/integraldisplaydx\n(2πa)2cos/parenleftbig√π(Θλ−Θλ′)/parenrightbig\ncos/parenleftBig√\n4π(Φλ+Φλ′)/parenrightBig\n+g5/integraldisplaydx\n(2πa)2cos/parenleftbig√π(Θλ−Θλ′)/parenrightbig\ncos/parenleftBig√\n4π(Φλ−Φλ′)/parenrightBig\n, (35)\nwhereg1=−2Jz\n⊥,λλ′,g2= 2Jz\n⊥,λλ′,g3= 2πJxy\n⊥,λλ′,g4=g5=πJxy\n⊥,λλ′. The Hamiltonian (31)\nof the next-to-nearest couplings has a similar form with a formal ch angeJz\n⊥,λλ′→J′z\n⊥,λλ′,\ng1→g′\n1=−2J′z\n⊥,λλ′,g2→g′\n2= 2J′z\n⊥,λλ′etc.\nFollowing the route of Refs.24,25it is convenient to introduce a symmetric mode Φ sand\nthree antisymmetric ones Φ a1, Φa2, Φa3\nΦs=1\n2(Φ1+Φ2+Φ3+Φ4),\nΦa1=1\n2(Φ1+Φ2−Φ3−Φ4),\nΦa2=1\n2(Φ1−Φ2−Φ3+Φ4),\nΦa3=1\n2(Φ1−Φ2+Φ3−Φ4).(36)\nIn terms of the new fields the quadratic part of the Hamiltonian (29) is diagonalized to\nH0=us\n2/integraldisplay\ndx/bracketleftbigg\nKsΠ2\ns+1\nKs(∂xΦs)2/bracketrightbigg\n+3/summationdisplay\ni=1uai\n2/integraldisplay\ndx/bracketleftbigg\nKaiΠ2\nai+1\nKai(∂xΦai)2/bracketrightbigg\n(37)\nwith\nus=u/parenleftbigg\n1+2KJz\n⊥\nuπ+KJ′z\n⊥\nuπ/parenrightbigg1\n2\n, K s=K/parenleftbigg\n1+2KJz\n⊥\nuπ+KJ′z\n⊥\nuπ/parenrightbigg−1\n2\n,\nua1=ua2=u/parenleftbigg\n1−KJ′z\n⊥\nuπ/parenrightbigg1\n2\n, K a1=Ka2=K/parenleftbigg\n1−KJ′z\n⊥\nuπ/parenrightbigg−1\n2\n,\nua3=u/parenleftbigg\n1−2KJz\n⊥\nuπ+KJ′z\n⊥\nuπ/parenrightbigg1\n2\n, K a3=K/parenleftbigg\n1−2KJz\n⊥\nuπ+KJ′z\n⊥\nuπ/parenrightbigg−1\n2\n.(38)\nThe relevant and marginally relevant terms of the interchain coupling s are given by\nHint= 2g12/summationdisplay\ni=1/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4πΦs/parenrightBig\ncos/parenleftBig√\n4πΦai/parenrightBig\n+2g′\n1/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4πΦs/parenrightBig\ncos/parenleftBig√\n4πΦa3/parenrightBig\n+2g22/summationdisplay\ni=1/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4πΦai/parenrightBig\ncos/parenleftBig√\n4πΦa3/parenrightBig\n+2g′\n2/integraldisplaydx\n(2πa)2cos/parenleftBig√\n4πΦa1/parenrightBig\ncos/parenleftBig√\n4πΦa2/parenrightBig\n+2g32/summationdisplay\ni=1/integraldisplaydx\n(2πa)2cos/parenleftbig√πΘai/parenrightbig\ncos/parenleftbig√πΘa3/parenrightbig\n+2g′\n3/integraldisplaydx\n(2πa)2cos/parenleftbig√πΘa1/parenrightbig\ncos/parenleftbig√πΘa2/parenrightbig\n+g4/integraldisplaydx\n(2πa)2/bracketleftBig\ncos/parenleftbig√π(Θa2+Θa3)/parenrightbig\ncos/parenleftBig√\n4π(Φs+Φa1)/parenrightBig19\n+cos/parenleftbig√π(Θa1−Θa3)/parenrightbig\ncos/parenleftBig√\n4π(Φs−Φa2)/parenrightBig\n+cos/parenleftbig√π(Θa2−Θa3)/parenrightbig\ncos/parenleftBig√\n4π(Φs−Φa1)/parenrightBig\n+cos/parenleftbig√π(Θa1+Θa3)/parenrightbig\ncos/parenleftBig√\n4π(Φs+Φa2)/parenrightBig/bracketrightBig\n+g′\n4/integraldisplaydx\n(2πa)2/bracketleftBig\ncos/parenleftbig√π(Θa1+Θa2)/parenrightbig\ncos/parenleftBig√\n4π(Φs+Φa3)/parenrightBig\n+cos/parenleftbig√π(Θa1−Θa2)/parenrightbig\ncos/parenleftBig√\n4π(Φs−Φa3)/parenrightBig/bracketrightBig\n. (39)\nTheg5terms are irrelevant and are omitted.\nThe Hamiltonian (37) describes four independent gapless spin-1 /2 chains coupled by the\ninterchain interaction in the form of Eq.(39). It is expected that th e interleg coupling results\nin the Haldane gap in the excitation spectrum. Note that in the vicinity of the single chain\nferromagnetic instability, ∆ = −1, the effective bandwidth collapses, u→0, and the effect\nof the interleg couplings becomes crucial. To find a detail behavior of the gap in the phase\ndiagram with antiferromagnetic ( J⊥,J′\n⊥>0) interleg coupling and ferromagnetic leg regime\n(∆<0) we use the renormalization group analysis.\nThe RG equations are derived through the standard technique (se e Ref.34, for example).\nThe result is\ndg1\ndl= [2−(Ks+Ka1)]g1,\ndg′\n1\ndl= [2−(Ks+Ka3)]g′\n1,\ndg2\ndl= [2−(Ka3+Ka1)]g2,\ndg′\n2\ndl= [2−2Ka1]g′\n2,\ndg3\ndl=/bracketleftbigg\n2−1\n4/parenleftbigg1\nKa1+1\nKa3/parenrightbigg/bracketrightbigg\ng3,\ndg′\n3\ndl=/bracketleftbigg\n2−1\n2Ka1/bracketrightbigg\ng′\n3,\ndg4\ndl=/bracketleftbigg\n2−/parenleftbigg\nKs+Ka1+1\n4Ka1+1\n4Ka3/parenrightbigg/bracketrightbigg\ng4,\ndg′\n4\ndl=/bracketleftbigg\n2−/parenleftbigg\nKs+Ka3+1\n2Ka1/parenrightbigg/bracketrightbigg\ng′\n4,\ndKs\ndl=−4g2\n1/parenleftbiggKs\n4πus/parenrightbigg2\n−2g′\n12/parenleftbiggKs\n4πus/parenrightbigg2\n−2g2\n4/parenleftbiggKs\n4πus/parenrightbigg2\n−g′\n42/parenleftbiggKs\n4πus/parenrightbigg2\n,\ndKa1\ndl=−1\n2g2\n1/parenleftbiggKa1\n2πua1/parenrightbigg2\n−1\n2g2\n2/parenleftbiggKa1\n2πua1/parenrightbigg2\n−1\n2g′\n22/parenleftbiggKa1\n2πua1/parenrightbigg2\n−1\n4g2\n4/parenleftbiggKa1\n2πua1/parenrightbigg220\n+1\n2/parenleftbiggg3\n4πua1/parenrightbigg2\n+1\n2/parenleftbiggg′\n3\n4πua1/parenrightbigg2\n+1\n4/parenleftbiggg4\n4πua1/parenrightbigg2\n+1\n4/parenleftbiggg′\n4\n4πua1/parenrightbigg2\n,\ndKa3\ndl=−1\n2g′\n12/parenleftbiggKa3\n2πua3/parenrightbigg2\n−g2\n2/parenleftbiggKa3\n2πua3/parenrightbigg2\n−1\n4g′\n42/parenleftbiggKa3\n2πua3/parenrightbigg2\n+/parenleftbiggg3\n4πua3/parenrightbigg2\n+1\n2/parenleftbiggg4\n4πua3/parenrightbigg2\n. (40)\nOne sees that the g1terms are relevant for Ks+Ka1<2; theg′\n1term is relevant for\nKs+Ka3<2; theg2terms are relevant for Ka3+Ka1<2; theg′\n2term is relevant for\nKa1<1; theg3term is relevant for K−1\na1+K−1\na3<8; theg′\n3term is relevant for Ka1>1/4.\nDespite the g4andg′\n4terms are irrelevant they are the most relevant terms which couple\nthe symmetric and antisymmetric modes24.\nUsing the RG equations the behavior of the gap in the whole phase diag ram can be\nestablished. Following to standard routine, we analyze the effect of the transversal ( Jxy\n⊥)\nand the longitudinal ( Jz\n⊥) parts of the interleg coupling separately.\n1. Transversal part of the interleg interactions\nIn this case Jxy\n⊥,J′xy\n⊥/ne}ationslash= 0 andJz\n⊥,J′z\n⊥= 0, the initial values of the coupling constants are\ngiven byg1(l= 0) =g′\n1(l= 0) = 0,g2(l= 0) =g′\n2(l= 0) = 0,g3(l= 0) = 2πJxy\n⊥,g′\n3(l= 0) =\n2πJ′xy\n⊥,g4(l= 0) =πJxy\n⊥andg′\n4(l= 0) =πJ′xy\n⊥. The bare Luttinger parameters are us=u,\nua1=ua2=ua3=u, andKs(l= 0) =K,Ka1(l= 0) =Ka2(l= 0) =Ka3(l= 0) =K.\nThe termg3is relevant for −1≤∆≤0 while the g4term is irrelevant. It is easily checked\nnumerically that g3,g′\n3grow whereas g4,g′\n4decrease under the RG. It means that Θ 1, Θ2, Θ3\nare locked in one of the vacuum states (Θ 1= Θ2= 0, Θ 3=√πor Θ1= Θ2=√π, Θ3= 0\nprovidedJxy\n⊥>J′xy\n⊥), fluctuations of the fields Θ ai(i= 1,2,3) are completely suppressed.\nTherefore arbitrary Jxy\n⊥> J′xy\n⊥/ne}ationslash= 0 generate a gap in the antisymmetric modes (Θ iare\npinned, disordered).\nAfter the fluctuations of the fields Θ iare stopped, the infrared behavior of the symmetric\nmode is governed by the term of the coupling with the antisymmetric m odes\n˜Hint= 2˜g42/summationdisplay\ni=1/integraldisplaydx\n(2πa)2cos(√\n4πΦs)cos(√\n4πΦai)\n+2˜g′\n4/integraldisplaydx\n(2πa)2cos(√\n4πΦs)cos(√\n4πΦa3). (41)21\nwhere\n˜g4= ¯g4/an}bracketle{tcos/parenleftbig√π[Θa1+Θa3]/parenrightbig\n/an}bracketri}ht= ¯g4/an}bracketle{tcos/parenleftbig√π[Θa2+Θa3]/parenrightbig\n/an}bracketri}ht,\n˜g′\n4= ¯g′\n4/an}bracketle{tcos/parenleftbig√π(Θa1+Θa2)/parenrightbig\n/an}bracketri}ht,\nand ¯g4, ¯g′\n4are renormalized couplings provided g3,g′\n3=O(1). Here, the invariance of ˜Hint\ngiven by Eq.(39) under Θ ai→ −Θaiyields/an}bracketle{tcos/parenleftbig√π(Θai−Θaj)/parenrightbig\n/an}bracketri}ht=/an}bracketle{tcos/parenleftbig√π(Θai+Θaj)/parenrightbig\n/an}bracketri}ht.\nDespiteeiΦaihas exponentially decaying correlations due to the Θ aiare pinned, a scrupulous\nanalysis24shows that the effective Hamiltonian for Φ spresents a standard sine-Gordon\nHamiltonian\n˜Heff=us\n2/integraldisplay\ndx/bracketleftbigg\n¯KsΠ2\ns+1\n¯Ks(∂xΦs)2/bracketrightbigg\n+g/integraldisplaydx\n(2πa)2cos/parenleftBig√\n16πΦs/parenrightBig\n,(42)\nwhere¯Ksis a renormalized value of K, andgis a new effective coupling constant. From\nthe correlation function /an}bracketle{texp/bracketleftbig\ni√\n16πΦs(x)/bracketrightbig\nexp/bracketleftbig\ni√\n16πΦs(y)/bracketrightbig\n/an}bracketri}ht= (a2/|x−y|2)4¯Ksit follows\nthat thegterm has a scale dimension 4 ¯Ks. Therefore, it is relevant for ¯Ks<1/2, when Φ s\nis pinned, i.e. becomes massive35.\nTo summarize, the transversal part of the interleg coupling suppo rts gapped antisymmet-\nric modes, the symmetric sector is gapped at ¯Ks<1/2, and remains gapless at ¯Ks>1/2.\nThe condition ¯Ks= 1/2 determines a boundary between the gapless Spin Liquid XY1\nphase36, and a generalization of the gapped Rung Singlets phase37for the for-leg spin tube\n(Fig.10). (Hereinafter, we retain names of phases used in the theo ry of spin ladders with\nferromagnetic legs.) In thelast case, spins onthe same rungs, or a longthe shortest diagonals\nform singlet pairs by a dynamical way.\n2. Longitudinal part of the interleg interactions\nFor the case of the longitudinal part of the interleg exchange, Jxy\n⊥,J′xy\n⊥= 0 andJz\n⊥,J′z\n⊥/ne}ationslash=\n0, the bare values of the coupling constants are given by g1(l= 0) =−2Jz\n⊥,g′\n1(l= 0) =\n−2J′z\n⊥,g2(l= 0) = 2Jz\n⊥,g′\n2(l= 0) = 2J′z\n⊥,g3(l= 0) =g′\n3(l= 0) = 0, and g4(l= 0) =g′\n4(l=\n0) = 0.\nThe strong-coupling phase diagramin the vicinity of ferromagnetic in stability point (∆ =\n−1andJz\n⊥,J′z\n⊥= 0)obtainedintheRGanalysisisshowninFig. 11. Inthesectordenot edas\nspin liquid II phase37theg1,2,g′\n1,2terms are irrelevant. The symmetric and antisymmetric\nmodes remain gapless. In the sector marked as a Haldane phase the termsg1,2,g′\n1,2are22\n/s48/s44/s48 /s48/s44/s53 /s49/s44/s48/s45/s49/s44/s48/s48/s45/s48/s44/s57/s56\n/s82/s117/s110/s103/s32/s83/s105/s110/s103/s108/s101/s116/s115\n/s74 /s47/s74/s32/s74 /s39/s47/s74 /s61/s48/s46/s50\n/s83/s112/s105/s110/s32/s76/s105/s113/s117/s105/s100/s32/s88/s89/s49/s32/s80/s104/s97/s115/s101\nFIG. 10: The ground-state phase diagram in the vicinity ∆ = −1 of the four-leg tube with\ntransverse coupling between legs.\nrelevant. Since all of the modes are coupled and locked together, b oth the symmetric and\nantisymmetric modes are gapped.\nThe phase of a ferromagnet with antiphase interchain order arises as a result of the\nferromagnetic instability with increasing interleg antiferromagnetic coupling. The boundary\nof the transition into the phase is obtained by studying the velocity r enormalization of the\ncorresponding gapless excitations. We mark the transition at uai= 0 (i= 1,2,3).\n3. Isotropic interleg exchange\nThe initial values of the coupling constants are g1(l= 0) =−2J⊥,g′\n1(l= 0) =−2J′\n⊥,\ng2(l= 0) = 2J⊥,g′\n2(l= 0) = 2J′\n⊥,g3(l= 0) = 2πJ⊥,g′\n3(l= 0) = 2πJ′\n⊥,g4(l= 0) =πJ⊥and\ng′\n4(l= 0) =πJ′\n⊥.\nFromtheRGequations(40)itisseenthatthemostrelevantoperat orsaretheg3,g′\n3terms.\nTherefore, the antisymmetric sector is gapped, and Θ aiare locked in the disordered phase.\nAs in the case of the transversal interleg interactions an effective sine-Gordon Hamiltonian\nfor the symmetric mode determines phase boundary ¯Ks= 1/2 between gapped and gapless\nphases. Numerical analysis shows that the ground state phase dia gram consists of the\ndisordered Rung Singlet gapfull phase and the stripe ferromagnet ic phase with dominating23\n/s48/s44/s48 /s48/s44/s53 /s49/s44/s48/s45/s49/s44/s48/s45/s48/s44/s56/s45/s48/s44/s54/s45/s48/s44/s52/s45/s48/s44/s50\n/s72/s97/s108/s100/s97/s110/s101/s32/s80/s104/s97/s115/s101\n/s83/s112/s105/s110/s32/s76/s105/s113/s117/s105/s100/s32/s73/s73/s32/s80/s104/s97/s115/s101/s32/s74\n/s122/s32/s39/s47/s74\n/s122/s32/s61/s32/s48/s46/s50\n/s74\n/s32/s47/s32/s74/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s32\n/s40/s97/s110/s116/s105/s112/s104/s97/s115/s101/s32/s105/s110/s116/s101/s114/s99/s104/s97/s105/s110/s32/s111/s114/s100/s101/s114/s41\n/s122\nFIG. 11: The ground-state phase diagram in the vicinity ∆ = −1 of the four-leg tube with\nlongitudinal coupling between legs.\nintraleg ferromagnetic ordering. The sector of the rung singlet ph ase increases at J′\n⊥→J⊥\n(see Fig. 12).\nA possible physical picture that reconciles the phase diagram with th e results of the\nprevious sections might look as follows. The system BIPNNBNO is an ar ray ofloosely\ncoupled ferrimagnetic chains in a presence of an extremely weak XY anisotropy and a strong\nfrustration, J′\n⊥∼J⊥, what corresponds to the region of the disordered Rung Singlet ph ase\nclose the point of the ferromagnetic instability, ∆ = −1.\nV. CONCLUSION\nWehave studied magnetizationprocess intwo-dimensional compoun dBIPNNBNO which\nexhibits ferrimagnetism of non-Lieb-Mattis type. The investigation is complicated by a lack\nof reliable information about exchange interactions in the system. F or a start, we proposed\nthenaive model ofnon-interacting ferrimagnetic chains andshowe d that anappearance both\n1/3 and 2/3 plateaus can be explained within the model. This provides u s the intrachain\nexchange couplings JAFandJ1. By setting these parameters in the exact diagonalization24\n/s48/s44/s48 /s48/s44/s49 /s48/s44/s50 /s48/s44/s51 /s48/s44/s52 /s48/s44/s53 /s48/s44/s54 /s48/s44/s55 /s48/s44/s56 /s48/s44/s57 /s49/s44/s48/s45/s49/s44/s48/s45/s48/s44/s57/s45/s48/s44/s56/s45/s48/s44/s55/s45/s48/s44/s54/s45/s48/s44/s53/s45/s48/s44/s52/s45/s48/s44/s51/s45/s48/s44/s50\n/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116\n/s40/s97/s110/s116/s105/s112/s104/s97/s115/s101/s32/s105/s110/s116/s101/s114/s99/s104/s97/s105/s110/s32/s111/s114/s100/s101/s114/s105/s110/s103/s41/s68/s105/s115/s111/s114/s100/s101/s114/s101/s100/s32\n/s82/s117/s110/s103/s32/s83/s105/s110/s103/s108/s101/s116/s115/s32\n/s74 /s32/s47/s74\nFIG. 12: The ground-state phase diagram in the vicinity ∆ = −1 of the four-leg tube with\nisotropic coupling between legs. The phase boundary betwee n the disordered rung singlet gapfull\nphase and the stripe ferromagnetic phase is shown by the dott ed and solid lines for J′\n⊥/J⊥= 0.2\nandJ′\n⊥/J⊥= 1.0, respectively.\nroutine the magnetization curve for the 32 and 40-sites clusters is numerically calculated.\nWe demonstrate that the magnetization curve similar to that obser ved in the experiment\nis obtained in the regime of weak interchain coupling, J1≫Jnn> Jnnn. Another revealed\nphenomenon is that a width of the singlet plateau increases with a gro wth of the antiferro-\nmagnetic frustrating coupling between the next-to-nearest cha ins. Following these results,\nwe apply on the tube lattice two low-energy theories which could expla in an appearance of\nthe singlet phase. The first one is based on the effective XXZ Heisenb erg model in a longitu-\ndinal magnetic field in the limit where the interchain coupling dominates, Jnn≥Jnnn≫J1.\nWe derive the critical field destroying the singlet plateau, it turns ou t that it does not\ndepend on the frustration parameter Jnnn. Another analytical strategy is realized via the\nabelian bosonization formalism which is relevant for the opposite limit J1≫Jnn≥Jnnn.\nWe demonstrate that the gapfull disordered Rung Singlets phase comes up when the XY\nexchange anisotropy may tilt the balance from the long-range orde r with an antiphase in-\nterchain arrangement of ferrimagnetic chains towards the spin liqu id phase. A role of the25\nanisotropy in a formation of the spin gap in the original two-dimension al system deserves a\nfurther study.\nAcknowledgments\nWe acknowledge Grant RFBR No. 10-02-00098-a for a support. We wish to thank Prof.\nD.N. Aristov for discussions.\nAppendix A: Wave functions of the ring\nThe states of the ring |S12S34SM/an}bracketri}htare composed from the functions |m1m2m3m4/an}bracketri}ht, where\nmi=±1/2 marks the spin-1/2 state of the separate i-th (1,1/2) block in the ring\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle11\n2;1\n2±1\n2/angbracketrightbigg\n=±/radicalbigg\n2\n3|1±1/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2∓1\n2/angbracketrightbigg\n∓1√\n3|10/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2±1\n2/angbracketrightbigg\n.\nThe basic functions of the singlet states are given by\n|00;00/an}bracketri}ht=1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n21\n2−1\n2/angbracketrightbigg\n−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n2−1\n21\n2/angbracketrightbigg\n−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n21\n2−1\n2/angbracketrightbigg\n+1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n2−1\n21\n2/angbracketrightbigg\n,\n|11;00/an}bracketri}ht=1√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2−1\n2−1\n2/angbracketrightbigg\n+1√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n2−1\n21\n21\n2/angbracketrightbigg\n−1\n2√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n21\n2−1\n2/angbracketrightbigg\n−1\n2√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n2−1\n21\n2/angbracketrightbigg\n−1\n2√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n21\n2−1\n2/angbracketrightbigg\n−1\n2√\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n2−1\n21\n2/angbracketrightbigg\n.\nThe triplet states read as\n|01;11/an}bracketri}ht=1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−1\n21\n21\n2/angbracketrightbigg\n−1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n21\n21\n21\n2/angbracketrightbigg\n,\n|10;11/an}bracketri}ht=1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n21\n2−1\n2/angbracketrightbigg\n−1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2−1\n21\n2/angbracketrightbigg\n.\nAppendix B: Saturation field in the limit of the strong ring coupling\nTo get the saturation field the functions of the ring with the total s pinsS= 5 andS= 6\n|ψ6/an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n4,\n|ψ5/an}bracketri}ht=−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n4+1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n426\n−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n4+1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n23\n2/angbracketrightbigg\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n21\n2/angbracketrightbigg\n4.\nwith the energies E5=JAF/2−Jnn/3+Jnnn/2,E6= 2JAF+Jnn+2Jnnnare needed.\nBy introducing the pseudo-spin operators in the restricted space similar to Eq.(23) the\noriginal spin operators are presented as follows ( S= 1,s= 1/2)\nSz\nin=23\n24+1\n12˜Sz\nn, sz\nin=5\n12+1\n6˜Sz\nn,\nS+\nin= (−1)i+11\n2/radicalbigg\n2\n3˜S+\nn, s+\nin= (−1)i1\n2/radicalbigg\n2\n3˜S+\nn,\nS−\nin= (−1)i+11\n2/radicalbigg\n2\n3˜S−\nn, s−\nin= (−1)i1\n2/radicalbigg\n2\n3˜S−\nn. (B1)\nThe effective XXZ spin chain Hamiltonian has the form (22) with the par ametersJeff\nxy=\n−2J1/3,Jeff\nz=J1/18 andBeff\nz=−7J1/9+B−Bc, whereBc=E6−E5.\nBy performing a Jordan-Wigner transformation followed by a partic le-hole\ntransformation20the new Hamiltonian of spinless holes is\nHh=−tN/summationdisplay\nn=1/parenleftbig\nd+\nidi+1+d+\ni+1di/parenrightbig\n+VN/summationdisplay\nn=1nd\nind\ni+1−µhN/summationdisplay\nn=1nd\ni,\nwheret=Jeff\nxy/2,V=Jeff\nzandµh=Jeff\nz−Beff\nz.\nThe saturation field Bsatcorresponds to the chemical potential where the hole band start s\nto fill up,µh=−2t. This yields Bsat=Bc+J1/6 = 3JAF/2+4Jnn/3+3Jnnn/2+J1/6.\n1F. Mila, Eur. J. Phys. 21, 499 (2000).\n2H. Kageyama, K. Yoshimura, R. Stern, N.V. Mushnikov, K. Oniz uka, M. Kato, K. Kosuge, C.P.\nSlichter, T. Goto, and Y. Ueda, Phys. Rev. Lett. 82, 3168 (1999).\n3S. Taniguchi, T. Nishikawa, Y. Yasui, Y. Kobayashi, M. Sato, T. Nishioka, M. Kotani, K. Sano,\nJ. Phys. Soc. Jpn. 64, 2758 (1995).\n4B.S. 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Pronin1,∗\n1Dresden High Magnetic Field Laboratory (HLD),\nHelmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Ger many\n2High Field Magnet Laboratory, Institute for Molecules and M aterials,\nRadboud University Nijmegen, 6525 ED Nijmegen, The Netherl ands\n3A. M. Prokhorov Institute of General Physics, Russian Acade my of Sciences, 119991 Moscow, Russia\n4Moscow Institute of Physics and Technology (State Universi ty), 141700 Dolgoprudny, Moscow Region, Russia\n51. Physikalisches Institut, Universit¨ at Stuttgart, Pfaffe nwaldring 57, 70550 Stuttgart, Germany\n6Moscow State Institute of Radio-Engineering, Electronics ,\nand Automation (Technical University), 117464 Moscow, Rus sia\n7Faculty of Physics, Southern Federal University, 344090 Ro stov-on-Don, Russia\n(Dated: May 24, 2022)\nWe report on an investigation of optical properties of multi ferroic CoCr 2O4at terahertz frequen-\ncies in magnetic fields up to 30 T. Below the ferrimagnetic tra nsition (94 K), the terahertz response\nof CoCr 2O4is dominated by a magnon mode, which shows a steep magnetic-fi eld dependence. We\nascribe this mode to an exchange resonance between two magne tic sublattices with different g-\nfactors. In the framework of a simple two-sublattice model ( the sublattices are formed by Co2+\nand Cr3+ions), we find the inter-sublattice coupling constant, λ=−(18±1) K, and trace the\nmagnetization for each sublattice as a function of field. We s how that the Curie temperature of the\nCr3+sublattice, Θ 2= (49±2) K, coincides with the temperature range, where anomalies of the\ndielectric and magnetic properties of CoCr 2O4have been reported in literature.\nPACS numbers: 75.85.+t, 76.50.+g\nCoCr2O4is a ferrimagnetic spinel compound with a\ncomplex network of competing magnetic interactions.1,2\nBoth, Co2+(A sites of the spinel structure) and Cr3+\nions (B1 and B2 sites), are magnetic. Below the Curie\ntemperature, TC= 94 K, the system exhibits a long-\nrange ferrimagnetic order.1AtTS= 26 K, a structural\ntransition occurs. Below this temperature, an incom-\nmensurate conical (i.e. uniform plus transverse spiral)\nmagnetic structure sets in. At Tlock−in= 15 K, the mag-\nnetic structure becomes commensurate – the period of\nthe spin spiral “locks” to the lattice parameter.2\nThe spiral order most likely survives above TS, but\non short ranges only.1,2AtTkink= 50 K, anomalies in\ndielectric and magnetic properties of CoCr 2O4have been\nreported3,4and tentativelyattributed to the formationof\nthis incommensurate short-range spiral magnetic order.4\nIn 2006, Yamasaki et al.have discovered the struc-\ntural transition at TS= 26 K to be accompanied by\nthe emergence of a spontaneous electric polarization,\nwhich direction can be reversed by applying a magnetic\nfield.5As multiferroics are appealing because of basic\nphysical interest as well as their potential technological\napplications,6the reported multiferroicity of CoCr 2O4\nhas triggered a number of experimental studies of the\ncompound.4,7–10To date, there is, however, no optical\ndata reported at terahertz frequencies. This is the fre-\nquency region where e.g.electromagnons have been ob-\nserved in some other multiferroic systems.11,12\nHere, we report on an optical study of CoCr 2O4at ter-\nahertz (or far-infrared) frequencies in magnetic fields upto 30 T. We do not find any evidence for electromagnons.\nHowever, below TC, we observe a resonance mode, which\nis highly sensitive to external magnetic field. We ascribe\nthis mode to a ferrimagnetic inter-sublattice exchange\nresonance, originally considered theoretically by Kaplan\nand Kittel.13Applying this model to CoCr 2O4allows us\nto separate the contributions of the Co2+and Cr3+sub-\nlattices to the total magnetization, to extract the mag-\nnetization for each sublattice as a function of field, and\nto find the inter-sublattice coupling constant. Further-\nmore, we show that the Curie temperature of the Cr3+\nsublattice coincides with Tkink= 50 K. Thus, the onset\nof the short-range spiral component must be related to\nthe ordering in the Cr3+sublattice.\nThe investigated CoCr 2O4samples have been synthe-\nsized from Co 3O4and Cr 2O3powders. CoCr 2O4pow-\nder has been pressed into pellets of 10 mm in diameter\nand 1 mm in thickness. X-ray diffraction measurements\nprove the cubic symmetry (space group Fd¯3m) with\nno indication of spurious phases. The lattice constant,\na= 8.328(2)˚A, is in good agreement with published\nresults.14,15More details on sample preparation, char-\nacterization, and thermodynamic properties of CoCr 2O4\nare given in our recent works.3,16\nThe optical measurements have been made in trans-\nmission mode, using two different setups. The first setup\nwas a spectrometer equipped with backward-wave os-\ncillators (BWOs) as sources of coherent and frequency-\ntunable radiation.17,18The measurements were done at\n(sub)terahertz frequencies ( ν=ω/2π= 8.5÷41 cm−1=2\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s76/s105/s110/s101/s32/s50/s32/s63/s84/s114/s97/s110/s115/s109/s105/s116/s116/s97/s110/s99/s101/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s51/s32/s84\n/s53/s32/s84\n/s56/s32/s84\n/s49/s48/s32/s84\n/s49/s50/s32/s84\n/s49/s53/s32/s84\n/s49/s55/s32/s84\n/s50/s48/s32/s84\n/s50/s53/s32/s84\n/s51/s48/s32/s84\n/s32 /s32\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s99/s109/s45/s49\n/s41/s67/s111/s67/s114\n/s50/s79\n/s52\n/s76/s105/s110/s101/s32/s49\n/s51/s48/s32/s75\nFIG. 1: (Color online) Examples of raw far-infrared transmi t-\ntance spectra of CoCr 2O4in Faraday geometry. The absorp-\ntion line, marked as “Line 1”, is discussed in the course of th e\narticle. At the higher-frequency end of the spectra, there i s\nan indication of another, very broad, line (“Line 2 ?”).\n250÷1230 GHz = 1 ÷5 meV) and at temperatures be-\ntween 4 and 300 K. The radiation was linearly polarized.\nThe absolute values of optical transmission (normalized\nto the empty-channel measurements) were obtained in\nessentially the same way as described, e.g., in Ref. 19.\nA commercial split-coil magnet, embedded into an op-\ntical cryostat, was utilized for measurements in magnetic\nfields up to 8 T. In these measurements, we used three\ndifferent geometries: k⊥H/bardbl˜h,k⊥H⊥˜h, and\nk/bardblH⊥˜h(hereHis the external static magnetic field;\nkand˜harethe wavevectorand the magnetic component\nof the probing electromagnetic radiation, respectively).\nIn the first geometry ( k⊥H/bardbl˜h), where electro-\nmagnons, if they exist, could be excited, we did not ob-\nserve any resonance absorption lines. Furthermore, these\nspectra show no detectable changes induced by applying\nmagnetic fields up to 8 T.\nIn the two other geometries (where H⊥˜h), we ob-\nserved absorption lines. We did not detect any differ-\nences between the results for these two geometries, al-\nthough the data obtained for k/bardblH(Faraday geometry)\nare somewhat less noisy due to peculiarities of the split-\ncoil magnet design. Thus, in the following we discuss the\nBWO data collected in the Faraday geometry.\nThe second optical setup was a commercial Fourier-\ntransform infrared (FTIR) spectrometer (Bruker\nIFS113v) combined with a continuous-field 33-Tesla\nBitter magnet, at the High Field Magnet Laboratory in\nNijmegen.20The measurements were performed in the\nFaradaygeometry( k/bardblH⊥˜h) at 4 and 30K. A mercury\nlamp was used as a radiation source. The far-infrared/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s48 /s49/s48 /s50/s48 /s51/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s32\n/s32/s66/s87 /s79/s32/s109/s101/s97/s115/s117/s114/s101/s109/s101/s110/s116/s115\n/s32/s70/s84/s73/s82/s32/s109/s101/s97/s115/s117/s114/s101/s109/s101/s110/s116/s115\n/s32/s77/s111/s100/s101/s108/s32/s99/s97/s108/s99/s117/s108/s97/s116/s105/s111/s110/s115/s32/s32\n/s51/s48/s32/s75/s32/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32\n/s48/s32/s40/s99/s109/s45/s49\n/s41/s67/s111/s67/s114\n/s50/s79\n/s52\n/s52/s32/s75/s32\n/s48/s72/s32/s40/s84/s41\n/s32/s32\nFIG. 2: (Color online) Field dependence of the magnon fre-\nquency for 4 and 30 K. Symbols are the experimental data,\nlines represent the results of simultaneous fits of the optic al\ndata with Eq. 3 and of magnetization data from Ref. 3 with\nEqs. 1 and 2. Note that the onset of the vertical scales is at\n10 cm−1.\nradiation was detected using a custom-made silicon\nbolometer operating at 1.4 K. For both temperatures,\nthe FTIR spectra were measured in 0, 3, 5, 8, 10,\n12, 15, 17, 20, 25, and 30 T. The optical data were\ncollected between 15 and 70 cm−1(450÷2100 GHz,\n1.9÷8.7 meV), using a 200- µm mylar beamsplitter and\na scanning velocity of 50 kHz. At each field, at least 100\nscans were averaged.\nIn the transmission spectra, obtained by use of both\nsetups, we reliably observed an intensive absorption line,\nwhich is highly sensitive to the applied magnetic field\nand temperature (“Line 1” in Fig. 1). At the higher-\nfrequency end of the FTIR spectra, we possibly observe\nanother, broad and week, absorption line (“Line 2 ?” in\nFig. 1). The position of this line shifts to higher frequen-\ncies (and thus out of the measurement-frequency win-\ndow) in applied magnetic field. We tentatively attribute\nthis line to an antiferromagnetic resonance within the\nCr2+sublattice. Hereafter, we solely discuss the “Line\n1”.\nThe field dependence of the frequency of “Line 1”, ν0,\nis shown in Fig. 2. The resonance frequency is basically\nproportional to the applied static field with a zero-field\ngap. At 4 K, the gap is 15.8 cm−1(475 GHz, 1.97 meV),\nand at 30 K, it is 16.3cm−1(489GHz, 2.02meV). At low\nfields[µ0H<∼10T],thedimensionlessslopesofthe ν0(H)3\ncurves,hν0/µBµ0H, reach 2.5(at 4 K) and 2.8 (at 30K).\nLet us note, that the typical slopes of the ν0(H) curves\nfor ferromagnetic resonances in the Co2+- or Cr3+-based\ncompounds are determined by the gyromagnetic ratios\nof the Co2+and Cr3+ions and amount to 2.2 and 1.95,\nrespectively.21\nResonance modes in ferrimagnetic substances have\noriginally been considered theoretically by Kaplan and\nKittel.13For a system with two magnetic sublattices,\nthey predicted the existence of two magnetic-resonance\nlines. The first of them is associated with the conven-\ntional spin precession. In CoCr 2O4, such mode has been\nobserved by electron-spin-resonance studies at frequen-\ncies below 100 GHz in fields up to 10 T.22,23The sec-\nond mode, the inter-sublattice exchange resonance, is\nsupposed to have a large zero-field gap defined by the\nexchange interaction between the sublattices, to harden\nin an applied magnetic field, and to have a steep field\ndependence.24Our observed mode demonstrates this\nvery behavior (Fig. 2). Thus, it is reasonable to asso-\nciate it with the Kaplan-Kittel inter-sublattice exchange\nresonance.\nIn the following, we show that using a simple two-\nsublattice Kaplan-Kittel model, we can consistently de-\nscribe our data on the magnon frequency and our earlier\ndata on magnetization (Ref. 3). From this description,\nwe obtain the magnetic moments of the sublattices and\nthe inter-sublattice exchange constant.\nIn our model, we consider two effective magnetic sub-\nlattices. Coupling within the first sublattice is provided\nby exchange interactions between Co2+and Cr3+ions,\nJAB. At Θ 1=TC= 94 K, this sublattice orders. Im-\nportant is that only non-collinearly ordered spins of the\nCo2+ions contribute to the magnetic moment, while the\nspins of the Cr3+ions do not. This is because of geo-\nmetrical frustration of the interaction between the Cr3+\nions,JBB.2Thus, one can consider the first sublattice to\nbe formed by Co2+ions only.\nWhen temperature decreases further, the Cr3+spins\nstart to contribute (negatively) to the net magnetiza-\ntion. Thus, as the second sublattice, one can consider\nthe sublattice formed by Cr3+ions. Let us note, that\nbecause of geometrical frustration, the effective coupling\nwithin the Cr3+sublattice is significantly smaller than\nJBB. The Curie temperature of this sublattice, Θ 2, is\nsomewhere below 94 K. This temperature will be one of\nour fit parameters (we take Θ 2to be field independent).\nFor a two-sublattice ferrimagnet, the net (measurable)\nmagnetization is the sum of the magnetizations of the\nsublattices, M(T,H) =M1(T,H)+M2(T,H).M1and\nM2have opposite signs; the inter-sublattice exchange\nconstant λis negative. Within the molecular-field ap-\nproximation, the reduced magnetization of each sublat-\ntice,yi≡Mi/M0\ni[here and thereafter i={1,2} ≡\n{Co2+,Cr3+}andM0\niis the zero-temperature magne-\ntization of the i-th sublattice], is given by the Brillouin/s48 /s52/s48 /s56/s48 /s49/s50/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s119/s111/s117/s108/s100/s45/s98/s101/s32/s99/s117/s114/s118 /s101/s115/s58\n/s32/s67/s114/s51/s43\n/s32/s119/s105/s116/s104/s111/s117/s116/s32/s99/s111/s117/s112/s108/s105/s110/s103\n/s98/s101/s116/s119/s101/s101/s110/s32/s116/s104/s101/s32/s115/s117/s98/s108/s97/s116/s116/s105/s99/s101/s115\n/s32/s67/s114/s51/s43\n/s32/s119/s105/s116/s104/s111/s117/s116/s32/s116/s97/s107/s105/s110/s103\n/s105/s110/s116/s111/s32/s97/s99/s99/s111/s117/s110/s116/s32/s116/s104/s101/s32/s115/s116/s114/s117/s99/s116/s117/s114/s97/s108\n/s116/s114/s97/s110/s115/s105/s116/s105/s111/s110/s32/s97/s116/s32/s50/s54/s32/s75/s115/s117/s98/s108/s97/s116/s116/s105/s99/s101\n/s99/s111/s110/s116/s114/s105/s98/s117/s116/s105/s111/s110/s115/s58\n/s32/s67/s111/s50/s43\n/s32/s67/s114/s51/s43/s32/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s32/s116/s111/s116/s97/s108/s32/s102/s105/s116\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110 /s32/s40\n/s66 /s32/s47/s32/s102/s46/s117/s46 /s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s48/s72/s32/s61/s32/s48/s46/s49/s32/s84/s67/s111/s67/s114\n/s50/s79\n/s52\nFIG. 3: (Color online) Example of a fit by use of Eqs. 1 and 2\nto the magnetization data obtained at 0.1 T [simultaneously ,\nthe magnon-frequency data were fitted with the same set of\nfree parameters, see Fig. 2]. Contributions of the Co2+and\nCr3+sublattices are shown with opposite signs. The contri-\nbution of the Cr3+sublattice diminishes at TC, rather than at\nΘ2= 49 K, because of the coupling between the sublattices.\nfunction:25,26\nyi=BSi(xi)≡2Si+1\n2Sicoth2Si+1\n2Sixi−1\n2Sicothxi\n2Si,\n(1)\nwhereSiis the spin magnetic moment for ions of the i-th\nsublattice and xiis defined as:\nxi=µB\nkBT/bracketleftbigg3Si\nSi+1/parenleftbiggkBΘiMi\nµBM0\ni+kBλMj\nµBM0\ni/parenrightbigg\n+HM0\ni/bracketrightbigg\n,(2)\ni/ne}ationslash=j.\nIn our case, Co2+and Cr3+ions have equal spin mo-\nments,S1=S2= 3/2.\nFor the resonance frequency ν0, we use a modified\nKaplan-Kittel equation:10,13\n/parenleftbigghν0\nµBg1+kB\nµBλM2+H/parenrightbigg\n×(3)\n/parenleftbigghν0\nµBg2−kB\nµBλM1+H/parenrightbigg\n+/parenleftbiggkB\nµBλ/parenrightbigg2\nM1M2= 0,\nwhereg1= 2.2 and g2= 1.95 are the gyromagneticratios\nfor the Co2+and Cr3+ions, respectively.21In Eqs. 2 and\n3, magnetization is in dimensionless units. In our model,\nwe neglect anisotropy, as our measurements have been\nperformed on powder samples.274\n/s48 /s49/s48 /s50/s48 /s51/s48/s45/s48/s46/s52/s45/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s67/s111/s67/s114\n/s50/s79\n/s52\n/s48/s72/s32/s40/s84/s41/s84/s32/s61/s32/s51/s48/s32/s75\n/s32/s32/s83/s117/s98/s108/s97/s116/s116/s105/s99/s101/s32/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40 /s32/s47/s32/s102/s46/s117/s46/s41/s67/s111/s50/s43\n/s67/s114/s51/s43\nFIG. 4: (Color online) Magnetization of the Co2+and Cr3+\nsublattices in CoCr 2O4, as obtained from the model fits with\nEqs. 1 – 3. The lowest-field points are at 0.1, 0.5, and 1 T.\nUnlike in collinear ferrimagnetics, in CoCr 2O4the vec-\ntor of the magnetization of each sublattice in an exter-\nnal magnetic field can change not only its orientation,\nbut also its size. In our calculations, we take this into\naccount. The corresponding increase in M0\n1andM0\n2is\nsupposed to persist until the spin-only values for Co2+\nand Cr3+ions of 3 µB/ion are reached. Recent experi-\nments in pulsed magnetic fields have shown that even in\nfields of up to 60 T, there are no signs of saturation in\nthe magnetization.28\nUsing Eqs. 1 – 3, we simultaneously fit the magneti-\nzation data, obtained in Ref. 3, and our results for the\nresonance frequency.29Results of these fits are shown in\nFigs. 2 and 3 together with experimental data. We find\nthat the best description of all our experimental data can\nbe reached with λ=−(18±1) K and Θ 2= (49±2) K.\nThe value found for the ordering temperature of the\nCr3+sublattice, Θ 2, coincides with Tkink– the tem-\nperature, which is possibly related to the formation of\nthe incommensurate short-range spiral magnetic order.4\nOur findings show that this short-range order appears to\nbe due to the involvement of the Cr3+ions. The rela-\ntively large value of the coupling constant, λ=−18 K,\nmay explain why the Tkink-related features appear to be\nbroad.2,4Figure 4 shows the magnetic-field dependence of the\ncalculated sublattice magnetizations. We show here the\nresults obtained at T= 30 K, i.e. just above the struc-\ntural transition. As it can be seen from Fig. 3, the\nlower-temperatureresults differ only marginallyfrom the\nshown in Fig. 4. We have found that within our model,\nthe observed behavior of the experimental data at TS\ncan be best described by a slight decrease in the absolute\nvalue ofMCratTS(theg-factors of the Co2+and Cr3+\nions are presumed to be independent of temperature).\nThe contribution of each sublattice to the total mag-\nnetic moment in zero field can be estimated from Fig. 4.\nWe findMCo(0 T) = 0.4 µB/f.u. and MCr(0 T) = – 0.33\nµB/f.u. We note, that although in fields smaller than\nthe coercive force (0.3 T, Ref. 3) the Mivalues are influ-\nenced by the magnetizing/demagnetizing processes, this\ninfluence hardly affects the above result [in Fig. 4, only\nthe lowest-field point (0.1 T) is below the coercive-force\nvalue, the next points are at 0.5 and 1 T].\nThe proposed model gives a natural explanation for\nthe steep slope of the ν0(H) curves [Fig. 2]. As can be\nseen from Eq. 3, this is due to the increase of Mias\na function of field. Similarly, the temperature evolution\nofν0(H) is explained as being due to the temperature\nevolution of the Mi(H) curves.\nSummarizing, the far-infrared optical response of\nCoCr2O4below the Curie temperature, TC= 94 K, is\ndominated by a magnon mode. Ascribing the magnon to\nan inter-sublattice exchange resonance (the sublattices\nare formed by the Co2+and Cr3+ions) and applying a\nmodified Kaplan-Kittel model of two interacting sublat-\ntices allows us to consistently describe our experimental\ndata on the magnon frequency and earlier magnetization\nmeasurements, to obtain the magnetization for each of\nthe sublattices, and to find the inter-sublattice exchange\nconstant, λ= – 18 K. The Curie temperature of the\nCr3+sublattice is found to coincide with the tempera-\nture, where the incommensurate short-range spiral mag-\nnetic order is believed to set in, Tkink= 50 K.\nWe thank M. D. Kuz’min for useful discussions and\nPapori Gogoi for assistance in measurements. Parts of\nthis work were supported by EuroMagNET II (EU con-\ntract No. 228043), by the Russian Foundation for Basic\nResearch (Grant No. 12-02-00151) and by the Ministry\nofEducation and Science ofthe Russian Federation (con-\ntract No. 14.A18.21.0740).\n∗Electronic address: a.pronin@hzdr.de\n1N. Menyuk, K. Dwight, and A. Wold, J. Phys. (Paris) 25,\n528 (1964).\n2K. Tomiyasu, J. Fukunaga, and H. Suzuki, Phys. Rev. B\n70, 214434 (2004).\n3A.V.Pronin, M.Uhlarz, R.Beyer, T.Fischer, J.Wosnitza,\nB. P. Gorshunov, G. A. Komandin, A. S. Prokhorov, M.\nDressel, A. A. Bush, and V. I. Torgashev, Phys. Rev. B85, 012101 (2012).\n4G. Lawes, B. Melot, K. Page, C. Ederer, M. A. Hayward,\nTh. Proffen, and R. Seshadri, Phys. Rev. B 74, 024413\n(2006).\n5Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima,\nand Y. Tokura, Phys. Rev. Lett. 96, 207204 (2006).\n6T. Kimura, T. Goto, H. Shintan, K. Ishizaka, T. Arima,\nand Y. Tokura, Nature 426, 55 (2003).5\n7Y. J. Choi, J. Okamoto, D. J. Huang, K. S. Chao, H. J.\nLin, C. T. Chen, M. van Veenendaal, T. A. Kaplan, and\nS.-W. Cheong, Phys. Rev. Lett. 102, 067601 (2009).\n8N. Mufti, A. A. Nugroho, G. R. Blake, and T. T. M. Pal-\nstra, J. Phys.: Condens. Matter 22, 075902 (2010).\n9L. J. Chang, D. J. Huang, W.-H. Li, S.-W. Cheong, W.\nRatcliff, and J. W. Lynn, J. Phys.: Condens. Matter 21,\n456008 (2010).\n10V. I. Torgashev, A. S. Prokhorov, G. A. Komandin, E. S.\nZhukova, V. B. Anzin, V. M. Talanov, L. M. Rabkin, A. A.\nBush, M. Dressel, and B. P. Gorshunov, Phys. Solid State\n54, 350 (2012).\n11A. Pimenov, A. A. Mukhin, V. Y. Ivanov, V. D. Travkin,\nA. M. Balbashov, and A. Loidl, Nat. Phys. 2, 97 (2006).\n12A. B. Sushkov, R. V. Aguilar, S.-W. Cheong, and H. D.\nDrew, Phys. Rev. Lett. 98, 027202 (2007).\n13J. Kaplan and C. Kittel, J. Chem. Phys. 21, 760 (1953).\n14B. Mansour, N. Baffier, and M. Huber, C. R. Hebd.\nSeances Acad. Sci., Ser. C 277, 867 (1973).\n15P. G. Casado and I. Rasines, Polyhedron 5, 787 (1986).\n16M. Uhlarz, A. V. Pronin, J. Wosnitza, A. S. Prokhorov,\nand A. A. Bush, Phys. Chem. Minerals 40, 203 (2013).\n17G. V. Kozlov and A. A. Volkov, in Millimeter and Submil-\nlimeter Wave Spectroscopy of Solids , edited by G. Gr¨ uner\n(Springer, Berlin, 1998), p. 51.\n18B. Gorshunov, A. Volkov, I. Spektor, A. Prokhorov, A.\nMukhin, M. Dressel, S. Uchida, and A. Loidl, Int. J. In-\nfrared Millim. Waves 26, 1217 (2005).\n19A. Mukhin, B. Gorshunov, M. Dressel, C. Sangregorio, andD. Gatteschi, Phys. Rev. B 63, 214411 (2001).\n20S. A. J. Wiegers, P. C. M. Christianen, H. Engelkamp, A.\nden Ouden, J. A. A. J. Perenboom, U. Zeitler, and J. C.\nMaan, J. Low Temp. Phys. 159, 389 (2010).\n21S. A. Altshuler and B. M. Kozyrev, Electron Paramagnetic\nResonance in Compounds of Transition Elements (Wiley,\nNew York, 1974).\n22J. J. Stickler and H. J. Zeiger, J. Appl. Phys. 39, 1021\n(1968).\n23S. Funahashi, K. Siratori, and Y. Tomono, J. Phys. Soc.\nJpn.29, 1179 (1970).\n24T. A. Kaplan, Phys. Rev. 109, 782 (1958).\n25S. V. Vonsovsky, Magnetism (Wiley, New York, 1974).\n26M. D. Kuz’min and A. M. Tishin, Cryogenics 32, 545\n(1992).\n27In practice, this approximation would also be valid for sin-\ngle crystals, as the magnitude of the anisotropy field in\nCoCr2O4is below 0.1 T [Ref. 23], i.e. significantly smaller\nthan all terms in Eqs. 2 and 3 for external fields above 0.1\nT.\n28V. Tsurkan, S. Zherlitsyn, S. Yasin, V. Felea, Y. Skourski,\nJ. Deisenhofer, H.-A. Krug von Nidda, J. Wosnitza, and\nA. Loidl, Phys. Rev. Lett. 110, 115502 (2013).\n29Since no experimental data on the temperature depen-\ndence of the magnetization at 30 T are available, we used\na linear extrapolation of the magnetization in accordance\nwith the M(H) measurements of Ref. 28."    },    {        "title": "1207.6771v1.Ferrimagnetism_of_the_Heisenberg_Models_on_the_Quasi_One_Dimensional_Kagome_Strip_Lattices.pdf",        "content": "arXiv:1207.6771v1  [cond-mat.str-el]  29 Jul 2012Ferrimagnetism of the Heisenberg Models on the Quasi-One-D imensional Kagome\nStrip Lattices\nTokuro Shimokawa1and Hiroki Nakano1\n1Graduate School of Material Science, University of Hyogo, K amigori, Hyogo 678-1297, Japan\n(Dated: May 27, 2018)\nWe study the ground-state properties of the S= 1/2 Heisenberg models on the quasi-one-\ndimensional kagome strip lattices by the exact diagonaliza tion and density matrix renormalization\ngroup methods. The models with two different strip widths sha re the same lattice structure in their\ninner part with the spatially anisotropic two-dimensional kagome lattice. When there is no mag-\nnetic frustration, the well-known Lieb-Mattis ferrimagne tic state is realized in both models. When\nthe strength of magnetic frustration is increased, on the ot her hand, the Lieb-Mattis-type ferrimag-\nnetism is collapsed. We find that there exists a non-Lieb-Mat tis ferrimagnetic state between the\nLieb-Mattis ferrimagnetic state and the nonmagnetic groun d state. The local magnetization clearly\nshows an incommensurate modulation with long-distance per iodicity in the non-Lieb-Mattis ferri-\nmagnetic state. The intermediate non-Lieb-Mattis ferrima gnetic state occurs irrespective of strip\nwidth, which suggests that the intermediate phase of the two -dimensional kagome lattice is also the\nnon-Lieb-Mattis-type ferrimagnetism.\nPACS numbers: 75.10.Jm, 75.30.Kz, 75.40.Mg\nI. INTRODUCTION\nFerrimagnetism is a fundamental phenomenon in the\nfield of magnetism. The simplest case of the conven-\ntional ferrimagnetism is the ground state of the mixed\nspin chain. For example, there is an ( s,S)=(1/2, 1)\nmixed spin chain with a nearest-neighbor antiferromag-\nnetic (AF) isotropic interaction. In this system, the\nso-called Lieb-Mattis (LM)-type ferrimagnetism1–6is re-\nalized in the ground state because two different spins\nare arranged alternately in a line owing to the AF in-\nteraction. Since this type of ferrimagnetism has been\nstudied extensively, the magnetic properties and the oc-\ncurrence mechanism of the LM ferrimagnetism are well\nknown. In particular, the ferrimagnetism in the quan-\ntum Heisenberg spin model on the bipartite lattice with-\nout frustration is well understood within the Marshall-\nLieb-Mattis (MLM) theorem1,2. In the LM ferrimagnetic\nground state, the spontaneous magnetization occurs and\nthe magnitude is fixed to a simple fraction of the satu-\nrated magnetization.\nIn recent years, a new type of ferrimagnetism, which is\nclearlydifferent fromLM ferrimagnetism, has been found\nin the ground state of several one-dimensional frustrated\nHeisenberg spin systems7–13. The spontaneous magneti-\nzation in this new type of ferrimagnetism changes grad-\nually with respect to the strength of frustration. The\nincommensurate modulation with long-distance period-\nicity in local magnetizations is also a characteristic be-\nhavior of the new type of ferrimagnetism. Hereafter, we\ncall the new type of ferrimagnetism the non-Lieb-Mattis\n(NLM) type. The mechanism of the occurrence of the\nNLM ferrimagnetism has not yet been clarified.\nOn the other hand, some candidates of the NLM ferri-\nmagnetism among the two-dimensional (2D) systems, for\nexample, the mixed-spin J1-J2Heisenberg model on the\nsquarelattice14andtheS= 1/2HeisenbergmodelontheUnion Jack lattice15, were reported. These 2D frustrated\nsystems have the intermediate ground state, namely, the\n“canted state”, in which the spontaneous magnetization\nis changed when the inner interaction of the system is\nvaried. It has not been, however, determined whether\nthe incommensurate modulation with long-distance pe-\nriodicity exists in the local magnetization of the canted\nstate owing to the difficulty of treating these 2D frus-\ntrated systems numerically and theoretically. Therefore,\nthe relationships between the canted states of these 2D\nfrustrated systems and the NLM ferrimagnetic state are\nstill unclear.\nUnder these circumstances, recently, another candi-\ndate of the NLM ferrimagnetism among the 2D systems\nwas reported in ref. VI in which the S= 1/2 Heisen-\nberg model on the spatially anisotropic kagome lattice\ndepicted in Fig. 1(a) was studied. A region of the\nintermediate-magnetization states is observed between\nthe LM ferrimagnetism that is rigorously proved by the\nMLM theorem1,2and the nonmagnetic state of the spa-\ntially isotropic kagome-lattice antiferromagnet in the ab-\nsence of magnetic field17–27. The local magnetization in\nthe intermediate state of the kagome lattice was investi-\ngated by the exact diagonalization method, and it was\nreported that the local magnetization greatly depends\non the position of the sites, although it is difficult to\ndetermine clearly whether the incommensurate modula-\ntion with long-distance periodicity is present. This result\nleads to the high expectation that the intermediate state\nof the spatially anisotropic kagome lattice is the NLM\nferrimagnetic state. Additional research is desirable to\nconclude that the intermediate state of this 2D system is\nthe NLM ferrimagnetism.\nIn this paper, we study the ground-state properties\nof theS= 1/2 Heisenberg models on the quasi-one-\ndimensional (Q1D) kagome strip lattices depicted in\nFigs. 1(b) and 1(c) instead of the 2D lattice depicted2\n+'\n+\u0013+\u0012\n+\u0012+\u0013\n\"` \"\n#\n$\n%\n\"` #` \n$` \"\n# +\u0012\n+\u0013+'\n$\tB\n \tC\n \tD\nFIG. 1: Structures of the lattices: the spatially anisotrop ic kagome lattice (a) and the quasi-one-dimensional kagome strip\nlattices (b) and (c) with different widths. An S= 1/2 spin is located at each site denoted by a black circle. Antif erromagnetic\nbondsJ1(bold straight line) and J2(dashed line), and ferromagnetic bond JF(dotted line). The sublattices in a unit cell of\nlattice (b) are represented by A, A′, B, B′, C, C′, and D. The sublattices in a unit cell of lattice (c) are repre sented by A, A′,\nB, and C.\nin Fig. 1(a). Note that the inner parts of the lattices in\nFigs. 1(b) and 1(c) are common to a part of the 2D lat-\ntice in Fig. 1(a). We also note that the lattice shapes of\nstripsinthepresentstudyaredifferentfromsomekagome\nstrips (chains) studied in refs. VI-VI, where the nontriv-\nial properties of kagome antiferromagnets were reported.\nAccordingtothestudyinref. VI,itwasalreadyknown\nthat the NLM ferrimagnetism is realized in the ground\nstate of the kagome strip lattice in Fig. 1(c). In the\npresent study, we show that both the lattice in Fig. 1(c)\nand the lattice in Fig. 1(b) reveal the NLM ferrimag-\nnetism in the ground state. Note also that the lattice\nshape at the edge under the open boundary condition\ndepicted in Fig. 1(c) is different from that in ref. VI [see\nFig. 1(b) in ref. VI]. Thus, one can recognize that the\nresults of the strip lattice with small width are irrespec-\ntive of boundary conditions. We also present clearly the\nexistence of the incommensurate modulation with long-\ndistance periodicity in the local magnetizations of both\nmodelsinFigs.1(b)and1(c). Ournumericalcalculations\nsuggest that the intermediate state of the 2D lattice in\nFig. 1(a) is the NLM ferrimagnetism.\nThis paper is organized as follows. In §2, we first\npresent our numerical calculation methods. In §3, we\nshow the ground-state properties of the lattice depicted\nin Fig. 1(c) in finite-size clusters. In §4, we show the\nground-state properties of the lattice depicted in Fig.\n1(b). Sections 5 and 6 are devoted to discussion and\nsummary, respectively.\nII. NUMERICAL METHODS\nWe employ two reliable numerical methods: the ex-\nact diagonalization (ED) method and the density matrixrenormalization group (DMRG) method32,33.\nThe ED method is used to obtain precise physical\nquantities of finite-size clusters. This method does not\nsuffer from the limitation of the cluster shape. It is ap-\nplicable even to systems with frustration, in contrast to\nthe quantum Monte Carlo (QMC) method coming across\nthe so-callednegative-signproblemforsystemswith frus-\ntration. The disadvantage of the ED method is the lim-\nitation that available system sizes are very small. Thus,\nwe should pay careful attention to finite-size effects in\nquantities obtained by this method.\nOntheotherhand,theDMRGmethodisverypowerful\nwhen a system is (quasi-)one-dimensionalunder the open\nboundary condition. The method can treat much larger\nsystemsthan theEDmethod. Notethatthe applicability\nof the DMRG method is irrespective of whether or not\nthe systems include frustrations. In the present research,\nwe use the “finite-system” DMRG method.\nIII. KAGOME STRIP LATTICE WITH SMALL\nWIDTH\nIn this section, we study the magnetic properties in\nthe ground state of the S= 1/2 Heisenberg model on the\nkagome strip lattice depicted in Fig. 1(c). The Hamilto-\nnian of this model is given by\nH=J1/summationdisplay\ni[Si,B·Si,C+Si,C·Si,A′+Si,C·Si+1,A+Si,C·Si+1,B]3\n+J2/summationdisplay\ni[Si,A·Si,B+Si,B·Si,A′]+JF/summationdisplay\ni[Si,A·Si+1,A+Si,A′·Si+1,A′], (1)\n\tB\n\tC\n0 10 20\nStotz–39.5–39–38.5–38E nergy kagome strip in Fig. 1(c) \nN=96, MS=900, SW=15\nJF/J 1=–1\nJ2/J 1=0.50J2/J 1=0.58\n0 0.5 1 1.5 2\nJ2/J 100.20.40.6M/M sN=24, periodic\nN=48, open\nN=96, open\nJF/J 1=–1kagome strip in Fig. 1(c) \nFIG. 2: (Color)(a) Dependence of the lowest energy on Sz\ntot.\nThe results of J2/J1= 0.5 and 0.58 for the system size of\nN= 96 are presented. Arrows indicate the values of the\nspontaneous magnetization Min eachJ2/J1. The position\nof an arrow is given by the highest Sz\ntotvalue among those\nthat give the common lowest energy. (b) J2/J1dependence\nofM/Msobtained from ED calculations for N= 24 (black\nsquare) under the periodic boundary condition and DMRG\ncalculation for N= 48 (red triangle) and 96 (blue inverted\ntriangle) under the open boundary condition. Note that for\nN= 96 in (a) and (b), we use MS= 900 and SW= 15 and,\nthat for N= 48 in (b), we use MS= 400 and SW= 15.\nwhereSi,ξistheS= 1/2spinoperatoratthe ξ-sublattice\nsite in the i-th unit cell. The positions of the four sub-\nlattices in a unit cell are denoted by A, A′, B, and C in\nFig. 1(c). Energies are measured in units of J1; we fixed\nJ1= 1 hereafter. In what follows, we examine the region\nof 0< J2/J1<∞in the case of JF=−1. Note that\nthe number of total spin sites is denoted by N; thus, the\nnumber of unit cells is N/4.\nWe examine the J2/J1dependence of the ratio M/Ms,\nwhereMandMsare the spontaneous and saturation\nmagnetizations, respectively. Let us explain the method\nusedtodetermine Masafunction of NandJ2/J1. First,\nwe calculate the lowest energy E(J2/J1,Sz\ntot,N), where\tB\n\tC\n0 10 20\ni–0.4 –0.2 00.20.4<S i, ξz>\nN=96 J F/J 1=–1 J 2/J 1=0.3 M=24\nAA’ BC\n0 10 20\ni–0.4 –0.2 00.20.4<S i, ξz>\nN=96 J F/J 1=–1 J 2/J 1=0.57 M=20 \nAA’ BC\nFIG. 3: (Color) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublattice\nξ; A (black square), A′(red triangle), B (blue square), and C\n(purple inverted triangle). Panels (a) and (b) show results for\nJ2/J1= 0.3 and 0.57, respectively. These results are obtained\nfrom our DMRG calculations for N= 96 (i= 1,2,···,24).\nSz\ntotvalue is the z-component of the total spin. For ex-\nample, the energies for various Sz\ntotin the two cases of\nJ2/J1are presented in Fig. 2(a); the results are obtained\nby our DMRG calculations of the system of N= 96 with\nthe maximum number of retained states ( MS) of 600\nand the number of sweeps ( SW) of 15. The spontaneous\nmagnetization M(J2/J1,N) is determined as the highest\nSz\ntotamong those at the lowest common energy [see ar-\nrows in Fig. 2(a)]. Our results of the J2/J1dependence\nofM/Msare depicted in Fig. 2(b). We find the existence\nof the intermediate magnetic phase of 0 < M/M s<1/2\nbetween the LM ferrimagnetic phase of M/Ms= 1/2\nand the nonmagnetic phase. In order to determine the\nspin state of this intermediate phase, we calculate the\nlocal magnetization /angbracketleftSz\ni,ξ/angbracketright, where/angbracketleftA/angbracketrightdenotes the expec-\ntation value of the physical quantity AandSz\ni,ξis the\nz-component of Si,ξ. Figure 3 depicts our results for a\nsystem size N= 96 on the lattice depicted in Fig. 1(c)\nunder the open boundary condition; Figs. 3(a) and 3(b)\ncorrespondtothe caseofthe LMferrimagneticphaseand4\nthat of the intermediate phase, respectively.\nThe results clearly indicate the existence of the incom-\nmensurate modulation with long-distance periodicity in\nthe intermediate phase. We also confirm that the period-\nicities of the local magnetizations in the NLM ferrimag-\nnetism in the present model depend on the J2/J1value\nbut not on the length of the system, as reported in the\ncase of ref. VI.\nItisworthemphasizingthe pointthatthe intermediate\nphase commonly exists irrespective of the shape at the\nedge of the strip lattice when one compares the result\nof the present lattice depicted in Fig. 1(c) and that in\nref. VI. Therefore, wecan concludethat the intermediate\nphase exists and that it is NLM ferrimagnetic.\nAlthough there exists an intermediate NLM ferrimag-\nnetic phase in the case of the kagome strip lattice de-\npicted in Fig. 1(c), we should note that there is a large\ndiscrepancy in dimensionality between the kagome striplattice depicted in Fig. 1(c) and the 2D kagome lattice\ndepicted in Fig. 1(a). In the next section, we treat the\nkagome strip lattice depicted in Fig. 1(b) whose width is\nlarger than that of the kagome strip lattice depicted in\nFig. 1(c).\nIV. KAGOME STRIP LATTICE WITH LARGE\nWIDTH\nA. Hamiltonian\nIn this section, we study the ground-state properties\nof theS= 1/2 Heisenberg model on the kagome strip\nlattice depicted in Fig. 1(b). The Hamiltonian of this\nmodel is given by\nH=J1/summationdisplay\ni[Si,B·Si,C+Si,C·Si,D+Si,C·Si+1,A+Si,C·Si+1,B\n+Si,C′·Si,B′+Si,C′·Si,A′+Si,C′·Si+1,D+Si,C′·Si+1,B′]\n+J2/summationdisplay\ni[Si,A·Si,B+Si,B·Si,D+Si,D·Si,B′+Si,B′·Si,A′]\n+JF/summationdisplay\ni[Si,A·Si+1,A+Si,A′·Si+1,A′]\n−h/summationdisplay\ni[Sz\ni,A+Sz\ni,A′+Sz\ni,B+Sz\ni,B′+Sz\ni,C+Sz\ni,C′+Sz\ni,D]. (2)\nHere, the positions ofseven sublattices are denoted by A,\nA′, B, B′, C, C′, and D in Fig. 1(b). Note that the last\nterm of eq. 2 is the Zeeman term. The number of spin\nsitesisdenotedby N. Thenumberofunitcellsis N/7; we\nconsider N/14 as an integer. We mainly use the DMRG\nmethod for investigating the magnetic properties in the\nground state of this Q1D system under the open bound-\nary condition. We also investigate the properties under\nthe periodic boundary condition by the ED method, al-\nthough the size treated by this method is only in the case\nofN= 28. Hereafter, we consider J1= 1 as an energy\nscale and we investigate the region of 0 < J2/J1<∞in\nthe case of JF=−1.\nB. Phase diagram\nFirst, let us examine the J2/J1dependence of the ratio\nM/Msin the absence of the external magnetic field h.\nThe procedure for determining Mis the same as that\nmentioned in §3 [see also Fig. 4(a)]. We present our\nresults of the spontaneous magnetization in Fig. 4(b).\nWe successfully observe the intermediate-magnetizationregion irrespective of the boundary conditions. A careful\nobservation of Fig. 4(b) enables us to observe the eight\nregions at least in the finite-size system. As a matter\nof convenience, hereafter, we call these regions R 1, R2,\n···, R7, and R 8. In the case of N= 112 under the open\nboundarycondition, forexample, Fig. 5(a)illustratesthe\nregions R 1to R8: R1is the region of M/Ms= 3/7, R2is\nthe region of 11 /28≤M/Ms<3/7, R3is the region of\n1/8< M/M s<11/28, R4is the region of M/Ms= 1/8,\nR5is the region of 0 < M/M s<1/8, R6is the region of\nM/Ms= 0, R 7is the region of 0 < M/M s<1/7, and R 8\nis the region of M/Ms= 1/7. Here, the dashed lines in\nFig. 5(a) indicate the boundaries of these regions.\nIt should be noted that the values of M/Msin the\nR4region and that at the lower edge of the R 2region\nchange with increasing N, as shown in Fig. 4(b); the for-\nmer value is M/Ms= (N−14)/7Nand the latter value\nisM/Ms= (3N−28)/7N. These changes due to the in-\ncrease in system size come from the finite-size effect. We\nfind that the value of M/Msin the R 4region under the\nopen boundary condition increases and approaches the\nvalue of M/Ms= 1/7 whenNincreases. Furthermore,\nthe magnetization value in the R 4region is M/Ms= 1/75\n\tB\n \n\tC\n 0 10 20\nStotz–20020 E nergy kagome strip in Fig. 1(b) \nN=56, MS=600, SW=15 \nJF/J 1=–1\nJ2/J 1=0.20J2/J 1=0.57\n0 0.5 1 1.5 2\nJ2/J 100.20.40.6M/M skagome strip in Fig. 1(b) \nN=28,  periodic \nN=56,  open \nN=112, open\nJF/J 1=–1\nFIG. 4: (Color)(a) Lowest energy in each subspace divided by\nSz\ntot. The results of the DMRG calculations obtained when\nthe system size is N= 56 for J2/J1= 0.20 and 0.57 are\npresented. Arrows indicate the spontaneous magnetization M\nfor a given J2/J1. (b)J2/J1dependence of M/Msobtained\nfrom the ED calculations for N=28 (blue triangle) under the\nperiodic boundary condition and the DMRG calculations for\nN=56 (red square) and 112 (black pentagon) under the open\nboundary condition. Note that for N= 56 in (a) and (b),\nwe useMS= 600 and SW=15, and that for N= 112 in\n(b), we use MS≥900 and SW=15. Here, MSdenotes the\nmaximum number of retained states and SWthe number of\nsweeps used in DMRG calculations.\nin the case of N= 28 under the periodic boundary condi-\ntion. Therefore, it is expected that the value of M/Msin\nthe R4regionis 1 /7in the thermodynamic limit. We also\nconfirm that the value of M/Msat the lower edge of the\nR2region gradually increases and approaches the value\nofM/Ms= 3/7 with increasing N. In addition, we can-\nnot confirm the R 2region in the case of N= 28 under\nthe periodic boundary condition. These circumstances\nindicate a possibility that the R 2region merges with the\nR1region of M/Ms= 3/7 in the thermodynamic limit.\nNext, to determine whether each region survives in\nthe thermodynamic limit, we study the size depen-\ndences of the boundaries between the regions under the\nopen boundary condition. Figure 5(b) shows the re-\nsults of N=42, 56, 84, and 112 from DMRG calcu-\nlations. Note here that we define R 2as the region of(3N−28)/7N < M/M s<3/7 and R 4as the region of\nM/Ms= (N−14)/7Nin the finite-size system. One\ncan find immediately from Fig. 5(b) that all regions, ex-\ncept the R 7region, survive in the limit N→ ∞. To\ndetermine whether the R 7region survives in the thermo-\ndynamic limit, we investigate the size dependence of the\nwidth of the R 7region in Fig. 6. This plot shows us that\nthe width of the R 7region decreases with increasing N.\nIt is difficult to determine whether the R 7region survives\nin the thermodynamic limit. The convex downward be-\nhavior is observed for large sizes so that the region might\nsurvive; however, the observed behavior may be one of\ntheseriousfinite-sizeeffects. Theissueofestablishingthe\npresence or absence of the R 7region should be clarified\nin future studies.\n\tB\n\tC\n JF/J 1=–1\n0 0.01 0.02 0.03\n1/N00.511.52J2/J 1\nR1–R 2R2–R 3R3–R 4R4–R 5R5–R 6R6–R 7R7–R 80 0.5 1 1.5 2\nJ2/J 100.20.40.6M/M skagome strip in Fig. 1(b) \nN= 112, open  \nR1 R2\nR3\nR4R5\nR6R7R8\nFIG. 5: (a) Definitions of the R 1-R8regions in the case of\nN= 112 underthe open boundary condition: R 1is the region\nofM/Ms= 3/7, R2is the region of 11 /28≤M/Ms<3/7,\nR3is the region of 1 /8< M/M s<11/28, R4is the region of\nM/Ms= 1/8, R5is the region of 0 < M/M s<1/8, R6is the\nregion of M/Ms= 0, R 7is the region of 0 < M/M s<1/7,\nand R 8is the region of M/Ms= 1/7. Dashed lines indi-\ncate the boundaries of these regions. (b) Size dependences\nof boundaries under the open boundary condition. The re-\nsults presented are those of N=42, 56, 84, and 112 from\nDMRG calculations. The curve line named R l-Rl+1indicates\nthe boundary line between the R land R l+1regions, where l\nis integer.6\n0 0.01 0.02 0.03 0.04\n1/N00.10.20.30.4Width of region R7 region\nFIG. 6: Size dependence of the width of the R 7region. It is\ndifficult to determine whether the R 7region survives in the\nthermodynamic limit.\nC. Magnetic properties in each region\nIn this subsection, we investigate the local magnetiza-\ntion/angbracketleftSz\ni,ξ/angbracketrightto study the magnetic structures in various\nregions except the R 2and R 6regions. Note that we cal-\nculate/angbracketleftSz\ni,ξ/angbracketrightwithin the subspace of the highest Sz\ntotcor-\nresponding to the spontaneous magnetization Mwhen\nJ2/J1is given. Considering the fact that the present\nlattice depicted in Fig. 1(b) has seven sublattices in the\nsystem, we will use different colors or symbols for each\nsublattice ξfor presenting our results of /angbracketleftSz\ni,ξ/angbracketright, as de-\npicted in the inset of Fig. 7(a); we use a black square for\nξ=A, a red triangle for ξ= A′, a blue cross for ξ= B, a\ngreen pentagon for ξ= B′, a purple inverted triangle for\nξ=C, an aqua diamond for ξ= C′, and a black circle for\nξ=D.\nFirst, we examine the R 1and R 8regions. We present\nour DMRG results of /angbracketleftSz\ni,ξ/angbracketrightof the system of N= 112 in\nFigs. 7(a) and 7(b) for J2/J1= 0.2 and 1.9, respectively.\nIn Fig. 7(a), we observethe uniform behavior of upward-\ndirection spins at the sublattice sites A, A′, B, B′, and\nD, and downward-direction spins at the sublattice sites\nC and C′. In Fig. 7(b), we also observe the uniform\nbehavior of upward-direction spins at the sublattice sites\nB, B′, C, and C′, and downward-direction spins at the\nsublatticesites A, A′, andD. Therefore, weconcludethat\nthe LM ferrimagnetic states are realized in the regions of\nR1and R 8.\nOur understanding of the origins of these LM ferri-\nmagnetic phases is based on the Marshall-Lieb-Mattis\n(MLM) theorem. In the case of J2/J1= 0, no frustra-\ntion occurs; thus, the spin state depicted in Fig. 8(a) is\nrealized. This state shows the LM ferrimagnetism with\nM/Ms= 3/7. The R 1region is directly connected to the\nLM ferrimagnetic state of J2/J1= 0. Therefore, the R 1\nregion of M/Ms= 3/7 is regarded as the LM ferrimag-\nnetic phase. In the limit J2→ ∞, on the other hand, the\npresent model becomes equal to a model of an S= 1/2\ndiamond chain depicted in Fig. 8(b). The value of mag-\tB\n\tC\n%\"\n# $\n#` $` \n\"` \n5 10 15\ni–0.2 00.20.4<S i, ξz>N=112 J F/J 1=–1 J 2/J 1=0.2 M=24\n5 10 15\ni–0.2 00.20.4<S i, ξz>N=112 J F/J 1=–1 J 2/J 1=1.9 M=8 \nFIG. 7: (Color) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublattice\nξ. Panels (a) and (b) show results for J2/J1=0.2 and 1.9,\nrespectively. These results are obtained from our DMRG cal-\nculations for N= 112 (i=1, 2,···, 16). The inset in the\npanel (a) presents the correspondence relationship betwee n\neach colored symbol and each sublattice ξused in Figs. 7, 9,\nand 10.\nnetization takes M/Ms= 1/7 in the ground state of this\ndiamond chain according to the MLM theorem. There-\nfore, the R 8region of M/Ms= 1/7 is regarded as the LM\nferrimagnetic phase.\nNext, we investigate the R 3, R5, and R 7regions. Our\nresultsobtainedfromthe DMRG calculationsof N= 168\nare depicted in Figs. 9(a)-9(c) for J2/J1= 0.57, 1.14,\nand 1.69, respectively. We clearly observe incommensu-\nratemodulationswithlong-distanceperiodicitiesinevery\ncase in Fig. 9. In addition, we confirm from Fig. 4(b)\nthat the ratio M/Mschanges gradually with the varia-\ntion inJ2/J1in the R 3, R5, and R 7regions. Since the\nwidths of the R 3and R 5regions survive in the thermo-\ndynamic limit as was clarified in the previous subsection,\nwe conclude that the R 3and R 5regions are NLM ferri-\nmagnetic phases. Although it is unclear whether the R 7\nregion survives in the thermodynamic limit, this region\nis an NLM ferrimagnetic phase if it survives.\nFinally, in this subsection, we examine the R 4region.\nOur result of /angbracketleftSz\ni,ξ/angbracketrightforJ2/J1= 1 in the system of\nN= 168 is depicted in Fig. 10. We do not detect the in-\ncommensuratemodulation in this R 4region. In addition,\nwe confirm from Fig. 4(b) that the ratio M/Msin the\nR4region does not change with the variation in J2/J1in\ncontrast to the cases in the R 3and R 5regions. These7\n%\n\"` #` \n$` \"\n#\n$\n$\n$` \tB\n\tC\nFIG. 8: (a) Kagome strip lattice depicted in Fig. 1(b) in the\nlimit of J2/J1= 0. White arrowheads denote the classical\nspin configuration in the LM ferrimagnetic state of M/Ms=\n3/7. (b) Kagome strip lattice depicted in Fig. 1(b) in the\nlimit of J2→ ∞. White circles represent the effective S=\n1/2 spins formed by five S= 1/2 spins at the sublattices\nξ=A, A′, B, B′, and D. The black bold and black dotted lines\nshow the antiferromagnetic and ferromagnetic interaction s,\nrespectively.\ncircumstances suggest that the R 4region is the LM ferri-\nmagnetic phase. However, one cannot speculate the spin\nstate from the result of /angbracketleftSz\ni,ξ/angbracketrightbecause it is difficult to\nclarify the spin state on the basis of the results of deter-\nmining whether the spins are up or down, as was success-\nfully observed in the R 8region. We further investigate\nthe magnetization curve in this region to know whether\nthe R4region is the LM ferrimagnetic phase. The mag-\nnetization curves of J2/J1= 1 forN= 56 and N= 112\ncalculated by the DMRG method are presented in Fig.\n11(a)34. Figure 11(b) is obtained by zooming the region\nofhnearh= 0. One can observe the existence of the\nmagnetization plateaus at the height of the spontaneous\nmagnetization for both system sizes. We also confirm\nthat the difference in the width of the plateau between\nN= 112 and 56 is very small. These features indicate\nthat the spin gap exists in the R 4region in the thermo-\ndynamic limit. If the R 4region is the NLM ferrimagnetic\nphase, no spin gap is present in this region. (It was re-\nported in ref. VI that the NLM ferrimagnetism is gapless\nas a response to a uniform magnetic field.) Therefore,\nwe conclude that the R 4region is the LM ferrimagnetic\nphase.\nV. DISCUSSION\nWe discuss the relationships between the interval 0 <\nJ2/J1≤1 of the kagome strip lattice depicted in Fig.\n1(b) and those of the spatially anisotropic kagome lat-\tB\n\tC\n\tD\n10 20 \ni–0.2 00.20.4<S i, ξz>\nN=168 J F/J 1=–1 J 2/J 1=0.57 M=28 \n10 20 \ni–0.2 00.20.4<S i, ξz>\nN=168 J F/J 1=–1 J 2/J 1=1.14 M=7\n10 20 \ni–0.2 00.20.4<S i, ξz>N=168 J F/J 1=–1 J 2/J 1=1.69 M=2\nFIG. 9: (Color) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublat-\nticeξ. The correspondence relationship between each colored\nsymbol and each sublattice ξis described in the inset in Fig.\n7(a). Panels (a), (b), and (c) show results for J2/J1=0.57,\n1.14, and 1.69, respectively. These results are obtained fr om\nour DMRG calculations for N= 168 (i=1, 2,···, 24).\ntice studied in ref. VI, while we consider the case of\nanother new lattice, with a larger but finite width along\nthe direction perpendicular to the bonds of interaction\nJFin Fig. 1(b), namely, the strip width.\nThe ratio M/Msof the R 1and R2regionsis commonly\nM/Ms= 3/7 in the thermodynamic limit of the lattice\ndepicted in Fig. 1(b). The difference of M/Ms= 3/7\nfromM/Ms= 1/3 in the case of the LM ferrimagnetic\nphase of the spatially anisotropic kagome lattice in ref.\nVIis attributedtothe finitenessofthestripwidth. Thus,\nthe ratio M/Msapproaches M/Ms= 1/3 when the strip8\n10 20 \ni–0.2 00.20.4<S i, ξz>\nN=168 J F/J 1=–1 J 2/J 1=1 M=11\nFIG. 10: (Color) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublattice\nξ. This result is obtained from our DMRG calculations for\nJ2/J1= 1 in the system of N= 168 (i=1, 2,···, 24). The\ncorrespondence relationship between each colored symbol a nd\neach sublattice ξis described in the inset in Fig. 7(a).\nwidth increases; the R 1and R 2regions of the present\nmodel depicted in Fig. 1(b) correspond to the LM ferri-\nmagnetic phase of the spatially anisotropic kagome lat-\ntice studied in ref. VI.\nThe ratio M/Ms= 1/7 atJ2/J1= 1 in the present\nmodel depicted in Fig. 1(b) may be related to the fact\nthat the model includes seven sublattices, namely, the\nstrip width is finite. At least at J2/J1= 1, on the other\nhand, it is widely believed that the spontaneous mag-\nnetization disappears in the limit of infinite width17–27.\nAlthough the relationship should be clarified in the ex-\namination of the case of the lattice with an even larger\nstrip width, there are two possibilities of the behavior\nnear the case of J2/J1= 1. One is the case when the\nratioM/Msdecreases with increasing strip width and fi-\nnally vanishes, while the LM ferrimagnetic phase, such\nas R4, survives in systems with larger strip widths. In\nthis case, the R 3region corresponds to the NLM phase\nofthe two-dimensionalmodel onthe spatiallyanisotropic\nkagome lattice. In the other case, the LM ferrimagnetic\nphase, such as R 4, becomes narrower with increasing\nstrip width, while the ratio M/Msdoes not vanish; fi-\nnally, the R 3and R 5regions merge with each other. In\nthis case, the value of J2/J1at the boundary between\nthe R5and R 6regions decreases across J2/J1= 1. In\nany cases, it is important to note that from our finding\nof an intermediate phase in all the three cases in Fig. 1\nbetween the ferrimagnetic phase and the nonmagnetic\nphase, the phase is considered to exist irrespective of the\nstrip width.\nFinally, one should note that the intermediate phase\nbetween the Lieb-Mattis ferrimagnetic and nonmagnetic\nstates is observed in other cases. However, this phase\nis not always ferrimagnetic. One of the cases observed is\nthecaseofthethree-legladdersystemformingastriplat-\ntice obtained by cutting off from the spatially anisotropic\ntriangular lattice in ref. VI, in which the properties of\ntheintermediatephasewereunclear. Sakai’sunpublished\tB\n\tC\n0 1 2 3\nh00.51M/M skagome strip in Fig. 1(b)   \nN=56  J F/J 1=–1 J 2/J 1=1 \nN=112 J F/J 1=–1 J 2/J 1=1 \n0 0.2 0.4\nh00.10.20.3M/M sN=56 \nN=112\nFIG. 11: Magnetization curve in the R 4region. Panel (a) is\nobtained from the DMRG calculations of J2/J1= 1 forN=\n56 (triangle) and N= 112 (square). Panel (b) is obtained by\nzooming the region near h= 0 in panel (a).\nstudy suggests that the intermediate phase is nematic38.\nCareful examinations are required to investigate such an\nintermediate phase if it is found.\nVI. SUMMARY\nWe have studied the ground-state properties of the\nS= 1/2 Heisenberg models on the kagome strip lattices\ndepicted in Figs. 1(b) and 1(c) by the ED and DMRG\nmethods. As a common phenomenon in the ground state\nofboth cases,wehaveconfirmedthe existenceofthe non-\nLieb-Mattis ferrimagnetism between the Lieb-Mattis fer-\nrimagnetic phase and the nonmagnetic phase. We have\nclearly found incommensurate modulations with long-\ndistance periodicity in the non-Lieb-Mattis ferrimagnetic\nstate. The occurrence of the non-Lieb-Mattis ferrimag-\nnetism irrespective of strip width strongly suggests that\nthe intermediate state found in the case of the spatially\nanisotropic kagome lattice in two dimensions is the non-\nLieb-Mattis ferrimagnetism.9\nAcknowledgments\nWe thank Prof. Toru Sakai for letting us know his\nunpublished results. This work was partly supported\nby Grants-in-Aid (Nos. 20340096, 23340109, 23540388,\nand 24540348) from the Ministry of Education, Culture,\nSports, Science and Technology of Japan. This work\nwas partly supported by a Grant-in-Aid (No. 22014012)\nfor Scientific Research and Priority Areas “Novel Statesof Matter Induced by Frustration” from the Ministry of\nEducation, Culture, Sports, Science and Technology of\nJapan. Diagonalization calculations in the present work\nwere carried out using TITPACK Version 2 coded by\nH. Nishimori. DMRG calculations were carried out us-\ning the ALPS DMRG application39. 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B78(2008) 014418.\n12T.ShimokawaandH.Nakano: J. Phys.Soc.Jpn. 80(2011)\n043703.\n13T.ShimokawaandH.Nakano: J. Phys.Soc.Jpn. 80(2011)\n125003.\n14N. B. Ivanov, J. Richter, and D. J. J. Farnell: Phys. Rev.\nB66(2002) 014421.\n15R. F. Bishop, P. H. Y. Li, D. J. J. Farnell, and C. E.\nCampbell: Phys. Rev. B 82(2010) 024416.\n16H. Nakano, T. Shimokawa, and T. Sakai: J. Phys. Soc.\nJpn.80(2011) 033709.\n17P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and\nP. Sindzingre: Phys. Rev. B 56(1997) 2521.\n18Ch. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuil-\nlier, P. Sindzingre, P. Lecheminant, and L. Pierre:\nEur. Phys. J. B 2(1998) 501.\n19K. Hida: J. Phys. Soc. Jpn. 70(2001) 3673.\n20D. C. Cabra, M. D. Grynberg, P. C. W. Holdsworth, and\nP. Pujol: Phys. Rev. B 65(2002) 094418.\n21J. Schulenberg, A. Honecker, J. Schnack, J. Richter, and\nH.-J. Schmidt: Phys. Rev. Lett. 88(2002) 0167207.\n22A.Honecker, J.Schulenberg, andJ.Richter: J.Phys.:Con-\ndens. Matter 16(2004) S749.\n23O. Cepas, C. M. Fong, P. W. Leung, and C. Lhuillier:\nPhys. Rev. B 78(2008) 140405(R).24P. Sindzingre and C. Lhuillier: Europhys. Lett. 88(2009)\n27009.\n25H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 79(2010)\n053707.\n26T. Sakai and H. Nakano: Phys. Rev. B 83(2011)\n100405(R).\n27H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80(2011)\n053704.\n28P. Azaria, C. Hooley, P. Lecheminant, C. Lhuillier, and A.\nM. Tsvelik: Phys. Rev. Lett. 81(1998) 1694.\n29Ch. Waldtmann, H. Kreutzmann, U. Schollwock, K.\nMaisinger, andH.-U. Everts: Phys. Rev.B 62(2000) 9472.\n30J. Schulenburg, A. Honecker, J. Schnack, J. Richter, and\nH.-J. Schmidt: Phys. Rev. Lett. 88(2002) 167207.\n31T. Shimokawa and H. Nakano: J. Phys.: Conf. Ser. 320\n(2011) 012007.\n32S. R. White: Phys. Rev. Lett. 69(1992) 2863.\n33S. R. White: Phys. Rev. B 48(1993) 10345.\n34One can find the macroscopic jump in this magnetization\ncurvenear the saturation fieldas well as in the case ofsome\nfrustrated systems30,35. It is known that the occurrence\nmechanism of the magnetization jump can be understood\nfrom the viewpoint of the localized-magnon picture (see\nrefs. VI, VI, and VI).\n35J. Schnack, H.-J. Schmidt, A. Honecker, J. Schulenburg,\nand J. Richter: J. Phys.: Conf. Ser. 51(2006) 43.\n36M. E. Zhitomirsky and H. Tsunetsugu: Phys. Rev. B 70\n(2004) 100403.\n37S. Abe, T. Sakai, K. Okamoto, and K. Tutsui: J. Phys.:\nConf. Ser. 320(2011) 012015.\n38T. Sakai: private communications.\n39A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A.\nFeiguin, L. Gamper, E. Gull, S. Gurtler, A. Honecker, R.\nIgarashi, M. Korner, A. Kozhevnikov, A. Lauchli, S. R.\nManmana, M. Matsumoto, I. P. McCulloch, F. Michel, R.\nM. Noack, G. Pawlowski, L. Pollet, T. Pruschke, U. Scholl-\nwock, S. Todo, S. Trebst, M. Troyer, P. Werner, and S.\nWessel: J. Magn. Magn. Mater. 310(2007) 1187 (see also\nhttp://alps.comp-phys.org)."    },    {        "title": "1803.00235v1.Calculating_the_Magnetic_Anisotropy_of_Rare_Earth_Transition_Metal_Ferrimagnets.pdf",        "content": "Calculating the Magnetic Anisotropy of\nRare-Earth|Transition-Metal Ferrimagnets\nChristopher E. Patrick,1,\u0003Santosh Kumar,1Geetha Balakrishnan,1Rachel\nS. Edwards,1Martin R. Lees,1Leon Petit,2and Julie B. Staunton1\n1Department of Physics, University of Warwick,\nCoventry CV4 7AL, United Kingdom\n2Daresbury Laboratory, Daresbury, Warrington WA4 4AD, United Kingdom\n(Dated: March 2, 2018)\nAbstract\nMagnetocrystalline anisotropy, the microscopic origin of permanent magnetism, is often ex-\nplained in terms of ferromagnets. However, the best performing permanent magnets based on rare\nearths and transition metals (RE-TM) are in fact ferrimagnets, consisting of a number of mag-\nnetic sublattices. Here we show how a naive calculation of the magnetocrystalline anisotropy of\nthe classic RE-TM ferrimagnet GdCo 5gives numbers which are too large at 0 K and exhibit the\nwrong temperature dependence. We solve this problem by introducing a \frst-principles approach\nto calculate temperature-dependent magnetization vs. \feld (FPMVB) curves, mirroring the exper-\niments actually used to determine the anisotropy. We pair our calculations with measurements on\na recently-grown single crystal of GdCo 5, and \fnd excellent agreement. The FPMVB approach\ndemonstrates a new level of sophistication in the use of \frst-principles calculations to understand\nRE-TM magnets.\n1arXiv:1803.00235v1  [cond-mat.mtrl-sci]  1 Mar 2018High-performance permanent magnets, as found in generators, sensors and actuators, are\ncharacterized by a large volume magnetization and a high coercivity [1]. The coercivity |\nwhich measures the resistance to demagnetization by external \felds | is upper-bounded\nby the material's magnetic anisotropy [2], which in qualitative terms describes a preference\nfor magnetization in particular directions. Magnetic anisotropy may be partitioned into\ntwo contributions: the shape anisotropy, determined by the macroscopic dimensions of the\nsample, and the magnetocrystalline anisotropy (MCA), which depends only on the mate-\nrial's crystal structure and chemical composition. Horseshoe magnets provide a practical\ndemonstration of shape anisotropy, but the MCA is less intuitive, arising from the relativistic\nquantum mechanical coupling of spin and orbital degrees of freedom [3].\nPermanent magnet technology was revolutionized with the discovery of the rare-earth/transition-\nmetal (RE-TM) magnet class, beginning with Sm-Co magnets in 1967 [4] (whose high-\ntemperature performance is still unmatched [5]), followed by the world-leading workhorse\nmagnets based on Nd-Fe-B [6, 7]. With the TM providing the large volume magnetization,\ncareful choice of RE yields MCA values which massively exceed the shape anisotropy con-\ntribution [8]. RE-TM magnets are now indispensable to everyday life, but their signi\fcant\neconomic and environmental cost has inspired a global research e\u000bort aimed at replacing\nthe critical materials required in their manufacture [9].\nIn order to perform a targeted search for new materials it is necessary to fully understand\nthe huge MCA of existing RE-TM magnets. An impressive body of theoretical work based\non crystal \feld theory has been built up over decades [10], where model parameters are\ndetermined from experiment (e.g. Ref. [11]) or electronic structure calculations [12{14]. An\nalternative and increasingly more common approach is to use these electronic structure\ncalculations, usually based on density-functional theory (DFT), to calculate the material's\nmagnetic properties directly without recourse to the crystal \feld picture [15{19].\nCalculating the MCA of RE-TM magnets presents a number of challenges to electronic\nstructure theory. The interaction of localized RE-4 felectrons with their itinerant TM\ncounterparts is poorly described within the most widely-used \frst-principles methodology,\nthe local spin-density approximation (LSDA) [12]. Indeed, the MCA is inextricably linked to\norbital magnetism whose contribution to the exchange-correlation energy is missing in spin-\nonly DFT [20, 21]. MCA energies are generally a few meV per formula unit, necessitating\na very high degree of numerical convergence [22]. Finally, the MCA depends strongly on\n2temperature, so a practical theory of RE-TM magnets must go beyond zero-temperature\nDFT and include thermal disorder [23].\nEven when these signi\fcant challenges have been overcome, there is a more fundamental\nproblem. Experiments access the MCA indirectly, measuring the change in magnetization of\na material when an external \feld is applied in di\u000berent directions. By contrast, calculations\nusually access the MCA directly by evaluating the change in energy when the material is\nmagnetized in di\u000berent directions, with no reference to an external \feld. These experimental\nand computational approaches arrive at the same MCA energy provided one is studying a\nferro magnet. However, the majority of RE-TM magnets (and many other technologically-\nimportant magnetic materials) are ferrimagnets, i.e. they are composed of sublattices with\nmagnetic moments of distinct magnitudes and orientations. Crucially the application of\nan external \feld may introduce canting between these sublattices, a\u000becting the measured\nmagnetization. Thus the standard theoretical approach of ignoring the external \feld is hard\nto reconcile with real experiments on ferrimagnets.\nIn this Letter, through a combination of calculations and experiments, we provide the\nhitherto missing link between electronic structure theory and practical measurements of the\nMCA. Speci\fcally, we show how to directly simulate experiments by calculating, from \frst\nprinciples (FP), how the measured magnetization ( M) varies as a function of \feld ( B) applied\nalong di\u000berent directions and at di\u000berent temperatures. We apply our \\FPMVB\" approach\nto the RE-TM ferro and ferrimagnets YCo 5and GdCo 5, which are isostructural to the\ntechnologically-important SmCo 5[24] and, in the case of GdCo 5, a source of controversy in\nthe literature [25{35]. Pairing FPMVB with new measurements of the MCA of GdCo 5allows\nus to resolve this controversy. More generally, FPMVB enables a new level of collaboration\nbetween theory and experiment in understanding the magnetic anisotropy of ferrimagnetic\nmaterials.\nThe electronic structure theory behind FPMVB treats magnetic disorder at a \fnite tem-\nperatureTwithin the disordered local moment (DLM) picture [36, 37]. The methodology\nallows the calculation of the magnetization of each sublattice i,M i(T) =Mi(T)^M i, and the\ntorque quantity @F(T)=@^M i, whereFis an approximation to the temperature dependent\nfree energy. @F(T)=@^M iaccounts for the anisotropy arising from the spin-orbit interaction,\nwhile the contribution from the classical magnetic dipole interaction is computed numer-\nically [38]. Many of the technical details of the DFT-DLM calculations [36, 39{43] were\n3FIG. 1. Data points and \fts of dF=d\u0012 calculated for GdCo 5(blue, empty symbols; Gd and Co\nmoments held antiparallel) and YCo 5(green, \flled symbols), at 0 and 300 K.\ndescribed in our recent study of the magnetization of the same compounds [44]; the exten-\nsions to calculate the torques are described in Ref. [37]. The Gd-4 felectrons are treated\nwith the local self-interaction correction [43], and we have also implemented the orbital\npolarization correction [20] following Refs. [45, 46] using reported Racah parameters [47].\nDetails are given as Supplemental Material (SM) [48].\nYCo 5and GdCo 5crystallize in the CaCu 5structure, consisting of alternating hexagonal\nRCo 2c/Co 3glayers [24]. Y is nonmagnetic, while in GdCo 5the large spin moment of Gd\n(originating mainly from its half-\flled 4 fshell) aligns antiferromagnetically with the Co\nmoments. We now consider a \\standard\" calculation of the MCA based on a rigid rotation\nof the magnetization. If the Gd and Co moments are held antiparallel, GdCo 5is e\u000bectively\na ferromagnet with reduced moment MCo\u0000MGd. Then, from the hexagonal symmetry we\nexpect the angular dependence of the free energy to follow \u00141sin2\u0012+\u00142sin4\u0012+O(sin6\u0012),\nwhere\u0012is the polar angle between the crystallographic caxis and the magnetization direc-\ntion. The constants \u00141;\u00142determine the change in free energy \u0001 F, calculated e.g. from the\nforce theorem [49] or the torque dF=d\u0012 [50].\nIn Fig. 1 we show dF=d\u0012 calculated for ferromagnetic YCo 5and GdCo 5at 0 and 300 K.\nFitting the data to the derivative of the textbook expression, sin 2 \u0012(\u00141+ 2\u00142sin2\u0012), \fnds\u00141\nand\u00142to be positive (easy caxis) with\u00141an order of magnitude larger than \u00142. Considering\nexperimentally measured anisotropy constants in the literature, for YCo 5our\u00141value of\n3.67 meV (all energies are per formula unit, f.u.) at 0 K compares favorably to the values of\n3.6 and 3.9 meV reported in Refs. [28] and [51]. At 300 K, our value of 2.19 meV exhibits a\nslightly faster decay with temperature compared to experiment (2.6 and 3.0 meV), which we\nattribute to our use of a classical spin hamiltonian in the DLM picture [36, 44]. However, for\nGdCo 5our calculated values of \u00141show very poor agreement with experiments [26, 29]. First,\n4at 0 K we \fnd \u00141to be larger than YCo 5(4.26 meV), while experimentally the anisotropy\nconstant is much smaller (1.5, 2.1 meV). Second, we \fnd \u00141decreases with temperature\n(2.39 meV at 300 K) while experimentally the anisotropy constant increases (2.7, 2.8 meV).\nTo understand these discrepancies we must ask how the anisotropy energies were actually\nmeasured. Torque magnetometry provides an accurate method of accessing the MCA [52],\nbut is technically challenging in RE-TM magnets, which require very high \felds to reach\nsaturation [53]. Singular point detection [54] and ferromagnetic resonance [55] has also\nbeen used to investigate the MCA of polycrystalline and thin-\flm samples. However, the\nmost commonly-used method for RE-TM magnets, employed in Refs. [26, 29], is based on\nthe seminal 1954 work by Sucksmith and Thompson [56] on the anisotropy of hexagonal\nferromagnets. This work provides a relation between the measured magnetization Mab\nand \feldBapplied in the hard plane in terms of \u00141,\u00142and the easy axis magnetization\nM0[48, 56]:\n(BM 0=2)=(Mab=M0)\u0011\u0011=\u00141+ 2\u00142(Mab=M0)2: (1)\nFurther introducing m= (Mab=M0), equation 1 shows that a plot of \u0011againstm2should\nyield a straight line with \u00141as the intercept. Even though this \\Sucksmith-Thompson\nmethod\" was derived for ferromagnets, the technical procedure of plotting \u0011againstm2can\nbe performed also for ferrimagnets like GdCo 5[26, 29]. In this case, the quantity extracted\nfrom the intercept is an e\u000bective anisotropy constant Ke\u000bso, unlike YCo 5, the anisotropy\nconstants reported in Refs. [26, 29] are distinct from the \u00141values extracted from Fig. 1. As\nrecognized at the time of the original experiments [27{30], the reduced value of Ke\u000bwith\nrespect to\u00141of YCo 5is a \fngerprint of canting between the Gd and Co sublattices.\nMaking contact with previous experiments thus requires we obtain Ke\u000b. To this end\nwe have developed a scheme of calculating \frst-principles hard-plane magnetization vs. \feld\n(FPMVB) curves, on which we perform the Sucksmith-Thompson analysis to directly mirror\nthe experiments. The central concept of FPMVB is that at equilibrium, the torques from\nthe exchange, spin-orbit and dipole interactions must balance those arising from the external\n\feld. Then,\nB=@F(T)\n@\u0012i1\nMicos\u0012i+P\njsin\u0012j@M j\n@\u0012i: (2)\nThe magnetization at a given B;T is determined by the angle set f\u0012Gd;\u0012Co1;\u0012Co2;:::gwhich\nsatis\fes equation 2 for every magnetic sublattice. The spin-orbit interaction breaks the\n5FIG. 2. Magnetization of GdCo 5vs. applied magnetic \feld shown on a standard plot (left panel)\nor after the Sucksmith-Thompson analysis (eq. 1, right panel). Crosses/circles are calculated with\nmethods (i)/(ii) discussed in the text, and the area between them shaded as a guide to the eye.\nNote the two methods are e\u000bectively indistinguishable in the left panel. The dashed/solid lines are\ncalculated from the model free energies F1andF2. The right panel also shows the geometry of the\nmagnetization and \feld with respect to the crystallographic c-axis (thick gray arrow).\nsymmetry of the Co 3gatoms such that altogether there are four independent angles to vary\nfor GdCo 5. The second term in the denominator of equation 2 re\rects that the magnetic\nmoments themselves might depend on \u0012i(magnetization anisotropy). We have tested (i)\nneglecting this contribution and (ii) modeling the dependence as Mi(\u0012i) =M0i(1\u0000pisin2\u0012i),\nwhereM0iandpiare parameterized from our calculations.\nFigure 2 shows FPMVB curves of GdCo 5calculated using equation 2 with methods (i)\nand (ii), (crosses and circles) which yield virtually identical values of Ke\u000b. TheMvs.B\ncurves in the left panel resemble those of a ferromagnet where, as the temperature increases,\nit becomes easier to rotate the moments away from the easy axis so that a given B\feld\ninduces a larger magnetization. However, plotting \u0011againstm2in the right panel tells a\nmore interesting story. The e\u000bective anisotropy constant Ke\u000b(y-axis intercept) at 0 K is\n1.53 meV, much smaller than \u00141of YCo 5. Furthermore Ke\u000bincreases with temperature,\nto 1.74 meV at 300 K. Therefore, in contrast to the standard calculations of Fig. 1, the\nFPMVB approach reproduces the experimental behavior of Refs. [26, 29].\n6Our FPMVB calculations provide a microscopic insight into the magnetization process.\nFor instance at 0 K and 9 T, we calculate that the cobalt moments rotate away from the\neasy axis by 6.1\u000e. By contrast the Gd moments have rotated by only 3.9\u000e, i.e. the ideal\n180\u000eGd-Co alignment has reduced by 2.2\u000e(the geometry is shown in Fig. 2). We also \fnd\ncanting between the di\u000berent Co sublattices, but not by more than 0.1\u000eat both 0 and 300 K\n(the calculated angles as a function of \feld are shown in the SM [48]). This Co-Co canting\nis small thanks to the Co-Co ferromagnetic exchange interaction, which remains strong over\na wide temperature range [44]. The temperature dependence of Ke\u000bcan be traced to the\nfact that the easy axis magnetization M0of GdCo 5initially increases with temperature [44].\nEven ifMabincreases with temperature at a given \feld, a faster increase in M0can lead to\nan overall hardening in Ke\u000b(equation 1).\nWe assign the canting in GdCo 5to a delicate competition between the exchange interac-\ntion favoring antiparallel Co/Gd moments, uniaxial anisotropy favoring c-axis (anti)alignment,\nand the external \feld trying to rotate all moments into the hard plane. We can quantify\nthese interactions by looking for a model parameterization of the free energy F. Crucially\nwe can train the model with an arbitrarily large set of \frst-principles calculations exploring\nsublattice orientations not accessible experimentally, and test its performance against the\ntorque calculations of equation 2. Neglecting the 0.1\u000ecanting within the cobalt sublattices\ngives two free angles, \u0012Gdand\u0012Co. Including Gd-Co exchange A, uniaxial Co anisotropy\nK1;Coand a dipolar contribution S(\u0012Gd;\u0012Co) [31, 48] leads naturally to a two-sublattice\nmodel [30],\nF1(\u0012Gd;\u0012Co) =\u0000Acos(\u0012Gd\u0000\u0012Co) +K1;Cosin2\u0012Co\n+S(\u0012Gd;\u0012Co): (3)\nThe training calculations showed additional angular dependences not captured by F1, so we\nalso investigated:\nF2(\u0012Gd;\u0012Co) =F1(\u0012Gd;\u0012Co) +K2;Cosin4\u0012Co\n+K1;Gdsin2\u0012Gd: (4)\nAs discussed below the training calculations showed no strong evidence of Gd-Co exchange\nanisotropy [31{34].\nThe dashed (solid) lines in Fig. 2 are the calculated Mvs.Bcurves obtained by mini-\nmizingF1(2)\u0000P\niM i\u0001B. The second term includes magnetization anisotropy on the cobalt\n7FIG. 3. Anisotropy constants Ke\u000bvs. temperature of YCo 5(green) and GdCo 5(blue). The left\npanel shows calculations using equation 2 at 0 and 300 K (stars), or using parameterized model\nexpressions F1(diamonds) and F2(circles), and from Ref. [57] (YCo 5, squares). For GdCo 5we\nalso show in red \u00141extracted from \\standard\" calculations where the Gd and Co moments were\nheld rigidly antiparallel (cf. Fig. 1). The experimental data in the right panel was measured by\nus for GdCo 5(crosses, with shaded background) or taken from Refs. [26], [29] and [58] (squares,\ndashed lines, circles) and Refs [28] and [51] (green diamonds and dashed lines, YCo 5).\nmoments [48, 57]. On the scale of the left panel both F1andF2give excellent \fts to the\ntorque calculations, especially up to moderate \felds. The plot of \u0011againstm2reveals some\ndi\u000berences with F2giving a marginally improved description of the data, but F1already\ncaptures the most important physics.\nWe also applied the FPMVB approach to YCo 5, using equation 2 and the model for F\nintroduced in Ref. [57]. Then, parameterizing the models [48] over the temperature range\n0{400 K, calculating Mvs.Bcurves and extracting Ke\u000busing the Sucksmith-Thompson\nplots gives the results shown in the left panel of Fig. 3. We also show \u00141of GdCo 5to\nemphasize the di\u000berence between FPMVB calculations and the \\standard\" ones of Fig. 1.\nComparing Ke\u000bto previously-published experimental measurements on GdCo 5raises\nsome issues. First, the three studies in the literature report anisotropy constants which di\u000ber\nby as much as 1 meV [26, 29, 58]. Indeed there was controversy over whether the observed\nresults were evidence of an anisotropic exchange interaction between Gd and Co [31, 32] or\n8an artefact of poor sample stoichiometry [33, 34]. Furthermore the only study performed\nabove room temperature [26] reports without comment some peculiar behavior where Ke\u000b\nof GdCo 5exceeds that of YCo 5at high temperature [28], despite conventional wisdom that\nthe half-\flled 4 fshell of Gd does not contribute to the anisotropy.\nOur calculations do in fact show an excess in the rigid-moment anisotropy of GdCo 5of\n16% at 0 K (Fig. 1) compared to YCo 5. The authors of Refs. [29, 31] \ftted their experimental\ndata with a much larger excess of 50%, while the high-\feld study of Ref. [33] found (11\n\u000615)%, with the authors of that work attributing the di\u000berence to an improved sample\nstoichiometry [34]. Our calculated excess at 0 K is formed from two major contributions:\nthe dipole interaction energy, which accounts for 0.31 meV/f.u., and K1;Gd(equation 4) which\nwe found to be 24% the size of K1;Co. The nonzero value of K1;Gdis due to the 5 delectrons,\nwhose presence is evident from the Gd magnetization (7.47 \u0016Bat 0 K). We did not \fnd a\nsigni\fcant contribution from anisotropic exchange, which we tested in two ways: \frst by\nattempting to \ft a term A(1\u0000p0sin2\u0012Co) cos(\u0012Gd\u0000\u0012Co) to our training set of calculations,\nand also by computing Curie temperatures with the (rigidly antiparallel) magnetization\ndirected either along the coraaxes. We found the magnitude of the anisotropy ( p0) to be\nsmaller than 0.5% and negative at 0 K, and to decrease in magnitude as the temperature\nis raised. Consistently the Curie temperature was found to be only 1 K higher for aaxis\nalignment, which we do not consider signi\fcant.\nHowever, our calculations do not predict the Ke\u000bvalue of GdCo 5to exceed YCo 5. Indeed,\nin Fig. 3\u00141of GdCo 5approaches that of YCo 5at high temperatures, which is signi\fcant\nbecause\u00141provides an upper bound for Ke\u000b[32]. To resolve this \fnal puzzle we performed\nour own measurements of Ke\u000bon the single crystal whose growth we reported recently [44].\nHard and easy axis magnetization curves up to 7 T were measured in a Quantum Design\nsuperconducting quantum interference device (SQUID) magnetometer, and the anisotropy\nconstants extracted from Sucksmith-Thompson plots [48]. The right panel of Fig. 3 shows\nour newly measured data as crosses. Previously reported measurements are shown in faint\nblue/green for GdCo 5[26, 29, 58]/YCo 5[28, 51].\nUp to 200 K, there is close agreement between the experiments of Ref. [26], our own ex-\nperiments, and the FPMVB calculations. Above this temperature our new experiments show\nthe expected drop in Ke\u000b, while the previously reported data show a continued rise [26]. We\nrepeated our measurements using di\u000berent protocols and found a reasonably large variation\n9in the extracted Ke\u000b[48]. Even taking this variation into account as the shaded area in\nFig. 3, the drop is still observed.\nWe therefore do not believe the high temperature behavior reported in Ref. [26] has\nan intrinsic origin. Possible extrinsic factors include the method of sample preparation,\ndegradation of the RCo 5phase at elevated temperatures [59], and potential systematic error\nwhen extracting Ke\u000b. We note that even the idealized theoretical curves in Fig. 2 show\ncurvature at higher temperature, making it more di\u000ecult to \fnd the intercept.\nIn conclusion, we have introduced the FPMVB approach to interpret experiments mea-\nsuring anisotropy of ferrimagnets, particularly RE-TM permanent magnets. We presented\nthe method in the context of our DLM formalism, but any electronic structure theory ca-\npable of calculating magnetic couplings relativistically [60{64] should be able to produce\nFPMVB curves, at least at zero temperature. However standard calculations which neglect\nthe external \feld should be used with care when comparing to experiments on ferrimagnets.\nSimilarly, the prototype GdCo 5serves as a reminder that a simple view of the anisotropy\nenergy does not fully describe the magnetization processes in ferrimagnets, which might have\nimplications in understanding e.g. magnetization reversal in nano-magnetic assemblies [65].\nOverall our work demonstrates the bene\ft of interconnected computational and experimen-\ntal research in this key area.\nThe present work forms part of the PRETAMAG project, funded by the UK Engineer-\ning and Physical Sciences Research Council (EPSRC), Grant no. EP/M028941/1. 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Lett. 81, 2029\n(2002).\n13"    },    {        "title": "1905.03521v1.Bidirectional_spin_wave_driven_domain_wall_motion_in_antiferromagnetically_coupled_ferrimagnets.pdf",        "content": "1 \n Bidirectional spin-wave -driven domain wall motion in \nantiferromagnetically coupled ferrimagnets  \nSe-Hyeok Oh1, Se Kwon Kim2, Jiang  Xiao3,4,5, and Kyung- Jin Lee1,6,7 * \n1Department of Nano -Semiconductor and Engineering, Korea University, Seoul 02841, Korea  \n2Department of Physics and Astronomy, University of Missouri, Columbia , Missouri 65211 , USA \n3Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shang hai \n200433, China  \n4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China  \n5Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, Shanghai 200433, \nChina  \n6Department of Materials Science and Engineering, K orea University, Seoul 02841, Korea  \n7KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, \nKorea  \n \n* corresponding email: kj_lee@korea.ac.kr   \n   2 \n Abstract  \nWe investigate  ferrimagnetic domain wall dynamics induced by circular ly polarized \nspin wave s theoretically and numerically . We find that the direction of domain wall \nmotion depends on both the circular polarization of spin  wave s and the sign of net spin \ndensity of ferrimagnet. Below the angular momentum compensation point, left - (right-) \ncircularly  polarized spin wave s push a domain wall toward s (away from ) the spin -wave \nsource. Above the angular momentum compensation point, on th e other hand, the \ndirection of domain wall motion is reversed . This bidirectional motion  originate s from  \nthe fact that the  sign of spin -wave -induced magnon ic torque depends on the circular \npolarization and the subsequent response of the domain wall to the magnonic torque is \ngoverned by the net spin density . Our finding provides a way to utilize a spin wave as a \nversatile driving force for bidirectional domain wall motion . \n \nI. Introduction \nOne class of ferrimagnet s of emerging interest is a  rare- earth (RE) -transition metal (TM) \ncompound where the RE and TM moments are coupled antiferromagnetically. Owing to  \ndifferent Land é-g factors between the RE and TM elements,  RE-TM ferr imagnet s exhibit two \nunique compensation temperatures : the magnetic moment compensation  temperature 𝑇𝑇𝑀𝑀 at \nwhich  net magnetic moment vanishes , and the  angular momentum compensation temperature \n𝑇𝑇𝐴𝐴 at which  net angular momentum vanishes  [1-3].  \nResearch on ferr imagnetic materials ha s focused on the understanding of their \nfundamental magnetism [4]  and optical switching of magnetization [5 -11]. Recently, RE -TM \nferrimagnet s attract renewed interest as they offer a material platform to investigate the \nantiferromagnetic spintronics [ 12-15]. Compared to ferromagnets that have served as  core \nmaterial s for spintronics research, antiferromagnets exhibit several distinct features such as \nthe immunity to external field perturbations and fast dynamics due to antiferromagnetic \nexchange interaction.  However, the external -field immunity of true antiferromagnets results \nin the experimental difficulty in both creating and controlling  antiferromagnetic textures. On \nthe other hand, RE -TM ferrimagnets have finite magnetic moment at the angular momentum 3 \n compensation point  at which the antiferromagnetic dynamics is realized. As a result, \npreviously established creation and detection schemes for ferromagnets is  directly appli cable  \nto RE -TM ferrimagnets. This simple but strong benefit of RE -TM ferrimagnets have recently \ninitiated extensive studies  on ferrimagnets, which include magnetization switching [ 16-19], \ndomain wall motion [ 20-24], skyrmion (or bubble domain) motion [ 25-28], low damping [ 29], \nand efficient spin -transfer and spin- orbit torque s due to antiferromagnetic alignment of \natomic spins  [30,31 ]. \nAmong the previous studies listed above, the low damping of RE -TM ferrimagnets [2 9] \nis of particular interest from the view point of magnonic applications based on ferrimagnets \nbecause it enables  a long -distance propagation of spin waves (SWs) . For ferromagnets [ 32-39] \nand antiferromagnets [ 40-43], it was reported that a SW can move a DW by transferring its  \nangular momentum  or linear momentum . Though the SW property in ferrimagnet s was \nestablished  [44-47], the effect of SWs  on ferrimagnetic domain wall  (DW) motion remains  \nunexplored. In comparison to ferromagnets and antiferromagnets, antiferromagnetically \ncoupled ferrimagnets exhibit  a distinguishing feature of  SW eigenmode s. In ferromagnet s, a \nspin wave (SW) with only one type of polarization is permitted , which drives a DW toward s \nthe SW source through the angular momentum trans fer [3 4-37]. In antiferromagnet s, however, \nboth the left - and right -circular ly polarized SWs are allowed and energetically degenerate,  \nwhich can transfer the linear momentum to a DW through the SW  reflection [40,41,43] , \nresulting in the  DW motion away from the SW source . In antiferromagnetically coupled \nferrimagnet s, on the other hand, the degeneracy of the  two circular ly polarized SWs can be  \nlifted  depending on the net spin density of ferrimagnet . Given that SW -induced DW motion \nin ferrimagnets has been unexplored, interesting and important question s remain unanswered : \nhow a SW m oves a ferrimagnetic DW  and what the role of circular polarization of SW is . \nIn this paper, we study the dynamics of a  ferrimagnetic DW induced by a SW in the \nvicinity of the ang ular momentum compensation temperature 𝑇𝑇𝐴𝐴. We inve stigate DW \ndynamics induced by left - and right -circular ly polarized SWs [see Fig. 1(a) for an illustration \nof the two eigenmodes ]. We begin with theoretical analysis based on the Lagrangian density \nand SW dispersion. We then conduct numerical  simulation based on the atomistic Landau -\nLifshitz -Gilbert (LLG) equation to confirm the analytical results . Our model system is shown  4 \n in Fig. 1(b).  \n \nII. Model  \nOur model system is a simple bipartite ferrimagnet which consists of two sublattices \nlabeled by A and B. We introduce the staggered vector 𝒏𝒏=(𝑨𝑨k−𝑩𝑩k)/2, and 𝒎𝒎=𝑨𝑨k+\n𝑩𝑩k, where  𝑨𝑨k and 𝑩𝑩k are the unit vectors of spin moment at a site k that belongs to the \nsublattice s A and B, respectively.  The Lagrangian density for the ferrimagnet is given by  \n[26,47- 49] \nℒ=[−s𝒏𝒏̇∙(𝒏𝒏×𝒎𝒎)−𝛿𝛿𝑠𝑠𝐚𝐚(𝒏𝒏)∙𝒏𝒏̇]−𝒰𝒰,                    (1) \nwhere 𝑠𝑠=(𝑠𝑠𝐴𝐴+𝑠𝑠𝐵𝐵)/2, 𝛿𝛿𝑠𝑠=𝑠𝑠𝐴𝐴−𝑠𝑠𝐵𝐵, 𝑠𝑠𝑖𝑖=𝑀𝑀𝑖𝑖/𝛾𝛾𝑖𝑖 is the angular momentum  density , 𝑀𝑀𝑖𝑖 \nis the magnetic moment, 𝛾𝛾𝑖𝑖 is the gyromagnetic ratio for sublattice 𝑖𝑖, 𝐚𝐚(𝒏𝒏) is the vector \npotential for the magnetic monopole . The total energy 𝒰𝒰 includes the exchange energy and \nanisotropy energy as  \n𝒰𝒰=𝑎𝑎\n2|𝒎𝒎|2+𝐴𝐴\n2(∇𝒏𝒏)2−𝐾𝐾\n2(𝒛𝒛�∙𝒏𝒏)2+𝜅𝜅\n2(𝒙𝒙�∙𝒏𝒏)2,                (2) \nwhere 𝑎𝑎 is the homogeneous exchange, 𝐴𝐴 is the inhomogeneous exchange, 𝐾𝐾 is the easy-\naxis anisotropy constant , and 𝜅𝜅 is the hard-axis anisotropy constant. The Rayleigh function \naccounting  for the dissipation is  given by  ℛ=𝛼𝛼𝑠𝑠𝒏𝒏̇2 where 𝛼𝛼 is the Gilbert damping \nconstant. From the  Lagrangian  density and the Rayleigh dissipation , we obtain the equation \nof motion in terms of staggered vector 𝒏𝒏 by integrating out  the net magnetization variable  \n𝒎𝒎 [26]: \n𝜌𝜌𝒏𝒏×𝒏𝒏̈+2𝛼𝛼𝑠𝑠𝒏𝒏×𝒏𝒏̇+𝛿𝛿𝑠𝑠𝒏𝒏̇=𝒏𝒏×𝒇𝒇𝒏𝒏,                      (3) \nwhere 𝜌𝜌=𝑠𝑠2/𝑎𝑎 paramet rizes the inertia  and 𝒇𝒇𝒏𝒏=−𝛿𝛿𝒰𝒰𝛿𝛿𝒏𝒏⁄ is the effective field.  After \nlinearizing the equations for small- amplitude fluctuations from the uniform state, w e consider \nthe SW ansatz as 𝒏𝒏 (𝑥𝑥,𝑡𝑡)=Re��𝑛𝑛𝑥𝑥exp[𝑖𝑖(𝜔𝜔𝑡𝑡−𝑘𝑘𝑥𝑥)],𝑛𝑛𝑦𝑦exp[𝑖𝑖(𝜔𝜔𝑡𝑡−𝑘𝑘𝑥𝑥)],1��, where \n𝑛𝑛𝑥𝑥,𝑛𝑛𝑦𝑦 are the amplitude s of SW (|𝑛𝑛𝑥𝑥|,�𝑛𝑛𝑦𝑦�≪1), 𝜔𝜔 is the SW frequency, and 𝑘𝑘 is the \nwavevector.  By solving the linearized equations with this ansatz, we obtain the d ispersion 5 \n relation as  \n𝜔𝜔±=±𝛿𝛿𝑠𝑠+�𝛿𝛿𝑠𝑠2+4𝜌𝜌(𝐴𝐴𝑘𝑘2+𝐾𝐾+𝜅𝜅/2)\n2𝜌𝜌.                   (4). \nHere the u pper (lower) sign corresponds to the left- (right -) circular ly polarized SW. The \nresonance frequencies for left - and right -circular ly polarized SWs are different except at the \nangular mom entum compensation point  𝑇𝑇𝐴𝐴 where the net spin density 𝛿𝛿𝑠𝑠 is zero . Figure 2 \nshows the agreement between the analytic SW dispersion relations  [Eq. (4) ; lines ] and \nnumerical results  that will be discussed below  (symbols) . Below  or above  𝑇𝑇𝐴𝐴, the energy of \nright -circular ly polarized SW differs from that of left- circular ly polarized SW [see Fig. 2(a) \nand Fig. 2(c) ]. At 𝑇𝑇𝐴𝐴 [Fig. 2(b) ], two circular ly polarized SW s are degenerate, which is \nanalogous with antiferromagnetic SW s.  \nWe n ext look into the dynamics of ferrimagnetic DW  induced by SW s. We consider \n𝒏𝒏 as 𝒏𝒏=𝒏𝒏𝟎𝟎+𝜹𝜹𝒏𝒏 with DW texture 𝒏𝒏𝟎𝟎 and small fluctuation 𝜹𝜹𝒏𝒏 (|𝜹𝜹𝒏𝒏|≪|𝒏𝒏𝟎𝟎|) with the \nconstraint 𝒏𝒏𝟎𝟎∙𝜹𝜹𝒏𝒏=0 to keep the unit length of 𝒏𝒏 to linear order in 𝜹𝜹𝒏𝒏. We introduc e two \ncollective coordinates  [50], the DW position 𝑋𝑋(𝑡𝑡) and center angle 𝜙𝜙(𝑡𝑡), and define  a DW \n𝒏𝒏𝟎𝟎 by Walker ansatz  [51], 𝒏𝒏𝟎𝟎(𝑥𝑥,𝑡𝑡)=(sin𝜃𝜃sin𝜙𝜙,sin𝜃𝜃cos𝜙𝜙,𝑐𝑐𝑐𝑐𝑠𝑠𝜃𝜃 ) where 𝜃𝜃=\n2tan−1[exp{(𝑥𝑥−𝑋𝑋)/𝜆𝜆}] and 𝜆𝜆 is the DW width . We consider  the magnonic torque  𝝉𝝉𝒎𝒎 \nthat is given by  [34,37,39,52]  \n𝝉𝝉𝒎𝒎=−𝐴𝐴[(𝑱𝑱𝒎𝒎∙𝜵𝜵)𝒏𝒏𝟎𝟎−(𝜕𝜕𝑥𝑥𝜌𝜌𝑚𝑚)𝒏𝒏𝟎𝟎×𝜕𝜕𝑥𝑥𝒏𝒏𝟎𝟎],                 (5) \nwhere magnon -flux density  𝐽𝐽𝑚𝑚𝑥𝑥=𝒏𝒏𝟎𝟎∙〈𝜹𝜹𝒏𝒏×𝜕𝜕𝑥𝑥𝜹𝜹𝒏𝒏〉, and magnon number density 𝜌𝜌𝑚𝑚=\n〈𝜹𝜹𝒏𝒏〉2/2. The first term in Eq. (5) represents the adiabatic magnonic torque  rooted in the \nmagnon current , and the second term  represents the non-adiabatic magnonic torque  caused by \nthe gradient of the magnon density . Inserting Eq. (5) into the staggered LLG equation Eq. (3), \nwe derive  two coupled equations of motion as  \n𝑀𝑀𝑋𝑋̈−𝐺𝐺𝜙𝜙̇+𝑀𝑀𝑋𝑋̇𝜏𝜏⁄=𝐹𝐹𝑚𝑚,                              (6) \n𝐼𝐼𝜙𝜙̈+𝐺𝐺𝑋𝑋̇+𝐼𝐼𝜙𝜙̇𝜏𝜏⁄=−𝜅𝜅𝜆𝜆sin2𝜙𝜙+𝑇𝑇𝑚𝑚,                    (7) \nwhere 𝑀𝑀=2𝜌𝜌𝜌𝜌𝜆𝜆⁄, 𝐼𝐼=2𝜌𝜌𝜆𝜆𝜌𝜌, 𝐺𝐺=2𝛿𝛿𝑠𝑠𝜌𝜌, and 𝜏𝜏=𝜌𝜌/𝛼𝛼𝑠𝑠 are the mass, the moment of 6 \n inertia, the gyrotropic coefficient, and the relaxation time , respectively, and 𝜌𝜌 is the cross \nsectional area of the DW . Here,  𝐹𝐹𝑚𝑚=(2𝐴𝐴𝜆𝜆⁄)∫𝑑𝑑𝑑𝑑[(𝜕𝜕𝑥𝑥𝜌𝜌𝑚𝑚)𝒏𝒏𝟎𝟎×𝜕𝜕𝑖𝑖𝒏𝒏𝟎𝟎] and 𝑇𝑇𝑚𝑚=\n−2𝐴𝐴∫𝑑𝑑𝑑𝑑[(𝑱𝑱𝒎𝒎∙𝛁𝛁)𝒏𝒏𝟎𝟎] correspond to the  magnon- induced force and torque , respectively. \nWe note that the sign of 𝑇𝑇𝑚𝑚 is different for left - and right -circular ly polarized SWs whereas \nthe sign of 𝐹𝐹𝑚𝑚 is independent of the circular  polarization of SW. It is because  the sign of 𝐽𝐽𝑚𝑚𝑥𝑥 \nis different for left - and right -circularly polarized SWs whereas the sign of 𝜕𝜕𝑥𝑥𝜌𝜌𝑚𝑚 is \nindependent of the circular  polarization of SW. From Eqs. (6) and (7), we finally obtain the \nsteady -state velocity of DW below the Walker breakdown [ 53] as \n𝑣𝑣𝐷𝐷𝐷𝐷=𝑠𝑠\n2𝜌𝜌(𝛼𝛼2𝑠𝑠2+𝛿𝛿𝑠𝑠2)�𝛼𝛼𝜆𝜆𝐹𝐹𝑚𝑚+𝛿𝛿𝑠𝑠\n𝑠𝑠𝑇𝑇𝑚𝑚�,                     (8) \nwhich is the central  result of this  work.  The first and second term s originate from non -\nadiabatic and adiabatic contributions , respectively. In Eq. (8), t he ratio 𝛿𝛿𝑠𝑠/𝑠𝑠 is an estimate  of \nthe degree how the dynamics of the system is close to that of ferromagnets. The condition of \n𝛿𝛿𝑠𝑠2𝑠𝑠⁄→±1 represent s the ferromagnetic limit, whereas that of 𝛿𝛿𝑠𝑠2𝑠𝑠⁄→0 represent s the \nantiferromagnet ic limit. In the ferromagnetic limit (𝛿𝛿𝑠𝑠2𝑠𝑠⁄→±1), the second term in Eq. (8) \nbecomes  dominant  for the DW motion. On the other hand, in the antiferromagnetic limit \n(𝛿𝛿𝑠𝑠2𝑠𝑠⁄→0), the second term vanishe s and the only first term  is responsible for DW  motion . \nTo verify Eq. (8), we perform micromagnetic simulation s with the atomistic LLG \nequation. We start with the initial condition that the DW  is located at the center of one -\ndimensional nanowire as shown in Fig. 1(b). SW  is excited  by an external AC field  𝐁𝐁𝐀𝐀𝐀𝐀 on \nthe left side of DW . The atomistic LLG equation including the external AC field is given by  \n𝜕𝜕𝑺𝑺𝒊𝒊\n𝜕𝜕𝑡𝑡=−𝛾𝛾𝑖𝑖𝑺𝑺𝒊𝒊×�𝐁𝐁𝐞𝐞𝐞𝐞𝐞𝐞,𝒊𝒊+𝐁𝐁𝐀𝐀𝐀𝐀�+𝛼𝛼𝑖𝑖𝑺𝑺𝒊𝒊×𝜕𝜕𝑺𝑺𝒊𝒊\n𝜕𝜕𝑡𝑡,                 (9) \nwhere 𝑺𝑺𝒊𝒊 is the normalized spin moment vector, 𝛾𝛾𝑖𝑖=𝑔𝑔𝑖𝑖𝜇𝜇𝐵𝐵ℏ⁄ is the gyromagnetic ratio,  \n𝜇𝜇𝐵𝐵 is the Bohr magneton, and 𝛼𝛼𝑖𝑖 is the damping constant at a lattice site 𝑖𝑖. The o dd (even) \nsite 𝑖𝑖 corresponds to the TM (RE) element. 𝐁𝐁𝐞𝐞𝐞𝐞𝐞𝐞,𝒊𝒊=−1\n𝜇𝜇𝑖𝑖𝜕𝜕ℋ\n𝜕𝜕𝑺𝑺𝒊𝒊 is the effective field  at each \nsite, where 𝜇𝜇𝑖𝑖 is the magnetic mome nt per atom , one dimensional discrete Hamiltonian \nℋ=𝐴𝐴𝑠𝑠𝑖𝑖𝑚𝑚∑𝑺𝑺𝒊𝒊∙𝑺𝑺𝒊𝒊+𝟏𝟏 𝑖𝑖 −𝐾𝐾𝑠𝑠𝑖𝑖𝑚𝑚∑(𝑺𝑺𝒊𝒊∙𝒛𝒛�)2\n𝑖𝑖 +𝜅𝜅𝑠𝑠𝑖𝑖𝑚𝑚∑(𝑺𝑺𝒊𝒊∙𝒙𝒙�)2\n𝑖𝑖 , 𝐴𝐴𝑠𝑠𝑖𝑖𝑚𝑚 and 𝐾𝐾𝑠𝑠𝑖𝑖𝑚𝑚 (𝜅𝜅𝑠𝑠𝑖𝑖𝑚𝑚) are the 7 \n exchange constant and easy  (hard) axis anisotropy for simulations , respectively . To excite the \nSW, an external AC field 𝐁𝐁𝐀𝐀𝐀𝐀=B0[cos𝜔𝜔𝑡𝑡𝒙𝒙�±sin𝜔𝜔𝑡𝑡𝒚𝒚�] is applied on two cells at \n252  nm away  from the DW. We use the following simulation parameters: 𝐴𝐴𝑠𝑠𝑖𝑖𝑚𝑚=\n1.64 meV , 𝐾𝐾𝑠𝑠𝑖𝑖𝑚𝑚=6.47 𝜇𝜇eV, 𝜅𝜅𝑠𝑠𝑖𝑖𝑚𝑚=0.02𝐾𝐾𝑠𝑠𝑖𝑖𝑚𝑚, B0=100  mT, the lattice constant is \n0.42 nm, and the Land é 𝑔𝑔–factor 𝑔𝑔𝑅𝑅𝑅𝑅=2 for rare earth  and 𝑔𝑔𝑇𝑇𝑀𝑀=2.2 for transition  \nmetal  [54]. We consider the damping constant is uniform for all site s, i.e., 𝛼𝛼RE=𝛼𝛼TM=5×\n10−4 for simplicity.  We use m agnetic moment s 𝑀𝑀RE and 𝑀𝑀TM as listed in TABLE 1.   \n \nIII. Results and Discussion  \nFigure 3(a) -(c) show the simulation results of  DW velocity as a function of the SW \nfrequency.  Figure 3(a) represents the results for the case below  the angular momentum \ncompensation point 𝑇𝑇𝐴𝐴. As the SW gap is different for left - and right -circular ly polarized  \nSWs [Fig. 2(a)], the threshold SW frequenc y for the DW  motion  is also different for left - and \nright -circular ly polarized SWs. An interesting observation for the DW motion is that the \nmoving  direction of DW  depends on the circular  polarization of SW . Left- (Right -) circular ly \npolarized SW moves the  DW toward s (away  from ) the SW source. This bi -directional DW \nmotion is understood by the fact that l eft- and right-circular ly polarized SWs  carry the \nangular momentum with opposite  signs. When the SW passes through the DW, the angular \nmomentum of SW is transferred to the DW so that the DW moving  direction depends on the \ncircular polarization of SW . This is directly related to the fact that the sign of 𝑇𝑇𝑚𝑚 is different \nfor left - and right -circular ly polarized SWs , whereas the sign of 𝐹𝐹𝑚𝑚 is independent of the \ncircular  polarization of SW. Given that  𝑇𝑇𝑚𝑚 and 𝐹𝐹𝑚𝑚 in Eq. (8) respectively correspond to \ncontributions from the adiabatic and non- adiabatic magnon ic torques , the bi -directional DW \nmotion depending on the circular polarization of SW evidences  that the adiabatic magnon ic \ntorque is dominant over the non- adiabatic one.  \nSolid and dashed lines in Fig. 3(a) are calculated from Eq. (8), with 𝐹𝐹𝑚𝑚 and 𝑇𝑇𝑚𝑚 \nobtained from numerical calculation s. We find that the numerically obtained bi -directional \nbehavior (symbols) is reasonably described by Eq. (8) in high- frequency ranges.  In low-\nfrequency  ranges , a discrepancy between Eq. (8) and numerical results appears possibly 8 \n because of nonlinear effects , which are  not captured by our current analytical models . The \nresults for the case above the angular momentum compensation point 𝑇𝑇𝐴𝐴 [Fig. 3(c) ] can be \nunderstood in a similar way . Contrary to the case below 𝑇𝑇𝐴𝐴, overall spin moment s in the \nsystem are reversed so that left - (right -) circular ly polarized SW makes DW move away from \n(toward s) the source.  \nTo further elucidate SW -induced ferrimagnetic DW motion  below and above  𝑇𝑇𝐴𝐴, we \ninvestigate the spin current 𝐽𝐽𝑆𝑆, which is defined as 𝑱𝑱𝒔𝒔=−𝐴𝐴〈𝒏𝒏×𝜕𝜕𝑥𝑥𝒏𝒏〉. Figure 4 shows the \nschematic of SW transmission through a DW (top panel) and the z  component of the spin \ncurrent 𝐽𝐽𝑆𝑆𝑧𝑧 along the propagation direction (i.e., the 𝑥𝑥 axis, bottom panel). For the left - \ncircular ly polarized SW (solid line) , the spin current in the left domain part decreases \ngradually due to the damping. After the SW passes through the DW, the spin current abruptly \nflips its sign due to overall reversal of spin moments. The spin -current  change is transferred \nto the DW, resulting in the DW motion. For the right -circular ly polarized SW (dashed line), \noverall sign of the spin current is reversed. It is the reason that  the direction of DW \npropagation is the opposite for left - and right -circularly polarized SWs.  The sign of the spin-\ncurrent change is the same below and above 𝑇𝑇𝐴𝐴, but 𝛿𝛿 𝑠𝑠 changes its sign  [see Eq. (8)]  \nbecause the spin directions in the domain part changes accordingly , which results in the sign \ndifference of the DW velocity below and above 𝑇𝑇𝐴𝐴.  \nFor the case at the angular momentum compensation point 𝑇𝑇𝐴𝐴 (i.e., 𝛿𝛿𝑠𝑠=0), both \nleft- and right-circular ly polarized SWs  drive the DW to the same direction ( i.e., toward s the \nSW source)  as shown in Fig. 3(b) . We note that this DW moving direction at 𝑇𝑇𝐴𝐴 is the \nopposite to the direction of the DW motion induced by circularly polarized spin waves  in true \nantiferromagnets [ 40,41]. In true antiferromagnets where the shape anisotropy is absent, \ncircular ly polarized SWs make the DW precess, which result s in the SW reflection. The \nreflected SWs transfer  linear momentum to DW, and push the DW away from the SW source. \nFor the case at 𝑇𝑇𝐴𝐴 of ferrimagnets, however, the net magnetic moment is finite so that the \nshape anisotropy does not vanish. As a result, the DW experiences the hard- axis anisotropy, \nwhich prevents the DW precession. Therefore, the ferr imagnetic DW still serves as a \nreflect ionless potential called the Pöschl -Teller potential [ 55] and its motion is governed by \nthe force from the  magnonic torque [ 𝛼𝛼𝜆𝜆𝐹𝐹𝑚𝑚 term in Eq. (8)]. As the sign of 𝐹𝐹𝑚𝑚 is 9 \n independent of the SW circular polarization, both left - and right -circular ly polarized SWs pull \nthe DW along the same direction , i.e., towards the SW source. This force -induced motion of \nthe ferrimagnetic DW toward the spin -wave source at 𝑇𝑇𝐴𝐴 is similar  to the motion of the \nantiferromagnetic DW toward the spin -wave source for lin early -polarized spin waves \nreported in Ref. [40] . \nDependence of the DW velocity on the SW circular polarization is summarized in \nFig. 3(d), which  shows the DW velocity as a function of the net spin density 𝛿𝛿𝑠𝑠 at a fixed \nSW frequency  (𝜔𝜔 = 0.7 THz) . The sign of DW velocity depends not only on the circular \npolarization, but also on the sign of the net spin density. We note that the DW velocity is not \nzero at 𝑇𝑇𝐴𝐴 because the adiabatic and non- adiabatic contributions are not compensated at 𝑇𝑇𝐴𝐴.  \n \nIV. Summary  \nWe have investigate d the SW circular -polarity dependence of  ferrimagnetic DW \ndynamics theoretically and numerically.  We find that the DW moves along  the opposite \ndirection depending on the circular polarization of SW. This bi -directional DW motion is \ncaused by the fact that  the signs of the spin current  and the angular momentum transferred  to \nDW are opposite for left - and right -circular ly polarized SWs . The o verall tendency of DW \nmoving  direction is reversed when the sign of the net spin density  𝛿𝛿𝑠𝑠 is reversed. At 𝑇𝑇𝐴𝐴 \nwhere the angular momentum vanishe s, the dissipative  non-adiabatic magnonic torque is the \nmain driving force so that DW moves along the same direction (towards the SW source) \nregardless of the SW circular polarization .  \nOur finding of bi-directional  ferrimagnetic DW driven by SWs can be generalized to \nother ferrimagnetic topological excitations  such as magnetic skyrmions and vortices. This bi-\ndirectionality  of ferrimagnetic DW motion depending either on the SW circular polarization \nor on the sign of the net spin density will be useful for magnonic spintronics [ 56] because \nsuch bi -directional motion, which makes the device functionality versatile, can be realized \nwithout moving the location of a  SW source.  \n 10 \n Acknowledgement  \nThis work was supported by the National Research Foundation of Korea (NRF) \n(Grants No. 2015M3D1A1070465 and No. 2017R1A2B2006119), the KIST Institutional \nProgram (Project No. 2V05750). S.K.K. was supported by the startup fund at the University \nof Missouri. J.X. was supported by the National Natural Science Foundation of China (Grants No. 11722430).  \n  11 \n Table 1. Used magnetic moments  MTM and MRE for transition metal and rare earth elements , \nrespectively , in simulation. Index 5 coincides  with the angular momentum compensation \npoint TA. \nIndex  1 2 3 4 5 6 7 8 9 \n𝑀𝑀𝑇𝑇𝑀𝑀(kA/m) 460 455 450 445 440 435 430 425 420 \n𝑀𝑀𝑅𝑅𝑅𝑅(kA/m) 440 430 420 410 400 390 380 370 360 \n \n  12 \n  \nFigure 1. (a) Illustration  of left- and right -circular ly spin wave s in an antiferromagnetically \ncoupled ferrimagnet. (b) Schematic graphic of one -dimensional ferrimagnet ic nanowire  with \na domain wall  (DW). Domain wall is positioned at the center of nanowire . Spin wave is \nexcited by an external AC field ( 𝐁𝐁𝐀𝐀𝐀𝐀) on the left side  (252 nm  apart from DW ). \n  \n13 \n  \nFigure 2. Spin- wave dispersion relations  (a) below  TA, (b) at  TA, and (c) above  TA. Symbols \nrepresent numerical simulation  results and lines represent Eq. (4). Solid  (Open ) triangular \nsymbols  correspond to left - (right -) circularly  polarized spin wave . \n  \n14 \n  \nFigure 3. Calculated domain wall velocity results  (a) below  TA, (b) at  TA, and (c) above TA \nwith various spin- wave frequencies . Symbols are the simulation results and lines are Eq. (8) \n(arbitrary unit). (d) Domain wall velocity as a function of the net spin density δs at a fixed \nfrequency ω = 0.7 THz . Negative  δs corresponds to the case below TA. \n  \n15 \n  \nFigure 4. Schematic of  transm itted SW (top)  and the z component of the spin current  Js along \nthe wire length. A DW is positioned at the atomic site i  = 2000, and SW source is at  i = \n1700. Assumed parameters are those with the index 3 (i.e., below TA) listed in the TABLE  1 \nand the SW  frequency is 0.6  THz. \n  \n16 \n Reference  \n1. R. Wangness, Phys. Rev. 91, 1085 (1953).  \n2. M. Binder et al., Phys. Rev. B 74, 134404 (2006).  \n3. C. Stanciu, A. Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, \nPhys. Rev. B 73, 220402 (2006).  \n4. J. M. D. 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Phys. 11, 453 \n(2015).  \n "    },    {        "title": "1005.5297v2.Haldane_Phases_and_Ferrimagnetic_Phases_with_Spontaneous_Translational_Symmetry_Breakdown_in_Distorted_Mixed_Diamond_Chains_with_Spins_1_and_1_2.pdf",        "content": "arXiv:1005.5297v2  [cond-mat.str-el]  10 Nov 2010Typeset with jpsj3.cls <ver.1.0> Full Paper\nHaldane Phases and Ferrimagnetic Phases with Spontaneous T ranslational\nSymmetry Breakdown in Distorted Mixed Diamond Chains with S pins1and1/2\nKazuoHida∗, Ken’ichi Takano1, and Hidenori Suzuki1†\nDivision of Material Science, Graduate School of Science an d Engineering,\nSaitama University, Saitama, Saitama, 338-8570\n1Toyota Technological Institute, Tenpaku-ku, Nagoya 468-8 51\n(Received February 28, 2018)\nThe ground states of two types of distorted mixed diamond cha ins with spins 1 and 1 /2\nare investigated using exact diagonalization, DMRG, and ma pping onto low-energy effective\nmodels. In the undistorted case, the ground state consists o f an array of independent spin-\n1 clusters separated by singlet dimers. The lattice distort ion induces an effective interaction\nbetween cluster spins. When this effective interaction is an tiferromagnetic, several Haldane\nphases appear with or without spontaneous translational sy mmetry breakdown (STSB). The\ntransition between the Haldane phase without STSB and that w ith (n+1)-fold STSB ( n= 1,\n2, and 3) belongs to the same universality class as the ( n+1)-clock model. In contrast, when\nthe effective interaction is ferromagnetic, the quantized a nd partial ferrimagnetic phases appear\nwith or without STSB. An effective low-energy theory for the p artial ferrimagnetic phase is\npresented.\nKEYWORDS: mixed diamondchain, distortion, frustration, H aldane phase, spontaneoustranslational sym-\nmetry breakdown, partial ferrimagnetism\n1. Introduction\nQuantum magnetism in frustrated spin systems is a\nrapidly developing field of condensed matter physics.1,2)\nAt first glance, one would expect that geometrical frus-\ntration enhances quantum fluctuation and drives an or-\ndered state into a disordered state. However, recent\nprogress in this field of physics has shown that this sim-\nple intuition is not always valid and that geometrical\nfrustration induces a variety of exotic quantum phenom-\nena,whicharenoteasilypredicted.Underanappropriate\ncondition, it even stabilizes an unexpected magnetic long\nrangeordersuch asthe frustration-inducedferrimagnetic\nand spin nematic orders.\nTo understand magnetism under the interplay of geo-\nmetrical frustration and quantum fluctuation, it is desir-\nable to begin with typical spin models with exact solu-\ntions. Among them, there exist a class of models whose\nground states are exactly written down as spin cluster\nsolid (SCS) states because of frustration. A SCS state is\natensorproduct stateofexactlocaleigenstatesofcluster\nspins. Well-known examples are the Majumdar-Ghosh\nmodel3)whose ground state is a prototype of sponta-\nneously dimerized phases in one-dimensional frustrated\nmagnets4)and the Shastry-Sutherland model5)which\ncorresponds to the material SrCu 2(BO3)2.6,7)In these\nmodels, the spin clusters are singlet dimers.\nThe diamond chain is another frustrated spin chain\nwith exact SCS ground states. The lattice structure is\nshown in Fig. 1. In a unit cell, there are two kinds of\nnonequivalent lattice sites occupied by spins with mag-\nnitudesSandτ; we denote the set of magnitudes by ( S,\nτ). One of the authors and coworkers8,9)introduced this\n∗E-mail address: hida@phy.saitama-u.ac.jp\n†Present address: Department of Physics, College of Humanit ies\nand Sciences, Nihon University, Setagaya-ku, Tokyo 156-85 50Sτ\nτ\nFig. 1. Structure ofthe diamond chain. Spin magnitudes ina u nit\ncell are indicated by Sandτ; we denote the set of magnitudes\nby (S,τ). The PDC is the case of S=τ, while the MDC is the\ncase ofS= 2τwith an integer or half-odd integer τ.\nlattice structure and generally investigated the case of\n(S,S), i.e., the pure diamond chain (PDC). Any PDC\nis shown to have at least one exact SCS ground-state\nphase where each spin cluster has spin 0. Particularly,\nin the case of (1/2, 1/2), they determined the full phase\ndiagram of the ground state by combining rigorous ar-\nguments with numerical calculations. After that, Nigge-\nmann et al.10,11)arguedabout a seriesofdiamond chains\nwith (S, 1/2). As for the special case of (1/2, 1/2), they\nreproduced the results of ref. 9.\nThe mixed diamond chain (MDC) is defined as a dia-\nmond chain with ( S,S/2) for the integer S.12)The spe-\ncial case of (1, 1/2) was first investigated by Niggemann\net al.10,11)They considered it asone of the seriesofmod-\nels with ( S,1/2).Recently, extensiveinvestigationonthe\nMDC has been carried out by the present authors.12–14)\nThe MDC is of special interest among diamond chains,\nbecause only the MDC has the Haldane phase in the ab-\nsence offrustration,so that wecan observethe transition\nfrom the Haldane phase to a SCS phase induced by frus-\ntration. In contrast, diamond chains of other types have\nferrimagnetic ground states for weak frustration.\nThe features common to all types of diamond chains\nare their infinite number of local conservation laws and\n12 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\nmore than two different types ofexact SCS groundstates\nthat are realized depending on the strength of frustra-\ntion. For example, S= 1/2 PDC has a nonmagnetic\nphaseaccompaniedbyspontaneoustranslationalsymme-\ntry breakdown (STSB) and a paramagnetic phase with-\nout STSB. This model also has a ferrimagnetic ground\nstate in the less frustrated region.9)On the other hand,\nthe MDC with spins 1 and 1 /2 has 3 different param-\nagnetic phases accompanied by STSB and one paramag-\nnetic phase without STSB. This model also has a non-\nmagnetic Haldane ground state in a less frustrated re-\ngion.10,12)The SCS structures of the ground states are\nalso reflected in characteristic thermal properties, as re-\nported in ref. 13.\nModifications of the PDC and MDC have been exam-\nined by many authors. Among them, the spin 1/2 PDC\nwith distortion has been thoroughly investigated by nu-\nmerical methods.15–17)It is found that azurite, a nat-\nural mineral, consists of distorted PDCs with spin 1/2\nand that the magnetic properties of this material have\nbeen experimentally studied in detail.18,19)Other mate-\nrials have also been reported.20,21)The diamond chain\nis one of the simplest models compatible with the 4-spin\ncyclic interaction. The effects of this type of interaction\non PDC have recently been investigated by Ivanov et\nal.22)The present authors also investigated the MDC\nwith bond-alternating distortion and found an infinite\nseries of ground states with STSB.14)In addition, as re-\nviewed in ref. 14, the MDC is related to other impor-\ntant models of frustrated magnetism such as the dimer-\nplaquette model,23–28)frustrated Heisenberg ladders,29)\nhybrid diamond chains consisting of Heisenberg bonds\nand Ising bonds,30,31)and an Ising model on a hierarchi-\ncaldiamondlattice.32)Amongthem,thedimer-plaquette\nchain with ferromagnetic interplaquette interaction re-\nduces to the MDC in the limit of strong interplaquette\ninteraction.28)\nThus far, in spite of the theoretical relevance of the\nMDC, no materials described by the MDC have been\nfound. Nevertheless, synthesizing MDC materials is not\nan unrealistic expectation in view of the success of the\nsynthesis of many low dimensional bimetallic magnetic\ncompounds33)and organic magnetic compounds.34)In\ngeneral, it is natural to expect that the lattice is pos-\nsibly distorted in real MDC compounds as in azurite.\nFrom this viewpoint, it is important to present theoret-\nical predictions on the ground state of distorted MDCs\nto widen the range of candidate materials of MDC and\nto raise the possibility of their synthesis.\nWe begin by classifying the distortion patterns by\nthe normal modes of each diamond unit. Excluding two\ntranslations and one rigid body rotation, we have 5 nor-\nmal modes as depicted in Fig. 2 within the diamond\nplane. A distorted MDC may be realized as a result of\nthe collective softening of these normal modes. In partic-\nular, the distortion patterns in (a) and (b) break the lo-\ncal conservation laws that hold in the undistorted MDC.\nHence, these distortions induce effective interactions be-\ntween the cluster spins in the whole lattice, and may\nform novel exotic phases. We investigate these interest-\ning cases in the present paper. In what follows, we name(a) (b) (c)\n(d) (e)\nFig. 2. Displacement modes of a diamond unit.\nSl1+δA\nλ\n1+δA1−δA\n1−δAτl(1)\nSl+1\nτl(2)(a)\nSl1+δB\nλ\n1−δB1+δB\n1−δBτl(1)\nSl+1\nτl(2)(b)\nFig. 3. Structures of MDC with S= 1 and τ(1)=τ(2)= 1/2\nwith (a) type A and (b) type B distortions.\nthe distortion patterns in (a) and (b) as type A and type\nB, respectively. The MDCs with type A and type B dis-\ntortions are depicted in Figs. 3(a) and 3(b), respectively.\nThe distortion patterns in Figs. 2(d) and 2(e) do not\nchange the geometry of the original undistorted MDC.\nThe distortion pattern in Fig. 2(c) is of another interest,\nsince it induces the bond alternation in the undistorted\nMDC without breaking the local conservation laws. This\ncase has been investigated separately and published in a\nprevious paper.14)\nThis paper is organized as follows. In §2, the Hamil-\ntonians for the MDCs with the type A and type B dis-\ntortions are presented, and the structure of the ground\nstatesoftheMDCwithoutdistortionissummarized.The\nground-statephasesfor the MDC with the type Adistor-\ntion are discussed in §3, and those for the MDC with the\ntype B distortion are discussed in §4. The last section is\ndevoted to summary and discussion.\n2. Hamiltonian\nThe MDCs with the type A and type B distortions are\ndescribed, respectively, by the following Hamiltonians:\nHA=N/summationdisplay\nl=1/bracketleftBig\n(1+δA)Slτ(1)\nl+(1−δA)τ(1)\nlSl+1\n+(1−δA)Slτ(2)\nl+(1+δA)τ(2)\nlSl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,\n(1)J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki 3\nHB=N/summationdisplay\nl=1/bracketleftBig\n(1+δB)Slτ(1)\nl+(1+δB)τ(1)\nlSl+1\n+(1−δB)Slτ(2)\nl+(1−δB)τ(2)\nlSl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,\n(2)\nwhereSlis the spin-1 operator, and τ(1)\nlandτ(2)\nlare\nthe spin-1/2 operators in the lth unit cell. The param-\neterδA(δB) represents the strength of type A (type B)\ndistortion, and is taken to be nonnegative without spoil-\ning generality. The number of unit cells is denoted by\nN, and then the total number of sites is 3 N. We will\nconsider these systems in the large Nlimit.\nForδA= 0 and δB= 0, both eqs. (1) and (2) reduce\nto the undistorted MDC Hamiltonian,\nH0=N/summationdisplay\nl=1/bracketleftbigg\nSlTl+TlSl+1+λ\n2/parenleftbigg\nT2\nl−3\n2/parenrightbigg/bracketrightbigg\n(3)\nwith the composite spin operators Tl≡τ(1)\nl+τ(2)\nl.\nBefore going into the analysis of the distorted MDC,\nwe briefly summarize the ground-state properties of the\nHamiltonian (3) reported in ref. 12 for convenience.\n(i)T2\nlcommutes with the Hamiltonian H0for anyl.\nTherefore, the composite spin magnitude Tldefined\nbyT2\nl=Tl(Tl+1) is a good quantum number that\ntakes the values 0 or 1. Hence, each energy eigen-\nstate has a definite set of {Tl}, i.e. a sequence of\n0’s and 1’s with length N. A pair of τ(1)\nlandτ(2)\nl\nwithTl= 0 is called a dimer. A cluster including\nnsuccessive Tl= 1 pairs bounded by two Tl= 0\npairs is called a cluster- n. The cluster- nis equiva-\nlent to an antiferromagneticspin-1 Heisenberg chain\noflength2 n+1withopen boundarycondition.Since\na cluster- nis decomposed into a sublattice consist-\ning ofn+1 sites with Sl’s and that consisting of n\nsites with Tl’s, the ground states of a cluster- nare\nspin triplet states with total spin unity on the basis\nof the Lieb-Mattis theorem.35,36)This implies that\neach cluster- ncarries a spin-1 in its ground state.\n(ii) There appear 5 distinct ground-state phases called\ndimer-cluster- n(DCn) phases with n= 0,1,2,3,\nand∞. The DC nstate is an alternating array\nof dimers and cluster- n’s. The phase boundary\nλc(n,n′) between DC nand DCn′phases are\nλc(0,1) = 3,\nλc(1,2)≃2.660425045542 ,\nλc(2,3)≃2.58274585704 ,\nλc(3,∞)≃2.5773403291 , (4)\nwhereλc(0,1)is obtainedanalyticallyandotherval-\nues are calculated numerically.\n(iii) In the DC ∞ground state realized for λ < λc(3,∞),\nTl= 1 for all l. This state is not accompanied by\nSTSB and is equivalent to the Haldane state of an\nantiferromagnetic spin-1 Heisenberg chain with infi-\nnite length.\n(iv) Each of the DC nstates with 0 ≤n≤3 realized forλ > λ c(3,∞) is a uniform array of cluster- n’s with\na common value of nand dimers in between. In the\nDCnphase with 1 ≤n≤3, (n+1)-fold STSB takes\nplace. In the DC0 phase, no translational symmetry\nis broken.\nIn what follows, we numerically examine various aspects\nof the type A and type B distortion effects on the MDC.\nBecause the DC3 phase is only realized within a very\nnarrow interval of λ, it is difficult to analyze the effect\nof distortion numerically in this phase. Hence, we do not\nconsider the DC3 phase in the following numerical anal-\nysis.\n3. Ground-State Properties of the MDC with\nType A Distortion\n3.1 Weak distortion regime\nWe now inspect the nature of the effective interaction\nbetween two cluster- n’s separated by a dimer consisting\nofτ(1)\nlandτ(2)\nlin the presence of the weak type A dis-\ntortion. For δA>0,Sl(Sl+1) tends to be antiparallel to\nτ(1)\nl(τ(2)\nl) rather than to τ(2)\nl(τ(1)\nl), as is known from\nFig. 3(a). The spins τ(1)\nlandτ(2)\nlare antiparallel to each\nother because they form a singlet dimer. Therefore, Sl\nandSl+1tend to be antiparallel to each other. In each\ncluster-n, the number of spins Sl’s is larger than the\nnumber of composite spins Tl’s by one. Hence, from the\nLieb-Mattistheorem,35)thetotalspinofthegroundstate\nof the cluster- npoints to the same direction as the Sl’s\nbelonging to that cluster- n. Therefore, the total spins of\nthe cluster- n’s on both sides of the dimer also tend to be\nantiparallel to each other. Thus, the effective coupling\nbetween the spins of neighboring cluster- n’s is antifer-\nromagnetic. This physical argument will be numerically\nensured below.\nIn general, the interaction between two spins with a\nmagnitude of 1 is the sum of bilinear and biquadratic\nterms. Therefore, the effective Hamiltonian for cluster-\nn’s in the phase that continues to the DC nphase in the\nlimit ofδA→0 is written as\nHeff=Nc/summationdisplay\ni=1Heff(i,i+1), (5)\nHeff(i,i+1) =JeffˆSiˆSi+1+Keff/parenleftBig\nˆSiˆSi+1/parenrightBig2\n,(6)\nwhereˆSiis the total spin of the i-th cluster- nwith a\nmagnitude of 1, Ncis the total number of cluster- n’s,\nandJeffandKeffare effective coupling constants. From\nsymmetry consideration, the signs of δAdoes not affect\nthe sign of the effective coupling constants. Hence, these\ncoupling constants are of the order of δ2\nAfor small δA.\nWe numerically calculated the ground-state energy of a\npair of cluster- n’s with total spin Stot, and compared\nit with the corresponding eigenvalues of Heff(i,i+ 1).\nThen we confirmed that Jeff/δ2\nAandKeff/δ2\nAare almost\nindependent of δAtypically for δA<0.002. The constant\nvaluesof Jeff/δ2\nAandKeff/δ2\nAareshowninFig.4forthree\nphases (n=0, 1, and 2), which will be explained below.\nBecause the effective coupling constants satisfy 0 <\nKeff/Jeff<1, the ground state is the Haldane state for4 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\n3 4 50510\n0.51\nλ λc(0,1)Keff/Jeff\nKeff/δA2Jeff/δA2Keff/JeffHDC0 Jeff/δA2\nKeff/δA2\n2.7 2.8 2.9 302040\n00.51\nλλc(0,1) λc(1,2)Keff/Jeff\nKeff/δA2Jeff/δA2Keff/Jeff HDC1Jeff/δA2\nKeff/δA2\n2.6 2.62 2.64 2.66050100\n00.51\nλc(1,2) λc(2,3)Keff/Jeff\nKeff/δA2Jeff/δA2Keff/Jeff HDC2 Jeff/δA2\nKeff/δA2\nλ\nFig. 4. Effective bilinear interaction ( Jeff) and biquadratic inter-\naction (Keff) between spin clusters for small δAin HDC0, HDC1,\nand HDC2 phases from top to bottom. The ratio Keff/Jeffis also\nshown.\nsmallδA.37)In the Haldane state, each spin-1 degree of\nfreedom is carried by a cluster- nrather than by a sin-\ngle spin. We call the state the Haldane DC n(HDCn)\nstate. In the HDC nstate with n≥1, the (n+ 1)-fold\ntranslational symmetry is spontaneously broken unlike\nthe conventional Haldane state without STSB. Both the\nHDC0 state for λ > λ c(0,1) and the HDC ∞state for\nλ < λ c(3,∞) are the Haldane states without STSB. In\nparticular, the HDC ∞state continues from the Haldane\nstate (DC ∞state) of the undistorted MDC mentioned\nin§2.12)Uniform Haldane\nλ>λc(1,0)\nHDC1\nHDC2\nHDC3Uniform Haldane\nλ>λc(1,0)\nUniform Haldane\nλ<λc(3,∞) or δA » 1(a)\n(b)\n(c)\n(d)\n(e)\nFig. 5. Valence bond structures of the ground states of all ph ases\nfor the MDC with type A distortion. A small filled circle repre -\nsents a spin with a magnitude of 1/2. An original spin with a\nmagnitude of 1 is represented by two decomposed 1/2 spins in\nan open circle indicating the symmetrization. A valence bon d is\nrepresented by a dashed oval.\n3.2 Connection to the strong distortion regime\nIn the strong distortion regime of δA→1 and small\nλ, the three spins τ(2)\nl−1,Sl, andτ(1)\nlform a singlet clus-\nter. Hence, the ground state is a state with spin gap and\nwithout STSB. This nature is common to the HDC0 and\nHDC∞phases in §3.1. Furthermore, the HDC ∞state\nis transformed into the HDC0 state only by rearranging\ntwo valence bonds within each diamond unit, as shown\nin Figs. 5(a) and 5(e). Therefore, the strong distortion,\nHDC0, and HDC ∞regimes are considered to be differ-\nent parts of a single phase. The continuity of the three\nregimes will be confirmed by the numerical analysis dis-\ncussed in §3.3. In what follows, we call this phase the\nuniform Haldane (UH) phase as a whole.\n3.3 Numerical phase diagram\nUnder the periodic boundary condition, even in the\nparameter region where STSB takes place, the ground\nstate of a finite chain is a superposition of the symmetry-\nbroken states, and the translational symmetry is re-\ncovered. Under the open boundary condition, however,\none of the symmetry broken states is selected by the\nboundary effect. Therefore, we employ the DMRG cal-\nculation with the open boundary condition to determine\nthe phase diagram for finite δA. The DMRG calculation\nis carried out using the finite-size algorithm up to 288\nsites keeping 200 states in subsystems. We calculate the\nground-state expectation values/angbracketleftbig\nT2\nl/angbracketrightbig\nand define the ef-\nfective spin magnitude Tlon thel-th diagonal bond by\nTl(Tl+1) =/angbracketleftbig\nT2\nl/angbracketrightbig\n. A typical ldependence of Tlis shown\ninFig.6ineachphase.Withtheincreasein δA,thetrans-\nlational symmetry is recoveredas expected. For finite δA,\nthe ground-state phase is identified from the periodicity\nin the oscillation of Tl. In the HDC nphase, the values ofJ. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki 5\n0 5001\nlTl\nδA=0.04\nδA=0.02λ=2.803N=288(a)\n0 5001\nlTl\nδA=0.009\nδA=0.007λ=2.623N=282(b)\nFig. 6. Profiles of Tlfor (a)λ= 2.8 and (b) λ= 2.62.\n0 0.2 0.4 0.600.20.40.6∆T\nN−1/8δA=0.0304\nδA=0.0306\nδA=0.0308(a)\n0 0.2 0.4 0.600.10.2∆T\nN−2/15δA=0.00824δA=0.00822\nδA=0.00826(b)\nFig. 7. System size dependences of ∆ Tat (a)λ= 2.85 and (b)\nλ= 2.62. The data are plotted against N−β/νwhereβandν\nare the critical exponents of the order parameter and correl ation\nlength, respectively, for the 2-dimensional (a) Ising and ( b) 3-\nclock model.\nTlfollow the sequence\n...TSTL···TL/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nnTS,TL···TL/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn..,(TL> TS).(7)\nThus, we define the order parameter of the HDC nphase\nby ∆T=TL−TS. In DMRG, ∆ Tis measured at the\nsites closest to the center of the chain.\nThe valence bond structures for the HDC nphases as2.6 2.8 300.020.04\nδA\nλλc(2,3)λc(3,∞)Z2STSB\n  Z3STSBNo STSB\nλc(1,2) λc(0,1)HDC1\nHDC2Uniform Haldane\nFig. 8. Phase diagram of the MDC with type A distortion. The\ntriangles indicate the position of the phase boundary for δA= 0.\nwell as the UH phase are shown in Fig. 5. We see the\ntranslational invariance of period n+ 1 in the HDC n\nground state in contrast to the period-1 invariance in\nthe UH ground state. Hence, the Zn+1STSB takes place\nat the HDC n-UH phase boundary. We expect that this\ntransition belongs to the 2-dimensional ( n+ 1)-clock\nmodel universality class. The system size dependence\nof ∆Tforλ= 2.85 is shown in Fig. 7(a) around the\nHDC1-UH phase boundary. Here, the data are plotted\nagainstN−β/νwith the order parameter critical expo-\nnentβ= 1/8 and the correlation length critical expo-\nnentν= 1 for the two-dimensional Ising universality\nclass.This showsthatthecriticalvalueof δAliesbetween\n0.0304 and 0.0308. A similar plot is shown in Fig. 7(b)\nforλ= 2.62 around the HDC2-UH phase boundary, as-\nsuming the critical exponents of two-dimensional 3-clock\nmodel(equivalently3-statePottsmodel38))withβ= 1/9\nandν= 5/6. This shows that the critical value of δA\nlies between 0.00822 and 0.00826. The critical points at\nother values of λare determined similarly. The results\nare shown in the phase diagram of Fig. 8. The error bars\nare within the size of the symbols.\nTo confirm the consistency of the universality class,\nthe finite size scaling plot for the order parameter ∆ Tis\ncarried out. According to the scaling hypothesis, the δA\ndependence of the order parameter ∆ Tof the finite size\nsystems near the critical point should obey the finite size\nscaling law39)\n∆TNβ/ν=f(N(δA−δc\nA)ν), (8)\nintermsofthescaledvariables∆ TNβ/νandN(δA−δc\nA)ν\nand the scaling function f(x). In Figs. 9(a) and 9(b),\n∆TNβ/νis plotted against N(δA−δc\nA)νaround the\nHDC1-UH and HDC2-UH phase boundaries assuming\nthe Ising and 3-clock universality classes, respectively.\nThe critical points δc\nA= 0.0307 (Fig. 9(a)) and 0.008248\n(Fig. 9(b)) are chosen so that all data fall on a single\nuniversal scaling curve as well as possible. These plots\nare consistent with the expected universality class.\nThe critical behavior at the HDC1-UH transition in6 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\n0 0.101∆TN1/8\n(δA−δAc)N3N=144\n216\n288(a)\n−0.2 0 0.201∆TN2/15\n(δA−δAc)5/6N3N=138\n210\n282(b)\nFig. 9. Finite-size scaling plot of ∆ Taround the critical points.\n(a) Plot around the HDC1-UH phase boundary at λ= 2.85.\nThe Ising critical exponents ν= 1 and β= 1/8 are assumed.\nThe critical point is set at δc\nA= 0.0307. (b) Plot around the\nHDC2-UHphase boundary at λ= 2.62. The3-state Potts critical\nexponents ν= 5/6 andβ= 1/938)are assumed. The critical\npoint is set at δc\nA= 0.008248.\n(a)\n(b)\nFig. 10. Valence bond structures of the ground states of spin -1\nbilinear-biquadraticchaininthe (a)Haldanephase and (b) dimer\nphase. The spins with a magnitude of unity represented by ope n\ncircles are decomposed into two spin-1/2 degrees of freedom rep-\nresented by smallfilled circles.The valence bonds are repre sented\nby dashed ovals. The spins belonging to disconnected cluste rs in\nthe dimer phase are connected by the valence bonds in the Hal-\ndane phase.\nour model should be compared with that of the S=\n1 bilinear-biquadratic chain at the Takhtajan-Babujian\npoint.40,41)Both transitions are accompanied by Z2-\nSTSB which contributes to the conformal charge by 1/2.\nFortheHDC1-UHtransitioninourmodel,therearrange-\nment of valence bonds take place only within each dia-\nmond unit, as shown in Fig. 5(a) and 5(b). In contrast,\nin theS= 1 bilinear-biquadratic chain, the spins be-\nlonging to disconnected clusters in the dimer phase are0 1 2 300.51\nλδB 3N=18\nUHFDC0\nm=1\nFDC1\nm=1/2\nFDC2\nm=1/3\nFig. 11. Phase diagram of the MDC with type B distortion with\n3N= 18. The triangles indicate the position of the phase bound-\nariesλc(n,n+1) for δB= 0.12)\nconnected by the valence bonds in the Haldane phase, as\nshown in Fig. 10. Apart from Z2-STSB, this is similar\nto the Gaussian criticality of the Haldane-dimer transi-\ntion in the spin-1 alternatingbond Heisenbergchain that\ncontribute to the conformal charge by 1.42–44)Therefore,\ntheS= 1 bilinear-biquadratic chain at the Takhtajan-\nBabujian point is described by the conformal field theory\nwithc= 1/2+1 = 3 /2, while the HDC1-UH transition\nin our model is described by the c= 1/2 Ising conformal\nfield theory.\n4. Ground-State Properties of the MDC with\nType B Distortion\nIn the case of type B distortion, the effective inter-\naction between the spins of two cluster- n’s separated by\nthedimerconsistingof τ(1)\nlandτ(2)\nlisferromagnetic,be-\ncause both SlandSl+1tend to be antiparallel to τ(1)\nl.\nTherefore,we expect the ferrimagneticgroundstate with\nspontaneous magnetization quantized as m= 1/(n+1)\nper unit cell for small δBin the range λc(n,n+ 1)<\nλ < λc(n−1,n). We call this phase a ferrimagnetic DC n\nphase (FDC nphase). In contrast, the ground state for\nλ < λ c(3,∞) will remain in the Haldane phase, since a\nnonmagnetic gapped phase is generally robust against a\nweakdistortion.Forfinite δB,we determined the ground-\nstate phase diagramby the numerical diagonalizationfor\nthe system size 3 N= 18, as shown in Fig. 11. Among\nsystem sizes tractable by numerical diagonalization, only\nthis size of 3 N= 18 is compatible with all the ground-\nstate structures with n= 0,1, and 2. As expected, the\nFDCnphases with m= 1/(n+ 1) are found for these\nvalues of n.\nBy inspecting numerical data for the 3 N= 18 system,\nwe also find other narrow ferrimagnetic phases between\nthe FDC nand FDC( n+ 1) phases with n= 0, 1, and\n2, although they are too narrow to be shown in Fig. 11.\nIn order to investigate these phases in detail, we employJ. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki 7\n2.4 2.6 2.800.51\nλm\n3N=18(ED)\n3N=72(DMRG)δB=0.2(a)\n1.7 1.800.51\nλm\nλm\n3N=18(ED)\n3N=72(DMRG)δB=0.6(b)\nFig. 12. Spontaneous magnetization for (a) δB= 0.2 and (b)\nδB= 0.6. The triangles on the vertical axes indicate the values\nof the spontaneous magnetization m= 1/(n+ 1) in the FDC n\nphases.\nthe DMRG calculation for 3 N= 72 keeping 120 states in\neach subsystem. Typical examples of the λdependence\nof spontaneous magnetization are shown in Fig. 12(a)\nforδB= 0.2 and Fig. 12(b) for δB= 0.6. Between the\nFDCnand FDC( n+1) phases with n= 0,1,2, we find\nthe partial ferrimagnetic phase in which the spontaneous\nmagnetization varies continuously with λ. The ferrimag-\nnetic phase of this kind has been found in various frus-\ntrated one-dimensional quantum spin systems.45–51)In\ncontrast, between the nonmagnetic phase and the FDC3\nphase, we find no partial ferrimagnetic phase for small\nδB.\nThis can be understood as follows: At λ=λc(n,n+1),\nthe cluster- nand cluster-( n+1) can coexist. As stated\nabove, it is physically evident that the effective magnetic\ninteraction between the clusters is ferromagnetic. There-\nfore, wecan restrict the statesof eachcluster to the max-\nimally polarized ground state with ˆSz\ni= 1. Hence, the\ngroundstateofthe wholechainisdescribedbyspecifying\nthe arrangement of cluster- n’s and cluster-( n+1)’s. We\nmap the two possible values of the length of i-th cluster,\nni=nandni=n+ 1, to two possible values of the\nspin-1/2 pseudospin, σz\ni= 1/2 andσz\ni=−1/2, respec-\ntively. Then, the total magnetization Mis equal to the\nnumber of clusters Nc. The total number of unit cells,N, is related to the pseudospins σz\nias\nN=Nc/summationdisplay\ni=1/parenleftbigg\nn+1+1\n2−σz\ni/parenrightbigg\n=Nc/parenleftbigg\nn+3\n2/parenrightbigg\n−Nc/summationdisplay\ni=1σz\ni.\n(9)\nTherefore, the ground-state magnetization per unit cell\nmis given by\nm=Nc\n/an}bracketle{tN/an}bracketri}ht=1\nn+3\n2−σ(10)\nwithσ≡/summationtextNc\ni=1/an}bracketle{tσz\ni/an}bracketri}ht/Nc.Thebracket /an}bracketle{t···/an}bracketri}htrepresentsthe\nground-state expectation value. In the presence of δB,\nthe length of neighboring clusters can exchange through\na second order process in δB. This corresponds to the\nspin exchange in terms of pseudospins. In this case, the\ninteraction between the pseudospins is approximated by\nthe spin-1 /2 XXZ Hamiltonian\nHXXZ=Nc/summationdisplay\ni=1HXXZ(i,i+1), (11)\nHXXZ(i,i+1) =Jeff\nzσz\niσz\ni+1+Jeff\n⊥(σx\niσx\ni+1+σy\niσy\ni+1)\n(12)\nuptothesecondorderin δB.Here,furtherneighborinter-\nactions are neglected. We estimate the effective exchange\nconstants by comparing the energy spectrum of the pair\nHamiltonian HXXZ(i,i+1) with that of the correspond-\ning pair of clusters as follows:\n(i)λ=λc(0,1)\nJeff\nz≃ −0.039δB2,\nJeff\n⊥≃0.087δB2. (13)\n(ii)λ=λc(1,2)\nJeff\nz≃ −0.0082δB2,\nJeff\n⊥≃0.069δB2. (14)\n(iii)λ=λc(2,3)\nJeff\nz≃ −0.0029δB2,\nJeff\n⊥≃0.018δB2. (15)\nThe details of the calculation are explained in Appendix.\nIn all cases (i) ∼(iii), we find that the effective cou-\npling constants satisfy the inequality −|Jeff\n⊥|< Jeff\nz≤\n|Jeff\n⊥|. As is well known, the ground state of the spin-\n1/2 XXZ chain in this parameter regime is nonmagnetic\nand gapless in the absence of a magnetic field. Roughly\nspeaking, ∆ λ≡λ−λc(n,n+1) corresponds to the effec-\ntive magnetic field heffconjugate to the total pseudospin/summationtext\niσz\ni, because the increase in λfavors cluster- nover\ncluster-(n+1); however, this correspondence should not\nbe taken literally. A more precise argument is also given\nin Appendix. When ∆ λtakes a large negative value,\nthe pseudospins are fully polarized downward to give\n/an}bracketle{tσz\nl/an}bracketri}ht=−1/2. This state corresponds to the FDC( n+1)\nstate with m= 1/(n+2). When heffreaches the critical\nvaluehc1≡ −(|Jeff\n⊥|+Jeff\nz), the magnetization starts to\nincrease continuously until all spins are fully polarized8 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\nupward at the critical effective field hc2≡ |Jeff\n⊥|+Jeff\nz.\nThis corresponds to the FDC nstate with m= 1/(n+1).\nOn the other hand, the magnetization jumps from 0 to\n1/4 at the phase boundary between the Haldane phase\nand the FDC3 phase for small δB. At this phase bound-\nary, no finite size clusters coexist with cluster-3. There-\nfore, no pseudospin degrees of freedom can be defined.\nConsequently, no partial ferrimagnetic phase can be re-\nalized. In contrast,for largervalues of δB, we numerically\nfind a partialferrimagnetic phasebetween the FDC3 and\nUHphases. This wouldbe ascribed tothe contributionof\nother finite-length clusters with low lying energies which\ncome into play through higher-order processes in δB.\n5. Summary and Discussion\nWeintroducedtwotypesofdistortion,typeAandtype\nB, into the MDC with spins 1 and 1/2, and investigated\nthe ground-state phases. The phase diagrams are charac-\nteristicofthetypeAandtypeBdistortions,respectively.\nFor the type A distortion, the effective interaction be-\ntween the cluster spins is antiferromagnetic with bilinear\nand biquadratic terms. The numerically estimated val-\nues of the effective couplings show that the DC nground\nstatesaretransformedintotheHDC ngroundstates.The\norder parameters characterizing the HDC nphases are\ndefined and the UH-HDC nphase boundaries are deter-\nmined using the DMRG data. From the valence bond\nstructure of each phase, we expect that the UH-HDC n\nphase transition belongs to the universality class of the\n2-dimensional ( n+1)-clock model. The finite size scaling\nplot of the order parameter is consistent with this iden-\ntification. For the type B distortion, the effective inter-\naction between the cluster spins is ferromagnetic. In ad-\ndition to the FDC nphases with quantized spontaneous\nmagnetization m= 1/(n+1), the partial ferrimagnetic\nphasesarealsofoundnumericallybetweentheFDC nand\nFDC(n+1) phases. A physical interpretation of the par-\ntial ferrimagnetic phase is given for small δBby mapping\nonto an effective pseudospin-1/2 XXZ chain.\nGenerally, the introduction of lattice distortion into\na physical model increases the possibility that a corre-\nsponding material is realized. In the MDC, there are\nthree types of distortion modes affecting the exchange\ninteractions. Among them, the two types investigated in\nthe present paper are of generic nature, because the lo-\ncal conservation laws that hold in the undistorted MDC\nare broken. This suggests that the observation of the ex-\notic phenomena predicted in the present paper is pos-\nsible even if the corresponding material is not exactly\ndescribed by the model Hamiltonians (1) and (2).\nIf a distorted MDC material is synthesized, the dis-\ntortion may be controlled by, e.g., applying pressure. If\nthe distortion is of type A, the Curie constant vanishes\nas the DC nground state turns into one of the HDC n\nground states. The magnetic susceptibility and magnetic\nspecific heat will have an activation-type temperature\ndependence with activation energy proportional to the\neffective coupling between the cluster spins, which is of\nthe order of δA2. These HDC nphases are not realized\nif the distortion δAexceeds∼0.03 even in the most ro-\nbust case of n= 1. In a real material, the STSB in thevalence bond structure manifests itself as a magnetic su-\nperstructure. It is also possible that it is accompanied\nby a lattice superstructure of corresponding periodicity\nif the spin-lattice coupling is present. Therefore careful\nmeasurements of magnetic and lattice superstructures\nwould help with the observation of HDC nphases with\n1≤n≤3.\nOn the other hand, if the distortion of the material\nis of type B, the ground sate is ferrimagnetic. At low\nbut finite temperatures, however, the spontaneous mag-\nnetization vanishes owing to the one-dimensionality. As\na precursor of ordering at T= 0, the low-temperature\nmagnetic susceptibility should diverge as T−2with a co-\nefficient proportional to the effective coupling ∼δB2be-\ntween the cluster spins.52–54)This means that even a\nweak magnetic field of the order of H∼T2/δB2derives\nthe finite-temperature magnetization up to the value of\nthe ground-state spontaneous magnetization. This en-\nables the experimental estimation of the spontaneous\nmagnetization in real materials. The quantized ferrimag-\nnetic behavior should be observed for wide ranges of the\nparameters λandδBas shown in Fig. 11, and should\nbe easily observed if an appropriate material is synthe-\nsized. The partial ferrimagnetic phases are limited to\nnarrow intervals of the parameters δBandλ. Therefore,\nthesecanonlybeobservedasatemperature-independent\ncrossoverbetween two quantized ferrimagnetic behaviors\nwith careful exclusion of the thermal effect.\nWe have demonstrated that various exotic ground\nstates and phase transitions between them are realized\nin the distorted MDC with spins 1 and 1/2, which has\na strong frustration. The physical pictures of these phe-\nnomenahavebecomeclear.Thisismadepossiblebecause\nthe ground state of the undistorted MDC is known ex-\nactly. Therefore, we expect that our model may provide\na means of understanding the similar exotic phenomena\nrealized owing to the interplay of spin ordering,quantum\nfluctuation, and strong frustration in more general frus-\ntrated quantum chains on a firm ground. For example,\npartial ferrimagnetic phases are found in various one-\ndimensional frustrated quantum spin models.45–51)How-\never, some of them are only numerically confirmed and\nno physical explanation has been given so far. We hope\nthat the present study paves the way to the general un-\nderstanding of these partial ferrimagnetic states.\nWe thank J. Richter for drawing our attention to ref.\n28andrelatedworks.Thenumericaldiagonalizationpro-\ngram is based on the package TITPACK ver.2 coded by\nH. Nishimori. The numerical computation in this work\nhas been carried out using the facilities of the Super-\ncomputer Center, Institute for Solid State Physics, Uni-\nversity of Tokyo and Supercomputing Division, Informa-\ntion Technology Center, University of Tokyo. KH is sup-\nported by a Grant-in-Aid for Scientific Research on Pri-\nority Areas, ”Novel States of Matter Induced by Frustra-\ntion” (20048003) from the Ministry of Education, Cul-\nture, Sports, Science and Technology of Japan and a\nGrant-in-Aidfor Scientific Research(C) (21540379)from\nthe Japan Society for the Promotion of Science. KT and\nHS are supported by a Fund for Project Research from\nToyota Technological Institute.J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki 9\nAppendix\nThe Hamiltonian HBwith the type B distortion is\nrewritten as\nHB=H0+δH, (A·1)\nwhere\nδH=δBN/summationdisplay\nl=1/parenleftbig\nSl+Sl+1/parenrightbig/parenleftbig\nτ(1)\nl−τ(2)\nl/parenrightbig\n.(A·2)\nFor small δB, the ground state around λ=λc(n,n+1)\nconsists almost entirely of cluster- n’s and cluster-( n+\n1)’s. Hence, as a good approximation, we consider HB\nonly in the restricted Hilbert space where each state\ninvolves no clusters except for cluster- n’s and cluster-\n(n+ 1)’s. Under the fixed cluster number Ncin this\nHilbert space, H0is equivalent to the following effective\nHamiltonian expressed in terms of pseudospin operators:\nH0\neff=E0\nG(n+1;λ)Nc/summationdisplay\ni=1/parenleftbigg1\n2−σz\ni/parenrightbigg\n+E0\nG(n;λ)Nc/summationdisplay\ni=1/parenleftbigg1\n2+σz\ni/parenrightbigg\n, (A·3)\nwhereσz\ni= 1/2 andσz\ni=−1/2 correspond to ni=n\nandni=n+ 1, respectively. E0\nG(n;λ) is the ground-\nstate energy of a cluster- nand a dimer in the absence of\ndistortion, and is given by\nE0\nG(n;λ) =˜E(2n+1)+λn\n4−3λ\n4,(A·4)\nwhere˜E(2n+1) is the ground-state energy of the spin-1\nantiferromagneticHeisenbergchainwith length2 n+1.12)\nTheapplicationof δHtotheunperturbed groundstate\ntransforms one of the Tl= 0 bonds to a Tl= 1 bond or\nviceversa.Thentheresultingstatescontainclusterswith\nlengths less than nor greater than 2 n. Since these states\nare outside the restricted Hilbert space, no correction to\nthe ground-state energy is present within the first order\ninδB. Hence, the lowest-order correction is of the order\nofδB2. Up to the second order in δB, the effective pseu-\ndospin Hamiltonian is given by\nHeff=EG(n+1;λ,δB)Nc/summationdisplay\ni=1/parenleftbigg1\n2−σz\ni/parenrightbigg\n+EG(n;λ,δB)Nc/summationdisplay\ni=1/parenleftbigg1\n2+σz\ni/parenrightbigg\n+HXXZ,(A·5)\nwhereEG(n;λ,δB) is the ground-state energy of a\ncluster-nand a dimer including the second order cor-\nrection in δB. This is simply expressed as\nHeff=Nc¯EG+∆EGNc/summationdisplay\ni=1σz\ni+HXXZ,(A·6)\nwith\n¯EG=1\n2(EG(n+1;λ,δB)+EG(n;λ,δB)),(A·7)\n∆EG=EG(n;λ,δB)−EG(n+1;λ,δB).(A·8)The effective coupling constants Jeff\nzandJeff\n⊥inHXXZ\nare also of the second order in δB. We determine Jeff\nzand\nJeff\n⊥soastoreproducethelow-lyingenergyspectrumofa\npairofcluster- n’sbythatoftwo-pseudospinHamiltonian\nHeff(i,i+1) = 2 ¯EG+∆EG(σz\ni+σz\ni+1)+HXXZ(i,i+1)\n(A·9)\nIn each of the subspaces σz\ni+σz\ni+1=±1,σz\ni=\nσz\ni+1=σ=±1/2. Therefore, the Hilbert space is one-\ndimensional and the eigenvalue of Heff(i,i+ 1) is sim-\nplyEσσ= 2¯EG+ 2∆EGσ+Jeff\nz/4 withσ=±1/2.\nIn the subspace σz\ni+σz\ni+1= 0, the Hilbert space is\ntwo-dimensional and the eigenvalues of Heff(i,i+1) are\nE±= 2¯EG−Jeff\nz/4±Jeff\n⊥/2.\nThe original Hamiltonian of the cluster consisting of a\ncluster-nand a cluster- n′is the distorted diamond chain\nwith length n+n′.\nH(n+n′) =n+n′+1/summationdisplay\nl=1/bracketleftBig\n(1+δB)Slτ(1)\nl+(1+δB)τ(1)\nlSl+1\n+(1−δB)Slτ(2)\nl+(1−δB)τ(2)\nlSl+1+λτ(1)\nlτ(2)\nl/bracketrightBig\n,\n(A·10)\nWe denote the α-th eigenvalue of H(n+n′) asE(n+\nn′;α). Comparing the corresponding expression for the\neigenvalues, we find\nE(2n;0) =E1\n2,1\n2= 2¯EG+∆EG+Jeff\nz\n4(A·11)\nE(2n+2;0) = E−1\n2,−1\n2= 2¯EG−∆EG+Jeff\nz\n4(A·12)\nE(2n+1;0) = E−= 2¯EG−Jeff\nz\n4−Jeff\n⊥\n2(A·13)\nE(2n+1;1) = E+= 2¯EG−Jeff\nz\n4+Jeff\n⊥\n2.(A·14)\nSolving these sets of equations, with respect to Jeff\nzand\nJeff\n⊥, we find\nJeff\n⊥=E(2n+1;1)−E(2n+1;0), (A·15)\nJeff\nz= 2[E(2n+2;0)+ E(2n;0)\n−E(2n+1;1)−E(2n+1;0)].(A·16)\nNote that the rhs’s of (A ·15) and (A ·16) vanish for\nδB= 0. We numerically evaluated E(2n;0),E(2n+1;0),\nE(2n+1;1), and E(2n+2;0) at λ=λc(n,n+1) (n=\n0,1,2) for small δB. Using these values in eqs. (A ·15) and\n(A·16), we determined Jeff\n⊥andJeff\nzas eqs. (13)-(15).\nForthewholeMDC,theground-stateenergyiswritten\nas\nE0=Nc¯EG+Nc∆EGσ+NcǫXXZ(σ) (A ·17)\nwhereǫXXZ(σ) is the ground-state energy per site of a\nmagnetized spin-1/2 XXZ chain with /an}bracketle{tσz\ni/an}bracketri}ht=σ. The\nnumber of unit cells, N, of the original MDC is given by\nthe expectation value of eq. (9) as N=Nc(n+3\n2−σ).\nTherefore, we have\nE0=N\nn+3\n2−σ/parenleftbig¯EG+∆EGσ+ǫXXZ(σ)/parenrightbig\n.(A·18)10 J. Phys. Soc. Jpn. Full Paper K.Hida, K.Takano , and H. Suzuki\n−4 −2 0 2−0.500.5\n∆λ/Jeff\n⊥σn=0n=1\nn=2\nFig. A·1. Relationship between σand ∆λforn= 0,1, and 2.\nMinimizing this with respect to σwith fixed N, we find\n∆λ=/parenleftbigg\nn+3\n2−σ/parenrightbigg∂ǫXXZ(σ)\n∂σ+ǫXXZ(σ),(A·19)\nwhere ∆ λ=λ−λc(n,n+ 1;δB) andλc(n,n+ 1;δB) is\ndefined by\n(n+2)EG(n;λc,δB)−(n+1)EG(n+1;λc,δB) = 0.\n(A·20)\nTo simplify the calculation, we replace ǫXXZ(σ) by the\nground-state energy of the spin-1/2 XY chain ǫXY=\n−(Jeff\n⊥/π)cosπσ, because |Jeff\n⊥|is substantially larger\nthan|Jeff\nz|in all cases. Then we find\n∆λ\nJeff\n⊥=/parenleftbigg\nn+3\n2−σ/parenrightbigg\nsinπσ−1\nπcosπσ. (A·21)\nThis relation is plotted in Fig. A ·1 forn= 0,1 and 2. 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B 57(1998) R14008."    },    {        "title": "1303.1372v1.Change_in_the_Magnetic_Domain_Alignment_Process_at_the_Onset_of_a_Frustrated_Magnetic_State_in_Ferrimagnetic_La2Ni_Ni1_3Sb2_3_O6_Double_Perovskite.pdf",        "content": "10-56 \n 1\nChange in the Magnetic Domain Alignment Process at the Onset of a \nFrustrated Magnetic State in Ferrimagnetic La 2Ni(Ni 1/3 Sb 2/3 )O6 \nDouble Perovskite \n(Revised ..) \n \nDiego G. Franco 1,2 , Raúl E. Carbonio 1, and G. Nieva 2,3 \n \n1INFIQC-CONICET, Depto. de Físico Química, Facultad de Cienc ias Químicas, Universidad Nacional de Córdoba, Ciudad \nUniversitaria. X5000HUA Córdoba, Argentina \n2Laboratorio de Bajas Temperaturas. Centro Atómico Baril oche –CNEA. 8400 Bariloche, R. N., Argentina. \n3Instituto Balseiro, CNEA and Universidad Nacional de Cuy o.  8400 Bariloche, R. N., Argentina. \n \nWe have performed a combined study of magnetization  hysteresis loops and time dependence of the magnet ization in a broad \ntemperature range for the ferrimagnetic La 2Ni(Ni 1/3 Sb 2/3 )O6 double perovskite.  This material has a ferrimagne tic order transition \nat ~100 K and at lower temperatures ( ~ 20 K) shows the signature of a frustrated state du e to the presence of two competing \nmagnetic exchange interactions. The temperature dep endence of the coercive field shows an important up turn below the point \nwhere the frustrated state sets in. The use of the magnetization vs. applied magnetic field hysteresis  data, together with the \nmagnetization vs. time data provides a unique oppor tunity to distinguish between different scenarios f or the low temperature \nregime. From our analysis, a strong domain wall pin ning results the best scenario for the low temperat ure regime. For \ntemperatures larger than 20K the adequate scenario seems to correspond to a weak domain wall pinning.   \n \nIndex Terms — Ferrimagnetic materials, Magnetic analysis, Magneti c domain walls, Magnetic hysteresis.  \n \nI.  INTRODUCTION  \nANY of the magnetic interactions found in transition \nmetal oxide perovskites are due to superexchange \nand/or super-superexchange interactions mediated through \nthe O2- p orbitals.  In some materials the relative strength of \nthese interactions determines the magnetic structure, range  \nof the ordering temperatures and the possibility of \nfrustration [1]-[5].  In the perovskite structure, the t ypical \nbond angles and distances usually favor antiferromagnetic \nsuperexchange interactions [6],[7]. However, in some \nspecial cases, due to disorder or differences in the magne tic \nstate of the cations, a ferrimagnetic state is develope d with \nmacroscopic characteristics similar to a ferromagnetic state. \nCoercivity and remanence are an indication of the \nmetastability in ferromagnetic samples. Their magnitudes \nindicate how far the system is from equilibrium. They ar e \nrelated, therefore, with the relaxation to the equilibriu m \nstate, the anhysteretic [8] curve in the ferromagnetic st ate.  In  \nbulk ferromagnets the energy barriers that determine the \ntime evolution of the magnetization are related to local \ninteractions within a domain, the nucleation and the \nmovement of domain walls (DW). The DW movement \ndepends on the applied magnetic force, wall thickness and \ntype and density of pinning centers.   \nIn bulk ferromagnetic samples, a local frustration is \nnormally hard to visualize due to the magnetic history \ndependence of the metastable states. However, the magneti c \nmoments alignment within a domain and the movement of \nthe DW have characteristic energies [9]  that could be \nmodified if some degree of magnetic frustration occurs at a  microscopic level. This normally results in a strong DW \npinning effect and causes an increase in the coercivity. \nThis article will present a detailed magnetic study of t he \nferrimagnetic double perovskite La 2Ni(Ni 1/3Sb 2/3)O 6.  We \nwill show that the material behaves as a ferrimagnet below  \n100 K and that there is a change in the magnetic domains \nalignment process at 20 K.  We will show that below 20 K \nthe hysteretic magnetic behavior is characteristic of a strong \ndomain wall pinning regime due to the onset of a frustrated \nmagnetic interaction. \nII.  RESULTS  \nWe prepared polycrystalline samples of \nLa 2Ni(Ni 1/3 Sb 2/3 )O6 by conventional solid-state reaction at \n1400 oC [10].  X ray diffraction data from powders at room \ntemperature showed the crystalline symmetry to be \nmonoclinic, space group P2 1/n. This space group \naccommodates a rock salt arrangement of BO 6 and B'O 6 \noctahedra described by the a -b-c+ system of three octahedral \ntilts in the Glazer's notation. The (Ni/Sb) 2dO6 and \n(Ni/Sb) 2cO6 octahedra are rotated in phase (along the \nprimitive c axis) and out-of phase (along the primitive a and \nb axes). We performed a Rietvelt refinement of the structur e \nusing the FULLPROF program [11], resulting in lattice \nparameters of a = 5.6051(3) Ǻ, b = 5.6362(3) Ǻ, c = \n7.9350(5) Ǻ and β = 89.986(4) o. We refined the two \ncrystallographic sites 2 d and 2 c with different occupancies \nNi 2+ /Sb 5+  to model the octahedral site disorder. The 2 d \ncation site is almost fully occupied by Ni 2+  whereas the 2 c \nsite has occupancy close to 1/3 of Ni 2+  ions and 2/3 of Sb 5+ . \nThe resulting crystallographic formula can be written as \nLa 2(Ni 0.976 Sb 0.024 )2d(Ni 0.357 Sb 0.643 )2cO6. M\nManuscript received January XX, 2013 (date on which  paper was \nsubmitted for review). Corresponding author: G. Nie va (e-mail: \ngnieva@cab.cnea.gov.ar). \nDigital Object Identifier inserted by IEEE 10-56 \n 2\nThe magnetic measurements were performed on \npolycrystalline pellets with a QD-MPMS SQUID \nmagnetometer in the range 2 to 300K and -5 to 5T. In the \nmain panel of Fig. 1 we show the magnetization, M, as a \nfunction of temperature, T, while cooling in a very low \napplied field, H.  There is a transition to a magnetic \npolarized state at TC = 98(2) K. \nWe extrated the low temperature value of the saturation \nmagnetization, Ms, from M vs. H curves, from the \nasymptotic extrapolation of the high field behavior with a \nLangevin function.  This saturation magnetization, Ms, has a \nlower value than the one expected for the complete \npolarization of the Ni 2+  magnetic moments, 2.67 µB/f.u.. \nInstead, the experimental Ms value was 1.19  µB/f.u., \nimplying that the system behaves as a ferrimagnet, with two \nNi 2+  magnetic sublattices antiferromagnetically coupled, one \nat the 2 d site and another at the 2 c site. The near 1/3 Ni 2+  \nrandom occupation of the 2 c sites sublattice give as a result \nuncompensated Ni 2+ magnetic moments that order at 100 K. \nFor a perfectly stoichiometric ferrimagnetic sample and ful l \nNi 2+  occupancy of the 2 d site Ms should be 1.33   µB/f.u., and \nlower values are expected if Sb 5+  partially occupies also the \n2d site. The expected value for Ms with the refined \noccupancies is 1.24   µB/f.u., very close to the experimental \none. \nWe measured hysteresis loops, M vs H, for several \ntemperatures below 100 K. We show in the inset of Fig. 1  a \ndetail of the loops for 2 K and 20 K.  \nFIG. 1 HERE \nWe have also measured the time evolution of the \nmagnetization at the coercive field (i.e. near the field f or \nzero magnetization)  after saturation at 1, 3 and 5 T for each \ntemperature. We show in Fig. 2 typical M vs time data, for \nthree different temperatures at their corresponding coercive \nfields. \n \nFIG. 2 HERE \nIII.  DISCUSION \nWe show in Fig. 3(a) and (b) the temperature dependence \nof the coercive field, Hc, and the ratio between remanent \nmagnetization and saturation magnetization, ( Mr / M s). The general feature observed in Fig. 3 is that Mr and Hc increase \nsteeply when the temperature is lowered below 20 K \nindicating an increase in the energy absorbed by the materi al \nto change the direction of M. \nThe measured values of the coercive field, Hc, display two \ndifferent regimes as can be seen in Fig. 3(a). For T > 20 K a \nlinear behavior of Hc was found. This linear behavior is \ncharacteristic of weak DW pinning (WDWP), produced by a \nrandom distribution of individual weak pinning sites [9].  In \nthis case the coercive field is given by \n \n\n\n\n\n\n\n\n− =2 031 25 1\nbTkH HB\nW cγ                       (1) \n \nwhere H0W  is the zero temperature extrapolated reversion \nfield, kB is the Boltzmann constant, γ is the DW energy per \nunit area and b is a measure of the DW thickness. The \nobtained values are shown in Table I. \nIn the low temperature regime, T < 20 K, two models \ndescribe reasonably well the data. One corresponds to stron g \nDW pinning (SDWP), \n \n23 / 2\n0475 1\n\n\n\n\n\n\n\n− =bf TkH HB\nS c                    (2) \n \nwhere H0S is the coercive field at zero temperature and f is \nthe magnetic force needed to depin a domain wall.  The \nfitted values are shown in Table I. \nThe other model corresponds to the freezing of single \ndomain large particles (SDLP) or clusters [11], [12]. In t his \nscenario, \n \n\n\n\n\n\n\n− =2/ 125 1KV TkH HB\nK c                              (3) \n \nwhere HK is the anisotropy field of a particle or cluster, V is \nits volume and K is the uniaxial anisotropy energy density. \nThe fitted values are shown in Table I.  \nFIG. 3 HERE \n \n \nFig. 1.  (color online) Magnetization as a function of temperature cooling \nwith an applied field of 1 Oe.  Inset: Magnetizatio n as a function of ap plied \nfield, detail of the magnetization loops for two fi xed temperatures T = 2 K \nand 20 K. \n \nFig. 2. (color online) Difference between the measured magnetization and \nthe initial one, M 0, as a function of time. The shown mag netization time \ndependence was taken at H c (M ~ 0) after saturation in the opposite \ndirection.  We indicate the fixed temperatures for each experiment.  10-56 \n 3\nIn Fig. 3(a) and the inset we show the lines (solid and \ndash doted) corresponding to each model. The best fit is \nobtained with the SDWP model but the freezing of SDLP \nmodel is also in fair agreement with the Hc data. \nThe time evolution of the magnetization could be used to \ndiscern between the two scenarios at low temperature.  I f a \ndistribution of activation energies is present in the mater ial \n[9] a logarithmic behavior is expected for M(t): \n \n)ln( 0 t S M M − =                                (4) \n \nwhere M0 is the starting value of the magnetization and S the \nmagnetic viscosity coefficient. The above relation holds  \napproximately for our polycrystalline pellets samples a s we \nshow in Fig. 2. \nIn a tipical ferromagnet, the time dependence of M is \nirreversible and this behavior has been connected with the \nirreversibility caused by a small change in field, the so \ncalled irreversible susceptibility, χirr . Both irreversibilities \nare related by a fictitious field, the fluctuation field,  Hf, in \nthe theory introduced by Neel [13] that represents an \naverage of the thermally activated, time dependent processes \n[14] leading to equilibrium by reversing the metastable  \nmagnetization. In terms of the magnetization derivatives, a t \na given field and temperature, \n \nf irr \nMi\nt iirr \nHirr HtH\nHM\ntMS\nirr iχ=\n\n\n\n∂∂\n\n\n\n\n∂∂=\n\n\n\n∂∂=)ln( )ln(       (5) \n \nwhere Mirr  is the irreversible magnetization and Hi is the \ninternal field. In the case of a time independent viscosity coeficient S, the fluctuation field is equivalent to the \nmagnetic viscosity parameter Sv, that can be written in terms \nof the activation energy, E, necessary for magnetization \nreversal [15], \n \nirr MB\nvS\ndH dE TkS\nirr χ=−=) /(.                       (6) \n \nTo determine the temperature dependence of Sv, a value of \nH equal to the coercive field is chosen [15] ( Mirr = 0). The \nactivation energy for the SDWP, WDWP and clusters or \nSDLP freezing are given by [16]: \n \n2/32/1\n01) 3 / 4 (\n\n\n\n\n\n\n\n− =\nSSHHbf E                   (7) \n\n\n\n\n\n\n\n− =\nWWHHb E\n021) 31 (γ                         (8) \n2\n1\n\n\n\n\n\n\n\n− =\nKFHHKV E                           (9) \n \nand in each case the magnetic viscosity parameters are g iven \nrespectively by, \n \n\n\n\n\n\n\n\n−\n\n\n=3/2 3/2\n0475 1475 \n75 4)(bf Tk\nbf TkH SB B\nS Sv         (10) \n2 031 25 \n25 1)(bTkH SB\nW Wvγ=                             (11) \n2/125 \n50 1)( \n\n=KV TkH SB\nK Fv.                        (12) \n \nFrom the experimental data (such as those of the inset of \nFig. 4) the values of χirr (H = Hc) can be extracted. They can \nbe approximated as those of the total χ at Hc, neglecting the \nreversible contribution to χ [14]. These values are shown in \nFig. 4. Also from the experimental data the S values can be \ncalculated from (4) at Hc, since the linear behavior holds, \nFig. 2. In this case we take Mirr  as the measured M, \nneglecting the reversible component. \nFIG. 4 HERE \nTABLE 1 HERE \nThe experimental values of Sv were obtained by using (6). \nThey are displayed in Fig. 5 together with the fits for \ndifferent models in different temperature ranges using (10) - \n(12). At low temperature, the best fit to the data is given  \nTABLE  I \n γb2 \n(10 -14 erg) H0W  \n(Oe)  4bf \n(10 -13  erg) H0S  \n(Oe) KV \n(10 -14  erg) H0F  \n(Oe) \nHc 1.26 53.5 3.07 780 7.4 760 \nSv 1.4 53.5 2.13  780 7.4 760  \n \nFitted parameters for the WDWP model above 20 K and  the two models \ncompared below 20K, SDWP and freezing of SDLP. The first column \nindicates whether the coercive field or the magneti c viscosity parameter \nwere used in the parameters determination.  In the case of the viscosity \nparameter, the zero temperature fields H0W , H0S  and H0F  were not fitted but \ntaken from the Hc fits. \n \n \nFi g. 3.  (color online) (a) Coercive field Hc vs T for La 2Ni(Ni 1/3Sb 2/3)O 6\npolycrystalline samples. The solid lines (SDWP mode l), dash- dotted lines \n(SDLP freezing model) at low temperature and the da shed line (WDWP \nmodel) at high temperature are fits described in th e text. Inset: H  c 1/2  vs T 2/3 \nshowing the linear behavior expected in the SDWP mo del, solid lines are \nthe SDWP model and dash- dotted lines are the SDLP freezing model. (b) \nNormalized remanent magnetization (at H = 0) vs T.  The different \nsymbols indicate different samples in both panels.  10-56 \n 4\nusing (10) (a SDWP scenario) provided the nonmonotonic \nbehavior of Sv. The parameters obtained are shown in Table \nI. Clearly no good agreement is found for the freezing of \nclusters or SDLP scenario. In the high temperature region \nthe experimental Sv vs T is in agreement with the linear \nbehavior calculated in (11). However, a non-zero Sv (T = 0) \nvalue was found, not present in the model. \nFIG. 5 HERE \nTherefore, based on combined data extracted from the \nhysteresis loops and time dependence of the magnetization,  \nwe depicted two regimes in La 2Ni(Ni 1/3Sb 2/3)O 6 pelletized \npolycrystalline samples: Weak pinning of DWs at T >2 0 K \nand strong pinning of DWs below that temperature. The \nmicroscopic origin of this change of regime could be related \nwith the onset of a super-superexchange antiferromagnetic \ninteraction among Ni 2+  via O 2- -Sb 5+  - O2- paths [10] that \ncreates a frustrated magnetic interaction. IV.  CONCLUSION  \nThe ferrimagnetic state in La 2Ni(Ni 1/3Sb 2/3)O 6 was found \nto be characterized by two different regimes for domain w all \nmovement, a strong and a weak domain wall pinning regime \nat low and high temperatures respectively. The temperature \nrange of the strong domain wall pinning regime coincides  \nwith that of the existence of a proposed frustrated stat e. The \nscenario of clusters or large single domain particles f reezing \nwas discarded based in the coercive field and magnetic \nviscosity parameter temperature dependence analysis.  \nACKNOWLEDGMENT  \nWe thank E.E. Kaul for fruitfull discussions.  R.E.C, and \nG.N. are members of CONICET. D.G.F. has CONICET \nscholarship. Work partially supported by ANPCyT PICT07-\n819, CONICET PIP 11220090100448 and SeCTyP-\nUNCuyo 06/C313.  R.E.C. thanks FONCYT (PICT2007-\n303), CONICET (PIP 11220090100995) and SECYT-UNC \n(Res. 214/10) for finantial support. \nREFERENCES  \n[1]  H.-J. Koo and M.-H. Whangbo, “Importance of the O-M -O bridges \n(M = V5+, Mo 6+) for the spin-exchange interactions in the magneti c \noxides of Cu 2+ ions bridged by MO 4 tetrahedra: Spin-lattice models of \nRb 2Cu 2(MoO 4)3,BaCu 2V2O8, and KBa 3Ca 4Cu 3V7O28 ,” Inorg. Chem.  \n45, 4440(2006).  \n[2]  M. del C. Viola, M. S. Augsburger, R. M. Pinacca, J . C. Pedregosa, \nR.E. Carbonio and R. C. Mercader, “Order-disorder at F e sites in \nSrFe 2/3 B” 1/3 O3 (B”=Mo, W, Te, U) tetragonal double perovskites,” J. \nSolid State Chem.  175, pp. 252-257 (2003). \n[3]  A. Maignan, B. Raveau, C. Martin and M. J. Hervieu, “L arge \nintragrain magnetoresitance above room temperature in the double \nperovskite Ba 2FeMoO 6,” J. Solid State Chem.  144, pp. 224-227 \n(1999). \n[4]  S. H. Kim and P.D. Battle, “Structural and electron ic properties of the \nmixed Co/Ru perovskites AA’CoRuO 6 (A, A’=Sr, Ba, La),” J. Solid \nState Chem.  114, pp. 174-183 (1995).  \n[5]  P. S. R. Murthy, K. R. Priolkar, P. A. Bhobe, A. Das,  P. R. Sarode \nand A. K. Niga, “Disorder induced negative magnetiz ation in \nLaSrCoRuO 6,” J. of Magn. and Magn. Mater . 322, pp. 3704-3709 \n(2010). \n[6]  J. B. Goodenough. Magnetism and the Chemical Bond , Interscience, \nNew York. N.Y. 1963. \n[7]  P. J. Hay, J. C. Thibeault and R. J. Hoffmann, “Orb ital interactions in \nmetal dimer complexes,”  J. Am. Chem. Soc.  97, pp. 4884-4899 \n(1975).  \n[8]  D. C. Jiles and D. L. Atherton, “Theory of ferromagn etic hysteresis”  \nJ. of Magn. and Magn. Mater.  61, pp. 48-60 (1986). \n[9]  P. Gaunt, “Ferromagnetic domain wall pinning by a r andom array of \ninhomogeneities,” Phil. Mag. B  48, pp. 261-276 (1983).  \n[10]  D. G. Franco, G. Nieva and R. E. Carbonio, to be pub lished \nelsewhere. \n[11]  X. -G. Li, X. J. Fan, G. Ji, W. B. Wu, K. H. Wong, C. L.Choy, and H. \nC. Ku, “Field induced crossover from cluster-glass t o ferromagnetic \nstate in La 0.7 Sr 0.3 Mn 0.7 Co 0.3 O3,” J. Appl. Phys.  85, pp. 1663-1666 \n(1999). \n[12]  J. García-Otero, A. J. García-Bastida and J. Rivas, “ Influence of \ntemperature on the coercive field of non-interactin g fine magnetic \nparticles,” J. Magn. Magn. Mater.  189, pp. 377-383 (1998). \n[13]  E. P. Wohlfarth, “The coefficient of magnetic visco sity,”  J. Phys. F. \nMet. Phys.  14, pp. L155-L159 (1984). \n[14]  O. V. Billoni, E. E. Bordone, S. E. Urreta, L. M. Fab ietti, H. R. \nBertorello, “Magnetic viscosity in a nanocrystalline  two phase \ncomposite with enhanced remanece,”  J. of Magn. Magn. Mater.  208, \npp. 1-12 (2000). \n[15]  D. C. Crew, P.G. McCormick, R. Street, “Temperature D ependence \nof the Magnetic Viscosity Parameter,”  J. of Magn. Magn. Mater.  177-\n181, pp. 987-988 (1998). \n[16]  J. F. Liu and H. L. Luo, “On the coercive force and  effective \nactivation volume in magnetic materials,”  J. of Magn. Magn. Mater.  \n94, pp. 43-48 (1991). \nFig. 5.  (color online) Magnetic viscosity paramete r Sv vs T for \nLa 2Ni(Ni 1/3Sb 2/3 )O 6 polycrystalline pellets. Different symbols indicate \ndifferent samples, the solid symbols corresponds to  Sv values cal culated \nwith the data displayed in Fig.2 . The lines represent the models described in \nthe text, SDWP model (solid line), freezing of SDLP  model (dash dot ted \nline) and WDWP model (dash line). In the inset , a zoom of the low \ntemperature region is shown.  \nFig. 4.  (color online) Total susceptibility χ= dM/dH measured at Hc vs T \nfor the polycrystalline pellets. Different symbols indicate different samples. \nThe line is a guide to the eye. The inset shows a t ypical χ vs H curve taken \nat 10 K.  "    },    {        "title": "1911.05270v3.Variety_of_order_by_disorder_phases_in_the_asymmetric__J_1_J_2__zigzag_ladder__From_the_delta_chain_to_the__J_1_J_2__chain.pdf",        "content": "Variety of order-by-disorder phases in the asymmetric J1\u0000J2zigzag ladder: From the delta chain\nto the J1\u0000J2chain\nTomoki Yamaguchi,1Stefan-Ludwig Drechsler,2Yukinori Ohta,1and Satoshi Nishimoto2, 3\n1Department of physics, Chiba University, Japan\n2Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany\n3Department of Physics, Technical University Dresden, 01069 Dresden, Germany\n(Dated: March 13, 2020)\nWe study an asymmetric J1-J2zigzag ladder consisting of two different spin-1\n2antiferromagnetic (AFM; J2,\n\rJ2>0) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; J1<0) inter-leg interaction. On the\nbasis of density-matrix renormalization group based calculations the ground-state phase diagram is obtained as\nfunctions of \randJ2=jJ1j. It contains four kinds of frustration-induced ordered phases except a trivial FM\nphase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather\nconventional order by disorder. Still, it is interesting to note that the VBS states possess an Affleck-Kennedy-\nLieb-Tasaki-type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which\nis distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic\norders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is\nbroken. In other words, the system lowers its energy via the FM inter-leg interaction by polarizing both of the\nAFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state\nappears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger-liquid\nbehavior in the AFM Heisenberg legs.\nI. INTRODUCTION\nLow-dimensional frustrated quantum magnets, in which a\nmacroscopic number of quasi-degenerate states compete with\neach other, provide an ideal playground for the emergence of\nexotic phenomena1. For instance, the interplay of frustration\nand fluctuations could lead to unexpected condensed matter\norders at low temperatures by spontaneously breaking some\nsort of symmetry order by disorder2. Long-range ordered\n(LRO) magnetic state with breaking a spatial symmetry as\nwell as valence bond solid (VBS) state with a formation of\ndisentangled-unit–like local spin-singlet pair are typical ex-\namples of order by disorder. Moreover, when quantum fluctu-\nations between the quasi-degenerate states prevent a selection\nof particular order, one ends up with spin liquids. Modern the-\nories have brought us new insight by identifying spin liquids\nas topological phases of matter3,4. In recent years the realiza-\ntion of topological phases in frustrated spin systems has been\none of the central topics in condensed matter physics5–7.\nIn one-dimensional (1D) and spin-1\n2case, quantum fluc-\ntuations are maximized so that we may place more expec-\ntations on the discovery of novel ground states by the co-\noperative effects with magnetic frustration. A most simply-\nstructured 1D frustrated system is the so-called spin-1\n2J1-\nJ2chain consisting of nearest-neighbor J1and next-nearest-\nneighborJ2couplings. When both J1andJ2are antiferro-\nmagnetic (AFM), the ground state is a VBS, the nature of\nwhich can be grasped by the Majumdar-Ghosh (MG) model8,\natJ2=J1>\u00180:249,10. The idea of MG model was generalized\nto the Affleck-Kennedy-Lieb-Tasaki (AKLT) model11exhibit-\ning spin- 1VBS ground state with a symmetry protected topo-\nlogical order12.\nMeanwhile, the J1-J2chain with ferromagnetic (FM)\nJ1and AFMJ2, which is known as a standard magnetic\nmodel for quasi-1D edge-shared cuprates such as Li 2CuO 213,\nLiCuSbO 414, LiCuVO 415, Li2ZrCuO 216, Rb 2Cu2Mo3O1217,and PbCuSO 4(OH) 218, encloses wider array of states of mat-\nter. Theoretically, this model has been extensively studied:\nOther than a trivial FM state in the dominant J1region, the\nground state is a topological VBS accompanied by sponta-\nneous multiple dimerization orders19,20(More details are de-\nscribed in Sec. III A of this article). It is also intriguing that\na vector chirality and multimagnon bound states are induced\nin the presence of magnetic field21–24. Especially, the detec-\ntion of nematic or higher multipolar phases is one of the most\nexciting experimental current issues14,25–29. Sensitive features\nto even tiny interchain couplings are another characteristic of\nthis system30–32.\nAnother typical example of frustrated 1D system is\nthe delta chain (or sawtooth chain). The lattice structure\nis a series of triangles, as shown in Fig. 1(b), which is\nsimilar to that of the J1-J2chain but certain parts of\nJ2bonds are missing. There have been several candi-\ndates for delta-chain and related materials: YCuO 0:2533,34,\n[Cu(bpy)(H2O)][Cu( bpy)(mal)(H2O)](ClO 4)235, Zn L2S4\n(L= Er, Tm, and Yb)36, Cu(AsO 4)(OH)\u00013H2O37,\nMn2GeO 438, Rb 2Fe2O(AsO 4)239, CuFe 2Ge240, Fe10Gd1041,\nCu2Cl(OH) 342, Fe2O(SeO 3)243, and V 6O1344. In these mate-\nrials, a wide array of complex phases has been experimentally\nobserved. Delta-chain systems may offer an outlook towards\npromising prospects on novel magnetic phenomena.\nThe magnetic properties of delta-chain systems are totally\ndifferent in different signs of J1andJ2. For the case of\nJ1<0andJ2>0(typically, referred as FM-AFM delta\nchain), only two corresponding materials have been recog-\nnized. One of them is malonatobridged copper complexes\n[Cu(bpy)(H2O)][Cu( bpy)(mal)(H2O)](ClO 4)235. The mag-\nnetic Cu2+ions with effective spin-1\n2form into a delta-chain\nnetwork. The base and the other exchange couplings in a tri-\nangle made of malonate were estimated, respectively, as AFM\n(J2= 6:0K) and FM ( J1=\u00006:6K) from the analysis of mag-\nnetic susceptibility \u001f(T); and, as AFM ( J2= 10:9K) and FMarXiv:1911.05270v3  [cond-mat.str-el]  12 Mar 20202\n(J1=\u000012:0K) from the fitting of magnetization curve (typ-\nically, referred as FM-AFM delta chain). In either case, the\nratio of AFM and FM couplings is close to 1. This means that\nthe material would be in the region of strong frustration. The-\noretically, the ground state was predicted to be a ferrimagnetic\nstate but the detailed spin structures are less understood45.\nIn fact, only qualitative behavior of measured magnetization\ncurve could be explained by assuming a ferrimagnetic ground\nstate46,47. A deeper understanding of the ferrimagnetic state\nis necessary to resolve the remaining discrepancy between ex-\nperiment and theory.\nThe second candidate of the FM-AFM delta-chain ma-\nterials is a mixed 3 d=4fcyclic coordination cluster system\nFe10Gd1041. In these days, the delta-chain physics is increas-\ningly attracting attention due to the synthesis of Fe 10Gd10.\nThis cluster consists of 10 + 10 alternating Gd and Fe ions.\nThe exchange couplings were estimated as FM ( J1=\u00001:0K)\nbetween Fe and Gd ions, AFM ( J2= 0:65K) between Fe\nions, and nearly zero between Gd ions; the magnetic ions\nform an FM-AFM delta chain short ring. The parameter ratio\nJ2=jJ1j= 0:65seems to be very close the FM quantum criti-\ncal pointJ2=jJ1j= 0:748Although the spin values of Fe and\nGd ions are higher than S= 1=2(S= 5=2andS= 7=2, re-\nspectively), quantum fluctuations would play important roles\nto determine the ground state because of the quantum critical-\nity49. This means that the magnetic properties can be drasti-\ncally changed upon even a small variation of external influ-\nences such as magnetic field, pressure, chemical means, and\ngating current. So, this delta chain material is drawing atten-\ntion also from the perspective of controlling magnetic states\nin molecular spintronics50.\nFor comparison, a few examples of delta-chain materials\nonly with AFM interactions ( J1;J2>0) have been also\nreported. With the help of MG-like projection method, the\nmagnetic properties of the AFM-AFM delta-chain are better\nunderstood than those of the FM-AFM one33,51–53. A pecu-\nliarly interesting feature is the dispersionless kink-antikink\ndomain wall excitations to the dimerized VBS ground state.\nA kink is highly localized only in the range of one triangle.\nThe first candidate of AFM-AFM delta-chain materials was\nthe delafossite YCuO 2:533. However, a first-principle calcula-\ntion revealed that the ratio of J2=J1in YCuO 2:5is out of the\nrange of the dimerized VBS ground state and additional in-\ntrachain FM interaction is significantly large34. Very recently,\nthe other candidate materials Cu 2Cl(OH) 3(S= 1=2)42and\nFe2O(SeO 3)2(S= 5=2) have been reported. They indeed\nexhibit characteristic features of AFM-AFM delta chain: a\nmagnetization plateau at half-saturation46,54in Cu 2Cl(OH) 3\nand an almost flat-band one-magnon excitation spectrum in\nFe2O(SeO 3)2.\nAs mentioned above, the research of frustrated 1D systems\nwithJ1-J2or delta-chain structures has become more and\nmore active. Interestingly, each of the J1-J2chain and the\ndelta chain is expressed as a limiting case of an asymmetric\nJ1-J2zigzag ladder, defined as two different AFM Heisen-\nberg chains coupled by zigzag-shaped interchain FM interac-\ntion [see Fig. 1(a)]. When one of the Heisenberg chains van-\nishes, it is the delta chain; and, when the Heisenberg chainsare equivalent, it is the J1-J2chain. However, it is known that\ntheir ground states are completely different. Then, one may\nsimply question how the two limiting cases are connected.\nOf particular interest is that the effect of exchange coupling\ntowards the J1-J2chain can be a likely perturbation in real\ndelta-chain compounds, e.g., the effect of tiny coupling be-\ntween Gd ions in Fe 10Gd10.\nIn this paper, we therefore study an asymmetric FM-AFM\nJ1-J2zigzag ladder using the density-matrix renormalization\ngroup (DMRG) technique. We first clarify the detailed spin\nstructure and low-energy excitations of ferrimagnetic state in\nthe delta chain limit. We suggest that the ferrimagnetic state is\na rare type of order by disorder, where the energy is lowered\nby FM fluctuation between two polarized AFM Heisenberg\nchains with spontaneous breaking of the global spin-rotation\nsymmetry. Then, we examine how the ferrimagnetic state is\ncollapsed and connected to the well-known incommensurate\nspiral state in the J1-J2chain. We also find there exist two\nkinds of VBS phases in the spiral region. Finally, we obtain\nthe ground-state phase diagram of asymmetric J1-J2zigzag\nladder with interpolating between the delta chain and J1-J2\nchain.\nThe paper is organized as follows: In Sec. II our spin\nmodel is explained and the applied numerical methods are de-\nscribed. In Sec. III we briefly mention to-date knowledge on\nthe ground state for two limiting cases of our spin model. In\nSec. IV we present our numerical results and discuss how the\ntwo limiting cases are connected. Finally we end the paper\nwith a summary in Sec. V .\n(a)\n(b)A\nB\nA\nB\n(c)\nA\nB\nFIG. 1. (a) Lattice structure of the asymmetric J1-J2zigzag ladder.\nThe indices ‘ A’ and ‘ B’ denote apical and basal chains, respectively.\nThe lattice spacing ais set as a distance between neighboring sites\nalong the chains. The AFM interaction in the apical chain is con-\ntrolled by\r. (b) Lattice structure of the so-called delta chain (or saw-\ntooth chain) which is realized in the limit of \r= 0. (c) Schematic\nrepresentation of the ferrimagnetic state with global spin-rotation-\nsymmetry breaking.3\nII. MODEL AND METHOD\nA. Model\nThe asymmetric J1-J2zigzag ladder is defined as two\nHeisenberg chains coupled by zigzag-shaped interchain inter-\naction. The lattice structure is sketched in Fig. 1(a). We call\na leg chain with larger interaction “basal chain” and the other\nwith smaller interaction “apical chain”. The Hamiltonian is\nwritten as\nH=J1X\niSA;i\u0001(SB;i+SB;i+1)\n+J2X\ni(SB;i\u0001SB;i+1+\rSA;i\u0001SA;i+1); (1)\nwhere SB;iis spin-1\n2operator at site ion the basal chain and\nSA;iis that on the apical chain. We focus on the case of\nFM interchain coupling ( J1<0) and AFM intrachain cou-\npling (J2>0). The intrachain interaction of apical chain is\ncontrolled by \r(0\u0014\r\u00141). The system (1) corresponds\nto the so-called delta chain (or sawtooth chain) at \r= 0\n[Fig. 1(b)] and the so-called J1-J2chain at\r= 1. The exist-\ning knowledge on the ground-state properties of these chains\nis briefly summarized in the next section. In our numerical\ncalculations the chain lengths of basal and apical chains are\ndenoted asLBandLA, respectively. The total number of\nsites isL=LB+LA. In this paper, we call the system\nfor0< \r < 1“asymmetric J1-J2zigzag ladder”, which has\nbeen little or not studied. In the case of J1>0andJ2>0,\nthere are a few studies55,56.\nB. DMRG methods\nIn order to examine the ground state and low-energy\nexcitations of asymmetric J1-J2zigzag ladder, we em-\nploy the DMRG techniques; namely, conventional DMRG\n(hereafter referred to simply as DMRG), dynamical DMRG\n(DDMRG), and matrix-product-state-based infinite DMRG\n(iDMRG) methods. They are used in a complementary fash-\nion to further confirm our numerical results.\nThe DMRG method is a very powerful numerical method\nfor various (quasi-)1D quantum systems57. However, some\ndifficulties are often involved in the DMRG analysis for\nstrongly frustrated systems like Eq. (1). First, the system\nsize dependence of physical quantities is usually not straight-\nforward. Therefore, relatively many data points are required\nto perform a reasonable finite-size scaling analysis. We thus\nstudy systems with length up to L= 161 (LB= 81;LA=\n80) under open boundary conditions (OBC) and systems with\nlength up to L= 64 (LB= 32;LA= 32 ) under periodic\nboundary conditions (PBC). Either OBC or PBC is chosen\ndepending on the calculated quantity. Second, a lot of nearly-\ndegenerate states are present around the ground state. To\nobtain results accurate enough, a relatively large number of\ndensity-matrix eigenstates mmust be kept in the renormal-\nization procedure. In this paper, we keep up to m= 8000density-matrix eigenstates, which is much larger than that kept\nin usual DMRG calculations for 1D systems, and extrapolate\nthe calculated quantities to the limit m!1 if necessary. In\nthis way, we can obtain quite accurate ground states within the\nerror of \u0001E=L = 10\u00008jJ1j.\nFor the calculation of dynamical quantities, we use the\nDDMRG method which has been developed for calculating\ndynamical correlation functions at zero temperature in quan-\ntum lattice models58. Since the DDMRG algorithm performs\nbest for OBC, we study a open cluster with length up to\nL= 129 (LB= 65;LA= 64 ). The DDMRG approach is\nbased on a variational principle so that we have to prepare\na ‘good trial function’ of the ground state with the density-\nmatrix eigenstates. Therefore, we keep m= 1200 to ob-\ntain the ground state in the first ten DMRG sweeps and keep\nm= 600 to calculate the excitation spectrum. In this way, the\nmaximum truncation error, i.e., the discarded weight, is about\n1\u000210\u00005, while the maximum error in the ground-state and\nlow-lying excited states energies is about 10\u00004jJ1j.\nThe iDMRG method is very useful because it enables us to\nobtain the physical quantities directly in the thermodynamic\nlimit59,60, if the matrix product state is not too complicated\nand the simulation can be performed accurately enough. In\nour iDMRG calculations, typical truncation errors are 10\u00008\nusing bond dimensions \u001fup to 6000 . In this way, the effective\ncorrelation length near criticality is less or at most equal to\n500, so that most of interesting parameter region of the system\n(1) can be reasonably examined by our iDMRG simulations.\nIII. PREVIOUS STUDIES FOR LIMITING CASES\nSo far, our system for two limiting cases, namely, J1-J2\nchain (\r= 1) and delta chain ( \r= 0), has been extensively\nstudied. In this section, we briefly summarize to-date knowl-\nedge on the ground state for the limiting cases.\nA.J1-J2chain (\r= 1)\nAt\r= 1, we are dealing with the J1-J2chain, which may\nbe also recognized as symmetricJ1-J2zigzag ladder. In the\nlimit ofJ2=jJ1j= 0, the system is a simple FM Heisenberg\nchain with FM ordered ground state. Increasing J2=jJ1j, the\nFM state persists up to J2=jJ1j= 1=4; then, a first-order\nphase transition from the FM to an incommensurate (“spiral”)\nstate occurs61. The total spin in the incommensurate phase\nis zero (Stot= 0). Since quantum fluctuations disappear at\nthe FM critical point, the critical value J2=jJ1j= 1=4can\nbe recognized similarly both in the quantum as well as in the\nclassical model62,63.\nAtJ2=jJ1j>1=4, the incommensurate correlations are\nshort ranged in the quantum model64,65. Instead, the sys-\ntem exhibits a spontaneous nearest-neighbor FM dimeriza-\ntion with breaking of translation symmetry, as a consequence\nof the quantum fluctuations typical of magnetic frustration,\ni.e, order by disorder. By regarding the ferromagnetically4\ndimerized spin-1/2 pair as a spin- 1site, the system is effec-\ntively mapped onto a spin- 1Heisenberg chain and an Affleck-\nKennedy-Lieb-Tasaki (AKLT)-like hidden topological order\nprotected by global Z2\u0002Z2symmetry is naively expected as\na Haldane state11,66(also see Sec. IV C 6). In fact, the hidden\norder has been numerically confirmed19,20.\nFurthermore, the existence of (exponentially small) singlet-\ntriplet gap at J2=jJ1j>\u00183:3was predicted by the field-theory\nanalysis67and its verification had been a longstanding issue.\nOnly recently, the gap was numerically estimated: with in-\ncreasingJ2=jJ1jit starts to open at J2=jJ1j= 1=4, reaches its\nmaximum\u00180:007jJ1jaroundJ2=jJ1j= 0:65, and exponen-\ntially decreases20. The ground state is a kind of VBS state with\nspin-singlet formations between third-neighbor sites. There-\nfore, the magnitude of gap basically scales to the strength of\nthird-neighbor valence bond.\nB. Delta chain ( \r= 0)\nAt\r= 0, the system is the delta chain consisting of a lin-\near chain of corner-sharing triangles [Fig. 1(b)]. The ground\nstate properties are less understood than those of the J1-J2\nchain. One main reason is that numerical investigation of the\ndelta chain is particularly difficult due to the strong magnetic\nfrustration and a number of nearly-degenerate states near the\nground state. Especially at the FM critical point is macroscop-\nically degenerate and consists of multi-magnon configurations\nformed by independent localized magnons and the special lo-\ncalized multi-magnon complexes48.\nNonetheless, a numerical study could identify the ground\nstate atJ2=jJ1j<1=2to be FM; that at J2=jJ1j>1=2\nto be ferrimagnetic45. The total spins of the ferromagnetic\nand ferrimagnetic phases are L=2andL=4, respectively. To\nunderstand the origin and the properties of this ferrimagnetic\nstate, the delta chain in the large limit of easy-axis exchange\nanisotropy was studied68. In this limit the system can be re-\nduced to a 1D XXZ basal chain under a static magnetic field\ndepending on the magnetic structure of apical chain. The\nground state was identified as ferrimagnetic with fully po-\nlarized apical spins and weakly polarized basal spins. It is\nexpected that some essential features may be inherent in the\nisotropic SU(2) limit. In fact, the spin structure agrees quali-\ntatively to the ferrimagnetic state determined in this paper [see\nFig. 1(c)].\nIV . RESULTS\nA. classical limit\nAs mentioned above, the FM critical point is known to be\nJ2=jJ1j= 1=4for theJ1-J2chain (\r= 1) andJ2=jJ1j= 1=2\nfor the delta chain ( \r= 0). To be examined first is how\nthe critical point changes between the limiting cases, i.e.,\n0< \r < 1. Since the quantum fluctuations vanish at the FM\ncritical point, the critical value can be exactly estimated by\nthe classical spin wave theory (SWT). The Fourier transform\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\n(a)\n(b)\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\n0.1\n0.4 0.6 0.800.2\n0.6 0.8FIG. 2. (a) Classical ground-state phase diagram of the asymmetric\nJ1-J2zigzag ladder. The phases are characterized by propagation\nvectorq=qminminimizingJq[Eq. (3)]: FM ( qmin= 0); commen-\nsurate (qmin=\u0019); incommensurate ( 0< qmin< \u0019). (b) Ground-\nstate phase diagram of the asymmetric J1-J2zigzag chain [Eq. (1)]\ndetermined by DMRG calculations. Inset: enlarged view around the\nPF phase.\nof our Hamiltonian (1) reads\nH=1\n2X\nqJq~Sq\u0001~S\u0000q (2)\nwith\nJq=(1 +\r)J2cosq\n\u0006q\n(1\u0000\r)2J2\n2(1\u0000cosq)2+ 4J2\n1cos2(q=2):(3)\nIf Eq. (3) has a minimum at q= 0, the system is in an FM\nground state. The FM critical point is thus derived as\nJ2;c\njJ1j=1\n2(1 +\r): (4)5\nAs shown in Fig. 2(a), the FM region is simply shrunk with\nincreasing\rbecause the AFM interaction is increased in the\napical chain. We have confirmed this FM critical boundary\nnumerically by calculating the total spin Stotof the whole\nsystem, which is defined as\nh~S2i=Stot(Stot+ 1) =X\ni;jh~Si\u0001~Sji: (5)\nIt can be also verified by finding the absence of LRO FM state\nin the spin-spin correlation functions. These results are shown\nin Appendix A.\nBy evaluating q(\u0011qmin) value to minimize Eq. (3), a\nclassical ground-state phase diagram is obtained as Fig. 2(a).\nThere are three kinds of LRO phases: FM phase with qmin=\n0, incommensurate phase 0< q min< \u0019 , and commensu-\nrate phase with qmin=\u0019. Since the ferrimagnetic state in\nFig. 1(c) is of commensurate with q=\u0019and the propagation\nnumber of the J1-J2chain is incommensurate, the SWT re-\nsults are consistent with those of the quantum system (1) in\nthe two limiting cases \r= 0 and\r= 1. Therefore, even in\nthe quantum system an incommensurate-commensurate phase\ntransition is naively expected at finite \rwithJ2=jJ1jfixed.\nB.\r= 0: delta chain\nAlthough the ground state of the delta chain is most prob-\nably ferrimagnetic at J2=jJ1j>1=2, the detailed magnetic\nstructure and properties have not been fully settled. To gain\nfurther insight into them, we here calculate the total spin, spin-\nspin correlation functions, and stabilization energy of ferri-\nmagnetic state for the delta chain. We need to pick through\nthe system-size dependence of the quantities to deal with non-\ntrivial finite-size effects of the delta chain.\n1. total spin\nInvestigating the total spin Stotis a simple way to explore\nthe possibility of ferrimagnetic state. As shown below, the\nsystem-size dependence of Stotis significantly different be-\ntween applying OBC and PBC. This obviously implies the\ndifficulty of performing numerical calculations for this sys-\ntem. Nevertheless, the fact that they should coincide in the\nthermodynamic limit L!1 makes the finite-scaling analy-\nsis even more reliable. We here use the DMRG method.\nLet us first see the case of OBC. In Fig. 3(a) the total spin\nper siteStot=Lis plotted as a function of inverse system size\n1=Lfor several values of J2=jJ1j. WhenJ2=jJ1jis order\nof1, the effect of strong frustration is explicitly embedded\nin the finite-size scaling; namely, due to the Friedel oscilla-\ntion,Stot=Lawkwardly oscillates with 1=L. However, as ex-\npected, such an oscillation no longer appears in a case of large\nJ2=jJ1j= 100 . WhenJ2=jJ1j\u001d1, a straightforward scaling\nis allowed since the frustration is much weaker. Eventually,\nfor all theJ2=jJ1jvalues,Stot=Lseems to be extrapolated to\n1=4in the thermodynamic limit. Although one may think that\n(a)\n(b)0 0.01 0.02 0.03 0.04 0.050.10.20.3\n0 0.02 0.04 0.06 0.0800.10.20.30.4FIG. 3. Finite-size scaling analysis of the total spin per site for the\ndelta chain ( \r= 0), where (a) OBC and (b) PBC are applied. The\ndotted lines are guide for eyes.\nthe PBC should be applied if the Friedel oscillation is prob-\nlematic, the situation is not so simple as explained below.\nWe then turn to the case of PBC. In Fig. 3(b) Stot=Lis plot-\nted as a function of 1=Lfor several values of J2=jJ1j. Unlike\nin the case of OBC, the oscillation is not seen in Stot=Lvs.\n1=L. Instead, there exists a ‘critical’ system size to achieve\nfiniteStot=Lin the ground state. This is caused by a kind\nof typical finite-size effects: Under the PBC, the basal chain\nforms a plaquette singlet and the basal spins are more or less\nscreened. This screening leads to the reduction of FM in-\nteraction between the basal and apical chains although the\nFM fluctuations between the two chains are essential to sta-\nbilize the ferrimagnetic state (see Sec. IV B 2). Besides, only\na small screening may be sufficient to collapse the ferrimag-\nnetic state because the stabilization gap of ferrimagnetic state\nis very small (see Sec. 5). Consequently, the ferrimagnetic\nstate can be readily prevented under the PBC. For reference,\nthe system-size dependence of energies for spin-singlet and\nferrimagnetic states is shown in Appendix B. Since the triplet\nexcitation gap of plaquette singlet roughly scales to J2=jJ1j\nwith a fixed system size, the critical system size is larger for\nlargerJ2=jJ1jas seen in Fig. 3(b). Once the system size goes\nbeyond the critical one, Stot=Lapproaches smoothly to 1=4\natL!1 . The fitting function is Stot=L= 1=4 + 1=Lfor\nanyJ2=jJ1j.6\n0 50 100\n|i−j|−0.10.00.10.2/angbracketleftBig\nSz\nα,iSz\nβ,j/angbracketrightBig\n(a)\nA-A\nA-BB-B\n0 50 100\n|i−j|−0.10.00.10.2/angbracketleftBig\nSz\nα,iSz\nβ,j/angbracketrightBig\n(b)\n0 50 100\n|i−j|−0.10.00.10.2/angbracketleftBig\nSz\nα,iSz\nβ,j/angbracketrightBig\n(c)\n0.6 1.0 1.5 2.0\nJ2/|J1|0.00.10.20.30.40.5\n/angbracketleftBig\nSz\nA,i/angbracketrightBig\n/angbracketleftBig\nSz\nB,i/angbracketrightBig(d)\nFIG. 4. Spin-spin correlation function hSz\n\u000b;iSz\n\f;jifor the delta chain\n(\r= 0) as a function of distance ji\u0000jjat (a)J2= 0:6, (b)1, and\n(c)2. The totalSzsector is set to be Sz\ntot=L=4. The legends\ndenote as A-A: (\u000b;\f) = (A;A), B-B: (\u000b;\f) = (B;B), and A-B:\n(\u000b;\f) = (A;B). (d) Averaged values of hSz\n\u000b;iion the apical and\nbasal sites as a function of J2=jJ1j.\nThus, we have confirmed that the total spin of the delta\nchain atJ2=jJ1j>1=2isStot=L=4, indicating the ferri-\nmagnetic state, and the ferrimagnetic phase persists up to the\nlargeJ2=jJ1jlimit.\n2. spin-spin correlation\nThen, in order to determine the magnetic structure of the\nferrimagnetic state, we examine spin-spin correlation func-\ntionshS\u000b;i\u0001S\f;jibetween apical sites: (\u000b;\f) = (A;A),\nbetween basal sites: (\u000b;\f) = (B;B), and between apical and\nbasal sites: (\u000b;\f) = (A;B)=(B;A). For convenience, we\nfix the z-component of total spin at the total spin retained in\nthe ferrimagnetic ground state, i.e., Sz\ntot=Stot=L=4. It\nlifts the ground-state degeneracy due to the SU(2) symmetry\nbreaking and the (spontaneous) magnetization direction is re-\nstricted to the z-axis direction. Hence, we need only to see\nhSz\n\u000b;iSz\n\f;jiinstead ofhS\u000b;i\u0001S\f;ji.\nIn Fig. 4(a)-(c) iDMRG results of the correlation function\nhSz\n\u000b;iSz\n\f;jiare plotted as a function of distance ji\u0000jjfor\nJ2=jJ1j= 0:6,1, and 2. We clearly see long-ranged cor-\nrelations indicating a magnetic order for all the J2=jJ1jval-\nues. If no magnetic order exists, all the correlation func-\ntions converge to (Sz\ntot=L)2= 1=16at long distance limit\nji\u0000jj ! 1 . The intrachain correlations hSz\nA;iSz\nA;jiand\nhSz\nB;iSz\nB;jiare both positive at the long distance, and the for-\nmer correlation is much larger than the latter one. This means\nthat the apical spins are nearly-fully polarized and the basalspins are only weakly polarized. In addition, a N ´eel-like stag-\ngered oscillation is clearly seen leastwise at the short distance\ninhSz\nB;iSz\nB;ji. These correlations immediately correspond\nto a ferrimagnetic state, which is schematically sketched in\nFig. 1(c). This picture seems to be valid in the whole region\nofJ2=jJ1j>0:5.\nNow, another question may arise: Is the N ´eel-like staggered\nspin alignment on the basal chain LRO? The answer is NO.\nAlthoughhSz\nB;iSz\nB;jiindeed exhibits an AFM oscillation at\nshort distancesji\u0000jj, it vanishes at the long distance limit.\nIn fact, the oscillating part of hSz\nB;iSz\nB;jiexhibits a power-\nlaw decay indicating a critical behavior of the Tomonaga-\nLuttinger liquid (TLL) (see Appendix C). This also excludes\nthe possibility of VBS formation in the basal chain because\nan exponential decay of the spin-spin correlation should be\nfound in a VBS state. It means that, very surprisingly, the\npresent ferrimagnetic order is an order by disorder but without\nany geometrical symmetry breaking. Instead, the magnetic\nfrustration is relaxed by a spontaneous breaking of the global\nspin-rotation symmetry. In other words, the system can gain\nenergy from the FM interaction between the apical and basal\nchains by polarizing both of the chains. This is a very rare\ntype of order by disorder.\nThis picture of order by disorder may be further convinced\nby looking at the relation between the polarization level of\nbasal spins and the stabilization of ferrimagnetic order. Thus,\nto see theJ2=jJ1j-dependence of polarization level in more\ndetail, we examine the expectation values of local spin mo-\nmentSz\nA;iandSz\nB;i. In Fig. 4(d) the averaged values of\nhSz\nA;iiandhSz\nB;iiare plotted as a function of J2=jJ1j. We\nfind that with increasing J2=jJ1j,hSz\nA;iiincreases and sat-\nurates at 1=2; while,hSz\nB;iidecreases and goes down to 0\nin the large J2=jJ1jlimit. The decrease of hSz\nB;iimay be\nnaively expected by the fact that the basal chain approaches\na 1D SU(2) Heisenberg model at large J2=jJ1j; while, the\nincrease ofhSz\nB;iiis a simple consequence of the condition\nhSz\nA;ii+hSz\nB;ii= 1=2. If the order-by-disorder picture asso-\nciated by interchain FM interaction is true, the ferrimagnetic\nstate at larger J2=jJ1jshould be fragile because of smaller po-\nlarization of basal spins. Actually, the stabilization energy of\nferrimagnetic order decreases with increasing J2=jJ1jas de-\nscribed in the following subsection.\n3. stabilization gap\nTo figure out the stability of the ferrimagnetic state, we\ncalculate a stabilization gap defined by energy difference be-\ntween the ferrimagnetic state and a lowest state in the Stot= 0\nsector:\n\u0001(L) =E0(0;L)\u0000E0(Sg:s:;L);\u0001 = lim\nL!1\u0001(L);(6)\nwhereE0(S;L)is the lowest-state energy of L-site periodic\nsystem in the Stot=Ssector, andSg:s:is a total spin of the\nground state. The energy of ferrimagnetic state can be sim-\nply obtained as a ground-state energy, though, as mentioned7\n0.5 1 1.5 21 210-310-20 0.05 0.1 0.1500.010.020.030.04\n00.0050.010.015(a)\n(b)\nFIG. 5. Energy difference between the lowest Stot= 0 state and\nferrimagnetic ground state for the delta chain, as a stabilization gap\nof the ferrimagnetic state. (a) Finite-size scaling and (b) the extrapo-\nlated values \u0001=jJ1jas a function of J2=jJ1j. Inset: Semi-log plot of\n\u0001=jJ1jas a function of J2=jJ1j. The dotted line is a fit for the large\nJ2=jJ1jregion: \u0001=jJ1j= 0:037 exp(\u00001:3J2=jJ1j).\nabove, the total spin Sg:s:=Lin the ferrimagnetic state can de-\nviate from 1=4for a finite cluster. Whereas, it is nontrivial\nto estimate the energy of a lowest state in the Stot= 0 sec-\ntor because it is an excited state and there are many quasi-\ndegenerate states near the ferrimagnetic ground state. A\nproper way needs to be provided to extract the Stot= 0state.\nWe therefore consider the following augmented Hamiltonian\nH0=H+\u0015~S2; (7)\nwhereHis our original Hamiltonian (1) and the additional\nterm corresponds to the total spin operator ( \u0015>0). By setting\n\u0015to be large enough, all the states with Stot>0are lifted.\nEventually, we can find a state with Stot= 0 as the lowest\nstate.\nIn Fig. 5(a) the energy difference \u0001(L)is plotted as a func-\ntion of 1=Lfor severalJ2=jJ1jvalues. Since the magnetic\nfrustration causes a sine-like oscillation in \u0001(L)vs.1=L20,\nthe finite-size scaling analysis is not very straightforward.\nStill, the plotted values show the sine-like oscillation with\nroughly more than one period, an acceptable scaling with lin-\near function \u0001(L)=jJ1j= \u0001=jJ1j+A=L(Ais fitting parame-\nter) may be possible. Actually, even if we assume a more gen-\neral fitting function \u0001(L)=jJ1j= \u0001=jJ1j+A=L\u0011, the extrap-\n012\n0\n00.10.20.3\n0\n00.10.20.30.4\n0\n00.10.20.3\n0(a) (b)\n(c) (d)0 1 Intensity (arb. units)FIG. 6. Dynamical spin structure factors for (a) apical and (b) basal\nchains atJ2=jJ1j= 0:6. (c)(d) The same spectra at J2=jJ1j=\n1. Finite broadening \u0011is introduced: \u0011= 0:03jJ1jin (a) and (c),\n\u0011= 0:02jJ1jin (b), and\u0011= 0:05jJ1jin (d). The dotted lines\nare approximate analytical expressions of the main dispersions (see\ntext).\nolated values of \u0001=jJ1jare almost unchanged because \u0011\u00191\nis always achieved. In Fig. 5(b) the extrapolated values of \u0001\nare plotted as a function of J2=jJ1j. At the FM critical point\nJ2=jJ1j= 1=2, the lowest energies for all Stotsectors are de-\ngenerate and it leads to \u0001 = 0 . As soon as the system goes\ninto the ferrimagnetic phase, the stabilization gap \u0001steeply\nincreases, reaches a maximum around J2=jJ1j= 1, and de-\ncreases with further increasing J2=jJ1j. The magnetic frustra-\ntion would be strongest at the maximum position J2=jJ1j\u00181\nbecause each triangle is fully frustrated. This is another in-\ndication of the fact that the ferrimagnetic state is originated\nfrom order by disorder. As shown in the inset of Fig. 5(b), the\nstabilization gap seems to decay exponentially with J2=jJ1jin\nthe largeJ2=jJ1jregion. It means that the ferrimagnetic state\nis rapidly destabilized in the large J2=jJ1jregime although it\npersists up to J2=jJ1j=1in a precise sense. This is con-\nsistent with the rapid decrease of hSz\nB;iiwithJ2=jJ1j. The\nsystem can gain only little energy from the interchain FM in-\nteraction in case where the basal spins are not really polarized.\nIn short, the quantum fluctuations between the apical and the\nbasal chains play an essential role to stabilize the ferrimag-\nnetic state.\n4. dynamical spin structure factor\nIn order to provide further insight into the ferrimagnetic\nstructure, we investigate the low-energy excitations of the8\ndelta chain. We calculate dynamical spin structure factors\nfor both the apical and basal chains with using the DDMRG\nmethod. The dynamical spin structure factor is defined as\nS\u000b(q;!) =1\n\u0019Imh 0jSz\n\u000b;q1\n^H+!\u0000E0\u0000i\u0011Sz\n\u000b;qj 0i\n=X\n\u0017jh \u0017jSz\n\u000b;qj 0ij2\u000e(!\u0000E\u0017+E0); (8)\nwhere\u000bdenotes either apical (A) or basal (B) chain, j \u0017iand\nE\u0017are the\u0017-th eingenstate and the eigenenergy of the system,\nrespectively ( \u0017= 0 corresponds to the ground state). Under\nOBC, we define the momentum-dependent spin operators as\nSz\n\u000b;q=r\n2\nL\u000b+ 1X\nleiqriSz\n\u000b;i; (9)\nwith (quasi-)momentum q=\u0019Zx=(L\u000b+ 1) for integers 1\u0014\nZx\u0014L. We use open clusters with LA= 31;LB= 32\nforSB(q;!)and withLA= 64;LB= 65 forSA(q;!). In\nFig. 6, DDMRG results of the dynamical spin structure factors\nare shown for J2=jJ1j= 0:6and1.\nLet us see the spectrum for the apical chain. Since the\nspins are fully polarized on the apical chain, the main dis-\npersion ofSA(q;!)is basically described by that of the 1D\nFM Heisenberg chain. For J2=jJ1j= 0:6, a precise fitting\nleads to!q=jJ1j=J0(cos(q)\u00001) +J00(cos(2q)\u00001)with\nnearest-neighbor FM coupling J0=\u00000:11and next-nearest-\nneighbor AFM coupling J00= 0:016. Interestingly, we find\nthat the dominant FM coupling is effectively induced on the\napical chain in spite of no direct interaction between apical\nsites. A similar fitting for J2=jJ1j= 1 givesJ0=\u00000:075\nJ00= 0:018. The reduction of jJ0jwith increasing J2=jJ1j\nis naturally expected because the apical spins are completely\nfree in the large J2=jJ1jlimit. This reduction also reflects the\nweakening of ferrimagnetic state.\nWe then turn to the spectrum for the basal chain. Although\nthe basal chain is weakly polarized, we expect that the funda-\nmental excitations could be at least qualitatively described by\nthose of the 1D SU(2) Heisenberg chain because the dom-\ninant short-range correlation is AFM. For J2=jJ1j= 0:6,\nthe magnon dispersion (lower bound of the continuum) of\nSB(q;!)is certainly sine-like function and the well-known\nshaped two-spinon continuum is seen. However, surpris-\ningly, such the weak spin polarization ( SB;tot=LB= 0:11)\ndrastically suppresses the dispersion width down to 0:15jJ1j\nfrom that of the 1D SU(2) Heisenberg chain \u0019jJ1j=269. It\nis interesting that the dispersion width is rapidly recovered\nto0:75jJ1jwhen the spin polarization is slightly reduced to\nSB;tot=LB= 0:085atJ2=jJ1j= 1. Another effect of the\nweak polarization on the dispersion is a shift of node. Due to\nthe dominant AFM fluctuation on the basal chain in the whole\nregion of ferrimagnetic phase, a largest peak always appears\nat(q;!) = (\u0019;0). If there is no spin polarization, the other\nnode should be at q= 0 but it is actually shifted to higher q\nvalue as seen in Fig. 6(b)(d). This behavior is similar to the\ncase in the presence of magnetic field. Namely, the node po-\nsition can be expressed as q= 2\u0019hSB;toti=LBwherehSB;toti\nis the total spin of the basal chain with length LB.\n0 0.01 0.02 0.0300.10.20.3\n0 0.1 0.200.10.20.3(a)\n(b)FIG. 7. (a) System-size dependence of total spin per site Stot=Lfor\nseveral\rvalues withJ2=jJ1j= 0:6fixed. The dashed line indicates\nthe value ofStot=Lfor the full ferrimagnetic state. (b) The L!1\nextrapolated values of Stot=Las a function of \r.\nC. Finite\r: asymmetric J1-J2zigzag ladder\nAs described above, we have confirmed that the ferrimag-\nnetic state is indeed stabilized at J2=jJ1j>1=2in the delta\nchain (\r= 0). Then, let us see what happens when the apical\nsites are connected by AFM interaction, the strength of which\ncan be controlled by \r. Since the system is in a singlet ground\nstate, i.e.,Stot= 0, in theJ1-J2chain (\r= 1), the collapse of\nferrimagnetic state is naively expected at some \r(<1). Inci-\ndentally, the ferrimagnetic state is trivially enhanced if an FM\ninteraction is introduced between apical sites.\n1. total spin\nSimply, we examine the \r-dependence of the total spin\nto identify when and how the ferrimagnetic state is destabi-\nlized. In Fig. 7(a) the total spin per site Stot=Lis plotted as\na function of 1=Lfor several\rvalues with J2=jJ1j= 0:6\nfixed. Due to the strong frustration, the value of Stot=Los-\ncillates with 1=L; however, we may perform a reasonable\nfinite-size scaling analysis with finer data points using large\nenough clusters. We here use open clusters with length up to\nL=LA+LB= 100+101 = 201 . TheL!1 extrapolated9\nvalue ofStot=Lis plotted as a function of \rin Fig. 7(b).\nAt\r= 0, the ground state is in the ferrimagnetic state\nwith nearly-fully polarized apical spins and the total spin per\nsite isStot=L=4. Hereafter, we call this state “ fullferri-\nmagnetic (FF) state” to discriminate it from another ferrimag-\nnetic state with Stot< L= 4(denoted as “ partial ferrimag-\nnetic (PF) state”) which appears below. When the AFM inter-\naction between apical spins is switched on, one may naively\nexpect a collapse or weakening of the full spin polarization in\nthe apical chain; nevertheless, interestingly, the FF condition\nStot=L=4survives up to \r\u00190:08. This can be interpreted\nas follows: Roughly speaking, since the system can gain more\nenergy from interchain FM interaction than AFM interaction\nbetween apical spins, the FM LRO in the apical chain is still\nmaintained at \r<\u00180:08.\nWith increasing \rfrom 0:08, the competition between in-\nterchain FM interaction and apical intrachain AFM interaction\nderives a new state. As seen in Fig. 7(a), at \r= 0:09the value\nofStot=Lseems to converge at a finite but smaller value than\n1=4as1=Ldecreases. This clearly suggests that some sort\nof collapse of the FF state happens around \r= 0:09. With\nfurther increasing \r, the value of Stot=Lappears to be con-\ntinuously reduced and reaches zero around \r= 0:14[see\nFig. 7(b)]. Surprisingly, we find that there exists a finite \r-\nrange exhibiting 0< S tot=L < 1=4. Since the basal chain\nbasically keeps its weakly polarized or nearly singlet state,\nit would be a good guess that spin polarization on the api-\ncal sites is gradually collapsed with increasing \rin this PF\nphase ( 0:08<\u0018\r<\u00180:14). In other words, the ferrimagnetic\norder by disorder in association with the global spin-rotation-\nsymmetry breaking disappears around \r= 0:14. Actually, as\nstated below, the system has the other order by disorder, i.e.,\ndimerization order, at \r>\u00180:14. The region of the PF phase\nis shown as a shaded area in the ground-state phase diagram\n[Fig. 2(b)].\nWe make some remarks on the existence of PF phase. Such\na ‘halfway’ magnetization 0< S tot=L < 1=4in a ferrimag-\nnetic state is generally prohibited by the Marshall-Lieb-Mattis\n(MLM) theorem70,71. There is an exception to this, however,\nwhen the ferrimagnetic order and a quasi-long-range order of\nTLL compete72. This corresponds to the competition between\nsmall FM polarization and dominant AFM fluctuations in the\nbasal chain of our system. As confirmed in Appendix C, the\nbasal chain in the FF state indeed exhibits a TLL behavior.\n2. spin-spin correlation\nWe then consider the evolution of spin-spin correlation\nfunction with \rin the FF phase. In Fig. 8(a)-(c) iDMRG\nresults of the spin-spin correlation function hSz\n\u000b;iSz\n\f;jifor\n\r= 0,0:04, and 0:08with fixedJ2=jJ1j= 0:6are plotted\nas a function of distance ji\u0000jj. We keepSz\ntot= 4=Las done\nin Fig. 4.\nAs far as the FF state is maintained up to \r\u00190:08, the cor-\nrelation functions seem to be almost independent of \r. Ac-\ncordingly, the expectation values of Sz\nA;iandSz\nB;iare un-\nchanged up to \r\u00190:08, as shown in Fig. 8(d). Perhaps, one\nFIG. 8. Spin-spin correlation function hSz\niSz\njiof the asymmetric\nJ1-J2zigzag ladder as a function of ji\u0000jjwith fixedSz\ntot=L= 1=4\nandJ2=jJ1j= 0:6for (a)\r= 0, (b)0:04, and (c) 0:08. (d) Averaged\nvalues ofhSz\niion the apical and basal sites as a function of \rfor\nJ2=jJ1j= 0:6. The circles and crosses denote iDMRG and DMRG\nresults, respectively.\nmay naively expect the reduction of hSz\nA;iiwith increasing\n\r, i.e., with increasing AFM coupling between apical spins.\nHowever, this is not actually the case. This can be understood\nas follows: as estimated by the fitting of low-energy excitation\nspectrum, an FM coupling with the magnitude \u00180:11jJ1jis\neffectively induced between neighboring apical sites at \r= 0.\nThus, the apical chain may be effectively mapped onto an FM\nchain withJe\u000b=\u00000:11jJ1j. Therefore, it would be rather\nnatural that the (nearly) full polarization is free of the influ-\nence of additional AFM coupling \rJ2until it reaches around\n\u00180:11jJ1j.\nNo\r-dependence of the spin structure up to \r\u00190:08then\nindicates that the total energy of FF state is simply lifted by\nthe AFM interaction \rJ2between nearly-fully polarized api-\ncal spins; it switches into the energy level with a metastable\nPF state at\r\u00190:09. Hence, the FF to PF phase transition\nis of the first order. It can be also confirmed by a steep (or\nalmost discontinuous) change of StotandhSz\n\u000b;iiat\r\u00190:09.\nOn the other hand, both of hSz\nA;iiandhSz\nB;iismoothly ap-\nproach to zero around \r\u00190:14. Thus, the transition from PF\nto the spiral singlet ( Stot= 0) region is of the second order or\ncontinuous.\n3. stabilization gap\nTo quantify the stability of ferrimagnetic state at finite \r,\nwe calculate the stabilization energy \u0001[Eq.(6)] as done in\nthe case of delta chain. In Fig. 9(a) the finite-size scal-10\n0 0.05 0.1 0.1500.010.020.03\n0 0.02 0.04 0.06 0.08 0.100.0020.0040.0060.008(a)\n(b)\nFIG. 9. (a) Finite-size scaling of energy difference between lowest\nStot= 0 state and ferrimagnetic ground state for the asymmetric\nJ1-J2zigzag ladder with fixed J2=jJ1j= 0:6. The solid and dotted\nlines show the fitting results with \u0001(L)=jJ1j= \u0001=jJ1j+ A=Land\n\u0001(L)=jJ1j= \u0001=jJ1j+ A=L\u0011, respectively. (b) Extrapolated val-\nues of \u0001=jJ1jas a function of \r. The width of error bar means the\ndifference of \u0001=jJ1jobtained by the two fitting functions.\ning analysis of \u0001(L)is performed for several \rvalues with\nJ2=jJ1j= 0:6fixed. Although a sine-like oscillation is\npresent as in the case of delta chain, an acceptable scaling may\nbe possible in the FF phase ( \r<\u00180:09). We here employ two\nkinds of fitting functions: \u0001(L)=jJ1j= \u0001=jJ1j+ A=Land\n\u0001(L)=jJ1j= \u0001=jJ1j+ A=L\u0011. Since the former (latter) func-\ntion seems to underestimate (overestimate) the extrapolated\nvalue of \u0001=jJ1j, their averaged value is plotted as a function\nof\rin Fig. 9(b). The width of error bar means the differ-\nnce between two values of \u0001=jJ1jobtained by the two fitting\nfunctions. The stabilization gap is approximately linearly re-\nduced by\rup to the critical point \r\u00190:08. This also sup-\nports the above speculation that the total energy of FF state is\nsimply lifted by \rJ2.\nIn the PF phase ( \r>\u00180:09), however, the scaling analysis\nof\u0001(L)is virtually impossible. As an example, \u0001(L)=jJ1j\nvs.1=Lfor\r= 0:1is shown in Fig. 9(a). This difficulty is\ncaused by the following several factors: (i) As shown in the\nnext subsection, an incommensurate oscillation is involved in\nthe PF state. (ii) The total spin per site Stot=Lis strongly de-\npendent on system size since the states in different Stotsectors\nare extremely quasi-degenerate around the ground state. (iii)\n0 0.1 0.2 0.30basal cha/g3444 n\nap/g3444 cal cha/g3444 n\n0 0.2 0.4 0.6 0.8 10\n0.06 0.1 0.14-12.7-12.6(a)\n(b)FIG. 10. (a) Static spin structure factor for J2=jJ1j= 0:6as a\nfunction of\r. The lattice spacing ais set as shown in Fig. 1(a). (b)\nEnlarged figure of (a) for 0\u0014\r\u00140:3. Inset: Ground-state energy\nas a function of \r.\nThe available system size is strictly limited because the sec-\nond term of Eq.(7) includes long-range interactions and a pe-\nriodic cluster must be used. Nevertheless, the stabilization gap\nshould be positive due to the nonzero total spin of the ground\nstate [Fig. 7(b)]. We can, at least, confirm that the PF order is\nvery fragile with the stabilization gap \u0001<4:3\u000210\u00004jJ1jat\n\r= 0:08. This small stabilization gap also tells us that there\nare a macroscopic number of quasi-degenerate states belong-\ning to different Stotsectors, since the total spin is continu-\nously varied from L=4to0with\rin the PF phase.\n4. Static spin structure factor\nIt is important to see how the intrachain spin modulation\nchanges with \r. A most significant quantity to know it would\nbe the propagation number which can be extracted as a max-\nimum position of static spin structure factor. Therefore, we\ncalculate the static spin structure factor for each of the apical\nand basal chains. It is defined as\nS\u000b(q) =1\nL2\u000bL\u000bX\ni:j=1eiq(ri\u0000rj)hS\u000b;i\u0001S\u000b;ji (10)\nfor the apical ( \u000b= A ) or basal (\u000b= B) chains. The lattice\nspacingais set as shown in Fig. 1(a). In Fig. 10 the propaga-11\ntion numbers qmaxforJ2=jJ1j= 0:6using a 64-site periodic\ncluster are plotted as a function of \r.\nAt\r= 0, the system is in the FF state whose spin structure\nis commensurate and simple as shown in Fig. 1(c). The apical\nspins are nearly-fully polarized and the propagation number\nisqmax= 0, whereas for basal chain the dominant correlation\nis AFM (qmax=\u0019=a) although it is slightly polarized. It is\nobvious that this spin structure persists in the whole region of\nthe FF phase at 0<\u0018\r<\u00180:08.\nIn the singlet ( Stot= 0) phase at\r>\u00180:14, the propagation\nnumbers of apical and basal chains are both incommensurate,\ni.e.,0< q max< \u0019=a . At\r= 1, they coincide and it is\nestimated as qmax= 0:958. This value is in good agreement\nwith our previous estimation qmax= 0:95873in the thermo-\ndynamic limit. With decreasing \r, the propagation number is\nreduced because short-range FM correlation is relatively en-\nhanced. Interestingly, they are equal or very close down to the\ncritical point at \r\u00190:14and obviously split for smaller \r. It\nwould be a good guess that the short-ranged spiral structure of\ntheJ1-J2chain (\r= 1) is approximately maintained down to\n\r\u00190:14. In a broad sense, this incommensurate region can\nbe referred to as a spiral singlet phase.\nIn Fig. 10(b) an enlarged figure around the PF phase\n(0:08<\u0018\r<\u00180:14) is shown. As shown in the inset of\nFig. 10(b) the phase boundaries are recognized by level cross-\ning of the ground state energies. It clearly indicates the exis-\ntence of an intermediate phase between the FF and spiral sin-\nglet phases, though the region ( 0:08>\u0018\r>\u00180:11) is a bit nar-\nrower than that in the thermodynamic limit ( 0:08>\u0018\r>\u00180:14)\ndue to finite size effects. The intermediate phase is the PF\nphase as described above.\nIn the PF phase, the dominant correlation of the basal chain\nseems to be incommensurate and the propagation number of\nthe apical chain keeps qmax= 0. It is a natural consequence of\nthe global spin-rotation-symmetry breaking because the total\nspin can be no longer finite if both of the propagation num-\nbers are nonzero. This incommensurate propagation is a con-\nsequence of the halfway magnetization so that the prohibition\nby the MLM theorem is also avoided by the TLL characteris-\ntic of basal chain. Similar incommensurate features have been\nreported for PF state in frustrated systems74,75.\n5. Dimerization order\nSo, let us see more about the magnetic properties of the\nspiral singlet ( Stot= 0) phase at larger \r. It is known that\nthe system has LRO with spontaneous dimerization in the J1-\nJ2chain (\r= 1)19,20. Therefore, as a starting point it would\nbe reasonable to examine the evolution of dimerization order\nparameters with decreasing \rfrom 1. The dimerization order\nparameter between sites distant \u000ealong theJ1zigzag chain is\ndefined as\nOdimer(\u000e) = lim\nL!1jhSA;i\u0001SB;i\u0000(\u000e\u00001)=2i\n\u0000hSA;i\u0001SB;i+(\u000e+1)=2ij;(11)\nδ\nδ = \nδ = \nδ = (a)δ = 1\nδ = 2\nδ = 3\n(b)FIG. 11. (a) Schematic pictures of possible dimerization order.\nThe states for \u000e= 2 and\u000e= 3 are characterized as VBS. A solid\n(dotted) ellipse denotes a spin-singlet (spin-triplet) dimer. (b) Aver-\naged dimerization order parameters Odimer(\u000e)as a function of \rat\nJ2=jJ1j= 1. Inset: each contribution to Odimer(2)from the apical\nand basal chains.\nfor odd\u000e, and\nOdimer(\u000e) = lim\nL!1jhS\u000b;i\u0001S\u000b;i+\u000e=2i\u0000hS\u000b;i\u0001S\u000b;i+\u000e=2ij;\n(12)\nfor even\u000e. IfOdimer(\u000e)is finite for\u000e, it signifies a long-range\ndimerization order associated with mirror-symmetry breaking\nfor odd\u000eor translation-symmetry breaking for even \u000e. We\nhere study the case of \u000e= 1,2, and 3. Schematic pictures of\nthe possible dimerization orders are shown in Fig. 11(a). A\nfiniteOdimer(\u000e)for\u000e= 2and3indicates a valence bond for-\nmation, i.e., spin-singlet formation, between two sites on the\ndimerized bond. In Fig. 11(b) the iDMRG results of dimeriza-\ntion order parameter Odimer(\u000e)for\u000e= 1,2, and 3are plot-\nted as a function of \ratJ2=jJ1j= 1. For confirmation, we\nalso estimateOdimer(\u000e)in the thermodynamic limit for some\n\rvalues using DMRG under OBC. We can find their excel-\nlent agreement with the iDMRG results. Note that the FM\ninteractionJ1at both edges in the open clusters is set to be\nzero. It enables us to perform the finite-size scaling analysis\nmore easily because the competing two translation-symmetry\nbreaking states are explicitly separated20. For confirmation,\nwe have checked that the ground state in the thermodynamic\nlimit does not depend on the choice of boundary conditions.\nAt\r= 1, two dimerization orders with \u000e= 1 and\u000e= 3\ncoexist20. With decreasing \r, interestingly,Odimer(3)is sig-\nnificantly enhanced and Odimer(1)is slightly increased down12\nto\r\u00190:195. With fixed J2=jJ1j= 1, we may deduce that\nthe magnetic frustration is largest in the limit of \r= 0where\nthe system is a series of isotropic triangles with uniform mag-\nnitude of interactions. The valence bond pair may be strength-\nened to screen spins more strongly for the relaxation of larger\nmagnetic frustration at smaller \r. SinceOdimer(2) = 0 down\nto\r\u00190:195, this state is dominantly characterized as a VBS\nstate with\u000e= 3 dimerization order (we call it “ D3-VBS”\nstate).\nEven more surprisingly, Odimer(2)exhibits a steep increase\n(almost jump) at \r\u00190:195. The other order parameters\nOdimer(1)andOdimer(3)are also not differentiable with \r\nat this point. This clearly indicates another first-order tran-\nsition at\r\u00190:195. We note that both of the apical and\nbasal chains are spontaneously dimerized along the chain di-\nrection in the region of Odimer(2). The order parameter for\neach chain is plotted in the inset of Fig. 11(b) (The main fig-\nure shows the averaged value). Since the value of Odimer(2)\nis much larger than the other dimerization order parameters at\n0:035<\u0018\r<\u00180:195, the state is recognized as a VBS one with\n\u000e= 2 dimerization order (we call it “ D2-VBS” state). Since\nthis\u000e= 2dimerization order is associated with a translation-\nsymmetry breaking, the magnetic structure consists of a su-\npercell with four sites. More detailed analysis is given in Ap-\npendix D.\nThe dimerization order can be also detected by studying\nthe topological properties of the system. Then, let us see\nthe entanglement spectrum (ES)76which can be obtained by\na canonical representation of the an infinite matrix-product-\nstate in the iDMRG calculations12. Using Schmidt decompo-\nsition, the ground-state wave function can be expressed as\nj i=X\nie\u0000\u0018i=2j\u001eA\nii\nj\u001eB\nii; (13)\nwhere the statesj\u001eS\niicorrespond to an orthonormal basis\nfor the subsystem S(either A or B). We study the ES for\nseveral kinds of splitting pattern between subsystems A and\nB. The splitting patterns are sketched in Fig. 12(a). In our\niDMRG calculations, the ES f\u0018\u000bgis simply obtained as \u0018\u000b=\n\u00002 ln\u0015\u000b, wheref\u00152\n\u000bgare the singular values of the reduced\ndensity matrices after the bipartite splitting. The low-lying\nfour ES levels are plotted as function of \rin Fig. 12(b)-(e).\nWhen the one dimerized singlet pair straddles the subsys-\ntems A and B [Fig. 12(b) and (d)], the lowest entanglement\nlevel is doubly degenerate as a reflection of the edge state.\nOn the other hand, when this is not the case [Fig. 12(c) and\n(e)], the lowest entanglement level is non-degenerate. Our re-\nsult for theD3-VBS state is consistent with previous research\non the symmetric case19. These facts would strongly support\nthe formation of long-range dimerization order. We can also\nfind a discontinuous change of @(\u00002 ln\u0015\u000b)=@\r at\r\u00180:195,\nwhich seems to correspond to the transition point from the\nD2-VBS toD3-VBS state. We note that the difference be-\ntween (b) and (d) as well as (c) and (e) in Fig. 12 comes from\nthe asymmetric nature of our system. More details about the\nasymmetric nature are discussed in Appendix D.\nIn the spiral singlet phase, the system is in either D2-VBS\norD3-VBS state. The phase boundary between them is shown\n(a)\n(e)(b) (c)\n(d)(b) (c) (d) (e)D2-VBS\nD3-VBS\nγ γ21210121\nα=λ λFIG. 12. (a) Schematic pictures of considered splitting of the sys-\ntem into two subsystems in the D2-VBS andD3-VBS state. A solid\nellipse denotes a spin-singlet pair. The number of singlet pair cross-\ning with each cut is shown in the green square. (b)-(e) Entangle-\nment spectrum for the corresponding splitting as a function of \rat\nJ2=jJ1j= 1.\nin Fig. 2(b). In general, the spin gap, namely, energy differ-\nence between spin-singlet ground state and spin-triplet first\nexcited state, is expected to be finite when the system is in a\nVBS state. In theD3-VBS phase, the spin gap simply scales to\nan energy to break a valence bond for \u000e= 3. In theD2-VBS\nphase, each of the apical and basal chains has a different va-\nlence bond. Nevertheless, it is easy to imagine that the valence\nbond in the apical chain is more fragile because of smaller\nAFM interaction, although Odimer(2)for the apical chain is\nlarger than that for the basal chain. Thus, the spin gap in the\nD2-VBS phase scales to an energy to break a valence bond in\nthe apical chain. This means that a larger energy than the spin\ngap is needed to break a valence bond in the basal chain. It\nwould provide a 1/2-plateau in the magnetization process with\nmagnetic field.13\n0.2 0.4 0.6 0.8 1.0\nγ0.000.010.020.030.040.050.060.070.08Ostring\nFIG. 13. String order parameter as a function of \ratJ2=jJ1j= 1\nusing iDMRG (circles) and DMRG (crosses) methods. The DMRG\nresults are extrapolated values to the thermodynamic limit.\n6. String order\nWe have confirmed the existence of nearest-neighbor ( \u000e=\n1) FM dimerization order in the whole spiral singlet region. A\nspin-triplet pair may be effectively formed in the each ferro-\nmagnetically dimerized bond: By relating three states j\"\"i ,\nj\"#i +j#\"i )=p\n2, andj##i toSz= 1,0, and\u00001states,\nrespectively, the resultant spin on the dimerized bond can be\nreduced to a spin- 1degree of freedom. Consequently, the sys-\ntem could be mapped onto a S= 1 Heisenberg chain ac-\ncompanied by the emergent effective spin- 1degrees of free-\ndom with the dimerized two spin-1\n2’s19. Furthermore, the\npresence of third-neighbor AFM dimerization order ensures\na valence bond formation between the neighboring effective\nS= 1 sites20. It leads to finite spin gap as a Haldane gap\nin symmetry-protected VBS state11,12. Although the spin gap\nis a good indicator to measure the stability of VBS state, it\nwould be too small to correctly estimate with DMRG method\nin most of the parameter region of system (1). Alternatively,\nthe stability of VBS state associated with the Haldane picture\ncan be evaluated by examining the string order parameter77:\nOz\nstring =\u0000lim\njk\u0000jj!1h(Sz\nA;k+Sz\nB;k)\nexp[i\u0019j\u00001X\nl=k+1(Sz\nA;l+Sz\nB;l)](Sz\nA;j+Sz\nB;j)i:\n(14)\nFor our system (1), Eq. (14) can be simplified as\nOz\nstring =\u0000lim\njk\u0000jj!1(\u00004)j\u0000k\u00002h(Sz\nA;k+Sz\nB;k)\nj\u00001Y\nl=k+1Sz\nA;lSz\nB;l(Sz\nA;j+Sz\nB;j)i:i (15)\nThe finite value of Oz\nstring suggests the formation of a VBS\nstate having a hidden topological long-range string order.Since two-fold degeneracy of the ground state due to the FM\ndimerization is lifted in our numerical calculations, jOz\nstringj\ncan have two different values depending on how to select k\nandj. We then take their average.\nIn Fig. 13 iDMRG and DMRG results for the string order\nparameter are plotted as a function of \ratJ2=jJ1j= 1. We\ncan see a good agreement between the iDMRG and DMRG\nvalues. With decreasing \rfrom 1,Oz\nstring is significantly\nincreased and has a pointed top at \r\u00190:195, which is\nthe first-order transition point between D3-VBS andD2-VBS\nphases. With further decreasing \r, it decreases and vanishes\nat\r= 0:035, which is the second-order transition point be-\ntweenD2-VBS and ferrimagnetic phases. We notice that the\noverall trend ofOz\nstring is similar to that of the \u000e= 3 dimer-\nization order parameter Odimer(3). It means that the stability\nof string order is dominated by the strength of valence bond\nwith\u000e= 3. In other words, a Haldane state is produced as\na structure where all the neighboring effective spin- 1sites are\nbridged by the \u000e= 3 valence bonds. Even though it is a\nHaldane state, the maximum value Oz\nstring\u00180:065is much\nsmaller thanOz\nstring =4\n9'0:4444 for the perfect VBS state\nfor the AKLT model11andOz\nstring'0:3743 for theS= 1\nHeisenberg chain78. This small value of Oz\nstring is interpreted\nas a sign of fragility of the D3-VBS state, which is however\ncomparable with the maximum value for the J1-J2chain at\nJ2=jJ1j\u00190:6(Oz\nstring\u00180:06)19,20.\nV . SUMMARY\nWe studied the asymmetric S=1\n2J1-J2zigzag ladder,\ndefined as two different AFM Heisenberg chains coupled by\nzigzag-shaped interchain FM interaction, using the DMRG-\nbased techniques. The AFM chain with larger (smaller) inter-\naction is referred to as apical (basal) chain.\nFirst, a classical phase diagram was obtained by the spin-\nwave theory. It contains three phases: FM, commensu-\nrate, and incommensurate phases. It offers the possibility of\ncommensurate-incommensurate phase transition by tuning the\nratio of AFM interaction of the apical and basal chains in the\nquantum case.\nNext, we revisited the ferrimagnetism in the so-called delta\nchain as the vanishing limit of AFM interaction in the apical\nchain. The ferrimagnetic state is characterized by total spin\nStot=L=4. By carefully examining the long-range spin-spin\ncorrelation functions and low-energy excitations, we pointed\nout that the origin of ferrimagnetic state is order by disor-\nder without geometrical symmetry breaking but with a global\nspin-rotation-symmetry breaking. Accordingly, the system\ncan gain energy from the FM interaction between the polar-\nized apical and basal spins. So to speak, FM fluctuations play\nan essential role to lower the ground state energy against the\nmagnetic frustration. This is a rare type of order by disor-\nder. And yet the basal chain is essentially a critical AFM\nHeisenberg chain as a TLL and its polarization is rather ill-\nconditioned as a state of chain itself. In this regards, one could\ninterpret this to mean that the ferrimagnetic order competes14\nwith a quasi-long-range AFM order of TLL.\nThen, we examined how the ferrimagnetic state is affected\nby AFM interaction of the apical chain, which is controlled by\n\r. We found that the ferrimagnetic state with Stot=L=4is\nmaintained up to a finite value of \r; and with further increas-\ning\rthe system goes into spiral singlet ( Stot= 0) phase at a\ncertain amount of \r. Of particular interest is the appearance of\nanother ferrimagnetic phase characterized by 0<Stot<L= 4\nbetween the Stot=L=4ferrimagnetic and Stot= 0 phases.\nSuch a ‘halfway’ magnetization, which is generally prohibited\nby the MLM theorem, is now allowed under the competition\nbetween ferrimagnetic and quasi-long-range order of TLL.\nFinally, the detailed magnetic properties of spiral singlet\nphase was investigated. With evaluating various dimerization\norder parameters, we confirmed that the Stot= 0 region is\ncovered by two kinds of VBS phases depending on the pa-\nrameter values. One is the D2-VBS phase where the valence\nbond is formed along the apical and basal chains; the other is\ntheD3-VBS phase where the valence bond is formed between\nthe apical and basal spins. Interestingly, a hidden topologi-\ncal long-range string order exists in the whole region of VBS\nphases and its strength scales only to the stability of D3-VBS\nstate.\nWe summarized the above numerical results as a ground-\nstate phase diagram, which contains a variety of frustration-\ninduced ordered phases. Especially, it is quite curious that a\nsimple frustrated system, like the asymmetric zigzag ladder,\nexhibits very different kinds of order by disorder – associated\nwith geometrical symmetry and global spin-rotation symme-\ntry.\nACKNOWLEDGEMENTS\nWe thank S. Ejima for fruitful discussions and U. Nitzsche\nfor technical assistance. This work was supported by Grants-\nin-Aid for Scientific Research from JSPS (Projects Nos.\nJP17K05530 and JP19J10805) and by SFB 1143 of the\nDeutsche Forschungsgemeinschaft (project-id 247310070).\nThe iDMRG simulations were performed using the ITensor\nlibrary79.\nAppendix A: Numerical confirmation of ferromagnetic critical\npoint\nIn the main text, we have obtained the \r-dependence of FM\ncritical point by classical SWT. It is given as\nJ2;c\njJ1j=1\n2(1 +\r)(A1)\nfor\r >0. This derivation can be easily extended to negative\n\r. Then, when \r <0we obtain the FM critical point as\nJ2;c\njJ1j=1\n2; (A2)\nwhich is independent of \r.\n-1 0 10.20.30.40.5\nDMRG\nsp/g3444 n-wave theory(a)\n(b)FIG. 14. (a) iDMRG results for spin-spin correlation function\nhSi\u0001Sjias a function of distance ji\u0000jjwith\r= 0:2fixed. (b)\nComparison of the \r-dependence of FM critical points J2;c=jJ1jes-\ntimated by DMRG and spin-wave theory.\nSince the quantum fluctuations vanish at the FM critical\npoint, Eqs. (A1) and (A2) should be exact. We here confirm\nthem numerically. A simplest way is to check the presence\nor absence of long-range FM order. In Fig. 14(a) we show\niDMRG result for the spin-spin correlation function hSi\u0001Sji\nat\r= 0:2as a function of distance ji\u0000jj. For convenience,\nthe site indices iandjare taken along the J1zigzag chain.\nAtJ2=jJ1j= 0:4we can clearly see an FM long-range or-\nder withhSi\u0001Sji=1\n4for any pair of iandjat the large\ndistance. However, the FM long-range order seems to be al-\nready destroyed at only a slightly larger J2=jJ1j= 0:42. In\nfact, this iDMRG result is consistent with the SWT estimation\nJ2;c=jJ1j= 0:417for\r= 0:2. Additionally, since the sys-\ntem is in theD3-VBS state at J2=jJ1j= 0:42, an exponential\ndecay ofhSi\u0001Sjiis thus expected.\nFurthermore, we also numerically estimate the FM critical\npoint for\r <0. In Fig. 14(b) we compare the \r-dependence\nofJ2;c=jJ1jvalues estimated by DMRG and SWT. We can\nfind a good agreement between them. The reason why the FM\ncritical point does not depend on \ris as follows: For any neg-\native\rthe apical chain is fully polarized and this FM order\naffects the basal chain as an external field via the interchain\ncouplingJ1. Therefore, the FM phase transition boils down15\nto the question of a competition between FM interchain inter-\nactionJ1and AFM interaction J2in the basal chain, namely,\nindependent of \r. Even intuitively, Eq. (A2) can be obtained\nby comparing the numbers of J1andJ2bonds.\nAppendix B: Artificial enhancement of spin-singlet state with\nshort chains\n0 0.05 0.1-0.1-0.09-0.08-0.07\nFIG. 15. Energies per site for lowest-lying spin-singlet and ferri-\nmagnetic states at J1=\u00001andJ2= 0:8as a function of inverse\nsystem size. The PBC is used.\nIn Fig. 3(b) of the main text, the total spin per site Stot=Lis\nplotted as a function of 1=Lunder PBC. With decreasing 1=L,\nwe see a drastic change of the total spin from Stot=L= 0 to\nStot=L= 1=4 + 1=Lat some value of 1=L. This is the conse-\nquence of a typical finite-size effect: If the system size is suf-\nficiently small, the basal chain forms a short AFM ring under\nPBC and a plaquette singlet is highly stabilized. On the other\nhand, the ferrimagnetic state may be readily prevented due\nto the screening of basal spins. This finite-size effect could be\neliminated by increasing the system size because the plaquette\nsinglet is rapidly destabilized for larger systems. To confirm\nit, we compare the energies for spin-singlet and ferrimagnetic\nstates in Fig. 15. When the system size is small, the energy\nfor spin-singlet state is much lower than that for ferrimagnetic\nstate. They approach with increasing system size and cross at\nsome value of L(\u0011Lc). Roughly speaking, one could inter-\npret this to mean that the energy gain from ferrimagnetic for-\nmation becomes larger than that from the plaquette singlet for-\nmation of basal chain around the critical system size L=Lc.\nTherefore, we take Stot=L= 1=4 + 1=Lfor the finite-size\nscaling and obtain Stot=L= 1=4in the thermodynamic limit\nL!1 .\nAppendix C: Tomonaga-Luttinger-liquid behavior of basal\nchain in full ferrimagnetic phase\nAs mentioned in the main text, the basal chain in the fer-\nrimagnetic states should be a TLL because it only allows the\nsystem to have an incommensurate ‘halfway’ magnetization\n100101102\n|i−j|10−810−710−610−510−410−310−210−1|/angbracketleftbig\nSz\nB,jSz\nB,i/angbracketrightbig\n−/angbracketleftbig\nSz\nB,j/angbracketrightbig/angbracketleftbig\nSz\nB,i/angbracketrightbig\n|\nγ= 0.0\nγ= 0.04\nγ= 0.08\n100 20010−510−310−1\nγ= 0.2FIG. 16. Spin-spin correlation function as a function of distance ji\u0000\njjwithJ2=jJ1j= 0:6.\nother than the commensurate one. We can check it by look-\ning at the spin-spin correlation in the basal chain. We plot\niDMRG results of the spin-spin correlations jhSz\nB;iSz\nB;ji\u0000\nhSz\nB;iihSz\nB;jijin the FF state are plotted as a function of dis-\ntanceji\u0000jjin Fig. 16. The z-component of total spin is fixed\natSz\ntot=L=4. We can see a power-law decay with distance,\ni.e.,jhSz\nB;iSz\nB;ji\u0000hSz\nB;iihSz\nB;jij\u00181=ji\u0000jj, as a signature of\nTLL. For comfirmation, we also plot the spin-spin correlation\nfor theD3-VBS state in the inset of Fig. 16. It clearly exhibits\nan exponential decay indicating a gapped VBS state.\nAppendix D: Dimerization order with translation symmetry\nbreaking\nIn the main text, we have estimated the dimerization order\nparameters defined as the difference between spin-spin corre-\nlations for the ‘neighboring’ bonds. To obtain further insight\ninto the dimerization structure, we here look at the bare spin-\nspin correlations. In Fig. 17(b)-(d) we plot the spin-spin cor-\nrelations for various pairs of spins as a function of \r. Each\nsymbol corresponds to a bond marked by the same symbol in\nFig. 17(a).\nAt\r>\u00180:195, the valence bond structure is relatively sim-\nple. The correlation h(Si\u0001Sj)Iihas two different values and\nthe difference between them provides the dimerization order\nparameter. This is the same for h(Si\u0001Sj)IIIi. No splitting of\nthe correlationh(Si\u0001Sj)IIiindicates the absence of dimeriza-\ntion order parameter for \u000e= 2.\nAt\r<\u00180:195, the situation is a bit more complicated.\nThe translation symmetry is broken due to the presence of\ndimerization order for \u000e= 2 and all the three dimerization\norders coexist. As a result, the two values of h(Si\u0001Sj)Iiand\nh(Si\u0001Sj)IIIiare further split into four values. 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B 48, 3844 (1993).\n79https://itensor.org/."    },    {        "title": "2303.02639v1.Electrical_detection_of_antiferromagnetic_dynamics_in_Gd_Co_thin_films_by_using_a_154_GHz_gyrotron_irradiation.pdf",        "content": "1 \n Electrical detection of antiferromagnetic dynamics  in Gd-Co thin films  \nby using a 154-GHz gyrotron irradiation  \n \nS. Funada1, Y . Ishikawa2, M. Kimata3, K. Hayashi2, T. Sano2, K. Sugi1, Y . Fujii2, \n S. Mitsudo2, Y . Shiota1,4, T. Ono1,4, and T. Moriyama *1,4,5 \n1 Institute for Chemical Research, Kyoto Univ ., Uji, Kyoto 611 -0011,  Japan  \n2 Research Center for Development of Far -Infrared Region, Univ . of Fukui,  Fukui, Fukui  910-\n8507,  Japan  \n3 Institute for Materials Research, Tohoku Univ ., Sendai, Miyagi  980-8577, Japan  \n 4 Center for Spintronics Research Network, Kyoto Univ., Uji, Kyoto 611 -0011, Japan  \n5PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 322 -0012, Japan  \n \n \nAbstract  \nTHz magnetization dynamics is a key property of antiferromagnets as well as \nferrimagnets that could harness the THz forefront and spintronics . While most of the \npresent THz measurement techniques  are for bulk materials whose sensitivities rely on \nthe volum e of the material, measurement techniques suitable for thin films are quite \nlimited. In this study, we explored and demonstrated electrical detection  of the \nantiferromagnetic dynamics in ferrimagnetic Gd -Co thin film s by using a 154 GHz \ngyrotron, a high-power electromagnetic wave  source . Captured resonant modes allow us \nto characterize the peculiar magnetization dynamics of the Gd-Co around the net angular \nmomentum compensation. As the gyrotron frequency is scalable up to THz, our \ndemonstration can b e an important milestone toward the THz measurements for  antiferro - \nand ferri - magnetic  thin films.  \n*Corresponding  to mtaka@scl.kyoto -u.ac.jp  \n \n  2 \n Antiferromagnet s are one of the promising  magnetic materials which can sustain \nand work at THz frequency1. Amid the antiferromagnetic -ferrimagnetic  spintronics2,3,4,5 \nwhere antiferromagne ts and ferrimagnets  are used  as active materials in spintronic \ndevices, the THz magnetization dynamics of antiferromagnets  and ferrimagnets  is a key \nproperty that could harness the THz forefront and spintronics. Recent advancements in \nTHz measurement techniques have realized some of the key experiments on  the \nantiferromagnetic dynamics  and the relevant phenomena,  such as control of THz spin \noscillation6,7,8, THz spin pumping  effect9, and ultra fast spin switching10,11.  \nHowever, the measurement principles used in those studies, such as the magneto -\noptical effect and resonant absorption, rely on the volume of the materials. Therefore, the \nsame principle can hardly be applied for characterizing the dynamics of the thin film s \nwhich is a central interest when considering any anti ferro - and ferri -magnetic integration \ndevices.  Indeed, due to the measurement difficulty, basic properties of magnetization \ndynamics  in antiferromagnetic thin films , such as resonant frequency and magn etic \ndamping constant, remain elusive while it is not perhaps legitimate  to assume that those \nproperties are same as those in bulk materials.  Therefore, establishment of the \nmeasurement technique for characterizing the antiferromagnetic dynamics in thin films \nis a pressing issue for THz spintronics.  \nElectrical detection of the magnetic resonance is one of the route s to \nestablishment of such a measurement technique. For ferromagnetic thin films , there are \nwide varieties of the electrical measurement t echniques that capture the magnetization \ndynamics . For instance, a homodyne detection technique taking advantage of a nonlinear \ncoupling between a rf current and a magnetoresistance oscillation12,13,14 is widely used to \ncharacterize the dynamics in ferroma gnetic thin films . The resultant electrical signal can 3 \n be expressed as 𝑉=(∆𝜌/𝐌2)(𝐉∙𝐌)𝐌−𝑅H𝐉×𝐌 , where ∆𝜌  is the resistance change \ndue to magnetoresistance, 𝐌  is the magnetization, J is the induction current by the \nelectromagnetic wave irradiation, 𝑅His the anomalous Hall effect constant. The technique \ncan not only characterize the ferromagnetic films as thin as sub -nano meters15 but also \ncan characterize various interesting nonequil ibrium phenomena such as spin -torque16 . \nAnother is  a spin -to-charge conversion technique taking advantage of a combination of \nthe spin pumping effect and the inverse spin Hall effect  in a bilayer of ferromagnet/spin \nHall metal , where the spin Hall metal is  Pt, Ta, etc17,18. Persistent magnetization dynamics \nin the ferromagnetic layer creates and pumps a spin current into the spin Hall metal layer \nby the spin pumping effect. The spin current injected in the spin Hall metal layer is \nconverted to a charge current  by which the electrical detection of the magnetization \ndynamics is realized.  The principle of the latter technique has recently been demonstrated \nin particular bulk-antiferromagnet/spin Hall metal interface s19 ,20 , i.e., MnF 2/Pt and  \nCr2O3/Pt, which strongly suggests the characterization of the thin films is possible.  \nAn electrical signal, or more precisely a dc voltage signal, emerg ing at the \nresonance for both techniques is in principle proportional to a power of the input \nelectromagnetic wave. Considering the case for ferro magnets, a sizable input  microwave \npower (>mWatt) i s generally  required in order to  obtain a comfortably detectable voltage  \nsignal  of the order of μV14,18. A concern when it comes to the THz frequency is that  such \nhigh-power  continuous wave (CW) THz sources are quite limited.  Gyrotron is one of the \nfew CW THz sources which can produce a power of >mWatt21. It consists of a linear -\nbeam vacuum tube in which the electron beam undergoes the cyclotron resonance in a \nstrong magnetic field. The cyclotron motion of the electrons emits electromagnetic waves \nat the resonan t frequency as well as  its higher harmonics. It is a lab -size equipment and 4 \n more accessible than other technologies such as free -electron laser22. \nIn this work,  we explore electrical detection of antiferromagnetic resonan t modes \nin Gd xCo1-x/Ta and GdxCo1-x/Pt thin films by  employing  a gyrotron  which can generate \n154 GHz with a nominal power of 500 mW . We clearly detected  the resonances as a \nvoltage signal over which the characteristic magnetization dynamics of the Gd -Co alloy \nand its temperature dependence are discussed.   \nBilayers  of Gd 0.17Co0.83 20 nm/Ta 3 nm  (Sample A) , Gd 0.16Co0.84 20 nm/Ta 3 nm  \n(Sample B),  and Gd 0.16Co0.84 20 nm/Pt 3 nm  (Sample C)  were deposited on a thermally \noxidized Si substrates by a magnetron sputtering with a base pressure of 1.5 × 10-6 Pa. \nGdxCo1-x layers were formed by co -sputtering of Gd and Co and the composition was \ncontrolled by the sputtering power  for each element . Ta and Pt capping layer is to protect \nthe Gd -Co layer from oxidation as well as  is to utilize the  inverse spin Hall effect.  Gd-Co \nalloys are one of the typical rare earth -transition  metal  ferrimagnets  which show both \nmagnetization and angular momentum compensation  temperatures  due t o the difference \nin the temperature dependence s of the magnetic moment  and the gyromagnetic ratio of \nthe Gd and Co . The Gd -Co alloys generally show two dynamic modes  due to the \nantiferromagnetically coupled  magnetic sublattices of Gd and Co23 ,24 . Peculiar \nmagnetization dynamics  in the vicinity of those compensation temperatures are \ncomprehended by non -zero angular momentum with zero magnetization and vice \nversa25,26,27,28.  \nFigure 1(a) shows the schematic of the measurement setup with a gyrotron.  154 \nGHz continuous  electromagnetic wave  with a nominal power of 500 mW is generated by \na gyrotron  and is guided  into a hollow waveguide  sticking out of  a cryostat with a \nsuperconducting magnet . The electromagnetic wave is linearly polarized by a wire grid  5 \n place d in a pathway . The sample shaped into a 1.5 mm × 5 mm rectangular  piece is placed  \nat the end of the waveguide  in the way that the Poynting vector of the electromagnetic \nwave is parallel both to the sample surface and the sample transverse direction. The \nlinearly polarized  rf h-field is aligned perpendicular to the sample plane.  V oltage is  \nmeasured across the sample longitudinal direction with sweeping external magnetic field \nHext colinear to the Poynting  vector. Field direction opposite to the Poynting vector is \ndefined  as the positive field as shown in Fig. 1 (a).  \nFigure  1(b) shows typical spectra for Sample A  with the Ta capping layer  at room \ntemperature with the positive and negative magnetic field. Clear dc voltage peaks are \nobserved and there are two resonant modes  at around ±2 T and ±4 T. The peaks at around \n±4 T (labeled as ⋆) are larger than those  at around ±2 T  (labeled as ∆) and the polarity of \nthe peak voltage is opposite to each other.  For both modes,  polarity of the peak voltage is  \nreversed when the magnetic field is reversed , which  suggests that the dc voltage is  from  \nmagnetic resonance s of the Gd-Co.  \nFirst,  we discuss the origin of the dc voltage peak . As discussed above, there are \ntwo possible detection mechanisms in our measurement configuration, i.e. , the homodyne \nmechanism and the spin pumping mechanism, that create the dc voltage at the resonance.  \nIf the spin pumping mechanism  is dominant , Sample B with the Ta capping layer and \nSample C with the Pt capping layer would show the opposite voltage polar ity at the \nresonance because sign of the spin Hall angle for Pt and Ta is opposite to each other29,30. \nAs shown in Fig. 1 (c), the voltage polarity at the resonance is the same for both Sample \nB and C, indicating a dominant mechanism is not the spin pumpin g but the homodyne  \ncaused by the nonlinear coupling  between the induction current and the \nmagnetoresistance of the Gd -Co film. A slight difference in the line shape between 6 \n samples could come from a misalignment of the  rf h-field with respect to the sample  \ngeometry as the line shape is quite sensitive to the alignment with the homodyne \nmechanism13,31.  \nNext, w e discuss  temperature dependence of the resonan t modes . Figure 2 (a) \nshows the spectra for Sample A at various temperature s. We observed a clear dc voltage \npeak shifting in the range between 4 and 2 T with decreasing temperature . Figure 2 (b) \nsummarizes the  resonan t field μ0Hres and resonant  linewidth μ0ΔHres, which are extracted \nby the Lorentzian  peak fitting, as a function of temperature . μ0Hres decreas es with \ndecreasing the temperature  and μ0ΔHres varies with respect to the temperature and took \nits maximum at round  100 K.  \nAssociating with the temperature dependent magnetization as well as the \ntemperature dependent angular momentum , Gd -Co alloys generally show a strong \ntemperature dependence of the  resonan t frequency.  We here define the  temperature \ndependent  net magnetization 𝑀(𝑇) as well as  net angular momentum  𝑠net(𝑇) as32,  \n𝑀(𝑇)=𝑀′Co(1−𝑇𝑇c⁄)𝛽Co−𝑀′Gd(1−𝑇𝑇c⁄)𝛽Gd (1) \n𝑠net(𝑇)= 𝑀′Co𝛾Co⁄ (1−𝑇𝑇c⁄)𝛽Co−  𝑀′Gd𝛾Gd⁄ (1−𝑇𝑇c⁄)𝛽Gd (2)\nwhere Tc is the critical temperature  for the ferrimagnetic -paramagnetic transition , and  \n𝑀′Gd,Co and 𝛽Gd,Co are the magnetization at 0 K  and the critical exponent, respectively, \nfor each element. 𝛾Gd,Co=𝑔Gd,Co𝜇B/ℏ is the gyromagnetic ratio  with the Bohr magneton \n𝜇B , the reduced Planck’s constant ℏ , and the g-factor  𝑔Gd,Co for each element. By \nobtaining the 𝑀′Gd,Co  and 𝛽Gd,Co  from measured 𝑀(𝑇) , 𝑠net(𝑇)  can be estimated using \nthe literature value of 𝑔Gd= 2.02 and 𝑔Co= 2.1133. \nFigure 3 shows  the measured  𝑀(𝑇)  for Sample A  with the magnetization \ncompensation temperature TM ~ 40 K.  The obtained 𝑠net(𝑇) is overlayed in Fig. 3. The 7 \n angular momentum  compensation temperature  TA is found  to be 130 K.  Resonan t \nlinewidths  in Gd -Co are theoretically predicted to increase as the temperature approaches \nto TA. As we saw in  Fig. 2 (b),  the μ0ΔH peaking at 100 K roughly corresponding to the \nestimated TA is consistent with the theory34  and the previous experimental \nobservations23,35. \nFerrimagnetic  dynamics is essentially  described by two sets of  Landau -Lifshitz -\nGilbert  (LLG)  equation s for each magnetic sublattice of Gd and Co.  When solving the \nsimultaneous equation using the Neel vector basis, one obtains the eigenfrequencies36,37, \n(𝑓±)2=𝑠net2+𝜌𝜇0𝑀(2𝐻ext+𝑀)±√𝑠net4+2𝜌𝑠net2𝜇0𝑀(2𝐻ext+𝑀)+𝜌2(𝜇0𝑀)2(𝑀)2\n2(2𝜋𝜌)2(3) \nwhere  𝑓−  and 𝑓+   indicate the low  frequency mode  and the high frequenc y mode , \nrespectively , and  𝜌  is the momen tum of inertia38. μ0 is the vacuum permeability. With  \nEq. 3, 𝑠net(𝑇) can also be obtained  at each temperature by using the obtained  Hres, 𝑀(𝑇), \nand 𝑓±=154  GHz with a literature value of  𝜌=2×10−19 kg・m-1 37.  \nFigure 4 shows  𝑠net(𝑇)  estimated from Eq. 3 and that  estimated  from Eq. 2  by \nthe measured 𝑀(𝑇)  (taken from Fig. 2) . Data points of the blue and red squares are \nobtained for the larger  resonant  peak s (labeled as ⋆) by setting 𝑓−=154  GHz and 𝑓+=\n154  GHz, respectively . There were no solutions of  𝑠net(𝑇) for the smaller peaks (labeled \nas ∆), indicating these peaks could be associated with non -uniform  magnetization modes, \nor spin wave modes.  𝑠net(𝑇) obtained by two different ways are roughly  consistent to \neach other . In addition, t his calculation identifies that observed resonant peak s are from \nthe low frequency mode  above  𝑇A and high frequency mode below  𝑇A. \nThese results invoke an  immediate question why there is only one mode at given \ntemperature s. In order to address this issue, we introduce a concept of  handedness for the \nmagnetization dynamics. When solving the set of LLG equations using the elliptical basis 8 \n of each sublattice magnetization , one then finds 𝑓−=𝑓𝑅  and 𝑓+=𝑓𝐿  at 𝑇>𝑇A  and \n𝑓−=𝑓𝐿   and 𝑓+=𝑓𝑅   at 𝑇M<𝑇<𝑇A  comparing with Eq. 324, where 𝑓𝑅   and 𝑓𝐿  are, \nrespectively, the resonant frequency for  the right -handed and left -handed precession \nmode with respect to a principal axis parallel to the net magnetization.  This a nalysis \nindeed reveals the resonant peaks we observed are always the right -handed mode s in all \nthe temperature range. We therefore find out that the left -handed modes are invisible in \nour experiment.  \nIn most of the previous experimental studies on the Gd -Co dynamics , on the \ncontrary to our results,  the two modes at a given temperature  have been clearly identified \n23,24. We suspect the reason why we do not see the left-handed modes might be rooted in \nhow the dynamics is excited. Those previous studies used eith er a pump -probe technique \nor the Brillouin light scattering (BLS). The pump -probe technique thermally excites the \nmagnetization dynamics and the BLS does by the energy and momentum transfer from \nphotons, the both excitation mechanisms of which are not dire ctly relevant to the \nmagnetic susceptibility. On the other hand, in the present measurement, the dynamics is \npurely driven by the rf h -field. Therefore, magnetic susceptibility is an important factor \nto efficiently excite the magnetization dynamics  in the present study . Indeed, the magnetic \nsusceptibility at  𝑓𝐿 is found to be  smaller than that at 𝑓𝑅 and they can differ by as large \nas a factor of 10 (see the supplementary material39,40,41,42,43,44,45), which could explain the \nresonant peak s of the left-handed mode  are too small to be seen in our experiment . \nFinally, we would like to note on the measurement configuration. Since the \nhomodyne mechanism seems to be dominant in our measurement, it is possible to further \nenhance and maximize the dc voltag e signal by looking into a right geometry of the \napplied field direction and the rf h -field directions with respect to the sample14,31 , which 9 \n in the present study cannot be realized due to various geometrical limitation in the \nmeasurement setup but will be optimized in the future work.  \nIn summary, we demonstrated the electr ical detection of magnetic resonance in \nGd-Co ferrimagnetic thin films  at 154 GHz  by employing  the gyrotron  as a high -power \nelectromagnetic wave source . It was found that t he dc voltage peaks are predominantly \nfrom the homodyne mechanism . The temperature d ependence of the resonant modes  is \nwell explained by the temperature dependence of the net angular momentum in the Gd -\nCo. Lack of the left -handed resonant modes in our observation could be related to the \nsignificant difference in the magnetic susceptibilit y of the right - and left -handed modes.   \nSince the homodyne detection is common effect in magnetic films  with appreciable \nmagnetoresistance  and the gyrotron frequency is scalable up to THz21, our demonstration \ncan be an important milestone toward the THz measurements for antiferro - and ferri- \nmagnetic thin films.  \n \n \nAcknowledgments  \nThis work was supported by JSPS KAKENHI  Grant Numbers  20H05665, 21H04562, \n21J10136, and 22H01936 , JST PRESTO Grant Number JPMJPR20B9, and the Mazda \nFoundation.  This work is partly supported by the Cooperative Research Program of the \nResearch Center for Development of Far -Infrared Region, University of Fukui (No. \nR03FIDG036A, R04FIRDM025B) .   10 \n Figure Captions  \n \nFigure 1  (a) Schematic illustration of the measurement  setup. 154 GHz electromagnetic \nwave  is generated by the gyrotron and irradiated  to the sample. D c voltage across the \nsample is measured with sweeping external magnetic field. (b) Typical dc  voltage spectra \nwith the positive and negative magnetic field for Sample A  (Gd 0.17Co0.83 20 nm/Ta 3 nm)  \nat room temperature . (c) Dc voltage spectra for Sample B  (Gd0.16Co0.84 20 nm/Ta 3 nm)  \nand Sample C (Gd0.16Co0.84 20 nm/Pt 3 nm)  at room temperature . ⋆ and ∆ are labeled on \nthe larger and smaller peaks, respec tively.  \n \nFigure 2 (a) Dc voltage spectra at various temperature s for Sample A . (b) Temperature \ndependence of the resonan t field μ0Hres and linewidth μ0ΔHres for the resonant peaks \nlabeled as ⋆ and ∆ . The data markers correspond to the labels  of Fig.  1 (b). \n \nFigure 3 Measured 𝑀(𝑇)  for Sample A under the in-plane magnetic field of 100 mT and \nthe estimated  𝑠net(𝑇) by Eq. 2. \n \nFigure 4  Comparison of  𝑠net(𝑇) estimated by Eq. 2  and Eq. 3.  The blue and red squares \nare obtained for the resonant peaks (labeled as ⋆ in Fig.  2 (a)) by setting 𝑓−=154  GHz \nand 𝑓+=154  GHz in Eq.  3, respectively . The black line shows the estimated 𝑠net(𝑇) by \nEq. 2. \n  11 \n    \n \nFig. 1 Funada et al.  \n  \n12 \n  \n \nFig. 2 Funada et al.  \n  \n13 \n  \nFig. 3 Funada et al.  \n  \n14 \n  \nFig. 4 Funada et al.  \n \n  \n15 \n References  \n \n1 D. M. Mittleman, Perspective: Terahertz science and technology, J. Appl. Phys. 122, 230901 \n(2017).  \n2 V . Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono and Y . 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"    },    {        "title": "1203.6220v1.Geometric_and_disorder____type_magnetic_frustration_in_ferrimagnetic__114__Ferrites__Role_of_diamagnetic_Li__and_Zn2__cation_substitution.pdf",        "content": " 1 Geometric and disorder – type magnetic frustration in ferrimagnetic “114” \nFerrites: Role of diamagnetic Li+ and Zn2+ cation substitution.  \n \nTapati  Sarkar, V. Caignaert, V. Pralong and B. Raveau * \n \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,  \n6 bd Maréchal J uin, 14050 CAEN, France  \n \nDedicated to Professor Jacques Friedel on the Occasion of His 90th Birthday.  \n \nAbstract  \n \nThe comparative study of the substitution of zinc and lithium for iron in the “114” ferrites, \nYBaFe 4O7 and CaBaFe 4O7, shows that these diamagne tic cations play a major role in tuning \nthe competition between ferrimagnetism and magnetic frustration in these oxides. The \nsubstitution of Li or Zn for Fe in the cubic phase YBaFe 4O7 leads to a structural transition to a \nhexagonal phase YBaFe 4-xMxO7, for M = Li (0.30 \n  x \n 0.75) and for M = Zn (0.40 \n  x \n \n1.50). It is seen that for low doping values i.e. x = 0.30 (for Li) and x = 0.40 (for Zn), these \ndiamagnetic cations induce a strong ferrimagnetic component in the samples, in contrast to \nthe spin glass behaviour of the cubic phase. In all the hexagonal phases, YBaFe 4-xMxO7 and \nCaBaFe 4-xMxO7 with M = Li and Zn, it is seen  that in the low doping regime (x ~ 0.3 to 0.5), \nthe competition between ferrimagnetism and 2 D magnetic frustration is dominated by the  \naverage valency of iron. In contrast, in the high doping regime (x ~ 1.5), the emergence of a \nspin glass is controlled  by the high degree of cationic disorder, irrespective of the iron \nvalency.  \n \n \n \nKeywords : “114” Ferrites, ferrimagnetism and magnetic frus tration.    \n \n \n \n \n* Corresponding author: Prof. B. Raveau  \ne-mail:  bernard.raveau @ensicaen.fr  \nFax: +33 2 31 95 16 00   \nTel:  +33 2 31 45 26 32   2 1. Introduction   \n \n Strongly  correlated electron systems, involving transition metal oxides with a “square” \ncrystal lattice, namely perovskites, have been the subject of numerous investigations during the \nlast thirty years, for their superconducting properties in cuprates as well as their magnetic \nproperties in CMR mangani tes. Oxides with a triangular lattice have also been studied , as \nshown for the spinel family , [1 – 3] that exhibits strong ferrimagnetism and unique magnetic \ntransitions, as for example in Fe 3O4, and for the pyrochlore family and for spinels with a \npyrochl ore sublattice , [4 – 6] which have been investigated for magnetic frustration. \nNevertheless, the number of oxides with a triangular lattice is much more limited, and the \nrecent synthesis of the “114” cobaltites and ferrites [7 – 11] with original structure s closely \nrelated to the spinel offers a new playground for the investigation of the competition between \nmagnetic ordering and geometric frustration in this class of materials [12 – 14]. The \nconsideration of the structure of the “114” ferrites, cubic LnBaF e4O7 (Fig. 1 a), and hexagonal \nCaBaFe 4O7 (Fig. 1 b), shows that these oxides, both consist of similar layers of FeO 4 \ntetrahedra, called triangular (T) and kagom é  (K), and that their “Fe 4O7” frameworks can be \ndeduced from each other by a translation of one  triangular layer out of two. Therefore, the \ncorresponding iron sublattice is very different in the two structural families: it consists of a \npure tetrahedral framework of corner sharing “Fe 4” tetrahedra  in the cubic LnBaFe 4O7 (Fig. 2 \na), whereas it is bui lt up of rows of corner – sharing “Fe 5” bipyramids running along \n , \ninterconnected through “Fe3” triangles in the (001) plane for hexagonal CaBaFe 4O7 (Fig. 2 b ). \nIt is this geometry of the iron framework which is at the origin of two different kinds of \ngeometric frustration. The pure tetrahedral iron sublattice of LnBaFe 4O7 oxides is similar to \nthat of pyrochlore [4] and consequently generates a 3 D magnetic frustration. In contrast, the \nmixed “ bipyramidal – triangular” sublattice of CaBaFe 4O7, allows a com petition between a 1 D \nmagnetic ordering along \n  and a 2 D magnetic frustration in the (001) plane.  \n Recently, we have shown the possibility of stabilizing the iron “bipyramidal – \ntriangular” lattice at the cost of the tetrahedral lattice by substitution of zinc for iron in the \ncubic LnBaFe 4O7 oxides [15]. Surprisingly, it was observed that this doping with a \ndiamagnetic cation destroys the spin glass behaviour of LnBaFe 4O7 and induces \nferrimagnetism. However, with progressive increase of the Zn concentrat ion, the ferrimagnetic \ninteraction starts to weaken, and we get a spin glass for very high doping concentration. In \norder to understand the role of the different factors which govern the magnetic properties of  3 these ferrites, we have studied the substituti on of lithium and zinc , two diamagnetic cations \nwith different valencies,  for iron, in the ferrites YBaFe 4O7 and CaBaFe 4O7. We discuss, herein, \nthe relative influence of valence effects and cationic disordering upon the competition between \nmagnetic orderin g and frustration in these systems.  We will specifically attempt to decouple \nthe role of the Fe valency, and that of the disorder on the Fe sites in order to determine how the \ntwo separately affect the magnetic ground state. This will allow us the possibil ity to tune and \ncustomize the magnetic properties of these oxides by understanding the role of the two \ngoverning factors – the average Fe valency, and the degree of disorder on the Fe sites.  \n \n2. Experimental  \n \n All the samples used in this study were prepa red by standard solid state reaction \ntechnique. The details of the synthesis procedure can be found in our earlier publications [ 15, \n16]. The samples were chemically monophasic, and the phase purity was checked from X -ray \ndiffraction patterns registered wi th a Panalytical X’Pert Pro diffractometer. The d. c. \nmagnetization measurements were performed using a superconducting quantum interference \ndevice (SQUID) magnetometer with variable temperature cryostat (Quantum Design, San \nDiego, USA). The a.c. susceptib ility, \n ac(T) was measured with a PPMS from Quantum Design \nwith the frequency ranging from 10 Hz to 10 kHz (H dc = 0 Oe and H ac = 10 Oe). All the \nmagnetic properties were registered on dense ceramic bars of dimensions ~ 4 \n  2 \n 2 mm3. \n \n3. Results and discus sion \n \n3.1. Zn substitution in YBaFe 4O7 and CaBaFe 4O7 \n \n In our previous work [15], we had synthesized the oxide series YBaFe 4-xZnxO7 with \nthe hexagonal symmetry , for x ranging between 0.4 – 1.5. For the sake of relevant \ncomparison, we have prepared CaBaFe 4-xZnxO7 samples with x = 0.5 and 1.5. We have \nspecifically chosen these two values of x so that we can investigate how the two factors, i.e. \nthe average Fe valency and the cationic disorder , affect the magnetic properties in two \nseparate regimes: the low do ping regime and the high doping regime.  The XRPD patterns of \nthese two samples clearly show that they are monophasic, keeping the hexagonal symmetry  of \nCaBaFe 4O7, and with cell parameters close to those of the virgin oxide i.e. a = 6.3527 (1)  Å \nand c = 10.3274 (2)  Å for x = 0.5, and a = 6.3668 (1)  Å and c = 10.29 75 (1)  Å for x = 1.5.   4  \n3.1.1.  Low doping regime: YBaFe 3.5Zn0.5O7 and CaBaFe 3.5Zn0.5O7 \n \n YBaFe 3.5Zn0.5O7 and CaBaFe 3.5Zn0.5O7 fall in the low doping regime. The degree of \ndisorder (measured in terms of the % of substituent cation) is the same and relatively small in \nthese two samples. We show the d. c. M vs T for the two samples in Fig. 3 . Both samples \nshow a ferrimagnetic transition (seen as a sharp rise in the M vs T curves) , similar to that \nobserve d for CaBaFe 4O7, [9] in accordance to the fact that the samples have been stabilized in \nthe hexagonal symmetry . However, their ordering temperatures , and importantly, their \nmagnetic moments are much smaller than those observed for CaBaFe 4O7 (TC = 270 K and  \nMFC(5K)  = 2.6 µ B/f.u.). Moreover,  the magnetic moments and the ordering temperatures  for the \ntwo samples  are very different , though they exhibit the same degree of disorder.  The \nCaBaFe 3.5Zn0.5O7 sample exhibits a much higher magnetic moment ( MFC(5K)  = 1.7 µB/f.u.) and \ntransition temperature (T C ~ 203.0 K) compared to the YBaFe 3.5Zn0.5O7 sample for which \nMFC(5K)  = 0.54 µ B/f.u. and T C ~ 119.5 K . Bearing in mind that both these oxides exhibit the \nsame “bipyramidal – triangular” iron sublattice ( Fig. 2 b ), the se results show that the \ncompetition between the 1 D ferrimagnetism that appears along \n  and the 2 D frustration in \nthe (001) plane of the hexagonal structure is strongly affected by the average valence of iron. \nIndeed, for the same degree of disorder (12. 5 % Zn), T C decreases and the magnetic \nfrustration increases significantly as the average value of Fe decreases from 2.57 for \nCaBaFe 3.5Zn0.5O7 to 2.29 for YBaFe 3.5Zn0.5O7. \nThe effect of the higher value of the average Fe valency of CaBaFe 3.5Zn0.5O7 is also  \nseen in  the M vs H loops of the two samples  (shown in the inset of Fig. 3 ). Not only is the \ncoercivity and remanence magnetization higher for CaBaFe 3.5Zn0.5O7, the shape of the M-H \nloops are also very different. CaBaFe 3.5Zn0.5O7 has a square loop, reminis cent of hard \nferrimagnets, while YBaFe 3.5Zn0.5O7 has a much softer M -H loop  signifying a weakening of \nthe ferrimagnetic interaction in YBaFe 3.5Zn0.5O7 compared to that in CaBaFe 3.5Zn0.5O7. Thus, \nin the low doping regime, the average Fe valency  of the ferri te is clearly the governing factor \nthat controls  the magnetic state and the strength of the magnetic interaction.  The degree of \ndisorder here plays a relatively minor role.  \n \n \n \n  5 3.1.2.  High  doping regime: YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 \n \nAn increase of  the doping concentration of the diamagnetic substituent subsequently \nleads to the appearance of spin glass behaviour in both YBaFe 4-xZnxO7 as well as CaBaFe 4-\nxZnxO7. This can be seen in Fig. 4 , where we have shown the d. c. M vs T  curves  for \nYBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7. In contrast to the sharp rise in the M(T) curves below \nthe ordering temperatures seen in the low doped samples (x = 0.5), the M(T) curves of the \nhigher doped samples (x = 1.5) show a more gradual rise in the magnetization with the \ndecrease in temperature terminating in a cusp -like behaviour at low temperature. The \ntemperature at which the ZFC M(T) curves of the two samples show cusps are T cusp = 35.5 K \nand 40.5 K for YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectively.  A. C. susceptib ility \nmeasurements \n '(T) of the two samples measured using different frequencies in the range 10 \nHz – 10 kHz  (Fig. 5 ) show that both samples show similar frequency dependent peaks with  Tg \n= 45 K and 50 K for YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectivel y. The two samples \nalso have very similar narrow S – shaped loops with almost the same values of the coercivity \nand remanence magnetization  (see inset of Fig. 4). Moreover , the important point to note here \nis that the average Fe valency of these two compou nds is very different (Fe val = 2.40 and 2.80 \nfor YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectively).  In spite of such a large difference in \nthe Fe valency, the two compounds behave strikingly similar to each other. We explain this \nresult as the dominating role of the cationic disorder in samples where the degree of disorder \nis large. Thus, for higher doped samples, the degree of disorder plays the deciding factor in \nstabilizing the magnetic ground state, and the Fe valency plays only a minor role.  \n Remarkab ly, the magnetic behaviour of these highly doped hexagonal phases is very \nsimilar to that of the virgin cubic sample YBaFe 4O7, which was shown to be a spin glass, with \na rather similar T g ≈ 50 K [10]. Thus, a high degree of cation disordering in the hexago nal \nphase has an effect similar to the pure geometric frustration of the tetrahedral iron sublattice \n(Fig. 2 a ) of the cubic phase i.e. it allows a complete magnetic frustration to be reached.  \n \n3.2. Li substitution in YBaFe 4O7 and CaBaFe 4O7 \n \n In the previo us sections, we have shown that the magnetic properties of hexagonal Zn \nsubstituted “114” ferrites  YBaFe 4O7 and CaBaFe 4O7 can be tuned on the basis of two doping \nregions – the low doping regime and the high doping regime. While the average Fe valency \nplays  a crucial factor in determining the magnetic state of the oxide in the low doping regime,  6 the degree of cationic disorder becomes the dominant factor controlling the magnetic state in \nthe regime of high doping.  In order to confirm that this effect is not specific to Zn, but is, in \nfact, a rather general phenomenon, we carry out similar studies on YBaFe 4O7 with a second \ndiamagnetic substituent, Li+, and we compare the magnetic behaviour of YBaFe 4-xLixO7 with \nthat previously observed for CaBaFe 4-xLixO7 [16].  \n Due to its univalent character, lithium has the advantage of inducing an average iron \nvalency which is different from that of the zinc compounds for the same substitution rate, \nthereby allowing the relative effects of  the Fe valence and the cationic dis order to be \ncompared further. The size of Li+, which is similar to that of Zn2+, and its ability to adopt the \ntetrahedral coordination , are favourable to such a substitution. In contrast to the case of zinc \nsubstitution, the maximum amount of lithium that was substituted was limited by the \nexperimental conditions of synthesis. Indeed, working in sealed tubes, and using the \nprecursors Y 2O3, BaFe 2O4, LiFeO 2, Fe 2O3 and Fe in order to avoid any reaction with the \nsupport and any Li 2O volatization, only the compo sitions YBaFe 4-xLixO7 with x \n  0.75 could  \nbe prepared , keeping the oxygen and lithium stoichiometry intact.  \n The first important point deals with the fact that lithium substitution, like zinc, \nstabilizes the hexagonal symmetry at the cost of the cubic phas e. Quite remarkably, a smaller \nlithium content, x = 0.30 only, is sufficient to stabilize the hexagonal form, instead of x = 0.40 \nfor Zn . In any case, the cell parameters of the Li substituted yttrium phase vary only slightly \nwith composition from a = 6.30 74 (1)  Å, c = 10.35 86 (2)  Å for x = 0.30 to a = 6.293 2 (1)  Å, c \n= 10.3 104 (1)  Å for x = 0.75.  \n The magnetic study of the compounds YBaFe 4-xLixO7 clearly show s that for the \nmaximum substitution rate i.e. x = 0.75, the complete spin glass behaviour cannot be  reached. \nThus, for the sake of comparison with other substituted phases, we discuss, in this section, the \nresults obtained in the low doping regime.  \n  \n3.2.1.  YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7 \n \n Li+ substitution in hexagonal CaBaFe 4O7 has been studied b y us before [16]. For the \npurpose of comparison with Li+ in YBaFe 4O7, we choose the samples with x = 0.3 from the \ntwo series. This is because, as we have stated before, x = 0.3 is the minimum amount of Li+ \nrequired to stabilize monophasic YBaFe 4-xLixO7 with the hexagonal symmetry.  \n The d. c. magnetization results  of YBaFe 3.7Li0.3O7 have been shown in Fig. 6 . In \naccordance with its hexagonal symmetry, YBaFe 3.7Li0.3O7 is ferrimagnetic. However,  7 considering the fact that the average Fe valency in YBaFe 3.7Li0.3O7 (Fe val = 2.35) is much less \nthan that in Ca BaFe 3.7Li0.3O7 (Fe val = 2.62), the ferrimagnetic interaction in YBaFe 3.7Li0.3O7 \nshould be weaker than that in Ca BaFe 3.7Li0.3O7. This is indeed the case as can be seen from  \nTable 1 , where we have compared the va lues of T C, M FC(T=5K)  and H C(T=5K)  for the two \nsamples.  The values for Ca BaFe 3.7Li0.3O7 in Table 1  have been quoted from reference 16.  \nCaBaFe 3.7Li0.3O7 has higher values of T C, M FC(T=5K)  and H C(T=5K)  than those of \nYBaFe 3.7Li0.3O7 , showing that the ferrim agnetic interaction is stronger and the magnetic \nfrustration is weaker in CaBaFe 3.7Li0.3O7.  \n \n3.2.2.  CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7 \n \n In all the above cases, we have compared samples with different average Fe valenc ies, \nand shown that in the regime of low doping, the Fe valency controls the magnetic properties \nof the oxide, while in the regime of high doping, the degree of cationic diso rder is the \ndeciding factor.  We have seen that in the low doping regime, samples with the same degree of \ndisorder, b ut with different Fe valency behave differently vis – à – vis their magnetic \nproperties. In our final section, we investigate what happens in the reverse case i.e. for \nsamples with different degree of disorder (but well within the low doping regime), but  having  \nthe same average Fe valency. For this purpose , we compare the  two samples CaBaFe 3.8Li0.2O7 \nand CaBaFe 3.5Zn0.5O7 which fall in the low doping regime, and have almost the same average \nFe valency (Fe val = 2.57 and 2.58 for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7 respectively).   \nThe d. c. magnetization results of CaBaFe 3.5Zn0.5O7 have been shown in Fig. 7. In accordance \nwith its hexagonal symmetry, CaBaFe 3.5Zn0.5O7 shows a sharp ferrimagnetic transition, and a \nlarge square hysteresis loop. However, what is more striking is the almost exact one – to – one \ncorrespondence of the magnetic parameters obtained for the two samples CaBaFe 3.8Li0.2O7 \nand CaBaFe 3.5Zn0.5O7. These are listed in Table 2 .  The values for CaBaFe 3.8Li0.2O7 have \nbeen quoted from reference 16.  This remarkable correspondence in the magnetic properties of \nthe two samples with the same average Fe valency proves unambiguously that in the low \ndoping regime the Fe valency is indeed the main factor controlling the magnetic state of the \noxide.   \n \n \n \n  8 4. Conclusion  \n \n This study shows the great impact of the substitution of diamagnetic cations such as \nzinc or lithium for iron upon the competition between ferrimagnetism and magnetic frustration \nin “114’ ferrites. The first effect is structural – it is seen t hat the substitution of these cations \nfor iron in the cubic phase, YBaFe 4O7, leads to a hexagonal symmetry, and consequently \ndestroys the 3 D geometric frustration at the benefit of a competition between a 2 D geometric \nfrustra tion and 1 D magnetic orderin g, thereby inducing ferrimagnetism. The second effect, \nobserved in the hexagonal phases such as CaBaFe 4O7, modifies the competition between the 2 \nD frustration and the 1 D magnetic ordering in two different ways, depending on the \nsubstituent concentration.  For low doping values, the modification of the average iron valency \nthat is induced by this substitution dominates the magnetism of these compounds leading to a \ndecrease of ferrimagnetism at the benefit of 2 D magnetic frustration, whereas in the high \ndoping regime, the disordering of the cations dominates, inducing a complete magnetic \nfrustration, irrespective of the iron valency. The nature of the ferrimagnetism of these ferrites, \ntill date, has not been completely elucidated, and in particular, it is st ill not known whether the \niron spins lie in plane or out of the triangular planes, so that a vast field is still open for the \ninvestigation and understanding of this new type of magnetic frustration.  \n \n5. Acknowledgements  \n \nWe acknowledge the CNRS and the Co nseil Regional of Basse Normandie for financial \nsupport in the frame of Emergence Program  and N°10P01391 . V. P. acknowledges support by \nthe ANR -09-JCJC -0017 -01 (Ref: JC09_442369).  \n \n \n \n \n \n \n \n \n \n \n  9 6. References  \n  \n      [1]    Walz , F.: J. Phys. Condens. Matter 14, R285  (2002)   \n      [2]    Verwey , E. J. W.,  Haayman, P. W.: Physica  8, 979  (1941 ) \n      [3]    Goodenough,  J. B. Magnetism and the Chemical Bond, Interscience Monographs on  \n               Chemistry , Vol. 1 5 Huntington, New York , 1976 )  \n      [4]    Greedan,  J. E.:  J. Alloys and Compounds  408 – 412, 444  (2006 ) \n      [5]    Muraoka,  Y., Tabata , H., Kawai,  T.: J. Appl. Phys. 88, 7223  (2000)  \n      [6]    Delgado,  G. E.,  Sagredo , V., Bolzoni,  F.: Cryst. Res. Technol . 43, 141  (2008)  \n      [7]    Valldor , M., Andersson,  M.: Solid State Sciences   4, 923  (2002 ) \n      [8]    Valldor,  M.: J. Phys.: Condens. Matter . 16, 9209  (2004)  \n      [9]    Raveau, B., Caignaert,  V., Pralong,  V., Pelloquin  D., Maignan,  A. : Chem. Mater. 20,  \n              6295  (2008)  \n      [10]   Caignaert,  V., Abakumov,  A. M., Pelloquin, D., Pralong,  V., Maignan,  A., Van   \n                Tendeloo , G., Raveau,  B.: Chem. Mater . 21, 1116  (2009)  \n      [11]   Pralong, V., Caignaert,  V.,  Maignan , A., Raveau,  B. : J. Mater. Chem . 19, 8335  \n                (2009)   \n      [12]   Chapon, L. C., Radaelli, P. G., Zheng , H., Mitchell, J. F.: Phys. Rev. B  74, 17240  \n               (2006)  \n      [13]   Manuel, P., Chapon,  L. C.,  Radaelli, P. G., Zheng , H., Mitchell, J. F.:  Phys. Rev.  \n                Lett. 103, 037202  (2009 ) \n      [14]   Caignaert, V., Pralong,  V., Hardy,  V., Ritter C., Raveau,  B.: Phys. Rev. B  81, 094417  \n                (2010 ) \n      [15]    Sarkar, T., Pralong,  V., Caignaert , V., Raveau,  B. : Chem. Mater . 22, 2885 (2010)  \n      [16]    Vijayanand hini, K., Simon, Ch., Pralong,  V., Caignaert V., Raveau,  B.: Phys. Rev. B  \n                79, 224407  (2009 ) \n \n \n \n \n \n \n \n  10 Figure Captions  \n \nFigure 1 : Structure of (a) cubic LnBaFe 4O7 and (b) hexagonal CaBaFe 4O7, built up of two \nsorts of layers of FeO 4 tetrahedra  called triangular (T) and kagomé (K).  \nFigure 2 : Schematic representation of the iron sublattice in (a) cubic LnBaFe 4O7 and (b) \nhexagonal CaBaFe 4O7. \nFigure 3 : MZFC(T) and M FC(T) curves for YBaFe 3.5Zn0.5O7 and Ca BaFe 3.5Zn0.5O7 measured at \nH = 0.3 T. The ins et shows the magnetization as a function of magnetic field at T = 5 K for \nthe two samples.  \nFigure 4 : MZFC(T) and M FC(T) curves for YBaFe 2.5Zn1.5O7 and Ca BaFe 2.5Zn1.5O7 measured at \nH = 0.3 T. The inset shows the magnetization as a function of magnetic field  at T = 5 K for \nthe two samples.  \nFigure 5 : Real (in -phase) component of a.c. susceptibilities  for (a) YBaFe 2.5Zn1.5O7 and (b) \nCaBaFe 2.5Zn1.5O7 as a function of temperature measured using a frequency range 10 Hz – 10 \nkHz.  \nFigure 6 : MZFC(T) and M FC(T) curves  for YBaFe 3.7Li0.3O7 measured at H = 0.3 T. The insets \n(a) show the magnetization as a function of magnetic field at T = 5 K and (b) dM/dT as a \nfunction of T for estimation of T C.  \nFigure 7 : MZFC(T) and M FC(T) curves for CaBaFe 3.5Zn0.5O7 measured at H = 0. 3 T. The insets \n(a) shows the plot of d. c. magnetic susceptibility as a function of temperature along with the \nCurie – Weiss fit, (b) the magnetization as a function of magnetic field at T = 5 K and (c) \ndM/dT as a function of T for estimation of T C. \n \nTabl e Captions  \n \nTable 1 : TC, M FC(T=5K)  and coercive field (H C) for YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7. \nTable 2 : T C, Curie -Weiss temperature (θ CW), effective paramagnetic moment (µ eff) and \ncoercive field (H C) for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7. \n \n \n \n \n \n  11 \n \n \nFig. 1. Structure of (a) cubic LnBaFe 4O7 and (b) hexagonal CaBaFe 4O7, built up of two sorts \nof layers of FeO 4 tetrahedra called triangular (T) and kagomé (K).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  12 \n  \nFig. 2 . Schematic representation of the iron sublattice in (a) cubic LnBaFe 4O7 and (b) \nhexagonal CaBaFe 4O7. \n \n \n \n \n  13 \n \n \nFig. 3. M ZFC(T) and M FC(T) curves for YBaFe 3.5Zn0.5O7 and Ca BaFe 3.5Zn0.5O7 measured at H \n= 0.3 T. The inset shows the magnetization as a function of magnetic field at T = 5 K for the \ntwo samples.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  14 \n \n \nFig. 4. M ZFC(T) and M FC(T) curves for YBaFe 2.5Zn1.5O7 and Ca BaFe 2.5Zn1.5O7 measured at H \n= 0.3 T. The inset s hows the magnetization as a function of magnetic field at T = 5 K for the \ntwo samples.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  15 \n \n \nFig. 5. Real (in -phase) component of a.c. susceptibilities  for (a) YBaFe 2.5Zn1.5O7 and (b) \nCaBaFe 2.5Zn1.5O7 as a function of temperature measured using a frequency range 10 Hz – 10 \nkHz.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  16 \n \n \nFig. 6. M ZFC(T) and M FC(T) curves for YBaFe 3.7Li0.3O7 measured at H = 0.3 T. The insets (a) \nshow the magnetization as a function of magnetic field at T = 5 K and (b) dM/dT as a \nfunction of T for estimation  of T C.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \n \nFig. 7. M ZFC(T) and M FC(T) curves for CaBaFe 3.5Zn0.5O7 measured at H = 0.3 T. The insets \n(a) shows the plot of d. c. magnetic susceptibility as a function of temperature along with the \nCurie – Weiss fit, (b) the magnetization as a  function of magnetic field at T = 5 K and (c) \ndM/dT as a function of T for estimation of T C.  \n \n \n \n \n \n \n \n \n \n \n \n \n  18 Table 1. TC, M FC(T=5K)  and coercive field (H C) for YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7. \n \n \nSample  \n  \nTC (K) \n  \nMFC(T=5K)   (µB/f.u.)  \n HC (at T = 5 K) (T)  \nYBaFe 3.7Li0.3O7  115.7  0.34 0.87 \nCaBaFe 3.7Li0.3O7 150.7  0.50 0.98 \n \nTable 2. T C, Curie -Weiss temperature (θ CW), effective paramagnetic moment (µ eff) and \ncoercive field (H C) for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7. \n \nSample  \n TC (K) \n θCW (K) \n µeff (µB/f.u.)  \n HC (at T = 5 K) \n(T) \nCaBaFe 3.8Li0.2O7 191.6  211.9  4.37 1.17 \nCaBaFe 3.5Zn0.5O7 203.0  218.8  4.05 1.12 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "2203.00818v1.Emergence_of_insulating_ferrimagnetism_and_perpendicular_magnetic_anisotropy_in_3d_5d_perovskite_oxide_composite_films_for_insulator_spintronic.pdf",        "content": " 1 Emergence of insulating ferr imagnetism and \nperpendicular magnetic anisotropy in 3d -5d \nperovskite oxide composite films for insulator \nspintronics  \n \nZeliang Ren1,2,3†, Bin Lao2,3†, Xuan Zheng2,3,4, Lei Liao5, Zengxing Lu2,3, Sheng Li2,3, \nYongjie Yang2,3, Bingshan Cao1,2,3, Lijie Wen2,3,6, Kenan Zhao2,3, Lifen Wang5, Xuedong  \nBai5, Xianfeng Hao6, Zhaoliang Liao7*, Zhiming Wang2,3,8* and Run -Wei Li2,3,8 \n \n1Nano Science and Technology Institute, University of Science and Technology of \nChina, Hefei 230026, Anhui, China  \n2CAS Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of Materials \nTechnology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China  \n3Zhejiang Province Key Laboratory of Magnetic Materials and Application Technolo gy, \nNingbo Institute of Materials Technology and Engineering, Chinese Academy of \nSciences, Ningbo 315201, China  \n4New Materials Institute, University of Nottingham Ningbo China, Ningbo 315100, \nChina  \n5Beijing National Laboratory for Condensed Matter Physics,  Institute of Physics, \nChinese Academy of Sciences, Beijing  100190, China  \n6Key Laboratory of Applied Chemistry, College of Environmental and Chemical \nEngineering, Yanshan University, Qinhuangdao 066004, China  \n7National Synchrotron Radiation Laboratory, Un iversity of Science and Technology of \nChina, Hefei 230026, Anhui, China  \n8Center of Materials Science and Optoelectronics Engineering, University of Chinese \nAcademy of Sciences, Beijing 100049, China  \n   2  \nAbstract : Magnetic insulators with strong perpendicular magnetic anisotropy (PMA) \nplay a key role in exploring pure spin current phenomena and developing ultralow -\ndissipation spintronic devices , thereby it is highly desirable to develop new material \nplatforms. Her e we report epitaxial growth of La 2/3Sr1/3MnO 3 (LSMO) -SrIrO 3 (SIO) \ncomposite oxide films (LSMIO) with different crystalline orientations fabricated by \nsequential two -target ablation process using pulsed laser deposition. The LSMIO films \nexhibit high crysta lline quality with homogeneous mixture of LSMO and SIO at atomic \nlevel. Ferr imagnetic and insulating transport characteristics are observed, with the \ntemperature -dependent electric resistivity well fitted by Mott variable -range -hopping \nmodel. Moreover, the  LSMIO films show strong PMA. Through further constructing \nall perovskite oxide heterostructures of the ferr imagnetic insulator LSMIO and a strong \nspin-orbital coupled SIO layer, pronounced spin Hall magnetoresistance (SMR) and \nspin Hall -like anomalous Hall effect (SH -AHE) were observed. These results illustrate \nthe potential application of the ferr imagnetic insulator LSMIO in developing all -oxide \nultralow -dissipation spintronic  devices . \n \nKeywords: Perovskite oxide, magnetic insulator, perpendicular magneti c anisotropy, \nspin hall magnetoresistance,  spintronics  \n   3 INTRODUCTION  \nIn order to satisfy the ever -increasing demanding of information storage capacity and \nprocessing speed, reducing the energy consumption of electronic devices becomes \nmore and more important in the microelectronics industry. In contrast to conventional \nmicroelectronics relying only the charge of electrons, spintronics use the additional spin \nof electrons to encode, st ore, process and transmit data, which is very promising to \nbring new capabilities to microelectronic devices1-3. Esp ecially, spintronics based on \nferromag netic/ferrimagnetic magnetic insulators  (FMI)  has attracted tremendous \nattention for developing ultralow -dissipation devices. The local magnetic moments in \nFMI act as ideal media for pure spin angular momentum  propagat ing while the thermal \nconsumptions such as Joule heating can be efficiently avoided by suppressing the \nmoving  of electrons4-6. Additionally, magnets with perpendicular magnetic anisotropy \n(PMA), where the magnetic easy axis preferentially points toward normal to the surface, \nis essentially required to reduce the devices size while maintaining high thermal \nstability7-11. Combination of FMI and PMA further provides exciting functionalities for \nlowering current threshold and enhancing mobilit y of domain wall displacement, as \nwell as achieving long decay length during spin -wave propagation12. Therefore, the \ndevelopment o f FMI with PMA is much meaningful for the development of high -\nperformance spintronic devices.  \n \nMagnetic insulators  are frequently found in transition metal oxides with various crystal \nstructures, such as rocksalt, perovskite, spinel, and garnet. Among them , perovskite \nABO 3 oxides are particularly promising candidates to achieve high -quality all -oxides \nspintronic devices because of their versatile functionalities, including spin source \ngenerator, spin current detector and so on13, 14. However, magnetic insulators  in \nperovskites often favor antiferromagneti sm instead of ferromagnetic ordering, due to \nthe super -exchange interaction between neighboring magnetic ions according to \nGoodenough -Kanamori -Anderson rules15. Moreover, FMI with strong PMA in \nperovskite oxides are rarely reported. Manganite La 1-xSrxMnO 3 (LSMO), as an \narchetypal colossal magnetoresistance material s, have been widely studied on account \nof its rich phase diagrams where multiple electric and magnetic properties are present16, \n17. Previous studies had demonstrated that PMA can be obtained in LSMO films and \nheterostructures through engineering the lattice and orbital degrees of freedom18-20. \nRecently, strong PMA can be triggered in LSMO when interfacing with strong spin -\norbital coupled SrIrO 3 (SIO)  21, 22. However, the LSMO maintained metallic  in most \ncases, unless large  strain was imposed 23-25. Through A - or/and B -site cation \nsubstitutions in ABO 3 the spin polarization of band structure can be directly controlled ,  4 leading to modif ied exchange interactions and associated magnetic and electrical \nground st ates26. Thanks to the advances in epitaxial synthesis techniques, composite \nfilms have emerged as powerful platforms for realizing FMI and strong PMA in LSMO -\nbased perovskite oxides since deliberating control of specific atoms can be realized27. \nGiven  these reasons, emergent FMI with strong PMA can be obtaine d via engineering \ncation substitution in LSMO and SIO, which enables to explore pure spin current \nphenomena in all -oxide heterostructures.  \n \nIn this letter, we reported a facile method to fabricate LSMO -SIO (LSMIO) composite \noxide films that are mixed homog enously at atomic level. The LSMIO films are FMI \nwith PMA, whose temperature -dependent resistivity was fitted well by Mott variable -\nrange -hopping model. By further constructing heterostructures of FMI LSMIO and \nstrong spin -orbital coupled SIO layers, we ex plore d pure spin current phenomena and \nobserved pronounced spin Hall magnetoresistance (SMR) and spin Hall -like anomalous \nHall effect (SH -AHE) . These results demonstrate that FMI LSMIO is promising for \nexploring insulator spintronics in all oxides heterost ructures and developing low -\ndissipation spintronic devices.  \n \nRESULTS AND DISCUSSION  \nFigure 1(a) shows the schematics of sequential two -target deposition process of LSMIO \nfilms using pulsed laser deposition (PLD) . During the growth, the LSMO and SIO \ntargets  were rotated repeatedly and periodically. In each repeat, sub -monolayer 0.3 unit \ncell (u.c.) LSMO and 0.2 u.c. SIO were deposited, ensuring homogeneous mixture of \ntwo ceramic targets at the atomic scale. The LSMIO films have a nominal stoichiometry \nof La 0.4Sr0.6Mn 0.6Ir0.4O3. The crystalline orientation of LSMIO films can be controlled \nby changing substrate orientation. The crystalline structures were characterized by X-\nRay diffraction ( XRD) 2θ-θ scans as shown in Fig. 1(b). The peaks of LSMIO films are \nlabeled and are locate d on the left side of SrTiO 3 (STO) substrates  closely . This result \nindicates that the films are slightly compressed  (<0.1%)  and of high -quality single \ncrystal without impurity phases. To further cha racterize the interface quality and \nstructural homogeneity, scanning transmission electron microscopy (STEM ) \nmeasurements on the (001) -oriented LSMIO films, as shown in Figs. 1(c) -1(h), were \nperformed. The sharp contrast across the LSMIO and STO substrates  in Fig. 1(c), \nsupplemented by a high -angle annular dark -field (HAADF) STEM image of the \nLSMIO films in Fig. 1(d), indicates high crystalline quality of the films. Figures 1(e) -\n(h) show the corresponding energy dispersive X -ray spectroscopy ( EDS) mappings of \nLa, Sr, Mn and Ir elements. All measured elements are distributed homogeneously at \natomic level in the composite films without observable clustering regions.   5  \nTo investigate the magnetic properties of LSMIO composite films, we measured \nmagnetic hystere sis ( M-H) loops and temperature -dependent magnetization ( M-T) \ncurves. Figures 2(a) -2(c) display the M-H loops observed with the magnetic field \napplied parallel ( H⊥c) and perpendicular ( H∥c) to the film plane for differently \noriented LSMIO composite films. The coercive field is found to be 0.67 T, which is \nmuch larger than that of soft ferromagnetic LSMO films. On the other hand, we note \nthat the saturation moments are determined to be around  2.4 μ B/u.c., which are \nconsiderably smaller than that in LSMO films28. The reduction of the saturation \nmoments  can be explained by the antiferromagnetic coupling in the Mn -O-Ir bonds as \nevidenced from the first -principle calculations shown below, which leads to the \nferrimagnetic ground state in LSMIO composite films.  Figure 2(d) show the \ncorresponding magnetization M as a function of temperature. T he Curie temperature \n(TC) becomes lower for all the LSMIO films  as compared to pure LSMO bulk material. \nIntriguingly, by comparing the magnetic hysteresis loops measured und er field parallel \nand perpendicular to films respectively, it is found that all easy magnetization axes are \nperpendicular to the film plane for all LSMIO films regardless of the crystal orientation \nwhile the magnetic anisotropy energy are 1.21 ×106, 1.55 ×106 and 1.56 ×106 erg/cm3 \nrespectively , which are comparable with the PMA reported in LSMO/SIO \nheterostructures22. Therefore, it is feasible to obtain perovskite FMI with strong PMA \nin LSMIO composite films.  Considering the negligible compressive strain imposed by \nthe SrTiO 3 substrates, the observed strong PMA  is not due to magneto -elastic \nanisotropy as been observed previously in LSMO films or heterostructures when under \nlarge compressive strain24, 25. The strong PMA is due to the magnetic crystalline \nanisotropy originated from strong spin -orbital coupling and Mn -O-Ir bonding21, 22, 29 \nTo further understand the electrical properties of LSMIO composite films, we have \nperformed the temperature -dependent electric resistivity measurements. A s shown in \nFig. 2(e), the resistivity increases monotonically as the temperature decreases, \nindicating that all the LSMIO composite films are insulators . This is in sharp contrast \nto individual LSMO and SIO films which are metallic and semi -metallic in the ir bulk \ncounterparts, respectively. To illustrate the origin of the emergent FMI state, we have \nfitted the resistivity data with thermal activation, Efros –Shklovskii variable -range -\nhopping (ES -VRH) and Mott variable -range -hopping (Mott -VRH) models, as show n \nin Figs. S3 (b) -(c) respectively. Figure 2(f) depicts the linear fitting curves using Mott -\nVRH model, which agrees well to the measured resistivity. Here, the Mott -VRH model \nis expressed as the following equation:   6 𝜌(𝑇)=𝜌(0)𝑒𝑥𝑝(𝑇𝑀𝑇⁄)(1𝑑+1⁄)\n \nWhere 𝜌0 is resistance coefficient, 𝑇𝑀 is characteristic temperature and the exponent \n𝑑 depends on the VRH mechanism. For the Mott VRH conduction, the exponent d is \ndimension dependent and has a value of 𝑑=3 in a three -dimensional system30-32. \nFrom the fitted characteristic temperature 𝑇𝑀, we can infer the Mott  hopping energy \n(𝐸𝑀) as: \n𝐸𝑀=1\n4𝑘𝐵𝑇(𝑇𝑀𝑇⁄)14⁄\n \nThe fitted hopping energy for (001) -, (110) - and (111) -oriented LSMIO composite \nfilms are determined to be 72.2, 93.7 and 117.2 meV at room temperature (300 K), \nrespectively.  The Mott -VRH model describes tha t the localized electron at the Fermi \nlevel moves to another localized state in an optimum hoping distance which is \ndetermined by the tradeoff between the lowest energy differences and the shortest \nhopping distances in a disordered system33. Given that the Mn -O-Ir bonds are \ndominated in  the LSMIO composite oxide films, the Mott -VRH behavior suggests that \nthe carriers hopping through the Mn -O-Ir bonds are suppressed and form localized \nstates. This situation is in sharp contrast with metallic Mn -O-Mn bonds in LSMO films, \nwhere the electron s can itinerantly hop between neighboring Mn3+ and Mn4+ ions \nthrough the double -exchange mechanism17, 34. \n \nTo gain fur ther insight into the electronic and magnetic properties of LSMIO \nnanocomposite films, we have performed first -principles calculations based on the \ndouble perovskite model LaSrMnIrO 6 (Fig. 3(a)). From the total energy results of the \ndistinct magnetic solutions ( schematically shown in Fig . 4S and Tab. S2 in \nSupplementary Materials ), we found that t he ferrimagnetic, uncompensated \nantiferromagnetic alignment, due to the unequal magnetic moments on Mn and Ir sites, \nis the most stable state for LSMIO, affir ming  the experimental observations.  Moreover, \nthe GGA+U schemes prefer quite stable magnetic solution with magnetic moments of \nMn (~ 4.3 µB) and Ir (~ -0.9 µB) ions, accompanied by considerable induced magnetic \nmoments ( ~ -0.1 µB) for all the surrounding O2- anions as demonstrated  in Fig . 3(b).  \nUnexpected, the magnetic moment at Mn sites is slightly larger than the ideal spin -only \n4 μB for Mn3+ cations, suggesting the special magnetic coupling mechanism behind.  \nFigure 3(c) shows that t he ferrimagnetic ground state is insulating with a band gap of \n0.2 eV, which agrees well with the experimental temperature -dependent electric  7 resistivity data. The antiferromagnetic coupling and unexpected large magnetic \nmoment on Mn3+ site can be interpre ted in the framework of superexchange interactions \nthrough the virtual hopping bridged by O 2 p spin-up electrons, as illustrated  in Fig. 3(d), \nwhich is derived from the partial density of states and spin density plots. We can see \nthat the virtual hopping f rom Ir 𝑡2𝑔↑to the empty Mn 𝑒𝑔↑via the bridged O  2𝑝↑states \naccounts for the antiferromagnetic coupling and the abnormal large computed Mn3+ \nmagnetic moment as well as the exclusively negative sign of the induced O2- magnetic \nmoments.  \n \nFinally, t o explore the application of FMI LSMIO in pure spin current phenomenon, \nwe fabricated a LSMIO/SIO heterostructure and performed temperature -dependent \nmagnetoresistance measurements. Figures 4(a)-4(b) show schematic structure of Hall \nbar devices and measure ment geometry. As illustrated in Fig. 4(a), the heterostructure \nconsists of the FMI LSMIO and a strong spin -orbital coupled SIO layer. The SIO layer \ncan efficiently convert the charge current into spin current35, 36, so a pronounced SMR \ncan be expected due to the asymmetry between absorption and reflection of the spin \ncurrent at the LSMIO/SIO interface. Figure 4(c) depi cts the angle dependent \nmagnetoresistance measurements where external field rotates in the y -z plane around \nangle β. It is necessary to distinguish the difference between SMR and anisotropic \nmagnetoresistance (AMR). SMR signal is known to be described as:  \n𝜌𝑥𝑥=𝜌0−∆ρ𝑚𝑦2 \nwhile AMR signal is described as:  \n𝜌𝑥𝑥=𝜌0−∆ρ𝑚𝑥2 \nIn these two formulas, the 𝜌0 is constant resistivity offset, ∆ρ is amplitude of the \nresistivity change as a function of the magnetization orientation and m i (i = x, y) is the \nproject ions of the magnetization orientation unit vector along x and y axis in the \ncoordinate system37-39. Note that 𝜌𝑥𝑥=𝑥𝑥𝑤𝑡/𝐽𝑞𝑙 and 𝜌𝑥𝑦=𝑥𝑦𝑡/𝐽𝑞, where w is \nchannel width ( w = 10 um), the l is the length of the separated voltage contacts ( l = 50 \num) and the t is the thickness of SIO layer ( t = 5 nm). During the measurement, the \nexternal field was fixed to 4 T which is much larger than the coercive  field of LSMIO \n(0.67 T) in order to obtain collinear magnetization. Figure 4(c) shows that the resistivity \ndata exhibit the cos2β dependence. This is in consistent with the SMR signal since my \nhas cosβ dependence, while AMR keep constant in angle β depend ent measurements  8 since mx equals to 0. These measurements demonstrate that the SMR can be detected in \nall perovskite oxide heterostructures.  \nFigure 4(d) shows the Hall resistance measurements with the magnetic field applied \nperpendicular to the sample and varied from -4 T to 4 T. By subtracting a linear \ncontribution from ordinary Hall effect, a clear hysteresis loop curve is visible, which \nresembles the AHE. Figure 4(e) shows the temperature -dependent Hall resistance after \nsubtracting a linear term. The coe rcive field inferred from the Hall hysteresis loop is \nabout 0.67 T at 10 K, which is similar as that of LSMIO films measured by SQUID \nshown in Fig. 2(a). Moreover, the coercive field decreases as a function of temperature \nand becomes invisible at above 120  K. These results demonstrate that the evolution of \nAHE with temperature resembles that of the magnetization versus temperature curve as \nshown in Fig. 2(d), indicating that the observed AHE is closely related to the PMA of \nLSMIO composite films. We note th at interfacial magnetism and AHE has been \nintensively studied in LSMO/SIO superlattices and heterostructures21, 40 -46. Our \nobserved AHE is quite different from that the AHE reported in ref.  41, where a large \nAHE is originated from SIO layers due to magnetic proximity effect. Our AHE stem s \nfrom the reflection of the spin current at the interface where an out -of-plane component \n(mz) of the LSMIO magnetization rotates the spin orientation of th e spin current and \ngenerates a transverse voltage via the inverse spin Hall effect (ISHE). Such AHE is \ntermed as SH -AHE47, 48. Therefore, both SMR and SH -AHE indicate occurrence of the \nspin transfer at the interface of the conduct or and ferrimagnetic insulator. The spin \ncurrent can be injected into insulators which gives a possibility of transmitting spin \ninformation through an insulator. Such behavior would have an advantage of ultralow \npower dissipation due to absence of Joule he ating.  \n \nCONCLUSION  \nIn conclusion, we have synthesized LSMIO composite oxide films with different \ncrystalline orientations by sequential two -target deposition process. The LSMIO \ncomposite films exhibit high crystalline quality with homogeneous mixture of L SMO \nand SIO at atomic level. The electric and magnetic measurements, complemented by \nfirst-principle calculations, indicate that the LSMIO composite films are FMI with \nstrong PMA . The temperature -dependent electric resistivity is well described by Mott -\nVRH model. Furthermore, t hrough fabricating heterostructures of the FMI LSMIO and \na strong spin -orbital coupled SIO layer, we observed pronounced SMR and SH -AHE. \nThese results illustra te the potential application of LSMIO in developing all -oxide \nultralow -dissipation spintronic  devices . Given the scarcity of perovskite FMI with \nstrong PMA, our result provides a promising real material candidate to boost the \ndevelopment of insulating spin tronics devices.   9  \nEXPRIMENT AL SECTION  \nSample  preparation : LSMIO composite films were synthesized by PLD, equipped with \nKrF (λ=248 nm) excimer laser and high reflection high energy electron diffraction \n(RHEED). Prior to the film deposition, (001) -, (110) - and (111) -oriented STO \nsubstrates were etched by NH 4F-buffered hydrofluoric acid followed by annealing at \n1100 ℃ for 90 minutes. During films deposition, the temperature of substrates, oxygen \npartial pressure and laser fluence were set to 700 ℃, 0.1 mbar an d 1.5 J/cm2 at the \nfrequency of 2 Hz, respectively. The film growth was monitored by RHEED and the \nthickness was determined to be 12 nm.  \nSample c haracterizations:  The quality and structure of the films was characterized by \nXRD (Bruker), STEM and EDS (JEOL  Grand ARM 300), respectively. The magnetic \nand electric properties were measured by Quantum Design Magnetic Property \nMeasurement System (MPMS -SQUID, Quantum Design) and Physical Property \nMeasurement System (PPMS, Quantum Design). Further investigations of  spin-related \nSMR and AHE were performed on Hall bar devices composed of LSMIO and SIO using \na home -build low -temperature and high -intensity magnetic field magneto -electric \nsystem. The LSMIO/SIO heterostructures were prepared by PLD as the same condition \nwith LSMIO films. The as grown (001) -oriented LSMIO/SIO heterostructure was \npatterned into Hall bars by photolithography and Ar ion etching.  \nFirst -principle calculations: In order to describe the structure of LSMIO composite \nfilms, we constructed the canoni cal perovskite -type model within 2× 2× 2 supercell \n(La 4Sr4Mn 4Ir4O24, 40 atoms).  The spin-polarized density functional theory (DFT) \ncalculations are achieved with the plane wave basis set as implemented in the Vienna \nab initio  simulation package49, 50. For the exchange correlation functional, the \ngeneralized gradient approximation (GGA) is used according to the Perdew -Burke -\nErnzerhof scheme51, and GGA+U approaches52, 53 are also performed for including the \nstatic electron correlation, with the Hubbard U = 4 eV (2 eV) and Hund J = 0.8 eV (0.4 \neV) for Mn  3d (Ir 5d) states. A  plane -wave energy cutoff of 400 eV was employed . The \nk-point meshes over the total Brillouin zone were sampled by 4×4×4 and 6 ×6×6 grids \nconstructed according to the Monkhorst -Pack scheme54, 55  for structural relaxations and \nelectronic calculations, respectively . We optimized the lattice vectors, volumes, as well \nas internal atomic with the relaxation terminated once the total energies and residual \natomic forces converged to < 1× 10-5 eV and < |0.01| eV/Å, respectively.  \n \n◼ AUTHOR INFORMATION  \nCorresponding Author   10 zliao@ustc.edu.cn . \nzhiming.wang@nimte.ac.cn . \nAuthor contribution  \n†Z.R. and B.L. contributed equally to this work . Z.W. designed the experiments. Z.R. \nperformed sample  growth. Structure characterization, magnetic and transport \nmeasurements were performed by Z.R., B.L., X.Z., Z.L., S.L., B.C., L.W., and K.Z. \nL.L.,  L.W.  and X.B.  performed STEM measurements. Z.R. and B.L. performed data \nanalysis. X.H. performed first -principle calculations and data analysis. Z.R., B.L., X.H., \nZ.L. and Z.W. wrote the manuscript. All authors participated in discussions and \napproved the submitted manuscript.  \n \n◼ ACKNOWLEDGMENTS  \nThis work was supported by the National Key  Research and Development Program of \nChina (Nos. 2017YFA0303600, 2019YFA0307800), the National Natural Science \nFoundation of China (Nos. 12174406, U1832102, 11874367, 51931011, 51902322), \nthe Key Research Program of Frontier Sciences, Chinese Academy of Sc iences (No. \nZDBS -LY-SLH008), the Thousand Young Talents Program of China, K.C.Wong \nEducation Foundation  (GJTD -2020 -11), the 3315 Program of Ningbo, the Natural \nScience Foundation of Zhejiang province of China (No. LR20A040001), the Beijing \nNational Laborat ory for Condensed Matter Physics.  \n \n◼ REFERENCES  \n1. Bader, S. D.; Parkin, S. S. P., Spintronics. Annu. Rev. Condens. Matter Phys. 2010,  1, 71. \n2. Shiraishi, M.; Ikoma, T., Molecular spintronics. Physica E Low Dimens. Syst. Nanostruct. 2011,  43, \n1295.  \n3. 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(a) -(c) Magnetic hysteresis loops measured with the magnetic field H applied along in -plane \n(black) and out -of-plane (red) for LSMIO composite films with different crystalline orientations. (d) -(e) \nThe corresponding magnetization M and electric r esistivity  as a function of temperature T for different \noriented LSMIO composite films. (f) The resistivity data are plotted with lnR~1/T^[1/(d+1)] (d = 3) to \nfit with the three -dimensional Mott -VRH model.  \n \n \n 16  \nFigure 3. (a) Double perovskite structure of L aSrMnIrO 6, in which the Mn atoms (medium purple spheres) \nand the Ir atoms (medium brown ones) are surrounded by O 6 octahedron denoted by small red spheres, \nwhile the light (dark) green spheres are Sr (La) atoms. (b) Isosurfaces of the spin density (0.02 e/ Å2) for \ndouble double perovskite LaSrMnIrO 6, within the ferrimagnetic magnetic ground state. Yellow and cyan \ncolor denote spin up and spin down, respectively. (c) Total and partial density of states of Mn -3d and Ir -\n5d orbitals for LaSrMnIrO 6. Majority and minority spin are presented in the top and bottom channels, \nrespectively. (d) Schematic representation of the superexchange interaction through the Mn -O-Ir bonds. \nThe virtual electron hopping results into the antiferromagnetic coupling between Mn -O-Ir bond s. \n \nFigure 4. (a) Illustration of the SMR in LSMIO/SIO heterostructure. (b) Schematic sample structure and \nmeasurement geometry. (c) The angle dependent SMR signal. The inset shows the measurement \ngeometry with external field rotating in the y -z plane around angle β . (d) Transverse Hall resistivity ρxy \nmeasured with the magnetic field applied perpendicular to the film plane; (e) Temperature dependent \nAHE sign  \n  \n 17 Suppl emental Informat ion \n1. Structural characterization of LSMIO and LSMIO/SIO heterostructure.  \n \n \n \nFigure S1.  XRD scans of LSMIO and LSMIO/SIO heterostructure.  \n \nAs shown in Fig. S1, the peak of SIO and LSMIO can be seen clearly in LSMIO/SIO \nheterostructure while there is only one peak of LMSIO in single LSMIO layer which \nmeans there is only one single crystal of LSMO -SIO composite films.  \n \n2. Structural characterization of LSMIO composite films by STEM  \n \n 18  \nFigure S2 (a) High -resolution HAADF -STEM image of the LSMIO films (b -f) EDS \nmaps of La, Mn, Sr,  O and Ir for LSMIO films, Scale bar:2nm.  \n \nWe can see the h omogen eously mixture of atoms in LSMIO films in Figs S2(b) -(f). \n \n \n3. The fittings  of measured resistivity of LSMIO.  \n \n \n \n 19  \nFigure S 3. (a) Temperature -dependent electric resistivity of LSMIO composite films with different \ncrystalline orientations. (b) -(d) The resistivity data are plotted with lnR~1/T ^[1/(d+1) ] (d = 0,1,3) \nto fitted with the thermal activation, Efros –Shklovskii -VRH and Mott -VRH model respectively.   \n \nIn this picture, we plot the resistivity with different physical model with thermal \nactivation (d=0), Efros–Shklovskii -VRH  (d=1) and 3 -dimentional Mott -VRH (d=3). \nHowever, the nonlinearity of the curves in the Fig . S3(b)-(c) indicate that the thermal \nactivation model cannot well fit the resistivities of LSMIO while only the Mott -VRH \nfits well to the measured resistivity which can be seen in Fig . S3(d).  \n \n \n4. DFT calculation s. \n \n \nIn order to describe the structure of LSMIO composit e films, we constructe d the \ncanonical perovskite -type model within 2× 2× 2  supercell (La 4Sr4Mn 4Ir4O24, 40 atoms), \nwith aid of the experimental characterizations, which can be pictured as a framework \nconsisting of alternative corners sharing MnO 6 and IrO 6 octahedra separated by \nLa3+/Sr2+ ions, due to the quite different charge states and ionic sizes of 3 d and 5 d \ncations. Concerning the La3+/Sr2+ ions distributions, the columnar La3+/Sr2+ cation order \nis taken (Fig. 3(a)), as it is prevailing among the nine pos sible double double AA ′BB′O6 \nperovskite1. Owning to the rotation and tilting of MnO 6 and IrO 6 octahedra, the Mn -O-\nIr bond angles significantly deviate from 180 º. \n \n \n \nFigure S4.  Different A -cations configurations of LSMIO composite considered in this \nstudy. (left) columnar type (middle) rocksalt type (right) layer type.  \n 20  \n \nTable S1. Relative energies (meV/ f.u.) obtained by GGA scheme within the magnetic \nground state, i.e. ferrimag netic ordering. The results suggest that the columnar type A -\ncations pattern is preferable in LSMIO composite films.  \n \n Columnar  Rocksalt  Layer  \nEnergy (meV)  0.0 15.7 86.7 \n \n \nFigure S5.  Different magnetic configurations of LSMIO composite considered in this \nstudy. Only Mn atoms (purple sphere s) and the Ir atoms (brown ones) are shown for \nsimplify. Green (yellow) arrows indicate spin up (down) character.  \n \n \nTable S 2. The calculated total energies (eV/ 2× 2× 2  supercell) obtained by GGA and \nGGA+U (UMn = 4.0 eV , UIr = 2.0 eV ) methods. The FIM (ferrim agnetic) is the ground \nstate.  \n \n FM FIM AFM1  AFM2  \nEnergy (eV)  GGA  -302.0302  -302.0391  -301.9835  -301.9835  \nGGA+U  -286.4180  -286.9023  -286.2655  -286.5611  \n \n \nAFM1 \n AFM2 \n FIM  \n FM  21 We performed electronic calculations on four distinct magnetic alignments: \nferromagnetic (FM), ferrimagnetic (FIM) (i.e., with Mn ions being antiferromagnetic \n(AFM) coupled to six neighboring Ir ions and vice versa) and two distinct AFM ones,  \nwithin the normal GGA/GGA+ U (UMn = 4.0 eV , U Ir = 2.0 eV ) prescriptions with the \ngoal of identifying the magnetic ground state. From the total energy results of the \ndistinct magnetic solutions, we found that t he ferrimagnetic, uncompensated \nantiferromagnetic alignment, due to the un equal magnetic moments on Mn and Ir sites, \nis the most stable state for LSMIO , irrespective of whether the Hubbard U values was \ntaken into account or not, affirming  the experimental magnetic observations.  \n The conspicuous feature of electronic structure is  that most of majority channel \nof Mn 3 d orbitals are occupied, forming a broad valence band ranged from -7.0 eV to \nthe Fermi level, while the corresponding minority channel is entirely empty centered at \n4.5 eV above the Fermi level, revealing that electronic configuration is close to the\n13\ng2etg\nstate and the oxidation state is 3+. Unexpected, the magnetic moment at Mn sites \nis slightly larger than the ideal spin -only 4 μB for Mn3+ cations, suggesting the special \nmagnetic coupling m echanism behind.  For the Ir states, on one hand, the occupied \nminority Os t2g states are completely separated from the empty eg states by an energy \ninterval of ~ 4 eV, accompanied by a moderate spin splitting of ~ 2 eV. On the other \nhand, the spin up t2g states are partially filled, splitting into a small band gap under the \nHubbard U correlation effect. As a consequence, electronic configuration is close to the \nnominal\n13\ng2t state and the oxidation state is 5+. In addition, there are some  Ir eg obitals \npresent around -6 eV below the Fermi level drowned in the O 2p manifold, unfolding \nthe strong covalency between the Ir 5d and O 2p states. The covalent bonding is so \nstrong that the computed Ir magnetic moment (~ 0.9 µB) is remarkable smaller than the \nideal spin -only value (~2 µB). \nThe antiferromagnetic  coupling and unexpected large magnetic moment on Mn3+ \nsite can be interpreted in the framework of superexchange interactions through the \nvirtual hopping bridged by O 2 p spin-up electrons . We can see that in contrast to the \nlarge energy separation of more than 5 eV between the down -spin Mn3+ and Ir5+ t2g \nstates, the up -spin Mn3+ eg and Ir5+ t2g states are separated by on the small band gap of \n0.2 eV, and their hybridization via O 2p up -spin orbitals can push the occupied Mn3+\n13\ng2etg\nand Ir5+\n13\ng2tbands downwards, consequently stabilizes the antiferromagnetic  \ninteraction and then the ferrimagnetic ground state. In other words, the virtual hopping \nfrom Ir5+\n\ng2tto the empty Mn3+\n\ngevia the bridged O\np2 states accounts for the \nantiferromagnetic cou pling and the abnormal large computed Mn3+ magnetic moment \nas well as the exclusively negative sign of the induced O2- magnetic moments.  \n \n "    },    {        "title": "2004.02443v2.Magnetic_and_vibronic_THz_excitations_in_Zn_doped_Fe___2__Mo__3_O__8_.pdf",        "content": "Magnetic and vibronic THz excitations in Zn doped Fe 2Mo 3O8\nB. Csizi,1S. Reschke,1A. Strini\u0013 c,1L. Prodan,1, 2V. Tsurkan,1, 2I. K\u0013 ezsm\u0013 arki,1and J. Deisenhofer1\n1Experimentalphysik V, Center for Electronic Correlations and Magnetism,\nInstitute for Physics, Augsburg University, D-86135 Augsburg, Germany\n2Institute of Applied Physics, MD-2028 Chi\u0018 sin\u0015 au, Republic of Moldova\n(Dated: May 23, 2022)\nWe report on optical excitations in the magnetically ordered phases of multiferroic\nFe1:86Zn0:14Mo3O8in the frequency range from 10-130 cm\u00001(0.3-3.9 THz). In the collinear easy-\naxis antiferromagnetic phase below TN= 50 K eleven optically active modes have been observed in\n\fnite magnetic \felds, assuming that the lowest-lying mode is doubly degenerate. The large number\nof modes re\rects either a more complex magnetic structure than in pure Fe 2Mo3O8or that spin\nstretching modes become active in addition to the usual spin precessional modes. Their magnetic\n\feld dependence, for \felds applied along the easy axis, re\rects the irreversible magnetic-\feld driven\nphase transition from the antiferromagnetic ground state to a ferrimagnetic state, while the number\nof modes remains unchanged in the covered frequency region. We determined selection rules for\nsome of the AFM modes by investigating all polarization con\fgurations and identi\fed magnetic-\nand electric-dipole active modes as well. In addition to these sharp resonances, a broad electric-\ndipole active excitation band, which is not in\ruenced by the external magnetic \feld, occurs below\nTNwith an onset at 12 cm\u00001. We are able to model this absorption band as a vibronic excitation\nrelated to the lowest-lying Fe2+electronic states in tetrahedral environment.\nI. INTRODUCTION\nMultiferroic compounds have been in the focus of con-\ndensed matter research for the last two decades [1, 2] and,\nmore recently, have attracted interest due to the possible\ndynamic magneto-electric e\u000bects, such as, for example,\nuni-directional light propagation [3] and potential appli-\ncations in spintronics based on antiferromagnetic mate-\nrials [4, 5].\nThe multiferroic compound Fe 2Mo3O8, which is known\nas the mineral kamiokite [6], adopts a hexagonal unit cell\nwith lattice constants a= 5:777\u0017A,c= 10:057\u0017A at room\ntemperature as depicted in Fig. 1(a) [7]. It belongs to the\npolar space group P63mcwith a polarization along the c\naxis [8, 9]. The Fe2+ions occupy two di\u000berent sites with\ntetrahedral (site A) and octahedral oxygen coordination\n(siteB) [10]. The two types of corner-sharing polyhe-\ndra form honeycomb-like layers in the ab-plane separated\nby breathing kagome layers of molybdenum octahedra as\nshown in Fig. 1(b). The Mo layers do not contribute to\nthe magnetism due to spin-singlet formation of structural\nMo3O13clusters [11, 12]. Fe 2Mo3O8exhibits a collinear\nantiferromagnetic (AFM) order of the iron ions below the\nN\u0013 eel temperature TN= 60 K, which is accompanied by a\nstrong increase of the polarization [10, 13], giving rise to\na type-I multiferroic state and strong magneto-optical ef-\nfects [14{16]. In addition, an applied magnetic \feld along\nthecaxis induces a ferrimagnetic (FiM) order [10, 13].\nRecently, the possibility of additional orbital ordering on\nthe Fe sites in both the AFM and the FiM state has been\nsuggested by ab-initio calculations [17].\nIsovalent substitution of Fe by nonmagnetic Zn\nin (Fe 1\u0000yZny)2Mo3O8has been reported to in\ruence\nthe magnetic phase diagram, leading to a stabilization\nof the ferrimagnetic phase with increasing Zn content\n[13, 18]. Previous studies revealed that the Zn2+ions\nFIG. 1. (a) Double unit cell of (Fe 1\u0000yZny)2Mo3O8, (b) top\nview onto the honeycomb-like layers stacked along the c-axis.\npredominantly occupy the tetrahedral sites for y\u00140:5\n[11, 19]. Both the antiferromagnetic and the ferrimag-\nnetic phases of (Fe 1\u0000yZny)2Mo3O8reportedly exhibit in-\nteresting electric- and magnetic-dipole active magnon ex-\ncitations and, furthermore, intricate magneto-optical ef-\nfects, e.g. gyrotropic birefringence [15] and the optical\ndiode e\u000bect [16]. In this study we investigated the low-\nenergy excitations in single crystals of Fe 1:86Zn0:14Mo3O8\nby THz spectroscopy. At this Zn concentration we have\nthe possibility to study the dynamic \fngerprints of the\nmagnetic-\feld induced transition regime and both phasesarXiv:2004.02443v2  [cond-mat.str-el]  23 Oct 20202\nin a suitable magnetic-\feld range: For y= 0:05 the FiM\nphase forT < 30 K can only be reached for H > 5 T\nand fory= 0:125 the antiferromagnetic phase report-\nedly occurs only as a metastable state [13]. Moreover,\nmagnetic-\feld dependent THz measurements in Faraday\ncon\fguration have not been reported previously for y= 0\nory= 0:125 [14], making a value 0 :05< y < 0:125\na suitable concentration to shed light on the dynamical\nproperties of both phases in a detailed THz transmission\nstudy.\nII. EXPERIMENTAL DETAILS\nPolycrystalline (Fe 1\u0000yZny)2Mo3O8samples were pre-\npared by repeated synthesis at 1000\u000eC of binary oxides\nFeO (99.999%), MoO 2(99%), and ZnO in evacuated\nquartz ampoules aiming for a concentration with x= 0:1.\nSingle crystals were grown by the chemical transport re-\naction method at temperatures between 950 and 900\u000eC.\nTeCl 4was used as the source of the transport agent.\nLarge single crystals up to 5 mm have been obtained\nafter 4 weeks of transport. The X-ray di\u000braction pat-\ntern of the crushed single crystals shown in Fig. 2(a) re-\nvealed a single-phase composition with a hexagonal sym-\nmetry using space group P63mcand a Zn content corre-\nsponding to x\u00190:07. The obtained lattice constants are\na=b= 5:773(2) \u0017A andc= 10:017(2) \u0017A.\nThe speci\fc heat was measured in a Quantum Design\nphysical properties measurement system from 1.8 to 300\nK and in \felds up to 9 T. Magnetization measurements\nwere performed using a superconducting quantum inter-\nference device magnetometer (Quantum Design MPMS-\n5).\nTransmission measurements in the frequency range\nfrom 10-130 cm\u00001were performed by THz time-domain\nspectroscopy using a Toptica TeraFlash spectrometer and\nan Oxford Instruments Spectromag cryomagnet in exter-\nnal magnetic \felds ranging from -7 T to 7 T in an ac-cut\nsample with a thickness of d= 1:48 mm and an ab-cut\nsample with d= 0:6 mm.\nIII. SAMPLE CHARACTERIZATION AND\nPHASE DIAGRAM\nThe temperature dependence of the speci\fc heat\nshown in Fig. 2(b) reveals a clear \u0015-like anomaly at the\nmagnetic ordering transition TN= 50 K, similar to the\nreported anomaly of pure Fe 2Mo3O8[10]. With an in-\ncreasing external magnetic \feld applied along the crys-\ntallographic c-axis the anomaly broadens and shifts to\nhigher temperatures. The transition temperatures were\nestimated using the maxima of the speci\fc heat.\nThe temperature dependence of the magnetic suscep-\ntibility measured in 1 T shown in Fig. 3(a) exhibits a\nsteep increase and a subsequent maximum at the same\ntemperature as the \u0015-anomaly in the speci\fc heat at\n02 04 06 08 01 00050100204 06 0-4000400800 \nµ0H = 0 T \n1 T \n3 T \n5 T \n7 T \n9 TCp(J/mol K)T\n (K)(b)(a)T\nN = 54 K Iobs \nIcal \nIobs- Ical \nBragg peaksI  (counts)2\nΘ (degree)FIG. 2. (a) Room temperature XRD spectra of\nFe1:86Zn0:14Mo3O8and the corresponding calculated pattern\nfor space group P63mc. (b) Temperature dependent speci\fc\nheatCpin the vicinity of the magnetic ordering transition in\nmagnetic \felds applied along the c-axis between 0-9 T.\nTN= 50 K con\frming that the transition corresponds\nto the magnetic ordering transition in agreement with\nprevious studies [7, 9, 13].\nThe temperature dependence of 1 =\u001fin the para-\nmagnetic state range from 100-300 K can be described\nby a Curie-Weiss law yielding \u0002 CW=\u000040 K and an ef-\nfective moment \u0016e\u000b= 5:5\u0016B, which is enhanced in com-\nparison to the spin-only value of \u0016e\u000b= 4:9\u0016Bexpected\nfor Fe2+ions with spin S= 2, assuming that all Zn ions\noccupy tetrahedral sites and that the e\u000bective g-factors\nare 2.1 [20] and 2.0 [21] for the tetrahedral and octahedral\nFe sites, respectively. The e\u000bective magnetic moment is\ncomparable to reported values of \u0016e\u000b= 5:7\u0016Bfor pure\nFe2Mo3O8[7]. The enhancement compared to the spin-\nonly value could be due to a sizeable orbital contribution\n[17, 22]. The obtained Curie-Weiss temperature is clearly\nreduced in comparison with the values \u0002 CW=\u0000110 K or\n100 K reported for Fe 2Mo3O8[7, 18], but in line with the\nreported tendency for increasing Zn substitution, where\n\u0002CWeventually becomes positive [18].3\n01 002 003 000.00.20.40.6-\n10001002000204060-\n4- 20 2 4 02468χ (emu/mol)T\n (K)TN(a)1\n/χ (mol/emu)T\n (K)µeff /Fe = 5.5 µBθ\nCW = - 40 Kθ\nCW1\n TM (µB/f.u.)µ\n0H (T)13 K20 K30 K(b)H || c-0.50 .00 .54\n0 K50 KH\nc1Hc2Hc*\nFIG. 3. (a) Temperature dependence of the magnetic sus-\nceptibility\u001fmeasured in 1 T for Hkc. The inset shows the\ntemperature dependence of \u001f\u00001and a Curie-Weiss \ft (solid\nred line); (b) Field dependent magnetization curves for several\ntemperatures below TN.\nThe magnetic-\feld dependence of the magnetization\nforHkcis shown for several temperatures below TNin\nFig. 3(b). At 13 K the transition from the antiferromag-\nnetic to the ferrimagnetic state is taking place via a two-\nstep feature characterized by the critical \felds Hc1= 3 T\nandHc2= 3:8 T as indicated in Fig. 3(b). This is in\nagreement with previous studies for samples with y= 0:1\n[13]. However, at 50 K, just below the magnetic transi-\ntion transition temperature, no hysteretic behavior and\nformation of a metastable state is observed, but an in-\ntermediate third step is observed at Hc\u0003. At 40 K a\nhysteretic behaviour appears, but the remanent magne-\ntization is still zero, while the width of the hysteresis is\nclearly increased at 30 K and the remanent magnetization\nalready corresponds to the metastable ferrimagnetic con-\n02 4 010203040506070 \nHc1 \nHc2 \nHc*m\nagnetic field (T)PMF\nMA\nF / FMtemperature (K)Fe1.86Zn0.14Mo3O8FIG. 4.H-Tphase diagram of Fe 1:86Zn0:14Mo3O8estab-\nlished from speci\fc heat and magnetization measurements.\nThe dashed areas indicate the metastable ferrimagnetic phase\nas described in the text.\n\fguration. At 20 K the intermediate step at Hc\u0003is not\ndiscernible anymore and an almost symmetric hysteresis\nhas evolved. In addition, the widths of the hysteretic\ncurves measured by the distance of the coercive \felds at\n30 K and 20 K are 0.27 and 1.3 T, respectively, seem\nto be enhanced in comparison with the widths shown in\nRef. 13 for y= 0:1. This is in agreement with a less\ndiluted system with a Zn concentration y= 0:07.\nIn Fig. 4 the critical \felds and temperatures obtained\nfrom speci\fc heat and magnetization measurements are\nsummarized in a H-T-diagram, which can be compared\nwith the phase diagrams for y= 0:05 andy= 0:1 re-\nported by Kuramaji and coworkers [13]. The intermedi-\nate magnetization step at Hc\u0003for temperatures T >20 K\nmay originate from a more delicate competition between\nthe exchange couplings and thermal \ructuations at this\nparticular Zn concentration y= 0:07. On account of the\nXRD results on crushed single crystals, we discard the\npossibility that the intermediate steps are due to impu-\nrity phases. While for y= 0:05 no remanent magneti-\nzation has been reported down to 13 K and the antifer-\nromagnetic ordering appeared close to 60 K, for y= 0:1\nthe metastable state with a \fnite remanent magnetiza-\ntion appears also in the range 40-30 K and at a similar\nordering temperature as in our case.4\nIV. EXPERIMENTAL RESULTS\nA. Temperature dependent polarized absorption\nspectra\nThe temperature dependence of the THz absorption\nspectra for the polarization con\fguration E!?c;H!?\nc(ab-cut) and the con\fgurations E!?c;H!kc\nandE!kc;H!?c(ac-cut) are shown in Fig. 5(a)-\n(c), respectively. Above TN, there are no detectable\nexcitations. The strong monotonous increase in ab-\nsorption with increasing frequency for E!?ccan\nbe attributed to the low-energy tail of the lowest-lying\ninfrared-active phonon of E1-symmetry with an eigen-\nfrequency of 130 cm\u00001[22, 23]. The decreasing absorp-\ntion of this contribution with decreasing temperature is\nin agreement with the strong narrowing of the IR-active\nphonon, when the system is cooled below TN[22]. Note\nthat the lowest lying IR-active phonon for E!kcis lo-\ncated at around 200 cm\u00001[22, 23] and, therefore, the\nfrequency range where transmission can be detected is\nwider than for E!?cpolarization. Upon cooling below\nTN, several new modes emerge or become observable due\nto the reduced absorption of the IR-phonon contribution\nin the investigated frequency range. Since all of these\nmodes seem to gain intensity and sharpen upon cooling\nwe assign them to new excitations of the magnetically\nordered state.\nmode!0 ab-cut ac-cut\nactivity [cm\u00001]E!?c E!?c E!kc\nH!?c H!kc H!?c\nV1 23 X X \u0002E!?c\nEM1 44 X X \u0002E!?c\nM1 68 X\u0002 XH!?c\nM2 90 X n.r. XH!?c\nM3 96 n.r. n.r. X ?\nM4 114 n.r. n.r. X ?\nEM 40 X X \u0002E!?c\nMM 1 90 X\u0002 XH!?c\nTABLE I. Upper part: Selection rules for the excitations\nfound in Fe 1:86Zn0:14Mo3O8for the di\u000berent polarization con-\n\fgurations as shown in Fig. 5. The notation Xand\u0002indi-\ncates the presence or absence of a mode. The cases when no\nobservation was possible are denoted as n.r. (not resolved).\nLower part: reported excitations in the AFM phase of pure\nFe2Mo3O8at 4.5 K taken from Ref. [14].\nNote that the high-frequency part of some spectra is\nnot shown, because the transmission values were too low\nto give reliable information in this region. For the same\nreason the absorption maxima of the most intense modes\ncould not be resolved. Whenever this was the case, the\neigenfrequency !0was obtained by assuming a symmetric\nlineshape and choosing the center of the absorption. For\nthe broad band V1the frequency of the intensity maxi-mum is given as a characteristic frequency.\nA comparison of the spectra measured at 2 K for the\nthree polarization con\fgurations is shown in Fig. 5(d).\nThe resulting selection rules and eigenfrequencies for the\ndetected modes are summarized in Table I together with\nthe modes reported for the AFM phase of pure Fe 2Mo3O8\n[14]. The lowest-lying broad asymmetric excitation band\nV1and modeEM 1are observed for the ab-cut as well\nas in theac-cut spectrum with E!?c; H!kcand\nare, therefore, assigned to be electric-dipole active for\nE!?c. The fact that the intensity of V1does not\nalter for these two con\fgurations implies that it is not\nmagnetic-dipole active. No feature similar to the V1-\nband has been reported for any other compound of the\n(Fe1\u0000yZny)2Mo3O8series. For EM 1our data does not\nallow to exclude a di\u000berence in intensity for the two con-\n\fgurations, because the maximum of EM 1was not re-\nsolved. However, the eigenfrequency and the selection\nrule forEM 1are in good agreement with a reported ex-\ncitation called EM in pure Fe 2Mo3O8[14] (see Table I),\nwhere the intensities for these two con\fgurations seem to\nshow no di\u000berences. Therefore, we also discard a possible\nmagnetic-dipole activity of EM 1. In addition, EM 1was\nnot observed for a Zn concentration of y= 0:125, where\nthe AFM phase is completely suppressed and the ground\nstate is ferrimagnetic [14].\nIn contrast to modes V1andEM 1, the four additional\nmodes,M1-M4, observed at higher frequencies can be\nexcited for H!?cand are thus considered to be mag-\nnetic dipole active. An additional electric-dipole activity\nis discarded for excitation M1, because it does not ap-\npear for the con\fguration E!?c; H!kc. For modes\nM2-M4our spectra for E!?c; H!kcdo not allow\nus to unambiguously exclude an electric-dipole activity,\nbut we expect any such contribution to be weak. Note,\nthat a magnetic-dipole active mode (named MM 1, see\nsee Table I) with an eigenfrequency close to M2has been\nreported for x= 0 andx= 0:125, previously [14] and de-\ntermined to be only magnetic-dipole active. Modes M1,\nM3, andM4have not been observed before, probably due\nto the low intensity of M1and the fact that the frequency\nrange ofM3andM4was not resolved in previous studies.\nB. Magnetic \feld dependent absorption spectra\nThe magnetic \feld dependence of the absorption spec-\ntra for the three possible polarization con\fgurations mea-\nsured forHkcis shown in Fig. 6(a)-(c). Since the transi-\ntion to the ferrimagnetic state occurs when the magnetic\n\feld is parallel to the c-axis, theab-cut sample was mea-\nsured in Faraday con\fguration ( Hkc; kkc) and the\nac-cut in Voigt con\fguration ( Hkc; k?c), wherek\nis the wave vector of the incoming THz beam. Spectra\nmeasured with H?c(not shown here) exhibited no \feld\ndependence of the absorption spectra.\nThe measurements were performed at 13 K, where the\ncritical \felds Hc1andHc2of the magnetization steps5\n050100150200250(\na)α (cm-1) 10K \n20K \n30K \n40K \n50K \n60K Hω, Eω ⊥ c (ab-cut)0\n20406080(\nb) Hω  ⊥ c, Eω  || c (ac-cut) 2K \n10K \n20K \n30K \n40K \n50K \n60Kα (cm-1)V\n12\n04 06 08 01 001 20020406080100120(\nc) Hω  || c, Eω  ⊥ c (ac-cut) 2K \n10K \n20K \n30K \n40K \n50K \n60Kα (cm-1)w\nave number (cm-1)204 06 08 01 001 200501001502002506\n8 cm-1114 cm-19\n6 cm-190 cm-14\n4 cm-1(d)M\n4M 3M2EM1M\n1 Hω, Eω ⊥ c (ab-cut) \nEω || c, Hω ⊥ c (ac-cut) \nEω ⊥ c, Hω || c (ac-cut)α (cm-1)w\nave number (cm-1)T = 2 K2\n3 cm-1\nFIG. 5. Absorption spectra for several temperatures above and below TNfor di\u000berent polarization con\fgurations: (a)\nE!?c;H!?c(ab-cut sample) , (b) E!kc;H!?c(ac-cut sample), (c) E!?c;H!kc(ac-cut sample), and (d) comparison\nof the absorption spectra at 2 K for the three di\u000berent con\fgurations.\nand the saturation magnetization (see Fig. 3(b)) of the\ncompound were accessible using our experimental setup.\nWhen increasing the magnetic \feld in the antiferromag-\nnetic phase (blue spectra) we observe a linear splitting\nof the modes M1\u0000M4as indicated by the black lines\nin Fig. 6(b) in Voigt con\fguration and for M1also in\nFaraday con\fguration (Fig. 6(c)). The slope of the lin-\near dependence was parametrized by an e\u000bective g-factor\nde\fned by ~!=~!0\u0006ge\u000b\u0016BHfor both branches. The\nvalues are indicated in Fig. 6(e) and (f). The behaviour\nofM2in Faraday con\fguration is more complex, as a\nlow-frequency satellite M5seems to emerge and soften\nwith increasing \feld up to Hc2. For the electric-dipole\nactive mode EM 1no shift or splitting with increasing\nmagnetic \feld could be resolved, but it looses inten-\nsity when approaching the transition region to the FiM\nstate and it \fnally disappears for H > H c2(spectra for\nHc1< H < H c2are shown in green, for H > H c2in\nred). This behaviour is in agreement with the corre-\nspondingEM-mode in pure Fe 2Mo3O8[14]. Above Hc2modesM1\u0000M4also vanish, and new modes appear in\nthe ferrimagnetic phase. We want to emphasize that the\ncoexistence of modes of the AFM and FiM phases for\nHc1< H < H c2indicates a two-phase regime in this\n\feld region. These new modes also show a \feld depen-\ndence and are labeled F1\u0000F11. A detailed comparison of\nthe splitting of the antiferromagnetic modes at 2 T and\nthe shape of the ferrimagnetic modes at 7 T is given in\nFig. 7. Notably, the broad band V1does not exhibit any\nsigni\fcant changes with increasing magnetic \felds and\ndoes not seem to be in\ruenced by the transition to the\nFiM state.\nThe resonance frequencies of the ferrimagnetic modes\nF1\u0000F11either increase or decrease linearly with increas-\ning magnetic \felds as shown in Fig. 6. Additionally, we\nincluded the resonance frequencies of modes F1\u0000F11\nmeasured upon lowering the magnetic \feld to zero (spec-\ntra not shown here) as open symbols in Fig. 6(d)-(f). The\neigenfrequencies of F1andF2were found to cross each\nother below Hc1. The eigenfrequencies of F1\u0000F11in6\n20 40 60 80 0 200 400 600 800 1000 1200 1400 1600 \n0 1 2 3 4 5 6 7 40 60 80 100 120 140 \n0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 40 60 80 100 120 140 F 11 F 10 \nEM1 \nabsorption coefficient  α (cm-1 ) \nwave num ber (cm-1 ) H || c, Eω ⊥ c, Hω || c ( ac -cut) (a) \nV 1 \n20 40 60 80 100 120 F 1 F 2 F 4 F 3 \nM4 M3 M2 M1 (b) H || c, Eω || c, Hω ⊥ c ( ac -cut) \nwave num ber (cm-1 ) 20 40 60 80 100 120 F 9 \nF 8 F 7 F 6 F 5 \nM5 \nM2 M1 EM1 V 1 \n(c) H || c, Eω , Hω ⊥ c ( ab -cut) \nwave num ber (cm-1 ) 0T 0.5T 1T 1.5T 2T 2.5T 3T 3.2T 3.4T 3.6T 3.8T 4T 4.2T 4.4T 5T 5.5T 6T 6.5T 7T \nFiM \nF 10 F 11 F 9 \nF 8 \nF 6 F 7 \nF 5 F 4 \nF 3 \nF 1 F 2 M3 M4 \nM2 M2 \nM1 \nM1 Hc2 (f) (e) eigenfrequency (cm-1 ) \nmagnetic field (T ) (d) Hc1 \nEM1 \nE ω ⊥ c, Hω || c ( ac -cut) E ω || c, Hω ⊥ c ( ac -cut) E ω , Hω ⊥ c ( ab -cut) AFM \ng eff  = 2.1 \ng eff  = 1.7 g eff  = 2.9 \ng eff  = 4.1 Hc2 Hc1 \nmagnetic field (T ) g eff  = 4.1 M5 \nEM1 Hc2 Hc1 \nmagnetic field (T ) \nFIG. 6. Upper panels: Magnetic \feld dependence of the absorption spectra with HkcatT= 13 K in the range of 0 T to\n7 T for (a) E!?c;H!kc(ac-cut), (b)E!kc;H!?c(ac-cut) and (c) E!;H!?c(ab-cut). All spectra are shown with a\nconstant o\u000bset of 80 cm\u00001with respect to the 0 T spectra. Spectra in the AFM phase are plotted in blue, for Hc1<H <H c2\nin green, and in the ferrimagnetic phase in red. The solid lines are drawn to guide the eyes. Lower panels: Magnetic \feld\ndependence of the eigenfrequencies for all detected modes (except A 1) for (d)E!?c;H!kc(ac-cut), (e)E!kc;H!?c\n(ac-cut) and (f) E!;H!?c(ab-cut). Filled symbols represent data obtained in increasing \feld starting from 0 T, while open\nsymbols were taken upon decreasing \felds. Dotted lines represent the critical \felds of the magnetic transition taken from the\nmagnetization data The inset in (d) depicts the suggested collinear spin con\fgurations of the AFM and FiM states.\nthe zero-\feld FiM state are given in Table II, together\nwith the e\u000bective g-factors. In the lower part of Ta-\nble II we list all reported modes of the FiM phase of\n(Fe1\u0000yZny)2Mo3O8for a comparison. For y= 0 and\ny= 0:125 a magnetic-dipole active mode (called \u00173or\nMM 2) with a similar eigenfrequency as F1has been re-\nported [14]. For y= 0:25 andy= 0:4 this mode \u00173\nseems to have developed an additional electric-dipole ac-\ntivity [15]. The additional modes \u00171,\u00172, and\u00174at lowerfrequencies cannot be directly related to the modes ob-\nserved in our study, as we expect a continuous evolution\nof the modes with increasing Zn content and, therefore,\nconsider the reported spectra for y= 0 andy= 0:125 as\nthe most relevant.7\nV. DISCUSSION\nA. Modes EM 1andM1\u0000M5of the AFM phase\nAs already mentioned above, modes EM 1andM2\nhave been reported already for pure Fe 2Mo3O8and inter-\npreted as precessional modes of the collinear AFM struc-\nture, both being doubly degenerate modes [14]. While\nthe degeneracy of M2is clearly lifted in the external\nmagnetic \feld with ge\u000b= 1:7 forH!ka, the possi-\nble doublet nature of EM 1remains to be con\frmed in\nhigher magnetic \felds. The respective electric and mag-\nnetic dipole activity of the two modes was assigned to\nan inverse Dzyaloshinskii-Moriya mechanism and di\u000ber-\nent single-ion anisotropies of tetrahedral and octahedral\nsites [14].\nAs our spectra reveal three additional magnetic modes\nM1;M3;M4, we have to take into account that the dis-\norder induced by substitution of iron by zinc may result\nin an increased number of nonequivalent tetrahedral Fe\nsites. In early M ossbauer studies for various zinc dopings\nat least four di\u000berent tetrahedral iron sites with di\u000berent\nhyper\fne \felds could be distinguished [11]. Given the\nabsence ofM1in the spectra of pure Fe 2Mo3O8[14] and\nthe low intensity of the mode, we assume that this mode\nis due to the Zn induced disorder. As the spectral range\nreported for Fe 2Mo3O8was limited to about 94 cm\u00001\n[14], it is not clear, whether M3(at 96 cm\u00001) andM4(at\n114 cm\u00001) are also disorder-induced modes or whether\nthey are present in pure Fe 2Mo3O8, too. Preliminary\nTHz measurements on pure Fe 2Mo3O8[24], however, in-\ndicate that an excitation with a similar eigenenergy as\nM4is present also in pure Fe 2Mo3O8and may be de-\nscribed as an intrinsic mode of Fe 2Mo3O8. Under these\nassumptions, we conclude that at least the three modes\nEM 1;M2;M4are to be regarded inherent to the magnetic\nstructure of the pure compound and the two doubly de-\ngenerate modes M1;M3may emerge due to the dilution\nof the magnetic iron sites by non-magnetic zinc. Since\nmodeM5shifts to lower eigenfrequencies with increasing\nmagnetic \feld and disappears above Hc2, its behavior\nmay be interpreted as a soft mode of the magnetic phase\ntransition. However, its relation to M2, from where it\nseems to originate, remains unresolved at present. Be-\nsides the in\ruence of di\u000berent Fe sites, the large number\nof magnetic modes may also be due to additional spin\nstretching modes [25, 26].\nB. Modes of the ferrimagnetic phase\nAt present we identify eleven modes F1\u0000F11in the\nspectra of the FiM phase for all measured polarization\ncon\fguration above Hc2(see Tab. II). In principle, this\ncorresponds to the number of excitation branches in the\nAFM phase below Hc1, if theEM 1mode is assumed to be\ndoubly degenerate and M5corresponds to a single mode.\nA direct comparison of the eigenfrequencies in zero-\n050100150200F\n11F\n10F6F7F8F9F 5F\n4F3F\n2F1M\n1(b)α (cm-1) Hω, Eω ⊥ c (ab-cut) \nEω ⊥ c, Hω || c (ac-cut) \nEω || c, Hω ⊥ c (ac-cut)(a)H\n = 7 TH\n = 2 TV12\n04 06 08 01 001 201 40050100150200M\n5M\n3M4M\n2EM1V1 Hω, Eω ⊥ c (ab-cut) \nEω ⊥ c, Hω || c (ac-cut) \nEω || c, Hω ⊥ c (ac-cut)α (cm-1)w\nave number (cm-1)FIG. 7. Comparison of the absorption spectra (a) at H= 7 T\nin the FiM state and (b) at H= 2 T in the AFM state for the\nthree polarization con\fgurations. The splitting of the AFM\nmodesM1\u0000M4is indicated in (b).\n\feld and the e\u000bective g-factors estimated by ~!=~!0+\nge\u000b\u0016BHis provided in Table II, together with the FiM\nmodes previously reported for (Fe 1\u0000yZny)2Mo3O8[14{\n16]. The corresponding e\u000bective g-factors of these modes\nwere determined by us using the published data.\nTo estimate the in\ruence of demagnetizing \felds for\nthe investigated samples we model the samples as ellip-\nsoids [27]. The ab-cut sample is assumed to have semi-\naxesa;b;c corresponding to its approximate dimensions\n2\u00021:5\u00020:6 cm3and experiences a demagnetization fac-\ntorNz= 0:623 along the c-axis. Theac-cut sample with\n2\u00021:6\u00021:48 cm3yieldsNz= 0:343. The resulting de-\nmagnetization \felds \u00160NzMS, whereMS'0:85\u0016B=f:u:\ndenotes the magnetization value measured at 13 K in a\n\feld of 5 T as shown in Fig. 3(b), correspond then to\n0.05 T and 0.03 T for the ab-cut andac-cut samples, re-\nspectively. Since the resulting \felds are about two orders\nof magnitude smaller than the applied external \felds in\nthe FiM state, we neglect demagnetization e\u000bects in the\nfollowing.8\nmode!0(Hdc= 0 T) ge\u000b ab-cut ac-cutactivity\n[cm\u00001] E!;H!?c E!?c;H!kc E!kc;H!?c\nF1 85 -2.0 \u0002 \u0002 X ?\nF2 79 3.4 \u0002 \u0002 X ?\nF3 102 2.7 \u0002 n.r. X ?\nF4 119 2.7 n.r. n.r. X ?\nF5 65 -0.6 X \u0002 \u0002 ?\nF6 83 -1.6 X \u0002 \u0002 ?\nF7 87 -1.7 X \u0002 \u0002 ?\nF8 91 -0.2 X n.r. \u0002 ?\nF9 102 2.3 X n.r. \u0002 ?\nF10 76 -1.7 \u0002 X \u0002 ?\nF11 83 -1.9 \u0002 X \u0002 ?\nMM 2 (y= 0) 87 -2.0 - \u0002 X H!?c\nMM 2;\u00173(y= 0:125) 87 -2.5 - \u0002 X H!?c\n\u00171(y= 0:25) 47 4.0 - \u0002 X H!?c\n\u00172(y= 0:25) 76 2.6 - \u0002 X H!?c\n\u00173(y= 0:25) 87 -2.6 - X X E!;H!?c\n\u00171(y= 0:4) 47 4.0 - X X E!;H!?c\n\u00172(y= 0:4) 73 3.6 - \u0002 X H!?c\n\u00173(y= 0:4) 87 -3.1 - X X E!;H!?c\n\u00174(y= 0:5) 42 - - \u0002 X H!?c\nTABLE II. Selection rules for the excitations F1\u0000F11found for the di\u000berent polarization con\fgurations at 13 K as shown\nin Fig. 5. The eigenfrequencies !0correspond to the zero-\feld values in the FiM phase. The sign of ge\u000bindicates the sign of\nthe linear slope with increasing magnetic \feld. The notation Xand\u0002indicates the presence or absence of a mode. The cases\nwhen no observation was possible are denoted as n.r. (not resolved). The lower part of the table lists reported modes of the\nFiM phase for di\u000berent Zn concentrations and their assigned optical activity taken from Refs. [14{16].\nUnfortunately, it is not possible to determine clear se-\nlection rules for the FiM modes, either because the ex-\ncitations could not be resolved experimentally or their\npresence or absence in the three polarization con\fgura-\ntion does not yield a consistent selection rule. For exam-\nple, modeF1agrees well both in eigenfrequency and g-\nfactor to the magnetic-dipole active mode MM 2reported\nfory= 0 andy= 0:125 [14], where the selection rule\nH!?cwas established by investigating an ac-cut sam-\nple in Voigt con\fguration, only. From our measurement\nin Voigt con\fguration alone, we would derive the same\nselection rule, but the absence of this mode in our ab-cut\nmeasurements with H!?cdoes not allow to establish\nthis selection rule. Similarly, modes F5\u0000F9should ap-\npear in one of the con\fgurations in the ac-cut measure-\nments to establish a selection rule, but there seem to be\nno corresponding excitations.\nIn principle, the symmetry of the FiM phase was found\nto allow the occurrence of gyrotropic birefringence as re-\nported for excitations \u00172and\u00173for Zn concentrations of\ny= 0:25 andy= 0:4 [15], which seem to have no di-\nrect correspondence to the mode observed in this study.\nHowever, we can not exclude that similar magnetoelectric\ne\u000bects might be present for some of these modes.C. Modelling the absorption band V1in terms of a\nvibronic mode\nIn the following we will analyze the nature of the\nelectric-dipole active excitation band V1, which emerges\nand gains intensity with decreasing temperatures below\nTN(see Fig. 8(b)). It can only be observed for E!ka\nand does not exhibit any changes in magnetic \felds up to\n7 T. The onset of the excitation occurs at about 10 cm\u00001,\nwhich is in agreement with the reported energy di\u000berence\nbetween the ground and \frst excited state of the Fe2+\nions in tetrahedral environment for pure Fe 2Mo3O8[11].\nStrong absorption features related to transitions within\nthe low-lying electronic d\u0000d-levels of Fe2+ions in tetra-\nhedral sites have been reported for diluted Fe2+on tetra-\nhedral sites in semiconductors [28] and also in several\ncompounds with concentrated Fe2+ions, e.g. in the\nspinels FeSc 2S4[29, 30] and FeCr 2S4[31] or in the system\nSr2FeSi 2O7[32]. Moreover, electron-phonon coupling can\nlead to vibronic excitations, where electronic and vibra-\ntional degrees of freedom cannot be separated. In partic-\nular, the fact that the electronic eigenfrequencies of the\ntetrahedrally coordinated Fe2+ions are of the same order\nof magnitude as possible involved phonon modes may fos-\nter vibronic e\u000bects [33{36]. A corresponding analysis of\nsuch low-lying vibronic excitations has been elaborated,\ne.g., by Testelin and coworkers [28].9\nHere, we follow this line and concentrate on modelling\nthe lineshape of the absorption by using the general ap-\nproach to describe vibronic excitations in case of a linear\nelectron-phonon coupling [37, 38]. The normalized ab-\nsorption spectrum at T= 0 can be described by\nI(!) =1Z\n\u00001f(t)ei!tdt (1)\nwith the generating function\nf(t) = exp(\u0000i!0t\u0000S+S(t)) (2)\nwhere\nS(t) =1Z\n0s(!)e\u0000i!td! (3)\nis determined by the coupling function s(!), which de-\nscribes the distribution of phonon modes in a given mate-\nrial andS=S(t= 0). Expanding f(t) in a power series\ninS(t) results in\nI(!) =e\u0000S1X\nn=0Sn\nn!\u0000n(!) (4)\nwith\n\u0000n(!) =1Z\n\u00001ei(!\u0000!0)t\u0014S(t)\nS\u0015n\ndt (5)\nwhich can be calculated iteratively for n>1 by the con-\nvolution\n\u0000n(!) =1Z\n\u00001\u0000n\u00001(!0)\u00001(!+!0\u0000!0)d!0(6)\nThe number ncorresponds to the number of phonons\ninvolved in the absorption process [37] and, hence, the\nzero-phonon-line with n= 0 describes the purely elec-\ntronic transition at !0, which is not visible in our spec-\ntra.\nIn order to simulate the vibrational lineshape the\nchoice of a suitable coupling function is important. Op-\ntically active phonons are usually at higher frequencies\nthan the observed absorption band and, therefore, we\nconsider a coupling to acoustic phonons, which are usu-\nally modelled by a Debye-like phonon density of states.\nUsing an exponential decay around the Debye frequency\n!1instead of a sharp cut-o\u000b [39] leads to the \frst term\nin our coupling function\ns(!) = exp\u0012\n\u0000!\n!1\u0013!2S1\n2!3\n1+ exp\u0012\n\u0000!\n!2\u0013!3S2\n6!4\n2:(7)\n020406080100120 (cm1)\n(a) 2 K\n15 K\n25 K\n30 K\n60 K\nphonon contribution\n10 20 30 40 50 60 70 80\nwave number (cm1)\n020406080100120 (cm1)\n(b)\nV1EM1\nM1n=1\nn=2\nn=3FIG. 8. (a) THz absorption data at various temperatures\nforE!?c;H!?c, and a Lorentzian phonon contribution of\nthe lowest-lying phonon as described in the text (black dashed\nline), (b) THz absorption spectrum at 2 K, where the phonon\ncontribution was subtracted, and a simulation using Eqs. (4)\nand (7), including the contributions for n= 1;2;3 (dashed\nlines).\nThis \frst Debye-like term alone, however, does not al-\nlow to fully capture the entire absorption band. There-\nfore, the second term with a cut-o\u000b frequency !2is intro-\nduced as an empirical modi\fcation of the phonon-density\nof states, satisfying the condition S=S1+S2as re-\nquired [38]. Note that this second term was suggested to\nmodel molecular vibronic spectra with the same approach\nused here [40]. The choice of the second term is some-\nwhat arbitrary and other terms satisfying S=S1+S2\nmight give the same result. As the experimental phonon-\ndensity of states is not available for comparison with the\ncoupling function, this term and its additional two pa-\nrameters deem, however, necessary to describe the entire\nabsorption curve successfully as we discuss in the follow-\ning.\nIn order to analyze the lineshape of the V1excita-\ntion, we use a spectrum obtained for E!?c;H!?c\n(ab-cut sample) and subtract the high-frequency contri-\nbution stemming from the lowest-lying infrared-active\nphonon modelled by a Lorentzian lineshape with eigen-\nfrequency!0= 128:6 cm\u00001, damping\r= 9:0 cm\u00001, and\nionic plasma frequency !p= 158:4 cm\u00001(see Fig. 8(a))10\nin agreement with reported data for pure Fe 2Mo3O8[22].\nAs the narrow electric-dipole excitation EM 1on top of\ntheV1-band is suppressed in the ferrimagnetic phase\naboveHc2(see Fig. 6) and M1is clearly magnetic in\norigin, while V1does not alter, we do not consider any\ncoupling of these excitations.\nTo restrict the parameters for this simulation, we \fxed\nthe zero-phonon-line to !0= 12 cm\u00001, a value close to\nexperimental [22, 23] and theoretical estimates [11] and\nminimized the number of phonons involved in the pro-\ncess ton= 3. Using these constraints, we could repro-\nduce the experimental lineshape with the \fve parameters\nS1= 0:02,S2= 1:1,!1= 12 cm\u00001,!2= 3:6 cm\u00001and\nthe amplitude A= 2:4\u0001103cm\u00001, which is de\fned by\n\u000b(!) =A\u0001I(!). The agreement between data and sim-\nulation is very good and, hence, we conclude that this\nsimple approach satisfactorily reproduces the V1band\nand supports a vibronic origin of the band. Note, that\nhigher convolutions for n > 3 do only weakly modify\nthe high-frequency tail. The values for !1and!2must\nbe regarded as a parametrization of the acoustic phonon\nbranches coupled to the electronic transitions. As such a\nvibronic band has not been reported for pure Fe 2Mo3O8\n[14], we assume that the disorder introduced by the Zn\nions and the corresponding impurity modes are responsi-\nble for the occurrence of this absorption band. It remains\nto be clari\fed why THz studies of comparable samples\nwithy > 0 do not report a similar feature [14{16] and\nwhy this mode can only be observed below TN.\nA possible explanation for the emergence below TN\ncould be that long range magnetic ordering triggers the\nsplitting of the lowest-lying iron orbitals as suggested in\nRef. [11]. Moreover, some of us recently reported the\nemergence of electronic excitations in the far- and mid-\ninfrared regime related to the higher-lying d\u0000d-transition\nof the Fe2+ions at tetrahedral sites in pure Fe 2Mo3O8\nbelowTN, where energy scales of 13 cm\u00001and 26 cm\u00001\noccur as a di\u000berence between the observed transitions.\nThis corroborates our interpretation that the vibronicband involves the lowest-lying electronic states of Fe2+\nions at tetrahedral sites.\nVI. SUMMARY\nWe observed ten magnetic modes in the AFM state\nin Fe 1:86Zn0:14Mo3O8in \fnite magnetic \felds, including\nboth magnetic- and electric-dipole active modes. Assum-\ning that the lowest-lying mode is a doublet with an un-\nresolved splitting in the magnetic \felds applied in this\nstudy, the actual number of AFM modes would increase\nto eleven. This is the number of modes observed in the\nmagnetic \feld induced FiM state in Fe 1:86Zn0:14Mo3O8.\nThe large number of modes, far exceeding the number of\nmagnetic sublattices in pure Fe 2Mo3O8, may imply that\nthe number of magnetic sublattices is increased due to\nZn substitution on the tetrahedral sites. Another pos-\nsible reason for the large number of excitations is that\nspin-stretching modes also become optically active, be-\nsides the precessional modes described by linear spin-\nwave theory. In the transition region between the two\nmagnetic phases a coexistence of AFM and FiM modes is\npresent, implying a coexistence of AFM and FiM phases\nforHc1< H < H c2. Additionally, a broad electric-\ndipole active excitation band was observed in the AFM\nphase and our analysis strongly suggests that it is of vi-\nbronic origin involving the lowest-lying electronic d-states\nof Fe in tetrahedral environment, which are split by about\n12 cm\u00001.\nACKNOWLEDGMENTS\nWe acknowledge support by the Deutsche Forschungs-\ngemeinschaft via TRR 80 (project no. 107745057) and\nvia the Institutional Project 20.80009.5007.19 (Moldova).\n[1] N. A. Spaldin and R. Ramesh, Nature Materials 18, 203\n(2019).\n[2] S. Dong, H. Xiang, and E. Dagotto, National Science\nReview 6, 629 (2019).\n[3] I. K\u0013 ezsm\u0013 arki, U. Nagel, S. Bord\u0013 acs, R. S. Fishman, J. H.\nLee, H. T. Yi, S. W. Cheong, and T. R~ o~ om, Phys. Rev.\nLett. 115, 127203 (2015).\n[4] T. Jungwirth, J. Sinova, A. Manchon, X. 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Phys. 141, 155102 (2014)."    },    {        "title": "2205.07541v1.Ferrimagnetism_in_stable_non_metal_covalent_organic_framework.pdf",        "content": "Ferrimagnetism in stable non -metal covalent organic framework  \nDongge Ma1,*, Yuhang Qian1, Mingyang Ji1, Jiani Li1, Jundan Li1, Anan Liu3,  \nYaohui Zhu2,* \n1College of Chemistry and Materials Engineering, Beijing Technology  and Business \nUniversity, Beijing, China  \n2Physics Department, School of Artificial Intelligence, Beijing Technology and \nBusiness University, Beijing, China  \n3Basic Experimental Centre for Natural Science, University  of Science and Technology \nBeijin g, Beijing, China  \n*Corresponding author:  madongge@btbu.edu.cn , yaohuizhu@gmail.com  \nAbstract:  We synthesized a pure organic non -metal crystalline  covalent organic \nframework TAPA -BTD -COF  by bottom -up Schiff base chemical reaction. And this \nimine -based COF is stable in aerobic condition and room -temperature. We discovered \nthat this TAPA -BTD -COF exhibited strong magneticity in 300 K generat ing magnetic \nhysteresis  loop in M -H characterization  and giant χ mol up to 0.028 . And we further \nconducted  zero-field cooling and field -cooling measurement of M -T curves . The as -\nsynthesized materials showed a large  χmol up to  0.028 in 300 K and increasing to 0.037 \nin 4.0 K with 200 Oe measurement field. The TAPA -BTD -COF 1/χ mol ~ T curve supported  its ferrimagnetism, with an intrinsic Δ temperature as -33.03 K by \nextrapolating the 1/χ mol ~ T curve . From the continuously increas ing slope of 1/χmol ~ T, \nwe consider that this TAPA -BTD -COF belongs to ferrimagnetic other than \nantiferromagnetic materials. And the large χmol value 0.028 at 300 K and  0.037 at 4.0 K  \nalso supported this, since common antiferromagnetic materials possess χ mol in the range \nof 10-5 to 10-3 as weak magnetics other than strong magnetic materials such as \nferrimagnetics and ferromagnetics. Since this material  is purely non -metal organic \npolymer, the possibility of d -block and f -block metal with unpaired -electron induced \nmagnetism can be excluded. Besides , since th e COF does not involve free -radical \nmonomer  in the processes of synthesis , we can  also exclude the origin of  free-radical \ninduced  magnetism. According to recent emerging flat-band  strong correlated exotic \nelectron property , this unconventional phenomenon may relate to n -type doping on the \nflat-band locat ing in the CBM  (conduction band minimum) , thus generating highly -\nlocalized electron with infinite  effective mass and  exhibiting st rong correlation , which \naccounts for th is non -trivial  strong and stable ferrimagneticity  at room -temperature and \naerobic atmospheric conditions .  \nI. Introduction  \nRecently, as the emerging of twisted bilayer magic -angle graphene  (TWBMG) (1, \n2), the exotic physics phenomena associated with flat -band induced strong electron \ncorrelation (3), such as superconductivity (4, 5), ferromagnet ism(6), topological states (7) \nand spin Hall  effect (8) have been  investig ated in  low temperature (9) (mK level ) and high magnetic field . Considering the difficulty to obtain a low -temperature condition to \nmK, it is  important to level up the T c of such flat -band materials. Inorganic  materials  \nsuch as TWBMG (10) are emerging as potential flat -band materials. However, due to \ntheir n arrow band -gap, and the considerable challenges to  precisely  tune and control \nthe twisting angle to several degrees, it is increasingly important to develop  tunable  \norganic flat -band materials with exotic electronic property such as magnetism.  \nAs a kind of novel emerging star materials, covalent organic framework (COF) is a \ntype of pure organic polymeric material connected  by strong covalent bond in two or \nthree -dimension other than metal -organic framework (MOF ), which is connected by \nweak coordinative bond (11). Besides, COF is crystalline polymer similar as MOF , \nmostly with crystalline domain as large as  several hundred nanometers to several \nmicrometers , even in certain structures can  be prepa red into  50-100 μm large  single \ncrystal (12). Apart from these , because of its covalent bond connection, COFs possess \nmuch higher chemical stability other than MOFs in aerobic humid, aqueous and organic \nsolvents condition (13). Moreover, due to its crystallinity, the COFs structure can  be \ncertainly confirmed via the assignment of XRD data with Pawley or Rietveld \nrefinement accompanying  theoretical modelling and calculation (14). Their  clear and \nunique structure facilitate s the demonstration of QSAR ( Quantitative structure –activity \nrelationship ) and guide s the design and synthesis of the materials with the optimal \nproperty as desired (15). Since COFs are synthesized via bottom -up organic synthetic \nroutes from organic small molecul e monomers, the main -chain, branch -chain  and pore \nstructures can all be precisely  tuned by choosing different monomers with diversifie d electronic and ster ic configurations. Besides, different dynamic chemical reactions such \nas boronate (16), Schiff -base(17), ketoenamine (18) and C=C (19, 20) formation  \nreactions  can be chosen to optimize COFs with best chemical, physical or biological \napplication performances.  \n  Since in 2005, Yaghi et al. discovered the first COF materials COF -1 and COF -5(16), \nCOFs materials have been utilized in various fields and applications, such as gas \nadsorption /separation (21), catalysi s(22), energy storage (23), proton conduction (24), \nfluorescence sensing (25) and optoelectronics (26). However, using COFs in magnetism \nas magnetic ordered materials is still rare. Jiang et al. reported in 2017 that an iodine -\ndoped C=C linked sp2c-COF can generate χmol up to 0.025 in 8 K (19). The authors \ndemonstrated  that it is a paramagnetic material other than ferromagnetic or \nferrimagnetic because of the absence of  a M-H hysterias loop  and spontaneous \nmagnetization phenomena , and this material exhibited paramagnetic property with \nmeasurable T c curie temperature . Wang (27), Cui (28),  Liu(29) and Br édas(30) \nindependently predicted the potential magneti sm in Lieb -lattice by theoretical \nmodelling and calculation. However, the macroscopic experimental realization of \nstrong magnetism such as ferrimagnetism in pure organic non -metal materials is still \nnot achieved. Herein, we demonstrated that a novel TAPA -BTD -COF, which is \nprepared via a Schiff-base condensation, exhibits large χ mol up to 0. 028 in 300 K and \n0.037 in 4.0 K in aerobic  atmospheric condition with 200 O e measurement field . \nMoreover, this COF exhibits measurably considerable large hysterias loop in M-H \ncurve in 300 K . By extrapolating and analyzing the 1/χ mol ~ T curve, we conclude that the TAPA -BTD -COF is a ferrimagnetics material with χmol ranging from 0.0 28 to 0.037 \nfrom  300 K cooling down  to 4.0 K with 200 Oe measurement field. This is the first \nroom -temperature strong magnetic organic materials exhibiting ferrimagnetism with \ngiant χmol = 0.028 at 300 K in aerobic atmospheric condition  which also possesses M -\nH hysteresis loop in room -temperature and large χ mol = 0.037 in 4.0 K, while previous \nmodel organometallic magnetic material  V(TCNE) xγ(CH2Cl2)  possesses a χmol equal \nto 8*10-6 in 4.2 K  and decompose in aerobic condition with short lifetime (31).  \nII. Experimental  and Methods Details  \nTAPA -BTD -COF was prepared  with a mixture of tris(4 -aminophenyl)amine (TAPA) \n(29.0 mg, 0.10 mmol), 4,4 -(benzothiadiazole -4,7-diyl)dibenzaldehyde  (BTD)  (51.4 mg, \n0.15mmol), n-butanol  (2.0 mL), and 1,4-dioxane  (1.0 mL), and acetonitrile (0.05 mL) \ncharging in a cylindrical glass tube (20 cm of length, ϕin = 0.8 cm, ϕout = 1.0 cm) and \nsonicated for 30 min to get a homogeneous dispersion solution. Then 0. 1 mL of 2 M \naqueous acetic acid was dropwise added to the solution, along with the color change to \ndark red. After it was degassed by the typical three freeze −pump −thaw cycles with \nliquid nitrogen, the tube was sealed and then heated at 60 °C for 3 days. The dark re d \nprecipitate was collected by  Soxhlet extraction and  washed continuously with acetone  \nand methanol . Final ly, solids were washed and immersed in acetone  for 24 h. After \nfiltration, samples were dried in vacuum oven under  120 °C overnight.  Yield:  90 %. Powd er X -ray diffraction data were collected using a Panalytical  Empyrean \ndiffractometer in parallel beam geometry employing Cu K α line focused radiation (λ \n=1.5405 Å) at 1600 W (40 kV , 40 mA) power . Sample powders were placing on glass \nsubstrate  recorded from  2θ =1.5° up to 30° with 0.02° increment.  \nSolid -state nuclear magnetic resonance (SS -NMR) spectra were collected on a Bruker \nA V ANCE III 400 NMR spectrometer using a standard Bruker magic angle -spinning \n(MAS)  probe with 4 -mm zirconia rotors. The magic angle  was adjusted by maximizing \nthe number and amplitudes of the signals of the rotational echoes observed in the 79Br \nMASFID signal from KBr. The transmitter frequency of 13C NMR is 100. 39MHz. The \nsolid -state 13C NMR spectra were acquired using cross -polariza tion (CP) MAS \ntechnique wit h the ninety -degree pulse of 1H with 4μs pulse width. The CP contact time \nwas 3 ms. High power two -pulse phase modulation (TPPM -15) 1H decoupling was \napplied during data acquisition. The decoupling frequency corresponded to 62.5 kHz. \nThe MAS sample spinning rates were 8 kHz. Recycle delays between scans were varied \nfrom 2 to 2.5 s. The 13C chemical shifts are given relative to neat tetramethylsilane as \nzero ppm, calibrated using the methylene carbon signal of adamantane assigned t o \n38.48 ppm a s secondary reference.  \nFourier -transform infrared (FTIR) spectra of starting monomer materials and COF \nsamples were recorded from 400 to 4000 cm–1 by using KBr pellets on a Bruker Tenso r-\n27 Fourier -transform infrared spectrometer  Solid -state UV -Vis electron absorption spectra  of starting materials  monomers  and COF \nsample were recorded on a Perkin -Elmer L amda  diffuse -reflectance -spectrometer.  \nScanning electron microscopy and its mapping -mode characterization of COF samples \nwere measured by dispersing the materials onto silica wafers attached to a flat \naluminum sample holder, which were further coated with platinum. Samples were \nanalyzed on a Zeiss  Sigma  300 field-emission SEM operating at 10 kV .  \nElement al analys es were carried out on a Thermo Flash Smart CHNOS organic element \nanalyzer.  \nElectron -paramagnetic -resonance (EPR)  measurements were carried out using an X -\nband spectrometer, Bruker E500  in 300 K under aerobic atmospheric conditions .  \nMagnetization measurements were  performed using a commercial  superconducting \nquantum interference device (SQUID) magnetometer  (MPMS -3, Quantum Design). All \nthe magnetization data of  TAPA -BTD -COF powder crystals were corrected by \nsubtracting diamagnetic sus ceptibility  of empty sample holder in atmospheric air \ncondition . \nModeling of crystal structures for target COFs were performed using the Vesta \n(Visualization and Electronic Structural Analysis) Version 3.5.2 software (K. Momma \nand F. Izumi, \"VESTA 3 for th ree-dimensional visualization of crystal, volumetric and \nmorphology data,\" J. Appl. Crystallogr., 1272 -1276 (2011) (32)). DFT calculation s were carried out within the  framework of the Perdew –Burke –Ernzerhof generalized gradient \napproximation  (PBE -GGA), as embedded in the Vienna ab initio simulation package \ncode (33). All the calculat ions were performed with a plane -wave cutoff energy of 500 \neV . For the  TAPA -BTD -COF, we adopted the experimental lattice constants and \ntransferred to the  primitive cell with a= 45.05700  Å; b= 45.05700 Å; c=3.65000  Å, α= \nβ= 90°, γ=120 ° [full structural relaxation would only change the lattice constants \nnegligibly  <0.1%)]. TAPA -BTD -COF  were  optimized to have AA stacking which will \nbe used for our study of  both bulk and bilayer systems. The geometric optimizations \nwere performed  without any constraint unti l the force on each atom is <0.01 eV ∙Å−1 \nand the  change of total energy is smaller than 10−4 eV per unit cell. The Γ centered \nBrillouin  zone k -point sampling was set with a spacing of 0.03 × 2π∙Å−1, which \ncorresponds  to 3 × 3 × 11  k-point meshes for bulk  unit cell .  \nIII．Results and discussions  \nWe condensed  the two  monomers t ris(4-aminophenyl)amine  (TAPA)  and 4,4' -\n(benzo (1,2,5 )thiadiazole -4,7-diyl)dibenzaldehyd e (BTD) to form imine -connected  \nTAPA -BTD -COF via a Schiff -base reaction. To improve its crystallinity and the \nreaction yield, w e optimized the preparation  conditions by screening multiple \nparameters. After tedious exploration, the optimum preparation conditions  were \nidentified, and we obtained the dark red crystalline powder with 90% yield ( the \nsynthetic route and structure details are shown in  Fig. 1). The successful preparation  of \nthe proposed COF was confirmed by a series of characterization techniques including powder X -ray diffraction (PXRD), solid -state-cross -polarization magic -angle -spinning  \n13C-NMR (SS -CP-MAS -13C-NMR), Fourier -transform infrared spectrometry (FT -IR), \nelement analysis  (EA) , scanning electron microscopy (SEM)  and element -mapping \n(See Fig . 2 and 3 ). The COF ’s structure was first characterized by PXRD comb ining  \nwith theoretical modelling and simulations  (see Fig 2A).  TAPA-BTD-COF exhibited \nsix diffraction peaks in PXRD spectrum, at 2θ = 2.20o, 3.86o, 4.46o, 5.93o, 7.82o and \n24.73o. With the proposed structure and the PXRD pattern at hand, we conducted the \ntheoretical calculation to simulate  the minimum -energy  geometry configuration via the  \nDFT methods  by modelling with VESTA 64  programs  and calculat ing with V ASP  \nsoftware package (see experimental  and methods  details part) . The theoretical \ncalculations indicated that the AA -stacking  structure possessing the following \ndiffraction peaks at 2 .26o, 3.92o, 4.53o, 5.99o, 7.84o and 24.36o well matched  the \nexperimental spectrum.  Then, the full -profile Rietveld refinement using GSAS \nsoftware was conducted. As sho wn in Fig.2 A, the refined AA -stacking structure \nprovided satisfactory  matching results between calculated curves and refined \nexperimental spectr a with small  Rwp=2.47% and Rp=1.80%. From the experimental \ncrystal lographic  results, t he TAPA -BTD -COF lattice parameters were determined as a= \n46.70 Å, b= 4 5.06 Å, c= 3.6 5 Å, α=β=  90.000°, γ= 120.000°. Furthermore,  the COF \nchemical structure  was i nvestigated by SS -CP-MAS -13C-NMR.  There appear four \nprominent  13C signals peaking at 153.7 8, 145.26, 136.74 and  129.37 ppm  of TAPA -\nBTD -COF  (see Figure 2B). Based on the previous reporte  TAPA and BTD -containing \nCOFs structure s, the corresponding carbons in different chemical environment s were assigned.  The peak at the most downfield (153.78 ppm) is belonged to the imine C=N \nand benzothiadiazole C=N carbons (marked as a and b in Figure 2 B inbox). The peak \nat 145.26 ppm origin ates from the quaternary carbons in the TAPA and phenyl rings. \nThe peak center ed at 136.74 ppm corresponds to the quaternary and tertiary carbon s of \nthe phenyl rings between T APA and BTD rings. The peak at the upper -most  field \n(129.37 ppm) corresponded to the ternary carbons in the T APA rings, benzothiadiazole \ncarbons and the phenyl g -position carbons signals. Moreover,  thorough disappearance \nof signals at 196.77 ppm  of C=O indicates that  the aldehy de is  totally  consumed up. \nWith the  new emerg ing signal at 153.78 ppm of imine C=N carbon together , the \nformation of the imine -linked TAPA -BTD -COF was further confirmed. The FTIR \nspectrometry was conduct ed to characterize  the formation of the imine functional group \nand the consummation of the amine and carbonyl  functional groups. As shown in Fig. \n2C, after condensation  reaction , the TAPA amine N -H stretching vibration signals at \n3336 and 3406 cm-1 wavenumbers  disappeared . Meanwhile, both the aldehyde H -C=O \nstretching vibration at 2720 and 2820 cm-1 and C=O stretching vibration at 1687 cm-1 \nwavelength of BTD precursor disappear  in newly formed  COF  FTIR spectrum . Besides , \nthe C=N imine stretching vibr ation at 1617 cm-1 appeared  in the COF  spectrum  whereas \nboth T APA and BTD precursors spectra lacked t his signal. These results further \nevidenced  the formation of imine COF from TAPA and BTD . The electronic absorption \nspectra of the three COFs studied in  this work were characterized via diffuse reflectance \nspectrometry. As shown in Fig. 2D, the obtained TAPA -BTD -COF exhibited red -shift \nin their absorption in comparison with the TAPA  and BTD monomers. This result demonstrated that the formation of imine-based COFs structures brought about the \nhigher degree of conjugation, which widened their UV -Vis absorption range. To obtain \nthe information of their microscopic morphology, the SEM  and element mapping  were \nundertaken (see Fig 3). From the SEM images, we obse rved that the TAPA -BTD -COF \nexhibited ordered spherical nanoparticle morphology with 500 -800 nm diameter  as \nshown in Fig  3A. The element -mapping images by SEM demonstrate s that the C, N and \nS elements distributed homogeneously in the whole COF nanocrystals  (see Figure 3B -\nF). Furthermore,  the chemical compositions of the as -synthesized T APA-BTD -COF,  \nwas confirmed by the element analysis (EA) technique . The obtained EA experimental \nresults well  agreed with the theoretical calculated value based on the proposed COF \nstructures (see Table 1). The experimental weight composition of  TAPA -BTD -COF is \nC (75.07%), H (4.39%), N (12.19%), and S (5.78%) , while the  infinite two-dimensional \nlayer model is (calculated for a single cell : C96H60N14S3) C 76.59%, H 3.99%, N 13.03%, \nand S 6.38%).    Figure 1. The schematic representation of the chemical structure s of TAPA-BTD -COF   \nFigure 2. Structural characterization of TAPA -BTD -COF (A)  X-ray-diffraction  \n(XRD) analysis of TAPA -BTD -COF with the experimentally observed pattern in \nblack, the Rietveld full -profile refined  curves in red, and the difference plot ted in blue. \n(B) Solid -state 13C CP -MAS NMR spectr um of TAPA -BTD -COF. (C) F ourier \ntransform -infrared  spectr um of (C) TAPA -BTD -COF, blue; TAPA monomer , red; \nBTD monomer, black. (D) Solid -state UV -Vis electron -absorption  spectrum of \nTAPA-BTD -COF, blue; TAPA monomer red; BTD monomer , black;  \n \n \n \n \nFigure 3.  Scanning electron microscopy of TAPA -BTD -COF (A) SEM image of TAPA -\nBTD -COF  (B) C, N and S element total mapping in SEM -mapping mode. (C) Separate \nC element distribution in SEM -mapping. (D) S element distribution in SEM -mapping. \n(E) S element distribution in SEM -mapping. (F) The energy spectrum in SEM -mapping.  \nTable 1.  Element analysis results of T APA-BTD -COF  \nTheoretical value  C (76.59%) H (3.9 9%),  N (13.03%)  S (6.38%)  \nExperimental value  C (75.07%)  H (4.39%)  N (12.19%)  S (5.78%)  \nFrom the deficiency of sulfur element as shown in element analysis results \nbetween experimentally obtained TAPA -BTD -COF material and theoretically modelled \nperfect  TAPA -BTD -COF crystal structure  (see Table 1) , we considered that th is S \ndeficiency also represents the deficiency in BTD unit compared with TAPA unit. As \nwell known, sulfur -contain ing BTD (benzothiadiazole) unit is extremely electron -poor, \nwhile TAPA (tris -4-aminophenyl -amine)  is an electron -rich group. The deficiency in \nBTD indicates that the COF  possesses excess electron s compared with its intrinsic state.          \n \n \n \n \n \n \nFigure 4. Electron -paramagnetic -resonance (ESR) spectr um of TAPA -BTD -COF  in \n300 K under aerobic atmospheric conditions.  \nBased on such consideration, we conducted the band -structure calculation and \nanalysis. To our delight, in the intrinsic TAPA -BTD -COF band structure, there appear \na set of 3 bands in including the conduction band bottom (CBM) just above the Fermi \nlevel. The  three bands include two Dirac bands with a crossing point and a flat band \n(conduction band) under these two Dirac bands (see Figure 5). The VBM bands  contain \nRight below the Fermi level is a set of two Dirac bands with a crossing point (Dirac \ncone). The a ppearance of non -dispersive flat band means the existence of infinite heavy \nelectron with zero -velocity. These  largely localized zero -velocity electrons is evidenced \nto belong to strong -correlated electron system. The strong correlation between electrons \nin flat band would generate various exotic non -trivial phenomena such as \nferromagneticity, ferrimagneticity, antiferromagneticity, superconductivity, topological \nstates, etc.  \n \n \n \n \n \nFigure  5. DFT calculated electronic band structure for k z =0 plane and the orbital -\nresolved  projected band.  And the projected density of states (PDOS) with blue , grey \nand yellow lines corresponding to nitrogen, carbon and sulfur element  respectively.  \n(Upper) The full -profile  and (bottom) the magnified band  structure s of VBM and CBM.  \n  \n \n \n \n \n \n \n \n \n \nFigure 6 . Hysteresis loop  in M-H curves  of TAPA -BTD -COF nanocrystal measured  by \nSQUID MPMS -3 at 300 K in aerobic atmospheric conditions. (Upper) the full -profile \nM-H spectrum (bottom) the magnified spectrum at low external field.  \nIn our system, because of the deficiency of electron -poor BTD unit, the whole \nCOF structure is electron excessive, which was prove n by the electron -paramagnetic -\nresonance  (EPR)  characterization in room temperature (300 K) under atmospheric \naerobic conditio n (see Fig ure 4). There appears  a strong signal of electron  with intensity \nabout 2000  in EPR spectrum. To investigate its magnetic property, we conducted the \nM-H magnetization experiments in 300 K under  aerobic  condition . To our surprise, the \nas-synthesize d TAPA -BTD -COF  exhibits pronounced magnetic hysteresis loop during \nM-H measurement  (See Fig ure 6). To further confirm this result, we conducted parallel \nmeasurements of different batches of our TAPA -BTD -COF materials synthesized and \nstored in different tim e. To our delight, the M -H hysteresis loop remained almost \nunchanged despite the choice of different batches COF s material s. To further explore \nthe intrinsic magnetic property of this COF material, the  M-T measurements under \nzero-field cooling and field co oling conditions were undertaken  (see Figure 7) . The M -\nT curves were transformed  to χmol ~ T curve  according  to the density and molar mass \nof TAPA -BTD -COF. As shown in Figure 7 , the χmol of TAPA -BTD -COF exhibits a very \nlarge value about 0.028  in 300 K. This value is considerably larger than common \nparamagnetic material with χmol about 10-3 to 10-5 level. With the continuously cooling, \nthe χ mol slowly reduces from 0.028 to 0.025. However, the curve demonstrates a steep \njump upwards at about 76. 4 K from 0.025 to 0.030 . The reason of this upwards jumping \nis not totally clarified . After the jumping to χ mol = 0.0 30 at 76. 4 K, the COF exhibits the \nsame trend as befor e the jumping. The χmol ~ T slope did not show  apparent change . \nAnd the χ mol remains near 0.030 from 76.7 K until cooling down to approximately 27.4 \nK. Further cooling from 27.4 K to 4. 0 K, the χ mol ~ T slope  value continues to increase  \nfrom 0.030 to 0.037 . And the χ mol value increased almost vertically from 8 K to 4.25 K , \nincreasing from 0.033 to 0. 037. This large χ mol is about the level of ferrimagnetic materials, which belong to strong magnetic materials similar as ferromagne tics with \nspontaneous magnetization  and coercive force  but magnetic domain antiparallel aligned \nwith different mag netic moment, which posses ses χmol in the range of 100 to 102. And \nthe 1/χ mol ~ T curve was also plotted to further analyze its magnetic property. From the \n1/χmol ~ T curve, we did not observe the T N (Néel temperature), which is critical to \nassign an antiferromagnetic materi al. Moreover, from the continuously  increasing \n1/χmol ~ T slope value, we can also ass ign it as a ferrimagnetic material other than \nantiferromagnetics . The relatively lower χ mol and coercive force differs it with \nferromagnetic materials . From the extrapolating  of 1/χmol ~ T curve to cross with x axis \n(y axis value equals zero), we obtained the critical temper ature Δ  equal ing to -33.03 K. \nThis negative value also differ entiates  it with  typical  ferromagnetic materials. Due to \nthese phenomena, we consider that our TAPA -BTD -COF belongs to strong magnetic \nferrimagnetic material.  This is the first stable pure organic non -metal material with \nconsiderable M -H hysteresis loop , coercive force  and giant χ mol up to  0.028 (previously \naerobic non -stable organometallic magnet mainly in the range of 10-6 to 10-4) at room \ntemperature in aerobic condition, and possess a large r χmol up to 0.037 at 4.0 K . \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \nFigure 7. (Upper ) Zero -Field cooled  (ZFC) and field cooled (FC)  magnetization curves  \nχmol ~ T of TAPA -BTD -COF nanocrystals measured at 200 Oe field.  (Bottom) Zero -\nField cooled  (ZFC) and field cooled (FC) curves of 1/χ mol ~T of TAPA -BTD -COF \nnanocrystals measured at 200 Oe field.  \nIV． Conclus ion \nIn summary, we reported the discovery of the first  non-metal organic ferrimagnetic \nmaterial TAPA -BTD -COF , which exhibited unconventional strong ferrimagnetism  and \neven keep its strong and stable magneticity in 300 K and aerobic atmosphere with \nconsiderable M-H hysteresis , coercive force and  χmol up to 0.028 . The synthesis, \ncharacterization and magnetic performance s were provided in detail. We discussed the \norigin of this non -trivial physics phenomena and attributed it to the population of excess \nelectrons  locating and localized  in CBM flat -band. The electron -popula ted flat band \nhosted the location for strong electron correlation. The strongly correlated electron  \nsystem s lead to  this unprecedented ferrimagneticity for our  non-metal and non -free-\nradical organic polymeric material. 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Physikalisches Institut, Universit¨ at G¨ ottingen, 37077 G¨ ottingen, Germany\n7Research Platform MMM Mathematics-Magnetism-Materials,\nUniversity of Vienna, Vienna, Austria\n8International Center for Advanced Studies of Energy Conversion (ICASEC),\nUniversit¨ at G¨ ottingen, 37077 G¨ ottingen, Germany\n1arXiv:2309.12956v1  [cond-mat.mtrl-sci]  22 Sep 2023Abstract\nFemtosecond laser excitation of materials that exhibit magnetic spin textures promises advanced\nmagnetic control via the generation of ultrafast and non-equilibrium spin dynamics. We explore such\npossibilities in ferrimagnetic [Fe(0.35 nm)/Gd(0.40 nm)] 160multilayers, which host a rich diversity\nof magnetic textures from stripe domains at low magnetic fields, a dense bubble/skyrmion lattice at\nintermediate fields, and a single domain state for high magnetic fields. Using femtosecond magneto-\noptics, we observe distinct coherent spin wave dynamics in response to a weak laser excitation\nallowing us to unambiguously identify the different magnetic spin textures. Moreover, employing\nstrong laser excitation we show that we achieve versatile control of the coherent spin dynamics\nvia non-equilibrium and ultrafast transformation of magnetic spin textures by both creating and\nannihilating bubbles/skyrmions. We corroborate our findings by micromagnetic simulations and by\nLorentz transmission electron microscopy before and after laser exposure.\nI. INTRODUCTION\nMagnetic spin textures are expected to be an important building block for future spintronic-\nand magnonic-based memory and logic devices, and even for unconventional computing\ntechniques such as neuromorphic computing [ 1–6]. Especially, localized magnetic solitons,\nknown as skyrmions, have attracted significant attention to be utilized in spin texture based\ndevices [ 7–9]. Potential applications of magnetic textures include the engineering of spin\nwave dispersions in magnonic crystals, using magnetic textures as spin wave phase shifters in\nlogic devices, and employing them as nanoscale spin wave emitters to name a few [ 10–12].\nObviously, the ability to actively control such devices by non-destructive and fast means is\nhighly desirable from an application point of view. Ultrafast optical manipulation of spin\ntextures is a potential avenue to achieve these goals. However, optical spectroscopy studies\nin spin-textured materials systems have so far either focused on precessional spin dynamics\npresent in low-temperature skyrmionic phases [ 13–16], or on statically observed laser-induced\ntransformations of spin textures [ 17–20]. Here we show that we can achieve ultrafast optical\ncontrol of spin textures at room temperature and thus laser-induced tuning of spin wave\nresonances to specific frequency ranges. Our work opens new avenues for future actively\ncontrolled magnetic spin texture-based devices.\n2II. RESULTS\nA. Static magnetic properties of [Fe( 0.35 nm )/Gd( 0.40 nm )]160thin films\nFIG. 1. Static magnetic sample properties: a) Out-of-plane M-Hhysteresis loop of the\n[Fe(0.35nm)/Gd(0 .40nm)]160thin film at room temperature. Solid lines correspond to M-Hdata\nfor increasing (red) and decreasing (blue) field, black squares mark assignments of LTEM images\n(I)–(V) in b) to magnetic field regions in the M-Hloop. The coloration of magnetic field regimes\nrepresent the underlying spin textures. b)LTEM images (grayscale) and micromagnetic simulations\n(color) of the magnetization for an upsweep (bottom, red arrow) and downsweep (top, blue arrow)\nof the magnetic field reveal the spin texture corresponding to different magnetic fields (I)-(V) in\nthe hysteresis loop. The direction of Min the micromagnetic simulation data is given by the color\ncode. c)Legend of different spin objects indicating the field-induced transformations possible for\na magnetic field upsweep with non-chiral and chiral domains walls, type-2 bubbles and clockwise\n(CW) and counter-clockwise (CCW) skyrmions.\nThe studied sample system is a multilayer stack of Fe and Gd with composition [Fe(0.35\nnm)/Gd(0.40 nm)] 160deposited on a thermally-oxidized Si(100) substrate. Static magnetic\ncharacterization of this sample was performed using Superconducting Quantum Interference\nDevice - Vibrating Sample Magnetometry (SQUID-VSM). Samples deposited on thin com-\n3mercial Si 3N4membranes were used for Lorentz Transmission Electron Microscopy (LTEM)\nstudies to image the static magnetic texture in applied out-of-plane (oop) magnetic fields.\nFigure 1a shows the M-Hhysteresis loop from SQUID-VSM for positive magnetic fields and\nFig. 1b displays the corresponding LTEM images (gray-scaled middle panels) for representative\nfield values together with micromagnetic simulations of the field-dependent magnetic texture\n(color-scaled top and bottom panels). Without applied field (Figs. 1a and 1b, left), the sample\nis in a demagnetized state, and the magnetic texture consists solely of oop stripe domains\nwith Bloch domain walls (region I). We find two different types of domain walls, similar to\nprevious studies [ 21,22]: Narrow stripes with domain walls that do not exhibit any chirality,\nand broader stripes with twice the periodicity featuring the same chirality. Applying an\nincreasing oop magnetic field (Fig. 1a, red part of M-Hloop), the oop magnetization of the\nsample initially increases linearly with the applied magnetic field. In LTEM, this process\nis visible as a decrease of the density of domain walls (Fig. 1b, II, bottom images) and the\nnucleation of cylindrical spin objects from collapsed narrow stripe domains (so called type-2\nmagnetic bubbles). For higher magnetic fields (region III), dipolar Bloch-type skyrmions\nstart to nucleate from the broad stripes. In region IV, in a field range of µ0H≈190−250mT,\na bubble and skyrmion (B/SK) lattice with characteristic spacing and groups of spin objects\nof the same type is formed. Close to saturation, the bubbles vanish and only skyrmions\nremain (see SI Fig. S4). For even higher applied fields (region V) magnetic saturation is\nachieved. Decreasing the magnetic field from saturation does not lead to the appearance of\na B/SK lattice in region IV, as reflected in the LTEM images (Fig. 1b, IV, top) and M-H\nloop (Fig. 1a, blue line) and corroborated by the micromagnetic simulation. The measured\noop magnetization in this region is larger than for the upsweep of the magnetic field, as\nno cylindrical spin objects reducing the oop magnetization exist. The small hysteresis in\nthis field region in the M-Hloop is thus indicative of the disordered B/SK lattice. For\nfurther decreasing field strengths (regions III to I), the evolution of the magnetic texture\nis again comparable to the upsweep, except that, at least in region III, the initial domain\nwall density is lower in the decreasing field case. Please note that small differences observed\nin the magnetic sample properties between SQUID-VSM and LTEM measurements, i.e.,\nhigher magnetic field values for the same spin texture in LTEM, arise from the fact that\ntwo different substrates are used for the sample preparation resulting in different growth\nconditions of the films. Micromagnetic simulations match the magnetic field values of the\n4corresponding SQUID-VSM data and are discussed further below.\nB. Coherent spin dynamics in response to weak optical excitations\nAfter the thorough characterization of the ground state magnetic spin textures, we now\nfocus on the magnetic response of the sample to an ultrashort optical excitation. In particular,\nwe seek correlations between the temporal response and the different spatial textures in Fig. 1.\nFor these measurements, we use a comparably weak perturbation (fluence F= 300 µJ cm−2,\npulse duration τp<40 fs) leading to a total demagnetization of the sample by only 3 .5 %.\nMagnetization dynamics M(t) were detected by changes in the light polarization upon\nreflection of the sample using the magneto-optical Kerr effect (MOKE) in polar geometry\nwith the magnetic field applied in oop direction. For details see the methods section.\nFIG. 2. Out-of-plane magnetization dynamics for a weak perturbation in dependence\nof the applied oop magnetic field. Magnetization dynamics for a)upsweep and b)downsweep\nof the magnetic field. The colors of the curves represent the regions of different spin texture as\ngiven in Fig. 1a: Stripe domains (blue), B/SK lattice (green), and magnetic saturation (red). Data\nis normalized to the Kerr signal in magnetic saturation in oop direction.\nFigure 2 depicts the magnetization dynamics in response to the optical excitation and\nin dependence of the applied oop magnetic field for both an upsweep (Fig. 2a) and a\ndownsweep of the magnetic field (Fig. 2b). Four processes can be identified: Immediately\nafter photoexcitation, the magnetization is quenched on a timescale of only 300 fs. This\nultrafast demagnetization process is followed by a fast remagnetization on a few picosecond\ntimescale. During the remagnetization, about half of the initial magnetization is recovered,\n5and the magnetization stays constant thereafter for up to ≈40 ps depending on the applied\nmagnetic field. Finally, the magnetization recovers on a ns-timescale by thermal transport\nout of the film. Most notably, in addition to this well-known and incoherent magnetic\nresponse [ 23,24], a coherent oscillatory signal component is clearly visible in the stripe\ndomain phase (blue transients) and the B/SK lattice phase for the upsweep of the magnetic\nfield (green transient).\nFIG. 3. Coherent magnetization dynamics in the experiment and micromagnetic\nsimulation in dependence of the applied oop magnetic field. Coherent magnetization\ndynamics for a)upsweep and c)downsweep of the magnetic field with the corresponding Fourier\nspectra b)andd), respectively. In case of the upsweep a),b)a unique spin wave mode is observed\nforµ0H= 190 −240 mT.\nThe main question now is to what extent this coherent response is indicative of the specific\nspin texture in the Fe/Gd sample. To answer this question, we performed micromagnetic\nsimulations of the spin dynamics after a magnetization quench of less than 10 % (see Methods).\nFigure 3 depicts a full comparison of the experimentally found and simulated coherent\noscillatory signal components after subtraction of the incoherent background for both the\nupsweep (Fig. 3a) and downsweep (Fig. 3c) of the magnetic field. The corresponding Fourier\ntransforms for experiment and micromagnetic simulation are shown in Fig. 3b and Fig. 3d,\nrespectively.\n6Evidently, experiment and simulation match well and show a clear correspondence of the\nunderlying spin textures with the observed coherent spin dynamics. The Fourier spectra for\nmagnetic fields of µ0H= 0−190 mT, i.e., the stripe domain state, show a strong oscillatory\nmode m1with f1≈2.3 GHz, which is identified as a breathing mode of stripe domains in the\nmicromagnetic simulation. A much weaker high-frequency component m2with f2≈5.4 GHz\nis additionally observed in the experiment, however, its origin needs to be further investigated.\nFor magnetic fields of µ0H≈190−240 mT the disordered B/SK lattice is present in the\nupsweep of the magnetic field only. Correspondingly, experiment and simulation show only\nin this case an oscillatory mode mbskwith fbsk≈1.4 GHz. This frequency is identified as\na breathing mode of bubbles and skyrmions [ 25] in the micromagnetic simulation in very\ngood agreement with previous studies on skyrmion dynamics [ 13,15,26]. We note that\nmicromagnetic simulations indeed predict a slightly lower frequency for magnetic bubbles\ncompared to skyrmions, however, we cannot resolve this difference with our experimental\nfrequency resolution. Last, in saturation, no oscillatory spin dynamics are observed in\nexperiment and simulation.\nWhile the agreement between experiment and simulation is very good in general, we\nnote that small differences do exist in the magnetic field range of µ0H≈150−200 mT\nfor both field up- and downsweep. Furthermore, there is an apparent phase shift in the\noscillatory dynamics with magnetic field in both experiment and simulation. For details on\nthese observations we refer to the Supplementary Information Section V A.\nC. Magnetic phase transformation in response to a strong optical excitation\nHaving shown a clear correlation between spin texture and spin dynamics, we now focus\non the response of the spin texture to a strong non-equilibrium excitation of the spin system.\nWe repeat the experiments from above for a strong laser excitation of F= 3mJ cm−2, i.e.,\nan order of magnitude larger fluence than before, and we find significant changes in the\nobserved dynamics. A much stronger incoherent demagnetization of about 45% occurs, i.e.,\nthe initial magnetic spin texture is strongly disturbed (see SI Figs. S6a and b). For a detailed\ninvestigation, we again subtracted the incoherent background (Fig. 4a and b) and performed\na Fourier transform (Fig. 4c and d).\nWe now observe the stripe domain phase mode m1in a magnetic field range µ0H=\n7FIG. 4. High-fluence coherent magnetization dynamics in time-resolved MOKE in\ndependence of the applied oop magnetic field. High fluence coherent magnetization dynamics\nfora)upsweep and b)downsweep of the magnetic field with the corresponding Fourier spectrum\nc)andd), respectively. For both directions of magnetic field sweep a single spin wave mode is\nobserved within µ0H= 100 −190 mT.\n0−100mTwith slightly lower frequency than before. A second, higher-frequency mode m2is\nweakly present at f2≈4.8GHz. The mode previously identified to be B/SK breathing now\nappears around fbsk≈1.2GHz in the magnetic field range µ0H= 100−190mT. In contrast\nto the weak perturbation limit, the mode now exhibits a clear frequency dependence, shifting\nfrom f≈1.4GHz tof≈1.0GHz from low to high magnetic fields. Above µ0H= 190 mT\nno oscillatory behavior is observed anymore. Most surprisingly, however, we now observe\nthembskmode even in the downsweep of the magnetic field (Fig. 4b and d), where it was\npreviously absent. This observation suggests that the strong laser excitation itself might\ninduce a B/SK lattice phase in the material. Indeed, skyrmion nucleation from a saturated\nstate by strong laser excitation was recently theoretically predicted [ 27] and experimentally\nobserved [ 18,19], which could explain our findings. However, our micromagnetic simulations\n8strongly suggest that a nucleation of skyrmions from saturation does not appear to be possible\nfor the Fe/Gd multilayers studied here.\nTo understand the observed behavior better, we performed LTEM measurements before\nand after femtosecond laser excitation. These measurements indeed allow us to identify\nthe changes in dynamics from Fig. 3 to Fig. 4 as a laser-induced transformation of the\nunderlying spin texture. We will show in the following that this transformation is not\ntrivial, and not dominantly given by a simple static laser-induced temperature shift of\nphase stability ranges [ 21] together with laser-induced skyrmion nucleation from magnetic\nsaturation [18, 19, 27].\nFIG. 5. Laser-induced transformation of magnetic textures in LTEM. LTEM experimental\ndata showing the magnetic texture before and after strong laser excitation of F= 6.4mJ cm−2.\nA)Unmodified stripe domain phase for low magnetic fields; B)transformation of a mixed phase\ninto a B/SK lattice phase; C)annihilation of bubbles and skyrmions; D)unmodified saturated\nmagnetic state; E)transformation of a stripe state in the downsweep of the magnetic field into a\nmixed state (see SI Fig. S9) and then further into a B/SK lattice state.\nUpon ultrafast and strong laser excitation, we distinguish five scenarios depending on\nthe field-dependent magnetic ground state of the system depicted in Fig. 5 starting with\nthe upsweep of the magnetic field: A) For low magnetic fields, a pristine stripe domain\nphase remains nearly unchanged after laser excitation. B) A former mixed stripe domain\nand B/SK state is turned into a pure B/SK lattice phase, as also observed in the simulation,\n9see SI Fig. S10. C) Starting within the B/SK lattice phase, bubbles and skyrmions are\nannihilated by the laser excitation, i.e., the system is driven into a fully saturated state\neven by a single laser shot. Thereafter, the new saturated magnetic state is not modified by\nfurther laser excitation. D) From saturation, we did not observe any laser-induced creation\nof skyrmions without irreversibly changing the sample properties, which is in contrast to\nliterature [ 18,19,27]. However, this observation concurs with the micromagnetic simulations,\nwhere it was also not possible to nucleate skyrmions from a saturated spin state. Note that\nclose to saturation in the downsweep of the magnetic field, a gradual transformation of the\nsystem starting from a single isolated stripe domain appears possible (see SI Fig. S7). E)\nFor the downsweep, a plain stripe domain phase with low stripe density is transformed into\na mixed phase. Here, all stripe domains convert into cylindrical spin textures after exposure\nto a single laser pulse and new stripe domains nucleate in regions that are initially saturated\n(see SI Fig. S9). Upon subsequent laser excitation the newly generated domains transform\ninto B/SKs. Hence, using ultrafast and strong laser excitation, a pure B/SK lattice phase\ncan be created. A schematic phase diagram of the magnetic texture depending on magnetic\nfield and the number of incident laser pulses can be found in SI Fig. S8.\nIn comparison to literature, for a Co/Pt multilayer sample, Gerlinger and coworkers [ 28]\nobserved laser-induced skyrmion nucleation from both i) magnetic saturation and ii) the\nstripe domain phase. Furthermore, they observed iii) an annihilation of skyrmions and stripe\ndomains close to saturation. In our study, we obviously find similar evidence for processes ii)\nand iii), but not for i), the laser-induced nucleation from saturation.\nHowever, taking the predictions from micromagnetic simulations and the observations from\nthe LTEM measurements into account, we can now understand the changes in the coherent\noscillatory component that come along with the strong excitation in Fig. 4: In the upsweep\nof the magnetic field, the former B/SK lattice phase completely vanishes before the actual\ndata recording starts due to continuous excitation by laser pulses in the time-resolved MOKE\nexperiment. Hence, we observe no oscillatory component, but a signal corresponding to\nmagnetic saturation in the magnetic field range 190 < µ 0H < 250mT(scenario C, Fig. 5). For\nlower magnetic fields, the initially mixed stripe domain and B/SK phase is laser-transformed\ninto a pure B/SK lattice phase, leading to the apparent shift of the mbskmode to lower\nfrequency at high fields, 150 < µ 0H < 190mT(scenario B, Fig. 5). The slightly higher\nfrequency observed in Fig. 4c for magnetic fields of about 110 < µ 0H < 150mToriginates\n10from a mixed phase that forms upon laser excitation out of a pristine stripe domain phase\nexhibiting none or only very few skyrmions, respectively. This means that no final pure B/SK\nlattice phase is created within this field range (see also SI Fig. S5). In the downsweep of the\nmagnetic field, for high fields (150 < µ 0H < 190mT) a pure B/SK lattice phase is formed\nout of a pure stripe phase (scenario E, Fig. 5), and, for lower fields (100 < µ 0H < 150mT),\na mixed phase is the final state. All these mechanisms lead to the generally more equal\noscillatory behavior observed for the strong perturbation case for magnetic field up- and\ndownsweep. However, the absence of an initially mixed phase in the downsweep hinders the\ngeneration of a B/SK lattice phase. For equal magnetic fields we observe less bubbles and\nskyrmions being created in the downsweep of the magnetic field resulting in a dominance of\nthe stripe’s oscillatory component in the generated mixed phase. Finally, the generally lower\nfrequencies of all modes can most likely be attributed to the transient laser-induced change of\nmacroscopic sample properties, e.g., sample temperature, magnetization and anisotropies [ 21].\nClosing the discussion, three further points are noteworthy: First, an asymmetry in the\nsystem response exists for the upsweep and downsweep of the magnetic field in the field\nregions where no B/SKs are observed in static studies. This asymmetry increases with a\nstronger perturbation. This is evident from the different shifts of magnetic oscillations for\nthe upsweep and downsweep of the magnetic field in Fig. 3a and c as well as Fig. 4a and b,\nwhich even shows a frequency shift between the up and down field sweeps (cf. Fig. 4c and d).\nHowever, SQUID-VSM and LTEM data from Fig. 1 do not suggest any specific macroscopic\ndifference in the low-field range for both field sweep directions, except possibly in domain wall\ndensity. Currently, we do not have an explanation for this behavior, but aim to investigate\nthis further in the future. Second, in contrast to Refs. [ 18,19,28], skyrmion creation in\nour case is only possible if there is an initially existing chiral spin texture. Even further,\nthe nucleation of skyrmions does not only have a lower but also an upper fluence threshold.\nAbove a certain fluence, a mixed phase is formed in the region of maximum excitation that\nis surrounded by a pure skyrmion phase due to the Gaussian intensity distribution of the\nlaser pulse, see Fig. S12 in the SI. Last, we want to note that no influence of spin texture on\nthe incoherent magnetization dynamics was observed, neither on the sub-ps timescale nor\nduring the remagnetization.\n11III. CONCLUSION\nIn summary, we studied the response of the spin-textured material Fe/Gd to weak\nand strong ultrafast laser excitation. In the weak excitation limit, unique coherent spin\ndynamics evolve that allowed us to directly identify the magnetic stripe domain phase, the\nbubble/skyrmion lattice phase, as well as magnetic saturation. In the strong excitation limit,\nwe observed significant changes in the dynamic magnetic phase diagram, which we attributed\nto a laser-induced transformation of the spin texture as verified by micromagnetics and LTEM\nmeasurements. Indeed, from the stripe domain state, we created both a bubble/skyrmion\nlattice and a mixed phase of bubbles, skyrmions and stripe domains, depending on the\nexternal magnetic field. These results highlight that ultrafast optical control of spin wave\nresonances is possible by optical transformation of the spin textures in the material. Moreover,\nthese rich spin physics are observed at room temperature and at quite moderate magnetic\nfields and laser fluence, making the Fe/Gd multilayer system an attractive model system\nfor fundamental studies of spin texture-based physics, as well as potential application in\nspin-electronics in the future.\nIV. METHODS\nA. Sample preparation and characterization\n[Fe(0.35 nm)/Gd(0.40 nm)] 160multilayer samples were prepared on both thermally-oxidized\nSi(100) substrates, as well as on 30-nm thick Si 3N4membranes required for LTEM imaging.\nThe films were deposited at room temperature by dc magnetron sputtering in one run.\nNevertheless, slightly varying magnetic properties were obtained due to different growth\nconditions during deposition on the two different substrates. The sputter process was carried\nout using an Ar working pressure of 3.5 µbar in an ultra-high vacuum chamber. For all\nsamples, a 5-nm-thick Pt seed layer and a 5-nm-thick Si 3N4capping layer were used to\nprotect the films from oxidation. The thickness of the layers was controlled by a calibrated\nquartz balance during layer deposition.\nThe magnetic properties of the samples were analyzed by Superconducting Quantum In-\nterference Device-Vibrating Sample Magnetometry (SQUID-VSM). M-Hhysteresis loops\nwere measured both in out-of-plane and in-plane configuration at room temperature. The\n12magnetic spin textures were imaged by Lorentz Transmission Electron Microscopy (LTEM)\nat room temperature using a JEOL NEOARM-200F system operated at 200 keVbeam energy\nin the Fresnel mode with an underfocus of 2 mm. Images are recorded with a Gatan OneView\ncamera in external out-of-plane magnetic fields.\nB. Time-resolved magneto-optical Kerr spectroscopy (TR-MOKE)\nTransient reflectivity ∆ R(t) and magnetization dynamics ∆ M(t) were measured simulta-\nneously using a bichromatic pump-probe setup. Sub- pslaser pulses are generated by a fiber\namplifier system operating at a repetition rate of 50 kHz. A pulse duration of less than 40 fs\nis achieved at the sample position via spectral broadening in a gas-filled hollow core fiber and\nsubsequent compression with chirped mirrors [ 29]. In the setup, the fundamental light pulses\nwith central wavelength of 1030 nmand their second harmonic at 515 nmare used as pump\nand probe pulses, respectively. External out-of-plane magnetic fields up to µ0H= 0.7 T can\nbe applied by mounting the sample in-between the poles of a variable-gap electromagnet.\nThe data is measured by a balanced bridge detector and collected by a 250 MHz digitizer\ncard enabling single pulse detection. Using a mechanical chopper to improve signal quality,\nboth the pumped and the unpumped signals can be recorded, allowing for maximum scope\nof data post processing. Data was acquired in polar Kerr geometry, where the magnetic field\nwas applied in oop direction. All M(t) data shown were recorded as the difference of M(t)\ntraces for two opposite magnetic fields to remove nonmagnetic signal components, taking\ncare to set the correct spin texture by approaching the current magnetic field either from\nbelow or above.\nC. Lorentz Transmission Electron Microscopy with in-situ optical excitation\nTo obtain a better understanding of the spin dynamics observed in TR-MOKE in the\nstrong excitation limit, we studied a sample on a thin commercial Si 3N4membrane by LTEM\nat the G¨ ottingen Ultrafast Transmission Electron Microscope, allowing for in-situ optical\nexcitation [ 17,30,31]. We recorded energy-filtered images using a CEOS CEFID equipped\nwith a TVIPS XF416 at a 10 eVwindow-width, an acceleration voltage of 200 keVand a\n6mmdefocus. In these experiments a laser source with a wavelength of 750 nmand a pulse\n13width of around 300 fsexcites the sample under a near normal angle of incidence with a\ndefined number of laser pulses starting from single shot. At various oop magnetic fields,\nmicrographs were acquired before, after a single and after multiple excitations to observe\nchanges of the samples magnetic texture. In order to obtain reproducible magnetic textures\nbefore laser excitation, we fully saturated the sample with an oop field before approaching a\nspecific field magnitude.\nIn addition to the somewhat different laser conditions, the LTEM sample on the thin\nTEM membrane has different thermal couplings and strain compared to the MOKE sample\non the rigid Si substrate. This also leads to slightly higher magnetic field ranges, where\nthe stripe phase, B/SK lattice phase, and magnetic saturation are present. Due to these\ndifferences in experimental conditions, a quantitative fluence matching between TR-MOKE\nand LTEM experiments was not attempted. However, the LTEM results, at a fluence about\na factor of 2 higher than in TR-MOKE, allow to fully understand the the changes to the\nmagnetic texture by laser excitation and allow to perfectly explain the observations in the\nTR-MOKE measurements. To obtain the fluences in the LTEM measurements from the laser\npulse energy, a 1 /e2beam diameter of 60 µmwas used, estimated from the maximum region\nof switched magnetic texture upon laser excitation.\nD. Static micromagnetic simulations\nWe use magnum.np [ 32], a GPU-enhanced micromagnetic simulation software, to per-\nform large-scale micromagnetic simulations. We simulate the Fe/Gd multilayers as a 3D\nferromagnet with low saturation magnetization Ms= 340 kA/m and low perpendicular\nmagnetic anisotropy Ku= 40 kJ/m3. The exchange constant is chosen Aex= 6pJ/m. Our\nfinite-difference micromagnetic solver discretizes the simulation box into 512 ×512×10\ncuboids, each with a constant volume of 10 ×10×11.2nm3. Note that all simulations are\nperformed without any thermal fluctuations. Basically, the system is simulated at T= 0 K,\nbut with effective magnetic parameters that were measured at room temperature. Similar\nmodeling was used in experimental studies on Fe/Gd-based multilayers [ 21,22]. To obtain\ntheM-Hhysteresis loops shown in Fig. 1, we start from a cellwise random magnetization\nstate. We relax the structure in zero field by solving the Landau-Lifshitz-Gilbert (LLG)\nequation numerically [ 33] at high damping α= 1.0, where we include the demagnetization,\n14exchange, anisotropy and Zeeman energies in the calculation of the effective field. We obtain\nthe stripe domain structure as our new initial state. Once the magnetic system is relaxed\nwith respect to its energy, we apply an oop field along the z-direction. We increase the\nmagnetic field by 2 mTafter each relaxation step in which we solve the LLG for 10 ns. The\ncomplete hysteresis is then simulated by raising and lowering the field between ±250mT.\nMagnetization states at each field are saved to be used as input for dynamic simulations to\nsimulate the resonance of the spin objects.\nE. Dynamic micromagnetic simulations\nWith the magnetization states obtained from the static simulations, we now simulate the\nintrinsic response of the spin textures. Commonly, ultrafast laser demagnetization can be\ndescribed micromagnetically by including a stochastic temperature-dependent term to the\nLLG, where the temperature in the magnetic system is calculated from the excitation energy\nvia the two-temperature model [ 34]. However, in our study we are not interested in the\ndemagnetization process itself, but rather in the excitation of the magnetic system and how\nthis relaxes into its equilibrium state. Thus, we employ a similar approach as applied for\nsimulating ferromagnetic resonance (FMR) [ 35]. That is, we assume that the laser pulse heats\nup the system and alters the temperature-dependent magnetic parameters. Excitation of the\nsystem occurs in the system trying to return to its original energetic minimum. Therefore,\nwe assume that during pumping Msis reduced with an excitation amplitude between 5%\nand 25%. KuandAexscale with M2\ns[36]. We start from the magnetization states from the\nzero-temperature hysteresis at a defined field. We solve the LLG for 1 nsat moderate damping\nα= 0.1 to get the magnetization near its new energetic minimum for the new set of magnetic\nparameters. We then abruptly set the initial magnetic parameters back and reduce the\ndamping to α= 0.02. The simulation time is 10 ns. The magnetization starts to oscillate back\ninto the original or new ground state, based on the strength of the excitation. Total resonances\ncan then be captured by first recording the averaged magnetization components for every\n∆t= 1psand performing an FFT. To probe the influence of a stroboscopic measurement\ntechnique, we repeat the excitation up to ten times and record the magnetization state\nafter each excitation, where changes in the magnetization states can be observed for high\nexcitation amplitudes. The resonance frequencies of the stripe domain and the skyrmion\n15are separated by averaging the magnetization in a smaller simulation box. For the skyrmion\nphase we average mzfor a single skyrmion and record it for each time step ∆ t= 10 ps. For\nthe stripe domain we first find an appropriately small section where only one stripe domain\nis stable and has almost parallel domain walls. We then sampled the magnetization along\na line that is 500 nmlong. The magnetization is then averaged for each component at the\nsame time step as used for the skyrmions.\nF. Data availability\nThe data that support the findings of this study are available from the authors upon\nreasonable request.\nG. Code availability\nMicromagnetic simulations were performed with the open source micromagnetic simu-\nlator magnum.np [ 32]. Simulations scripts to reproduce the results can be requested from\ncorresponding authors.\nH. Acknowledgements\nS.K. and C.A. gratefully acknowledge the Austrian Science Fund (FWF) for support\nthrough Grant No. P34671 (Vladimir). S.K. and D.Su. acknowledge the Austrian Science\nFund (FWF) for support through Grant No. I 6267 (CHIRALSPIN). T.S. and M.A. grate-\nfully acknowledge funding from Deutsche Forschungsgemeinschaft (DFG, German Research\nFoundation) grant no. 507821284. D.St., S.M. and T.T. acknowledge funding by the DFG,\ngrant no. 217133147/SFB 1073, project A02. The computational results presented have been\nachieved in part using the Vienna Scientific Cluster (VSC).\nI. Author contributions\nD.St., M.A., and S.M. conceived the project. T.T. performed the time-resolved Kerr\neffect studies and analyzed the time-resolved magnetism data with supervision from D.St.\nSample growth, static SQUID-VSM and LTEM analysis were performed by T.S. LTEM\n16with femtosecond laser excitation was conceived by C.R., measured by M.M. and T.T.,\nand evaluated by T.T. S.K., F.B., and C.A. wrote and improved the micromagnetic code.\nS.K. performed the micromagnetic simulations under the supervision of C.A. and D.Su. The\nsimulations were evaluated by S.K. and T.T. with support from D.St. T.T., D.St. and\nS.M. wrote the paper with input from all authors.\nJ. Corresponding authors\nCorrespondence to Daniel Steil or Stefan Mathias\nK. Competing interests\nThe authors declare no competing interests.\n[1]Finocchio, G., B¨ uttner, F., Tomasello, R., Carpentieri, M. & Kl¨ aui, M. Magnetic skyrmions:\nfrom fundamental to applications. Journal of Physics D: Applied Physics 49, 423001 (2016).\n[2]Fert, A., Reyren, N. & Cros, V. 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Micromagnetics and spintronics: models and numerical methods. The European\nPhysical Journal B 92, 120 (2019).\n[34] Atxitia, U. et al. Micromagnetic modeling of laser-induced magnetization dynamics using the\nLandau-Lifshitz-Bloch equation. Applied Physics Letters 91, 232507 (2007).\n[35]Baker, A. et al. Proposal of a micromagnetic standard problem for ferromagnetic resonance\nsimulations. Journal of Magnetism and Magnetic Materials 421, 428–439 (2017).\n[36]Moreno, R. et al. Temperature-dependent exchange stiffness and domain wall width in Co.\nPhys. Rev. B 94, 104433 (2016).\n19[37] Bayer, C., Schultheiss, H., Hillebrands, B. & Stamps, R. L. Phase shift of spin waves traveling\nthrough a 180◦bloch-domain wall. IEEE Transactions on Magnetics 41, 3094–3096 (2005).\nV. SUPPLEMENTARY\nA. Weak perturbative regime: Extended discussion of the spin dynamics of stripe\ndomains, bubbles, and skyrmions\nThis section provides an extended comparison of the experimental results and micromag-\nnetic simulations in the weak perturbation limit. We also provide further information on the\ndifferent observed breathing modes by comparison of experiment and simulation.\n1. Field-dependent differences in coherent spin dynamics between experiment and simulation\nFor magnetic fields <150mTthe results obtained from modelling and experiment in\nFig. 3 are in very good agreement. The mode m1with frequency f1≈2.3GHz is observed\nin both cases. However, the higher-frequency mode m2as well as the opposite phase shift\nobserved in experiment for up- and downsweep of the magnetic field are not reproduced by\nthe simulation. We attribute these differences to sample properties not accounted for in the\nsimulation, e.g., sample inhomogeneities and defects as well as the fact that the simulation is\na strongly discretized situation.\nFurthermore, in the magnetic field range 150 mT< µ 0H < 200mTa sharp transition\nexists between the two modes m1andmbskin the experiment, while in the simulation this\ntransition appears to be more continuous. Additionally, a transition area for mode m1is\nalso present for the downsweep of the magnetic field in the simulation. We explain this\ndifference to the experiment with the high amount of cylindrical-like spin objects in the\nsimulated ground state of the mixed phase for both magnetic field up- and downsweep, see\nmicromagnetic simulation data in Fig. 1b, image III. In contrast, according to the LTEM\ndata, the ground state in the experiment is mainly governed by stripe domains. Due to the\nstrong spatial discretization within the simulation, as well as the absence of defects likely\nacting as nucleation sites for domain walls in the actual sample, the evolution of spin texture,\nespecially starting from a formerly saturated state, can be expected to deviate from the\n20experiment.\n2. Breathing mode of a single skyrmion\nA snapshot of the spin texture is collected for each time step of the micromagnetic\nsimulation. We select one skyrmion ( µ0H= 200 mT, upsweep) to demonstrate the time\nevolution of the skyrmion breathing mode for a single period in Fig. S1.\nFig. S1. Breathing mode of a single skyrmion. One period of the breathing mode of a single\nskyrmion obtained from micromagnetic simulations. The black circle indicates the initial skyrmion\nsize as a guide to the eye.\n3. Comparison of stripes, bubble/skyrmion breathing modes in simulation and experiment\nThe width of a stripe domain and the size of B/SKs change periodically as a response\nto the excitation. This breathing mode causes a periodic oscillation of the magnetization\ncomponent along the oop direction, which can be observed both in our TR-MOKE experiment\nand in micromagnetic simulations as depicted in Fig. S2. Note that the experimental data\ncorresponds to the integrated collective spin response over the probed sample area, whereas\nthe simulation (solid lines) shows the integrated Mz-component of the magnetization in a\nsmall area containing a single skyrmion or a single stripe domain. Especially at early times,\ndeviations are to be expected, because the excitation in experiment and the micromagnetic\nsimulations are different (see Methods section). Furthermore, in the stripe breathing the\nhigher frequency mode m2is present in the experiment, but not in the simulation. Overall,\nconsidering these differences, experiment and theory match closely.\n21Fig. S2. Breathing modes of stripes and B/SK. Low-fluence coherent magnetization dynamics\nwithin the stripe and B/SK phase that can be linked to the collective breathing modes of stripes\nand skyrmions, respectively. Data points are experimental data corresponding to the collective\nresponse of the Mz-component of the spin system in the probed sample area, solid lines depict the\nMz-component of the magnetization from the micromagnetic simulations for an area of 300x300 nm2\ncontaining a single skyrmion (bottom) or 300x500 nm2for part of a stripe domain (top).\n4. Phase and frequency shifts in oscillatory spin dynamics\nIn general, the spin wave dynamics observed in Fig. 3 and Fig. 4 exhibit a field-dependent\nphase and/or frequency shift both in the stripe domain and B/SK lattice phases. The\nexpected direction of these shifts is currently not well understood by the authors, but is\nbelieved to partially result from the density of domain walls that will influence the spin wave\nphase [ 37]. We discuss below two particular examples from Fig. 3 and outline our current\nunderstanding of the observed effect.\na. B/SK lattice phase shift—weak perturbation limit The frequency of mbskdoes not\nvisibly depend on the oop magnetic field in experiment and simulation. However, we\nexperimentally observe a shift of the oscillation onset by ∆ t≈100pswith increasing\nmagnetic field and a phase shift is also visible in the simulation. Here, this phase shift\ninterestingly appears together with the magnetic field-dependent vanishing of bubbles and\n22shrinking of skyrmions. As depicted in Fig. S3, the micromagnetic simulation predicts a drop\nin both size and number of cylindrical spin objects with respect to an increasing applied\nmagnetic field. As a result, the amount of Bloch-like domain walls is reduced. However, the\nphase of a spin wave is modified when transmitting through a Bloch-like domain wall [ 37].\nSince there are fewer domain walls at higher magnetic fields, the phase shift must change,\nwhich we attribute to the observed phase shift of the mbskmode.\nFig. S3. Micromagnetic simulation of field-dependent B/SK evolution. Evolution of the\nsize of cylindrical spin objects with magnetic field in the micromagnetic simulation. The direction\nofMis given by the color code.\nExperimentally, we find that skyrmions are more stable with respect to an increasing\nmagnetic field than bubbles. As depicted in Fig. S4, the bubbles continuously vanish with\nincreasing field while the skyrmions remain almost unaffected. As a result, within a specific\nmagnetic field range, only skyrmions are present. Again, the density of domain walls is\nreduced, supporting the above argument.\nb. Coherent magnetization dynamics within the mixed phase Considering the upsweep of\nthe magnetic field in the experiment, we observe a non-monotonous behavior of the observed\noscillation shift within the stripe domain phase, e.g., the oscillations shift to the right for\nmagnetic fields µ0H > 150mT(see Fig. 3a). We attribute this behavior to the coexistence\nof stripes and B/SKs within this field range. As a result, the coherent dynamics can be\nmodelled as a superposition of the stripe domain and B/SK breathing mode. In Fig. S5 we\nexemplarily constructed the coherent dynamics for µ0H= 193 mTby using the breathing\nmodes shown in Fig. S2 with a B/SK to stripe domain ratio of 1 .4. For the strong excitation\nregime given in Fig. 4c) and d), this effect manifests in a resolvable frequency shift in the\nmixed phase region between 110 < µ 0H < 150 mT.\n23Fig. S4. LTEM measurement of field-dependent B/SK evolution. Experimentally measured\nspin texture in dependence of the magnetic oop field for upsweep of the magnetic field. Bubbles are\nobserved to be less stable than skyrmions.\nFig. S5. Coherent magnetization dynamics in the mixed phase state. Low-fluence coherent\nmagnetization dynamics within the mixed phase with experimental data as red squares. The\ndynamics are modelled (solid line) by a superposition of the coherent dynamics within stripe domain\nand B/SK lattice phase as depicted in Fig. S2\n.\n24B. Strong perturbative regime: Extended discussion of the laser-induced transfor-\nmation of spin textures\n1. Magnetization dynamics: Strong optical excitation\nFigure S6 depicts the magnetization dynamics for a laser excitation of F= 3mJ cm−2.\nIn comparison to the weak excitation shown in Fig. 2, we observe a much stronger inco-\nherent demagnetization of about 45%, which is indicative for the strong optically-induced\nperturbation of the sample.\nFig. S6. Field-dependent TR-MOKE for strong optical excitation. Out-of-plane magnetiza-\ntion dynamics in dependence of the applied oop magnetic field for upsweep a)and downsweep b)of\nthe magnetic field in the high fluence regime. The colors of the curves represent the attribution to\ndifferent static spin textures: Stripe domains (blue), bubble/skyrmion lattice (green), and magnetic\nsaturation (red).\n2. Skyrmion nucleation from an isolated chiral domain wall\nThe process of skyrmion nucleation starting from an almost fully saturated state in\nthe downsweep of the magnetic field is depicted in Fig. S7 a-i. The LTEM images were\nsignificantly post-processed to obtain better visibility of the magnetic texture. In more detail,\nthe contours of chiral spin objects associated with black and white LTEM contrast were\nextracted by running a computer vision algorithm. Small contours that are referred to image\nnoise were disregarded. The remaining contours were then filled and framed with color to\nincrease the visibility of the spin objects. The color indicates the underlying chirality of\n25the spin object, with blue objects and red objects originating from black and white LTEM\ncontrast, respectively.\nFig. S7. Skyrmion nucleation from an isolated chiral domain wall. From a)toi): Evolution\nof spin texture in dependence of the number of laser pulses with fluence of 6 .4mJ cm−2. The\nmagnetic field is chosen to be at the transition from saturation to the stripe domain phase during a\ndownsweep of the magnetic field. Note that the LTEM images were significantly post-processed to\nenhance the visibility of magnetic skyrmions and their chirality.\nFor the chosen magnetic field of µ0H= 210 mTin the downsweep of the magnetic field\nonly a single stripe domain is present. This stripe domain transforms into a local skyrmion\nlattice upon excitation by the first laser pulse. However, multiple shots are required to\nnucleate a new chiral stripe domain pointing towards a rather stochastical nature of this\nprocess. During the following five shots, we observe a subsequent growth of the generated\nskyrmion phase that remains almost unchanged after ten additional shots. Interestingly,\n26skyrmions nucleate directly after single shot excitation, however, only in a well defined area\nadjacent to already existing skyrmions. We assume that this area is preferred for chiral\nstripe domain nucleation upon a slight decrease of the magnetic field, i.e., the local spin\ntexture is not completely saturated. Laser-induced demagnetization then results in a strong\nperturbation of this pre-stripe state. However, since the chirality of the pre-stripe can be\nseen as an additional energy barrier, we assume that chirality is conserved upon excitation.\nInstead, the spins along the stripe domains are affected and cause local inhomogeneity and a\npartition of the stripe domains in various parts (cf. Fig. S11). This spin alignment then leads\nto a preferred forming of a skyrmion phase during the relaxation process. Therefore, the\nchirality of all generated skyrmions is identical.\n3. Optically-induced phase transformations: Schematic phase diagram\nFig. S8. Spin texture phase diagram. Schematic phase diagram of the spin texture in dependence\nof magnetic field and number of laser pulses for up- a) and downsweep b) of the magnetic field for a\nfixed fluence of 6 .4mJ cm−2. Stars correspond to the LTEM data depicted in Fig. 5. The magnetic\nfield ranges correspond to the spin textures for the sample on the membrane, which are higher than\nfor the MOKE sample on a Si substrate, see also Fig. 1\n.\nFigure S8 shows a schematic phase diagram of the magnetic texture depending on both\nthe magnetic field and the number of incident laser pulses. Depending on the magnetic\n27state of the system, we observe different scenarios upon laser excitation including unchanged\nmagnetic texture (A,D), B/SK creation from stripe domains (B,E), and B/SK annihilation\n(C).\nIn case of scenario E (see also Fig. 5), laser excitation leads to a gradual transformation\nof a pristine stripe domain phase in case of the downsweep of the magnetic field, as depicted\nin Fig. S9. Upon excitation by a single laser pulse the initial stripes transform into bubbles\nand skyrmions, respectively. In addition to that, we observe the creation of stripe domains\nin formerly saturated regions. These generated stripe domains transform in turn into\nbubbles/skyrmions, finally yielding a pure bubble/skyrmion lattice phase in the laser excited\nregion.\nFig. S9. Transformation of stripe state into B/SK lattice state via a mixed state. Left:\nA stripe state in the downsweep of the magnetic field transforms into a mixed state upon excitation\nby a single laser pulse with fluence of 6 .4mJ cm−2. Right: Excitation by multiple laser pulses leads\nto a B/SK lattice state.\n4. Skyrmion nucleation in the micromagnetic simulations\nIn Fig. S10 the simulated spin texture is depicted for three magnetic field regimes\nconsidering multiple subsequent excitations with an MSreduction amplitude of 25 %.\nFor low magnetic fields (here 50 mT), the spin texture, predominantly consisting out\nof stripe domains, is stable, i.e., it remains unaffected by laser excitation. Increasing the\nmagnetic field to 150 mT transforms a part of the stripe domains into skyrmions, leading to\na mixed phase ground state. Subsequent excitations transform additional stripe domains to\nskyrmions. A stable equilibrium between stripe domains and skyrmions emerges after few\nexcitation cycles. Thus, a metastable state is created during the very first laser excitations.\n28Fig. S10. Skyrmion nucleation in the micromagnetic simulations. Spin texture obtained\nfrom micromagnetic simulations after laser excitation for different magnetic field regimes. For high\nmagnetic fields ( >150mT) we observe the transformation of stripe domains into skyrmions upon\nsubsequent excitation, which is absent in case of low magnetic fields (50 mT).\nAs a consequence, it is not possible to capture this process with our stroboscopic pump-probe\nmeasurement scheme. For slightly higher magnetic fields (160 mT) the amount of stripe\ndomains in the ground state is further reduced, facilitating complete transformation of the\nremaining stripes into skyrmions upon additional excitation.\nWe find that this kind of skyrmion nucleation is mediated by the induced motion of the\nstripe domains and therefore strongly depends on the excitation amplitude. The process of\nskyrmion nucleation from a stripe domain is depicted in Fig. S11. Note that no skyrmion\nnucleation is observed in micromagnetic simulation upon excitation with an MSreduction\namplitude of 5 %.\n29Fig. S11. Temporal evolution of stripe to skyrmion transformation. Time evolution\nof skyrmion nucleation from stripe domains from micromagnetic simulations. We observe the\ntransformation of stripe domains into skyrmions upon subsequent excitation for high magnetic\nfields ( >150 mT).\n5. Fluence-dependence of B/SK nucleation from LTEM studies\nMultiple laser excitation of a pristine stripe phase imaged by LTEM is shown in Fig. S12.\nUpon the first excitation with a fluence of 12 .8mJ cm−2the stripe phase is transformed.\nInterestingly, a mixed phase appears in the region of highest excitation, while a pure skyrmion\nphase emerges in its surrounding. Further excitation with the same pulse energy does not\nmacroscopically change the spin texture. However, we observe that the chirality of the\nskyrmions at the edge of the mixed and the pure skyrmion phase can be inverted by\nsubsequent laser excitation (see offset image zooms). For subsequent weaker excitation, the\ncentral mixed phase is transformed into a pure B/SK lattice phase as well. We conclude\nthat the laser pulse has to induce a specific amount of demagnetization in order to nucleate\nthe B/SK lattice phase thereafter. Above a certain threshold of pulse energy this process\ngets more improbable. Our explanation is that due to the strong demagnetization both the\nspin texture and the underlying chirality completely vanish. As a result, during the recovery\nprocess, the formation of skyrmions is suppressed and single skyrmions only nucleate if by\nchance the local spin alignment is correct.\n30Fig. S12. Fluence-dependence of spin texture transformation. Spin texture in dependence of\nthe number of laser pulses. During a downsweep of the magnetic field, the field was set to 195 mT.\nNote that the LTEM images were significantly post-processed to enhance the visibility of magnetic\nskyrmions and their chirality. For details on post-processing see the description of Fig. S7.\n31"    },    {        "title": "1410.5745v2.Magnetic_switching_dynamics_in_a_ferrimagnetic_two_sub_lattice_model_including_ultrafast_exchange_scattering.pdf",        "content": "arXiv:1410.5745v2  [cond-mat.mtrl-sci]  25 Mar 2015Magnetic Switching Dynamics due to Ultrafast Exchange\nScattering: A Model Study\nAlexander Baral and Hans Christian Schneider∗\nPhysics Department and Research Center OPTIMAS,\nUniversity of Kaiserslautern, 67663 Kaiserslautern, Germ any\n(Dated: March 19, 2021)\nAbstract\nWe study the heat-induced magnetization dynamics in a toy mo del of a ferrimagnetic alloy, which\nincludes localized spins antiferromagnetically coupled t o an itinerant carrier system with a Stoner\ngap. We determine the one-particle spin-density matrix inc luding exchange scattering between\nlocalized and itinerant bands as well as scattering with pho nons. While a transient ferromagnetic-\nlike state can always be achieved by a sufficiently strong exci tation, this transient ferromagnetic-like\nstate only leads to magnetization switching for model param eters that also yield a compensation\npoint in the equilibrium M(T) curve.\nPACS numbers: 72.25.Rb, 75.78.-n, 75.78.Jp, 77.80.Fm\n1I. INTRODUCTION\nHeat-induced reversal of magnetization in ferrimagnetic alloys1,2and multilayers3is a\nrecent development that has generated a lot of attention becaus e it realizes an ultrafast\ndeterministic switching without magnetic fields that may pave the way to a faster m ag-\nnetic logic. After the first phenomenological analysis of this switchin g process using the\nBaryakhtar equation,4a more microscopic understanding of this effect is now being devel-\noped using models that in one way or another include classical models of localized spins on\ndifferent sub-lattices, which are coupled by an exchange interactio ns between the different\nlattices5–7The atomistic models usually involve a thermal bath averaging, which is also\nimplicit in Landau-Lifshitz-Bloch type calculations.8–10Ref. 7 does include itinerant carrier-\nphonon scattering, but this model is based on a non-standard elec tron-phonon Hamiltonian\nand separates the charge degrees of electrons from their spins s o that its two spin systems\nessentially are also two types of localized spins coupled to a bath. Alth ough existing theoret-\nical approaches have established the importance of an exchange c oupling between localized\nmagnetic moments on different sublattices for heat-induced magne tic switching and the\noccurrence of transient ferromagnetic-like states, the microsc opic picture of magnetization\nswitching isstillunclear, asthephenomenologicalmodel4impliesacounterintuitive interplay\nbetween spin-orbit (“relativistic”) and exchange scattering. Fur ther, there are differences\nbetween existing atomistic calculations, some of which stress6thattransverse magnetization\ndynamics occurs during switching, while others do not find such a tra nsverse magnetization\ncomponent after bath averaging.5\nII. MODEL AND EQUILIBRIUM CONFIGURATION\nIn this paper we put forward a different microscopic model for magn etization dynamics—\ninspired by theories11,12for magnetic semiconductors—that (i) includes an exchange inter-\naction between delocalized and localized electrons in a band picture and (ii) that is capable\nof including the coupling of the carriers to the environment, i.e., phon ons, in a microscopic\nfashion. The model used in the following contains two bands of differe nt carrier species\nwhich have different spin and are designed to resemble itinerant 3 delectrons in iron and lo-\ncalized 4felectrons in gadolinium, respectively. For simplicity, we assume a spin o fs= 1/2\n2and a parabolic dispersion for the itinerant carriers, as well as a spin S= 1 and a com-\npletely flat band for the localized states; we thus ignore complication s that arise froma finite\nwidth of the bands which is present in real materials. Denoting the loc alized spin states\nby|ν/angbracketright=±1,0 and the electron states by |/vectorkσ/angbracketright, whereσ=±, we calculate dynamically the\nspin-resolved one-particle density matrices of the itinerant electr onsρσσ′\nkand the localized\nspinsρνν′\nlocwith an equation of motion technique. Using the spin-dependent red uced density\nmatrices we do not separate the charge and the spin of the itineran t electrons, as done in\nmodels that work with three temperatures, most notably with differ ent spin and electron\ntemperatures. “Magnetic” contributions are an antiferromagne tic exchange interaction J\nbetween itinerant and flat bands and a Stoner-like on-site coupling Uamong itinerant car-\nriers. The latter is treated in mean-field approximation and favors a ferromagnetic electron\nspin polarization. Long and short range ( U) contributions to electronic Coulomb scattering\nare neglected because (i) they do not change the itinerant spin pola rization, and (ii) for the\nconditions studied here, the electronic distributions never deviate much from Fermi-Dirac\ndistributions.13By contrast, we include both mean field and scattering contribution s from\nthe exchange interaction between both localized and itinerant spins , as well as a coupling\nof the itinerant carriers to acoustic phonons at the level of Boltzm ann scattering integrals.\nThroughout, we work with single-particle states that are obtained from a diagonalization\nincluding the exchange and Stoner mean-field contributions. The re sulting mean-field ener-\ngies are denoted by Eνfor the localized spins and ǫ/vectorkσfor the spin-split electron bands. The\nrelevant equations of motion then take the form\n∂\n∂tρνν′\nloc=i\n/planckover2pi1(Eν−Eν′)ρνν′\nloc+∂\n∂tρνν′\nloc/vextendsingle/vextendsingle\nxc(1)\n∂\n∂tρσσ′\n/vectork=i\n/planckover2pi1(ǫ/vectorkσ−ǫ/vectorkσ)ρσσ′\n/vectork+∂\n∂tρσσ′\n/vectork/vextendsingle/vextendsingle\nxc+∂\n∂tρσσ′\n/vectork/vextendsingle/vextendsingle\ne-pn−ρσσ′\n/vectork−δσσ′Fσ\n/vectork\nτsf(2)\nThe first terms in Eq. (1) and (2) are coherent contributions, whic h describe precessional\nmotion of one spin around the mean-field of the spin of the other spe cies. All the other\nterms are incoherent terms. The incoherent exchange scatterin g contributions at the level\nof Boltzmann scattering integrals are given by\n3∂\n∂tρν1ν2\nloc/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nxc=i\n/planckover2pi1/summationdisplay\n/vectork/vectork′/summationdisplay\nσ1...σ4/summationdisplay\nν3ν4ν5Wν5ν1\n/vectork′σ1/vectorkσ2(Wν3ν4\n/vectork′σ3/vectorkσ4)∗\n·δν4ν2ρν5ν3\nlocρσ1σ3\n/vectork′(δσ4σ2−ρσ4σ2\n/vectork)−δν5ν3ρν4ν2\nlocρσ4σ2\n/vectork(δσ1σ3−ρσ1σ3\n/vectork′)\nǫ/vectorkσ4−ǫ/vectork′σ3+Eν4−Eν3−iγ+(ν1↔ν2)∗\n(3)\n∂\n∂tρσ1σ2\n/vectork/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nxc=i\n/planckover2pi1/summationdisplay\n/vectork′/summationdisplay\nν1...ν4/summationdisplay\nσ3σ4σ5Wν1ν2\n/vectork′σ5/vectorkσ1(Wν3ν4\n/vectork′σ3/vectorkσ4)∗\n·δν4ν2ρν1ν3\nlocρσ5σ3\n/vectork′(δσ4σ2−ρσ4σ2\n/vectork)−δν1ν3ρν4ν2\nlocρσ4σ2\n/vectork(δσ3σ5−ρσ3σ5\n/vectork′)\nǫ/vectorkσ4−ǫ/vectork′σ3+Eν4−Eν3−iγ+(σ1↔σ2)∗\n(4)\nHere,γdenotes an infinitesimal broadening and we use the abbreviation Wνν′\n/vectorkσ/vectork′σ′≡\nJ/angbracketleft/vectorkσ|ˆ/vector s|/vectork′σ′/angbracketright/angbracketleftν|ˆ/vectorS|ν′/angbracketright, whereˆ/vector sandˆ/vectorSare the spin-operators of the localized and the itinerant\nelectrons. At this point, a comment regarding the exchange param eterJmay be in order. In\nour calculation, Jis a Coulomb matrix element that occurs in the hamiltonian and directly\nenters the equation of motion for quantum-mechanical correlatio n functions; it could be\ncalculated directly from the true electronic Bloch or Wannier functio ns. In atomistic spin\nmodels the Js are the parameters of an effective classical Heisenberg model that are ex-\ntracted from ab-initio electronic structure calculations by comput ing the (exchange) energy\nchanges for a small tilting of the magnetic moments in adjacent unit c ells.14\nThe electron-phononscattering (e-pn) contribution inEq. (2) isa standardexpression, an\nexplicit derivation of which with special attention to spin splitting is give n in Ref. 15. It is\nan important property of scattering with long-wavelength longitud inal phonons that it does\nnotlead to a transfer of angular momentum from the itinerant carriers to the lattice, it only\ncools down the itinerant carrier system and increases the tempera ture of the phonon system\nin accordance with energy conservation. The transfer of angular momentum to the lattice\nis left to a relaxation-time expression in Eq. (2) because the fundam ental mechanism is not\nthe important point of this paper. Likely it is a combination of electron -phonon/electron-\nelectron scattering and spin-orbit coupling.13,16We generally assume spin-flip processes to\nbe faster than the heat transfer to the phonons, but slower tha n the exchange scattering. In\nEq. (4),Fσ\n/vectork=f(ǫ/vectorkσ−µF) denotes a Fermi-Dirac distribution with the same energy as the\nactualρσσ′\n/vectork, but equal chemical potentials µFfor both spins.\n4We determine the equilibrium configuration self-consistently, assum ing that the equilib-\nriumreduced density matrix ρσσ′\n/vectork=δσ,σ′f(ǫσ/vectork−µ0)isspindiagonalandgiven byFermi-Dirac\ndistributions with temperature T0andequal chemical potentials µ0for bothspin states. The\naverage spins /vectorSand/vector sof the two species are parallel to the mutual exchange field. Depen d-\ning on the value of the coupling constants JandUthe itinerant system is either partially\nor fully spin polarized. In the numerical calculations below, as done in R ef. 15, we employ\ntwo-dimensional /vectorkvectors because it reduces the numerical complexity of propagat ing the\nscattering calculations over long times. This simplification also change s the equilibrium spin\npolarization, exchange splitting and the Curie temperature compar ed to three-dimensional\n/vectorkvectors. In the following numerical calculations we always assume a c ommon initial tem-\nperature of all three sub-system T0= 10K, which is far lower than the Curie temperature\nTC.\nIII. RESULTS FOR MAGNETIZATION DYNAMICS\nWe model the excitation of a short linearly-polarized laser pulse as an instantaneous\nheating of the itinerant carriers at t= 0. We assume that the localized spin system is not\nexcited optically.17Immediately after the excitation the itinerant spin density-matrix ρσσ′\n/vectork\nis assumed to be spin diagonal with the spin dependent distributions d etermined by Fermi\nfunctions with the same spin-expectation value but with an elevated initial temperature T(0)\ne\nthat usually exceeds TC. Even though the spin polarization of the carriers does not change ,\nthe chemical potentials µσbecome different.\nFigure 1 presents the key dynamical quantities for an excitation ch aracterized by T(0)\ne=\n1500K as well as a short and infinitely long spin-flip time, respectively. The parameters\nJ= 100meV and U= 400meV give rise to a Curie temperature of TC= 480K. These\nparameters together with the initial carrier temperature T(0)\nelead to a demagnetization\nscenario, regardless of the spin-flip time τsf, as shown in the top panel. The components\nof both localized and itinerant spins parallel to the mutual exchange field,S/bardblands/bardbl, show\nanultrafast symmetric decrease due to exchange scattering on a timescale of s everal ten\nfemtoseconds. The exchange scattering conserves the total a ngular momentum and energy\nof the combined system of spin-split itinerant carriers and localized s pins, so that such an\nultrafast drop does not occur for a ferromagnetic exchange cou pling.4We have checked this\n50246−0.500.51\ntime (ps)S/bardbl&s/bardbl\n0246−0.500.51\ntime (ps)S/bardbl&s/bardbl\n0246050010001500\ntime (ps)T(Kelvin)\n  \n0246050010001500\ntime (ps)T(Kelvin)\n  \n0246180220260\ntime (ps)µσ(meV)\n  \n0246180220260\ntime (ps)µσ(meV)\n  \nlower band\nupper bandlower band\nupper bandlattice\nlocalized\nitinerantlattice\nlocalized\niItinerantb) a)\nFIG. 1. Computed results for demagnetization scenario with J= 100 meV, U= 400 meV, and\nT(0)\ne= 1500 K. The spin-flip times are τsf= 200 fs (a) and τsf→ ∞ (b). Top: Dynamics of the aver-\naged localized spin S/bardbl(solid curve) and itinerant spin s/bardbl(dashed curve); middle: quasi-equilibrium\ntemperatures and TC(thin line); bottom: quasi-chemical potentials for the two itinerant bands.\nalso for our model. We will analyze the ultrafast dynamics due to the e xchange scattering\nin more detail below.\nComparedtotheintrinsictimescaleoftheexchange scattering, sp in-flipscattering, which\ndissipates only carrier angular momentum, and carrier-phonon sca ttering, which only trans-\nfers heat from the itinerant carriers to the phonon system, act o n much longer time scales of\nhundreds of fs and several ps, respectively. During the compara tively slow remagnetization\nprocess, the angular momentum and energy transfer between th e localized and itinerant\nsub-systems due to the exchange scattering is limited to the times s cales of these slower\nmechanisms. Changing the spin-flip scattering time in Fig. 1(b) as com pared to (a) does\nnot change the qualitative behavior.\nIn the middle panel of Fig. 1 we show the quasi-equilibrium temperatur es of the itinerant\n60 2.5 5−0.501\ntime (ps)S/bardbl&s/bardbl\n0 2.5 5−0.501\ntime (ps)S/bardbl&s/bardbl\n0 2.5 5150200250\ntime (ps)µσ(meV)\n  \n0 2.5 5150200250\ntime (ps)µσ(meV)\n  \nlower band\nupper bandlower band\nupper banda) b)\nFIG. 2. Top: Computed dynamics of the average localized spin S/bardbl(solid curve) and the average\nitinerant spin s/bardbl(dashed curve) for a TFS scenario with T(0)\ne= 2000 K and U= 400 meV (a)\nas well as a switching scenario with T(0)\ne= 2500 K and U= 500 meV (b). Bottom: Electronic\nquasi-chemical potentials. J= 100 meV is unchanged.\ncarriers (or “electrons”) Te, the localized spins Tlocand the phonons TL, which are obtained\nfrom the computed dynamics of the spin density matrix as the tempe ratures of thermalized\ndistributions with the same energy as the non-equilibrium distribution s. Note, in particular,\nthat the excited electrons have a quasi-equilibrium temperature Te(t) anddifferent chemical\npotentials µσ(t) for each spin species, as shown in the bottom panel of Fig. 1. With a non-\nvanishing spin-flip rate τ−1\nsfas in Fig. 1(a), the temperatures of the itinerant electrons Teand\nof the localized spins Tlocessentially converge on the time scale of the spin-flip relaxation τsf.\nThe phonon temperature TLapproaches these two on the time scale of the electron-phonon\nscattering. Figure 1(b) shows that without dissipation of angular m omentum, viz. τsf→ ∞,\nTlocdoes not get close to Te. This makes clear that exchange scattering neither simply\nequalizes the temperatures TlocandTe, nor the chemical potentials µσ.\nWe next model stronger excitations by raising T(0)\nein Fig. 2. The parameter Uin Fig. 2\n(a) is the same as before so that we have TC≃480K. In addition, we relax the lattice\ntemperature toward T0with a time constant of 10ps to include the effect of heat diffusion.\nThis guarantees a final remagnetization without affecting the fast er dynamics. For Fig. 2(b)\nwe increase Uto 500meV, which leads to TC≃1200K. For the same material parameters\n700.10.20.30.4−0.500.51\ntime (ps)S/bardbl&s/bardbl\n  \n00.10.20.30.4−0.500.51\ntime (ps)S/bardbl&s/bardbl\n  a) b)\nFIG. 3. Computed dynamics of average localized spin S/bardbl(red, starting from 1) and average itinerant\nspin (blue, starting from −0.5) after excitation with T(0)\ne= 1500 K (solid lines), T(0)\ne= 2000 K\n(dashed lines) and T(0)\ne= 2500 K (dashed-dotted lines) for U= 400 meV (a), U= 500 meV (b),\nandJ= 100 meV.\nas in Fig. 1(a), Fig. 2 exhibits a TFS starting around 50fs and persist ing up to about 2ps.\nDuring the TFS, the localized spins experience a populationinversion a nd the corresponding\nspin-temperature Tlocis no longer well defined. As the coupling to the phonons cools down\nthe itinerant carriers, the system either returns, as in Fig. 2(a), into its initial state or,\nas in Fig. 2(b), remagnetizes with an inverse orientation and thus un dergoes magnetization\nswitching(SW).NotethatinFig.2(b)theitinerantcarriersbecome fullyspin-polarized, s/bardbl≃\n1/2, around 2.5ps so that the exchange scattering cannot further remagnetize the localized\nspins. We label the eigenstates according to their energy, so that this label changes if the\neffective mean-field splitting changes its sign. Comparing the bottom panels of Figs. 1 and 2\nindicates that the difference between the chemical potentials does not drive the switching\nor demagnetization dynamics.\nIn Fig. 3 we come back to the role of the exchange scattering in realiz ing the TFS. We\nplotherefordifferentinitialtemperatures T(0)\nethecomputeddynamicsobtainedbyincluding\nonly exchange scattering. While this leads to an unphysical steady s tate, it illustrates the\nway in which exchange scattering works. In Fig. 3(a), for U= 400meV, an initial itinerant\ntemperature T(0)\ne= 1500K leads to a state with reduced localized spin, i.e., demagnetizat ion\nin each spin system. Increasing T(0)\neleads to a reversal of the itinerant spin s/bardbland thus a\nTFS on the timescale of the exchange scattering. This behavior occ urs because the exchange\nscattering redistributes the deposited energy as well as angular m omentum between the\nitinerant and localized spins all the while satisfying both conservation of total energy and\n8300 400 500 6000\n1000\n2000\n3000\nU(meV)T(0)e(K)\nFIG. 4. Color-coded dynamics: demagnetization (black), tr ansient ferromagnetic-like state (gray)\nand switching (white) as a function of Stoner parameter Uand excitation temperature T(0)\ne. The\nexchange coupling constant is J= 100 meV. The arrow marks the “critical value” of U= 445 K,\nabove which a compensation point in the equilibrium M(T) relation occurs.\ntotal angular momentum. In Fig. 3(b) the Stoner parameter is incr eased to U= 500meV,\nwhich is indicative of a more rigid itinerant carrier magnetism. Corresp ondingly, for the\nsmaller initial temperatures T(0)\nethe demagnetization of each spin system due to exchange\nscattering is reduced compared to Fig. 3(a). For the highest excit ation, i.e., T(0)\ne= 2500K\nnot only a TFS occurs, but the average itinerant spin s/bardblbecomes larger than the average\nlocalized spin S/bardbl. Thus the redistribution of the deposited energy during the ultraf ast\nexchange scattering obviously is decisive for the following behavior. Note that the exchange\nscattering alone, i.e., without spin-flip scattering, yields the initial ult rafast demagnetization\nand the subsequent TFS. This seems to be a more realistic scenario t han that described in\nRef. 4 where spin-orbit (“relativistic”) relaxation dominates the su b-picosecond dynamics,\nand the exchange interaction acts on a picosecond timescale.\nIn Fig. 4 we collect the results from individual calculations as shown in F ig. 1(a) and 2\nby plotting the type of dynamics: demagnetization, transient ferr omagnetic-like state (TFS)\nand switching vs. excitation temperature T(0)\neand the strength of the itinerant ferromag-\nnetic coupling constant Uin the carrier system. We find that there is a threshold for the\ntemperature T(0)\ne, below which the system only de- and remagnetizes into its initial orien -\ntation (without entering a TFS). Above this threshold it depends on the Stoner parameter\nUwhether “only” a TFS occurs or whether heat-induced switching is a chieved. Fig. 4\nclearly shows that switching only occurs for Stoner parameters U >(450±25)meV. With\n9the help of the Stoner parameter Uwe can, in the framework of our mean-field model,\nput the criterion for switching in correspondence with the tempera ture dependence of the\nequilibrium magnetization M(T): We find that a compensation point occurs in the M(T)\nrelation only for Stoner parameters larger than U≃445meV. Within the accuracy of our\nnumerical study these values are identical, and one can speculate a bout the importance of\nthe existence of a compensation point for heat-induced magnetic s witching (assuming that\none starts below the compensation temperature). The present r esults should be taken into\naccount when analyzing optically induced magnetization dynamics in fe rrimagnetic alloys,\nwhere both heat-induced and all-optical processes may be importa nt.18,19\nIV. CONCLUSION\nWe presented a dynamical calculation of exchange scattering afte r spin-conserving in-\nstantaneous heating (designed to model ultrafast optical excita tion) in a simple quantum-\nmechanical mean-field model of a ferrimagnetic alloy. Wefound that the exchange scattering\nprovides the essential contribution to the ultrafast switching dyn amics and established the\nimportance of the Stoner parameter Ufor the occurrence of a transient-ferromagnetic state\nand/or switching. We speculated that the occurrence of ultrafas t magnetization switching\nmay be related to the existence of a compensation point in the equilibr ium magnetization.\n∗hcsch@physik.uni-kl.de\n1I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. D¨ urr, T. A. Ostler, J. Barker,\nR. F. Evans, R. W. Chantrell, et al. , Nature 472, 205 (2011).\n2T. A. Ostler, J. Barker, R. F. Evans, R. W. Chantrell, U. Atxit ia, O. Chubykalo-Fesenko,\nS. El Moussaoui, L. Le Guyader, E. Mengotti, L. J. 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Chubykalo-Fesenko,\nPhys. Rev. B 77, 184428 (2008).\n11T. Jungwirth, J. Sinova, J. Maˇ sek, J. Kuˇ cera, and A. MacDon ald, Reviews of Modern Physics\n78, 809 (2006).\n12/suppress L. Cywi´ nski and L. J. Sham, Physical Review B 76, 045205 (2007).\n13B. Y. Mueller, A. Baral, S. Vollmar, M. Cinchetti, M. Aeschli mann, H. C. Schneider, and\nB. Rethfeld, Physical Review Letters 111, 167204 (2013).\n14A. I. Liechtenstein, M. Katsnelson, V. Antropov, and V. Guba nov, Journal of Magnetism and\nMagnetic Materials 67, 65 (1987).\n15A. Baral, S. Vollmar, and H. C. Schneider, Phys. Rev. B 90, 014427 (2014).\n16M. Krauß, T. Roth, S. Alebrand, D. Steil, M. Cinchetti, M. Aes chlimann, and H. C. Schneider,\nPhys. Rev. B 80, 180407 (2009).\n17A. Manchon, Q. Li, L. Xu, and S. Zhang, Physical Review B 85, 064408 (2012).\n18S. Alebrand, M. Gottwald, M. Hehn, D. Steil, M. Cinchetti, D. Lacour, E. E. Fullerton,\nM. Aeschlimann, and S. Mangin, Phys. Rev. B 101(2012).\n19S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V. Uhl´ ır, L . Pang, M. Hehn, S. Alebrand,\nM. Cinchetti, Y. Malinowski, G. Fainman, M. Aeschlimann, an d E. E. Fullerton, Nature ma-\nterials13(2014).\n11"    },    {        "title": "2102.11143v1.The_role_of_density_dependent_magnon_hopping_and_magnon_magnon_repulsion_in_ferrimagnetic_spin__1_2___S___chains_in_a_magnetic_field.pdf",        "content": "arXiv:2102.11143v1  [cond-mat.str-el]  22 Feb 2021The role of density-dependent magnon hopping and magnon-ma gnon repulsion in\nferrimagnetic spin-(1/2, S) chains in a magnetic field\nW. M. da Silva1,2and R. R. Montenegro-Filho2\n1Secretaria de Educa¸ c˜ ao da Para´ ıba, 58015-900 Jo˜ ao Pess oa-PB, Brasil\n2Laborat´ orio de F´ ısica Te´ orica e Computacional, Departa mento de F´ ısica,\nUniversidade Federal de Pernambuco, 50760-901 Recife-PE, Brasil\nWe compare the ground-state features of alternating ferrim agnetic chains (1 /2,S) withS=\n1,3/2,2,5/2 in a magnetic field and the corresponding Holstein-Primako ff bosonic models up to order/radicalbig\ns/S, withs= 1/2, considering the fully polarized magnetization as the bos on vacuum. The single-\nparticle Hamiltonian is a Rice-Mele model with uniform hopp ing and modified boundaries, while\nthe interactions have a correlated (density-dependent) ho pping term and magnon-magnon repul-\nsion. The magnon-magnon repulsion increases the many-magn on energy and the density-dependent\nhopping decreases the kinetic energy. We use density matrix renormalization group calculations to\ninvestigate the effects of these two interaction terms in the bosonic model, and display the quanti-\ntative agreement between the results from the spin model and the full bosonic approximation. In\nparticular, we verify the good accordance in the behavior of the edge states, associated with the\nferrimagnetic plateau, from the spin and from the bosonic mo dels. Furthermore, we show that the\nboundary magnon density strongly depends on the interactio ns and particle statistics..\nI. INTRODUCTION\nThe gapped phases of magnetic insulators are respon-\nsible for magnetization ( m) plateaus in the mvs. mag-\nnetic fieldcurves[1]. In onedimension, theseincompress-\nible phases satisfy the topological Oshikawa-Yamanaka-\nAffleck [2] criteria, and exhibit associated edge states in\nopen spin chains[3]. The gappedphasesareseparatedby\ngapless phases that have a low-energy physics described\nby the Tomonoga-Luttinger-liquid theory [4]. Thus, the\nmagneticfield hinducesquantumphasetransitionsinthe\nspin chain, with quantum critical points at the plateau\nextremes [5, 6]. In the vicinity of the quantum critical\npoints, the magnons are in a high-dilute regime and can\nbe treated as a gas of hard-core bosons [7, 8] or fermions\n[9]. In this approximation, the energy is comprised only\nby asimple hopping term and the uniform Zeeman term,\nwhich plays the role of chemical potential in the effective\nmodel. Following this approach, we can show that the\nm(h) curve (magnon density) presents a square-root sin-\ngularity in the gapless side of the transition. The first\ncorrection to this law is linear and obtained by taking\ninto account magnon-magnon interaction through a phe-\nnomenological scattering length [10–14].\nAnother kind of hopping term is known as corre-\nlated(ordensity-dependent )hopping and is essential\nin the modeling of a variety of quantum systems [15].\nOne of these terms, the bond-charge interaction [16],\nis used to model electrons in strongly correlated ma-\nterials [17–21] and was particularly investigated in the\ncontext of high- Tcsuperconductivity. Besides, the Hub-\nbard model with this term has an exact solution in a\nspecial point of the parameter space [22–25]. On the\nother hand, the extended boson Hubbard model with a\ndensity-dependent hopping is an effective Hamiltonian\nfor bosonic molecules, typically polar species [26–28], in\noptical lattices [29–36]. Further, general correlated hop-ping hard-core bosonic Hamiltonians are investigated to\nunderstand the physics of frustrated insulating magnetic\nmaterials [29–31, 37].\nThe ground-state of ferrimagnetic chains satisfy the\nLieb-Mattis theorem [38] and exhibit ferromagnetic and\nantiferromagnetic long-range orders [39]. and was inves-\ntigatedintheisotropic[40–42]andanisotropic[41,43,44]\ncases. Ferrimagnetic systems ˙In particular, the behavior\nof the edge states associated with the 1/3 magnetization\nplateau of the AB 2anisotropic chain was recently inves-\ntigated [45]. Furthermore, rich phase diagrams are ob-\nserved through doping [46–50] or adding geometric frus-\ntration [51–58] to the ferrimagnetic models. In partic-\nular, ferrimagnetic spin-(1/2, S) chains under an ap-\nplied magnetic field present magnetization plateaus at\nm=S−1/2 (ferrimagnetic plateau) and m=S+ 1/2\n(saturation plateau), where mis the magnetization per\nunit cell [59–64]. On the experimental side, the one-\ndimensional magnetic phase of a variety of bimetallic\ncompounds was shown to be modeled by spin-(1/2, S)\nferrimagnetic chains [65–70]. Recently, the full magne-\ntization curve of the charge-transfer salt (4-Br- o-MePy-\nV)FeCl 4was experimentally investigated [71] and shown\nto be a spin-(1/2, 5/2) chainabove the three-dimensional\nordering temperature.\nSince spin-( s,S) ferrimagnetic chains have a long-\nranged ordered ground state, linear and interacting spin-\nwave theory [72] from the classical ferrimagnetic state\nwas used to characterize their low-energy magnetic ex-\ncitations, mainly through the Holstein-Primakoff formal-\nism [59, 60, 73–78]. Furthermore, linear spin-wavetheory\nfrom the fully polarized state [64] gives good results for\nthe gapped and gapless phases of spin-(1/2, S) chains in\na magnetic field.\nHereweshowthattheHolstein-PrimakoffHamiltonian\nup to order/radicalbig\ns/Sgives almost exactly results to the\nground-state phase diagram of ferrimagnetic spin-(1/2,2\nS) chains in a magnetic field. We use the density matrix\nrenormalization group (DMRG) [79, 80] to obtain the\nmagnetization curves, besides local properties, from the\nHolstein-Primakoff Hamiltonian and the spin Hamilto-\nnian. In addition to the simple hopping and thenearest-\nneighbor interacting terms, a correlated hopping term is\nessential to obtain a good accord between the numerical\nresults from the spin and bosonic models. In Sec. II we\npresent the Holstein-Primakoff Hamiltonian and the an-\nalytical formulas for the hard-core boson approximation.\nIn Sec. III we compare the DMRG results for the magne-\ntization and local properties from the Holstein-Primakoff\nHamiltonian and the spin model. In Sec. IV we present\na summary of the main results of the manuscript.\nII. HOLSTEIN-PRIMAKOFF BOSONIC\nHAMILTONIAN FROM THE FULLY\nPOLARIZED VACUUM\nThe alternating mixed-spin ( s= 1/2,S) chain with L\nunit cells has the Hamiltonian\nHspin=JL/summationdisplay\nj=1(sj·Sj+sj·Sj−1)−BSz\ntot,(1)\nwhere\nSz\ntot=/summationdisplay\nj(sz\nj+Sz\nj) (2)\nis thez-component of the total spin, and we consider\nthe magnetic field Bin thezdirection, with gµB≡1,\nwhereµBis the Bohr magneton and gis theg-factor.\nThe spin −1/2 are attached to asites, while spin −Sto\nbsites along the chain, and we study chains for which\nS= 1,3/2,2,5/2, as schematically shown in Fig. 1(a).\nThe ground-state total spin for B= 0 isS−sand has\na copy in each sector in the range −(S−s)≤Sz\ntot≤\n(S−s), as expected from the Lieb-Mattis theorem [38],\nwith a ferrimagnetic (FRI) long-range ordered state. If\na little magnetic field is applied, the ground state with\nSz\ntot= (S−s) is chosen. Further, the ground state has\na finite gap to spin excitations carrying a spin ∆ Sz=\n+1, which induces a magnetization plateau at mFRI=\n(S−s). Also, a second magnetization plateau is the fully\npolarized plateau at mFP=S+s.\nThefully polarized state is an exact ground state of\nthe spin Hamiltonian, and we build the spin-wave the-\nory considering it as the magnon vacuum. Making the\nHolstein-Primakoff mapping on asites\nsz\nj=s−na\nj,\ns−\nj=√\n2sa†\nj/parenleftBig\n1−na\nj\n2s/parenrightBig1/2\n≈√\n2sa†\nj/parenleftBig\n1−na\nj\n4s/parenrightBig\n,(3)and onbsites:\nSz\nj=S−nb\nj;\nS−\nj=√\n2Sb†\nj/parenleftBig\n1−nb\nj\n2S/parenrightBig1/2\n≈√\n2Sb†\nj/parenleftBig\n1−nb\nj\n4S/parenrightBig\n,(4)\nFIG. 1. (a) Schematic representation of the alternating spi n\nmodel, with spin-1 /2 atasites, and spin- Satbsites, with\nS= 1,3/2,2,and 5/2. Thez-direction is the direction of an\napplied magnetic field Band the superexchange coupling is J.\n(b) Holstein-Primakoff bosonic Hamiltonian up to O(S−1/2),\nwith a hopping term t=J√\nsS, local potentials εa=−2SJ\nandεb=−2sJ, nearest-neighbor interaction V=J, and\ndensity dependent correlated hopping process X=J/radicalbig\ns/S.\nTheasites have a hard-core constraint, na≤1, while the\nconstraint nb≤2Sis imposed on bsites, with na(nb) as the\nnumber of bosons in one a(b) site. The magnetic field Bacts\nas a chemical potential µin the bosonic model: µ=−B.\n(c) The bosonic approximations investigated: t,t−V, and\nt−X−V. In thetandt−Vapproximations, there is a hard-\ncore constraint on all sites; while in the t−X−Vmodel, the\nbsites can accommodate up to two magnons.\nwherena\nj=a†\njajandnb\nj=b†\njbj, wearrivein the following\nspin-wave Hamiltonian3\nHSW=J/summationdisplay\nj/braceleftBigg/parenleftBig\nS−nb\nj/parenrightBig/parenleftBig\ns−na\nj/parenrightBig\n+√\nsS/bracketleftBigg/parenleftBig\n1−nb\nj\n4S/parenrightBig\nbja†\nj/parenleftBig\n1−na\nj\n4s/parenrightBig\n+\n+b†\nj/parenleftBig\n1−nb\nj\n4S/parenrightBig/parenleftBig\n1−na\nj\n4s/parenrightBig\naj/bracketrightBigg\n+√\nsS/bracketleftBigg/parenleftBig\n1−na\nj\n4s/parenrightBig\najb†\nj+1/parenleftBig\n1−nb\nj+1\n4S/parenrightBig\n+\n+a†\nj/parenleftBig\n1−na\nj\n4s/parenrightBig/parenleftBig\n1−nb\nj+1\n4S/parenrightBig\nbj+1/bracketrightBigg\n+/parenleftBig\ns−na\nj/parenrightBig/parenleftBig\nS−nb\nj+1/parenrightBig/bracerightBigg\n+\n−B/summationdisplay\nj/parenleftBig\nS+s−na\nj−nb\nj/parenrightBig\n+O(S−1). (5)\nDropping the classical energy of the ferromagnetic state\nEclass= 2JLsS−B/parenleftbig\nS+s/parenrightbig\nL,therelevantmagnonHamil-\ntonian is:\nHSW=Ht+HX+HV+O(S−1),(6)\nwithHSW=HSW−Eclass.\nAs sketched in Fig. 1(b), the Htterm comprises a\nhopping process and distinct local potentials on aandb\nsites:\nHt=t/summationdisplay\nj/parenleftBig\nb†\njaj+1+b†\nj+1aj+1+H. c./parenrightBig\n+/summationdisplay\nj[(εa−µ)na\nj+(εb−µ)nb\nj] (7)\nwith\n\n\nt=J√\nsS, hopping parameter;\nεa=−2SJ, local potential on asites;\nεb=−2sJ, local potential on bsites;\nµ=−B.(8)\nIn an open chain, if aorbis a boundary site, the local\npotential is half of the above value:\n/braceleftBigg\nε(boundary)\na =−SJ\nε(boundary)\nb=−sJ.(9)\nConsidering the local potentials, we see that the magnon\nhas a higher probability to be found on asites, and this\nprobability increases with S. However, since the hopping\nparameter tincreaseswith√\nS, quantum fluctuations are\nrelevant for moderate values of S, and the magnons can\novercome the potential barrier between aandbsites.\nWe observe that the bulk Hamiltonian Htis a partic-\nular case of the Rice-Mele model [81]:\nHRice-Mele=/summationdisplay\nj/parenleftBig\nt2b†\njaj+1+t1b†\nj+1aj+1+H. c./parenrightBig\n+/summationdisplay\nj(εana\nj+εbnb\nj), (10)\nputtingµ= 0 and considering the general case of al-\nternating hopping: t1(t2) for intra-cell (inter-cell) hop-\nping. The Rice-Mele Hamiltonian was originally pro-\nposed to model the physics of electrons in polymers, butis a paradigmatic model to the understanding of topolog-\nical insulators, and can be realized by atoms in optical\nlattices [82]. The model presents the bulk-boundary cor-\nrespondence [83], and an interacting version was recently\ninvestigated [83] to probe the connection between topol-\nogyandparticle-particleinteractions[83]. TheRice-Mele\nmodel recovers the bulk Hamiltonian Htfort1=t2.\nHowever, for an open chain, the mapping of the spin\nmodel requires local potentials on the boundary sites,\nEq. (9).\nThe second term in the bosonic Hamiltonian (6) is a\ndensity-dependent or correlated hopping term given by\nHX=−X/summationdisplay\nj/bracketleftbig\n(a†\nj+a†\nj+1)nb\njbj+H. c./bracketrightbig\n,(11)\nwith\nX=J\n4/radicalbiggs\nS. (12)\nSinces= 1/2 andna\nj≤1, a hard-core constraint must\nbe imposed on asites for all models considered. Hence,\nwe discard a term similar to the Xterm but with aandb\nvariablesexchanged, with X′=J\n4/radicalbig\nS/sas the correlated\nhopping amplitude. As sketched in Fig. 1(b), the energy\nof the system is lowered by the hopping of a magnon to a\nsite that is alreadyoccupied. In other words, the magnon\nprobability to overcome the potential barrier between a\nandbsites increases if there is one magnon on the bsite.\nThis term becomes relevant for higher magnon densities,\nsince it is an interaction term, and X→0 asSincreases.\nThe last term in the Hamiltonian (6), see Fig. 1(b),\nis a repulsive term between magnons in nearest neighbor\nsites:\nHV=V/summationdisplay\njna\nj+1(nb\nj+nb\nj+1), (13)\nwithV=J, and increases the energy for higher magnon\ndensities. For one magnon per unit cell, this term favors\nthe magnon localization on alternating asites, since the\nlocalpotentials( ε)favorsthe occupationof asites. Thus,\nε(anydensity) and V(higher densities) favormagnonlo-\ncalization on asites, while quantum fluctuations (tunnel-\ning between aandbsites) are favored by t(any density)\nandX(higher densities).4\nIn this work, we compare data from three approxima-\ntions of Hamiltonian (6), as summarized in Fig. 1(c),\nand from the spin Hamiltonian (1). The first, which we\nidentify as the tapproximation, is an analytical solution\nto the free hard-core model. In this approximation, all\nsites have a hard-core constraint and can be occupied by\nonly one magnon: na\nj≤1 andnb\nj≤1, for any j. This\nconstraint implies that there is not an energy contribu-\ntion from the Xterm, and we drop the nearest-neighbor\ninteraction V. In the second approximation, t−V, we\nkeep the hard-coreconstraint but add the Vcontribution\ntoHt. The last approximation, t−X−V, has the three\ntermsHt,HX, andHV. We consider a hard-core con-\nstraint on asites and a constraint nb\nj≤2 onbsites. As\nwe present below, the relaxation of the hard-core con-\nstraint on bsites and the consequent activation of the\ncorrelated hopping term implies an excellent agreement\nbetween the results of this approximation and the spin\nmodel.\nA. Free hard-core magnons [ t-approximation]: Ht\nandL→ ∞\nThe first approximation to the magnetization curve, a\nmany-magnon state, is to consider the magnons as free\nhard-core bosons or free fermions. This mapping is exact\nin the high-dilute regime ofmagnons, near the saturation\nfield. Here, we extend this approach for the full range of\nthe magnetization, ( S−s)≤m≤(S+s), and compare\ntheir results to more precise calculations considering the\ninteraction terms.\nThe single-magnon energies are given by the term Ht,\nEq. (7), in Hamiltonian (6). Using the following Fourier\ntransforms:\n/braceleftBigg\naj=1√\nL/summationtext\nke−ik/4eikjak;\nbj=1√\nL/summationtext\nke+ik/4eikjbk,(14)\nwhere a phase e±ik/4is included to ease the calculation,\nHtbecomes:\nHt=/summationdisplay\nk/parenleftbig\na†\nkb†\nk/parenrightbig/parenleftbigg\nεaγk\nγkεb/parenrightbigg/parenleftbigg\nak\nbk/parenrightbigg\n,(15)\nwhere\nγk= 2tcos(k/2). (16)\nAfter diagonalization, the Hamiltonian is written as\nHt=/summationdisplay\nkω(−)\nkn(−)\nk+ω(+)\nkn(+)\nk, (17)\nwhere the dispersion relations are\nω(±)\nk=εa+εb\n2±ωk=−J(s+S)±ωk,(18)with\nωk=/radicalBigg/parenleftbiggεb−εa\n2/parenrightbigg2\n+γ2\nk\n=J/radicalbig\n(S−s)2+4sScos2(k/2),(19)\nandn(±)\nk=α(±)†\nkα(±)\nk, where\n/parenleftbigg\nα(−)\nk\nα(+)\nk/parenrightbigg\n=/parenleftbigg\ncosθk−sinθk\nsinθkcosθk/parenrightbigg/parenleftbigg\nak\nbk/parenrightbigg\n,(20)\nwith\n/braceleftBigg\ncos2θk=1\n2+S−s\n2ωk;\nsin2θk=1\n2−S−s\n2ωk.(21)\nA magnetic field (or chemical potential) adds a + Ben-\nergy term to the two bands.\n-1 0 1\nk /  π-2(S+s)-2S-2s = -10Energy / Jinactive\none - magnon\n       states-1-0.500.51\nk /  π02s = 1\n-1-0.500.51\nk /  π-2s = -10 B=2(s+S)=−µ\nB=2SFP\nFRIB=0vacuum\none magnon\nper unit cell\nFIG. 2. One-magnon states from the fully polarized (FP)\nvacuum and B= 0 as a function of lattice wave-vector k.\nIn thet-approximation, the high energy band is inactive. For\nB= 2(s+S) =−µ, the exact ground state is the FP vacuum;\nwhile there is one magnon per unit cell in the system for\nB= 2S, and the ground state is ferrimagnetic (FRI).\nThe magnon bands are shown in Fig. 2 for (i) B= 0;\n(ii)B= 2(s+S), the saturation field, for which m=\nmFP= (S+s); and (iii) B= 2S, the critical field of the\nferrimagnetic plateau, for which m=mFRI= (S−s), or\n1 magnon per unit cell: mFP−mFRI= (S+1/2)−(S−\n1/2) = 1. As expected from the Lieb-Mattis theorem,\nthe magnetization in the null field is S−s. In the free\nhard-corebosonorfreefermionapproximation,wefillthe\nsingle-particle states following a Fermi distribution up to\nthe effective Fermi wave-vector kF. As shown in Fig. 2,\nif we follow this procedure the two bands should be filled\nforB= 0(twomagnonsperunitcell)andthemagnetiza-\ntion curves of the spin model would not be reproduced.\nWe have shown in Ref. [64] that this problem can be\novercome, even for finite T, by introducing an effective\nchemical potential µto the upper band, in a way similar\nto Takahashi’s solution to the ferromagnetic linear chain\n[84]. In particular, µ→ −J=−2sJasT→0, such\nthat the overall effect of µatT= 0 is the suppression\nof the upper band. Hence, in the free hard-core approx-\nimation, we must consider only the lower band in the5\ncalculations, as schematically indicated in Fig. 2. Thus,\nfor example, the energy per unit cell in the free hard-core\napproximation is written as\nE\nL=1\nLkF/summationdisplay\nk=−kFω(−)\nk(22)\nfor a magnon density per unit cell n, where\nkF=πn, (23)\nandn=mFP−m.\n1. Average local magnetizations\nIf the chain has a magnon density per unit cell n\nand considering the hard-core approximation, the aver-\nage magnetizations of aandbsites are given by\n/braceleftBigg\n/angbracketleftsz/angbracketright=s−1\nL/summationtextkF\nk=−kF/angbracketleftna\nk/angbracketright(asites);\n/angbracketleftSz/angbracketright=S−1\nL/summationtextkF\nk=−kF/angbracketleftnb\nk/angbracketright(bsites).(24)\nUsingEqs. (20)anddiscardingtermsinvolvingtheupper\nband, we obtain\n/braceleftBigg\n/angbracketleftna\nk/angbracketright=n(−)\nkcos2θk;\n/angbracketleftnb\nk/angbracketright=n(−)\nksin2θk.(25)\nWe, thus, have\n/braceleftBigg/angbracketleftbig\nsz/angbracketrightbig\n=s−1\n2−S−s\n2/summationtextkF\nk=−kF1\nωk;/angbracketleftbig\nSz/angbracketrightbig\n=S−1\n2+S−s\n2/summationtextkF\nk=−kF1\nωk,(26)\nafter the substitution of Eqs. (21) in Eqs. (25) and the\nresults in Eqs. (24).\nIII. DMRG RESULTS FOR THE SPIN MODEL\nAND THE BOSONIC HAMILTONIANS\nA. Methodology\nWe use the density matrix renormalization group to\nobtain the magnetization curves and local properties of\nthe spin, t−V, andt−X−Vmodels; and also com-\npare this data with the analytical results from the free\nhard-core model ( t-model) in the thermodynamic limit:\nL→ ∞. These approximations are summarized in Fig.\n1(c). All DMRG results (boson and spin models) are ob-\ntained through the Algorithms and Libraries for Physics\nSimulations (ALPS) project [85] for chains with L= 128\nunit cells, with one asite at one extreme and a bsite\nat the other. If the system has an asite at the left\nextreme and a bin the right ( a-bboundaries), the renor-\nmalization process for the magnetization step inside the\nferrimagnetic plateau becomes trapped in a metastablestate for S= 3/2,2,and 5/2. In these cases, the global\nenergyminimum is reachedbythe algorithmif the chains\nhave absite at the left extreme and an asite at the right\nextreme ( b-aboundaries). In the other magnetizations,\nthis is irrelevant, i. e., the same state is calculated for\nthea-bor theb-aboundaries. We retain a maximum of\n243 states per block and the maximum discarded weight\nless than 10−9.\nFor the spin model, the magnetization curves are cal-\nculated from the lowest energy state for a fixed Szand\nB= 0:E(Sz); since for B/negationslash= 0 we need only to add\nthe Zeeman term, such that EB(Sz) =E(Sz)−BSz. In\na gapless (non-plateau) phase, the magnetization curves\nare made of finite steps in a finite-size system. Defining\nthe extreme points of these finite steps as B−andB+,\nwe have\nB±=±[E(Sz±1)−E(Sz)] (27)\nfor the step at Sz. In a gapless phase, B−→B+as\nL→ ∞, while in a thermodynamic-limit plateau state\nB−/negationslash=B+forL→ ∞. In the last case, B±are quantum\ncritical fields separatingthe plateau insulating state from\na gapless critical Luttinger liquid phase. For the bosonic\nmodels, the magnon density per unit cell nas a function\nof chemical potential µis obtained with a similar pro-\ncedure. We calculate the lowest energy state for a fixed\nnumber of bosons N, withN=nLandµ= 0. The value\nof the chemical potential at the extremes ( µ−andµ+) of\nthe step at Nare given by\nµ±=±[E(N±1)−E(N)]. (28)\nA gapless phase has µ+→µ−asL→ ∞, while in a\nplateau insulating phase µ+/negationslash=µ−in the thermodynamic\nlimit. The transformation from the boson to the spin\nlanguage is performed through the following equations:\n/braceleftBigg\nn=m−mFP,and\nB=−µ.(29)\nB. Magnetization curves and local magnetizations\nThe magnetization curves from the spin model and the\nboson Hamiltonians are shown in Fig. 3(a). The models\npresent two magnetization plateaus, one at the ferrimag-\nnetic magnetization mFRI=S−1/2 and the fully polar-\nized state mFP=S+1/2. The saturation field BFP, end\npoint of the FP plateau, is obtained through the closing\nof the single-particle magnon gap with B, as sketched in\nFig. 2, at B=BFP= 2(S+s). The saturation field\nBFPfrom any bosonic approximation is rigorously equal\nto its exact value since the fully polarized state, an ex-\nact eigenstate of the spin model, is the vacuum for the\nbosonic models.\nThe Oshikawa-Yamanaka-Affleck topological criteria\n[2] states that a plateau can appear in the magnetiza-\ntion curve of spin systems if\nSu−m=integer, (30)6\n0 1 2 3 4 5 6\nB / J0.30.40.50.60.70.80.91m / ms(1/2, 5/2)\n(1/2, 2)\n(1/2, 3/2)\n(1/2, 1)Spin, DMRG (L=128)\nt, L →∞\nt - X - V, DMRG (L=128)(a)\nFRI plateau\nFRI plateau\nFRI plateau\nFRI\nplateau arrowsedge states\n0 1 2 3 4 5 6\nB / J00.51n(1/2, 5/2)\n(1/2, 2)\n(1/2, 3/2)\n(1/2, 1)Spin, DMRG (L=128)\nt, L →∞\nt - X - V, DMRG (L=128)(b)\nt - V, DMRG (L=128)13/22 5/2\nGapless\nLuttinger\nliquidFRI plateau\n     (n = 1)\nFIG. 3. (a) Magnetization per unit cell ( m) of (1/2,S) chains,\nwithS= 1,3/2,2,and 5/2, normalized by its saturation\nvalue (ms) and (b) magnon densities per unit cell nas func-\ntions ofBin units of J. DMRG data for (full lines) the spin\nmodel and for ( •) thet−X−Vapproximation. We present\nin both figures the results for the t-model (dashed lines) and\nL→ ∞ . In (b), we also show the magnon density as a func-\ntion ofBunits ofJfor (•) thet−Vapproximation, calculated\nwith DMRG for a system with L= 128. In the thermody-\nnamic limit, the ferrimagnetic plateau is observed for n= 1,\nwhile the gapless Luttinger liquid phase for 0 < n < 1. On\nboth figures, we indicate (arrows) the magnetic field at which\nthe edge states are occupied by one magnon.\nwhereSuis the maximum spin of a unit cell, unless the\nground state spontaneously break translation symmetry.\nThis corresponds to a number of magnons per unit cell\nn= 0,1,2,...,S u(Su−1/2) for integer (half-integer) Su,\nfrom the fully polarized state. Since Su=S+1/2 (sat-\nurated magnetization) for spin-(1/2, S) chains, plateaus\ncould appear at m=S+1/2,(S−1/2),...,0(1/2), for\nhalf-integer (integer) S. In the spin-(1/2, S) chains, the\ndata shows that there are two magnetization plateaus,\none at the fully polarized ( n= 0) magnetization and the\nother at one magnon per unit cell ( n= 1), the ferrimag-\nnetic plateau. The other possible magnetization plateaus\nare inhibited by the magnon-magnon interaction term V\n[Eq. (13)]. The magnetization curves of ferrimagnetic\nspin chains with 1 /2< s < S can exhibit other plateaus\nbetween the ferrimagnetic and the fully polatized ones.\nThis is observed, for example, in spin-(1,2) and spin-\n(1,3/2) chains [86]. For these chains, the ferrimagnetic\nplateau state has two magnons per unit cell, n= 2, andthemagnetizationcurvesalsoexhibitsaplateauat n= 1.\nWe also observe the occupancy of the edge states of the\nferrimagnetic plateau by one magnon at the indicated\nmagnetic fields. These edge states appears in finite-size\nopen systems associated with topological aspects [3, 45]\nof the ferrimagnetic state.\nThe non-interacting Rice-Mele model (10) does not\npresent edge states for uniform hopping. However, re-\ncently [83], it was shown that an interacting fermionic\nsystem, with a local Coulomb interaction, presents ef-\nfective edge states, and that a fraction of the boundary\ncharge is, in fact, related to the bulk properties. Our\nnon-interacting Hamiltonian (7) has the modified local\npotentials in the boundaries (9), required from the spin\nmapping, and thus localized edge states. Furthermore,\nour interacting model is a bosonic system having the cor-\nrelated hopping and nearest-neighbor coulomb repulsion.\nIn the Sec. IV, we study the boundary magnon den-\nsity and compare it from relevant interacting and non-\ninteracting bosonic models.\nThe gapless phase between the magnetization plateaus\nis a Luttinger liquid phase with power-law decay of the\ntransverse spin correlation functions [4] and has a dy-\nnamical exponent z= 1.\nWe notice in Fig. 3(a) that as Bdecreases from\nBFP, the magnetization from the free hard-core model,\nt-approximation, departs from that of the spin model\nat roughly half filling of the lower magnon band ω(−),\nEq. (18). At this filling, the interaction effects start to\nbecome relevant. The critical field of the ferrimagnetic\nplateau from the free hard-core model: 2 S, see Fig. 2,\nbecomes more near its value for the spin model as Sin-\ncreases, as also confirmed in Table I. This effect can be\nattributed to the local energy of the asites that becomes\ndeeper in comparison with the local energy of the bsites,\nas can be seen in the energy term (7) and the sketch in\nFig. 1. Thus, the magnons become more localized on\nasites asSincreases, and the XandVenergy terms,\nEqs. (11) and (13), respectively, become lesser relevant.\nTheXterm, due to the low occupancy probability of\nonebsite by two magnons, and also because X→0\nasS→ ∞, see Eq. (12). The Vterm, on the other\nhand, because the probability of finding two magnons\nin nearest neighbor sites is also very low. Further, the\nt−X−V-approximation, which has all energy terms in\nHamiltonian(6)active, is inexcellentagreementwith the\nresults for the spin model. Even the location of the edge\nstates is well reproducedby the t−X−V-approximation.\nIn Fig. 3(b) we show the average magnon density per\nunit cell/angbracketleftn/angbracketright. Besides the models presented in (a), we add\nthet−V-approximation. The presence of the magnon-\nrepulsionmakestheaccordancewith thespin modelgood\nfor/angbracketleftn/angbracketright>1/2, while in the t-approximation this agree-\nment is good up to a value of /angbracketleftn/angbracketrightless than 1/2. As\n/angbracketleftn/angbracketright →1, reaching the ferrimagnetic plateau, the repul-\nsionVincreases much the energy of the system, and,\nthus, implies a lower value of the critical magnetic field\nBFRI, compared to the spin model. The location of the7\nTABLE I. Critical field of the ferrimagnetic plateau in units\nofJfor the spin-(1/2, S) chains in the free hard-core boson\napproximation t-approximation and the spin model.\nS t -approx.: 2 S spin model: β2S−β\nβ\n1 2 1.76 0.13\n3/2 3 2.85 0.05\n2 4 3.88 0.03\n5/2 5 4.91 0.02\nTABLE II. Average spins at a(/angbracketleftsz/angbracketright) andb(/angbracketleftSz/angbracketright) sites for\nthe spin-(1/2, S) chains in the free hard-core boson approxi-\nmationtand the spin model: ( /angbracketleftsz/angbracketright,/angbracketleftSz/angbracketright), atm=S−s, the\nferrimagnetic magnetization.\nS t -approx. spin model\n1 ( −0.27,0.77) ( −0.29,0.79)\n3/2 ( −0.34,1.34) ( −0.36,1.36)\n2 ( −0.38,1.88) ( −0.39,1.89)\n5/2 ( −0.40,2.40) ( −0.41,2.41)\nedge states in the finite-size system is also different be-\ntween the spin and the t−V-approximation. Further,\ntheX-term, the density-dependent hopping term, lowers\nthe energy, and the full /angbracketleftn/angbracketright-curve of the spin model is\nwell reproduced by the t−X−Vapproximation. We\nnotice, however, that only for the finite size spin-(1/2, 1)\nchain studied, the lower critical field of the ferrimagnetic\nplateau is B≈0.06J. In Sec. IV, we present the magnon\ncurve for N >128.\nThe average spin and magnon density at aandbsites\nare shown in Fig. 4. Notice that the average spin at\nbsites, Fig. 4(b), is normalized by S. Also shown are\nthe probability of occupancy of aandbsites. We use\nthe expressions (26) to obtain the average spins from the\nt-approximation, while we calculate the average magnon\ndensity from the spin model with\n/braceleftBigg\n/angbracketleftna/angbracketright=1\n2−1\nL/summationtextL\nl=1/angbracketleftsz\nl/angbracketright,\n/angbracketleftnb/angbracketright=S−1\nL/summationtextL\nl=1/angbracketleftSz\nl/angbracketright,(31)\nfor a finite spin chain of size Land open boundaries.\nThe value of the average spin at asites from any of the\nconsidered bosonic approximations is in good agreement\nwith its value from the spin model. A relevant departure\nbetween the approximations and the spin model occurs\nin the average spin on bsites as the ferrimagnetic mag-\nnetization plateau is approached. However, even the t-\napproximationprovidesgoodvaluesfortheaveragespins,\nas shown in Table II for the ferrimagnetic magnetiza-\ntion. Also, the results become indistinguishable, even on\nbsites, as Sincreases or as the fully polarized plateau\nis approached. Furthermore, as in the magnetization re-\nsults, the t−X−Vmodel is an excellent approximation\nto the spin Hamiltonian. The data in Figs. 4(c) and (d)\nconfirm that due to the higher value ofthe localpotential\nonasites, theprobabilityofoccupancyof asitesishigher0.40.60.8 1\nm / ms-0.2500.250.5 <Sz\na>\n(1/2, 1)\n(1/2, 3/2)\n(1/2, 2)(1/2, 5/2)\n0.40.60.8 1\nm / ms0.80.91\n<Sz\nb> / S\n(1/2, 1)(1/2, 3/2)(1/2, 2)(1/2, 5/2)(a) (b) Spin, DMRG(L=128)\nt, L→∞\nt - X - V, DMRG (L=128)\n0.40.60.8\nm / ms00.20.40.60.81<na>(1/2, 1)(1/2, 3/2)(1/2, 2)\n(1/2, 5/2)\n0.40.60.81\nm / ms00.10.2\n<nb>(1/2, 1)\n(1/2, 3/2)\n(1/2, 2)\n(1/2, 5/2)\n(c) (d)\nFIG. 4. Average spin at (a) a,/angbracketleftSz\na/angbracketright, and (b) b,/angbracketleftSz\nb/angbracketright,\nsites, in this case normalized by S, for (1/2, S) chains with\nS= 1,3/2,2,and 5/2, as a function of the normalized mag-\nnetization m/ms, wheremsis the saturation magnetization\nof each chain. The hard-core boson tapproximation in the\nthermodynamic limit (dashed lines), DMRG results for the\nspin model (full lines) and the t−X−Vapproximation ( •),\nboth for systems with L= 128. Average magnon densities at\n(c)a,/angbracketleftna/angbracketright, and (d) b,/angbracketleftnb/angbracketright, sites as a function of m/msand\nthe same legend of (a) and (b).\nthan that on bsites, as discussed in the context of the\nmagnetization curves in Fig. 3. In particular, quantum\nfluctuations are reduced as Sincreases, since /angbracketleftnb/angbracketright →0 in\nthis limit, as the data in Fig. 4(d) suggests.\nIn Figs. 5(a) and (b) we present the local magnon\ndensities in finite chains with open boundaries for the\nspin and the t−X−Vmodels. The magnon densities\nfrom the spin model are calculated through\n/braceleftBigg\n/angbracketleftna\nl/angbracketright=1\n2−/angbracketleftsz\nl/angbracketright;\n/angbracketleftnb\nl/angbracketright=S−/angbracketleftSz\nl/angbracketright,(32)\nand/angbracketleftncell\nl/angbracketright=/angbracketleftna\nl/angbracketright+/angbracketleftnb\nl/angbracketright. We show the data for the spin-\n(1/2, 1) and spin-(1/2,5/2) chains, for two low magnon\ndensities ( n= 8/128,n= 16/128), near the fully polar-\nized magnetization plateau, and two high magnon densi-\nties, more near the ferrimagnetic magnetization plateau\n(n= 72/128,n= 116/128), in a chain with L= 128 unit\ncells. For the lower magnon densities (hard-core limit)\nthe results for t−X−V-approximation departs from the\nspin model data near the boundaries for the spin-(1/2,\n1). However, the accordance between the two models is\nexcellent in the case of the spin-(1/2, 5/2), even near the\nboundaries, for the two lower magnon densities shown.8\n00.020.040.060.08\n00.050.10.15\n00.5\n0 64 12800.20.40.60.8n = 8/128 = 0.0625\nn = 16/128 = 0.12500\nn = 72/128 = 0.56250spin: a site\nspin: b sitespin: cell\nt - X - VAverage magnon densities along the chain\nln = 116/128 = 0.90625(a) (1/2, 1) chain\n00.020.040.060.08\n00.050.10.15\n00.5\n0 64 12800.20.40.60.8n = 8/128 = 0.0625\nn = 16/128 = 0.12500\nn = 72/128 = 0.56250spin: b sitespin: cell\nt - X - V\nln = 116/128 = 0.90625(b) (1/2, 5/2) chain\nspin: a site\n00.20.40.60.81\n00.20.40.60.81\n00.20.40.60.81\n0 1000.20.40.60.81(1/2, 1)\n(1/2, 3/2)\n(1/2, 2)\nl(1/2, 5/2)(c) edge statesDensity change along the chain δ<n>FRI+1→FRIcell: spin\na site: spin\nb site: spin\nFIG. 5. Average magnon densities along (a) the (1 /2,S= 1) chain and (b) the (1 /2,S= 5/2) withL= 128 unit cells. Magnon\ndensities at ( •)asites,/angbracketleftna,l/angbracketright, at (•)bsites,/angbracketleftnb,l/angbracketright, and (•) total magnon density, /angbracketleftncell,l/angbracketright, for unit cell lfrom the spin model\nand (/trianglesolid) thet−X−Vapproximation, both calculated with DMRG for systems with L= 128. The following total average\nmagnon densities are shown: n= 8/128,16/128,72/128,and 116 /128, from top to bottom, corresponding, respectively, to\nthe magnetizations (a) m= 184/128,176/128,120/128,and 76/128; and (b) m= 376/128,368/128,312/128,and 268/128.\n(c) Average magnon distribution change δ/angbracketleftn/angbracketrightFRI+1→FRI,l =/angbracketleftn/angbracketrightN=L,l− /angbracketleftn/angbracketrightN=L−1,lalong the chain as a function of unit cell\nlshows the presence of the edge states for the four chains stud ied. (d) The average magnon probability density /angbracketleftn/angbracketrightalong a\nchain with L= 128 at (circles) aand (diamonds) bsites for (white symbols) N=L−1 and (red symbols) N=Lbosons from\nthet−X−Vapproximation of the (1/2,1) chain. (e) Sketch of the ground states for N=L−1 andN=Lbosons. Average\nmagnon densities are shown as grey circles, fluctuations bet weenaandbsites are indicated by a blue stripe, and we use a red\nfilled curve to represent the localized orbital in the bounda ry unit cell.\nFor the two higher magnon densities, there is an excel-\nlent agreement between the spin model and the t−X−V\napproximation for the two chains.\nIn Fig. 5(c), we show the magnon distribution in\nthe edge localized state occupied by one magnon for the\nfour chains studied. To calculate it, we notice that the\nedge state appears between the two magnetization steps:\nSz=L(S−s) andSz=L(S−s) + 1, which will join\nin the thermodynamic limit, and make the S−sferri-\nmagnetic plateau, see Fig. 3(a) and (b). Thus, to visu-alize the spatial extent of the edge state, we consider the\nmagnon distribution change, δ/angbracketleftn/angbracketrightFRI+1→FRI, between a\ntotal number of magnons N=L−1 [Sz=L(S−s)+1]\nandN=L[Sz=L(S−s)]:\nδ/angbracketleftn/angbracketrightFRI+1→FRI,l=/angbracketleftn/angbracketrightN=L,l−/angbracketleftn/angbracketrightN=L−1,l.(33)\nWe notice in the data shown in Fig. 5(c) that a tiny dis-\ncrepancyisobservedbetweentheresultsforthe t−X−V-\napproximationand the spinmodel inthe caseofthe (1/2,\n1)chain,whilefortheotherchainstheagreementisexcel-9\nlent. Theholeaddedtothemany-magnonstateat N=L\nbecomes well localized in the boundary cell, mainly on\nthe boundary asite, thus implying the presence of this\nempty orbital in the N=L−1 [Sz=L(S−s)] state.\nThe localization in the boundary increases with Sas ex-\npected from the increasing of the bulk gap. In fact, the\nfluctuations of the magnon density in the boundary be-\ncome neglegible as S→ ∞, and the boundary hosts one\nmagnon in this limit. We can attibute this behavior to\nthe increasing potential barrier between bulk and edge,\nEqs. (8) and (9), respectively, as S→ ∞.\nAs an example of the data used to make the results\nshown in Fig. 5(d), we exhibit in Fig. 5(d) the average\nmagnon density along the chain, /angbracketleftn/angbracketright, foraandbsites,\nin thet−X−Vapproximation of the (1/2,1) chain.\nWe notice that the asite at the left boundary is almost\nempty for N=L−1, while its ≈0.6 forN=L, implying\nthe value shown in (c). We also notice a sizable change\nin the occupation of the bsite at the left boundary for\nthe two fillings, while /angbracketleftn/angbracketrightdoes not change at the right\nboundary. A sketch of the two ground states is shown in\nFig. 5(e). For N=L−1, we have an insulating state\nwith themagnonaveragehigheron asitesthan on bsites.\nQuantum fluctuations between aandbsites in nearby\nunit cells are expected, given the excellent accordance\nbetween the results from the t−X−V-approximation\nandthe spinmodel in the high-densitylimit, m→mFRI.\nAdding one magnon to the ( L−1)-magnon state, we\nobtain the magnetization at m=S−sin the finite-\nsize system, which is also an insulating state. The added\nmagnon occupies the empty localized orbital state in the\nunit cell of the boundary asite. The penetration of this\nedge state in the gapped bulk is very tiny and decreases\nwith increasing S, see Fig. 5(c), as can be estimated\nthrough the average magnon distribution.\nIV. THE RELEVANCE OF THE BOUNDARY\nPOTENTIALS AND INTERACTIONS IN THE\nBOUNDARY MAGNON DENSITY OF THE\nBOSONIC MODEL\nWith the help of Fig. 6, we discuss the relevance of the\ndifference between the local potential in the boundaries\nand the bulk sites on the boundary magnon densities,\nas well as the importance of interactions in it. We use\nthe parameters of the spin-(1/2, 1) chain, but focus on\nthe bosonic Hamiltonian, for three models in finite-size\nchains: (1) the interacting model considering the map-\nping to the spin system ( X/negationslash= 0,V/negationslash= 0) - the t−X−V\nmodel; (2)thecorrespondingnon-interactingmodel,i. e.,\nthet−X−Vmodel with X= 0 and V= 0, which has\na hard-core constraint on asites and the bsites can host\nup to 2 bosons; and (3), the tapproximation, which has\na hard-core constraint on aandbsites. One of the sets,\nFigs. 6(a) and (c), shows data for the boundary poten-\ntials distinct from the bulk potentials, with the values in\nEqs. (9) and (8), respectively. The other set, Figs. 6(b)00.511.522.5\nB / J122124126128130132134136138\n00.511.522.5\nB / J122124126128130132134136138 N(b) Bound. pot. = Bulk pot.\nt - X - V (a) Bound. pot. ≠ Bulk pot. \nB = 0.06 J     with\n X=0, V=0t - X - V t - X - V \nt, finite L t, finite Lt - X - V \n     with\n X=0, V=0\n0 10\nl00.20.40.60.81\n0 10\nl00.20.40.60.81δ <n>Ni→Nf, l\n050100\nl00.1\n120\nl(d) Bound. = Bulk (c) Bound. ≠ Bulk \nFIG. 6. DMRG results for finite-size modified Rice-Mele\nmodels with L= 128. In (a) and (c), the boundary potentials\nhave the values in Eq. (9); while in (b) and (d), the boundary\npotentials have the same values of the bulk local potentials\nonaandbsites,εaandεbin Eq. (8). Taking the parameters\nfrom the spin-(1/2, 1) chain, we show data for the full inter-\nacting model, t−X−V, thet−X−Vmodel with X= 0 =V,\nand the hard-core approximation t. In (a) and (b), we present\nthe number of magnons Nas a function of magnetic field (in\nunits of J)B=−µ, where µis the chemical potential in\nthe bosonic system. In (c) and (d), we show the change in\nmagnon density per unit cell δ/angbracketleftn/angbracketrightNi→Nf,l=/angbracketleftn/angbracketrightNf,l−/angbracketleftn/angbracketrightNi,l\nbetween states N=NiandN=Nfas a function of unit cell\nl, where the arrows in (a) and (b) identify NiandNffor each\nmodel.\nand (d), takes the value of the boundary potentials equal\nto the bulk ones, Eq. (8). The arrows in Figs. 6(a) and\n(b) indicate the edge states whose boson density change\nδ/angbracketleftn/angbracketrightNi→Nf,l=/angbracketleftn/angbracketrightNf,l−/angbracketleftn/angbracketrightNi,l (34)\nis shown in Figs. 6(c) and (d), respectively, where Nf\n(Ni) is the highest (lowest) value of Nbetween the two\nmagnon density steps separated by the occupancy of the\nedge state.\nFor the bulk different from the boundaries, Fig. 6(a),\nwe see that the edge state is very robust, appearing even\nin thetapproximation. This does not occur for the t\napproximation if the boundary is equal to the bulk, Fig.\n6(b), which is an expected result for the Rice-Mele model\n[83]. However, we notice in Fig. 6(a) that, in comparison\nwith the spin model, the plateau width is very underesti-\nmatedinthe tapproximation,sincewearenotdiscarding\nthe single-particle states in the upper band shown in Fig.\n2. We observethat this upper band wasdiscarded for the\nL→ ∞results shown in Figs. 3 and 4, due to the in-\ntroduction of the effective chemical potential connected\nwith the Takhashi’s constraint [64]. Considering also the10\nspin model, the error in the plateau width for t−X−V\ninteracting case is very tiny, with a lower critical field\n≈0.06J. Also in Fig. 6(a), bulk and boundaries dis-\ntinct, we note that the edge state in the non-interacting\nt−X−Vmodel appears only for two bosons over the\nN= 127 state, Fig. 6(a), with one of the bosons oc-\ncupying an extended state and the other an edge state,\nas shown in Fig. 6(c). This shows that the distinction\nbetween boundaries and bulk is not sufficient to a good\nrepresentation of the edge state in the spin model, the\ninteractions are essential for it. In the case of the tap-\nproximation, the particle statistics provides a change in\nthe single particle quantum states that mimic the effect\nof theXandVinteractions in the t−X−Vmodel. In\nFig. 6 (b), where the boundary is equal to the bulk, we\nshow that edge states appear in the t−X−Vmodel in\nthe interacting and non-interacting cases. As mentioned\nabove, the edge state of the non-interacting case does\nnot appear in the tapproximation. Further, the bound-\nary density, Fig. 6 (d), in the interacting t−X−Vcase\nappears in the right extreme and has a value approxi-\nmately equal to that in Fig. 6 (c). From these results,\nwecanassertthat theinteractions XandV, and the par-\nticle statistics, are essential to understand the boundary\nmagnon density.\nV. SUMMARY AND DISCUSSION\nWe have investigated the Holstein-Primakoff bosonic\nHamiltonian, up to order/radicalbig\ns/S, of the alternating spin-\n(s= 1/2,S) chains in a magnetic field, and considering\nthe fully polarized as the vacuum. Three bosonic Hamil-\ntonians were considered: the first has only a hopping\nterm (t-approximation) and distinct local potentials ( ε)\non the spin-1 /2 (asite) and the spin- S(bsite) sites , this\napproximation is a Rice-Mele Hamiltonian with bound-\naries different from the bulk and uniform hopping term.\nThesecondonehasthetermsofthe t-approximationplus\namagnon-magnonrepulsion V(t-V-approximation); and\nthe last approximation shows the tandVterms, and a\ndensity-dependent correlatedhopping term X(t−X−V-\napproximation). In the t−X−Vapproximation, bsites\ncan accommodate up to two magnons, while in the oth-\ners a hard-core constraint is imposed on aandbsites.\nThe local potentials, at any density, and Vfor higherdensities favor magnon localization on the asites. On\nthe other hand, quantum fluctuations, magnon tunneling\nbetween aandbsites, are favored by t, at any density,\nandXfor higher densities. We use the density matrix\nrenormalization group to investigate the spin model and\nthe bosonic Hamiltonians t−Vandt−X−Vin finite-\nsize open systems, while for the t-approximation we have\nconsidered its analytical solution. We compare the mag-\nnetization and magnon densities per unit cell as a func-\ntion of a magnetic field, average bulk density and local\ndensities along the chains from the spin model and the\nbosonic approximations. From the ferrimagnetic plateau\n(one magnon per unit cell) to saturation (empty chain),\nthet−X−V-approximation is in excellent agreement\nwith the spin model, while the tandt−Vresults de-\npart from that of the spin model as the magnon density\nincreases. This, thus, showstherelevanceofbothinterac-\ntion terms, magnon-magnonrepulsion and the correlated\nhopping term X, and its associated particle fluctuations,\nto describe the many-body system near the ferrimagnetic\nplateau. The edge state associated with the insulating\nferrimagnetic plateau is well reproduced by the bosonic\nmodel. In particular, we have shown that the magnon\nboundary densities are strongly dependent on the inter-\nactions and the particle statistics.\nThe use of the fully polarized state as the vacuum en-\nables a better understanding of the underlying quantum\nprocesses in the spin Hamiltonian, compared to a ferri-\nmagnetic vacuum, which is not an exact eigenstate of the\nspin Hamiltonian. 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Mates, H. Matsuo, O. Parcol-\nlet, G. Paw/suppress lowski, J. D. Picon, L. Pollet, E. Santos,\nV. W. Scarola, U. Schollw¨ ock, C. Silva, B. Surer, S. Todo,\nS. Trebst, M. Troyer, M. L. Wall, P. Werner, and S. Wes-\nsel, The ALPS project release 2.0: open source software\nfor strongly correlated systems, J. Stat. Mech.: Theory\nExp.2011 (05), P05001.\n[86]T. Sakai and S. Yamamoto, Coexistent quantum\nand classical aspects of magnetization plateaux in\nalternating-spin chains, Journal of Physics Condensed\nMatter12, 9787 (2000)."    },    {        "title": "2402.12112v1.Magnetic_anisotropy_and_GGG_substrate_stray_field_in_YIG_films_down_to_millikelvin_temperatures.pdf",        "content": "1\nMagnetic anisotropy and GGG substrate stray field\nin YIG films down to millikelvin temperatures\nRostyslav O. Serha1,2∗, Andrey A. V oronov1,2, David Schmoll1,2, Roman Verba3, Khrystyna O. Levchenko1,\nSabri Koraltan1,2,4, Kristýna Davídková1,2, Barbora Budinska1,2, Qi Wang5, Oleksandr V . Dobrovolskiy1,\nMichal Urbánek6, Morris Lindner7, Timmy Reimann7, Carsten Dubs7, Carlos Gonzalez-Ballestero8,\nClaas Abert1,4, Dieter Suess1,4, Dmytro A. Bozhko9, Sebastian Knauer1, and Andrii V . Chumak1\n1Faculty of Physics, University of Vienna, 1090 Vienna, Austria.\n2Vienna Doctoral School in Physics, University of Vienna, 1090 Vienna, Austria.\n3Institute of Magnetism, Kyiv 03142, Ukraine.\n4Research Platform MMM Mathematics – Magnetism – Materials, University of Vienna, Vienna, Austria.\n5Huazhong University of Science and Technology, Wuhan, China.\n6CEITEC BUT, Brno University of Technology, 61200 Brno, Czech Republic.\n7INNOVENT e.V . Technologieentwicklung, 07745 Jena, Germany.\n8Institute for Theoretical Physics, Vienna University of Technology, 1040 Vienna, Austria.\n9Department of Physics and Energy Science, University of Colorado Colorado Springs, Colorado Springs,\nColorado 80918, USA.\nEmail:∗rostyslav.serha@univie.ac.at\nAbstract —Quantum magnonics investigates the quantum-\nmechanical properties of magnons such as quantum coherence or\nentanglement for solid-state quantum information technologies\nat the nanoscale. The most promising material for quantum\nmagnonics is the ferrimagnetic yttrium iron garnet (YIG),\nwhich hosts magnons with the longest lifetimes. YIG films of\nthe highest quality are grown on a paramagnetic gadolinium\ngallium garnet (GGG) substrate. The literature has reported\nthat ferromagnetic resonance (FMR) frequencies of YIG/GGG\ndecrease at temperatures below 50 K despite the increase in YIG\nmagnetization. We investigated a 97 nm-thick YIG film grown\non 500 µm-thick GGG substrate through a series of experiments\nconducted at temperatures as low as 30 mK, and using both\nanalytical and numerical methods. Our findings suggest that the\nprimary factor contributing to the FMR frequency shift is the\nstray magnetic field created by the partially magnetized GGG\nsubstrate. This stray field is antiparallel to the applied external\nfield and is highly inhomogeneous, reaching up to 40 mT in\nthe center of the sample. At temperatures below 500 mK, the\nGGG field exhibits a saturation that cannot be described by\nthe standard Brillouin function for a paramagnet. Including the\ncalculated GGG field in the analysis of the FMR frequency\nversus temperature dependence allowed the determination of\nthe cubic and uniaxial anisotropies. We find that the total\nanisotropy increases more than three times with the decrease\nin temperature down to 2 K. Our findings enable accurate\npredictions of the YIG/GGG magnetic systems behavior at low\nand ultra-low millikelvin temperatures, crucial for developing\nquantum magnonic devices.\nI. I NTRODUCTION\nMagnonics is the field of science that deals with data carried\nand processed by spin waves and their quanta, magnons, in\nmagnetically ordered media [1]. The ferrimagnet yttrium iron\ngarnet (YIG) Y 3Fe5O12is the material with the lowest known\nmagnetic damping as bulk material [2], [3], [4] and in formof thin films [5], [6], [7], [8], [9], [10], [11]. Thus, YIG is the\nmedium with the longest lifetimes of magnons up to one mi-\ncrosecond [12]. Therefore, YIG has emerged as a preeminent\nmaterial in RF technologies and magnonic experiments, show-\ning promise for quantum magnonic applications. The field of\nquantum magnonics is a rapidly growing and highly promising\nresearch area that operates with quantum magnonic states, e.g.\nsingle magnons, and hybrid structures [13], [14], [15], [16],\n[17], [18], [19], [20], [21], [22]. These investigations have to\nbe performed at millikelvin temperatures to ensure that there is\nminimal thermal noise, allowing for precise observation and\nmanipulation of the quantum states of magnons, which are\nextremely fragile to any kind of distortion.\nNote, YIG was already the material of choice in experiments\nat low kelvin and ultralow millikelvin temperature magnonics\nfor coupling to superconducting resonators [13], [23], [24],\n[19] and propagating spin-wave spectroscopy [25], [26], [20].\nFurthermore, the first demonstration of magnon control and\ndetection at the single magnon level [17] and the first measure-\nments of the Wigner function of a single magnon [18] were\nperformed also using a YIG sphere as a magnetic medium.\nProposals for applications in quantum computing have also\nemerged, in particular the use of YIG spheres as magnonic\ntransducers for qubits [19]. A pressing question in the quantum\nmagnonics community is how to bring the more complex,\nflexible, and property-rich structures employed in classical\nmagnonic nanodevices to the quantum regime. Two dimen-\nsional geometries are a paramount example, for instance YIG\nfilms down to tens of nanometers thick grown on gadolinium\ngallium garnet (GGG) Gd 3Ga5O12substrates [6], [7], [8], [5],\n[10], [11] enables the development of nanoscale magnetic\ndevices and circuits [14]. As the temperature decreases, thearXiv:2402.12112v1  [cond-mat.mes-hall]  19 Feb 20242\nspin-wave damping in YIG/GGG increases up to tenfold due to\nvarious effects associated with impurities in YIG and parasitic\ninfluence of the paramagnetic GGG substrate [27], [28], [29],\n[30], [20], [31]. Our experimental investigations of spin-wave\ndamping agree well with the results reported in the literature\n(but are out of the scope of this article).\nSeveral experimental reports have demonstrated that low-\nering the temperature below about 50 K in YIG/GGG shifts\nthe ferromagnetic resonance (FMR) frequency or the fre-\nquency of propagating spin waves [32], [26], [33], [34],\n[30]. Our experimental results show the same behavior. The\ninterpretation given by the authors in [32], agrees with our\nanalysis. However, at the time of the study, the measurements\nand calculations were not performed below 4.2 K, which is\nparticularly interesting for quantum magnonics and is the\ntemperature regime in which GGG shows a complex magnetic\nphase behavior [35], [36]. Additionally, the role of the strong\nnon-uniformity of the field induced by the partially magnetized\nGGG was not explored as well as the change of crystallo-\ngraphic anisotropy in YIG with decreasing temperature.\nHere, through experiments, theory, and numerical simula-\ntions, we studied the stray field induced by the GGG substrate\nat temperatures as low as 30 mK. Using vibrating-sample mag-\nnetometry (VSM), we measured the magnetization of GGG as\na function of the applied field and temperature down to 2 K.\nTo extrapolate values for lower temperatures, we utilized the\nBrillouin function. We utilized analytical theory and numerical\nsimulations to determine the GGG stray field and the cubic\nand uniaxial crystallographic anisotropies of YIG. Our findings\nrevealed that the GGG field is strongly non-uniform, ranging\nfrom 10 mT to 70 mT for the YIG/GGG sample of (5x5) mm2,\nat a temperature of 2 K, and an applied magnetic field of\n600 mT when the magnetization of GGG was 238 kA/m. It has\nbeen experimentally confirmed by FMR measurements that\nthe magnetization of the GGG substrate does not change with\ntemperature below 500 mK [36] in contrast to the Brillouin\nmodel of a paramagnet.\nII. M ETHODOLOGY\nA. Experimental methods\nIn our research, we studied a (5x5) mm2and 97 nm-thick\nYIG film grown on a 500 µm-thick GGG substrate, using liquid\nphase epitaxy [5], [6]. We conducted stripline ferromagnetic-\nresonance (FMR) spectroscopy using a vector network ana-\nlyzer (VNA), within a Physical Property Measurement System\n(PPMS), at temperatures ranging from 2 K to 300 K. FMR\nspectroscopy was conducted up to 40 GHz. The experiments at\nultralow temperatures were conducted in a dilution refrigerator\nthat can reach a base temperatures of 10 mK.\nThe following measurements were taken with the external\nmagnetic field in the in-plane orientation and applied along\nthe FMR stripline antenna. To determine the FMR spectrum\nfor a specific field, S12and S21parameters were measured\nusing a VNA not only at the target field but also at reference\nfields adjusted to approximately 15 mT to 40 mT, both above\nand below the desired value [37]. By subtracting the averaged\nsignals of the reference fields from the measured FMR signal,we obtained the FMR absorption spectrum in YIG that was\nnot affected by GGG (see Fig. 2 (b) as an example). This\ndual reference measurement approach enabled to obtain the\nbest results when working with kelvin and sub-kelvin tem-\nperatures, since the GGG parasitic signal is greatly affected\nby the change in the applied field. To obtain the resonance\nfrequency and full linewidth at half maximum, the background\nis first analyzed using a 1D cubic spline model [38]. The\nresonance shape is then fitted using the double Lorentzian\nmodel, which individually describes the left and right sides\nof the asymmetric absorption peaks.\nTo accurately determine the magnetization of GGG for our\nanalytic calculations and numerical simulations, we utilized\nvibrating-sample magnetometry (VSM) on a pure GGG slab\nin the temperature range from 2K to 300K. The Gd+3ions\nin GGG have a relatively large spin (S = 7/2), resulting in\na saturation magnetization that is notably higher than that\nof YIG. Specifically, the saturation magnetization of GGG,\ndenoted by Ms\nGGG, is equal to 805 kA/m, as shown in Fig. 2 (a).\nBased on established practices, experimental FMR data can\nbe used to extract the effective magnetization, the anisotropy\nfields, and the Gilbert damping parameter of a magnetic mate-\nrial (see [39] and the supplementary materials therein). In this\nstudy, the temperature-dependent saturation magnetization of\nYIG, Ms\nYIG, is taken from the analytical calculation performed\nin [40]. Thus, we can use this information to more accurately\ndetermine the anisotropy fields of YIG using them as a fitting\nparameter.\nAs reported in the literature [32], [20] and supported by our\nFMR analysis, the paramagnetic GGG becomes sufficiently\nmagnetized at temperatures below about 100 K, together with\nan external magnetic field applied. This magnetization in-\nduces a magnetic stray field BGGG in the YIG layer, which\ncauses a shift of the YIG FMR frequencies [32]. For the in-\nplane applied magnetic field, the FMR shift is toward lower\nfrequencies because BGGG and the applied bias field B0are\nantiparallel. Conversely, in an out-of-plane geometry, the stray\nfield BGGG aligns parallel to the field B0, resulting in a shift\nof the resonance frequency to higher values [32]. The positive\nshift was also confirmed by our experimental results, but in\nthis paper we focus only on the in-plane configuration.\nThe magnitude of this inhomogeneous stray field BGGG is\ninfluenced by both the temperature and the strength of the\nexternal magnetic field. At lower temperatures and higher\nexternal fields, the GGG-induced stray field becomes more\npronounced, which is crucial for determining the FMR fre-\nquency fFMR. The modified Kittel formula tailored for in-plane\nmagnetization geometry describes the influence of this field on\nthe FMR frequency:\nfFMR =γ·p\n(B0−BGGG)·\n·q\n(B0−BGGG +Beff\nani+µ0Ms\nYIG),(1)\nwhere γis the reduced gyromagnetic ratio, B0is the applied\nexternal field, Beff\nanithe effective crystallographic anisotropy\nfield [39], which is a fit parameter as described in detail below,\nBGGG is the GGG-induced stray field defined analytically\nin the next section and Ms\nYIGthe theoretical value for the3\nsaturation magnetization of YIG at any temperature taken\nfrom [40].\nB. Analytical calculation of GGG field\nAt low temperatures below 10 K and in the presence of\nmagnetic fields measuring several hundred millitesla, GGG\nattains a significant magnetization exceeding hundreds of\nkA/m (see Fig. 2 (a)). In this state, the GGG substrate becomes\na magnet and emits a stray magnetic field that expands\nbeyond its volume. To accurately determine the strength and\ncharacteristics of the stray field generated by the GGG, two\ncrucial parameters are required: MGGG, which denotes the\nmagnetization of GGG, and ˜Nxx, which represents the averaged\nmutual in-plane demagnetization factor of GGG and YIG\nslabs. Then, one can accurately determine the strength and\ncharacteristics of the stray field generated by the GGG in the\ndirection of the external field BGGG as\nBGGG =µ0MGGG(T,B0)˜Nxx. (2)\nBGGG is crucial for understanding its overall magnetic influ-\nence, particularly at low temperatures, on the internal field of\nthe YIG film. Net GGG magnetization MGGG is given by the\nimplicit equation [41]:\nMGGG =Ms\nGGG·\n·B7\n2\u00127\n2·gµB·(B0−µ0MGGGNxx+λ µ0MGGG)\nkBT\u0013\n,(3)\nwhere Ms\nGGG= 805 kA/m denotes the saturation magnetization\nof GGG, g= 2 the Landé factor, µBthe Bohr magneton, B7\n2\nthe Brillouin function for the angular momentum J=7\n2,λ\nthe coefficient of the molecular field and Nxxis the standard\ndemagnetization factor (i.e. self-demagnetization) of GGG\nsample.\nIn our case of thin GGG cuboid of the sizes 2 a×2a×2c,\nwhere c≪a, self demagnetization factor can be approximated\nas [42]\nNxx≈c\nπa\u0010\n0.726−logc\na\u0011\n. (4)\nFor the mutual demagnetization factor, we account that YIG\nfilm is much thinner than GGG, and that we deal with\nnonuniform FMR in YIG, because the lateral sizes of YIG\nare much larger than spin-wave free path, so that standing\neigenmodes are not formed. In this case, the FMR peak\nposition is mostly determined by central YIG area, for which\nmutual demagnetization factor is\n˜Nxx=1\nπ·arctan\"√\n2c√\na2+2c2#\n. (5)\nFull expressions for arbitrary prisms are available in [42].\nNote, both approximations for self and mutual demagnetiza-\ntion factors may not be adequate for small external fields B0,\nas it neglects the potential nonuniform magnetization of GGG.\nTo ensure that our calculations and numerical micromag-\nnetic simulations would be most precise, the magnetization\nof GGG MGGG was measured for each temperature within\nthe same field range used in the FMR experiments. Forcalculations at temperatures below 2 K, we relied on the\nVSM data obtained earlier. The data was fitted using Eq. 3\nby adjusting the molecular field λ= -0.854 and saturation\nmagnetization Ms\nGGG= 756 kA/m as fitting parameters. This\nfitting method was used to extrapolate the values of MGGG\nwith respect to temperature Tand external magnetic field B0.\nIt allowed us to compare our theoretical and simulation work\nwith experimental results even at temperatures below 2 K.\nC. Numerical simulations\nTo gain a better understanding of the stray magnetic field\npresent in the YIG layer on the GGG substrate, we also\nconduct micromagnetic simulations using magnum.np [43] and\nmumax3[44]. These simulations offer a more precise depiction\nof the inhomogeneity of the field.\nThe simulations for the GGG substrate were conducted\nusing the same geometric parameters as in the experimental\nstudy. The saturation magnetization, denoted as MGGG, was\ntake from the experimental results measured using VSM. Al-\nthough micromagnetic solvers are not intended for simulations\nof a paramagnetic and cannot give a rigorous description\n(it requires other numerical approaches [45], which are too\ncumbersome in 3D), static paramagentic behavior of the\nmaterial can be reasonably simulated by setting the exchange\nconstant Aexto zero. Due to the small magnetic susceptibility\nχ(see Ref. [36]) in the linear regime, it is assumed that the\nmagnitude of the local magnetization does not change by\ncause of self-demagnetizing effects. The external magnetic\nfield B0was applied in-plane, as depicted in Fig. 1 (a). To\nensure saturation within a simulation time of 2 ns, a Gilbert\ndamping factor of α= 1 was utilized, and the resulting demag-\nnetization field was recorded as a projection in the direction\nof the external bias field BGGG. The simulation employed a\nmesh of 1000 ×1000×26 cells. Given that the simulation’s\nobjective was to examine the magnetization’s relaxed state\nin the absence of exchange interaction, a cell dimension of\n(5×5×20) µm3was deemed adequately sufficient. In the end,\nto determine the strength of the demagnetization field BGGG\nat the interface between YIG and GGG, the mean-field value\nfrom the last two layers along the z-axis was calculated.\nIII. R ESULTS\nTo simulate the strength and profile of the stray field induced\nby the paramagnetic substrate over the YIG layer, we first\nmeasured the magnetization of a bare GGG substrate, as shown\nin Fig. 2 (a). In the partially magnetized state, GGG has a\nsmall value of magnetic susceptibility χ≈0.3 and creates a\nhighly inhomogeneous y-field component By\nGGGon its surface,\nas seen in Fig. 1 (b). As an example, at 2 K, this induced field\nBy\nGGGopposes the external magnetic field of 600 mT. It varies\nin strength from 13 mT at the center to 70 mT at the edges,\nantiparallel to the direction of the external magnetic field.\nThis field is crucial in investigating YIG/GGG systems at low\ntemperatures.\nFig. 2 (b) displays two spectra that exemplify the impact\nof cryogenic temperatures on the FMR signal of the YIG\nfilm in an external magnetic field of 425 mT. The FMR peak4\nFig. 1. (a) Depiction of the experimental system of a YIG film grown on\nGGG. The sample is in-plane magnetized by an external magnetic field,\nand at temperatures approaching 0 K the paramagnetic GGG spin system\nsaturates. The substrate creates an inhomogeneous stray field, that becomes an\nadditional component to the internal magnetic field of the YIG but is oriented\nantiparallel to the external field. For measuring FMR the YIG film sample is\nplaced on a CPW antenna, through which the magnetic system is excited. (b)\nMicromagnetic simulation of the highly inhomogeneous By\nGGGstray field y-\ncomponent at the interface between YIG and GGG layers. The inhomogeneity\ncan also be seen for the xand y-axis for the center of the sample by the\ntwo cross-section plots marked in the 2D map with the red dotted lines.\nThe simulation is performed using mumax3[44] for the temperature 2 K and\nthe strength of the applied external magnetic field of 600 mT at which the\nmagnetization of GGG was 238 kA/m.\nis almost entirely Lorentzian-shaped at room temperature.\nHowever, when the temperature decreases below 100 K, the\npeak becomes asymmetric on one side, as shown in Fig. 2 (b)\nfor 52 K, and then broadens significantly with a lower reso-\nnance frequency, as demonstrated for 2 K. Additionally, the\namplitude of the FMR peak decreases as the temperature\nreduces. Note the shape and width of the FMR peak will be\nanalyzed quantitatively and discussed in future work. For now,\nthis manuscript exclusively focuses on the internal field andthe FMR frequency of the YIG film.\nFig. 2 (c) clearly shows the impact of the stray field induced\nby the GGG substrate. The graph displays the FMR frequen-\ncies fFMR obtained for the YIG film at various temperatures,\nranging from 300 K to 30 mK. As the temperature decreases,\nthe FMR frequency increases due to the rise in the saturation\nmagnetization of YIG. The theoretical curves, represented by\ndashed lines, were calculated using Eq. 1, with the gyromag-\nnetic ratio γand effective anisotropy field Beff\nanitaken from the\nKittel fit at room temperature, but neglecting the contribution\nof the GGG-induced stray field ( BGGG= 0).\nAt temperatures below 50 K, the experimental FMR fre-\nquencies deviate from the theoretical values due to the GGG-\ninduced stray field. The data indicate that for an external\nfield B0of 925 mT, the FMR frequency begins to decrease\nat approximately 50 K, while for B0values of 325 mT, this\noccurs at around 25 K. At temperatures below 2 K and high\nexternal fields of 925 mT, there is a notable deviation of over\n0.5 GHz between the experimental and theoretical results. This\ndifference is attributed to the GGG stray field, which opposes\nthe external magnetic field and lowers the FMR frequency\nfFMR of YIG. The FMR frequency shift, dependent on MGGG,\nbecomes more pronounced as the temperature decreases and\nthe external magnetic field increases, as shown in Fig. 2 (c).\nThe results of comparing three different magnetic fields at\n30 mK shows that the impact is more pronounced at higher\nexcitation frequencies, which is associated with stronger mag-\nnetic fields.\nBelow 500 mK, the frequency fFMR displays unusual be-\nhavior as its decline stabilizes, showing negligible change\ndown to 30 mK, despite the varied magnetization of GGG in\nthis temperature range as predicted by the Brillouin function\n(Eq.,3). This phenomenon is due to the complex nature of\nGGG, which possesses a geometrically highly frustrated spin\nsystem [46], [47], leading to a complex phase diagram for\ntemperatures below 1 K [35], [36]. Understanding the sub-\nkelvin temperature behavior of GGG requires considering\ncompeting interactions among loops of spins, trimers, and\ndecagons, along with the interplay between antiferromagnetic,\nincommensurate, and ferromagnetic orders [36]. Consequently,\nthe Brillouin function fails to describe the magnetization of\nGGG below 500 mK and can be seen even clearer in later\ndescribed Fig. 3 (c). This finding aligns with previous experi-\nmental studies [36] using different techniques, such as single-\ncrystal magnetometry and polarized neutron diffraction. In-\ncorporating the analytically-calculated GGG stray field BGGG\ninto the Kittel formula Eq. 1 is necessary for determining the\nFMR frequency. To accurately identify the FMR frequencies\nfFMR e.g. shown in Fig. 2 (b), it is essential to account for\nthe GGG-induced stray field BGGG at the sample center. At\nthis location, the gradient is zero, and the region of the same\nfield magnitude is the largest, most significantly affecting the\narea excited by the microwave stripline and determining the\nposition of the FMR peak. The red vertical dashed line in\nFig. 1 (b) approximately depicts the position of the microstrip\nFMR antenna. Once incorporated into the Kittel equation,\nthe gyromagnetic ratio γand the effective anisotropy field\nBeff\nanibecome fitting parameters. Obtaining Beff\naniallows us to5\nFig. 2. (a) VSM measurements of the magnetization of GGG MGGG as\nfunction of the temperature Tand the external magnetic field B0.MGGG\nsaturates at a value of 805 kA/m. The graphs represent the GGG magneti-\nzation over the range available in the experiment. (b) Example spectra of\nFMR at the temperatures of 2 K, 52 K, and room temperature (RT) for an\nexternal magnetic field of 425 mT. (c) FMR frequency as function over the\ntemperature, plotted in a x-axis logarithmic scale, for 3 different magnetic\nfields. Measurements are depicted as points and were performed with the\nsample magnetized in the ⟨110⟩direction. The dotted lines are portraying the\nanalytical calculation for the frequency of the FMR by the Kittel formula,\nwhich is neglecting the induced GGG stray field. The parameters for the\ngyromagnetic ratio γand effective anisotropy field Beff\naniare obtained by fitting\nroom temperature measurements.gain insight into and make predictions about the internal field\nof thin YIG films at low temperatures. Fig. 3 (a) displays\nthe fitting outcomes for the magnetization orientations of\n⟨110⟩(red) and ⟨112⟩(black), with the effective anisotropy\nfield Beff\naniplotted against temperature on a x-axis logarithmic\nscale. The error bars are taken from the root-mean-square\ndeviation of the fit. At room temperature, the anisotropy field\nis relatively small, measuring approximately 5 mT, with a\nvariation of about 0.6 mT between the two orientations due\nto cubic anisotropy of the YIG single crystal.\nHowever, as the temperature decreases, both the strength of\nBeff\naniand the difference between ⟨110⟩and⟨112⟩magnetization\ndirections increase significantly. At a temperature of 2 K, the\neffective anisotropy field Beff\naniis more than 3.2 and 3.5 times\nlarger for the ⟨110⟩and⟨112⟩directions, respectively, than at\nroom temperature. The inset in Fig. 3 (a) displays the second\nfitting parameter, the gyromagnetic ratio γ, as a function of\ntemperature on a x-axis logarithmic scale. It is known from\nprevious research on YIG that γis considered to be weakly\ntemperature dependent [48], [49], [31]. The behavior of γ\nshows a very weak decrease at lower temperatures, changing\nfrom 28.13 GHz/T to 28.0 GHz/T. γis nearly identical for\nboth magnetization directions, and the values fall within each\nother’s error bars.\nTo get a better understanding of the changes in the effective\nanisotropy field Beff\nani, the field was divided into its two main\ncomponents: the cubic anisotropy field Bcand the uniaxial\nanisotropy field Bu. This separation is achieved by considering\nthe two magnetization directions and applying two separate\nequations for the FMR frequency, as described in [50].\nf⟨110⟩\nFMR =γ⟨110⟩·p\n(B0−BGGG)·\n·q\n(B0−BGGG−Bu−Bc+µ0Ms)−2B2c,(6)\nf⟨112⟩\nFMR =γ⟨112⟩·p\n(B0−BGGG)·\n·p\n(B0−BGGG−Bu−Bc+µ0Ms).(7)\nUsing Eq. 6 and Eq. 7 to fit the FMR frequency fFMR data,\nand utilizing the gamma values from Fig. 3 (a), we can obtain\nvalues for the two distinct anisotropy fields, Bc(green) and\nBu(blue) and their errors by root-mean-square deviation.\nThe resulting plot in Fig. 3 (b) shows the anisotropy field\nBanias a function of temperature Ton a x-axis logarithmic\nscale. Our experimental findings at room temperature indicate\n(-6.9 ± 2) mT for Bcand (2.1 ± 2) mT for Bu, which agree\nwell with previously reported values for thin YIG films [6].\nNotably, the cubic anisotropy increases to (-11.6 ± 0.8) mT\nat temperatures as low as 2 K, almost doubling the room\ntemperature value. The uniaxial anisotropy exhibits a unique\ncharacteristic of changing its positive-to-negative sign at cryo-\ngenic temperatures and reaching a peak value of (-5 ± 0.7) mT\nat 2 K.\nWith this understanding of the low-temperature anisotropy\nand the stray field caused by GGG, we can make accurate6\npredictions about the FMR behavior in YIG films even at\ntemperatures as low as 2 K. Note, that the anisotropy increase\nreaches a saturation point below 10 K, as demonstrated in\nFig. 3 (a). As a result, we can assume that the anisotropy\nremains constant down to the millikelvin temperatures and can\nbe treated as equivalent to the 2 K values.\nHowever, in the absence of magnetization MGGG measure-\nments below 2 K, we cannot utilize experimental data to com-\npute the stray field BGGG using Eq. 2. Therefore, we must rely\non the Brillouin fit presented in Sec. II-A (Eq. 3) and extrapo-\nlate MGGG as an estimation. Additionally, a second method\nwas used to determine the values for BGGG at millikelvin\ntemperatures in the center of the YIG sample by measuring the\nFMR position at these temperatures and rearranging Eq. 1 to\nsolve for BGGG. The results are shown in Fig. 3 (c). It depicts\nthe stray field induced by GGG at the center of the sample\nas a function of the externally applied magnetic field B0.\nThe solid lines represent the calculated values of BGGG for\ntemperatures of 2 K and above, while the solid points represent\nthe obtained BGGG from the FMR measurements. Both data\nsets match perfectly. This alignment, in Eq. 1, highlights the\naccuracy of the fitted effective anisotropy Beff\nani, affirming the\nprecision and appropriateness of the fit. The data points shown\nby the hollow-center-dot illustrate the BGGG values acquired\nvia FMR measurements in the dilution refrigerator, spanning\na temperature range of 30 mK to 2 K. It is worth noting that\nthe values of BGGG at 2 K obtained from both the dilution\nrefrigerator and the PPMS measurements are consistent.\nThe extrapolated data obtained through the Brillouin fit is\nshown as dotted lines for both 500 mK and 30 mK, in com-\nparison to the experimental data (Fig.,3 (c)). At 500 mK, the\nextrapolation matches at fields above 1 T and below 300 mT\nwith the experimental data but diverts in-between (purple\ncircle points and dashed line). At 30 mK, it does not match\n(yellow circle points and dashed line). The extrapolated field\nstrength of GGG (dashed yellow) experiences a sharp increase\nand then saturates above 1.1 T. While the extrapolated curves\ndeviate, the measured data for 30 mK and 500 mK overlap.\nThese results show that the method of fitting the GGG mag-\nnetization MGGG and the stray magnetic field BGGG effectively\ndescribes the inter- and extrapolations only at temperatures\nabove 500 mK. This limitation confirms the conclusion above.\nMGGG is solely dependent on the externally applied magnetic\nfield below 500 mK, which is supported by the behavior of\nthe FMR frequency in Fig. 2 (c) and previous research on\nthe complex behavior of GGG phase states [36]. At these\ntemperatures, GGG was observed to transition through various\nmagnetic phases, such as spin glass and antiferromagnetic,\ndepending on the external magnetic field. [35].\nIV. C ONCLUSION\nWe found that YIG films grown on GGG substrates are im-\npacted by stray fields originating from the partially magnetized\nparamagnetic GGG at low temperatures and under externally\napplied magnetic fields. The strength and configuration of\nthese fields depend on the shape of the GGG substrate (the\nratio between the width, length and thickness of the sample)\nFig. 3. (a) Effective anisotropy field Beff\naniof the YIG film as a function of\nthe temperature in a x-axis logarithmic scale for two different crystallographic\nmagnetization directions — ⟨110⟩and⟨112⟩. The inset depicts the gyromag-\nnetic ratio γas a function of the temperature for the same magnetization\ngeometries respectively. The points are obtained as fitting parameters from\nEq. 1. (b) Cubic and uniaxial anisotropiy fields Bc,Buvs temperature T.\nValues for BcandBuwere obtained from according Eq. 6 and 7 for the Kittel\nfits [50]. (c) GGG-induced stray field as function of the externally applied\nmagnetic field B0. Solid lines are the values calculated from the measured\nMGGG by Eq. 2 of the VSM and solid lines are values calculated from the\nFMR peaks measured in the PPMS by converting Eq. 1 for BGGG. The hollow\npoints are BGGG values calculated from FMR peak measurements performed\nin the dilution refrigerator. The dotted lines are calculated via Eq. 2 by taking\nthe extrapolated values of MGGG by the Fit from Eq. 3.7\nand are highly inhomogeneous across the YIG layer. In the\nin-plane magnetization geometry, the stray field can reach up\nto 40 mT in the center and increases five-fold at the edges of\nthe sample.\nWe used an analytical approach validated by micromag-\nnetic simulations to calculate the stray field BGGG induced\nby GGG. This approach allowed us to integrate BGGG into\nthe Kittel-fit formula and accurately determine the effective\nanisotropy field Beff\naniin the crystallographic directions ⟨110⟩\nand⟨112⟩of the YIG film for temperatures as low as 2 K.\nMoreover, we were able to extract the crystallographic cubic\nand uniaxial anisotropy fields, BcandBu, respectively. These\nfields increase in magnitude from -6.9 mT and 2.1 mT at room\ntemperature to -11.6 mT and -5 mT at 2 K. The anomalous\nbehavior of the FMR frequency of YIG, which is constant\nfor temperatures below 500 mK, can be explained by the\nabsence of the variation of the GGG magnetization MGGG with\ndecreasing temperature, and therefore by the GGG-induced\nmagnetic field BGGG. This behavior can be described by the\nproperty of GGG as a geometrically highly frustrated magnet,\nresulting in the complex phase transition diagram of GGG\nat these temperatures and fields [46], [47], [35], [36]. Our\nfindings enable accurate predictions of the YIG/GGG magnetic\nbehavior at low and ultra-low temperatures, which is a key\nelement for successfully implementing YIG/GGG quantum-\nmagnonic networks.\nACKNOWLEDGEMENTS\nA VC acknowledges the Austrian Science Fund FWF for the\nsupport by the project I-6568 \"Paramagnonics\". SK acknowl-\nedges the support by the H2020-MSCA-IF under Grant No.\n101025758 (\"OMNI\"). DS acknowledges the Austrian Science\nFund FWF for the support by the project project I 4917-N\n\"MagFunc\". KOL acknowledges the Austrian Science Fund\nFWF for the support through ESPRIT Fellowship Grant ESP\n526-N \"TopMag\". Work by DAB was supported by the U.S.\nDepartment of Energy (DOE), Office of Science, Basic Energy\nSciences (BES) under Award DE-SC0024400. RV acknowl-\nedges support of the NAS of Ukarine (project 0123U104827).\nThe work of ML was supported by the German Bundesmin-\nisterium für Wirtschaft und Energie (BMWi) under Grant No.\n49MF180119. CD thanks R. Meyer (INNOVENT e.V .) for\ntechnical support. CGB acknowledges the Austrian Science\nFund FWF for the support with the project PAT-1177623\n\"Nanophotonics-inspired quantum magnonics\". The authors\nthank Prof. Dr. G. A. Melkov for valuable scientific discus-\nsions.\nAUTHOR CONTRIBUTIONS\nROS conducted all measurements, processed and analyzed\nthe data, and authored the initial draft of the manuscript.\nAA V executed the micromagnetic simulations, implemented\nthe method for fitting the data, and contributed to data analysis.\nSAK, CA and DS developed the software used for micro-\nmagnetic simulations. DAS, SAK, DAB and SK constructed\nthe experimental setup and supported the experimental inves-\ntigations. KOL, KD, QW and MU assisted in interpreting theexperimental data and provided insights into the measurement\nresults. ML, TR and CD synthesized the YIG film. RV and\nCGB provided theoretical support. BB and OVD facilitated\nsupport in conducting PPMS measurements. SK supervised\nthe millikelvin experiments. A VC led the project. 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K´ ezsm´ arki2, J. Viirok3, U. Nagel3, T.\nR˜ o˜ om3, A. Puri4, U. Zeitler4, Y. Tokunaga5,6, Y. Taguchi5, and Y. Tokura5,7\n1Materials Science and Technology Division, Oak Ridge Natio nal Laboratory, Oak Ridge, Tennessee 37831, USA\n2Department of Physics, Budapest University of Technology a nd Economics and MTA-BME\nLend¨ ulet Magneto-optical Spectroscopy Research Group, 1 111 Budapest, Hungary\n3National Institute of Chemical Physics and Biophysics, Aka deemia tee 23, 12618 Tallinn, Estonia\n4High Field Magnet Laboratory (HFML-EMFL), Radboud Univers ity Nijmegen,\nToernooiveld 7, 6525 ED Nijmegen, The Netherlands\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Sait ama 351-0198, Japan\n6Department of Advanced Materials Science, University of To kyo, Kashiwa 277-8561, Japan and\n7Department of Applied Physics, University of Tokyo, Hongo, Tokyo 113-8656, Japan\n(Dated: September 17, 2021)\nCompetingexchangeinteractions canproducecomplexmagne ticstates together withspin-induced\nelectric polarizations. With competing interactions on al ternating triangular and kagome layers,\nthe swedenborgite CaBaCo 4O7may have one of the largest measured spin-induced polarizat ions of\n∼1700 nC/cm2below its ferrimagnetic transition temperature at 70 K. Upo n rotating our sample\naboutc= [0,0,1] while the magnetic field is fixed along [1 ,0,0], the three-fold splitting of the spin-\nwave frequencies indicates that our sample is hexagonally t winned. Magnetization measurements\nthen indicate that roughly 20% of the sample is in a domain wit h theaaxis along [1 ,0,0] and that\n80% of the sample is in one of two other domains with the aaxis along either [ −1/2,√\n3/2,0] or\n[−1/2,−√\n3/2,0]. Powder neutron-diffraction data, magnetization measur ements, and THz absorp-\ntion spectroscopy reveal that the complex spin order in each domain can be described as a triangular\narray of bitetrahedral c-axis chains ferrimagnetically coupled to each other in the abplane. The\nelectric-field dependence of bonds coupling those chains pr oduces the large spin-inducedpolarization\nof CaBaCo 4O7.\nPACS numbers: 75.25.-j, 75.30.Ds, 75.50.Ee, 78.30.-j\nI. INTRODUCTION\nCompeting exchange interactions produce complex\nmagnetic states with a wide range of interesting behav-\nior found in spin glass [1], spin ice [2], and magnetic\nskyrmions [3]. In multiferroic materials, complex spin\nstates can exibit a spin-induced electric polarization P\ndue to either the spin current, p-dorbital hybridization,\nor magnetostriction [4,5]. Because the coupling between\nthe electrical and magnetic properties in multiferroic ma-\nterials is both scientifically and technologically impor-\ntant, the effects of competing exchange interactions have\nbeen investigated in a wide range of multiferroic materi-\nals such asRMnO3[6], CoCrO 4[7], CuCrO 2[8], CuFeO 2\n[9], and MnWO 4[10]. While the first four materials [6–\n9] are geometrically frustrated due to competing interac-\ntions on a triangular lattice, MnWO 4[10] exhibits long-\n∗Copyright notice: This manuscript has been authored by UT-\nBattelle, LLC under Contract No. DE-AC05-00OR22725 with th e\nU.S. Department of Energy. The United States Government re-\ntains and the publisher, by accepting the article for public ation,\nacknowledges that the United States Government retains a no n-\nexclusive, paid-up, irrevocable, world-wide license to pu blish or re-\nproduce the published form of this manuscript, or allow othe rs to\ndo so, for United States Government purposes. The Departmen t of\nEnergy will provide public access to these results of federa lly spon-\nsored research in accordance with the DOE Public Access Plan\n(http://energy.gov/downloads/doe-public-access-plan ).range competing interactions [11] on a highly-distorted\nmonoclinic lattice.\nCompounds in the “114” swedenborgite family [12]\nRBaM4O7(M= Co or Fe) contain alternating triangu-\nlar and kagome layers, both of which are geometrically\nfrustrated when undistorted. The “114” cobaltites [13–\n15] were initially studied to find charge ordering among\nthe Co2+and Co3+ions. An important member of this\nfamily, YBaCo 4O7exhibits antiferromagnetic ordering\n[16,17] below 110 K and diffuse scattering [13,14] indica-\ntive of spin disorder below 60 K. The magnetic state be-\ntween 110 K and 60 K is stabilized by a structural tran-\nsition [18] that relieves the geometric frustration. Both\nstructural and magnetic transitions are quite sensitive\nto excess oxygen and no magnetic order [19,20] appears\nin YBaCo 4O7+δforδ≥0.12. Another family member,\nYbBaCo 4O7undergoes a structural transition at 175 K\nthatstabilizesanantiferromagneticstatebelow80K[21].\nAparticularlyinteresting“114”cobaltite, CaBaCo 4O7\nundergoes an orthorhombic distortion [22,23] that re-\nlieves the geometric magnetic frustration on both the\nkagome and triangular layers sketched in Fig. 1 above\nthe magnetic transition temperature Tc= 70 K. Below\nTc, CaBaCo 4O7develops a very large spin-induced po-\nlarization ∼1700nC/cm2[24], secondonly to the conjec-\ntured [25] spin-induced polarization ∼3000 nC/cm2of\nBiFeO 3. Also unusual, CaBaCo 4O7displays a substan-\ntial ferrimagnetic moment of about 0.9 µBper formula\nunit (f.u.) [26], which could allowmagnetic controlof the2\n!\"#$\n%\"&'($)$ !*#$\n+,) $\n+,- $+.$!\"#\n+.$\n+,- $\n$\"#+.$\n+.$$\"#+.$\n%\"#\n%\"#&\"#\n&'#+.$\n(\"#(\"#)\"#+/$\n+/$\n+/$+/$+/$\n%\"#*\"#\n%\"#\n!\"#)\"#)\"#\n*\"#+.$+0$\n+0$+,, $+,) $+,, $\n+,, $+,- $\n+,- $+,, $+,) $+,, $+,) $+,- $\n+0$+,) $*\"#\n!\"#\n$\"#$\"#\n+0$+0$\n*\"#+\"#\n+\"# +,) $+,, $+,) $+,, $\n+,, $+,- $\n+,- $*$\n\"$%\"&'($,$\n+,- $+,- $%#!#\n$#\n$#!#\n+,- $\n+.$+.$\n+.$\n&#\n&#\n+/$+/$+/$\n&#+,) $\n+.$*#+.$\n%#(#\n+#\n+#)#\n)#)#\n+0$+/$+,- $\n+,, $+,) $+,, $+,) $\n+0$+/$\n+,, $+,) $+,, $\n+,, $+,) $+,, $+,) $+,, $+,) $\n+,- $+,) $&#\n(# (#\n!#(#\n*#\n+0$\n!#!#\n!#+0$\n+0$\nFIG. 1: (Color online) (a) and (b) The predicted spin config-\nuration for layers 1 and 2 in zero field. Spins 1 and 5 lie on\na triangular layer above the first kagome layer in (a); spins 1′\nand 5′lie on a triangular layer above the second kagome layer\nin (b). Layers are arranged so that spins 1′and 5′lie directly\nabove spins 1 and 5.\nelectric polarization. Although its ferroelectrictransition\nis inaccessible and its permanent electric polarization is\nnot switchable [27], applications of CaBaCo 4O7might\nutilize the large spin-induced polarization produced by a\nmagnetic field just below Tc[24].\nThis paper examines the magnetic properties of\nCaBaCo 4O7basedonaHeisenbergmodelwith12nearest\nneighbor interactions and associated anisotropies. The\nmagnetic state of CaBaCo 4O7can be described as a tri-angular array of ferrimagnetically aligned, bitetrahedral\nc-axis chains with net moment along b. Competing in-\nteractions within each chain produce a non-collinear spin\nstate. The strongelectric polarizationofCaBaCo 4O7be-\nlowTcis induced by the displacement of oxygen atoms\nsurrounding bonds that couple those chains.\nThis paper has six sections. Section II proposes a mi-\ncroscopic model for CaBaCo 4O7. New magnetization\nand optical measurements are presented in Section III.\nFitting results are discussed in Section IV. In Section V,\nwe predict the spin-induced electric polarization. Section\nVI contains a conclusion.\nII. MICROSCOPIC MODEL\nEach magnetic unit cell of CaBaCo 4O7contains 16 Co\nions on two kagome and two triangular layers with or-\nthorhombic lattice constants a= 6.3˚A,b= 11.0˚A, and\nc= 10.2˚A. Four crystallographically distinct Co ions\nhave three different valences [22,28]. Triangular layers\ncontain mixed-valent Co3+/Co2+L(Lis a ligand hole)\nspins1, 5, 9, and13withmoments M1= 2.9µB. Kagome\nlayers contain Co2+spins 2, 3, 6, 7, 10, 11, 14, and\n15 with moments M2=M3= 2µBand mixed-valent\nCo3+/Co2+Lspins 4, 8, 12, and 16 with M4= 2.4µB.\nBecause adjacent kagome or triangular layers are related\nby symmetry, Si′=Si+8on layer two is identical to Si\non layer one. With Si=Si(cosφi,sinφi,0) constrained\nto theabplane, the ferrimagnetic moment lies along bif\nφi+4=π−φi(i= 1,...,4).\nThe 12 different nearest-neighbor exchange couplings\nJiare drawn in Figs. 1(a-b) and 2. Six of these ( J1\nthroughJ6) couple the kagome and triangular layers as\nshown in Fig. 2; the other six ( J7throughJ12) couple\nthe spins within a kagome layer as shown in Figs. 1(a)\nand (b). The dominance of nearest-neighbor exchange\nover next-nearest neighbor exchange [27] justifies setting\nthe exchange interactions between spins on the triangu-\nlar layers to zero. Our model also includes easy-plane\nanisotropies D, easy-axis anisotropies Cwithin both\nkagome and triangular layers, and hexagonal anisotropy\nAon the triangular layers.\nWith magnetic field Balongm, the Hamiltonian is\nH=−/summationdisplay\n/angbracketlefti,j/angbracketrightJijSi·Sj+Dtri/summationdisplay\ni,triSic2+Dkag/summationdisplay\ni,kagSic2\n−Ckag/summationdisplay\ni,kag(oi·Si)2−Ctri/summationdisplay\ni,tri(ni·Si)2\n−AtriRe/summationdisplay\ni,tri/parenleftbig\nSia+iSib/parenrightbig6−gµBB/summationdisplay\nim·Si,(1)\nwhereSiis a spinSoperator on site i. For simplicity,\nwe setg= 2 for all spins.\nThe easy-axis anisotropy terms proportional to Ckag\nandCtriinvolve unit vectors oialong the “bowtie” di-\nrectionsφi=π/2 (spins 2 and 6), 5 π/6 (spins 3 and 8)3\n!\"\n#\"$\"\n!\"\n%&\"%'\" %(\"\n%)\"%*\"\n#\"%+\"!$\"\n%$\"\n#$\"\n!$\"%&\"%)\"\n%+\"\n%'\"%(\"%*\"%\"&$\"\n&\"%*\"\n%)\"\n'\"(\")$\"\n%&\"%'\"\n%+\"%(\"($\"\n($\"%&\" %+\"%)\"\n%'\"%*\"%(\")\"'$\"\n*$\"\n*\"\n$,#-.\"α $,#-.\"β\nFIG. 2: (Color online) A sideways view of the zero-field spin\nconfiguration showing bitetrahedral c-axis chains αandβ.\nand 7π/6 (spins 4 and 7) for the kagome layers and ni\nalong theφi=π/6 (spin 1) and −π/6 (spin 5) directions\nfor the triangular layers. The hexagonal anisotropy on\nthe triangular layers has expectation value\n−AtriS16/summationdisplay\ni,trisin6θicos6φi.\nAll anisotropy terms may act to constrain the spins to\ntheabplane.\nSpin amplitudes Snare fixed at their observed val-\nuesMn/2µBafter performing a 1 /Sexpansion about the\nclassical limit. Alternatively, the spins Sncould all have\nbeentakenas3/2butwithdifferent g-factorsfordifferent\nsets of spins. As discussed below, that would reduce the\nestimated exchange coupling Jijby a factor of 4 SiSj/9.\nStaticpropertiesareobtainedbyminimizingtheclassi-\ncal energy /an}bracketle{tH/an}bracketri}ht(the zeroth-order term in this expansion)\nwith respect to the 16 spin angles. The eigenvalues and\neigenvectors of a 32 ×32 equations-of-motion matrix [29]\nproduced by the second-order term in the 1 /Sexpan-\nsion give the optical mode frequencies and absorptions,\nrespectively.\nIII. MAGNETIZATION AND OPTICAL\nMEASUREMENTS\nPerhaps due to excess or deficient oxygen [30] or dif-\nferent domain populations (see below), previous magne-\ntization measurements [22,26,31–33] on CaBaCo 4O7are\nrather scattered. Consequently, new magnetization mea-\nsurementswereperformedat4Konhexagonally-twinned00.25 0.5 0.75 11.25 1.5 \n0 4 8 12 16 20 24 28 32 M ( µ\nB/f.u.) \nB (T) m = [1,0,0] \n[0,1,0] \n[0,0,1] \nFIG.3: (Color online)Themeasured(circles andsquares)an d\npredicted (solid curves) magnetizations for field along [1 ,0,0],\n[0,1,0], or [0,0,1].\ncrystals with a common c=z= [0,0,1] axis. In domain\nI,alies along the laboratory direction x= [1,0,0] andb\nlies along y= [0,1,0]; in domain II, a= [−1/2,√\n3/2,0]\nandb= [−√\n3/2,−1/2,0]; and in domain III, a=\n[−1/2,−√\n3/2,0] andb= [√\n3/2,−1/2,0]. Ifplare the\ndomain populations, then the magnetizations Mxand\nMymeasured with fields along xandyonly depend on\np1andp2+p3= 1−p1. Of course, Mzmeasured with\nfield along zis independent of pl. Fig. 3 indicates that\nall three magnetizations increase monotonically up to at\nleast 32 T.\nPrevious optical measurements [34] at the ordering\nwavevector Qfound two conventional spin-wave modes\nthat couple to the ground state through the magne-\ntization operator M= 2µB/summationtext\niSi. These magnetic-\nresonance (MR) modes are degenerate in zero field with\na frequency of1.07 THz and split almost linearly with in-\ncreasingfield along y, as shown in Fig. 4. For m=y, the\nMR modes are excited in two geometries: ( i) with THz\nfieldsEω||xandBω||zand (ii) withEω||zandBω||x.\nThose measurements also found an electromagnon (EM)\nthat couples to the ground state through the polarization\noperator P. The EM with zero-field frequency 1.41 THz\nis only excited in geometry ii.\nBecause the exchange couplings already break every\ndegeneracy in the unit cell, the 16 predicted modes for a\nsingle domain are non-degenerate. Therefore, the split\nMR modes must come from different domains. This\nwas verified by measuring [35] the MR mode frequen-\ncies as a function of the rotation angle θfor field B=\nB(cosθ,sinθ,0) =B(xcosθ+ysinθ) in the laboratory\nreference frame. In practice, this is accomplished by ro-\ntating the sample about cwhile keeping the field fixed4\nTABLE I: Exchange and anisotropy parameters in meV.\np1J1=J5J2=J4J3=J6J7J8J9J10J11=J12DkagDtriCkagCtriS14Atri\n0.185 −91.5−10.8 41.4−29.9 187.8 7.9 108.0−6.7−0.67−1.24 3.70 0.77 0.0064\nerror±0.071±3.6±0.5±0.7±1.9±7.5±0.1±4.5±0.1±0.06±0.03±0.15±0.09±0.0004\n1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0510 15 \nν(THz) B (T) MR EM \nFIG. 4: (Color online) The predicted MR (solid) and EM\n(dashed) modes for domain I (thick) and domains II and III\n(thin). Measured modes are indicated by solid points.\nalongx. As shown in Fig. 5 for 12 and 15 T, each hexag-\nonal domain then contributes one MR branch with a pe-\nriod ofπ.\nWith field Bloc=B(cosψ,sinψ,0) =B(acosψ+\nbsinψ) in the domain reference frame, the upper MR\nmode in Fig. 4 corresponds to the ψ=π/2 mode for\ndomain I while the lower MR mode corresponds to the\ndegenerate ψ=±π/6 modes for domains II and III. Pre-\nviously measured MR frequencies plotted in Fig. 4 at 12\nT correspond to the diamond and triangular points in\nFig. 5(b) at θ=π/2. Cusps in the MR curves for each\ndomain at ψ= 0 andπare caused by flipping the b\ncomponent of the magnetization (see inset to Fig. 5(b)).\nIV. FITTING RESULTS\nFits for the coupling parameters utilize the field de-\npendence of M, the zero-field powder-diffraction data\n[22], the field dependence of the MR and EM modes at\nθ=π/2 [34], and the MR mode frequencies at θ= 0\nandπ/3 for 7, 12 and 15 T. The resulting exchange and\nanisotropy constants are provided in Table I and the cor-\nresponding zero-field spin state is plotted in Figs. 1(a-b).\nIncontrasttothepreviouslyproposed[22]spinstatewith\n!\"#$ %&' !(#$ %)' **# ***# *#\n!\"ψ+#0\u0001 !\"\nψ+#0+\nFIG. 5: (Color online) The measured (solid circles) and pre-\ndicted (blue, red, and green curves for domains I, II, and III ,\nrespectively) angular dependence of the MR mode frequen-\ncies for 12 and 15 T. The inset to (a) sketches the angular\ndependence of BSF(dashed curve), which separates low-field\n(LF) and high-field (HF) states. The inset to (b) shows the\nnet magnetization of any domain for angles ψon either side\nof 0. Flips of the b-axis spin at ψ= 0 andπproduce cusps in\nthe mode frequencies.\nzig-zagchainsinthe abplanecontainingspins2, 3, 6, and\n7, our spin state can be better described as an array of\nc-axis chains or connected bitetrahedra [16,36] contain-\ning spins {1,2,3,4}(chainα) or{5,6,7,8}(chainβ) as\nsketched in Fig. 2. Chains are coupled by exchanges J9,5\nJ10, andJ12in theabplane.\nWhat explains the wide range of Jivalues? An or-\nthorhombic distortion [22,23] with b/(a√\n3)−1→0.018\nasT→0 breaks the hexagonal symmetry of the abplane\nand explains the difference between the pairs {J1,J2},\n{J8,J11}, and{J10,J12}. The difference between cou-\nplings like {J7,J8}is caused by charge ordering: whereas\nJ7couples moments 2 and 3 with M2=M3,J8couples\nmoments 2 and 4 with M2/ne}ationslash=M4. Charge ordering also\nexplains the difference between the pairs {J2,J3}and\n{J9,J10}. Although not demanded by symmetry, we set\nJ1=J5,J2=J4,J3=J6, andJ11=J12because the\nspin state and excitations at Qonly depend on their av-\nerages [37].\nGiven other conditions, our fit chooses the spin state\nthat matches the powder-diffraction data [22] as closely\nas possible. At zero field, the predicted spin state has\nanglesφ1=−0.83π,φ2= 0.40π,φ3=−0.23π, and\nφ4= 0.62π. Basedexclusivelyonpowder-diffractiondata\nand symmetry constraints, the previously proposed spin\nstate [22] had φ1=−0.24π,φ2=φ3= 0.67π, andφ4=\n−0.44π. Inbothcases, φi+4=π−φi(i= 1,...,4)sothat\nthe moment Mblies along the baxis. As shown in Table\nII, our spin state does not satisfy the powder diffraction\ndata quite as well as the earlier state, primarily because\nit underestimates the powder diffraction peak I(112).\nFor the previous spin state, χ2is minimized by\nLorentzian form factors with Q1/4π= 0.088˚A−1,\nQ2/4π=Q3/4π= 0.095˚A−1, andQ4/4π= 0.088˚A−1\nfor spinsSn. Forthe new spin state, Q1/4π= 0.052˚A−1,\nQ2/4π=Q3/4π= 0.224˚A−1, andQ4/4π= 0.102˚A−1.\nAll are smallerthan the scale Q0/4π≈0.3˚A−1measured\nby Khan and Erickson [38] for Co2+in CoO.\nOur results indicate that the exchange coupling J8≈\n188 meV between moments 2 (Co2+,S2= 1) and 4\n(Co3+/Co2+L,S4= 1.2) is strongly ferromagnetic and\nlarger in magnitude even than the 155 meV antiferro-\nmagnetic coupling found in the cuprate Nd 2CuO4[39].\nThe strength of this coupling might be explained by the\ndouble-exchange mediated hopping of ligand holes L[19]\nfrom site 4 to 2. Bear in mind, however, that the es-\ntimated exchange parameters would be significantly re-\nduced if the fits were performed with S= 3/2 for all Co\nspins. In particular, J8would then fall from 188 to 100\nmeV.\nExceptforJ10, thefivelargestexchangecouplings J1=\nJ5≈ −92 meV,J3=J6≈41 meV, and J8≈188 meV\nlie within connected bitetrahedral, c-axis chains. Inside\neach chain, competing interactions between spins 1, 2,\nand 4 produce a non-collinear spin state.\nAlthough occupying a triangular lattice, chains α\nandβare magnetically ordered with moments Mch=\n(±1.18,1.33,0)µB/f.u.. These chains are primarily cou-\npled by the strongly ferromagnetic interaction J10≈108\nmeV between nearly parallel spins {4,6}(φ4= 0.62π,\nφ6= 0.60π) and{2,8}(φ2= 0.40π,φ8= 0.38π). Above\nTc= 70 K, the short-range order within each chain may\nbe responsible for the large, negative Curie-Weiss tem-perature Θ CW≈ −1720 [26] or −890 K [31], the larger\nthan expected Curie constant [26], and the susceptibility\nanomaly[31] at 360K suggestiveof short-rangemagnetic\norder far above Tc.\nComparison between the theoretical and experimen-\ntal results for the magnetization in Fig. 3 suggests that\nroughly 20% of the sample is in domain I. Different do-\nmainpopulationsorevenorthorhombictwinningin other\nsamples may explain the discrepancies between the re-\nported magnetization measurements [22,26,31–33].\nEasy-axis anisotropies AandCfavor ferrimagnetic\nalignment along brather than a. The spin-flop (SF) field\nrequired to flip the spins towards the adirection must\nincrease as the field along bincreases [40]. As shown in\nthe inset to Fig. 5(a), BSF(ψ) then increases with ψ. If\nBSF(ψ= 0)<15 T, then the MR spectrum for 15 T\nwould show a discontinuity at the transition from a low-\nfield (LF) to a high-field (HF) state below some critical\nvalue ofψ. Since the MR mode frequencies in Fig. 5(a)\ndo not exhibit any discontinuities as a function of ψ, we\nconclude that BSF(ψ= 0) exceeds 15 T and probably,\nbased on the smooth dependence of the magnetizations\non field, exceeds 32 T as well. The apparent small size\nofBSF[24,41] must reflect the net magnetization of all\nthree domains.\nPredicted modes below 5 THz are plotted in Fig. 4.\nThe Goldstone modes for all three domains are lifted by\nin-planeanisotropiestobecometheMRmodeswith zero-\nfield frequencies of 1.07 THz. As remarked earlier, the\nlower MR mode comes from domains II and III while the\nupper MR mode comes from domain I. Below 3.5 THz,\none EM mode is produced in domain I and another in\ndomains II and III. The degenerate EM modes from do-\nmains II and III dominate the optical absorption. The\npredictedfielddependenceoftheupperMRmodeisquite\nclose to the observed dependence. But the predicted cur-\nvatures of the lower MR mode and the EM mode, both\nfrom domains II and III, is not observed.\nV. SPIN-INDUCED ELECTRIC\nPOLARIZATION\nBelow the ferrimagnetic transition, CaBaCo 4O7is re-\nported [24] to develop a very large spin-induced polar-\nization∼1700 nC/cm2, which is surpassed in type I\nmultiferroics only by the conjectured [25] spin-induced\npolarization ∼3000 nC/cm2of BiFeO 3. Other mea-\nsurements indicate that the spin-induced polarization of\nCaBaCo 4O7rangesfrom320nC/cm2[32] to 900nC/cm2\n[42].\nThe electric-field dependence of any interaction term\nin the spin Hamiltonian Hcan induce an electric po-\nlarization below Tc. However, the electric-field depen-\ndence of the easy-plane anisotropy Dcannot explain the\nspin-induced polarization along cbecause the expecta-\ntion value of Pi=κSic2withκ=−∂D/∂E cwould van-\nish in zero magnetic field when all the spins lie in the6\nTABLE II: Ratios of powder-diffraction peak intensities\nI(002)/I(101)I(012)/I(101)I(111)/I(101)I(112)/I(101)I(121)/I(101)I(122)/I(101)Mb(µB/f.u.)χ2\nexperimental [22] 0.344 0.326 0.322 0.477 0.262 0.404\nprevious [22] 0.286 0.414 0.384 0.411 0.232 0.427 0.88 0.021\ncurrrent 0.360 0.380 0.378 0.286 0.313 0.449 1.33 0.047\nabplane. Easy-axis anisotropy AorCin theabplane\ncould produce a spin-induced electric polarization per-\npendicular to c. But the EM mode would then become\nobservable for a THz electric field in the abplane, con-\ntrary to measurements.\nAs conjectured previously [24], the spin-induced po-\nlarization in CaBaCo 4O7must then be generated by the\ndependence of the exchange interactions Jijon an elec-\ntric field, called magnetostriction. Coupling constant\nλij=∂Jij/∂Ecfor bond {i,j}is associated with a spin-\ninduced polarization [43] per site of Pij\nc=λijSi·Sj/4,\nwhich accounts for the four equivalent bonds per unit\ncell. Expanding in the electric field Ecyields an interac-\ntion term −EcλijSi·Sj, linear in the electric field and\nquadratic in the spin operators.\nTaking|0/an}bracketri}htas the ground state and |n/an}bracketri}htas the excited\nspin-wave state, the MR matrix element /an}bracketle{tn|Ma|0/an}bracketri}htmixes\nwith the EM matrixelement /an}bracketle{tn|Pij\nc|0/an}bracketri}htfor domainsII and\nIII but not for domain I. Therefore, our model can ex-\nplain the strong asymmetry [44] ∼Re/braceleftbig\n/an}bracketle{tn|M·Bω|0/an}bracketri}ht/an}bracketle{t0|P·\nEω|n/an}bracketri}ht/bracerightbig\nin the absorption of counter-propagating light\nwaves [34] for the lower observed MR mode in geometry\niiwithEω||c. But it cannot explain the observed asym-\nmetry of this mode in geometry iwithEω⊥cif only\n/an}bracketle{t0|Pc|n/an}bracketri}htis significant.\nHow can we estimate the coupling constants λijand\nthe spin-induced electric polarization? The optical ab-\nsorption of any mode in domain lis proportional to pl2\nbecauseeachmatrixelementisseparatelyproportionalto\npl. At nonzero field, the EM mode absorption is propor-\ntional top22+p32while the upper MR mode absorption\nis proportionalto p12. At zero field, all domains have the\nsame mode spectrum so that both the MR and EM mode\nabsorptions are proportional to p12+p22+p32. Experi-\nmentally, the ratio rof the absorptionof the EM mode to\nthe absorption of the upper MR mode rises from r= 7.5\nat 0 T tor= 35 at 10 T. This growth is explained by the\nB >0ratio(p22+p32)/p12≈10±5forp1= 0.185±0.071\nandp2=p3= (1−p1)/2.\nAt both 0 and 10 T, the only sets of bonds that gen-\nerate spin-induced polarizations of the right magnitude\nare{2,7}and{3,4}. Each of those bonds couples adja-\ncentc-axis chains through pairs of spins that are almost\nanti-parallel. From the relative absorptions rat 0 or\n10 T, we estimate that /an}bracketle{tP27\nc/an}bracketri}ht ≈2350 or 1960 nC/cm2\nand/an}bracketle{tP34\nc/an}bracketri}ht ≈2110 or 1730 nC/cm2. Results for both\nsets of bonds are consistent with the recently observed[24] polarization of 1700 nC/cm2. By contrast, density-\nfunctional theory [27] predicts that the spin-induced po-\nlarization along cis 460 nC/cm2. The spin-induced\npolarization should remain fairly constant with applied\nmagnetic field, decreasing by about 1% for a 10 T field\nalongb.\nVI. CONCLUSION\nWe have presented a nearly complete solution for the\nmagnetization, spin state, and mode frequencies of the\nswedenborgite CaBaCo 4O7. An orthorhombic distor-\ntion aboveTcpartially relieves the geometric frustration\non the kagome and triangular layers and allows ferri-\nmagnetism and ferroelectricity to coexist below Tc. Al-\nthough occupying a triangular lattice, bitetrahedral c-\naxis chains are ferrimagnetically ordered in the abplane.\nCompeting interactions within each chain produce non-\ncollinear spin states. Sets of bonds coupling those chains\nare responsible for the large spin-induced polarization of\nCaBaCo 4O7.\nDespite its fixed permanent electric polarization, this\nswedenborgite may yet have important technological ap-\nplications utilizing the large changes [24] in the spin-\ninduced polarizationwhen a modest magnetic field <1T\nis applied along bjust belowTc. Abig jump in the polar-\nization should also be produced just below Tcby rotating\na fixed magnetic field about the caxis. Above all, our\nwork illuminates a pathway to develop other functional\nmaterials with sizeable magnetic moments and electrical\npolarizations.\nAcknowledgments\nResearchsponsoredby the U.S. Department ofEnergy,\nOffice of Science, Basic Energy Sciences, Materials Sci-\nences and Engineering Division (RF), by the Hungarian\nResearch Funds OTKA K 108918, OTKA PD 111756,\nand Bolyai 00565/14/11 (SB, VK, and IK), and by the\ninstitutional research funding IUT23-3 of the Estonian\nMinistry of Education and Research and the European\nRegional Development Fund project TK134 (TR and\nUN).7\n1K. Binder and A.P. Young, Rev. Mod. Phys. 58, 801\n(1986).\n2M.J. Harris, S.T. Bramwell, D.F. McMorrow, T. Zeiske,\nand K.W. Godfrey, Phys. Rev. Lett. 79, 2554 (1997).\n3S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. B¨ oni, Science323, 915\n(2011).\n4D.I. Khomskii, J. Magn. Magn. Mater. 306, 1 (2006).\n5S.-W. Cheong and M. Mostovoy, Nat. 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B 81094417 (2010).\n23S.N. Panja, J. Kumar, S. Dengre, and S. Nair, J. Phys.:\nCond. Mat. 16, 9209 (2016).\n24V. Caignaert, A. Maignan, K. Singh, Ch. Simon, V. Pra-\nlong, B. Raveau, J.F. Mitchell, H. Zheng, A. Huq, andL.C.\nChapon, Phys. Rev. B 88, 174403 (2013).\n25J.-H. Lee and R.S. Fishman, Phys. Rev. Lett. 115, 207203(2015).\n26V. Caignaert, V. Pralong, A. Maignon, and B. Raveau,\nSol. St. Comm. 149, 453 (2009).\n27R.D. Johnson, K. Cao, F. Giustino, and P.G. Radaelli,\nPhys. Rev. B 90, 045192 (2014).\n28S. Chatterjee and T. Saha-Dasgupta, Phys. Rev. B 84,\n085116 (2011).\n29R.S. Fishman, J.T. Haraldsen, N. Furukawa, and S. Miya-\nhara,Phys. Rev. B 87, 134416 (2013).\n30Md. M. Seikh, V. Caignaert, V. Pralong, and B. Raveau,\nJ. Phys. Chem. Sol. 75, 79 (2014).\n31Z. Qu, L. Ling, L. Zhang, and Y. Zhang, Sol. St. Comm.\n151, 917 (2011).\n32H.Iwamoto, M. Ehara, M. Akaki, and H. Kuwahara, J.\nPhys.: Conf. Ser. 400, 032031 (2012).\n33Md. M. Seikh, T. Sarkar, V. Pralong, V. Caignaert, and\nB. Raveau, Phys. Rev. B 86, 184403 (2012).\n34S. Bord´ acs, V. Kocsis, Y. Tokunaga, U. Nagel, T. R˜ o˜ om,\nY. Takahashi, Y. Taguchi, and Y. Tokura, Phys. Rev. B\n92, 214441 (2015).\n35The THz spectra of CaBaCo 4O7were measured at 2.5 K\nwith linear light polarization using Fourier-transform sp ec-\ntroscopy [34]. The reference spectrum was taken in zero\nfield.\n36M. Valldor, J. Phys.: Cond. Mat. 16, 9209 (2004).\n37Spin excitations away from q=Qdepend separately on\nthese exchange constants. The stability of the coplanar\nground state requires that each pair of exchange constants\nbe sufficiently close to one another.\n38D.C. Khan and R.A. Erickson, Phys. Rev. B 1, 2243\n(1970).\n39P. Bourges, H. Casalta, A.S. Ivanov, and D. Petitgrand,\nPhys. Rev. Lett. 79, 4906 (1997).\n40While the coplanar LF spin state has a lower energy than\nthe HF state below 32 T, it becomes locally unstable to a\nbuckledstate with spins cantedoutof the abplane between\n16 and 32 T for field along a. Based on the smooth depen-\ndence of the magnetizations on field, the transition from\nthe coplanar to the buckled state must be second order.\n41V. Pralong, V. Caignert, T. Sarkar, O.I. Lebedev, V. Duf-\nfort, and B. Raveau, J. Sol. St. Chem. 184, 2588 (2011).\n42V. Koscis, S. Bord´ acs, and I. K´ ezsm´ arki, (unpublished).\n43Based on the symmetry of adjacent layers, it is easy to\nshow that the polarization operator P12associated with\nbond{1,2}has components\nP12\na∝C12−C56−C1′2′+C5′6′,\nP12\nb∝C12+C56−C1′2′−C5′6′,\nP12\nc∝C12+C56+C1′2′+C5′6′,\nwhereCij=Si·Sj. Similar relations hold for other bonds.\nOnly theccomponent Pij\nccan produce a static polariza-\ntion but all three components may contribute to the off-\ndiagonal polarization matrix elements ∝an}bracketle{t0|Pij\nα|n∝an}bracketri}ht(n∝ne}ationslash= 0).\n44S. Miyahara and N. Furukawa, J. Phys. Soc. Japan 80,\n073708 (2011)."    },    {        "title": "2104.14819v1.Current_Induced_Magnetization_Control_in_Insulating_Ferrimagnetic_Garnets.pdf",        "content": " 1 Current-Induced Magnetization Control in Insulating Ferrimagnetic Garnets Can Onur Avci Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus de la UAB, Bellaterra, 08193, Spain  The research into insulating ferrimagnetic garnets has gained enormous momentum in the past decade. This is partly due to the improvement in the techniques to grow high-quality ultrathin films with desirable properties and the advances in understanding the spin transport within the ferrimagnetic garnets and through their interfaces with conducting materials. In recent years, we have seen remarkable progress in controlling the magnetization state of ferrimagnetic garnets by electrical means in suitable heterostructures and device architectures. These advances have readily placed ferrimagnetic garnets in a favorable position for the future development of insulating spintronic concepts. The purpose of this article is to review recent experimental results of the current-induced magnetization control and associated phenomena in ferrimagnetic garnets, as well as to discuss future directions in this rapidly evolving area of spintronics.             2 1. Introduction The discovery of ferrimagnetic garnets (FMGs) dates back to 1950s1,2) followed by several decades of extensive investigations to understand their structural and magnetic properties.3) In the early 1970s, the development of liquid phase epitaxy has allowed the growth of micrometer-thick FMG films such as yttrium iron garnet (Y3Fe5O12, YIG) with ultrahigh quality and low production cost.4) Consequently, FMGs have become a model system for engineering magnetic and magneto-optical properties with high precision through their composition. FMGs with suitable domain properties and large magneto-optic response have led to technological developments such as magnetic bubble memory,5) magnetic bubble display,6) and magneto-optic printers.7) These advances marked the first integration of magnetic materials into modern electronic technologies in the late 1970s and early 1980s. The 1980s have marked a paradigm shift in magnetism research. The discovery of giant magnetoresistance has initiated the spintronics revolution and boosted the magnetic recording industry.8,9) For the next two decades, other prominent phenomena such as interfacial perpendicular magnetic anisotropy10) (PMA) and spin-transfer torques11-13) (STTs) have spearheaded spintronics research and related applications.14) All these effects exclusively required conducting ferromagnetic structures. During this period, FMGs have been set aside because of their insulating character, and metallic multilayers have become the preferential materials in this exciting field. Magnetic insulators, and in particular FMGs, possess extraordinary properties that could be beneficial in spintronics. They are often characterized by low damping and long magnon diffusion lengths, enabling efficient spin current generation and its nonlocal transport.15-19) They can now be grown in nanometer form, in a highly ordered single-crystal structure, desirable for low-pinning domain wall20-26) and skyrmion motion.27,28) They offer multiple degrees of freedom for tuning relevant magnetic properties such as saturation magnetization, magnetocrystalline anisotropy, etc., through composition and strain engineering.3,29,30) They are naturally protected and highly robust against heat, oxidation, aging, and degradation. Finally, the recently discovered transport phenomena such as spin pumping,31,32) spin Seebeck effect,33) and spin Hall magnetoresistance34 suggest FMGs as highly efficient spin current generating and detecting materials.   3 Recently, a better understanding of electrical transport in bulk and interfaces of certain material systems characterized by large spin-orbit coupling revealed that relativistic spin-dependent effects could generate pure spin currents as a result of a charge current.35) The resulting spin-orbit torques (SOTs) can lead to magnetization switching,36-40) domain wall (DW) motion,21,22,24,26) magnon generation/suppression,41) etc., in suitable magnetic heterostructures. In the SOTs scheme, in contrast to the STTs, charge and spin currents travel orthogonally, which provides a simple means to generate spin-torques, among others, on FMGs neighboring to a SOT-source, usually a nonmagnetic metal (NM). The intersection of FMGs and the SOT-driven magnetization control have created fertile grounds in spintronics. We have seen some remarkable results of SOT-induced magnetization control in FMGs in less than half a decade. In this Review, we will give an overview of some of the pioneering experiments on the SOTs-driven control of FMGs. We will first discuss the characteristics of FMGs, SOTs and other relevant physical phenomena. We will then review some key results and discuss them in a broader context. We will finally discuss the potential directions and future opportunities in this rapidly evolving and exciting area of spintronics.  2. General Properties of Ferrimagnetic Garnets 2.1 Composition and structure FMGs constitute a large family of magnetic oxides with the general formula of X3Fe5O12, where X is either a non-magnetic rare-earth element such as yttrium (Y) or a magnetic rare-earth element such as gadolinium (Gd), terbium (Tb), thulium (Tm), etc. The lattice structure consists of two octahedral Fe3+ ions, three tetrahedral Fe3+ ions, and three X3+ ions at the dodecahedral sites, per formula unit. The octahedral and tetrahedral Fe3+ ions are coupled antiparallel to each other while X3+, if magnetic, is aligned parallel to the octahedral Fe3+. All magnetic interactions are mediated via superexchange interaction through O2- anions. In the case of YIG, there is no magnetic compensation, and it preserves ferrimagnetic order up to 559 K. In iron garnets with magnetic rare-earth elements, the total magnetic moment depends sensitively on the rare-earth; hence the magnetism differs significantly depending on the rare-earth choice, composition, and temperature. Notably, there exists a temperature at which the opposing magnetic sublattices are equal to each other and the film shows zero net magnetization, thereby acts like a collinear antiferromagnet. Nonetheless, the ordering temperature is mainly determined by the magnetic moment of Fe, independent of the rare-earth choice, and hence is about the same (530-580 K)  4 in all FMGs,42) which makes the entire family of FMGs suitable for room temperature research and applications  2.2 Growth Ultrathin films of FMGs are typically grown on lattice-matched gadolinium gallium garnet (Gd3Ga5O12, GGG) or similar garnet substrates by pulsed laser or magnetron sputter deposition using stoichiometric targets.43-45) For the sputter deposition, an off-axis method (i.e., the target and substrate are held at 90° with respect to each other) is generally preferred due to the enhanced stoichiometry of the deposited films.46) Typically, the substrate is kept at a high temperature (650-850°C) during deposition and slowly cooled down to room temperature to obtain the desired high crystallinity, homogeneity and surface smoothness. Alternatively, room temperature deposition and post-growth annealing are also reported to yield similar results.47) By employing the mentioned methods, films of thicknesses ranging from a few to tens of nm can be obtained with sub-nm surface roughness.  2.3 Perpendicular magnetic anisotropy Of particular interest to this Review, some FMGs possess bulk PMA, which is an essential property in spintronic devices since it enhances the thermal stability of magnetic elements and ensures long-term data retention. The PMA in FMGs is mainly of magneto-elastic origin, mediated by lattice strain. When grown epitaxially, a small lattice mismatch between the substrate and the FMG film leads to compressive or tensile strain, giving rise to negative magneto-elastic energy and consequently to PMA.48) Thus far, PMA has been reported for thin films of TmIG,49-52) TbIG53, EuIG,52,53) GdIG,54) YIG,55) and Bi substituted YIG18) (Bi:YIG) grown on GGG or similar substrates, polycrystalline EuIG grown on a quartz substrate,30) and DyIG grown on a SiO2 substrate by pulsed laser deposition.56) There were also successful efforts to grow TmIG and YIG films by r.f. magnetron sputtering with similar magnetic and structural properties compared to those obtained by pulsed laser deposition.46,47,57) Recent studies suggest that there is an additional interface contribution to the PMA from the metal capping of FMGs.58) For the complementary metal-oxide-semiconductor (CMOS) technology integration, magnetron sputtering is the preferred method since films with high uniformity can be grown on large-scale wafers. Tunable PMA through material and interface engineering and the use of magnetron sputtering together make FMGs very attractive for spintronics.  5 3. Spin-Orbit Torques and Electrical Detection of Magnetization in Ferrimagnetic Garnets SOTs are presently the state-of-the-art current-induced magnetic manipulation method in spintronic devices, including the ones based on FMGs.35) Their experimental discovery dates back to the end of the 2000s and early 2010s, first in semiconducting heterostructure,59) and then in metallic multilayers.36,60) Initially, the SOTs research has mostly focused on heavy metal/ferromagnet bilayers such as Pt/Co,36,37,60,61) Ta/CofeB38,40,62) and W/CoFeB63) due to their historical significance, technological relevance, the convenience of deposition and fabrication, and large torque-to-current ratios reported. Later on, topological insulators,64,65) semimetals,66,67) oxidized metals68-70) and intermetallic alloys71,72) combined with appropriate magnetic materials, have been considered. Many SOTs generating mechanisms have been discovered thus far, including the spin Hall effect63) (SHE), interfacial Rashba-Edelstein effect,60) topological surface states,73) bulk crystal asymmetry,74) anomalous Hall effect,75) (AHE), planar Hall effect76) (PHE), interface spin filtering,77) etc. Among these, the SHE is primarily considered as a SOT source for current-induced magnetization manipulation in FMGs. The SHE emerges in the bulk of the materials characterized by strong spin-orbit coupling (Pt, W, Ta, etc.). It describes the spin-dependent scattering of conduction electrons which generates a pure spin current transverse to the charge current injection direction. A magnetic layer in interfacial contact with a SHE material can then absorb the resulting spin current, which acts as a spin torque on the magnetization and enables its reversal (Fig. 1). In the other listed SOT sources, while the spin current generation mechanism differs, the applied torque has comparable symmetry. 3.1 Symmetry and characterization of spin-orbit torques  SOTs consist of damping-like and field-like components with the symmetries 𝛕\"#∝𝐦×(𝐲×𝐦) and 𝛕)#∝𝐦×𝐲, respectively (see Fig. 2a). Here m is the magnetization unit vector and y is the in-plane axis perpendicular to the current flow direction. 𝛕\"# has the suitable geometry to induce 180° switching of the in-plane magnetization along the axis perpendicular to the current injection as well as the perpendicular magnetization; though in the latter, an in-plane magnetic field is required to break the rotational symmetry of 𝛕\"# effective field (Fig. 2a). Furthermore, 𝛕\"# can move homochiral Néel-type DWs and skyrmions in the PMA systems with high efficiency and large velocities.  6 A handful of electrical and optical techniques have been developed to characterize the SOTs.78) Perhaps the simplest and most accessible method is the harmonic Hall voltage detection of the SOT effective fields.61,79) It consists of applying an a.c. current through Hall bar devices and detecting and analyzing the harmonic Hall voltage response proportional to m (Fig. 2b). The Hall voltage in metallic magnetic systems predominantly consists of AHE and PHE where the former (latter) is proportional to the out-of-plane (in-plane) component of the magnetization vector: 𝑉+=𝑅.+/𝐼cos𝜃+𝑅6+/𝐼sin9𝜃sin2𝜑. Here 𝜃 and 𝜑 represent the polar and azimuthal angle of m and I is the injected current (see Fig. 3a). Upon an a.c. current injection, m oscillates about its equilibrium position due to oscillating SOTs and generates a harmonic response in the Hall voltage, twice the frequency of the input current, hence generating a second harmonic Hall voltage (𝑉9<). 𝑉9< contains contributions from SOTs but also thermoelectric effects (anomalous Nernst and spin Seebeck effects) driven by Joule heating-induced temperature gradients.80) Self-consistent analysis of the data taken in suitable experimental geometries, detailed in Refs.61,79,80), leads to vector quantification of SOTs in both in-plane and PMA systems, and the proper separation of the non-SOT originating signals.80,81) 3.2 Spin Hall magnetoresistance in magnetic insulators A charge current cannot pass through FMGs henceforth cannot generate AHE and PHE. However, it has been discovered that FMGs, and in general magnetic insulators, can influence the electrical resistance of the material that is in interfacial contact with it and give rise to spin Hall magnetoresistance34,82-84) (SMR). SMR, first discovered in YIG/Pt bilayers, describes the changes in the electrical resistivity of a nonmagnetic metal adjacent to a magnetic insulator depending on the magnetization orientation. The effect arises due to the asymmetry in the scattering of the SHE-induced spin current at the FMG/NM interface, which depends on the relative direction of the magnetization with respect to the spin polarization (collinear or orthogonal). The back-scattered spin current contributes to the charge current via the inverse SHE, which ultimately manifests itself as a modification in the longitudinal and transverse resistivity. In a Hall voltage measurement, like the AHE and PHE in conducting magnets, the SMR gives rise to signals proportional to the in-plane and out-of-plane components of the magnetization vector due to the real and imaginary part of the interface spin mixing conductance,85) respectively. The Hall voltage then reads: 𝑉+=𝑅6+/=>?@𝐼sin9𝜃sin2𝜑+𝑅.+/=>?@𝐼cos𝜃+𝑅A+/𝐼𝐻C (1).  7 Note, however, that the amplitude of the 𝑅6+/=>?@ is generally much larger than 𝑅.+/=>?@, contrary to the situation in the metallic systems. Moreover, there is a non-negligible ordinary Hall effect signal (𝑅A+/) linearly proportional to the out-of-plane component of the external field (𝐻C), but unrelated to the magnetization vector, which is generally neglected in all-metal systems due to much larger AHE and PHE contributions. Figure 3 shows a typical characterization of the SMR and ordinary Hall effect signals using a Hall bar device made of TmIG/Pt.86)  The harmonic Hall voltage method can be used also for quantifying the SOTs in insulating magnets, e.g., in FMG/Pt. Due to the dominant 𝑅6+/=>?@ contribution, the in-plane oscillations of m induced by SOTs produce the largest 𝑉9<, henceforth, in FMG/Pt the measurement geometry differs from all-metal systems. Furthermore, thermoelectric contributions to 𝑉9< should be treated with utmost care. In metallic systems, the main thermoelectric contribution is found to originate from the anomalous Nernst effect produced within the magnetic layer itself, whereas in magnetic insulators the spin Seebeck effect (SSE), the subject of the next section, gives the dominant thermoelectric contribution to 𝑉9<. The detailed description of how to extract the damping-like and field-like SOTs and thermoelectric signals in FMG/NM systems with PMA is given in Refs.87,88) 3.3 Spin Seebeck effect The SSE describes spin current generation by applying a temperature gradient (∇T) across a magnetically ordered material33. In a simplified picture, the temperature difference along the gradient direction creates an imbalance in the magnon population, generating a pure spin current flow along ∇T with polarization parallel to m. The SSE-originated spin current can diffuse into an adjacent metal with a large SHE and generate an inverse SHE voltage or alternatively called a spin Seebeck voltage (𝑉>>/). Additional to the SMR, the in-plane magnetization vector can be probed by measuring 𝑉>>/ in FMGs by simply using current-induced Joule heating as a means to generate ∇T perpendicular to the layers.89,90) In contrast to the SMR, the SSE can be used to sense 180° rotation of m since 𝑉>>/ is expected to change sign upon the reversal of m. With an a.c. current injection, the SMR and SSE together can be measured by 𝑉< and 𝑉9<, respectively, for effectively characterizing the magnetization vector in the 3D space and other relevant properties such as coercivity (𝐻F), magnetic anisotropy (𝐻G), etc. (see Fig. 4). We note that, in addition to the SMR and SSE, it is also possible to detect the magnetization vector orientation in FMGs through spin-torque ferromagnetic resonance  8 measurements91-93. However, such measurements rely on a different and more complex experimental setup and signal analysis, hence not depicted in Fig. 4. 4. Spin-Orbit Torque Switching in Ferrimagnetic Garnets Over the past decade, the characterization of various spin-dependent transport phenomena such as SSE, spin pumping, and SMR has shown a significant spin current transmission through FMG/NM interfaces. However, the first demonstration of robust SOT-induced magnetization switching in an FMG came only in 2017. Avci et al. have shown that the magnetization of an 8 nm-thick TmIG can be switched by SOTs generated by a 5 nm-thick Pt overlayer.86) Through Hall effect measurements, the magnetization was found to switch between the up and down states after consecutive millisecond-long pulses of alternating polarity in the presence of an in-plane field (Fig. 5a-b). The switching polarity was found to change upon reversing the in-plane field, expectedly of the SOT-induced switching.  Furthermore, the critical current density was characterized with a variety of in-plane fields and found to be inversely proportional to the external field strength (Fig. 5c-d) similar to previous reports.38) The switching current densities through Pt were comparable to those used in metallic Pt/Co(~1nm) bilayers despite the large magnetic volume (8 nm) of TmIG. However, after considering the difference in the saturation magnetization -which is about one order of magnitude lower in FMGs with respect to Co- the switching efficiency in TmIG/Pt was found to be comparable to Pt/Co and other Pt-based metallic heterostructures.  Later on, the SOT-induced switching in TmIG/Pt was also demonstrated with a few nanosecond-long pulses and with a low in-plane field requirement of only 2 Oe in 2 µm-wide Hall bars94) (Fig. 5e-f). Following these initial demonstrations, several other works have reported SOT-induced switching in TmIG/Pt, TbIG/Pt, YIG/Pt and Bi:YIG/Pt bilayers with similar or higher efficiencies.57,95-98) In particular, Vélez et al. have reported local switching of magnetization in a continuous TmIG film by sending current through patterned Pt structures.96) Local control of the magnetization in extended films is unique to magnetic insulators, and may find interest in the implementation and in-situ reconfiguration of synthetic magnetic structures for magnonic applications.  The SOT switching of TmIG is not limited to the use of Pt. Shao et al. have shown efficient switching in TmIG/W bilayers with TmIG thickness varying between 3.2 and 15 nm.99 The switching polarity was found the opposite to that obtained in TmIG/Pt for a fixed current  9 polarity, verifying the predominant SHE origin of the switching torques from Pt and W. Note that the SHE in Pt and W has an opposite sign. Switching in thick, bulk-like TmIG has demonstrated the significant potential of SOTs for controlling the magnetization in a wide thickness range. This study has also revealed that the SOT switching efficiency strongly depends on the TmIG thickness and that above 10 nm of TmIG, the SOT reaches the maximum efficiency (Fig. 6). It was argued that the spin current absorption is less efficient in thinner films due to the reduced magnetic moment density, which effectively reduces the exchange interaction between the current-induced accumulated spins and the local magnetic moments of TmIG near the interface, producing smaller torques. When the Ms is close to the bulk limit, though, the SOT efficiency was maximized. Similar conclusions have been reached in another study by using different SHE metals grown on TmIG.100) Interestingly, while switching of FMGs with PMA has been demonstrated in several different materials, SOT-induced switching of in-plane FMGs has remained little explored.101) Limitations in suitable device fabrication with strong in-plane anisotropy and magnetization detection may be some of the reasons for the lack of such experiments. Overall, the various reports on FMG-based heterostructures have demonstrated that the SOTs act similarly upon magnetization in insulating magnets and hence revealing a significant potential for consideration in electrically-controlled magnetic devices. 5. Chiral Domain Walls and Their Current-Driven Motion in Ferrimagnetic Garnets Magnetic DWs and their current-driven dynamics have been a prominent subject in magnetism and spintronics due to their intriguing physics and potential use in data storage and logic devices.102,103) Earlier experiments of current-driven DW motion relied on less efficient STTs. Recently though, SOTs emerged as more efficient alternative to displace DWs, which, moreover, does not require the magnetic layer to be conducting.  SOT-induced DW motion is most effective in PMA layers possessing Néel type DWs due to the orthogonal alignment of DW internal spins and 𝛕\"#, which maximizes the effective field on the DW. Earlier, it was believed that in the standard PMA systems like Pt/Co, the DWs are of Bloch type governed by magnetostatics. Lately, though, it was understood that the interfacial Dzyaloshinskii-Moriya interaction (DMI) in Pt/Co and other similar NM/ferromagnet layers, gives rise to homochiral Néel DWs in structurally asymmetric multilayers.104) Therefore, SOTs have been shown to move DWs in metallic heterostructures effectively.21,22)  10 The efficient SOT-induced switching in FMGs, as well as the increased interest in the interfacial DMI, have triggered research into DWs and their current-induced dynamics in perpendicular FMGs. In 2019, Avci et al. have shown, through measurements of SOT-induced DW motion, that TmIG and TbIG possess DMI-stabilized left-handed Néel DWs.105) A series of measurements with Pt and Cu/Pt capping layers on TmIG and TbIG indicated that the DMI predominantly originates from the FMG interface with the GGG substrate and not from the interface with the metal on top. This was the first direct demonstration of chiral magnetism occurring at a magnetic oxide interface. Shortly after, Vélez et al. and Ding et al. have reached the same conclusion that the GGG/TmIG substrate interface gives rise to the interfacial DMI and left-handed DWs in TmIG.95,96) Vélez et al., moreover, have found, through nitrogen-vacancy center magnetometry measurements, that the Pt capping contributes to the DMI with an opposite sign; thereby concluded that the DMI contributions from the two interfaces compete with each other (see Fig. 7). Nevertheless, the reported DMI effective fields were only in the order of a few mT but strong enough to turn the equilibrium DW configuration from the Bloch-type to the partially or fully Néel-type.  Later on, Ding et al. and Caretta et al. have confirmed the interfacial origin of the DMI through TmIG thickness-dependent experiments.100,106) Furthermore, Caretta et al. have reported that the DMI originates from the rare-earth orbital magnetism since the effect was absent in perpendicular Bi:YIG films but present in TmIG and TbIG grown on GGG substrates (Fig.8). Given the wide tunability of FMGs through composition and substrate choice, understanding the origins of DMI is an important step for designing materials with chiral spin textures. Despite experimental progress, though, the theoretical foundations of the interfacial DMI at oxide-oxide garnet interfaces remains yet to be established.  In the study mentioned above by Avci et al.,105) it was also shown that the DMI-stabilized chiral DWs in TmIG can be moved in a DW track (Fig. 9a) by current injection through the Pt overlayers as fast as 800 m/s with current densities of the order of 1.2x108 A/cm2 (Fig. 9b). The fast domain wall dynamics were a consequence of ferrimagnetic ordering and the absence of the precessional dynamics. In ferromagnetic systems such as Pt/Co, the DW velocity in the flow regime linearly increases as a function of current and saturates when the DL-SOT effective field becomes comparable to the DMI effective field.104) This is due to the rotation of the DMI-stabilized DW spins towards the transverse direction (parallel to 𝛕\"#) at large excitations. However, in ferrimagnets, the antiferromagnetic coupling between the two  11 sublattices prevents the net moment from precessing; hence the limiting factor is not anymore the DMI effective field but rather the much larger exchange field.25) This intriguing physics makes the ferrimagnetic materials appealing for DW-based ultrafast spintronic devices.  In compensated FMGs, additional to the magnetic compensation temperature, there exists a second temperature at which the angular momentum is compensated and the magnetization dynamics resembles to that of antiferromagnets, allowing fast domain wall motion.25,107) A recent study has shown that out-of-plane field assisted current-induced domain wall motion in GdIG with PMA can be as fast as 6000 m/s near the angular momentum compensation.54) This result marks the highest domain wall velocity reported to date and demonstrates yet other degrees of freedom offered by FMGs for ultrafast spintronics. Very recently, Caretta et al. have reported a fascinating feature of current-induced DW motion in FMGs98. They have shown that in Bi:YIG, DWs can be propelled at velocities in excess of 4300 m/s (Fig.9 c), the highest reported in any material so far without the need of an out-of-plane assisting field. The velocity was found to saturate to a universal limit above a critical current density and applied in-plane fields. Supported by analytical and atomistic modeling, this saturation anomaly was attributed to Lorentz contraction, a consequence of special relativity on the extremely fast-moving magnetic solitons. The reported velocities were calculated to be within 10% of the relativistic limit. These experiments open doors to realizing relativistic phenomena such as spin-wave Cherenkov radiation, higher-dimensional relativistic dynamics, and a host of related phenomena that previously seemed out of reach. And to achieve these behaviors in room-temperature, table-top experiments will hopefully lead to significant advances. Importantly, understanding the fundamental limits to the speeds of magnetic DWs and solitons, and the relation to the underlying material parameters can be used to maximize their speeds in future devices.  6. Signatures of Skyrmions in Ferrimagnetic Garnets The exciting discovery of chiral magnetism at FMGs’ interfaces and ultrafast DW motion have also accelerated the research into magnetic skyrmions in these materials. A skyrmion is a topologically protected magnetic texture stabilized by the DMI.108) Skyrmions can be as small as a few nm in diameter up to hundreds of nm determined by the material parameters, and can be moved by SOTs,27) Henceforth, they offer desirable prospects for efficient data storage, processing, and logic applications.108)  12 So far, the direct observation of skyrmions in FMGs has been elusive. Though, strong evidence of skyrmions in TmIG and YIG has been found through the measurements of the topological Hall effect109,110) (THE). The THE emerges due to the transverse deflection of electrons passing through the electromagnetic field generated by a skyrmion111) (Fig. 10a). It gives a Hall signal contribution proportional to the skyrmion density in the (Hall cross area of the) film and is generally characterized by additional bumps overlapping the standard Hall resistance curves within a characteristic out-of-plane applied field range (Fig. 10b). Characterization of the THE as a function of FMG thickness, temperature and applied magnetic fields has revealed that there is a phase pocket in which the THE emerges (Fig. 10c). The common observation in such studies is that the THE occurs in ultrathin FMG films (<6 nm), particularly at higher temperatures, when the PMA becomes weaker. The collective understanding of these findings pinpoints the existence of skyrmions in ultrathin FMGs at the right combination of different material and experimental parameters.  Independently of the above studies, bubble domains of only a few hundred nm of diameter (similar to typical skyrmion size) have been observed in relatively thick (25.6 nm) TmIG by scanning transmission x-ray microscopy.112) Photoemission electron microscopy analysis further showed that the DWs at the boundaries of the observed bubbles are of Bloch-type with arbitrary chirality. Such bubbles have been known since 1970s in micrometer-thick films,113) but the current study marked their first nucleation, observation and detailed characterization in nanometer-thick FMGs. The established sample fabrication procedure and experimental scheme for observing such small magnetic textures, combined with the efforts of electrically characterizing skyrmions in ultrathin FMGs holds promise for the direct observation and engineering of insulating skyrmions in the near future.  7. Conclusions The results summarized in this Review were obtained using only several FMGs and with a few years of experimental efforts. Yet, some impressive results such as highly competitive SOT-induced switching, interfacial chiral magnetism, ultrafast current-driven DW motion and relativistic current-induced dynamics have been achieved. The vast family of FMGs with many parameter tuning knobs combined with alternative SOT source materials creates a fertile ground for future studies and application prospects. Thus far, the SOT source has been limited to the SHE in Pt and W. There exist many other SOT materials and mechanisms including Weyl semimetals, topological insulators, two-dimensional electron gas interfaces, the interfacial  13 Rashba-Edelstein effect and more recently orbital Hall currents114) are to be considered for more effective magnetic manipulation of FMGs in suitable devices.  The advances in the deposition of FMGs shall inevitably open the doors to investigations of magnetic coupling phenomena in multilayers. Thus far, only single FMG layers have been considered in the studies summarized here. Sequential deposition of FMGs with varying magnetic and structural properties may lead to novel properties and functionalities that may not be achievable by individual layers. For instance, a recent report showed antiferromagnetic coupling between YIG and GdIG.115) Similarly coupled FMGs with PMA could reveal novel device concepts and architectures in electrically-controlled insulating spintronic devices.   The lack of in-depth understanding of the SOT-induced switching characteristics in FMGs opens new arenas for exploration. In all-metal systems, a significant effort has been spent to understand the SOT switching dynamics, device size and shape dependence, speed limits, etc. in the past couple of years, revealing outstanding results.116-118) For instance, the state-of-the-art switching speed in all-metal systems are in the order of a few picoseconds,119) some three orders of magnitude lower than that obtained in FMGs. Due to smaller Gilbert damping in FMGs in comparison to metallic ferromagnets, the switching dynamics in short timescales will be considerably different. FMGs are best suited to explore and understand the effect of the Gilbert damping on the switching, domain wall and skyrmion dynamics since it can be effectively tuned by the rare-earth choice. Efforts for understanding fast switching dynamics, device size and shape dependencies and the effect of the Gilbert damping in FMG-based devices are lagging behind and could reveal unpredictable twists. Taken together, FMGs, particulary those possessing PMA, have emerged as highly promising materials for electrical manipulation of magnetization in spintronic and magnonic devices. We are only starting to become aware of the vast potential of FMGs for spintronic research and future applications. It is beyond doubt that FMG-based materials and associated physical phenomena will keep scientists busy for many years to come. Acknowledgements The author thanks Dr. Lucas Caretta and Martin Testa Anta for fruitful discussions. The author acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (project MAGNEPIC, grant agreement No. 949052).  14 Author email address: cavci@icmab.es  References  1 F. Bertaut and F. Forrat,  Acad. Sci. Paris 242, 382 (1956). 2 S. Geller and M. A. Gilleo,  Acta Cryst. 10, 239 (1957). 3 G. Winkler, Magnetic Garnets. (Braunschweig, Vieweg, 1981), p.735. 4 S. L. Blank and J. W. 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A charge current through the normal metal (e.g., Pt) can generate a spin accumulation at the interface with the FMG, which then generates a damping-like SOT on the magnetization. The resulting effective field (HDL) can switch the magnetization between the up and down state in the presence of an in-plane field applied along the current injection direction that breaks the rotational symmetry of HDL. (Adapted from Ref. 87)          \n 23   Figure 2 – (Color online) (a) Damping-like and field-like SOTs (𝝉IJ, 𝝉KJ), and their associated effective fields (𝐻IJ, 𝐻KJ) for the given magnetization and current injection geometry. Note that the torque/field signs and amplitudes are arbitrary as they are material-dependent. (b) A visualization of the harmonic measurements. The magnetization vector oscillates with an a.c. current injection due to the oscillating SOTs and produce harmonic Hall voltage signals. (Adapted from Ref. 38)       \n 24  Figure 3 – (Color online) (a) Schematic of a setup for the Hall effect measurement on a device made of TmIG/Pt. (b) Typical Hall resistance signal for TmIG/Pt layers with PMA, recorded during an out-of-plane field sweep, consisting of the anomalous and ordinary Hall effect contributions as shown. (c) Hall resistance measured with an in-plane field (HIP) sweep applied at various angles. The curves at 45° and 135° represent the typical SMR behavior where a large positive and negative signal appears, and m is fully rotated in-plane. Orange curves are macrospin simulations based on experimental parameters and estimated PMA of about 2700 Oe. (d) Hall resistance signal amplitude measured at various in-plane angles reveals the expected sin2j behavior (see Eq. 1). (Adapted from Ref. 86) \n 25  Figure 4 – An electrical detection toolbox for magnetic insulators having out-plane (top row) or in-plane (bottom row) equilibrium magnetization orientation.        \n 26  Figure 5 – (Color online) Current-induced magnetization switching experiments in TmIG/Pt bilayers on Hall bar devices similar to the one in Fig. 3a. (a) Reference Hall resistance signal after subtraction of the OHE slope. (b) Magnetization switching experiments by consecutive application of 5 ms-long pulses through the device in the presence of an in-plane field of +/-500 Oe. The switching polarity reverses upon inverting the field polarity, as expected of the SOT-induced switching behavior. (c) A current sweep in the presence of a constant in-plane field, showing reversible switching at critical current density of about 1.8x1011 A/m2. (d) Switching phase diagram constructed based on measurements reported in (c) with different in-plane fields. (e) Partial switching of TmIG by electrical pulses of 1-5 ns length and 25-35 V \n 27 amplitude. 100% corresponds to the full switching of the Hall cross area and 0% corresponds to no switching. (f) In-plane field dependence of the switching characterized by stepping the field and applying positive and negative pulses and measuring RH consecutively. These measurements have revealed an in-plane field requirement as low as 2 Oe for switching TmIG.  (Adapted from Refs. 86 and 94) \n Figure 6 – (Color online) TmIG saturation magnetization (Ms) dependence of the damping-like SOT efficiency in TmIG/W bilayers. The enhanced torque efficiency at larger Ms (which is obtained at larger TmIG thicknesses) is attributed to the stronger exchange coupling between the SHE-induced interface spin accumulation and the local magnetization in TmIG. (Adapted from Ref. 99)  \n 28  Figure 7 – (Color online) Nitrogen-vacancy center microscopy characterization of domain wall internal magnetization in TmIG and TmIG/Pt bilayers. (a) Schematics of the experimental setup and the DW types. (b) Color map of the magnetization orientation measured by magnetic stray fields (BNV) showing a straight vertical domain wall crossing the Pt edge. (c,d) BNV measured along the line scans on the TmIG/Pt and bare TmIG regions. Fits showing that the domain wall in TmIG/Pt is a Bloch-type while in TmIG only it is a left-handed Néel type.  (Adapted from Ref. 96) \n 29  Figure 8 – (Color online) The interfacial DMI energy measured in various FMG-based systems. (Adapted from Ref. 106)          \n 30  Figure 9 – (a) (Color online) A typical DW device with integrated Hall arms. The magnetization can be probed either by a laser spot, as shown, or magnetic imaging by using magneto-optic Kerr effect.  DW velocity measurements as a function of the current density in TmIG/Pt (b) and in Bi:YIG/Pt (c). (Adapted from Refs. 98 and 105) \n 31  Figure 10 – (Color online) (a) Schematics of a magnetic sykrmion and the emergence of the topological Hall effect due to polarization-dependent electron deflection. (b) The transverse Hall resistivity in TmIG(1.85 nm)/Pt at room temperature as a function of an out-of-plane field sweep and the extracted topological Hall effect signal (bottom panel). (c) Colored phase diagram showing the temperature and field dependence of the emergent topological Hall signal in layers with different TmIG thickness. (Adapted from Refs. 110 and 111)  \n"    },    {        "title": "1911.12103v1.Spin_lattice_coupling_in_a_ferrimagnetic_spinel__The_exotic__H___T__phase_diagram_of_MnCr__2_S__4__up_to_110_T.pdf",        "content": "arXiv:1911.12103v1  [cond-mat.str-el]  27 Nov 2019Spin-lattice coupling in a ferrimagnetic spinel: The exoti cH–Tphase diagram of\nMnCr 2S4up to 110 T\nA. Miyata,1H. Suwa,2,3T. Nomura,4L. Prodan,5V. Felea,4,5,6Y. Skourski,4J. Deisenhofer,7\nH.-A. Krug von Nidda,7O. Portugall,1S. Zherlitsyn,4V. Tsurkan,5,7J. Wosnitza,4,6and A. Loidl7\n1Laboratoire National des Champs Magn´ etiques Intenses,\n(LNCMI-EMFL), CNRS-UGA-UPS-INSA, 31400 Toulouse, France\n2Department of Physics and Astronomy, The University of Tenn essee, Knoxville, TN 37996, USA\n3Department of Physics, The University of Tokyo, Tokyo 113-0 033, Japan\n4Hochfeld-Magnetlabor Dresden (HLD-EMFL) and W¨ urzburg-D resden Cluster of Excellence ct.qmat,\nHelmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Ger many\n5Institute of Applied Physics, MD 2028, Chisinau, R. Moldova\n6Institut f¨ ur Festk¨ orper- und Materialphysik, TU Dresden , 01069 Dresden, Germany\n7Experimental Physics 5, Center for Electronic Correlation s and Magnetism,\nInstitute of Physics, University of Augsburg, 86159, Augsb urg, Germany\n(Dated:)\nInantiferromagnets, theinterplayofspinfrustration and spin-latticecouplinghasbeenextensively\nstudied as the source of complex spin patterns and exotic mag netism. Here, we demonstrate that,\nalthough neglected in the past, the spin-lattice coupling i s essential to ferrimagnetic spinels as\nwell. We performed ultrahigh-field magnetization measurem ents up to 110 T on a Yafet-Kittel\nferrimagnetic spinel, MnCr 2S4, which was complemented by measurements of magnetostricti on and\nsound velocities up to 60 T. Classical Monte Carlo calculati ons were performed to identify the\ncomplex high-field spin structures. Our minimal model incor porating spin-lattice coupling accounts\nfor the experimental results and corroborates the complete phase diagram, including two new high-\nfieldphasetransitions at75and 85T. Magnetoelastic coupli nginducesstrikingeffects: Anextremely\nrobust magnetization plateau is embedded between two uncon ventional spin-asymmetric phases.\nFerrimagnetic spinels provide a new platform to study asymm etric and multiferroic phases stabilized\nby spin-lattice coupling.\nINTRODUCTION\nFerrimagnets, thanks to their finite spontaneous mag-\nnetization, have been widely used in engineering and\ntechnology, for example, as strong permanent magnets,\noptical isolators, and circulators for optical communi-\ncations. Unlike ferromagnets, by tuning the antiferro-\nmagnetic exchanges in ferrimagnets, additional function-\nalities, e.g., ferroelectricity and skyrmion controllability\n[1, 2], may emerge as a consequence of a noncollinear\nspin structure. The realization of such multifunctional-\nity opens the door for next-generation technologies.\nNoncollinear ferrimagnets have been well studied, for\ninstance, in spinels, AB2X4, where the AandBcations\nform a bipartite diamond lattice and a pyrochlore lat-\ntice, respectively, with X= O, S, or Se. The key\nessence is the competition (or frustration) of magnetic\nexchanges within or between the two AandBlattices\n(JA-A,JB-B, andJA-B). Yafet and Kittel (YK) proposed\na model for triangular-structure ground states, where\nJA-A∼JA-B[3], while Lyons, Kaplan, Dwight, and\nMenyuk (LKDM) proposed another model for conical-\nstructure ground states, where JB-B∼JA-B[4]. Both\nnoncollinear ground states have been widely observed in\nferrimagnetic spinels [5, 6].\nTo realize unconventional ferrimagnetic structures be-\nyond both the YK and the LKDM models, one can con-\nsider that spontaneous lattice deformation will modulatethese main antiferromagnetic exchanges, i.e., through a\nspin-lattice coupling mechanism [7]. Such lattice defor-\nmation has been expected in strongly frustrated antifer-\nromagnets to lift the macroscopic degeneracy (or lower\nthe free energy of the system), resulting in unconven-\ntional magneto-structural phases [8, 9]. On the other\nhand, this kind of spin-lattice coupling mechanism has\nnot been taken into account in previoustheoretical works\non ferrimagnetic spinels, because they have been consid-\nered to be less frustrated. Here, our experimental and\ntheoretical studies demonstrate that the spin-lattice cou-\npling is essential in ferrimagnetic spinels as well.\nMnCr2S4(A= Mn2+with spin S= 5/2, and B=\nCr3+withS= 3/2 in the spinel structure, see Fig. 1a)\nis a representative material for the YK model, where\nJMn-Mn∼JMn-Cr[10–14]. The interaction between the\nCrspins, JCr-Cr,isferromagneticandmuchstrongerthan\nthe other spin interactions, which has been evidenced\nby two consecutive magnetic phase transitions at TC≈\n65 K (ferromagnetic order of Cr spins) and at TYK≈5\nK (triangular-like magnetic order of Cr and Mn spins)\n[10, 14]. Recent high-field ultrasound and magnetization\nexperiments on MnCr 2S4up to 60 T revealed a complex\nphase diagram and novel magneto-structural phases [15].\nAn extremely robust magnetization plateau at 6 µB/f.u.\nbetween 25 and 50 T was observed. In analogy to the\nmodels proposed by Matsuda and Tsuneto [16] and Liu\nand Fisher [17], Tsurkan et al. proposed that the mag-2\nnetic orders of the Mn spin below and above the magne-\ntization plateau should be interpreted as spin supersolid-\nlike phases [15].\nFurthermore, it has been revealed very recently that\nthe supersolid and superfluid-like phases of MnCr 2S4are\nferroelectric, i.e., multiferroic [18]. It is, thus, impor-\ntant to clarify the magnetic structures of MnCr 2S4in\nthe sense of multiferroicity as well. The understanding\nof the phases could pave the way to exploit ferrimagnetic\nmultiferroicity, where finite magnetization moments can\nbeutilized, e.g., the electric-fieldreversalofferrimagnetic\nmoments [19].\nIn this paper, we report on the magnetization process\nof MnCr 2S4in magnetic fields up to 110 T and show\nthecomplete phase diagram of the YK-type ferrimag-\nnetic spinel. The magnetization experiments were com-\nplemented by magnetostriction and ultrasound measure-\nments up to 60 T, which evidence strongspin-lattice cou-\nplings. We propose a minimal Hamiltonian for describ-\ning the magnetic and structural properties of MnCr 2S4,\nincorporating spin-lattice coupling between the Mn and\nCr spins. Using the classical Monte Carlo method we\ncompare the theoretical and experimental data regard-\ning both the spinandlatticedegrees of freedom. Our\nanalysisevidencesthatthespin-latticecouplingstabilizes\nthe magnetization plateau and unconventional phases of\nasymmetric magnetic structures. The obtained phase di-\nagramisremarkablysymmetricwithrespecttothecenter\nof the magnetization plateau ( ∼40 T), where the exter-\nnal field perfectly cancels out the internal exchange field\nacting on the Mn spins.\nMETHODS\nMagnetization measurements were performed at the\nLNCMI-EMFL in Toulouse in pulsed magnetic fields up\nto 110 T along H∝bardbl[110] using a single-turn-coil tech-\nnique [20]. The magnetization was measured using the\ninduction method with a compensated pair of coils as\ndescribed in ref. [21].\nThe optical fibre Bragg grating (FBG) method was\nused to measure the magnetostriction [22] up to 60 T at\nthe HLD-EMFL in Dresden. The relative length change\n∆L/Lis obtained from the shift of the Bragg wavelength\nof the FBG. The magnetostriction was measured along\nH∝bardbl[110].\nUltrasound measurements were performed using the\nstandard pulse-echo technique to investigate the elastic\nproperties [23] up to 60 T at the HLD-EMFL in Dres-\nden. The longitudinal ( c11+ 2c12+ 4c44)/3 mode was\nstudied with the alignment of H∝bardblk∝bardblu∝bardbl[111], where\nkanduare propagation and polarization vector, respec-\ntively. Polyvinylidene fluoride (PVDF) film transducers\nwere used to excite and to detect ultrasonic waves at the\nfrequency of 65 MHz.Classical (Markov chain) MC simulations were per-\nformed for the system described by eq. (1). The spin\nand lattice configurations were sampled from the Boltz-\nmann distribution at finite temperatures. The number of\nsites and bonds in the simulations are 5,184 and 34,560,\nrespectively. We confirmed that the results for 5,184 and\n1,536 sites are consistent within error bars: the size ef-\nfectisnegligibleandthepresentresultsarerepresentative\nfor the thermodynamic limit. We adopted the overrelax-\nation technique both for the spin and lattice degrees of\nfreedom, which enables rejection-free updates with the\nenergy constant. MC updates with the Metropolis algo-\nrithm were combined for thermalization. We repeated\nthe whole process consisting of one Metropolis and five\noverrelaxation steps. We took the average over 221MC\nsamples after 216thermalization steps, which are much\nlonger than the autocorrelation time. The error bars\nare comparable to or smaller than the linewidths in the\npresent figures.\nRESULTS\nMagnetization measurements under ultrahigh\nmagnetic fields\nWemeasuredthemagnetization MofMnCr 2S4at4, 7,\nand14Kin pulsed fieldsup to 110T shownin Fig. 1b. In\nFig. 1c, the field derivatives of the magnetization curves\ndM/dH are presented. Focusing on the 4 K derivative\nin Fig. 1c, we easily identify five distinct steps labeled\nasH1–H5, which characterize field-induced transitions.\nThe observation of an extremely robust magnetization\nplateaubetween25( H2) and50T( H3) isconsistentwith\nthe previous study up to 60 T [15]. From the magneti-\nzation plateau at 6 µB/f.u. and the strong ferromagnetic\ninteractions between the Cr spins JCr-Cr, one can expect\nthat the observed moment of 6 µB/f.u. is attributed to\nthe full ferromagnetic moment of the Cr spins and zero\nnet moment of the antiparallel Mn spins. The spin state\nbelow 10 T ( H1) corresponds to the well-known coplanar\nYK-type structure, where the two Mn-sublattice spins\nare canted and their net magnetization moment is an-\ntiparallel to the Cr spins.\nTo connect continuously from these canted Mn spins\n(in the YK phase) to the antiferromagnetically collinear\nMn spins (in the plateau phase) without magnetization\njumps, one can assume that the canted Mn spins rotate\nin theplane with changingthe anglebetween twospinsin\nthe intermediatephasebetween H1andH2. As discussed\nin the following section, this asymmetric spin structure\nappearingin the intermediate phase is supported by clas-\nsical Monte Carlo (MC) calculations (see Fig. 2d).\nAbove the plateau phase, we found two additional\nphase transitions at approximately 75 ( H4) and 85 T\n(H5) in the ultrahigh-field experiments up to 110 T. Re-3\n—-!!\"#!\"!$%#$% \n!$%#!\"\n$%& $%' !\"\n0.3 \n0.2 \n0.1 \n0.0 \n-0.1 \n-0.2 \n-0.3 dM/dH (µB/T f.u.) \n100 80 60 40 20 0\nMagnetic field (T) H1H2\nH3H4\nH512 \n10 \n8\n6\n4\n2\n0\n-2 Magnetization ( µB/f.u.) \n100 80 60 40 20 0\nMagnetic field (T) H || [110] \n 4 K \n 7 K \n 14 K Exp. Exp. ()* (+* (,*\n—-\nFIG. 1. (a) Crystal structure of MnCr 2S4and the three main magnetic interactions, JMn-Mn,JMn-Cr, andJCr-Cr. (b)\nMagnetization and (c) its field derivative as a function of th e magnetic field along the [110] direction. The curves at 4 K ar e\nshown on the original scale, while the curves at 7 and 14 K are v ertically shifted for clarity. In (c), the arrows indicate t he phase\ntransitions, H1–H5, at which the step-like anomalies corresponding to the sudd en changes of the slope in the magnetization\nwere observed. The magnetization saturates in the fully pol arized state above 85 T.\nmarkably, the behavior observed below and above the\nplateau is symmetric: the plateau is sandwiched by two\nphases with steep slopes of the magnetization curve and\nthen by two phases with more gentle slopes. The broad-\neneddM/dH data at 14 K in Fig. 1c also show a rea-\nsonably symmetry with respect to the magnetic field of\n40 T. Based on this symmetry, one can assume that the\ntwo Mn spins rotate again through a spin-asymmetric\nphase between H3andH4and then a spin-canted phase\nappears between H4andH5. Finally, full polarization\nof 11µB/f.u. is found above H5. Our MC calculations\nsupport this assumption as shown below.\nSpin-lattice coupling model\nSpin-lattice coupling is expected to play an essential\nrole in magnetic properties of MnCr 2S4. The previous\nultrasound experiments showed strong anomalies in the\nvelocity of the longitudinal sound waves at H1,H2, and\nH3[15]. This observation indicates a significant spin-\nlattice coupling in the material.\nSpin-lattice coupling has been discussed for antiferro-\nmagnetic chromium spinel oxides, CdCr 2O4[24, 25] and\nHgCr2O4[26, 27], where only Cr is the magnetic ion.\nPencet al. [7] theoretically predicted that a collinear\nspin configuration is favored and a robust magnetization\nplateau appearsas aresult ofspin-lattice coupling, which\nintroduces biquadratic spin interactions after tracing out\nthe lattice degrees of freedom. In order to render the\nplateau robust for MnCr 2S4, where both Mn and Cr ions\nare magnetic, it is reasonableto assume collinear Mn and\nCr spins. We thus take into account a biquadratic term\nbetween Mn and Cr spins, bMn-Cr/parenleftbig\nSMn·SCr)2.Although the importance of the biquadratic terms in\nMnCr2S4has been pointed out in the past [28, 29], this\nMn-Cr biquadratic term has been neglected so far.\nWe propose the following minimal model:\nH=HMM+HCC+HMC+HZ, (1)\nwhere\nHMM=/summationdisplay\n/angbracketleftij/angbracketrightJMn-Mn/parenleftbig\nSMni·SMnj),\nHCC=/summationdisplay\n/angbracketleftij/angbracketrightJCr-Cr/parenleftbig\nSCri·SCrj),\nHMC=/summationdisplay\n/angbracketleftij/angbracketrightJMn-Cr(1−αρij)/parenleftbig\nSMni·SCrj)+K\n2ρ2\nij,\nHZ=−gµBH·(/summationdisplay\niSMni+/summationdisplay\njSCrj).\nIn this Hamiltonian, ∝angbracketleftij∝angbracketrightruns over the nearest neighbor\nwithin or between the Mn and Cr lattices, ρijdescribes\nthe lattice displacement between neighboring Mn and Cr\nions,Kandαarethespringconstantand thespin-lattice\ncoupling, g≈2 andµBare the g-factor and the Bohr\nmagneton, respectively.\nWe expect the interaction between Mn and Cr spins\nto be more sensitive to the positions of atoms than that\nbetween Mn spins. It is because the superexchange path\nof a Mn-Cr bond is involved with a single sulfur atom,\nwhile the path of a Mn-Mn bond is involved with multi-\nple sulfur atoms. The angle formed by Mn-S-Cr atoms,\nwhich is naturally parameterized by the bond phonon,\nshould affect the interaction between Mn and Cr spins\nsignificantly. It is, therefore, reasonable to include the4\n—-0.4 \n0.2 \n0.0 \n-0.2 dM/dH (µB/T f.u.) \n100 80 60 40 20 0\nMagnetic field (T) H1H2\nH3H4\nH512 \n10 \n8\n6\n4\n2\n0\n-2 Magnetization ( µB/f.u.) \n100 80 60 40 20 0\nMagnetic field (T) H || [110] \n 4 K \n 7 K \n 14 K 150 \n100 \n50 \n0Angle (deg) \n100 80 60 40 20 0\nMagnetic field (T) 4 K \n Mn1 \n Mn2 \n Cr Calc. Calc. Calc. !\"# !\"# !\"#\n!\"\n#$% #$& !\"\n#$% #$& \n#'($)*+,-+)./ !\"01234'5) 6.'*)'7234'5) \n#$% #$& !% !&\n859::)*\"+,234'5) !;\n859::)*\"+,234'5) !<\n=>$?)\"5)@201234'5) \n#$% #$& !\"!A\n!\"\n#$% #$& B/C\n—-\nFIG. 2. (a) Magnetization and (b) its field derivative as a fun ction of the magnetic field applied along the [110] direction\ncalculated from classical Monte Carlo simulations. The cur ves at 4 K are shown on the original scale, while the curves at 7\nand 14 K are vertically shifted for clarity. The arrows ( H1–H5) indicate the transitions similar to Fig. 1c. (c) Angles of t he\nCr, Mn1, and Mn2 spins with respect to the external-field dire ction. (d) Typical magnetic structures formed by Cr, Mn1, an d\nMn2 spins for each phase.\nspin-lattice coupling only in the Mn-Cr spin interaction\nas a minimal model.\nAfter tracing out ρij, we can exactly rewrite the inter-\nactions between the Mn and Cr lattices as\nHMC=/summationdisplay\n/angbracketleftij/angbracketrightJMn-Cr/parenleftbig\nSMni·SCrj)−bMn-Cr/parenleftbig\nSMni·SCrj)2,\nwhere the biquadratic term is introduced with the co-\nefficient bMn-Cr=J2\nMn-Crα2/2K, which we use as a pa-\nrameter for the spin-lattice coupling below. Despite the\nintegrability of lattice displacements, we intentionally re-\ntain them in the model (1) to compare theoretical results\nto experimental data regarding not only spin(magneti-\nzation) but also lattice(magnetostriction and sound ve-\nlocity) degrees of freedom.\nNote that the magneto-crystalline anisotropy of\nMnCr2S4is weak [30], because Cr ions have a half-filled\nt2gshell (3d3) and Mn ions have a half-filled t2gandeg\nshell (3d5) with zero orbital momentum.Classical Monte Carlo calculations: Magnetization\nand spin angles\nWe performed classical Monte Carlo simulations for\nthe model (1) and calculated the magnetization, M, and\nthe field derivatives of the magnetization, dM/dH, at 4,\n7, and 14 K as shown in Figs. 2a and 2b. Indeed, our\nsimple model already accounts for the experimental data\n(Figs. 1b and 1c). The five distinct steps, H1–H5, are\nall reproduced at 4 K. The two intermediate phases from\nH1toH2and from H3toH4, which are adjacent to\nthe magnetization plateau, have the steep slopes of the\nmagnetization curve in a similar way as the experimental\ndata.\nWe optimized the parameters in the model,\nJMn-Mn= 3.4 K, JMn-Cr= 3.1 K, JCr-Cr= -9.1 K,\nandbMn-Cr= 0.04 K, by fitting the calculated magne-\ntization to the experimental data. Some parameters\nwere simultaneously optimized to reproduce the two\ntransitions at TC≈65 K and at TYK≈5 K observed in\nprevious experiments without magnetic field [10, 14] (see\nAppendix A). Our estimates differ only by a few percent5\n!\" #$% \n#$& '$()*+\"\",-./$+()01,\"2+\"13 !%4\n5+\"\",-./$+()01,\"2+\"13 !&4!\"0.4 \n0.3 \n0.2 \n0.1 \n0.0  ∆L/L (10 -3 )\n60 40 20 0\nMagnetic field (T)  2.7 K \n-0.10 -0.05 0.00 0.05 0.10 \nq\n100 80 60 40 20 0\nMagnetic field(T) 4 K q1q2\nq\n-0.05 0.00 0.05 \n∆v/v\n60 40 20 0\nMagnetic field (T)  3.9 K \n 6.2 K \n 11 K -0.10 -0.05 0.00 0.05 \n∆v/v\n100 80 60 40 20 0\nMagnetic field (T)  4 K \n 7 K \n 14 K \n HYK \nHYK Exp.\nExp.Calc.\nCalc.\nH1H2H3\nH1H2H3\nH4H53.4 364\n304 3243+4\nFIG.3. Magnetostriction andsoundvelocityobtainedbyexp erimentandtheory. (a)Experimentallyobtainedmagnetost riction\nalongH/bardbl[110]. (b) Exchange-modification parameters q1andq2, which are proportional to the Cr-Mn1 and Cr-Mn2 bond\nlengths, respectively. The magnetostriction measurement ∆L/Lshould be proportional to q= (q1+q2)/2. In (b), q1andq2\nare identical below 20 T and above 60 T. (c, d) Relative change s of the sound velocity obtained in experiment and theory.\nThe longitudinal acoustic mode ( c11+2c12+4c44)/3 propagating along H/bardbl[111] was measured. (e) Schematic illustration of\nantiferromagnetically ordered spins with q1and ferromagnetically ordered spins with q2in the plateau phase.\nfrom the previous estimates [15], JMn-Mn= 3.1 K and\nJMn-Cr= 3.24 K [31]. In our spin-lattice model, the\neffective exchange coupling between Mn and Cr spins\nbecomes JMn-Cr(1−αρij)≈JMn-Mnaround zero mag-\nnetic field, which is consistent with the YK model [13].\nThe internal magnetic field that is created by the Cr\nspins and acting on the Mn spins is estimated to be\nabout 42 T using JMn-Cr= 3.1 K. This is consistent with\nthe observation of the symmetric behavior with respect\nto the field of 40 T.\nFromthequantitativepointofview, thetwointermedi-\nate phasesobtainedtheoreticallyarenarrowerthan those\nobserved experimentally. We incorporated additional bi-\nquadratic terms, bMn-Mn/parenleftbig\nSMn·SMn)2, but they do not\nimprove the results (not shown here). A more complex\nmodel is required for a quantitative comparison to the\nexperimental data.\nTo investigate the magnetic structures of the nontriv-\nial phases induced by magnetic field, we calculated the\nanglesofthe Cr and Mn spins from the external-fieldaxis\nat 4 K, shown in Fig. 2c. Using this data, we sketched\ntypical magnetic structures for each phase in Fig. 2d.\nThe spin-lattice coupling stabilizes not only the magne-\ntization plateau with a collinear magnetic structure but\nalso the two intermediate phases, where a triangular-like\nstructure formed by one Cr and two Mn spins rotates\ncontinuously with magnetic fields variation. The inter-\nmediate phases give rise to unconventional asymmetricmagnetic structures.\nMagnetostriction and sound velocity\nMCcalculationsforthe model(1) accountforthe mag-\nnetostriction and the sound velocity as well. The ex-\nperimentally obtained magnetostriction ∆ L/Lis shown\nin Fig. 3a. Figure 3b shows the exchange-modification\nparameters q1=αρMn1−Cr,q2=αρMn2−Cr, andq=\n(q1+q2)/2 obtained by the MC calculations.\nThe averaged displacement qis proportional to the\nchange of the sample length, which is measured in the\nmagnetostriction experiment.\nThe relative change of the sound velocity ∆ v/vob-\ntained experimentally and theoretically are shown in\nFigs. 3c and 3d, respectively. In the MC calculations,\nwe deduced the change of the sound velocity from the\neffective spring constant Keff, which differs from the\noriginal constant Kowing to the spin-lattice coupling.\nWe calculated the compressibility κin the MC simula-\ntions:κ≡βVar[q]∼β/integraltext\ndqq2e−βKeff\n2q2//integraltext\ndqe−βKeff\n2q2=\n1/Keff, where βis the inverse temperature. Although\nacousticphononmodesarenotincluded inourmodel(1),\ntheeffectivespringconstantshouldbecommontovarious\nphonon modes. The sound velocity measured in experi-\nments should be related to the effective spring constant:\nv∝√Keff. Therefore, we can estimate the velocity using6\nthe relation v∝1/√κ.\nOur theoretical model accounts for all the character-\nistic features related to the phase transitions ( H1–H3)\nobserved in the experiments up to 60 T. Moreover, the\nlow-field anomalies labelled as HYK(2 T at 6.2 K and\n9 T at 11 K in Fig. 3c), which corresponds to a phase\nboundary between the YK phase and the ferrimagnetic\nphase, where only the Cr spins order [14], are also re-\nproduced in the MC simulations as shown in Figs. 3c\nand 3d. The agreement between experiment and theory\nclearly shows the validity of our model and the relevance\nof the magnetoelastic coupling between the Mn and Cr\nions.\nThe theory reproduces the magnetostriction data very\nwell and suggests that the spin-lattice coupling produces\nstrong (q1<0) and weak ( q2>0) bonds between the\nMn and Cr spins (see sketch in Fig. 3e). In the plateau\nphase, the Cr and Mn spins are antiferromagnetically\n(ferromagnetically) ordered on the strong (weak) bonds.\nIn other words, this Z2symmetry of the Mn-Cr bond\nis spontaneously broken. Thus, a domain-wall excitation\nrelatedtothe Z2symmetryisexpectedtoappearatsome\nfinite temperatures due to the entropic effect. The broad\nminimum observedaround40Tintheultrasounddataat\nhigher temperatures (6.2 K and 11 K), which is absent in\nthe theoretical results, might be attributed to dynamics\nof the domain wall.\nPhase diagram\nCombining the present ultrahigh magnetic-field exper-\niments with previousdatain ref. [15], weobtain the com-\nplete phase diagram of MnCr 2S4in Fig. 4. The magnetic\nstructures for all phases, identified by our MC calcula-\ntions, are schematically shown in the figure. To identify\nthe higher temperature phase above TYKat zero mag-\nnetic field, we calculated the spin correlations between\nthe Mn1 and Mn2 spins using the model (1) with the\nsame parameters (see Appendix B). While the correla-\ntion between the Mn spin components parallel to the Cr\nspins develops below TC, the perpendicular components\nare almost independent down to TYK. This indicates\nthat the transverse components of the Mn spins remain\nparamagnetic in the intermediate temperature phase be-\ntweenTCandTYK, and the antiferromagnetic order of\nthe transverse components eventually takes place at TYK\nas shown in Fig. 4.\nWith increasing temperatures, both spin-asymmetric\nphases diminish on the expense of the expanding spin-\ncollinear phase. This finding indicates that thermal fluc-\ntuations favor the collinear magnetic structure rather\nthan noncollinear ones [32, 33]. In the phase diagram,\nthe higher-field phases mirror the lower-field phases; the\nphase diagram is symmetric with respect to approxi-\nmately 40 T, where the external fields cancel out theinternal exchange fields acting on the Mn spins.\n!\"#$% \n#$& #$!\"%'(100 \n80 \n60 \n40 \n20 \n0Magnetic field (T) \n20 15 10 5 0\nTemperature (K) \nFIG. 4. Magnetic-field-temperature phase diagram of\nMnCr 2S4. Thephaseboundarieswere obtainedfrom themag-\nnetization, ultrasound propagation and magnetostriction ex-\nperiments in pulsed fields. The closed circles with error bar s\nof±2 T and ±1 K were obtained in the present ultrahigh-\nfield experiments. The open squares were taken from ref.\n[15]. Typical magnetic structures revealed by the MC calcu-\nlations are illustrated in each phase. The Cr spins (orange\narrows) are almost perfectly aligned with the external field .\nThe Mn spins (two blue arrows) show a complex, strongly\nfield-dependent order.\nDISCUSSIONS\nThanks to our measurements in ultrahigh magnetic\nfields, we unraveled the complete H–Tphase diagram\nof the YK ferrimagnetic spinel compound. The experi-\nmental results are accounted for by a quite simple model\nincorporating spin-lattice coupling between the Mn and\nCr ions. Our findings indicate that the spin-lattice cou-\nplingiscrucialforestablishingthe extremelyrobustmag-\nnetization plateau. The effect of spin-lattice couplings\nhas been extensively investigated both experimentally\nand theoretically in antiferromagnetic Cr spinel oxides,\nZnCr2O4[34], CdCr 2O4[24], and HgCr 2O4[27], where\nthe pyrochlore lattice reveals strong geometrical frustra-\ntion. Here, with only weak frustration caused by the\ncompetition between JMn-MnandJMn-Crand with two\ndifferent magnetic ions, Mn and Cr, still a robust mag-\nnetization plateau was observed. Our spin-lattice model\nappears to be universally applicable to a broad range of\nferrimagnetic andantiferromagnetic spinel compounds.\nIn fact, strongspin-latticecouplingshavebeen widely ex-7\npected to exist in spinel compounds, such as in CoCr 2O4\n[35] and ZnCr 2S4[36], where unconventional magneto-\nstructural phase transitions have been observed. They\nmight be understood incorporating spin-lattice couplings\nto a simple spin model in a similar way as done in the\npresent study.\nAnother important finding is that the unconventional\ntriangular-like magnetic structures in the two intermedi-\nate phases are formed by one Cr and two Mn spins rotat-\ning continuously as magnetic fields increase. This identi-\nfication finally gives a clear solution to the long-standing\nproblemofthehigh-fieldmagneticstructuresofMnCr 2S4\n[15, 28, 29, 37]. The origin of the intermediate phases is\nthecompetitionbetweentheYK-typeinteractions,which\nfavors a noncollinear triangular structure, and the spin-\nlattice coupling, which favorsa collinear structure. Here,\nwe gain the straightforward insight that incorporation of\nspin-lattice couplings in ferrimagnetic spinel compounds\nleads to a variety of unconventional magnetic phases,\nsuchasfractionalmagnetizationplateausandspin-driven\nmultiferroic phases.\nLastly, we discuss possible improvements in our spin-\nlattice model. In the ultrasound experimental data\n(Figs. 3c and d), anomalous features were observed\naroundthe middle ofthe magnetizationplateauat higher\ntemperatures, which is absent in our theoretical con-\nsideration. This anomaly is even more pronounced in\nthe ultrasound attenuation as shown in previous experi-\nments [15]. It was speculated that this anomalousbehav-\nior might be attributed to a frustration in the Mn dia-\nmond lattice induced by next-nearest-neighbour interac-\ntions, as recently discussed for another spinel compound,\nMnSc2S4[38, 39]. The inclusion of farther spin interac-\ntions to our minimal model (1) might be appropriate.\nAnother possibility to improve the agreement may lie\ninmorecomplexinteractionsofthelatticedegreesoffree-\ndom. In our minimal model, lattice displacements are\nindependent of each other, forming a flat phonon band.\nA phonon band structure formed by connected lattice\ndegrees of freedom could more accurately describe rel-\nevant lattice displacements and reproduce the anomaly\nobserved in experiment.\nAlthough it has been believed for a long time that a\nmajority of magnetic properties of ferrimagnetic spinels\ncanbeunderstoodwithintheYKandLKDMmodels,our\nstudies have revealed that additional perturbations, such\nas the spin-lattice coupling, are crucial to explain their\nproperties under magnetic fields. Incorporating spin-\nlattice coupling, the YK and LKDM models reopen a\nnew platform to study unconventional magnetic phases,\nsuchasfractionalmagnetizationplateausandspin-driven\nmultiferroic phases.ACKNOWLEDGEMENTS\nThis research has been supported by the DFG via\nTRR 80 (Augsburg - Munich), by SFB 1143 (Dresden),\nthrough the W¨ urzburg-Dresden Cluster of Excellence on\nComplexity and Topology in Quantum Matter - ct.qmat\n(EXC 2147, project-id 39085490), and by the BMBF via\nDAAD (project-id 57457940). We acknowledge support\nby the project 16.80012.02.03F (ASM) and by HLD at\nHZDR and LNCMI at CNRS, both members of the Eu-\nropean Magnetic Field Laboratory (EMFL).\nAPPENDIX A: MAGNETIC SUSCEPTIBILITY\nAND SOUND VELOCITY\nWe calculated the inverse magnetic susceptibility χ−1\nand the relative change of the sound velocity ∆ v/vat\ntemperatures ranging from 2.5 to 400 K using the clas-\nsical Monte Carlo simulation for the model (1) in the\nmain text. The parameters were set to the same val-\nues as those in the main text: JMn-Mn= 3.4 K, JMn-Cr\n= 3.1 K, JCr-Cr= -9.1 K, and bMn-Cr=J2\nMn-Crα2/2K\n= 0.04 K. The quantum-classical crossover of the spin\nsystem needs to be correctly considered [40]: We treat\nthe spins as vectors and set the spin (vector) lengths to\n|S|cl=Sbelow 15 K and |S|cl=/radicalbig\nS(S+1) above 20 K.\nThe classical Monte Carlo simulations qualitatively re-\nproduce experimental data observed by Tsurkan et al. in\nref. [15].\nAPPENDIX B: SPIN CORRELATION\nTo investigate a magnetic phase between T YKand TC,\nwecalculatedspincorrelationsoftheMn1andMn2spins,\nS/bardbl\nMn1S/bardbl\nMn2(parallel component along the external field)\nand S⊥\nMn1S⊥\nMn2(perpendicular component against the ex-\nternal field).8\n—-\n0.9 \n0.8 \n0.7 χ-1  (mol/emu) \n15 10 5\nTemperature (K) 50 \n40 \n30 \n20 \n10 \n0χ-1  (mol/emu) \n400 300 200 100 \nTemperature (K) θCW  ~ -29 K 50 \n40 \n30 \n20 \n10 \n0χ-1  (mol/emu) \n400 300 200 100 \nTemperature (K) θCW  ~ 12 K \n1.4 \n1.3 \n1.2 \n1.1 \n1.0 \n0.9 χ-1  (mol/emu) \n15 10 5\nTemperature (K) Exp. Calc. \nExp. Calc. TYK TYK TC TC!\"# !$#\n!\"# !$#\n—-\nFIG. 5. Inverse magnetic susceptibility χ−1of MnCr 2S4ob-\ntained in the experiments [15] and the theory. An external\nmagnetic field of 1 T is applied along the [111] direction. (a)\nand (b) show the data at high temperatures of 20–400 K. (c)\nand (d) show the data at low temperatures of 2.5–15 K.\n—-\n30 \n20 \n10 \n0\n-10 \n-20 ∆v/v (10 -3 )\n15 10 5\nTemperature (K) 10 \n5\n0\n-5 ∆v/v (10 -3 )\n15 10 5\nTemperature (K) Exp. Calc. \nTYK TYK !\"# !$#\n—-\nFIG. 6. Relative changes of elastic constant ∆ c/cof MnCr 2S4\nobtained in the experiments (a) [15] and the theory (b).\nREFERENCE\n[1] T. Kimura, et al.,Nature426, 55 (2003).\n[2] S. Seki, X.Z. Yu, S. Ishiwata, and Y. Tokura, Science\n336, 198 (2012)..\n[3] Y. Yafet, and C. Kittel, Phys. Rev. 87, 290 (1952).\n[4] D.H. Lyons, T.A. Kaplan, K. Dwight, and N. Menyuk,\nPhys. Rev. 126, 540 (1962).\n[5] J. M. D. Coey, Can. J. Phys. 65, 1210 (1987).\n[6] T.A. Kaplan, and N. Menyuk, Philos. Mag. 87, 3711\n(2007).\n[7] K. Penc, N. Shannon, and H. Shiba, Phys. Rev. Lett. 93,\n197203 (2004).—-\nCr !\nMn1 ! Mn2 !\nTemperature Cr !\nMn1 ! Mn2 !Paramagnetic state !TYK  ~ 5 K ! 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With Co substitution,\ndepending on Co concentration ( x) Mn 3Ga prefers tetragonal (cubic) phase when x\u00140:5 (x\u00150:5). Ferrimag-\nnetism is robust regardless of xin both phases. While magnetic moments of two Mn do not vary significantly\nwith x, Co magnetic moment in two phases exhibit different behaviors, leading to distinct features in total mag-\nnetic moment (M tot). When x\u00140:5, in tetragonal phase, Co magnetic moment is vanishingly small, resulting in\na decrease of M totwith x. In contrast, when x\u00150:5, in cubic phase, Co magnetic moment is roughly 1 mB, which\nis responsible for an increase of M tot. Electronic structure is analyzed with partial density of states for various x.\nTo elucidate the counterintuitively small Co moment, the magnetic exchange interaction is investigated where\nexchange coefficient between Co and Mn is much smaller in x\u00140:5 case than x\u00150:5 one.\nI. Introduction\nFor decades, Heusler compounds with plethora of material\nclasses have drawn continuously renewed attentions [1, 2].\nRich physics has been platforms for various research. Numer-\nous functionalities and properties, such as superconductivity,\nmagnetism, multiferroicity, half-metallicity, topological insu-\nlators, and so forth [3–6], have attracted for applications in\nmany areas. In particular, in the context of rapid progress\nin magnetic devices and spintronics, tetragonal Heusler com-\npounds have been intensively and extensively explored. Prop-\nerties of tetragonal Heusler compounds, perpendicular mag-\nnetic anisotropy (PMA) and low saturation magnetization\n(MS) [7–9], are regarded advantageous in applications with\nlow switching current ( IS) [10–12].\nMn 3Ga belonging to Heusler family in tetragonal structure,\npossesses aforementioned magnetic properties: strong PMA\naround 0.89 MJ/m\u00003, high Curie temperature ( TC) of 740 K\n[13, 14], low MSaround 200 emu/cm3, and approximately\n58% of spin polarization [15]. Low MSstems from the intrin-\nsic ferrimagnetism of antiparallel alignment of two Mn mo-\nments with unequal magnitudes. Efforts have been devoted\nto enhance the magnetic properties of Mn 3Ga. One is to re-\nplace Mn by other transition metal (TM) [16] where V-shape\nmagnetization with respect to TM content ( x) is quite distinct.\nUpon substitution of nonmagnetic TM, M totdecreases with\nxas nonmagnetic TM replaces one Mn, moment is reduced\nbut the antiparallel moment with the other Mn is retained.\nMtot=0 is achieved for certain xabove which the survived\nantiparallel moment dominates hence M totincreases.\nThe V-shape of magnetization is also prominent with mag-\nnetic TM substitution such as Fe, Co, and Ni [17, 18]. How-\never, M totnearly vanishes above x=0:5 and is not compen-\nsated. Interestingly, Co substitution accompanies a structural\ntransition from tetragonal to cubic while the substitution by\nFe and Ni retains the tetragonal structure. Moreover, the mag-\nnetic behavior of magnetic TM in two structural phases is dif-\nferent. Magnetic moments in tetragonal phase are smaller than\n\u0003sonny@ulsan.ac.krthe bulk counterpart but those in cubic phase are comparable\nto the bulk ones. While the substitution of Fe results in hard\nmagnet and that of Ni exhibits the shape memory phenom-\nena, the property of Co substitution depends on the Co con-\ntent. It has been reported earlier that Co-poor (0 <x<0:5)in\ntetragonal alloy is well suited for spin-transfer torque (STT)\nwhereas Co-rich (0:5<x<1)favors cubic phase with half-\nmetallicity. As STT and half-metallicity are favorable compo-\nnents in spintronic application, further study would shed light\non uncovering Mn 3\u0000xCoxGa.\nIn this paper, employing ab initio calculations, the effect\nof Co substitution on Mn 3\u0000xCoxGa is investigated. Magnetic\nproperty of Mn 3\u0000xCoxGa is illustrated along with total as well\nas atomic resolved magnetic moments. Analysis follows to re-\nveal the electronic structure of Mn 3\u0000xCoxGa. To elucidate the\ncontrasting behavior of Co magnetism in two phases, tetrag-\nonal and cubic, the exchange coefficients are discussed in the\nframework of the magnetic force theorem using the Heisen-\nberg model.\nII. Structure and computational Methods\nTetragonal ( x=0) and cubic phases ( x=1) are presented\nin Fig. 1, which correspond to D022andL21structure, respec-\ntively. Both tetragonal and cubic phase have two inequivalent\nMn atoms distinguished by Mn(I) and Mn(II). Space group\nof tetragonal phase is I4=mmm (No. 139) and that of cubic\nphase is Fm¯3m(No. 225). Wyckoff positions of both phases\nare listed in Table I.\nTABLE I. Wyckoff positions of constituent atoms of tetragonal\nand cubic phases, whose space groups are I4=mmm (No:139)and\nFm¯3m(No:25), respectively. Co takes either 2 cor 2dsite.\nTetragonal Cubic\nMn(I) 2 b 4a\nMn(II) 4 d 4c\nGa 2 a 4b\nCo 2c/2d 4 darXiv:2209.10216v1  [cond-mat.mtrl-sci]  21 Sep 20222\nFIG. 1. Crystal structure of (a) tetragonal Mn 3Ga and (b) cubic\nMn2CoGa. Blue, red, green and grey spheres denote Mn(I), Mn(II),\nCo and Ga atoms, respectively.\nFirst-principles calculations are performed using Vienna\nAb initio Simulation Package (V ASP) [19, 20] with projec-\ntor augmented-wave basis [21]. The exchange-correlation\npotential is treated using generalized gradient approxima-\ntion (GGA) with Perdew, Burke, and Ernzerhof (PBE)\nparametrization [22, 23]. Cutoff energy of 500 eV is em-\nployed for wave function expansion. For Brillouin zone in-\ntegration, 15\u000215\u00027 and 15\u000215\u000211kmeshes are used\nfor tetragonal and cubic phases, respectively, wherep\n2=2\u0002p\n2=2\u00021 unit cell is adopted for the primitive cell.\nFrom lattice constant optimization, we obtain a=3:78˚A\n(c=a=1:88) for tetragonal phase and a= 4.12 ˚A for cubic\nphase, respectively, which are used for further calculations\nThe exchange coefficients are calculated using the Heisenberg\nmodel in the framework of the magnetic force theorem (MFT)\n[24]. To do so, additional self-consistent calculations are per-\nformed using OpenMX [25], where atomic basis sets for Mn,\nGa, and Co are s3p2d1,s2p2d1, and H- s3p2d1, respectively\nwith energy cutoff of 300 Ry and cutoff radii of 10.0 ˚A(Mn\nand Co) and 9.0 ˚A(Ga), respectively. Then, Jxpackage is em-\nployed [26–28] to extract the exchange coefficients ( J) in the\nframework of MFT.\nIII. Results and Discussion\nA. Magnetic Moments\nMn 3Ga is either tetragonal or cubic depending on Co con-\ncentration x. When x<0:5, tetragonal phase with c=a=1:88\nis favored whereas when x>0:5 cubic phase is preferred. As\nsuch, x=0:5 is a phase boundary of two phaes [29].\nTotal magnetic moment (M tot) as function of xis plotted\nin Fig. 2, where each moment of individual Co, Mn(I), and\nMn(II) is also shown. For all x, Mn 3\u0000xCoxGa is ferrimagnetic:\nMoments of Mn(I) and Mn(II) are antiparallel with different\nmagnitudes. Without Co, i.e. x=0, Mn(I) and Mn(II) have\nmoments of 2.87 and -2.32 mB, respectively, exhibiting indeed\nferrimagnetism. Total magnetic moment (M tot) is 1.72 mBper\nFIG. 2. (a) Magnetic moment (M tot) per unit cell with respect to x,\nwhere white and grey area are tetragonal and cubic phases, respec-\ntively. Right axis is in emu/cm3converted from mBon the left axis.\n(b) Atom-resolved magnetic moment: blue, red, and green circles\ndenote Mn(I), Mn(II), and Co, respectively. Open (solid) circles rep-\nresent unstable (stable) phase.\nformula unit, consistent with previous works [14, 30], which\ncorresponds to magnetization of 314 emu/cm3. As shown in\nFig. 2(a), M totdecreases (increases) with xin tetragonal (cu-\nbic) phase. Notably, at the phase boundary when x=0:5, M tot\nof cubic phase is larger by 0.4 mBthan tetragonal phase. This\ndifference is due to different Co magnetism in both phases.\nAcknowledging ferromagnetism of Co, Co moment in tetrag-\nonal phase is quite small [31], which is against to the conven-\ntional wisdom. This is discussed later in Sec.III B and III C.\nMagnetic moments of individual atoms are shown in\nFig. 2(b) [32]. For all x, Mn(I) and Mn(II) have almost con-\nstant moments with 2.90 and -2.32 mB, respectively. However,\nCo moment in tetragonal phase nearly vanishes while in cubic\nphase is around 1.0 mB. Consequently, M totdecreases with x\nin tetragonal phase: M tot=1:72mBforx=0 drops to 0.72 mB\nforx=0:5. On the other hand, in cubic phase, more Co sub-\nstitution results in increase of M totfrom 1.00 to 2.00 mB. The\nvanishingly small Co magnetic moment in tetragonal phase is\nalso reported in previous work which has been also reported in\nother theoretical work [33]. As mentioned earlier, this coun-\nterintuitive feature will be visited in III C.\nB. Electronic Structure\nPartial density of states (PDOS) is presented in Fig. 3.\nLeft (right) upper panel is tetragonal (cubic) phase for x=0\n(x=1). Lower panel is for x=0:5 of tetragonal and cubic\nphases, respectively. Important peaks are labelled with tri-\nangle marks: 1, 2, and Cstand for Mn(I), Mn(II), and Co,\nrespectively; + (-) for the majority (minority) spin state, re-\nspectively. Different peaks in the same spin state of particular\natom are distinguished by primes. PDOS’s of x=1=3 and 2 =3\nare shown in Fig. S3, where the vertical line serves to manifest\nthe overall trend upon x. The shift of EFby Co substitution is\nevidence as Co has two more electrons than Mn[34].\nIn tetragonal phase with x=0 [Fig. 3(a)], Mn(I) and Mn(II)\nexhibit distinct features. Mn(I) has two main peaks: 1+ (1-)\nin occupied (empty) states around -2.7 eV (+1.0 eV) in the3\nFIG. 3. Partial density of states (PDOS) of dorbitals. Left and\nright panels for tetragonal and cubic phases, respectively. Tetragonal\nphase with (a) x=0 and (b) x=0:5. Cubic phase with (c) x=1:0 and\n(d)x=0:5. Lower panel is comparison between tetragonal and cubic\nphases with x=0:5. Blue, red, and green lines denote dorbitals\nof Mn(I), Mn(II), and Co, respectively. Peaks are emphasized with\ntriangles. Fermi energy ( EF) is set to zero.\nmajority (minority) spin state. On the other hand, Mn(II) has\nrather broader peak with smaller height than Mn(I): 2+ around\n+1.3 eV in the majority spin state and 2-, 20-, and 200- around\n-2.5, -1.2 eV , and +1.0 eV in the minority spin state, respec-\ntively.\nForx=0:5 in tetragonal phase [Fig. 3(b)], Mn(I) and\nMn(II) PDOS do not change much with respect to x=0 but\nshifted by +0.20 eV due to Co. [See vertical line in Fig. S3].\nOn the other hand, several peaks emerge in Co PDOS. As Co\nreplaces Mn(II), most Co peaks overlap with Mn(II): C- co-\nincides with 20- and small hump just above EFoverlap with\n20+. Nonetheless, the overlap with Mn(I) in the majority spin\nstate around -2.8 eV (marked with double triangles) is evi-\ndent. Broad C+ appear between -1.4 and 0.4 eV owing to hy-\nbridization with Ga pstate [See Fig. S2]. In particular, there\nare small humps around +0.2 and +0.6 eV , where the former,\njust mentioned earlier, overlaps with 20+ but the latter has no\nsuch feature. As the sum of occupied PDOS of two spin states\nis almost equal, Co magnetic moment nearly vanishes, which\nis prominent in tetragonal phase.\nNow we switch to cubic phase with x=1 [Fig. 3(c)]. Com-\npared to x=0 of tetragonal phase, 1+ changes little but 1-\nshifts by +0.3 eV . However, changes in Mn(II) and Co are ap-\nparent. In the minority spin state, 2- shifts by 0.2 eV and 20-\nis replaced by C-. Overlaps between 200- (not labeled) and C0-\nare prominent around EF+0.8 eV . On the other hand, in the\nmajority spin state, C+ appears almost around the same place\nasC-. Overlaps between C0+ and 2+ around EF+0.6 eV are\nFIG. 4. Calculated exchange coefficients ( J) for nearest neighbor in-\nteractions for various x(0\u0014x\u00141).J0,J1, and J2, represent the ex-\nchange coefficients of Mn(I)-Mn(II), Co-Mn(I), Co-Mn(II) in black,\nblue, and red, respectively. Tetragonal and cubic regions are divided\nby white and grey regions. Solid and dashed lines for tetragonal and\ncubic phases, respectively.\nmuch closer to EFthan x=0. Notably, 20+ is occupied in\ncontrast to x=0.\nCubic phase with x=0:5 is shown in Fig. 3(d). Here, we\ndiscuss changes with respect to x=0:5 of tetragonal phase\nandx=1 of cubic phase. Hence in this paragraphs x=0:5\n(x=1:0) refers to that in tetragonal (cubic) phase. Mn(I) is lit-\ntle affected: 1+ and 1- change negligibly with respect to x=1\nwhereas 1- shifts by +0.3 eV with respect to x=0:5. On the\nother hand, Mn(II) is quite affected with negative (positive)\nshifts with respect to x=0:5 (x=1). More specifically, com-\npared to x=1, peaks of 2+ and 20+ shift by +0.30 and +0.20\neV , respectively; that of 2- by +0.35 eV . Compared to x=0:5,\nshifts of 2+ and 20+ are -0.60 and -0.40 eV , respectively; 2- by\n-0.50 eV . In particular, 20+ accompanies occupation change.\nFor Co PDOS, C+ and C- shift by -0.4 and -0.1 eV , respec-\ntively, with respect to x=1, and 0 and -0.1 eV with respect to\nx=0:5. Positive shift of Mn(II) with respect to x=1 is well\nexpected, as mentioned earlier, as Co has more electrons than\nMn.\nC. Magnetic Exchange Interaction\nSo far PDOS analysis is presented. While Co magnetic mo-\nment in cubic phase ( x\u00150:5) is around 1 mB, that in tetrag-\nonal phase x\u00140:5 nearly vanishes, which is against the con-\nventional wisdom. In order to clarify this puzzling Co mag-\nnetism, the exchange coefficients, J, are discussed, employing\nthe Heisenberg model, ˆH=\u0000åi jJi jSi\u0001Si, where Ji j, the ex-\nchange coefficients, with positive (negative) sign implies par-\nallel (antiparallel) magnetic coupling. As briefly mentioned4\nin II, J0s are calculated in the framework of MFT [24–26, 28].\nFig.4 shows calculations of the exchange coefficients for three\ncases, Mn(I)-Mn(II), Co-Mn(I), and Co-Mn(II), denoted as J0,\nJ1, and J2, respectively.\nJ0, the exchange coefficients for two Mn types, is nega-\ntive for all cases. This indicates the antiparallel magnetism\nbetween two Mn sites confirming the ferrimagnetism. More-\nover,jJ0jis larger thanjJ1jorjJ2jroughly by an order of mag-\nnitude. In tetragonal phase ( x\u00140:5),J0=\u00009:89 meV for\nx=0 whose magnitude increases with xranging 18 – 21 meV .\nIn cubic phase ( x\u00150:5),J0=\u000014:9 meV for x=0:5, whose\nmagnitude also increases with xup to 20 meV . From this, ad-\ndition of Co introduces stronger tendency of the antiparallel\nmagnetism between Mn(I) and Mn(II).\nThe exchange coefficients between Co-Mn(I) and Co-\nMn(II), namely J1and J2, exhibit opposite sign behavior\nwhere magnitudes in cubic phase are slightly larger. In con-\ntrast to J0, bothjJ1jandjJ2jare little influenced by Co. In\ntetragonal phase,jJ1jandjJ2jare approximately 1 and 5 meV ,\nrespectively, while in cubic phase jJ1jandjJ2jare approxi-\nmately 5 and 7 meV , respectively. The opposite signs of J1\nandJ2is due to the antiparallel magnetism between Mn(I)\nand Mn(II). From jJ2j>jJ1j, the antiparallel coupling be-\ntween Co-Mn(II) is stronger than Co-Mn(I). Moreover, J1and\nJ2change signs between tetragonal and cubic phases, hence-\nforth Co shows the opposite sign of magnetic moment in both\nphases. From the analysis of the exchange coefficients, Mn(I)\nand Mn(II) exhibit rather rigid magnetic moments. More-\nover, the exchange coefficients with Co in tetragonal phase are\nweaker than cubic phase. As a result, Co moments in tetrago-nal phase is smaller than cubic one.\nIV . Conclusion\nIn summary, we have investigated the effect of Co substi-\ntution on Mn 3Ga for structural and magnetic properties. The\nferrimagnetic feature is robust regardless of Co concentration\n(x). When x\u00140:5, tetragonal phase is favored over cubic and\nvice versa, where the magnetic behavior of each phase is dif-\nferent. In tetragonal phase, M totdecreases with xwhile in\ncubic phase M totincreases with x. Electronic structure upon\nCo substitution is investigated with PDOS, whose peak struc-\ntures are analyzed. Except Co, magnetic moments of indi-\nvidual atom change little with x. Co magnetic moment in\ntetragonal phase nearly vanishes in contrast to 1 mBof cubic\nphase. To get more insight on this contrasting Co moment in\ntwo phases, the exchange coefficients are estimated using the\nHeisenberg model. Indeed, the antiparallel magnetism, more\nprecisely, the ferrimagnetism between Mn(I)-Mn(II) with dif-\nferent magnitudes is well manifested. On the other hand, the\nexchange coefficients of Co-Mn(I) and Co-Mn(II) exhibit op-\nposite signs in tetragonal and cubic phase.\nAcknowledgments\nWe are grateful to Soon Cheol Hong for fruitful discussions\nand comments. This work was supported by the National Re-\nsearch Foundation (NRF) of Korea (2019R1I1A3A01059880,\n2020R1A2C3008044).\n[1] K. Manna, Y . Sun, L. Muechler, J. K ¨ubler, and C. Felser,\nHeusler, Weyl, and Berry, Nat. Rev. Mater. 3, 244 (2018).\n[2] L. Wollmann, A. K. Nayak, S. S. Parkin, and C. Felser, Heusler\n4.0: tunable materials, Annu. Rev. Mater. Res. 47, 247 (2017).\n[3] Y . Kurtulus, R. Dronskowski, G. D. Samolyuk, and V . P.\nAntropov, Electronic structure and magnetic exchange coupling\nin ferromagnetic full Heusler alloys, Phys. Rev. B 71, 014425\n(2005).\n[4] T. Graf, C. Felser, and S. S. P. Parkin, Simple rules for the un-\nderstanding of Heusler compounds, Prog. Solid State Chem. 39,\n1 (2011).\n[5] M. Sargolzaei, M. Richter, K. Koepernik, I. Opahle, H. 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B 70, 184421 (2004).\n[29] See Supplemental Material at http://link.aps.org/xxx for com-\nparison tetragonal and cubic phases for Mn (II) sites. DE=\nET\u0000ECis plotted in Fig. S2 as function of Co concentration\n(x).\n[30] W. S. Yun, G.-B. Cha, I. G. Kim, S. H. Rhim, and S. C. Hong,\nStrong perpendicular magnetocrystalline anisotropy of bulk and\nthe (001) surface of D0 22Mn3Ga: a density functional study, J.\nPhys.: Condens. Mater. 24, 416003 (2012).\n[31] See Supplemental Material at http://link.aps.org/xxx for mag-\nnetic moments using GGA+U calculations. The validity of ap-\nplying on-site Coulomb interaction to Mn 3Ga is not justified\nyet. However, two cases are considered: (i) Ufor all magnetic\natoms and (ii) only for Co atom, with U=0\u00183 eV . 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Wang1,† \n1Department of Electrical and Computer Engineering, University of California, Los Angeles, \nCalifornia 90095, USA  \n2Shanghai Engineering Research Center of Ultra -Precision Optical Manufacturing, Department \nof Optical Science and Engineering, Fudan University, Shanghai 200433, China  \n*Corresponding author : wuhaophysics@ucla.edu  \n†Corresponding author : wang@ee.ucla.edu  \n  2 \n Abstract:  \nSpintronic s provide s an efficient platform for realizing non -volatile memory and logic \ndevices. In these systems, data is stored in the magnetization of magnetic materials , and  \nmagnetization is switched in the writing process . In conventional spintronic devices, ferromagnetic \nmaterials are used which have a magnetization dynamics timescale of around the nanosecond s, \nsetting a limit for the switching speed . Increasing the magnetization switching speed has been one \nof the challenges  in spintronic research . In this work we take advantage of the ultrafast \nmagnetization dynamics in ferrimagnetic materials instead of ferrom agnets, and  we use \nfemtosecond laser pulses and a photoconductive Auston switch to create picosecond current pulses \nfor switching the ferrimagnet.  By anomalous Hall and magn eto-optic Kerr (MOKE) measurement, \nwe demonstrate the robust picosecond  SOT driven magnetization switching of ferrimagnetic \nGdFeCo. The time-resolved  MOKE shows more than 50 GHz magnetic resonance frequency of \nGdFeCo, indicating faster  than 20 ps spin dynamics and tens of picosecond SOT switching speed.  \nOur work provides a promisi ng route  to realize picosecond operation speed for non -volatile \nmagnetic memory and logic applications.  \nMain Text : \nElectrically manipulat ion of  the magnetization is both fundamentally interesting and \npractically important  for spintronic devices1,2. To date, two crucial issues need to be addressed : (1) \nreducing the energy consumption3 and (2) in creasing the switching speed.  There have been  several \napproaches to reduce the energy consumption, such as voltage -controlled magnetization4,5, the \ngiant spin -orbit torque in topological insulators6-11, and so on. As for the switching speed, up to \nnow, most spintronic devices are still operated in several nanoseconds’  timescale, which \ncorresponds to the GHz magnetic resonance frequency of ferromagnetic materials.  Ferrimagnets 3 \n have antiferromagnetically -coupled spin sublattices, as a result, the exchange coupling -induced \nmagnetic resonance frequency is on the order of 100 GHz near their compensation point12, which \ncan enable picosecond magnetization switching speed.  \nThere have been efforts  for realizing ultrafast all -optical switching of ferrimagnetic \nmaterials13, with circularly -polarized femtosecond laser pulses resulting in switching speeds on \nthe order o f tens of picoseconds14-17. Similarly, laser-induced hot-electron pulses have been \nexploited to induce ultrafast switching dynamics in ferrimagnetic materials18-20. However, these \nmethods mainly  rely on heating effects and have relat ively low energy efficiencies compared to \nelectrical methods . Therefore, picosecond electrical switching of ferrimagnets provides a \npromising route towards  ultrafast spintronic devices with high energy -efficiency , which has not \nbeen experimentally explored yet.  \nIn this work, a photoconductive Auston  switch is employed to generate the picosecond \nelectrical pulse from a femtosecond laser pulse21, which is used to provide  the picosecond spin-\norbit torque (SOT) switching22-24 in Ta/GdFeCo heterostructures . The magnetization switching is \ndetected by both the anomalous Hal l effect (AHE) and the magneto -optic optical Kerr effect \n(MOKE) , which demonstrate robust magnetization switching by the picosecond SOT  with a peak \ncurrent density of 107 A/cm2. Furthermore, 50 GHz spin dynamics of GdFeCo is detected by the  \ntime-resolved MOKE  (TR-MOKE) , further confirming the tens of picosecond  switching speed  of \nferrimagnets .  \nFigure 1a shows the schematic of the device structure and the measurement method, where a \nphotoconductive Auston switch is connected to the magnetic  strip of Ta/GdFeCo heterostructures. \nWhen the femtosecond laser pulse is focused on the gap of the Auston sw itch, the charge carriers \nwill be excited in the conduction band of low temperature grown GaAs (L T-GaAs), which are 4 \n subsequently accelerated  by the bias voltage . Based upon simulations in the Sentaurus software \npackages, the induced photocurrent response t ime is estimated  to be around 1 ps.  The \nmagnetization of GdFeCo is detected either by the AHE or the MOKE , where a n in-plane magnetic \nfield is applied to break the inversion symmetry between ±M during the SOT switching.  Figure \n1b shows the microscopic picture of the Auston switch+ magnetic  strip device , where the width of \nthe magnetic  strip is 20  μm. The Rxy-Hz curve in Figure 1c  demonstrates  the strong  perpendicular \nmagnetic anisotropy (PMA)  of GdFeCo.  \n \nFIG. 1. a, Schematic of the Auston switch+magnetic strip device and the measurement set up. \nThe THz lase r pulse is focused  on the gap of the voltage -biased photoconductive Auston switch \nto excite the picosecond current, which  then provide s a picosecond spin -orbit torque  (SOT) on \n5 \n the ferrimagnetic GdFeCo.  b, SEM picture of the patterned Auston switch+magnetic str ip \ndevice.  c, Rxy-Hz curve shows the strong perpendicular magnetic anisotropy (PMA) of GdFeCo.  \nTo test the device , we first perform all -electrical SOT switching of the Ta/GdFeCo \nheterostructure  using relatively long electrical pulses . Figure 2a,b show the current -driven SOT \nswitching, where the magnetization is detected by the AHE resistance. In this measurement scheme, \na 1-ms writing current pulse is applied to the switch the magnetization  firstly, and is followed one \nsecond later by another  1-ms reading current pulse to detect the AHE resistance.  Under ±50 Oe  \nin-plane magnetic field Hx, the current -induced switching polarity changes, which is the typical \nSOT characteristic.  The switching current density  Jc is around 1.1 ×107 A/cm2 for the 1 -ms writing \npulse, which is similar to th e previous report25. MOKE microscope is employed to detect the \nmagnetic domain directly , which also clearly show the robust SOT -driven magnetic domain \nswitching , as shown in Figure 2c . In order to figure out the current -induced Joule heating effect on \nthe SOT switching, we cha nge the writing pulse width from 0.5 s to 10 ns , as shown in Figure 2d . \nJc significantly increases for the writing pulse width from 0.5 s to 1  ms, and then slightly changes \nfrom 1 ms to 10 ns, indicat ing Joule heating is not a domina nt effect  with short er writing pulse s. 6 \n  \nFIG. 2. a and b show the current -driven SOT switching of the Ta/GdFeCo heterostructure  under \nHx = ±50 Oe . c shows the SOT -driven magnetic domain switching measured by the MOKE \nmicroscope . d, Switching current density Jc as a function of the pulse width of the writing \ncurrent.  \nThe magnetic precession period of the GdFeCo samples is more than 3 orders of magnitude \nshorter than the duration of the writing pulses in Fig ure 2d. It is therefore reasonable to expect that \nthe SOT switching of the GdFeCo could  be achieved in devices subjected to significantly shorter \nwriting pulses. Previous time resolved magnetization studies of GdFeCo structures report full \nmagnetization switching in structures subjected to ps and sub -ps length optical  excitation. \nHowever, t hese previous results are both rooted in thermal effects, therefore , the demonstration of \n7 \n deterministic, ultrafast, electrical switching of GdFeCo magnetization is highly desirable from a \ntechnological perspective.  \nTo test the temporal limitations of SOT  switching in the Ta/GdFeCo  heterostructure , we \nperform  the experiments on devices subjected to ps duration electrical pulses. These ultrashort \nelectrical excitations are generated by illuminating a DC biased photoconductive switch with the \nfs output of a mode locked Ti:Sapphire laser. A typical device used for these experiments is \ndepicted in Fig ure 1b. Given the duration of the laser pulse (~150  fs) and the known carrier  lifetime \nwithin the switch (~300  fs), electrical pulses on the order of 1  ps duration are predicted by \nsimulations in the Sentaurius software package. By tuning  the applied laser power (100 mW ) and \nthe bias voltage  (10 V) , average currents as high as 10  μA are observed in these devices, equating \nto a peak densit y of 107 A/cm2 for the picosecond current . When testing the viability of ultra fast \nSOT switching in our devices, we appl y an in -plane magnetic field to the device under test  to break \nthe inversion symmetry between ±M. Experiments are performed by illuminating the switch for \napproximately a minute, and then reading out the magnetization state through measurements of \nthe AHE  resistance (Fig ure 3a) and the Kerr rotation angle (Figure 3b). Significantly, we observe \nthat the magnetization state s of device are indeed changed when subject to this procedure. Unlike \nthe previous report of ultrafast electrical switching in GdFeCo, however, we observe that the final \nmagnetization state of our devices is dependent upon the polarity of the bias voltage across the \nswitc h as seen in Fig ure 3a,b. That the switching is dependent upon bias polarity indicates that this \nmagnetization switching  primarily comes from picosecond current -induced SOT, rather than  as a \nconsequence of thermal effects.  8 \n  \nFIG. 3. a, THz laser pulse is applied at the gap the photoconductive Auston switch to excite the \npicosecond current under the bias voltage, and then provides a SOT on the Ta/GdFeCo \nheterostructure, where the magnetization is detected by the AHE resistance. When the b ias \nvoltage polarity is switched from +10 V to -10 V, the sign change of the AHE resistance \ndemonstrate s the magnetization switching from up to down states, which indicates SOT \nswitching rather than thermal -induced switching. b, Picosecond -SOT driven magne tization \nswitching detected by the MOKE . \nWhile deterministic switching is observed in our ultrafast measurements, the measured Hall \nand Kerr signals provided in Fig ure 3a,b do not fully return to their initial state upon magnetization \nreversal, and appear to drift slightly over many switching cycles. The failure to recover the Hall \n9 \n resistivity or Kerr rotation observed at full magnetization is likely an indicator of only par tial SOT \nswitching in these devices, resulting in a multi -domain final state. That the Hall resistance and \nKerr rotation of the final state drift following several switching cycles stems from an asymmetry \nin the current generated in our Auston switch upon the reversal of the external bias.  \nWe have show n that the picosecond SOT can efficiently switch the ferrimagnetic GdFeCo, \nwhile another important question is how fast of the magnetization switching speed? In order to \nanswer this question, we perform the TR-MOKE  measurement26-28 to pick up the spin dynamics \nof the ferrimagnetic GdFeCo.  Figure 4a shows the typical TR -MOKE curve of the Ta/GdFeCo  \nheterostructure  under H = 15 kOe. To extract precession frequency f, the dynamic Kerr signals are \nfitted by using the following expression29: \n \n0 1 0 2 exp( / ) sin(2 )exp( / )= + + + −kc c t t c ft t      (1) \nHere c0 is the background signal. The second term is an exponential decaying signal representing \nthe slow recovery process, where c1 is the amplitude and t0 is the characteristic relaxation time. c2, \nf, φ, and τ in the third term represent the  magnetization preces sion parameters of amplitude, \nfrequency, initial phase, and decay time, respectively. The fitted frequencies under different H are \nshown in  Figure 4b , which is consistent with our expectations.  From the fitting, we can get as large \nas 52.4 GHz magnetic resonance frequency  of GdFeCo , which corresponds to a faster  than 20 ps \ntimescale of spin dynamics , and therefore, indicating the tens of picosecond speed  of SOT \nswitching . 10 \n  \nFIG. 4. a, Normalized dynamic TR-MOKE signals for  the Ta/GdFeCo heterostructure . b, \nMagnetic resonance frequency and damping factor as a function of the magnetic field.  \nIn summary, by combing the photoconductive Auston switch with the Ta/GdFeCo  \nheterostructure , a picosecond writing current pulse is applied to provide the SOT, where both the \nAHE and MOKE measurements demonstrate the robust picosecond SOT-induced magnetization \nswitching . The TR-MOKE  measurement demonstrate s faster  than 20 ps  spin dynamics, indicating \ntens of picosecond SOT switching speed of ferrimagnetic GdFeCo.  These results demonstrate the \npicosecond SOT switching of ferrimagnets, which can increase the speed of spintronic devices \nfrom current ns to ps  ranges . \nExperimental Section  \nPicosecond electrical pulses  are generated using a  photoconductive  Auston switch  based on \nthe low-temperature -grown gallium arsenide  (LT-GaAs) . LT-GaAs has a high mobility  and a  high \nintrinsic resistivity, and the  photogenerated carriers exhibit a short lifetime ( 300 fs), which leads \nto the short photocurrent response time (1 ps) . A mode locked Ti:Sapphire laser operating at 80 \nMHz is used to excite the Auston switch. In transport experiments, 800  nm pulses are focused on \nthe gap of Auston switch using a microscope objective. After illuminating the Auston switch for \napproximately one minute, the beam is blocked, and the AHE  resistance is read out using a four-\n11 \n contact  measu rement. All -optical experiments are conducted using the same protocol of minute -\nlong illumination of the Auston switch, followed by a measurement of the Kerr rotation  at the \nmagnetic strip . In these experiments, a n 820 nm-wavelength laser beam  is used to excite \nphotoconductive Auston switch, while the magnetization is extracted from the Kerr signal of a \nreflected 410  nm-wavelength laser  beam.  In both instances, the pulse duration is between 150 -\n200 fs. \nThe dynamic  TR-MOKE measurements are achieved by using a pulsed Ti:sapphire laser with \na central wavelength of 800 nm, a pulse duration of 150 fs, and a repetition rate of 1000  Hz. An \nintense pump pulse beam with a fluence of approximately 1.0 mJ/cm2 is used to excit e the dynamic \nmagnetization behaviors, and the transient MOKE signal is detected by a weak probe pulse of \napproximately 0.0 5 mJ/cm2, which is time delayed with respect to the pump beam. The pump and \nprobe laser beams are at almost perpendicular incidence, with spot diameters of about 1.0 and 0.2 \nmm, respectively. During the TR -MOKE measurement, an external field  H is applied in the x-z \nplane , with a polar angle of θH=71 to drive the magnetization orientation away from the \nperpendicular easy axis.  \nAcknowledgements  \nThis work is supported by the NSF Award No. 1611570, the  Nanosystems  Engineering \nResearch Center for Translational Applications  of Nanoscale Multiferroic Systems (TANMS), and \nthe Spins and Heat in  Nanoscale Electronic Systems (SHINES), an Energy Frontier Research  \nCenter funded by the US Department of Energy (DOE). The authors are  also grateful for the \nsupport from the Function Accelerated nanomaterial  Engineering (FAME) Center, and a \nSemiconductor Research Corporation  (SRC) program sponsored by Microelectronics Advanced \nResearch  Corporation (MARCO) and Defense Advanced Research Projects Agency  (DARPA).  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Laser -Induced Magnetization Dynamics in \nInterlayer -Coupled [Ni/Co] 4/Ru/[Co/Ni] 3 Perpendicular Magnetic Films for Information Storage. \nACS Applied Nano Materials  2, 5140 -5148 , doi:10.1021/acsanm.9b01028 (2019).  \n "    },    {        "title": "1401.1539v1.Complex_magnetic_behavior_of_the_sawtooth_Fe_chains_in_Rb___2__Fe___2__O_AsO___4______2__.pdf",        "content": "Complex magnetic behavior of the sawtooth Fe chains in Rb 2Fe2O(AsO 4)2\nV. Ovidiu Garlea,1,\u0003Liurukara D. Sanjeewa,2Michael A. McGuire,3\nPramod Kumar,4Dino Sulejmanovic,2Jian He,5and Shiou-Jyh Hwu2\n1Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\n2Department of Chemistry, Clemson University, Clemson, South Carolina 29634-0973, USA\n3Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n4Indian Institute of Information Technology, Allahabad U.P. 211012, India\n5Department of Physics and Astronomy, Clemson University, Clemson, South Carolina 29634-0978,USA\n(Dated: March 2, 2022)\nResults of magnetic \feld and temperature dependent neutron di\u000braction and magnetization mea-\nsurements on oxy-arsenate Rb 2Fe2O(AsO 4)2are reported. The crystal structure of this compound\ncontains pseudo-one-dimensional [Fe 2O6]1sawtooth-like chains, formed by corner sharing isosce-\nles triangles of Fe3+ions occupying two nonequivalent crystallographic sites. The chains extend\nin\fnitely along the crystallographic b-axis and are structurally con\fned from one another via dia-\nmagnetic (AsO 4)3\u0000units along the a-axis, and Rb+cations along the c-axis direction. Neutron\ndi\u000braction measurements indicate the onset of a long range antiferromagnetic order below approxi-\nmately 25 K. The magnetic structure consists of ferrimagnetic chains which are antiferromagnetically\ncoupled with each other. Within each chain, one of the two Fe sites carries a moment which lies\nalong the b-axis, while the second site bears a canted moment in the opposite direction. Exter-\nnally applied magnetic \feld induces a transition to a ferrimagnetic state, in which the coupling\nbetween the sawtooth chains becomes ferromagnetic. Magnetization measurements performed on\noptically-aligned single crystals reveal evidence for an uncompensated magnetization at low mag-\nnetic \felds that could emerge from to a phase-segregated state with ferrimagnetic inclusions or from\nantiferromagnetic domain walls. The observed magnetic states and the competition between them\nis expected to arise from strongly frustrated interactions within the sawtooth chains and relatively\nweak coupling between them.\nPACS numbers: 75.50.Gg, 75.60.Jk, 75.25.-j, 71.27.+a, 61.05.F-\nI. INTRODUCTION\nLow-dimensional magnetic materials have drawn con-\ntinued attention in condensed matter physics, owing to\ntheir distinct electronic and magnetic properties. In par-\nticular, the oxyanion systems based on transition metal\n(M) oxides sublattices that are magnetically isolated\nby closed shell nonmagnetic oxyanions (SiO4\u0000\n4, PO3\u0000\n4,\nAsO3\u0000\n4) show great potential for exploring and charac-\nterizing new emergent phenomena.1This class of com-\npounds has been shown to exhibit a variety of magnetic\nground states ranging from ferromagnetism, to ferrimag-\nnetism and antiferromagnetism.2On the other hand, due\nto their impressive structural diversity in terms of archi-\ntecture these materials have shown potential applications\nin the \felds of energy storage, chemical sensors, catalysis\nand biomedical research.3{6\nRecent e\u000borts have been directed towards the synthe-\nsis of new members of the family of oxy-arsenates with\ngeneral formula A2M2O(AsO 4)2, where A= K, Rb, and\nMis a transition metal. These compounds exhibit in-\ntriguing one-dimensional triangular-shaped chains made\nof edge-sharing MO 6octahedra (see Fig. 1). Antifer-\nromagnetic sawtooth chain model (or \u0001-chain), as that\nrealized in these systems, is known to be highly frus-\ntrated and has drawn considerable attention from a the-\noretical standpoint.7{9For example, the ground state of\ntheS=1/2 sawtooth chain is two-fold degenerate witheither left pair or right pair of spin at each triangle form-\ning spin-singlet states. The lowest excitation in a chain\nwith periodic boundary conditions is given by a kink-\nantikink pair which has a dispersionless gap \u0001E '0.234J\nwhereJis the coupling between pairs of spins.7,8Exper-\nimental realization of magnetic lattices corresponding to\nthe sawtooth chain is only found in a very small num-\nber of compounds, including the delafossite YCuO 2:5,10\neuchroite Cu 2(AsO 4)(OH)\u00013H2O11and ZnL2S4olivines,\nwithL=Er, Tm, Yb.12\nWe report here on the magnetic behaviour of oxy-\narsenite Rb 2Fe2O(AsO 4)2, containing sawtooth chains of\nmagnetic Fe3+ions in high-spin state S=5/2. Although\nthis compound was \frst reported \fve years ago,13a thor-\nough investigation of its magnetic properties has not been\nconducted so far. The development of well-controlled\nmethods for the growth of high quality single crystals\nhas allowed us for detailed orientation-dependent mag-\nnetic measurements. Magnetization measurements have\nbeen complemented with x-ray and neutron di\u000braction\nmeasurements. Temperature and magnetic \feld depen-\ndent neutron di\u000braction measurements demonstrate con-\nclusively the development of a non-collinear antiferro-\nmagnetic ground state below approximately 25 K, which\nunder relatively low magnetic \felds, transforms to a ferri-\nmagnetic state. These transitions are also observed in dc\nmagentization measurements, which reveal strong mag-\nnetic anisotropy and complex \feld and temperature de-\npendent behaviors associated with the multiple magneticarXiv:1401.1539v1  [cond-mat.str-el]  7 Jan 20142\nFIG. 1: (Color online) (top) Polyhedral view of the extended\nstructure of Rb 2Fe2O(AsO 4)2showing the structural con-\n\fnement of the chains. (bottom) Partial structure of the\n[Fe4O14]1sawtooth chain de\fned by unequal base-base ( Jbb)\nand base-vertex ( Jbv) exchange interactions.\nstates.\nII. EXPERIMENTAL DETAILS\nSingle crystals of Rb 2Fe2O(AsO 4)2were grown using\n\rux methods in the RbCl/RbI (50:50 wt%) molten-salt\nmedia. The reactants KO 2(Alfa Aesar, 96.5%), Fe 2O3\n(Alfa Aesar, 99.945%) As 2O5(Alfa Aesar, 99.9%) were\nmixed and ground together in a nitrogen-purged dry box.\nThe reaction mixture was sealed in an evacuated fused-\nsilica ampoule and then heated to 650\u000eC for one day,\nfollowed by another heating to 800\u000eC at 1\u000eC/min for\ntwo days, slowly cooled to 300\u000eC at 0.05\u000eC/min, and\nthen furnace-cooled to room temperature. Brown colum-\nnar crystals (up to about 2.6 mm x 0.6 mm x 0.4 mm)\nwere recovered by washing the product with de-ionized\nwater using vacuum \fltration method. Single crystals\nwere physically examined and selected under an optical\nmicroscope equipped with a polarizing light attachment.\nThe crystals were then mounted on a glass \fber and sin-\ngle crystal x-ray di\u000braction data was collected using a\nRigaku Mercury CCD detector on an AFC-8S di\u000brac-\ntometer equipped with graphite monochromated Mo K \u000b\nradiation (\u0015= 0.7107 \u0017A). The structure was solved by di-rect methods using the SHELX-9714program and re\fned\non F2by least-squares, full-matrix technique.\nPolycrystalline samples were prepared by the solid\nstate reaction of stoichiometric amounts of Rb 2CO3\n(Alfa Aesar, 99.8%), Fe 2O3(Alfa Aesar, 99.945%) and\n(NH 4)H2AsO 4(Alfa Aesar, 99.9%). The mixture was\nloaded into an alumina crucible, heated to 800\u000eC at a\nrate of 2\u000eC/min, and held at that temperature for 2 days\nbefore being furnace cooled to room temperature. All\npolycrystalline samples were characterized using Rigaku\nUltima IV multipurpose X-ray di\u000braction system.\nTemperature and \feld-dependent magnetic measure-\nments were carried out with a Quantum Design Magnetic\nProperty Measurement System (MPMS). The measure-\nments were taken on collections optically co-aligned sin-\ngle crystals at temperatures from 2 K to 300 K in the\napplied \felds up to 50 kOe. Additional magnetization\nmeasurements were performed on polycrystalline sample\nusing a physical property measurement system (PPMS)\nat \felds up to 120 kOe.\nNeutron di\u000braction measurements were carried out us-\ning the HB2A high-resolution powder di\u000bractometer at\nthe High Flux Isotope Reactor (HFIR). Di\u000braction pat-\nterns were collected using the 2.408 \u0017A wavelength pro-\nduced by the (113) re\rection of a vertically focusing ger-\nmanium wafer-stack monochromator. The beam collima-\ntion was 120{400{60. More details about the HB2-A in-\nstrument and data collection strategies can be found in\nRef. 15. For the measurements, polycrystalline samples\nwith a total mass of approximately 5 grams were com-\npacted in pellets to prevent grain reorientation by the\napplied \feld. The pellets were loaded into a cylindrical\nvanadium can, and placed in a vertical-\feld cryomagnet.\nData were collected at various temperatures from 40 K\nand down to 4 K, and magnetic \felds up to 40 kOe.\nRietveld re\fnements were carried out with the FullProf\nSuite program.16\nIII. RESULTS AND DISCUSSION\nA. Crystal structure\nThe single crystal X-ray and neutron powder di\u000brac-\ntion data were well \ft by using an orthorhombic struc-\nture of space group Pnma , similar to that proposed\nby Chang et al. .13The re\fned structural parameters of\nRb2Fe2O(AsO 4)2, such as atomic coordinates and dis-\nplacement parameters are listed in Table I, while se-\nlected bond lengths and angles are given in Table II. Ri-\netveld analysis indicate that the powder samples contain\na Fe 2O3impurity phase in an amount of less than 3%\nweight.\nThe structure of Rb 2Fe2O(AsO 4)2consists of iron-\narsenate layers parallel to the abplane, separated by\nRb+cations. In each layer, there are two crystallo-\ngraphically distinct Fe3+sites (Fe1 and Fe2) which are\nbridged together to form triangle-based chains, as shown3\nTABLE I: Re\fned structural parameters of Rb 2Fe2O(AsO 4)2\nfrom single-crystal x-ray data collected at room temperature.\nAtom (Wyck.) x y z Ueq\nRb1 (4 c) 0.8888(1) 1/4 0.8628(1) 0.023(1)\nRb2 (4 c) 0.0379(1) 1/4 0.2679(1) 0.028(1)\nFe1 (4 c) 0.2844(1) 1/4 0.4347(1) 0.009(1)\nFe2 (4 b) 0 0 1/2 0.009(1)\nAs1 (4 c) 0.8204(1) 1/4 0.6426(2) 0.010(1)\nAs2 (4 c) 0.6758(1) 1/4 0.4440(2) 0.008(1)\nO1 (4 c) 1.0580(1) 1/4 0.4375(3) 0.009(1)\nO2 (4 c) 0.9610(1) 1/4 0.5760(3) 0.010(1)\nO3 (8 d) 0.7666(1) 0.0108(1) 0.4785(2) 0.015(1)\nO4 (4 c) 0.4923(1) 1/4 0.4744(4) 0.014(1)\nO5 (8 d) 0.7043(1) 0.0127(1) 0.6338(2) 0.018(1)\nO6 (4 c) 0.6843(1) 1/4 0.3556(3) 0.018(1)\nO7 (4 c) 0.9130(2) 1/4 0.7197(3) 0.025(1)\nPnma ,a= 8.5331(2) \u0017A,b= 5.7892(2) \u0017A,c= 18.611(4) \u0017A\nR1= 0.0347, wR 2= 0.0936, GOF = 1.038\nTABLE II: Selected bond distances and angles in\nRb2Fe2O(AsO 4)2, as determined from x-ray analysis.\nFe1 - O1 (x 1) 1.932(6) \u0017A Fe2 - O1 (x 2) 1.922(4) \u0017A\nFe1 - O3 (x 2) 2.254(4) \u0017A Fe2 - O2 (x 2) 2.051(4) \u0017A\nFe1 - O4 (x 1) 1.923(6) \u0017A Fe2 - O3 (x 2) 2.032(4) \u0017A\nFe1 - O5 (x 2) 1.986(4) \u0017A\nd(Fe1 - Fe2) = 3.076(1) \u0017A,Jbvvia O1, O3\nFe1 - O3 - Fe2 = 91.5(1)\u000eFe1 - O1 - Fe2 =105.9(1)\u000e\nd(Fe2 - Fe2) = 2.895(1) \u0017A,Jbbvia O1, O2\nFe2 - O1 - Fe2 = 97.7(1)\u000eFe2 - O2 - Fe2 = 89.7(1)\u000e\nin Fig. 1. These [Fe 4O14]1chains extend along the crys-\ntallographic b-axis, and are structurally con\fned from\none another via diamagnetic As5+cations in AsO 4tetra-\nhedral units along the crystallographic a-axis (Fig. 1).\nDue to the non-magnetic nature of As5+cation, the mag-\nnetic interactions between adjacent [Fe 4O14]1\u0001-chains\nvia Fe-O-As-O-Fe connection are expected to be subdom-\ninant compared to the interactions within the chains.\nA comparison of the re\fned bonds and angles in\nRb2Fe2O(AsO 4)2indicates di\u000berent interactions occur-\nring between spins on the base ( Jbbcoupling) and be-\ntween the base-vertex ( Jbv) of the triangles. The Fe1-Fe2\nand the Fe2-Fe2 distances are 3.076(1) and 2.895(1) re-\nspectively. The bond angles Fe2{O1{Fe2 and Fe2{O2{\nFe2 de\fning the super-exchange paths along the base\nof triangles ( Jbb) are 97.7(3)\u000eand 89.7(7)\u000e, respectively,\nwhile the base-vertex angles ( Jbv) Fe1-O1-Fe2 and Fe1-\nO3-Fe2 are 105.9(1)\u000eand 91.5(1)\u000e.\nFIG. 2: Evolution of the magnetic susceptibility (M/H) as a\nfunction of temperature, for H = 100 Oe applied parallel and\nperpendicular to the b-axis. The inserted picture displays the\nsingle crystals assembly used in the measurements. The inset\nshows the inverse susceptibility versus temperature above the\ntransition temperature, and a linear (Curie-Weiss) \ft.\nB. Macroscopic magnetic measurements\nMagnetization measurements were performed on single\ncrystals with the b-axis (chain direction) aligned parallel\nand perpendicular to the magnetic \feld. Due to the small\nsize of the crystals, several specimens were optically co-\nmounted on a Kapton tape as displayed in Fig. 2. Two\nsamples were prepared with total masses of 4.0 and 8.4\nmg. The measured magnetization was con\frmed to be\nconsistent between the two samples. The temperature\ndependence of magnetic susceptibility ( \u001f=M=H ) mea-\nsured with an applied \feld of 100 Oe, is shown in Fig. 2.\nThe sharp rise in magnetization data reveals the onset\nof long range magnetic order below approximately 25 K.\nThere is strong anisotropy between the two crystal ori-\nentations near the transition temperature. The inverse\nsusceptibilty (H/M) for temperature well above this tem-\nperature is shown in the inset, along with a \ft using the\nCurie Weiss model, M/H = C/(T - \u0002 CW). The data are\nnot strictly linear over this wide temperature range. A\nbetter \ft can be obtained by including a temperature in-\ndependent term; however, such a \ft returns unreasonable\nvalues for the \ftting parameters. The observed devia-\ntion from Curie Weiss behavior may indicate some short\nranged or low dimensional magnetic ordering above the\nlong-ranged ordering transition at 25 K. Using the \ft\nshown in Fig. 2, an e\u000bective moment of 6.7(1) \u0016Band a\nWeiss temperature of -510 K are determined. Restrict-\ning the \ftted range to the more linear region between\n250 and 350 K gives an e\u000bective moment of 6.3 \u0016Band\na Weiss temperature of -426 K. These e\u000bective moments\nare somewhat larger than the spin only value for high4\nFIG. 3: (Color online) Magnetization curves for co-align\nsingle crystals of Rb 2Fe2O(AsO 4)2measured on warming at\nthe indicated \felds after \feld-cooling from 50 K. Results are\nshown for H parallel and perpendicular to the b-axis along\nwhich the sawtooth chain of Fe are oriented.\nFIG. 4: Zero-\feld-cooled magnetization curves measured in\n10 Oe magnetic \feld (H kb) upon warming from 2 to 50 K,\nafter having reduced the \feld to zero at 50 K from (a) 10 kOe,\nwhich leaves a small negative residual \feld, and (b) -10 kOe\nwhich leaves a small positive residual \feld.\nspind5, 5.9\u0016B. The large and negative Weiss tempera-\ntures indicates strong antiferromagnetic interactions.\nAdditional magnetization measurements were per-\nformed near the ordering temperature. Data were col-\nlected upon warming from 2 to 50 K in zero-\feld-cooled\n(ZFC) and \feld-cooled (FC) modes. Figure 3 displays\nthe FC results for the four selected \felds of 10 Oe, 100\nOe, 1 kOe and 10 kOe. The temperature dependence\nchanges signi\fcantly as the applied \feld is varied, and\nthe data show strong anisotropy between the two mea-\nsurement directions at all \felds. For H along the direc-tion of the sawtooth chains (the b-axis), a sharp cusp,\nsuggesting a canted antiferromagnetic order, is seen at\nthe magnetic ordering transition for H \u00141 kOe, while\nat 10 kOe the susceptibility increases and then saturates\nupon cooling, similar to the behavior expected at a ferro-\nmagnetic transition. For H aligned perpendicular to the\nb-axis, a smaller cusp is seen at all of the \felds shown.\nThe qualitative di\u000berence between the behavior in the\ntwo directions at H = 10 kOe is due to a \feld induced\nmagnetic transition which occurs at a di\u000berent \feld in\nthe di\u000berent orientations and is discussed in more detail\nbelow.\nNegative magnetic susceptibilities were observed in\nsome ZFC measurements at low \felds. An example is\nshown in Figure 4(a), where the magnetization is neg-\native at low temperatures, and switches sign abruptly\nupon warming as the phase transition is approached.\nThis was observed at \felds up to 100 Oe and in both\norientations of the \feld. Temperature induced magne-\ntization reversal phenomena is well known to occur in\nferrimagnetic systems with strong magnetic anisotropy\ndisplaying a partial cancellation of antiferromagnetically\ncoupled magnetic sublattices with di\u000berent magnitudes\nof magnetic moments and/or di\u000berent temperature de-\npendence of magnetization.17,18This scenario is certainly\nplausible in our system where Fe3+occupies two non-\nequivalent sites and are exposed to di\u000berent molecular\n\felds. However, a negative ZFC magnetization curve has\nalso been attributed in some cases to the existence of a\nsmall trapped \feld in the superconducting magnet.19,20\nSuch an experimental artefact is likely to occur in some\nferromagnetic materials with strong anisotropy and a\nsigni\fcant coercive \feld that can drastically a\u000bect the\nZFC process. To investigate this possibility we per-\nformed experiments in which we reducing the \feld to\nzero from both 10 kOe and -10 kOe using the MPMS\n\"no-overshoot\" mode at 50 K, then cooling in \"zero \feld\"\nto 2 K, applied 10 Oe, and then measured upon warm-\ning. As discussed in Ref.19, the sign of the trapped\n\feld is opposite in sign when reducing the \feld to zero\nfrom a positive or negative \feld, and hence, a starting\n\feld of 10 kOe is expected to generate a small nega-\ntive trapped \feld, while -10 kOe will create a small pos-\nitive trapped \feld. Our measurements on a Pd reference\nsample using similar conditions con\frm this and indicate\nthe magnitude of the \feld is several Oe. As apparent\nin Fig. 4, the ZFC magnetization curves are indeed fol-\nlowing the direction of the trapped \feld in the super-\nconducting magnet. This prompts us to conclude that\nthe observed negative magnetization under ZFC condi-\ntion is not a ferrimagnetic e\u000bect, but rather a feature\ninduced by the presence of pinned uncompensated spins\noriginated from inhomogeneities,20or from the antifer-\nromagnetic domain walls,21,22or slight canting in the\nantiferromagnetic structure, although no canting could\nbe resolved in the neutron di\u000braction results presented\nbelow. We note that several ZFC measurements using\na \feld of 10 Oe were also conducted after resetting or5\nFIG. 5: (Color online) Magnetic \feld dependence of magne-\ntization for oriented single crystals of Rb 2Fe2O(AsO 4)2with\nHkbat four di\u000berent temperatures. Shown in the insert is\nthe isothermal magnetization curve at 2 K for H ?b.\nFIG. 6: Isothermal magnetization data measured on a com-\npacted powder sample at T = 4 K. The upper inset shows\na blow up of the region near the metamagnetic transition (\n\u00194 kOe). The lower insert displays the magnetization data\nmeasured upon reducing the \feld from 120 kOe at 5 K.\nquenching the magnet at 50 K. Some of these measure-\nments resulted in behavior like that shown in Fig. 4(a),\nand others like Fig. 4(b). This indicates that the residual\n\feld, which is expected to be very small ( <1 Oe) after\nresetting the magnet had unpredictable polarity and that\nthe low temperature state that is adopted by the mate-\nrial upon cooling through the transition temperature is\nextremely sensitive to even very small magnetic \felds.\nThe isothermal magnetization curves at 2 K, 10 K and50 K in magnetic \felds up to 15 kOe applied along baxis\nare shown in Fig. 5. The presence of hysteresis in the\nmagnetization data demonstrates the presence of a fer-\nroic component in the high-\feld magnetic phase. Apart\nfrom a hysteresis separating the ascending and descend-\ning branches, there is a readily noticeable stepwise mag-\nnetic behavior. The metamagnetic transition occurs near\n3 kOe at T = 2 K and shifts to 6 kOe at T = 10 K.\nWhen the \feld is applied perpendicular to the baxis the\nmetamagnetic phase transition occurs at a much higher\n\feld of 30 kOe, at T = 2 K (as shown in the inset of\nFig. 5). As expected, the magnetization measured on\ncompacted powder sample shows a more gradual \feld\ntransition, arising at approximately 4 kOe at a tempera-\nture of 4 K (see Fig. 6). It is also worth noting that the\nmagnetization curves does not reach the full saturation\neven for the highest applied external \felds of 120 kOe.\nThe largest absolute value of the magnetization, of about\n0.3\u0016B/Fe, is smaller by over an order of magnitude than\nthe predicted value corresponding to a parallel alignment\nof the Fe3+magnetic moments ( gS= 5\u0016B).\nC. Neutron di\u000braction\n1. Measurements in zero magnetic \feld\nNeutron di\u000braction patterns collected in the paramag-\nnetic state at 40 K are well described by the structural\nmodel described in the previous section. Upon cooling\nbelow 25 K, magnetic Bragg peaks (e.g. (100), (102))\nstart to appear at low angles (see Fig. 7) indicating a\ntransition to an antiferromagnetic long-rage order. The\nevolution with temperature of (100) magnetic peak inten-\nsity is shown in Fig. 8(upper panel). Near the transition\ntemperature the order parameter follows a power law be-\nhaviour.\nRepresentational analysis was conducted to determine\nof the symmetry allowed magnetic structures that can\nresult from a second-order magnetic phase transition\ngiven the crystal structure and the propagation vec-\ntork=(0, 0, 0) of the magnetic ordering. These cal-\nculations were carried out using the program SARA h-\nRepresentational Analysis.23There are eight possible\nirreducible representations (IRs) associated with the\nPnma space group and k=(0, 0, 0). The decompo-\nsition of the magnetic representation for the Fe1 site\n(:715; :25; :565) is \u0000 Mag = 1\u00001\n1+ 2\u00001\n2+ 2\u00001\n3+ 1\u00001\n4+\n1\u00001\n5+ 2\u00001\n6+ 2\u00001\n7+ 1\u00001\n8, while for the Fe2 site (0 ;0; :5) is\n\u0000Mag= 3\u00001\n1+0\u00001\n2+3\u00001\n3+0\u00001\n4+3\u00001\n5+0\u00001\n6+3\u00001\n7+0\u00001\n8. The\nlabeling of the IRs follows the scheme used by Kovalev.24\nTable III presents only two of representations (\u00001, \u00005)\nthat are relevant for our discussion here.\nThe 4 K di\u000braction data was \ft by a magnetic struc-\nture model based on the representation \u0000 1, de\fned in\nTable III. \u0000 1representation is equivalent to the Shub-\nnikov magnetic space group Pnma . Within this model,\nthe Fe1 moments are constrained to lie along the b-axis6\nFIG. 7: (Color online) Upper panel: Rietveld plot of neuron\npowder di\u000braction data collected at 4 K in zero magnetic \feld.\nThe inset displays a comparison of the low- Qdata, near the\n(100) magnetic peak, measured at 4 K and 30 K. Lower panel:\nRietveld plot of neuron data measured at 4 K in magnetic \feld\nof 40 kOe.\ndirection and are ferromagnetically coupled to each other\ninside the chain. In contrast, the Fe2 moments have\ncomponents in all three coordinate directions, and form\nan undulating pattern with the relative angles between\nneighboring spins of approximately 60\u000e. The spin canting\noccurs in such a way that they remain mostly con\fned\nto the plane of the sawtooth chain. The projection along\nthe chain axis of the Fe2 moment is oriented in oppo-\nsite direction to the Fe1 moment, thus producing a non-\nzero net moment. The polarity of the chains alternates\nalong thec-axis direction, giving an overall antiferromag-\nnetic arrangement. A stereographic view of the magnetic\nstructure is illustrated in Fig. 9(a).\nThe re\fned magnitudes of the ordered moments for\nFe1 and Fe2 sites are 3.64(6) and 3.19(8) \u0016B, respec-\ntively. The projections of Fe2 magnetic moment on the\nthree axes of coordinates are: \u0016a= 1.19(8)\u0016B,\u0016b=\nFIG. 8: (Color online) Upper panel: Temperature depen-\ndence of the (100) and (101) peaks intensities measured in\nzero magnetic \feld and 40 kOe, respectively. The lines repre-\nsent a guide to the eye indicating power law behavior. Lower\npanel: Variation with magnetic \feld of the (100) and (101)\nmagnetic peaks at 4 K. The dashed line is a guide to the\neye. The vanishing of (100) re\rection at the expense of (101)\nmagnetic is caused by an induced transition from antiferro-\nmagnetic to ferrimagnetic state.\n2.88(4)\u0016Band\u0016c= 0.68 (9)\u0016B. The re\fned values are\nsigni\fcantly lower than those expected for the Fe3+ions\nwith the e\u000bective spin S=5/2. Considering the high value\nof the frustration parameter f=j\u0002CWj=TNof approx-\nimately 18, one can infer that the reduced static mag-\nnetic moment is due to the presence of spin frustration.\nThe frustration is reasonably expected in this system due\nto competing superexchange interactions within the tri-\nangular chains. Furthermore, the non-collinear spin ar-\nrangement is reminiscent of the predicted non-collinear\nspin con\fgurations in triangular-lattice Heisenberg anti-\nferromagnet where the frustration of the three nearest-\nneighbor spins on a triangular plaquette is resolved by a\n120\u000erotation of neighboring spins.25\nPerhaps the most important aspect that needs to be\nemphasized here is the fully compensated character of\nthe magnetic structure, which makes the magnetiza-\ntion reversal process observed in the ZFC magnetiza-\ntion data di\u000ecult to comprehend. Ferrimagnetic-like\nresponse has previously been observed only in a hand-\nful of antiferromagnetic compounds where a weak fer-\nromagnetic moment is due to the tilting of sublattice\nmagnetizations.26{29One of the mechanisms that has\nbeen proposed to explain the magnetization reversal in7\nTABLE III: Basis vectors (BV) of the IRs \u0000 1and \u0000 5, of the\nspace group Pnma with k= (0, 0, 0). The equivalent Fe1\natoms of the nonprimitive basis are de\fned according to 1:\n(:715; :25; :565), 2: ( :215; :25; :934), 3: ( :284; :75; :434), 4:\n(:784; :75; :065). The Fe2 atoms are de\fned as follows: 1:\n(0;0; :5), 2: ( :5; :5;0), 3: (0 ; :5; :5), 4: ( :5;0;0).\nIR BV Atom Fe1 Fe2\nmkamkbmkcmkamkbmkc\n\u00001 1 1 0 1 0 0 1 0\n2 0 -1 0 0 -1 0\n3 0 1 0 0 1 0\n4 0 -1 0 0 -1 0\n 2 1 1 0 0\n2 1 0 0\n3 -1 0 0\n4 -1 0 0\n 3 1 0 0 1\n2 0 0 -1\n3 0 0 -1\n4 0 0 1\n\u00005 4 1 0 1 0 0 1 0\n2 0 1 0 0 1 0\n3 0 1 0 0 1 0\n4 0 1 0 0 1 0\n 5 1 1 0 0\n2 -1 0 0\n3 -1 0 0\n4 1 0 0\n 6 1 0 0 1\n2 0 0 1\n3 0 0 -1\n4 0 0 -1\nantiferromagnets relies on the competition between sin-\ngle ion anisotropy and the DzyaloshinskyMoriya (D-M)\ninteraction. For our system, the antisymmetric DM ex-\nchange coupling can exist between the magnetically or-\ndered Fe2 ions located at 4 bsite ofPnma , and the mag-\nnetic spin corresponding to that site has indeed been de-\ntermined to be canted. However, because the chains are\nantiferromagnetically coupled, any ferromagnetic compo-\nnent due to canting is compensated by the nearest neigh-\nbor chain. These results come to con\frm our conjecture\nin the previous section that the negative magnetization\nis not an inherent ferrimagnetic feature, but likely an ef-\nfect of an uncompensated magnetization originating from\nantiferromagnetic domain walls or from other magnetic\ninhomogeneities present in the sample.\n(a)\n(b)\nFIG. 9: (Color online) (a) Schematic view of the magnetic\nstructure of Rb 2Fe2O(AsO 4)2at 4 K and zero magnetic \feld.\nThe Fe1 magnetic moments are collinear with the b-axis, while\nthe Fe2 moments are forming undulating patterns, running in\nopposite direction with respect to the Fe1 moments. The fer-\nrimagnetic chains are coupled antiferromagnetically to neigh-\nbouring chains in the cdirection (b) Magnetic structure at 4\nK and applied magnetic \fled of 40 kOe. In contrast to the\nzero-\feld magnetic order, the polarity of the ferrimagnetic\nchains remains the same throughout the lattice yielding an\noverall ferrimagnetic ordering.\n2. Measurements in external magnetic \felds\nTo understand the e\u000bect of magnetic \feld on the mag-\nnetic structure of Rb 2Fe2O(AsO 4)2, polycrystalline sam-\nple in the form of pressed pellets has been measured un-\nder di\u000berent magnetic \felds up to 40 kOe. The di\u000braction\ndata measured at 4 K reveals a progressive decrease in\nthe amplitude of (100) magnetic peak with increasing the\napplied magnetic \feld, and at the same time, an increase\nin intensity of the (101) peak. The change in peaks in-8\ntensities only manifests at low momentum transfer ( Q),\nin accordance with the Fe3+magnetic form factor. The\nfact that the high- Qre\rections are not a\u000bected by the\nmagnetic \feld gives assurance that the sample grains do\nnot re-align in the direction of the \feld. As displayed\nin Fig. 8(lower panel), the (100) peak intensity starts\nto decrease just above 3 kOe, and it nearly vanished at\n40 kOe. The (101) peak follows a very similar trend to\nthat seen for the isothermal magnetization curve mea-\nsured for the powder sample in the magnetically ordered\nstate at 4 K, in Fig. 6. The intensity of the (101) peak\nhas also been monitored as a function of temperature in a\nconstant magnetic \feld of 40 kOe. The order parameter\nversus temperature shown in Fig. 8(upper panel) follows\na power-law dependence, similar as that observed for the\n(100) peak in zero magnetic \feld.\nRietveld re\fnement has been performed using the\ndi\u000braction data collected at 40 kOe and 4 K, and the\n\ft result is presented in Fig. 7. The magnetic scattering\nhas been well reproduced by a ferrimagnetic model based\non the irreducible representation \u0000 5given in Table III.\nThis representation is equivalent to the Shubnikov mag-\nnetic space group Pn0ma0. In this new magnetic state\nall chains exhibit the same polarity, with all symmetry\nequivalent Fe ions having the b-axis component of mag-\nnetic moment aligned in the same direction. On the other\nhand, Fe1 and Fe2 sub-lattices maintain opposite mag-\nnetizations while not compensating each other. Another\ne\u000bect of the magnetic \feld is the change in the canting di-\nrection of Fe2 spins with respect to the local plane of the\nsawtooth-chain. The spins rotate around the b-axis from\nan in-plane canting to an out-of-plane canting. The rel-\native angle between Fe2 and Fe1 moments remains close\nto 120\u000e. The high-\feld magnetic structure is shown in\nFig. 9(b).\nThe magnitude of the ordered magnetic moments at\n40 kOe is determined to be very close to that observed\nfor the antiferromagnetic state in zero magnetic \feld.\nThe re\fned Fe1 moment is 3.5(1) \u0016B, while the ampli-\ntude of the Fe2 moment is 3.1(1) \u0016Bwith the three axes\ncomponents \u0016a=0.5(1)\u0016B,\u0016b= 2.76(6)\u0016Band\u0016c=\n1.3(1)\u0016B. These values appear to rule out the possibility\nthat the external magnetic \feld may polarize additional\ndisordered Fe spins into the novel ordered state. On the\nother hand, the net magnetization resulted from such am-\nplitudes is about 0.37 \u0016Bper Fe3+, which is reasonably\nconsistent to what has been determined from magnetiza-\ntion measurements (see Figs. 5 and 6). Interestingly, the\nmagnetization data shown in Fig. 5 suggest that at su\u000e-\nciently low temperatures, the \feld induced ferrimagnetic\nstate can be maintained even after reducing the \feld back\nto zero.\nIt appears that the metamagnetic transition corre-\nsponds to the transition from the antiferromagnetic to\nferrimagnetic state where all magnetic couplings between\nadjacent sawtooth chain become exclusively ferromag-\nnetic while the intra-chain exchange interactions remain\nnearly unaltered. The \feld induced ferrimagnetic order-ing provides an interesting insight into the tendency of\nthe system to produce a net ferromagnetic component,\nand opens up the question whether local ferrimagnetic\ncluster may start to form within the antiferromagnetic\nhost even at very low magnetic \felds. The contribu-\ntion to the scattering of such weak uncompensated mag-\nnetic order could be well below the detection limit of our\nneutron scattering measurements. Magnetic clustering\ncould plausibly explain the observed negative magnetiza-\ntion but it implies the existence of inhomogeneities in the\nsample. Note that in our case the inhomogeneities would\nlikely have their origin in oxygen vacancies, which could\nnot be either con\frmed or ruled out by our structural\nre\fnement. As mentioned before, the uncompensated\nspins might also result from antiferromagnetic domain\nwalls, as demonstrated in the case of the HoMnO 3multi-\nferroic system by means of polarized small angle neutron\nscattering experiments.21\nIV. SUMMARY\nThe magnetic properties of the new oxy-arsenate\nRb2Fe2O(AsO 4)2have been studied by means of magne-\ntization and neutron powder di\u000braction measurements.\nThe system structure consists of triangle-based magnetic\nchains made of S= 5/2 Fe3+ions occupying two crys-\ntallographically distinct tetrahedral sites (Fe1 and Fe2).\nMagnetization measurements indicate a complex mag-\nnetic behavior with a temperature induced magnetization\nreversal at low magnetic \felds, and a step-like metamag-\nnetic transition. Neutron di\u000braction experiments per-\nformed in zero magnetic \feld reveal an antiferromag-\nnetic long-range magnetic order below the N\u0013 eel tem-\nperature of about 25 K. The magnetic structure con-\nsists of ferrimagnetic sawtooth chains which are mutually\ncompensating each-other along the c-direction. Within\neach chain, the Fe1 moments are collinear lying along\nthebdirection, while the Fe2 moments are reversely\ncanted by approximately 30\u000eforming a zigzag pattern.\nA reduced static magnetic moments of Fe has been at-\ntributed to a magnetic frustration caused by the trian-\ngle geometry. Neutron data collected under externally\napplied magnetic \feld reveal a transition from an an-\ntiferromagnetic to a ferrimagnetic state. In the ferri-\nmagnetic state the individual chains remain ferrimag-\nnetic with similar spin topology but their coupling be-\ncomes exclusively ferromagnetic. The ferrimagnetic or-\ndering at at magnetic \felds greater than 3 kOe cannot\nbe invoked to explain the negative ZFC magnetization\nobserved in the macroscopic study. Instead, the present\ndata suggests that the negative magnetization could be\nan e\u000bect of a phase-segregated state with ferrimagnetic\ninclusions, or of an uncompensated magnetization arising\nwithin antiferromagnetic domain walls. On the basis of\nour results, the sawtooth-like lattice model displayed by\nthe Rb 2Fe2O(AsO 4)2system deserves further experimen-\ntal and theoretical investigations to fully understand its9\nmagnetic behaviour. While the magnetic phase segrega-\ntion antiferromagnetic domain walls are at present only\nhypothetical scenarios, future polarized neutron scatter-\ning experiments able to provide increased sensitivity to\nweek ferromagnetism may shed light on the origin of the\nnegative magnetization phenomenon.\nAcknowledgments\nWork at the Oak Ridge National Laboratory, was spon-\nsored by the Scienti\fc User Facilities Division (neutrondi\u000braction) and Materials Sciences and Engineering Divi-\nsion (magnetization measurements), O\u000ece of Basic En-\nergy Sciences, US Department of Energy (DOE). The au-\nthors acknowledge the \fnancial support by grants from\nthe National Science Foundation: DMR-0706426, CHE-\n9808165 and CHE-9808044. L.D.S. and D.S. travel to\nORNL was supported by DOE through the EPSCoR\nGrant, DE-FG02-08ER46528.\n\u0003Electronic address: garleao@ornl.gov\n1S-J. Hwu, Chem. Mater. 10, 2846 (1998).\n2W. L. Qeen, S.-J. Hwu, L. Wang, Angew. Chem. Int. 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Kimishima, Y. Chiyanagi, K. Shimizu and T. Mizuno\nJ. Magn. Magn. Mater. 210, 244 (2000)."    },    {        "title": "2103.04105v1.Anomalies_in_the_dynamics_of_ferrimagnets_near_the_angular_momentum_compensation_point.pdf",        "content": "1 \n JMMM 509 (2020) 166876  \nCorrected version  \nAnomalies in the dynamics of ferrimagnets near the angular momentum \ncompensation point  \n \nA.K. Zvezdin1,2, Z.V. Gareeva  3*, K.A. Zvezdin1,4  \n  \n1Prokhorov General Physics Institute, Russian Academy of Sciences, 119991, Moscow, Russia  \n2P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninskiy Prospekt  53, \n119991, Moscow, Russia  \n3Institute of Molecule and Crystal Physics, Subdivision of the Ufa Federal Research Centre of the \nRussian Academy of Sciences , 450075 , Ufa, Russia  \n4Moscow Institute of Physics and Technology (State University), 141700, Dolgoprudny, Russia  \n \nCorresponding authors: gzv@anrb.ru , zvezdin@gmail.com  \n \nAbstract  \nIn this paper, we elaborate analytical theory of domain wall dynamics close to  the angular \nmomentum compensation point based  on non-linear dynamic equations derived from the effective  \nLagrangian of a ferrimagnet. In the framework of the proposed m odel, we explore dynamic \nprocesses in the Walker and post Walker regimes . Analysis of the precession angle and domain \nwall’s velocity oscillations in post Walker regime in a ferrimagnet is performed. We show that \nalthough spin oscillations quench the dynam ics of domain walls near the Walker breakdown field, \na further increase of the driving magnetic field increases domain wall speed and mobility. An \nanomalous behavior of domain wall dynamic properties near the angular momentum compensation \npoint in ferrimag nets is discussed . 2 \n Keyword s: magnetization dynamics, ferrimagnets , domain walls  \n \n \n \n1. Introduction  \nUltrafast spin dynamics in antiferromagnets (AFMs) and ferrimagnets (FiMs) near \ncompensation temperature are receiving much attention owing to the potential applications for spin \nelectron devices [1 - 15]. Recent a dvances in antiferromagnetic spintronics and femtosecond lasers \nsources operating with ultrafast speeds and ultrashort times open new horizons for future \ntechnologies based on high -speed nanoscale elements in memory devices.  \nAn important attributes of spin dynamics in magnets are velocity and mobility of domain \nwalls (DWs). Magnetic DWs, active nanoscale elements, are well – suited for the concept of \nracetrack memory, logic units and spintronic memristors [7 - 15]. In this sense, a lot of expectations \nare related with antiferromagnetic materials where DW dynamics is extremely fast. As known [16 \n- 19], in antiferromagnetic orthof errites (YFeO 3) DW velocity attains 20 km/s and DW mobility is \nof the order of 0.1 km/s Oe.  \nDespite the undoubted advantages of antiferromagnetic dynamics, the difficulties \nassociated with the detection of DWs and their manipulation in AFMs force research ers to turn to \nferrimagnetic materials. However, the Walker breakdown velocities in FiMs are much lower (an \norder of magnitude) than those in AFMs.  \nThe exchange enhancement of dynamic parameters and giant DWs velocities are expected \nin FiMs near the subla ttice compensation point Tcomp [20]. However, there are still no reliable \nexperimental data in this direction. In particular, this can be explained by the fact that the saturation \nmagnetization vanishes \n0sM  near the compensation poin t \ncomp TT , and the corresponding spin \ntorque, which one can call the Zeeman torque, also tends to zero.  \nAnomalous spin dynamics in FiMs with strong sublatices coupling can occur at two \ntemperatures:  \ncompT , sublatti ce compensation temperature where M1=M2, and TA , angular 3 \n momentum compensation temperature where \n12\n12MM\n , \ni is the gyromagnetic ratio for the \nspecific sublattice, Mi  is the sublattice magnetization ( i=1,2) [5 - 8].  \nNear the angular momentum compensation point, the situation to study spin dynamics is \nmore favorable. Recent experiments on domain wall dynamics in ferrimagnetic GdFeCo alloy \nwere reported in [1, 2]. Authors of Ref. [1] measured enhancement  of DW velocity close to the \nangular momentum compensation point. The phenomenon was qualitatively explained in terms of \ncollective coordinates and atomistic spin model simulations.   \nIn our work, we elaborate analytical model based on the effective Lagran gian of FiMs [21] \nto describe dynamics of FiMs near the angular momentum compensation point TA. Proposed \napproach allows us to explore the structure of a moving DW and its featured dynamic properties: \nDW speed, mobility and the angle of precession both for  steady state motion and post – Walker \noscillatory regime.   \n2. Model  \nTo describe dynamics of a  ferrimagnet  (FiM) close to the angular momentum compensation \npoint we consider FiM with two magnetic sublattices  M1, M2 whose magnetization have different \ntemperature dependences. Such situation is realized in intermetallics, the most of rare earth \ntransition metals alloys (e.g. GdFeCo, TbFeCo, CoGd)  and weak ferromagnets (e.g. YFeO 3, \nRFeO 3) [1, 16 –23]. Approaching the FiMs compensation temperature  AFM ordering tends to be \nestablished ; and antiferromagn etic \n12L M M  and ferromagnetic vectors \n12M M M  are \nconsidered as  order paramet ers.  \nDynamics of FiM is used to be described in terms of Landau Lifshitz  pair equations  for each \nof Mi sublattices  [24, 25]. However,  to consider  domain wall (DW) motion it is more convenient \nto use Lagrange formalism in the spherical coordinates. The Lagrangian and the dissipation \nRayleigh function of two –sublatticed  FiM read \n2\n11 cosi\ni i i\ni iMLW   \n         (1) 4 \n \n 2\n22\n1sini\ni i i i\ni iRM \n        (2) \nwhere Mi is the magnitude of the i-th sublattice magnetization ;\n,ii  are the polar and azimuthal \nangles characterizing the orientation of the i-th sublattice magnetization ; \ni and \ni  are the i-th \nsubllatice damping parameter  and gyromagnetic ratio ; Wi is the  i-th subllatice thermodynamic \npotential.     \nTo characterize the canting of magnetic sublattices we introduce additional variables \n,  \ndefined as \n12 ,            , \n12 ,              , where \n,  are the polar and \nazimuthal angles of antiferromagnetic vector L in the spherical coordinate frame with polar axis \noriented along the direction of applied magnetic field.  In our problem  magnetic field is oriented \nalong the normal to a film  (see inset in Figure 1 ), uniaxial magnetic anisotropy of a film is \nsupposed to be strong . In the vicinity of compensation temperature ( TA) the canting angles \n,  \nare assumed to be sufficiently small  \n1, 1   that is valid at H<<Hex, Hex ~50 T (TbFeCo, \nGdFeCo).  \nThe effective Lagrangian of two - sublatticed uniaxial ferrimagnets was suggested and \ndescribed in details in Ref. [21]. We include into consideration the non – uniform exchange energy, \nthe in -plane magnetic anisotropy and rewrite the effective Lagrangian [21] as follows  \n22\n2\n22\n2 2 2 2cos sin22\nsin sin sin sineff\neff eff eff\nuL m H H\nddK K Adx dx      \n   \n     \n                    \n            \n    (3) \nwhere \n21\n2MMm , \n12\n2MMM ;\nexM\nH  is the transverse magnetic susceptibility, Hex is the \nexchange magnetic field acting between sublattices; \n,uKK are the constants of  perpendicular and  \nin-plane magnetic anisotropies, here we assume that \nu\neffHK\n \n ; A is the exchange stiffness 5 \n constant, \n and   are the polar and azimuthal angles of the ferromagnetic vector m,  H=(0,0, Hz) \nis magnetic field applied along the “easy magnetization axis ” (geometry of a problem is shown in \ninsert to Fig.1) . The effective Rayleigh dissipation function is taken in a form [21]  \n 2 2 2sin2eff\neff\neffMR    \n           (4) \nwhere  \n0effm\nmm\n, \n0effm\nmm ,\n12\n12 , \n121 1 1 1\n2  \n , \n12\n0\n12mM\n , \n1\n0\n21effmm\nM\n \nwhere m0 is the magnetization in the compensation point of angular momentum  TA.   \nLet’s consider for the beginning DW dynamics θ(t) in the Walker approach, i.e. assume that \n0\n,  \n0d\ndx , then t he Euler – Lagrange equation for the function θ(x, t) reads  \n2\n2\n222 sin 2 sin 02u\neffdMA K H mHdx             \n     (5) \nAt the boundary conditions \n 0,           we find the particular solution  of eq. ( 5)  \n0222arctg exp\n1/x qt\nqc \n,               (6) \nwhere  \n2\n21qmHM q\nс\n\n\n,           (7) \n2\n2uA\nKH\n\n,  \n2\neffAc  \nwhere  q is the coordinate of DW centre  \nEqs. (6), (7) represent  the exact solution of eq.  (5). One can check  it by the direct substitution 6 \n of eq s. (6), (7) into eq.( 5) [16].  Eq. (6) describe s the spin distribution in side the 1800 domain wall \n(DW) moving with the velocity \nq\n . Eq. ( 5), the double sin -Gordon equation with dissipation, and \nits solution ( 6) were obtained and used for description and investigation of DW dynamics in the \ndifferent systems [16, 2 6, 27].  \nIn the general case\n0\n . We solve the system of Euler -Lagrange equations using  the \nperturbation theory for solitons  \n\n01\n1, , ...,     \n, , ...x t x qt x t\nx t t x t  \n     \n  \n       (8) \nwhere Euler Lagrange equation for φ in which we substitute eq. ( 6) reads  \n0 0 0 0 0 0 22sin sin 2 sin sin cos cos 0\neff eff eff effmMKH                \n\n      \n   (9) \n Accounting ( 8) we rewrite eq. ( 5) in operator form \n1ˆLf  where \n2\n0 21ˆ cos 2x L     , function \nf depends on \n0x qt\n  and the system parameters. Solvability condition of such equations is \ndetermined by the requirement of orthogonality between functions f and \n0d\ndx : \n00df dxdx\n  \n(Fredholm alternative). Implementing the described procedure for eq. ( 5) and integrating ( 9) over \n, x  \n, we obtain the system of dynamic equations similar to the Slonczewski equations [2 5] \nat \n/1qc\n   \n2sin 2 0eff\neff effMq m H\nm q MK\n\n    \n\n\n   \n       (10) \nNote that at \n0 K  eqs. ( 10) give the exact solutions of eqs. ( 5), (9). As follows from eq. ( 10) \nthe DW velocity \nq\n  is proportional to the magnetic torques, the precession rate \n\n  depends on a \npressure on DW exerted by driving field, damping and magnetic anisotropy.  7 \n  \n3. Domain wall dynamics in the steady state (Walker) and post Walker regimes  \n To understand the characteristic features of DW  dynamics close to the angular momentum \ncompensation point we make several steps.   \nI) We start with the case when in -plane magnetic anisotropy is absent\n0 K . Since \n1  and \n10\neff\n close to the angular momentum compensation point \n0 mm  one can neglect the first \nterm in eq. ( 5). In this case in accordance with eqs. ( 10) DW velocity is determined as in Ref. [1]  \nby equation  \n2\n2 0qmHM mm\nM\n \n                        (11) \nThe dependences of DW velocity on the parameter \n/mM  at various magnitudes of \nmagnetic field H are shown in Figure 1 . For calculations we used the parameters of  GdFeCo [1, \n21, 22 ]  \n5 6 3 7\n3~ 1 10 , 1 10 , ~ 2 10 , 1100 , ~ 0.02, ~ 2 10 , 2.2, 2u d ferg ergK A M G g gcm cm  \n        \n 8 \n \nm/M0.00 0.05 0.10 0.15 0.20Domain wall velocity\n          V, m/s\n02004006008001000120014001600\nH=1000 Oe\nH=400 OeH=600 Oe\n \nFigure 1 . Domain wall velocity as the function of the \n12\n12MMm\nM M M  (dimensionless FiM ’s \nmagnetization)  in the case of non – stationary motion  at different values of driving magnetic field: \nblack line stands for H=400 Oe, green line stands for H=600 Oe, red line stands for H=1000 Oe . \nGeometry of t he problem is shown in the inse t. \n \nAs seen in Fig.1 the DW velocity enhances approaching the angular momentum compensation \npoint  \n0  (\n0 mm ) and tends to zero at magnetization compensation temperature \n(M1(Tc)=M2(Tc)) \n0 (\n0 m ). Our results ( Figure 1 ) are in consistence with the findings of \nRef. [1], the dependences shown in Figure 1  can be reconstructed in terms of T accounting the \ntemperature dependences of the sublattices  magnetizations \n12( ),  ( )M T M T . \nII) We a nalyse the dynamic impact given by in -plane magnetic anisotropy \n0 K .  \ni) At first we consider the steady state motion occurring in magnetic fields lower than the \nWalker field \n0w HH . In this case \n00\n  the DW moves with the velocity \nqH\n  (see eq. 9 \n (10)), where \n  is the DW mobility.  \nSteady – state motion velocity linearly depends on magnetic field up to the Walker velocity  \nwwmVHM\n\n           (12) \nestablished at the Walker field  \n22eff\nw effH K M Mm\n\n .                        (13) \n if we take \n22 Km   \nm/M0.03 0.06 0.09 0.12 0.15 0.180.00.51.01.52.0\n=0.0476\n \nFigure 2 . The Walker field and the effective mass as functions of \n12\n12MMm\nM M M (dimensionless \nFiM’s magnetization ).  \n. \nAs seen from eq.(13)  the Walker field diverges  close to the angular momentum \ncompensation point \n0 mm  (Figure 2 ).  \nii) We turn to the non – stationary regime \n0\n  in magnetic fields higher than the Walker field 10 \n \nw HH.  In the case \n1  one can neglect the \n2\neff\n  term in eqs. (10) and  obtain the  solution  \n22 2\n22tan 1 tan 1w w w\neff\neffH H HHtH H H                \n    (14) \n22\n22sin 2eff\neff w\neff w effqHHH      \n     (15) \nEquations (1 4), (15) describe the oscillations of precession angle \n  and  DW velocity \nq\n  . That \nmeans that in post -Walker regime at\nw HH  the angle of in -plane precession \n and DW velocity \nq\n become the periodical functions dependent on system parameters . The dependences of \noscillations amplitude and period on parameter \n/mM  are shown in Figure 3 . One can see \nthat DW velocity tends to be constant in a time close to the angular momentum compensation point \nTA. However, due to an unlimited increase in the oscillation frequency approaching \n0 mm , a \nrapid change in the angle of pre cession at \ntT (green line in Figure  3a) manifests itself in the \nsharpening “beak - like” areas occurring at \ntT  on the dependence \n()qt\n  (green line in Figure  \n3b), where T is the oscillations period  11 \n \nt, ns0.0 0.5 1.0 1.5 2.0Precession angle\n, deg.\n0200400600800 \nt, ns0.0 0.5 1.0 1.5 2.0Domain wall velocity\n           V , m/s\n020040060080010001200\n \nFigure 3  a) Oscillations of in – plane precession angle as the function of a time, b) domain wall \nvelocities as the functions of a time  at different values of the \n12\n12MM\nMM : black lines stand for \n0.025\n, red lines stand for \n0.035 , green lines stand for \n0.04 , H=800 Oe.  \n \nAveraging of DW velocity (1 5) on the precession movement of azimuthal angle \n given 12 \n by eq. (1 4) yields  \n22\n22sin 2eff\neff w\neff w effHqHH       \n     (16) \nDriving magnetic field\n               H, Oe0 200 400 600 800 1000Averaged domain wall velocity\n                  <V>, m/s\n02004006008001000120014001600\n=0.047\n=0.035\n=0.025\n=0.04\n \nFigure 4 . Average domain wall velocity as the function of driving magnetic field H in the Walker \nand post – Walker regimes calculated at different values of  the \n12\n12MM\nMM (dimensionless \nFiM’s magnetization) : black line stands for \n0.025 , red line stands for \n0.035 , green line \nstands for \n0.04 , blue line stands for angular momentum compensation point  \n00.0475 . \n \nDependences of the average DW velocity on magnetic field at different ν values are shown in \nFigure 4 where  the Walker and post Walker regimes are distinguished. In the steady state Walker \nregime DW velocity linearly increases with the driving field H and attains the maximum value \nmax qV\n at the Walker field \nw HH . In the post Walker regime close to the critical point \nw HH\n jump – like  decrease of DW velocity  occurs, however with the enhancement of magnetic 13 \n field \nw HH average DW velocity \nq\n  increases. Close to the angular momentum compensation \npoint \n00.0475  the DW velocity permanently increases.  \nIn contrast to Ref. [1] we show the differences in slope angles of the curves < V>(H) (Figure 4 ) \nboth in Walker and post – Walker  regimes and present more detailed consideration of post Walker \nDW dynamics.  \nEq. ( 7) describes the quasi - relativistic compression of the DW width, which occurs with \nan increase of the DW velocity. This effect, which is typical for the AFMs dynamics, in particular, \nleads to saturation of DW velocity with an increase of the driving magne tic field; in AFMs, the \nDW reaches its ultimate velocity which can be about 20 km/s [16 - 19]. As we showed, in FiMs \nthe situation becomes essentially different: in the vicinity of angular momentum compensation \npoint \n0 mm  the Walker f ield \nWH and ultimate velocity diverge \nc  (see eq. ( 7)).  \n \n4 Discussion and c onclusion  \nTo conclude, we present the analytical model with integrable dynamic equations on the \nbase of effective ferrimagnet Lagrangian given in Ref. [21]. We obtain the exact solutions of \ndynamic equations allowing  describe DW dynamics in ferrimagnets close to the angular \nmomentum compensation point TA. The processes in -between stationary Walker and non – \nstationary oscillatory  regimes have been explored.  \nWe compare results of our calculations with experimental data on DW dynamics and their \nexplanation given in Ref. [1].  In the case when in -plane magnetic anisotropy is absent the \ncalculated DW velocity and its temperature depe ndences ( Figure 1 ) coincide with the findings of \nRef. [1]. We show the principal agreement with results of Ref. [1] on the behavior of DW velocity \nin the driving magnetic field. Magnetic field applied along the “easy magnetization axis” increases \nDW veloci ty up to the Walker breakdown, afterwards the DW quenches due to the transition into \noscillatory post – Walker regime typical for ferromagnets. Further enhancement of a field increases 14 \n DW velocity. The highest values of velocity (up to 1.5 km per second in  a field around 1000 Oe) \nare attained at the temperatures close to the angular momentum compensation point.  \nOur analysis also reveals the differences from findings of Ref. [1]. We show that DW \nvelocity is also the function of the net angular momentum  \n2 1 2 1 ( ) ( ) / ( )T M M M M    related \nwith  magnetization of specific sublattices dependent on the temperature. In contrast to Ref. [1] \nwe include in our consideration in – plane magnetic anisotropy typical e.g. for uniaxial crystals \n(GdFeCo) and demonstrate osci llations of the precession azimuthal angle and DW velocity in the \npost Walker regime. Actually, the in – plane magnetic anisotropy favors outcome of magnetic \nvector from the DW plane described by azimuthal angle  φ. The competition between impacts given \nby the driving magnetic field and magnetic anisotropy results in the oscillations of φ angle and as \na consequence DW velocity close to the Walker breakdown field.   \nOur findings give additional information on the oscillatory post Walker regime. We show \nthat t he amplitude and period of oscillations depend on temperature and material parameters.  \nElaborated model can be used as a tool to study plethora of spin dynamics phenomena in \nferrimagnetic materials such as magnetic – dipole radiation of electromagnetic wa ves excited by \noscillations of the moving DW, phenomenon theoretically predicted in Ref. [ 31]; current induced \nmotion of DWs, experimentally observed in AFM coupled bi -layers [2]; accelerating and focusing \nmagnetic vortices and skyrmions [ 32] attractive for spintronic applications.  \nDeclaration of Competing Interest  \nThe authors declare that they have no known competing financial  interests or personal \nrelationships that could have appeared to influence  the work reported in this paper.  \nAcknowledgments  \nThis work is supported by the Russian Science Foundation grant No. 17-12-01333.  \n \n \n \n 15 \n References  \n \n [1] K. -J Kim, S.K. Kim, Y. Hirata, S.H. Oh, T. Tono, D.H. 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K. Zvezdin, J. Phys.: Condens. Matter  \n2019 , 32 (1), 01LT01.  \n[22] D.-H. Kim, et al. Phys. Rev. Lett . 2019 , 122 (12), 127203  \n[23] S. K. Kim, Nature Electronics  2020 , 3 (1), 18 –19. \n[24] C. Kittel, Phys. Rev. 1951 , 82, 565  \n[25] A. P. Malozemof f, J. C. Slonczewski, Magnetic Domain Walls in Bubble Materials: \nAdvances in Materials and Device Research (vol.1) , Academic Press, 2016 . \n[26] A.K. Zvezdin, A.F. Popkov, Pis’ma Zh. Eksp. Teor. Fiz . 1985 , 41, 90. \n[27] A.K. Zvezdin, A.A. Mukhin, Pis’ma Zh. Eksp. Teor. Fiz . 1985 , 42, 129.  \n[28] W. Doering. Zs. Naturforsch . 3a, 373 (1948)  \n[29] H. E. Khodenkov, Physica status solidi  (a) 1979 , 53 (1), K103.  \n[30] X.R. Wang, P. Yan, J.Lu, C.He, Annals of Physics  324 (2009) 1815   \n[31] A.K. Zvezdin, Pis’ma Zh. Eksp. Teor. Fiz . 1980 , 31, 508 -510,  \n[32] S. K. Kim, K.J. Lee, Y. Tserkovnyak, Phys. Rev. B 2017,  95, 140404(R)  \n \n "    },    {        "title": "1802.07113v3.Magnon_photon_coupling_in_a_non_collinear_magnetic_insulator_Cu__2_OSeO__3_.pdf",        "content": "arXiv:1802.07113v3  [cond-mat.mes-hall]  15 Feb 2019Magnon-photon coupling in a non-collinear magnetic insula tor Cu 2OSeO 3\nL.V. Abdurakhimov,1,∗S. Khan,1N.A. Panjwani,1J.D. Breeze,2,3M.\nMochizuki,4S. Seki,5Y. Tokura,5,6J.J.L. Morton,1and H. Kurebayashi1,†\n1London Centre for Nanotechnology, University College Lond on, London WC1H 0AH, United Kingdom\n2Department of Materials, Imperial College London,\nExhibition Road, London SW7 2AZ, United Kingdom\n3London Centre for Nanotechnology, Imperial College London ,\nExhibition Road, London SW7 2AZ, United Kingdom\n4Department of Applied Physics, Waseda University, Okubo, S hinjuku-ku, Tokyo 169-8555, Japan\n5RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0 198, Japan\n6Department of Applied Physics and Quantum Phase Electronic s Center (QPEC), University of Tokyo, Tokyo 113-8656, Japan\n(Dated: February 18, 2019)\nAnticrossing behavior between magnons in a non-collinear c hiral magnet Cu 2OSeO 3and a two-\nmode X-band microwave resonator was studied in the temperat ure range 5–100K. In the field-\ninduced ferrimagnetic phase, we observed a strong coupling regime between magnons and two mi-\ncrowave cavity modes with a cooperativity reaching 3600. In the conical phase, cavity modes are\ndispersively coupled to a fundamental helimagnon mode, and we demonstrate that the magnetic\nphase diagram of Cu 2OSeO 3can be reconstructed from the measurements of the cavity res onance\nfrequency. In the helical phase, a hybridized state of a high er-order helimagnon mode and a cavity\nmode — a helimagnon polariton — was found. Our results reveal a new class of magnetic systems\nwhere strong coupling of microwave photons to non-trivial s pin textures can be observed.\nIntroduction. Strong coupling between microwave\nphotons and particle ensembles is a general phenomenon\nin light-matter interactions that has been observed in\na broad range of condensed-matter systems, including\nensembles of magnetically ordered spins[1–6], param-\nagnetic spins[7–10], and two-dimensional electron sys-\ntems[11–13]. A common feature of ensemble coupling is\nthat the coupling strength between a photon and Npar-\nticlesscaleswiththesquarerootof N,gN=g0√\nN, inac-\ncordancewith the Dickemodel[14–16]. Studies on strong\ncoupling in spin systems are particularly interesting due\nto possible applications of hybrid spin-ensemble-photon\nsystems for quantum information processing as quantum\nmemories[17, 18] and quantum transducers[19]. The\nspin-ensemble coupling strength can be extremely large\nin magnetically ordered systems due to their high spin\ndensities, and extensive studies of strong coupling to\nmagnons – the quanta of spin wave excitations in mag-\nnetically orderedsystems – have been performed recently\nin experiments on ferrimagnetic insulators. In particu-\nlar, new magnon-cavity-coupling phenomena have been\nobserved in yttrium iron garnet (YIG), such as coherent\ncoupling between a magnon and a superconducting qubit\n[20], microwave-to-optic-light conversion[21, 22], cavity-\nmediated coherent coupling between multiple ferromag-\nnets[23, 24], spin pumping in a coupled magnon-photon\nsystem[25], and other phenomena[26–30].\nSo far, most studies of strong coupling in magnetic\nmaterials have focused on ferrimagnetic materials with\ntheSi·Sj-like Heisenberg exchange interaction between\nneighbor spins SiandSj. In these materials, all spins\nare collinear in the ground state, and mainly the uni-\nform precession ferromagnetic mode, or the Kittel mode,has been used in the studies of magnon-photon coupling\nin those systems. However, there is a growing interest\nin the coupling of photons to non-collinear and other\nnon-trivial spin systems[31, 32]. In chiral magnets, the\nspin-spin exchange interaction consists of two terms; be-\nsides the symmetric Heisenberg interaction which favors\ncollinear spin structures, there is an additional antisym-\nmetricSi×Sj-like Dzyaloshinskii-Moriya (DM) interac-\ntion which tends to twist neighbor spins. As a result\nof the interplay between Heisenberg and DM exchange\ninteractions, various non-collinear spin textures can be\nformed in chiral magnets, such as helical, conical, and\nSkyrmion spin structures (see Fig.1(a)). Studies of the\ncouplingbetweenmicrowavephotonsandnon-trivialspin\ntextures is a potentially rich and largely unexplored area.\nA chiral magnetic insulator copper-oxoselenite\nCu2OSeO3crystallizes in a non-centrosymmetric cubic\nstructure with 16 copper ions Cu2+per unit cell (space\ngroupP213, lattice constant a≈8.93˚A[33]). The basic\nmagnetic building block of Cu 2OSeO3is a tetrahedral\ncluster formed by four Cu2+spins in a 3-up-1-down spin\nconfiguration, which behaves as a spin triplet with the\ntotal spin S= 1[34]. This magnetic structure has been\nvisualized elsewhere[34, 35]. Due to a combination of\nHeisenberg and DM exchange interactions, the system\nofS= 1 clusters forms helical, conical, ferromagnetic\nand Skyrmion magnetic phases in an applied external\nmagnetic field below the Curie temperature TC≈60K,\nas shown in Fig.1(b). Above TC, the system is para-\nmagnetic. Besides being a chiral magnet, Cu 2OSeO3\nis also of particular interest due to its multiferroic and\nmagnetoelectric properties[35–39].\nIn this Letter, we report a study of magnon-photon2\ncoupling in a Cu 2OSeO3system. In the collinear ferri-\nmagnetic phase, we observed a strong coupling regime\nbetween a microwave cavity mode and a uniform Kit-\ntel magnon mode, and the temperature dependence of\nthe coupling strength was found to follow that of the\nsquarerootofthenetmagnetization. Inthenon-collinear\nconical phase, a dispersive coupling regime between he-\nlimagnons and microwave photons was observed, and we\ndemonstrate that the magnetic phase diagram can be de-\nterminedbythe measurementofthe frequencyofacavity\nmode. In the non-collinear helical phase, normal-mode\nsplittingwasdetectedbetweenahigher-orderhelimagnon\nmode and a cavity mode.\nhelicalconical ara-\nagnetic\nSkyrmion(b) (a)\nSkyrmionconicalhelical\n(c) (d)\nFIG. 1. (color online) (a) Schematic view of non-collinear\nspin textures. (b) Magnetic phase diagram of Cu 2OSeO 3. (c-\nd) Strong coupling between ferrimagnetic mode of Cu 2OSeO 3\nand multiple microwave cavity modes: (c) experimental data\nof microwave reflection |S11|2as a function of the applied ex-\nternal field and the microwave probe frequency (temperature\nT≈5K, microwave input power P≈3mW), (d) microwave\nreflection |S11|2calculated using Eq.(2) and parameters de-\nscribed in the text.\nExperimental details. We performed microwave X-\nband spectroscopy studies of Cu 2OSeO3in the temper-\nature range 5–100K using a helium-flow cryostat[40].\nA sample of single crystal Cu 2OSeO3was inserted into\na commercial Bruker MD5 microwave cavity consisting\nof a sapphire dielectric ring resonator mounted inside\na metallized plastic enclosure. The sample mass was\nm≈60mg, corresponding to a total effective spin num-\nberN≈7×1019. The shape of the sample was close\nto semi-ellipsoidal, with the lengths of semi-axes being\n1.5mm, 1.5mm, and 2mm, and a flat plane being ori-\nented along the long ellipsoid axis. The orientation of\ncrystallographic axes of the sample relative to the cavity\naxis was chosen arbitrarily. The cavity supported two\nmicrowave modes[40, 41]: the primary mode TE 01δwith\nthe resonance frequency of about ω1/2π≈9.74GHz and\nthe hybrid mode HE 11δwith the resonance frequency ofaboutω2/2π≈9.24GHz. We tuned the qualityfactor Q1\nof the primary-mode resonance by adjusting the position\nof a coupling loop antenna. In our measurements, we\nused a slightly under-coupled cavity with Q1≈5×103.\nThe quality factor of the hybrid-mode resonance did not\ndepend on the position of the coupling antenna, and was\naboutQ2≈100. In our experiments, the microwave re-\nflection S-parameter |S11|2was measured as a function of\nexternal magnetic field and microwave probe frequency.\nResults. Figure1(c) shows typical data from mi-\ncrowave reflection measurements at temperature\nT≈5K, obtained from raw experimental data by\nbackground-correction processing[40]. The input mi-\ncrowave power was P≈3mW. Two avoided crossings\nare visible at the degeneracy points where two cavity\nmodes would otherwise intersect a magnon mode. The\nmagnon mode corresponds to a uniform spin precession\n(Kittel mode) with frequency ωm/2π=γ(H0+Hdemag),\nwhereH0is the applied magnetic field, Hdemagis the\ndemagnetizing field, and γ≈28GHz/T is the electron\ngyromagnetic ratio. Here, we assume that anisotropy\nfields are small and can be neglected.\nThe interaction between two cavity modes and a\nmagnon mode can be described by the following Hamil-\ntonian in the rotating-wave approximation (RWA):\nH0//planckover2pi1=ω1a†\n1a1+ω2a†\n2a2+ωmm†m+\n+g1(a†\n1m+a1m†)+g2(a†\n2m+a2m†),(1)\nwherea†\n1(a1) is the creation (annihilation) operator for\nmicrowave photons at frequency ω1,a†\n2(a2) is the cre-\nation (annihilation) operator for microwave photons at\nfrequency ω2,m†(m) is the creation (annihilation) op-\nerator for magnons at frequency ωm, andg1(g2) is the\ncouplingstrengthbetweenthemagnonmodeandthefirst\n(second) cavity mode.\nIn order to extract numerical values of coupling\nstrengths g1andg2and other parameters from the ex-\nperimental data, we used the followingequationobtained\nfrom input-output formalism theory[40]:\n|S11|2=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1+κ(c)\n1F2+κ(c)\n2F1−2/radicalBig\nκ(c)\n1κ(c)\n2F3\nF1F2−F2\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,(2)\nwhere\nF1=i(ω1−ω)+/parenleftBig\nκ1+κ(c)\n1/parenrightBig\n/2+g2\n1(i(ωm−ω)+γm/2)−1,\nF2=i(ω2−ω)+/parenleftBig\nκ2+κ(c)\n2/parenrightBig\n/2+g2\n2(i(ωm−ω)+γm/2)−1,\nF3=/radicalBig\nκ(c)\n1κ(c)\n2/2+g1g2(i(ωm−ω)+γm/2)−1,\nandκ1(κ2) is the damping rate of the first (second) cav-\nity mode, κ(c)\n1(κ(c)\n2) is the coupling rate between the\nfirst (second) cavity mode and the output transmission\nline, and γmis the damping rate of the magnonic mode.\nDamping rates represent linewidths (FWHM) of the cor-\nresponding modes.3\n0.1 0.4 0.7 1\nMagnetic Field (kOe)9.759.769.779.78Frequency (GHz)\n0.10.20.3\n0.1 0.4 0.7 1\nMagnetic Field (kOe)9.759.769.779.78Frequency (GHz)\n0.10.20.3\n0 20 40 60 80\nTemperature (K)00.20.40.60.81Magnetic Field (kOe)Hc2\nHc1(a) Hc1\n2Q/2field-up field-down(b) (c)ferrimagnetic\nconical\nhelicalHc2\n|S11|2|S11|2\nFIG. 2. (color online) Magnon-photon coupling in non-colli near magnetic phases. (a) Microwave reflection |S11|2at 5K. The\nmagnetic field sweep was performed from low to high field value s (“field-up”). Dashed line corresponds to a higher-order\nhelimagnon k=±2Qmode, and an avoided crossing between the helimagnon mode an d the cavity mode is clearly visible.\nHc1andHc2are critical magnetic fields of helical-to-conical and coni cal-to-ferrimagnetic phase transitions, respectively. ( b)\nMicrowave reflection |S11|2at 5K. The magnetic field sweep was performed from high to low v alues (“field-down”). The\ntransition from ferrimagnetic to conical phase demonstrat ed hysteretic behavior. (c) The magnetic phase of Cu 2OSeO 3sample\nreconstructed from the “field-up” measurements of Hc1andHc2. The dashed lines are for eye guidance.\nWe reproduce the data shown in Fig.1(c) by using\nEq.(2) with the followingparameters: g1/2π≈600MHz,\ng2/2π≈450MHz, κ1/2π≈1MHz,κ(c)\n1/2π≈1MHz,\nκ2/2π≈60MHz, κ(c)\n2/2π≈10MHz, Hdemag≈280Oe\nandγm/2π≈50MHz (see Fig.1(d)). Thus, the cou-\npling strengths are much greater than the damping rates\nof both cavity and magnon modes, g1≫(κ1+κ(c)\n1),γm\nandg2≫(κ2+κ(c)\n2),γm, and strong coupling regimes\nare realized for both avoided crossings. From the ob-\ntained value of the ferrimagnetic resonance linewidth\nγm, we estimate the Gilbert damping parameter α≈\n2.6×10−3at 5K which is consistent with the litera-\nture[42, 43]. Cooperativity parameters are much higher\nthan unity: C1=g2\n1//parenleftBig\nγm/parenleftBig\nκ1+κ(c)\n1/parenrightBig/parenrightBig\n≈3600 and\nC2=g2\n2//parenleftBig\nγm/parenleftBig\nκ2+κ(c)\n2/parenrightBig/parenrightBig\n≈60. Moreover, ratios\ng1/ω1≈0.06 andg2/ω2≈0.05 are close to the condition\nof the ultra-strong coupling regime ( g/ω≥0.1), where\nthe coupling strength is comparable with the frequency\nof the degeneracy point of an avoided crossing, and new\nphysics beyond RWA can be explored[3, 4].\nThe obtained values of coupling strengths g1andg2\nbetweenmagnonsandcavitymodesareinrelativelygood\nagreement with the theoretical estimates\ng(th)\ni=ηi\n2γ/radicalbigg\nµ0/planckover2pi1ωi\nVi√\n2SN,(i= 1,2) (3)\nwhereµ0is the vacuum permeability, V1≈230mm3\n(V2≈290mm3) is the mode volume of the primary\n(hybrid) cavity resonance, and the coefficient η1≈0.85\n(η2≈0.83) describes the spatial overlap between the pri-\nmary (hybrid) cavity mode and the magnon mode[40].\nSubstituting other known parameters into the equations,\nwe obtain g(th)\n1/2π≈825MHz and g(th)\n2/2π≈700MHz.\nSlight discrepancies between theoretical and experimen-\ntal values of coupling strengths can be caused by theexcitation of additional k∝negationslash= 0 spin-wave modes in the\nsample which are visible as additional faint narrow lines\nin the experimental data, and the resulting reduction of\nthe effective number of spins involved in the uniform\nKittel mode precession[44]. The difference in coupling\nstrengths can be also related to the fact that the cou-\npled system of magnons and photons is close to the\nultra-strong coupling regime mentioned above where the\nHamiltonian (1) is not valid.\nCoupling strengths g1andg2depended strongly on\ntemperature[40]. In the ferrimagnetic phase, the effec-\ntive number of spins is proportional to the net magneti-\nzation, and we found that the temperature dependence\nof the coupling rate could be fitted by square-root func-\ntion of the net magnetization which is in good agreement\nwith the results of studies of strong coupling in collinear\nferrimagneticsystems[45]. Inourexperiments, anticross-\ning behavior was independent of microwaveprobe power,\nconsistent with observations of strong coupling in other\nsystems[1, 5, 7].\nIn order to study magnon-photon coupling in non-\ncollinear spin textures, we performed measurements at\nlow magnetic fields where the system exhibits helical and\nconical magnetic phases (see Fig.2,3). We found that\nthe frequency of a cavity resonance depended on the ap-\nplied external magnetic field not only in the ferrimag-\nnetic phase, but also in the helical and conical states\n(Fig.2(a-b)). We identify two features in microwave re-\nsponse at magnetic field values Hc1andHc2which can\nbe attributed to helical and conical magnetic phase tran-\nsitions in Cu 2OSeO3, respectively (see Fig.2(c)). The\nobserved transition from helical to conical phases is rel-\natively smooth, which can be related to the fact that\nin the helical phase the spin system forms a multido-\nmain structure of flat helices[35], and, since the external\ndc magnetic field was not aligned along the high sym-\nmetry directions of the crystal structure in our measure-4\n0.1 0.2 0.3 0.4 0.5\nMagnetic Field (kOe)9.7559.7609.7659.7709.775Frequency (GHz)\n0.10.20.3\n0.1 0.2 0.3 0.4 0.5\nMagnetic Field (kOe)9.7559.7609.7659.7709.775Frequency (GHz)\n0.10.20.3 (a)\n2Q/2(b)\n2Q/2\n|S11|2|S11|2\nFIG. 3. (color online) Normal-mode splitting between a cav-\nity mode and a higher-order helimagnon mode in the helical\nphase. (a) Microwave reflection |S11|2at 5K. (b) Results of\nthe numerical calculation of |S11|2by using equations and\nparameters described in the text.\nments, domains gradually reoriented themselves with the\nincrease of the magnetic field[46].\nWe suppose that the changes in the frequencies of cav-\nity modes are caused by dispersive coupling between mi-\ncrowave photons and magnonic modes in the helical and\nconicalphases—so-calledhelimagnons[46,47]. Frequen-\ncies of the fundamental n=±1 helimagnon modes ω±Q\nlie well below the cavity resonance frequencies and de-\npend weakly on the applied magnetic field. Since a full\ntheoretical model of helimagnon-photon coupling would\nrequire detailed calculation of spectral weights of heli-\nmagnon modes[46] which is outside the scope of this\npaper, here we present a qualitative description of the\ndispersive magnon-photon coupling in Cu 2OSeO3. In\nthe helical phase H < H c1, the net magnetization Mis\nsmall[35], and helimagnon-photon coupling is very weak.\nIn the conical phase Hc1< H < H c2, the net magneti-\nzation is substantial, and the cavity mode is dispersively\ncoupled to the fundamental helimagnon mode which re-\nsults in the shift ofthe cavityresonancefrequency. In the\nferrimagnetic phase H > H c2, the net magnetization is\nclose to its maximum value (saturation magnetization),\nthe coupling strength to the Kittel mode is large, and the\nshift ofthe cavityresonancewith the increaseofthe mag-\nnetic field is large. It should be noted that the magnetic-\nfielddependenceofthecavityresonancefrequencycannot\nbe explainedbythe variationofadc magneticpermeabil-\nity of the material[40].\nThe value of the magnetic field Hc2of the conical-to-\nferrimagnetic transition was found to be dependent on\nthe direction of magnetic field sweep (Fig.2(a-b)). This\nhysteretic behavior was observed only at low tempera-\nturesT/lessorsimilar40K, and it can be related to either the com-\npetition betweenthe conicalstateandtheunusual“tilted\nconical” and skyrmion states recently reported in[48, 49]\nor the extension of the conical n= 1 mode into the\ninduced-ferrimagnetic phase (and vice versa)[47].\nIn the helical magnetic phase, we observed hybridiza-\ntion between a cavity mode and a higher-order heli-\nmagnon mode (see Fig.3). By analogy with cavitymagnon polaritons[6, 25, 26, 28], a hybrid helimagnon-\nphoton state can be called a helimagnon polariton. In\ncontrast to the ferrimagnetic mode, the dispersion curve\nof a helimagnon mode ωnQ(H) exhibits a negative slope\n(dωnQ/dH <0). The helical phase is characterized by a\nmultidomain structure of flat helices, where the propaga-\ntion vectors of the different helices are pinned along the\npreferred axes of the system. The accurate description\nof helimagnons in the helical phase requires taking into\naccount the cubic anisotropies of Cu 2OSeO3[46], which\nis outside the scope ofthis paper. Instead, wecan use the\nanalyitcal equation used for the description of magnons\nin the conical phase[47]:\nωnQ=|n|γBc2\n1+Nd·χ/radicalbig\nn2+(1+χ)(1−(B0/Bc2)2),\n(4)\nwhereNdis the demagnetization factor along the direc-\ntion ofthe Qvector,χis the internalconicalsusceptibiliy\n(χ≈1.76 for Cu 2OSeO3[46]),B0is the applied exter-\nnal field, and Bc2is the critical field for the transition\nbetween conical and ferrimagnetic phases ( Bc2≈0.08T\nin our experiments). The mode number ndescribes the\nrelation between the helimagnon wave vector kand the\nwave vector of the helical spiral Q,k=±nQ. By adjust-\ning the parameters nandNd, we identify the observed\nhelimagnon mode as an n=±2 mode (the demagnetiza-\ntion factor Nd≈0.1), which is shown by the dashed line\nin Fig.3.\nIn order to characterize the observed avoided crossing\nquantitatively, we use the following equation for the mi-\ncrowave reflection[3]:\n|S11|2=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1+κ(c)\n1\ni(ω1−ω)+κ1+κ(c)\n1\n2+g2\n2Q\ni(ω2Q−ω)+γ2Q\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,\n(5)\nwhereω2Qandγ2Qare the frequency and the linewidth\nof the helimagnon mode, respectively, and g2Qis the\ncoupling strength between the cavity mode and the\nhelimagnon mode. We found that experimental data\nshown in Fig.3(a) can be reproduced by using the\nfollowing parameters: g2Q/2π≈8.5MHz,κ1/2π≈\n1.5MHz,κ(c)\n1/2π≈1.5MHz, and γ2Q/2π≈60MHz (see\nFig.3(b)). Therefore, the observed normal-mode split-\nting can be attributed to the Purcell effect ( κ < g2Q<\nγm), where the decay of microwave cavity photons is en-\nhanced due to their interaction with lossy magnons[3].\nWe could not detect the higher-order helimagnon mode\nω2Qat temperatures T/greaterorsimilar30K which is consistent with\nthe literature[47].\nAccording to our numerical simulations, the mode\noverlapping between a cavity mode and higher-order\n|n|= 2 helimagnon modes is suppressed as compared\nto the one for the Kittel ferromagnetic mode[40]. It\nshould be noted that, according to the work[47], the di-5\nrect interaction between a uniform microwave mode and\n|n|= 2 helimagnons is negligible, but microwavephotons\ncan couple to |n|= 2 helimagnons indirectly via funda-\nmental|n|= 1 helimagnons. Indeed, in the coordinate\nframe co-rotatingwith the spins around the helical-spiral\nwave vector Q∝bardblz, the magnetic component of the TE 01δ\ncavity mode — which is spatially uniform within the\nsample volume in the laboratory frame of coordinates —\ncorresponds to an effective microwave field with compo-\nnentsheff\n1x∝cos(Qz), andheff\n1y∝sin(Qz). Therefore,\nthe applied microwave magnetic field can excite directly\nonly|n|= 1 helimagnons which are characterized by the\nwave vector |k|=Q. However, in a cubic crystal, due\nto the fourth-order magnetic anisotropy m4\nx+m4\ny+m4\nz,\nwheremis a unit vector in the direction of the mag-\nnetization, n=∓1 andn=±2 helimagnon modes are\nhybridized[47]. Thus, the observed avoided crossing is\ncaused by the double hybridization between microwave\nphotons, |n|= 1 helimagnons, and |n|= 2 helimagnons.\nWe were not able to resolve coupling to magnetic ex-\ncitations in the Skyrmion phase in the measurements in\nthe correspondingtemperature range, presumably due to\nlarge detuning between the microwave cavity mode and\nSkyrmion modes[50].\nConclusions. We performed a study of magnon-\nphoton coupling in helical, conical and ferrimagnetic\nphases of a chiral magnet Cu 2OSeO3. We achieved a\nstrong coupling regime between a ferrimagnetic magnon\nmode and multiple microwave cavity modes. In the\nnon-collinear conical phase, we observed the dispersive\ncoupling between cavity modes and a fundamental he-\nlimagnon mode which allowed us to use a cavity mode\nas a probe for the sensing of magnetic phase transi-\ntions in Cu 2OSeO3. In the non-collinear helical phase,\nwe detected a normal-mode splitting between microwave\nphotons and high-order helimagnons in the Purcell-effect\nregime. These findings establish a new area of stud-\nies of strong coupling phenomena in multiferroic chiral\nmagnetic systems, paving the way for new hybrid sys-\ntems consisting of non-trivial spin textures coupled to\nmicrowave photons via magnetic and magnetoelectric in-\nteractions.\nThis work was partly supported by the Grants-In-\nAid for Scientific Research (Grants Nos. 18H03685,\n17H05186 and 16K13842) from JSPS, and by the Eu-\nropean Union’s Horizon 2020 programme (Grant Agree-\nment Nos. 688539 (MOS-QUITO) and 279781 (AS-\nCENT)); as well as the Engineering and Physical Sci-\nence Research Council UK through UNDEDD project\n(EP/K025945/1).\n∗leonid.abdurakhimov.nz@hco.ntt.co.jp; Present address :\nNTT Basic Research Laboratories, NTT Corporation, At-sugi, Kanagawa 243-0198, Japan\n†h.kurebayashi@ucl.ac.uk\n[1] H. Huebl, C. 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S1-1. (color online) Schematic view of the experimenta l setup.\nAn example of raw experimental data is shown in FigureS1-2(a). We c an clearly see two avoided crossings and a\nbackground pattern consisting of horizontal stripes. The backg round is caused by the presence of a standing wave\nin coaxial cables between the microwave resonator and the vector network analyzer due to impedance mismatch at\nthe input of the microwave resonator. To eliminate that backgroun d, we reconstructed the standing-wave profile by a\npiecewise-defined function with different pieces being taken at differ ent values of magnetic field, where the standing-\nwave background was not affected by the avoided crossings. Back ground-corrected data was obtained by subtracting\nthe standing-wave profile from the raw experimental data, and ad ding an offset value to keep the minimum and\nmaximum levels of |S11|2at the original level. An example of background-corrected data is s hown in FigureS1-2(b).\n1000 2000 3000 4000 5000 6000 7000\nMagnetic Field (Oe)8.599.51010.5Frequency (GHz)\n0.050.10.150.20.250.30.350.40.45\n|S11|2\n1000 2000 3000 4000 5000 6000 7000\nMagnetic Field (Oe)8.599.51010.5Frequency (GHz)\n0.050.10.150.20.250.30.350.40.45\n|S11|2(a) (b)\nFIG. S1-2. (color online) The standing-wave background cor rection. (a) An example of raw experimental data ( T= 5K). (b)\nThe same data after the correction.\nThe strong coupling regime between a ferrimagnetic mode and two ca vity modes was manifested as splitting of two\ncavity resonances into three hybrid resonant modes (see Fig.S1- 3).3\n1 2 3 4 5\nMagnetic Field (kOe)8.599.51010.5Frequency (GHz)\n 0.10.20.30.4\n8.5 9 9.5 10 10.5\nFrequency (GHz)0.340.360.380.40.420.440.46|S11|2(b) (a)\n|S11|2H=3120 Oe\nm/2\n1/2\n2/2\nFIG. S1-3. (color online) Normal-mode splitting due to stro ng coupling between a ferrimagnetic mode and two cavity mode s\n(see the main text for other details). (a) Microwave reflecti on|S11|2as a function of the applied external field and the microwave\nprobe frequency (temperature T≈5K). (b) The frequency scan at the magnetic field H0= 3120Oe. The splitting of two\ncavity modes into three hybrid resonance modes is clearly vi sible. The additional fine structure of the low-frequency re sonance\nis due to excitation of additional standing spin wave resona nces.\nMICROWAVE TE 01δAND HE 11δMODES OF A DIELECTRIC RESONATOR\nWe performed numerical simulations of the sapphire dielectric ring re sonatorcavity by using CST MicrowaveStudio\nsoftware(seeFig.S1-4). Thesamplewasmodeledasadielectricsph erewiththeradius r= 2mm, therelativemagnetic\npermeability µr= 1, and the relative electrical permittivity εr≈12.5[S1-1]. We found that the cavity supports two\nmicrowave modes: a primary TE 01δmode with the resonance frequency of about ω1/2π≈9.8GHz (Fig.S1-4(a-b))\nand a hybrid HE 11δmode with the resonance frequency of about ω2/2π≈9.56GHz (Fig.S1-4(c-d)). The microwave\nmagnetic field of the primary mode is parallel to the axis of the dielectr ic resonator, and magnetic energy density\nWmis concentrated inside the cavity bore (Fig.S1-4(a)). The effective magnetic mode volume for the primary cavity\nresonance can be estimated as\nV1=/integraltext\n(1\n2ε0εrE2+1\n2µ0µrH2)dV\nµ0µrH2max=/integraltext\n(We+Wm)dV\n2Wm,max, (S1-1)\nwhereε0is the vacuum permittivity, εr=εr(x,y,z) is the relative electrical permittivity at a given point, E=\nE(x,y,z) is the microwave electric field, µ0is the vacuum permeability, µr=µr(x,y,z) is the relative permeability,\nH=H(x,y,z) is the microwave magnetic field, Hmaxis the maximum value of the microwave magnetic field, Weand\nWmare microwave electrical and magnetic energy densities, and the int egration is taken over the total volume of the\nsystem. From numerical simulations, we obtain V1≈230mm3.\nNext, we introduce an overlap coefficient η1to take into account a slight non-uniformity of the distribution of th e\nmicrowave magnetic field across the sample which affects our calculat ions of coupling strengths described in the main\ntext:\nη1=/radicalBigg/integraltext\nVsµ0µH2dV\nµ0µH2maxVs=/radicalBigg/integraltext\nVsWmdV\nWm,maxVs, (S1-2)\nwhereVs=4\n3πr3is the volume of the sample used in the numerical model, and the integr ation is taken over the\nsample volume. Numerically, we find η1≈0.85 for the primary mode.\nThe hybrid HE 11δmode is a double-degenerate axially-asymmetric mode. At the site of the sample, the magnetic\ncomponent lies in the plane perpendicular to the axis of the dielectric r esonator. The HE 11δmode is degenerate, and\ntwo mutually perpendicular orientations of microwave magnetic field — two polarizations — are possible. Typical\ndistributions of microwave magnetic and electric energy densities fo r one of two possible polarizations are shown in4\nFig.S1-4(c-d)). In calculations of the coupling strength describe d in the main text, we take into account only one\npolarization of the HE 11δmode for which the microwave magnetic field at the site of the sample is perpendicular to\nthe external dc magnetic field (a hybrid mode with another polarizat ion does not interact with a ferrimagnetic mode).\nBy using equations similar to Eq.(S1-1–S1-2), we estimate that the effective magnetic mode volume is V2≈290mm3,\nand the overlap coefficient is η2≈0.83 for the magnetic component of the hybrid mode.\nTE01 mode\n-5 0 5\nx (mm)-5\n0\n5y (mm)\n123456\nWm (J/m3)106 TE01 mode\n-5 0 5\nx (mm)-5\n0\n5y (mm)\n123456\nWe (J/m3)106\nHE11 mode\n-5 0 5\nx (mm)-5\n0\n5y (mm)\n123456\nWm (J/m3)106 HE11 mode\n-5 0 5\nx (mm)-5\n0\n5y (mm)\n123456\nWe (J/m3)106(a)\nmagnetic\n(c)\nmagnetic(b)\nelectric\n(d)\nelectric\nFIG. S1-4. (color online) Results of numerical simulations of the dielectric resonator cavity. Distributions of magne tic (Wm) and\nelectric ( We) energy densities in the center horizontal plane, which is p erpendicular to the axis of the dielectric ring resonator,\nare shown for the primary cavity mode (a-b) and the hybrid cav ity mode (c-d). Dashed white circles represent the contour o f\nthe dielectric ring resonator.5\nINPUT-OUTPUT FORMALISM: STRONG COUPLING BETWEEN A MAGNON M ODE AND A\nTWO-MODE CAVITY\nFollowing input-output formalism theory [S1-2, S1-3], we can write th e following Heisenberg-Langevin equations:\n˙a1(t) =−i\n/planckover2pi1[a1(t),HSYS]−κ(cpl)\n1\n2a1(t)−/radicalBig\nκ(cpl)\n1κ(cpl)\n2\n2a2(t)+/radicalBig\nκ(cpl)\n1a(in)(t),\n˙a2(t) =−i\n/planckover2pi1[a2(t),HSYS]−κ(cpl)\n2\n2a2(t)−/radicalBig\nκ(cpl)\n1κ(cpl)\n2\n2a1(t)+/radicalBig\nκ(cpl)\n2a(in)(t),\n˙m(t) =−i\n/planckover2pi1[m(t),HSYS],\nain(t)+aout(t) =/radicalBig\nκ(cpl)\n1a1(t)+/radicalBig\nκ(cpl)\n2a2(t),\nwhereainandaoutare an external input and output fields, κ(cpl)\n1andκ(cpl)\n1are coupling constants between cavity\nfields and the external fields. The Hamiltonian HSYS=H0+HBdescribes the cavity modes and the magnon mode,\nwhere the Hamiltonian H0is defined in the main text, and HBdescribes the dissipation inside the cavity due to the\ninteraction with a heat bath.\nain\naoutma1\na2g1\ng2\nFIG. S1-5. (color online) A schematic representation of the two cavity fields, the magnon mode, and the input and output\nfields for a single-sided cavity.\nBy assuming that solutions should be in the form of a1(t) =a1e−iωt,a2(t) =a2e−iωt, etc., we can obtain the\nfollowing equations:\n(−iω)a1=/parenleftBig\n−iω1−κ1\n2/parenrightBig\na1−ig1m−κ(cpl)\n1\n2a1−/radicalBig\nκ(cpl)\n1κ(cpl)\n2\n2a2+/radicalBig\nκ(cpl)\n1a(in),\n(−iω)a2=/parenleftBig\n−iω2−κ2\n2/parenrightBig\na2−ig2m−κ(cpl)\n2\n2a2−/radicalBig\nκ(cpl)\n1κ(cpl)\n2\n2a1+/radicalBig\nκ(cpl)\n2a(in),\n(−iω)m=−ig1a1−ig2a2+/parenleftBig\n−iωm−γm\n2/parenrightBig\nm,\nain+aout=/radicalBig\nκ(cpl)\n1a1+/radicalBig\nκ(cpl)\n2a2.\nwhereω1,ω2, andωmare resonant frequencies of the cavity modes and the magnon mod e, respectively, κ1andκ2\nare dissipation rates of the cavity modes, and γmis the dissipation rate of the magnon mode.\nBy solving those equations, we find the following expression for S11:\naout\nain=−1+κ(cpl)\n1F2+κ(cpl)\n2F1−2/radicalBig\nκ(cpl)\n1κ(cpl)\n2F3\nF1F2−F2\n3,6\nwhere the following notations are used:\nF1=i(ω1−ω)+κ1+κ(cpl)\n1\n2+g2\n1\ni(ωm−ω)+γm\n2,\nF2=i(ω2−ω)+κ2+κ(cpl)\n2\n2+g2\n2\ni(ωm−ω)+γm\n2,\nF3=/radicalBig\nκ(cpl)\n1κ(cpl)\n2\n2+g1g2\ni(ωm−ω)+γm\n2.7\nTEMPERATURE DEPENDENCE OF THE MAGNON-PHOTON COUPLING STRE NGTH IN THE\nFERRIMAGNETIC PHASE\nThe coupling strengths g1andg2depended strongly on temperature. In particular, the temperat ure dependence\nof the coupling strength g1(T) is shown in Fig.S1-6 (measurements were performed in the zero-fi eld-cooled regime).\nAssuming that the effective total number of spins is proportional t o the net magnetization M,N(T)∝M(T), we find\nthat the experimental data can be well described by the/radicalbig\nM(T) function normalized to the value of coupling strength\nat low temperatures (see Fig.S1-6(c)), where M(T) is calculated using the Weiss model of ferromagnetism described\nbelow. In our experiments, it was not possible to measure the magne tization profile M(T) directly. However,\nin Fig. S1-6(c), we plot a theoretical estimate of the temperature dependence of the coupling strength based on\nthe magnetization profile of a Cu 2OSeO3sample of isotropic shape which we measured at the magnetic field of\nH0= 3000Oe, H0∝bardbl[110], using a standard SQUID measurement system. A slight discre pancy between the SQUID\ndata and our results is caused by 1) different demagnetization fact ors of the two samples, and 2) a difference between\nthe actual temperature of the sample and the value measured by a temperature sensor mounted on the wall of a\nhelium-flow cryostat used in our experiments.\n2000 4000 6000\nMagnetic Field (Oe)9.49.69.810Frequency (GHz)\n 0.10.20.30.4\n2000 4000 6000\nMagnetic Field (Oe)9.49.69.810Frequency (GHz)\n0.10.20.3\n0 25 50 75 100\nTemperature (K)0200400600\n|S11|2|S11|2FMPMT  64 K\ng1/2  400 MHz g1/2  200 MHzT  70 K\nTC(b) (a) (c)\nFIG. S1-6. (color online) (a) Microwave response |S11|2at 64K. Dashed lines correspond to theoretical estimates ca lculated\nby using equations described in the main text and g1/2π≈400MHz. (b) Microwave response |S11|2at 70K. Dashed lines\ncorrespond totheoretical estimates calculated byusingth e valueg1/2π≈200MHz. (c)Temperature dependenceofthe coupling\nstrength g1in ferrimagnetic (FM) and paramagnetic (PM) phases: experi mental data (squares) obtained in the zero-field-cooled\nregime and theoretical estimations for g1∝/radicalbig\nM(T) forH= 0Oe (solid blue line) and H= 3000Oe (solid red line). The\ndashed black line corresponds to an estimation of the coupli ng strength based on the magnetization profile of a Cu 2OSeO 3\nsample of isotropic shape measured at the magnetic field of H0= 3000Oe, H0∝bardbl[110].\nWe will use the Weiss model of ferromagnetism[S1-4] to calculate th e net magnetization Min the ferrimagnetic\nphase as a function of the temperature Tand the external magnetic field B,M=M(T,B). The ferromagnetic\ninteraction is modeled by introducing the molecular field\nBmf=λM,\nwhereλis a constant parameter. For a ferromagnet, λ >0.\nThe magnetization Mcan be found by solving self-consistently the following non-linear equ ations:\nM=MsBJ(y), (S1-3)\ny=gJµBJ(B+λM)\nkBT, (S1-4)\nwhere the saturation magnetization Msis\nMs=ngJµBJ,\nBJ(y) is the Brillouin function given by\nBJ(y) =2J+1\n2Jcoth/parenleftbigg2J+1\n2Jy/parenrightbigg\n−1\n2Jcoth/parenleftBigy\n2J/parenrightBig\n,8\n0 20 40 60 80 100\nTemperature (K)00.20.40.60.811.2M/MsB = 0 T\nB = 0.3 T\nFIG. S1-7. (color online) Temperature dependence of magnet ization at different values of the external magnetic field: 0 T (blue\ncurve) and 0.3 T (red curve) according to numerical calculat ions.\nandgJis ag-factor, µBis the Bohrmagneton, Jis aspin number, kBis the Boltzmannconstant, Tisthe temperature,\nandnis the spin density.\nIt can be shown that the parameter λis related to the Curie temperature TCas [S1-4]\nλ=3kBTC\ngJµB(J+1)Ms.\nFor Cu 2OSeO3,J= 1,gJ≈2,n= 4/a3(lattice constant a= 8.93˚A), and, by assuming that TC≈67K at\nthe magnetic field B0≈0.3T, equations(S1-3-S1-4) can be solved numerically. Results of nu merical calculations for\nmagnetic field values 0T and 0.3T are shown in Fig.S1-7.9\nEFFECT OF THE MAGNETIC PERMEABILITY OF THE MATERIAL ON THE CA VITY RESONANCE\nFREQUENCY\nIn SI units, the relative dc magnetic permeability can be calculated fr om the field dependence of the net magneti-\nzation as µdc= 1+M/H. In our experiments, the direct measurement of the magnetizatio nM(H) was not possible,\ntherefore in our analysis we will use the data from[S1-5] which was ob tained in the experiments with H∝bardbl[111]. In\nour measurements, the orientation between the applied magnetic fi eldHand crystal axes was arbitrary, therefore\nthe magnetic fields of the magnetic phase transitions observed in ou r experiments were different from ones shown in\nFig.S1-8. However, the dependence of the dc permeability on the a pplied magnetic field in our experiments should\nbe qualitatively similar to the one presented in Fig.S1-8(b).\nObviously, there is a significant difference between the experimenta l data on the magnetic field dependence of cavity\nresonance frequencies presented in the main text and the one sho wn in Fig.S1-8(d). In our experiments, the bare\ncavity mode frequency did not depend on the magnetic field — values o f cavity resonance frequencies at very low\nmagnetic fields and at very high magnetic fields were very close to eac h other. Therefore, we can make a conclusion\nthat the cavity mode was not affected by the dc permeability.\n0 1000 2000 3000\nH (Oe)00.10.20.30.40.50.6M (B /Cu2+)\n0 1000 2000 3000\nH (Oe)1.41.61.82dc\n1 1.25 1.5 1.75 2\neff9.49.59.69.79.8Frequency (GHz)\n0 1000 2000 3000\nH (Oe)9.49.59.69.7Frequency (GHz)(a)\nH C FM(b)\nH C FM\n(c) (d)\nH C FM\nFIG. S1-8. (color online) Simulation of the effect of the dc pe rmeability on the cavity resonance frequency in the case H∝bardbl[111].\n(a) Magnetic field dependence of the magnetization at 5K (dat a is taken from[S1-5]). (b) The dc magnetic permeability\nµdccalculated from the magnetization data. (c) The dependence of the cavity resonance frequency on the effective magnetic\npermeability µeffaccording to numerical simulations in CST. (d) The calculat ed dependence of the cavity resonance frequency\non the magnetic field caused by the variation of the dc permeab ility (under assumption µeff=µdc). The calculated data is\nnot consistent with the behavior of the cavity mode observed in our experiments. Thus, experimental data presented in th e\nmain text cannot be explained by the effect of the dc permeabil ity. Letter symbols H, C, and FM denote helical, conical, and\nferrimagnetic states, respectively. Vertical lines corre spond to the transitions between magnetic phases.\nHowever, it should be emphasized that microwave response of magn etic materials can be described using an ac\npermeability[S1-6, S1-7], and many aspects of magnon-photon cou pling phenomena can be explained in terms of an1\neffective ac permeability[S1-8].\n∗leonid.abdurakhimov.nz@hco.ntt.co.jp; Present address : NTT Basic Research Laboratories, NTT Corporation, Atsugi ,\nKanagawa 243-0198, Japan\n†h.kurebayashi@ucl.ac.uk\n[S1-1] E. Ruff, P. Lunkenheimer, A. Loidl, H. Berger, and S. Kr ohns, Scientific Reports 5, 15025 (2015).\n[S1-2] C. Gardiner and P. Zoller, Quantum Noise , 2nd ed. (Springer, Berlin, Heidelberg, 2000) Chapter 5.\n[S1-3] D. Walls and G. Milburn, Quantum Optics , 2nd ed. (Springer, Berlin, Heidelberg, 2008) Chapter 7.\n[S1-4] S. Blundell, Magnetism in Condensed Matter (Oxford University Press, 2001) Chapter 5.\n[S1-5] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 (2012).\n[S1-6] P. Queffelec, S. Mallegol, and M. LeFloc’h, IEEE Trans actions on Microwave Theory and Techniques 50, 2128 (2002).\n[S1-7] J. Krupka, B. Salski, P. Kopyt, and W. Gwarek, Scienti fic Reports 6, 34739 (2016).\n[S1-8] P. Hyde, L. Bai, M. Harder, C. Dyck, and C.-M. Hu, Phys. Rev. B95, 094416 (2017).2\nSupplemental Material for “Magnon-photon coupling in a non -collinear magnetic\ninsulator Cu 2OSeO 3”. Part 2: Calculations of the overlap integral ηm.\nMasahito Mochizuki\nDepartment of Applied Physics, Waseda University, Okubo, S hinjuku-ku, Tokyo 169-8555, Japan\nNumerical calculations of the overlap integrals ηmfor magnon modes in Cu 2OSeO3.\nMAGNETISM OF THE CHIRAL CUBIC COPPER OXOSELENITE\nA spin model for Cu 2OSeO3is constructed as follows. The crystal structure of this compoun d belongs to the chiral\ncubic P2 13 space group, while the magnetic structure is comprised of tetrah edra composed of four Cu2+(S=1/2) ions\nat their apexes. The Cu spins in each tetrahedron form a three-up and one-down ferrimagnetic ( S=1) configuration\nbelowTc∼58 K, and this spin assembly can be regarded as a magnetic unit becau se the intra-tetrahedron spin\ncouplingsaremuch strongerthan the inter-tetrahedronspin cou plings. Thecrystallographicunit cell ofthis compound\nis a cube whose volume is a3with the lattice constant of a=8.93˚A. Because the unit cell contains four tetrahedra,\neach tetrahedron occupies a spatial volume of a3\n0wherea0=a/rwithr= 41/3.\nSPIN MODEL AND COARSE GRAINING\nThe magnetism of Cu 2OSeO3is described by a classical Heisenberg model on a cubic lattice. The Ha miltonian is\ngiven by,\nH0=−J0/summationdisplay\ni,γ=ˆx,ˆy,ˆzmi·mi+ˆγ\n−D0/summationdisplay\ni,γ=ˆx,ˆy,ˆz(mi×mi+ˆγ)·ˆγ\n−gµBSH0·/summationdisplay\nimi\n+K0/summationdisplay\ni/parenleftbig\nm4\nia+m4\nib+m4\nic/parenrightbig\n(S2-1)\nwheremiis a normalized classical magnetization vector for the ith tetrahedron. The first term describes the fer-\nromagnetic exchange interactions, while the second term depicts t he Dzyaloshinskii-Moriya interactions with ˆx,ˆy,\nandˆzbeing the directional vectors along the x,yandzaxes. The third term represents the Zeeman interactions\nassociated with an external magnetic field H0whereg(= 2) is the Lande’s g-factor, µ0is the magnetic permeability\nfor vacuum, and S(= 1) is the spin amplitude per tetrahedron. The last term denotes t he fourth-order magnetic\nanisotropy allowed in the cubic crystals where a,bandcare the orthogonal coordinates in the cubic setting.\nFor slowly varying spin textures in the helical, conical and skyrmion ph ases, the spins are nearly decoupled from\nthe background lattice structure. It justifies our theoretical t reatment based on a spin model on the cubic lattice for\nsimplicity without consideringcomplicated crystalstructures ofre almaterials. This spin Hamiltonian exhibits a phase\ntransition between the paramagnetic and the helical states at kBTc=1.43J0whenH0=0. From the experimentally\nobserved Tc∼58 K,J0is evaluated as J0=3.50 meV.\nAs far as the slowly varying spin textures are concerned, this lattic e spin model is rewritten in a continuum form\nas,\nH1=/integraldisplay\nd3rJ0\n2a0(∇m)2+D0\na2\n0m·(∇×m)\n−gµBS\na3\n0H0·m+K0\na3\n0/summationdisplay\ni/parenleftbig\nm4\na+m4\nb+m4\nc/parenrightbig\n(S2-2)\nThis expression indicates that we can reconstruct a lattice spin mod el by regarding an assembly of magnetizations in\na cubic cell with volume of a3=(ra0)3(instead of a single spin or a single magnetic unit) as one magnetization . This3\nTemperature kBT/J= 1T=J0/kB=40.6 K\nMagnetic field gµBH/J= 0.001H0=H/r3=120 Oe\nFrequency f=ω/2π/planckover2pi1ω/J= 0.001 f=1.35 GHz\nTime tJ//planckover2pi1= 1 t=0.12 ps\nTABLE I. Unit conversion table when J=rJ0=5.56 meV and S=1.\ncoarse graining again gives a classical Heisenberg model on a cubic lat tice as,\nHc.g.=−J/summationdisplay\ni,γ=ˆx,ˆy,ˆzmi·mi+ˆγ\n−D/summationdisplay\ni,γ=ˆx,ˆy,ˆz(mi×mi+ˆγ)·ˆγ\n−gµBSH·/summationdisplay\nimi\n+K/summationdisplay\ni/parenleftbig\nm4\nia+m4\nib+m4\nic/parenrightbig\n. (S2-3)\nHeremiis a normalized classical vector describing a representative magnet ization in the ith cubic cell. We find that\nfollowing relationships hold between coupling constants of the origina l spin model H1and those of the coarse-grained\nspin model Hc.g.,\nJ=rJ0, D=r2D0,H=r3H0, K=r3K0. (S2-4)\nWhen we adopt the crystallographic unit cell ( a=8.93˚A) as a cubic cell for the coarse graining, r=a/a0becomes\n41/3because the unit cell contains four tetrahedra.\nIn the following calculations, we adopt J=rJ0=5.56 meV as the energy units, and we use D/J=-0.09 and K/J=-\n0.003. This Dzyaloshinskii-Moriya parameter reproduces the exper imentally observed helical periodicity of λH=63\nnm. The ground-state energy of the helical order is given as a func tion of the spin rotation angle θas\nE(θ)/N=−Jcosθ−Dsinθ. (S2-5)\nSolving the saddle-point equation\ndE(θ)\ndθ=Jsinθ−Dcosθ= 0, (S2-6)\nwe obtain a relationship tan θ=D/Jfor the helical and conical states with a minimum energy, which gives θ=5.142◦\nandλH=(360◦/θ)a=63 nm when D/J=0.09 and a=8.93˚A. This set of parameters also reproduces the experimentally\nobserved critical magnetic field of ∼750 Oe for the phase transition between the conical and the ferrim agnetic states\nat low temperatures. In the numerical simulations, we found that t he phase transition occurs at gµBSHc=0.006J,\nwhich corresponds to H0c=Hc/r3=720 Oe. The unit conversions when J0=3.50 meV ( J=rJ0=5.56 meV) and S=1\nare summarized in Table I.\nWe define the x,yandzaxes oriented as x∝bardbl[1¯21],y∝bardbl[10¯1], andz∝bardbl[111]. With these orthogonal coordinates,\nthe fourth-order magnetic anisotropy term is rewritten in the for m,\nK/summationdisplay\ni/parenleftBigg\n1\n2+m2\niz−7\n6m4\niz−2√\n2\n3miz(m3\nix−3mixm2\niy)/parenrightBigg\n(S2-7)\nFor the external magnetic field H, we consider both the static magnetic field Hext= (0,0,Hz) applied along the z\naxis and the time-dependent magnetic field H(t).\nRESULTS FOR RESONANCE MODES\nWe calculate dynamical magnetic susceptibilities to evaluate the reso nance frequencies of magnon modes. We first\nperformed Monte-Carlothermalizationsfor the Hamiltonian Hc.g.to obtain stable magnetizationconfigurationsat low4\n00.20.40.60.81\n0 5 10 15 20 \nf (GHz) 00.20.40.60.81Im /g70y (a.u.) \n0 5 10 15 20 \nf (GHz)(a) (b)\nHz=240 OeHz=600 Oe\nHz=360 OeHz=480 OeConical Phase\n(Hz<720 Oe)Ferromagnetic Phase\n(Hz>720 Oe)\nHz=720 Oe Hz=1080 Oe \nHz=840 Oe Hz=960 Oe n=1 \n0246810 12 \n300 600 900 1200 fR (GHz) \nHz (Oe)Conical Ferromagnetic(c)\nn=2 mode\nn=1 modeIm /g70y (a.u.) n=2 \nFIG. S2-1. (a), (b) Calculated imaginary parts of the dynami cal magnetic susceptibilities in (a) the conical phase and ( b)\nthe ferromagnetic phase for various magnetic-field strengt hsHz. For the conical phase, the resonance peaks of the n=2 mode\nappear in addition to the large peaks of the lower-lying n=1 mode. (c) Resonance frequencies of the conical n=1 and n=2\nmodes and the ferromagnetic resonance mode as functions of Hz.\ntemperatures ( kBT/J=0.01) for various values of Hz. Then we further relaxed them in the micromagnetic simulations\nusing the Landau-Lifshitz-Gilbert (LLG) equation to obtain the gro und-state magnetization configurations. The LLG\nequation was solved by the fourth-order Runge-Kutta method. A system of 20 ×20×140 sites with periodic boundary\nconditions were used for the numerical calculations. The LLG equat ion is given by,\ndmi\ndt=−γmi×Heff\ni+αG\nmmi×dmi\ndt. (S2-8)\nHereαGis the Gilbert-damping coefficient. The effective magnetic field Heff\niacting on the local magnetization mi\non theith cell is calculated from the Hamiltonian Hc.g.in the form\nHeff\ni=−1\nγ/planckover2pi1r3∂Hc.g.\n∂mi. (S2-9)\nThedynamicalmagneticsusceptibilities foramicrowavefield H(t)∝bardblyperpendiculartotheexternalstaticmagnetic\nfieldHextis given by,\nχy(ω) =∆My(ω)\nHy(ω)(S2-10)\nHereHy(ω) and ∆My(ω) are Fourier transforms of the time-dependent magnetic field H(t) and the simulated time-\nprofile of the total magnetization ∆ M(t) =M(t)−M(0) with M(t) =1\nN/summationtextN\ni=1mi(t). For these calculations, we\nuse a short rectangular pulse for the time-dependent field H(t) whose components are given by,\nHy(t) =/braceleftBigg\nHpulse 0≤tJ//planckover2pi1≤1\n0 others(S2-11)\nAn advantageofusing the shortpulse is that for a sufficiently short duration ∆ twithω∆t≪1, the Fouriercomponent\nHy(ω) becomes constant being independent of ωup to the first order of ω∆t. The Fourier component is calculated as\nHy(ω) =/integraldisplay∆t\n0Hpulseeiωtdt=Hpulse\niω/parenleftbig\neiω∆t−1/parenrightbig\n∼Hpulse∆t. (S2-12)\nAs a result, we obtain the relationship χy(ω)∝∆My(ω).\nIn Fig. S2-1(a) and (b), we show calculated imaginary parts of the d ynamical magnetic susceptibilities Im χyfor the\nconical phase and the ferromagnetic phase respectively. We find t hat in the conical phase, a tiny peak corresponding\nto then=2 mode appears in each spectrum in addition to the large peak of the lower-lying n=1 mode. In Fig. S2-1(c)\nwe plot the calculated resonance frequencies as functions of Hz.5\n-0.00600.006\n0 2 4 6 8 10 12 14 0.5 1\n00.5\ntims (ns)/g39My(t)/g75(t)\n/g75(t)-0.0200.02 /g39My(t)(a)\n(b)(d)\n(e)\nHz=240 OeHz=600 Oe\nHz=360 OeHz=480 Oe\nHz=720 OeHz=1080 Oe\nHz=840 OeHz=960 Oe/g75(t)\n00.4000.400.4\n0.6 10.6 10.6 1\n0.40.6 1 (c)\n(f)Conical n=2 mode (Hz=480 Oe, fR=12.5 GHz)\nFerromag. resonance ( Hz=840 Oe, fR=5.06 GHz)\n0 2 4 6 8 10 12 14 \ntims (ns)\n0 2 4 6 8 10 12 14 \ntims (ns)0 2 4 6 8 10 12 14 \ntims (ns)\nFIG. S2-2. (a), (d) Simulated time profiles of oscillating co mponent of the net magnetization ∆ M(t) =M(t)−M(0) with\nM(t) = (1/N)/summationtext\nimi(t) under application of the y-polarized microwave field with a resonant frequency for (a) the conical n=2\nmode when Hz=480 Oe and (b) the ferromagnetic resonance mode when Hz=840 Oe. (b), (e) Calculated time evolutions of\nthe microwave-magnon overlap integral ηfor (b) the conical n=2 mode when Hz=480 Oe and (e) the ferromagnetic resonance\nmode when Hz= 840 Oe. (c), (f) Time evolutions of ηfor various magnetic-field strengths Hz. Those for the conical n=2\nmode always converge at a very tiny value of ∼0.01 after a sufficient duration, whereas those for the ferrom agnetic resonance\nare constantly unity.\nRESULTS FOR THE OVERLAP INTEGRALS\nFollowing works[S2-1, S2-2], we calculate the overlap integral ηwhich describes the extent of an overlap between a\nmicrowave and a magnon mode. This quantity is given by,\nη(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraltext\nH(r,t)·∆m(r,t)dr\nmax|H(r,t)|max|∆m/bardbl(r,t)|V/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (S2-13)\nwhereH(r,t) is a microwave magnetic field, ∆ m(r,t)≡m(r,t)−m(r,0) is an oscillating component of the local\nmagnetization m(r,t), ∆m/bardbl(r,t) is a component of ∆ mparallel to H(r,t), andVis the system volume. Here\nmax|H(r,t)|and max |∆m/bardbl(r,t)|denote maximum values in the spatial volume at each time t. In the case of\ncontinuous excitations, this quantity convergesat a constant va lue and becomes time independent when the excitation\nbecomes steady after a sufficiently long duration.\nTo calculate this quantity, we simulate time profiles of ∆ m(r,t) under application of a sinusoidally oscillating\nmicrowave field H(t)∝bardblyby numerically solving the Landau-Lifshitz-Gilbert equation using the fourth-order Runge-\nKutta method. The y-axis component of H(t) is given by\nHy(t) =Hωsinωt. (S2-14)\nHere the amplitude of microwave field is fixed at gµBSHω/J= 0.00001 when calculating ηfor the conical n=1\nmode and the ferromagnetic resonance mode, whereas at gµBSHω/J= 0.0002 for the conical n=2 mode. The6\n01/g75\n300 600 900 1200 \nHz (Oe)Conical\nFerromagnetic00.020.04\n300 600 Hz (Oe)\nn=2 mode0.20.6\n0.40.8\n/g75\nFIG. S2-3. Calculated values of microwave-magnon overlap i ntegrals η(t) at the moment of time t=15.36 ns for the conical\nn=2 mode and the ferromagnetic resonance mode as functions o f the magnetic-field strength Hz. The inset magnifies the plots\nfor the conical n=2 mode.\nGilbert-damping coefficient is fixed at αG= 0.02. We performed the calculation for various resonance modes and\nmagnetic-field strengths Hzby substituting the corresponding resonance frequencies ωRintoω.\nIn Fig. S2-2(a) and (d), we show simulated time profiles ofoscillating c omponent ofthe net magnetization∆ M(t) =\nM(t)−M(0) with M(t) = (1/N)/summationtext\nimi(t) under application of the y-polarized microwave field with a resonant\nfrequency for the conical n=2 mode when Hz=480 Oe and the ferromagnetic resonance mode when Hz=840 Oe,\nrespectively. We find that steady oscillations occur after a duratio n of∼4 ns. Accordingly, the time evolution of η\nfor the conical n=2 mode in Fig. S2-2(b) is time-dependent, whereas that for the fe rromagentic resonance mode in\nFig. S2-2(e) takes constantly unity. These mode-specific behavio rs do not alter even when the magnetic-field strength\nHzvaries as seen in Fig. S2-2(c) and (f) which summarize the simulated t ime profiles of ηfor various values of Hz.\nAlthough it is difficult to evaluate the precise saturation value of η(t) by the exponential fitting, the time profiles of\nη(t) shown in Fig. S2-2(c) indicate that the mode overlappingfor the n= 2 helimagnon mode is significantly suppressed\nas compared to the one for the ferromagnetic resonance mode. H owever, the value is still expected to be finite as\nevidenced by the time profile of ∆ My(t) in Fig. S2-2(a), which shows a finite amplitude of the steadily oscillatin g\nn=2 mode, indicating the finite coupling between the microwave driving fi eld and the n=2 helimagnon mode. We\nexpect that ηfor then=2 helimagnon mode is typically of the order of 0.01 according to the plo t in the Fig. S2-3\nwhich shows the field-dependence of η(t) after sufficient duration ( t=15.56 ns).\n∗leonid.abdurakhimov.nz@hco.ntt.co.jp; Present address : NTT Basic Research Laboratories, NTT Corporation, Atsugi ,\nKanagawa 243-0198, Japan\n†h.kurebayashi@ucl.ac.uk\n[S2-1] N.J. Lambert et al., J. Appl. Phys. 117, 053910 (2015) .\n[S2-2] X. Zhang et al., J. Appl. Phys. 119, 023905 (2016)."    },    {        "title": "1104.2399v1.Optimized_Effective_Potential_Model_for_the_Double_Perovskites_Sr2_xYxVMoO6_and_Sr2_xYxVTcO6.pdf",        "content": "arXiv:1104.2399v1  [cond-mat.mtrl-sci]  13 Apr 2011Optimized Effective Potential Model for the Double\nPerovskites Sr 2−xYxVMoO 6and Sr 2−xYxVTcO 6\nI V Solovyev\nNational Institute for Materials Science, 1-2-1 Sengen, Tsukuba , Ibaraki 305-0047,\nJapan\nE-mail:solovyev.igor@nims.go.jp\nAbstract. In attempt to explore half-metallic properties of the double perovs kites\nSr2−xYxVMoO 6and Sr 2−xYxVTcO 6, we construct an effective low-energy model,\nwhich describes the behavior of the t2g-states of these compounds. All parameters\nof such model are derived rigorously on the basis of first-principles electronic structure\ncalculations. In order to solve this model we employ the optimized effe ctive potential\nmethod and treat the correlation interactions in the random phase approximation.\nAlthough correlation interactions considerably reduce the intraat omic exchange\nsplitting in comparison with the Hartree-Fock method, this splitting s till substantially\nexceeds the typical values obtained in the local-spin-density appro ximation (LSDA),\nwhich alters many predictions based on the LSDA. Our main results ar e summarized\nas follows: (i) all ferromagnetic states are expected to be half-me tallic. However, their\nenergies are generally higher than those of the ferrimagnetic orde ring between V- and\nMo/Tc-sites (except Sr 2VMoO 6); (ii) all ferrimagnetic states are metallic (except fully\ninsulating Y 2VTcO 6) and no half-metallic antiferromagnetism has been found; (iii)\nmoreover, many of the ferrimagnetic structures appear to be un stable with respect to\nthe spin-spiral alignment. Thus, the true magnetic ground state o f the most of these\nsystemsisexpected tobemorecomplex. Inaddition, wediscusssev eralmethodological\nissues related to the nonuniqueness of the effective potential for the magnetic half-\nmetallic and insulating states.\nPACS numbers: 75.25.-j, 72.80.Ga, 71.10.-w, 75.47.-mOptimized Effective Potential Model for Double Perovskites 2\n1. Introduction\nRecently double perovskites A2−xA′\nxBB′O6, where AandA′are the di- and three-\nvalent (typically alkali-earth or rare-earth) elements, respective ly, and B(B′) are\nthe transition-metal elements, have attracted considerable att ention [1], following\nexperimental discoveries of the room-temperature ferromagne tism and the colossal-\nmagnetoresistance effect in Sr 2FeMoO 6[2], which make these materials promising in\nthe field of spin electronics. The extraordinary properties of the d ouble perovskites are\nfrequently attributed to their half-metallic electronic structure, although the situation\ncan be rather subtle [3]. Nevertheless, the first discoveries spark the new research\nactivity aiming at the systematic search of half-metallic double-pero vskite materials.\nMany such predictions has been made on the basis of first-principles electronic structure\ncalculations. Besides ferromagnetic (FM) compounds, a particular attention was paid\nto the half-metallic antiferromagnets – the systems, where the ha lf-metallicity coexists\nwith the zero net magnetization [4, 5]. However, even apart from th e technological\nissues, related to the defects in the double-perovskite structur e, there is a number of\nfundamental problems, which call many predictions into question:\n(i) Most of the calculations are based on the local-spin-density appr oximation (LSDA)\nor its refinement – the generalized gradient approximation, which ha ve many\nlimitations for the transition-metal oxides. Particularly, all these t heories employ\nthe functional dependence of the exchange-correlation energy obtained in the limit\nof homogeneous electron gas. However, many unique properties o f the double\nperovskites are related to the alternation of two different transit ion-metal sites\nBandB′, and in this sense the systems are essentially inhomogeneous. Thus , even\nthough the double-perovskite compounds may be classified as itiner ant electron\nmagnets, it does not necessary mean that their properties are we ll described within\nLSDA;\n(ii) In the vast majority of calculations, it is believed that the LSDA+ Utechnique,\nwhich incorporates the physics of on-site Coulomb interactions [6, 7], can serve as\na reasonable alternative to LSDA and provide more superior descrip tion for the\nproperties of double perovskites. However, the LSDA+ Uscheme for the double\nperovskites is ill-defined: there are many uncertainties related to t he values of the\non-site Coulomb repulsion Uat the sites BandB′, which are frequently taken\nas adjustable parameters, the subspace of “localized orbitals”, a nd the double-\ncounting term [8]. Moreover, the total energy in LSDA+ Uis taken in the ad hoc\nform, which is valid only in the atomic limit and equivalent to some constra ined\ndensity-functional theory [7]. In such a situation, all predictions o f the half-metallic\nbehavior for the double perovskites should be taken very cautious ly.\nIn this paper we will illustrate how these problem could be resolved by a pplying\nthe optimized effective potential (OEP) method, and what the cons equences on the\nelectronic andmagnetic propertiesofthedoubleperovskites could be. The OEPmethodOptimized Effective Potential Model for Double Perovskites 3\nis formulated in the spirit of Kohn-Sham density functional theory [9 ], by introducing\nan auxiliary one-electron problem, which is specified by some local effe ctive potential\nand which is used for the search of the true spin density through th e minimization of the\ntotal energy of the system [10, 11, 12, 13]. In this work we will apply the OEP technique\nforthe solutionof thelow-energy model, where allthe parameters arederived rigorously,\non the basis of first-principles electronic structure calculations. B esides simplicity, such\na model analysis typically allows one to concentrate directly on the mic roscopic picture,\nunderlying the considered phenomena. For the first applications of the OEP model, it is\nmore convenient to consider the t2g-systems, whose electronic and magnetic properties\nare predetermined by the behavior of the t2g-states of the sites BandB′located near\nthe Fermi level. In this case, the effective potential is specified by o nly three parameters,\nwhich should be used for the minimization of the total energy.\nWeselect two prototypical examples Sr 2−xYxVMoO 6andSr 2−xYxVTcO 6, for which\nwe adopt the undistorted double perovskite structure with the lat tice parameters a=\n7.846 and 7.899 ˚A, respectively. In fact, a= 7.846 ˚A is the experimental lattice\nconstant for SrLaVMoO 6[14], which is close to theoretical values obtained in the first-\nprinciples calculations [15], and a= 7.899 ˚A is the theoretical lattice constant for\nLa2VTcO 6[16]. Deviations from the ideal double perovskite structure are exp ected to\nbe small [16] and were not considered here. The band-filling depende nce due to the\nsubstitution of Sr by Y was treated in the virtual-crystal approxim ation. According to\nthe electronic structure calculations in the local-density approxima tion (LDA), the basic\ndifference between Sr 2VMoO 6and Sr 2VTcO 6is in the relative position of the V- and\nMo(Tc)-states: in Sr 2VMoO 6, the V- and Mo-states are separated in energy and form\ntwo nonoverlapping bands, while in Sr 2VTcO 6these states form one common band\n(Fig. 1). Both Sr 2−xYxVMoO 6and Sr 2−xYxVTcO 6(and other isoelectronic to them\n/s49/s48 /s56 /s54 /s52 /s50 /s48/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s68/s101/s110/s115/s105/s116/s121 /s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41/s32\n/s32/s32\n/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s85/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s87/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s88/s32/s32/s75/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s88/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s76\n/s49/s48 /s56 /s54 /s52 /s50 /s48\n/s68/s101/s110/s115/s105/s116/s121 /s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41/s32\n/s32/s32\n/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\n/s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s85/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s87/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s88/s32/s32/s75/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s88/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s76\nFigure 1. Electronic structure near the Fermi level in the local-density appr oximation\n(total and partial densities of states, and the energy dispersion of thet2g-bands) for\nSr2VMoO 6(left) and Sr 2VTcO 6(right). The position of the Fermi level is shown by\nthe dot-dashed line.\ncompounds compounds) were intensively discussed in the literature in the context of the\nhalf-metallic applications. For instance, SrLaVRuO 6, SrLaVMoO 6, and La 2VTcO 6were\nproposed to be the half-metallic antiferromagnets [14, 16, 17, 18]. On the other hand,\nthe metallic ferrimagnetic (i.e., non spin-compensated) behavior was also suggested for\nSrLaVMoO 6[15, 19]. In general, such conclusions are sensitive to the values of t heOptimized Effective Potential Model for Double Perovskites 4\non-site Coulomb repulsion U, which are used in the calculations.\nThe rest of the paper is organized as follows. The method is discusse d in Sec. 2.\nParticularly, in Sec. 2.1 we briefly describe construction of the low-e nergy model for\nthe double perovskites and in Sec. 2.2 – the basic idea of the OEP meth od. Results of\nthe model OEP calculations are shown in Sec. 3. Particularly, the obt ained electronic\nband structure is discussed in Sec. 3.1, the screening of on-site int eractions due to\nthe correlation effects and its consequences on the properties of Sr2−xYxVMoO 6and\nSr2−xYxVTcO 6is considered in Sec. 3.2, and the behavior of the total energy and\nstability of the ferrimagnetic states are discussed in Sec. 3.3. Finally , brief summary of\nthe work is presented in Sec. 4.\n2. Method\n2.1. Construction and parameters of the low-energy model\nThe first step of our approach is the construction of the effective low-energy model for\nthet2g-states of the double perovskites:\nˆH=/summationdisplay\nij/summationdisplay\nσ/summationdisplay\nmm′tmm′\nijˆc†\nimσˆcjm′σ+1\n2/summationdisplay\ni/summationdisplay\nσσ′/summationdisplay\nmm′m′′m′′′Ui\nmm′m′′m′′′ˆc†\nimσˆc†\nim′′σ′ˆcim′σˆcim′′′σ′,(1)\nwhich is derived in the Wannier basis, by starting from the LDA band st ructure. The\nWannier functions are specified by the indices m=xy,yz, orzx, andσstands for the\nspin indices ↑or↓. The one-electron part tmm′\nijof the model Hamiltonian was derived\nby using downfolding method, and the Coulomb (and exchange) inter actionsUi\nmm′m′′m′′′\nwere obtained by combining the constrained density-functional th eory with the random-\nphase approximation (RPA). The method was discussed in the literat ure, and for details\nthe reader is referred to the review article [20]. There are three s ets of parameters in\nthe model Hamiltonian (1):\n(i) the “charge transfer” energies, or the energy splitting betwe en “atomic” t2g-levels of\nthe V and Mo (Tc) sites, which are described by the site-diagonal pa rt oftmm′\nij. The\nvalues of these parameters are −391 and 94 meV for the VMo and VTc compounds,\nrespectively. Thus, as expected from the LDA band structure (F ig. 1), for the VMo\ncompounds, the atomic V-states are located lower in energy and we ll separated\nfrom the Mo-states, while for the VTc compounds, the situation is r eversal and the\nsplitting is considerably smaller;\n(ii) the transfer integrals, which are described by the off-diagonal elements of tmm′\nijwith\nrespect to the site indices. As expected, the largest parameter is theddπ-transfer\nintegral between nearest-neighbor V- and Mo(Tc)-sites, which is 271 (245) meV.\nAnother important parameter is the ddσ-transfer integral between next nearest\nneighbors of the same atomic type. Its values in the V- and Mo-subla ttices of\nthe VMo compounds are −72 and−116 meV, respectively. Similar values were\nobtained forthe VTc compounds: −73 and−113 meV forthe V- andTc-sublattice,\nrespectively;Optimized Effective Potential Model for Double Perovskites 5\n(iii) intraatomic Coulomb and exchange interactions in the t2g-band, which are screened\nby all other states. At each site of the system, these interaction s are fully\nspecified by two Kanamori parameters [21]: the intraorbital Coulom b interaction\nU=Ummmmand the exchange interaction J=Umm′m′mform/negationslash=m′. In the perfect\ncubic environment, the third Kanamori parameter U′=Ummm′m′(the interorbital\nCoulomb interaction) can be related to UandJasU′=U−2J[20]. Other types\nof matrix elements of /bardblUmm′m′′m′′′/bardblare equal to zero. The band-filling dependence\nofU,U′, andJ, as obtained in the virtual crystal approximation, is explained in\nFig. 2. Generally, as the number of t2g-electrons increases, the effective Coulomb\n/s48/s49/s50/s51/s52\n/s74/s32/s77/s111/s85\n/s32/s39/s77/s111/s85/s32/s77/s111\n/s74/s32/s86/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s101/s86/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54/s85/s32/s86\n/s85\n/s32/s39/s86\n/s48/s49/s50/s51/s52\n/s74/s32/s84/s99/s85\n/s32/s39/s84/s99/s85/s32/s84/s99\n/s74/s32/s86/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s101/s86/s41\n/s32/s32\n/s32/s32/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s84/s99/s79\n/s54/s85/s32/s86\n/s85\n/s32/s39/s86\nFigure 2. Intraorbital Coulomb interaction U, interorbital Coulomb interaction U′,\nand the exchange interaction Jfor the Sr 2−xYxVMoO 6(left) and Sr 2−xYxVTcO 6\n(right) compounds.\ninteractions also increase, reflecting the tendencies of the RPA sc reening [20]. On\nthe other hand, the effective exchange interaction Jis less sensitive to the band\nfilling. The Coulomb repulsion UVis larger for the VMo-compounds, due to\nthe specific electronic structure of Sr 2−xYxVMoO 6(Fig. 1): since the V-band is\nseparated and located in the low-energy part of the spectrum, it w ill be populated\nin the process of the band-filling. It will effectively reduce the numbe r of channel\navailable for the RPA screening of UV, for example, due to the transitions from the\noccupied oxygen band to the unoccupied part of the V-band [20]. Th us,UVwill be\nless screened in the VMo-systems in comparison with the VTc-ones, where the V-\nand Tc-states form one common band (Fig. 1).\n2.2. Optimized effective potential method\nInorder tosolve themodel andfindthemagneticgroundstateoft hedoubleperovskites,\nwe employ the optimized effective potential method. The OEP model w orks in the spirit\nof the Kohn-Sham density functional theory [9] on the double-per ovskite lattice, where\nthe kinetic energy\nEkin=/summationdisplay\nσocc/summationdisplay\nn1\nΩBZ/integraldisplay\ndk/angbracketleftcσnk|ˆtk|cσnk/angbracketright,Optimized Effective Potential Model for Double Perovskites 6\nthe number of electrons\nnτ=/summationdisplay\nσ/summationdisplay\nmocc/summationdisplay\nn1\nΩBZ/integraldisplay\ndkcmτ†\nσnkcmτ\nσnk,\nand the spin magnetic moment\nmτ=/summationdisplay\nmocc/summationdisplay\nn1\nΩBZ/integraldisplay\ndk/parenleftBig\ncmτ†\n↑nkcmτ\n↑nk−cmτ†\n↓nkcmτ\n↓nk/parenrightBig\nat the site τare constructed from the one-electron orbitals cσnk, which are obtained\nfrom the solution of the auxiliary one-electron problem\n/parenleftbigˆtk+ ˆvσ\neff/parenrightbig\n|cσnk/angbracketright=εσnk|cσnk/angbracketright (2)\nwith the potential ˆ vσ\neff. In the above notations, ˆtkis the Fourier image of ˆtij=/bardbltmm′\nij/bardbl,\neach eigenvector |cσnk/angbracketrightis specified by its components cmτ\nσnkin the basis of t2g-orbitals\nm=xy,yz, orzxof the atomic type τ= V or Mo(Tc), nis the band index, and kis\nthe position in the first Brillouin zone with the volume Ω BZ.\nThe parameters of the potential ˆ vσ\neff, which is supposed to be “local” (or diagonal\nwith respect to the site indices), are obtained from the minimization o f the total energy\nE=Ekin+EC+EX+Ecorr, (3)\nwhere the Coulomb ( EC) and exchange ( EX) energies are given by\nEC=1\n2/summationdisplay\nτ/summationdisplay\nσσ′/summationdisplay\nmm′m′′m′′′Uτ\nmm′m′′m′′′nστ\nmm′nσ′τ\nm′′m′′′\nand\nEX=−1\n2/summationdisplay\nτ/summationdisplay\nσ/summationdisplay\nmm′m′′m′′′Uτ\nmm′′′m′′m′nστ\nmm′nστ\nm′′m′′′,\nrespectively, in terms of the density matrix:\nnστ\nmm′=occ/summationdisplay\nn1\nΩBZ/integraldisplay\ndkcmτ†\nσnkcm′τ\nσnk.\nThe expression for the correlation energy, which is treated in the r andom phase\napproximation, is given by [22, 23, 24]:\nEcorr=−1\n4ΩBZ/integraldisplay\ndqTr/braceleftBig\nln/bracketleftBig\n1−ˆP(iω,q)ˆU/bracketrightBig/bracketleftBig\n1−ˆUˆP(iω,q)/bracketrightBig\n+2ˆP(iω,q)ˆU/bracerightBig\n,(4)\nwhereˆUare the diagonal matrices /bardblUτ\nmm′m′′m′′′/bardblwith respect to the site indices τ,\nˆP=/bardblPττ′\nmm′m′′m′′′/bardblis the polarization in the imaginary frequency:\nPττ′\nmm′m′′m′′′(iω,q) =/summationdisplay\nσocc/summationdisplay\nnunocc/summationdisplay\nn′1\nΩBZ/integraldisplay\ndk2(εσnk−εσn′k+q)\nω2+(εσnk−εσn′k+q)2cmτ†\nσn′k+qcm′τ\nσnkcm′′τ′†\nσnkcm′′′τ′\nσn′k+q,(5)\nand matrix multiplication in (4) implies the convolution over two orbital in dices:\n(ˆUˆP)ττ′\nmm′m′′m′′′=/summationtext\nll′Uτ\nmm′ll′Pττ′\nll′m′′m′′′.Optimized Effective Potential Model for Double Perovskites 7\nFor the cubic systems (and neglecting the orbital polarization effec ts), the effective\npotential is specified by three parameters: ˆ v↑V\neff=−∆V\nex/2, ˆv↓V\neff= ∆V\nex/2, ˆv↑Mo(Tc)\neff=\n∆ct−∆Mo(Tc)\nex/2, and ˆv↓Mo(Tc)\neff= ∆ct+∆Mo(Tc)\nex/2. In the other words, ∆V\nexand ∆Mo(Tc)\nex\nis the exchange-correlation splitting between minority- and majorit y-spin states at the\nsites V and Mo(Tc), respectively, and ∆ ctis the “charge transfer energy” between sites\nV and Mo(Tc). All these parameters are obtained by minimizing the to tal energy (3).\nIn practice, first we calculate the total energy on some coarse me sh of these parameters,\nidentify the minimum, then continue the calculations around the minimu m by using\nfiner mesh, and so on until reaching necessary accuracy for the t otal energy minimum\n(less than 1 meV).\n3. Results and Discussions\n3.1. Electronic structure and arbitrariness of the OEP proc edure for half-metallic and\ninsulating states\nIn the following, we consider two magnetic solutions: ferromagnetic (∆V\nex>0 and\n∆Mo(Tc)\nex>0) and ferrimagnetic (∆V\nex>0, while ∆Mo(Tc)\nex<0), and find parameters of\nthe effective potential by minimizing the total energy for each case . Total and partial\ndensities of states for Sr 2−xYxVMoO 6, obtained from the solution of the Kohn-Sham\nequations (2) with the optimized potential, are shown in Fig. 3. All FM s olutions are\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114/s89/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s32/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s50/s48/s49/s48/s48/s49/s48/s50/s48/s89\n/s50/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48\n/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s83/s114\n/s50/s86/s77/s111/s79\n/s54\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114/s89/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s89\n/s50/s86/s77/s111/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s77/s111\nFigure 3. Total and partial densities of states as obtained in the optimized eff ective\npotential method for the ferromagnetic (upper panel) and ferrim agnetic (lower panel)\nconfigurations of Sr 2−xYxVMoO 6. The Fermi level is at zero energy.Optimized Effective Potential Model for Double Perovskites 8\nhalf-metallic: the Fermi levels crosses the majority-spin band, while the minority-spin\nband is completely empty. In the case of Sr 2VMoO 6and SrYVMoO 6, there is a strong\nhybridization between V- and Mo-states, which form the common ma jority-spin band.\nHowever, in Y 2VMoO 6the majority-spin band is split into the Mo- and V-subbands,\nwhich are separated by an energy gap. This behavior is related to th e large Coulomb\nrepulsion UVin Y2VMoO 6(Fig. 2), which additionally shifts the V-states to the higher-\nenergy part of spectrum. Any deviation from the half-metallic beha vior and partial\npopulation of the minority-spin band increases the total energy of Sr2−xYxVMoO 6and\nthus makes this state unstable.\nThe choice of the parameters of the effective potential in the half- metallic regime is\nnot unique. For example, any redistribution of states in the unoccu pied minority-spin\nband does not affect the ground-state properties. Moreover, a ny change of ∆V\nex, ∆Mo\nex,\nand ∆ ct, which does not deform the partially populated majority-spin band ( apart from\nthe constant shift), does not affect the ground-state propert ies either. Quantitatively,\nthis condition can be formulated as follows: the ground-state prop erties in the half-\nmetallic regime are controlled by the single parameter a= ∆ct+(∆V\nex−∆Mo\nex)/2. Any\nchange of ∆V\nex, ∆Mo\nex, and ∆ ct, which keeps a, does not affect the ground-state properties\nin the half-metallic regime.\nAll ferrimagnetic configurations of Sr 2−xYxVMoO 6are metallic. In this case all\nthree parameters of the effective potential are uniquely defined. The degree of the spin\npolarization ( P) in the ferrimagnetic state is sensitive to the concentration xand the\ndefinition of P[25]. For example, using the simplest “density of states” definition (s ee\nFig. 3), one obtains P= 0.3, 0.1, and 0.2 for x= 0, 1 and 2, respectively. On the other\nhand, the “Bloch-Bolzmann transport theory” definition yields P= 0.1, 0.1, and 0.5 for\nx= 0, 1 and 2, respectively. The experimental value of Pfor SrLaVMoO 6, measured\nby using the point-contact Andreev reflection technique, is about 0.5 [14]. Nevertheless,\nas we will see below, many of these ferrimagnetic states appear to b e unstable and\ncalculated spin polarization cannot be directly compared with the exp erimental data.\nSimilar behavior was obtained for Sr 2VTcO 6and SrYVTcO 6, where again half-\nmetallic FM state competes with the normal metallic ferrimagnetic sta te (Fig. 4)\nY2VTcO 6requires a special attention: it has six electrons in the t2g-band. The\nsystem is insulating and all metallic configurations have higher energie s. Therefore,\nin the FM state, the parameters ∆V\nex, ∆Tc\nex, and ∆ ctcan be chosen in an arbitrary\nway. Indeed, any combination of these parameters (provided tha t the system remains\ninsulating) yields the same number of electrons nτ= 3, spin magnetic moments\nmτ= 3µB, andthetotal energyof thesystem. Thecorrelationenergy is ex actly equal to\nzero for this particular case. The arbitrariness with the choice of t he effective potential\nremains also in the antiferromagnetic state of Y 2VTcO 6(Fig. 5), although the number\nof arbitrary parameters decreases. Indeed, the splitting betwe en V- and Mo-states with\nthe same spin affects the ground-state properties. However, th e bands with different\nspins can be rigidly shifted relative to each other and, as long as the s ystem remains\nin the insulating state, this shift does not affect the ground-state properties. Thus,Optimized Effective Potential Model for Double Perovskites 9\n/s51 /s50 /s49 /s48 /s49 /s50 /s51/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\n/s51 /s50 /s49 /s48 /s49 /s50 /s51/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114/s89/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s32/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s50/s48/s49/s48/s48/s49/s48/s50/s48/s83/s114/s89/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\nFigure 4. Total and partial densities of states as obtained in the optimized eff ective\npotential method for the ferromagnetic (upper panel) and ferrim agnetic (lower panel)\nconfigurations of Sr 2VTcO 6and SrYVTcO 6. The Fermi level is at zero energy.\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s50/s48/s49/s48/s48/s49/s48/s50/s48/s89\n/s50/s86/s84/s99/s79\n/s54/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s41\n/s32/s32\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s116/s111/s116/s97/s108\n/s32/s86\n/s32/s84/s99\nFigure 5. Total and partial densities of states as obtained in the optimized eff ective\npotentialmethod forthe antiferromagneticconfigurationofY 2VTcO 6. The Fermi level\nlies in the band gap.\nthe properties of the antiferromagnetic Y 2VTcO 6are defined by only two parameters:\n∆ctand (∆V\nex−∆Tc\nex), while the third parameter (∆V\nex+ ∆Tc\nex) can be taken arbitrarily.\nFinally, wewouldlike toemphasize thatantiferromagneticY 2VTcO 6becomesinsulating\ndue to the large exchange splitting (both at the V- and Tc-sites) ob tained in the OEP\ncalculations. If the splitting were smaller, the majority-spin V- and T c-bands would\neventually merge, yielding the half-metallic antiferromagnetic state , which was obtained\nin the LSDA calculations for the isoelectronic La 2VTcO 6[16]. Thus, such half-metallicOptimized Effective Potential Model for Double Perovskites 10\nantiferromagnetic state is probably an artifact of LSDA, and more rigorous treatment\nof the correlation effects will open the band gap in the both spin chan nels.\n3.2. Screening and the effective Stoner model\nSince the effective potential is local (or diagonal with respect to th e site indices) and\ndoes not affect the orbital degrees of freedom, the OEP scheme h as many similarities\nwith the Stoner model of magnetism, where the strength of the int raatomic splitting ∆τ\nex\nbetween the minority- and majority-spin states is controlled by the effective parameter\nIτ: ∆τ\nex=Iτmτ. Thus, using the intraatomic exchange splitting, obtained from the\nminimizationofthetotalenergy, andcorrespondingvaluesofthes pinmagneticmoments\nin the ground state (Fig. 6), one can find the effective Stoner para meterIτ\nOEP. All such\n/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s77/s111/s77/s97/s103/s110/s101/s116/s105/s99/s32/s77/s111/s109/s101/s110/s116/s115/s32/s40\n/s66 /s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54/s86\n/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s84/s99/s77/s97/s103/s110/s101/s116/s105/s99/s32/s77/s111/s109/s101/s110/s116/s115/s32/s40\n/s66 /s41\n/s32/s32\n/s32/s32/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s84/s99/s79\n/s54/s86\nFigure 6. Local spin magnetic moments at the V- and Mo(Tc)-sites, as obtain ed in\ntheoptimizedeffectivepotentialmethodforSr 2−xYxVMoO 6(left)andSr 2−xYxVTcO 6\n(right).\nconsiderations are valid only for the ferrimagnetic metallic state, wh ere ∆τ\nexare uniquely\ndefined. The results for Sr 2−xYxVMoO 6are explained in Fig. 7. The effective Stoner\nparameters derived from the OEP approach are of the order of 1.3 -1.6 eV at the V-sites\nand 1.0-1.1 eV at the Mo-sites. Similar parameters were obtained for Sr2−xYxVTcO 6:\nIV\nOEP= 1.27 eV and ITc\nOEP= 1.06 eV in the case of Sr 2VTcO 6, andIV\nOEP= 1.28 eV\nandITc\nOEP= 1.18 eV in the case of SrYVTcO 6. All of them are systematically larger\nthat the typical values of Iτobtained in the local-spin-density approximation (typically,\nclose 1 eV for the 3 d-elements and even smaller for the 4 d- and 5d-elements [26]).\nThis example clearly shows the importance of the rigorous treatmen t of the exchange-\ncorrelation interactions (beyond LSDA) in the physics of double per ovskites.\nAnother limiting case is the Hartree-Fock (HF) approximation, which neglects the\ncorrelation interactions. Then, corresponding Stoner paramete r describing the splitting\nin the system of three t2g-orbitals, are given by Iτ\nHF= (Uτ+ 2Jτ)/3. They are also\nplotted in Fig. 7. The difference between Iτ\nHFandIτ\nOEPis the measure of correlation\ninteractions and their impact on the magnetic properties of the dou ble perovskites. One\ncan see that Iτ\nOEPis reduced by about 10-20% in comparison with Iτ\nHF, which indicates\nthe importance of correlation interactions.\nFinally, it is instructive to consider the effect of the static screening alone. TheOptimized Effective Potential Model for Double Perovskites 11\n/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s73/s32/s86\n/s82/s80/s65/s73/s32/s77/s111\n/s82/s80/s65/s73/s32/s77/s111\n/s79/s69/s80/s73/s32/s77/s111\n/s72/s70/s73/s32/s86\n/s79/s69/s80/s73/s32/s86\n/s72/s70/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s83/s116/s111/s110/s101/s114/s32/s80/s97/s114/s97/s109/s101/s116/s101/s114/s115/s32/s40/s101/s86/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54\nFigure 7. EffectiveStoner parametersforSr 2−xYxVMoO 6asderivedin the optimized\neffective potential (OEP) method for the ferrimagnetic state, in t he Hartree-Fock(HF)\napproximation, and using only static values of Coulomb and exchange interactions in\nthe random phase approximation (RPA).\ncorresponding on-site interaction parameters in RPA are given by\nˆURPA=1\nΩBZ/integraldisplay\ndq/bracketleftBig\n1−ˆUˆP(0,q)/bracketrightBig−1ˆU,\nwhich yields the Stoner parameter Iτ\nRPA= (Uτ\nRPA+ 2Jτ\nRPA)/3. Such a picture\ncan be regarded as the static limit of the GWapproximation [27, 28], where the\nfrequency-dependent screened Coulomb interaction ˆURPA(iω) is replaced by its static\ncounterpart ˆURPA≡ˆURPA(0)[29]. Typically, it issufficient toconsider onlysite-diagonal\ncontributions to ˆURPA, because if ˆUis site-diagonal, the off-diagonal contributions to\nˆURPAare expected to be small [30]. These results are also shown in Fig. 7. O ne can\nclearly see that the parameters Iτ\nRPAare overscreened and strongly underestimated in\ncomparison with Iτ\nOEP. This indicates the importance of dynamical screening effects in\nthe construction of the optimized effective potential.\n3.3. Total energy and stability of the ferrimagnetic state\nThe behavior of the total energy and of its partial contributions is explained in Fig. 8.\nFor all considered compounds, except Sr 2VMoO 6, the ferrimagnetic configuration has\nlower energy. The tendencies towards ferrimagnetism increase wit h the band-filling.\nThe main factor stabilizing the ferrimagnetic state is the gain of the k inetic energy. The\nmain factor destabilizing it is the loss of the exchange energy.\nIn order to investigate the local stability of the ferrimagnetic phas e, we calculate\nthe spin stiffness D, which is related to the second derivative of the total energy withOptimized Effective Potential Model for Double Perovskites 12\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s107/s105/s110/s101/s116/s105/s99\n/s32/s67/s111/s117/s108/s111/s109/s98\n/s32/s101/s120/s99/s104/s97/s110/s103/s101\n/s32/s99/s111/s114/s114/s101/s108/s97/s116/s105/s111/s110\n/s32/s116/s111/s116/s97/s108/s69/s110/s101/s114/s103/s121/s32/s100/s105/s102/s102/s101/s114/s101/s110/s99/s101/s32/s40/s101/s86/s47/s102/s111/s114/s109/s117/s108/s97/s32/s117/s110/s105/s116/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s107/s105/s110/s101/s116/s105/s99\n/s32/s67/s111/s117/s108/s111/s109/s98\n/s32/s101/s120/s99/s104/s97/s110/s103/s101\n/s32/s99/s111/s114/s114/s101/s108/s97/s116/s105/s111/s110\n/s32/s116/s111/s116/s97/s108/s69/s110/s101/s114/s103/s121/s32/s100/s105/s102/s102/s101/s114/s101/s110/s99/s101/s32/s40/s101/s86/s47/s102/s111/s114/s109/s117/s108/s97/s32/s117/s110/s105/s116/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s84/s99/s79\n/s54\nFigure 8. Total energy difference between ferrimagnetic and ferromagnet ic states and\npartial contributions to this energy difference, calculated for Sr 2−xYxVMoO 6(top)\nand Sr 2−xYxVTcO 6(bottom).\nrespect to the spin-spiral vector qin the ferrimagnetic state:\nE(q)≈E(q0)+D(q−q0)2,\nwhereq0= (π/a,π/a,π/a ) andais the cubic lattice parameter. More specifically,\nthe numerical calculations of E(q), were performed by employing the generalized Bloch\ntheorem [31] and the so-called local-force theorem for the infinites imal spin rotations.\nThe latter states that the total energy change for the small rot ations can be replaced by\nthe change of the one-electron energies, obtained from the Kohn -Sham equations (2) for\nthe rigid spin rotations of the effective potential ˆ vσ\neff[32]. The theorem is rather general\nand can be proven by considering the rotational invariance of the e xchange-correlation\nfunctions [8]. Thus, the range of the applicability of this theorem is no t limited by the\nlocal-spin-density approximation, for which it was originally proven, b ut can be also\nextended to the OEP approach [8], provided that the effective pote ntial ˆvσ\neffis uniquely\ndefined. Particularly, since the choice of ˆ vσ\neffis not unique in the case of insulating\nY2VTcO 6, this strategy cannot be directly applied to this system.\nThe behavior of Din the OEP model is explained in Fig. 9. Particularly, Dis\nnegative in the case of Sr 2VMoO 6, Sr2VTcO 6, and SrYVMoO 6. Thus, the ferrimagnetic\nstate in these compounds is unstable with respect to the incommens urate spin-spiral\nstate. Since FM state has lower energy in the case of Sr 2VMoO 6(Fig. 8), the spin-spiral\ninstability of the ferrimagnetic state may be finally resolved in the fav or of the collinearOptimized Effective Potential Model for Double Perovskites 13\n/s45/s50/s45/s49/s48/s49/s83/s114\n/s50/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s84/s99/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s84/s99/s79\n/s54/s83/s112/s105/s110/s32/s83/s116/s105/s102/s102/s110/s101/s115/s115/s32/s40/s101/s86/s32/s65/s50\n/s41\n/s32/s32\n/s32/s83/s114\n/s50/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s32/s83/s114/s89/s86/s77/s111/s79\n/s54/s32/s32/s32/s32/s32/s32/s89\n/s50/s86/s77/s111/s79\n/s54/s111\nFigure 9. Spin stiffness in the ferrimagnetic state calculated using optimized eff ective\npotentials for Sr 2−xYxVMoO 6(filled circles) Sr 2−xYxVTcO 6(empty squares).\nFM arrangement, to make it the true magnetic ground state of Sr 2VMoO 6. However,\nin Sr2VTcO 6and SrYVTcO 6already ferrimagnetic state has lower energy. Moreover,\nthis ferrimagnetic state itself appears to be unstable with respect to the noncollinear\nspin-spiral alignment. Thus, the ground state in this case will be neit her ferro- nor\nferrimagnetic.\n4. Summary and Conclusions\nIn summary, using first-principles electronic structure calculation s, we have constructed\nthe effective low-energy model for the t2g-states of double perovskites Sr 2−xYxVMoO 6\nand Sr 2−xYxVTcO 6. Then, the ground-state properties of the obtained model have\nbeen investigated on the basis of the optimized effective potential m ethod, by treating\ncorrelation interactions in the random-phase approximation. The r eason for using\nsuch strategy was twofold. On the one hand, such approach allows one to improve\nthe commonly used local-spin-density approximation for the double p erovskites and\nincorporate the physics of electron correlations beyond the homo geneous electron gas\nlimit. On the other hand, since OEP is formulated as the ground-stat e method,\nwhere all the parameters are derived rigorously by minimizing the tot al energy of\nthe system, it gets rid of many ambiguities inherent to the LSDA+ Umethod.\nParticularly, although Sr 2−xYxVMoO 6, Sr2−xYxVTcO 6, and other isoelectronic to\nthem compounds are frequently proposed to exhibit half-metallic fe rromagnetic or\nantiferromagnetic properties, the situation is probably not such o ptimistic because of\nthe electron correlation effects, which affect the intraatomic exch ange splitting andOptimized Effective Potential Model for Double Perovskites 14\nmakes it substantially larger than in LSDA. Of course, this will revise m any predictions\nbased on the LSDA. For example, Y 2VTcO 6, which was expected to be half-metallic\nantiferromagnet, appears to be insulator. The behavior of other materials is even more\ncomplicated. For the most of them (except Sr 2VMoO 6), the half-metallic ferromagnetic\nstate is energetically less favorable than the (normal) metallic ferrim agnetic one.\nNevertheless, this ferrimagnetic state appears to be also unstab le with respect to the\nspin-spiral alignment. Thus, the true magnetic ground state of th ese double perovskites\nwill be the spin spiral or some more complicated non-collinear magnetic structure.\nFinally, we have argued that the effective potential (and therefor e the one-electron\nband structure) for the half-metallic and insulating ferro- or antif erromagnetic state is\nnot uniquely defined. Generally, there is a bunch of parameters, wh ich yield the same\ntotal energy, occupation numbers and spin magnetic moments at t he transition-metal\nsites.\nAcknowledgments\nI wish to thank the support from the Federal Agency for Science a nd Innovations,\ngrant No. 02.740.11.0217, during my stay at the Ural State Technic al University\n(Ekaterinburg, Russia).\nReferences\n[1] Serrate D, De Teresa J M and Ibarra M R 2007 J. Phys.: Condens. Matter 19023201\n[2] Kobayashi K-I, Kimura T, Sawada H, Terakura K and Tokura Y 19 98Nature395677\n[3] Solovyev I V 2002 Phys. Rev. B65144446\n[4] van Leuken H and de Groot R A 1995 Phys. Rev. Lett. 741171\n[5] Pickett W E 1998 Phys. Rev. B5710613\n[6] Anisimov V I, Zaanen J and Andersen O K 1991 Phys. Rev. B44943\n[7] Solovyev I V, Dederichs P H and Anisimov V I 1994 Phys. Rev. B5016861\n[8] Solovyev I V and Terakura K 1998 Phys. Rev. B5815496\n[9] Kohn W and Sham L J 1965 Phys. Rev. 140A1133\n[10] Talman J D and Shadwick W F 1976 Phys. Rev. A1436\n[11] Kotani T and Akai H 1996 Phys. Rev. B5416502\n[12] Kotani T 1998 J. Phys.: Condens. Matter 109241\n[13] Engel E and Schmid R N 2009 Phys. Rev. Lett. 103036404\n[14] Gotoh H, Takeda Y, Asano H, Zhong J, Rajanikanth A and Hono K 2009Applied Physics Express\n2013001\n[15] Park M S and Min B I 2005 Phys. Rev. B71052405\n[16] Wang Y K, Lee P H and Guo G Y 2009 Phys. Rev. B80224418\n[17] Park J H, Kwon S K and Min B I 2002 Phys. Rev. B65174401\n[18] Uehara M, Yamada M and Kimishima Y 2004 Solid State Commun. 129385\n[19] Song W, Wang J, Meng J and Wu Z 2010 J. Comp. Chem. 311795\n[20] Solovyev I V 2008 J. Phys.: Condens. Matter 20293201\n[21] Kanamori J 1963 Prog. Theor. Phys. 30275\n[22] Pines D 1999 Elementary Excitations in Solids (Oxford: Westview Press)\n[23] von Barth U and Hedin L 1972 J. Phys. C51629\n[24] Aryasetiawan F, Miyake T and Terakura K 2002 Phys. Rev. Lett. 88166401Optimized Effective Potential Model for Double Perovskites 15\n[25] Mazin I I 1999 Phys. Rev. Lett. 831427\n[26] Gunnarsson O 1976 J. Phys. F6587\n[27] Hedin L 1965 Phys. Rev. 139A796\n[28] Aryasetiawan F and Gunnarsson O 1998 Rep. Prog. Phys. 61237\n[29] Aryasetiawan F, Imada M, Georges A, Kotliar G, Biermann S and L ichtenstein A I 2004 Phys.\nRev.B70195104\n[30] Solovyev I V 2006 Phys. Rev. B73155117\n[31] Sandratskii L M 1998 Adv. Phys. 4781\n[32] Bruno P 2003 Phys. Rev. Lett. 90087205"    },    {        "title": "1505.00583v1.Spiral_ferrimagnetic_phases_in_the_two_dimensional_Hubbard_model.pdf",        "content": "Spiral ferrimagnetic phases in the two-dimensional\nHubbard model\nJ. D. Gouveia\u0003, R. G. Dias\nDepartamento de F\u0013 \u0010sica, I3N, Universidade de Aveiro, Campus de Santiago, Portugal\nAbstract\nWe address the possibility of spiral ferrimagnetic phases in the mean-\feld phase\ndiagram of the two-dimensional (2D) Hubbard model. For intermediate values\nof the interaction U(6.U=t.11) and doping n, a spiral ferrimagnetic phase is\nthe most stable phase in the ( n;U) phase diagram. Higher values of Ulead to a\nnon-spiral ferrimagnetic phase. If phase separation is allowed and the chemical\npotential\u0016replaces the doping nas the independent variable, the ( \u0016;U) phase\ndiagram displays, in a considerable region, a spiral (for 6 .U=t.11) and non-\nspiral (for higher values of U) ferrimagnetic phase with \fxed particle density,\nn= 0:5, re\recting the opening of an energy gap in the mean-\feld quasi-particle\nbands.\nKeywords: A. Magnetically ordered materials, C. Crystal structure and\nsymmetry, D. Electron-electron interactions, D. Electronic band structure, D.\nPhase transitions\n1. Introduction\nThe 2D Hubbard model remains the most important open theoretical prob-\nlem in the \feld of the strongly correlated electronic systems, despite all e\u000borts\nfuelled by the advent of the high- Tcsuperconductivity.[1, 2] At half-\flling, the\nspin dynamics of the 2D Hubbard model is described by the Heisenberg anti-\nferromagnetic exchange term.[3] Away from half-\flling, the movement of holes\nthrough the spin background generates additional spin mixing. The competition\nbetween the Heisenberg exchange and the spin con\fguration mixing generated\nby hole hopping in the 2D Hubbard model is still far from understood.[4, 5, 6] In\nparticular, there is no consensus regarding the ground state magnetic phase dia-\ngram of the 2D Hubbard model and di\u000berent authors obtain di\u000berent mean-\feld\n(MF) phase diagrams depending on the magnetic phases allowed.[7] Tradition-\nally, one considered ferromagnetism, antiferromagnetism and paramagnetism\n\u0003Corresponding author. Tel. +351 227 828 940\nEmail address: gouveia@ua.pt (J. D. Gouveia)\nPreprint submitted to Solid State Communications October 30, 2018arXiv:1505.00583v1  [cond-mat.str-el]  4 May 2015B D \nC A Figure 1: The 2D lattice and its four sublattices A, B, C and D. We consider two situations:\n(i)mA=mD=m1andmB=mC=m2; (ii)mA=mC=m1andmB=mD=m2.\nphases.[8, 9, 10, 11, 12] The complexity of the MF phase diagram was increased\nwith the introduction of spiral phases[13], which appear between the \"usual\"\nmagnetic phases in the diagram. This complexity was further increased by the\nconsideration of spatial phase separation.[14, 15, 16]\nIn this paper, we extend the results above mentioned, by introducing the\npossibility of a spiral ferrimagnetic phase, that is, a ferrimagnetic phase such\nthat the orientation of magnetic moments changes along the lattice (see Fig.\n1). More precisely, we study the 2D Hubbard model using the Hartree-Fock\napproximation in a square lattice decomposing the lattice in four square sub-\nlattices (A, B, C and D as in Fig. 1) and allowing di\u000berent amplitudes for\nmagnetizations of the spiral phases in the sublattices. Note that, even under\nthe MF approximation, when four sublattices are considered, it is not possible\nto obtain the analytical form of the spectra of the 2D Hubbard model. Our MF\napproach to the 2D Hubbard model follows that of Dzierzawa and Singh.[17, 18]\n2. Calculations\nIntroducing a di\u000berent creation operator in each sublattice, Ay,By,Cyand\nDy, the tight-binding term of the Hubbard Hamiltonian, is\nHt=P\nx;yAy\nx;yBx;y+Ay\nx;yCx;y\n+By\nx;yDx;y+Cy\nx;yDx;y\n+Ay\nx;yBx;y\u00001+Ay\nx;yCx\u00001;y\n+By\nx;yDx\u00001;y+Cy\nx;yDx;y\u00001+H:c:; (1)\nwhere we set the hopping constant equal to 1.\nWe consider for now only the sublattice A (we add the other sublattice\nterms later on). The interaction term of the Hubbard Hamiltonian is, as usual,\nHU=UP\nrAy\nr\"Ar\"Ay\nr#Ar#. We assume that the magnetic moments align in the\nx-yplane, so thathSzi= 0 and the Hartree term becomesU\n4Phni2, wherehni\n2is the density of electrons on each sublattice (here assumed to be the same on\nall of them).\n0.0 0.2 0.4 0.6 0.8 1.005101520\nU\nnFerro\nParam(q,q)\nAF(0,q)\n(0,π)\n(q,π)\n(a) (n;U) phase diagram\nU\nnFerro\nS\nParam(0,π)\n(q,π)(0,q)(q,q)\nU\nn0.0 0.2 0.4 0.6 0.8 1.005101520\nAFPS (b) (n;U) phase diagram with\nphase separation\nn=1 AF\n5 10 15/Minus4/Minus2024\nFerro\nParam(q,π)\n(0,π)\n(0,q)(q,q)\nU(c) (U;\u0016) phase diagram with\nphase separation\n(d)m(n;U)\n (e)qx(n;U)\n (f)qy(n;U)\n (g)E(n;U)\n (h)\u0016(n;U)\nFigure 2: (a) Mean-\feld phase diagram for the usual 2D Hubbard model: The system displays\nantiferromagnetism (AF), ferromagnetism (F), paramagnetism (P) or spiral phases ( q6= 0;\u0019).\nThe antiferromagnetic state ~ q= (\u0019;\u0019) only occurs for n= 1 (half-\flling). (b) ( n;U) and (c)\n(U;\u0016) mean-\feld phase diagrams for the 2D Hubbard model, allowing for phase separation\n(yellow region). (d) m, (e)qx, (f)qy, (g)EMFand (h)\u0016as functions of the doping nand\nCoulomb interaction U, for the 100\u0002100 2D Hubbard model.\nThe Fock term includes averages like hAy\n\"A#i=hS+\nAi=hSAx+iSAyi, whose\nvalues depend on the magnetic phase. Let us assume the average spin in the\nsublattice A is\nh~S~ rAi=mA\n2[cos(~ q\u0001~ rA);sin(~ q\u0001~ rA);0]: (2)\nThe vector ~ q= (qx;qy) de\fnes the magnetic phase of the system. In ~k-space we\nhave\nhS+\nA~ki=1p\nLX\n~k0hAy\n~k0;\"A~k0\u0000~k;#i=mApLu:c:\n2\u000e~k;\u0000~ q; (3)\nwhereLu:c:is the number of unit cells, which gives\nhAy\n~k;\"A~k+~ q;#i=mA\n2; (4)\nwhile all the other mean values in the summation of Eq. 3 vanish. The Fock\nterm in Fourier space is\n\u0000mU\n2X\n~k\u0010\nAy\n~k+~ q;#A~k;\"+Ay\n~k;\"A~k+~ q;#\u0011\n+ULu:c:\n4m2\nA: (5)\n3Adding the tight-binding, Hartree and Fock terms, the Hamiltonian HMFreads,\nin thefA~k;B~k;C~k;D~k;A~k+~ q;B~k+~ q;C~k+~ q;D~k+~ qgbasis,\n \nHt(~k)Hm\nHy\nmHt(~k+ 2~ q)!\n; (6)\nplus the diagonal term\nULu:c:\n4(m2\nA+m2\nB+m2\nC+m2\nD) +ULhni2\n2: (7)\nHere,Ht(~k) is the tight-binding term (Eq. 1) of the Hamiltonian in ~k-space,\n0\nBB@0 1 + eiky 1 +eikx 0\n1 +e\u0000iky 0 0 1 + eikx\n1 +e\u0000ikx 0 0 1 + eiky\n0 1 + e\u0000ikx1 +e\u0000iky 01\nCCA; (8)\nHmis the diagonal matrix, Hm= diag(\u0001 A;\u0001B;\u0001C;\u0001D) with\n\u0001A=\u0000UmA\n2; \u0001B=\u0000UmB\n2eiqy;\n\u0001C=\u0000UmC\n2eiqx;\u0001D=\u0000UmD\n2eiqx+iqy:(9)\n3. Results and discussion\nBy settingmA=mB=mC=mD=m, we recover the MF magnetic phase\ndiagram of the usual 2D Hubbard model, consistent with the ones obtained by\nseveral authors[17, 13, 15] for zero temperature, as presented in Fig. 2a. In order\nto obtain such a diagram, one minimizes either the MF energy EMFusing the\nelectronic density nas an independent variable, or the thermodynamic potential\n\nMFusing the chemical potential \u0016, with respect to the site magnetization\namplitudemand the order parameter ~ q= (qx;qy). These parameters de\fne the\nmagnetic phase of the system.\nA solution with m= 0 is paramagnetic and is usually ~ q-degenerate, while\nsolutions for m6= 0 are in general unique. In the latter case, the wave vector ~ q\nspeci\fes the type of magnetic ordering. For instance, ~ q= (0;0) for the ferromag-\nnetic phase, ~ q= (\u0019;\u0019) for the antiferromagnetic phase and all other choices for\nspiral phases. In the example shown in Fig. 1, we have qx=\u0019=18 andqy=\u0019=6.\nAdditionally, in the same example, the magnetization amplitudes (denoted by\nthe size of the arrows) are mA=mD=m1andmB=mC=m2<m 1. Com-\nparing, for each pair ( n;U) or (U;\u0016), the data obtained for m(Fig. 2d), qx\n(Fig. 2e) and qy(Fig. 2f), the MF magnetic phase diagram displayed in Fig. 2a\nensues. For some values of \u0016, there is more than one pair ( ~ q;m) which minimizes\nthe thermodynamic potential. In those cases, a \frst-order phase transition in\n40.0 0.2 0.4 0.6 0.8 1.005101520\nU\nn\nAF\n/LParen4m2m1\nm1m2/RParen4/LParen4m2m2\nm1m1/RParen4 /LParen4m2m2\nm1m1/RParen4\n0.0 0.2 0.4 0.6 0.8 1.005101520\nU\nnFerro\nParamAF (q,q)\n(0,π)(q,π)(0,q)Spiral \nFerriFerri Ferri Ferri\nSpiral \nFerri(a) (n;U) phase diagram\n5 10 15/Minus4/Minus2024\nFerroParamn=1 AF\nn=0.5 Ferrin=0.5 Spiral \n          Ferri\nU/LParen4m2m1\nm1m2/RParen4 (b) (U;\u0016) phase diagram with phase\nseparation\n(c)m1;m2(i)\n (d)m1;m2(ii)\nU=9.0\n/Minus6/Minus4/Minus20\n0\n    .5\n   1U=19.0\n/Minus6/Minus4/Minus20 (e)E(n;U)\n (f)\u0016(n;U)\nFigure 3: Top \fgures: (a) Mean-\feld phase diagram for the 2D Hubbard model with sublat-\ntices with independent magnetization amplitudes. In the central (green) region of the phase\ndiagram, con\fguration (i) is the one that minimizes the MF energy, while con\fguration (ii) is\nthe most stable in the red regions. (b) ( U;\u0016) mean-\feld phase diagram with phase separation\noccurring on the borders of the green and red regions (thick solid lines). This phase diagram\ndisplays spiral and non-spiral ferrimagnetic phases (green regions) with \fxed particle density,\nn= 0:5, re\recting the opening of an energy gap in the MF quasi-particle bands. Bottom\n\fgures:m1(n;U) andm2(n;U) for the MF 2D Hubbard model with two sublattices in (c)\ncon\fguration (i) and (d) con\fguration (ii). (e) Ground state MF energy for usual 2D (blue),\ncase (i) (green) and case (ii) (red) for U= 9 andU= 19. (f)\u0016as a function of the doping n\nand Coulomb interaction U. All \fgures represent numerical results obtained for the 100 \u0002100\n2D Hubbard model with two sublattices with di\u000berent magnetization amplitudes.\nthe order parameters occurs. When using nas a basic variable (and posteriorly\ncalculating \u0016=@E=@n\u0019\u0001E=\u0001nusing the data in Fig. 2g), nseems to be\nmultiply de\fned for some values of \u0016, which implies instability (e. g. of the\nspiral phase for U= 15). The use of \u0016as a basic variable solves this ambiguity\nand leads to plateaus in the chemical potential \u0016(n;U) in the regions where\nphase separation (PS) occurs (see Fig. 2h). In each PS region of the diagram,\ntwo spatially separated phases occur: the ones immediately to the left and to\nthe right of the PS region in question (see Fig. 2b). The two phases have dif-\nferent electronic densities, such that the electronic density of the whole system\namounts to n. In Fig. 2c, we show the same phase diagram as in Fig. 2a, but\nusing\u0016as the independent variable. The colors of corresponding regions are\nthe same for easier reading. The thick solid line indicates a discontinuity in n.\n5In this work, the magnetic phase diagram for the Hubbard 2D model com-\nprising four sublattices is obtained by \fnding the magnetization amplitudes\n(mA;mB;mC;mD) and the vector ~ qwhich minimize the energy. We consider\ntwo situations: (i) mA=mD=m1andmB=mC=m2; (ii)mA=mC=m1\nandmB=mD=m2.\nThe ground state magnetization amplitude of the usual 2D Hubbard model\nis proportional to nfor each value of Uin the ferromagnetic phase (see Fig. 2d).\nWhen a spiral ferrimagnetic phase is allowed, it was found that, near zero \flling\n(n= 0) and half-\flling ( n= 1), the ground state magnetization remains the\nsame as in the usual 2D case (see Figs. 3c and 3d). This means that in these\nregions, the ground state magnetization is still constant throughout the whole\nlattice. However, as one moves to intermediate n, one \fnds that m1andm2\nbecome distinct, as shown in Figs. 3c and 3d for cases (i) and (ii) respectively,\nwherem1andm2are displayed as a function of nandU. These \fgures show\ntwo sheets re\recting the separation of the magnetization amplitudes. The colors\ngreen for case (i) and red for (ii) are used on all plots of Fig. 3. For intermediate\n\flling, the system is able to lower its energy by adopting di\u000berent magnetization\namplitudes on sublattices 1 and 2 in both cases (i) and (ii). This is shown in Fig.\n3e forU= 19 andU= 9. Depending on the region of the phase diagram one\nanalyses, con\fguration (i) or (ii) may have the lowest energy, as shown in Fig.\n3a. In this \fgure, we added another layer on top of the usual 2D MF magnetic\nphase diagram, showing which of the two-sublattice con\fgurations considered\nhas the lowest energy in the ferrimagnetic region: green for case (i) and red for\ncase (ii). Furthermore, the energy was minimized with respect to qxandqy,\nwhile using the new magnetization values, but it was found that only very small\nchanges in ~ qoccur, i.e., despite the changes in magnetization amplitudes, the\nmagnetic phases in the diagram remain the same. For this reason, the magnetic\nphases are shown as being the same as those of the usual 2D model.\nThe mean-\feld energy dispersion relation of the usual 2D Hubbard model\ndisplays two bands. Electrons occupy the lowest band until half-\flling ( n=\n1) and then proceed to occupying the higher band. As can be seen in Fig.\n2h, the fermionic density increases with the chemical potential until the phase\nseparation region is reached. In this region, the chemical potential is constant\ndespite any increase in the number of particles, up to half-\flling. At this point,\nany increase in ninduces a jump in the value of \u0016, equal to the energy separation\nbetween the two energy bands (called the energy gap). As the plot in Fig. 2h\nonly goes up to half-\flling, we see \u0016increasing smoothly until it reaches the\nphase separation region, followed by a plateau and a jump at n= 1. In both\ncases studied in this work, with the lattice divided into two sublattices, the\nenergy bands open a gap at quarter \flling ( n= 0:5), as shown in Fig. 3f.\nAnother gap appears at three quarter \flling ( n= 0:75), but only the phases\nwithn\u00141 are shown in Fig. 3b. The plot in Fig. 3b is again the same as\nFig. 3a, but using \u0016as the basic variable. In this diagram, the green region\ncorresponds to n= 0:5, therefore only con\fguration (i) for the spiral and non\nspiral ferrimagnetic phases is present in the phase diagram. The dashed line\nseparates the ferrimagnetic region from the spiral ferrimagnetic one and the\n6thick solid lines denote again discontinuities in n.\n4. Conclusion\nHaving addressed the possibility of a spiral ferrimagnetic phase in the mean-\n\feld phase diagram of the 2D Hubbard model, we conclude that, for interme-\ndiate values of the interaction Uand doping n, the spiral ferrimagnetic phase\nis the most stable phase in the ( n;U) phase diagram. Higher values of Ulead\nto non-spiral ferrimagnetic phases. We emphasize the case of intermediate n\nand higher U, for which the ground state does not appear to be purely ferro-\nmagnetic, contrasting with results by other authors.[15] Additionally, allowing\nphase separation and replacing nby\u0016as the independent variable, the ( \u0016;U)\nphase diagram displays, in a considerable region, spiral (for intermediate values\nofU) and non-spiral (for higher values of U) ferrimagnetic phases with \fxed\nparticle density, n= 0:5, re\recting the opening of an energy gap in the mean-\n\feld quasi-particle bands. We further note that generalizing the ferrimagnetic\nphase to cases where more than two di\u000berent magnetization amplitudes are al-\nlowed should lead to even more stable ferrimagnetic phases in certain regions\nof the phase diagram. Preliminary results with three di\u000berent amplitudes are\nconsistent with this conjecture. All these results provide strong evidence on\nthe stability of the spiral ferrimagnetic phase in the mean \feld magnetic phase\ndiagram of the 2D Hubbard model.\nAcknowledgements\nR. G. Dias acknowledges the \fnancial support from the Portuguese Science\nand Technology Foundation (FCT) through the program PEst-C/CTM/LA0025/2013.\nJ. D. Gouveia acknowledges the \fnancial support from the Portuguese Science\nand Technology Foundation (FCT) through the grant SFRH/BD/73057/2010.\nReferences\n[1] J. G. Bednorz, K. A. M uller, Zeitschrift fur Physik B Condensed Matter\n64 (1986) 189{193. doi: 10.1007/BF01303701 .\n[2] M. A. Kastner, R. J. Birgeneau, G. Shirane, Y. 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Phys. 38 (1992) 211{217.\n8"    },    {        "title": "2110.13644v1.Optical_interface_for_a_hybrid_magnon_photon_resonator.pdf",        "content": "arXiv:2110.13644v1  [physics.app-ph]  23 Oct 2021Optical interface for a hybrid magnon-photon resonator\nBanoj Kumar Nayak*,1Cijy Mathai*,1Dekel Meirom,1Oleg Shtempluck,1and Eyal Buks1\n1Andrew and Erna Viterbi Department of Electrical Engineeri ng, Technion, Haifa 32000 Israel\n(Dated: October 27, 2021)\nWe study optical detection of magnetic resonance of a ferrim agnetic sphere resonator, which is\nstrongly coupled to a microwave loop gap resonator. Optical fibers are employed for coupling the\nsphere resonator with light in the telecom band. We find that m agnetic resonance can be optically\ndetected in the region of anti-crossing between the loop gap and the ferrimagnetic resonances. By\nmeasuring the response time of the optical detection we rule out the possibility that microwave\ninduced heating is responsible for the optical detectabili ty.\nMagnons are widely employed in a variety of devices\n[1–7], includingnarrowbandoscillators[8], filters[9], and\nparametric amplifiers [10]. Magnons can couple with mi-\ncrowave (MW) photons [11, 12], optical photons [13–23],\nphonons [24, 25], and with superconducting qubits [26–\n29]. Hybrid magnon devices may help developing optical\nchannels linking remote quantum computers [30–32].\nHere we study a hybrid system composed of a MW\nloopgapresonator(LGR) stronglycoupledtoaferrimag-\nneticsphereresonator(FSR)madeofyttrium irongarnet\n(YIG) [33, 34]. Optical fibers are employed for transmit-\nting light in the telecom band through the sphere. The\nfrequency of the hybrid FSR-LGR system is controlled\nusing an externally applied magnetic field (generated by\na magnetized Neodymium). We explore magneto-optic\n(MO) coupling and Faraday rotation of optical polariza-\ntion, and demonstrate optical detection of magnetic reso-\nnance (ODMR) of the hybrid FSR-LGR system. ODMR\nof FSR has been demonstratedbefore in [35], by coupling\na tapered optical fiber to whispering gallery modes of an\nFSR. However, the ODMR method that has developed in\n[35]isbasedonheatinginduced byMWdriving, andcon-\nsequently the response time of this method is relatively\nlong (on the order of a second). As is shown below, the\nresponse time of our ODMR method, which is not based\non heating, is significantly shorter (limited by the ring\ndown time of the FSR, which is about 1 µs).\nThe experimental setup, which is schematically shown\nin Fig. 1, is designed to allow exploring the MO coupling\nbetween MW and optical photons, which is mediated by\nFSR magnons. In Fig. 1, optical components and fibers\nare red colored, whereas blue color is used to label MW\ncomponents and coaxial cables.\nA MW cavity made of an LGR allows achieving a rela-\ntively large coupling between magnons and MW photons\n[36–38]. The LGR is fabricated from a hollow concentric\naluminium tube. A sapphire strip of 260 µm thickness is\ninserted into the gap in order to increase its capacitance,\nwhich inturn reducesthe frequency fcofthe LGRfunda-\nmental mode [39]. An FSR made of YIG having radius\nofRs= 125µm is held by two ceramic ferrules inside\n*These authors contributed equally to this work.\nFIG. 1: Experimental setup. Optical components [ErS (Er-\nbium source), TL (tunable laser), PC (polarization con-\ntroller), OSA (optical spectrum analyzer) and PD (photode-\ntector)] and fibers are red colored, and MW components [LA\n(loop antenna), S (splitter), C (circulator), VNA (vector n et-\nwork analyzer), SA (spectrum analyzer), SG (signal genera-\ntor), and LIA (lockin amplifier)] and coaxial cables are blue\ncolored. The LA is weakly coupled to the FSR-LGR system.\nOptical fibers are installed on both sides of the FSR for trans -\nmission of light through the sphere. Components outlines by\na thick black line (TL, PD and LIA) are used only for the\nmeasurements presented in Fig. 3.\nthe LGR. The applied static magnetic field His con-\ntrolled by adjusting the relative position of the magne-\ntized Neodymium using a motorized stage. The LGR-\nFSR coupled system is encapsulated inside a metallic\nrectangularshield made ofaluminum (represented by the\nblack colored rectangle in Fig. 1). The cavity is weakly\ncoupled to a loop antenna (LA). More information about\nthe FSR-LGR hybrid system, including its fabrication\nand magnetic energy density distribution, can be found\nin Ref. [38].\nAvectornetworkanalyzer(VNA) is employedformea-\nsuring the MW reflection coefficient |S11|2. The plot\nshown in Fig. 2(b) exhibits |S11|2in dB units as a\nfunction of the externally applied magnetic field Hand\nVNA frequency f. The measurement is performed in2\nFIG. 2: Continuous wave measurements. In both (a) and (c) the ErS is employed as a source, and the optical transmission\nmeasurement is performed using the OSA. (a) Measured optica l spectrum data as a function of SG frequency fwith SG power\nof 20 dBm relative to measured optical spectrum correspondi ng to smallest SG frequency in the plot, with a fixed magnetic\nfield ofH= 140.13mT, showing Fabry–P´ erot oscillation near the FSR-LGR re sonance. (b) VNA reflection |S11|2in dB units\nas a function of frequency fand magnetic field Hwith input power of −30 dBm. (c) Optical intensity (in arbitrary units)\nmeasured at a specific wavelength of λ= 1524.292nm as a function of SG frequency fand magnetic field H, with SG power of\n20 dBm.\nthe regionofanti-crossingbetween the LGRfundamental\nmode at frequency fc= 3.9235GHz and the Kittel mode\nFSR frequency fs, which is given by fs=γgH/(2π),\nwhereγg/2π= 27.98 GHz T−1is the gyromagnetic ratio\n[40, 41].\nThe frequencies f±of the hybrid FSR-LGR eigen\nmodes are given by [42]\nf±=fc+fs\n2±/radicalBigg/parenleftbiggfc−fs\n2/parenrightbigg2\n+g2.(1)\nwheregis the FSR-LGR coupling coefficient [43, 44].\nThe overlaid black dashed lines in Fig. 2(b) are calcu-\nlated using Eq. (1). A fitting procedure yields the value\ng/(2π) = 16MHz. Note that in general, gis proportional\nto the FSR volume.\nIn the telecom band (wavelength λ≃1.5µm) YIG\nhas an optical absorption coefficient αof about α=\n(0.5m)−1[45], a Verdet constant AVof about AV=\n5×10−5mT−1µm−1[46, 47], and a polarization beating\nlengthlPin magnetic saturation of about lP= 7.0mm\n[46, 48, 49]. YIG spherescan be employed formaking op-\ntical circulators and isolator in the telecom band [50, 51].\nAn optical cavity can be constructed to enhance Faraday\nrotation [52, 53].\nIn our setup telecom light is transmitted through the\nFSR using single mode optical fibers. The FSR serves\nas a thick optical lens having focal length of FFSR=\n(1/2)n0Rs/(n0−1), where n0= 2.19 is YIG refractive\nindex in the telecom band [51, 54], A graded index fiber\n(GIF) is attached to the tip of one the the fibers that are\ninstalled near the FSR (see Fig. 1). The length of the\nGIF is 0.1pg, wherepg= 1mmis the GIF pitch. Focusingis achieved by displacing both fibers along the optical\naxis and maximizing the fiber to fiber transmission TF,\nwhich for the current device under study is TF= 0.59.\nSpontaneous emission from an Erbium doped fiber is\nusedastheopticalsourceforthemeasurementspresented\nin Fig. 2. The Erbium source (ErS) intensity peaks near\nwavelength of 1530nm. A polarization controller (PC)\nis employed to manipulate the light transmitted through\nthe FSR. An optical spectrum analyzer (OSA) having\nresolution of 0 .004nm is used to probe the transmitted\nlight. All fibers are single mode having 125 µm clad di-\nameter and 9 µm core diameter.\nThe OSAis employedforprobingthe transmitted light\nin the range 1520nm to 1540nm, as a function of MW\ndriving frequency fapplied to the LA, with a fixed mag-\nnetic field of 140 .13mT [see Fig. 2(a)]. In this mea-\nsurement, a signal generator (SG) operating at 20 dBm\nserves as a source [see Fig. 1, and note that a circula-\ntor (C) and a MW spectrum analyzer (SA) are used to\nprobethe backreflectedMWsignal]. The measuredopti-\ncal transmission shown in Fig. 2(a) reveals Fabry–P´ erot\noscillation near the FSR resonance fs. The wavelength\nperiod of the Fabry–P´ erot oscillation is observed to be\n1.77nm. The oscillation is attributed to an optical cavity\nformed between both fibers coupled to the FSR due to\nFresnel reflection at the fibers’ tips. The measured spac-\ning of 1.77nm allows estimating the distance between the\nfibers to be 700 µm.\nTo study the dependence on both the MW frequency\nfas well as magnetic field H, the optical intensity is\nrecorded at wavelength 1524 .292nm [see Fig. 2(c)], at\nwhich the transmission is maximized [see Fig. 2(a)]. The\nSG frequency is varied from 3 .8GHz to 4 .05GHz, and the3\nFIG. 3: LIA measurements. (a) VNA reflection |S11|2in dB units as a function of frequency fand magnetic field H. The\nVNA input power is −10 dBm. (b) LIA measured voltage amplitude (in arbitrary uni ts) as a function of SG frequency fand\nmagnetic field H, with SG power of −10 dBm. In both (a) and (b) the TL wavelength is 1530 .87nm and power is −2.6 dBm.\n(c) The dependence of LIA measured voltage amplitude VLIA(in arbitrary units) on LIA modulation frequency fAM. For this\nmeasurement, the TL wavelength is set to 1538 .556nm and TL power is set to 6 dBm.\npower is set to 20 dBm. The measured optical intensity\npeaks near the Larmorresonance, i.e. when f≃fs. Note\nthat the splitting between f+andf−cannot be resolved\nin Fig. 2(c) due to anisotropy-induced Kerr nonlinearity\n[38].\nNext we explore the response time of the above-\ndiscussed ODMR method. This is done in order to deter-\nmine the role played by MW induced heating, which has\na relatively long time scale [35]. To that end, we perform\nexperiments using a lockin amplifier (LIA). Components\noutlines by a thick black line in the setup sketchshown in\nFig. 1 (TL, PD and LIA) are used only for the LIA mea-\nsurements presented in Fig. 3. A tunable laser (TL) is\nused instead of the high bandwidth ErS. The OSA is re-\nplaced with a photodetector (PD) to measure the optical\nintensity. The LIA reference signal is used to amplitude\nmodulate (AM) the SG signal at a modulation frequency\nfAM, and the PD signal output is fed into the LIA in-\nput port. For LIA measurement shown in Fig. 3(c),\nthe tunable laser wavelength is set to 1538 .556nm, which\ncorresponds to the second highest optical intensity in the\nspectrum shown in Fig. 2(a).\nBoth the VNA measurement shown in Fig. 3(a) andthe LIA measurement shown in Fig. 3(b) are performed\nwith MW power of −10 dBm and TL optical power of\n−2.6 dBm. Figure 3(c) shows a plot of LIA voltage am-\nplitudeVLIA(in arbitraryunits) as a function of modula-\ntion frequency fAM. The measured dependency on fAM\nindicates that the ODMR response time is on the order\nof a microsecond. This observation suggests that the re-\nsponse time is limited by FSR damping, and it rules out\nthe possibility that heating plays a dominant role in the\nunderlying mechanism allowing the ODMR.\nIn summary, ODMR of FSR is demonstrated in the\ntelecomband, and thepossibility that MWinduced heat-\ningistheunderlyingmechanismisruledout. 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Noda, “Optical circulator consist-\ning of a yig spherical lens, panda-fibre polarisers and a\nfibre-optic polarising beam splitter,” Electronics Letters ,\nvol. 21, no. 17, pp. 746–748, 1985."    },    {        "title": "1909.11381v1.Stability_and_Absence_of_a_Tower_of_States_in_Ferrimagnets.pdf",        "content": "Stability and Absence of a Tower of States in Ferrimagnets\nLouk Rademaker,1Aron Beekman,2and Jasper van Wezel3\n1Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland\n2Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan\n3Institute for Theoretical Physics Amsterdam, University of Amsterdam,\nScience Park 904, 1098 XH Amsterdam, The Netherlands\n(Dated: September 26, 2019)\nAntiferromagnets and ferromagnets are archetypes of the two distinct (type-A and type-B) ways\nof spontaneously breaking a continuous symmetry. Although type-B Nambu{Goldstone modes arise\nin various systems, the ferromagnet was considered pathological due to the stability and symmetry-\nbreaking nature of its exact ground state. However, here we show that symmetry-breaking in\nferrimagnets closely resembles the ferromagnet. In particular, there is an extensive ground state\ndegeneracy, there is no Anderson tower of states, and the maximally polarized ground state is\nthermodynamically stable. Our results are derived analytically for the Lieb{Mattis ferrimagnet and\nnumerically for the Heisenberg ferrimagnet. We argue that these properties are generic for type-B\nsymmetry-broken systems, where the order parameter operator is a symmetry generator.\nI. INTRODUCTION\nSpontaneous symmetry breaking (SSB) is the phe-\nnomenon that the thermal equilibrium state of a many-\nbody system has lower symmetry than the Hamilto-\nnian that governs it. For a continuous symmetry, there\nis a multitude of degenerate symmetry-breaking ground\nstates in the thermodynamic limit where the number of\nconstituents Ngoes to in\fnity. Conversely when Nis \f-\nnite, most such systems possess a unique and symmetric\nground state. This was explicitly shown for the Heisen-\nberg antiferromagnet by Marshall, and Lieb and Mat-\ntis [1, 2], and is now understood to be quite general [3, 4].\nHowever, these unique ground states are not stable, in the\nsense that adding even a small symmetry-breaking per-\nturbation\u000fwill lead to a symmetry-broken ground state;\nin the thermodynamic limit an in\fnitesimal perturbation\nsu\u000eces to break the symmetry. Examples of this type of\nsymmetry breaking include the breaking of Z2(up{down)\nsymmetry in Ising models, of U(1) (phase rotation) sym-\nmetry inXY-models, of SU(2) (spin-rotational) symme-\ntry in Heisenberg antiferromagnets, and of translational\nsymmetry breaking in crystals [5].\nThe instability of the symmetric ground state of \fnite-\nsized systems may be intuitively understood by realizing\nit is actually a type of `Schr odinger cat state', namely\na superposition of macroscopically distinct states which\neach break the symmetry di\u000berently [5, 6]. There must\nthen be some observable Awith an extensive expectation\nvaluehAi\u0018O (N) whose variance scales as Var A\u0018N2,\nviolating the cluster decomposition property [7]. In other\nwords, the symmetric ground state contains macroscopic\nuncertainty of an extensive observable, making it exceed-\ningly susceptible to local perturbations. The symmetry-\nbroken state, on the other hand, does not contain macro-\nscopic uncertainties and is thermodynamically stable. All\nthe while it is not an energy eigenstate; instead, it is a\nsuperposition of the ground state and zero-wavenumber\nlow-energy states. If the broken symmetry is continuous,\nthe gap of these states is of the order O(1=N). The ex-istence of this low-energy tower of states was observed\nfor quantum antiferromagnets by Anderson [8], and sub-\nsequently shown to be generic in SSB systems with a\nsymmetric ground state [3, 4, 9{17]. To understand the\nphysics of \fnite-size systems with SSB, stability is at least\nas important as the energy spectrum [18]. The existence\nof the tower of states can be an important numerical di-\nagnostic to show the propensity to SSB even in very small\nsystems [19, 20].\nThis behavior is however not completely general. In-\ndeed it has long been known that the Heisenberg fer-\nromagnet has degenerate, symmetry-breaking ground\nstates for systems of any size. Furthermore its order\nparameter is a symmetry generator itself and hence a\nconserved quantity [21], there is only a single Nambu{\nGoldstone modes while two symmetry generators are bro-\nken, and this Goldstone mode has a quadratic dispersion.\nIt has recently been cleared up that these two features\ngo hand-in-hand: whenever the commutator of two bro-\nken symmetries has a non-vanishing expectation value,\ntwo Goldstone \felds conspire to form a single, quadrati-\ncally dispersing gapless mode accompanied by a gapped\npartner mode.[22{26] Such Goldstone modes have been\ndubbed type-B, while ordinary, linearly dispersing Gold-\nstone modes are called type-A.\nIt is now the question whether the other ferromagnet\nphenomenology|degenerate, thermodynamically stable\n\fnite-size ground states and no tower of states|also car-\nries over to any type-B SSB system. A natural starting\npoint is the ferrimagnet , a state with antiferromagnetic\ncorrelations between two unequal-size spin species, which\nimplies in addition to antiferromagnetic order alsoferro-\nmagnetic order. Earlier, one of the authors suggested\nthat ferrimagnets would feature a tower of states, since\ntheir classical ground states are not eigenstates of the\nquantum Hamiltonian [27]. On the other hand, it has\nlong been known that spin systems with any non-zero\nmagnetization have macroscopically degenerate ground\nstates [2, 28].\nHere we show that the Heisenberg ferrimagnet is fararXiv:1909.11381v1  [cond-mat.str-el]  25 Sep 20192\nmore akin to a ferromagnet than to an antiferromagnet:\nwe demonstrate explicitly that there exists a thermody-\nnamically stable \fnite-size ground state, and that there\nis no tower of states separated from the ground state by\nan excitation gap of order O(1=N). This stable ground\nstate can be understood to be a classical (product) state\nsupplemented by quantum corrections, in the same way\nthat the SSB states of type-A systems are [8, 21]. We pro-\nvide an analytic derivation of the stability of this state in\nthe simpli\fed case of the Lieb{Mattis model, and provide\nnumerical evidence for the stability in the full Heisenberg\nHamiltonian.\nWe furthermore argue that this behavior is general\nfor any system with exclusively type-B SSB. This paints\na comprehensive picture of SSB: if the ground state is\nunique, it must be accompanied by a tower of states in\norder for thermodynamically stable SSB states, as a su-\nperposition of very-closely spaced energy eigenstates, to\nexist. In type-B systems such a tower of states is absent,\nbut SSB is possible because thermodynamically stable,\nsymmetry-breaking exact ground states exist even for\n\fnite-size systems.\nThis article is organized as follows. In Section II we\nbrie\ry outline the Lieb{Mattis argument which leads to\nthe tower of states in antiferromagnets, the archetype\nfor type-A SSB. In Section III we show that a tower of\nstates in absent in Heisenberg ferrimagnets, while there\nis a ground state degeneracy. These ground states all\nbreak theSU(2) spin-rotation symmetry as is shown in\nSection IV. In Section V we calculate the overlap of the\nSSB ground state with the classical N\u0013 eel state and com-\npare with the situation in the antiferromagnet. The cen-\ntral part of this work is the demonstration that the two\nground states which have maximal positive or negative\nmagnetization are thermodynamically stable. In Sec-\ntion VI this is shown analytically for the Lieb{Mattis\nmodel. Numerical evidence of the stabily of small 1D\nferrimagnets is provided in Section VII by exact diago-\nnalization. We conclude with a comprehensive picture of\nSSB and directions for further research in Section VIII.\nII. ANTIFERROMAGNETS\nTo set the stage, we shall \frst recall some well-known\nfacts about Heisenberg antiferromagnets on bipartite lat-\ntices. The Heisenberg Hamiltonian is\n^HH=JX\nhiji^~Si\u0001^~Sj: (1)\nHere^~Siis a spin-soperator[29] on site i, the sum is over\nnearest-neighbor lattice sites, and J > 0 is a coupling\nconstant. This Hamiltonian is invariant under global\nSU(2)-spin rotations. If the lattice is bipartite it can\nbe divided in A- andB-sublattices such that each site\nhas neighbors only on the other sublattice. The classi-\ncal ground states are N\u0013 eel states with spins anti-alignedon the two sublattices, breaking the SU(2) symmetry\ntoU(1); the direction of N\u0013 eel ordering is spontaneously\nchosen. If furthermore the number of sites of each sub-\nlattice is the same (for instance in square and hexagonal\nlattices), there is no net magnetization hP\ni^~Sii= 0.\nThe N\u0013 eel states are not eigenstates of Eq. (1), and will\nbe a\u000bected by quantum corrections. But even stronger,\na \fnite-size system governed by this Hamiltonian has a\nunique ground state with total spin value S= 0, which\ntherefore does not break any symmetry. This was shown\nby Marshall [1], and can be understood due to an elegant\nargument by Lieb and Mattis [2]: consider the following\nHamiltonian (\\Lieb{Mattis model\")\n^HLM=2J\nN^~SA\u0001^~SB=J\nN(^S2\u0000^S2\nA\u0000^S2\nB); (2)\nwhere^~S=P\ni^~Siis the total spin of the system,^~SA;B=\nP\ni2A;B^~Sithe total sublattice spin, and ^S2=^~S2etc.\nNote that^~S=^~SA+^~SB. In this model each spin on the A-\nsublattice interacts with all spins on the B-sublattice and\nvice versa. This Hamiltonian simultaneously commutes\nwith ^S2\nA,^S2\nB,^S2and^Sz, and eigenstates can therefore be\ndesignated by the quantum numbers jSASBSMzi, with\nenergiesE=J\nN\u0000\nS(S+ 1)\u0000SA(SA+ 1)\u0000SB(SB+ 1)\u0001\n.\nClearly the energy is minimal when Sis minimal and\nbothSAandSBare maximal. The minimal value is\nS= 0, and therefore the ground state is a total spin\nsinglet, is unique, and does not break any symmetry.\nIt can be easily seen that the Lieb{Mattis model is\nequal to only the ~k=~0 and~k=~Q= (\u0019;\u0019;:::;\u0019 ) con-\ntributions of the Fourier-transformed Heisenberg Hamil-\ntonian Eq. (1). Lieb and Mattis have shown that for\nany \fniteNthe overlap between the ground state of\nEq. (2) and the ground state of Eq. (1) is non-vanishing.\nTherefore, these two states must have the same quantum\nnumbers, and also the ground state of the Heisenberg\nantiferromagnet is a total spin singlet.\nThe correspondence between the two Hamiltonians\ngoes further. Excitations that keep SAandSB\fxed\nwhile increasing Scost an energyO(J=N). For large\nN, these energy levels are almost degenerate with the\nground state. There is therefore a tower of extremely\nlow-lying states with SAandSBmaximal,Mz= 0, and\ndi\u000beringS, and this carries over to the Heisenberg an-\ntiferromagnet by the same argument. Note that excita-\ntions in the Lieb-Mattis model that change SAorSBcost\nenergy of at least O(J), just as local excitations such as\nspin \rips, while Nambu{Goldstone modes (spin waves) in\nthe Heisenberg model have lowest energy O(J=L) withL\nthe linear size of the system.\nThe variance of the local N\u0013 eel order parameter ^Nz\ni=\n(\u00001)i^Sz\niin the symmetric ground states is of order\none [4], so that variance of the total N\u0013 eel order parameter\nscales asN2, indicating that this state is not thermody-\nnamically stable. This is easy to see, when one realizes\nthe ground state of the Lieb{Mattis model is equal to3\nthe equal-weight superposition of classical N\u0013 eel states in\nall magnetization directions [6]. This is a Schr odinger\ncat state, which is extraordinarily sensitive to external\nperturbations.\nIII. THE ABSENCE OF A TOWER OF STATES\nWe will now begin demonstrating the di\u000berences with\nthe picture painted in Section II, for the case where the\nmagnetization is \fnite. For concreteness, we study the\nHeisenberg ferrimagnet governed by Hamiltonian Eq. (1),\nbut now the spins on A- andB-sublattices are sAandsB\nrespectively, with sA6=sB. Without loss of generality\nwe choosesA> sB. On a bipartite lattice with equal\nnumber of sublattice sites, the classical N\u0013 eel state has\na \fnite magnetization h^Szi=N(sA\u0000sB)=2, and stag-\ngered magnetization h^Nzi=N(sA+sB)=2. Everything\nwe say here also holds for antiferromagnets where the\nnumber ofA-sublattice sites is di\u000berent from number the\nB-sublattice sites; the imporant feature is that the total\nspin isS=jSA\u0000SBj>0 whereSA;B=P\ni2A;BsA;B.\nIt is known that the number of ground states of this\nmodel is equal to 2 S+ 1 = 2jSA\u0000SBj+ 1 =NjsA\u0000\nsBj+ 1 [2, 28], which can again be inferred from the\noverlap of these states with the ground states of the\ncorresponding Lieb{Mattis model. This number is ex-\ntensive (proportional to N) sinceSis extensive. More-\nover, the lowest excitation according to Eq. (2) has spin\nS=jSA\u0000SBj\u00061 whileSAandSBare the same, with\nenergy gap \u0001 E=J\u0000\n(sA\u0000sB) +2\nN\u0001\n. Crucially, this en-\nergy gap isO(J) instead of order O(J=N) sincesA6=sB.\nThere is therefore no tower of states with energy gap\nO(1=N) that would vanish in the thermodynamic limit.\nIndeed, the exchange energy Jis typically of order 1{10\nmeV, which is certainly not neglible, possibly even mea-\nsurable.\nRecall that the Lieb{Mattis Hamiltonian is the ~k=\n~0;~Qpart of the full Heisenberg Hamiltonian. The tower\nof states consists precisely of the zero-wavenumber exci-\ntations, and therefore these states and their energies are\nidentical for both models. We can therefore conclude that\nthe lowest excitations in Heisenberg ferrimagnets will be\nlow (non-zero) wavenumber collective excitations: spin\nwaves with quadratic dispersion whose energy scales as\nO(J=L2) withLthe linear system size (so N=Ld).\nWhile this energy can get arbitrarily low as L!1 , it\nwill not be as low as the gap in a putative tower of states\nind>2 dimensions. We will con\frm this numerically in\nSection VII.\nIn Ref. [4], Tasaki provides a proof of the existence of\na tower of states with gaps of order O(1=N), based on\nseveral assumptions. One of these assumptions is that\nthe ground state be unique andbe an eigenstate of a\nsymmetry generator with eigenvalue M. In the present\ncase, although the ground states are eigenstates of ^Sz,\nthey are degenerate and not unique. Below we will argue\nthat whenM > 0, the ground state is always degenerate.Tasaki's derivation applies therefore only to the case M=\n0 and the results of this section are not in contradiction\nwith the proof. The alternative is when the symmetry\ngenerator is broken itself in the type-A way, which we\nshortly discuss in Section VIII.\nIV. SPONTANEOUS SYMMETRY BREAKING\nFrom the standard viewpoint of SSB, the absence of\na tower of states naively poses a conundrum. In type-\nA systems, all ~k=~0 energy eigenstates, including the\nsymmetric ground state, are thermodynamically unsta-\nble, and a tiny perturbation will be able to break the\nsymmetry. The existence of a tower of states is nec-\nessary to be able to construct the dynamically stable,\nsymmetry-breaking superpositions of energy eigenstates,\nas the energy \ructuations of these superpositions fall o\u000b\nasO(J=N). We have just seen the smallest energy gap\ntowards total spin excitations in ferrimagnets is instead\nO(J). What does this imply for the symmetry breaking?\nThe answer is in fact quite simple: the exact ground\nstates already break the symmetry themselves. This is\neasy to see from the Lieb{Mattis model. We have shown\nthat its spectrum can be assigned de\fnite quantum num-\nbersSandMz(where the z-axis is chosen arbitrarily).\nThe only such state which has full SU(2) spin-rotation\nsymmetry is the one with S=Mz= 0. (It is not su\u000e-\ncient to have only Mz= 0. Recall for instance that a two-\nspin-1\n2system in the triplet state s= 1;mz= 0 breaks\nSx- andSy-rotation symmetry.) The ground states of\nthe ferrimagnet instead have S=jSA\u0000SBj>0. In fact,\nthe symmetric state with S= 0 has quite a high energy\ncompared to this state.\nWe can also reach this result more formally, by con-\nsidering the usual SSB procedure of adding an external\nstaggered magnetic \feld Bcoupling to the order param-\neter, to the Lieb{Mattis model:\n^HLM=J\nN\u0010\n^S2\u0000^S2\nA\u0000^S2\nB\u0011\n\u0000B\u0010\n^Sz\nA\u0000^Sz\nB\u0011\n: (3)\nHere ^Sz\nA;B=P\ni2A;B^Sz\nirepresent the z-component of\nthe total spin on sublattices AandB. The matrix ele-\nments of the symmetry-breaking \feld in the basis of Lieb{\nMattis eigenstates, hSASBSMzj(^Sz\nA\u0000^Sz\nB)jS0\nAS0\nBS0M0\nzi,\nare known exactly (reproduced in Appendix A for com-\npleteness) [30]. For zero \feld B= 0, the ground states\nhave maximal SA, andSB, minimal S=jSA\u0000SBj,\nand are degenerate for any value of Mz. For non-zero\n\feld,B > 0, the degeneracy is lifted, and the state\nwith the lowest expectation value of the energy has mag-\nnetizationMz=S. This state must be a superpo-\nsition of states with di\u000berent values of the total spin,\njSA\u0000SBj<S < (SA+SB), because the staggered mag-\nnetization operator Sz\nA\u0000Sz\nBcouples state with total spin\nStoS\u00061 states.\nIn the limit of large Nthe matrix elements of the\nHamiltonian can be conveniently expressed in terms of4\nthe shifted total spin ~S=S\u0000jSA\u0000SBj. The Hamilto-\nnian is then, up to order O(1\nN):\n^H\u0019E0+X\n~S;~S0jSASB~SMzi\u0010\nf~S+1\u000e~S;~S0\u00001\n+a~S\u000e~S;~S0+f~S\u000e~S;~S0+1\u0011\nhSASB~S0Mzj;\nwithE0=J\n4(sA\u0000sB)2N+J\n2(sA\u0000sB)\n\u0000B\n2(sA+sB)N\u00002BsB\nsA\u0000sB;\na~S=~S\u0012\nJ(sA\u0000sB) + 2BsA+sB\nsA\u0000sB\u0013\n;\nf~S=\u00002BpsAsB\nsA\u0000sB~S: (4)\nThis expression for the Hamiltonian is a special case of\na tridiagonal matrix discussed in Appendix B. As shown\nthere, the ground state in the large Nlimit is a superpo-\nsition of states with di\u000berent ~S=S\u0000jSA\u0000SBj,\nj 0(B;N )i=X\n~S (~S)j~Si; (5)\nwith weights given by an exponentially decaying function\n (~S) =ce\u0000~S=\u0015. The decay `length' \u0015is given by\n\u0015= 1=log \n1\n2\u0000\n1 +p\n1\u00004\u000f2\u0001\nj\u000fj!\n; (6)\nwith\u000f=\u00002BpsAsB\nJ(sA\u0000sB)2+ 2B(sA+sB): (7)\nIn the limit of zero staggered \feld B!0, the decay\nlength vanishes, and the Lieb{Mattis ground state j 0iis\nthe total-spin eigenstate with S=jSA\u0000SBjandMz=S.\nContrary to the case of type-A SSB systems, the ther-\nmodynamic limit and the limit of vanishing \feld com-\nmute for the ferrimagnet:\nlim\nB#0lim\nN!1j 0(B;N )i= lim\nN!1lim\nB#0j 0(B;N )i:(8)\nJust as for the ferromagnet, a particular orientation (i.e.\nthe choice of the z-axis) for the ferrimagnetic ground\nstate can be singled out from the ground state manifold\nby an in\fnitesimal \feld even for \fnite system size. There\nis thus a discontinuity in the ground state as a function\nof applied \feld for any system size:\nlim\nB#0j 0(B;N )i6= lim\nB\"0j 0(B;N )i: (9)\nNumerical evaluation of the ground state of the ferri-\nmagnetic Lieb{Mattis Hamiltonian for large but \fnite N\nandB > 0 con\frms that it is an exponentially decaying\nfunction in ~S, as shown in Fig. 1. The decay length\nfollows the analytical result of Eqns. (6), (7).\nWe conclude that although the ferrimagnet has 2 S+ 1\ndegenerate ground states, any small external staggered\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★●N=20\n◆N=200\n★N=2000\nEqs.(6)-(7)0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070.00.10.20.30.40.50.60.7\nB/JλFIG. 1. The ground state of the ferrimagnetic Lieb{Mattis\nmodel with staggered magnetic \feld Bis a superposition of\nstates with di\u000berent ~S=S\u0000jSA\u0000SBj, with weights given\nby e\u0000~S=\u0015. Here, a numerically determined decay length \u0015is\nplotted as a function of B=J for increasing system size up to\nN= 2000. The dashed black line is the exact result in the\nthermodynamic limit following Eqs. (6)-(7). It is important\nto observe that in the thermodynamic limit N!1 the decay\nlength\u0015remains smooth near B= 0.\n\feld is su\u000ecient to lift the degeneracy, upon which the\nground state will be remain dominated by the spin state\nwithS=jSA\u0000SBj, and the magnetization direction\nparallel to the external \feld.\nV. THE CLASSICAL N \u0013EEL STATE\nBesides the total (ferromagnetic) magnetization Mz,\nferrimagnets also have a nonzero staggered (antiferro-\nmagnetic) magnetization n=h^Sz\nA\u0000^Sz\nBi 6= 0. Using\nthe matrix elements in Appendix A, the staggered mag-\nnetization expectation value for the maximally polar-\nized ground state of the Lieb{Mattis ferrimagnet, with\nMz=SandS= (sA\u0000sB)N=2, and no applied \felds, is\nfound to be:\nn=NsA+sB\n2\u00002sB\njsA\u0000sBj+O(N\u00001) (10)\nTo leading order, this agrees with the expectation from\nthe classical limit for a ferrimagnet, which is the N\u0013 eel\nproduct statej N\u0013 eeliwith all spins on the Asublattice\npointing maximally up and on the Bsublattice maxi-\nmally down. That state has maximal staggered magne-\ntizationn=NsA+sB\n2.\nThe N\u0013 eel state itself can be written as a superpo-\nsition of states with di\u000berent total spin Sbut \fxed\nMz=SA\u0000SB, andSA;B=sA;BN=2. Becausej N\u0013 eeli\nis the ground state of the Hamiltonian in Eqn. (3) with\nJ= 0 andB > 0, the N\u0013 eel state wavefunction has the\nsame exponentially decreasing form  (~S) =ce\u0000~S=\u0015as\nthe ferrimagnetic ground state. For J= 0, however,\nthe decay length \u0015in Eqn. (6) is independent of Nand\nB. This means that the N\u0013 eel state has nonzero overlap5\nwith the ground state of the Lieb{Mattis Hamiltonian for\nany system size, and even in the thermodynamic limit\nN!1 . Its value can be found by normalising the N\u0013 eel\nstate wavefunction:\nh N\u0013 eeljS=jSA\u0000SBji=p\n1\u0000e\u00002=\u0015(J=0) (11)\nFor the special case of sA= 1 andsB= 1=2, we have\n\u000f(J= 0) =\u0000p\n2\n3and the overlap is found to be1p\n2.\nBecause the N\u0013 eel state is a superposition of total spin\nstates, which do not form a tower of states in the ferri-\nmagnet, the N\u0013 eel state cannot be become energetically\ndegenerate with the ground state of the Lieb{Mattis\nHamiltonian for J > 0 andB= 0. The energy di\u000ber-\nence follows directly from\nh N\u0013 eelj^HLMj N\u0013 eeli=J\n4NjsA\u0000sBj2+J(s2\nA+s2\nB):(12)\nThis should be compared with the Lieb{Mattis ground\nstate energy E0=J\nN(SA\u0000SB)(SA\u0000SB+1) =J\n4NjsA\u0000\nsBj2+J\n2(sA\u0000sB). The energy di\u000berence between j N\u0013 eeli\nand the actual ground state is thus:\n\u0001EN=J\u0012\ns2\nA+s2\nB\u00001\n2sA+1\n2sB\u0013\n: (13)\nFor the speci\fc case with sA= 1 andsB= 1=2, the\nenergy di\u000berence is exactly \u0001 EN=J. In all cases, it is\nof orderO(J), and does not vanish in the thermodynamic\nlimit. This is to be contrasted with the antiferromagnet,\nwhere the N\u0013 eel state becomes exactly degenerate with\nthe ground state in the thermodynamic limit of the Lieb{\nMattis model.\nThe fact that the staggered magnetization in Eqn. (10)\ndi\u000bers from its classically expected value at sub-leading\norder, may be interpreted as indicating that the Lieb{\nMattis ground state involves zero-wavenumber quantum\ncorrections on top of the classical N\u0013 eel state. These quan-\ntum corrections correspond precisely to the suppression\nof total-spin components outside the ground state man-\nifold. Going towards more realistic models, one should\nnote that the ground state of the Heisenberg ferrimagnet\nis not the same as that of the Lieb{Mattis Hamiltonian.\nBecause the Heisenberg model includes only local inter-\nactions, its ground state will have quantum corrections\nat all wave numbers as compared to the N\u0013 eel state, and\ntheir overlap vanishes in the thermodynamic limit (con-\n\frmed numerically below), even though both states will\nagree to leading order on the expectation value of stag-\ngered magnetization. The vanishing overlap is in fact\na generic property for ground states of distinct models\nin an exponentially large Hilbert space, and is expected\nalso for example for the overlap between the ground state\nof the Heisenberg antiferromagnet and the classical N\u0013 eel\nstate.VI. STABILITY\nIn Section IV we addressed one part of the conundrum\nthat the absence of a tower of states poses, namely how\nthe ground state breaks the symmetry. However, zero-\nwavenumber energy eigenstates are global|they are not\ntensor products of local states|and typically unstable as\nwe have shown for instance for the ground state of the\nantiferromagnet in Section II. The question is whether\nthe symmetry-breaking exact ground states of the \fnite\nsize ferrimagnet are stable.\nIn this section we show that the exact ground states of\nthe Lieb-Mattis model with maximal polarization, Mz=\n\u0006S=\u0006jSA\u0000SBjare thermodynamically stable. In\nthe next section we provide numerical evidence that the\nmaximally polarized ground states in the full Heisenberg\nmodel are also stable.\nRecall that a state is de\fned to be unstable if there\nexists an extensive observable ^Awhose variance Var ^A=\nh^A2i\u0000h ^Ai2scales asN2[4, 5, 18]. This de\fnition is\nequivalent to saying there exists a connected correlation\nfunction that does not satisfy the cluster decomposition\nproperty [5, 7].\nTo prove that a state is stable one thus needs to com-\npute the variance of all possible extensive observables. In\nthe case of the Lieb{Mattis Hamiltonian in Eqn. (3), all\nstates are global, and we can su\u000ece by computing the\nvariance of the transverse total spin, Var ^Sx, and of the\ntotal sublattice magnetization, Var( ^Sz\nA\u0000^Sz\nB).\nThe variance of the transverse total spin ^Sxis inde-\npendent of SAandSB. BecausehSMzj^SxjSMzi= 0, we\n\fnd\nVarh\n^Sxi\n=hSMzj(^Sx)2jSMzi\n=1\n4hSMzj(^S++^S\u0000)2jSMzi\n=1\n4hSMzj(^S+^S\u0000+^S\u0000^S+)jSMzi: (14)\nThe maximally polarized states, with Mz=\u0006S, are an-\nnihilated by ^S\u0006, and we then \fnd the variance to be\nS=2\u0018O(N). On the other hand, for jMzj<Swe \fnd:\nVarh\n^Sxi\n=1\n2S(S+ 1)\u00001\n2M2\nz: (15)\nAs long asjMzjdoes not scale with system size in exactly\nthe same way as S, the variance of the total transver-\nsal spin scales as O(N2), implying instability. Therefore\nonly the states with jMzj=S!1 are thermodynamically\nstable with respect to the total transversal spin.\nThe variance of the sublattice magnetization ^Sz\nA\u0000^Sz\nB\ncan be computed directly using the matrix elements in\nAppendix A. For the maximally polarized ground state\nwithMz=SandsA>sB, it is to leading order:\nVarh\n^Sz\nA\u0000^Sz\nBi\n= 4sAsB\n(sA\u0000sB)2+:::\u0018O(1):(16)6\n◆◆◆◆◆◆★★★★★★(L2)\n(L)Mz=0 or 1/2\nMz=S\n4 6 8 10 12140.5125\nLVarS x\n◆◆◆◆◆◆★★★★★★(L)\n(L)Mz=0 or 1/2\nMz=S\n4 6 8 10 12142468\nLVar(Sz)A-(Sz)B\nFIG. 2. A numerical comparison of the stabilities of the max-\nimally (Mz=S=L=2, green stars) and minimally ( Mz= 0\nor 1=2, red diamonds) polarized ground states of the 1D fer-\nrimagnetic sA= 1,sB=1\n2Heisenberg model. The left panel\nshows Var ^Sx, and the results exactly follow Eq. (15) (shown\nas dashed lines). The variance in the maximally polarized\nstate scales asO(L) and thus signal stability, whereas the\nO(L2) scaling in the minimally polarized state implies in-\nstability. The right panel shows the variance of ^Sz\nA\u0000^Sz\nB.\nHere, the results do not match the variance found for the\nground states of the Lieb{Mattis model, because SAandSB\nare not symmetries of the ferrimagnetic Heisenberg Hamil-\ntonian. Nevertheless, the variances still scales as O(L) (the\ndashed lines represent a \ft) which implies that all ground\nstates are stable with respect to the uncertainty of the anti-\nferromagnetic correlations.\nThis shows that the maximally polarized ground state\nis thermodynamically stable with respect to all exten-\nsive observables even in \fnite-sized ferrimagnets. For\nstates with non-maximal polarization, M\u0018 O (1), we\nhave Varh\n^Sz\nA\u0000^Sz\nBi\n=sAsB\nsA\u0000sBN+:::, which also suggests\nstability. However, since we have already seen that these\nstates with non-maximal polarization are unstable with\nrespect to total transverse spin ^Sx, the two maximally\npolarized ground states are the only thermodynamically\nstable ground states.\nVII. NUMERICAL RESULTS\nThe Mermin{Wagner{Hohenberg{Coleman theorem\ndoes not prohibit type-B SSB from occurring in one-\ndimensional systems at zero temperature [5, 31, 32]. We\ncan therefore con\frm the generality of the analytic re-\nsults of the Lieb{Mattis model in Section VI using numer-\nical results for a one-dimensional ferrimagnetic Heisen-\nberg chain. The Hamiltonian is given by:\n^H=JX\ni^~Si\u0001^~Si+1: (17)\nFor concreteness, we take all the even sites to have sA= 1\nspins and the odd sites sB= 1=2 spins. We consider\nchains up to lengths L= 14 using exact diagonaliza-\ntion, and evaluate degeneracies and the stability of the\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★ ★ ★ ★◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆\nk≠0 states\nk=0 states, S>|S A-SB|\nk=0 states, S<|S A-SB|●\n◆\n★(1)\n(1/L2)\n(1/L2)6 8 10 12 140.20.512\nLΔE/JFIG. 3. An overview of all low-energy excited eigenstates\nwith their energy gap in units of J, in the 1D ferrimagnetic\nsA= 1,sB=1\n2Heisenberg chain, as a function of system\nsizeL. Black dots represent states with \fnite wave number,\nk6= 0, while red stars indicate states with zero wave number\nk= 0 andS <jSA\u0000SBj, and green diamonds have k= 0 and\nS >jSA\u0000SBj. HerejSA\u0000SBj=L=4 is the ground state total\nspin. In one dimension, the excitation gap towards Goldstone\nmodes with nonzero wave number (black dots) is O(1=L2)\n(indicated by the black dashed line), because \u000fk\u0018k2and\nthe smallest momentum scales as k\u0018O(1=L). States with\nk= 0 andS <jSA\u0000SBjare two-Goldstone mode states (red\nstars), which follows from the fact that their magnetization\neigenvalueMzis smaller than the ground state value. Finally,\nthe states that would constitute the tower of states in an\nantiferromagnet, namely those with k= 0 andS >jSA\u0000SBj\n(green diamonds), have a gap that is independent of system\nsize (green dashed line).\nground states. Also note that because our system is one-\ndimensional, the linear size Land total system size N\nare equal.\nWe con\frm that the sA= 1,sB=1\n2Heisenberg ferri-\nmagnet has a L=4+1-dimensional ground state manifold\nwith total spin S=L=4, just like the Lieb{Mattis ground\nstates.\nNext we analyze the stability of the states in the\nground state manifold. Following Sec. VI, we compute\nthe variance of the transverse total spin and the stag-\ngered magnetization, shown in Fig. 2. The variance of\nstaggered magnetization scales with the system size, in-\ndependent of the magnetic number Mz. All states in the\nground state manifold are therefore stable with respect\nto staggered magnetization. However, the variance of\ntransverse spin scales as O(L2) for the state with mini-\nmalMzbut it scales asO(L) for the maximally polarized7\nNéel/Lieb-Mattis\nHeisenberg/Lieb-Mattis\nNéel/Heisenberg\n4 6 8 10 12 140.20.40.60.81.0\nLOverlaps|<ψ|ϕ>2\nFIG. 4. The overlap-squared between ground states of vari-\nous models shown on a log-log scale. The overlap between the\nmaximally polarized ground state of the Lieb{Mattis model\nstate and the classical N\u0013 eel state follows Eqn. (11) and stays\nnonzero even in the thermodynamic limit. On the other hand,\nthe presence of spin \rips or quantum corrections in the maxi-\nmally polarized ground state of the ferrimagnetic Heisenberg\nmodel causes its overlaps with both the N\u0013 eel state and the\nLieb{Mattis state to vanish with increasing system size.\nstateMz=S. In fact, the results are exactly equal to\nEq. (15). We therefore con\frm that the only thermody-\nnamically stable ground state is the maximally polarized\nstate withMz=S.\nThe absence of a tower of states is con\frmed by an\nanalysis of the low-lying eigenstates, shown in Fig. 3.\nFor each low-energy eigenstate we computed its energy,\nmomentum k, total spin Sand polarization Mz. If a\ntower of states would be present, the energy gap towards\nstates with k= 0,Mz=jSA\u0000SBjandS >jSA\u0000SBj\nwould vanish as O(1=L). This is not the case, as we\nargued in Sec. III: in Fig. 3 we see that these states (green\ndiamonds) have a gap of order O(1) (green dashed line).\nNonetheless, there are states whose gap vanishes as\nO(1=L2). We identify here the excitation of a Goldstone\nmode, with dispersion \u000f\u0018k2and the smallest momen-\ntum scaling as k\u0018O(1=L). Goldstone modes necessarily\nhave nonzero momentum, shown in Fig. 3 as black cir-\ncles. The excitation of two Goldstone modes with oppo-\nsite momentum now leads to states with zero momentum\nbut still a gap that scales as O(1=L2), shown as red stars.\nWe thus have analyzed the full eigenvalue spectrum and\nexcluded the possibility of a tower of states.\nFinally, Fig. 4 shows the overlap of the Lieb{Mattis\nand Heisenberg ground states with the classical N\u0013 eel\nstate. While the former have a nonzero overlap in the\nthermodynamic limit, the overlap of the latter decreases\nwith system size. This can be understood as a conse-\nquence of the extensive number of single and few-spin\n\rips contained in the Heisenberg ground state relative to\nthe N\u0013 eel state. These can be seen as quantum corrections\nto the classical state at all wave numbers, and although\nthey hardly a\u000bect the macroscopic staggered magneti-zation, and do not a\u000bect the magnetization at all, they\ndo cause the overlap with the N\u0013 eel state to vanish in\nthe thermodynamic limit. The latter also occurs in the\n(type-A) antiferromagnet.\nVIII. OUTLOOK\nWe found the ferrimagnet to have an extensive ground\nstate degeneracy and no tower of states. Furthermore,\nwe found that within the ground state manifold, the\nmaximally polarized states are thermodynamically sta-\nble. The ferrimagnet shares these features with the fer-\nromagnet, which turns out to be less unique than often\nassumed [24, 27, 31].\nAlthough we found these results for the speci\fc case\nof the ferrimagnet, we hypothesize that these conclusions\napply to type-B systems in general. The de\fning prop-\nerty of such systems is that the the expectation value of\nthe commutator of two broken symmetry generators does\nnot vanish. Apart from some pathological cases [22], such\na commutator is a linear combination of symmetry gen-\nerators itself, implying that the order parameter operator\ncommutes with the Hamiltonian , i.e. the order parameter\noperator is a symmetry generator.\nLet us consider the Lie algebra structure of the Hamil-\ntonian and its eigenstates. The symmetry generators ^Qa,\nwhich by de\fnition are Hermitian and commute with the\nHamiltonian, can be expressed in the Cartan{Weyl basis,\nin which the Cartan subalgebra is spanned by a maximal\nset ofrmutually commuting generators ^Fi, whereris\ncalled the rank of the Lie algebra. The remaining gen-\nerators can be expressed in pairs of Hermitian-conjugate\nroot generators ^E\u000b;^E\u0000\u000b, with\u000bcalled the root vector\nand commutation relations [ ^E\u000b;^E\u0000\u000b] =\u000bi^Fi. Watan-\nabe and Brauner have shown that the Cartan subalgebra\ncan be chosen in such a way that only the Cartan gener-\nators ^Fican obtain an expectation value, and hence lead\nto type-B SSB [23]. We can simultaneously diagonalize\nthe Hamiltonian and the Cartan generators. Eigenstates\nof the Cartan generators ^Fiareweight statesj\u0016iwith\neigenvalues \u0016i, collected in a weight vector \u0016.\nNow we recall several important theorems from Lie\ngroup theory [33, 34]. First, irreducible representations\nof a semisimple Lie algebra are completely classi\fed by\nspecifying the highest weight \u0016 \u0016. Second, a Lie algebra\nof rankrcontainsrCasimir operators, which commute\nwith the entire Lie algebra. Third, by Schur's lemma,\nany operator that commutes with all generators of the\nLie algebra is proportional to the unit matrix in any ir-\nreducible representation.\nFor discussing the ground states and the tower of\nstates, we only need to consider the k= 0-part of the\nHamiltonian, which therefore only depends on internal\ndegrees of freedom which transform under the Lie group\nof symmetry transformations. Consequently we are go-\ning to assume that the k= 0 Hamiltonian can be com-\npletely speci\fed in terms of Lie algebra generators, in8\nother words it is a spectrum-generating algebra [34]. It\nfollows that the k= 0 Hamiltonian, which commutes\nwith the entire algebra, consists only of Casimir opera-\ntors.\nFor type-B SSB, a ground state is an eigenstate of sym-\nmetry generators in the Cartan subalgebra, i.e. a weight\nstate, where at least one weight component is non-zero.\nSince a symmetry generator is an extensive operator, this\nweight component is extensive. Because the irreducible\nrepresentation of this generator is speci\fed by its high-\nest weight, the highest weight must also be extensive,\nwhich implies that the representation space has exten-\nsive dimensions. And since the Hamiltonian consists of\nCasimir operators, which are proportional to the iden-\ntity matrix in any irreducible representation, this implies\nthat there is a ground state degeneracy of the extensive\ndimension of this representation. This proves that un-\nder mild assumptions type-B SSB involves an extensive\nground state degeneracy.\nAs for the tower of states, we shall discuss the\ncase where the k= 0 Hamiltonian is proportional to\nthe quadratic Casimir operator ^C2, which consists of\nquadratic combinations of generators ^Qa. This com-\nprises most models of interest, including the Heisen-\nberg ferrimagnet and (anti)ferromagnet. Because the\n^Qaare extensive, the Hamiltonian must be of the form\n^Hk=0=1\nN^C2, up to factors of order O(1), so that the en-\nergy is extensive. In the Cartan-Weyl basis, the quadratic\nCasimir operator can be expressed as [35]\n^C2=X\ni^Fi^Fi+X\n\u000b2\u0001+(^E\u000b^E\u0000\u000b+^E\u0000\u000bE\u000b); (18)\nwhere \u0001 +is the set of positive root vectors. Since\nCasimir operators are proportional to the identity ma-\ntrix, we can \fnd the proportionality constant by acting\non the highest weight state for which ^E\u000bj\u0016\u0016i= 0. Then,\nacting on this state, the term in brackets is equal to\n[^E\u000b;^E\u0000\u000b] =\u000bi^Fi. We therefore \fnd the energy of all\nstates in the irreducible representation \u0016 \u0016to be\nE=1\nNX\ni\u0016\u0016i\u0000\n\u0016\u0016i+X\n\u000b2\u0001+\u000bi\u0001\n: (19)\nWith all this, we can analyze the putative tower of\nstates. Denoting the representation to which the ground\nstate belongs by \u0016 \u00160, it consists of k= 0 states in dif-\nferent irreducible representations \u0016 \u00160, but with the same\neigenvalues \u0016for the Cartan generators, which implies\nthat \u0016\u00160>\u0016\u00160\u0015\u0016. From Eq. (19) we see that, since \u0016 \u00160\nis extensively non-zero, the energies of any excited state\ncontain at least a factor of1\nN\u0016\u00160\nj, and therefore are at least\nO(1). There is no tower of states with gaps O(1=N).\nThere is one caveat: it could be that the weight state\nhas eigenvalue \u0016j= 0 for one (or more, but not all) Car-\ntan generators. Then Eq. (19) would allow for a tower of\nstates with energy gaps O(1=N). But if a Cartan genera-\ntor has eigenvalue 0, there are two possibilities: \frst, theSSB# ground\nstatestower of\nstates# NG\nmodesNG\ndispersionlower critical\ndimension\ntype-A 1 yes n linear 1\ntype-BO(N) non\n2quadratic 0\nTABLE I. Comparison between type-A and type-B SSB phe-\nnomenology. They di\u000ber in the number of ground states;\nthe existence of a tower of states; the number of Nambu{\nGoldstone modes in terms of the number of broken symmetry\ngenerators n; their dispersion relation; and the lower criti-\ncal dimension, which is the lowest dimension at which zero-\ntemperature SSB is possible (the lower critical dimension at\n\fnite temperature is 2 for both type-A and type-B SSB).\nstate could be invariant under one or more SU(2) sub-\ngroups, in which case the root generators which would\nconstruct the tower of states also annihilate the state.\nSecond, the symmetry is broken in the type-A way, in\nwhich case a tower of states is expected. We leave de-\ntailed investigation of systems with both type-A and\ntype-B breaking for future work.\nWe therefore conclude that states which feature type-B\nSSB exclusively, have an extensive ground state degener-\nacy and no tower of states with energy gaps O(1=N).\nThe distinction between symmetry breaking where the\nexpectation value of the Casimir operator is minimal or\nmaximal has been discussed before [36], although these\nauthors do not consider the case where \u0016 < \u0016\u0016as is the\ncase for almost all type-B SSB [27].\nThe nonzero overlap between the ground state of the\nferrimagnetic Lieb{Mattis Hamiltonian and the classical\nN\u0013 eel state suggests it may be possible to \fnd a simple ex-\nact representation of this ground state, which we leave for\nfuture work. It also opens up the question of the entan-\nglement structure of the ferrimagnet, which unlike that of\nthe product state Heisenberg ferromagnet, is quite subtle.\nIt would be interesting to see whether the entanglement\nin type-B SSB systems exhibits the same Goldstone mode\ncounting as type-A SSB systems.[16, 17]\nConcluding, the distinction between type-A and type-\nB SSB seems to go much further than the counting of\nGoldstone modes [24, 25], as is summarized in Table I.\nType-A, ordinary, SSB has a unique symmetric ground\nstate and a tower of low-lying states with energy gap\n\u0018 O (1=N). There is a linearly dispersing Goldstone\nmode for each broken symmetry. Furthermore it has\nquantum corrections to the classical SSB state, and due\nto the Coleman theorem type-A SSB in one dimension\nat zero temperature is forbidden in the thermodynamic\nlimit. Conversely, here we have found that type-B SSB\nis accompanied by an extensive ground state degeneracy\nand has no tower of low-lying states. Instead, at least one\nof the ground states is thermodynamically stable. Two\nbroken symmetry generators lead to one quadratically\ndispersing Goldstone mode and a gapped partner mode.\nFinally, type-B systems do not su\u000ber from the Cole-\nman theorem and are stable in one dimension [5, 31, 32]9\n(although both type-A and type-B systems are subject\nto the Mermin{Wagner{Hohenberg theorem that for-\nbids SSB in two or lower dimensions at \fnite tempera-\nture [5, 31]). There seems to be only one essential di\u000ber-\nence between general (\\ferri\") type-B SSB and the pecu-\nliar case of the ferromagnet: the latter is the same as the\nclassical ferromagnet, whereas the ferrimagnet is a clas-\nsical N\u0013 eel state dressed with quantum corrections [27].\nAnother di\u000berence is that the ferromagnet does not con-\ntain a gapped mode partnered with the gapless Goldstone\nmode; this can be interpreted as the gap being pushed\nto in\fnity [5, 37]. Nevertheless, here we have seen that\neven if quantum corrections are present in type-B sys-\ntems, the lowest energy gap of zero-wavenumber states is\nnotO(1=N) butO(1) and they do not constitute a tower\nof states in the sense of Refs. [3, 4, 8{11].\nACKNOWLEDGMENTS\nWe thank Hal Tasaki for inspiring discussions and for\nsharing a draft version of Ref. [28]. This work is sup-\nported by the Swiss National Science Foundation via an\nAmbizione grant (L. R.); by the MEXT-Supported Pro-\ngram for the Strategic Research Foundation at Private\nUniversities \\Topological Science\" (Grant No. S1511006)\nand by JSPS Grant-in-Aid for Early-Career Scientists\n(Grant No. 18K13502) (A. J. B.);\nAppendix A: Matrix elements\nFor completeness, we reproduce here the matrix ele-\nments of the staggered magnetic \feld in the basis of Lieb{\nMattis eigenstates jSASBSMzi, as described in Ref. [30]:\nhSASBSMzj(Sz\nA\u0000Sz\nB)jS0\nAS0\nBS0M0\nzi=\u000eSA;S0\nA\u000eSB;S0\nB\n\u0002\u000eMz;M0z[fS+1\u000eS;S0\u00001+gS\u000eS;S0+fS\u000eS;S0+1];(A1)\nHere, we de\fned the functions:\nfS\u0011r\n(S2\u0000(SA\u0000SB)2)((SA+SB+1)2\u0000S2)(S2\u0000M2z)\n(2S+1)(2 S\u00001)S2 ;\ngS\u0011(SA\u0000SB)(SA+SB+ 1)Mz\nS(S+ 1):(A2)\nFor the speci\fc case of SA=B=sA=BN=2,Mz= 0, and\nlargeN, the matrix elements can be conveniently ex-\npressed in terms of the shifted total spin ~S=S\u0000jSA\u0000\nSBj, up to orderO(1=N):\nf~S\u00192psAsB\nsA\u0000sB~S\ng~S\u00191\n2(sA+sB)N\u00002sB+ (sA+sB)~S\nsA\u0000sB\nS(S+ 1)\u00191\n4(sA\u0000sB)2N2\n+1\n2(sA\u0000sB)(2~S+ 1)N (A3)Appendix B: Ground state of a tridiagonal matrix\nConsider a symmetric tridiagonal matrix, where the\nonly nonzero elements are on the diagonal and just below\nand above it:\nM=0\nBBB@a1b1\nb1a2b2\nb2a3b3\n.........1\nCCCA: (B1)\nWe are interested in the ground state eigenvector and\neigenvalue for the special case where both axandbxin-\ncrease linearly with x > 0. By rescaling the matrix we\nde\fne:\nax=x; bx=\u000fx: (B2)\nAs an ansatz for the ground state eigenvector, with\neigenvaluev, we choose:\n x= (\u0000sgn[\u000f])xe\u0000x=\u0015: (B3)\nThe eigenvalue equation now reads, for each row x:\nj\u000fj\u0010\n(x\u00001)e1=\u0015+xe\u00001=\u0015\u0011\n=x\u0000v: (B4)\nBy looking at the \frst row, with x= 1, we can relate \u000f,\n\u0015andv:\nj\u000fje\u00001=\u0015= 1\u0000v: (B5)\nInserting this back into the equation for general x > 1,\nwe \fnd:\nj\u000fj2x\u00001\n1\u0000v+x(1\u0000v) =x\u0000v; (B6)\nwhich can be simpli\fed to:\n(x\u00001)(j\u000fj2\u0000v(1\u0000v)) = 0: (B7)\nBecause the second factor must equal zero for all x, we\nfound an expression for vin terms of\u000fthat is independent\nofx. This proves that  xis indeed an eigenvector of M\nwith eigenvalue:\nv=1\n2\u0010\n1 +p\n1\u00004j\u000fj2\u0011\n: (B8)\nThe exponential decay length is given by:\n\u0015= 1=log0\n@1\n2\u0010\n1 +p\n1\u00004j\u000fj2\u0011\nj\u000fj1\nA: (B9)10\n[1] W. Marshall, Antiferromagnetism, Proc. Roy. Soc. Lon-\ndon A , 48 (1955).\n[2] E. H. Lieb and D. Mattis, Ordering Energy Levels of\nInteracting Spin Systems, J. Math. Phys. 3, 749 (1962).\n[3] T. Koma and H. Tasaki, Symmetry breaking and \fnite-\nsize e\u000bects in quantum many-body systems, J. Stat.\nPhys. 76, 745 (1994).\n[4] H. Tasaki, Long-range order, \\tower\" of states, and sym-\nmetry breaking in lattice quantum systems, J. Stat. Phys.\n174, 735 (2019).\n[5] A. Beekman, L. Rademaker, and J. van Wezel,\nAn Introduction to Spontaneous Symmetry Breaking,\narXiv:1909.01820 (2019).\n[6] L. Rademaker, Exact Ground State of Lieb-Mattis\nHamiltonian as a Superposition of N\u0013 eel states,\narXiv:1909.09663 (2019).\n[7] A. Shimizu and T. 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D 92, 045028 (2015)."    },    {        "title": "2206.01586v1.Engineering_the_spin_orbit_torque_efficiency_and_magnetic_properties_of_Tb_Co_ferrimagnetic_multilayers_by_stacking_order.pdf",        "content": "Engineering the spin-orbit torque efficiency and magnetic properties of Tb/Co\nferrimagnetic multilayers by stacking order\nMickey Martini,1, 2Can Onur Avci,1Silvia Tacchi,3Charles-Henri Lambert,1and Pietro Gambardella1\n1Department of Materials, ETH Zürich, Hönggerbergring 64 CH-8093 Zürich, Switzerland\n2Department of Physics, Politecnico di Milano, P.za Leonardo da Vinci 32, I-20133 Milano, Italy\n3Istituto Officina dei Materiali del CNR (CNR-IOM), Sede Secondaria di Perugia,\nc/o Dipartimento di Fisica e Geologia, Università di Perugia, I-06123 Perugia, Italy\n(Received June 6, 2022)\nWe measured the spin-orbit torques (SOTs), current-induced switching, and domain wall (DW)\nmotion in synthetic ferrimagnets consisting of Co/Tb layers with differing stacking order grown\non a Pt underlayer. We find that the SOTs, magnetic anisotropy, compensation temperature and\nSOT-induced switching are highly sensitive to the stacking order of Co and Tb and to the element in\ncontact with Pt. Our study further shows that Tb is an efficient SOT generator when in contact with\nCo, such that its position in the stack can be adjusted to generate torques additive to those generated\nby Pt. With optimal stacking and layer thickness, the dampinglike SOT efficiency reaches up to 0.3,\nwhich is more than twice that expected from the Pt/Co bilayer. Moreover, the magnetization can be\neasily switched by the injection of pulses with current density of about 0:5\u00002\u0002107A=cm2despite the\nextremely high perpendicular magnetic anisotropy barrier (up to 7.8 T). Efficient switching is due to\nthe combination of large SOTs and low saturation magnetization owing to the ferrimagnetic character\nof the multilayers. We observed current-driven DW motion in the absence of any external field,\nwhich is indicative of homochiral Néel-type DWs stabilized by the interfacial Dzyaloshinkii-Moriya\ninteraction. These results show that the stacking order in transition metal/rare-earth synthetic\nferrimagnets plays a major role in determining the magnetotransport properties relevant for spintronic\napplications.\nI. INTRODUCTION\nMagnetization switching and domain wall (DW) mo-\ntion driven by spin-orbit torques (SOTs) have attracted\nextensive research interest due to their potential applica-\ntions in nonvolatile memory and logic technologies [ 1–14].\nTo improve the scalability and thermal stability of de-\nvices in these applications, materials with perpendicular\nmagnetic anisotropy (PMA) are often preferred. Ultra-\nthin films of 3dtransition metals and alloys (e.g., Co or\nCoFeB) have been widely investigated for such purposes,\nwhere the interfacial spin-orbit coupling gives rise to the\nPMA [15–18]. Recently, however, increasing interest has\nbeen devoted to ferrimagnetic alloys of rare earth metals\nand transition metals possessing bulk PMA [ 19–37]. In\nthese compounds, the rare-earth-metal and transition-\nmetal magnetic sublattices are coupled antiparallel to\neach other, resulting in a reduced total magnetic moment\nrelative to the individual sublattices [ 38,39]. The low\nmagnetization allows for efficient SOT-induced switching,\nwhereas the large PMA increases the thermal stability\nof the devices. Moreover, the possibility of obtaining\nangular momentum compensation allows for antiferro-\nmagnetlike magnetization dynamics that, e.g., leads to\nultrafastswitchingandDWmotion[ 31–33]. Todate, stud-\nies of SOTs and related phenomena have mainly focused\non the alloy form of the rare-earth-metal–transition-metal\nsystems [ 19–33], whereas multilayer structures have been\nless explored [ 34–37]. In such structures, the stacking\norder and the thickness of each layer offer additional de-\ngrees of freedom to tune the SOTs and understand the\nrole of each element in interface-driven phenomena [ 13].Theoretical [ 40] and experimental [ 34,37,41–44] stud-\nies suggest that the rare-earth elements with partially\nfilled forbitals are candidates for generating large spin\nHall effect, and hence can be used as a source of SOTs.\nThe strong spin-orbit interaction in the rare earth metals\ncombined with the inversion symmetry breaking at their\ninterfaces with transition metals can also produce sizeable\nRashba-Edelstein effect and Dzyaloshinkii-Moriya inter-\naction (DMI). The latter stabilizes homochiral Néel-type\ndomain walls with efficient and ultrafast current-induced\ndynamics [9–11, 45–47].\nMotivated by the above considerations, we investigate\nthe influence of the stacking order and layer thickness\non current-induced SOTs, switching, and DW motion\nin Pt/Co/Tb-based multilayers. Harmonic Hall and\nmagneto-optic Kerr effect measurements reveal strong\nSOTs in the perpendicularly magnetized Tb/Co multi-\nlayers, which lead to efficient current-induced switching\nand DW motion, comparable to those of TbCo alloys in\ncontact with Pt. Moreover, the SOTs and other mag-\nnetic properties, such as the saturation magnetization,\nmagnetic anisotropy, compensation temperature, and the\nanomalous Hall effect (AHE) are highly sensitive to the\nstacking order as well as to the element in contact with the\nPt underlayer. We find that Tb can be an effective source\nof SOTs with additive contributions to the ones due to Pt\nwhen placed on top of Pt/Co. These results show that the\nstacking order in rare-earth-metal/transition-metal multi-\nlayers offers an additional degree of freedom to tune the\nefficiency of SOT-controlled switching and DW motion.arXiv:2206.01586v1  [cond-mat.mtrl-sci]  3 Jun 20222\nII. SAMPLE PREPARATION AND\nCHARACTERIZATION\nWe grow Pt(4)/Tb( tTb)/[Co(0.5)/Tb(0.65)/Co(0.5) ]\nand Pt(4)/ [Co(0.5)/Tb(0.65)/Co(0.5) ]/Tb(tTb) multilay-\ners with PMA as well as Pt(5)/Tb( tTb)/Co(5) and\nPt(5)/Co(5)/Tb( tTb) trilayers with in-plane anisotropy\non thermally oxidized silicon substrates by dc magnetron\nsputtering (numbers indicate the thickness in nm). For all\nlayers, aTicappingof 5 nmandaTaseedlayerof 3 nmare\nused. The base pressure and Ar partial pressure of the de-\nposition chamber are 6\u000210\u00008mbarand4\u000210\u00003mbar,\nrespectively. The first two series, schematically illustrated\ninFig.1(a), areusedfortheSOTs, current-inducedswitch-\ning, and DW motion studies, whereas in the last two series\nonly the SOTs are quantified. In the samples with PMA\nwe fix the layers between square brackets and change the\nTb thickness ( tTb) and position relative to them; similarly,\nin the trilayers with in-plane anisotropy we changed the\nTb thickness and position relative to Co. Hall bar devices\nare fabricated by patterning the substrates using optical\nlithography before deposition and subsequent lift-off.\nIn Fig. 1(b) we show representative magnetization\ncurves measured by a superconducting quantum inter-\nference device at room temperature with an out-of-plane\nfield sweep for the two different positions of the Tb(1)\nlayer with respect to the fixed [Co/Tb/Co] trilayer. Both\nsets of data show clear hysteresis and nearly 100 %rema-\nnence at zero field, indicative of strong PMA. However,\ntheir saturation magnetization ( Ms) differs significantly\neven though the amount of Tb and Co is the same in\nboth multilayers. Measurements of all the six samples\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\n(a)\n-200 -100 0 100 2000Hz(mT)-1-0.500.51M(105 A/m)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)\n1 1.5 2tTb(nm)00.40.81.2Ms (105 A/m)(c)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)\nFIG. 1. (a) Schematic illustration of the Co/Tb/Co multilay-\ners with different stacking order. (b) Magnetization curves of\nPt/Tb(1)/[Co/Tb/Co] and Pt/[Co/Tb/Co]/Tb(1). (c) Satu-\nration magnetization versus Tb thickness.with PMA [Figure 1(c)] show that Msis larger when Co\nis in direct contact with Pt except for the thickest Tb\nsamples. In both multilayer systems, Msis significantly\nsmaller relative to that expected for Co (>106A/m), as\nexpected due to the antiferromagnetic coupling of Co and\nTb. The large difference in Msand its dependence on Tb\nthickness indicate that the net magnetization is highly\nsensitive to the stacking order.\nFigure 2(a) shows the device layout with the electrical\nconnections and the coordinate system. Figure 2(b) re-\nports the sheet resistance Rsmeasured using a four-point\ngeometry in the structures with PMA. Rsis about 20 %\nlarger in samples with Pt/Tb interface compared with\nthose with Pt/Co interface. The larger Rsand its in-\ncrease with tTbin Pt/Tb(tTb)/[Co/Tb/Co] might be due\nto increased interfacial electron scattering and/or inter-\nmixing between Tb and Pt in this system. In contrast, in\nPt/[Co/Tb/Co]/Tb( tTb),Rsdoesnotsignificantlychange\nwith the Tb thickness, suggesting that the electric con-\nductivity is predominantly dictated by the Pt and Co\nlayers. The resistivity \u001aof all six multilayers with PMA,\nestimated by considering only the conduction through\nPt, Co, and Tb and neglecting the contribution from the\nbuffer and capping layers is reported in Table II along\nwith the other properties measured in this study.\nIn alloys of rare earth metals and transition metals and\nin multilayers, the magnetotransport properties are domi-\nnatedbythe3 d4sorbitalsofthetransition-metalelements\nnear the Fermi level; thus the AHE is mainly driven by\nthe transition-metal sublattice [ 20,34,48]. In Fig. 2(c) we\nreport the Hall resistance RHas a function of out-of-plane\nmagnetic field. As evident from the data, the AHE is\npositive and large in the Pt/[Co/Tb/Co]/Tb series as op-\nposed to the Pt/Tb/[Co/Tb/Co] series, where the AHE is\nnegative and relatively small. This means that i) at room\ntemperature the total magnetization is dominated by the\nCo (Tb) sublattice for the layers with Pt/Co (Pt/Tb) in-\nterface and that the magnetic compensation temperature\nTMis below (above) room temperature; ii) the Pt/Co\ninterface has a significant contribution to the AHE, in\nagreement with literature data [ 49]. Our data clearly show\nthat the stacking order plays a crucial role in determining\nthe magnetic compensation point as well as the AHE,\nin addition to MsandRsdiscussed earlier. We further\nstudy the coercive field ( Hc) as a function of temperature\nto characterize TMof all six samples. As the temperature\ncrossesTM, the anomalous Hall resistance RHchanges\nsign; further, Hcdiverges because a stronger external\nfield is required to reverse the vanishing net magnetiza-\ntion [50], as shown for Pt/Tb(1; 1.5 nm)/[Co/Tb/Co] in\nFig. 2(d).TMdetermined from the change of sign of RH\nis shown in Fig. 2(e) for all the six samples. The stack-\ning order gives rise to a substantial difference of more\nthan 100 K in TMbetween Pt/Tb( tTb)/[Co/Tb/Co] and\nPt/[Co/Tb/Co]/Tb( tTb), which is likely related to the\nlargerMsof Pt/[Co/Tb/Co]/Tb( tTb).\nFinally, we characterize the effective magnetic ansitropy\nfieldHanby measuring RHin the samples with PMA as3\na function of an external magnetic field applied nearly in\nplane at polar and azimuthal angles \u0012H=88°and'H=\n90°[see Fig. 2(a) for the definition of the angular coordi-\nnates]. We use a macrospin approximation to calculate\nHan=H\u0000sin\u0012H\nsin\u0012\u0000cos\u0012H\ncos\u0012\u0001\n, where\u0012=arccos\f\fRH\u000e\nRAHE\f\f\n[51], as reported in Fig. 2(f). We find \u00160Han= 7:8\u00060:5T\nin Pt/Tb(1)/[Co/Tb/Co], which rapidly decreases withincreasing Tb thickness due to the shift of TMto higher\ntemperatures. In the Pt/[Co/Tb/Co]/Tb(1) series, on\nthe other hand, Hanis relatively constant, which is consis-\ntent with the behavior of TMin these samples. We note\nthat extremely large anisotropy fields in excess of 3 T\nare measured in all of our samples, with the exception of\nPt/Tb(2)/[Co/Tb/Co].\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(tTb)/[Co/Tb/Co][Co/Tb/Co]/Tb(tTb)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n(a)\n-100 -50 0 50 1000Hz (mT)-5000500RH(c)tTb = 1 nm\n-100 -50 0 50 1000Hz (mT)tTb = 1.5 nm\n-100 -50 0 50 1000Hz (mT)tTb = 2 nmPt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\n1 1.5 2tTb(nm)90100110120130Rs(b)Tb(t)/[Co/Tb/Co][Co/Tb/Co]/Tb(t)01002003000Hc (mT)(d)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]\n300 315 330 345T (K)-0.10.1RAHE1 1.5 2tTb (nm)0100200300400TM (K)(e)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)1 1.5 2tTb (nm)024680Han (T)(f)Pt/Tb(t)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(t)\nFIG. 2. (a) Schematics of a Hall bar device and coordinate system. The Hall bar channel is 100µmlong and 5µmwide. (b)\nSheet resistance versus thickness of Tb layer in Pt/Tb( tTb)/[Co/Tb/Co] and Pt/[Co/Tb/Co]/Tb( tTb). (c) Hall resistance\nas a function of out-of-plane magnetic field in samples with different Tb thickness. (d) Coercive field versus temperature in\nPt/Tb(1,1.5)/[Co/Tb/Co]. The regions shaded in blue and magenta represent temperature ranges in which the magnetization is\nstrongly compensated. (e) Magnetization compensation temperature as a function of Tb thickness. The black arrow indicates\nthatTM>370 K, which is the largest temperature reached in our setup; the green solid line indicates room temperature. (f)\nAnisotropy field at room temperature versus Tb thickness.\nIII. RESULTS AND DISCUSSION\nA. SOTs in multilayers with PMA\nThe SOT-induced effective fields are quantified by per-\nforming harmonic Hall effect measurements in the PMA\nsamples [ 52,53]. An alternating current with frequency\n!=2\u0019=10 Hzand amplitude j= 0:3\u00001\u0002107A=cm2is ap-\nplied along the xaxis, and the first- and second-harmonic\nHall resistances ( R1!\nHandR2!\nH) were measured during an\nin-plane field sweep along the xand yaxes (Hx;Hy). The\ncurrent density is estimated by dividing the current by\nthe total cross section of the Pt, Co, and Tb layers. We\nneglect the conductivity of the highly resistive Ta buffer\nand Ti capping layers. In multilayers with inversion sym-\nmetry along the zaxis and current injection along x, the\ndampinglike and fieldlike SOT effective fields are given\nbyHDL=HDL(y\u0002^M)andHFL=HFLy, respectively,\nwhere Mis the unit vector of the magnetization and\nthe current-induced spin accumulation is polarized along\ny[13]. In order to define the sign of the field ampli-\ntudesHDL;FL, we adopt the convention that HDL(HFL)\nis positive (negative) when the field points in the samedirection as in Pt/Co (opposite direction to the Oersted\nfield), where Pt is deposited before Co [ 52]. Due to the\nlarge PMA in our samples, we opt for the small-angle ap-\nproximation method where the magnetization is slightly\ntilted away from equilibrium by the applied field. In this\nregime, the effective fields can be estimated using the\nfollowing relations [54, 55]:\n\u0001Hx(y)= \n@R2!\nH\n@Hx(y)\u001e@2R1!\nH\n@H2\nx(y)!\n; (1)\nHDL(FL) =\u00002\u0001Hx(y)\u0006\u0017\u0001Hy(x)\n1\u00004\u00172; (2)\nwhere\u0017is the ratio of the planar Hall resistance RPHEto\nthe anomalous Hall resistance RAHEand\u0006corresponds\nto the measurements taken when the magnetization is\npointing along the \u0006zdirection. To characterize RAHE\nandRPHE, we use the first-harmonic signal given by R!\nH=\nRAHEcos\u0012+RPHEsin2\u0012sin 2'. Since the magnetization\ncannot be fully saturated in plane by the available field,\nwe quantify RPHEby measuring R!\nHwith an external field\napplied at (\u0012H;'H)\u0011(88°;45°). We then separate the\nantisymmetric and symmetric signal contributions (with4\nrespect to the field reversal), which originate from the first\nand second terms in the above equation, respectively. The\nantisymmetric signal is then plotted as a function of sin2\u0012,\nwhere\u0012is estimated using the macrospin approximation\nas\u0012=arccos\f\fR1!\nH\u000e\nRAHE\f\f. Finally the slope of the linear\nfit represents RPHE. The details of this approach can be\nfound in Ref. [52].\nFigure 3 shows representative R1!\nHandR2!\nHmea-\nsurements taken from the Pt/[Co/Tb/Co]/Tb(1.5) and\nPt/Tb(1)/[Co/Tb/Co] samples. R1!\nHhas a parabolic field\ndependence due to the coherent rotation of the magneti-\nzation. Note that we only plot the data corresponding\nto theHxsweep since the Hysweep shows an identical\nbehavior. The sign of R!\nHis opposite in these two samples\nfor a given magnetization orientation due to the opposite\nAHE [see Fig.2(c)]. R2!\nHvaries linearly with both Hx\nandHyas expected from the action of SOTs. We note\nthat the slopes of R2!\nHare opposite in Co-like and Tb-like\nsamples. Considering also the opposite sign of R1!\nHin\nthese two samples we conclude that the SOT effective\nfields have the same sign in both systems. Thus, the SOTs\nact on the net magnetization, in agreement with previous\nworks [21,23]. The SOT effective fields normalized by the\ncurrent density, \u001fDL(FL) =\u00160HDL(FL)\u000e\nj, are summarized\nin Figs. 4(a)-(4b) as a function of Tb thickness. The\nvalues of\u001fDL(FL)are obtained by performing a linear\nfit ofHDL(FL)versusjfor both magnetization orienta-\ntions (not shown). The fieldlike component includes alsothe Oersted field contribution due to the current flowing\nthrough Pt, which we estimate as \u00160HOe\u000e\nj=0:3 mT\n10\u00007A\u00001cm2by Ampere’s law [ 51]. The Oersted field\npoints in the opposite direction to the fieldlike component\nof the SOT field, and can be neglected due to its small am-\nplitude compared with HFL. We observe that \u001fDLis very\nstrong in Pt/Tb(1)/[Co/Tb/Co], reaching up to +72 mT\n10\u00007A\u00001cm2but rapidly decreasing as the Tb thickness\nis increased. Also \u001fFLreaches remarkable values, about\n+25 mT 10\u00007A\u00001cm2in both Pt/Tb(1)/[Co/Tb/Co] and\nPt/[Co/Tb/Co]/Tb(1). To compare the SOTs in dif-\nferent layers on an equal footing we calculate the SOT\nefficiencies \u0018j\nDL(FL)=2e\n~MstM\u001fDL(FL)[56], whereeis\nthe electron charge, ~is the reduced Planck’s constant,\nandtMis the magnetic layer thickness. The SOT ef-\nficiencies are plotted in Figs. 4(c,-d) as a function of\nTb thickness. We observe that \u0018j\nDLand\u0018j\nFLvary be-\ntween +0.1 and +0.3 and between +0.03 and +0.2, respec-\ntively. Whereas \u0018j\nDLand\u0018j\nFLin Pt/[Co/Tb/Co]/Tb(1)\nand Pt/Tb(1)/[Co/Tb/Co] are comparable to those mea-\nsured in Pt/Co bilayers [ 13,52], we observe that both\nefficiencies increase strongly as a function of Tb thickness\nin Pt/[Co/Tb/Co]/Tb( t), reaching values that are about\ntwo times larger compared with Pt/Co. Moreover, \u0018j\nFLis\nsignificantly larger in Pt/[Co/Tb/Co]/Tb( tTb) compared\nwith that in Pt/Tb( tTb)/[Co/Tb/Co], and has a different\nthickness dependence in the two series. Similar trends are\nalso observed if we consider the SOT efficiency per unit\nelectric field \u0018E\nDL(FL)=\u0018j\nDL(FL)\u000e\n\u001a(see Table II).\n(a)updown\n-400 0 400Hx (mT)RH-430-420-410410420430\n-400 0 400Hx (mT)-2000200RH-400 0 400Hy (mT)-40040RH+Mz-Mz\n(b)updown\n-800 0 800Hx (mT)RH-124-123123124\n-800 0 800Hx (mT)-15015RH-800 0 800Hy (mT)-505RH\nFIG. 3. First and second harmonic of the Hall resistance ( R1!\nHandR2!\nH, respectively) versus in-plane longitudinal and transverse\nmagnetic fields ( HxandHy) measured using j=0:8\u0002107A=m2in (a) Pt/[Co/Tb/Co]/Tb(1.5) and (b) Pt/Tb(1)/[Co/Tb/Co].5\n0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)0255075DL (mT 10-7A-1cm2)(a)\n0255075FL (mT 10-7A-1cm2)(b)Pt/Tb(tTb)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(tTb)\n1 1.5 2tTb (nm)00.10.20.3DLj(c)\n1 1.5 2tTb (nm)00.10.20.3FLj(d)\nFIG. 4. (a) Dampinglike and (b) fieldlike SOT effective field per unit current density \u001fDL;FLversus Tb thickness in\nPt/[Co/Tb/Co]/Tb( tTb) and (b) Pt/Tb( tTb)/[Co/Tb/Co]. (c) Dampinglike and (d) fieldlike SOT efficiency \u0018DL;FLver-\nsus Tb thickness. The error bars take into account the errors of the linear and parabolic fits of R1!\nHandR2!\nHrespectively,\naveraged according to Eq. 2 for each current density, propagated in the linear fits of HDL(FL) (j)for the +Mzand\u0000Mz\nconfigurations.\nWe note that \u0018j\nDLand\u0018j\nFLare obtained by normalization\nto the average current density jflowing in the multilayers,\nsuch that even larger efficiencies would be obtained by\nnormalization to the current density flowing in Pt, as\noften done in the literature. The dependence of \u0018j\nDL(FL)\non the stacking order and Tb thickness can have several\norigins. First, the structural details of the multilayers\nand of the interface with Pt generally change with the\nstacking order, which may ultimately lead to changes\nof the SOTs. This point is supported by the resistivity\nmeasurements, which suggest that Tb deposited on Pt\ninduces intermixing between the two species rather than\nfavoring sharp interfaces. Second, the Tb layer can be a\nsource of dampinglike and fieldlike SOTs in itself, with\na contribution opposite to that of Pt. Such a scenario,\nwhich agrees with a previous study of Pt/[Co/Ni]/Co/Tb\nmultilayers [ 37], is consistent with the enhancement of\n\u0018j\nDL(FL)when Pt and Tb are placed on opposite sides of\nthe [Co/Tb/Co] trilayer. Third, placing the Tb layer on\ntop of the stack might provide an efficient sink for the spin\ncurrent generated by Pt and the Pt/Co interface [ 57]. The\nmonotonic increase of the dampinglike and fieldlike SOTs\nin this series is consistent with both the second and third\nexplanation taking into account the finite spin diffusionlength of Tb. Overall, even if the exact mechanism giving\nrise to the enhancement of \u0018j\nDLand\u0018j\nFLcannot be singled\nout with certainty, our data indicate that Tb and the\nstacking order of Co/Tb multilayers play a crucial role in\ndetermining the strength of the SOTs.\nB. SOTs in Pt/Tb/Co and Pt/Co/Tb trilayers\nwith in-plane magnetic anisotropy\nWe quantify the dampinglike SOT efficiency in samples\nwith in-plane anisotropy following the method outlined in\nRef. [58]. Injection of a relatively small alternating cur-\nrent (!=2\u0019=10 Hz,j= 0:3\u00001\u0002107A=cm2) gives rise to\nperiodic oscillations of the magnetization about its equi-\nlibrium position. The second-harmonic Hall resistance\nrelated to the current-induced fields and the thermoelec-\ntric signal due to Joule heating can be written as:\nR2!\nH=\u0012\nRAHEHDL\nHext+I0\u000brT\u0013\ncos'+\n2RPHE(2 cos3'\u0000cos')HFL+HOe\nHext+Hdem(3)6\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 50 100 150 200 250 300 350(°)-50-40-30-20-1001020304050RH(b)\n0 0.1 0.2 0.31/0(Hext + Hdem)-2002040RDL+RT\n12345tTb (nm)00.020.040.060.080.1DL(c)Pt/Tb(tTb)/CoPt/Co/Tb(tTb)0 50 100 150 200 250 300 350-100-50050100RH(a)0.60 T0.75 T0.90 T1.05 T1.20 T1.35 T1.50 T1.65 T1.80 T\n!(deg)!(deg)cos(!) contribution\nFIG. 5. (a) In-plane angular dependence of the second-harmonic Hall resistance of Pt/Co/Tb(3) for j=0:8\u0002107A=cm2. (b)\nDampinglike and thermal contribution to the second-harmonic Hall resistance. Inset: linear fit of the cosine component of\nthe second-harmonic Hall resistance versus the inverse of the external field and the demagnetizing field acting against the the\ncurrent-induced field. (c) Dampinglike SOT efficiency versus Tb thickness. The error bars take into account the uncertainty of\nthe linear fit of HDLversusj.\nwhereI0is the amplitude of the injected current, \u000bthe\nanomalous Nernst coefficient, rTthe perpendicular ther-\nmal gradient and Hdemthe demagnetizing field. Figure\n5(a) shows a representative angle scan measurement of\nR2!\nHin Pt/Co/Tb(3) for different external in-plane fields\nalong with the fitting curves (dashed lines) given by Eq. 3.\nIn Fig. 5(b) we plot the cos'component of R2!\nH, which in-\ncludes the combined contributions of dampinglike torque\nand the Nernst (and/or Seebeck) effect. The amplitude\nof the cos'component of R2!\nHdecreases on increasing\nthe external field. The latter tends to rigidly align the\nmagnetization along its direction, suppressing the peri-\nodic oscillations. This results in a reduction of the SOT\nand therefore of R2!\nHand its cos'component. This field\ndependence can be exploited to further separate the damp-\ninglike torque and the thermoelectric effects in the cos'\ncomponent of R2!\nH. The anomalous Nernst effect depends\nonly on the magnetization direction and not on the exter-\nnal field amplitude (provided that the magnetization is\nsaturated), whereas the SOT decreases at higher field, as\nalready discussed. Thus, the dampinglike effective field\nHDLcan be quantified by performing a linear fit of the\ncos'component of R2!\nHversus the inverse of the external\nfield plus the demagnetizing field Hdem[58], as shown\nin the inset of Fig. 5(b). This experiment is repeated\nfor all the samples for several current densities to esti-\nmate\u001fDLas the slope of HDLversusj, as done for the\nmultilayers with PMA in Sect. IIIA. The values of \u001fDL\nare reported in Table I. Compared with the multilayers\nwith PMA, the dampinglike effective field in the samples\nwith in-plane anisotropy is 2 orders of magnitude lower.\nThis is due to the large magnetization on which the SOT\nacts, that is dominated by the bulk magnetic moments\nof the Co layer (the ferromagnetic layer is now thick).\nThe dampinglike SOT efficiency, calculated also here as\n\u0018j\nDL=2e\n~MstM\u001fDL, is plotted in Fig. 5(c) as a function of\ntTb. Although \u0018j\nDLis significantly lower in these samplescompared with the samples with PMA, it scales in the\nsame way with tTb, increasing (decreasing) in the trilayer\nwith Pt/Co (Pt/Tb) interface. The comparison between\nPt/Tb/Co and Pt/Co/Tb structures provides further ev-\nidence that the SOTs are very sensitive to tTband the\nstacking order of Co/Tb.\nC. Current-induced magnetization switching\nThe large SOTs evidenced by the harmonic Hall effect\nmeasurements in Sect. IIIA can be exploited to realize\ncurrent-induced magnetization switching with enhanced\nefficiency relative, e.g., to Pt/Co bilayers. In order to\nstudy SOT-induced switching in our multilayers with\nPMA, we injected 10- ms-long current pulses of variable\namplitude through the Hall bar devices in the presence of\nan in-plane bias field Hxapplied parallel to the current\nline. This field is required to break the SOT symmetry\nand uniquely define the switching polarity [ 1,13]. Figure\n6(a) shows representative AHE hysteresis loops as a func-\ntion of current measured for Pt/Tb(2)/[Co/Tb/Co] and\nPt/[Co/Tb/Co]/Tb(2). In both samples, the net mag-\nTABLE I. Saturation magnetization, demagnitizing field,\ndampinglike effective field per unit current density and SOT\nefficiency in trilayers with in-plane anisotropy. Errors are a\nfraction of the least significant digit.\nMs\u00160Hdem \u001fDL \u0018j\nDL\n(106A m\u00001)(T) ( mT 10\u00007A\u00001cm2)\nPt/Tb(1)/Co 1.13 1.80 0.19 0.040\nPt/Tb(3)/Co 0.81 1.59 0.12 0.023\nPt/Tb(5)/Co 0.48 1.48 0.10 0.014\nPt/Co/Tb(1) 1.22 0.75 0.27 0.061\nPt/Co/Tb(3) 0.81 0.72 0.37 0.073\nPt/Co/Tb(5) 0.68 0.56 0.44 0.0917\nnetization switches from +Mzto\u0000Mzwhen the current\nis parallel to the in-plane field and from \u0000Mzto+Mz\nwhen it is antiparallel to it. The loops of Fig. 6(a) have\nopposite polarity because the AHE has opposite sign in\nthese two samples at room temperature [see Fig. 2(c)].\nWe repeate the measurements summarized in Fig. 6(a)\nfor all of the multilayers with PMA and several in-plane\nmagnetic fields to characterize the threshold current den-\nsityjthrequired to switch the magnetization. The data\nare shown in Fig. 6(b) as a function of \u00160Hx. In the\nmacrospin approximation, the threshold switching cur-\nrent is given by [29, 59]\njth=2e\n~\u00160MstM\n\u0018j\nDL\u0012Hk\n2\u0000Hxp\n2\u0013\n; (4)\nwheretMis the thickness of the magnetic layer. Although\nEq. 4 does not apply to switching by domain nucleation\nand propagation, which is the preferred mode of magneti-\nzationreversalinmesoscopicsamples[ 47], onestillexpects\njthto scale with HkandHxgiven that these two fields\n-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)-4 -2 0 2 4Ip(mA)-400-2000200400RH(a)Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(2)\n0 0.2 0.4 0.6 0.8 10Hx (T)00.511.522.5jth(107 A/cm2)(b)Pt/Tb(1)/[Co/Tb/Co]Pt/Tb(1.5)/[Co/Tb/Co]Pt/Tb(2)/[Co/Tb/Co]Pt/[Co/Tb/Co]/Tb(1)Pt/[Co/Tb/Co]/Tb( 1.5)Pt/[Co/Tb/Co]/Tb(2)\nFIG. 6. (a) Change of the anomalous Hall resistance\ndue to SOT-induced switching in Pt/Tb(2)/[Co/Tb/Co]\nwith constant in-plane field \u00160Hx=182 mT and in\nPt/[Co/Tb/Co]/Tb(2) with \u00160Hx=333 mT. The Hall resis-\ntance is plotted versus the amplitude Ipof 10- ms-long current\npulses. (b) Threshold current density for switching as function\nof the in-plane field \u00160Hx.determine the domain nucleation barrier. Figure 6(b)\nshows that, despite the very large Hkof our multilayers\n(see Sect. II), we are able to switch all of the samples with\ncurrent densities below 2:5\u0002107A=cm2and in-plane field\nHx\u001cHk. This is due to the combined effect of strong\ndampinglike SOTs and low M sowing to the ferrimagnetic\ncharacter of our multilayers. The strongest dependence\nofjthonHxis found in the sample with weakest mag-\nnetic anisotropy, namely Pt/Tb(2)/[Co/Tb/Co], as ex-\npected. The samples with strongest anisotropy, where\n\u00160Hk\u00156T, present a much weaker dependence of jth\nonHx, suggesting that the in-plane field is not the main\ncause of reduction of the domain nucleation barrier in\nsuch a case. On the other hand, jthscales with \u0018DLand\nMsin samples with comparable anisotropy, such as in the\nPt/[Co/Tb/Co]/Tb( tTb) series.\nTo investigate the role of Joule heating in the switch-\ning experiments, we measure the Hall effect by injecting\ndifferent current densities up to j'2\u0002107A=cm2. We\nfind that the coercivity decreases by about 30% upon\nincreasingjfrom 0:5to2\u0002107A=cm2(not shown). Like-\nwise, we also estimate the anisotropy field \u00160Hkin all the\nmultilayers and record a drop smaller than 25% from the\ncase ofj <1\u0002107A=cm2to the case of j= 2\u0002107A=cm2.\nTherefore we conclude that the Joule heating does not\nplay a major role in determining jthforj <1\u0002107A=cm2,\nwhereas it partially assists the switching at higher current\nby decreasing the magnetic anisotropy energy barrier and\nfavoring DW depinning. Indeed, jthis determined by the\ninterplay of several factors beyond Hk,Hx,\u0018DL, andMs,\nwhich include Joule heating as well as \u0018FL, the DMI and\nDW pinning properties of the multilayers. Overall, these\ndata show that SOTs can switch ferrimagnetic multilayers\nwith very strong PMA at a current density of the order of\n107A=cm2, similarly to what is observed in CoTb alloys\n[28].\nD. Current-induced domain wall motion\nSOTs can efficiently move DWs in suitable devices\nand heterostructures [ 13]. Compensated ferrimagnets\nare particularly interesting for the SOT-induced DW mo-\ntion because the DWs can be propelled at very high\nspeeds with moderate currents owing to their partic-\nular dynamical properties near the magnetic and an-\ngular momentum compensation [ 11,27,31,32,60–62].\nWe investigate the current-induced DW motion in the\nPt/[Co/Tb/Co]/Tb( tTb) samples. We use a wide-field\nKerr microscope configured in the polar geometry to\ndetect the out-of-plane magnetization component in\nPt/[Co/Tb/Co]/Tb( tTb) racetracks and study the DW\nresponse to external fields and currents. The experimental\nprocedure can be briefly described as follows. First, the\nmagnetization is saturated along +zor\u0000zby applying\nan out-of-plane magnetic field larger than the coercivity.\nNext, a domain with the opposite magnetization is nu-\ncleated by the Oersted field produced by a short current8\npulseInucinjected into an electrically insulated wire cross-\ning the racetrack [Fig. 7(a)]. Finally, the DW is displaced\nby injecting current pulses along the DW track. We per-\nform two types of experiments: i) determination of the\ncritical current for depinning a single DW in the presence\nor absence of an out-of-plane field ( Hz);ii) determination\nof the DW velocity in the absence of any applied field.\nFigure 7(b) shows the critical current density jdepin\nrequired to depin a DW as function of Hzfor both the\nup-down and down-up DWs in Pt/[Co/Tb/Co]/Tb(1).\nThe linear variation of jdepin(Hz)shows that the current\ninduces a dampinglike SOT that acts as an effective mag-\nnetic field along\u0006z, assisting or hindering Hzto depin\nthe DWs [ 9,11,63,64]. In the one-dimensional DW\nmodel, neglecting the spin-transfer and fieldlike torques,\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jcr (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8j (107 A/cm2)-200200Hzeff (mT)(c)0Hd UP-DN DW0Hd DN-UP DW/20HDL  +Mz/20HDL  -Mz\n(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW/20HDL  +Mz/20HDL  -Mz(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW-/20HDL  +Mz-/20HDL  -Mz(a)\nDomainWalljInucl10 μm\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jcr (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8j (107 A/cm2)-200200Hzeff (mT)(c)0Hd UP-DN DW0Hd DN-UP DW/20HDL  +Mz/20HDL  -Mz\n(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW/20HDL  +Mz/20HDL  -Mz(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW-/20HDL  +Mz-/20HDL  -Mz(a)\nDomainWalljInucl10 μm\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jcr (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8j (107 A/cm2)-200200Hzeff (mT)(c)0Hd UP-DN DW0Hd DN-UP DW/20HDL  +Mz/20HDL  -Mz\n(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW/20HDL  +Mz/20HDL  -Mz(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW-/20HDL  +Mz-/20HDL  -Mz(a)\nDomainWalljInucl10 μm\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jcr (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8j (107 A/cm2)-200200Hzeff (mT)(c)0Hd UP-DN DW0Hd DN-UP DW/20HDL  +Mz/20HDL  -Mz\n(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW/20HDL  +Mz/20HDL  -Mz(a)\n-4 -2 0 2 40Hz (mT)-0.3-0.2-0.10jdepin (107 A/cm2)(b)DN-UP DWUP-DN DW\n0 0.2 0.4 0.6 0.8 1j (107 A /cm2)-200200Hzeff (mT)(c)0Hdepin UP-DN DW0Hdepin DN-UP DW-/20HDL  +Mz-/20HDL  -Mz(a)\nDomainWalljInucl10 μm\nFIG. 7. (a) Differential Kerr microscopy image showing\nthe nucleation of a magnetic domain (white region) in a\nPt/[Co/Tb/Co]/Tb(1) racetrack delimited by the red dotted\nline. (b) Critical current density required to depin up-down\nand down-up DWs as a function of applied out-of-plane mag-\nnetic field in Pt/[Co/Tb/Co]/Tb(1). (c) Effective out-of-plane\nmagnetic field corresponding to the depinning field \u0001Hdepin\n(squares) and dampinglike SOT effective field - \u0019=2\u0002HDL(cir-\ncles) versus current density for both DWs and magnetic states\nin Pt/[Co/Tb/Co]/Tb(1).\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 ns\n(a)\n012345jp (107 A/cm2)01234vDW (m/s)(b)p = 15 nsp = 20 nsp = 30 nsp = 50 nsp = 100 nsFIG. 8. (a) Series of Kerr differential images showing current-\ndriven DW motion in Pt/[Co/Tb/Co]/Tb(2) as hundreds of\n20-ns-long current pulses with jp=3:6\u0002107A=cm2amplitude\nare injected along the track. (b) DW velocity as a function\nof current density for different pulse widths \u001cp=15-100 ns\nin Pt/[Co/Tb/Co]/Tb(2). The errors in the DW velocity are\ndue to the arrival time uncertainty, which is larger when a\nsmall number of pulses is used.\nthe effective field induced by the current is given by\nHe\u000b\nz= \u0001Hdepin, where \u0001Hdepinis the change in the\nDW depinning field under dc current injection. Alterna-\ntively, we estimate He\u000b\nz=\u0000\u0019\n2HDLcos\t, whereHDLis\nthe dampinglike SOT effective field obtained by the har-\nmonic Hall effect measurements in Sect. IIIA and \tis the\nangle between the DW magnetization and the xaxis [65].\nFigure 7(c) compares He\u000b\nzobtained using \u0001Hdepinand\n-\u0019\n2HDLfor different current densities. The two measure-\nments are in excellent agreement, which implies that the\nDWs are of purely homochiral Néel type (i.e., \t = 0 deg).\nMoreover, the DW motion along the current flow indicates\nthat they have left-handed chirality. This is consistent\nwith the interfacial DMI occurring at the Pt/Co inter-\nface with possible additional contributions from the inner\ninterfaces between Co and Tb and the top Tb/Ti inter-\nface [66]. Qualitatively the same results are obtained\nfor Pt/[Co/Tb/Co]/Tb(1.5) and Pt/[Co/Tb/Co]/Tb(2),\nshowing that the current-induced DW depinning measure-\nments can be effectively used to identify the DW type\nand estimate the DL-SOT effective field in this system.\nNext, we characterize the current-induced DW velocities\nin Pt/[Co/Tb/Co]/Tb(1, 1.5, 2). To do so, we capture a\nsequence of Kerr images while injecting a train of current\npulses through the track to move a pre-nucleated DW, as\nshown in Fig. 8(a). To estimate the DW velocity ( vDW),\nwe divide the total displacement by the total pulse length.9\nFigure 8(b) shows the results from Pt/[Co/Tb/Co]/Tb(2),\nwherevDWis plotted as function of current density jp\nfor different pulse lengths \u001cp. The data exhibit several\nimportant features. First, the current required to initi-\nate the DW motion, i.e., the onset of the linear regime,\nis strongly dependent on the pulse width and is lower\nfor longer pulses. This can be understood by the effect\nof thermal activation on the DW depinning, which is\nstronger for longer pulses. Second, in the linear regime\nthe slopevDW=jis approximately the same for all pulse\nwidths, suggesting that Joule heating plays a minor role\ninvDWunlike in the depinning process. In all of the\ncases, the depinning current in the Pt/[Co/Tb/Co]/Tb\nmultilayers is comparable to that found in Pt/Co 1-xTbx\nferrimagnetic alloys [ 27]. Third, the extrapolation of the\nlinear fits yields velocities in the range of 10-15 m/s for\njp= 108A=cm2. Thesevaluesaremoderateincomparison\nwith those of other ferrimagnetic [ 27,31,67] and synthetic\nantiferromagnetic [ 68] systems despite the strong SOTs.However, we refrain from drawing definitive conclusions\non the DW velocity behavior of our films based on the\navailable data because we cannot conclude whether the\nupturn of the vDWis due to the onset of a wide creep\nregime or the flow regime. In the former case, it is not\npossible to extrapolate the velocities at higher current\ndensity.\nFinally, we attempt to quantify the effective DMI field\nHDMIin all the samples studied above. To do so, we\nmeasurejdepinas function of applied in-plane field up\ntoHx=40 mT, the maximum in-plane field available in\nthe Kerr microscope. If Hxis applied opposite to HDMI,\nwe expect that the DW magnetization rotates first along\ntheyaxis, i.e., the DW becomes Bloch-like for jHxj'\njHDMIj, and then Néel-like with right-handed chirality\nforjHxj>jHDMIj[11]. In the applicable in-plane field\nrange, however, we do not observe any change in the sign\nofjdepin, indicating that the chirality is not affected by\nHxand thatjHDMIj>40 mT. These measurements show\nthatHDMIis relatively strong in Pt/[Co/Tb/Co]/Tb.\nTABLE II. Summary of the magnetic properties and SOT efficiency of Pt/Tb( tTb)/[Co/Tb/Co] and Pt/[Co/Tb/Co]/Tb( tTb)\nmultilayers with PMA. All values are measured in ambient conditions. Errors are a fraction of the least significant digit, except\nforTM, whose error is of the order of 1 K.\n\u001a M sTMRAHERPHE\u00160Hk\u001fDL\u001fFL\u0018j\nDL\u0018j\nFL\u0018E\nDL\u0018E\nFL\n(µ\n cm) (105(K) (m\n) (m\n) (T) (mT 10\u00007(mT 10\u00007(105(105\nA m\u00001) A\u00001cm2) A\u00001cm2) \n\u00001m\u00001)\n\u00001m\u00001)\nPt/Tb(1)/[Co/Tb/Co] 72.5 0.24 310 129 18 7.8 +72 +23 +0.14 +0.04 +1.92 +0.61\nPt/Tb(1.5)/[Co/Tb/Co] 80.3 0.22 313 154 22 5.2 +55 +21 +0.12 +0.04 +1.44 +0.56\nPt/Tb(2)/[Co/Tb/Co] 91.9 0.50 >370 120 16 1.1 +17 +7 +0.10 +0.04 +1.02 +0.41\nPt/[Co/Tb/Co]/Tb(1) 63.5 0.74 193 375 89 3.2 +22 +20 +0.13 +0.12 +2.09 +1.94\nPt/[Co/Tb/Co]/Tb(1.5) 67.0 0.85 187 425 101 3.5 +22 +20 +0.18 +0.17 +2.63 +2.53\nPt/[Co/Tb/Co]/Tb(2) 74.4 0.53 212 341 62 2.7 +50 +35 +0.29 +0.21 +3.93 +2.79\nIV. CONCLUSIONS\nOur systematic study of synthetic ferrimagnets based\non Co/Tb layers shows that the magnetic properties, SOT\nefficiency, and switching depend strongly on the stack-\ning sequence of Co and Tb in the heterostructures as\nwell as on the thickness of Tb. In Pt/[Co/Tb/Co]/Tb,\nthe dampinglike SOT efficiency reaches up to 0.3, which\nwe attribute to the additive SOT contributions from the\nbottom Pt and the top Tb layer. Our conclusion is also\nsupported by harmonic Hall measurements performed in\nthe Tb/Co multilayers with in-plane anisotropy. The\nhigh value of the SOT efficiency in Pt/[Co/Tb/Co]/Tb is\nabout two and three times larger than that in Pt/Co and\nPt/Tb/[Co/Tb/Co], respectively. The PMA in these mul-\ntilayers is very strong with effective magnetic anisotropy\nfields estimated between 1 and 8 T. Despite the large\nPMA, the critical current for magnetization switching\nis relatively low, of the order of 0:5\u00002:5\u0002107A=m2, de-pending on the assisting in-plane field. Current-induced\nDW motion measurements reveal that the DWs are of\nNéel type with left-handed chirality, stabilized by inter-\nfacial DMI similar to Pt/Co bilayers. Our experiments\nshow that synthetic ferrimagnetic heterostructures have\ncomparable or improved PMA, SOT, and DMI relative\nto ferromagnetic systems and ferrimagnetic alloys. The\nstacking order offers an additional degree of freedom to\ntune and maximize these parameters in spintronic devices.\nV. ACKNOWLEDGMENTS\nThis work was supported by the Swiss National Sci-\nence Foundation through grants no. 200020_200465 and\nPZ00P2_179944. The European Metrology Programme\nfor Innovation and Research (EMPIR), under the Grant\nAgreement 17FUN08 TOPS, is also acknowledged.\n[1]I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. 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Nanotech. 10,\n221 (2015)."    },    {        "title": "1201.0654v3.Kondo_Metal_and_Ferrimagnetic_Insulator_on_the_Triangular_Kagomé_Lattice.pdf",        "content": "arXiv:1201.0654v3  [cond-mat.str-el]  16 Jan 2012Kondo Metal and Ferrimagnetic Insulator on the Triangular K agom´ e Lattice\nYao-Hua Chen1, Hong-Shuai Tao1, Dao-Xin Yao2,∗and Wu-Ming Liu1†\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2State Key Laboratory of Optoelectronic Materials and Techn ologies,\nSun Yat-sen University, Guangzhou 510275, China\n(Dated: May 24, 2018)\nWe obtain the rich phase diagrams in the Hubbard model on the t riangular Kagom´ e lattice as a\nfunction of interaction, temperature and asymmetry, bycom bining the cellular dynamical mean-field\ntheorywith the continuoustime quantumMonte Carlo method. The phase diagrams showthe asym-\nmetry separates the critical points in Mott transition of tw o sublattices on the triangular Kagom´ e\nlattice and produces two novel phases called plaquette insu lator with an obvious gap and a gapless\nKondo metal. When the Coulomb interaction is stronger than t he critical value Uc, a short range\nparamagnetic insulating phase, which is a candidate for the short rang resonating valence-bond spin\nliquid, emerges before the ferrimagnetic order is formed in dependent of asymmetry. Furthermore,\nwe discuss how to measure these phases in future experiments .\nPACS numbers: 71.30.+h, 75.10.-b, 05.30.Rt, 71.10.Fd\nGeometrically frustrated systems have shown a lot of\ninteresting phenomena, such as the spin liquid and spin\nice [1–6]. The charge and magnetic order driven by in-\nteraction is still one of the central issues in the field\nof strongly correlation systems [7–12]. Recently, a new\nclass of two-dimensional materials Cu9X2(cpa)6·xH2O\n(cpa=2-carboxypentonic acid, a derivative of ascorbic\nacid; X=F, Cl, Br) has been found [13–15], which is\nformed by an extra set of triangles (B-sites in Fig. 1)\ninsides of the Kagom´ e triangles (A-sites in Fig. 1). The\nCu spins form a new type of geometrically frustrated lat-\ntice called triangular Kagom´ e lattice (TKL). This lattice\ncan also be realized by cold atoms in the optical lattices,\nin which the interaction between trapped atoms can be\ntuned by Feshbach resonance and the kinetic energy can\nbe adjusted by the lattice depth [16–22]. Although the\neffective spin models on this system have been investi-\ngated [7, 23], the real charge dynamics with spins and\nthe phase diagram on this lattice have not been stud-\nied. Different with other frustrated system, such as the\ntriangular lattice and Kagom´ e lattice, the “triangles-in-\ntriangles” structure on the TKL induces two different\nsublattice. Therefore, it is desirable to investigate the\nMott and magnetic transition on the TKL under the in-\nfluence of asymmetry which is induced by different hop-\npings between two sublattices.\nRecently, manyanalyticalandnumericalmethodshave\nbeen developed to investigate the strongly correlatedsys-\ntem [24–31], such as the dynamical mean-field theory\n(DMFT) [32]. However, the DMFT works ineffectively\nin the frustrated systems because the nonlocal correla-\ntion and spatial fluctuations are ignored. Therefore, a\nnew method called cellular dynamical mean-field theory\n(CDMFT) has been developed to incorporate spatially\nextended correlations and geometrical frustration in the\nframeworkofDMFT [33–36] by mapping the originallat-\nticeproblemontoaneffectiveclustermodelcoupledtoaneffective medium. The continuous time quantum Monte\nCarlo method (CTQMC) [37] is employed as an impu-\nrity solver in the CDMFT loop, which is more accurate\nthan the translational QMC method due to the absence\nof Trotter decomposition. So, the CDMFT combining\nwith the CTQMC can provide useful numerous insights\ninto the phase transitionsin the frustrated system [8, 34].\nIn the present Letter, we employ the CDMFT com-\nbining with the CTQMC to investigate the influence of\nasymmetry on the Mott and magnetic phase transition\non the TKL based on Hubbard model. Two novel phases\ncalled plaquette insulator and Kondo metal induced by\nasymmetry is found in the phase diagram and their prop-\nerties is studied by determining the momentum resolved\nspectrums. By defining a magnetic order parameter, we\ncan characterize a ferrimagnetic order for the large inter-\nactionontheTKL.The phasediagramwiththe competi-\ntion between interaction and asymmetry is provided. In\naddition, the spectrum functions on the Fermi surface for\nthe different interaction and asymmetry are presented.\nThese interesting phases can be probed by the angle-\nresolved photoemission spectroscopy (ARPES) [38], neu-\ntron scattering, nuclear magnetic resonance (NMR) and\nother experiments.\nWe considerthe standardHubbard model on the TKL,\nH=−/summationdisplay\n<ij>σtijc+\niσcjσ+U/summationdisplay\nini↑ni↓+µ/summationdisplay\niσniσ,(1)\nwheretijis the nearest-neighborhopping energy, Uis the\nCoulomb interaction, µis the chemical potential, c+\niσand\nciσdenote the creation and the annihilation operators,\nandniσ=c+\niσciσcorresponds to the density operator.\nThe asymmetry factor is defined as λ=tab/tbb, which\ncan be adjusted by suppressing the samples in experi-\nments. For the convenience, we use tbb= 1.0 as the en-\nergy unit. As shown in Fig. 1(a), there are two different\nsublattices on the TKL called A-sites and B-sites. The2\n(a1)\n(c)\nA-sites \nB-sites 3\n1ȑ Γ(b)\n3ȑ 1A-sites\nB-sitestab tbb 01 23\n45\n6\n7 8(a2)\n(a3)\nl>1.0\nl<1.0l=1.0\nFIG. 1: (Color online) (a1) The unit cell of the triangular\nKagom´ e lattice (TKL) without asymmetry ( λ= 1.0). The\nopen circles denote A-sites and the solid circles denote B-\nsites. The blue lines represent the hopping between A-sites\nand B-sites. And the red lines denote the hopping between\nB-sites. (a2) When λ >1.0, the TKL is similar with the\nKagom´ e lattice. (a3) When λ <1.0, the TKL is trans-\nformed into a system composed of many triangular plaque-\nttes. (b) The thick red lines show the first Brillouin zone of\nthe triangular Kagom´ e lattice. The thin black lines corre-\nspond to the Fermi surface for the non-interacting case. The\nΓ,K,M,K′,M′pointsdenotethepointsinfirstBrillouin zone\nwith different symmetry. (c) The density of states of the tri-\nangular Kagom´ e lattice for the A- and B-sites in the case of\nU= 0 and λ=tab/tbb= 1.0.\nspace group of the TKL is p6msame as the honeycomb\nlattice in the case of λ= 1, as show in Fig. 1(a1). Fig.\n1(a2) shows the TKL is similar as the Kagom´ e lattice\nwhenλ >1. We find the TKL can be transformed into\na system composed of many triangular plaquettes in the\ncase ofλ <1, as shown in Fig. 1(a3). The first Brillouin\nzone and Fermi surface of the symmetric TKL ( λ= 1)\nwithU= 0 is shown in Fig. 1(b). The density of states\n(DOS) for the different sublattices are shown in Fig. 1(c)\nin the case of U= 0 and λ= 1. They have the similar\nDOS around the Fermi surface.\nWe improve the CDMFT by combining it with the\nCTQMC to investigate the correlation effects on the\nTKL. In the CDMFT, the originallattice is mapped onto\nan effective cluster model via a standard DMFT proce-\ndure. Thus, the effective medium ˆ gcan be obtained by\nFIG. 2: (Color online) Phase diagram of the triangular\nKagom´ e lattice at λ= 0.6 withtbb= 1.0 as the unit of energy.\nThe black solid lines with square points show the transition\nline of the A-sites, and the red dashed lines with circle poin ts\nshow the transition line of the B-sites. Two kinds of coexist -\ning phases between red lines and black lines are the plaquett e\ninsulating phase and Kondo metal phase. Inset: The phase\ndiagram of the symmetric TKL, i.e. λ= 1, in which there are\nno coexisting phases.\na Dyson equation:\nˆg−1(iω) = (/summationdisplay\n/vectork1\niω+µ−ˆt(/vectork)−ˆΣ(iω))−1+ˆΣ(iω),(2)\nwhereˆt(/vectork) is the hopping matrix element in the cluster,\n/vectorkis the wave vector within the reduced Brioullion zone\nbased on clusters, ˆΣ(iω) is the self-energy matrix, and ω\nistheMatsubarafrequency. Then,weintroduceanimpu-\nrity solver, such as the CTQMC, to calculate the cluster\nGreen function ˆG(iω). The new self-energy ˆΣ(iω) can be\ncalculatedbyDysonequation ˆΣ(iω) = ˆg−1(iω)−ˆG−1(iω)\nto close the self-consistent iterative loop. This CDMFT\nloop is repeated until the ˆΣ(iω) converges to a desired\naccuracy. The CDMFT contains the important informa-\ntion of nonlocal correlation and geometrical frustration,\nwhich is absent in the general dynamical mean-field the-\nory (DMFT). In addition, the absence of Trotter decom-\nposition in the CTQMC makes this method more accu-\nrate than the translational auxiliary-field QMC.\nThe phase diagram obtained at λ= 0.6 shows two\ncoexisting phases and a reentrant behavior of the Mott\ntransition when half-filling. The asymmetry can cause\nthe separation of the phase transition points of the A-\nand B-sites. As shown in Fig. 2, with an increasing in-\nteraction at T= 0.5, the A-sites are found to translate\nfrom a metallic phase into an insulating phase and the\nB-sites stay in the insulating phase. This phase is called\nthe plaquette insulating phase shown in Fig. 2. In this\nplaquette insulating phase, electrons are localized on the\nA-sites and the absence of next nearest neighbor hop-3\nFIG. 3: (Color online) The momentum resolved one-particle\nspectrum Ak(ω) atλ= 0.6 in the units of tbb= 1. (a) The\nmetallic phase in the weak interaction case, such as U= 6.0\nandT= 0.5. (b) The Mott insulating phase in the strong\ninteraction case, such as U= 9.0 andT= 0.5. An obvious\nsingle particle gap shows up around the Fermi energy. (c)\nThe plaquatte insulating phase in the intermediate interac -\ntion case, in which the A-sites are insulating but B-sites ar e\nmetallic, such as U= 7.6 andT= 0.5. A small gap shows up\nin this phase. (d) The Kondo metallic phase at U= 7.0 and\nT= 0.2, in which A-sites are metallic, B-sites are insulating.\nThe single particle gap vanishes in this phase.\npings causes the electrons can only be itinerant within\nthe B-sites. When T <0.34, the B-sites translates into\nthe insulating phase, however the A-sites remain in the\nmetallic phase, see Fig. 2. This coexisting phase corre-\nsponds to a Kondo metal. The localized electrons on the\nB-sites act as the magnetic impurity, and the electrons\non the A-sites are still highly itinerant. In this phase,\nthe system shows the strong Kondo effect because of the\nhigh density of magnetic impurities. When the U > U c,\nfor example, Uc= 7.6 atT= 0.25, both the A-sites and\nB-sites translate into insulators. In Fig. 2, we find a\nreentrant behavior in the Mott transition of the A-sites\ncaused by the frustration and the asymmetry, which is\nalso found in the anisotropic triangular lattice [10]. This\nreentrant behavior divides the coexisting phases into two\nparts: the plaquette insulating phase and Kondo metal-\nlic phase. The inset figure in Fig. 2 shows the phase\ndiagram of the TKL at λ= 1.0. There are no coexisting\nphase and reentrant behavior found on the TKL when\nthe asymmetry is absent, i.e. λ= 1.\nIn order to find the properties of the different phases\nin Fig. 2, we investigate the momentum resolved one-\nparticle spectrum Ak(ω) by using the Maximum Entropy\nMethod (MEM) [39]. The results are shown in Fig. 3.\nThe Γ,K,MandM′points have been defined in Fig. 1\n(b). In the case of T= 0.5 andU= 6.0, we find two\nobvious quasi-particles near ω= 4.0 andω= 2.0, see\nFig. 3 (a). The gapless behavior shows that the system\nFIG. 4: (Color online) The evolution of double occupancy D\non the A-sites as a function of interaction Uwith different\ntemperatures at λ= 0.6 in the units of tbb= 1. The inset\nfigure shows the evolution of double occupancy on the B-sites .\nThe arrows with different colors show the phase transition\npoints at different temperatures.\nstays in the metallic phase shown in Fig. 2. When the\ninteraction U increases, we find an obvious gap near the\nFermi energy (see Fig. 3 (b)), which indicates the sys-\ntem becomes an insulator denoted in Fig. 2. Ak(ω) of\nthe plaquette insulating phase is shown in Fig. 3 (c).\nIn Fig. 3 (d), we find there is no obvious gap near the\nFermi energy when the system stays at the Kondo metal\nstate in Fig. 2. The momentum resolved one-particle\nspectrum function can be obtained by the ARPES ex-\nperiment, and the plaquette insulator and Kondo metal\nphases can be found in certain samples. We define the\ndouble occupancy Don the TKL as\nD=∂F/∂U=1\nN/summationdisplay\ni/angbracketleftni↑ni↓/angbracketright. (3)\nThe decreasing of Don both sublattices shows that the\nsuppressingoftheitinerancyofelectrons,whichisachar-\nacteristic behavior of the Mott transition. When tem-\nperature drops, D increases due to the enhancing of the\nspin fluctuation at low temperature. The arrowsindicate\nthe phase transition point at different temperatures. The\nsmoothlydecreasing DcharacterizesasecondorderMott\ntransition when the interaction Uincreases.\nIn Fig. 5, we show the evolution of spectrum function\non Fermi surface as a function of /vectorkfor different inter-\nactionsU, temperature Tand asymmetry λ, which is\ndefined as\nA(k;ω= 0)≈ −lim\nω→0ImGk(ω+i0)/π, (4)\nwhere\nGk(ω) =1\nN/summationdisplay\nγ,δeik·(rγ−rδ)[ω+µ−ˆt(k)−ˆΣ(ω)]−1\nγδ.(5)\nandkis the wave vector in the original Brillouin zone. A\nlinear extrapolation of the first two Matsubara frequen-\ncies is used to estimate the self-energy to zero frequency4\nFIG. 5: (Color online) The evolution of the spectrum functio n\non Fermi surface in the units of tbb= 1. (a) λ= 0.6, (b)\nλ= 1.0, (c)λ= 1.25.\n[36]. In the case of λ= 1.0, the spectrum function on\nFermi surface shows six points near the M point in Fig.\n1 (b), similar as the non-interacting case. When inter-\nactionUincreases, the Fermi surface shrinks because of\nthe localization of electrons. Fig. 5 (a1) shows the Fermi\nsurface is similar as the system composed of many trian-\ngular plaquettes at λ= 0.6, due to the suppression of the\nhopping between the A-sites and the B-sites. The Fermi\nsurfaces at λ= 1.0 andλ= 1.25 are similar as a Kagom´ e\nlattice. When the interaction Ukeeps increasing, the\nFermi surface is developing into a flat plane showing the\nlocalization of electrons, see Fig. 5 (a2), (b2), and (c2).\nThe spectrum function can be measured by the ARPES\nexperiment.\nIn order to investigate the evolution of magnetic order\non the TKL, we define a ferrimagnetic order parameter\nby:\nm=1\nN/summationdisplay\nisign(i)(/angbracketleftni↑/angbracketright−/angbracketleftni↓/angbracketright), (6)\nwheresign(i) = +1, for i=0, 2, 6; and sign(i) =−1, for\ni=1, 3, 4, 5, 7, 8. Fig. 6 shows the evolution of the single\nparticle gap ∆ Eand m as a function of interaction Uin\nthe case of λ= 1.0,T= 0.2. The Mott transition points\nof the A-sites and B-sites coexist when the asymmetry is\nabsent. A gap opens in the case of U= 8.5. The ferri-\nmagnetic order parameter m= 0 atU < U c= 13.8 indi-\ncates the system is in a paramagnetic insulating phase.\nWe argue that this paramagnetic insulating phase is a\ncandidate for the short range RVB spin liquid due to the\nabsence of long range correlations. When U > U c, the\nfinitemmeans the system enters a ferrimagnetic state.\nThe inset picture of Fig. 6 shows the evolution of single\nparticle gap ∆ Eatλ= 0.6 andT= 0.5, in which a gap\nopens on the A-sites first.\nFinally, we provide a phase diagram about the asym-\nmetryλand interaction UatT= 0.2, see Fig. 7.\nAs the asymmetry factor λincreases, a coexisting zone\nFIG. 6: (Color online) The evolution of single particle gap\n∆Eand ferrimagnetic order parameter matλ= 1.0 and\nT= 0.2 in the units of tbb= 1.0. A paramagnetic metallic\nphase is found when the interaction is weak with ∆ E= 0\nandm= 0. As the interaction Uincreases, a gap is opened\nand no magnetic order is formed with ∆ E/negationslash= 0 and m= 0.\nThis paramagnetic insulating phase can be a short range RVB\nspin liquid. An obvious magnetic order is formed when the\ninteraction is strong enough with ∆ E/negationslash= 0 and m/negationslash= 0. The\ninsert picture shows the evolution of ∆ Eatλ= 0.6 and\nT= 0.5. A plaquette insulator is found when the A-sites are\ninsulating and the B-sites are metallic.\ncontaining the plaquette insulator and the Kondo metal\nshows up. This zone is suppressed from λ= 0.9 to\nλ= 1.11. When U > U c, such as Uc= 13.8 in the\ncase ofλ= 1.0, the system becomes a ferrimagnetic in-\nsulator with m/negationslash= 0. Before entering the ferrimagnetic\nphase, a paramagnetic insulator state is found. We ar-\ngue this paramagnetic insulator state is a candidate of a\nshort range RVB spin liquid due to the absence of any\nmagnetic order and long range correlations.\nIn summary, we haveobtained the rich phase diagrams\nas the function of interaction U, temperature T, and\nasymmetry factor λ. The asymmetry introduced in this\nstrongly frustrated system can change the shape of the\nmetal-insulator translation line and induces several in-\nteresting phases, such as plaquette insulator and Kondo\nmetal. Intheplaquetteinsulatorphase, anobviousgapis\nfound nearthe Fermi energyby investigatingthe momen-\ntum resolvedspectrum function. The Kondometal phase\nis an interesting phase where the partial sites work as the\nmagneticimpuritiesinthemetallicenvironment. Wealso\nfind an interesting paramagnetic insulating phase which\nis related to the short range RVB spin liquid state before\nthe ferrimagnetic order is formed. And the increasing of\nasymmetry does not suppress the emerging of this ferri-\nmagnetic order. In addition, we have presented all kind\nof spectrum which can be used to detect these phases\nin real materials, such as Cu9X2(cpa)6·xH2O, under\napplied pressure in experiments. Our studies provide a\nhelpful step for understanding the coexisting behavior in\nmetal-insulator transition, the formation of magnetic or-\nder and the emerging of spin liquid in frustrated systems5\nFIG. 7: (Color online) The phase diagram of the triangular\nKagom´ e lattice represents the competition between intera c-\ntion and asymmetry for T= 0.2 in the units of tbb= 1. The\nregion between the black lines with square points and the red\nlines with circle points denotes the coexisting zone which c on-\ntains the plaquette insulator and the Kondo metal parts. A\nwide paramagnetic insulating region is found with an inter-\nmediate U. The blue lines with triangular points show the\ntransition point to the ferrimagnetic insulator with a clea r\nmagnetic order. Insert: (a) Dimers formed in the paramag-\nnetic insulator, which is a candidate for the short range RVB\nspin liquid, (b) Spin configuration of ferrimagnetic insula tor.\nwith asymmetry.\nWe would like to thank N. H. Tong and S. Q. Shen\nfor valuable discussions. This work was supported\nby the NKBRSFC under grants Nos. 2011CB921502,\n2012CB821305, 2009CB930701, 2010CB922904, and\nNSFC under grants Nos. 10934010, 60978019, 11074310,\nand NSFC-RGC under grants Nos. 11061160490\nand 1386-N-HKU748/10, and RFDPHE of China\n(20110171110026).\n∗Electronic address: yaodaox@mail.sysu.edu.cn\n†Electronic address: wmliu@iphy.ac.cn\n[1] Z. Y. Meng et al., Nature (London) 464, 847 (2010).\n[2] S. T. Bramwell et al., Nature (London) 461, 956 (2009).\n[3] E. Mengotti et al., Nature Physics 7, 68 (2011).\n[4] S. H. Lee et al., Nature Materials 6, 853 (2007).\n[5] B. K. Clark, D. A. Abanin, and S. L. Sondhi, Phys. Rev.\nLett.107, 087204 (2011).[6] M. Wang et al., Phys. Rev. B 84, 094504 (2011).\n[7] Y. L. Loh et al., Phys. Rev. B 77, 134402 (2008).\n[8] T. Yoshioka, A. Koga, and N. Kawakami, Phys. Rev.\nLett.103, 036401 (2009).\n[9] Y. Kato et al., Nature Physics 4, 614 (2008).\n[10] T. Ohashi, N. Kawakami, and H. Tsunetsugu, Phys. Rev.\nLett.97, 066401 (2006).\n[11] M. Jarrell, and H. R. Krishnamurthy, Phys. Rev. B 63,\n125102 (2001).\n[12] W. Wu et al., Phys. Rev. B 82, 245102 (2010).\n[13] M. Gonzalez, F. Cervantes-Lee, and L. W. ter Haar, Mol.\nCryst. Liq. Cryst. Sci Technol. 233, 317 (1993).\n[14] S. Maruti and L. W. ter Haar, J. Appl. Phys. 75, 5949\n(1993).\n[15] M. Mekata et al., J. Magn. Magn. Mater. 177, 731\n(1998).\n[16] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998).\n[17] W. Hofstetter et al., Phys. Rev. Lett. 89, 220407 (2002).\n[18] M. Greiner et al., Nature (London) 415, 39 (2002).\n[19] L. M. Duan, E. Demler, and M. D. Lukin, Phys. Rev.\nLett.91, 090402 (2003).\n[20] P. Soltan-Panahi et al., Nature Physics 7, 434 (2011).\n[21] N. Gemelke et al., Nature (London) 460,995 (2009).\n[22] Y. H. Chen et al., Phys. Rev. A 82, 043625 (2010).\n[23] A. Yamada et al., Phys. Rev. B 83, 195127 (2011).\n[24] D. Galanakis, T. D. Stanescu, and P. Phillips, Phys. Rev .\nB79, 115116 (2009).\n[25] K. Aryanpour, W. E. Pickett, and R. T. Scalettar, Phys.\nRev. B74, 085117 (2006).\n[26] C. C. Chang and S. W. Zhang, Phys. Rev. Lett. 104,\n116402 (2010)\n[27] W. Yao, and Q. Niu, Phys. Rev. Lett. 101, 106401\n(2008).\n[28] D.Xiao, W.Yao, andQ.Niu,Phys.Rev.Lett. 99, 236809\n(2007).\n[29] H. Lee, G. Li, and H. Monien, Phys. Rev. B 78, 205117\n(2008).\n[30] H. Terletska et al., Phys. Rev. Lett. 107, 026401 (2011).\n[31] F. Zhou, Phys. Rev. Lett. 87, 080401 (2001).\n[32] A. Georges et al., Rev. Mod. Phys. 68, 13 (1996).\n[33] T. Maier et al., Rev. Mod. Phys. 77, 1027 (2005).\n[34] H. Park, K. Haule, and G. Kotliar, Phys. Rev. Lett. 101,\n186403 (2008).\n[35] L. DeLeo, M. Civelli, and G. Kotliar, Phys. Rev. Lett.\n101, 256404 (2008).\n[36] O. Parcollet, G. Biroli, and G. Kotliar, Phys. Rev. Lett .\n92, 226402 (2004).\n[37] A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein,\nPhys. Rev. B 72, 035122 (2005).\n[38] A. Damascelli, Z. Hussain, and Z. X. Shen, Rev. Mod.\nPhys.75, 473 (2003).\n[39] M. Jarrell, and J. E. Gubernatis, Phys. Rep. 269, 133\n(1996)."    },    {        "title": "1806.07719v2.Magneto_optic_Kerr_effect_in_a_spin_polarized_zero_moment_ferrimagnet.pdf",        "content": "Magneto-optic Kerr e\u000bect in a spin-polarized zero-moment ferrimagnet\nKarsten Fleischer,1, 2Naganivetha Thiyagarajah,3Yong-Chang Lau,3Davide Betto,3Kiril\nBorisov,3Christopher C. Smith,1Igor V. Shvets,1J.M.D. Coey,3and Karsten Rode3\n1School of Physics, Trinity College Dublin\n2Dublin City University\n3CRANN and School of Physics, Trinity College Dublin\n(Dated: November 10, 2021)\nThe magneto-optical Kerr e\u000bect (MOKE) is often assumed to be proportional to the magnetisation\nof a magnetically ordered metallic sample; in metallic ferrimagnets with chemically distinct sublat-\ntices, such as rare-earth transition-metal alloys, it depends on the di\u000berence between the sublattice\ncontributions. Here we show that in a highly spin polarized, fully compensated ferrimagnet, where\nthe sublattices are chemically similar, MOKE is observed even when the net moment is negligible.\nWe analyse the spectral ellipsometry and MOKE of Mn 2RuxGa, and show that this behaviour is\ndue to a highly spin-polarized conduction band dominated by one of the two manganese sublattices\nwhich creates helicity-dependent re\rectivity determined by a broad Drude tail. Our \fndings open\nnew prospects for studying spin dynamics in the infra-red.\nI. INTRODUCTION\nSpin valves, where angular momentum can be trans-\nferred from a reference layer to a free layer, are expected\nto form the basis of future memory devices and oscil-\nlators operating in the far infra-red frequency domain.1\nThe key materials properties necessary to achieve these\nfrequencies are a high Fermi-level spin polarization P;\nlow net magnetization Msand a high e\u000bective magnetic\nanisotropy \feld \u00160Hk. A compensated ferrimagnetic half\nmetal where negligible magnetization can be obtained,\ncoupled with full ( P= 100 %) spin polarization, would\nbe an ideal choice.2\nSuch a material was suggested by van Leuken and\nde Groot3in 1995, but the \frst experimental evidence\nof one was only provided by the work of Kurt et al.4\nin 2014. We have recently shown that the material,\nMn2RuxGa (MRG), shows negligible Msover a wide\ntemperature and thermal treatment range;5,6high spin\npolarization4as re\rected in the anomalous Hall e\u000bect;7\nproduces sizeable magnetoresistive e\u000bects in spin elec-\ntronic structures8such as magnetic tunnel junctions;9\nand proposed an ab initio model reconciling the ex-\nperimental observations with density functional theory\n(DFT) band structure calculations.10\nTime-resolved magneto-optical Kerr/Faraday e\u000bect is\nparticularly suited to determine the Larmor frequency\nof thin-\flm samples, especially when the resonance fre-\nquency exceeds the experimental range of cavity- or\nnetwork-analyser-based techniques. The unique proper-\nties of MRG indicate that the material may be a candi-\ndate for low-power all-optical switching as the magnetic\ncompensation temperature Tcomp can be tuned to lie just\nabove room temperature,11but these measurements on\nnew materials are complicated by the lack of an accepted,\nwavelength-dependent optical model.\nFerrimagnets with two chemically distinct sublattices,\nnotably the rare-earth transition metal alloys have been\ninvestigated by spectroscopic MOKE.12,13The spectralresponse of the 4 fand 3delements allow the temperature\ndependence of the Kerr e\u000bect of each sublattice to be\ndetermined. The new aspect we present here is that in\na ferrimagnet with chemically identical, but structurally\ninequivalent, sublattices it is possible to track the spin\npolarization associated with only one of the sublattices.\nWe \frst show that even at magnetic compensation,\nMRG exhibits a clear Kerr rotation ( \u0012K). Spectroscopic\nellipsometry, in combination with the spectroscopic polar\nMOKE14allow us to determine the diagonal (~ \"xx) and\no\u000b-diagonal (~ \"xy) dielectric tensor components as a func-\ntion of Ru concentration x. Polar MOKE is often used\nsimply to measure hysteresis loops using monochromatic\nlight, and the spectroscopic variant provides more spe-\nci\fc information on the sites responsible for the magnetic\nresponse. Our measurements show a clear correlation\nbetween the infrared Kerr rotation and the electrically-\nmeasured Hall angle, providing contactless, optical mea-\nsurements of this quantity. The experimentally deter-\nmined, optical model developed here will facilitate mea-\nsurements in the time- and frequency-domains, that can\nadvance the development of spin electronic devices oper-\nating in the far infra-red frequency range.\nII. SAMPLE DETAILS\nThe thin-\flm samples studied here were grown on\nMgO (001) substrates by dc magnetron sputtering at\n250\u000eC substrate temperature in a 'Shamrock' system\nwith a base pressure of 2 \u000210\u00008Torr. The \flms were\nco-sputtered from Ru and Mn 2Ga targets. The Ru con-\ncentration was controlled by varying the Mn 2Ga target\nplasma power while keeping that of the Ru \fxed. The\nsamples were then capped with a protective 2 nm layer\nof amorphous aluminium oxide. The crystal structure,\nlattice parameters and strain of the \flms were charac-\nterised by 2 \u0012\u0000\u0012X-ray di\u000braction and reciprocal space\nmapping using a Bruker D8 Discover instrument, with aarXiv:1806.07719v2  [cond-mat.mtrl-sci]  3 Sep 20182\nFIG. 1. Schematic of the MRG crystal structure. There are\nthree \flled fcc sublattices (red: Mn-site (4 a), grey: Ga-site\n(4b), green: Mn-site (4 c)). Ruthenium occupies the remaining\nfcc sublattice (blue: Ru-site (4 d)). The two Mn sites are\nnon equivalent in the structure as the 4 a-site has only Ga as\nnearest neighbour, while the 4 c-site Ru, or vacancies (dashed\ncircles) depending on the Ru content x.\nmonochromatic copper K \u000b1source. We found that the\nin-plane lattice constant ais 0:5956 nm =p\n2aMgO for\nall samples, indicating that MRG grows epitaxially on\nMgO with its basal plane rotated by 45\u000ewith respect\nto the substrate. The out-of-plane parameter cvaries\nbetween 0:598 nm and 0 :618 nm, depending on Ru con-\ncentration and \flm thickness ( \u001826 to 81 nm). This con-\n\frms a slight, substrate-induced, tetragonal distortion of\nthe cubic structure. An illustration of the crystal is given\nin FIG. 1.\nIn order to determine the Ru concentration, we de-\nposited four MRG samples with varying Mn 2Ga tar-\nget power and a pure Ru \flm. The density and thick-\nness were then measured using X-ray re\rectivity (XRR).\nBased on the measured density and lattice parameters\nof these \fve control samples, we established a rela-\ntion between X-ray density and Ru concentration x,\nagainst which all the samples were calibrated. The trans-\nport properties of the thin \flms were measured on non-\npatterned \flms in a Quantum Design physical properties\nmeasurement system (PPMS) at temperatures ranging\nfrom 10 to 400 K in an applied magnetic \feld, \u00160H, of\nup to 12 T.\nIII. EXPERIMENTAL DETAILS\nBoth ellipsometry and magneto-optical Kerr spec-\ntroscopy were used to determine the optical and\nmagneto-optical properties of the \flms. Spectroscopic\nMOKE was recorded at near-normal incidence with a in-\nhouse built re\rectance anisotropy spectrometer, based\non the design of Aspnes.15Light from a Xe lamp passes\nthrough a Rochon Mg 2F polariser and is re\rected from\nthe sample. The beam then passes through a photo-\nelastic modulator (PEM), an analysing polariser and a\nmonochromator before \fnally reaching a diode detector\nsystem. All measurements were performed from 0 :35 to\n5 eV, using a Bentham TMc300 triple-grating monochro-mator and three individual photo-diodes (InAs, InGaAs,\nSi). The procedure for extracting the MOKE rotation\nhas been discussed elsewhere.16{18All samples where\nmeasured at room temperature in remanence (no exter-\nnal \feld). Samples were magnetised before these mea-\nsurements in \felds of up to 12 T at room temperature.\nSelected sample have also been magnetised at 200 K.\nTo correct for any set-up or non-magnetic anisotropy,\nwe recorded spectra for two orthogonal in-plane direc-\ntions as well as for samples magnetized in opposite di-\nrections. The spectra obtained by reversal of ~M, were\nstrictly equal to those derived from a rotation of the sam-\nple of 90\u000earound the optical axis.\nDue to larger errors in the Kerr-ellipticity measure-\nment in a PEM setup without in-situ magnetisation re-\nversal, only the Kerr rotation ( \u0012K) will be discussed.\nA Sopra GESP 5 spectroscopic ellipsometer was used\nover an energy range from 1 :5 to 5 eV with incidence\nangles of 70\u000e, 75\u000eand 80\u000e. Both, the thin \flm dielectric\nfunction (~\"xx, ~\"xy), as was derived from the raw measure-\nments by minimising the deviation between measured el-\nlipsometric data (tan \t ;cos \u0001) over the whole spectral\nrange, as well as all three angles. Full transfer matrix\ncalculations of the air/AlO x/MRG/MgO layer stack have\nto be evaluated, as in contrast to simple bulk samples no\nanalytical expression for the relationship between mea-\nsured tan \t ;cos \u0001, and \u0012Kexists. The spectral shape of\n~\"was simulated by a standard Drude-like component and\nthree harmonic oscillators (Drude-Lorentz oscillator). To\nreduce uncertainties in ~ \"xx, only the MRG layer was mod-\nelled. The sample thickness and interface roughness, as\nmeasured by XRR was used as input parameters together\nwith the known bulk dielectric functions of the MgO sub-\nstrate and the AlO xcapping layer.\nIn thin \flm structures, the \u0012Kspectra depend not\nonly on the o\u000b-diagonal dielectric tensor components\n~\"xy=iQmz~\"xxof the magnetic material but also on the\noverall re\rectance of the stack. Knowing the diagonal\ncomponents ~ \"xxfrom the ellipsometric model, the mea-\nsured\u0012Kspectral signature can then also be modelled,\nand hence ~\"xydetermined using the same methodology\nwithout skewed results caused by variations in ~ \"xxand\nsample thickness/roughness and modulation of the over-\nall re\rectivity.19{21\nFor all optical models, the MRG and AlO xthicknesses\ndetermined by XRR were treated as \fxed parameters.\nAll interfaces have been treated as sharp, while the di-\nelectric function of the AlO xcoating has been modelled\nby a Bruggeman e\u000bective medium of crystalline Al 2O3\nand air with a \fll factor set to the density ratio of the\nXRR \ftted capping layer and crystalline Al 2O3. We treat\nthe MRG to be perfectly cubic, as the deviation from cu-\nbic symmetry due to epitaxial strain of \u00181 %22is well\nbelow the sensitivity of our instrument.3\n−50510152025\n180 200 220 240 260 280 300 320 340 360051015\n200 250 300 350\nM(kA m−1)\nT(K)µ0Hc(T)\nT(K)\nFIG. 2. SQUID magnetometry of sample with Tcomp close\nto RT. The net magnetisation of the sample was measured at\nremnance on warming after saturation at low temperature. In\nthe inset we show an estimate of the coercive \feld, \u00160Hc, as\na function of temperature. Around room-temperature, even\n15 T is insu\u000ecient to saturate the magnetisation.\nIV. MOKE: DIRECT MEASUREMENTS\nDue to the di\u000berent crystallographic sites occupied by\nthe two magnetic sublattices in ferrimagnets, the temper-\nature dependence of the magnetisation of the two sub-\nlattices di\u000ber. Generally, perfect compensation and zero\nnet moment occurs only at a speci\fc temperature ( Tcomp)\nwhere the magnitudes of the two magnetic sublattices are\nequal but opposite in sign. For MRG this temperature\ndepends on both Ru content xand the biaxial substrate-\ninduced strain.\nUpon crossing Tcomp in zero applied \feld the net mag-\nnetisation changes sign whereas the spin-dependent DOS\nremains unchanged because the individual sublattices\nhave not changed direction. We therefore expect that the\nsign of any property that depends on the spin-resolved\nDOS such as MOKE, tunnel magneto-resistance (TMR)\nand anomalous Hall e\u000bect, will exhibit opposite signs de-\npending on whether the sample was magnetised below or\naboveTcomp.7,9\nIn FIG. 2 we show the temperature dependence of the\nnet magnetic moment of an MRG sample with Tcomp\u0018\n295 K. The inset gives an estimate of the coercive \feld as\na function of temperature. Close to Tcomp, the anisotropy\n\feld\u00160Hkdiverges, and as the coercive \feld is directly\nrelated to the anisotropy \feld, the sample cannot be mag-\nnetically saturated in this temperature range.\nWe now compare MOKE spectra of two samples with\nTcomp of 295 K and 250 K, respectively. The data are\nshown in FIG. 3. When magnetised at RT, i.e.above\nTcomp, both samples exhibit clear MOKE negative in\nsign. The sample with Tcomp closest to RT appears to\nhave a reduced magnitude. We then magnetise both\nsamples well below their respective Tcomp,Tsat= 200 K,\n−1.5−1−0.500.511.5\n0 1 2 3 4 5\nθK(mrad)\nhν(eV)Tcomp∼250 K\nTcomp∼295 KFIG. 3. Room-temperature Kerr rotation as a function of\nphoton energy for samples with Tcomp\u0018295 K and 250 K, re-\nspectively. The samples were \frst magnetised at RT in a \feld\nof\u00160H= 2 T (above Tcomp, open symbols). Subsequently,\nthe samples were remagnetised, this time well below Tcomp\n(Tsat= 200 K, \flled symbols). Although all measurements\nare done at RT, the sign of the MOKE signal is reversed\ndepending on whether magnetisation is carried out above or\nbelowTcomp. WhenTcomp\u0018295 K, full saturation is only\nachieved when magnetised well below Tcomp.\nand subsequently warm to RT in zero applied \feld. This\nprocedure ensures that while the net magnetic moment\nchanges sign when crossing Tcomp, the spin-dependent\ndensity of states (DOS) remains unchanged. We then\nrecord MOKE spectra at RT. As can be seen from FIG. 3,\nboth samples exhibit a change of sign as expected, and\ncrucially, the sample with Tcomp\u0018295 K and thus\u0018zero\nnet moment now displays as strong or stronger MOKE\nsignal as the sample measured further away from Tcomp.\nThis set of measurements proves that MOKE in MRG\ndepends on the spin-dependent DOS, and not the net\nmagnetic moment; and furthermore even when the net\nmagnetic moment is zero, the MOKE signal is clearly\nobserved.\nWe note here that disproportionally between the rem-\nnant magnetization and the polar MOKE measured dur-\ning a hysteresis loop have been observed earlier.23,24This\nwas attributed coupling, during switching , of the in-plane\nmagnetization and the polar MOKE through higher or-\nder terms. It was only observed in highly anisotropic\nexchange-biased materials and only during switching.\nThe samples measured here do not exhibit uniaxial in-\nplane anisotropy, and are all measured at remanence.\nIn order to pinpoint the origin of MOKE in MRG, we\nrecorded angle and energy dependent ellipsometry, which\nin combination with the measured Kerr rotation allow us\nto determine all matrix elements of the dielectric tensor.4\n354045505560\n0 1 2 3 4 5(a)\n0 1 2 3 4 5 6−4−202468\n(b)\n−2−1012\n0 1 2 3 4 5 6(c)\nσ1(1014s−1)\n/epsilon11x= 0.44\nx= 0.51x= 0.61\nx= 0.76x= 1.19θK(mrad)\nhν(eV)\nFIG. 4. (a) and (b): Imaginary and real components of\nthe diagonal tensor elements ~ \"xxof the dielectric function\nof MRG as a function of photon energy and Ru concentration\nx. The imaginary part is shown in terms of optical conduc-\ntivity (\u001b1=\"2!=4\u0019)25, to highlight the small di\u000berences be-\ntween samples.The behaviour is dominated by the Drude-like\nmetallic response, with smaller deviations caused by inter-\nband transitions. (c) MOKE spectra for the same series of\nsamples used to infer the o\u000b-diagonal ~ \"xytensor elements.\nV. OPTICAL MODEL\nWe have shown that MOKE is not determined by the\nnet magnetic moment, but depends on the spin depen-\ndent DOS in MRG. MOKE measurements alone are in-\nsu\u000ecient to determine ~ \"xxand ~\"xy. We therefore \frst\nuse spectroscopic ellipsometry to determine ~ \"xxand sub-\nsequently develop the optical model including the o\u000b-\ndiagonal tensor elements ~ \"xyfor MRG.\nIn FIG. 4 we show the dielectric function ~ \"xxas ex-\ntracted from least square \fts of the measured tan \t and\ncos \u0001 for a set of samples with varying Ru concentration\nx. Our optical model, used for both ~ \"xxand ~\"xyis based\non the following considerations:\nFor metallic materials such as MRG the dielectric re-\nsponse is usually dominated by a strong Drude-like tail\ncaused by free electron absorption, also known as intra-\nband transitions.26This contribution is the reason why\nthe imaginary part of ~ \"xxof MRG does not tend towards\nzero (\u001b\u001845\u00021014s\u00001ath\u0017= 1 eV) in the infra-red\nregion. Furthermore, there are absorption features in\nthe visible (VIS - peaks \u00182:3 eV) which we attribute\nto interband transitions on both Mn sites, ultra-violet\n(UV\u00183:2 eV) due to Ga, and a deep ultra-violet con-\ntribution (DUV >5:5 eV) necessary to account for the\nline shape but that cannot be assigned to a speci\fc el-\nement. The assignment of the VIS structure to Mn in-terband transitions is motivated by several experimental\nand theoretical \fndings, including similar spectral fea-\ntures in bulk Mn,25a strong peak in the unoccupied den-\nsity of states in ab-initio calculations for bulk Mn,27and\n\fnally the similarity in the shift of the energetic posi-\ntion of the VIS peak with increasing Ru concentration,\nfollowing the same trend as that seen in site-selective\nXAS/XMCD measurements on Mn 2RuxGa.5\nTo minimize the number of free parameters, we there-\nfore describe the diagonal part of the dielectric function\nvia one Drude-like oscillator at != 0 eV, and three broad\nharmonic oscillators (\u0000 >6 eV) in the VIS, UV and DUV\nranges, respectively.\nThe Kerr spectra show strong features in the IR and\nthe VIS regions, associated with the Drude-tail and the\nMn, respectively, but no separate features in the UV\nand DUV. As the magnetism in MRG is dominated by\nMn in two antiferromagnetically coupled sub-lattices, ~ \"xy\ncan be described in more detail. We associate the sin-\ngle, broad, VIS oscillator observed in ellipsometry at\nh\u0017\u00182:3 eV with two individual components in the Kerr\nspectra (NIR and VIS in FIG. 5, bottom panel), which\nwe assign to the spin-dependent transitions on the Mn-4 c\nand Mn-4asite. In agreement with our assignment above\nof the UV and DUV transitions to non-magnetic compo-\nnents, no separate UV and DUV oscillators are required\nto describe ~ \"xy.\nThis assignment is motivated by our earlier X-ray ab-\nsorption measurements5where we \fnd that the \fnal\nstates of the 2 p!3dtransitions of Mn in the two sites\nare only separated by approximately 1 eV, and no circu-\nlar magnetic dichroism on the Ga L2;3edges nor on the\nRuM4;5edges, as well as the dielectric functions of Mn25\nand Ga28.\nIn FIG. 5 we show an example of a \ft to the model\ndescribed above. Having established the diagonal tensor\nelements ~\"xxwe then model the o\u000b-diagonal terms which\nwe plot as the expected Kerr rotation \u0012K(the real part\nof the complex Kerr rotation). The agreement between\nthe experimental data and the optical model is excellent,\nindicating that our model is well justi\fed as well as high-\nlighting the predominant role of Mn in MRG magnetisa-\ntion. The (extrapolated) steep increase in MOKE in the\nIR and towards DC ( h\u0017!0 eV) is clearly an e\u000bect of\nthe highly spin-polarised conduction band, as re\rected by\nthe Drude contribution to both diagonal and o\u000b-diagonal\ntensor elements.\nOur modelling allows us to study in \fner detail the\nbehaviour of the spectral components of the ellipsometry\nand MOKE as a function of Ru concentration x(FIG. 4).\nWhen the Ru concentration xchanges from 0 :44 to\n1:19, the oscillator associated with Ga (UV) changes po-\nsition only slightly from 3 :4 to 3:2 eV whereas the oscilla-\ntor associated with Mn (VIS), exhibits a stronger trend\nand moves from 3 :0 to 2:2 eV with increasing x. With\nincreasing Ru content, the electronic occupation of Mn\nis increasing linearly, wheres that of Ga remains con-\nstant. The (small) changes associated with Ga are thus5\n00.511.522.5\n0 1 2 3 4 5 6\nNIR\nVIS0102030405060\nθK(mrad)\nhν(eV)Exp\nFitσ1(1014s−1) Drude\nVIS\nUV\nDUV\nExp\nFIG. 5. Complete optical model and experimental data for an\nMRG sample with x= 0:61. (Top) Experimentally observed\noptical conductance (line) and the \ftted contributions from\nthe four harmonic oscillators (areas) described in the text.\n(Bottom) Experimental Kerr rotation (points) and \ft to the\ndata (line) based on the optical model described in the main\ntext.\ndue to increasing tetragonality of the crystal structure\nwhile those associated with Mn correlate directly with\nthe band \flling. As described above, the MOKE in MRG\nis dominated by the contributions of Mn, both through\nthe spin polarisation of the conduction band as well as\nsingle-ion transitions. The two oscillators used to de-\nscribe these transitions in MOKE can therefore be asso-\nciated with Mn in the 4 cposition (\u00181:7 eV - NIR) and in\nthe 4aposition (\u00182:5 eV - VIS). Due to the half-metallic\nnature of the sample, when Ru content is increased, the\nelectronic occupation of Mn in the 4 cposition increases\nleading to a blue shift (from 1 :4 to 1:8 eV) of the oscilla-\ntor associated with this position. Simultaneously, adding\nRu decreases the spin-gap10thereby inducing a red shift\n(from 2:7 to 2:4 eV) of the oscillator associated with Mn\nin the 4aposition. More details on the optical model pa-\nrameter as function of Ru concentration xare found in\nthe supplemental information.\nIn FIG. 6 we show the o\u000b-diagonal component of the\ndielectric tensor as a function of photon energy, extracted\nfrom the \ft of the experimental ellipsometry and MOKE\ndata, in terms of \u001bxy. Unlike the raw MOKE spectra, dif-\nferences between di\u000berent compositions are now clearly\nvisible, and highlight, independently of our discussion fo-\ncussed on MRG, the need to employ robust optical mod-\nelling when reporting MOKE data on thin \flm samples\nof even slightly varying thickness.\nWe \fnally turn to a comparison of the DC limit of the\nmagneto-optical properties with the anomalous Hall an-\ngle measured by standard electronic magnetotransport.\nAs outlined above, our optical model describes the be-\n−1.5−1−0.500.511.52\n0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\nσxy(1014s−1)\nhν(eV)x= 0.44\nx= 0.61\nx= 0.76FIG. 6.\u001bxyextracted from the optical model for three dif-\nferent Ru concentrations x. The real and imaginary parts are\nplotted in solid and broken lines, respectively. In the light\nand dark grey areas, the curves are extrapolated from the\nexperimentally available range in ellipsometry and MOKE,\nrespectively. Please not that Im( \u001bxy) is only determined up\nto a potential integrational constant\n00.511.522.5\n0 1 2 3 4 5 6 7\nRe(σxy)/Re(σxx) (%)\nρxy/ρxx(%)\nFIG. 7. Ratio of the real part of the o\u000b-diagonal and on-\ndiagonal terms of the dielectric tensor extrapolated to 0 :4 eV\nplotted as a function of the electrically measured anomalous\nHall angle. The straight line is a \ft to the data, and provides\na guide to the eye. As expected, the DC limit of the magneto-\noptical characterisation agrees very well with the values ob-\ntained through electronic transport measurements.\nhaviour of MRG fully in the spectral region measured,\nand we therefore expect that extrapolation outside this\nregion should accurately predict the behaviour at di\u000ber-\nent photon energies. The lower limit is of course DC\nwhere optical measurements are not possible, but where\nwe can record magnetotransport, in particular anoma-\nlous Hall e\u000bect.7The ratio\u001axy=\u001axxis the Hall angle, the\nDC limit of the ratio Re( \u001bxy)/Re(\u001bxx). In FIG. 7 we plot\nthe ratio determined from the extrapolation of our opti-\ncal model as a function of the Hall angle determined by6\nmagnetotransport measurements as described in II. The\nratio was evaluated at h\u0017= 0:4 eV as we only need to\nextrapolate \u001bxxand not\u001bxyat this energy. As expected\nwe observe a strong correlation between the two mea-\nsurements, most certainly because the two e\u000bects share a\ncommon origin; the high spin polarisation of the conduc-\ntion band. The deviations from a perfectly proportional\nbehaviour is due to small di\u000berences in the scattering\nmatrices that have a big in\ruence in transport measure-\nments whereas they do not in\ruence the values obtained\nby optical means. We also emphasize that whereas elec-\ntronic transport probes the entire thickness of a metallic\n\flm, MOKE and ellipsometry only probe the skin-depth,\nwhich varies with photon energy, found here to be 5 nm\nat 2 eV and 40 nm at 0 :4 eV.\nVI. CONCLUSION\nWe have shown that MOKE in zero-moment half met-\nals persists, even when the net magnetic moment crosses\nzero, and that its origin is dominated by the highly spin-\npolarised conduction band associated in MRG with Mn in\nthe 4csite.5The sign of the MOKE depends on whether\nthe sample was magnetised above or below its compensa-\ntion temperature, as the process of magnetisation sets the\ndirection of the axis of quantization for the spin-polarised\ndensity of states.\nBy combining spectroscopic ellipsometry and MOKE\nit was possible to construct an optical model that identi-\n\fes the optically active components of MRG, and we use\nthis to infer the full parameterized dielectric tensor as a\nfunction of photon energy in the range 0 to 5 eV. The\nmodel is deliberately constructed to minimize the num-\nber of free parameters, yet it captures with reasonableaccuracy both the nonmagnetic and the magnetic contri-\nbutions for MRG in the spectral range measured. Based\non the model, we compare the results obtained by optical\nmeans with the anomalous Hall angle recorded by elec-\ntronic transport measurements. Although the DC limit\nis far beyond our experimentally available range, the ex-\ntrapolated values of \u001bxyagree remarkably well with those\nobtained by transport measurements.\nDi\u000berences between magnetic properties observed by\nanomalous Hall e\u000bect and MOKE are often used nowa-\ndays as a \fngerprint of topological electronic transport.29\nAn extension of this study with a more in-depth compar-\nison between the two in high and zero applied magnetic\n\felds, may allow an unambiguous determination of the\nscattering coe\u000ecients that lead to topologically driven\nspin structures in for example non-collinear ferrimagnets,\nas a complete optical model is necessary to disentangle\ne\u000bects that depend solely on di\u000berences in sample thick-\nness and re\rectivity from those that are purely magnetic\nin nature.\nFinally, this result interestingly clears a path to study\nthe behaviour of magnetic domains in antiferromagnets\nby MOKE microscopy, while the shown spectral depen-\ndence highlights photon energies most suitable for pump-\nprobe based time dependent measurements.30\nACKNOWLEDGMENTS\nThis project has received funding from the Euro-\npean Union's Horizon 2020 research and innovation pro-\ngramme under grant agreement No 737038. 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Gensch, Applied Physics Letters 109,\n032403 (2016), https://doi.org/10.1063/1.4958855."    },    {        "title": "1005.0480v1.Competition_between_Ferrimagnetism_and_Magnetic_Frustration_in_Zinc_Substituted_YBaFe4O7.pdf",        "content": " 1 Competition between Ferrimagnetism and Magnetic Frustration in Zinc \nSubstituted YBaFe 4O7 \n \nTapati Sarkar*, V. Pralong, V. Caignaert and B. Raveau  \n \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,  \n6 bd Maréchal Juin, 14050 CAEN, France  \n \nAbstract  \n \nThe substituti on of zinc for iron in YBaFe 4O7 has allowed the oxide series YBaFe 4-\nxZnxO7, with 0.40 \n  x \n 1.50, belonging to the “114” structural family to be synthesized. These \noxides crystallize in the hexagonal symmetry ( P63mc), as opposed to the cubic symmetry ( F-\n43m) of YBaFe 4O7. Importantly, the d.c. magnetization shows that the zinc substitution \ninduces ferrimagnetism, in contrast to the spin glass behaviour of YBaFe 4O7. Moreover, a.c. \nsusceptibility measurements demonstrate  that concomitantly these oxides exhibit  a spin glass \nor a cluster glass behaviour, which increases at the expense of ferrimagnetism, as the zinc \ncontent is increased. This competition between ferrimagnetism and magnetic frustration is \ninterpreted in terms of lifting of the geometric frustration , inducing the magnetic ordering , and \nof cationic disordering , which favours the glassy state.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n* Corresponding author: Dr. Tapati Sarkar  \ne-mail: tapati.sarkar @ensicaen.fr  \nFax: +33 2 31 95 16 00   \nTel:  +33 2 31 45 26 32  \n \n  2 Introduction  \n \nTransiti on metal oxides, involving a mixed valence of the transition element , present a \ngreat potential for the generation of strongly correlated electron systems. An extraordinary \nnumber of studies have been carried out in the last two decades on systems such as \nsuperconducting cupr ates, colossal magnetoresistive  manganites, and magnetic cobaltites, \nwhose oxygen lattice exhibits a “square” symmetry derived from the perovskite structure. In \ncontrast, the number of oxides with a triangular symmetry of the oxygen fra mework is much \nmore limited, if one excludes the large family of spinels which exhibit exceptional \nferrimagnetic properties and unique magnetic transitions  [1 – 3] as for Fe 3O4, but also magnetic \nfrustration [4 – 5] due to the peculiar triangular geometry of their framework as for  spinel \nferrite thin films.  \nThe recent discovery of the new series of cobaltites (Ln,Ca) 1BaCo 4O7 [6 – 8] and \nferrites (Ln,Ca) 1BaFe 4O7 [9 – 11], termed “114 ” oxides , have opened up a new field for the \ninvestigation of strongly corre lated electron systems. These oxides are closely related to spinels \nand barium hexaferrites, by their close packing of “O 4” and “BaO 3” layers. Importantly, they \ndiffer from all the other mixed valent transition metal oxides by the fact that cobalt or iron \nexists only in the tetrahedral coordination. Moreover, the symmetry of the structure may vary \naccording to the temperature as well as  the nature of the transition element, cobalt or iron, from \nhexagonal to cubic or orthorhombic. This change of symmetry see ms to dramatically  influence  \nthe magnetic properties, ranging from spin glass behaviour for cubic LnBaFe 4O7 oxides [10 – \n11] to ferrimagnetism for orthorhombic CaBaCo 4O7 [8] and hexagonal CaBaFe 4O7 [9] oxides. \nVery complex magnetic transitions are observed , as exemplified for YBaCo 4O7, for which a \nspin glass behaviour was first reported around 66 K  [7], whereas long range magnetic order \nwas observed below T N = 110 K [12], and a magnetic transition with short range correlations \nwas revealed above T N [13]. M oreover, such magnetic transitions are often induced by \nstructural transitions (T S), as shown by the lifting of the magn etic frustration at T S in \nLnBaCo 4O7 oxides [6, 7, 14].  \nThe remarkable feature of the “114” ferrites is the existence of the cubic symme try of \nLnBaFe 4O7 oxides [10 – 11] with Ln = Dy, Ho, Er, Yb, Lu, which has never been observed in \nthe cobaltite series, whereas the hexagonal symmetry is observed for both types of oxides, \nLnBaCo 4O7 [6 – 7] oxides whatever Ln, as well as  CaBaFe 4O7 [9] and L nBaFe 4O7 [11] with \nLn = Tb, Gd. Understanding the relative stability of these two structural types of ferrites is of \ngreat importance since it appears that it governs their magnetic properties, leading to spin glass  3 behaviour and ferrimagnetism for cubic a nd hexagonal LnBaFe 4O7 oxides respectively [11]. In \nfact, the two structures are closely related, i.e. built up of an ordered 1:1 stacking of identical \ntetrahedral layers, i.e. kagomé (K) and triangular (T) layers. The cubic structure ( Fig. 1(a) ) can \nthen be deduced from the hexagonal one ( Fig. 1 (b) ) by translation of one T layer out of two by \n2a\n. Bearing in mind that the structural transition from cubic to hexagonal symmetry is very \nsensitive to geometric factors, as shown from the ef fect of the size of Ln3+ cations [11], we \nhave explored the effects of cationic substitutions upon the structural and magnetic properties \nof the cubic spin glass YBaFe 4O7. In this stud y, we show that the substitution of Zn2+ for Fe2+ \nin this phase allows t he hexagonal symmetry to be stabilized for the series YBaFe 4-xZnxO7 with \n0.40 \n  x \n 1.50. Moreover, it is observed that this substitution by a diamagnetic cation induces \na ferrimagnetic ground state, in agreement with the change of symmetry. However, since  Zn2+ \nis a diamagnetic cation, it does not participate in the magnetic interactions , but rather dilutes \nthe magnetic sublattice and introduces cationic disordering, which in turn is responsible for the \nappearance of magnetic frustration. The combined measu rements of a.c. and d.c. magnetic \nsusceptibility throw light on the competition between ferrimagnetic interactions and magnetic \nfrustration in this series, suggesting a possible phase separation.  \n \nExperimental  \n \nPhase -pure samples of YBaFe 4-xZnxO7  [x = 0. 40 – 1.50] were prepared by solid state \nreaction technique. The precursors used were Y 2O3, BaFe 2O4, ZnO, Fe 2O3 and metallic Fe \npowder. First, the precursor BaFe 2O4 was prepared from a stoichiometric mixture of BaCO 3 \nand Fe 2O3 annealed at 1200°C for 12 hrs in air. In a second step, a stoichiometric mixture of \nY2O3, BaFe 2O4, ZnO, Fe 2O3 and metallic Fe powder was intimately ground and pressed in the \nform of rectangular bars. The bars were then kept in an alumina finger, sealed in silica tubes \nunder vacuum and annealed at 1100°C for 12 hrs. Finally, the samples were cooled to room \ntemperature at a rate of 200°C / hr.  \nThe X -ray diffraction patterns were registered with a Panalytical X’Pert Pro \ndiffractometer under a continuous scanning mode in the 2\n  range 10° - 120° and step size \n 2\n \n= 0.017°. The d.c. magnetization measurements were performed using a superconducting \nquantum interference device (SQUID) magnetometer with variable temperature cryostat \n(Quantum Design, San Diego, USA). The a.c. susceptibility, \n ac(T) was measured with a \nPPMS from Quantum Design with the frequency ranging from 10 Hz to 10 kHz (H dc = 0 Oe  4 and H ac = 10 Oe). All the magnetic properties were registered on dense ceramic bars of \ndimensions ~ 4 \n  2 \n 2 mm3. \n \nResults and discussion  \n \nFor the above synthesis conditions, a monophasic system was obtained for YBaFe 4-\nxZnxO7 with 0.40 \n  x \n 1.50. The EDS analysis of the samples ( Table 1 ) allowed the cationic \ncompositions to be confirmed.  \nStructural characterization  \nThe X -ray diffraction (XRD) patte rns show that all the samples with 0.40 \n  x \n 1.50 \nexhibit the hexagonal symmetry, as exemplified for the two extreme members of the series, x = \n0.40 and x = 1.50 ( Fig. 2 ), whose fits have been achieved by  Rietveld analysis using the \nFULLPROF refinement p rogram [15]. The extracted cell parameters ( Table 1 ) show that “ a” \nincreases slightly as x increases, whereas “ c” decreases so that the cell volume does not vary \nsignificantly, in agreement with the size of Zn2+ (\n2Znr\n  0.60 Å) being  very similar to that of \nFe2+ (\n2Fer\n  0.63 Å). Thus, the stabilization of the hexagonal phase with respect to the cubic \none is probably not due to the smaller size of the Zn2+ cation . The rather different distortion of \nthe ZnO 4 tetrahedra comp ared to the FeIIO4 tetrahedra may be at the origin of this structural \ntransition allowing a different tuning of the kagomé and triangular layers in the hexagonal \nphase compared to the cubic one.  \nA prefer ential occupancy of one of the  two different kinds of  sites by Zn2+ in the \nhexagonal structure is possible, but cannot be detected here from XRD data  due to the very \nclose values of the scattering factors of Zn and Fe. In any case, the c / a ratio of the hexagonal \ncell of the series YBaFe 4-xZnxO7 (Table 1 ) clearly shows that the introduction of zinc in the \nstructure tends to destroy the close packing of the “BaO 3” and “O4” layers, leading to a c / a \nvalue significantly larger than the ideal value for a perfect close packed structure (1.6333),  and \nin the same way ruling out a possible cubic close packed symmetry. Note, however, that the \ndecrease of c / a ratio as the zinc content increases is  not yet understood.  \nIt is also quite remarkable that attempts to synthesize phase pure samples for 0 < x < \n0.40 were uns uccessful , instead leading to a mixture of the cubic and hexagonal oxides. This \nobservation is important, since it demonstrates that in this region of cross -over from cubic to \nhexagonal  symmetry , no cubic solid solution can be obtained , confirming that Zn2+ cannot \naccommodate the unique tetrahedral site of the cubic structure of YBaFe 4O7, most likely due to \nthe different distortion of ZnO 4 tetrahedra compared to FeO 4 tetrahedra.   5  \nD. C. magnetization studies  \nThe temperature dependence of d.c. magnetization w as registered according to the \nstandard zero field cooled (ZFC) and field cooled (FC) procedures. A magnetic field of 0.3 T \nwas applied during the measurements. The measurements were done in a temperature range of \n5 K to 400 K. The M ZFC(T) and M FC(T) curve s of all the samples are shown in Fig. 3 . The \nhighest value of the magnetization was observed for the sample with the lowest zinc content, x \n= 0.40. For this sample, the magnetization increases sharply as the sample is cooled below 150 \nK, as is seen from t he FC and ZFC curves. The FC curve shows saturation at about 0.78 \n B/f.u. \nat 5 K, with the typical shape of ferro (or ferri) magnetic behaviour, whereas the ZFC curve \nexhibits a maximum at ~ 80 K  reflecting the existence of strong irreversibility. This beh aviour \nis in contrast to the spin glass behaviour of YBaFe 4O7 [10]. Rather, i t is comparable to \nCaBaFe 4O7 which was found to be ferrimagnetic with a hexagonal symmetry. Nevertheless, \nthe T C value of YBaFe 3.6Zn0.4O7 is signi ficantly smaller than the value  obtained for CaBaFe 4O7 \n(270 K). Also, the Curie -Weiss temperature (\n CW) for YBaFe 3.6Zn0.4O7, obtained by \nextrapolation of the linear \n-1(T) region, is 63.7 K. This is substantially lower than the value \nthat was obtained for CaBaFe 4O7 (268.3 K).  This sugges ts that the dilution introduced by the \ndiamagnetic cation Zn2+ weakens the ferrimagnetic interaction in this system, and the final \nground state of the oxide may not be a simple ferrimagnet, but a more complex system \ninvolving magnetic frustration.  \nWith an increase in the substitution level (x), the value of the magnetization attained by \nthe sample at the lowest measured temperature of 5 K decreases monotonically. (This decrease \nis shown in the inset of Fig. 3 .) This feature is again a reflection of the fact  that the Zn2+ ions \ndo not carry any magnetic moment, and thus, serve to dilute the ferrimagnetic coupling. This \ndilution of the ferrimagnetic interaction is also seen in the fact that the temperature below \nwhich the magnetization starts rising, as well as  the sharpness of the transition keeps \ndecreasing with an increase in the zinc content (x)  (Fig. 3 ). This has been shown quantitatively \nin Fig. 4 , where we have plotted the first derivative of magnetization (dM/dT) versus \ntemperature curves. The inflection  points of the curves (T inf) can be roughly considered to be \nthe temperatures at which the paramagnetic to ferrimagnetic transition takes place. (Later on \nwe will show that the ground state obtained at low temperature in these oxides is not a simple \nferrim agnet. Instead, it is rather complex , as will become clear from the a.c. magnetic  6 susceptibility measurements, which we detail in the next section ). Inset (a) of Fig. 4  shows the  \nmonotonic decrease of T inf with increase in the substitution level (x).  \nIn in set (b) of Fig. 4 , we show the variation of \n T1/2 with the doping concentration (x). \nT1/2 is the full width at half maximum (F.W.H.M.) of the dM/dT vs T  curves. It gives a \nmeasure of the width of the transition (a lower value of \n T1/2 indicates a sharper transition). As \nis seen in the figure, the width of the transition shows a systematic increase with an increase in \nthe substitution level (x), thereby indicating the dilution of the ferrimagnetic interactions.  \nIn view of the observations on the irreversibi lity of the FC and ZFC M(T) curves seen \nat low temperature, detailed investigations were carried out on the d.c. magnetization M(H) \ncurves of the various samples at 5 K. The isothermal magnetization curves recorded at 5 K \n(Fig. 5 ) show that the x = 0.40 sa mple exhibits the largest hysteresis loop i.e. the largest values \nof the coercive field (HC ~ 1 T) and of the remanent magnetization (M r ~ 0.8 \n B/f.u.), \nindicating that YBaFe 3.6Zn0.4O7 is, like CaBaFe 4O7, a hard ferrimagnet. Importantly, the shape \nof the M (H) loop of this phase is much smoother than for CaBaFe 4O7, and rather similar to that \nobserved for CaBaFe 4-xLixO7 oxides, that were shown to be glassy ferrimagnets [16]. These \nsamples show similar magnetic hysteresis loops, characterized by  a lack of magn etic \nsaturation. Moreover, one observes that the coercive field H C (inset (a) of Fig. 5 ) and the \nremanent magnetization M r (inset (b) of Fig. 5 ) exhibit a monotonic decrease as the \nsubstitution level increases.  \nThe lack of magnetic saturation in these samp les can be quantified by taking the ratio of \nthe value of the magnetization attained at a field of 5 T and M r [M (H = 5 T) / M r]. This is \nshown in Fig. 6  (a). As can be seen from the figure, the quantity M (H = 5 T) / M r increases by \nnearly 5 times as the doping concentration is increased from x = 0.40 to x = 1.50. This again \nindicates the strong weakening of the ferrimagnetic interaction with increasing zinc \nconcentration.  The spontaneous magnetization (M at H = 0, as read out from the virgin curve) \nshows a monotonic decrease with an increase in the substitution level (x) ( Fig. 6 (b) ). As x \nincreases from 0.40 to 1.50, the spontaneous magnetization decreases by 4 orders.  \nThus, these results suggest that in these  hexagonal oxides, ferrimagnetism competes \nwith magnetic frustration due to the triangular geometry of the kagomé layers. Such a \nstatement is also supported by the recent study of the isostructural “114” hexagonal cobaltite \nYBaCo 4O7 [13] which shows that 1 D magnetic ordering competes with 2 D magneti c \nfrustration.  \n  7 A. C. magnetic susceptibility studies  \nThe measurements of the a.c. magnetic susceptibility \n ac(T,f) were performed in zero \nmagnetic field (H dc = 0) at different frequencies ranging from 10 Hz to 10000  Hz, using a \nPPMS facility. The amplitu de of the a.c. magnetic field was ~ 10 Oe. The evolution of the real \npart \n(T) and the imaginary part \n (T) of the susceptibility versus temperature at different \nfrequencies for the two end members of the series, x = 0.40 ( Fig. 7 ) and x = 1.50 ( Fig. 8 ) sheds \nlight on the complex nature of the magnetic ground state of these oxides. The \n (T) curves \n(Fig. 7 (a)  and Fig. 8 (a) ) exhibit a maximum at T max(a.c.) which is slightly higher than the peak \nvalues  obtained from d.c. magnetic measurements  (Tmax(d.c.) ) i.e. T max(a.c.)  = 10 5 K at 10 Hz \n(Fig. 7  (a) and Table 2 ) against T max(d.c.)  = 83 K (Fig. 3) for the x = 0.40 sample, and T max(a.c.) = \n45 K  (Fig. 8 (a)  and Table 2 ) against T max(d.c.)  = 36  K for the x = 1.50 sample.  Both samples \nshow frequency dependent pe aks in the a.c. susceptibility measurements i.e. as the frequency is \nincreased, T max(a.c.) shifts to higher values associated with a decrease in the magnitude of \n ac. \nThis behaviour strongly suggests a spin glass [17] or superparamagnetic behaviour, which  will \nbe discussed below. The peak observed in the real part (in -phase component) of the magnetic \nsusceptibility (\n ac(T)) at a frequency of 10 Hz decreases from 10 5 K to 45 K as the doping \nconcentration is increased from x = 0.4 0 to x = 1.5 0. \nThe plot of the imaginary part of the a.c. susceptibility ( Fig. 7 (b)  and Fig. 8 (b) ) show \nthat the maximum of \n ac(T) appears at a lower temperature than the maximum of \n ac(T), and \nis also frequency dependent. We note here that for the x = 0.40 sample, the magnitud e of the \npeak in \n ac is largest for the lowest frequency measured (11 Hz). On the other hand, for t he x \n= 1.5 sample , the magnitude of the peak in \n ac is largest for the highest frequency measured \n(10 kHz). This indicates that the inverse mean relaxati on time of the spin system for the x = \n0.40 sample is lower than that for the x = 1.50 sample. This is in accordance with the values of \nthe spin relaxation time \n 0 obtained later from the fitting of the a.c. susceptibility data  (\n0(x = 0.40)  \n> \n 0(x = 1.50) )  (see Table 2).  \nTo check the existence of a spin glass behaviour or of cluster glass behaviour, we have \nanalyzed the frequency dependence of the peak in \n ac(T) using the method previously  \ndeveloped by Bréard  et.al. [18], starting from the dynamic scalin g theory [17] which predicts a \npower law of the form \nz\nSGSG f\nTTT\n0 , where, \n 0 is the shortest relaxation time available \nto the system, TSG is the underlying spin -glass transition temperature determined by the \ninteractions in the system, z is the dynamic critical exponent and \n  is the critical exponent of  8 the correlation length. The actual fittings were done using the equivalent form of the power \nlaw: \nSGSG f\nTT Tzln ln ln0\n . The remarkable linearity obtained for the plots of \nln\nversus \nSGSG f\nTTTln  using the fit parameters , shown in Fig. 9 , indicate  that these samples \nwell obey the behaviour expected for a spin glass like system. Similar fittings were performed \nfor all the other samples of the series, and the extracted  parameters of this study have been \nlisted in Table 2 .  \nThe parameter \nfTT\npff\nlog\n  is known to  vary from 0.004 to 0.02 for canonical  spin \nglass systems. The value of p that we obtain  for the two end members of the series under \ninvestigation are p = 0.015 (for the x = 0.40 sample) and p = 0.020 (for the x = 1.50 sample). \n[Note: For superparamagnetic materials, the value of p is higher (~ 0.1), and so we discard the \npossibility of these oxides being superparamagnetic.]  In spite of the fact that the  value of p is \nquite similar for the two end members of the series, the fact that the shape of the a.c. \nsusceptibility versus temperature is quite different in the two cases gets reflected in the values \nof the parameters \n 0 and z\n. In fact, for the lower z inc concentrations (x = 0.40 to x = 0.90), the \nspin relaxation time \n 0 lies in the range of values typical of canonical spin glasses (10-10 – 10-12 \nsec) [19 – 21], and the z\n values lie between 6 and 10. However, as the zinc concentration is \nincreased beyo nd x = 1, the \n 0 values are much smaller, and the z\n values obtained are above \n10.  Thus, the power law model provides a satisfactory description of our a.c. susceptibility \ndata. \n \nConclusion  \nThe present study of the zinc substituted “114” oxides YBaFe 4-xZnxO7 shows that the \nsubstitution of zinc for iron in the YBaFe 4O7 oxide induces a ferrimagnetic behaviour, in spite \nof the diamagnetic character of Zn2+, but that these magnetic interactions compete with a spin \nglass or a cluster glass behaviour as the zinc  content is further increased.  In fact, the shape of \nthe M -T curve obtained for this series of oxides (sharp decrease in the ZFC  curve at lower \ntemperatures with  saturation of FC curve) is very similar to that reported for various alloys \nshowing mictomagne tism[22,23,24] , i.e. coexistence of frustrated spin clusters and \nferrimagnetic spin region s. This competition between ferrimagnetism and magnetic frustration  9 can be understood by considering the structure, or more exactly the iron lattice of these oxides. \nThe cubic structure of YBaFe 4O7 (Fig. 1 (a) ) forms a tetrahedral [Fe 4]\n lattice of Fe 4 tetrahedra \nshari ng apices ( Fig. 10  (a)), very similar to the [Ln 4]\n lattice observed in pyrochlores. As \npreviously shown for these compounds, this tetrahedral framework exhibits a complete 3 D \ngeom etric frustration and consequen tly, one observes, like for the pyrochlores [2 5], a spin glass \nbehaviour [10].  As soon as a part of Zn2+ is substituted for Fe2+, i.e., for YBaFe 3.6Zn0.4O7 (x = \n0.40), the structure becomes hexagon al (Fig. 1 (b) ), forming a different [Fe 8]\n lattice ( Fig. 10  \n(b)), built up of rows of corner – sharing “Fe 5” trigonal bipyramids running along “ c”, and \ninterconnected through “Fe 3” triangles in the “ a b” plane. As a consequence, the zinc \nsubstitution indu ces a lifting of the geometric frustration, favouring the appearance of magnetic \nordering, as previously observed for hexagonal CaBaFe 4O7 which is known to be ferrimagnetic \n[9]. In fact, such a hexagonal framework could be regarded as composed of ferri (or  ferro) \nmagnetic rows running along “ c” formed by the “Fe 5” bipyramids, whereas in the “ a b” plane, \nthe triangular geometry of the iron lattice is maintained so that a bidimensional magnetic \nfrustration is still possible, and may compete with the 1 D magne tic ordering along “ c”. \nHowever, t he presence of zinc on the iron sites tends to weaken the magnetic interactions due \nto its diamagnetic character by dilution effect , and the cationic disordering on the Fe / Zn sites \nfavours the spin glass or cluster glass  behaviour. As a consequence the ferrimagnetic behaviour \ndecreases at the benefit of the spin glass or cluster glass behaviour as the zinc content increases \nbeyond x = 0.40.  \nA neutron diffraction study of these glassy ferrimagnets will be necessary to det ermine \nthe exact distribution of zinc on the two different sites of the structure and to understand the \ncomplex nature of the magnetic states that arise from the competition between ferrimagnetism \nand magnetic frustration. The possibility of phase separati on should also be considered.  \n \nAcknowledgements  \n \nWe gratefully acknowledge Dr. Vincent Hardy, CRIS MAT, ENSICAEN, France for \nhelpful discussions. We also acknowledge the CNRS and the Conseil Regional of Basse \nNormandie for financial support in the frame of Emergence Program . \n \n \n \n \n  10 References  : \n      [1]    Friedrich Walz, J. Phys.: Condens. Matter ., 2002 , 14 , R285  \n      [2]     E. J. W. Verwey and P. W. Haayman, Physica , 1941 , 8, 979  \n      [3]    J.B. Goodenough, Magnetism and the Chemical Bond , Interscience  Monographs on  \n            Chemistry  vol. 1 , Huntington, New -York (1976)  \n      [4]     Yuji Muraoka, Hitoshi Tabata and Tomoji Kawai, J. Appl. Phys ., 2000 , 88, 7223  \n      [5]     G. E. Delgado, V. Sagredo and F. Bolzoni, Cryst. Res. Technol . 2008 , 43, 141  \n      [6]     Martin Valldor and Magnus Andersson, Solid State Sciences , 2002 , 4, 923  \n      [7]     Martin Valldor, J. Phys.: Condens. Matter ., 2004 , 16, 9209  \n      [8]     V. Caignaert, V. Pralong, A. Maignan and B. Raveau, Solid State Communications ,  \n             2009 , 149, 453  \n      [9]     B. Raveau, V. Caignaert, V. Pralong, D. Pelloquin and A. Maignan, Chem. Mater .,  \n             2008 , 20, 6295  \n      [10]  V. Caignaert, A. M. Abakumov, D. Pelloquin, V. Pralong, A. Maignan, G. Van  \n             Tendel oo and B. Raveau, Chem. Mater ., 2009 , 21, 1116  \n      [11]  V. 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H. Hsieh and   \n              C.  T. Chen, J. Phys.: Condens. Matter, 2007 , 19, 216212  \n      [19]     J. Souletie and J. L. Tholence, Phys. Rev. B , 1985 , 32, 516  \n      [20]     K. Gunnarsson, P. Svedlindh, P. Nordblad, L. Lundgren, H. Aruga and A. Ito, Phys.  \n             Rev. Lett ., 1988 , 61, 754  \n      [21]    K. H. Fischer and J. A. Hertz, Spin Glasses , Cambridge University Press,   11                Cambridge, 1993  \n      [22]     Yuntao Wang, Huaiyu Shao, Xingguo Li, Lefu Zhang and Seiki Takahashi, J. \n             Magn. M agn. Mater. , 2004 , 284, 13 \n      [23]      N Saito, H Hiroyoshi, K Fukamichi and Y Nakagawa, J. Phys. F: Met. Phys.  1986 ,  \n            16, 911  \n      [24]      Hidetoshi Hiroyoshi, Kou Noguchi, Kazuaki Fukamichi and Yasuaki Nakagawa, J. \n           Phys. Soc . Japan , 1985 , 54, 3554  \n      [25]        John E. Greedan, Journal of Alloys and Compounds , 2006 , 408 – 412, 444  \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  12 Table caption s \n \nTable 1 : Refined crystal structure parameters as obtained from the Rietveld refinement of X -\nray powde r diffraction data.  \nTable 2 : Parameters extracted from fitting of a.c. magnetic susceptibility data.   \n \nFigure captions  \n \nFigure 1:  Relative positions of the T layers in (a) cubic YBaFe 4O7 and (b) hexagonal \nCaBaFe 4O7 (adapted from Ref. 10). For details, see text. \nFigure 2:  X-ray diffraction patterns along with the fits for (a) YBaFe 3.6Zn0.4O7 and (b) \nYBaFe 2.5Zn1.5O7. \nFigure 3:  MZFC(T) and M FC(T) curves of YBaFe 4-xZnxO7 measured at H = 0.3 T. The empty \nsymbols are for MZFC(T) and the solid symbols are for MFC(T). The inset shows the variation of \nMFC at T = 5 K with the substitution level (x).  \nFigure 4 : dM/dT vs T for YBaFe 4-xZnxO7. Inset (a) shows the variation of T inf with x, inset (b) \nshows the variation of \n T1/2 with x (see text for details).  \nFigure 5:  M (H)  for YBaFe 4-xZnxO7 at T = 5 K . Inset (a) shows the variation of HC with x, \ninset (b) shows the variation of M r with x.  \nFigure 6:  (a) M (H = 5 T) / M r vs x, (b) M (H = 0 T) vs x for YBaFe 4-xZnxO7. \nFigure 7 : (a) Real (in -phase) and (b) Imaginary (out -of-phase) component of a.c. \nsusceptibilities  for YBaFe 3.6Zn0.4O7 as a function of temperature measured using a frequency \nrange 10 Hz – 10 kHz.  \nFigure 8 : (a) Real (in -phase) and (b) Imaginary (out -of-phase) component of a.c. \nsusceptibilities  for YBaFe 2.5Zn1.5O7 as a function of temperature measured using a frequency \nrange 10 Hz – 10 kHz.  \nFigure 9 : Plot of ln \n  vs \nSGSG f\nTTTln  for (a) YBaFe 3.6Zn0.4O7 and (b) YBaFe 2.5Zn1.5O7 (for \ndetails, see text).  \nFigure 10 : Schematic representation of (a) cubic YBaFe 4O7 and (b) hexagonal CaBaFe 4O7 \n(adapted from Ref. 11). For details, see text.  \n \n \n  13 Table 1  \nDoping \nconcentration \n(x)  \nCrystal \nsystem \n(Space \ngroup)  \n  \nUnit cell parameters  \n c / a Reliability \nfactor (R F) x as obtained \nfrom EDS \nanalysis  a (Å) c (Å) \n0.40 Hexa gonal \n(P63mc)  \n6.317(2)  \n 10.380(2)  1.6432  3.11 0.36(0.02)  \n0.50 Hexagonal \n(P63mc)  \n6.319(2)  \n 10.379(2)  1.6425  8.31 0.53(0.03)  \n0.60 Hexagonal \n(P63mc)  \n6.321(1)  \n 10.377(1)  1.6417  5.30 0.65(0.04)  \n0.80 Hexagonal \n(P63mc)  \n6.321 (2) \n 10.371 (2) 1.6407  9.28 0.80(0.04)  \n0.90 Hexagonal \n(P63mc)  \n6.323 (1) \n 10.367 (1) 1.6396  7.97 0.90(0.04)  \n1.25 Hexagonal \n(P63mc)  \n6.325 (3) \n 10.355 (3) 1.6372  8.31 1.27(0.03)  \n1.50 Hexagonal \n(P63mc)  \n6.328 (1) \n 10.348 (1) 1.6353  8.24 1.56(0.03)  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  14 Table 2   \nDoping \nconcentration \n(x) Tmax  from \n(T) at f = \n10 Hz (K)  p Spin relaxation time \n 0 \n(sec)   \nSpin glass \ntransition \ntemperature \nTSG (K) \n z\n \n \n0.40 \n 104.5 0.015  (1.95 \n  0.04) \n  10-10 100.44 (7)  6.2 \n1) \n \n0.50 \n 95.4 0.014  (8.14 \n  0.04) \n  10-11 92.30 (3)  6.6 \n2) \n \n0.60 \n 91.8 0.013  (3.09 \n  0.08) \n  10-11 88.04 (2)  6.9 \n2) \n \n0.80 \n 79.8 0.011  (1.11 \n  0.04) \n  10-11 76.05 (2)  7. 3 \n 1) \n \n0.90 \n 75.8 0.011  (2.93 \n  0.01) \n  10-12 72.96 (6)  7.4 \n1) \n \n1.25 \n 56.0 0.015 (8.75 \n  0.09) \n  10-13 54.26 (2)  10.8 \n2) \n \n1.50 \n 44.8 0.020  (3.75  \n 0.04) \n 10-15 40.63 (6)  13.6 \n4) \n \n \n \n \n \n \n \n \n \n \n \n \n \n  15 \n \n \nFigure 1  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  16 \n \nFigure 2  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \nFigure 3  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  18 \n \nFigure 4  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  19 \n \nFigure 5  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  20 \n \nFigure 6  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  21 \n \nFigure 7  \n \n \n \n \n \n \n \n  22 \n \nFigure 8 \n \n \n \n \n \n  23 \n \nFigure 9  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  24 \n \n \nFigure 10  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "2102.03020v1.Inter_valence_charge_transfer_and_charge_transport_in_the_spinel_ferrite_ferromagnetic_semiconductor_Ru_doped_CoFe__2_O__4_.pdf",        "content": " 1 Inter-valence charge transfer and charge transport in the spinel ferrite ferromagnetic semiconductor Ru-doped CoFe2O4  Masaki Kobayashi1,2,*, Munetoshi Seki1,2, Masahiro Suzuki3, Miho Kitamura5, Koji Horiba5,  Hiroshi Kumigashira5,6, Atsushi Fujimori3,4, Masaaki Tanaka1,2, and Hitoshi Tabata1,2 1Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Center for Spintronics Research Network, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 3Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 4Department of Applied Physics, Waseda University, Okubo, Shinjuku, Tokyo 169-8555, Japan 5Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba 305-0801, Japan 6Insitute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University,  Sendai 980–8577, Japan *Email: masaki.kobayashi@ee.t.u-tokyo.ac.jp  Abstract Inter-valence charge transfer (IVCT) is electron transfer between two metal M sites differing in oxidation states through a bridging ligand: Mn+1 + M’m à Mn + M’m+1. It is considered that IVCT is related to the hopping probability of electron (or the electron mobility) in solids. Since controlling the conductivity of ferromagnetic semiconductors (FMSs) is a key subject for the development of spintronic device applications, the manipulation of the conductivity through IVCT may become a new approach of band engineering in FMSs. In Ru-doped cobalt ferrite CoFe2O4 (CFO) that shows ferrimagnetism and semiconducting transport properties, the reduction of the electric resistivity is attributed to both the carrier doping caused by the Ru substitution for Co and the increase of the carrier mobility due to hybridization between the wide Ru 4d and the Fe 3d orbitals. The latter is the so-called IVCT mechanism that is charge transfer between the mixed valence Fe2+/Fe3+ states facilitated by bridging Ru 4d orbital: Fe2+ + Ru4+ ↔ Fe3+ + Ru3+. To elucidate the emergence of the IVCT state, we have conducted x-ray absorption spectroscopy (XAS) and resonant photoemission spectroscopy (RPES) measurements on non-doped CFO and Co0.5Ru0.5Fe2O4 (CRFO) thin films. The  2 observations of the XAS and RPES spectra indicate that the presence of the mixed valence Fe2+/Fe3+ state and the hybridization between the Fe 3d and Ru 4d states in the valence band. These results provide experimental evidence for the IVCT state in CRFO, demonstrating a novel mechanism that controls the electron mobility through hybridization between the 3d transition-metal cations with intervening 4d states.    I. Introduction Inter-valence charge transfer (IVCT) occurs in mixed-valence coordination complexes, and is an electron transfer between two metal M sites differing in oxidation states through a bridging ligand: Mn+1 + M’m à Mn + M’m+1, and is usually related to photoexcitation phenomena in these compounds [1, 2]. Additionally, magneto-optical properties of ferrites or iron oxides that seem to originate from IVCT such as magneto-optical Kerr rotation and photo-induced magnetization have been reported so far [3, 4, 5, 6, 7, 8]. In semiconducting or insulating materials with mixed-valence states, especially for metal-organic frameworks, IVCT is considered to be related to the transport properties with hopping conduction [6, 9, 10, 11, 12, 13]. Then, it is expected that one can control magneto-optical and transport properties of mixed-valence semiconductors or insulators through IVCT.  Ferromagnetic semiconductors (FMSs) having both semiconducting and ferromagnetic properties are key materials for spintronics, which is a research field to use both the charge and spin degrees of freedom for functional electronic devices [14, 15, 16]. Controlling the conductivity of FMSs with keeping their ferromagnetic properties is useful for utilizing FMSs to spintronics devices such as magnetic random-access memories (MRAMs) and spin transistors. Spinel ferrites MFe2O4 [M = 3d transition metals (TMs)] are promising magnetic materials for spintronics because of their chemical stability and high Curie temperatures [17]. Indeed, spinel ferrite layers in magnetic tunnel junctions have been demonstrated to act as spin filters [18, 19]. CoFe2O4 (CFO) with the inverse spinel structure shows ferrimagnetism with the Curie temperature (TC) of 793 K and CFO epitaxial thin films have been studied for the application to spintronics devices [10, 20, 21, 22, 23]. In CFO, the stoichiometric valence states of the constituent Co and  3 Fe ions are Co2+ at the octahedral crystal-field (Oh) site and Fe3+ at both the Oh and the tetrahedral crystal-field (Td) sites, as shown in Fig. 1. In contrast to the semiconducting CFO, magnetite Fe3O4 shows metallic transport property because of double-exchange interaction between the Fe2+ and Fe3+ sites. Recently, Iwamoto et al. [10] have succeeded in increasing the conductivity of CoFe2O4 by Ru doping, where the Ru4+ ions are considered to preferentially occupy the Co2+ sites in CFO [7, 10]. Figure 2 shows the transport properties of Ru-doped CoFe2O4 thin films [10]. The results indicate that the Ru doping increases the carrier density and the electron mobility. The doped Ru4+ ions in CFO are expected to act as double donors and that the Ru doping increases not only the carrier concentration but also the hopping probability though hybridization between the Fe 3d and Ru 4d orbitals. It is considered that the carriers supplied by Ru doping generate the mixed valence state of Fe2+/Fe3+ in CFO, leading to the improvement of the conductivity through double-exchange interaction like Fe3O4. First-principles band calculation suggests that the increase of the hopping conduction occurs through hybridization between the Fe 3d and Ru 4d orbitals following the IVCT mechanism Fe2+ + Ru4+ ↔ Fe3+ + Ru3+.   To prove the existence of such an IVCT state in CRFO, two key factors should be confirmed experimentally: (1) the Fe2+/Fe3+ mixed-valence state and (2) the hybridization between the Fe 3d and Ru 4d orbitals. In this letter, we have conducted x-ray absorption spectroscopy (XAS) and resonance photoemission spectroscopy (RPES) studies of non-doped CFO and Ru-doped CoFe2O4 thin films in order to obtain experimental evidence for the IVCT state in CRFO. XAS and RPES enable us to probe element- and orbital-specific electronic structures. The XAS spectra reveal the valence states of Fe and Co in CRFO. The RPES spectra indicate that the partial density of states (PDOS) of Fe 3d change with Ru doping. These experimental findings suggest that the IVCT state is realized in Ru-doped CFO. The manipulation of the conductivity through IVCT may become a new approach of band engineering in FMS materials.  II. Experimental CoFe2O4 (CFO) and Co0.5Ru0.5Fe2O4 (CRFO) epitaxial thin films were grown on single crystal a-Al2O3(0001) substrates using the pulsed laser deposition (PLD) technique with an ArF-excimer laser with the wavelength was 193 nm, the frequency was 5 Hz, and the  4 fluence was E = 60 mJ. The details of the thin-film growth are described elsewhere [10]. The PES and XAS measurements were performed at BL-2A of Photon Factory (PF), High Energy Acceleration Research Organization (KEK) [24]. The total energy resolution was set to be 100-250 meV for PES measurements using photon energy of 400-1200 eV . The binding energies were calibrated by measuring the EF of a gold foil which was electrically contacted to the samples. The PES measurements were conducted with an SES2002 electron analyzer at room temperature under the base pressure below 2.0 ´ 10-8 Pa. The XAS spectra were measured in the total electron yield (TEY) mode.   III. Results and discussion Basically, XAS spectra reflect the local electronic structures of specific elements. We have performed XAS measurements on CRFO thin films to elucidate the valence state of each element. Figure 3 shows the Co L2,3 XAS spectra of parent CFO and Ru-doped one. The spectra of CFO and CRFO show multiplet structures that are different between them. Comparing the observed Co L2,3 spectra with those of reference compounds CoO (Co2+ Oh) and LaCoO3 (Co3+ Oh) [25], the spectrum of CRFO is similar to that of CoO and the spectrum of CFO seems to be a superposition of them. Here, the Co3+ Oh spectrum represents cation inversion or Co-antisite defects (the inter-site cation exchange between the A Td and B Oh sites, and the on-site cation replacement at the A sites) because the XAS spectra of the Co3+ Oh is similar to that of the Co3+ Td [26]. Here, the cation inversion defects is represented by the inversion parameter y defined by the chemical formula [Fe1-yCoy]Td[Fe1+yCo1-y]OhO4. To estimate quantitatively the ratio of these oxidation (valence) states, the Co L2,3 XAS spectra of CFO and CRFO are fitted by a linear combination of the reference spectra of CoO and LaCoO3. Figures 3(b) and 3(c) show such decomposition analyses for the Co L2,3 XAS spectra. Figure 3(b) shows that the Co L2,3 XAS spectrum of CRFO is fitted by the spectrum of CoO and that there is nearly no contribution of the Co3+ state to the experimental spectrum within the accuracy of the analysis. In contrast, the Co L2,3 XAS spectrum of CFO is decomposed into the ~75% Co2+ and the ~25% Co3+ components, as shown in Fig. 3(c), indicating that part of the Co2+ ions at the B site becomes Co3+ at the B site or move to the A site. The latter case means the existence of antisite defects, namely, Co ions at the A site and excess Fe ions at the B site. This is consistent with that non-negligible cation inversion defects usually exist in spinel ferrite thin films [22, 23]. In CRFO, on the other hand, electron-rich  5 environment induced by Ru doping suppresses the occurrence of Co3+ ions and possibly the cation inversion. It should note here that the Co2+ components possibly include both the Co2+ at the B site and the antisite defect of the Co2+ at the A site. The chemical compositions are discussed below considering the decomposition analysis of the Fe L2,3 XAS spectra.  Figure 4 shows the Fe L2,3 XAS spectra of the parent CFO and CRFO thin films. The Fe L2,3 XAS spectra show multiplet structures and the spectral line shapes are similar to that of Fe2O3, as shown in Fig. 4(a). In contrast to the Co L2,3 XAS spectra, the Fe XAS spectrum of CRFO film looks nearly identical to that of CFO one. To elucidate the Fe components precisely, the Fe L2,3 XAS spectra are decomposed into various valence and crystal-field states using linear combinations of reference spectra of Fe2O3 (Fe3+ Oh) [27], YBaCo3FeO7 (Fe3+ Td) [28], and FeO (Fe2+ Oh) [29]. As show in Figs. 4(b) and 4(c), the linear combinations of the reference spectra well reproduce the experimental spectra of CRFO and CFO samples. The ratios of the components are listed in Table I. As expected from the line shapes, the ratio among the Fe components of CRFO is similar to that of CFO. Considering the possible existence of the cation inversion defects in the CFO thin film observed in the Co L2,3 XAS spectrum, the Fe2+ component in the CFO film likely originates from the Fe-antisite defect, where the excess Fe ions substitute for Co2+ ions at the Oh sites, as in the case for the Co3+ component tentatively assigned to cation inversion defects [Fig. 3(c)]. Based on the analysis, the chemical composition of CFO is estimated as [Fe0.67Co0.33]Td[Fe1.33Co0.67]OhO4 (y = 0.33). On the other hand, the presence of the Fe2+ component in the CRFO film would be induced by the electron doping caused by the Ru substitution because there will be few cation inversion defects in the CRFO film. Although the ratio of Fe2+ is expected to be 25% in CRFO from the nominal stoichiometry assuming Ru4+, the deduced value is ~10.5%. Since there is nearly no Co3+ component in CRFO, this quantitative difference may come from Ru-antisite defects substituting for the Fe sites. These results provide spectroscopic evidence for the Fe2+/Fe3+ mixed-valence state in the CRFO thin film caused by the Ru doping, even though there is a quantitative difference.   To elucidate the possible hybridization between the Fe 3d and Ru 4d orbitals, which is another key factor for the IVCT, the Fe and Co 3d PDOS in the valence band (VB) that  6 arise from the different valence states have been obtained using RPES. Figure 5 shows the RPES spectra of the CFO and CRFO thin films. When incident photon energy hn corresponds to that for the core-hole excitation, the photoemission intensity of the orbitals into which the excited electron orbital is resonantly enhanced. Difference between the on-resonance spectrum and off-resonant spectrum reflects the 3d PDOS corresponding to those orbitals. Since the 2p-3d excitation energy for each component is different, one can obtain the PDOS dominated by each component by tuning hn (it is difficult to separate these PDOS perfectly because of the overlap of the XAS spectra between these components). For instance, as shown in Fig. 5(a), the Fe2+ and Fe3+ PDOS of the thin films are obtained from the on-resonance spectra measured at hn of 708.5 eV and 710 eV , respectively. Figure 5(b) shows the Fe and Co 3d PDOS and off-resonance spectra of CFO and CRFO films. Here, the on-resonance spectra are taken at the core-hole excitation energies for the Co2+, Co3+, Fe2+, and Fe3+ states, and the spectra are normalized to the spectral area. For the Co-3d PDOS, the differences of the spectra between the samples reflect the effect of Ru substitution for the Co sites. Since the Co 3d spectral intensity near the valence-band maximum (VBM) decreases with Ru doping, we conclude that the Co 3d PDOS hardly contributes to the increase of the conductivity. Similarly, the Fe3+ PDOS shows nearly the same spectral changes as Co 3d with Ru doping, suggesting few direct contributions of the Fe3+ component to the improvement of the conductivity.   It should note here that the differences of the Co off-resonance spectra between the CFO and CRFO films reflect the appearance of the Ru 4d PDOS and the decrease of the Co-3d PDOS with Ru doping. Considering the cross section of the atomic orbitals around this incident-photon energy region, the VB spectra without resonance mainly reflect the Ru 4d PDOS [30]. Since the spectral intensity near the VBM increases with Ru doping, the Ru 4d PDOS predominantly contributes to the DOS near the VBM, qualitatively consistent with the increase of the conductivity with Ru doping. To reveal the 3d component hybridized with the Ru 4d orbitals, the additional 3d PDOS near VBM induced by Ru doing are examined. Figure 5(c) shows comparison of the various PDOS in the vicinity of VBM that have been normalized to the intensity at EB~1.5 eV . The difference in the off-resonance spectra indicates that the Ru 4d PDOS is located near the VBM. Note also that additional Fe2+ PDOS appears with Ru doping and the position of the Fe2+ PDOS is nearly identical to that of the Ru 4d PDOS [see arrows in Fig. 5(c)]. In  7 contrast, the Co3+, Co2+, and Fe3+ PDOS have the same slopes irrespective to the Ru doping. These observations are consistent with the first-principle calculation that the B-site Fe 3d orbitals hybridizes with the Ru 4d orbital except for the opening of the conductivity gap in experiment [10]. The band gap may come from Coulomb interaction, i.e., a Mott gap. Thus, the present result is consistent with hybridization between the Fe 3d and Ru 4d orbitals in CRFO.   The present findings, that is, the mixed-valence state of Fe2+/Fe3+ and the hybridization between the Fe 3d and Ru 4d orbitals, provide spectroscopic evidence for the IVCT state in Ru-doped CFO. In CRFO, the O 2p orbitals that are located at the nearest neighbor atoms of the B sites make bonding states directly with the Fe 3d eg and Ru 4d eg orbitals. It is likely that the ligand O 2p orbitals bridge electron hopping between the Fe and Ru sites such as Fe2+ + Ru4+ ↔ Fe3+ + Ru3+. Here, while IVCT for photoexcitation is usually expressed with a single-headed arrow between M atoms, we use a double-headed arrow to IVCT for transport. In contrast to the M-M’ distances for IVCT in coordination complexes reportedly in the range of 10~25 Å [2], the effective distance for the IVCT possibly becomes longer in single crystalline TM compounds like spinel ferrite oxides because the ligand bands bridging the IVCT are expanded the entire crystal. Compared with the M-M’ distance in coordination complexes, in which the photo-excitation induces charge transfer in the environment of highly insulating nature, the longer effective distances for the IVCT in the TM compounds lead to the increase of the hopping probability (or electron mobility) through IVCT as a consequence of the reduction of energy barrier for the charge transfer. This may explain the increase of the conductivity by Rh doping in Fe2O3 (Fe1.8Rh0.2O3) without mixed-valence states [31], where the doped Rh3+ ions isovalently substitute for the Fe3+ sites. The same mechanism may also explain the increase of the hopping probability through IVCT of FeTiO3 ilmenite [9], Fe2+ + Ti4+ ↔ Fe3+ + Ti3+, without double-exchange interaction [since the Ti4+ ion (d0 configuration) is non-magnetic]. As in the cases for Fe3O4 and (La,Sr)MnO3, double-exchange interaction between magnetic ions with the mixed-valence states stabilizes the ferromagnetism and contributes to the increase the metallic conductivity [32]. In contrast, the conduction mechanism through IVCT in solids will be applicable for ferromagnetic semiconducting materials having low carrier concentrations with hopping conduction. Furthermore, as described in the introduction, the IVCT has the capability to modify the  8 optical properties of semiconducting materials. It follows from the above arguments that control of the IVCT in FMSs with mixed-valence states will be a new approach manipulating the electron mobility and magneto-optical properties through hybridization between the TM d orbitals with maintaining the ferromagnetism.   IV. Summary In this study, we have investigated the electronic structure of Ru-doped CoFe2O4 using XAS and RPES to elucidate the emergence of IVCT in semiconducting crystal. The Co L2,3 XAS spectra suggest that although the 25% of the total amount of Co atoms exist as Co3+ in the CFO thin film, there are only Co2+ ions in the Ru-doped CFO thin film. The result that there is almost no Co3+ ions in the Ru-doped CFO thin film provides an important aspect to control defect level for materials growth of spinel ferrites. The Fe L2,3 XAS spectra indicate that the Fe2+/Fe3+ mixed-valence state is realized in CRFO. The observation using RPES demonstrates that the Ru 4d PDOS appears near the VBM and hybridizes with the Fe2+ 3d state, and not with the Co ones. These findings provide spectroscopic evidence for the IVCT state between the Ru 4d and Fe 3d orbitals. Controlling the IVCT through the hybridization will open a new way to manipulate both the magneto-optical properties and the carrier mobility of FMSs.   Acknowledgment This work was supported by a Grant-in-Aid for Scientific Research (Nos. 15H02109,16H02115, 17H04922, 18H05345) and Core-to-Core Program from the Japan Society for the Promotion of Science (JSPS), CREST (JPMJCR1777), and the MEXT Elements Strategy Initiative to Form Core Research Center. This work was partially supported the Spintronics Research Network of Japan (Spin-RNJ) and Basic Research Grant (Hybrid AI) of Institute for AI and Beyond for the University of Tokyo. 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Tabata, Appl. Phys. Express 5, 115801 (2012). [32] C. Zener, Phys. Rev. 82, 403 (1951).         11 Table Table I. Decomposition analysis for the Co L2,3 and Fe L2,3 XAS spectra. It should note here that the Co2+ components include both the Co2+ at the B site and the possible antisite defect of the Co2+ at the A site.    A(Td) site B(Oh) site Co3+ Fe3+ Co2+ Fe3+ Fe2+ CoFe2O4 25% 33.6% 75% 57.9% 8.5% Co0.5Ru0.5Fe2O4 0% 37.2% 100% 52.3% 10.5%     12 Figures  \n   FIG 1: Crystal structure of inverse spinel CoFe2O4. (a) Unit cell of CoFe2O4. The magnetic sublattice of the A(Td) site antiferromagnetically couples with that of the B(Oh) site. (b) Spin configurations of the A and B sites.      \n 13  \n FIG. 2: Transport properties of Co1-xRuxFe2O4 thin films [10]. (a) Temperature (T) dependence of resistivity r. (b), (c) Compositional dependence of the carrier density and the Hall mobility, respectively.     10-210-1100101102103Resistivity ρ (Ωcm)\n350300250200150100Temperature (K)\n681020246810212Carrier density (1/cm-3)\n765431000/T (K-1)6810-2246810-124Hall mobility (cm2/Vs)\n765431000/T (K-1) x = 0.2 x = 0.5 x = 0.8(a)Co1-xRuxFe2O4\n(b) x = 0.2 x = 0.5 x = 0.8(c) x = 0.2 x = 0.5 x = 0.8 14   \n FIG. 3: Co L2,3 XAS spectra of Co1-xRuxFe2O4 thin films. (a) The XAS spectra of parent CFO and CRFO thin films. The spectra of CoO (Co2+) and LaCoO3 (Co3+) are also shown as references [25]. The arrows denote excitation energies for RPES, i.e., 775 eV for the off-resonance, 778.7 eV for the Co2+ on-resonance, and 780 eV for the Co3+ on-resonance. (b), (c) Decomposition of the Co L2,3 spectra for the CRFO and CFO thin films, respectively.    Intensity (arb. units)800795790785780775Photon Energy (eV)Intensity (arb. units)784780776772Photon Energy (eV)784780776772Photon Energy (eV)Co2+ [CoO]Co3+ [LaCoO3] CRFO CFOCo L2,3 XAS(a)\n(b) Exp. Fitted Co2+ Co3+CRFOCFO(c) Exp. Fitted Co2+ Co3+ 15   FIG. 4: Fe L2,3 XAS spectra of Co1-xRuxFe2O4 thin films. (a) The XAS spectra of parent CFO and CRFO thin films. The reference spectra of Fe2O3 (Fe3+ Oh) [27], YBaCo3FeO7 (Fe3+ Td) [Hollmann_PRB_09], and FeO (Fe2+ Oh) [29] are also shown. The arrows denote excitation energies for RPES, i.e., 705 eV for the off-resonance, 708.5 eV for the Fe2+ on-resonance, and 710 eV for the Fe3+ on-resonance. (b), (c) Decomposition of the Fe L2,3 spectra for the CRFO and CFO thin films, respectively.    Intensity (arb. units)730725720715710705Photon Energy (eV)Intensity (arb. units)716712708704Photon Energy (eV)716712708704Photon Energy (eV)Fe L2,3 XAS CRFO CFOFe3+ Oh [Fe2O3]Fe3+ Td [YBaCo3FeO7]Fe2+ Oh [FeO] Exp. FittedFe3+ OhFe3+ TdFe2+OhCRFO(a)\n(b) Exp. FittedFe3+ OhFe3+ TdFe2+ Oh(c)CFO 16  \n FIG. 5: Resonant photoemission spectra taken at Fe L3 and Co L3 edges of Co1-xRuxFe2O4 thin films. (a) On- and off-resonance spectra at the Fe L3 edge of CFO. (b) Partial density of states of the Fe 3d and Co 3d states. The shaded areas are difference of PDOS between CFO and CRFO films. The spectra have been normalized to the intensity integrated from EF to EB ~ 11 eV . (c) Fe 3d and Co 3d PDOS near the valence band maximum. The spectra have been normalized to the intensity at EB = 1.5 eV in order to emphasizing the spectral change near the VBM.    1086420Binding Energy (eV)Intensity (arb. units)\nNormalized Intensity (arb. units)\n1086420Binding Energy (eV)Normalized Intensity (arb. units)1.51.00.50Binding Energy (eV)(a)Fe L3 RPES On-Reso. Fe3+ On-Reso. Fe2+ Off-Reso.(b)\nCRFOCFOPDOS\nCo2+Co3+\nOffFe2+Fe3+CRFOCFOCo3+Co2+Fe2+Fe3+Off(c)"    },    {        "title": "2303.17302v1.Ferrimagnetic_Regulation_of_Weyl_Fermions_in_a_Noncentrosymmetric_Magnetic_Weyl_Semimetal.pdf",        "content": "Ferrimagnetic Regulation of Weyl Fermions in a\nNoncentrosymmetric Magnetic Weyl Semimetal\nCong Li1;2;];\u0003, Jianfeng Zhang3;], Yang Wang1;], Hongxiong Liu3;], Qinda\nGuo1, Emile Rienks4, Wanyu Chen1, Bertran Francois5, Huancheng Yang6,\nDibya Phuyal1, Hanna Fedderwitz7, Balasubramanian Thiagarajan7, Maciej\nDendzik1, Magnus H. Berntsen1, Youguo Shi3, Tao Xiang3, Oscar Tjernberg1\u0003\n1Department of Applied Physics, KTH Royal\nInstitute of Technology, Stockholm 11419, Sweden\n2Department of Applied Physics, Stanford University, Stanford, CA 94305, USA\n3Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n4Helmholtz-Zentrum Berlin f ur Materialien und Energie,\nElektronenspeicherring BESSY II, Albert-Einstein-Stra\u0019e 15, 12489 Berlin, Germany\n5Synchrotron SOLEIL, L'Orme des Merisiers,\nD\u0013 epartementale 128, 91190 Saint-Aubin, France\n6Department of Physics and Beijing Key Laboratory of\nOpto-electronic Functional Materials &Micro-nano Devices,\nRenmin University of China, Beijing 100872, China\n7MAX IV Laboratory, Lund University, 22100 Lund, Sweden\n]These people contributed equally to the present work.\n\u0003Corresponding authors: conli@kth.se, oscar@kth.se\n1arXiv:2303.17302v1  [cond-mat.str-el]  30 Mar 2023The study of interaction between electromagnetism and elementary particles\nis a long-standing topic in physics. Likewise, the connection between particle\nphysics and emergent phenomena in condensed matter physics is a recurring\ntheme and condensed matter physics has often provided a platform for inves-\ntigating the interplay between particles and \felds in cases that have not been\nobserved in high-energy physics, so far. Here, using angle-resolved photoemis-\nsion spectroscopy, we provide a new example of this by visualizing the electronic\nstructure of a noncentrosymmetric magnetic Weyl semimetal candidate NdAlSi\nin both the paramagnetic and ferrimagnetic states. We observe surface Fermi\narcs and bulk Weyl fermion dispersion as well as the regulation of Weyl fermions\nby ferrimagnetism. Our results establish NdAlSi as a magnetic Weyl semimetal\nand provide the \frst experimental observation of ferrimagnetic regulation of\nWeyl fermions in condensed matter.\nIn recent decades, considerable e\u000borts have been made to identify topological quasi-\nparticles in condensed matter physics that follow the same physical laws as elementary\nparticles[1{7]. The discovery of topological semimetals has made this possible[8{20]. Weyl\nsemimetals is an important class of topological semimetals, which hosts emergent relativistic\nWeyl fermions in the bulk, and Fermi arc surface states connect two Weyl points with op-\nposite chirality on the boundary of a bulk sample[1, 6, 7, 13{15]. These emergent particles\nare the result of inversion symmetry (IS) [6, 7, 13{15, 21, 22] or time-reversal symmetry\n(TRS)[1, 2, 23{32] breaking. The TRS-breaking Weyl semimetals are usually derived from\nmagnetic materials[1, 2, 23{32], compared to the Weyl semimetals with IS-breaking, which\nprovide a platform for the study of the interplay between magnetism, electron correlation\nand topological orders. In the magnetic Weyl semimetals, a nonvanishing Berry curvature\ninduced by TRS-breaking can give rise to rich novel phenomena, such as a anomalous[33{35]\nand spin[36, 37] Hall e\u000bect, and chiral anomalous charge[17{20]. These exotic physical prop-\nerties make magnetic Weyl semimetals potential candidates for a wide range of applications\nin spintronics.\nRecently, a rare case of magnetic Weyl semimetal candidates, the RAlX (R: Rare earth; X:\nSi or Ge) family, has attracted attention. This family displays both IS and TRS breaking[38{\n63]. From the perspective of crystal structure, RAlX is a non-centrosymmetric crystal with\nmagnetism derived from R (Rare earth) atoms[38]. The magnetic structure of the RAlX\n2family is easy to regulate and presents diverse magnetic ordering. For example, the magnetic\nstructure of RAlX can be nonmagnetic[64{66], ferromagnetic[38{49], antiferromagnetic[49{\n56], ferrimagnetic[57{60] and even spiral magnetic[57, 61] by rare earth element substitution.\nSuch a rich magnetic structure makes the RAlX family an appropriate candidate to study\nthe interaction between magnetism and Weyl fermions. The interaction between magnetism\nand Weyl fermions includes two parts: the \frst part is the mediating e\u000bect of Weyl fermions\non magnetism[57], and the second part is magnetism regulation of the Weyl fermions and\nfurther e\u000bects on the topological ordering. To data, the mediating e\u000bect of Weyl fermions\non magnetism has been con\frmed by neutron di\u000braction measurements[57]. However, un-\nambiguous and direct experimental con\frmation of the regulation of Weyl fermions by mag-\nnetism remains unobserved until now.\nHere, we present angle-resolved photoemission spectroscopy (ARPES) measurements and\nband structure calculations to systematically investigate the electronic structure and topo-\nlogical properties of NdAlSi and how they are regulated by ferrimagnetism. We observe\nFermi arcs and bulk Weyl fermion dispersion in the paramagnetic state, after determining\na surface state associated with a Nd terminated surface cleaved at the Nd-Al plane. In ad-\ndition, we also observe that a new Weyl fermion dispersion corresponding to the new Weyl\nfermion generated in the ferrimagnetic state. These observations are in good agreement\nwith the prediction of theoretical calculations and con\frm the existence of Weyl fermions\nregulated by ferrimagnetism in NdAlSi. The results provide key insights into the interplay\nbetween magnetism and topological orders.\nNdAlSi was predicted to be a magnetic Weyl semimetal that crystallizes in the tetragonal\nstructure with the space group I41md(no. 109)[57], as shown in Fig. 1a. The crystal struc-\nture of NdAlSi has two vertical mirror planes, mxandmy, as well as two vertical glide mirror\nplanes,mxyandmxy, but lacks the mirror symmetry of the horizontal plane which induces\nthe inversion symmetry breaking. The corresponding three-dimensional (3D) Brillouin zone\n(BZ) of NdAlSi is shown in Fig. 1b. In order to determine the magnetic structure of NdAlSi\nat low temperature, magnetic susceptibility measurements were performed (for details see\nFig. S1 in the Supplemental Material). Based on the magnetic susceptibility (Fig. S1)\nand previous neutron scattering measurements[57], it can be inferred that the spontaneous\nmagnetization of NdAlSi appears as a up-up-down ( \"\"#, UUD) type ferrimagnetism at low\ntemperature (T <T com\u00183.4 K), as shown in Fig. 1c. The corresponding 3D BZ of NdAlSi in\n3the UUD ferrimagnetic state is shown in Fig. 1d. In order to study the electronic structure\nand topological properties of NdAlSi and its regulation by ferrimagnetic order, we performed\ndensity functional theory (DFT) calculations in the paramagnetic and the UUD ferrimag-\nnetic states, respectively. In the paramagnetic state, we \fnd a total of 40 Weyl nodes whose\nlocations within the \frst 3D BZ are shown in Fig. 1e-1f. These Weyl points are mainly\ndistributed in the k z\u00180\u0019=c(Fig. 1g) and k z\u0018\u00060.67\u0019=c(Fig. 1h) planes. In the UUD\nferrimagnetic state, the BZ is reconstructed relative to the paramagnetic state, resulting in\nthe folding of the electronic structure and Weyl points. Fig. 1i shows the distributions of all\nWeyl fermions in the ferrimagnetic state, to facilitate direct comparison with Weyl fermions\nin the paramagnetic state, we fold the Weyl fermions back into the \frst BZ in the paramag-\nnetic state (gray red and blue dots in Fig. 1i). It is found that the folded back Weyl nodes\nare mainly distributed in the k z\u00180\u0019=c(Fig. 1j), k z\u0018\u00060.33\u0019=c(Fig. 1k) and k z\u0018\u00060.67\n\u0019=c(Fig. 1l) planes of the \frst BZ of the paramagnetic state. Fig. 1m and 1n show the DFT\nband structure calculations along \u0000-X-Z 2-M-\u0000-Z directions in the paramagnetic (Fig. 1m)\nand ferrimagnetic (Fig. 1n) states. This elucidates the in\ruence of the UUD ferrimagnetic\norder on the electronic structure and topological properties of NdAlSi.\nAccording to the above DFT calculations, the UUD ferrimagnetic order not only regulates\nthe position of the existing Weyl nodes, but also generates new Weyl nodes in the 3D BZ.\nARPES measurements provide a tool to con\frm or refute this prediction. In order to do\nthis, we \frst study the electronic structure and topological properties of the paramagnetic\nstate in detail, and then observe the in\ruence of the UUD ferrimagnetic order induced by\nlowering the temperature.\nFigure 2 displays the overall constant energy contours and band structures of NdAlSi\nmeasured by ARPES in the paramagnetic state. The evolution of constant energy contours\nof the electronic bands at di\u000berent binding energies (Fig. 2a) exhibit sophisticated structures.\nA simple comparison with the DFT calculations of the bulk Fermi surface (Fig. S2 in the\nSupplemental Material) shows that the ARPES measurements (Fig. 2b) display a more\ncomplex structure, which may be related to the presence of surface states. In order to\ndetermine the contribution from surface states, we performed surface projected DFT band\ncalculations on six di\u000berent possible (001) termination surfaces of NdAlSi (for the detailed\nband structure calculations see Fig. S3 in the Supplemental Material). The results show\nthat the band structure calculations along the Y-\u0000-Y(Fig. 2g) and M-\u0000-M(Fig. 2h)\n4directions on the Nd atom terminated surface of the Nd-Al cleavage plane can capture most\nof the measured band features in the corresponding direction (Fig. 2e and Fig. 2f) except\nthe M shaped band at 0.8 eV binding energy and some very small details in Fig. 2e. When\nthe two orthogonal domain structures leading to superposition of bands along the X-\u0000-X\nandY-\u0000-Ydirections are taken into account (Fig. 2i and 2j), all the band features can be\nunderstood. The surface projected DFT Fermi surface calculations on the Nd terminated\nsurface of the Nd-Al cleavage plane is shown in Fig. 2c and agrees well with the measured\nconstant energy contours (Fig. 2b) and when the two orthogonal domain structures are\ntaken into account (Fig. 2d) it com\frms that the measured Fermi surface (Fig. 2b) is\nmainly from the surface states (for detailed analysis see the Supplemental Material).\nWe now proceed to explore Fermi arc candidates. By surface Green function calculations,\nwe \fnd two Fermi arc surface states connecting two pairs of Weyl nodes at the energy of 38\nmeV (Fig. 2k) and 56 meV (Fig. 2l) above the Fermi level (for details see the Supplemental\nMaterial). Similar arc features can also be found in the Fermi surface mapping in the\nsame position, as shown in Fig. 2b. To con\frm that the observed arc features in Fig. 2b\nare the Fermi arc surface states. We carried out photon energy dependent band structure\nmeasurements (Fig. 2m-2p) at 15 K along the direction of Cut3 ( ky=0.34 \u0017A\u00001) in Fig.\n2b. For further quantitative analysis, the extracted photon energy dependent momentum\ndistribution curves (MDCs) at a binding energy of 10 meV from Fig. 2m-2p are plotted in\nFig. 2q. The MDCs exhibit negligible photon energy dependence, suggesting that it is a\nsurface state. To further con\frm that the arc features observed in Fig. 2b are indeed the\nWeyl Fermi arcs, we examine the signatures of chiral charge in NdAlSi based on the bulk-\nboundary correspondence between the bulk Weyl fermions and surface Fermi arcs[1, 67]. In\norder to do so, band dispersion was measured along the straight Cut3 in Fig. 2b. Along\nthis cut (Fig. 2m-2p), a left-moving (Chern number n l=-1) and right-moving (Chern numbe\nnr=+1) edge mode related by a mirror plane kx=0 is clearly observed and associated with\na 2D momentum-space slice carrying a total Chern number n tot=0. In addition, we also\nexamine the signatures of chiral charge in the close loop cuts (Fig. 2b). For a closed loop\nin the surface BZ where the bulk band structure is everywhere gaped we add up the signs\nof the Fermi velocities of all surface states along this loop, with Chern number n=+1 for\nright movers and n=-1 for left movers[67]. Fig. 2r and 2s show the unrolling of the closed\nloop cuts along Loop1 (Fig. 2r, L1 in Fig. 2b) and Loop2 (Fig. 2s, L2 in Fig. 2b). Loop\n5L1 shows a left-moving chiral mode while a right-moving chiral mode is observed for loop\nL2. These results unambiguously show that the arc features in Fig. 2b are the topological\nFermi arcs.\nAfter having identi\fed the topological Fermi arc, we next search for the characteristic\nbulk Weyl fermion dispersion. Looking at the results from DFT calculations, we \fnd a total\nof 40 Weyl nodes in the paramagnetic state, with 24 of them (W2, W3 and W4) distributed\nin the k z\u00180\u0019=cplane (Fig. 3c) and the others (W1) are situated in the k z\u0018\u00060.67\u0019=c\nplane (Fig. 3b). In order to precisely identify the k zmomentum locations of the Weyl\nnodes, we performed broad-range (30 to 90 eV) photon energy dependent ARPES measure-\nments on a cleaved sample with a poor surface so that the surface states are completely\nsuppressed (for details see Supplemental Material). Due to the absence of interference from\nsurface states, bulk bands can be clearly distinguished and characterized. These results were\nfurther con\frmed using bulk sensitive soft X-ray ARPES (SX-ARPES) measurements (for\ndetails see Supplemental Material). Fig. 3a shows the photon energy dependent ARPES\nspectral intensity map (k x-kzFermi surface) at a binding energy of 0.2 eV along the X-\u0000-X\ndirection, measured at 30 K. By analyzing the periodic structure along the k zdirection, the\ncorrespondence between the high symmetry points of the BZ along the k zdirection and the\nphoton energy is determined as shown in Fig. 3a. Based on this, the k z\u0018\u00060.67\u0019=cplane\nhosting W1 type of Weyl fermions corresponds to a photon energy of \u001834 eV and the k z\u00180\n\u0019=cplane hosting W2, W3 and W4 types of Weyl fermion corresponds to a photon energy\nof\u001841 eV.\nFigure 3d and 3h show the Fermi surface mapping of NdAlSi measured at 15 K with\nthe photon energy of 34 eV (Fig. 3d) and 41 eV (Fig. 3h) under Positive Circular (PC)\npolarization on a high quality cleaved surface. The experimental data and the calculated bulk\nFermi surface (Fig. 3e and 3l) are distinctively di\u000berent due to the admixture of the surface\nstates. The Fermi surface mapping of the low quality surface (Fig. 3p), on the other hand,\nshows good overall agreement with the bulk calculations (Fig. 3l) due to the suppression\nof the surface states. In order to expose the Weyl fermion dispersion, we conducted band\ndispersion measurements along 4 cuts (Fig. 3b-c). The corresponding dispersion data is\ndisplayed for cuts 1-4 in Fig. 3f,i-k for the high quality surface and for cuts 2-4 in Fig.\n3q-s for the low quality surface. The corresponding surface projected DFT calculations are\nplotted in Fig. 3g and 3m-3o for comparison with the corresponding DFT bulk calculations\n6overlayed. It is observed that the measured dispersions on the low quality surface (Fig. 3q-\n3s) show good overall agreement with DFT bulk calculations and the measured dispersions\non the high quality surface (Fig. 3f and 3i-3k) are also consistent with the surface projected\nDFT calculations (Fig. 3g and 3m-3o). The observation of the topological Fermi arc and\nbulk Weyl fermions with linear dispersions, as well as the consistency between theoretical\ncalculations and experimental observations, establishes NdAlSi as a Weyl semimetal.\nWith the topological properties of the paramagnetic state well established, we turn to\nthe question of how these topological properties are regulated by the ferrimagnetism. The\nresults from the DFT bulk calculations show that the distribution of Weyl fermions in the\nUUD ferrimagnetic state (Fig. 1i) represent a folding into 1/3 of the paramagnetic state\nalong the in-plane diagonal direction. Based on the previous analysis, if the Weyl fermions\nare folded back into the \frst BZ of the paramagnetic state, the back folded Weyl nodes are\nmainly distributed in the k z\u00180\u0019=c(Fig. 1j), k z\u0018\u00060.33\u0019=c(Fig. 1k) and k z\u0018\u00060.67\u0019=c\n(Fig. 1l) planes of the \frst BZ of the paramagnetic state. By comparing the distribution of\nWeyl fermions to the paramagnetic state, it can be found that the folding of the BZ caused\nby ferrimagnetism generates 4 pairs of W2, 2 pairs of W3 and 2 pairs of W4 type Weyl\nfermions in the k z\u00180\u0019=cplane (Fig. 1j, Fig. 4a) as well as 4 pairs of W1 type of Weyl\nfermions in the k z\u0018\u00060.67\u0019=cplane (Fig. 1l) of the \frst BZ of the UUD ferrimagnetic\nstate. Furthermore, 4 pairs of W1 type of Weyl fermions are generated in the k z\u0018\u00060.33\n\u0019=cplane (Fig. 1k) of the \frst BZ of the paramagnetic state. Therefore, we can use the\ngeneration of the Weyl fermions mentioned above as a criterion to judge the presence of\nWeyl fermions regulated by the UUD ferrimagnetic order.\nFigure 4b and 4c show the constant energy contours at the binding energy of 70 meV\nmeasured with a photon energy of 41 eV at 15 K (Fig. 4b) and 2K (T <T com\u00183.4 K) (Fig.\n4c). As the temperature is lowered below Tcom, a new feature (marked by a green arrow\nin Fig. 4c) can be observed. This feature is not present at 15 K (Fig. 4b). In terms of\nshape, it is mirror-symmetric to the features marked by the red arrow and can be related to\nthe folding of the electronic structure caused by the UUD ferrimagnetic order (green lines\nin Fig. 4f). To further con\frm this conjecture, we conducted temperature dependent band\ndispersion measurements (Fig. 4e-4h) along Cut1 in Fig. 4a, and compared them to the\ncorresponding surface projected DFT band calculations in the paramagnetic state (Fig.4d).\nAll the band features measured at 15 K (Fig. 4e) are captured by the band calculations in\n7the paramagnetic state (Fig. 4d). However, the features marked by a green arrow in Fig.\n2h can not be found in Fig. 4d. It appears to be a duplicate of the band marked by the red\narrow in Fig. 2h (for detailed MDCs analysis see Fig. S10 in the Supplemental Material).\nThe noteworthy feature is that they are mirror-symmetric exactly around the boundary of\nthe BZ of the ferrimagnetic state and break at 15 K (for more comparison of band dispersion\ncuts see Fig. S11 in the Supplemental Material). Therefore, we infer that the new features\ngenerated at 2 K are the folded electronic structures caused by ferrimagnetism (green lines\nin Fig. 4f). It also implies that new Weyl fermions are generated due to the regulation of\nthe UUD ferrimagnetism. A cartoon picture of this process is shown in Fig. 4i.\nThe present observations establish NdAlSi as a ferrimagnetic Weyl semimetal and pro-\nvide strong evidence for the interaction between ferrimagnetism and Weyl fermions. This\nhelps us understand why there is an incommensurate ferrimagnetic order mediated by Weyl\nfermions between 3.4 K and 7.3 K[57]. In the ferrimagnetic state, ferrimagnetism not only\ndrives Weyl fermion shifting in the 3D BZ due to the breaking of TRS, but also generate\nnew Weyl fermions. This, in turn, results in an increase of the density of states near the\nFermi level, leading to higher conductivity. The expected increase in conductivity is consis-\ntent with transport measurements[57, 58]. In addition, new Weyl fermions generated in the\nferrimagnetic state can also impact the thermal conductivity of ferrimagnetic Weyl semimet-\nals. The increasing number of the Weyl nodes and Fermi arcs in the 3D BZ maybe lead\nto a reduction in the thermal conductivity due to the increased scattering of heat-carrying\nphonons. This reduction in thermal conductivity can result in improved thermoelectric per-\nformance, making ferrimagnetic Weyl semimetals potentially useful for energy conversion\napplications. Future heat transport measurements are needed to con\frm this.\nIn summary, ARPES measurements combined with band structure calculations paint a\nclear physical picture of the ferrimagnetic regulation of Weyl fermions. This interplay is\nkey to understanding the electrical and thermal transport properties of ferrimagnetic Weyl\nsemimetals and could turn to be important for the development of new technologies based\non these materials. Our study provides the \frst spectral evidence for the interplay between\nmagnetism and Weyl fermions in condensed matter physics and opens up a new avenue to\nstudy the magnetic regulation of topological ordering, a topic of great interest in topological\nphysics.\n8Methods\nSample Single crystals of NdAlSi were grown from Al as \rux. Nd, Al, Si elements were\nsealed in an alumina crucible with the molar ratio of 1: 10: 1. The crucible was \fnally\nsealed in a highly evacuated quartz tube. The tube was heated up to 1273 K, maintained\nfor 12 hours and then cooled down to 973 K at a rate of 3 K per hour. Single crystals were\nseparated from the \rux by centrifuging. The Al \rux attached to the single crystals were\nremoved by dilute NaOH solution.\nARPES Measurements High-resolution ARPES measurements were performed at the\nbeamline UE112 PGM-2b-13at BESSY II (Berlin Electron Storage Ring Society for Syn-\nchrotron Radiation) synchrotron, equipped with a SCIENTA R8000+DA30L electron ana-\nlyzer and at the CASSIOPEE beamline of the SOLEIL synchrotron, equipped with a SCI-\nENTA R4000 electron analyzer. The total energy resolution (analyzer and beamline) was\nset to 5\u001820 meV for the measurements. The angular resolution of the analyser was \u00180.1 de-\ngree. The samples were cleaved in situ and measured at di\u000berent temperatures in ultrahigh\nvacuum with a base pressure better than 1.0 \u000210\u000010mbar. Soft X-ray (SX) ARPES mea-\nsurements were performed at the VERITAS beamline of the MaxIV synchrotron equipped\nwith a R4000 electron analyzer.\nDFT calculations The electronic structure calculations for NdAlSi were performed based\non the density functional theory (DFT)[68, 69] as implemented in the VASP package[70,\n71]. The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE)\ntype[72] was chosen for the exchange-correlation functional. The projector augmented wave\n(PAW) method[73, 74] was adopted to describe the interactions between valence electrons\nand nuclei. In the calculation of the high-temperature paramagnetic phase of NdAlSi, the\nNd pseudopotential was chosen without the 4 felectrons. The kinetic energy cuto\u000b of the\nplane-wave basis was set to be 350 eV. A 16 \u000216\u000216 Monkhorst-Pack grids[75] was used\nfor the BZ sampling. For describing the Fermi-Dirac distribution function, a Gaussian\nsmearing of 0.05 eV was used. For structure optimization, both lattice parameters and\ninternal atomic positions were fully relaxed until the forces on all atoms were smaller than\n0.01 eV/ \u0017A. The relaxed lattice constants are: a0= 4.22 \u0017A and c0= 14.59 \u0017A, which agree well\nwith the experimental measurements[58]. The spin-orbit coupling e\u000bect was also included for\nstudying the non-trivial band topological. The positions and chirality of the Weyl points were\nstudied by using WannierTools package[76], which interfaces the Wannier90 package[77]. For\n9further studying the surface states of NdAlSi with di\u000berent terminal surfaces, we employed\na 36 atomic layers slab system with 20 \u0017A vacuum. The orbital weights of the top three\natomic layers were calculated as the surface states with respect to the corresponding terminal\nsurfaces.\nAs for the calculation of the low-temperature ferrimagnetic phase, we adopted the pseu-\ndopotential of Nd including the 4 felectrons. The kinetic energy cuto\u000b of the plane-wave\nbasis was set to be 450 eV. A ferrimagnetic supercell was constructed the same with Ref.[57],\nand a 6\u00026\u00026 Monkhorst-Pack grids was used for its folded BZ sampling. The electronic\ncorrelation e\u000bect among Nd 4 felectrons was incorporated by using the GGA+U frame-\nwork of Dudarev formalism[78]. The e\u000bective U value on Nd 4 felectrons was set to be 6\neV, which is the same with the previous study[57]. For checking the in\ruence of the FIM\nstate on the Weyl point distribution around the Fermi level, we performed band unfolding\ncalculations by using the VASPKIT package[79].\n[1] X. G. Wan et al., Topological semimetal and Fermi-arc surface states in the electronic structure\nof pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).\n[2] G. Xu et al., Chern Semimetal and the Quantized Anomalous Hall E\u000bect in HgCr 2Se4. Phys.\nRev. Lett. 107, 186806 (2011).\n[3] Z. J. 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Y.G.S. acknowledges the National Natural Science Foundation of China\n(Grants No. U2032204), and the Informatization Plan of Chinese Academy of Sciences\n(CAS-WX2021SF-0102).\nAuthor Contributions\nC.L. proposed and conceived the project. C.L. carried out the ARPES experiments with the\nassistance from Y.W., Q.D.G., W.Y.C. and D.P.. J.F.Z., H.C.Y. and T.X. contributed to\nthe band structure calculations and theoretical discussion. H.X.L. and Y.G.S. contributed\nto NdAlSi crystal growth. C.L. characterized the single crystals. C.L. contributed to soft-\nware development for data analysis and analyzed the data with the assistance from Y.W..\nC.L. wrote the paper. E.R., B.F., H.F. and B.T. provided the beamline support. M.D.,\nM.H.B. and O.T. contributed to the scienti\fc discussions. All authors participated in and\ncommented on the paper.\n15FIG. 1. Crystal structure and band structure calculations for NdAlSi. (a) The crystal structure\nof NdAlSi with the space group I41md(no. 109). (b) The 3D BZ of the original unit cell of NdAlSi, and the\ncorresponding two-dimensional BZ projected on the (001) plane (green lines) in the pristine phase in (a).\n(c) A three dimensional lattice view of the NdAlSi spin structure. (d) The corresponding 3D BZ of NdAlSi\nwith the UUD ferrimagnetic order which exhibit the BZ folding relative to the paramagnetic state. (e) The\ndistribution of Weyl fermions in the 3D BZ in the paramagnetic state, while the red dots representing nodes\nwith chiralities +1 and blue dots representing -1. (f) The top view of (e). (g-h) The top view of (e) at the\nkz\u00180\u0019=cplane (g) and\u0018\u00060.67\u0019=cplane (h). (i) Distributions of all Weyl fermions from the top view of\nthe BZ in ferrimagnetic state. The gray lines show the boundary of the BZ in the ferrimagnetic state. The\nbright blue and red dots represent the Weyl fermions in the folded BZ due to the ferrimagnetic order. The\nshaded blue and red dots are the Weyl fermions in the ferrimagnetic state after folding back into the \frst BZ\nof the paramagnetic state (Weyl nodes folded back that are distributed in the \u0018\u0006\u0019=cplane are not shown\nhere). (j-l) The top view of (i) at the k z\u00180\u0019=cplane (j),\u0018\u00060.33\u0019=c(k) and\u0018\u00060.67\u0019=cplane (l). (m-n)\nCalculated band structures of NdAlSi along high-symmetry directions across the BZ in paramagnetic (m)\nand UUD type ferrimagnetic (n) state. The high symmetry points are de\fned in (b), while Z 2is the Z point\nin the second BZ in the k z=0\u0019=cplane. The blue lines are the bands corresponding to the paramagnetic\nstate, and the red lines are the folded bands due to the UUD type ferrimagnetic order.\n16FIG. 2. Determination of the surface termination and observation of topological Fermi arcs\nand chiral charges in NdAlSi. (a) Stacking plots of constant energy contours at di\u000berent binding energies\n(Eb), obtained from ARPES, display complex band structure evolution as a function of energy. (b) Fermi\nsurface and constant energy contours at di\u000berent binding energies measured with a photon energy of 41\neV at 15 K under positive circular (PC) polarization. (c) Surface projected DFT calculations of the (001)\nFermi surface and constant energy contours on the Nd atom terminated surface. (d) Surface projected DFT\ncalculations for two domain structures derived from (c) and their superposition after rotation of 90 degrees.\n(e-f) Band dispersions along Y-\u0000-Y[Cut1, (e)] and M-\u0000-M[Cut2, (f)] directions. The momentum directions\nof the cuts are show in the leftmost panel of (b). (g-h) The corresponding surface projected DFT calculations\nof the bands along Y-\u0000-Y(g) andM-\u0000-M(h) directions. (i-j) The surface projected DFT calculations of\nthe bands along Y-\u0000-Y(i) andM-\u0000-M(j) directions, considering the two domain structures. The spectral\nintensity is illustrated by the size and transparency of the markers. (k-l) Surface Green function calculation\nof constant energy contours, taking into account the double domain structures, at an energy of 38 meV (k)\nand 56 meV (l) above the Fermi level. (m-p) Photon energy dependent band dispersions along Cut3 in (b).\n(q) The momentum distribution curves (MDCs) extracted from (m-p) at a binding energy of 10 meV. The\nred and green diamond shaped markers indicate the peaks positions corresponding to two di\u000berent Fermi\narcs. (r-s) Band dispersions along Loop 1 [L1, (r)] and Loop 2 [L2, (s)].\n17FIG. 3. Weyl fermions in NdAlSi. (a) Photon energy dependent ARPES spectral intensity map at\nthe binding energy of 0.2 eV along the X-\u0000-Xdirection measured at 30 K. For the sake of comparison, the\nspectral intensities are normalized. (b-c) The distribution of Weyl fermions in the BZ at the k z\u00180.67\u0019=c\nplane (b) and\u00180\u0019=cplane (c) in which the paths of Cut1 to Cut4 and the types of Weyl fermion W1,\nW2, W3 and W4 are de\fned. (d) The Fermi surface map measured at 15 K with photon energy of 34 eV\nunder PC polarization. (e) Calculated bulk Fermi surface at the k z\u00180.67\u0019=cplane. The BZ is de\fned\nby orange solid lines. (f) Band dispersion along Cut1 measured with a photon energy of 34 eV at 15 K\nrelated to Weyl fermions W1, with DFT bulk calculations overlaid. (g) Corresponding surface projected\nDFT calculations for (f). The spectral intensity is denoted by the size and transparency of the markers.\n(h) The Fermi surface map measured at 15 K with a photon energy of 41 eV under PC polarization. (i-k)\nBand dispersion measured along Cut2 to Cut4 with a photon energy of 41 eV under PC polarization at 15\nK related to Weyl fermions W2 (i), W3 (j) and W4 (k), with DFT bulk calculations overlaid. (l) Calculated\nbulk Fermi surface at the k z\u00180\u0019=cplane. The BZ is de\fned by orange solid lines. (m-o) Corresponding\nsurface projected DFT calculations for (i-k). (p) The Fermi surface map measured on a low quality surface\nat 30 K with a photon energy of 41 eV under LH polarization. (q-s) Similar measurements as (i-k) but\nmeasured on the low quality surface sample at 30 K with photon energy of 41 eV under LH polarization.\n18FIG. 4. Ferrimagnetic regulation of Weyl fermions in NdAlSi. (a) The distribution of Weyl fermions\nin the BZ at the k z\u00180\u0019=cplane in the ferrimagnetic state, in which the path of Cut1 is de\fned. (b-c) The\nconstant energy contours at the binding energy of 70 meV measured with a photon energy of 41 eV at 15 K\n(b) and 2 K (c). (d) The surface projected DFT calculations of the bands in the paramagnetic state along\nthe path of Cut1 in (a). The spectral intensity is denoted by the size and transparency of the markers. (e-f)\nThe band dispersions (after symmetrization) along the path of Cut1 in (a) measured with photon energy of\n41 eV under PC polarization at 15 K (e) and 2 K (f), with DFT bulk calculations overlaid. (g-h) Zoom in\non the areas in (e-f) enclosed by the pink lines. The blue, green and red dashed lines are guides to the eye.\n(i) Cartoon picture of the regulation of Weyl fermions by ferrimagnetism in NdAlSi in the k z\u00180\u0019=cplane,\ncorresponding to the area enclosed by dashed pink lines in (a).\n19"    },    {        "title": "2402.06258v2.Level_attraction_in_a_quasi_closed_cavity.pdf",        "content": "Level attraction in a quasi-closed cavity\nGuillaume Bourcin,1, 2,∗Alan Gardin,1, 2, 3Jeremy Bourhill,4Vincent Vlaminck,1, 2and Vincent Castel1, 2,†\n1IMT Atlantique, Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France\n2Lab-STICC (UMR 6285), CNRS, Technopole Brest-Iroise, CS 83818, 29238 Brest Cedex 3, France\n3School of Physics, The University of Adelaide, Adelaide SA 5005, Australia\n4Quantum Technologies and Dark Matter Labs, Department of Physics,\nUniversity of Western Australia, 35 Stirling Hwy, 6009 Crawley, Western Australia.\n(Dated: March 29, 2024)\nWe provide a comprehensive analytical description of the effective coupling associated with an\nantiresonance within a hybrid system comprised of a quasi-closed photonic cavity and a ferrimag-\nnetic material. Whilst so-called level attraction between a resonant system inside an open cavity\nis well understood, the physical underpinnings of this phenomena within quasi-closed cavities have\nremained elusive. Leveraging the input-output theory, we successfully differentiate between the\nrepulsive and attractive aspects of this coupling. Our proposed model demonstrates that by under-\nstanding the phase-jump at the resonances and the studied antiresonance, we can predict the nature\nof the effective coupling of the antiresonance for a given position of the ferrimagnet in the cavity.\nI. INTRODUCTION\nCavity spintronics is an expanding field of research\ndedicated to the precise control and exploration of\nthe interactions between light and magnetic materials.\nNotably, the interaction between cavity photons and\nmagnons, the quanta of spin waves, has gained signifi-\ncant prominence since 2010 [1–3]. When a magnon is\ntuned coincident in frequency with a photonic resonance,\nand the coupling rate between the two systems is faster\nthan their individual loss mechanisms, the two separate\nsystems become hybridised into what is referred to as a\ncavity magnon polariton (CMP).\nIn cavity magnonics, two distinct types of resonators\nexist [3]: open cavities, characterized by an antireso-\nnance (dip) in the transmission spectra at their mode\nfrequencies, and quasi-closed cavities, identified by a res-\nonance (peak) in the transmission at their mode frequen-\ncies. While only the level repulsion is possible between\na cavity resonance and a magnon mode, the effective in-\nteraction between a cavity antiresonance and a magnon\nmode can lead to either level repulsion or level attrac-\ntion, which is characterized by a crossing or anticrossing\nin the dispersion spectrum, respectively [3–5]. The ob-\nservation of level attraction in cavity magnonics systems\nwas first reported by Harder et al. (2018) [6]. The phys-\nical processes governing level attraction in open cavities\nhas been explained by employing traveling waves [6–11].\nIn contrast, the attractive character of the coupling\nin a quasi-closed cavity was solely experimentally iden-\ntified by Rao et al. (2019) [12]. To explain this observa-\ntion, a phenomenological model based on RLC circuits\nwas employed. However, this approach did not provide\na clear understanding of the mechanisms underlying the\nemergence of level attraction. A more comprehensive un-\n∗guillaume.bourcin@imt-atlantique.fr\n†vincent.castel@imt-atlantique.frderstanding of the origins of this phenomenon is a valu-\nable insight for the design of cavities in various appli-\ncations, such as magnon-magnon interactions [13–15], or\nnon-reciprocal wave propagation [16–21] (e.g. backscat-\ntering isolation or unidirectional signal amplification).\nIn this paper, we focus on the coupling between an\nantiresonance and a magnon mode by applying input-\noutput theory. This approach provides a deeper under-\nstanding of the parameters that govern the coupling be-\nhavior, whether it is repulsive or attractive. We begin\nby introducing the general S-matrix derived from input-\noutput theory and proceed to explore the two distinct\npathways leading to the occurrence of antiresonance.\nThis analysis provides significant insight into the factors\ncontributing to the antiresonance coupling phenomenon,\nparticularly focusing on the phase-jump, which is the key\nfeature of the effective antiresonance coupling behaviour.\nFinally, the agreement between finite element method\n(FEM) simulations and our proposed model prove the\npossibility to precisely control the coupling behavior\nwhen positioning a ferrimagnetic sphere at different lo-\ncations in a quasi-closed cavity.\nII. PHYSICAL MODEL\nThe model Hamiltonian for a closed cavity, denoted as\nˆHsys, encompasses pinternal bosonic modes, which can\nbe either photons or magnons in the context of this study.\nThe Hamiltonian is expressed as follows:\nˆHsys\nℏ=X\np\n˜ωpˆa†\np(t)ˆap(t) +1\n2X\nq̸=p\u0000\ngqpˆa†\np(t)ˆaq(t) + h.c.\u0001\n.\n(1)\nThe first term represents the unperturbed Hamiltonian of\na single oscillator, where ωp/2πis the eigenfrequency, and\nˆa†\np(ˆap) is the creation (annihilation) operator of mode p.\nThe second term is the interaction Hamiltonian between\ntwo internal modes ˆ apand ˆaq, with their mutual couplingarXiv:2402.06258v2  [quant-ph]  28 Mar 20242\n00\n01\ng0\nˆc0ˆb0ˆb1ˆm/uni0000000b/uni00000044/uni0000000c\n10\n00\n11\n01\nˆc1\nˆc0ˆb0ˆb1/uni0000000b/uni00000045/uni0000000c\n10\n00\n11\n01\ng0g1ˆc1\nˆc0ˆb0ˆb1ˆm/uni0000000b/uni00000046/uni0000000c\nFIG. 1. Schemes of classical oscillators representing a quasi-closed cavity composed of (a) a photonic mode ˆ c0coupled with\nstrength g0to a magnon mode ˆ m; (b) two photonic modes ˆ c0and ˆc1; and (c) two photonic modes ˆ c0and ˆc1coupled with\nstrength g0andg1respectively to a magnon mode ˆ m.ˆb0andˆb1are the ports, and κij=√γijeiϕijthe external couplings.\nassessed by gqp, and h.c. indicating the hermitian con-\njugate. It is worth noting that the fast oscillating terms\n(i.e. ˆa†\npˆa†\nqand ˆapˆaq) are neglected in this approximation,\nwhich is known as the Rotating Wave Approximation\n(RWA) [22].\nIn a quasi-closed cavity, the cavity modes are coupled\nto a common photon bath [23, 24]. For each port n\n(i.e. a probe), an associated photon bath is represented\nby a continuum of photonic oscillators, named external\nmodes, with eigenfrequency ω. The related Hamiltonian\ncan be expressed as follow:\nˆHbath\nℏ=X\nnZ\nRdω ωˆb†\nω,n(t)ˆbω,n(t). (2)\nHere, ˆb†\nω,n(ˆbω,n) represents the creation (annihilation)\noperator of the external mode associated with port n\nand having the frequency ω.\nThe interaction between the bath and the system can\nbe described by the following model Hamiltonian (under\nthe RWA):\nˆHint\nℏ=i√\n2πX\np,nZ\nRdω\u0010\nκpn(ω)ˆb†\nω,n(t)ˆap(t)−h.c.\u0011\n,(3)\nwhere κpn(ω) is the external coupling strength between\nthe external mode ˆbn,ωand the internal mode ˆ ap.\nIn the first Markov approximation, the external cou-\npling strength is assumed to be independent of the fre-\nquency:\nκpn(ω) =κpn=√γpneiϕpn, γ pn∈R. (4)\nHere, γpnis real and represents the external photonic\ndamping rate. Additionally, a phase contribution ϕpnis\nintroduced to the external coupling κpn, as previously\ndiscussed in [25, 26]. The coupling phase is contingent\non the phase of the electric (magnetic) field injected or\nprobed within the cavity. At a given time t, the first\nprobe injects a field with a certain phase. Consequently,\neach excited cavity mode shares the same phase at this\nlocation in their field distribution within the cavity. How-\never, the phase of the modes at the second probe maydiffer, according to their field distribution, giving rise to\nboth constructive and destructive interferences. A probe\nsenses the field only along one axis, thereby resulting in\nan external coupling phase of either 0 or π.\nThe complete Hamiltonian is given by:\nˆH=ˆHsys+ˆHbath+ˆHint. (5)\nIt is important to note that, owing to the RWA, this\nHamiltonian is no longer applicable to systems oper-\nating beyond the Strong-Coupling (SC) regime, where\ngqp/√ωqωp<0.1 and κpn(ω)/√ωp<0.1 [27–29]. Both\ninternal and external modes obey to the bosonic relation\n[30]. The derivation of the S-matrix using the input-\noutput theory is given in Appendix A.\nIII. PHYSICS OF AN ANTIRESONANCE\nThe following section will discuss the two contributions\nto the existence of an antiresonance with two internal\nmodes: when both modes are hybridized Cavity-Magnon\nPolaritons (CMP), and when both modes are photonic\nmodes. These scenarios are illustrated in the two system\nschemes in Fig. 1 (a), and (b), respectively. Fig. 1 (c),\na combination of Fig. 1 (a) and (b), will be discussed in\nSec. IV.\nA. One photon mode & one magnon mode\nThe depicted system in Fig. 1 (a) consists of a sin-\ngle photonic mode ˆ c0with a frequency of ω0, interacting\nwith two ports (i.e. two probes) denoted as ˆb0andˆb1.\nThe coupling strengths for these interactions are charac-\nterized by κ00andκ01, respectively. Additionally, the\nmagnon ˆ mwith a frequency of ωmis only coupled to the\nphotonic mode with a coupling strength of g0. Accord-\ning to eq. (A12) and eq. (A14), the transmission of this\nsystem is expressed as follows:\nS21=−i√γ00γ01∆meiΦ0\n˜∆0∆m−g2\n0, (6)3\nwhere ∆ m=ω−ωm,˜∆0=ω−˜ω0, ˜ω0=ω0−i\n2(γ00+γ01),\nand Φ 0=ϕ01−ϕ00.\nMinimizing the denominator in eq. (6) gives rise to the\npolaritonic frequencies:\nω±=1\n2\u0014\n˜ω0+ωm±q\n(ωm−˜ω0)2+ 4g2\n0\u0015\n.(7)\nMinimizing the numerator of eq. (6) leads to the deter-\nmination of the antiresonance frequency ωararising from\nthe interaction between a photon and a magnon:\nωar=ωm. (8)\nFor this case, regardless of the coupling phase between\nthe cavity photon and the ports, the antiresonance fre-\nquency will always coincide with the magnon frequency,\nwhich is the externally non-excited oscillator, as high-\nlighted in the transmission spectra of the system depicted\nin Fig. 2.\nNote that eq. (6) can be expressed as a sum of\nLorentzians:\nS21=−i√γ00γ01\nω+−ω−\u0012ωm−ω−\nω−ω−+ω+−ωm\nω−ω+\u0013\neiΦ0.(9)\nThe terms within the parentheses correspond to the res-\nonances of the lower and upper polaritons. Numerators\nand the shared factor term (except for eiΦ0) are all posi-\ntive. As a result, the phase of both resonances shift from\nπ/2 to−π/2 when Φ 0= 0 (referred to as negative phase-\njump in all subsequent discussions), and from −π/2 to\nπ/2 when Φ 0=π(positive phase-jump). The common\nphase factor eiΦ0for both polaritons indicates that these\nresonances undergo the same phase-jump.\nω0ωm/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040ω0ω/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040\n|S21|/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni00000050/uni00000044/uni0000005b\n/uni00000050/uni0000004c/uni00000051\nFIG. 2. Transmission spectrum of a system composed of a\ncavity photon and a magnon, sketched in Fig. 1 (a), using\neq. (6), and the polariton frequency dependence in dashed\nwhite line using eq. (7).To emphasize the phase-jump of the antiresonance, the\ntransmission can be rewritten in a different manner:\nS21=−i√γ00γ01∆meiΦ0\n(ω−ω−)(ω−ω+). (10)\nIn a frequency range between the two polariton frequen-\ncies, the given equation characterizes the antiresonance.\nIn this frequency range, all terms maintain the same sign\nexcept for ∆ m, which influences the phase-jump. To en-\nsure a meaningful comparison of the phase in eq. (9), it\nis crucial that all terms remain positive (excluding ∆ m\nandeiΦ0); otherwise, the phase-jump would be affected.\nHowever, ω−ω+is negative around the antiresonance\nfrequency. Therefore, to render all terms positive in the\nexpression of the transmission, eq. (10) becomes:\nS21=−i√γ00γ01∆meiΦar\n(ω−ω−)(ω+−ω), (11)\nwhere Φ ar= Φ 0+πrepresents the phase factor of the an-\ntiresonance, and its phase factor is π-dephased compared\nto the phase factors of the polaritons, meaning that the\nphase-jump is positive when Φ 0= 0, and negative when\nΦ =π. Further discussion of this phase signature will be\npresented in the following sections.\nB. Two photon modes\nThe system illustrated in Fig. 1 (b) comprises two pho-\ntonic modes ˆ c0and ˆc1with frequencies ω0andω1, respec-\ntively. These two photon modes interact with two ports,\nˆb0andˆb1, characterized by external coupling strengths\ndenoted as κ00andκ01with respect to mode ˆ c0, and κ10\nandκ11with respect to mode ˆ c1. From eq. (A12) and\neq. (A14), the transmission of such a system reads:\nS21=−i√γ00γ01˜∆1eiΦ0+√γ10γ11˜∆0eiΦ1+i\n2Γ1\n˜∆0˜∆1+|Γ|\n4,\n(12)\nwhere ˜∆1=ω−˜ω1, ˜ω1=ω1−i\n2(γ10+γ11), Φ 1=\nϕ11−ϕ10, Γ =√γ00γ10ei(ϕ10−ϕ00)+√γ01γ11ei(ϕ11−ϕ01),\nand Γ 1=√γ00γ11Γei(ϕ11−ϕ00)−√γ01γ10Γ∗ei(ϕ10−ϕ01).\nThe resonance frequencies of the transmission reads:\nω±=1\n2h\n˜ω0+ ˜ω1±p\n(˜ω1−˜ω0)2− |Γ|i\n. (13)\nIn the context of a cavity exhibiting non-degenerate\nmodes, the following hypothesis applies:\n|ω1−ω0| ≫ |Γ|. (14)\nThis inequality reflects the condition that the frequency\ndifference between the two modes is significantly larger4\nthan the magnitude of Γ, interpreted as a crosstalk be-\ntween the two photonic modes. In this approximation,\nthe resonance frequencies are equal to the photon mode\nfrequencies:\nω−= ˜ω0, ω += ˜ω1. (15)\nThe antiresonance frequency of such a system is given\nby:\nωar=ω1+δeiΦω0\n1 +δeiΦ, (16)\nwhere δ=p\nγ10γ11/γ00γ01represents the external dis-\nsipation ratio between the mode ˆ c1and ˆc0at the two\nprobes, and Φ = Φ 1−Φ0=ϕ00+ϕ11−ϕ01−ϕ10rep-\nresents the phase factor difference between the two pho-\ntonic modes.\nWe highlight three distinct antiresonance frequency\nregimes based on the values of δand Φ, summarized in\ntable I, and discussed just below.\nConsidering the hypothesis given in eq. (14), the trans-\nmission from eq. (12) can also be expressed as a sum of\nLorentzians:\nS21=−i p\nγ∗\n00γ01eiΦ0−iΓ2\nω−˜ω0+p\nγ∗\n10γ11eiΦ1−iΓ2\nω−˜ω1!\n,\n(17)\nwhere Γ 2=Γ∗\n˜ω1−˜ω0Γ1, is negligible far from the an-\ntiresonance frequency. By neglecting Γ 2, the two terms\nrepresent isolated photon modes.\nWhen Φ = 0, the antiresonance frequency ωaris sit-\nuated between the two photon mode frequencies ω0and\nω1, as depicted in Fig. 3 (a). This corresponds to the\nscenario where the two Lorentzians in eq. (17) share the\nsame phase factor, i.e. Φ 0= Φ 1, therefore the same\nphase-jump. Thereby, the two Lorentzian are π-dephased\nand destruct themselves for frequencies between ω0and\nω1. This aligns with the case discussed in Sec. III A in\neq. (9); when two eigenmodes share the phase factor,\nthe antiresonance frequency lies between their respective\neigenfrequencies.\nWhen Φ = π, the antiresonance frequency ωarlies out-\nside the two photon mode frequencies ω0andω1. This\ncorresponds to the scenario where the phase factor of\nthe two Lorentzians in eq. (17) are π-dephased, therefore\nopposed phase-jumps. Two cases arise: when δ >1, indi-\ncating that the photon mode ˆ c1is more coupled with the\nprobes than the photon mode ˆ c0, and the antiresonance\nfrequency ωaris lower than ω0, as illustrated in Fig. 3\nTABLE I. Antiresonance behavior\nΦ = 0 Φ = π\nδ >1ωar∈[ω0;ω1]ωar∈[0;ω0]\nδ <1 ωar∈[ω1; +∞[(b); when δ <1, indicating that the photon mode ˆ c0is\nmore coupled with the probes than the photon mode ˆ c1,\nandωaris higher than ω1, as shown in Fig. 3 (c).\nAs in the previous section, the transmission can be\nrewritten to emphasize the phase of the antiresonance:\nS21=−i|1 +δ|(ω−ωar)eiΦar+i\n2Γ3\n(ω−ω−)(ω−ω+), (18)\nwhere Γ 3=√γ00γ01(γ10+γ11)+√γ10γ11(γ00+γ01)+Γ 1,\nis negligible far from the antiresonance frequency, and\nΦar= arg(1 + δeiΦ).\nWhen Φ = 0, the term ω−ω+becomes negative be-\ncause the antiresonance frequency is situated between the\ntwo photon frequencies, and Φ ar→Φar+π. Similar to\nthe case of a system composed of one photon and one\nmagnon, as discussed in Sec. III A, Φ ar=π. This in-\ndicates an opposing phase-jump compared to the phase-\njumps of the two resonances.\nIn the case where Φ = Φ 1=π(hence Φ 0= 0), and\nδ >1, the antiresonance has the same phase-jump as the\nsecond photon mode with phase factor Φ ar= Φ 1=π.\nConversely, when δ <1, the antiresonance has the same\nphase-jump as the first photon mode with phase factor\nΦar= Φ 0= 0. Understanding the phase-jump of the\nresonances and antiresonance is highly valuable in cavity\nengineering, as demonstrated in the following sections.\nTo conclude on the occurrence of antiresonances, we\nhave explored two simplified scenarios involving only two\ninternal modes. In a real cavity containing an infinity of\nmodes, obtaining an analytical expression for the antires-\nonance frequency becomes a complex task. Nevertheless,\nit is feasible to obtain a reasonable approximation of the\nantiresonance frequency in a real cavity by considering\nthe nearest and most attractive modes numerically, as\nwill be shown later.\n/uni0000000b/uni00000044/uni0000000c Φ=0|S21|/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni0000000b/uni00000045/uni0000000c Φ=π\nδ>1\nω0ω1ω/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni0000000b/uni00000046/uni0000000c Φ=π\nδ<1\nFIG. 3. Three cases of the frequency dependence of the an-\ntiresonance of the system composed of two cavity photons,\nsketched in Fig. 1 (b). δand Φ are defined in eq. (16).5\nω0ω1ωm/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040ω0ω1ω/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni0000000b/uni00000044/uni0000000cg1=0\nω0ω1ωm/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni0000000b/uni00000045/uni0000000cg0=0\nω0ω1ωm/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni0000000b/uni00000046/uni0000000cg1=0\nω0ω1ωm/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni0000000b/uni00000047/uni0000000cg0=0\n|S21|/uni00000003/uni0000003e/uni00000044/uni00000011/uni00000058/uni00000011/uni00000040/uni00000050/uni00000044/uni0000005b\n/uni00000050/uni0000004c/uni00000051δ>1 δ<1\nFIG. 4. Transmission spectra of the system sketched in Fig. 1 (c) considering the 4 different cases related in table II and using\neq. (19) showing either an effective level repulsion or level attraction between the antiresonances, where δand Φ are defined in\neq. (16), and the effective antiresonance frequency dependences in dashed white lines using eq. (21). Φ = πin all four cases.\nIV. COUPLING BEHAVIOR\nHere, we clarify all the contributions and conditions\nrequired to observe either antiresonance level repulsionor level attraction between two cavity photon modes and\none magnon mode. The system is illustrated in Fig. 1\n(c). From eq. (A12) and eq. (A14), the transmission of\nsuch a system reads:\nS21=−i√γ00γ01(˜∆1∆m−g2\n1)eiΦ0+√γ10γ11(˜∆0∆m−g2\n0)eiΦ1+ Γ4g0g1−i\n2Γ2∆m\n˜∆0˜∆1∆m−g2\n0˜∆1−g2\n1˜∆0+|Γ|\n4∆m+i\n2(Γ + Γ∗)g0g1, (19)\nwhere Γ 4=√γ00γ11ei(ϕ11−ϕ00)−√γ01γ10ei(ϕ10−ϕ01).\nUnder the assumption outlined in eq. (14), the system\nfeatures three resonance frequencies given by:\nΩ−=ω−\n0, Ω0=ω+\n0+ω−\n1−ωm, Ω+=ω+\n1,(20)\nwhere ω±\n0=ω±(˜ω0), and ω±\n1=ω±(˜ω1), from eq. (7).\nThe minima in transmission of the system are determined\nby the following antiresonance frequencies:\nω±\nar=1\n2\u0014\nωar+ωm±q\n(ωar−ωm)2+ 4|gar|2eiΦar\u0015\n,\n(21)\nwhere ωaris defined in eq. (16). The effective coupling\nstrength between the two antiresonances reads as:\ngar=s\ng2\n1+δeiΦg2\n0+Cg0g1\n1 +δeiΦ, (22)\nwhere C=p\nγ11/γ01ei(ϕ11−ϕ01)+p\nγ10/γ00ei(ϕ00−ϕ10),\ng0andg1are the coupling strengths of each cavity mode\nto the magnon, and δand Φ are the same as in eq. (16).\nWhen Φ = π,garcan be either real or imaginary, de-\npending on the values of δ,g0,g1, and C. In eq. (21), we\nchose to explicitly introduce the effective coupling phase\nΦar, justifying the absolute value of the effective couplingstrength. This choice implies that Φ aris equal to 0 ( π)\nwhen garis real (imaginary), leading to level repulsion\n(attraction).\nDrawing an analogy with the resonance frequencies of a\nsingle photon coupled with a magnon described in eq. (7),\nwe can derive the Hamiltonian that governs the antires-\nonance:\nˆHar\nℏ=ωarˆc†\narˆcar+ωmˆm†ˆm+gar(ˆc†\narˆm+eiΦarˆcarˆm†),(23)\nwhere ˆ c†\nar(ˆcar) represents the effective creation (annihila-\ntion) operator of the cavity antiresonance with an eigen-\nfrequency of ωar. Note that the eigenfrequencies resulting\nfrom the diagonalization of this Hamiltonian correspond\nto the antiresonance frequencies in eq. (21).\nIt is worth noting that while the effective Hamilto-\nnian in eq. (23) and the frequencies of the effective an-\ntiresonances in eq. (21) have been previously introduced\nbased on phenomenological considerations, the effective\ncoupling strength garlacked a clear physical explanation\n[12]. The transmission spectrum of the antiresonance ex-\nhibits a level repulsion when the coupling strength gar\nis real. Conversely, the transmission spectrum shows a\nlevel attraction when garis imaginary.\nIn the case where only one photonic mode is coupled\nwith the magnon mode and the phase-jump of the two6\nmodes are opposed, e.g. their phase factors are Φ = Φ 1=\nπand Φ 0= 0, the spectrum will exhibit either a repulsive\nor an attractive signature, and this is determined by the\nvalue of δ. The various situations based on the values of\ng0,g1, and δ, when Φ = πare illustrated in table II.\nAs mentioned earlier, when δ > 1,ωar≤ω0. In\nthis case, the phase-jump of the lower cavity mode and\nthe cavity antiresonance ˆ carare opposed. As a result,\nthe lower CMP resulting from the hybridization of the\nmagnon and the lower cavity mode exhibits level repul-\nsion with the antiresonance mode, as depicted in Fig. 4\n(a). However, the the upper cavity mode and the cav-\nity antiresonance exhibit the same phase-jump. Conse-\nquently, the lower CMP resulting from the hybridization\nof the magnon and the upper cavity mode exhibits level\nattraction with the antiresonance modes, as shown in\nFig. 4 (b).\nConversely, when δis less than 1, ωar≥ω1. In this sce-\nnario, the phase-jumps ot the upper mode and the cav-\nity antiresonance are opposed, leading to level repulsion\nat the antiresonance coupling, as depicted in Fig. 4 (c).\nHowever, the lower cavity mode and the cavity antireso-\nnance exhibit the same phase-jump, leading to level at-\ntraction at the antiresonance coupling, as shown in Fig. 4\n(d).\nIn summary, when there is an opposed phase-jump be-\ntween a cavity mode and an antiresonance mode, it leads\nto level repulsion in the coupling between the antireso-\nnance and the CMP resulting from the hybridization of\nthe cavity mode and a magnon. Conversely, when there\nis a similar phase-jump between a cavity mode and an\nantiresonance mode, it results in level attraction in the\ncoupling between the antiresonance and the CMP.\nAs mentioned earlier, in contrast to this two-mode cav-\nity, a real cavity is characterized by an infinite amount\nof modes coupled to the same magnon mode. This re-\nsults in a highly complex system, making it challenging\nto derive a straightforward analytic equation with easy\ninterpretability. Nevertheless, the analysis of the phase\nof the resonances and the cavity antiresonance proves to\nbe valuable for engineering cavities and predicting the\ncoupling behavior of the antiresonance.\nTABLE II. Effective coupling behavior\ng1= 0 g0= 0\nδ >1 gar=g0√\n1−δ−1∈R gar=ig1√\nδ−1∈iR\nδ <1 gar=ig0√\nδ−1−1∈iR gar=g1√\n1−δ∈RV. SIMULATION\nA. Model comparison\nThis section, focusing on FEM simulation, aims to put\ninto practice the concepts from the previous section ap-\nplied not only to just two-mode cavity but to a cylindrical\ncavity exhibiting seven modes in the Ku-band coupled to\none magnon mode, represented by a YIG sphere placed\nat different location.\nIt has previously been concluded that proximity to a\nnode of the RF H-field of a cavity antiresonance was\nan essential condition for observing the level attraction\nof an antiresonance [12]. This effect was attributed to\nthe Lenz effect produced by the cavity, generating a mi-\ncrowave current that hinders the dynamics of the mag-\nnetization. It was concluded that the CMP solely repul-\nsively interacted with the antiresonances. Consequently,\nin an antinode of the RF H-field of a cavity antireso-\nnance, the strong repulsive coupling would prevent the\nmanifestation of the Lenz effect. However, CMP can in\nfact exert either repulsive or attractive influence on the\ncoupling with the antiresonance. This behavior depends\non the phase-jumps of both the antiresonance and the\nhybridized cavity mode.\nA plausible initial approach to modeling a cavity with\ninfinite modes involves considering the minimum number\nof modes necessary to achieve the same antiresonance fre-\nquency. Identifying the phase-jump of an antiresonance\nprovides insights into whether a cavity mode, through the\nCMP resulting from its coupling with a magnon, behaves\nrepulsively or attractively with the antiresonance.\nB. Cavity features\nIn this study, a cylindrical cavity described in Rao et al.\n(2019) [12] is utilized, featuring a height of 35 mm and a\nradius of 12.5 mm. Positioned at each side of the cavity\nare two electrical probes, arranged in parallel as displayed\nin Fig. 5 (a). These probes are situated at 10 mm from\nthe cavity’s bottom. A YIG sphere, having a radius of\n0.5 mm, is positioned at two distinct locations: position\nA (x, y, z = 11.9, 0, 0.6 mm from the bottom center of\nthe cavity), inducing level repulsion; and position B (x,\ny, z = 0, 0, 0.6 mm), inducing level attraction.\nIn Fig. 5 (b) and (c), the amplitude and the phase of\nthe transmission are respectively depicted. Seven cav-\nity modes were considered in the input-output model\nto match the same |S21|trace from the Frequency Do-\nmain (FD) simulations. To accomplish this, we solve\nfor the eigenmodes of the system first to extract various\nparameters, including the phases of the E-field located\nat each probes, the quality factor, and the cavity mode\nfrequencies, as detailed in table III. It is worth noting\nthat our assumption involved equal coupling of a cavity\nmode with each probe, resulting in equal external cou-\npling strengths between an internal mode and ports. As7\n/uni0000003c\n/uni0000003b/uni0000003d/uni00000024/uni00000025\n/vectorH0/uni0000000b/uni00000044/uni0000000c\n/uni00000014/uni00000013/uni00000013\n/uni00000018/uni00000013\n/uni00000013|S21|/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040TE211TM012TE212TE113TM111TE013TE311\n/uni0000000b/uni00000045/uni0000000c\n/uni00000014/uni00000015 /uni00000014/uni00000016 /uni00000014/uni00000017 /uni00000014/uni00000018 /uni00000014/uni00000019\nω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040−π−π/2/uni00000013π/2πΦ21/uni00000003/uni0000003e/uni00000055/uni00000044/uni00000047/uni00000040\n/uni0000000b/uni00000046/uni0000000c\n/uni00000014\n /uni00000013 /uni00000014\n(ωIO−ωFD)/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000051/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni00000050/uni00000052/uni00000047/uni00000048/uni00000056\nTM012+TE212TE211TE113TM111TE013TE311\n+++++/uni0000000b/uni00000047/uni0000000c ∆=11/uni00000003/uni00000030/uni0000002b/uni0000005d\n/uni00000029/uni00000028/uni00000030/uni00000003/uni00000036/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051\n/uni0000002c/uni00000051/uni00000053/uni00000058/uni00000057/uni00000010/uni00000032/uni00000058/uni00000057/uni00000053/uni00000058/uni00000057/uni00000003/uni00000030/uni00000052/uni00000047/uni00000048/uni0000004f\nFIG. 5. (a) Scheme of the cylindrical cavity, whith Athe YIG position to observe effective repulsive coupling, and Bthe YIG\nposition to observe effective attractive coupling; (b) |S21|, and (c) Φ 21versus the frequency of the empty cavity, where the\ninput-output model in dashed red line is compared to the FEM simulation in solid blue line. In green areas are the modes\nexhibiting a negative phase-jump (from π/2 to−π/2), and in purple areas are the modes exhibiting a positive phase-jump (from\n−π/2 toπ/2), as observed for the phase-jump of the antiresonance at 13.59 GHz in the same color. (d) Graphe illustrating the\nconvergence of the antiresonance frequency between the FEM simulation frequency ωFDand the input-output model frequency\nωIObased on the number of considered modes. Colored dots (associated to the added mode) hold the same significance as the\ncolored areas in (c).\ndepicted in the inset of Fig. 5 (b), there is a frequency\nshift of ∆ = 11 MHz observed for the antiresonance\naround 13.59 GHz between the FD simulation and the\ninput-output model, corresponding to the frequencies of\n13.589 GHz and 13.600 GHz respectively.\nNote that we need to consider a sufficient number of\nmodes to approach the FD simulation closely as illus-\ntrated in Fig. 5 (d), which shows the dependence of the\nmismatch between FD simulation and input-output as\nfunction of the number of modes considered. The input-\noutput model demonstrates consistent phase, resembling\nthe FEM simulation, except for a constant phase shift\nproportional to the frequency in the FEM simulation,\nas shown in Fig. 5 (c). The absence of this shift in\nthe input-output model has no significant impact on the\nstudy. Furthermore, with the inclusion of more than 7\nmodes, the model fails to converge for the antiresonance\nfrequency and cannot accurately match the frequency ob-\ntained from FEM simulations. This phenomenon may be\nattributed to the first Markov approximation, where the\nexternal coupling is assumed to be independent of fre-\nquency [31, 32]. Consequently, there is a significant in-\nfluence from far-detuned modes on the antiresonance fre-\nquency. However, this shifts the antiresonance frequency\nwithout altering its phase-jump, which remains the key\nfeature for predicting whether the effective antiresonance\ncoupling would be repulsive or attractive.\nAround the antiresonance at a frequency of 13.59 GHz,\nthe phase undergoes a positive phase-jump, from −π/2\ntoπ/2. On the contrary, the resonances may exhibit a\npositive or negative phase-jumps, depicted in purple or\ngreen areas respectively in Fig. 5 (c), and are summarized\nin table III. As mentioned earlier, a resonance with the\nsame phase-jump contributes attractively, while a reso-nance with an opposite phase-jump acts repulsively with\nthe antiresonance.\nThe TM θrzand TE θrzmodes are characterized by the\nnumber of field nodes in the θ(spanning 180 °),r, and z\ndirections, where rrepresents the radial direction and θ\nsymbolizes the azimuthal direction in the xy-plane. Re-\nferring to Fig. 5, for a specified rvalue (in this case,\nequal to 1), only the θnumber will determine the cou-\npling phase dependence associated with each mode since\nthe probes are oriented in the radial direction. In this\nscenario, modes with an odd θnumber induce a repul-\nsive effect, while modes with an even θnumber generate\nan attractive influence.\nTheH-field and E-field distributions of the repulsive\nmode TE 212is illustrated in Fig. 6 (a) and (c), while\nthe field distributions of the attractive mode TE 113is\ndepicted in Fig. 6 (b) and (d). For each mode in the Ku-\nband, we calculated the coupling strengths [33] for two\ndistinct locations of the YIG sphere, A and B, as sum-\nmarized in table III. Here, we present only two modes\nTABLE III. Cavity Modes characteristics\nCouplingΦ21-Modeω/2πQgA/2πgB/2π\njump [GHz] [MHz] [MHz]\nrepulsive -TE211 12.4 1525 13.5 0.0\nTM012 12.5 4441 39.3 0.0\nTE212 14.4 912 30.1 0.0\nTM013 15.8 9228 43.3 0.0\nattractive +TE113 14.6 7501 3.6 50.0\nTM111 15.2 12023 3.2 72.2\nTE311 16.6 739 2.5 4.58\n/uni00000025 /uni00000024/uni0000000b/uni00000044/uni0000000c/uni00000037/uni00000028212\n/uni00000025 /uni00000024/uni0000000b/uni00000045/uni0000000c/uni00000037/uni00000028113\n/uni00000025\n/uni00000024/uni0000000b/uni00000046/uni0000000c\n/uni00000035/uni00000048/uni00000053/uni00000058/uni0000004f/uni00000056/uni0000004c/uni00000059/uni00000048/uni00000003/uni00000030/uni00000052/uni00000047/uni00000048\n/uni00000025\n/uni00000024/uni0000000b/uni00000047/uni0000000c\n/uni00000024/uni00000057/uni00000057/uni00000055/uni00000044/uni00000046/uni00000057/uni0000004c/uni00000059/uni00000048/uni00000003/uni00000030/uni00000052/uni00000047/uni00000048\n/uni00000014/uni00000016/uni00000011/uni00000017 /uni00000014/uni00000016/uni00000011/uni00000018 /uni00000014/uni00000016/uni00000011/uni00000019 /uni00000014/uni00000016/uni00000011/uni0000001a\nωm/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni00000014/uni00000016/uni00000011/uni00000018/uni00000013/uni00000014/uni00000016/uni00000011/uni00000018/uni00000018/uni00000014/uni00000016/uni00000011/uni00000019/uni00000013/uni00000014/uni00000016/uni00000011/uni00000019/uni00000018/uni00000014/uni00000016/uni00000011/uni0000001a/uni00000013ω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000048/uni0000000c\n∆=11/uni00000003/uni00000030/uni0000002b/uni0000005d/uni0000003c/uni0000002c/uni0000002a/uni00000003/uni00000056/uni00000053/uni0000004b/uni00000048/uni00000055/uni00000048/uni00000003/uni0000004c/uni00000051/uni00000003/uni00000053/uni00000052/uni00000056/uni0000004c/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000024\n/uni00000014/uni00000016/uni00000011/uni00000018/uni00000013/uni00000014/uni00000016/uni00000011/uni00000018/uni00000018/uni00000014/uni00000016/uni00000011/uni00000019/uni00000013/uni00000014/uni00000016/uni00000011/uni00000019/uni00000018/uni00000014/uni00000016/uni00000011/uni0000001a/uni00000013ω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000049/uni0000000c\n∆=11/uni00000003/uni00000030/uni0000002b/uni0000005d\n/uni0000003c/uni0000002c/uni0000002a/uni00000003/uni00000056/uni00000053/uni0000004b/uni00000048/uni00000055/uni00000048/uni00000003/uni0000004c/uni00000051/uni00000003/uni00000053/uni00000052/uni00000056/uni0000004c/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000025\n/uni00000014/uni00000017/uni00000013\n/uni00000014/uni00000015/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni0000001b/uni00000013\n/uni00000019/uni00000013\n/uni00000017/uni00000013\n/uni00000015/uni00000013\n|S21|/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040H/uni00000010/uni00000049/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000003xz/uni00000010/uni00000053/uni0000004f/uni00000044/uni00000051/uni00000048 E/uni00000010/uni00000049/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000003xy/uni00000010/uni00000053/uni0000004f/uni00000044/uni00000051/uni00000048\nFIG. 6. Norm of the H-field in the xz-plane for (a) TE 212and (b) TE 113. Norm of the E-field in the xy-plane at the height of\nthe probes for (c) TE 212, and (d) TE 113. In (a)-(d), the circles illustrate the positions (A or B) of the YIG sphere mentioned\nin Fig. 5, and the black arrows in (c) and (d) indicate the polarization of the E-field at the probe locations. The transmission\nspectra from FD simulation are shown in (e) when the YIG sphere is placed at position A, and in (f) when the sphere is\nplaced at position B. Utilizing the parameter values from table III, including the mode frequencies, the phase of the E-field\npolarization at each probe, the quality factors, and the coupling strengths of the 7 modes, the white dashed line in (e) and (f)\nrepresents the antiresonance frequency obtained by fitting the resulting spectra from the input-ouput model sketched in Fig. 7\nin Appendix B 1.\nto illustrate the main message of the paper: regarding\nthe field distribution of modes that act repulsively or\nattractively to the generated antiresonance permit in de-\ntermining the optimal positioning of the YIG sphere as\nfunction of the needs, level attraction or repulsion be-\nhavior with the antiresonance. The conclusion of the\nfield distribution of the two illustrated mode also applies\nfor the other repulsive and attractive modes, as shown in\nAppendix B 2.\nAt the bottom right side in the xz-plane of the cavity\n(position A), the YIG sphere is positioned at an anti-\nnode of the H-field for the repulsive modes and at a\nnode of the H-field for the attractive modes. In con-\ntrast, at the bottom center position (position B), the\nYIG sphere is placed at a node of the H-field for the re-\npulsive modes and at an anti-node of the H-field for the\nattractive modes.\nThe modes are either even (for repulsive modes) or odd\n(for attractive modes) along the azimuthal direction, de-\ntermining the sign of the probed E-field in transmission.\nThis relationship is depicted in Fig. 6, which shows the\nE-field distribution in the xy-plane. In Fig. 6 (e) and\n(f), the FD simulation of the transmission spectrum ver-\nsus the magnon frequency and the applied RF frequency\nreveals the effective level repulsion of the antiresonancewhen the YIG sphere is positioned at position A, and\nthe effective level attraction of the antiresonance when\nthe YIG sphere is positioned at position B, respectively.\nBoth types of coupling are effectively replicated by the\ninput-output model, depicted in white dashed lines in\nFig. 6 (e) and (f). Note that the effective antiresonance\nfrequencies were obtained by fitting the resulting spec-\ntra from the input-output model. These frequencies are\nillustrated in B 1 in B 1.\nDespite the similarity in coupling behavior, there ex-\nists a notable disparity in the coupling strengths between\nthe simulation and the model. Specifically, the effective\nrepulsive (attractive) coupling strength is of 26 MHz (15\nMHz) in the simulation, contrasting with 14 MHz (24\nMHz) in the model when the YIG sphere is positioned at\nlocation A (B). This discrepancy may arise from the ap-\nproximations made for the external coupling strength and\nthe neglected internal photonic damping rates. Addition-\nally, limitations in mode truncation may contribute to an\ninadequate value. Nevertheless, our findings demonstrate\nthat modes can exhibit either repulsive or attractive ef-\nfective coupling, and by strategically engineering cavities\nto spatially separate the modes, it becomes possible to\ncontrol the coupling behavior when positioning the YIG\nsphere at different locations.9\nVI. CONCLUSION\nThis study has provided valuable insights into the in-\ntricate interaction between cavity modes and magnons\nwithin quasi-closed cavities. Through the incorporation\nof the input-output formalism enhanced with a crucial\nphase factor in external couplings, we have not only\nsuccessfully replicated antiresonance behavior in simu-\nlations but also provided explanations for the intriguing\nphenomena of antiresonances, encompassing both their\nlevel repulsion and level attraction, as observed in the\ntransmission spectrum. Furthermore, we provided and\ndemonstrated the physical underpinnings of the effective\nantiresonance coupling gar. Understanding the phase-\njumps of the antiresonance and the different mode fami-\nlies (e.g. characterized by their symmetries), enables the\nprediction of the behavior of garregarding the coupling\nof mode families.The reconfigured model introduced here carries\npromising implications for cavity design, offering a versa-\ntile tool to tailor these structures for a wide range of ap-\nplications, from metrology [34, 35] to RF devices [36, 37]\nand the domain of quantum devices [14, 38–40].\nVII. ACKNOWLEDGMENTS\nWe acknowledge financial support from Brest\nM´ etropole for the PhD funding of Guillaume Bourcin.\nThis work is part of the research program supported\nby the European Union through the European Regional\nDevelopment Fund (ERDF), by the Ministry of Higher\nEducation and Research, and Brittany region, through\nthe CPER SpaceTech DroneTech, and the ANR projects\nICARUS and MagFunc. Jeremy Bourhill is funded by\nthe Australian Research Council Centre of Excellence\nfor Engineered Quantum Systems, CE170100009 and the\nCentre of Excellence for Dark Matter Particle Physics.\nAppendix A: Input-Output Theory\n1. Heisenberg Equation of Motion\nFrom eq. (5) and considering the first Markov approximation given in eq. (4), Heisenberg Equation of Motion (EoM)\nfor the external modes reads [41]:\n˙ˆbω,n(t) =−iωˆbω,n(t) +1√\n2πX\npκpnˆap(t). (A1)\nThe solution of this differential equation reads as:\nˆbω,n(t) =ˆbτ\nω,ne−iω(t−τ)\n+1√\n2πX\npκpntZ\nτdt′ˆap(t′)e−iω(t−t′),(A2)\nwhere τis a time reference.\nSubsequently, we define the polychromatic bosonic operator for each port by considering all the bosonic operators of\nthe same port across all frequencies:\nˆbτ\nn(t) =1√\n2πZ\nRdωˆbτ\nω,ne−iω(t−τ). (A3)\nFrom this equation, we define the incoming and outgoing wave operators at each port:\n(ˆbin\nn(t) =ˆbt0\nn(t), t 0=−∞\nˆbout\nn(t) =ˆbt1\nn(t), t 1= +∞. (A4)10\n2. Input-Output Relation\nIntegrating over ωon both sides, eq. (A2) becomes [42]:\n\n\n1√\n2πZ\nRdωˆbω,n(t) =1√\n2πZ\nRdωˆbt0\nω,ne−iω(t−t0)+1\n2πX\npκpntZ\n−∞dt′ˆap(t′)Z\nRdω e−iω(t−t′)\n1√\n2πZ\nRdωˆbω,n(t) =1√\n2πZ\nRdωˆbt1\nω,ne−iω(t−t1)−1\n2πX\npκpn+∞Z\ntdt′ˆap(t′)Z\nRdω e−iω(t−t′). (A5)\nwhere the first term on the right hand side of the first (second) equation is equal to ˆbin\nn(t) (ˆbout\nn(t)), and the second\nterm is equal to1\n2P\npκpn(−1\n2P\npκpn), according to the following properties:\n\n\nZ\nRdω e−iω(t−t′)= 2πδ(t−t′)\ntZ\n−∞dt′ˆap(t′)δ(t−t′) =+∞Z\ntdt′ˆap(t′)δ(t−t′) =1\n2ˆap(t). (A6)\nThis results in the input-output relation:\nˆbout\nn(t) =ˆbin\nn(t) +X\npκpnˆap(t). (A7)\n3. Quantum Langevin Equation\nAssuming g∗\npq=gqp, the Quantum Langevin Equation (QLE) reads as [41]:\n˙ˆap(t) =−iωpˆap(t)−iX\nq̸=pgqpˆaq(t)−1√\n2πX\nnZ\nRdω κ∗\npn(ω)ˆbωn(t). (A8)\nSubstituting the value of ˆbω,nfrom eq. (A2) for τ=t0in the QLE gives rise to:\n˙ˆap(t) =−iωpˆap(t)−iX\nq̸=pgqpˆaq(t)−1√\n2πX\nnκ∗\npn\nZ\nRdωˆbt0\nω,ne−iω(t−t0)+κpnX\nqtZ\nt0dt′Z\nRdω e−iω(t−t′)ˆaq(t′)\n.(A9)\nThe properties given in eq. (A6) lead to:\n˙ˆap(t) =−iωpˆap(t)−iX\nq̸=pgqpˆaq(t)−X\nnκ∗\npn \nˆbin\nn(t) +X\nqκqn\n2ˆaq(t)!\n. (A10)\nTaking the Fourier transform, eq. (A10) can be expressed as:\n \nω−˜ωp+i\n2X\nn|κpn|2!\nˆap(ω) +X\nq̸=p \ni\n2X\nnκ∗\npnκqn−gqp!\nˆaq(ω) =−iX\nnκ∗\npnˆbin\nn(ω). (A11)\nIn matrix form:\n\n\nΩ·ˆa=−iK∗·ˆbin\nΩqp= (ω−ωp)δqp+i\n2X\nn(κ∗\npnκqn)−gqp, (A12)11\nwhere ˆais the vector containing all ˆ ap(ω) operator components, ˆbinis the vector containing all ˆbin\nn(ω) operator\ncomponents, Kis the p×mmatrix with κpmas components, Ω qpare the components of the p×pmatrix Ω, and δqp\nis the Kronecker delta.\n4. S-parameters\nSubstituting the solution of eq. (A12) into eq. (A7) results in:\nˆbout=S·ˆbin, (A13)\nwhere the S-matrix reads as:\nS=1−iKt·Ω−1·K∗, (A14)\nwhere 1is the identity matrix.\nTheS-matrix simplifies the computation of reflection and transmission for complex systems with multiple internal\nmodes and ports.\nAppendix B: Cavity\n1. Model fitting\nIn Fig. 7, the transmission spectra generated by the\ninput-output model for two distinct positions of the YIG\nsphere (A and B) are illustrated in (a) and (b) respec-\ntively. To generate these spectra, the model was provided\nwith necessary parameter values, including the mode fre-\nquencies, the phase of the E-field polarization at each\nprobe, the quality factors, and the coupling strengths of\nthe 7 modes as detailed in table III. Due to the com-\nplexity of obtaining an analytical solution for the polari-\nton frequencies, the antiresonance frequencies were de-\ntermined by fitting the two spectra.\n2. Field Distribution of cavity modes\nFig. 8 illustrates the magnitudes of the H- and E-fields\nfor the seven considered cavity modes in the input-output\nmodel, as discussed in Sec. V B.\nForr= 1, the repulsive modes exhibit an even θnum-\nber, leading to a minimum in the H-field at position B.\nAdditionally, the E-field polarization at the two probes\nisπ-dephased, a condition necessary for observing level\nrepulsion in the effective coupling of the antiresonance at\n13.59 GHz with the coupled magnon-photon mode.\nConversely, with r= 1, the attractive modes exhibit an\noddθnumber, resulting in a minimum of the H-field at\nposition A. Furthermore, the E-field polarization at the\ntwo probes is in phase, a condition for observing level\nattraction in the effective coupling of the antiresonanceat 13.59 GHz with the coupled magnon-photon mode.\n/uni00000014/uni00000016/uni00000011/uni00000018/uni00000013/uni00000014/uni00000016/uni00000011/uni00000018/uni00000018/uni00000014/uni00000016/uni00000011/uni00000019/uni00000013/uni00000014/uni00000016/uni00000011/uni00000019/uni00000018/uni00000014/uni00000016/uni00000011/uni0000001a/uni00000013ω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000044/uni0000000c\n/uni00000014/uni00000016/uni00000011/uni00000017 /uni00000014/uni00000016/uni00000011/uni00000018 /uni00000014/uni00000016/uni00000011/uni00000019 /uni00000014/uni00000016/uni00000011/uni0000001a /uni00000014/uni00000016/uni00000011/uni0000001b\nωm/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni00000014/uni00000016/uni00000011/uni00000018/uni00000013/uni00000014/uni00000016/uni00000011/uni00000018/uni00000018/uni00000014/uni00000016/uni00000011/uni00000019/uni00000013/uni00000014/uni00000016/uni00000011/uni00000019/uni00000018/uni00000014/uni00000016/uni00000011/uni0000001a/uni00000013ω/2π/uni00000003/uni0000003e/uni0000002a/uni0000002b/uni0000005d/uni00000040/uni0000000b/uni00000045/uni0000000c\n/uni00000014/uni00000017/uni00000013\n/uni00000014/uni00000015/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni0000001b/uni00000013\n/uni00000019/uni00000013\n/uni00000017/uni00000013\n/uni00000015/uni00000013\n|S21|/uni00000003/uni0000003e/uni00000047/uni00000025/uni00000040\nFIG. 7. Transmission spectra generated by the input-output\nmodel for two different positions of the YIG sphere: (a) when\npositioned at A, and (b) when positioned at B. The parameter\nvalues injected into the model, including mode frequencies,\nthe phase of the E-field polarization at each probe, quality\nfactors, and coupling strengths of the 7 modes, are detailed\nin table III. The antiresonance frequencies, indicated to the\ndashed black lines, were determined by fitting the spectra.\n[1] A. Barman, G. Gubbiotti, S. Ladak, A. O. Adey-\neye, M. Krawczyk, J. Gr¨ afe, C. Adelmann, S. Coto-fana, A. Naeemi, V. I. Vasyuchka, B. Hillebrands,12\n/uni0000000b/uni00000044/uni0000000c/uni00000037/uni00000028211\n/uni0000000b/uni00000045/uni0000000c/uni00000037/uni00000030012\n/uni0000000b/uni00000046/uni0000000c/uni00000037/uni00000028212\n/uni0000000b/uni00000047/uni0000000c/uni00000037/uni00000030013\n/uni0000000b/uni00000048/uni0000000c\n /uni0000000b/uni00000049/uni0000000c\n /uni0000000b/uni0000004a/uni0000000c\n /uni0000000b/uni0000004b/uni0000000c\n/uni0000000b/uni0000004c/uni0000000c/uni00000037/uni00000028113\n/uni0000000b/uni0000004d/uni0000000c/uni00000037/uni00000030111\n/uni0000000b/uni0000004e/uni0000000c/uni00000037/uni00000028311\n/uni0000000b/uni0000004f/uni0000000c\n /uni0000000b/uni00000050/uni0000000c\n /uni0000000b/uni00000051/uni0000000c/uni00000035/uni00000048/uni00000053/uni00000058/uni0000004f/uni00000056/uni0000004c/uni00000059/uni00000048/uni00000003/uni00000030/uni00000052/uni00000047/uni00000048/uni00000056\n/uni00000024/uni00000057/uni00000057/uni00000055/uni00000044/uni00000046/uni00000057/uni0000004c/uni00000059/uni00000048/uni00000003/uni00000030/uni00000052/uni00000047/uni00000048/uni00000056H/uni00000010/uni00000049/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000003xz/uni00000010/uni00000053/uni0000004f/uni00000044/uni00000051/uni00000048 E/uni00000010/uni00000049/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000003xy/uni00000010/uni00000053/uni0000004f/uni00000044/uni00000051/uni00000048\nH/uni00000010/uni00000049/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000003xz/uni00000010/uni00000053/uni0000004f/uni00000044/uni00000051/uni00000048 E/uni00000010/uni00000049/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000003xy/uni00000010/uni00000053/uni0000004f/uni00000044/uni00000051/uni00000048\nFIG. 8. Norm of the H-field in the xz-plane for (a)-(d) repulsive modes and (i)-(k) attractive modes. Norm of the E-field in\nthexy-plane at the height of the probes for (e)-(h) repulsive modes, and (l)-(n) attractive modes. The circles illustrate the\npositions (A or B) of the YIG sphere mentioned in Fig. 5, and the black arrows in (e)-(h) and (l)-(n) indicate the polarization\nof the E-field at the probe locations.\nS. A. Nikitov, H. Yu, D. Grundler, A. V. Sadovnikov,\nA. A. Grachev, S. E. Sheshukova, J.-Y. Duquesne,M. Marangolo, G. Csaba, W. Porod, V. E. Demidov,\nS. Urazhdin, S. O. Demokritov, E. Albisetti, D. Petti,13\nR. Bertacco, H. Schultheiss, V. V. Kruglyak, V. D.\nPoimanov, S. Sahoo, J. Sinha, H. Yang, M. M¨ unzenberg,\nT. Moriyama, S. Mizukami, P. Landeros, R. A. Gallardo,\nG. Carlotti, J.-V. Kim, R. L. Stamps, R. E. Camley,\nB. Rana, Y. Otani, W. Yu, T. Yu, G. E. W. Bauer,\nC. Back, G. S. Uhrig, O. V. Dobrovolskiy, B. Budinska,\nH. Qin, S. van Dijken, A. V. Chumak, A. Khitun, D. 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Velu,b Kalaivani Thangavel,b Francesco Orsini,b \nKonstantinos Brintakis,a,c Stylianos Psycharakis,a,d Anthi Ranella,a Lorenzo Bordonali,e Alexandros \nLappas,*a Alessandro Lascialfari* b \n \na Institute of Electronic Structure and Laser, Foundation for Research and Technology - Hellas, \nVassilika Vouton,71110 Heraklion, Greece. Fax: +30 2810 301305; Tel: +30 2810 391344; E-\nmail: lappas@iesl.forth.gr  \nb Dipartimento di Fisica,  Università  degli studi di Milano and INSTM, via Celoria 16, I -20133 \nMilano, Italy. Fax: +39 02 50317208; Tel: +39 02 50317383; E -mail: \nalessandro.lascialfari@unimi.it  \nc Department of Physics, Aristotle  University of Thessaloniki, 54124 Thessaloniki, Greece.  \n d Department of Medicine, University of Crete, Voutes, 71003 Heraklion, Greece.  \ne Dipartimento di Fisica, Università degli studi di Pavia and INSTM, via Bassi 6, I -27100, Pavia, \nItaly  \n \n \n \n† Electronic Supplementary Information (ESI) available: [Fluorescence microscopy images, \nCalculation of the magnetic material volume fraction, Table of transverse relaxivities of different \nferrite -based samples]. See DOI:  10.1039/b000000x/  2 \n Abstract  \nElevated -temperature polyol -based colloidal -chemistry approach allows for the development of \nsize-tunable (50 and 86 nm) assemblies of maghemite iso -oriented nanocrystals, with enhanced \nmagnetization. 1H-Nuclear Magnetic Resonance ( NMR ) relaxometric experiments sh ow that the \nferrimagnetic cluster -like colloidal entities exhibit a remarkable enhancement (4 ÷ 5 times) in the \ntransverse relaxivity, if compared  to that of the superparamagnetic contrast agent Endorem, over \nan extended frequency range (1-60 MHz ). The marked increa se of the transverse relaxivity  r2 at a \nclinical magnetic field strengt h (~1.41 T), which is 405.1 and  508.3 mM-1 s-1 for small and large \nassemblies  respectively, allows to relate  the observed response to the raised intra -aggregate \nmagnetic ma terial volume fraction. Furthermore, cell tests with murine fibroblast culture medium \nconfirmed the cell viability in presence of the clusters. We discuss the NMR dispersion profiles on \nthe basis of relaxivity models to highlight the magneto -structural cha racteristics of the materials for \nimproved T 2-weighted magnetic resonance images.  \n 3 \n A Introduction  \nOver the past decade, there has been a considerable progress in the synthesis of colloidal inorganic \nnanocrystals and in their exploitation for applications in various fields , straddling from electronics \nand optoelectronics1 to nanomedicine and clinical practice2. An interesting development in this \ndirection involves the synthesis of colloidal nanocrystals made of multiple subunits arranged in a \ncontrolled topological fashion3, 4 or self -assembled in cluster -like structures (e.g. of -Fe2O3, Fe 3O4, \nPbS, ZnO, TiO 2 etc)5-9. Their synthesis is eit her the result of direct specific interactions (e.g. van der \nWaals attractions, steric repulsions, attractive depletion or capillary forces, Coulomb forces etc.) of \ntheir discrete components, or of some external stimulus, such as light, magnetic or electric fields.10 \nThe coexistence within the same nanoscale entity of distinct material sections, directly \ninterconnected through inorganic interfaces, or secondary structur es communicating through strong \ninteractions, enables multifunctionality. This is because coupling between mechanisms is \nestablished across the interfaced material domains,11,12,13 while the physical properties may also \nhave  a collective nature.14 Based on these appealing features, such multifunctional systems are \npromising for  technological areas entailing biology and biomedicine. Furthermore, those materia ls \nfeaturing a soft magnetic state in conjunction with some other non -homologous property (e.g. \nplasmonic or excitonic emission) and with appropriate surface biocompatible co atings, occupy a \nprominent position in this scenario: 4, 9 they can be exploited  for in-vivo biomedical applications, \nsuch as magnetically driven contrast enhancement,15, 16 selective hyperthermia treatment17, and even \ntargeted drug delivery18. \n Specifically, the afore -mentioned iron -oxides based nanomaterials emerge as promising probes in \nbionanotechnology as they allow for a novel approach to the diagnosis of various diseases with \nmagnetic resonance imaging (MRI). During the last two decades, several commercial MRI contrast \nagents have been developed  for clinical studies. The majority of such systems are  based on \nsuperparamagnetic iron -oxide nanoparticles, coated with hydrophilic polymers  or sugars , including \nEndorem, Resovist, and Combidex. However, poly -dispersity and poor crystallinity  are often 4 \n the most common limitations of such systems, originating from the synthesis protocol; as a result,  \nmagnetic properties and contrast efficiencies vary from one batch to another. 19, 20 \n To develop new magnetic compounds for improved MR -based diagnosti cs, it is desirable to \nadjust the magnitude of the particle’s net magnetic moment μ (because R 2 = 1/T 2  μ2, where R2 is \nthe transverse relaxation rate and T2 is the spin -spin relaxation time) by controlling the struct ural \ncharacteristics of the nano -object, i.e. the nanoparticle’s size, shape, composition and crystallinity.21 \nRapid advances in the chemical routes for  the synthesis of size and shape controlled, surface \nfunctionalised magnetic nanoparticles,22, 23 lead to increased particle magnetizations and enhanced \nimage contrast, preserving biocompatibility at the same time. As individual Superparamagnetic Iron \nOxide Nanocrystals ( SPIONs ) tend to give relatively low magnetization,21 to improve the MRI \ncontrast, larger colloidal entities composed of primary nanocrystals (PNCs), which are self -\nassembled24 or incorporated into polymer matrices, have been investigated. The latter approach, for \ninstance, entails SPIONs encapsulated in polyacrylamine25, silica26, phosph olipid micelles27, block \npolymers28 or amphiphilic polymers29, 30, while the former assembling approaches lead to efficient \n“clustering” of individual PNCs, and  promote close co ntact. In turn, this pathway further boosts the \nmagnetization of such nanostructures, while  the proximity of nanoparticles within the cluster \npromotes the coupling of the neighboring magnetic moments and results in a stronger perturbation \nof the local magn etic field in their vicinity. Indeed, the hydrogen nuclear moments relax faster , \nleading to a strong contrast in the MRI images . Magnetic nanoclusters of this type have been \ndeveloped with capping agents of different nature, including oleylamine/oleic acid ,31, 32 citrate,33-35 \npolymers7, 9, 24, 36-44 or block copolymers45-47. When the enlarged cluster -like entity contains a \nbiocompatible polymer -coating such as polyvinyl alcohol (PVA), poly(ethylene glycol) - co \nfumarate (PEGF) or cross linked PEGF, wide -spread opportunities in biomedical f ield become \npossible.24 These colloidal nanoclusters agents feature an improved contrast efficiency with re spect \nto commercial compounds, or at least their MRI contrast response is comparable to that of the \nwidely available compound  Endorem.  5 \n  In the present work we report on size -controlled assemblies of pure maghemite ( γ-Fe2O3) \nPNCs, which are crystallographically  aligned within the hydrophilic colloidal nanoclusters (CNCs). \nOur system displays a weak ferrimagnetic behavior, in contrast to the majority of the known \nnanoparticle -based aggregates devised for MR -based applications. We suggest that careful \nengineering of the CNCs’ magneto -structural characteristics produces a remarkable improvement of \nr2 (about four -to-five times higher) compared to Endorem. From the point of view of design \ncriteria we suggest that an increased intra -aggregate magnetic material volume fraction and the \nproper choice of the surfactant, allowing the water proton penetration, is a necessary condition for \nthe superior MR -related contrast enhancing features of such low -cytotoxicity nanoarchitectures.  \n \nB Experimental  \nB.1 Materials.  \nAll reagen ts were used as received without further purification. Anhydrous iron chloride (FeCl 3, \n98%), was purchased from Alfa Aesar  (United States ). Anhydrous Sodium hydroxide (NaOH, \n98%), polyacrylic acid (PAA, M w= 1800), were purchased from Sigma Aldrich  (United States ), \nwhile Diethylene glycol (DEG, (HOCH 2CH 2)2O) of Reagent ( <0.3% ) and Laboratory ( <0.5% ) \ngrades were purchased from Fisher  (United States) . The absolute Ethanol was purchased from \nSigma Aldrich.  \nCell cultures.  \nNIH/3T3 murine fibroblasts (ATCC - Amer ican Type Culture Collection) were suspended at a \nconcentration of 105 cells/mL in Dulbecco’s modified Eagle’s medium (DMEM) supplemented \nwith 10% fetal bovine serum (FBS) and 1% antibiotic solution (GIBCO, Invitrogen, Karlsruhe, \nGermany); cells were then cultured at 37 °C, in an atmosphere of 5% CO 2, and detached when \ngrowth reached 75% confluence using 0.05% trypsin/EDTA (GIBCO, Invitrogen, Karlsruhe, \nGermany).  \n 6 \n B.2 Synthesis of γ -Fe2O3 Nanoparticle Clusters.  \nThe CNCs were synthesized by a modified high -temperature colloidal chemistry protocol based on \na one -step polyol process first reported by Yin and co -workers.48 All syntheses were carried out \nunder argon atmosphere in 100 -mL round -bottom three -neck flasks connected via a reflux \ncondenser to standard Schlenk line setup, equipped with immersion temperature probes and \ndigitally -controlled heating mantles. The reactants, FeCl 3, NaOH, polyacrylic acid (PAA, M w= \n1800), diethylene glycol (DEG), except Eth anol, were stored and handled under anaerobic \nconditions in an Ar -filled glove -box (MBRAUN, UNILab). Crucially, utilising two types of DEG \ngrades allowed for assemblies of iron -oxide nanocrystals with small and large diameter to be \nprepared upon increasing  water content in the reaction mixture. Observation of the purified \nnanocluster samples by transmission electron microscopy, at regular time -intervals over a period of \na year, have shown that they are stable with no aggregation among their units.  Details o f the \nsynthetic protocol can be found elsewhere.49  \n \nB.3 Characterization techniques.  \na) Transmission electron microscopy (TEM).  \nLow-magnification and high -resolution TEM images were recorded on a LaB 6 JEOL 2100 e lectron \nmicroscope operating at an accelerating voltage of 200 kV. For the purposes of the TEM analysis, a \ndrop of a diluted colloidal nanoparticle aqueous solution was deposited onto a carbon -coated copper \nTEM grid. All the images were recorded by the Gat an ORIUSTM SC 1000 CCD camera and the \nstructural features of the nanostructures were studied by two -dimensional (2D) fast Fourier \ntransform (FFT) analysis.  \nb) Magnetic Measurements.  \nThe magnetic properties of the samples were studied by means of a Superc onducting Quantum \nInterference Device  (SQUID) magnetometer (Quantum Design MPMS XL5). The magnetic \nmeasurements were performed with the dried powder of the nanomaterials. The magnetic data have 7 \n been normalized to the mass of the γ -Fe2O3, as derived by Inductively Coupled Plasma Atomic \nEmission Spectroscopy ( ICP-AES ) analysis ([Fe] 50.2 nm CNCs  = 44.9 ± 0.3 mM, [Fe] 85.6 nm CNCs  = 42.3 \n± 0.3 mM).  \nc) Nuclear Magnetic Resonance (NMR).  \nTwo solutions of CNCs were prepared for the NMR investigation, having an iron concentration C= \n1.225 and 1.208 mmol/L, as determined by ICP-AES  measurements for small and large CNCs, \nrespectively. NMR longitudinal and transverse relaxation times were measured by using two \ndifferent pulsed Fourier Transform (FT) -NMR spectrometer s: i) a Smartracer Stelar relaxometer \n(using Fast -Field -Cycling, FFC, technique) for frequencies in the range 0.01   10 MHz, and ii) a \nStelar Spinmaster relaxometer for  10 MHz.  We used a common saturation recovery sequence to \nmeasure the longitudinal (spin -lattice, T 1) nuclear relaxation time. Standard radio frequency \nexcitation pulse sequence s, CPMG for  > 3 MHz and pre-polarized Hahn echo sequence  for  < 3 \nMHz , were used to measure  the  transverse (spin -spin, T 2) nuclear relaxation time.  The investigated \nfrequency range 0.01   60 MHz corresponds to an external magnetic field H= 0.00023 - 1.41 \nTesla (ω= γ Η, where ω is the Larmor frequency and γ/2π= 42.58 MHz/T). The magnetic field range \nwas chosen in order to cover the typical fields used in both clinical  and research laboratory MRI \ntomography (H= 0.2, 0.5 and 1.5 Tesla). The measurements at room and physiological temperatures \ngave the same results within 10 %.  \nThe efficiency of the  CNCs as MRI contrast agents was determined by calculating the nuclear \nlongitudinal and transverse  relaxivities, r 1 and r 2 respectively , defined as  \nCdiam)i(1/T m eas)i(1/T\nir\n, i = 1, 2  (1) \nwhere (1/T i)meas is the value measured for the solutions with concentration C (mmol/L) of the \nmagnetic center and (1/T i)diam represents the nuclear relaxation rate of the diamagnetic host solution, \nwhich in our case is water.  \n 8 \n d) Cell Viability Assay.  \nThe interaction of the cells with the CNCs was evaluated using the live -dead cell staining kit \n(Biovision). The assay protocol has been described in previous studies.50, 51 Cell suspension ( 105 \ncells/mL) was deposited in 24 -well plates and cultivated for 1, 3, 5 and 7 days. In parallel, different \nconcentration of CNCs ( 25, 50, 100 and 200 μg/mL  Fe) was  added in the cell suspension for every \ndifferent culture period. Cells that remained alive after certain days of cultivation were stained \ngreen whilst dead cells were stained yellow -red. The images were recorded in a Zeiss fluorescence \nmicroscope equipped with a Laser scanning system Radi ance 2100 (400 -700 nm) and a Carl -Zeiss \nAxio Camera HR. Experiments were repeated at least three times for each size and available \nconcentration of the CNCs.  \ne) Cell Proliferation Assay.  \nQuantification of live cells after each day of culture was performed  by using ATP -Glo TM \nbioluminometric cell viability assay kit (Biotium, Inc.). This highly sensitive homogenous assay for \nquantifying Adenosine Triphosphate  (ATP ) involves an addition of ATP -GloTM detection cocktail \nto cells cultured in a serum -supplemented medium. Since ATP is an indicator of metabolically \nactive cells, the number of viable cells can be assessed based on the amount of ATP available. This \nATP detection kit takes advantage of Firefly luciferase’s use of ATP to oxidize D -Luciferin and the \nresulting production of light in order to assess the amount of ATP available. Luminescence \nmeasurements were performed using the Synergy™ HT Multi -Mode Microplate Reader (BioTek). \nThe different CNCs concentrations were tested in duplicates in each individual cell culture  \nexperiment and these  experiments were repeated  at least three times . For statistical analysis, the \ndata were subjected to one way ANOVA followed by Tukey tests for multiple comparisons between \npairs of means, using commerciall y available software (SPSS 21, IBM).  \n \n \n 9 \n C Results and Discussion  \nC.1 Morphology and Crystal Structure  \nIron-oxide colloidal assemblies with two different sizes were prepared  on the basis of a high -\ntemperature wet chemistry polyol pathway.  The influence of different synthetic parameters49 on the \nstructural characteristics of the CNCs were carefully examined in order to attain a significant degree \nof size/shape homogeneity. 57Fe Mössbauer spectra (MS ) recorded for the two nanocrystal \nassemblies at 10, 77 and 300 K show similar features, confirming the maghemite (γ -Fe2O3) nature49 \nof these iron -oxides. Furthermore, bright -field TEM images for two representative  samples are \nshown in Figures 1a, d . Both CNCs samples have a flower -like, almost spherical shape without \naggregation between units. Each nanocluster is an assembly of small PNCs, with no isolated \nnanoparticles left out of the aggregate. The average diamet ers of the entities are 50.2 ±5.4 and 85.6 \n±13.3 nm (Figure 1a, c). Further evidence for the topological arrangement and the crystallographic \norientation of the assembled nanocrystals is provided by HRTEM images and the calculated FFT \npatterns taken from i ndividual CNCs (Figure 1b, d). The observed diffraction spots suggest that the \nindividual nanocrystals are iso -oriented. Analysis of the FFT patterns allowed us to identify that the \ncrystal symmetry of the CNCs as a whole is similar to that of the individu al, small PNCs, which \ncrystallize in the cubic spinel structure. The attractive magnetic dipolar interactions (vide infra) of \nthe PNCs are strong enough to counter -balance the electrostatic repulsions and allow for the \ncontrolled assembly of the PNCs in a secondary structure. During the growth, the as -formed PNCs \nre-orient in the colloid and minimize their surface energy by aggregating through oriented \nattachment, and form a single -crystalline -like secondary nanostructure.38  \n 10 \n  \nFig. 1 Low- (a, c) and high - (b, d) resolution TEM images of colloidal nanoclusters (CNCs) of maghemite. Insets to \npanels a, c show the size distribution for CNCs with diameter of 50 and 86 nm. Insets to panels b, d show the calculated \nFFT patterns from an isolat ed cluster -like structure; the reflections are indexed on the basis of a cubic spinel (fcc) iron -\noxide crystal structure. The corresponding zone -axis is marked by B.  \n \nC.2 Magnetic Properties  \nIn Figures 2a,b the room temperature hysteresis loops indicate t hat the dried powders of the CNCs \nwere ferr imagnetic (FiM) with coercive fields (H c) of 5.4 and 2.1 Oe, for the small and the large \nCNCs, respectively, while the saturation magnetizations (M S) were of 71.2 and 73.4 emu/g γ-Fe2O3, \nvery close to the bulk val ue of 74 emu/g for γ-Fe2O3.  \n \n11 \n  \n \nFig. 2 Magnetic properties of the small (50 nm) and large (86 nm) CNCs: (a) the temperature evolution of the ZFC and \nFC susceptibility, as well as hysteresis loops at 300 K (b) and 5 K (c).  \n \n The hysteretic behavior  in the temperature dependent zero-field and field -cooled ( ZFC-FC) \nsusceptibil ity (χ) indicates blocked particles  below 300 K for both cases  (Figure 2a). Since no \nobvious difference between the Mossbauer spectra of the dried powder and those of the frozen \nsolutions of the CNCs could be resolved, we conclude that ferrimagnetism of the CNCs can be \nattributed to intra -cluster dipolar interactions only.49 \n \nC.3 Cytotoxicity  \nCNCs appear to satisfy the first prerequisite of tailored magnetic properties for improved MR \nproperties. In addition, the purposeful choice of the PAA as surfactant renders the assemblies \nnegatively charged (z -potential, was -65.4±10.8 and -50.0±6.5 mV for the small and the large \nCNCs, respectively) with good colloidal stability. However, a second condition for their \n12 \n implementation in the MR -based diagnostics requires verification of their toxicity on living cells. \nFor this purpose, murine fibroblast cells (NIH/3T3) were exposed to the CNCs to thoroug hly check \nthe cell -nanocluster interactions. An ATP -based GloTM cell viability assay was used for the \nluminometric measurement of the NIH/3T3 cells growth. Since the ATP indicates the metabolically \nactive cells, we quantified the number of viable cells bas ed on the concentration of the avai lable \nATP. ATP concentrations, recorded over a regular period of cultivation time (1, 3, 5 or 7 days), are \nplotted in Figures 3a, b. A series of different concentrations (25, 50, 100 and 200 μg/mL  Fe) was \nused for both sm all and large CNCs.  \n Specifically, the cell growth for different concentrations of the CNCs, for the same number of \ndays of culture, displayed no statistical difference when it was compared to the respective control \nsamples (cells without CNCs).  The signif icance levels  (p) were estimated using one way ANOVA  \nfollowed by Tukey tests (p<0.05).  Additionally, the difference in th e size of the CNCs did not affect \nthe cell proliferation after 1, 3, 5 or 7 days of culture. The unaffected proliferation of NIH/3T3 cells \nunder the presence of different concentrations of the CNCs provides valid evidence  for the low \ntoxicity of the present nanomaterial.  \n 13 \n  \nFig. 3 Cytotoxicity profile of the 50 (a) and 86 nm (b) CNCs samples. NIH/3T3 fibroblast cells were incubated with \nCNCs at the indicated doses for 1, 3, 5 and 7 days. ATP -Glo TM Bioluminometric cell via bility assay kit was used for \nluminometric measurement of the fibroblasts’ growth. Each histogram reflects the luminescence (mean value in RLU - \nrelative luminescence units - ± Standard Deviation -SD) as derived from three independent experiments. Fluoresce nce \nmicroscopy images of  live (green) and dead (orange -red) fibroblasts, cultured without CNCs (control) (c, e) and with \nsmall (d) or large (f) CNCs, after 5 days of culture.  \n \n The viability of fibroblasts was also tested by using a live -dead cell stainin g kit (BioVision). Cell \nviability was assessed after 1, 3, 5 and 7 days of culture with and without CNCs. Figures 3c -f show \nrepresentative results from a 5 -days cell culture and a concentration of 200 μg/mL  Fe. The 3 and 7 \ndays cultures, for all the CNCs c oncentrations ( 25-200 μg/mL  Fe) show a similar behavior (Figure \nS1). In particular, these images  depict the healthy fibroblasts (stained green), while dead cells \n(stained with PI, red) could only rarely be detected even on cell cultures with the highest me asured \nconcentration ( 200 μg/mL  Fe) and the large CNCs (Figure 3f). In su mmary, fibroblast cells seem to \ntolerate the CNCs, showing high levels of viability, cell -cell interactions and low levels of toxicity, \ngiving further support towards the potential use of the presently developed nanoa rchitectures in \nMR-based diagnostics.  \n \n14 \n C.4 The NMR Relaxation Properties and the Influence of the Intra -cluster Magnetic Material \nVolume Fraction.  \nLet us now examine how the unique magnetostructural attributes of these nanocrystal assemblies \ncan mediate t he MR -based properties in view of their perspective imaging capability.  The CNCs \nwere shown to be ferrimagnetic, thus they are highly magnetized under an applied magnetic field  \nand, consequently, they induce a substantial local perturbation of the dipolar magnetic field in their \nvicinity.21 However, the possible increase of nuclear relaxation rates, giving a better efficiency in \ncontrasting the MRI images, depends also on the dynamic electronic properties of the ensemble.  To \nevaluate the physical properties of the system that influence the nuclear relaxation , we performed \nproton NMR relaxometry measurements. The longitudinal r1 and transverse r 2 relaxivities  (eq. (1)) , \nwere studied as a function of the measuring frequency for the two CNCs samples and compared \nwith those obtained for Endorem (Figure 4). Irrespective of the CNCs’ size, the longitudinal r 1 \nrelaxivity (~200 mM-1 s-1) is higher than that of Endor em (10 mM-1 s-1) at low frequencies (10 kHz \n- 9 MHz), but falls a bit below the commercial product’s values at higher frequencies (10 MHz - 60 \nMHz) (Figure 4a). The frequency behavior of r 1 is a consequence of the physical mechanisms \nresponsible for the n uclear relaxation.19, 52 In the low frequency (<1 MHz) range, the very high \nvalues of r 1 () and its flattening reflect the dominant role of the ferrimagnetic CNCs’ high \nmagnetic anisotropy. The absence of a maximum in r 1(), at   1 MHz, suggests that the diffusion \nprocess in this frequency range is poorly effective in shortening the nuclear relaxation of water \nprotons ( see comparison  with Endorem, Figure 4a).  15 \n  \nFig. 4 Room temperature longitudinal r 1 (a) and transverse r 2 (b) relaxivities as a function of proton Larmor frequency \n(or the external magnetic field, H) for the 50 (circles) and 86 nm (triangles) CNCs. The corre sponding data for the until \nrecently commercial contrast agent Endorem® (squares) are also shown.  \n \n On the other hand, the transverse r 2 relaxivity, i.e. the crucial relaxivity in T 2-relaxing materials, \nis enhanced by a factor 4 (5 for large CNCs) with r espect to Endorem (where r 2 ~ 100 mM-1 s-1), \nover almost all the investigated frequency range (Figure 4b). It is worth considering that a recent \nscaling -law approach has shown that the transverse relaxivity depends on three parameters, namely, \nthe hydrodynamic diameter, the magnetization of the whole nanoparticle aggregate and the volume \nfraction of the magnetic material.53 In that respect the r 2 values may be predicted by models \n16 \n assuming that the spin -spin relaxation occurs either in the motional a veraging regime (MAR), \nwhere ΔωτD< 1 or in the static dephas ing regime (SDR), where Δω τD> 5; within the first regime \nthe transverse relaxivity increases with the size, while in the second it remains almost unchanged.54  \nTo calculate the expected Larmor frequency shift we first estimated the values of the intra -cluster \nvolume fraction, φ intra, resorting to the following relation (refer to S2):  \nργ-Fe2O3 ( 1 - fmγ-Fe2O3 ) / (ρPAA  fmγ-Fe2O3) =  ( 1 - φintra ) / φintra     (2) \nwhere ρ γ-Fe2O3 = 4900 kg/m3 is the density of bulk maghemite, ρ PAA= 1150 kg/m3 is the density of \nthe polyacrylic acid, and f mγ-Fe2O3 is the weight fraction of the iron -oxide in a known mass of dried \nnanocluster powders, estimated from the thermogravimetric (TGA) measurements.2 Eq. (2) yields \nφintra=0.60 and 0.72 for small and large CNCs, respectively.  \nOn the other hand, the Larmor frequency shift is expressed as  \nΔω = 0 M*\nV / 3         (3) \nwhere M* V is the normalized magnetization (see Table I ), while the translational diffusion time τ D  \nis defined as  D = d2 / 4D, where d is the diameter of the CNCs and D = 3 ×10-9 m2/s is the water  \ntranslational diffusion coefficient. τ D amounts to ~0.2 and ~0.6 μs, for the small and large CNCs.  \nThus, finally:  \nΔωτD = 4.9     and     Δω τD = 18.2       (4) \nfor small (50 nm) and large (86 nm) CNCs respectively, indicating that the investigated samples fall \nat the extremes of the SDR regime (5 < Δωτ D < 20). We note that, within the SDR regime, the \nstraddling water molecules feel a relatively constant dipolar magnetic field in their vicinity and it is \nnot surprising that the experimentally observe d r 2(ν) are marginally dependent on the CNCs’ \naverage size. The small difference in the sample volume magnetization, M*\nV, and φ intra, is an \nadditional reason behind the little variation at   4 MHz (Figure 4b).  \nThe result of eq. (4) suggests that the transverse relaxivity (r 2) values for both CNCs samples may \nbe predicted by models assuming that the spin -spin relaxation occurs in the SDR regime.  17 \n  \nFirstly, if we calculate the expected r 2 value with the theoretical (approximate) expression found in \nref. [53],  \n39/ 2*\nmat 0 2VPtheoM r \n \nwhere ν mat is a volume fraction to iron concentration conversion factor (ν mat = 1.57x10-5 m3mol-1 for \nmaghemite), and compare it to the experimental result r 2exp, we observe a reasonable agreement, \nsince r 2theo/ r2exp = 1.11 a nd 1.10. Conversely, the expected normalized relaxivity r 2* = r2 φintra /M*\nv2 \nfor 50 nm and 86 nm clusters is about 2 10-8 and 810-8 s-1mM-1m2A-2, respectively, but the real \nvalues characterizing the two samples are 5.5 10-9 and 5.410-9 s-1mM-1m2A-2, i.e. an order of \nmagnitude different. Such a marked difference evidently hints at a poor agreement between theory \nand experiment ; on the other hand, one may argue that samples located at the edges of the SDR \nregime are prone to misbehavior: the 50 nm sampl e is neither in the motional averaging regime \n(MAR) nor completely in the SDR range, i.e. in a region not described by any existing model; the \n86 nm sample is very close to the Δωτ D ~ 20 limit, where the refocusing pulses used in the T 2 \nsequence become eff ective, an effect which is not accounted for by the SDR model.  The \nexperimental evidence proves, on the other hand , that Endorem is not affected by these  issue s at \nall, since for the commercial compound Δωτ D = 4.4 and  r2*= 7.810-8 s-1mM-1m2A-2, which is \nperfectly in line with the theory.  \n Finally, a justification for the significant  (4 to 5 times) increase of r 2 with respect to the one of \nEndorem, can be inferred by considering the almost two -times higher volume magnetization in \nconjunction with the roug hly three -times larger intra -cluster volume fraction (Endorem: M*\nV= \n0.77 105 A/m, φ intra= 0.23; Table 1).  \nThe above discussed points suggest that the φintra parameter has a crucial role in the improvement  of \nthe MR properties.  \n  18 \n Table 1 . Parameters utilized for the calculation of the intra -cluster magnetic material volume fraction and the \ndetermination of the transverse relaxivity regime to which CNCs belong. These entail: D hydro, the hydrodynamic \ndiameter of the CNCs; f m γ-Fe2O3, the weight fraction of the iron -oxide in a known mass of dried nanocluster powder as \nderived by the thermogravimetric (TGA) measurements; φ intra, the intra -cluster volume fraction of the magnetic \nmaterial; M S, the saturation magnetization of the CNCs at room temperature; M*\nV, the volume magnetization; Δω, the \nLarmor frequency shift; τ D, the translational diffusion time.  Note: parameter values for Endorem® tabulated in this table \nwere taken from reference [53] . \nSample  Dhydro  (nm)  fmγ-Fe2O3  (%) φintra  M*\nV (105 A/m)  ΔωτD Regime  \nEndorem®1 80 63.8 0.23 0.77 4.4 SDR  \n50.2 nm CNCs  78.6 86.8 0.60 1.95 4.9 SDR  \n85.6 nm CNCs  121.8  92.1 0.72 2.37 18.2 SDR  \n \nC.5 The CNCs Relaxivities  and their Relevance for other Ferrite Nanoachitectures  \nTwo broad categories, depending on the nature of their magnetic state, entailing either \nsuperparamagnetic (SPM) or ferrimagnetic (FiM) functional structures, are considered. A clear \nenhancement of the  transverse relaxivity r 2 is presented against individual SPM nanocrystals with \ndifferent capping agents such as, polymers 19, dendrons 55 or DHAA 56. Their size -only dependent \nrelaxivity in the MAR regime renders them less efficient T 2-relaxation agents compared to the \nCNCs, whose improved faster reduction of T 2 is determined by the synergetic action of M s, size and \nφintra.53 However, a progressive increase of the size of indi vidual nanoparticles can lead to FiM \nnanoarchitectures which may provide enhanced relaxivites with the advantages of the SDR regime. \nThe only FiM systems of large size and enhanced magnetic anisotropy are the iron -oxide \nnanocubes, with an edge length of 22  nm, encapsulated in PEG -phospholipids, with good colloidal \nstability; these systems have shown a value r 2 = 761 mM-1s-1, higher than our system 57 (see Table \nS2 for a more comprehensive comparison).  \n Alternatively, controlled aggregation of nanoparticles in secondary structures c an further boost 19 \n the relaxivities with an outcome equivalent to the SDR. In this respect, to the best of our \nknowledge, the only ferrimagnetic secondary structure that has been studied so far for its \noutstanding visualizing (T 2-weighted contrast properties ) as well as drug -delivering actions, is a \nnewly designed liposome -encapsulated magnetic nanoparticle cluster. 58 This system shows an \nunprecedented r 2 of 1286 mM-1s-1 but at a higher magnetic field of 2.35 T, thus leaving the CNCs a \nvalid alternative.  \n Nevertheless, the strategy to embed nanoparticles in matrices/polymers 44 59 or aggregating them \nin well -controlled morphologies 32, 33, 36, 40, 45, 46, 60-62 can also afford SPM nanoarchitectures and \nshows relaxivities explainable within the SDR regime. In this case, our system shows larger r 2 (405 \nand 510 mM-1s-1 for the small and large assemblies, respectively) compared to most of the SPM \ncluster -type particles,  such as those capped with citrate 33, PVP 36, polystyrene 63, Dextran 44, PEG \n39, (Mal)mPEG -PLA copolymer 46, amine 32, TREG 62. An exception is the SPM magnetite -based \ncluster -analogue of higher r 2 relaxivity, which is covered by the same surfactant as our s, but \nsynthesized in an autoclave, through a polyol process for an extended period of time (12 hr).60, 61 \nAmongst them, those clusters of diameter 34 and 63 nm (with r 2 of 540 and 630 mM-1s-1, \ncorrespondingly) have a φ intra of 0.30 and 0.50 and M V of 1.23 105 and 1.79  105 A/m, \nrespectively, both smaller than the value for CNCs of the present study. The enhanced values of \ntheir relaxivities suggest that r 2 is not mediated only by the latter two magnetostructural parameters, \nbut it must also be a function of the surface  properties of the assemblies (e.g. thickness of surface \ncoating, L, over the hydrodynamic diameter, D hydro, of the assembly), as the interactions between \nthe water protons and the assemblies could occur primarily on their surface.64 65  \n A further observation is that, in general, r 2 can be thought to originate from the variation of the \ndiffusion length of the water molecules, relative to the dimensions of the surfactant -coordinated \ninorganic entities themselves. T hen, the r 2 relaxivity is expected to decrease as the molecular weight \nof the capping agent increases 64 , but this appears not to be the case of our system since the clusters \nare capped with a surfactant (PAA) of lower molecular weight (M w= 1800 versus M w= 500060). The 20 \n possible lower steric hindrance amongst shorter chains,  allows for a higher packing density of a \nlarger number of polyacrylate chains to coordinate the nanoclusters’ surface. In turn, this complies \nwith the h igher z -potential attained by the CNCs in the present case ( -65.4 and -50.0 mV for the \nsmall and the large clusters, compared to -38 and -43 mV, for 34 and 63 nm clusters by Li et al60). \nSuch a dense surface coverage may impede the penetration ability (diffusivity) of the water \nmolecules. As a consequence, the water -proton nuclear moments are less strongly perturbed by the \nlocal magnetic field generated by the inorganic entities at the surface of the nanoclusters themselves \nand generates a lower r 2.  \nOther difference s may also be attributed to the likely modification of the chemical bonding \n(entailing the surface functionalised nanocrystals of the assembly) in the two cases. This bonding \ncan mediate the particles’ magnetic anisotropy or even enhance their surface spin  disorder, thus \nhaving an impact in lowering the relaxivity values 66, 67.  \n \nConclusions  \nThe suggested colloidal chemistry pathway allows to obtain ferrimagnetic assemblies of \ncrystallographically oriented primary nanocrystals of maghemite. They are not only well -dispersed \nin aqueous media and have low cytotoxicity, but show an enhanced transverse  NMR relaxivity, r 2. \nThe results suggest that the pronounced enhancements in the magnitude of r 2 have their origin in the \nability to guide the assembly of individual nanocrystals so that they become crystallographically \noriented, allowing for an increased magnetic material volume fraction (φ intra) within the larger -\ngrown nanocluster en tities. The favourable high intra -cluster volume fraction of the magnetic \nmoments appears to make them efficiently coupled to each other in the CNCs. In turn, this permits a \ncoherent and intense perturbation of the dipolar magnetic field in the near vicini ty of the straddling \nwater molecules, a condition that enhances the relaxation of the associated proton nuclear spins. \nThe polyacrylate surface coordinating groups of the CNCs further mediate the efficient penetration \nof the water protons and in conjunctio n with the surface spin disorder of the inorganic nanocrystals 21 \n determine the magnitude of r 2. Such nanoarchitectures appear to have the potential for improved \ndiagnostic quality in the T 2-weighted MR imaging techniques.  \nFinally, the observations presented in Section 3.4 lead to the conclusion that a definitive theory for \nr2 relaxivity still has to be formulated, and a further effort needs to be sustained to account for \noverlooked effects that may indeed have a strong impact on the relaxometric properties of  a \nnanoparticle -based contrast agent. Specifically, we would like to highlight the existence of classes \nof compounds that do not strictly obey the universal scaling law envisioned by Vuong et al53 for \nclusters, since when the ΔωτD is at lower and upper li mits of the SDR range, predictions based on \nref. [53] cannot be reliable.  \n \nAcknowledgments  \nThis work was supported by the European Commission through the Marie -Curie Transfer of \nKnowledge program NANOTAIL (Grant no. MTKD -CT-2006 -042459) . SKPV, KVT, FO, LB and \nAL thank the Italian projects INSTM -Regione lombardia \"Mag -NANO\", FIRB \"Riname\"and \nFondazione Cariplo n. 2010 -0612.  We thank Tomas Orlando and Paolo Arosio for experimental \nhelp.  \n \nNotes and references  \n1. D. V. Talapin, J. S. Lee, M. V . Kovalenko and E. V. Shevchenko, Chem Rev , 2010, 110, 389 -\n458. \n2. P. Zrazhevskiy, M. Sena and X. Gao, Chem. Soc. Rev. , 2010, 39, 4326 -4354.  \n3. L. Carbone and P. D. Cozzoli, Nano Today , 2010, 5, 449 -493. \n4. N. C. Bigall, W. J. 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Velu, b Kalaivani Thangavel, b Francesco Orsini, b \nKonstantinos Brintakis, a,c Stylianos Psycharakis, a,d Anthi Ranella, a Lorenzo Bordonali, b \nAlexandros Lappas,* a Alessandro Lascialfari*  b \n \naInstitute of Electronic Structure and Laser, Founda tion for Research and Technology-Hellas, \nVassilika V outon, 71110 Heraklion, Greece \nbDipartimento di Fisica, Università degli studi di M ilano and INSTM, via Celoria 16, I-20133 Milano, \nItaly \ncDepartment of Physics, Aristotle University of Thes saloniki, 54124 Thessaloniki, Greece \ndDepartment of Medicine, University of Crete, V outes , 71003 Heraklion, Crete, Greece \n \nCorresponding Authors \n*lappas@iesl.forth.gr \n*alessandro.lascialfari@unimi.it \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n S1. Toxicity experiments \n \n \nFigure S1.  Fluorescence microscopy images of live (green) and  dead (orange-red) fibroblast cells \ncultured without CNCs (a), with 85.6 nm CNCs (I, II I), and 50.2 nm CNCs (II, IV) under different \nconcentrations (25 μg/mL-b, 50 μg/mL-c, 100 μg/mL-d, 200 μg/mL-e), after 3 (I,II) and 7 (III,IV) days \nof culture. The scale bar corresponds to 50 μm. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \nS2. Calculation of the intra-cluster volume fractio n of the magnetic material 1 \n \nTable S1.  (Copy of Table 1) Parameters utilized for the calc ulation of the intra-cluster magnetic \nmaterial volume fraction and the determination of t he transverse relaxivity regime where CNCs \nbelong. These entail: D hydro , the hydrodynamic diameter of the CNCs; f m γ-Fe2O3 , the weight fraction of \nthe iron-oxide in a known mass of dried nanocluster  powder as derived by the thermogravimetric \n(TGA) measurements; φintra , the intra-cluster volume fraction of the magnetic  material; M S, the \nsaturation magnetization of the CNCs at room temper ature; M V, the volume magnetization; Δω , the \nLarmor frequency shift; τD, the translational diffusion time. \n \nSample Dhydro \n(nm) fm γ-Fe2O3 \n(%)  φintra MV \n(10 5 A/m) Δω  τD Regime \nEndorem ®1 80 63.8 0.23 0.77 4.4 SDR \n50.2 nm CNCs 78.6 86.8 0.60 1.95 4.9 SDR \n85.6 nm CNCs 121.8 92.1 0.72 2.37 18.2 SDR \n \nThe transverse relaxivity (r 2) regime in which the CNCs belong is determined mai nly by three \nquantities, which are material-dependent characteri stics, namely: the volume magnetization (M V), the \nvolume fraction of the magnetic material ( φintra ) and the hydrodynamic size (D). 1 \nThe intra-cluster volume fraction ( φintra ; Table S1) is calculated by the formula: \n   \nintra intra \nOFe  γm PAA OFe  γm OFe γ\nφφ1\nfρ) f(1 ρ\n3232 32 −=−\n−− −    (1) \nwhere ρ γ-Fe2O3 = 4900 kg/m 3, f m γ-Fe2O3 the weight fraction of the iron-oxide in a known ma ss of dried \nnanocluster powders was estimated from the thermogr avimetric (TGA) measurements and ρPAA = 1150 \nkg/m 3 the density of the polyacrylic acid. 2 \nThe  normalized volume magnetization M V*, in A/m, can be calculated by the formula: \n    \n32OFe γ S intra V Mφ*M−× × = ρ      (2) \nwhere the CNCs’ room temperature saturation magneti zations are M S= 71.2 and 73.4 emu/g for the \n50.2 nm and 85.6 nm CNCs samples, respectively. \nThe Larmor frequency shift, Δω  is calculated by the formula: \n    3*M γμΔωV 0=       (3) where γ= 2.67513× 10 8 rad s -1T-1 is the proton gyromagnetic ratio and μ0= 4 π×10 -7 Τ m Α-1 the \nmagnetic permeability of the vacuum. \nThe estimated Δω  for the 50.2 nm and 85.6 nm CNCs samples are 2.37 ×10 7 s-1 and 2.95 ×10 7 s-1, \nrespectively. \nThe translational diffusion time τD is given by the formula: \n    D4Dτ2\nhydro \nD=      (4) \nwhere D hydro  is the hydrodynamic diameter of the CNCs, as calcu lated from the Dynamic Light \nScattering (DLS) experiments and D= 3 ×10 -9 m2/s the water-proton translational diffusion time . \nThe calculated values of the τD are 0.2 and 0.6 μs  for the 50.2 nm and 85.6 nm CNCs samples, \nrespectively. \n \nBased on the magnitude of the parameters derived fr om equations (3) and (4), ~5 < Δω  τD < ~20 for \nthe CNCs (Table S1). For this reason we are of the opinion that the transverse relaxivity (r 2) values for \nboth CNCs samples can be predicted by models assumi ng that the spin-spin relaxation occurs in the \nstatic-dephasing-regime (SDR).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Table S2.  Compilation of in-vitro transverse relaxivities, r 2, that allow comparison of the maghemite \nnanoclusters (CNCs) developed herein against other nanoarchitectures with potential in MR imaging. \nThe materials were divided in two major categories depending on the nature of their magnetic state, \nnamely, superparamagnetic (SPM) and ferrimagnetic ( FiM). The regime, SDR or MAR, based on the \nouter sphere relaxation theory, in which they fall in is shown. The frequency, ν [= (γμ/2 π) H], at which \nthe relaxivity was measured, is also given. \n \nMagnetic \nState Type of \nnanoarchitecture Sample surface \nfunctionalisation  Regime r 2 (mM -1s-1) ν (MHz) \nSPM Individual NCs capped with polymer 3 MAR 11 60 \nNCs capped with dendrons 4 MAR 349 64 \nNCs capped with DHAA 5 MAR 121 64 \nEndorem a  SDR 99 64 \nEncapsulated in \nmatrices NCs encapsulated in \nDextran 6 SDR 312 64 \nNCs encapsulated in \nliposomes 7 SDR 116 20 \nClusters Citrate-stabilized multi-core \nparticles 8 SDR 365  9.25  \nPVP-stabilized nanospheres \n9 SDR 94  64  \npolystyrene-capped \nnanoclusters 10  SDR 435  128  \nDextran-coated \nnanoclusters 6 SDR 312  64  \nPEG-stabilized nanoflowers \n11  SDR 238  64  \n(Mal)mPEG-PLA \ncopolymer-capped \nnanoclusters 12  SDR 465  60  \namine-stabilized \nnanoassemblies 13  SDR 315  85  \nTREG-stabilized \nnanoclusters 14  SDR 295  400  \n34 nm and 63 nm PAA \ncapped nanoclusters 15, 16  SDR 540 \n630 64 \nFiM Individual nanocubes encapsulated in \nPEG-phospholipids our \nsystem 17  SDR 761 128 \nEncapsulated in \nmatrices liposome-encapsulated \nmagnetic nanocluster 18 .  SDR 1286  100 \nNanocrystal \nassemblies a PAA capped 50 nm \nnanoclusters a SDR 405 60 \nPAA capped 86 nm \nnanoclusters a SDR 508 60 \na results of the present work \n \n  \n1. Q. L. Vuong, J. F. Berret, J. Fresnais, Y. Gossu in and O. Sandre, Advanced Healthcare \nMaterials , 2012, 1, 502-512. \n2. F. Xu, C. Cheng, F. Xu, C. Zhang, H. Xu, X. Xie,  D. Yin and H. Gu, Nanotechnology , 2009, 20 , \n405102. \n3. M. F. Casula, P. Floris, C. Innocenti, A. Lascia lfari, M. Marinone, M. Corti, R. A. Sperling, W. J.  \nParak and C. Sangregorio, Chem Mater , 2010, 22 , 1739-1748. \n4. B. Basly, D. Felder-Flesch, P. Perriat, C. Billo tey, J. Taleb, G. Pourroy and S. Begin-Colin, Chem \nCommun , 2010, 46 , 985-987. \n5. L. Xiao, J. Li, D. F. Brougham, E. K. Fox, N. Fe liu, A. Bushmelev, A. Schmidt, N. Mertens, F. \nKiessling, M. Valldor, B. Fadeel and S. Mathur, Acs Nano , 2011, 5, 6315-6324. \n6. E. K. Lim, E. Jang, B. Kim, J. Choi, K. Lee, J. S. Suh, Y. M. Huh and S. Haam, J Mater Chem , \n2011, 21 , 12473-12478. \n7. M. S. Martina, J. P. Fortin, C. Menager, O. Clem ent, G. Barratt, C. Grabielle-Madelmont, F. \nGazeau, V. Cabuil and S. Lesieur, J Am Chem Soc , 2005, 127 , 10676-10685. \n8. L. Lartigue, P. Hugounenq, D. Alloyeau, S. P. Cl arke, M. Lévy, J. C. Bacri, R. Bazzi, D. F. \nBrougham, C. Wilhelm and F. Gazeau, Acs Nano , 2012, 6, 10935-10949. \n9. S. Xuan, F. Wang, Y. X. Wang, J. C. Yu and K. C.  F. Leung, J Mater Chem , 2010, 20 , 5086-5094. \n10. Y. T. Wang, F. H. Xu, C. Zhang, D. Lei, Y. Tang , H. Xu, Z. Zhang, H. Lu, X. Du and G. Y. Yang, \nNanomed-Nanotechnol , 2011, 7, 1009-1019. \n11. F. Hu, K. W. MacRenaris, E. A. Waters, E. A. Sc hultz-Sikma, A. L. Eckermann and T. J. Meade, \nChem Commun , 2010, 46 , 73-75. \n12. C. Zhang, X. Xie, S. Liang, M. Li, Y. Liu and H . Gu, Nanomed-Nanotechnol , 2012, 8, 996-1006. \n13. K. C. Barick, M. Aslam, Y. P. Lin, D. Bahadur, P. V. Prasad and V. P. Dravid, J Mater Chem , \n2009, 19 , 7023-7029. \n14. D. Maity, P. Chandrasekharan, P. Pradhan, K. H.  Chuang, J. M. Xue, S. S. Feng and J. Ding, J \nMater Chem , 2011, 21 , 14717-14724. \n15. F. Xu, C. Cheng, D. X. Chen and H. Gu, Chemphyschem , 2012, 13 , 336-341. \n16. M. Li, H. Gu and C. Zhang, Nanoscale Res Lett , 2012, 7, 204. \n17. N. Lee, Y. Choi, Y. Lee, M. Park, W. K. Moon, S . H. Choi and T. Hyeon, Nano Lett , 2012, 12 , \n3127-3131. \n18. G. Mikhaylov, U. Mikac, A. A. Magaeva, V. I. It in, E. P. Naiden, I. Psakhye, L. Babes, T. \nReinheckel, C. Peters, R. Zeiser, M. Bogyo, V. Turk , S. G. Psakhye, B. Turk and O. Vasiljeva, \nNature Nanotechnology , 2011, 6, 594-602. \n \n "    },    {        "title": "2107.06597v1.Ultrafast_Optomagnonics_in_Ferrimagnetic_Multi_Sublattice_Garnets.pdf",        "content": "1 \n \n Ultrafast O ptomagnonics in F errimagnetic Multi-Sublattice G arnet s \nAndrzej Stupakiewicz1* and Takuya Satoh2† \n1 Faculty of Physics, University of Bialystok, 15 -245 Bialystok, Poland  \n2 Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152- 8551, Japan \n \nThis review discusses the ultrafast m agnetization  dynamics within the gigahertz  to terahertz  \nfrequency range in ferrimagnetic  rare-earth iron garnets  with different substitutions . In these \ngarnets , the roles of spin– orbit and exchange interactions have been detected  using femtosecond \nlaser pulses via the i nverse Faraday effect. The all-optical control of spin -wave and Kaplan –\nKittel exchange resonance modes in different frequency ranges is shown.  Generation and \nlocalization of the electric field distribution inside the garnet through the metal- bound surface \nplasmon-polariton strongly  enhance the amplitude of the exchange resonance modes.  The \nexchange resonance mode in yttrium iron garnets was observed using circularly polarized \nRaman spectroscopy. The results of this study may be utilized in the development of a wide \nclass of optomagnonic  devices in the gigahertz  to terahertz frequency range.  \n \n1. Introduction \nLaser -induced magnetization dynamics is a flourishing area of research concerned with a \ndeep understanding of previously unexplored and often strongly non- equilibrium phenomena, \nenabling the coherent control of magnetization  on ultrashort  time scales [ 1]. Using modern laser \nlight sources, one can easily witness processes occurring on timescales down to femtoseconds. \nThis field offers an unprecedented capability to probe light –matter  interactions and manipulate \nmagnetism at deep levels. Fundamental magnetic interactions, such as the exchange interaction \nresponsible for magnetic ordering and the spin–orbit coupling responsible for magnetic anisotropy, have  characteristic interaction times ranging from 1 ps to a few femtoseconds [2]. \nThe practical applications include solving the most pressing issues related to m odern \ntechnological developments , future generations of electronic devices, ultrafast processing of \ninformation , and development of spintronics and photonics . \nModern optical and ma gneto- optical methods offer new possibilities to study magnetization \ndynamics  with spatial and temporal resolutions  over a w ide frequenc y range  [3–6]. The laser -\ninduced magnetization dynamics  represent  a fundamentally challenging physics, since they are \ntypically strongly non- equilibrium , and the proper theoretical understanding is complicated by \nthe multitude of couplings between the microscopic constituents of matter. Additionally, there is a fundamental distinction between the mechanisms allowing the m anipulation of the \nmagnetization by means of ultrashort laser pulses, a nd these phenomena can be classified  into \ntwo types. First, the thermal effects, which are  based on the ultrafast demagnetization with \nheating of the medium leading to the change in  its magnetic properties, observed usually in \nmetallic systems [7,8]. Second, non- thermal effects, which operate without heating the medium , \nare usually observed in the form of the  inverse Faraday effect  (IFE)  [9,10] and the \nphotomagnetic effect [11–13] in magnetic dielectrics, such as iron garnets with different dopant s. The non- thermal effect are the most intriguing due to the dissipation- free mechanism \nof magnetization dynamics  and all -optical switching of magnetization with lowest heat load . \nThe non- thermal photomagnetic effect is based on absorption of photons during optical \nexcitation of the strongly anisotropic ions using linear polarization of light. Such excitation allows one to generate an effective field of photo- induced magnetic anisotropy, which in 2 \n \n general can have lifetime much longer than the laser pulse width itself and magnitude \ncomparable to the intrinsic anisotropy; it is sufficient to trigger angle precession of the \nmagnetization. On the contrary, the IFE does not require the  absorption of the pump light in the \nmaterials . The lifetime of the effective magnetic field arising from the IFE is only as long as \nthe pulse width, which is typically couple of tens to hundred femtoseconds. This fact allows the  \nuse of IFE as a mechanism of laser -induced control of magnetization  for fundamental research \nand applications.  \nThe first explanation of all -optical helicity -dependent magnetization switching [ 14] was \nbased on the IFE using femtosecond circularly polarized laser pulses  [15]. However , further \ninvestigations showed that the helicity  dependence of all -optical switching, previously regarded  \nas the confirmation of the non-thermal origin of effects, occurs due to circular dichroism, \nresulting in different heating efficiencies [ 16]. Difficulties with manipulation of magnetization \nin metals by IFE are associated with high absorption in the visible and near -infrared regions of \nspectra and as a result, weak IFE amplitude. However, the idea of using an impulsive magnetic \nfield created by laser pulses for non -thermal manipulation of magnetization by IFE seems to be \npromising for highly transparent and magneto- optically active materials , such as ferrimagnetic \ndielectrics (e.g. , garnets).  \nThe IFE is the creation of an effective magnetic field in magnetic media during the action of \na circularly polarized laser pulse. This effect was originally predicted by Faraday in the second \npart of his work concerning the interaction between light and magnetization  [17]. In later \ntheoretical work s [18,19] , it was shown that circularly polarized light can induce the “ DC” \nmagnetic field  along the propagation direction of the incident light : \n𝐇𝐇(0)=𝑎𝑎[𝐄𝐄(𝜔𝜔)×𝐄𝐄∗(𝜔𝜔)],                                (1) \nwhere  a is the magneto -optical constant . The s ame constant  a is used in the direct Faraday effect \nand indicates magneto- optical coupling. Thus, the process of DC magnetic field generation via \nphotons is called the IFE, which underlines the same magneto -optical constant  presented in Eq. \n(1). The microscopic mechanism of the IFE was explained using impulsive stimulated Raman \nscattering [ 20,21].  This mechanism is based on a laser pulse stimulation of an optical transition \nfrom the ground state to a virtual excited state, which is split due to the spin–orbit coupling. \nThen the same laser pulse stimulates the transition back to the ground state, in which the \nmagnetic state is changed.  \nThe present review consists of five parts: (1) Introduction, (2) Experimental details \ndescrib ing the magnetic ordering in rare earth  (RE)  iron garnets and pump- probe magneto -\noptical method to laser -induced magnetization dynamics ; (3) L aser-induced wide frequency \nrange of magnetization dynamics  in iron garnets , present ing the results of the excitation  of \nprecession of magnetization at different frequency range s corresponding to the spin- wave and \nexchange resonance modes in RE  iron garnets by IFE; (4) Generation and localization of \nmagnetization precession under excitation of the exchange resonance mode in magneto-\nplasmo nic Au/garnets , dedicated to the amplification of amplitude of magnetization  precession  \nwith a nanoscale localization in depth of the garnet  during excitation of exchange resonance \nmode via surface plasmon- polariton (SPP) resonance in Au/garnet structure; (5) Exchange \nresonance mode i n yttrium iron garnet (YIG) , devoted to study of the exchange resonance mode \nin YIG using c ircularly -polarized Raman spectroscopy. \n \n2. Experimental D etails  \n2.1 RE iron garnets  3 \n \n The magnetic order in a pure YIG (Y3Fe5O12) is attributed  to the localized spins of its \nmagnetic ions and their super -exchange interaction with  the surrounding oxygen ions. Due to \nthe spatial overlap of the electron wave functions of the neighboring ions, the oxygen’s 2p states \nmediate the interaction between 3 d5 states of the iron ions close  by [22]. The super -exchange \nof the overlapping orbital s brings about  an extra energy term, whose nature depends  on the \nrelative arrangement of the electron’s spins , the strongest being the super -exchange interaction \nbetween inequivalent tetrahedral  (d) and octahedral  [a] sites [Fig. 1(a )]. As the Fe3+ are arranged \nin two separate tetra - and octahedral sublattices and the spins within each sublattice are \nferromagnetically ordered, one can consider the sublattices to have own magnetizations (of \nvarious magnitude s). However, the iron sublattices themselves are coupled \nantiferromagnetically and the total magnetization is a sum of each sublattice contribution, producing a ferrimagnetic ordering. Various possibilities in garnet substitution, especially by \nmagnetic RE ions in dodeca hedral  {c} sublattice provide the opportunity to vary properties over \na wide range.  \nGenerally, t he f shell can have more electrons , and in some cases , the RE ions produce larger \nmagnetic moments than transition metal (TM ) ions . However, because of their larger ionic radii \nand consequently longer RE -RE distances, the ir coupling is weaker than that of TM ions. This \nleads to differences in the temperature dependence of magnetization of sublattices. Doping with ions having large spin –orbit splitting parameter into dodec ahedral sites can produce an increase \nin the Faraday rotation angle. Generally, the Faraday rotation in Bi -substituted RE iron garnets  \nis strongly dependent on the wavelength in the visible range  [23]. An enhanced Faraday rotation \nis usually accompanied by a large absorption [2 4,25]. In particular, a giant Faraday rotation is \ncaused by an increase in the spin –orbit splitting in the excited states because of the formation \nof hybrid molecular orbitals between the 3d orbital in Fe\n3+ and the 2p orbital in O2-, mixed with \nthe 6p orbital in Bi3+, which has a large spin –orbit interaction parameter. In such materials, the \nmagnetic moment of the RE  sublattice originates from 4f  localized electrons, in contrast to the \niron sublattice, where the magnetism occurs by means of 3 d delocalized conduction electrons. \nConsequently, the RE  sublattice is polarized by the iron sublattice, and the magnetic moments \nbecome antiparallel through exchange interactions. One c an expect that the exchange \ninteraction between  iron sublattices will be stronger than the exchange interaction between iron  \nand RE (e.g., Gd and Yb ions) sublattices. We note here that the exchange between RE \nsublattices will be negligible in comparison to the above.  \nIn our experiments, we used 380- µm-thick  Gd4/3Yb2/3BiFe 5O12 (GdYb -BIG), 200-µm-thick \nGd2BiFe 5O12 (Gd-BIG) , and 160- µm-thick YIG  single crystals  [26–28] . In this review , we also \nconsider 7-µm-thick  Lu1.69Y0.65Bi0.66Fe3.85Ga1.15O12 (LuIG) garnet  grown on GGG substrate \n[29]. In these garnets, Gd, Yb, Lu, Y, and Bi ions are surrounded by eight oxygen ions in a \ndodecahedral sublattice {c}. 4 \n \n  \nFig. 1. (Color online) (a) Garnet lattice with octahedral -[a], dodecahedral -{c} and tetrahedral -(d) sites. \n(b) Visualization sketch of the magnetization precessi on and s chematic illustration of the samples  with \nexperimental geometry.  \n \n2.2 Time -resolved magneto- optical method  \nAs shown in Fig. 1(b), t ime-resolved pump- probe measurements were performed in a \ntransmission geometry to investigate the ultrafast magnetization  dynamics in ferrimagnetic RE \niron garnets  [26,27] . Pump pulses with a duration of 50 fs from an amplifier at a 500 Hz \nrepetition rate were directed at an angle of incidence of approximately a few degrees from the \nsample normal . The probe laser pulses at a repetition rate  of 1 kHz were incident normal to the \nsample. In order to keep off influence of most intens e pump pulse on detector , the propagation \ndirection of the pump pulse is generally not identical to that of the probe puls e. The wavelength \nof the pump beam was adjustable b y an optical parametric amplifier in the near -infrared range \nof 1000–1500 nm, whereas the probe wavelength was fixed at 800 nm. The pump beam with \nan energy density of ~40 mJ/cm2 was focused to a spot approximately 100 μm in diameter , \nwhile the 50 times weaker probe beam on the sample was twice as small  in diameter . The delay \ntime ∆𝑡𝑡 between the pump and probe pulses can be adjusted up to 2.5 ns. The polarization of \nthe pump pulse was adjusted between circular polarization with different helicities. The \npolarization plane of the probe beam was linear. An external in-plane magnetic field 𝐻𝐻∥ was \napplied to the sample. We measured the time- resol ved Faraday rotation angle 𝜃𝜃F of the probe \nas a function of the delay time ∆𝑡𝑡. The angle of the Faraday rotation 𝜃𝜃F is proportional to the \nout-of-plane component of magnetization 𝑀𝑀𝑥𝑥 in a sample.  \n \n3. Laser- Induced Wide F requency R ange of M agnetization D ynamics  in Iron Garnets  \nHere , we will consider  the results of  the non -thermal laser-induced ultrafast magnetization \nprecession in Lu -iron garnet  [29], and our results obtained for  Gd- [27], and GdYb-  [26] iron \ngarnets . The magnetization precession in  the picosecond and nanosecond time scales was \noptically excited via the IFE using the circular polarization of pump  pulses. Two types of \n5 \n \n magnetization  precession with different time scales were observed [ Fig. 1(b )]: a low -frequency \n(ferromagnetic resonance ( FMR) and spin- wave ) mode with a frequency in the gigahertz (GHz) \nrange for a duration of approximately 2.5 ns , and a high- frequency (HF)  (exchange resonance ) \nmode with a frequency in the sub- terahertz (sub-THz) range, which was present immediately  \nafter the initiation of the magnetization precession and lasted for approximately 30 ps. \n \n3.1 Magnetization precession in ferrimagnets  \nThe collective oscillations of excited spins in magnetic materials, such as the magnetization \nprecession or spin waves, can be used to extract the information about the essential magnetic \nproperties  and phenomena . A clear understanding of the oscillations in the ferrimagnet can be \nobtained from the analysis of the equation of motion of each sublattice under an external magnetic field. Generally, the number of possible self -modes of oscillation is equal to the \nnumber of sublattices , which is in turn related to the number of different magnetic ions. The \nsimplest case is a ferrimagnet with two sublattices. Neglecting the anisotropy  field, \ndemagnetizing field, and damping, the equations of motion for the two- sublattice systems 𝐌𝐌\n𝟏𝟏 \nand 𝐌𝐌2 under the external magnetic field 𝐇𝐇  are \n𝑑𝑑𝐌𝐌1\n𝑑𝑑𝑡𝑡=−𝛾𝛾1[𝐌𝐌1×(𝐇𝐇1+𝐇𝐇)], \n𝑑𝑑𝐌𝐌2\n𝑑𝑑𝑡𝑡=−𝛾𝛾2[𝐌𝐌2×(𝐇𝐇2+𝐇𝐇)]. \n(2) \nThe exchange field s are 𝐇𝐇1=−𝜆𝜆𝐌𝐌2 and 𝐇𝐇2=−𝜆𝜆𝐌𝐌1, where 𝜆𝜆 is the molecular field \ncoefficient between the two sublattices. 𝛾𝛾 1,𝛾𝛾2(>0) are the gyromagnetic ratio. We set 𝐌𝐌1=\n�𝑚𝑚1𝑥𝑥𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖,𝑚𝑚1𝑦𝑦𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖,𝑀𝑀1�, 𝐌𝐌2=�𝑚𝑚2𝑥𝑥𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖,𝑚𝑚2𝑦𝑦𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖,−𝑀𝑀2�, and 𝐇𝐇=(0,0,𝐻𝐻∥) and solve the \nequations of motion for 𝐌𝐌1 and 𝐌𝐌2. Here, 𝑀𝑀1>𝑀𝑀2>0 and 𝐻𝐻∥>0. Two solutions of the \nsecular equation are possible, representing two types of oscillati ons [30,31] : FMR  with a \npositive frequency  𝜔𝜔FMR =𝛾𝛾eff𝐻𝐻∥, and exchange resonance  with a negative frequency  \n \n𝜔𝜔ex=−[𝜆𝜆(𝛾𝛾2𝑀𝑀1−𝛾𝛾1𝑀𝑀2)−𝛾𝛾eff′𝐻𝐻∥].                            ( 3) \n \nHere, we also assume that the molecular fields 𝐇𝐇1 and 𝐇𝐇 2 are much stronger than the external \nfield 𝐇𝐇. Further, assum ing a thin film  with the thickness direction parallel to x  axis, uniaxial \nanisotropy field 𝐻𝐻A along x axis, demagnetizing field (demagnetization factor 𝐍𝐍), and in -plane \nmagnetic field 𝐻𝐻∥(>𝐻𝐻A) along z axis, the FMR frequency can be obtained by  the Kittel formula \n[32] \n \n𝜔𝜔FMR =𝛾𝛾eff�[4𝜋𝜋(𝑀𝑀1−𝑀𝑀2)(𝑁𝑁𝑥𝑥−𝑁𝑁𝑧𝑧)+𝐻𝐻∥−𝐻𝐻A]𝐻𝐻∥.   (4) \n \nFor zero -orbital contribution to the magnetic moment of the ion, the gyromagnetic ratio for each \nsublattice can be the same:  𝛾𝛾1=𝛾𝛾2=𝛾𝛾 and 𝛾𝛾eff=𝛾𝛾eff′=𝛾𝛾. These types of precession are \nillustrated in Fig.  2. The parameter  𝜆𝜆dc responsible for the exchange resonance mode is a \nmolecular -field parameter between (d) and {c} [ 22]. 𝜆𝜆dc is typically higher than 𝜆𝜆ac, which 6 \n \n implies a stronger interaction.  A positive  𝜆𝜆dc corresponds to an antiferromagnetic interaction. \nThus , the exchange resonance modes have an antiferromagnetic nature.  \n \nFig. 2. (Color online)  Schematic of (a) FMR  (a) and (b) exchange resonance modes  for a two -sublattice \nferrimagnet . \n \n3.2 Spin- wave mode excitation via the IFE  \nIn order to determine the mechanism and characteristics of the l aser-induced magnetization \nprecession , we carried out measurements as function s of the pump polarization and the static \napplied magnetic field. Such measurements allow one to determine the excite d magnetization \nprecession mode , such as that under  the change in frequency of the precession . The Bi -\nsubstituted iron garnets have the giant static Faraday rotation . Thus , all experimental data, \nwhich are presented below, were obtained using the GdYb -BIG sample at room temperature. \nHowever, the behaviors of the main intrinsic parameters of magnetization dynamics  in the Gd-\nBIG sample at room temperature were similar. Figure 3 shows the experimental geometry and \nthe magnetization precession as a time-resolved Faraday rotation angle for left -handed (σ+) and \nright -handed (σ−) circularly polarized pump  pulses with  in-plane external magnetic field  𝐻𝐻∥=\n1.2 kOe .  \n \nFig. 3. (Color online)  Time -resolved Faraday rotation for different helicities of the circular \npolarizations of the pump beam in an applied magnetic field 𝐻𝐻 ∥=1.2 kOe at room temperature for the \nGdYb -BIG sample.  Inset s show the graphical illustration of the excitation of magneti zation precession \nby effective IFE field H IFE.  \n \n7 \n \n Figure 4(a) shows the linear dependence of amplitude of magnetization  precession for \ndifferent pump laser pulse s. Such behavior is typical for excitation of magnetization dynamics \nvia IFE because this effect does not require photon absorption during laser excitation. For \nsignificant ly higher pulse energies , the effect saturates. The dependence of the magnetization \nprecession amplitude , which was extracted from time- traces measurements,  as a function of a \nquarter -wave plate angle (compensator polarization) for the pump polarization is shown in Fig. \n4(b). The magnetization precessions with opposite phases were triggere d by σ+ and σ− \npolarizations. Generally, the opposite phase precession of magnetization for different pump \npolarization (especially for different helicity of circular polarization) in ferrimagnets is the \nfingerprint of the non- thermal excitation. This result is consistent with a laser -induced IFE \nmagnetic field  HIFE along the direction perpendicular to the sample plane during a femtosecond \nlaser pulse excitation in an iron garnet [9]. In this case, the magnetization vector precesses \naround the effective magnetic field ( Heff) as a result of the action of different contributions: the \nexternal in -plane magnetic field  and magnetic anisotropy field (inset in Fig. 3) . \n \nFig. 4. (Color online)  Dependence of the amplitude of magnetization  precession as a function of (a) \npump fluence and (b) compensator pump polarization  (45° - left-handed cirlular  polarization σ−, 135° - \nleft-handed cirlular polarization σ+, and 90° - linear  polarization  E) at 𝐻𝐻∥=1.2 kOe in the GdYb -BIG \nsample.  \n \nIn order to interpret the nature of the magnetization  precession with respect to FMR mode,  \ntime-resolved measurements of magnetization dynamics have been performed  with different \namplitudes of external magnetic field larger than value of the magnetic anisotropy field HA = \n620 Oe. The transient Faraday rotation as a function of delay time with various 𝐻𝐻∥ are shown \nin Fig. 5. Behavior of these curves suggest the beating effect which can be determined using \nthe analysis of  spectrum  frequenc ies of the  magnetization  precession. For this, w e have \nperformed the fast Fourier transform (FFT) which shows the spectrum of distinct freque ncies \nof the magnetization precession (see the inset in Fig. 5).  \n8 \n \n  \nFig. 5. (Color online)  Time -resolved Faraday rotation as a function of the delay time for different \namplitudes of external magnetic field 𝐻𝐻∥. The red solid lines are fit ted using Eq. ( 5). The inset is the \nFFT spectrum for 𝐻𝐻∥=1.2 kOe.  \n \nAs presented in Ref. 10  for a similar garnet, the excitation and propagation of spin- wave \nmodes using IFE ha s been shown. In such case , the spin- wave excitation in this garnet can \ninduce a spatial distribution of the magnetization orientation within the spot of the pump beam, \nthus leading to a beating precession process. The two frequency components 𝑓𝑓1 and 𝑓𝑓2 \noriginated from the backward- volume m agnetostatic waves (BVMSWs) with wavevectors k. \nThus, the results of laser- induced magnetization  precession can be described as following \nfunction [10]:  \n \n𝜃𝜃F(Δ𝑡𝑡)=𝐴𝐴∫𝑑𝑑𝐤𝐤ℎ(𝐤𝐤)sin(𝐤𝐤⋅𝐫𝐫−2𝜋𝜋𝑓𝑓(𝐤𝐤)Δ𝑡𝑡)exp(−2𝛼𝛼𝜋𝜋𝑓𝑓(𝐤𝐤)Δ𝑡𝑡),    (5)  \n \nwhere 𝐴𝐴 is the amplitude and ℎ(𝐤𝐤) is the FFT of the intensity distribution at the pump spot, and \n𝛼𝛼 is Gilbert damping . The results of the numerical calculations using Eq. ( 5) confirmed the \nresults of the experiment s and the FFT spectrum (inset in Fig. 5).  In addition, e xperimental \ntime-traces datasets were fitted using Eq. ( 5) and are shown in Fig. 5 (red solid lines). With \nincreasing external magnetic field, significant  changes were observed in the two frequencies 𝑓𝑓1 \nand 𝑓𝑓2 over the entire range of applied fields as shown in Fig. 6 (a). In orde r to verif y BVMSW \nmodes , the dispersion dependence for frequency 𝑓𝑓(𝐤𝐤) have been calculated [ Fig. 6(b)]. In \naddition, as shown, BVMSW mode can exist for 𝐤𝐤 perpendicular to the in -plane orientation of \nmagnetic field 𝐻𝐻∥ [33]. \n9 \n \n  \nFig. 6. (Color online) (a) Experimental data for the frequencies correspond ing to spin-wave modes \n(circle dots) and FMR  data (square dots) as a function of the external magnetic field. Red solid line \nrepresents the ferromagnetic resonance frequency results from the calculations using the Kittel formula \nEq. ( 4). (b) Dispersion spectra of the spin -wave mode at 𝐻𝐻∥=4.2 kOe.  Here, 𝑘𝑘 is defined as the inverse \nof the wavelength.  \n \nTo compare the obtained spin- wave mode s, we measured and calculated  the frequency  of \nFMR mode with the  Kittel formula using Eq. ( 4), using 𝐻𝐻A=620  Oe and 4𝜋𝜋𝑀𝑀S=\n4𝜋𝜋(𝑀𝑀Fe−𝑀𝑀RE)=1140  G. The magnetization of RE sublattices 𝑀𝑀RE was defined by the \nmagnetization of Gd sublattice due to the zero contribution of the magnetization Yb ions at \nroom temperature. In Fig. 6(a), it is clearly seen that the frequencies of laser -induced \nmagnetization precession are lower than  the measured and calculated FMR frequency  [Fig. \n6(a)]. \n \n3.3 Laser- induced exchange resonance between iron sublattices in garnets  \nBased on the Néel  theory of ferrimagnetism, antiparallelly oriented non- equal magnetic \nmoments reside in the exchange field created by themselves. In this case, the precession of sublattice magnetizations in the exchange field is the so -called Kaplan –Kittel exchange \nresona nce [ 30]. Typically, in comparison with the FMR, a high value of the exchange field \nleads to a high frequency for such oscillations, which corresponds to the far -infrared (FIR) \nrange. The FIR absorption and transmission spectra were measured for a number o f \nferrimagnets [34,35] , and the exchange resonance frequency was determined. Nevertheless, \nthere is poor information on the optical excitation of the exchange resonance in the visible and NIR regions of the spectrum.  \nThe possibility of the excitation of ultrafast magnetization precession using the IFE without \nheating in LuIG has been reported [ 29]. In this case, the  exchange  resonance mode was excited  \n10 \n \n between iron sublattices (tetrahedral  and octahedral ) 𝐌𝐌d and 𝐌𝐌a in LuIG with a frequency \nrange of 400–700 GHz.  \n \nFig. 7 (Color online) (a) Exchange resonance mode excited  by σ+ and σ− circularly polarized pump \nlight at room temperature . The inset graphs show the amplitude dependence on the pump fluence and \nthe spectral dependence of this mode. (b) Schematic illustration of the excitation of the exchange \nresonance.  The figure is reproduced with permission from Ref. 29  (@2010 American Physical Society).  \n \nThe nature of this mode is  related to the canting of the two sublattices away from their mutual \nexchange interactions. The exchange resonance mode shows a 180°- phase shift between the \nexcitations with σ+ and σ− circularly polarized light [Fig. 7(a) ]. No variations were observed \nin the amplitude or frequency of changes in a sufficiently small external field u p to 3 kOe.  \nIn this case, femtosecond pump pulses with circularly polarized light were used to excite the \nmagnetic -dipole forbidden exchange resonance between the magnetizations of two iron \nsublattices via the IFE. The tetrahedral and octahedral iron sublattices experience inequivalent \noptically induced effective magnetic fields because of the different magneto -optical \nsusceptibilities, resulting in a canting between magnetizations of the sublattices [Fig. 7(b)] . In \nRef. 29, the contribution of dodecahedral ions  was negligible , which play an important role, \nespecially in Gd-substituted iron garnets.  \n \n3.4 Laser -induced exchange resonance between RE and iron sublattices in garnets  \nWe experimentally demonstrated the laser -induced precessional motion of the magnetization \nwith a frequency of approximately 410 GHz in GdYb- BIG at room temperature via the IFE \n[26]. In this HF mode , we observed a precession  of oppotiste phases with respect to the pump \nhelicity  [Fig. 8(a) ]. Figure 8(b) shows the precession dependencies for different amplitudes of \nmagnetic field. In addition, the spin-wave mode was excited along with the HF mode . In a \nduration of  approximately 20 ps, we observed the  simultaneous relaxation of the HF  mode  and \nthe excitation of the spin -wave mode at  a frequency of a few GHz .  \n11 \n \n  \nFig. 8. (Color online) (a) Time -resolved Faraday rotation as a function of delay time for an external \nmagnetic field of 1.2 kOe  induced by σ+ (open points) and σ− (full points) circularly polarized pump \nlight at room temperature  in the GdYb -BIG sample. (b) Exchange resonance frequen cy as a function of \nthe external magnetic field measured at room temperature.  \n \nThe frequency of the HF mode  [Fig. 8(b)] linearly decreased  with increasing intensity of the \nexternal magnetic field. Such dependence agrees with Eq.  (3); however, the maximum \nachievable change in frequency was only 3.8%. The value of slope 1.74 MHz/Oe is lower than \nthat for free electron  motion, i.e., 𝛾𝛾=2.8 MHz/Oe. This dynamic behavior is a consequence \nof the exchange resonance between the magnetic ions in the RE  and iron sublattices  𝐌𝐌RE and \n𝐌𝐌Fe. In this case, the frequency of exchange resonance is lower than the  frequency result ing \nfrom exchange resonance between the iron sublattices (650 GHz at room temperature [29]). In \ncontrast to our results for GdYb -BIG garnet, in LuIG garnet , changes were not observed in the \nexchange frequency for changes in the external field up to 3 kOe [29 ]. In RE garnets with non-\nzero contribution to the magnetization  from the dodecahedral sites at room temperature, the \nexchange field between the iron and RE  sublattices is weaker than that between the octahedral \nand tetrahedral iron sublattices. Therefore, influence of external magnetic field in relation to \nexchange re sonance between RE and iron  sublattices will be weaker than in case of the spi n-\nwave or FMR modes [ Fig. 6(a) ]. \n \n3.5 Temperature dependence of  the exchange resonance mode  \n12 \n \n Because of the high value of the exchange field, the  dependence on the external magnetic \nfield is not strong. Hence , the variation in temperature seems to be a suitable way to confirm \nthe excitation of such modes as a consequence of the temperature dependence of the exchange \nfield.  \nTo confirm the prediction related to the excitation of the exchange resonance mode, \ntemperature experiments were performed. Figure 9 shows the time- resolved Faraday rotation \ncurves measured with the Gd -BIG sample at the picosecond time scale for different \ntemperatures. This sample was simple for temperature investigation owing to the single RE  \nsublattice of Gd. For temperatures l ower  than T = 100 K, the modulation of the spin- wave mode \nin the HF sub -THz mode  can be observed. The time decay of the HF mode over temperatures \nranging from 10 K to 70 K was approximately 20 ps. After 20  ps, the HF precession relaxed, \nand the spin- wave mode was simultaneously excited, decaying at approximately 100 ps at 12 K.  \nThe initial phases of the  HF and spin -wave modes heavily depend on the helicity of the circular -\npump polarization, which is typical for excitations via the IFE  (Fig. 2 in Ref. 27) . \n \nFig. 9. (Color online)  Time -resolved Faraday rotation for different temperatures  in Gd -BIG sample.  \n \nIn general, RE iron garnets  can be treated as two -sublattice ferrimagnets, one made up of Fe  \nions occupying both tetrahedral (d) and octahedral [a] sites ( 𝑀𝑀d−𝑀𝑀a=𝑀𝑀Fe) and the other of \nRE ions occupying dodecahedral {c} sites  (𝑀𝑀c=𝑀𝑀Gd). In our case, total magnetization is given \nby 𝑀𝑀Fe−𝑀𝑀Gd. We obtained the temperature dependence of the Gd and Fe sublattice \nmagnetizations using molecular field theory with a single molecular -field coefficient [ 27]. The \nGd ions exhibit antiferromagnetic exchange coupling with the Fe ions , and the Gd ions tend to \nalign their spins against the net moment of the Fe ions. Because this coupling is relatively weak, \nthe Gd ion ordering is significant only at low temperatures (<60  K), at which the Gd sublattice \nis much more strongly magnetized than the Fe sublattice [ 27].  \nAs the temperature decreases below 𝑇𝑇comp , the frequency of the HF mode tends to increase \nwith an increasing 𝑀𝑀s, which suggests an excitation of the Kaplan–Kittel exchange resonance. \nIn the Gd- BIG sample, this resonance exists becau se of the exchange interaction between the \n13 \n \n Gd and Fe sublattices. T he magnetization s of these sublattices are noncollinear (inset in Fig. \n10) during the entire precession (Fig. 9). Below 60 K, the Fe ions are strongly coupled, and \nexcitations between them are difficult [22]. Thus, in garnets , the Gd –Fe coupling is relatively \nweak, and the Gd–Gd coupling is even weaker and hence neglected. The m agnetization \nprecession in our garnets occurs in a molecular field close to 500 kOe  [36,37].   \nBy fitting the time -resolved Faraday rotation data (Fig. 9) with two frequencies, we were \nable to determine the temperature dependence of both mode frequencies. As the total \nmagnetization increased , the HF mode increased in frequency within the  GHz –THz  range via \nexchange  resonance excitation (see Fig.  10). \n \nFig. 10. (Color online)  Temperature dependence of the exchange resonance mode (points) for the Gd -\nBIG sample. The solid line 𝑓𝑓ex(𝑇𝑇) was obtained using Eq. ( 3). \n \nThe temperature dependence of the HF mode provides direct evidence of this resonance. We \ncan calculate the temperature dependence of the HF mode using Eq. ( 3) for the exchange \nresonance between the Fe and Gd sublattices. The gyromagnetic ratio 𝛾𝛾 is the same for these \nsublattices ( 𝛾𝛾=𝛾𝛾Gd=𝛾𝛾Fe) because for Gd ions, 𝐿𝐿=0. The red solid line in Fig. 10  is calculated \nusing Eq. ( 3). The experimental data and theoretical calculations show a good agreement . \n \n4. Generation and L ocalization  of M agnetization  Precession under E xcitation of the \nExchange R esonance M ode in M agneto- Plasmonic Au/ Garnets  \nMagneto -plasmonics is one of the most promising manipulation methods of spins in the \nnanometer localization in space [3 8,39]. In magneto- plasmonic dielectric -based \nheterostructures , the non -magnetic (noble) metallic layer provides high -quality collective \nresonances of the free electron gas, whereas magnetic properties are introduced usually by the \niron garnets with different substitutions  [6]. The la tter enables external control of the interface \nresonances by means of either static magnetic field or the laser -induced effective magnetic field \ngenerated via the IFE [15,20,21,26,27] . In Au/garnet heterostructure , two materials with very \ndistinct timescal es are combined. The transient electronic response of Au is fast, on the sub -\npicosecond timescale, whereas garnet lacks free electrons and its relatively slow magneto -\noptical response is tied to the spin and phonon subsystems.  \n14 \n \n In the previous part of this  review , we dem onstrated that a circularly polarized light pulse \nimpulsively excite s magnetization precession in GdYb- BIG crystal with HF  of exchange \nresonance mode  at about 410 GHz . We assumed that the IFE-induced magnetization dynamics  \ncan be enhanced with a localization of the electromagnetic field in such garnet upon the SPP \nexcitation.  For that, w e studied the SPP- assisted magnetization dynamics in a magneto -\nplasmonic crystal consisting of 380 μm-thick GdYb -BIG crystal which was comp lemented with \na periodically perforated 50 nm-thick Au overlayer (800 nm period), allowing for the excitation \nof the SPP  mode at the Au/air and Au/garnet interfaces  [inset in Fig. 11(a) ]. In experiments we \nused the magneto- optical pump- probe technique with 60 fs temporal resolution. The pump \nbeam wavelength was tuned in the near -IR range  [Fig. 11(a) as a purple stripe] , while the probe \nbeam wavelength was fixed at 800 nm  [red spot on Fig. 11(a) ]. The SPP dispersion is show n in \nFig. 11(a). By tuning the wave length and the incidence angle, SPPs can be excited,  thus \ncontrolling the localization of the hotspot [ Fig. 1 1(a)]. Once the magnetization precession in a \ngarnet  is excited via the IFE, the delayed probe beam meas ures the transient Faraday rotation . \nFigure  11 (b) shows the dependences of HF  precession of exchange resonance mode at \ndifferent values of the pump wavelength, and the angle of rotation of the quarter -wave plate for \npump pulse . The corresponding values are marked with colored circles in Fig. 11(c) . Figure \n11(c) show s the dependences of the amplitude and phase of the magnetization  precession on \nthe angle of rotation of the quarter -wave plate for two pump wavelengths: 1300 nm (no \nresonance) and 1380 nm (the vicinity of the SPP resonance at Au/garnet interface).  In the non-\nresonant case (black curves), magnetization  precession should be excited in the volume of the \nferrite garnet due to the ellipticity of the pump pulse . Accordingly, the precession amplitude \nwill be zero with linear pump polarization , and will increase with a deviation from the zero \nposition of the quarter -wave plate, while the phase will change by π for opposite angles of \nrotation of the plate [ Fig. 11(c) ]. On the other hand, upon excitation SPP, the precession of \nmagnetization will be  excited in the subsurface layer due to the effective plasmon field. Indeed, \nat a pump wavelength of 1380 nm (red curves), the character istics of the dependences change \n[Fig. 11(c) ]. This behavior can be explained by the fact that when ellipticity is added to the \npolarization of the pump pulse, in addition to the magnetization precession in the near -surface \nlayer, the magnetization  precession is also excited in the garnet. The phases of these excitations \nare different;  therefore, the resulting angle of the Faraday rotation is determined by their \ninterference.  \n 15 \n \n  \nFig. 11. (Color online) (a) SPP dispersion map of an Au/GdYb-BIG garnet heterostructure. The purple \nstripe and the red spot show the spectral tunability of the pump and probe pulses, respectively . (b) Time -\nresolved Faraday rotation with the background removed,  together with the fit lines for linear pump \npolarization at different pump wavelengths. The green data points show the magnetization dynamics  \nexcitation with circularly polarized light via IFE. (c) The amplitude and phase of the magnetization  \nprecession as a function of the incident pump polarization, where λ/4 wave plate angle of 0° corresponds \nto the p -polarized excitation. The colored circles correspond to the data shown in (b) with respecti ve \ncolors.  The figure is reproduced with permission from Ref. 40 (@2018 American Chemical Society).  \nTo clearly  demonstrate the effect of SPP excitation on the amplitude and phase of \nmagnetization  precession, we measured their dependence on the pump wavelength. By \nmeasuring the variations in the Faraday rotation of the probe pulse  with the time delay between \nthe pump and probe, we extract ed the amplitude and phase of the exchange resonance mode at \ndifferent pump -pulse wavelengt hs. The amplitude spectrum of the exchange resonance mode is \nshown in Fig. 12 for circular and linear pump polarizations.  \nSpectral dependences of the circularly polarized pump pulse we re monotonic and show ed \nno resonance features. On the contrary, with a l inear ly polariz ed pump pulse both in the \namplitude and phase of precession, strongly pronounced features  were observed near the SPP \nexcitation (pump wavelength of 1400 nm, see Fig. 12). The precession phase at resonance \nchanges by π, which is equivalent to a change in the sign of the amplitude (solid and dashed \ncurves in Fig. 12). We note that in near 1200 nm wavelength, a feature is also observed in the \nprecession phase and amplitude. In this case, the excitation of two different SP Ps should occur \nat different interfaces [ 40]. \n16 \n \n  \nFig. 12. (Color online)  Phase and amplitude of exchange resonance mode versus the wavelength for C-\ncircular (black curve) and P -linear (red curve) pump pulse polarizations.  The phase shift of π is identical \nto the amplitude sign changes, as shown by red dashed curves.  The figure is reproduced with permission \nfrom Ref. 40 (@2018 American Chemical Society).  \nHere, unlike all the previously demonstrated examples, the SPP affects the sub -THz coherent \nmagnetization  dynamics  by the spatial confinement of the IFE in a metal –dielectric hybrid \nstructure. As we demonstrated experimentally and through modeling, the SPP electric field \nrotates in the incidence plane as the SPP propagates along the Au/GdYb- BIG interface and \ninduces a static transversal IFE magnetic field localized on the scale of 100 nm  [40]. In the case \nof circular pump polarization, magnetization  precession is excited in depth of a garnet crystal \nwith a thickness of 𝑑𝑑bulk = 380 μm (absorption in this wavel ength range is low). On the other \nhand, as shown by numerical calculations [ 40], with SPP- excitation at the Au/garnet interface, \nprecession should be efficiently excited in a near -surface layer with a depth of about 𝑑𝑑SPP≈ \n100 nm. Taking into account the d ifference in precession amplitudes (Faraday rotation  ∆𝜃𝜃) with \nresonant and non- resonant excitation ∆𝜃𝜃SPP/∆𝜃𝜃bulk≈0.1, we obtain following relation \nbetween the precession excitation efficiencies with and without SPP excitation: \n𝑑𝑑bulk𝑑𝑑bulk\n𝑑𝑑SPP∆𝜃𝜃SPP\n∆𝜃𝜃bulk≈400.  Taking advantage of the tunability of the pump wavelength, the \nexcitation efficiency at the SPP resonance was enhanced by two orders of magnitude.  \n \n5. Exchange R esonance M ode in YIG  \nAmong other rare -earth iron garnets, YIG single crystal possesses superior properties, such \nas the narrowest  FMR linewidth ( α≈10−5) and a Curie temperature T c of 560 K, and has been \nregarded as an indispensable magnetic material [4 1–45]. Y3+ ions in the dodecahedral {c} sites \nare non -magnetic. Therefore, YIG form s a two-sublattice ferrimagnet . Previous studies on YIG \nhave mostly focused on FMR or acoustic magnons  in the GHz  range. Limited information has \nbeen provided about the Kaplan–Kittel exchange resonance mode in the THz frequency range \n[41]. Recently, se veral theoretical investigations have revealed full magnon band structures \n[46–50]. Only inelastic neutron scattering spectroscopy has been experimentally identified for \nTHz magnons in YIG [5 1–54]. Therefore, the THz -magnon should be investigated using Raman \nspectroscopy.  \n17 \n \n Raman spectroscopy was performed in a 180°  backscattering geometry with a 785 nm \nexcitation laser. The polarization of the incident and scattered light is denoted as parallel ( ∥) \nand crossed ( ⊥) for linearly polarized configurations and RL, LR, RR, and LL for circularly \npolarized configurations. The hand of the circularly polarized light (R or L) was defined as the \nsense of polarization rotation in the sample plane, regardless of the direction of propagation.  \n \nFig. 1 3. (Color online)  Raman spectra in the 240 –270 cm−1 range with (a) RR - and (b) LL -polarized \nconfigurations in the 80– 140 K temperature range in YIG  (111). The red, green, and purple curves fit \nthe magnon and two T 2g phonons, respectively. The blue curve denotes the sum of t he red, green, and \npurple curves. The figure is reproduced with permission from Ref. 28 (@2020 American Physical \nSociety).  \n \nThe Raman spectra in the 240 –270 cm−1 range are depicted in Figs. 13(a) and 13(b) for the \nRR- and LL -polarized configurations, respectively [ 28]. In both configurations, a tiny signal \nwas observed and assigned to the resonance exchange mode because of the significant temperature dependence of the frequency shift. At 80 K , the Raman shift was 260 cm\n−1, which \ncorresponds to 7.8 THz. \nIn Fig. 14(a), the frequency shifts of the THz magnon in the RR - and LL -polarized \nconfigurations are plotted as a function of temperature [ 28] and compared with the results \nobtained from the neutron scattering measurements [5 1–54 ] and simulations [ 49,50].  The \ntendency is the same, but a frequency deviation exists between the Raman results and the others. \nRaman spectroscopy provides a more precise frequency resolution than other techniques. A \nfrequency of 8.0 THz at 4 K was obtained by extrapolating the magnon frequency using the \ntemperature dependence of magnetization [ 55]. Equation ( 3) is equivalent to (12𝑆𝑆d−\n8𝑆𝑆a)|𝐽𝐽ad|=10|𝐽𝐽ad|, where a - and d- site spins are 𝑆𝑆a=𝑆𝑆d=5/2 and 𝐽𝐽ad is the exchange \nconstant between the a-  and d- site sublattice. This yields 𝐽𝐽ad=−38 K according to 10|𝐽𝐽ad|=\n8.0 THz.  \n18 \n \n  \nFig. 1 4. (Color online)  Temperature-dependent frequency shift in the resonance exchange mode  (a). The \nred and black lines with error bars denote the exchange resonance mode observed in our Raman spectra \nwith RR - and LL -polarized configurations, respectively. S election rule for the resonance exchange mode \nof YIG observed with crossed -, RR-, and LL-polarized configurations at 260 cm−1 (7.8 THz) and 80 K  \n(b). The figure is reproduced with permission from Ref. 28 (@2020 American Physical Society).  \nThe Raman selection rule was investigated  at 80 K , as shown i n Fig. 14(b). The Raman \nintensity ratio of the resonance exchange mode was [𝐼𝐼∥:𝐼𝐼⊥:𝐼𝐼RL:𝐼𝐼RR]=[0:1:0:1]. This finding \nis consistent with the magnon Raman tensor in antisymmetric form as \n𝑅𝑅=�0𝑖𝑖𝑚𝑚𝑧𝑧\n−𝑖𝑖𝑚𝑚𝑧𝑧 0�,      (7) \nwhich corresponds to linear (first -order) magnetic excitations [56–60]. The consistency of the \nexchange resonance mode to the FMR mode excited through light scattering indicates that these \nmodes possess the same symmetry. In the exchange resonance mode, all the atomic spins \nbelonging to the same sublattice simultaneously rotate with their spins always parallel and with \nthe same amplitude, which is analogous to the FMR mode. The spins in the other sublattice s \nbehave in the same manner. Therefore, the exchange resonance and FMR modes correspond to \nthe A2 mode [ 61]. However , unlike the FMR mode, the net magnetization 𝐌𝐌a+𝐌𝐌d of the \nexchange resonance mode did not occur. Therefore, the exchange resonance mode is considered \nto be unobservable via optical methods [ 62]. Nevertheless, the exchange resonance mode was \nobserved in the Raman method as a linear magnetic excitation, which usually results from the \n19 \n \n precession of 𝐌𝐌. This is because the Faraday rotation is more sensitive to the a -site Fe sublattice \nthan the d- site Fe sublattice. The exchange resonan ce mode resulted from the precessional \nmotion of 𝐌𝐌 a [29]. \n \n6. Conclusions  \nIn this review, we have presented the experimental studies and theoretical approaches in RE  \niron garnets by using ultrafast time -resolved magneto -optical spectroscopy and Raman \nspectroscopy . Technology for fabricating high- quality garnets at thicknesses of  micro - to nano -\nscales provides an extremely wide variety of compositions. Using magnetic RE-ions \nsubstitution for YIG at dodecahedral sites (e.g. Gd, Yb, Lu) in the garnet matrix enables the \nmagnetic ordering and modes of magnetization  precession to be varied. We reviewed the non-\nthermal optically excited precession  of magnetization via the IFE and Raman process in RE \niron garnet single crystals.  \nWe show ed that different frequency modes of the magnetization  precession in the GHz and \nTHz ranges strongly depend on the polarization of the pump laser and amplitude of external \nmagnetic field. The low frequency modes in the GHz  regime correspond to the spin wave  in \nBi-substituted iron garnets excitation via the IFE. The HF  mode is related to a Kaplan –Kittel \nexchange resonance mode between magnetic sublattices in ferrimagnetic garnets. The \nmechanism underlying the exchange interaction plays an important role in understanding the \nphysical origin of magnetism in materials on the atomic level.  \nA novel approach was using of the garnet crystal with exchange resonance mode at  range of \nsub-THz frequency to study of SPP -driven of magnetization  precession with a nanoscale \nlocalization.  Our experiments and modeling verify the localization of the magnetization \ndynamics  excitation within a 100 nm -thin layer of a transparent dielectric garnet. Taking \nadvantage of the tunability of the pump wavelength, we demonstrated two orders of magnitude \nenhancement of the excitation efficiency at the SPP resonance. Combining spatial localization \nwith the non -thermal spin –photon coupling, our results open the way toward high- density, low -\nloss ultrafast optomagn onics . \nA wide frequency range of magnetization  precession from  GHz up to THz and large spatial \npropagation variations may be useful for wavenumber, magnetic resonance, and dispersion control, as well as offering options for tuning the ultrafast dynamic response with a giant Faraday rotati on. Our results demonstrated near -infrared  pulse -induced phenomena that are \nusually observed in the far -infrared range of energies corresponding to atomic interactions in \ncondensed matter. Many related issues remain open for further investigation in optoma gnonics . \n \nAcknowledgment  \nA.S. acknowledge s support from the grant s of the  Foundation for Polish Science \nPOIR.04.04.00- 00-413C/17- 00 and the National Science Centre of Poland (Grant DEC -\n2017/25/B/ST3/01305). 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"    },    {        "title": "2401.08235v1.Spin_Waves_in_Ferrimagnets_near_the_Angular_Magnetization_Compensation_Temperature__A_Micromagnetic_Study.pdf",        "content": "SW in FiMs near the AMC Temperature AIP/123-QED\nSpin Waves in Ferrimagnets near the Angular Magnetization Compensation\nTemperature: A Micromagnetic Study\nLuis Sánchez-Tejerina,1David Osuna Ruiz,2Eduardo Martínez,3Luis López Díaz,3and\nÓscar Alejos1\n1)Department of Electricity and Electronics, University of Valladolid,\nSpain.\n2)Departament of Electrical, Electronical and Communications Engineering,\nPublic University of Navarra, Spain.\n3)Department of Applied Physics, University of Salamanca,\nSpain.\n(*Electronic mail: oscar.alejos@uva.es.)\n(Dated: 17 January 2024)\nSpin wave propagation along a ferrimagnetic strip with out-of-plane magnetization is stud-\nied by means of micromagnetic simulations. The ferrimagnetic material is considered as\nformed by two antiferromagnetically coupled sub-lattices. Two critical temperatures can\nbe defined for such systems: that of magnetization compensation and that of angular mo-\nmentum compensation, both different due to distinct Landé factors for each sub-lattice.\nSpin waves in the strip are excited by a spin current injected at one of its edges. The\nobtained dispersion diagrams show exchange-dominated forward volume spin waves. For\na given excitation frequency, Néel vector describes highly eccentric orbits, the eccentric-\nity depending on temperature, whose semi-major axis are oriented differently at distinct\nlocations on the FiM strip.\n1arXiv:2401.08235v1  [cond-mat.mtrl-sci]  16 Jan 2024SW in FiMs near the AMC Temperature\nMagnon propagation in antiferromagnetic (AFM) and/or ferrimagnetic (FiM) has drawn atten-\ntion recently and is the subject of very recent works1–9due to the variety of benefits as opposed to\nusing ferromagnetic (FM) materials, such as greater speeds, marginal sensitivity to external fields\nand the wider range of materials available, many being electrical insulators. Spin wave propaga-\ntion in insulating AFM materials connects magnonics with spintronics since spin current (a key\nconcept in spintronics) is mainly transported by magnons in such materials, due to the absence\nof conducting electrons as opposed to FM materials, where electrons are the main spin carriers.\nThe use of insulators also points to the importance of thermal effects due to local heating. The\ndynamics of spins in AFM materials have been widely investigated theoretically. For example,\ncharacterising the fundamental k=0 resonance modes2–4, propagating magnons5,6or AFM do-\nmain walls (DWs) and their interactions with magnons7,8, but always focused on their fundamental\nphysics, as AFM magnonics is a field still in its infancy2,9,10. Additionally, FiM materials at the an-\ngular momentum compensation point ( TA) can mimic the dynamical response of AFM11–13while\nbeing easier to manipulate and detect by electric and/or optical means14,15.\nIn this work, micromagnetic simulations have been used to investigate the propagating high-\nfrequency ( ≈THz) AFM modes in a FiM strip at different temperatures. Simulations consider\nFiMs as formed by two sub-lattices (indexes i=1,2 will be used to refer to each sub-lattice).\nAccordingly, a pair of coupled Landau-Lifshitz-Gilbert equations are used to model the material:16\n˙mi=−γimi×Heff,i+αimi×˙mi+τSOT ,i (i=1,2), (1)\nwhere mi,γiandαiare the local orientation of the magnetization, the gyromagnetic factor, and\nthe damping parameter for each sub-lattice, respectively. Heff,iis the effective field including all\nrelevant interactions within the system, and τSOT ,icorresponds to the spin-orbit torque (SOT) due\nto spin currents. In our model, the FiM is formed by computational elementary cells, in a finite\ndifference scheme. We have two magnetic moments within each cell, one for each sub-lattice, and\nboth sub-lattices are coupled by an interlattice exchange interaction (see details in Ref.16).\nSimulations have been carried out by means of a homemade code17implemented on graphic\nprocessing units. Details of the modeled device are depicted in Fig. 1. FiM strip parameters\nconsidered are those that can be found in the literature for a prototypical FiM as GdFeCo11. Sat-\nuration magnetization for each sub-lattice varies with temperature according to the law Ms,i=\nM0\ns,i\u0010\n1−T\nTC\u0011bi,TC=450 K being the Curie temperature, M0\ns,ibeing the saturation magnetization\nat 0K: M0\ns,1=1.71MA\nmM0\ns,2=1.4MA\nm, and b1=0.76, and b2=0.5, resulting in the tempera-\n2SW in FiMs near the AMC Temperature\nX YZ\n32 nm𝐽𝑡=𝐽0sinc 2𝜋𝑓𝑐𝑡−𝑡𝑐8 nm8 nmprobes\n𝑡=0𝐽0=1TA\nm2\n𝑡\n𝑡𝑐=0.5 ns\n𝑓𝑐=1 THz\n𝐉 =𝐽𝑡𝐮𝑦\n(a)\n(b)\n(c)\n𝐉s\nFIG. 1. Geometry and properties of the modeled device: (a) Temperature dependence of the saturation\nmagnetization for each sub-lattice, (b) dimensions of the FiM strip and of the exciting spin current injection\narea, (c) time dependence of the exciting current to obtain the dispersion diagrams.\nture dependence plotted in Fig. 1(a). According to these parameters, magnetization compensation\noccurs at TM=241 K. The rest of significant parameters are: intralattice exchange Ai=70pJ\nm, in-\nterlattice exchange Bi j=−9MJ\nm3, damping constant αi=0.001, anisotropy constant Ku,i=1.4MJ\nm3,\nDzyaloshinskii-Moriya constant Di=0.12mJ\nm2, and spin-Hall angle θSH=0.15. Finally, each sub-\nlattice is characterized by distinct Landé factors: g1=2,g2=2.05, which result in a temperature\nof angular momentum compensation TA=260K, slightly greater than TM.\nThe device consists of a FiM strip and a current line of a heavy metal (HM) running perpendicu-\nlar to the strip at one end as shown in Fig. 1(b). The FiM material is initially uniformly magnetized\nin the out-of-plane direction. The FiM strip area is 8192 nm ×32 nm, and it is 6-nm thick, while\nthe HM strip is 32-nm wide. This defines a 32 nm ×32 nm SW excitation area on the leftmost end\nof the FiM strip due to an electric current flowing through the HM line, then exciting spin waves\nvia SOT. Beside this area, a set of 254 probes are regularly spaced on the FiM strip to monitor the\nvalue of the local magnetization, i.e., Néel’s vector n=m1−m2. Probes are 3 nm ×8 nm in size\nand 5-nm distant from each other, then occupying a total length of 2032 nm.\nA spin current arising from an electric current carried by the HM strip is injected through the\nexcitation area. This spin current is polarized along the ±uxdirection depending on the instanta-\n3SW in FiMs near the AMC Temperature\nneous direction of the electric current along the HM strip. Figure 1(c) presents the time dependent\ncurrent considered to achieve a uniform excitation across a desired frequency range in the dis-\npersion diagram. In this first case, an (unnormalized) sinc-shaped electric current pulse has been\nused, J(t) =J0sinc[2πfc(t−tc)], where fcis the excitation cut-off frequency, which was set to\n1 THz, J0is the pulse amplitude, initially set to 1TA\nm2, and tcis a delay time which has been chosen\nas one half of the total simulation time, set to 1 ns. Using this activation, we ensure that each mode\nis equivalently fed. This current is sufficiently small to remain in the linear regime of activation of\ndamped, stable oscillations3and to avoid any static changes in the AFM phases of the sample. The\ndelay time also provides a reasonable offset to the peak of the pulse, allowing a gradual increase\nof the amplitude from the beginning of the simulation. However, when later analyzing the time\nevolution of the magnetic signal, a monochromatic wave (MW) excitation is to be applied with\na current at a specific frequency f0, i.e., J(t) =J0sin(2πft). Equivalently to the previous case,\nJ0is chosen to be 1TA\nm2to remain in the same modes regime and to obtain a good magnetization\ncontrast for m1,2(t). Finally, temperature is taken here as an additional parameter, i.e., a way to\nbalance the value of the saturation magnetization of both sub-lattices to achieve both magnetiza-\ntion compensation and angular momentum compensation. This means that only coherent magnons\nare considered in our simulations.\nSW spectra in AFM materials have revealed ultra-high frequency (in the THz regime), field-\ndependent modes that are related to each of the sub-lattices (strongly coupled by exchange) usu-\nally called ‘optic’ or higher frequency mode, and ‘acoustic’ or lower frequency mode2,3. For\neach mode, there are opposite senses of precession for each sub-lattice, either right-handed (RH)\nor left-handed (LH). The AFM resonance frequencies ( k=0 magnons) depend on the different\nequilibrium states of the coupled sub-lattices, or so-called AFM phases2, which can also be tuned\nby external fields. These modes can be locally excited via spin currents Js, as represented in\nFig. 1(b). The localized perturbation allows AFM magnons with k̸=0 to propagate in the material\nvia exchange interactions, consequently possessing very short wavelength. To investigate the SW\npropagation in the frequency domain, we analyze the Néel vector since it transports coherently\nexcited magnons, such as those generated by Js18.\nWe focus on the in-plane components of the Néel vector nxandnywhich account for the\ndynamic (differential) x-component and y-component respectively from the two sub-lattices at\nthe closest position to the excitation region. Figure 2(a) shows the SW spectra (FFT intensity of\nnx) and fundamental resonances at k=0 for different strip temperatures ranging from below TMto\n4SW in FiMs near the AMC Temperature\n(a)(b)\nFIG. 2. (a) SW spectra (FFT intensity of nx) and fundamental resonances at k=0 for different strip temper-\natures, and (b) dispersion relations obtained for nyat four different temperatures in the strip ranging below\nTMto above TA. The regularly spaced interruptions in the dispersion curves are related to the geometry of\nthe excitation zone (see text).\nabove TA, as a response to the sinc-shaped pulse current. The simulated (normalised) spectra are in\ngood agreement with recent work characterising such modes in FiMs19, where the two distinctive\npeaks overlap at TA. Similar results are obtained for ny(not shown), but with greater amplitude\nthan nxatTA, and generally wider peaks, suggesting an elliptical precession of m1,22. In contrast\ntony,nxis more attenuated at TAthan at any other temperature. In particular, Fig. 2(b) shows the\ndispersion relations obtained for nyat four different temperatures in the strip ranging below TM\nto above TA. The color scale is the same for all color plots, so intensities are directly comparable\nbetween different temperatures. The exchange-dominated character of the SWs is clear, in the\nform of a exchange-dominated k2-dependent mode frequency20. Solid curves show the fitting to\nthe data from Kalinikos and Slavin’s theory20, when each sub-lattice magnetization is modeled as\nof separate effective FM, i.e. as in a synthetic antiferromagnet (SAF), and MS,i(Tk). A Forward\nV olume SW (FVSW) configuration is assumed, which corresponds well to the initial out-of-plane\nconfiguration, where m1,2are perpendicular to both kand the strip plane at equilibrium. When\nthe AFM resonance frequencies ( k=0) for each sub-lattice (see Fig. 2(a)) are included as the cut-\nofffrequencies into the FVSW equations, the agreement with the simulated dispersion relations\nis remarkable (green curve for i=1 and red curve for i=2). Such dispersion relations can be\nanalytically described as2\nωi=γi\u0012\nHc,i+2Ai\nµ0MS,ik2\nx\u0013\n(i=1,2), (2)\nwith Hc,i=q\n2Hanis ,iHexch ,i+H2\nanis ,i,Hexch ,i=|Bi j|\nµ0MS,ibeing the interlattice exchange field, and\n5SW in FiMs near the AMC Temperature\nT<TM\nTM<T<TA\n0 0.5 1 1.5 2T≈TA\nnx,ny(a.u)nx\nnynx,ny(a.u)nx\nnynx,ny(a.u)\nx(µm)nx\nny\nFIG. 3. Typical instantaneous values of nxandnyat the location of each probe. The graphs shows the\nresults obtained at t=1 ns at different temperatures: T<TM,TM<T<TAandT≈TA.\nHanis ,i=2ku,i\nµ0MS,ibeing the anisotropy field. According to these expressions, the pair of dispersion\ncurves merges as the temperature approaches TA, as Fig. 2(b) confirms. It should be mentioned at\nthis point that the dispersion curves are regularly interrupted for the various multiples of a certain\nkx≈100rad\nµm. This modulation is caused by the spatially limited excitation zone 32 nm-wide. Note\nthat at the edges of the excitation zone, the spin accumulation vanishes giving a half-wavelength\nofλ\n2=32 nm corresponding to a k-vector with kx=2π\nλ.\nWith the purpose of analyzing the time evolution of the magnetic signal under a MW excitation,\nthe following results presented correspond to an excitation via a MW signal of frequency f=\n360 GHz. Although other frequency values have been considered, this one has been found to be\noptimal for revealing magnon propagation characteristics along the FiM strip. In particular, this\nvalue is slightly higher than the cut-off frequency ( k=0 magnons) for all of the temperatures\nconsidered. The intersection of this excitation frequency with the dispersion curves shows how\nthe wave vector is in general different for each of the two sub-lattices. Accordingly, SWs travel\nwith different phase velocities through each sub-lattice. This fact is illustrated in the graphs in\nFig. 3. The plots show projections along the x- and y-directions of the Néel vector orbits. Plots\nare obtained at the end of the excitation time to ensure that the stationary regime is achieved. The\ncoupled oscillations between both sublattices but with different kxgenerate patterns of nodes and\nanti-nodes of nxandnyalong the FiM strip. All patterns possess anti-nodes of nyat the closest\nzone to the excitation area in agreement with the injected spin current. In addition to the oscillatory\n6SW in FiMs near the AMC Temperature\nbehavior of the two components, their amplitudes are also modulated by the progressive vanishing\nof the amplitude of the oscillations as the SWs travel along the strip. To get a clearer idea of the\nmagnon propagation, a set of videos showing the evolution in time of the plots in Fig 3 has been\nadded as part of the supplementary material.\nWe can further analyze the consequences of the phase velocity difference between the two sub-\nlattices. Fig.4 represents the in-plane components of the Néel vector at different positions along\nthe strip for three different temperatures: below TM, in-between TMandTA, and at approximately\nTA. These trajectories are always elliptic, their eccentricity being determined by temperature and\nreaching a maximum at TA. Fig.4 also shows that the strip position plays no role in the eccentricity,\nthough the semi-major axis takes different orientations depending on location. The Néel vector\ntrajectory is composed of the two magnetization sub-lattice precessions. Because the phase ve-\nlocity of each sub-lattice differs, the relative phase between the two precessions is modified from\npoint to point, thus modifying the semi-major axis orientation. Consequently, different positions\nare subjected to different torques, not only by the amplitude fading but because of this reorienta-\ntion effect. Nevertheless, the reorientation vanishes as the temperature approaches TA, which can\nbe a touchstone to detect this temperature for FiM materials under test.\nIn conclusion, we have numerically explored the excitation and propagation of SWs along\nFiM strips as a function of temperature, below and above the angular and the magnetization com-\npensation points. The analysis of results could be extrapolated to FiM materials with different\ncompositions at room temperature. The fact that the eccentricity of the orbits can be controlled\nby temperature, in addition to the gradual rotation of the semi-major axis along the strip, provides\nan unprecedented local control of magnetization oscillations, which can be exploited to imple-\nment novel functionalities such as sensing in the emerging field of THz magnonics10. Our results\nmay also be relevant to spintronics for making very localized excitation positions given the sen-\nsitivity to distance of the different exerted torques along the magnetic strip. Besides, a potential\nreconfigurability can be envisaged by using controlled heat sources along the strip, such as laser\npulses.\nThis work was supported by Projects No. SA114P20 from Junta de Castilla y León, No.\nPID2020117024GB-C41 funded by MCIN/AEI/10.13039/501100011033, and Project MagnEFi,\nGrant Agreement No. 860060, (H2020-MSCA-ITN-2019) funded by the European Commission.\nThe data that support the findings of this study are available from the corresponding author\nupon reasonable request.\n7SW in FiMs near the AMC Temperature\n𝑇𝑀<𝑇<𝑇𝐴\n𝑇≈𝑇𝐴\n𝑇<𝑇𝑀\nSingle frequency  excitation  \n𝑓=360 GHz\n𝑡=0𝐽0=1TA\nm2\n𝑡\nFIG. 4. Trajectories of the in-plane component of the Néel vector in the stationary regime under a MW\nexcitation. Trajectories are obtained at different locations along the FiM strip and at different temperatures.\nREFERENCES\n1C. Chen, C. Zheng, J. Zhang, and Y . 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Slavin, “Theory of dipole-exchange spin wave spectrum for ferro-\nmagnetic films with mixed exchange boundary conditions,” Journal of Physics C: Solid State\nPhysics 19, 7013–7033 (1986).\n10"    },    {        "title": "0909.2182v1.Spin_models_of_quasi_1D_quantum_ferrimagnets_with_competing_interactions.pdf",        "content": "arXiv:0909.2182v1  [cond-mat.str-el]  11 Sep 2009\n/BV/D3/D2/CS/CT/D2/D7/CT/CS /C5/CP/D8/D8/CT/D6 /C8/CW/DD/D7/CX\r/D7/B8 /BR/BR/BR/BR/B8 /CE /D3/D0/BA /BR/B8 /C6/D3 /BR/B4/BR/BR/B5/B8 /D4/D4/BA /BD /AL /BR/BR/CB/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2/D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7 /DB/CX/D8/CW\r/D3/D1/D4 /CT/D8/CX/D2/CV /CX/D2/D8/CT/D6/CP\r/D8/CX/D3/D2/D7/C6/BA /BU/BA /C1/DA/CP/D2/D3/DA\n/BD/B8/BE/BD/C1/D2/D7/D8/CX/D8/D9/D8/CT /D3/CU /CB/D3/D0/CX/CS /CB/D8/CP/D8/CT /C8/CW/DD/D7/CX\r/D7/B8 /BU/D9/D0/CV/CP /D6/CX/CP/D2 /BT \r/CP/CS/CT/D1/DD /D3/CU /CB\r/CX/CT/D2\r/CT/D7/B8 /BJ/BE /CC /DE/CP 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/D5/D9/CP/D2 /D8/D9/D1 /DA /CP/D6/CX/CP/D2 /D8\r/circlecopyrt /C6/BA /BU/BA /C1/DA/CP/D2/D3/DA /BD/C6/BA /BU/BA /C1/DA/CP/D2/D3/DA/D3/CU /D8/CW/CX/D7 /D1/D3 /CS/CT/D0 /B4S1= 1,S2= 1/2 /B5 /CX/D7 /D9/D7/CT/CS /CX/D2 /CB/CT\r/D8/CX/D3/D2 /BE /D8/D3 /D7/D9/D6/DA /CT/DD /D7/D3/D1/CT /CQ/CP/D7/CX\r /CV/D6/D3/D9/D2/CS/B9/D7/D8/CP/D8/CT /CP/D2/CS /D8/CW/CT/D6/B9/D1/D3 /CS/DD/D2/CP/D1/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7/BA /CC/CW/CT /CT/D1/D4/CW/CP/D7/CX/D7 /CX/D2 /CB/CT\r/D8/CX/D3/D2 /BF /CX/D7 /D3/D2 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1/D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1/D7 /D3/CU /CP /CU/CT/DB /CV/CT/D2/CT/D6/CX\r /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /CS/CT/D7\r/D6/CX/CQ/CX/D2/CV /BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7 /DB/CX/D8/CW/D7/D8/D6/D3/D2/CV \r/D3/D1/D4 /CT/D8/CX/D2/CV /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/B8 /D7/D9\r /CW /CP/D7 /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS J1−J2\n\r /CW/CP/CX/D2 /DB/CX/D8/CW /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /B4/BD/B8/BD/BB/BE/B5 /D7/D4/CX/D2/D7/B8/D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /CS/CX/CP/D1/D3/D2/CS \r /CW/CP/CX/D2 /DB/CX/D8/CW /CU/D3/D9/D6/B9/D7/D4/CX/D2 \r/DD\r/D0/CX\r \r/D3/D9/D4/D0/CX/D2/CV/D7/B8 /CP/D2/CS /D7/D3/D1/CT /CQ/CP/D7/CX\r /D8 /DD/D4 /CT/D7 /D3/CU /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2/D0/CP/CS/CS/CT/D6/D7 /DB/CX/D8/CW /CV/CT/D3/D1/CT/D8/D6/CX\r /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2/BA /CC/CW/CT /D0/CP/D7/D8 /CB/CT\r/D8/CX/D3/D2 \r/D3/D2 /D8/CP/CX/D2/D7 /D7/D3/D1/CT \r/D3/D2\r/D0/D9/D7/CX/D3/D2/D7/BA\nNOCNi\nCu\nab\nc❋ ❋ ❋\n❋ ❋\n❋ ❋❋\n❋❋J(a) (b)\nS1,nS2,nSS\n1,n+12,n+1/BY/CX/CV/D9/D6/CT /BD/BA /B4/CP/B5 /CB/D8/D6/D9\r/D8/D9/D6/CT /D3/CU /D8/CW/CT /D5/D9/CP/D7/CX/B9/D3/D2/CT/B9/CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /CQ/CX/D1/CT/D8/CP/D0/D0/CX\r \r/D3/D1/D4 /D3/D9/D2/CS/C6/CX/BV/D9/B4/D4/CQ/CP/B5/B4/C0 2\n/C7/B53· /BE/C02\n/C7 /DB/CX/D8/CW /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /D7/CX/D8/CT /D7/D4/CX/D2/D7S1=SNi= 1 /CP/D2/CSS2=SCu=1\n2/CP/D0/D3/D2/CV /D8/CW/CT /CP/DC/CX/D7b /BA /C0/DD/CS/D6/D3/CV/CT/D2 /CP/D8/D3/D1/D7 /CP/D6/CT /D3/D1/CX/D8/D8/CT/CS /CU/D3/D6 \r/D0/CP/D6/CX/D8 /DD /CJ /BL ℄/BA /B4/CQ/B5 /CC/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /C6 /A1 /CT/CT/D0 /D7/D8/CP/D8/CT /D3/CU /D8/CW/CT/D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /BT/BY/C5 \r /CW/CP/CX/D2 /D1/D3 /CS/CT/D0 /B4/BD/B5 /CS/CT/D7\r/D6/CX/CQ/CX/D2/CV /D8/CW/CT /CQ/CX/D1/CT/D8/CP/D0/D0/CX\r \r/D3/D1/D4 /D3/D9/D2/CS /C6/CX/BV/D9/B4/D4/CQ/CP/B5/B4/C0 2\n/C7/B53· /BE/C02\n/C7/BA/BE/BA /CC/CW/CT /CP/D2/D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /C0/CT/CX/D7/CT/D2/CQ /CT/D6/CV \r/CW/CP/CX/D2/BT /CV/CT/D2/CT/D6/CX\r /D7/D4/CX/D2 /D1/D3 /CS/CT/D0 /D3/CU /D8/CW/CT /BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8 /CX/D7 /D8/CW/CT /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /D7/D4/CX/D2 \r /CW/CP/CX/D2 /DB/CX/D8/CW /BT/BY/C5/D2/CT/CP/D6/CT/D7/D8/B9/D2/CT/CX/CV/CW /CQ /D3/D6 /CT/DC\r /CW/CP/D2/CV/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /B4J >0 /B5 /CP/D2/CS /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /D7/D4/CX/D2/D7S1\n/CP/D2/CSS2\n/B4S1> S2\n/B5/B8 /CS/CT/D7\r/D6/CX/CQ /CT/CS/CQ /DD /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2\nH=JL/summationdisplay\nn=1(S1,n+S1,n+1)·S2,n−µBHL/summationdisplay\nn=1/parenleftbig\ng1Sz\n1,n+g2Sz\n2,n/parenrightbig\n. /B4/BD/B5/CC/CW/CT /CX/D2 /D8/CT/CV/CT/D6/D7n /D2 /D9/D1 /CQ /CT/D6 /D8/CW/CT /D9/D2/CX/D8 \r/CT/D0/D0/D7/B8 /CT/CP\r /CW \r/D3/D2 /D8/CP/CX/D2/CX/D2/CV /D8 /DB /D3 /D0/CP/D8/D8/CX\r/CT /D7/D4/CP\r/CX/D2/CV/D7 /CP/D2/CS /D8 /DB /D3 /CZ/CX/D2/CS/D7 /D3/CU /D5/D9/CP/D2 /D8/D9/D1/D7/D4/CX/D2 /D3/D4 /CT/D6/CP/D8/D3/D6/D7 S1,n\n/CP/D2/CSS2,n\n\r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /CQ /DD /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /D2 /D9/D1 /CQ /CT/D6/D7 S1\n/CP/D2/CSS2\n/B4S1> S2\n/B5/B8/D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA g1\n/CP/D2/CSg2\n/CP/D6/CT /D8/CW/CTg /B9/CU/CP\r/D8/D3/D6/D7 /D3/CU /D8/CW/CT /D7/CX/D8/CT /D1/CP/CV/D2/CT/D8/CX\r /D1/D3/D1/CT/D2 /D8/D7/B8 µB\n/CX/D7 /D8/CW/CT /BU/D3/CW/D6 /D1/CP/CV/D2/CT/D8/D3/D2/B8/CP/D2/CSH /CX/D7 /CP/D2 /CT/DC/D8/CT/D6/D2/CP/D0 /D9/D2/CX/CU/D3/D6/D1 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /CP/D4/D4/D0/CX/CT/CS /CP/D0/D3/D2/CV /D8/CW/CTz /CS/CX/D6/CT\r/D8/CX/D3/D2/BA /BT/D7 /CP/D2 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT/CU/D3/D0/D0/D3 /DB/CX/D2/CV /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /B4/BD/B5 /CW/CP /DA /CT /CQ /CT/CT/D2 /CT/DC/D8/D6/CP\r/D8/CT/CS /CU/D6/D3/D1 /D1/CP/CV/D2/CT/D8/CX\r /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/D3/D2 /D8/CW/CT /D6/CT\r/CT/D2 /D8/D0/DD /D7/DD/D2 /D8/CW/CT/D7/CX/DE/CT/CS /D5/D9/CP/D7/CX/B9/BD/BW /CQ/CX/D1/CT/D8/CP/D0/D0/CX\r \r/D3/D1/D4 /D3/D9/D2/CS /C6/CX/BV/D9/B4/D4/CQ/CP/B5/B4/BW 2\n/C7/B53· /BE/BW2\n/C7 /CJ /BL ℄ /DB/CX/D8/CW /CP/D7/D8/D6/D9\r/D8/D9/D6/CT /DB/CW/CX\r /CW /CX/D7 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D8/CW/CT /D3/D2/CT /D7/CW/D3 /DB/D2 /CX/D2 /AS/CV/D9/D6/CT /BD /BM(S1,S2)≡(SNi,SCu) = (1,1/2) /B8J/kB=\n121K /B8g1≡gNi= 2.22 /B8g2≡gCu= 2.09 /BA /C0/CT/D2\r/CT/CU/D3/D6/D8/CW /DB /CT /D7/D9/CV/CV/CT/D7/D8g1=g2≡g /CP/D2/CS /D9/D7/CT /D8/CW/CT /C8/D0/CP/D2\r /CZ\r/D3/D2/D7/D8/CP/D2 /D8 ¯h /B8 /D8/CW/CT /BU/D3/D0/D8/DE/D1/CP/D2/D2 \r/D3/D2/D7/D8/CP/D2 /D8 kB\n/B8 /CP/D2/CS /D8/CW/CT /D0/CP/D8/D8/CX\r/CT /D7/D4/CP\r/CX/D2/CVa0\n/CP/D7 /D9/D2/CX/D8/CT/D7/BA/BE/BA/BD/BA /BZ/D6/D3/D9/D2/CS/B9/D7/D8/CP/D8/CT /D4 /D6/D3/D4 /CT/D6/D8/CX/CT/D7 /CP/D2/CS /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7/BE/BA/BD/BA/BD/BAH= 0/C4/CT/D8 /D9/D7 /D7/D8/CP/D6/D8 /DB/CX/D8/CW /D8/CW/CT \r/CP/D7/CTH= 0 /BA /BT \r\r/D3/D6/CS/CX/D2/CV /D8/D3 /C4/CX/CT/CQ/B9/C5/CP/D8/D8/CX/D7/B3 /D8/CW/CT/D3/D6/CT/D1 /CJ /BD/BI ℄/B8 /CU/D3/D6H= 0 /D8/CW/CT/CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /D3/CU /D8/CW/CT /CQ/CX/D4/CP/D6/D8/CX/D8/CT /D1/D3 /CS/CT/D0 /B4/BD/B5 /CW/CP/D7 /CP /D8/D3/D8/CP/D0 /D7/D4/CX/D2ST= (S1−S2)L /B8L /CQ /CT/CX/D2/CV /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU/BE/CB/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2/D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7/D9/D2/CX/D8 \r/CT/D0/D0/D7/B8 /D7/D3 /D8/CW/CP/D8 /CX/D8 /CX/D7 /D2/CT\r/CT/D7/D7/CP/D6/CX/D0/DD /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT /D3/D6/CS/CT/D6/CT/CS/BA /CB/D9\r /CW /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/D7 /D1/CP /DD /CP/D0/D7/D3/CQ /CT /D6/CT/CU/CT/D6/D6/CT/CS /D8/D3 /CP/D7 /D5/D9/CP/D2/D8/CX/DE/CT /CS /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT/D7 /D7/CX/D2\r/CT /D8/CW/CT /BY/C5 /D1/D3/D1/CT/D2 /D8 /CX/D7 /D5/D9/CP/D2 /D8/CX/DE/CT/CS /CX/D2 /CX/D2 /D8/CT/CV/D6/CP/D0 /B4/D3/D6/CW/CP/D0/CU/B9/CX/D2 /D8/CT/CV/D6/CP/D0/B5 /D1 /D9/D0/D8/CX/D4/D0/CT/D7 /D3/CU /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D9/D2/CX/D8 \r/CT/D0/D0/D7L /BA /BT/D7 /CT/DC/D4/D0/CX\r/CX/D8/CT/D0/DD /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/CS /CQ /CT/D0/D3 /DB /B4/CB/CT\r/D8/CX/D3/D2/BF/B5/B8 /D7/D8/D6/D3/D2/CV /CT/D2/D3/D9/CV/CW \r/D3/D1/D4 /CT/D8/CX/D2/CV /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 \r/CP/D2 /D7/D9/D7/D4 /CT/D2/CS /D8/CW/CX/D7 /D5/D9/CP/D2 /D8/CX/DE/CP/D8/CX/D3/D2 /D6/D9/D0/CT /CX/D2 /D7/D3/D1/CT /D6/CT/CV/CX/D3/D2/D7 /D3/CU/D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CT /DB/CW/CT/D6/CT /D8/CW/CT /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D3/D6/CS/CT/D6 /D7/D8/CX/D0/D0 /D7/D9/D6/DA/CX/DA /CT/D7/BA /CB/CX/D2\r/CT /D8/CW/CT /D1/D3 /CS/CT/D0 /CW/CP/D7 /CP/D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /D3/D6/CS/CT/D6/CT/CS /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/B8 /D8/CW/CX/D7 /D1/CP/CZ /CT/D7 /D8/CW/CT /BD/BW /D4/D6/D3/CQ/D0/CT/D1 /CP/D1/CT/D2/CP/CQ/D0/CT /D8/D3 /D8/CW/CT /D7/D4/CX/D2/B9/DB /CP /DA /CT /D8/CW/CT/D3/D6/DD/B4/CB/CF/CC/B5 /CP/D4/D4/D6/D3/CP\r /CW /CJ /BD/BJ ℄/BA/CE /CP/D0/D9/CP/CQ/D0/CT /D5/D9/CP/D0/CX/D8/CP/D8/CX/DA /CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CP/D2/CS /D0/D3 /DB/B9/D0/DD/CX/D2/CV /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 \r/CP/D2 /CQ /CT /CT/DC/B9/D8/D6/CP\r/D8/CT/CS /CP/D0/D6/CT/CP/CS/DD /CU/D6/D3/D1 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /D7/D4/CX/D2/B9/DB /CP /DA /CT /D8/CW/CT/D3/D6/DD /B4/C4/CB/CF/CC/B5 /CJ /BD/BK /B8 /BD/BL /B8 /BE/BC /B8 /BE/BD ℄/BA /CC/CW/CT /D3/D2/B9/D7/CX/D8/CT /D1/CP/CV/D2/CT/B9/D8/CX/DE/CP/D8/CX/D3/D2/D7 m1=∝an}bracketle{tSz\n1,n∝an}bracketri}ht /CP/D2/CSm2=−∝an}bracketle{tSz\n2,n∝an}bracketri}ht /B4m1−m2=S1−S2\n/B5 /CP/D6/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1/CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /CZ /CT/CT/D4/CX/D2/CV /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT /D7/D4/CX/D2 \r/D3/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7/BA /C4/CB/CF/CC /CX/D1/D4/D0/CX/CT/D7/D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CT/DC/D8/D6/CT/D1/CT /D5/D9/CP/D2 /D8/D9/D1 \r/CP/D7/CT(S1,S2) = (1,1/2) /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /D6/CT/CS/D9\r/CT /D7/D9/CQ/B9/D7/D8/CP/D2 /D8/CX/CP/D0/D0/DD /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D3/D2/B9/D7/CX/D8/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/D7 /B4m2/S2≈0.39 /B5/BA /C6/D3/D8/CX\r/CT/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CP/D8 /CS/D9/CT /D8/D3 /D8/CW/CT/CQ/D6/D3/CZ /CT/D2 /D7/D9/CQ/D0/CP/D8/D8/CX\r/CT /D7/DD/D1/D1/CT/D8/D6/DD /B8 /CX/D2 /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D8/CW/CT/D6/CT /CP/D4/D4 /CT/CP/D6 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 \r/D3/D6/D6/CT\r/B9/D8/CX/D3/D2/D7 /D8/D3m1\n/CP/D2/CSm2\n/D6/CT/D7/D9/D0/D8/CX/D2/CV /CU/D6/D3/D1 /D8 /DB /D3/B9/CQ /D3/D7/D3/D2 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /CQ /D3/D7/D3/D2/CX\r /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CJ /BE/BE ℄/BA /CD/D4/D8/D3 /D7/CT\r/D3/D2/CS /D3/D6/CS/CT/D6 /CX/D21/S2\n/B4/CX/D2 /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B5/B8 /CB/CF/CC /D7/CT/D6/CX/CT/D7 /CV/CX/DA /CT/D7 /D8/CW/CT /D4/D6/CT\r/CX/D7/CT /D6/CT/B9/D7/D9/D0/D8m2= 0.29388 /CJ /BE/BF ℄/B8 /D8/D3 /CQ /CT \r/D3/D1/D4/CP/D6/CT/CS /DB/CX/D8/CW /D8/CW/CT /CS/CT/D2/D7/CX/D8 /DD/B9/D1/CP/D8/D6/CX/DC /D6/CT/D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/B9/CV/D6/D3/D9/D4 /B4/BW/C5/CA /BZ/B5/CT/D7/D8/CX/D1/CP/D8/CT m2= 0.29248 /CJ /BE/BC ℄/BA /BT/D2/D3/D8/CW/CT/D6 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D4 /CT\r/D9/D0/CX/CP/D6/CX/D8 /DD /D3/CU /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2 /CX/D7 /D8/CW/CT /CT/DC/B9/D8/D6/CT/D1/CT/D0/DD /D7/D1/CP/D0/D0 \r/D3/D6/D6/CT/D0/CP/D8/CX/D3/D2 /D0/CT/D2/CV/D8/CW /B4/D7/D1/CP/D0/D0/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D9/D2/CX/D8 \r/CT/D0/D0/B5 /D3/CU /D8/CW/CT /D7/CW/D3/D6/D8/B9/D6/CP/D2/CV/CT /D7/D4/CX/D2 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7/CJ /BD/BL /B8 /BE/BC ℄/BA /CC/CW/CX/D7 /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /CT/DC/D4/D0/CP/CX/D2/D7 /D8/CW/CT /CV/D3 /D3 /CS /D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/D3 /CS/CT/D0 /CP\r /CW/CX/CT/DA /CT/CS /DB/CX/D8/CW/D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2/CP/D0 /CP/D4/D4/D6/D3/CP\r /CW /D9/D7/CX/D2/CV /D1/CP/D8/D6/CX/DC/B9/D4/D6/D3 /CS/D9\r/D8 /D7/D8/CP/D8/CT/D7 /CJ /BE/BG /B8 /BE/BH ℄/BA\n00.511.522.53\n0 0.2 0.4 0.6 0.8 1E\nπkk+\n−\n2k/EExcitation energies\n+●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●/BY/CX/CV/D9/D6/CT /BE/BA /BW/CX/D7/D4 /CT/D6/D7/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D3/D2/CT/B9/D1/CP/CV/D2/D3/D2 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 /B4E±\nk\n/B5 /CX/D2 /D8/CW/CT /D7/DD/D7/D8/CT/D1(S1,S2) = (1,1/2)\r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D4 /D8/D3 /D7/CT\r/D3/D2/CS /D3/D6/CS/CT/D6 /CX/D2 /D8/CW/CT1/S2\n/D6/CT/D0/CP/D8/CX/DA /CT /D8/D3 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CJ /BE/BJ ℄/BA /CC/CW/CT /D4 /D3/CX/D2 /D8/D7/CX/D2E±\nk\n/D7/CW/D3 /DB /D2 /D9/D1/CT/D6/CX\r/CP/D0 /BX/BW /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6 /D4 /CT/D6/CX/D3 /CS/CX\r \r /CW/CP/CX/D2/D7 /B4/CU/D3/D6E+\nk\n/B5 /CP/D2/CS /C9/C5/BV /D6/CT/D7/D9/D0/D8/D7 /B4/CU/D3/D6E−\nk\n/B5 /CJ /BE/BD ℄/BA/CC /D9/D6/D2/CX/D2/CV /D8/D3 /D8/CW/CT /CT/DC\r/CX/D8/CP/D8/CX/D3/D2 /D7/D4 /CT\r/D8/D6/D9/D1/B8 /CB/CF/CC /D4/D6/CT/CS/CX\r/D8/D7 /D8 /DB /D3 /D8 /DD/D4 /CT/D7 /D3/CU /D0/D3 /DB/B9/D0/DD/CX/D2/CV /D1/CP/CV/D2/D3/D2 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7\nE±\nk\n/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /B8 /CX/D2 /D8/CW/CT /D7/D9/CQ/D7/D4/CP\r/CT/D7 /DB/CX/D8/CW /DE \r/D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /D7/D4/CX/D2Sz\nT=ST±1 /B4/D7/CT/CT /AS/CV/D9/D6/CT /BE/B5/BA/C1/D2 /D8/CW/CT /D0/D3/D2/CV /DB /CP /DA /CT/D0/CT/D2/CV/D8/CW /D0/CX/D1/CX/D8k≪1 /B8 /D8/CW/CT /BY/C5 /D1/D3 /CS/CTE−\nk\n/D8/CP/CZ /CT/D7 /D8/CW/CT /C4/CP/D2/CS/CP/D9/B9/C4/CX/CU/D7/CW/CX/D8/DE /CU/D3/D6/D1\nE−\nk=̺s\nM0k2+O(k4), /B4/BE/B5/DB/CW/CT/D6/CT̺s=JS1S2\n/CX/D7 /D8/CW/CT /D7/D4/CX/D2 /D7/D8/CX/AR/D2/CT/D7/D7 \r/D3/D2/D7/D8/CP/D2 /D8 /CJ /BE/BI ℄ /CP/D2/CSM0= (S1−S2)/2 /CX/D7 /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CS/CT/D2/D7/CX/D8 /DD /D3/CU/D8/CW/CT /D2/CT/D8 /BY/C5 /D1/D3/D1/CT/D2 /D8/BA /CC/CW/CX/D7 /CU/D3/D6/D1 /D3/CU /D8/CW/CT /BZ/D3/D0/CS/D7/D8/D3/D2/CT /D1/D3 /CS/CT/D7 /CX/D7 /D8 /DD/D4/CX\r/CP/D0 /CU/D3/D6 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/D7/B8 /CP/D2/CS/D6/CT/AT/CT\r/D8/D7 /D8/CW/CT /CU/CP\r/D8 /D8/CW/CP/D8 /D8/CW/CT /BY/C5 /D3/D6/CS/CT/D6 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /CX/D7 /CX/D8/D7/CT/D0/CU /CP \r/D3/D2/D7/D8/CP/D2 /D8 /D3/CU /D8/CW/CT /D1/D3/D8/CX/D3/D2/BA /BT/D7 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/CS/CQ /CT/D0/D3 /DB/B8 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 ̺s\n/CP/D2/CSM0\n/D4/D0/CP /DD /CP /CQ/CP/D7/CX\r /D6/D3/D0/CT /CX/D2 /D8/CW/CT /D0/D3 /DB/B9/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r/D7 /D3/CU/D8/CW/CT /D1/D3 /CS/CT/D0/BA /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /D8/CW/CT /BT/BY/C5 /CQ/D6/CP/D2\r /CWE+\nk\n/CX/D7 /CV/CP/D4/D4 /CT/CS/B8 /DB/CX/D8/CW /CP /D1/CX/D2/CX/D1 /D9/D1 /CP/D8k= 0 /CV/CX/DA /CT/D2/CQ /DD∆+= 2(S1−S1)J /CX/D2 /CP /C4/CB/CF/CC /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/BA /C6/D3/D8/CT/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CP/D8 /D8/CW/CT /C4/CB/CF/CC /CT/D7/D8/CX/D1/CP/D8/CT/D7 /CU/D3/D6\n̺s\n/CP/D2/CS /D8/CW/CT /BT/BY/C5 /CV/CP/D4∆+/CP/D6/CT /CU/D9/D6/D8/CW/CT/D6 /D6/CT/D2/D3/D6/D1/CP/D0/CX/DE/CT/CS /CQ /DD /D8/CW/CT /CQ /D3/D7/D3/D2/B9/CQ /D3/D7/D3/D2 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CJ /BE/BJ ℄/BM /C1/D2 /D8/CW/CT/BF/C6/BA /BU/BA /C1/DA/CP/D2/D3/DA/CT/DC/D8/D6/CT/D1/CT /D5/D9/CP/D2 /D8/D9/D1 /D7/DD/D7/D8/CT/D1(S1,S2) = (1,1/2) /B8 /CP/D0/D6/CT/CP/CS/DD /D8/CW/CT /AS/D6/D7/D8 /D3/D6/CS/CT/D6 \r/D3/D6/D6/CT\r/D8/CX/D3/D2/D7 /CX/D21/S2\n/CV/CX/DA /CT /D8/CW/CT/D6/CT/D7/D9/D0/D8/D7̺s/(Js1s2) = 0.761 /CP/D2/CS∆(+)= 1.676J /BA /CC/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /C5/D3/D2 /D8/CT/B9/BV/CP/D6/D0/D3 /B4/C9/C5/BV/B5 /D6/CT/D7/D9/D0/D8 /CU/D3/D6 /D8/CW/CX/D7/D4/CP/D6/CP/D1/CT/D8/CT/D6 1.759 /CJ /BE/BD ℄ \r/D0/CT/CP/D6/D0/DD /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT /CX/D1/D4 /D3/D6/D8/CP/D2\r/CT /D3/CU /D8/CW/CT1/S2\n\r/D3/D6/D6/CT\r/D8/CX/D3/D2/D7/BA /C8/D6/CT\r/CX/D7/CT /CT/D7/D8/CX/D1/CP/D8/CT/D7/CU/D3/D6 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT(1,1/2) \r /CW/CP/CX/D2 /CW/CP /DA /CT /CP/D0/D7/D3 /CQ /CT/CT/D2 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /D9/D7/CX/D2/CV \r/D0/D9/D7/D8/CT/D6 /D7/CT/D6/CX/CT/D7 /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /CJ/BE/BK ℄/BMm2= 0.292487(6) /B8∆+/J= 1.7591(6) /CP/D2/CS̺s/(Js1s2) = 0.831(5) /BA/BE/BA/BD/BA/BE/BAH/negationslash= 0/CC/CW/CT /CX/D2 /D8/D6/D3 /CS/D9\r/D8/CX/D3/D2 /D3/CU /CP /CI/CT/CT/D1/CP/D2 /D8/CT/D6/D1 /CX/D2 /BX/D5/BA /B4/BD/B5 /D0/CT/CP /DA /CT/D7 /CP/D0/D0 /CT/CX/CV/CT/D2/D7/D8/CP/D8/CT/D7 /DB/CX/D8/CW /CP /CV/CX/DA /CT/D2 /DE \r/D3/D1/D4 /D3/D2/CT/D2 /D8/D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /D7/D4/CX/D2Sz\nT\n/CX/D2 /DA /CP/D6/CX/CP/D2 /D8/B8 /DB/CW/CX/D0/CT /D7/CW/CX/CU/D8/CX/D2/CV /D8/CW/CT /CT/CX/CV/CT/D2/CT/D2/CT/D6/CV/CX/CT/D7 /CQ /DDhSz\nT\n/B8 /DB/CW/CT/D6/CTh=gµBH /BA /C1/D2/D4/CP/D6/D8/CX\r/D9/D0/CP/D6/B8 /D8/CW/CT /D3/D2/CT/B9/D1/CP/CV/D2/D3/D2 /D7/D8/CP/D8/CT/D7E±\nk\n/B8 \r/CP/D6/D6/DD/CX/D2/CV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/D7 ±1 /B8 \r /CW/CP/D2/CV/CT /D8/D3E±\nk(h) =E±\nk∓h /B8/CX/BA/CT/B8 /D8/CW/CT /CI/CT/CT/D1/CP/D2 /D8/CT/D6/D1 /CX/D2 /D8/D6/D3 /CS/D9\r/CT/D7 /CP /CV/CP/D4∆−(h) =h /CU/D3/D6 /D8/CW/CT /BY/C5 /CQ/D6/CP/D2\r /CW /CP/D2/CS /D6/CT/CS/D9\r/CT/D7 /D8/CW/CT /CV/CP/D4 /D3/CU /D8/CW/CT/BT/BY/C5 /CQ/D6/CP/D2\r /CW/B8∆+(h) = ∆+−h /BA /C1/CU /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /CT/DC\r/CT/CT/CS/D7 /D8/CW/CT \r/D6/CX/D8/CX\r/CP/D0 /DA /CP/D0/D9/CTh=hc1= ∆+/B8 /D8/CW/CT/BT/BY/C5 /CV/CP/D4 \r/D0/D3/D7/CT/D7 /CP/D2/CS /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CT/D2 /D8/CT/D6/D7 /CP \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CT /DB/CW/CX\r /CW /CX/D7 /CT/DC/D4 /CT\r/D8/CT/CS /D8/D3 /CQ /CT /CP /CZ/CX/D2/CS /D3/CU /C4/D9/D8/D8/CX/D2/CV/CT/D6/D0/CX/D5/D9/CX/CS/B8 /CX/D2 /CP/D2/CP/D0/D3/CV/DD /DB/CX/D8/CW /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D3/D8/CW/CT/D6 /CV/CP/D4/D4 /CT/CS /D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D0/CX/CZ /CT /D8/CW/CT /D7/D4/CX/D2/B9/BD \r /CW/CP/CX/D2 /CP/D2/CS /D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /D0/CP/CS/CS/CT/D6 /CJ /BE/BL /B8 /BF/BC /B8 /BF/BD ℄/BA /CC/CW/CT \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CT /D8/CT/D6/D1/CX/D2/CP/D8/CT/D7 /CP/D8 /CP /D7/CT\r/D3/D2/CS \r/D6/CX/D8/CX\r/CP/D0 /AS/CT/D0/CSh=hc2= 3J /CP/D8/DB/CW/CX\r /CW /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CQ /CT\r/D3/D1/CT/D7 /CU/D9/D0/D0/DD /D4 /D3/D0/CP/D6/CX/DE/CT/CS /CJ /BF/BE ℄/BA /BT/D2 /CP\r\r/D9/D6/CP/D8/CT /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /B41,1/2 /B5/C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /B4H∝ne}ationslash= 0) /CX/D2 /D8/CW/CT \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CThc1< h < h c1\n/CW/CP/D7 /CQ /CT/CT/D2 /CP\r /CW/CX/CT/DA /CT/CS /CQ /DD /CP /D1/CP/D4/D4/CX/D2/CV /D8/D3/CP/D2 /CT/AR/CT\r/D8/CX/DA /CT /D7/D4/CX/D2/B9/BD/BB/BE /CG/CG/CI \r /CW/CP/CX/D2 /CX/D2 /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /DB/CW/CX\r /CW /D9/D7/CT/D7 /DA /CP/D6/CX/CP/D8/CX/D3/D2/CP/D0 /D1/CP/D8/D6/CX/DC/B9/D4/D6/D3 /CS/D9\r/D8/D7/D8/CP/D8/CT/D7 /CJ /BE/BH ℄/BA /C1/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV/D0/DD /B8 /D7/D9\r /CW /CP \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CT /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CU/D3/D6 /D8/CW/CT /CT/D2 /D8/CX/D6/CT \r/D0/CP/D7/D7 /D3/CU/D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 (S1,S2) /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX\r /CG/CG/CI /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2/D7 /DB/CX/D8/CW /CT/CP/D7/DD/B9/D4/D0/CP/D2/CT /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CP/D2/CS /D7/CW/D3/D6/D8/B9/D6/CP/D2/CV/CT/CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/BA /BT/D2 /CT/DC/D8/CT/D2/D7/CX/DA /CT /D2 /D9/D1/CT/D6/CX\r/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7/B8 /D9/D7/CX/D2/CV /CT/DC/CP\r/D8/B9/CS/CX/CP/CV/D3/D2/CP/D0/CX/DE/CP/D8/CX/D3/D2 /B4/BX/BW/B5 /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6(1,1/2)/CP/D2/CS(3/2,1/2) /D4 /CT/D6/CX/D3 /CS/CX\r \r /CW/CP/CX/D2/D7/B8 /D7/D9/CV/CV/CT/D7/D8/D7 /D9/D2/CX/DA /CT/D6/D7/CP/D0 \r/D6/CX/D8/CX\r/CP/D0 /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /CU/D3/D6 /D8/CW/CT/D7/CT \r /CW/CP/CX/D2/D7 /CX/D2 /D8/CW/CT /CT/D2 /D8/CX/D6/CT/CX/D2 /D8/CT/D6/DA /CP/D0 /CU/D6/D3/D1 /D8/CW/CT /BY/C5 /D8/D3 /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /CX/D7/D3/D8/D6/D3/D4/CX\r /D4 /D3/CX/D2 /D8/D7 /CJ /BF/BF ℄/BA /BT/D0/D3/D2/CV /D8/CW/CX/D7 /D4/CW/CP/D7/CT/B8 /D8/CW/CT \r/D6/CX/D8/CX\r/CP/D0/AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /CP/D6/CT /D6/D9/D0/CT/CS /CQ /DD /CP \r/D3/D2/CU/D3/D6/D1/CP/D0 /AS/CT/D0/CS /D8/CW/CT/D3/D6/DD /D3/CU /BZ/CP/D9/D7/D7/CX/CP/D2 /D8 /DD/D4 /CT /DB/CX/D8/CW /CP /D8/D3/D4 /D3/D0/D3/CV/CX\r/CP/D0 \r /CW/CP/D6/CV/CTc= 1 /BA/BW/CX/D7\r/D9/D7/D7/CT/CS /D3/D2/CT/B9/D1/CP/CV/D2/D3/D2 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 \r/D3/D2 /D8/D6/D3/D0 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D4/D6/D3 \r/CT/D7/D7 /D3/CU /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2/BA/CB/CX/D2\r/CT /D8/CW/CT /D0/D3 /DB /CT/D7/D8 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2 /CX/D2\r/D6/CT/CP/D7/CX/D2/CV /D8/CW/CT /BY/C5 /D1/D3/D1/CT/D2 /D8 M=∝an}bracketle{tSz\n1,n+Sz\n2,n∝an}bracketri}ht /CT/DC/CW/CX/CQ/CX/D8/D7 /CP/D2 /CT/D2/CT/D6/CV/DD /CV/CP/D4/B8/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 \r/D9/D6/DA /CTM(H) /CW/CP/D7 /CP /D4/D0/CP/D8/CT/CP/D9 /CP/D8M=S1−S2\n/BA /CC/CW/CX/D7 /D4/D0/CP/D8/CT/CP/D9 /D4/CW/CP/D7/CT/B8 /CW/D3 /DB /CT/DA /CT/D6/B8/CP/D4/D4 /CT/CP/D6/D7 /CX/D2 /D8/CW/CT /D6/CT/D0/CP/D8/CT/CS \r/D0/CP/D7/D7/CX\r/CP/D0 /D7/DD/D7/D8/CT/D1/B8 /CP/D7 /DB /CT/D0/D0/BA /C1/D2 /D8/CW/CX/D7 \r/D3/D2/D2/CT\r/D8/CX/D3/D2/B8 /CX/D8 /CX/D7 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D8/D3 /CT/DC/CP/D1/CX/D2/CT /D8/CW/CT/D6/D3/D0/CT /D3/CU /CS/CX/AR/CT/D6/CT/D2 /D8 /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX/CT/D7 /DB/CW/CX\r /CW /D1/CP /DD /CP/D4/D4 /CT/CP/D6 /CX/D2 /D6/CT/CP/D0 /D1/CP/D8/CT/D6/CX/CP/D0/D7/BA /BT/D7 /D8/D3 /D8/CW/CT /CT/DC\r /CW/CP/D2/CV/CT /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /B8/CX/D8 /D3 \r\r/D9/D6/D6/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /CT/AR/CT\r/D8/D7 /D7/CX/D1/D4/D0/DD /D7/D8/CP/CQ/CX/D0/CX/DE/CT /D8/CW/CT /D4/D0/CP/D8/CT/CP/D9 /CP/D8M=S1−S2\n/CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /CG/CH/CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /B8 /CX/BA/CT/BA/B8 /D8/CW/CT /CT/AR/CT\r/D8 /CX/D7 /D6/CT/CS/D9\r/CT/CS /D8/D3 /D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT \r /CW/CP/D2/CV/CT/D7 /D3/CU /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D6/CT/D7/D9/D0/D8 /CJ /BF/BG ℄/BA /BT \r/D0/CT/CP/D6/CX/D2/CS/CX\r/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D3/D6/CX/CV/CX/D2 /D3/CU /D8/CW/CT /CS/CX/D7\r/D9/D7/D7/CT/CS /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D4/D0/CP/D8/CT/CP/D9 /DB /CP/D7 /D4/D6/D3 /DA/CX/CS/CT/CS /CU/D3/D6 /D8/CW/CT\n(1,1/2) \r /CW/CP/CX/D2 /DB/CX/D8/CW /CP /D7/CX/D2/CV/D0/CT/B9/CX/D3/D2 /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CU/D3/D6 /D7/D4/CX/D2/B9/BD /D7/CX/D8/CT/D7/B8D/parenleftbig\nSz\n1,n/parenrightbig2/B8 /DB/CW/CX\r /CW /CX/D7 /CT/DA /CT/D2 /D1/D3/D6/CT /D6/CT/CP/D0/CX/D7/D8/CX\r/CU/D3/D6 /D8/CW/CT /CT/DC/CX/D7/D8/CX/D2/CV /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D1/CP/D8/CT/D6/CX/CP/D0/D7 /CJ /BF/BH ℄/BA /C1/D8 /DB /CP/D7 /D6/CT/DA /CT/CP/D0/CT/CS/B8 /CX/D2 /D4/CP/D6/D8/CX\r/D9/D0/CP/D6/B8 /D8/CW/CP/D8 /D8/CW/CT /D1/CT\r /CW/CP/D2/CX/D7/D1 /D3/CU/D8/CW/CTM= 1/2 /D4/D0/CP/D8/CT/CP/D9 \r/CP/D2 \r /CW/CP/D2/CV/CT /CU/D6/D3/D1 /CP /C0/CP/D0/CS/CP/D2/CT /D8/D3 /CP /D0/CP/D6/CV/CT/B9D /D8 /DD/D4 /CT /D8/D6/D3/D9/CV/CW /CP /BZ/CP/D9/D7/D7/CX/CP/D2 /D5/D9/CP/D2 /D8/D9/D1\r/D6/CX/D8/CX\r/CP/D0 /D4 /D3/CX/D2 /D8/B8 /CT/D7/D8/CX/D1/CP/D8/CT/CS /CP/D7D/J= 1.114 /B8 /DB/CW/CX\r /CW /CX/D7 /CP /CZ/CX/D2/CS /D3/CU /CY/D9/D7/D8/CX/AS\r/CP/D8/CX/D3/D2 /CU/D3/D6 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D3/D6/CX/CV/CX/D2/D3/CU /D8/CW/CTM= 1/2 /D4/D0/CP/D8/CT/CP/D9 /CX/D2 /D8/CW/CX/D7 /D7/DD/D7/D8/CT/D1/BA /CC /DB /D3 /D4/D0/CP/D8/CT/CP/D9 /D4/CW/CP/D7/CT/D7 /B4M= 1/2 /B5 /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8 /D4/CP/D6/CX/D8/CX/CT/D7/CW/CP /DA /CT /CP/D0/D7/D3 /CQ /CT/CT/D2 /D7/D8/D9/CS/CX/CT/CS /CX/D2 /CP /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2 /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8 /D2/CT/DC/D8/B9/D2/CT/CP/D6/CT/D7/D8/B9/D2/CT/CX/CV/CW /CQ /D3/D6 /BT/BY/C5/CQ /D3/D2/CS/D7 /B4/D7/CT/CT /AS/CV/D9/D6/CT /BG/CP/B5 /CJ /BF/BI ℄/BA/BE/BA/BE/BA /CC/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r/D7/C5/D3/D7/D8 /D3/CU /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D3/D2 /D5/D9/CP/D7/CX/B9/BD/BW /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D7/DD/D7/D8/CT/D1/D7 \r/CP/D6/D6/CX/CT/CS /D3/D9/D8 /CX/D2 /D8/CW/CT /D4/CP/D7/D8 \r/D3/D2\r/CT/D6/D2/CT/CS/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r \r /CW/CP/CX/D2/D7 /DB/CX/D8/CW /D6/CP/D8/CW/CT/D6 /CQ/CX/CVS1\n/D7/D4/CX/D2/D7 /B4/D8 /DD/D4/CX\r/CP/D0/D0/DD /B8 S1= 5/2) /B5/B8/D1/CP/CZ/CX/D2/CV /D8/CW/CT /D7/DD/D7/D8/CT/D1 /D6/CP/D8/CW/CT/D6 \r/D0/CP/D7/D7/CX\r/CP/D0 /CJ /BF/BJ ℄/BA /BT/D7 /CP /D1/CP/D8/D8/CT/D6 /D3/CU /CU/CP\r/D8/B8 /D5/D9/CP/D2 /D8/D9/D1 /CT/AR/CT\r/D8/D7 /CP/D6/CT /D1/D3/D7/D8 /D4/D6/D3/D2/D3/D9/D2\r/CT/CS/CX/D2 /D8/CW/CT /CT/DC/D8/D6/CT/D1/CT /D5/D9/CP/D2 /D8/D9/D1 \r/CP/D7/CT(S1,S2) = (1,1/2) /B8 /D6/CT/CP/D0/CX/DE/CT/CS/B8 /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /CX/D2 /D8/CW/CT /D5/D9/CP/D7/CX/B9/BD/BW /CQ/CX/D1/CT/D8/CP/D0/D0/CX\r/D1/CP/D8/CT/D6/CX/CP/D0 /C6/CX/BV/D9/B4/D4/CQ/CP/B5/BW 2\n/C7/B53· /BE/BW2\n/C7 /CJ /BL℄/BA /BT/D7 /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CP/CQ /D3 /DA /CT/B8 /D8/CW/CX/D7 /D1/CP/D8/CT/D6/CX/CP/D0 /CW/CP/D7 /CP/D2 /CT/DC\r /CW/CP/D2/CV/CT /CX/D2 /D8/CT/D6/CP\r/B9/D8/CX/D3/D2 /D3/CU /CP/CQ /D3/D9/D8 /BD/BE/BD /C3/B8 /CP/D2/CS /D7/CW/D3 /DB/D7 /CP /D8/CW/D6/CT/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/CP/D0 /BT/BY/C5 /D3/D6/CS/CT/D6/CX/D2/CV /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CP/D8 /BJ /C3 /B4/D7/CT/CT /AS/CV/D9/D6/CT /BF/B5/DB/CW/CX\r /CW /D7/D3/D1/CT/DB/CW/CP/D8 /D3/CQ/D7\r/D9/D6/CT/D7 /D8/CW/CT /DA /CT/D6/DD /D0/D3 /DB /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /B4T /B5 /CU/CT/CP/D8/D9/D6/CT/D7 /D3/CU /D8/CW/CT /BD/BW /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/BA /CB/D8/CX/D0/D0/B8 /D0/D3 /DB/B9/AS/CT/D0/CS /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /CP/D6/CT /CX/D2 /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0/D0/DD /CT/DC/D4 /CT\r/D8/CT/CS /D6/CT/D7/D9/D0/D8/D7/BA /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6/CW/CP/D2/CS/B8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/D7 /D2/CT\r/CT/D7/D7/CP/D6/DD /D8/D3 /D6/CT/CP\r /CW /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0/D0/DD /D1/D3/D7/D8 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV \r/D6/CX/D8/CX\r/CP/D0 /D4/CW/CP/D7/CT /DB /D3/D9/D0/CS/CQ /CT /CU/D3/D6 /D8/CW/CX/D7 \r/D3/D1/D4 /D3/D9/D2/CS /DB /CT/D0/D0 /CQ /CT/DD /D3/D2/CS /BD/BC/BC /CC/B8 /D1/CP/CZ/CX/D2/CV /D8/CW/CT /D7/CT/CP/D6\r /CW /CU/D3/D6 /D3/D8/CW/CT/D6 \r/D3/D1/D4 /D3/D9/D2/CS/D7 /DB/CX/D8/CW /DB /CT/CP/CZ /CT/D6/CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CP/D2 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV \r /CW/CP/D0/D0/CT/D2/CV/CT /CU/D3/D6 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/CX/D7/D8/D7/BA/BG/CB/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2/D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7/C4/D3 /DB/B9/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r/D7 /B4T≪̺sM0\n/B5 /D3/CU /D8/CW/CT /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2/D7 /CX/D7 \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS /CQ /DD /D8/CW/CT/BY/C5 /D1/CP/CV/D2/D3/D2/D7/B8 /D7/D3 /D8/CW/CP/D8 /CX/D2 /DE/CT/D6/D3 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /CX/D8 \r/CP/D2 /CQ /CT /CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /DD /D9/D7/CX/D2/CV /CC /CP/CZ /CP/CW/CP/D7/CW/CX/B3/D7 \r/D3/D2/D7/D8/D6/CP/CX/D2/CT/CS/CB/CF/CC /CU/D3/D6 /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/D7 /CJ /BG/BC /B8 /BG/BD ℄/BA /BT /CU/CT/DB /DA /CP/D6/CX/CP/D2 /D8/D7 /D3/CU /D8/CW/CX/D7 /D8/CW/CT/D3/D6/DD /CW/CP /DA /CT /CP/D0/D7/D3 /CQ /CT/CT/D2 /CP/D4/D4/D0/CX/CT/CS /D8/D3 /BD/BW /CU/CT/D6/D6/CX/B9/D1/CP/CV/D2/CT/D8/D7 /CJ /BF/BK /B8 /BG/BE /B8 /BG/BF ℄/BA /C1/D2 /D8/CT/D6/D1/D7 /D3/CU /D8/CW/CT /CV/D6/D3/D9/D2/CS/B9/D7/D8/CP/D8/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 ̺s\n/CP/D2/CSM0\n/B8 /D8/CW/CT /CT/DC/D4/D0/CX\r/CX/D8/CT /CU/D3/D6/D1 /D3/CU/D8/CW/CT /D7/CT/D6/CX/CT/D7 /CX/D2 /D4 /D3 /DB /CT/D6/D7 /D3/CUt≡T/̺sM0\n/CU/D3/D6 /D8/CW/CT /D9/D2/CX/CU/D3/D6/D1 /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD χ /CP/D2/CS /D8/CW/CT /D7/D4 /CT\r/CX/AS\r /CW/CT/CP/D8Cv\n/D6/CT/CP/CS/CP/D7 /CJ /BG/BG ℄\nχT\nM0=2\n3t−1−ζ(1\n2)√πt−1\n2+ζ2(1\n2)\n2π+O/parenleftBig\nt1\n2/parenrightBig\n,\nCv\nM0=3\n8ζ(3\n2)√πt1\n2−t\n2+/bracketleftbigg\nconst−15ζ(1\n2)\n32√π/bracketrightbigg\nt3\n2+O/parenleftbig\nt2/parenrightbig\n. /B4/BF/B5/C0/CT/D6/CT/B8ζ(z) /CX/D7 /CA/CX/CT/D1/CP/D2/D2/B3/D7 /DE/CT/D8/CP /CU/D9/D2\r/D8/CX/D3/D2/B8 /CP/D2/CSconst= 15(S2\n1+S1S2+S2\n2)ζ(5\n2)/128√π /BA/BT/D7 /D1/CP /DD /CQ /CT /CT/DC/D4 /CT\r/D8/CT/CS/B8 /D8/CW/CT /CP/CQ /D3 /DA /CT /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /D6/CT/D4/D6/D3 /CS/D9\r/CT /CC /CP/CZ /CP/CW/CP/D7/CW/CX/B3/D7 /D3/D6/CX/CV/CX/D2/CP/D0 /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT/C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /BY/C5 \r /CW/CP/CX/D2 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /CQ /DD /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 M0=S /CP/D2/CS̺s=JS2/BA /BT/D4/CP/D6/D8 /CU/D6/D3/D1 /D8/CW/CT\r/D3 /CTꜶ\r/CX/CT/D2 /D8 /D3/CUt3/2/CX/D2 /D8/CW/CT /CT/DC/D4/CP/D2/D7/CX/D3/D2 /CU/D3/D6Cv\n/B8 /D8/CW/CT /CP/CQ /D3 /DA /CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /D6/CT/D4/D6/D3 /CS/D9\r/CT /D4/D6/CT\r/CX/D7/CT/D0/DD /D8/CW/CT /D8/CW/CT/D6/D1/D3/B9/CS/DD/D2/CP/D1/CX\r /BU/CT/D8/CW/CT/B9/CP/D2/D7/CP/D8/DE \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /BY/C5 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /CJ /BG/BH /B8 /BG/BI ℄/BA /C1/D8 /CX/D7 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV/D8/D3 /D2/D3/D8/CT /D8/CW/CP/D8 /DB/CX/D8/CW/D3/D9/D8 /D8/CW/CT /CU/CP\r/D8/D3/D6const /CX/D2 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6Cv\n/B8 /CQ /D3/D8/CW /CT/DC/D4/CP/D2/D7/CX/D3/D2/D7 /CU/D9/D0/AS/D0/D0 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0/CW /DD/D4 /D3/D8/CW/CT/D7/CX/D7 /CP\r\r/D3/D6/CS/CX/D2/CV /D8/D3 /DB/CW/CX\r /CW /CX/D2 /BD/BW /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/D7 /CP/D0/D0 /D3/CQ/D7/CT/D6/DA /CP/CQ/D0/CT/D7 /D7/CW/D3/D9/D0/CS /CQ /CT /D9/D2/CX/DA /CT/D6/D7/CP/D0/CU/D9/D2\r/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /CQ/CP/D6/CT \r/D3/D9/D4/D0/CX/D2/CV/D7 M0\n/B8̺s\n/B8 /CP/D2/CSh /B8 /D6/CT/CP/D0/CX/DE/CX/D2/CV /CP /D2/D3/B9/D7\r /CP/D0/CT/B9/CU/CP\r/D8/D3/D6 /D9/D2/CX/DA/CT/D6/D7/CP/D0/CX/D8/DD /CJ /BG/BJ /B8 /BG/BK ℄/BA /C1/D2/D4/CP/D6/D8/CX\r/D9/D0/CP/D6/B8 /D8/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /CU/D6/CT/CT/B9/CT/D2/CT/D6/CV/DD /CS/CT/D2/D7/CX/D8 /DD /D7/CW/D3/D9/D0/CS /CW/CP /DA /CT /D8/CW/CT /CV/CT/D2/CT/D6/CX\r /CU/D3/D6/D1\nF\nN=TM0ΦF/parenleftbiggρsM0\nT,h\nT/parenrightbigg\n, /B4/BG/B5/DB/CW/CT/D6/CTΦF(x,y) /CX/D7 /CP /D9/D2/CX/DA /CT/D6/D7/CP/D0 /D7\r/CP/D0/CX/D2/CV /CU/D9/D2\r/D8/CX/D3/D2 /DB/CX/D8/CW /D2/D3 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /D7\r/CP/D0/CT /CU/CP\r/D8/D3/D6/D7/BM /CU/D3/D6 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT/D7/CX/D8/CT /D7/D4/CX/D2S /CT/D2 /D8/CT/D6/D7 /D3/D2/D0/DD /CX/D2/CS/CX/D6/CT\r/D8/D0/DD /B8 /D8/CW/D6/D3/D9/CV/CW ̺s\n/CP/D2/CSM0\n/BA /BT/D4/D4 /CT/CP/D6/CP/D2\r/CT /D3/CU /D8/CW/CT /CU/CP\r/D8/D3/D6 /DA/CX/D3/D0/CP/D8/CX/D2/CV /D8/CW/CT/CP/CQ /D3 /DA /CT /D7\r/CP/D0/CX/D2/CV /CW /DD/D4 /D3/D8/CW/CT/D7/CX/D7 /CX/D7 /D4/D6/D3/CQ/CP/CQ/D0/DD /CP/D2 /CP/D6/D8/CX/CU/CP\r/D8 /D3/CU /CC /CP/CZ /CP/CW/CP/D7/CW/CX/B3/D7 /CB/CF/CC/BA /BT/D7 /CU/CP/D6 /CP/D7 /DB /CT /CZ/D2/D3 /DB/B8 /D9/D4 /D8/D3/D2/D3 /DB /D9/D2/CX/DA /CT/D6/D7/CP/D0 /D0/D3 /DB/B9T /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7 /CW/CP /DA /CT /D2/D3/D8 /CQ /CT/CT/D2/D7/D8/D9/CS/CX/CT/CS/BA\n0.51.01.5\n0 50 100 150 200χT (emu K/mol)\nTemperature (K)AFMFM(a)\n0.0 0.5 1.0\nTemperature0.00.51.0Specific heat(b)\nAFM contribution\nspin−1/2 ferromagnet(1,1/2) ferrimagnet/BY/CX/CV/D9/D6/CT /BF/BA /B4/CP/B5 /BX/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /B4χ·T /DA/D7T /B5 /CU/D3/D6 /D8/CW/CT(1,1/2) /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2 /C6/CX/BV/D9/B4/D4/CQ/CP/B5/BW 2\n/C7/B53·/BE/BW2\n/C7 /CJ /BL℄/BA /B4/CQ/B5Cv\n/DA/D7T /D3/CU /D8/CW/CT(1/1/2) /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2 /B4/BW/C5/CA /BZ/B5 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3Cv(T) /D3/CU /D8/CW/CT/D7/D4/CX/D2/B9/BD/BB/BE /BY/C5 \r /CW/CP/CX/D2 /B4/BW/C5/CA /BZ/B5 /CP/D8h= 0.05J /BA /CC/CW/CT /CS/CX/AR/CT/D6/CT/D2\r/CT /D3/CU /CQ /D3/D8/CW /D7/D4 /CT\r/CX/AS\r /CW/CT/CP/D8/D7 \r/CP/D2 /CQ /CT /CX/CS/CT/D2 /D8/CX/AS/CT/CS/CP/D7 /BT/BY/C5 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CJ /BF/BE ℄/BACv\n/CP/D2/CST /CP/D6/CT /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /D8/CW/CT /D9/D2/CX/D8/CT/D7 /D3/CUJ /BA/CF/CW/CT/D6/CT/CP/D7 /D0/D3 /DB/B9T /D8/CW/CT/D6/D1/D3 /CS/DD/D2/CP/D1/CX\r/D7 /D6/CT/AT/CT\r/D8/D7 /D8/CW/CT /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /D6/CT/D0/CP/D8/CX/D3/D2 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χ(T)∝T−1/CX/D2 /D8/CW/CT /CW/CX/CV/CW/B9T /D6/CT/CV/CX/D3/D2/BA/BY/CX/CV/D9/D6/CT /BF/CP /CS/CX/D7/D4/D0/CP /DD/D7 /CP /D8 /DD/D4/CX\r/CP/D0 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 \r/D9/D6/DA /CT /B4χT /DA/D7T /B5 /DB/CW/CX\r /CW /CX/D7 /D9/D7/D9/CP/D0/D0/DD /D9/D7/CT/CS /D8/D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT/D8/CW/CT \r /CW/CP/D6/CP\r/D8/CT/D6 /D3/CU /D8/CW/CT /D7/CW/D3/D6/D8/B9/D6/CP/D2/CV/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/BA /BY /D3/D6 /CP /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /D7/DD/D7/D8/CT/D1/B8 /BV/D9/D6/CX/CT/B3/D7 /D0/CP /DB /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CT/CQ /CT/CW/CP /DA/CX/D3/D6 χT=const /D3 /DA /CT/D6 /D8/CW/CT /DB/CW/D3/D0/CTT /D6/CP/D2/CV/CT/BA /C1/CU /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D7/DD/D7/D8/CT/D1 /CW/CP/D7 /CS/D3/D1/CX/D2/CP/D2 /D8 /BY/C5 /B4/BT/BY/C5/B5/CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/B8 χT /CX/D2\r/D6/CT/CP/D7/CT/D7 /B4/CS/CT\r/D6/CT/CP/D7/CT/D7/B5 /DB/CW/CT/D2T /CX/D7 /CS/CT\r/D6/CT/CP/D7/CT/CS/BA /CC/CW /D9/D7/B8 /CX/D2 /D8/CW/CT /CX/D2 /D8/CT/D6/DA /CP/D070K < T < 300K/D8/CW/CT /CS/D3/D1/CX/D2/CP/D8/CX/D2/CV \r/D3/D9/D4/D0/CX/D2/CV /CQ /CT/D8 /DB /CT/CT/D2 /D2/CT/CP/D6/CT/D7/D8/B9/D2/CT/CX/CV/CW /CQ /D3/D6 /D7/D4/CX/D2/D7 /CX/D7 /BT/BY/C5/B8 /DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT /CX/D2\r/D6/CT/CP/D7/CT /D3/CUχT /CQ /CT/D0/D3 /DB/D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CP/D6/D3/D9/D2/CST≈70 /C3 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CQ /CT/CW/CP /DA /CT/D7 /D0/CX/CZ /CT /CP /BY/C5 \r /CW/CP/CX/D2 /CP/D8 /D0/D3 /DBT /BA /CC/CW/CT/D7/D8/CT/CT/D4 /CS/CT\r/D6/CT/CP/D7/CT /CQ /CT/D0/D3 /DB10 /C3 /D1/CP /DD /CQ /CT /CP/D8/D8/D6/CX/CQ/D9/D8/CT/CS /D8/D3 /CP/D2 /CT/D7/D8/CP/CQ/D0/CX/D7/CW/CT/CS /BF/BW /BT/BY/C5 /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT /D3/D6/CS/CT/D6/BA/CC /D9/D6/D2/CX/D2/CV /D8/D3 /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /D7/D4 /CT\r/CX/AS\r /CW/CT/CP/D8/B8 /CX/D8 \r /CW/CP/D2/CV/CT/D7 /CU/D6/D3/D1Cv(T)∝T1/2/CX/D2 /D8/CW/CT /CT/DC/D8/D6/CT/D1/CT /D0/D3 /DB/B9T/D6/CT/CV/CX/D3/D2 /CJ/D7/CT/CT /BX/D5/BA 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/CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2/B8/CQ /D3/D8/CW /CU/D3/D6 /CW/CP/D0/CU/B9/CX/D2 /D8/CT/CV/CT/D6 /CP/D2/CS /CX/D2 /D8/CT/CV/CT/D6 /D7/D4/CX/D2/D7 /CJ /BH/BC /B8 /BH/BD /B8 /BH/BE /B8 /BH/BF /B8 /BH/BG ℄ /B4/D7/CT/CT /CP/D0/D7/D3 /D8/CW/CT /D6/CT/DA/CX/CT/DB /CP/D6/D8/CX\r/D0/CT /CJ /BD/BH ℄/B5 /D7/CT/D8 /D9/D4/CP/D2 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D4/CP/D6/D8 /D3/CU /D8/CW/CX/D7 /D6/CT/D7/CT/CP/D6\r /CW/BA /C1/D8 /CX/D7 /D6/CT/D1/CP/D6/CZ /CP/CQ/D0/CT /D8/CW/CP/D8 /D9/D4 /D8/D3 /D2/D3 /DB /CP /D6/CT/D0/CP/D8/CX/DA /CT/D0/DD /D7/D1/CP/D0/D0 /CP/D1/D3/D9/D2 /D8 /D3/CU/D6/CT/D7/CT/CP/D6\r /CW /D3/D2 /BD/BW /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /CW/CP/D7 /CQ /CT/CT/D2 /D4/D9/CQ/D0/CX/D7/CW/CT/CS/BA /CC/CW/CT/D7/CT /D1/D3 /CS/CT/D0/D7 /D8 /DD/D4/CX\r/CP/D0/D0/DD /CT/DC/CW/CX/CQ/CX/D8/CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/D7 /DB/CX/D8/CW /CP /D2/CT/D8 /BY/C5 /D1/D3/D1/CT/D2 /D8/B8 /D7/D3 /D8/CW/CP/D8 /CP /D2 /D9/D1 /CQ /CT/D6 /D3/CU /CX/D2 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/D3/D6/CX/D2/CV /D7/D4/CX/D2/D7Q/2 =π /CX/D7 /D7/D8/CP/CQ/D0/CT /D9/D4 /D8/D3 /D8/CW/CT /D4/CW/CP/D7/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8\nJ2c=S1S2J1/[2(S2\n1+S2\n2)] /BA /C1/D2 /D8/CW/CT /D7/D8/D6/D3/D2/CV/D0/DD /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D6/CT/CV/CX/D3/D2J2> J2c\n/B8 /D8/CW/CT /D7/D8/CP/CQ/D0/CT /D7/D8/CP/D8/CT /CX/D7 /CP /D7/D4/CX/D6/CP/D0/DB/CX/D8/CW /CP/D2 /D3/D6/CS/CT/D6/CX/D2/CV /DB /CP /DA /CT /DA /CT\r/D8/D3/D6 /CV/CX/DA /CT/D2 /CQ /DDcos(Q/2) =−S1S2J1/[2J2(S2\n1+S2\n2)] /BA /C1/D2 /D8/CW/CT /D0/CX/D1/CX/D8J2→ ∞ /B8\nQ=π /CP/D2/CS /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 \r/D3/D1/D4 /D3/D7/CT/CS /D3/CU /D8 /DB /D3 /CS/CT\r/D3/D9/D4/D0/CT/CS /BT/BY/C5 \r /CW/CP/CX/D2/D7 /DB/CX/D8/CW /D7/CX/D8/CT /D7/D4/CX/D2/D7S1\n/CP/D2/CSS2\n/BA/C6/CT/DC/D8/B8 /D0/CT/D8 /D9/D7 /CS/CX/D7\r/D9/D7/D7 /D8/CW/CT /CT/DC/D8/D6/CT/D1/CT /D5/D9/CP/D2 /D8/D9/D1 \r/CP/D7/CT(S1,S2) = (1,1/2) /BA /BT/D0/D6/CT/CP/CS/DD /CP /D5/D9/CP/D0/CX/D8/CP/D8/CX/DA /CT /D7/CT/D1/CX/B9\r/D0/CP/D7/D7/CX\r/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D8 /D2/CT/CP/D6 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CP/D8J2= 0.2J1\n/D8/CW/CT /BY/C5 /D1/CP/CV/D2/D3/D2/CQ/D6/CP/D2\r /CWE−\nk\n/B4/D7/CT/CT /AS/CV/D9/D6/CT /BE/B5 /CX/D7 /D7/D8/D6/D3/D2/CV/D0/DD /AT/CP/D8/D8/CT/D2/CT/CS/B8 /DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT /D3/D4/D8/CX\r/CP/D0 /BT/BY/C5 /CQ/D6/CP/D2\r /CW /D7/CW/D3 /DB/D7 /D3/D2/D0/DD /CP/D7/D1/D3 /D3/D8/CW /CX/D2\r/D6/CT/CP/D7/CT /D3/CU /D8/CW/CT /BT/BY/C5 /CV/CP/D4∆+/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /D3/D4/D8/CX\r/CP/D0 /D1/CP/CV/D2/D3/D2/D7 /CS/D3 /D2/D3/D8 /D4/D0/CP /DD /CP/D2 /DD /CX/D1/B9/D4 /D3/D6/D8/CP/D2 /D8 /D6/D3/D0/CT /CX/D2 /D8/CW/CT /D1/CT\r /CW/CP/D2/CX/D7/D1 /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/BA /C7/D2 /CP/D4/D4/D6/D3/CP\r /CW/CX/D2/CV /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8/B8 /D8/CW/CT/CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /CP/D2/CS /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /D4/D0/CP /DD /D7/D3/D1/CT/DB/CW/CP/D8 /CS/CX/AR/CT/D6/CT/D2 /D8 /D6/D3/D0/CT/D7/BM /CF/CW/CT/D6/CT/CP/D7 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r/CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D7/D8/D6/D3/D2/CV/D0/DD /D6/CT/CS/D9\r/CT/D7 /D8/CW/CT /D7/CW/D3/D6/D8/B9/D6/CP/D2/CV/CT /D7/D4/CX/D2 \r/D3/D6/D6/CT/D0/CP/D8/CX/D3/D2 /D0/CT/D2/CV/D8/CW /CX/D2 /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT/B8 /D8/CW/CT/D5/D9/CP/D2 /D8/D9/D1 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /D7/D8/CP/CQ/CX/D0/CX/DE/CT /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D3/D6/CS/CT/D6 /D9/D4 /D8/D3 /D8/CW/CT /D4 /D3/CX/D2 /D8J2= 0.231J1\n/CQ /CT/DD /D3/D2/CS /D8/CW/CT\r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CP/D8J2= 0.2J1\n/BA /BT/D2/D3/D8/CW/CT/D6 /CT/AR/CT\r/D8 /D3/CU /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /CX/D7 /D6/CT/D0/CP/D8/CT/CS /D8/D3/D8/CW/CT \r /CW/CP/D2/CV/CT /D3/CU /D8/CW/CT \r /CW/CP/D6/CP\r/D8/CT/D6 /D3/CU /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/BM /BT /CS/CT/D8/CP/CX/D0/CT/CS /BW/C5/CA /BZ /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD/D0/CT/DA /CT/D0/D7 /CP/D6/D3/D9/D2/CS /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 \r/D0/CT/CP/D6/D0/DD /CX/D2/CS/CX\r/CP/D8/CT/D7 /CP /D0/CT/DA /CT/D0 \r/D6/D3/D7/D7/CX/D2/CV/B8 /CX/BA/CT/BA/B8 /CP /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D5/D9/CP/D2 /D8/D9/D1/D4/CW/CP/D7/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D8/D3 /CP /D7/CX/D2/CV/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CP/D8J2= 0.231J1\n/BA /BY /D9/D6/D8/CW/CT/D6/B8 /BW/C5/CA /BZ /CW/CP/D7 /CP/D0/D7/D3 /CX/D2/CS/CX\r/CP/D8/CT/CS /D8/CW/CP/D8/CP/D8 /D0/CT/CP/D7/D8 /CU/D6/D3/D1J2= 0.25J1\n/D9/D4 /DB /CP/D6/CS/D7 /D8/CW/CT /CS/CX/D7\r/D9/D7/D7/CT/CS /D1/D3 /CS/CT/D0 /CT/DC/CW/CX/CQ/CX/D8/D7 /CP /D7/CX/D2/CV/D0/CT/D8 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/AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7/B8 /D3/D2/CT /CX/D7 /CT/D2/CU/D3/D6\r/CT/CS /D8/D3/D0/D3 /D3/CZ /CU/D3/D6 /D4 /D3/D7/D7/CX/CQ/D0/CT /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CS/CX/D7/D3/D6/CS/CT/D6/CT/CS /D7/D8/CP/D8/CT/D7/BA /BT /DA /CP/D0/D9/CP/CQ/D0/CT /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 \r/CP/D2 /CQ /CT /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1/D8/CW/CT /C4/CX/CT/CQ/B9/CB\r /CW /D9/D0/D8/DE/B9/C5/CP/D8/D8/CX/D7 /D8/CW/CT/D3/D6/CT/D1 /CJ /BI/BF ℄ /CP/CS/CP/D4/D8/CT/CS /D8/D3 /D1/CX/DC/CT/CS/B9\r /CW/CP/CX/D2 /D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /CJ /BI/BG ℄/BA /CC/CW/CT /D8/CW/CT/D3/D6/CT/D1 /CX/D7/CP/D4/D4/D0/CX\r/CP/CQ/D0/CT /D8/D3 /D7/DD/D7/D8/CT/D1/D7 /DB/CX/D8/CW /CP /CW/CP/D0/CU/B9/CX/D2 /D8/CT/CV/CT/D6 \r/CT/D0/D0 /D7/D4/CX/D2 /B4S1+S2\n/B5 /CP/D2/CS /D7/CP /DD/D7 /D8/CW/CP/D8 /D8/CW/CT /D1/D3 /CS/CT/D0 /CT/CX/D8/CW/CT/D6 /CW/CP/D7/CV/CP/D4/D0/CT/D7/D7 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 /D3/D6 /CT/D0/D7/CT /CW/CP/D7 /CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT/D7/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D3/D2/CT /D1/CP /DD /D0/D3 /D3/CZ /CU/D3/D6 /D4/CW/CP/D7/CT/D7 /DB/CX/D8/CW/D4/D6/CT/D7/D9/D1/CP/CQ/D0/DD /D7/D3/D1/CT /CQ/D6/D3/CZ /CT/D2 /CS/CX/D7\r/D6/CT/D8/CT /D7/DD/D1/D1/CT/D8/D6/DD /BA /C1/D2 /D4/CP/D6/D8/CX\r/D9/D0/CP/D6/B8 /D8/CW/CT /D0/D3/D2/CV/B9/D6/CP/D2/CV/CT/CS \r /CW/CX/D6/CP/D0 /D4/CW/CP/D7/CT /CU/D3/D9/D2/CS /CX/D2/D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D7/D4/CX/D2/B9/BD/BB/BE /BY/C5J1−J2\n/D1/D3 /CS/CT/D0 /CX/D2 /D8/CW/CT /D7/D1/CP/D0/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D6/CT/CV/CX/D3/D2 \r/D3/D9/D0/CS /CQ /CT /CP /D4 /D3/D7/D7/CX/CQ/D0/CT\r/CP/D2/CS/CX/CS/CP/D8/CT /CJ /BI/BD ℄/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /DA /CP/D6/CX/CT/D8 /DD /D3/CU /D4 /D3/D7/D7/CX/CQ/D0/CT /D2/D3/D2/B9/D1/CP/CV/D2/CT/D8/CX\r /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT/D7 /D1/CP /DD /CQ /CT \r/D3/D2/D7/CX/CS/CT/D6/CP/CQ/D0/DD/CT/D2/D0/CP/D6/CV/CT/CS /CS/D9/CT /D8/D3 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2\r/CT /D3/CU /D8 /DB /D3 /CZ/CX/D2/CS/D7 /D3/CU /D7/CX/D8/CT /D7/D4/CX/D2/D7 /CX/D2 /D8/CW/CT /D1/CX/DC/CT/CS /DA /CP/D6/CX/CP/D2 /D8 /D3/CU /D8/CW/CT /D4/CP/D6/CT/D2 /D8 /D1/D3 /CS/CT/D0/BA/BF/BA/BE/BA /BY /D6/D9/D7/D8/D6/CP/D8/CT/CS /CS/CX/CP/D1/D3/D2/CS \r/CW/CP/CX/D2/D7/CC/CW/CT /CS/CX/CP/D1/D3/D2/CS /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /CX/D7 /CP/D2/D3/D8/CW/CT/D6 /CV/CT/D2/CT/D6/CX\r /D1/D3 /CS/CT/D0 /D3/CU /BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7 \r/D3/D2/B9/D7/D8/D6/D9\r/D8/CT/CS /CU/D6/D3/D1 /D3/D2/CT /CZ/CX/D2/CS /D3/CU /D7/D4/CX/D2/D7 /D0/CX/DA/CX/D2/CV /D3/D2 /D7/D9/CQ/D0/CP/D8/D8/CX\r/CT/D7 /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8 /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D7/CX/D8/CT/D7 /B4/D7/CT/CT /AS/CV/D9/D6/CT /BG/CQ/B5/BA/CC/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D7/DD/D1/D1/CT/D8/D6/CX\r /CS/CX/CP/D1/D3/D2/CS \r /CW/CP/CX/D2 /B4/CB/BW/BV/B5 /B4J1=J′\n1\n/B8J= 0 /B5 /DB/CX/D8/CW /BT/BY/C5 /DA /CT/D6/D8/CX\r/CP/D0 /CQ /D3/D2/CS/D7\nJ⊥>0 /CX/D7 /D4/D6/D3/CQ/CP/CQ/D0/DD /D8/CW/CT /AS/D6/D7/D8 /D7/D8/D9/CS/CX/CT/CS /D1/D3 /CS/CT/D0 /D3/CU /BD/BW /D5/D9/CP/D2 /D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8 /DB/CX/D8/CW \r/D3/D1/D4 /CT/D8/CX/D2/CV /CX/D2 /D8/CT/D6/B9/BJ/C6/BA /BU/BA /C1/DA/CP/D2/D3/DA/CP\r/D8/CX/D3/D2/D7 /CJ /BI/BH /B8 /BI/BI ℄/BA /BT /D4/CP/D6/D8/CX\r/D9/D0/CP/D6 /DA /CP/D6/CX/CP/D2 /D8 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/D3 /CS/CT/D0/B8 /D8/CW/CT /CS/CX/D7/D8/D3/D6/D8/CT/CS /B4J1∝ne}ationslash=J′\n1\n/B5 /D7/D4/CX/D2/B9/BD/BB/BE/CS/CX/CP/D1/D3/D2/CS \r /CW/CP/CX/D2/B8 /CW/CP/D7 /D6/CT\r/CT/CX/DA /CT/CS /CX/D2\r/D6/CT/CP/D7/CX/D2/CV /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /CJ /BI/BJ /B8 /BI/BK ℄ /CP/D7 /DB /CT/D0/D0 /CP/D7 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /CX/D2 /D8/CT/D6/CT/D7/D8 /CX/D2 /D8/CW/CT/D4/CP/D7/D8 /CS/CT\r/CP/CS/CT /CS/D9/CT /D8/D3 /CX/D8/D7 /D6/CX\r /CW /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /CJ /BI/BL ℄ /B4/D7/CT/CT /AS/CV/D9/D6/CT /BH/CP/B5 /CP/D2/CS /D8/CW/CT /D6/CT/D0/CT/DA /CP/D2\r/CT /CU/D3/D6 /D8/CW/CT/D6/CT/CP/D0 /D1/CP/D8/CT/D6/CX/CP/D0 /BV/D93\n/B4/BV/C73\n/B52\n/B4/C7/C0/B52\n/B4/CP/DE/D9/D6/CX/D8/CT/B5 /CJ /BJ/BC ℄/BA /CF/CX/D8/CW/D3/D9/D8 /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/B8 /D8/CW/D6/CT/CT /D5/D9/CP/D2 /D8/D9/D1/D4/CW/CP/D7/CT/D7 /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CT /B4J1/J⊥,J′\n1/J⊥\n/B5 /CW/CP /DA /CT /CQ /CT/CT/D2 /CS/CX/D7\r/D9/D7/D7/CT/CS /CU/D3/D6 /D8/CW/CT /CS/CX/D7/D8/D3/D6/D8/CT/CS /D1/D3 /CS/CT/D0/BM /BY /D3/D6\nJ1/J⊥,J′\n1/J⊥≪1 /B8 /D8/CW/CT /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD /D7/CT\r/D8/D3/D6 /CX/D7 /CV/D3 /DA /CT/D6/D2/CT/CS /CQ /DD /CP/D2 /CT/AR/CT\r/D8/CX/DA /CT /D7/D4/CX/D2/B9/BD/BB/BE /BT/BY/C5 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV/D1/D3 /CS/CT/D0/B8 /DB/CW/CX\r /CW /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT /CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /D4/CW/CP/D7/CT /B4/CB/BY/B5 /DB/CX/D8/CW /D7/D3/D1/CT /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /CW/CX/CV/CW/B9/CT/D2/CT/D6/CV/DD /D1/D3 /CS/CT/D7 /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D0/D3 \r/CP/D0 /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /DA /CT/D6/D8/CX\r/CP/D0 /CS/CX/D1/CT/D6/D7/BA /BY /D3/D6 /CX/D2 /D8/CT/D6/D1/CT/CS/CX/CP/D8/CT J1/J⊥\n/CP/D2/CSJ′\n1/J⊥\n/B8/D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CS/CX/D1/CT/D6/CX/DE/CT/D7/B8 /CU/D3/D6/D1/CX/D2/CV /CP /D8 /DB /D3/CU/D3/D0/CS /CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /D7/CT/D5/D9/CT/D2\r/CT /D3/CU /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /D8/CT/D8/D6/CP/D1/CT/D6/D7 /CP/D2/CS/CS/CX/D1/CT/D6/D7 /B4/CC/BW1\n/D4/CW/CP/D7/CT/B5/BA /BY/CX/D2/CP/D0/D0/DD /B8 /CU/D3/D6 /CQ /D3/D8/CWJ1/J⊥\n/CP/D2/CSJ′\n1/J⊥\n/D7/D9Ꜷ\r/CX/CT/D2 /D8/D0/DD /D0/CP/D6/CV/CT/B8 /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CX/D7 /CP/BD/BW /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/BA /CC/CW/CT/D7/CT /D4/CW/CP/D7/CT/D7 \r/CP/D2 /CQ /CT \r/D0/CT/CP/D6/D0/DD /CX/CS/CT/D2 /D8/CX/AS/CT/CS /CP/D0/D6/CT/CP/CS/DD /CX/D2 /D8/CW/CT /CB/BW/BV/BM /CB/BY /B4J1<0.5J⊥\n/B5/B8 /CC/BW1/B4<0.5J⊥< J1<1.10J⊥\n/B5/B8 /CP/D2/CS /BD/BW /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8 /B4J1>1.10J⊥\n/B5/BA\n1.0 2.0 0.01.02.0\nFerrimagnet\nJ1’J1__\nSFTD1J =1(a)\n 1 2 3\n−2 −1  0 0\n 1  2  3L=8\nL=10\nL=12J\n1TD\nTD1\n2\nFerrimagnetFM\nFM1\n2 ?|\nKSF\nDM(b)/BY/CX/CV/D9/D6/CT /BH/BA /B4/CP/B5 /C9/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /B4T= 0 /B5 /D3/CU /D8/CW/CT /CS/CX/D7/D8/D3/D6/D8/CT/CS /CS/CX/CP/D1/D3/D2/CS \r /CW/CP/CX/D2 /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/D4/CP\r/CTJ1\n/DA/D7/BAJ′\n1\n/B4J⊥= 1 /B5 /CJ /BI/BL℄/BA /CB/BY /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /D4/CW/CP/D7/CT/B8 /DB/CW/CT/D6/CT/CP/D7 /CC/BW1\n/D1/CP/D6/CZ/D7 /D8/CW/CT/D8/CT/D8/D6/CP/D1/CT/D6/B9/CS/CX/D1/CT/D6 /D4/CW/CP/D7/CT \r/D3/D2/D7/D8/D6/D9\r/D8/CT/CS /CU/D6/D3/D1 /CP/D0/D8/CT/D6/D2/CP/D8/CX/D2/CV /CS/CX/D1/CT/D6/D7 /CP/D2/CS /D8/CT/D8/D6/CP/D1/CT/D6/D7 /CX/D2 /D0/D3 \r/CP/D0 /D7/CX/D2/CV/D0/CT/D8 /D7/D8/CP/D8/CT/D7/BA/B4/CQ/B5 /C9/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /B4T= 0 /B5 /D3/CU /CP /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /CB/BW/BV /DB/CX/D8/CW \r/D3/D1/D4 /CT/D8/CX/D2/CV /CU/D3/D9/D6/B9/D7/D4/CX/D2 \r/DD\r/D0/CX\r/CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CTJ⊥\n/DA/D7/BAK /B4J′\n1=J′\n1= 1 /B5 /CJ /BJ/BD ℄/BA /C1/D2 /D8/CW/CX/D7 /D7/DD/D7/D8/CT/D1/B8 /D8/CW/CT/D6/CT /CP/D4/D4 /CT/CP/D6 /CP/D8/D0/CT/CP/D7/D8 /AS/DA /CT /D2/CT/DB /D4/CW/CP/D7/CT/D7 /CS/CT/D2/D3/D8/CT/CS /CP/D7 /CU/D3/D0/D0/D3 /DB/D7/BM /BY/C51\n/B4/CU/D9/D0/D0/DD/B9/D4 /D3/D0/CP/D6/CX/DE/CT/CS /BY/C5 /D4/CW/CP/D7/CT/B5/B8 /BY/C52\n/B4/CP/D2/D3/D8/CW/CT/D6 /D4/CW/CP/D7/CT/DB/CX/D8/CW /CP /D2/CT/D8 /BY/C5 /D1/D3/D1/CT/D2 /D8/B5/B8 /BW/C5 /B4/CU/D3/D9/D6/CU/D3/D0/CS /CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /D7/CX/D2/CV/D0/CT/D8 /D4/CW/CP/D7/CT/B5/B8 /CC/BW2\n/B4/CP/D2/D3/D8/CW/CT/D6 /D8/CT/D8/D6/CP/D1/CT/D6/B9/CS/CX/D1/CT/D6/D4/CW/CP/D7/CT /DB/CX/D8/CW /D8/CT/D8/D6/CP/D1/CT/D6/D7 /CX/D2 /D0/D3 \r/CP/D0 /D8/D6/CX/D4/D0/CT/D8 /D7/D8/CP/D8/CT/D7/B5/B8 /CP/D2/CS /CP/D2 /CT/DC/D3/D8/CX\r /D2/D3/D2/B9/C4/CX/CT/CQ/B9/C5/CP/D8/D8/CX/D7 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT/CS/CT/D2/D3/D8/CT/CS /CQ /DD /CP /D5/D9/CT/D7/D8/CX/D3/D2 /D1/CP/D6/CZ/BA /CC/CW/CT /CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7 /D8/D6/CP\r/CT /D8/CW/CT /D4/CW/CP/D7/CT /CQ /D3/D9/D2/CS/CP/D6/CX/CT/D7 /D3/CU /D8/CW/CT /D7/CX/D2/CV/D0/CT/B9/CS/CX/CP/D1/D3/D2/CS/D7/DD/D7/D8/CT/D1/BA/CC/CW/CT /CS/CX/CP/D1/D3/D2/CS /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /CX/D7 /CP/D0/D7/D3 /D3/D2/CT /D3/CU /D8/CW/CT /D7/CX/D1/D4/D0/CT/D7/D8 /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /CP/CS/D1/CX/D8/D8/CX/D2/CV /CP/CU/D3/D9/D6/B9/D7/D4/CX/D2 \r/DD\r/D0/CX\r /CT/DC\r /CW/CP/D2/CV/CT \r/D3/D9/D4/D0/CX/D2/CV/BA /BU/CT/D0/D3 /DB /DB /CT /CS/CX/D7\r/D9/D7/D7 /D8/CW/CT /CX/D1/D4/CP\r/D8 /D3/CU /D8/CW/CX/D7 \r/D3/D1/D4 /CT/D8/CX/D2/CV /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D3/D2/D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /B4T= 0 /B5 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /CB/BW/BV /CJ /BJ/BD ℄/BA /CC/CW/CT /D7\r /CW/CT/D1/CP/D8/CX\r /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2/D3/CU /D8/CW/CT /D1/D3 /CS/CT/D0 /CX/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /AS/CV/D9/D6/CT /BG /CQ/B8 /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /CU/D3/D9/D6/B9/D7/D4/CX/D2 \r/DD\r/D0/CX\r /CT/DC\r /CW/CP/D2/CV/CT \r/D3/D9/D4/D0/CX/D2/CV /CJ /BJ/BE ℄ /CX/D2/CP /D7/CX/D2/CV/D0/CT /CS/CX/CP/D1/D3/D2/CS /CQ /CT/CX/D2/CV \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS /CQ /DD /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 K /BA /C1/D8 /CX/D7 /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 /D8/D3 /D2/D3/D8/CX\r/CT /D8/CW/CP/D8 /D8/CW/CT \r/DD\r/D0/CX\r\r/D3/D9/D4/D0/CX/D2/CV /CS/D3 /CT/D7 /D2/D3/D8 /DA/CX/D3/D0/CP/D8/CT /D8/CW/CT /D0/D3 \r/CP/D0 /D7/DD/D1/D1/CT/D8/D6/DD /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /B4J⊥>0 /B5 /CB/BW/BV /D1/D3 /CS/CT/D0 /D9/D2/CS/CT/D6 /D8/CW/CT/CT/DC\r /CW/CP/D2/CV/CT /D3/CU /D4/CP/CX/D6/D7 /D3/CU /D3/AR/B9\r /CW/CP/CX/D2 /D7/D4/CX/D2/D7S1a,n\n/CP/D2/CSS1b,n\n/CX/D2 /D8/CW/CT /CS/CX/CP/D1/D3/D2/CS/D7/BA /CC/CW /D9/D7/B8 /CX/D2 /D8/CW/CT /CX/D1/D4 /D3/D6/D8/CP/D2 /D8 \r/CP/D7/CT/D3/CU /D7/D4/CX/D2/B91/2 /D3/AR/B9\r /CW/CP/CX/D2 /D3/D4 /CT/D6/CP/D8/D3/D6/D7/B8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /B4/CP/D7 /CX/D2 /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/D3 /CS/CT/D0 /DB/CX/D8/CW/D3/D9/D8\r/DD\r/D0/CX\r /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/B5 /CQ /DDL /D0/D3 \r/CP/D0 /CV/D3 /D3 /CS /D5/D9/CP/D2 /D8/D9/D1 /D2 /D9/D1 /CQ /CT/D6/D7 sn= 0,1 /B4n= 1,2,...,L /B5 /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT\r/D3/D1/D4 /D3/D7/CX/D8/CT /D3/AR/B9\r /CW/CP/CX/D2 /D7/D4/CX/D2/D7S1,n=S1a,n+S1b,n\n/BMS2\n1,n=sn(sn+1) /BA /CD/D7/CX/D2/CV /D8/CW/CX/D7 /D0/D3 \r/CP/D0 /D7/DD/D1/D1/CT/D8/D6/DD /B8 /D8/CW/CT/C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /D3/CU /D8/CW/CT /CB/BW/BV \r/CP/D2 /CQ /CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /D8/CW/CT \r/D3/D1/D4/CP\r/D8 /CU/D3/D6/D1\nHc=E0+L/summationdisplay\nn=1/bracketleftbigg\nJ1S1,n·(S2,n+S2,n+1)+JnS2,n·S2,n+1+K\n2{S1,n·S2,n,S1,n·S2,n+1}/bracketrightbigg\n. /B4/BH/B5/C0/CT/D6/CTE0=J⊥/summationtextL\nn=1[sn(sn+1)−3/4] /CX/D7 /CP /AS/DC/CT/CS /D2 /D9/D1 /CQ /CT/D6 /CU/D3/D6 /CT/DA /CT/D6/DD /D7/CT\r/D8/D3/D6 /CS/CT/AS/D2/CT/CS /CP/D7 /CP /D7/CT/D5/D9/CT/D2\r/CT /D3/CU /D8/CW/CT/BK/CB/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2/D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7/D0/D3 \r/CP/D0 /D5/D9/CP/D2 /D8/D9/D1 /D2 /D9/D1 /CQ /CT/D6/D7 [s1,s2,...,s L] /B8Jn=J+K/4−snK /B8 /CP/D2/CS{A,B} /CX/D7 /D8/CW/CT /CP/D2 /D8/CX\r/D3/D1/D1 /D9/D8/CP/D8/D3/D6/D3/CU /D8/CW/CT /D3/D4 /CT/D6/CP/D8/D3/D6/D7 A /CP/D2/CSB /BA/C4/CT/D8 /D9/D7 /CQ/D6/CX/CT/AT/DD \r/D3/D1/D1/CT/D2 /D8 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /CU/D3/D6 /D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /CB/BW/BV /B4/AS/CV/D9/D6/CT /BH/CQ/B5/B8 /CQ /DD /D9/D7/CX/D2/CV /D7/D3/D1/CT/D7/DD/D1/D1/CT/D8/D6/CX/CT/D7 /D3/CU 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1.5\n℄/CT/DC/CW/CX/CQ/CX/D8/CX/D2/CV /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7/B8 /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D7/D8/D6/D9\r/D8/D9/D6/CT/D7/CW/D3 /DB/D2 /CX/D2 /AS/CV/D9/D6/CT /BG /CS /CJ /BJ/BH ℄/BA /CF /CT /CP/D6/CT /D2/D3/D8 /CP /DB /CP/D6/CT /D3/CU /CP/D2 /DD /D6/CT/CP/D0 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r \r/D3/D1/D4 /D3/D9/D2/CS /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT/D0/CP/CS/CS/CT/D6 /D7/D8/D6/D9\r/D8/D9/D6/CT /D7/CW/D3 /DB/D2 /CX/D2 /AS/CV/D9/D6/CT /BG /CT/BA /C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7/B8 /CX/D2 /DA/CX/CT/DB /D3/CU /D8/CW/CT /CT/CP/D7/DD \r/D3/D2 /D8/D6/D3/D0 /D3/CU /D8/CW/CT /D1/D3/D0/CT\r/D9/D0/CP/D6/B9/D9/D2/CX/D8/D4 /D3/D7/CX/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /D1/D3/D0/CT\r/D9/D0/CP/D6 \r /CW/CT/D1/CX/D7/D8/D6/DD /B8 /CX/D8 /D1/CP /DD /CQ /CT /CT/DC/D4 /CT\r/D8/CT/CS /D8/CW/CP/D8 /D3/D8/CW/CT/D6 /D1/CP/D8/CT/D6/CX/CP/D0/D7/B8 /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /CV/CT/D2/CT/D6/CX\r/D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /AS/CV/D9/D6/CT /BG/B8 /DB/CX/D0/D0 /CQ /CT /D7/DD/D2 /D8/CW/CT/D7/CX/DE/CT/CS /CX/D2 /D8/CW/CT /D2/CT/CP/D6 /CU/D9/D8/D9/D6/CT/BA/C7/D2 /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /D7/CX/CS/CT/B8 /D9/D2/CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /DA /CP/D6/CX/CP/D2 /D8/D7 /D3/CU /D8/CW/CT /D0/CP/CS/CS/CT/D6 /D1/D3 /CS/CT/D0/D7 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /AS/CV/D9/D6/CT /BG /CW/CP /DA /CT/CP/D0/D6/CT/CP/CS/DD /CQ /CT/CT/D2 /CP/D2/CP/D0/DD/DE/CT/CS /CX/D2 /CP /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D4/D9/CQ/D0/CX\r/CP/D8/CX/D3/D2 /CJ /BJ/BI/B8 /BJ/BJ /B8 /BJ/BK /B8 /BJ/BL /B8 /BK/BC /B8 /BK/BD ℄/BA /CC/CW/CT /D9/D2/CU/D6/D9/D7/D8/D6/CP/D8/CT/CS\r /CW/CT\r /CZ /CT/D6/CQ /D3/CP/D6/CS /D0/CP/CS/CS/CT/D6 /B4 /AS/CV/D9/D6/CT /BG\r/B5 /DB/CX/D8/CW /BT/BY/C5 /CQ /D3/D2/CS/D7 /B4J,J1>0 /B5 /CT/DC/CW/CX/CQ/CX/D8/D7 /CP /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /CV/D6/D3/D9/D2/CS/D7/D8/CP/D8/CT/BA /C1/D8/D7 /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 \r/D0/D3/D7/CT/D0/DD /D6/CT/D4/D6/D3 /CS/D9\r/CT /D8/CW/CT /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /D8/CW/CT /CV/CT/D2/CT/D6/CX\r /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 \r /CW/CP/CX/D2/CS/CX/D7\r/D9/D7/D7/CT/CS /CX/D2 /CB/CT\r/D8/CX/D3/D2 /BE /CJ /BJ/BL ℄/BA /C6/D3/D8/CT /D8/CW/CP/D8 /CP /DA /CP/D6/CX/CP/D2 /D8 /D3/CU /D8/CW/CX/D7 /D7/D8/D6/D9\r/D8/D9/D6/CT /DB/CX/D8/CW /BY/C5 /D0/CT/CV/D7 /B4J1<0 /B5 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/D7/BL/C6/BA /BU/BA /C1/DA/CP/D2/D3/DA\nCB AM =3/2\nGapless spin fluid\nM =1/2\n012345h/J\n0.10.20.30.4 0.5 0.60.7\nJ /J211\nNon−Lieb−Mattisferrimagnet\nD(a)\n0\n0\nM\n00.20.40.6\n0.30 0.34 0.38 0.42J /J120(S ,S )=(1,1/2)\nh=01 2(b)/BY/CX/CV/D9/D6/CT /BI/BA /B4/CP/B5 /C8/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /D3/CU /D8/CW/CT \r /CW/CT\r /CZ /CT/D6/CQ /D3/CP/D6/CS /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 (1/1/2) /D0/CP/CS/CS/CT/D6 /B4/AS/CV/D9/D6/CT /BG \r/B5 /CX/D2 /D8/CW/CT\n(J2/J1,h/J1) /D4/D0/CP/D2/CT /CJ /BK/BE ℄/BA /CC/CW/CT /D4/CW/CP/D7/CT /CQ /D3/D9/D2/CS/CP/D6/DD /D3/CU /D8/CW/CT /CU/D9/D0/D0/DD /D4 /D3/D0/CP/D6/CX/DE/CT/CS /D4/CW/CP/D7/CT /B4M0= 3/2 /B5 /CX/D7 /CT/DC/CP\r/D8/BA/C0/CT/D6/CTM0\n/CX/D7 /D8/CW/CT /BY/C5 /D1/D3/D1/CT/D2 /D8 /D4 /CT/D6 /D6/D9/D2/CV/BA /CC/CW/CT /CQ /D3/D9/D2/CS/CP/D6/DD /B4/BT/BU/BV/BW/B5 /D3/CU /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r M0= 1/2/D4/D0/CP/D8/CT/CP/D9 /D4/CW/CP/D7/CT /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /BX/BW /D3/CU /D4 /CT/D6/CX/D3 /CS/CX\r \r/D0/D9/D7/D8/CT/D6/D7 /DB/CX/D8/CWL= 10 /B4/D7/D5/D9/CP/D6/CT/D7/B5 /CP/D2/CS\nL= 12 /B4\r/D6/D3/D7/D7/CT/D7/B5 /D6/D9/D2/CV/D7/BA /CC/CW/CT /D4 /D3/CX/D2 /D8/D7 /BU/B8 /BV/B8 /CP/D2/CS /BW /D1/CP/D6/CZ/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /B8 /D8/CW/CT \r /CW/CP/D2/CV/CT /D3/CU /D8/CW/CT /D0/D3 /DB /CT/D7/D8/CT/DC\r/CX/D8/CT/CS /D7/D8/CP/D8/CT/D7/B8 /D8/CW/CT /D8/CX/D4 /D3/CU /D8/CW/CT /D0/D3/CQ /CT/B8 /CP/D2/CS /D8/CW/CTh= 0 /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CP/D8J2/J1= 0.399 /CU/D6/D3/D1 /D8/CW/CT\nM0<1/2 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /D8/D3 /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /D4/CW/CP/D7/CT/D7/BA /CC/CW/CT /D0/CP/D8/D8/CT/D6 /D4/CW/CP/D7/CT /D3 \r\r/D9/D4/CX/CT/D7 /D8/CW/CT /D6/CT/D7/D8/D3/CU /D8/CW/CT /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1/BA /B4/CQ/B5M0\n/DA/D7J2/J1\n/CU/D3/D6 /D8/CW/CT /D2/D3/D2/B9/C4/CX/CT/CQ/B9/C5/CP/D8/D8/CX/D7 /B4/CX/BA/CT/BA/B8M0<1/2 /B5 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r/D4/CW/CP/D7/CT/B8 /CP/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D2 /D9/D1/CT/D6/CX\r/CP/D0 /BX/BW /D3/CUL= 12 \r/D0/D9/D7/D8/CT/D6/D7 /B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/B5/BA /CC/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT \r/D3/D2/D2/CT\r/D8/D7/D8/CW/CT /D1/CX/CS/D4 /D3/CX/D2 /D8/D7 /D3/CU /D8/CW/CT /D7/D8/CT/D4/D7/CT /CJ /BK/BF ℄/BA\r/D3/D1/D4/D0/CT/D8/CT/D0/DD /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/CT/CP/D8/D9/D6/CT/D7/BM /CC/CW/CT /BY/C5 /D0/CT/CV \r/D3/D9/D4/D0/CX/D2/CV /CS/D6/CX/DA /CT/D7 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D2 /D8/D3 /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /CV/D6/D3/D9/D2/CS/D7/D8/CP/D8/CT /DB/CW/CX\r /CW /CX/D7 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CU/D3/D6 /D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /BT/BY/C5 /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV \r /CW/CP/CX/D2 /CJ /BK/BD ℄/BA /CB/CX/D1/CX/D0/CP/D6 /CV/CP/D4/D0/CT/D7/D7 /D4/CW/CP/D7/CT/CP/D4/D4 /CT/CP/D6/D7 /CX/D2 /D8/CW/CT /D9/D2/CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /DA /CP/D6/CX/CP/D2 /D8 /D3/CU /D8/CW/CT /D1/D3 /CS/CT/D0 /CS/CX/D7/D4/D0/CP /DD /CT/CS /CX/D2 /AS/CV/D9/D6/CT /BG /CT /CJ /BJ/BJ /B8 /BJ/BK ℄/B8 /DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT /D1/D3 /CS/CT/D0/D7/CW/D3 /DB/D2 /CX/D2 /AS/CV/D9/D6/CT /BG /CS /D4 /D3/D7/D7/CT/D7/D7/CT/D7 /CP /CV/CP/D4/D4 /CT/CS /D2/D3/D2/B9/CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /D6/D9/D2/CV/B9/D7/CX/D2/CV/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CJ /BK/BC ℄/B8 /DB/CW/CX\r /CW /CX/D7 /D8/CW/CT\r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /D4/CW/CP/D7/CT /D3/CU /D8/CW/CT /D9/D2/CX/CU/D3/D6/D1 /D7/D4/CX/D2/B9/BD/BB/BE /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV /D0/CP/CS/CS/CT/D6/BA/CC /D3 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS /D8/CW/CT /D6/D3/D0/CT /D3/CU /D8/CW/CT /CV/CT/D3/D1/CT/D8/D6/CX\r /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /CX/D2 /D7/D9\r /CW /D1/CX/DC/CT/CS/B9/D7/D4/CX/D2 /D1/D3 /CS/CT/D0/D7/B8 /D0/CT/D8 /D9/D7 /D8/D9/D6/D2/D8/D3 /D8/CW/CT /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /CX/D2 /AS/CV/D9/D6/CT /BI/CP /D4/D6/CT/D7/CT/D2 /D8/CX/D2/CV /D8/CW/CT /D4/CW/CP/D7/CT/D7 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS (1,1/2) \r /CW/CT\r /CZ /CT/D6/CQ /D3/CP/D6/CS/D0/CP/CS/CS/CT/D6 /B4/AS/CV/D9/D6/CT /BG\r/B5 /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 /D7/D4/CP\r/CT(J2/J1,h/J1) /CJ /BK/BE ℄/BA /CF /CT /CP/D6/CT /D2/D3/D8 /CP /DB /CP/D6/CT /D3/CU /CP/D2 /DD /D4/D9/CQ/D0/CX\r/CP/D8/CX/D3/D2/D7/D7/D8/D9/CS/DD/CX/D2/CV /D8/CW/CT /D3/D8/CW/CT/D6 /D8 /DB /D3 /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS /D1/D3 /CS/CT/D0/D7 /CS/CX/D7/D4/D0/CP /DD /CT/CS /CX/D2 /AS/CV/D9/D6/CT/D7 /BG /CS/B8 /CP/D2/CS /CT/BA /BT/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT/CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 J2/J1\n/B8 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CT/CS \r /CW/CT\r /CZ /CT/D6/CQ /D3/CP/D6/CS /D1/D3 /CS/CT/D0/CT/DC/CW/CX/CQ/CX/D8/D7 /D8/CW/D6/CT/CT /D4/CW/CP/D7/CT/D7/B8 /DB/CW/CX\r /CW \r/CP/D2 /CQ /CT /CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /DD /D8/CW/CT /CP/D2/CV/D0/CT/D7(θ,φ) /AS/DC/CX/D2/CV /D8/CW/CT /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT\r/D0/CP/D7/D7/CX\r/CP/D0 /D7/D4/CX/D2/D7S1,n\n/CP/D2/CSS2,n\n/CX/D2 /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r \r/D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 (θ,φ) = (0,0)/DB/CX/D8/CW /D9/D4/B9S1,n\n/CP/D2/CS /CS/D3 /DB/D2/B9S2,n\n/D7/D4/CX/D2/D7/BA /C1/D2 /D8/CW/CT /D7/D4 /CT\r/CX/CP/D0 \r/CP/D7/CT(S1,S2) = (1,1/2) /B8 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 \r/CP/D2 /D8/CT/CS/D7/D8/CP/D8/CT /D7/CW/D3 /DB/D2 /CX/D2 /AS/CV/D9/D6/CT /BG\r /CX/D7 /D7/D8/CP/CQ/D0/CT /CX/D2 /D8/CW/CT /CX/D2 /D8/CT/D6/DA /CP/D0 0.3219J1< J2<0.4606J1\n/BA /BY /D3/D6 /D0/CP/D6/CV/CT/D6J2\n/B8 /CP\r/D3/D0/D0/CX/D2/CT/CP/D6 \r/D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /DB/CX/D8/CW(θ,φ) = (π/2,π/2) /CX/D7 /D7/D8/CP/CQ/CX/D0/CX/DE/CT/CS/BA /C1/D2 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /D0/CX/D1/CX/D8/B8 /D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/D7/CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT \r/CP/D2 /D8/CT/CS /D7/D8/CP/D8/CT /CP/D2/CS /D8/CW/CT /D3/D8/CW/CT/D6 /D8 /DB /D3 /D4/CW/CP/D7/CT/D7 /CP/D6/CT \r/D3/D2 /D8/CX/D2 /D9/D3/D9/D7/BA /CC /D9/D6/D2/CX/D2/CV /D8/D3 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D1/D3 /CS/CT/D0/B8/D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV \r /CW/CP/D2/CV/CT/D7 /CX/D2 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 /B4/D0/CX/D2/CTh= 0 /CX/D2 /AS/CV/D9/D6/CT /BI /CP/B5 \r/CP/D2 /CQ /CT /CX/D2/CS/CX\r/CP/D8/CT/CS/BA/CF/CW/CT/D6/CT/CP/D7 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /D7/D9/D6/DA/CX/DA /CT/D7 /D5/D9/CP/D2 /D8/D9/D1 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7/B8 /D8/CW/CT \r/D3/D0/D0/CX/D2/CT/CP/D6 /D1/CP/CV/D2/CT/D8/CX\r/D7/D8/CP/D8/CT /CX/D7 \r/D3/D1/D4/D0/CT/D8/CT/D0/DD /CS/CT/D7/D8/D6/D3 /DD /CT/CS/BA /C1/D2/D7/D8/CT/CP/CS/B8 /CU/D3/D6J2>0.399J1\n/D8/CW/CT/D6/CT /CP/D4/D4 /CT/CP/D6/D7 /CP /CV/CP/D4/D0/CT/D7/D7 /D7/D4/CX/D2/B9/AT/D9/CX/CS /D4/CW/CP/D7/CT/BA/C1/D2 /D8/CW/CT /CV/CT/D2/CT/D6/CP/D0 \r /CW/CP/D7/CT/B8 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT /CX/D7 /CT/CX/D8/CW/CT/D6 \r/D6/CX/D8/CX\r/CP/D0 /B4/CU/D3/D6 /CW/CP/D0/CU/B9/CX/D2 /D8/CT/CV/CT/D6 S1+S2\n/B5/B8/D3/D6 /CV/CP/D4/D4 /CT/CS /B4/CU/D3/D6 /CX/D2 /D8/CT/CV/CT/D6S1+S2\n/B5/BA/CC/CW/CT /D1/D3/D7/D8 /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV \r /CW/CP/D2/CV/CT/D7 /CP/D4/D4 /CT/CP/D6 /CX/D2 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 \r/CP/D2 /D8/CT/CS /D7/D8/CP/D8/CT/BA /C1/D2 /AS/CV/D9/D6/CT /BI/CQ /DB /CT /D7/CW/D3 /DB /D2 /D9/D1/CT/D6/CX\r/CP/D0/BX/BW /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6 /D8/CW/CT /BY/C5 /D1/D3/D1/CT/D2 /D8 /D4 /CT/D6 /D6/D9/D2/CVM0\n/CP/D7 /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 J2/J1\n/BA /CF /CT /D7/CT/CT/D8/CW/CP/D8 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /D4/CW/CP/D7/CT /DB/CW/CX\r /CW /D7/D9/CQ/D7/D8/CX/D8/D9/D8/CT/D7 /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 \r/CP/D2 /D8/CT/CS /D4/CW/CP/D7/CT /CX/D7 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /CQ /DD /CP /AS/D2/CX/D8/CT/BY/C5 /D1/D3/D1/CT/D2 /D8 /DB/CW/CX\r /CW /CX/D7/B8 /CW/D3 /DB /CT/DA /CT/D6/B8 /D0/CT/D7/D7 /D8/CW/CP/D8 /D8/CW/CT /D5/D9/CP/D2 /D8/CX/DE/CT/CS /BY/C5 /D1/D3/D1/CT/D2 /D8 /D4 /CT/D6 /D6/D9/D2/CV /B4S1−S2= 1/2 /B5/D3/CU /CP /D7/D8/CP/D2/CS/CP/D6/CS /C4/CX/CT/CQ/B9/C5/CP/D8/D8/CX/D7 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/BA /C6/D3/D8/CX\r/CT /D8/CW/CP/D8 /D8/CW/CX/D7 /D4/CW/CP/D7/CT /CT/DC/CX/D7/D8/D7 /D3/D2/D0/DD /CP/D8h= 0 /BA /BY /D3/D6h >0/CX/D8 /D1/CT/D6/CV/CT/D7 /CX/D2 /D8/D3 /D8/CW/CT /C4/D9/D8/D8/CX/D2/CV/CT/D6 /D0/CX/D5/D9/CX/CS /D4/CW/CP/D7/CT /B4/D7/CT/CT /AS/CV/D9/D6/CT /BI/CP/B5/BA /BT/D2 /CT/DC/D8/D6/CP/D4 /D3/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /BX/BW /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6\nL= 8,10 /CP/D2/CS12 /D6/D9/D2/CV/D7 /CS/CT/AS/D2/CX/D8/CT/D0/DD /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT /D4/D6/CT/D7/CT/D2\r/CT /D3/CU /D8/CW/CX/D7 /D4/CW/CP/D7/CT /CX/D2 /D8/CW/CT /CX/D2 /D8/CT/D6/DA /CP/D00.341J1<\nJ2<0.399J1\n/BA /BT/D0/D6/CT/CP/CS/DD /CP /D5/D9/CP/D0/CX/D8/CP/D8/CX/DA /CT /D7/CT/D1/CX\r/D0/CP/D7/D7/CX\r/CP/D0 /CP/D2/CP/D0/DD/D7/CX/D7 /B4/D7/D9/D4/D4 /D3/D6/D8/CT/CS /CQ /DD /BX/BW /D6/CT/D7/D9/D0/D8/D7/B5 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D8/D3/D2 /CP/D4/D4/D6/D3/CP\r /CW/CX/D2/CV /D8/CW/CT /D4/CW/CP/D7/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CP/D8J2= 0.341J1\n/CU/D6/D3/D1 /D8/CW/CT /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT/B8 /D8/CW/CT/D0/D3 /DB /CT/D6 /D1/CP/CV/D2/D3/D2 /CQ/D6/CP/D2\r /CWE−\nk\n/D7/D3/CU/D8/CT/D2/D7 /CX/D2 /D8/CW/CT /DA/CX\r/CX/D2/CX/D8 /DD /D3/CUk=π /B8 /CP/D2/CS /D8/CW/CT /CV/CP/D4 /CP/D8k=π /DA /CP/D2/CX/D7/CW/CT/D7 /CP/D8/BD/BC/CB/D4/CX/D2 /D1/D3 /CS/CT/D0/D7 /D3/CU /D5/D9/CP/D7/CX/B9/BD/BW /D5/D9/CP/D2/D8/D9/D1 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/D7/D8/CW/CT /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /D8/D3 /D8/CW/CT \r/CP/D2 /D8/CT/CS /D4/CW/CP/D7/CT/BA /CC/CW /D9/D7/B8 /D8/CW/CT/D6/CT /CP/D4/D4 /CT/CP/D6/D7 /CP /D0/CX/D2/CT/CP/D6 /BZ/D3/D0/CS/D7/D8/D3/D2/CT /D1/D3 /CS/CT /DB/CW/CX\r /CW /CX/D7\r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /D3/CU /D8/CW/CT \r/D0/CP/D7/D7/CX\r/CP/D0 \r/CP/D2 /D8/CT/CS /D4/CW/CP/D7/CT/BA /C1/D8 /D1/CP /DD /CQ /CT /D7/D9/CV/CV/CT/D7/D8/CT/CS /D8/CW/CP/D8 /D8/CW/CX/D7 \r/D6/CX/D8/CX\r/CP/D0 /D1/D3 /CS/CT /D7/D9/D6/DA/CX/DA /CT/D7/D5/D9/CP/D2 /D8/D9/D1 /AT/D9\r/D8/D9/CP/D8/CX/D3/D2/D7 /CJ /BH/BH ℄/B8 /DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT /D7/D4/CX/D2 /D6/D3/D8/CP/D8/CX/D3/D2 /D7/DD/D1/D1/CT/D8/D6/DD U(1) /CX/D2 /D8/CW/CTxy /D4/D0/CP/D2/CT /D7/CW/D3/D9/D0/CS /CQ /CT/D6/CT/D7/D8/D3/D6/CT/CS/B8 /CX/BA/CT/BA/B8∝an}bracketle{tSx\n1,n∝an}bracketri}ht=∝an}bracketle{tSx\n2,n∝an}bracketri}ht= 0 /BA /CC/CW/CX/D7 /D7\r/CT/D2/CP/D6/CX/D3 /DB/CX/D8/CW /CP /D4 /D3 /DB /CT/D6/B9/D0/CP /DB /CS/CT\r/CP /DD /D3/CU /D8/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /D7/D4/CX/D2/B9/D7/D4/CX/D2\r/D3/D6/D6/CT/D0/CP/D8/CX/D3/D2/D7 /CX/D7 /D7/D9/D4/D4 /D3/D6/D8/CT/CS /CQ /DD /D8/CW/CT /D6/CT/D2/D3/D6/D1/CP/D0/CX/DE/CP/D8/CX/D3/D2/B9/CV/D6/D3/D9/D4 /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D7/CX/D1/CX/D0/CP/D6 /D4/CW/CP/D7/CT/D7 /CX/D2 /D5/D9/CP/D2 /D8/D9/D1 /D6/D3/D8/D3/D6/D1/D3 /CS/CT/D0/D7 /CJ /BH/BH ℄ /CP/D7 /DB /CT/D0/D0 /CP/D7 /CQ /DD /CP /D6/CT\r/CT/D2 /D8 /BW/C5/CA /BZ /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /CP /D7/CX/D1/CX/D0/CP/D6 /D4/CW/CP/D7/CT /CU/D3/D9/D2/CS /CX/D2 /CP /CV/CT/D2/CT/D6/CP/D0/CX/DE/CT/CS /CB/BW/BV/D1/D3 /CS/CT/D0 /DB/CX/D8/CW /CP/D2 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 \r/D3/D1/D4 /CT/D8/CX/D2/CV /BT/BY/C5 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D3/AR/B9\r /CW/CP/CX/D2 /D7/D4/CX/D2/D7 /CX/D2 /AS/CV/D9/D6/CT /BG /CQ /CJ/BK/BG ℄/BA /CC/CW/CT /AS/D6/D7/D8 /D7/D8/D9/CS/CX/CT/CS /D5/D9/CP/D2 /D8/D9/D1 /D7/D4/CX/D2 /D1/D3 /CS/CT/D0 /CT/DC/CW/CX/CQ/CX/D8/CX/D2/CV /D7/D9\r /CW /CP/D2 /CT/DC/D3/D8/CX\r /D5/D9/CP/D2 /D8/D9/D1 /D7/D8/CP/D8/CT /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT/D8/CW/CT /D7/D4/CX/D2/B9/BD/BB/BE /D8 /DB /D3/B9/D0/CT/CV /D0/CP/CS/CS/CT/D6 \r/D3/D2/D7/D8/D6/D9\r/D8/CT/CS /CU/D6/D3/D1 /CS/CX/AR/CT/D6/CT/D2 /D8 /B4/D3/D2/CT /BY/C5 /CP/D2/CS /CP/D2/D3/D8/CW/CT/D6 /BT/BY/C5/B5 /D0/CT/CV/D7 /CP/D2/CS /BT/BY/C5/D6/D9/D2/CV/D7 /CJ /BK/BH ℄/BA /C9/D9/CX/D8/CT /D6/CT\r/CT/D2 /D8/D0/DD /B8 /D8/CW/CT/D6/CT /CW/CP/D7 /CQ /CT/CT/D2 /CP /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D6/CT/D4 /D3/D6/D8/D7 /CX/D2/CS/CX\r/CP/D8/CX/D2/CV /D7/CX/D1/CX/D0/CP/D6 /BD/BW /D5/D9/CP/D2 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Zhou,1† T. Seki,1,2,* T. Kubota,1,2 G. E. W. Bauer1-4, and K. Takanashi,1,2 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 3AIMR, Tohoku University, Sendai 980-8577, Japan 4Zernike Institute for Advanced Materials, University of Groningen, The Netherlands   * e-mail: go-sai@imr.tohoku.ac.jp †Present affiliation: National Institute for Materials Science, Tsukuba, Japan    Page 2 Abstract: We present the Co-Gd composition dependence of the spin-Hall magnetoresistance (SMR) and anisotropic magnetoresistance (AMR) for ferrimagnetic Co100-xGdx / Pt bilayers. With Gd concentration x, its magnetic moment increasingly competes with the Co moment in the net magnetization. We find a nearly compensated ferrimagnetic state at x = 24. The AMR changes sign from positive to negative with increasing x, vanishing near the magnetization compensation. On the other hand, the SMR does not vary significantly even where the AMR vanishes. These experimental results indicate that very different scattering mechanisms are responsible for AMR and SMR. We discuss a possible origin for the alloy composition dependence.   Page 3 1. Introduction  Antiferromagnetic spintronics [1-6] is an emerging research field that has attracted much attention because of the unique properties of antiferromagnets: zero net magnetization, small magnetic susceptibility [7], and magnetization dynamics characteristically different from ferromagnets [8-11]. Antiferromagnets have great potential for the development of novel spintronic devices such as crosstalk-free and ultrahigh-density non-volatile memories because they do not generate and are robust against magnetic stray fields [12]. However, several problems have to be solved before exploiting the aforementioned functionalities in practical devices. A major issue is the efficient control of the antiferromagnetic order. The small magnetic susceptibility of antiferromagnets renders magnetic field control difficult. Current-induced spin transfer phenomena may be a possible solution for this problem [13,14]. Recent studies demonstrate that spin current (Js) can be generated by an antiferromagnet [4,15] and also interacts with its magnetic moments [2,3,5,11,14,16-19]. However, the coupling phenomenon between antiferromagnetic order and spin currents has not yet been fully understood. It is a complex problem involving the spin-dependent scattering in the bulk and at interfaces to electric contacts. Here we focus on the latter by revealing details of the spin mixing at the interface between platinum and a ferrimagnet around the compensation point.  Co-Gd amorphous alloys are ferrimagnets, in which the Co and Gd moments (mCo and mGd) are coupled antiferromagnetically [20]. The net magnetic moment of Co-Gd (mCo-Gd) is given by | mCo - mGd |, Page 4 meaning that dominance of one of them in the mCo-Gd magnetization strongly depends on the alloy composition as shown in Figs. 1(a) and 1(b). The ferrimagnetic state with zero magnetization at the Co-Gd compensation point resembles an antiferromagnet. By exploiting ferrimagnetic materials such as Co-Gd and Co-Fe-Gd, several studies recently reported an interaction between Js and magnetizations near the compensation points. Ham et al. [21] found an enhanced damping-like component of the spin-orbit torque (SOT) near the compensated perpendicularly magnetized Gd25Fe65.6Co9.4 / Pt. This was explained by the reduction of the net magnetic moment. Although Mishra et al. [22] also observed a substantial increase of the SOT effective field and switching efficiency in perpendicularly magnetized Co-Gd / Pt, they conclude that the negative exchange interaction in the ferrimagnet enhances the SOT near compensation. Co-Gd alloys can also display angular momentum compensation, which is beneficial for ultra-fast magnetization dynamics: Kim et al. [23] demonstrated magnetic field-driven fast domain wall motion in a ferrimagnetic Co-Fe-Gd wire at the angular momentum compensation temperature. Even though Co-Gd is an important material class, both from fundamental and application point of view, the detailed mechanism of spin-dependent transport in this material is not well understood. Here we present a systematic study of magnetotransport of Co-Gd alloy / Pt thin films that accesses spin-dependent scattering parameters and sheds light on the interaction between Js and the ferrimagnetic order. Our analysis separates the contributions from the spin-Hall magnetoresistance (SMR) and Page 5 the anisotropic magnetoresistance (AMR) that occur simultaneously in all-metal magnetic bilayers, which should help to establish microscopic models for both effects.  We report the different composition dependences of SMR and AMR for the Co-Gd / Pt bilayers with in-plane magnetization. The sign of the AMR monotonically changes from positive to negative by increasing the Gd concentration and vanishes near the magnetization compensation composition. On the other hand, the SMR remains finite even when the AMR vanishes, which is a direct proof for different physics. We interpret the composition dependence of the SMR in terms of a spin mixing conductance that, in contrast to the conventional wisdom, depends on the magnet.   2. Experimental Procedure  Thin films were deposited on a thermally oxidized Si substrate using an ultrahigh vacuum compatible magnetron sputtering system with the base pressure below 2 × 10-7 Pa. First, a 4 nm-thick Cr buffer was deposited on the Si-O substrate. Then Co and Gd were co-deposited to form the Co100-xGdx layers with a thickness of 30 nm. Finally, a 4 nm-thick Pt layer was deposited. All the layers were deposited at room temperature. By tuning the sputtering powers of Co and Gd targets, the Gd concentration x (at. %) was widely varied from x = 0 to x = 45. Except for x = 0, i.e. pure cobalt, the Co-Gd layers were amorphous alloys, as confirmed by reflection high energy electron diffraction (RHEED) in Fig. 1(c). In contrast to the amorphous Page 6 phase of Co-Gd, the RHEED pattern of Fig. 1(d) indicates that the top Pt layer crystallizes on the amorphous Co-Gd. The top Pt layer serves as not only the capping layer to prevent the Co-Gd from oxidation, but also as converter of a charge current (Jc) to a transverse spin current Js by the spin-Hall effect (SHE) [25]. We also prepared reference samples consisting of Al / Co100-xGdx / Al, for which we anticipated negligible SMR because of the small spin-orbit coupling in Al. Magnetic properties were measured by a superconducting quantum interference magnetometer, a vibrating sample magnetometer and a longitudinal magneto-optical Kerr effect (L-MOKE) set-up with laser wavelength of 680 nm, all at room temperature.  The thin films were patterned into a 10 µm-wide Hall-cross by photolithography and Ar ion milling. In order to separate the contributions of AMR and SMR as depicted in Fig. 2, we measured the magnetoresistance as a function of direction of an applied magnetic field (H) in two configurations. In the g scan, H rotates in the x - z plane (Fig. 2(a)), anticipating an AMR since Jc flows along the x direction. The SMR is accessed by the b scan in which H rotates in the y - z plane (Fig. 2(b)) [26]. In these measurements we apply a large magnetic field |H|=70 kOe such that the magnetization and field are collinear to a good approximation. All measurements were carried out at room temperature.   3. Experimental Results and Discussion A. Composition Dependence of the Magnetic Properties Page 7  Figs. 3(a) - 3(d) show the magnetization versus magnetic field (M - H curves) of Cr / Co100-xGdx / Pt with (a) x = 12, (b) 24, (c) 25 and (d) 37 and Figs. 3(e) - 3(h) the corresponding L-MOKE loops. H was applied in the film plane. When x is increased from 12 to 37, the net magnetization is suppressed at x = 24 and 25 and accompanied by an increased coercivity (Hc). M is observed to increase again for x = 37. In contrast to the M - H curves, the L-MOKE loops show a gradual decrease in the magnitude of Kerr rotation angle (qK) with x. A remarkable feature of the L-MOKE loops is the sign reversal of qK  between x = 24 and 25.   The solid circles in Fig. 4 summarize (a) M, (b) the maximum qK and (c) its absolute value as a function of x for Cr / Co100-xGdx / Pt. As x is increased, M shows a local minimum at x = 24 and qK  reverses its sign at x = 25. |qK| slowly and monotonically decreases with x. We conclude that the room-temperature magnetization compensation point of Co-Gd lies between x = 24 and 25. This composition is close to the reported magnetization compensation point at room temperature [20]. We attribute the sign reversal of qK to the change of the dominant magnetic component from mCo to mGd. The present L-MOKE system is equipped with a 680-nm-wavelength semiconductor laser, which selectively probes mCo. At the Co-rich composition (x = 12), mCo dominates and is parallel to H, resulting in a positive qK. On the other hand, when mGd dominates, mCo is antiparallel to H and qK is negative. The results for the Al / Co100-xGdx / Al reference samples, denoted by open squares in Fig. 4, exhibited magnetic properties very similar to Cr / Co100-xGdx / Pt.  Page 8 B. Field Angular Dependence of SMR and AMR  Here we present results of the high field (70 kOe) SMR and AMR measurements for Cr / Co100-xGdx / Pt. The g scans of the longitudinal resistance (Rxx) are displayed in Figs. 5(a) - 5(d) ((a) x = 12, (b) 24, (c) 25 and (d) 45), and the corresponding b scans in Figs. 5(e) - 5(h). Jc was set at 0.1 mA or current density of 2.6 × 104 A cm-2. As shown in Fig. 2, the magnetization of the g (b) scan lies in the x - z (y – z) plane, which corresponds to the AMR (SMR), respectively. The AMR ratio AMR=Rxxγ=90º−Rxxγ=0º()Rxxγ=0º{}×100=ρ//−ρ⊥()ρ⊥{}×100 in terms of the longitudinal (r//) and transverse (r⊥) resistivities. The definition of SMR=Rxxβ=90º−Rxxβ=0º()Rxxβ=0º{}×100=Δρρ0{}×100, where r0 is the resistivity at the SMR maximum and Dr the resistivity modulation. This definition agrees with previous ones for metallic bilayers [27], but differs from that used for magnetic insulators / Pt [26]. Therefore, a negative SMR here corresponds to the “normal” situation in Ref. [28]. The longitudinal electric fields along Jc,x due to AMR (ExxAMR) and SMR (ExxSMR) are well described by [26, 27]: ExxAMR=ρ⊥+ρ//−ρ⊥()sin2γ{}Jx,   (1) and  ExxSMR=ρ0+Δρsin2β{}Jx.    (2) The solid lines in Fig. 5 denote fits by Eqs. (1) and (2), which is a strong evidence that the angular dependences indeed are caused by AMR and SMR.  Page 9  AMR is the dependence of the resistance on the angle between current and magnetization and defined to be positive when r// > r⊥. The SMR is generated by the spin-orbit interaction in the normal metal layer and the exchange interaction at the interface. Here, we find for Gd-Co alloys an AMR > 0 for x = 12, while AMR < 0 for x = 45, and AMR = 0 at x = 25. In contrast, the SMR ratio is negative regardless of x even at the composition for which the AMR vanishes. As mentioned above, a negative sign of SMR in the convention of Ref. [27] implies a net magnetization of Co-Gd is parallel to the external magnetic field.  Figure 6 summarizes (a) the composition dependence of the longitudinal resistance R (on this scale the dependence on magnetization direction is negligibly small), (b) AMR ratio and (c) SMR ratio for Cr / Co100-xGdx / Pt. The experimental values of R, AMR ratio and SMR ratio are also summarized in Table 1. Compared with pure Co, the alloy scattering of amorphous Co-Gd strongly increases the resistivity. Figure 6(b) clearly demonstrates the sign change of the AMR from positive to negative as x increases (see the inset of Fig. 6(b)). Pure Co shows a positive AMR [29] while a negative AMR has been reported for a Gd single crystal [30]. Our results suggest that the d-electrons of Co and the f-electrons of Gd contribute oppositely to the AMR phenomenon. Hence, the effect of local s-d (s-f) scattering appears to cancel (to the experimental accuracy) exactly at the compensation point. However, the microscopic mechanism of the AMR is much more complicated, being governed by the full electronic structure, see e.g. Ref. [31]. Nevertheless, the vanishing of the AMR at the compensation point is most likely not a coincidence and our results should help to develop better theoretical Page 10 models for spin and charge transport in magnetic metals. While the AMR changes sign at x = 25, the SMR is negative regardless of x, which implies that the scattering mechanisms of AMR and SMR are very different.   In order to shed light on this matter we carried out g and b scan magnetoresistance measurement for the Al / Co100-xGdx / Al reference samples as shown in Fig. 7. The Co-rich Co-Gd with x = 7 (Fig. 7a) exhibits a positive AMR whereas in the Gd-rich Co-Gd with x = 42 (Fig. 7b) AMR < 0. This sign change is consistent with the results for Cr / Co100-xGdx / Pt, and proves that the AMR is qualitatively not affected by the normal metal and the interfaces. The absence of a clear SMR for the reference samples in Figs. 7c and 7d can be attributed to the small spin-orbit coupling and negligibly small spin Hall effect in Al as anticipated. We therefore may conclude that (i) the sign change in the g scan with increasing x originates from the bulk scattering in the Co-Gd layer, and (ii) the SMR for the Cr / Co100-xGdx / Pt is dominantly caused by the direct and inverse spin Hall effects without a significant bulk contribution. A contribution from the transverse AMR [32] to the b scans of Rxx can be excluded because the Al / Co100-xGdx / Al sample resistance does not change in the b scans.  C. Discussion  We now discuss the composition dependence of SMR for the Cr / Co100-xGdx / Pt. In metallic bilayers of a nonmagnet (N) with large spin-orbit coupling and a ferromagnet (F) [27]  Page 11  Δρρ0~−θSH()2λNtNtanh2tN2λN()1+ξgR1+gRcothtNλN()−gF1+gFcothtNλN()⎧⎨⎪⎩⎪⎫⎬⎪⎭⎪,  (3) in terms of  ξ≡ρNtFρFtN,   (4)  gR≡2ρNλNReGMIX[],  (5)  gF≡1−P2()ρNλNρFλFcothtFλF(),  (6) where rN(F) ,lN(F) ,tN(F) are the resistivity, spin diffusion length, thickness of the N (F) layer, respectively, qSH is the spin-Hall angle of the N layer, P is the current spin polarization of the F layer, and GMIX is the spin mixing conductance of the interface. The first term in the curly bracket of Eq. (3) coincides with the expression for the SMR for a ferromagnetic insulator. The second term takes the absorption of the longitudinal spin current by the ferromagnetic metal into account (and is not to be confused with the unidirectional SMR [33]).   Calculating the SMR ratio Eq. (3), i.e. Δρρ0()×100, requires values of many parameters. The Gd concentration dependence of rF is displayed in Fig. 8a. rN for Pt was experimentally measured using the 4 nm-thick Pt single layer film, and was obtained to be 4 × 10-7 W m. We cannot simply measure lF in our Co-Gd, but it is a few nm at most and we assume lF ≈ 2 nm in the following. P = 0.3 has been derived from the tunneling spin polarization of Co-Gd [34]. GMIX is a measure of the transverse spin current absorption that we initially choose to not depend on x, e.g. GMIX = 1 × 1015 W-1 m-2. qSH ≈ 0.08 and lN ≈ 3 nm reported for the Pt [Ref. 35] are chosen. Figure 8b compares the observed absolute values of the experimental and model SMR Page 12 ratios for each alloy composition, where the small scatter of calculated SMR reflects that in the measured resistivities. Those agree quite well, but in contrast to the experimental observation, the absolute value of the calculated SMR ratios increases with increasing x. This calculated tendency as a function of composition does not agree with the experimental trend, suggesting an alloy concentration-dependence of material parameters that we assumed constant, such as lF and GMIX. P affects the SMR because a spin current can penetrate the metallic ferromagnets when polarized parallel the magnetization. However, because the alloy resistance is relatively high, the calculated SMR does not change much for P ≤ 0.4.  We can turn the table and calculate the parameter dependence on Gd concentration. The dependence would reproduce the experiments. Here, we focus on GMIX. The Gd concentration dependence of GMIX that results from inverting Eq. (3) is shown in Fig. 8c. GMIX is seen to decrease strongly with increasing x, implying that the ratio of mCo versus mGb at the interface to Pt plays an important role in the spin mixing. This is surprising, since theory predicts that spin mixing is mainly governed by the electron density [36] or the dynamical spin susceptibility [37] of the normal metal. However, these theories do not take the spin-orbit interaction into account, which might importantly modify the spin-mixing conductance of the CoGd / interface and cause its suppression with increasing Gd concentration. M. A. Schoen et al. [38] report a compositional dependence of GMIX for the NixCo1-x, NixFe1-x, and CoxFe1-x, even for such 3d transition-metal binary alloys. M. Tokaç et al. [39] report that GMIX depends on the crystal structure even for elemental Co. Page 13  D. In-plane Field Angular Dependence  The dependence of Rxx on the magnetic field angle a (left panel in Fig. 9) is plotted in Fig. 9 for x = 25 at which the AMR vanishes. The a scan should therefore give identical results with the b scan, as confirmed by comparison with Fig. 5(g). In a Co-Gd alloy with x = 25 the AMR and associated planar Hall effect vanish, which could be a technical advantage. For example, in attempts to measure the spin Hall angle by the spin pumping technique, spurious contribution from the planar Hall effect must be subtracted [40]. This technical difficulty can be overcome by choosing Co-Gd (or in fact any other ferrimagnet) at its compensation composition or temperature to accurately measure the spin-Hall effect, even in the case of metallic bilayer system.    4. Summary  We systematically investigated the Co-Gd composition dependence of the SMR and AMR of in-plane magnetized Co100-xGdx / Pt layers. As x increases, the dominant magnetization changes from the mCo to the mGd sublattices. We realized a nearly compensated ferrimagnetic structure at x = 24 (at room temperature). We find distinctly different composition dependences of the SMR and AMR. The AMR decreases monotonically with increasing x and changes sign near the compensation composition while SMR remain constant when the Page 14 AMR sign chances. The composition dependence of the AMR suggests a local picture of the AMR in which the magnetic moments of the Co and Gd contribute with opposite sign with canceling contributions at the compensation. The observed composition dependence of the SMR can be explained by an exchange interaction of the conduction electrons in Pt that is dominated by the Co magnetic moments or the spin-orbit interaction at the interface. Our findings contribute to a better understanding of an important material class.  Acknowledgement The authors thank J. Barker and S. Takahashi for valuable discussions. Y. M u r a k a m i a n d  I . N a r ita  provided technical support during the structural characterization. This work was supported by the Grant-in-Aid for Scientific Research B (16H04487) and Innovative Area “Nano Spin Conversion Science”(26103006）as well as the Research Grant from the TEPCO Memorial Foundation. The device fabrication was partly carried out at the Cooperative Research and Development Center for Advanced Materials, IMR, Tohoku University.  References [1] T. Jungwirth, X. Marti, P. Wadley and J. Wunderlich, Nature Nano. 11, 231 (2016). [2] Z. Wei, A. Sharma, A. S. Nunez, P. M. Haney, R. A. Duine, J. Bass, A. H. MacDonald, and M. Tsoi, Phys. Rev. Lett., 98, 116603 (2007). Page 15 [3] P. M. Haney and A. H. MacDonald, Phys. Rev. Lett., 100, 196801 (2008). [4] W. Zhang, M. B. Jungfleisch, W. Jiang, J. E. Pearson, A. Hoffmann, F. Freimuth, and Y. Mokrousov Phys. Rev. Lett., 113, 196602 (2014). [5] T. Moriyama, S. Takei, M. Nagata, Y. Yoshimura, N. Matsuzaki, T. Terashima, Y. Tserkovnyak, and T. Ono, Appl. Phys. Lett., 106, 162406 (2015). [6] T. Moriyama, N. Matsuzaki, K.-J. 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Page 19  Table 1 Experimental values of R, AMR ratio and SMR ratio for Cr / Co100-xGdx / Pt. x (at. %) R (W) AMR ratio (%) SMR ratio (%) 0 24.1 ± 0.1 1.5 ± 0.002 -0.200 ± 0.001 12 202.5 ± 0.1 0.050 ± 0.001 -0.045 ± 0.003 19 257.8 ± 0.7 0.022 ± 0.002 -0.043 ± 0.001 24 257.9 ± 0.1 0.012 ± 0.002 -0.035 ± 0.001 25 272.3 ± 0.6 0 ± 0.002 -0.036 ± 0.001 30 280.7 ± 3.3 0 ± 0.002 -0.020 ± 0.001 34 272.8 ± 0.3 -0.003 ± 0.002 -0.030 ± 0.002 37 274.6 ± 0.1 -0.007 ± 0.001 -0.025 ± 0.002 39 295.3 ± 0.1 -0.011 ± 0.003 -0.020 ± 0.001 45 301.0 ± 0.1 -0.012 ± 0.001 -0.019 ± 0.004       Page 20 Figure Captions Figure 1 (a,b) Relationship between net magnetization, Co magnetic moment, and Gd magnetic moment for (a) Co-rich Co-Gd and (b) Gd-rich Co-Gd. (c,d) Reflection high-energy electron diffraction patterns of the (c) Co-Gd surface (x = 12) and (d) Pt surface.  Figure 2 Measurement configurations for the angular dependence of (a) anisotropic magnetoresistance (AMR) and (b) spin-Hall magnetoresistance (SMR) of the Co100-xGdx / Pt bilayer. A charge current (Jc) was applied along the x direction. The AMR is observed when the external magnetic field (H) is rotated in the x-z plane by the angle of g. The SMR is the resistance change when H is rotated in the y-z plane by an angle b.  Figure 3 Magnetization curves for the Co100-xGdx / Pt bilayer films with (a) x = 12, (b) 24, (c) 25 and (d) 37, at room temperature and magnetic field H in the film plane. The longitudinal magneto-optical Kerr effect (L-MOKE) loops for (e) x = 12, (f) 24, (g) 25 and (h) 37.  Figure 4 (a) Gd concentration (x) dependence of magnetization (M), (b) the maximum Kerr rotation angle (qK), and (c) the absolute value of qK (|qK|). The solid circles represent the data for Cr / Co100-xGdx / Pt while the open squares are those of the reference samples Al / Co100-xGdx / Al.  Page 21  Figure 5 Angular (g) dependence of the AMR of the Cr / Co100-xGdx / Pt with (a) x = 12, (b) 24, (c) 25 and (d) 45, and angular (b) dependence of SMR for the Cr / Co100-xGdx / Pt with (e) x = 12, (f) 24, (g) 25 and (h) 45. Jc was set at 0.1 mA, and H = 70 kOe was applied. The solid curves are fits by Eqs. (1) and (2).    Figure 6 (a) Gd concentration (x) dependence of device resistance (R), (b) AMR and (c) SMR for the Cr / Co100-xGdx / Pt. The inset of (b) is a magnified plot of the AMR versus x.  Figure 7 Angular (g) dependence of AMR of the reference samples Al / Co100-xGdx / Al with (a) x = 7 and (b) 42, and angular (b) dependence of SMR for the Al / Co100-xGdx / Al with (c) x = 7 and (d) 42. Jc was set to 0.1 mA, and H = 70 kOe was applied. The solid curves are fits by Eq. (1).  Figure 8 (a) Gd concentration (x) dependence of resistivity (rF) of the present Co-Gd. (b) Comparison of the absolute values of the SMR ratios between the experiment (solid circles) and the model (solid line). (c) GMIX calculated from the SMR ratio as a function of x for the Cr / Co100-xGdx / Pt by inverting Eq. (3).  Figure 9 Illustration of the in-plane field angular (a) dependence of magnetoresistance, and a scan for the Cr / Page 22 Co100-xGdx / Pt with x = 25. Jc was set at 0.1 mA, and H = 70 kOe was applied. The solid curve is the fit by Eq. (2) with an offset of 90 degree for a.    Page 23    \n   Figure 1    \nPage 24    \n  Figure 2    \nPage 25    \n  Figure 3    \nPage 26   \n   Figure 4    \nPage 27   \n  Figure 5     \nPage 28    \n  Figure 6    \nPage 29     \n  Figure 7    \nPage 30   \n  Figure 8    \nPage 31      \n  Figure 9  \n"    },    {        "title": "1911.05393v1.Unveiling_domain_wall_dynamics_of_ferrimagnets_in_thermal_magnon_currents__competition_of_angular_momentum_transfer_and_entropic_torque.pdf",        "content": "Unveiling domain wall dynamics of ferrimagnets in thermal magnon currents:\ncompetition of angular momentum transfer and entropic torque\nAndreas Donges,1Niklas Grimm,1Florian Jakobs,1, 2Severin\nSelzer,1Ulrike Ritzmann,1, 2Unai Atxitia,1, 2and Ulrich Nowak1\n1Fachbereich Physik, Universit¨ at Konstanz, Universit¨ atsstraße 10, DE-78457 Konstanz, Germany\n2Dahlem Center for Complex Quantum Systems and Fachbereich Physik,\nFreie Universit¨ at Berlin, Arnimallee 14, DE-14195 Berlin, Germany\n(Dated: November 14, 2019)\nControl of magnetic domain wall motion holds promise for efficient manipulation and transfer of\nmagnetically stored information. Thermal magnon currents, generated by temperature gradients,\ncan be used to move magnetic textures, from domain walls, to magnetic vortices and skyrmions. In\nthe last years, theoretical studies have centered in ferro- and antiferromagnetic spin structures, where\ndomain walls always move towards the hotter end of the thermal gradient. Here we perform numerical\nstudies using atomistic spin dynamics simulations and complementary analytical calculations to\nderive an equation of motion for the domain wall velocity. We demonstrate that in ferrimagnets,\ndomain wall motion under thermal magnon currents shows a much richer dynamics. Below the\nWalker breakdown, we find that the temperature gradient always pulls the domain wall towards\nthe hot end by minimizating its free energy, in agreement with the observations for ferro- and\nantiferromagnets in the same regime. Above Walker breakdown, the ferrimagnetic domain wall can\nshow the opposite, counterintuitive behavior of moving towards the cold end. We show that in\nthis case, the motion to the hotter or the colder ends is driven by angular momentum transfer and\ntherefore strongly related to the angular momentum compensation temperature, a unique property\nof ferrimagnets where the intrinsic angular momentum of the ferrimagnet is zero while the sublattice\nangular momentum remains finite. In particular, we find that below the compensation temperature\nthe wall moves towards the cold end, whereas above it, towards the hot end. Moreover, we find that\nfor ferrimagnets, there is a torque compensation temperature at which the domain wall dynamics\nshows similar characteristics to antiferromagnets, that is, quasi-inertia-free motion and the absence of\nWalker breakdown. This finding opens the door for fast control of magnetic domains as given by the\nantiferromagnetic character while conserving the advantage of ferromagnets in terms of measuring\nand control by conventional means such as magnetic fields.\nI. INTRODUCTION\nFundamental interest in the understanding of the inter-\naction of thermal stimuli and magnetic domains has been\npropelled by its potential to impact recording and pro-\ncessing technologies for magnetically stored information\n[1,2]. Control of magnetic states by thermally generated\nstimuli is hence a growing field of research. Prominent\nexamples include the fields of spin caloritronics [ 3,4], e.g.\ndomain wall ( DW) motion by temperature gradients [ 5–12],\nand the field of ultrafast spin dynamics, e.g. thermally-\ninduced magnetic toggle-switching by ultrafast heat load\nin ferrimagnets ( FIs) [13–22]. Other coherent means of ma-\nnipulating magnetic textures, such as heat-assisted mag-\nnetic recording ( HAMR ) [23,24] and helicity-dependent\nall-optical switching ( HD-AOS ) for instance [ 25–30]—albeit\nnot primarily induced thermally—are facilitated tremen-\ndously by additional application of an ultrashort thermal\nexcitation und subsequent demagnetization [31–36].\nA key ingredient for the theoretical description of\nthe aforementioned magnetothermal effects, especially\nthermally-induced DWmotion, lies in the understanding\nof transport processes for energy and angular momentum.\nWhile in metals thermal spin currents are also transported\nby electrons, in insulators magnons, low energy magnetic\nexcitations, are responsible for the transport of angularmomentum via the spin Seebeck effect ( SSE) [37]. No-\ntably, thermal magnons can be used to move magnetic\ntextures, such as DWs, vortices, and skyrmions [ 38–40]. In\nprevious works the DWmotion of ferromagnets ( FMs) and\nantiferromagnets ( AFM s) induced by temperature gradi-\nents has been investigated thoroughly [ 7–11]. For instance,\nboth, experimental [ 5,41] and theoretical [ 7,8,10,12]\nstudies on FMs, have shown that a DWin a temperature\ngradient moves towards the hotter end of the sample.\nOn a microscopic level, the hot sample region acts as\na magnon source. Since ferromagnetic magnons carry\nspin, angular momentum conservation dictates that a\nmagnon which is transmitted through a DWexerts an\nadiabatic spin transfer torque ( STT) onto the wall. As a\nconsequence, the DWmoves in opposite direction to the\nmagnon propagation direction, i.e. towards the source\n[7,42,43]. Differently to the mechanism based on angu-\nlar momentum conservation, an alternative explanation\nbased on thermodynamic arguments has been suggested.\nSince the DW-free energy decreases as the temperature\nincreases [ 44], the so-called non-adiabatic entropic torque\nacts on the magnetization pulling the magnetic texture to-\nwards the hotter region of the sample, thereby maximizing\nthe entropy and minimizing the free energy [ 8,10]. The\ngenerality of the latter picture makes it also applicable to\nDWsinAFM s, in which thermal magnons do on averagearXiv:1911.05393v1  [cond-mat.mtrl-sci]  13 Nov 20192\nnot carry angular momentum [ 9,11], but also to more\ncomplex systems such as spin-spirals and skyrmions [ 39].\nDomain wall motion by thermal gradients in AFM sof-\nfers complementary properties to the motion in FMs. On\nthe one hand, AFM DW motion can be faster due to the\nalmost complete lack of inertia and the missing Walker\nbreakdown, which limits the maximum velocity. On the\nother hand, a disadvantage of AFM DW sis the difficulty to\nmanipulate, control and measure by conventional means,\nsuch as external magnetic fields. This kind of conven-\ntional magnetization control is only possible in a subclass\nofAFM s, so-called weak ferromagnets ( WFM s) such as\nthe rare earth ( RE)-orthoferrites for instance, in which\nthe Dzyaloshinskii-Moriya interaction ( DMI) induces a\nsmall net-magnetic moment, perpendicular to the N´ eel\norder parameter [ 45,46]. So-called “pure” AFM s, such as\nNiO for instance, in which there is no net-magnetization\nin bulk, require more sophisticated means of excitation\n[47,48].FIscan be seen as a generalization of both sys-\ntems, FMsand AFM s, since one may selectively tune the\nrelevant magnetic properties by modifying for instance\nthe sample temperature or composition [ 49,50]. This\nallows for an enhanced control of the ferromagnetic- or\nantiferromagnetic-like character of the spin dynamics and\nenables to potentially exploit the characteristically fast\nspin-dynamics of an AFM [49–51], while at the same time\none can easily manipulate them by using magnetic fields,\nand measure it by conventional detection methods such as\nthe magneto-optical Kerr effect ( MOKE ) or x-ray magnetic\ncircular dichroism ( XMCD ).\nNaturally, the larger parameter space of the FI, which\nemerges from the (at least) two non-equivalent magnetic\nsublattices, also implies that its magnetization dynamics\nbecomes more complex to understand, i.e. the properties\nof thermal magnon currents strongly depend on the un-\nderlying microscopic spin structure [ 52,53]. Thus, DW\nmotion in FIdriven by temperature gradients has been\nscarcely investigated so far [ 6] and previous works on DW\nmotion in FIs(and synthetic AFM s) were focused on more\ncontrollable stimuli, such as electric currents [ 51,54] and\nmagnetic fields [49].\nIn this work we study DW-dynamics in FIsdriven by\nthermal magnon currents in constant temperature gra-\ndients [ 53,55]. We use an atomistic spin model based\non the stochastic Landau-Lifshitz-Gilbert ( LLG) equation,\nto simulate ferrimagnetic DWsin a temperature gradient.\nOur simulation results will be compared to the previously\ndeveloped theory for DWmotion in FMs[56–58], based on\nthe collective coordinates approach. Depending on the\nstrength of the thermal gradient and the base temper-\nature, we find similarities in the DWdynamics to both,\nthe FMand AFM. For instance we can find a Walker\nbreakdown as observed for FMs[7,8], but we also find the\nquasi-inertia-free motion observed in AFMs [9]. However,\nin addition we find a completely new feature that is unique\nto the FIand has so far neither been reported for the FM\nnor the AFM: a motion towards the cold sample region in\nthe case of a FIbelow angular momentum compensationandabove Walker breakdown. Using a theoretical model\nbased on linear spin-wave theory ( LSWT ) we show that\nthis peculiar motion is due to angular momentum transfer\nand not linear momentum transfer.\nII. METHODS\nA. Atomistic spin model\nWe model the most simple kind of FI, that is a two-\nsublattice FIwith a rock salt structure (G-type magnetic\nordering) as depicted in Fig. 1. Our atomistic spin model\nis based on an extended Heisenberg Hamiltonian\nH=−/summationdisplay\ni<jJijST\niSj−/summationdisplay\niST\niKiSi (1)\nfor normalized magnetic moments Si=µi/µi.Jijde-\nnotes the isotropic Heisenberg exchange coupling and\nKi=dz\niˆ zˆ zT+dy\niˆ yˆ yTis the biaxial on-site anisotropy with\neasyz-axis and hard x-axis (0≤dy\ni< dz\ni=0.5 meV ).\nWe use the following exchange parameters for the interac-\ntion between spins located on the same sublattice A/B,\nJAA=16 meV ,JBB=0.5 meV , and between spins on\ndifferent sublattices, JAB=−6 meV . These values are\ncharacteristic for ferrimagnetic RE-transition metal ( TM)\nalloys [ 59], which are testbed materials in the field of ultra-\nfast spin dynamics [ 13–17], and are receiving an increasing\nattention in the field of spintronics [54].\nThe time evolution of the spins is computed with the\nstochastic LLGequation [60]\n(1 +α2\nG)∂Si\n∂t=−γi\nµiSi×(Hi+αGSi×Hi) (2)\nwith the effective field Hi=−∂H/∂Si+ζi, containing\nboth the deterministic field from the spin Hamiltonian H,\nEq.(1), and the stochastic field ζiin the form of Gaußian\nwhite noise, with\n/angbracketleftζi/angbracketright= 0 and/angbracketleftζi(0)ζT\nj(t)/angbracketright=2αGµikBTi\nγiδijδ(t).(3)\nFor the atomic magnetic moments we use µA= 4µBohr\nandµB= 5µBohr. Here, for simplicity, we assume that\nthe Gilbert damping and gyromagnetic ratios are the\nsame for both sublattices. The gyromagnetic ratios are\nγi= 2µBohr/~=1.76×1011s−1T−1, and the Gilbert\ndamping is set to αG= 0.01. By numerical integration\nof Eq. (2), for a range of temperatures we calculate the\nthermal average of the sublattice-specific as well as the\nnet angular momentum (Fig. 1). The angular momen-\ntum compensation temperature, at which the net angular\nmomentum is zero, is found at TA=107 K (Fig. 1). More-\nover, our numerical calculations allow us to determine\nthe Curie temperature of the system, TC=616 K , in the\nrange of ferrimagnetic RE-TMalloys [ 61,62]. We assume\na lattice constant of a=250 pm . Similar models have3\n0 100 200 300 400 500 600 700\ntemperature (K)0.00.51.01.52.02.5angular momentum ( )\nlA\nlB\nlfu\na2a\nTA(M)\nFigure 1. Sublattice-specific and net thermal average angular\nmomentum as a function of temperature (points). Solid lines\nare fits to the simulation data. At the angular momentum\ncompensation temperature, TA, the net angular momentum\nvanishes. Note that TA=TM, due to our choice of γA=\nγB. The sketch shows the G-type magnetic ordering of the\nunderlying atomic spin model of the ferrimagnet.\nbeen already used in the literature to model the spin\ndynamics of ferrimagnetic systems, the most prominent\nexample being GdFeCo alloys used for ultrafast toggle\nswitching [ 13–16,18–20] and HD-AOS [25,29,30]. Despite\ntheir potential key role on such a switching process, the\nstudy of DWmotion under thermal gradients of such kind\nof materials [6] has gained far less attention.\nB. Computation of domain wall dynamics\nIn our simulations a DWis placed in a constant temper-\nature gradient and the magnetization is relaxed to a base\ntemperature of T0. The base temperature determines the\nremanent angular-momentum and thus enables us to tune\nthe magnetic properties of the FI. During this relaxation\nphase (t<0) we setαG= 1 which efficiently suppresses\nany DWdynamics. At t= 0 we set αGto 0.01, which\nreleases the DWinstantaneously. The wall coordinates,\ni.e. angles Φ νand positions Zν(ν=A,B), are tracked\nby fitting the wall profiles\nm⊥\nν(z) =mν(z) exp(iΦν)\ncosh((z−Zν)/∆ν),and (4)\nmz\nν(z) =mν(z) tanh((z−Zν)/∆ν) (5)\nto the simulation data. ∆ νis hereby the wall width\nandmν(z) is the saturation magnetization for which we\nassume a linear correction in zto improve the fitting\naccuracy, compared to a spatially constant saturation\nmagnetization. Absorbing boundaries in the form of\nenhanced Gilbert damping are applied in the longitudinal\ndirection, whereas the transverse boundaries are periodic\nin order to have bulk-like properties. Due to the sizable\ninter-sublattice coupling JAB, the deviation of the DW\n6080100120T (K)\nT0=92 KdT/dz=0.82 K/nm\nZDW(t0)\nZDW(tmax)\n1.0\n0.5\n0.00.51.0m(t0)\n020406080100120\nposition (nm)1.0\n0.5\n0.00.51.0m(tmax)\nmA\nx\nmB\nxmA\ny\nmB\nymA\nz\nmB\nzFigure 2. Top panel shows the temperature profile used in\nthe simulation setup, together with the initial (solid circle)\nand final (open circle) DWpositions;T0indicates the base\ntemperature at the DWcenter at the start of the simulation\nt0. Central and bottom panels show the wall profile of the\nnormalized A (circles) and B (crosses) sublattice magnetiza-\ntions. Black lines correspond to the fits according to Eqs. (4)\nand (5). The data shown corresponds to Fig. 3 (b).\ncoordinates of the two sublattices A and B from each\nother are relatively small, such that for the tracking of\ntheDWcoordinates it is not necessary to distinguish the\nDWvariablesZ,Φ,∆ for the two sublattices A and B. A\nsimulation setup of a typical DWprofile in a temperature\ngradient is shown in Fig. 2.\nWe use a comparably large grid cross section of 96 ×192\nspins, with a length of 480spins in the direction of the\ntemperature gradient, to reduce thermal fluctuations of\nthe data [ 63]. To handle the large computational effort of\nalmost 107spins in total with simulation times of several\nhundreds of picoseconds, we use a previously developed,\nhighly efficient GPU accelerated atomistic spin dynamics\nsimulation routine based on the Nvidia CUDA C-API[64].\nIII. THEORY OF DOMAIN WALL MOTION\nA. Adiabatic and non-adiabatic spin transfer\ntorques\nThe theory of DWmotion driven by spin-polarized\nelectric currents in FMsis well established. Initial works4\nhave suggested that thermal magnon currents can be\nviewed as spin currents, whose amplitude is proportional\nto the temperature gradient [ 8]. For FIs, the question is\nto what extend a similar picture holds and how theory\nhas to be modified in order to account for the particular\nproperties of FIs.\nIn a temperature gradient, the spatial variation of the\nstochastic noise ζi(Ti), Eq. (3), in the effective field of the\nLLGequation (2)can be interpreted as sources of thermal\nmagnons. This thermal magnon current acts on a DWin a\nsimilar way as the STTused to describe DWmotion under\nspin-polarized electric currents in micromagnetic models\n[56,57]. In those models, the LLGequation is augmented\nby two additional torque terms to take into account the\ninteraction of a spin-polarized electron current on the\nmagnetization. The so-called adiabatic torque\nTad=−(u·∇)m∝−(∇T·∇)m (6)\nand the non-adiabatic torque\nTnad=m×(βeffu·∇)m∝m×(∇T·∇)m. (7)\nThe parameter βeffis the dimensionless non-adiabaticity\nand by definition specifies the ratio between the two STTs.\nThe adiabatic torque can be related to angular mo-\nmentum conservation [ 7,42,43]: when a magnon current\n(or spin-polarized electron current) passes through the\nwall, their polarization is continuously rotated by 180◦\nthereby changing the magnons’ angular momentum. To\nobey angular momentum conservation in the combined\ndomain+magnon system, one domain has to grow in size,\ni.e. the DWhas to move—the direction depending on the\nrelative polarization of the particle current and magne-\ntization with respect to the DW. The adiabatic torque\namplitude in this case is simply given by (cf. [43])\nu=Ja3/lfu (8)\nwhereJ∝∂T/∂z (see also Ref. [ 65] and Appendix A and\nB) is the spin current density and lfu= (lA−lB)/2 the\nangular momentum per unit volume ( lν=µνmν/γν), see\nFig. 1. The difficulty here is to find an expression for the\nspin current Jfor a FI.\nOn the other hand, Schlickeiser et al. [8]introduced the\nconcept of non-adiabatic entropic torque due to the spa-\ntially varying exchange stiffness ∇Aeff= (∂Aeff/∂T)∇T\nin a FM. Here, we adapt their model for the FMto the FI,\nand we find the following expression for the non-adiabatic\nentropic torque strength (see Appendix C for the temper-\nature dependence of the exchange stiffness Aeff)\nβeffu=−2a3\n|lfu|dAeff\ndT∂T\n∂z. (9)\nOne of the main result of the present work is the demon-\nstration of the validity of these relations, Eqs. (8)and\n(9), for FI, by comprehensive comparison to atomistic\nspin dynamics simulations of the DWmotion under a\ntemperature gradient.B. Dynamics of the domain wall\nThe conceptual idea behind Eqs. (8)and(9)is that\nthe dynamics of a FIcan be viewed as an effective FM\nwith angular momentum given by lfu—a model which has\nrecently been employed in a similar fashion by Kim et al.\n[49]for field-driven FI-DWmotion. Such a model should\nbe valid for the low wall velocities in thermal gradients;\nalthough some deviations might occur close to TAsince\nwe do not take into account inertial effects proportional\nto¨Z, and ¨Φ [48, 66].\nThe dynamics of a rigid, ferromagnetic DWcan then be\ndescribed by the two collective coordinates that are the\nwall position ZDWand tilting angle Φ DW[57,58]. We can\nadapt the corresponding equations of motion and rewrite\nthem for the ferrimagnetic DW, leading to\n˙ZDW=βeffu\nα⊥\neff−∆/prime\nDW˙ΦDW\nα⊥\neff(10)\n˙ΦDW=α⊥\neff\n1 + (α⊥\neff)2/bracketleftbiggβeff−α⊥\neff\n∆/prime\nDWα⊥\neffu+K⊥\n|lfu|sin 2Φ DW/bracketrightbigg\n(11)\nwhere K⊥=Kyy−Kxxis the in-plane anisotropy, α⊥\neffis\nthe transverse Gilbert damping parameter, known from\nmagnetic resonance for instance [ 67,68], and ∆/prime\nDW=\n−sign(lfu)∆DWis the signed wall width [ 69]. The coupled\nequations (10) and(11) have two kinds of steady state\nsolution.\nBelow the Walker breakdown “current”\nuW=K⊥\n|lfu|∆DWα⊥\neff\n|βeff−α⊥\neff|(12)\nthe driving stimulus is insufficient for Φ DWto overcome\nthe potential barrier, scaling with K⊥, hence the wall\nangle becomes stationary and consequently the motion\nlinear. In this case, the DWvelocity reads\nVDW=βeffu\nα⊥\neff=−4a3\nαG|lA+lB|dAeff\ndT∂T\n∂z, (13)\nwhere in the second equality we used the relation [70]\nα⊥\neff≈αG|lA+lB|\n|lA−lB|, (14)\ncommonly derived under magnetic resonance conditions\nand related to its linewidth in FIs. We note that Eq.\n(13) is a generalization of the already known relations\nfor the velocity of the DWsforFMs[8] and AFM s[9].\nImportantly, the difference to those cases is that the\neffective damping in FIshas a non-monotonous behavior.\nThe relatively simple expression in Eq. (14) holds if the\nmagnetization is not too close to the compensation point,\nat which an apparent divergence occurs, and where more\nsophisticated damping models would be necessary [ 67,68].\nThus, very close to this point we expect some deviations\nin the wall velocity and precession which we derive from5\nthe ferromagnetic model here in this section. Though\nqualitatively, this singular behavior accelerates the spin\ndynamics around TAand is the reason why FIswith\nangular momentum compensation points are becoming so\nrelevant for applications and functionalities related to the\nspeed of the spin dynamics [ 49,51,54]. Additionally, while\nthe micromagnetic exchange stiffness and its temperature\ndependence for FMsand AFM sis somehow known [ 71], the\nspecifics ofAeffinFIsremains an open problem—especially\nat elevated temperatures.\nAccordingly, above the Walker breakdown, the wall\nangle Φ DWprecesses continuously as the repelling force\nis limited by K⊥/|lfu|. The corresponding wall velocity\nthen reads [58]\nVDW=βeffu\nα⊥\neff±|βeff−α⊥\neff|/radicalbig\nu2−u2\nW\nα⊥\neff(1 +α⊥2\neff)(15)\nwhere the positive sign is for the case ( βeff−α⊥\neff)u<0.\nIn the limit of a vanishing Walker threshold uW(∼K⊥)\n[Eq. (12)] or high driving currents u/greatermuchuW, this solution\nconverges to\nlim\nuW→0VDW= lim\nu→∞VDW=1 +βeffα⊥\neff\n1 +α⊥2\neffu. (16)\nInterestingly, the velocity in this limit and in the small\ndamping regime α⊥\neff/lessmuch1 can be approximated by the sim-\nple relation, VDW≈u=Ja3/lfu. We note that the small\ndamping condition, αG/lessmuch|lA−lB|/|lA+lB|holds in a\nwide range of temperature, when the system temperature\nis not too close to the compensation temperature.\nTo evaluate the equations presented in this section, we\nwant to refer to Appendix A-C. First of all, in Appendix\nA we derive the dispersion relation of the FI, from the\nLLG-equation. In Appendix B this dispersion relation is\nused to calculate the thermal spin current density J, from\nwhich we can then derive the adiabatic STTparameteru.\nFinally, in Appendix C we compute the non-adiabatic STT\nfor this system using the effective exchange stiffness of the\nFI. Altogether this allows us to compute the effective DW\nvelocity and precession in a self-consistent way from the\nLLG-equation, (2), and the spin Hamiltonian, (1), without\nthe need of additional parameters.\nIV. ATOMISTIC SPIN DYNAMICS\nSIMULATIONS\nA. Overview of the domain wall dynamics\nFigure 3 shows the wall velocity VDWwe obtained from\nour atomistic spin dynamics simulations, as a function\nof the temperature gradient ∂T/∂z , i.e. the amplitude\nuof the driving STT(the calculation of the steady-state\nwall velocity is described in detail in Appendix D). We\nstudy a range of temperature gradients such that we can\ninvestigate the dynamics below and above the Walker\n100\n0100VDW(m/s)\nT= 58+6\n16K\n100\n0100200VDW(m/s)\nT= 92+12\n11K\n0100200VDW(m/s)\nT= 127+15\n0K\n0.0 0.2 0.4 0.6 0.8 1.0\ntemperature gradient (K/nm)0100VDW(m/s)\nT= 174+21\n0K\nbiaxial ferrimagnetFigure 3. DWvelocity as a function of temperature gradient\nfor various temperatures. Filled (open) symbols denote walls\nbelow (above) Walker breakdown. Error margins indicate the\nmaximum thermal drift of the wall towards the hot (plus) and\ncold (minus) sample regions throughout the simulation time,\ni.e. the maximum deviation from the starting temperature.\nThe intermediate anisotropy is dy\nA=12.5µeV. Black lines are\nfits to Eqs. (13) and (15).\nbreakdown. Additionally, we consider four different base\ntemperatures T0, ranging from below to above the com-\npensation temperature TA=107 K that will allow us\nto assess the role of the net angular momentum in the\ndynamics of the DW.\nBelow the Walker breakdown (filled symbols) we find\nthat the wall velocity VDWscales linearly with the ther-\nmal torque amplitude uas it is expected according to\nEq.(13). The positive sign of the velocity hereby indicates\na motion towards the hotter end as previously predicted\nforFMsand AFM s[7–10]. Above the Walker breakdown\nthe situation is vastly different: here we find that below\nthe compensation point the wall moves towards the cold\nregion (VDW<0) in the limit u/greatermuchuW, and above the\ncompensation point to the hot one ( VDW>0) for any\nvalue ofu. Nevertheless, the theoretical wall velocity\npredicted by Eqs. (13) and(15) nicely traces our simula-\ntions results for all four temperatures displayed in Fig. 3.\nThus, different propagation directions of the DWmotion\nare found for temperatures below and above the angular\nmomentum compensation temperature TA. This implies6\nthat around TA(i) the adiabatic torque parameter u(T)\nchanges sign due to the sign change of J/lfuin Eq. (8)\nand (ii) the non-adiabaticity βeff(T) changes sign likewise\nsince the product βeffu, Eq. (9), is strictly positive (for\n∂T/∂z> 0).\nAnother intriguing observation is the apparent increase\nin the Walker threshold uWfor the upper three pan-\nels (a)-(c), where in (c) the threshold was actually too\nhigh to be determined by our simulations. This is highly\ncounterintuitive since the critical current in a biaxial\nmagnet is determined by the in-plane anisotropy K⊥\n[see Eq. (11)] which decreases quickly with temperature\n/angbracketleftKν/angbracketright(T)∼Kν(0)m3\nν(T) [72]. Thus, the Walker threshold\nuW(T) is expected to decrease monotonically with Tlike-\nwise which we observe only at even higher temperatures\nshown in panel (d). However, in the FI, the expression\nuW∝K⊥/|lfu|in Eq. (12)increases, since the net-angular\nmomentum lfu(T) decreases faster than the individual\nsublattice order parameters mν(T) and hence faster than\n/angbracketleftK⊥/angbracketright(T). These insights about the Walker threshold could\nhave impact in the design of ferrimagnetic devices with\nimproved functionalities, as the temperature dependence\nof the individual order parameters can be readily tuned\nby material engineering techniques, e.g. by modification\nof sample composition.\nB. Diversity of temperature dependence of the\ndomain wall dynamics\nThe DWdynamics below the Walker breakdown ( u<\nuW) behave as one would expect from previous works in\nFMbut with effective parameters accounting for the fact\nthat the FIis composed of two antiferromagnetically cou-\npled sublattices. Thus, it is worth to further investigate\nthe range of validity of this idea. As shown above, the\nWalker threshold can be controlled via the perpendicular\nanisotropy parameter, dy\nA. Therefore, in order to investi-\ngate the regime below Walker breakdown ( u<u W) we\nhave to consider systems with biaxial anisotropy. On the\nother hand, one of the main results of this work is the\ndemonstration that a DWmotion towards the cold end\nof the sample is possible above the Walker breakdown\n(u/greatermuchuW). Since for a uniaxial FI(dy\nν= 0) the wall mo-\ntion is always above the Walker breakdown ( uW= 0), we\ncan investigate the validity of Eq. (16)for the wall velocity\nwithout mixing effects coming from the presence of per-\npendicular anisotropy. To study the DWdynamics below\nand above the Walker breakdown more thoroughly, we\ncomputed the temperature dependence of the steady state\nDWvelocity for the biaxial ( dy\nA=25µeV) and uniaxial\n(dy\nA= 0) FI, shown in Fig. 4 (a,b), for a fixed temperature\ngradient of ∂T/∂z = 260 K µm−1.1. Domain wall velocity in biaxial systems\nAs we could already expect from the data in Fig. 3,\nthe wall velocity below the Walker breakdown is only\nweakly sensitive to the base temperature, as can be seen\nin the panel (a) of Fig. 4. The wall velocity decreases\nonly slightly at higher temperatures. However, this ef-\nfect is already known from previous studies and can be\nrelated to the changing equilibrium magnetic properties\n[11,71]. We can estimate the expected DWvelocity by\nevaluating Eq. (13) (see also Appendix C). For the sim-\nulation parameters described in Sec. II A, this yields an\nSTTofβeffu=8.6 m s−1and an expected DWvelocity of\nabout 95 m s−1(black cross) and is in good agreement\nwith the simulation data (colored squares), especially\nwhen we consider the rough approximations we applied.\nNote that although the entropic torque (9)alone is pro-\nportional to 1 /|lfu|and hence expected to diverge at the\nangular momentum compensation point (white marker\ncolor) where lfu→0, the DWvelocity Eq. (13) remains\nfinite since the damping coefficient diverges in the same\nfashion:α⊥\neff∝1/|lfu|.\n2. Domain wall velocity in uniaxial systems\nMore fascinating dynamics can be found for the uniax-\nialFI, where predominantly the adiabatic STTdrives the\nDW, see Fig. 4 (b). Here we can make two clear obser-\nvations: (i) the direction of motion of the wall changes\nsign very close to the compensation temperature (white\nmarker color). Below the compensation point the wall\nmoves to the colder sample regions, i.e. co-propagates\nwith the magnon current, whereas above the compensa-\ntion point the regular motion towards the thermal source\nis obtained. (ii) The absolute value of the wall veloc-\nity drastically increases towards the compensation point,\nwhich is supported by Eq. (15) for a magnon current\ndensityJwhich depends only weakly on temperature.\nThis assumption, in particular that Jdoes not change\nsign atTA, is hereby motivated by its derivation from\nthe dispersion relation (see Appendix A), which does not\ndepend on temperature in a qualitative way, as long as\nT/lessmuchTC[73].\nHowever, to fully understand the DWdynamics for the\nfreely precessing wall, far above the Walker breakdown,\nwe first need a better understanding of the origin of the\nspin current density J. Until now, it is not even clear\nwhat the sign of the spin current density Jis in the FI\n[52]. For a single sublattice FMthe magnetic moment\nof the magnon is given by the reduction of the Szspin\ncomponent with respect to the saturation value ( mz=±1)\nand is hence antiparallel to the ground state magnetization\n[7,42]. In the FIthe net-momentum of the magnon will be\ndetermined by the ratio of the two components µASz\nA/γA\nandµBSz\nB/γBrelative to each other.\nThe classical spin-wave amplitudes for the uniaxial\nFIatT=0 Kcan be calculated from LSWT , follow-7\nTT(b) velocity\n(d) precession\nfrequencyuniaxial ferrimagnet\n(c) tilting angle(a) velocitybiaxial ferrimagnet\nTT(e)\nTT\nFigure 4. Top panels show the DWvelocityVDWfor a biaxial (a) and a uniaxial (b) FIas a function of temperature for a\nconstant thermal gradient of ∂T/∂z =260 K µm−1. Bottom panels show the corresponding tilting angle of the wall (c) for the\nbiaxial FIbelow Walker breakdown and the DWprecession frequency for the uniaxial magnet (d), respectively. The inset (e)\nshows the non-adiabaticity parameter βeff(markers) calculated from Eq. (18) in comparison to the effective damping coefficient\nα⊥\neff(dash-dotted line), corresponding to the magnetic resonance linewidth, Eq. (14). The black crosses in (a) and (e) correspond\nto the entropic torque (9). Solid black lines in (b) and (d) correspond to Eqs. (11) and(16), using the spin current Eq. (17)\nas input and a constant non-adiabaticity parameter |βeff|=0.55 (see text). Dashed vertical lines mark the so-called torque\ncompensation temperature TT(see text). The marker color indicates the net-angular momentum lfu; horizontal error bars\nindicate the wall drift.\ning Refs. [ 55,74] (see Appendix A). We find that the\nlow-frequency branch ( σ=−1) in the dispersion rela-\ntion carries momentum parallel to mz\nA= +1 and the\nhigh-frequency branch ( σ= +1) parallel to mz\nB=−1.\nThe question now is which of these two branches dom-\ninates the net spin current? Lower frequency implies\nhigher thermal population—in the classical model that\nis a population according to a Rayleigh-Jeans distribu-\ntionn0\nk,σ=kBT/~ωk,σ—and at the same time longer life\ntimesτk,σ. However, the high-frequency branch has a\nmuch steeper dispersion relation and hence much higher\ngroup velocities vk,σ=∂ωk,σ/∂kand propagation lengths\nξk,σ=|vk,σ|τk,σ.\nTo answer this question, we calculate the thermal\nmagnon current density Jquantitatively, by solving the\nfollowing k-space integral (see Appendix A and B for the\nderivation):\nJ=/summationdisplay\nσ=±1kB∂T\n∂zσ/integraldisplay\nvz\nk>0d3k\n(2π)3∂lnωk,σ\n∂kzξk,σcosϑk.(17)\nThe dispersion relation ωk,σand the magnon propagation\nlengthsξk,σcan be written in closed form expression,\nandϑkis simply the angle between the k-vector and the\nz-direction. Thus the numerical solution of Eq. (17)poses\nonly minimal computational effort, and we find that the\nhigh-frequency branch of the dispersion clearly dominates\nthe net-magnon current density J. Thus, we have the\npeculiar situation in which below the compensation point\nTA, the net-magnon current has a polarization parallel\nto the ground state angular momentum lfu—a situationopposite to the case of a simple FM—leading to the op-\nposite direction of DWmotion, that is, towards the cold\nsample regions for T <T A(see Fig. 5 left). We can use\nthe spin current Eq. (17) to compute the adiabatic STT\nquantitatively, yielding u=−15.7 m s−1atT= 0. More-\nover, we can now calculate the non-adiabaticity parameter\nβeffusing our previously determined non-adiabatic torque\nparameter βeffu= +8.6 m s−1yieldingβeff=−0.55. In\ncombination with Eq. (16) we can now predict a wall\nvelocity of about VDW(T= 0) = −14.8 m s−1for the\n260 K µm−1thermal gradient at T= 0, which as already\nexpected, can be well approximated by VDW≈u. This\nvalue matches quite well with the results from our atom-\nistic spin dynamics simulation, although we are unable\nto simulate T= 0 exactly.\nNow that we have gained some insight on the role of\nthe adiabatic STTon the DWmotion at low temperatures,\nwe can address the domain wall velocity in the uniaxial\nFIat finite temperature. In particular, the DWvelocity\nin Fig. 4 (b) presents an apparent asymmetry close to\nthe angular momentum compensation point TA. As we\ndiscussed above, the magnon current density Jabove and\nbelowTAshould not change drastically as it is derived\ndirectly from the dispersion relation, cf. [ 73]. Thus, we\nargue that the temperature dependence of u=Ja3/lfu\nis mostly due to lfu(T). Consequently, we expect the\nadiabatic STT to act approximately antisymmetrically\non the DWin the vicinity of TA, implying min[VDW]≈\n−max[VDW], as indicated in Fig. 5. However, the wall\nvelocity from our numerical simulations clearly does not\nfollow this symmetry—we find min[VDW] =−54 m s−18\nZDW(t)\nZDW(t+Δt)lA\nlB\nhot coldthermal\nmagnonsJ+\nJ–T < TA\nZDW(t+Δt)lA\nlB\nhot coldthermal\nmagnonsT > TA\nZDW(t)J+\nJ–\nFigure 5. Schematics of DWmotion in a thermal gradient due to a magnon current Jσfor base temperatures T <T A(left) and\nT >T A(right). The magnon current density Jσof the two branches ( σ=±1) of the dispersion relation carry opposite angular\nmomentum. We find that for both base temperatures T, the high frequency branch σ= +1 (light blue) dominates over the low\nfrequency branch σ=−1 (orange), see Appendix A. This leads to a polarization of the net-magnon current density J=J++J−\nparallel to the lBsublattice angular momentum (in the source region), i.e. |J+|>|J−|. Due to the change in the magnon\npolarization when passing through the wall, to satisfy angular-momentum conservation in the combined domain+magnon system,\nthe domain with the net-angular momentum lfuin the “down” direction has to grow in size. Below TAthis is the domain on the\nhot side (left), whereas above TAit is the domain on the cold magnon side (right). Hence, for the same spin current density J,\nwe obtain different DWpropagation directions above and below TA.\nand max[VDW] = +142 m s−1.\nWe can trace back the origin of such asymmetry by\nconsidering Eq. (16)for the domain velocity in the temper-\nature regime close to TA. We know that both the adiabatic\n(8)and non-adiabatic STTs(9)scale via∼1/|lfu|, thus,\nthe temperature dependence of βeffshould be mostly due\nto the softening of the exchange stiffness Aeff(T), i.e. weak\nforT/lessmuchTC(cf. Appendix C). This allows us to assume\n|βeff|∼const and only include a temperature dependence\nin the form of the necessary sign change at TA. The tem-\nperature dependence of VDW(T) in Eq. (16)is thus due to\nα⊥\neff(T) [Eq. (14)] andu(T) =Ja3/lfu(T) [Eq. (8)]—the\nformer being only relevant close to TA. Using these as-\nsumptions we can calculate VDW(T) from our theoretical\nmodel, shown as the black line in Fig. 4 (b). This model\nexcellently describes the temperature dependence of the\nwall velocity over the full temperature range, including\nits asymmetry.\nWe conclude that below TA, the entropic torque and\nthe angular momentum transfer work against each other,\nwhereas above TAthey act in the same direction. Natu-\nrally, the angular momentum transfer becomes less impor-\ntant if angular momentum conservation is broken, that\nis, whenα⊥\neffbecomes large in the vicinity of TA. At the\nsame time, the contribution of the non-adiabatic term\n∼βeffα⊥\neff/(1 +α⊥2\neff) increases. These findings demon-\nstrate for the first time that a ferrimagnetic DWcan be\npushed away from a thermal magnon source by angular\nmomentum transfer—an effect which in FMsand AFM s\ncan be achieved only by a less efficient linear momentumtransfer, i.e. magnon reflection [12, 75–77].\nC. Emergence of torque compensation temperature\nAside from the DWvelocity, Fig. 4 (c) and (d) also show\nthe temperature dependence of the dynamics of both the\nwall tilting (biaxial) and precession (uniaxial). For the\nbiaxial FIthe steady state tilting angle Φ DWgradually\ndecreases with temperature until it reaches zero at a tem-\nperature of about 125 K after which it increases again\nwith opposite sense of rotation. A similar behavior is\nfound for the wall precession Ω DW=˙ΦDWin the uniaxial\nFI. The wall precession changes sign at the very same\ntemperature of 125 K , suggesting that this phenomenon\nis independent of the in-plane anisotropy K⊥. In the\nfollowing we define this point of completely suppressed\nDWtilting and precession the torque compensation tem-\nperatureTT. However, unlike in the biaxial FI, in the\nuniaxial FIthere is an additional rapid increase of the\nwall precession frequency Ω DWat the angular momentum\ncompensation point. Such an increased wall precession\nclose toTAwas recently predicted for field-driven DWmo-\ntion in ferrimagnetic GdFeCo by Kim et al. [49], however,\nin their case, the sign change of the wall precession coin-\ncides with the angular momentum compensation point.\nFor the case of thermal magnon current driven DWmotion\nTTdiffers from TAimplying that field driven DWmotion\nis fundamentally different from thermally induced motion.\nWe can understand the existence of the torque com-9\n(a)\n(b)\n(c)(d)\n(e)\n(f)(g)\n(h)\n(i)(j)\n(k)\n(l)38 m/s 31 m/s31 m/s\n28 m/s108 m/s92 m/s\n-45 m/s\n187 m/s\n71 m/s9 m/s10 m/s36 m/s\nFigure 6. DWdisplacement ∆ ZDWfor the biaxial FIat different temperatures [(a)-(i); dy\nA=12.5µeV] in comparison to the FM\n[(j)-(l);dy\nA=62.5µeV]. Note the much faster acceleration of the ferrimagnetic walls (left and center panels), compared to the\nferromagnetic ones (right panels). Labels indicate the steady state velocities.\npensation by comparison to the AFM. In the AFM, the\nsymmetry of the non-adiabatic STTcan only lead to prop-\nagationVDWof the wall along the temperature gradient.\nA rotation of the wall angle Φ DWon the other hand does\nnot occur, since both sublattices try to rotate in opposite\ndirections, canting the sublattice magnetizations instead\nof tilting the DWangle [ 9]. In the FIthis is also the case,\nbut unlike in the AFM the torques from the two sublattices\nwill in general not have equal magnitude and are thus not\nfully compensated.\nAnother explanation of these results is provided by the\ncoupled equations of motions for the collective coordinates\nZDWand Φ DW, Eqs. (10) and(11). Crucial is hereby the\nrole of the non-adiabaticity parameter βeff(T), shown in\nFig. 4 (e). In the previous section we already determined\na crude estimate of βeff≈0.55 by calculating the non-\nadiabaticity at low temperature from Eq. (9). We can use\nthis number again, to solve Eq. (11) for the steady-state\nprecession frequency, shown in Fig. 4 (d), where we find\nexcellent agreement with the simulation results.\nA more rigorous approach to compute the non-\nadiabaticity βeff(T), including its temperature de-\npendence, can be computed from the ratio R=\nVbiaxial\nDW/Vuniaxial\nDW of the wall velocity Vbiaxial\nDW below\nWalker breakdown, Fig. 4 (a), and the one for the freely\nprecessing wall Vuniaxial\nDW Fig. 4 (b). By dividing Eqs. (13)\nand(16) we can then solve for βeff(T) usingR(T) fromthe numerical simulations:\nβeff=α⊥\neffR\n1 +α⊥2\neff(1−R). (18)\nThe magnonic torque udrives the wall precession via\nΩDW∝(βeff−α⊥\neff)u. Thus, the DWprecession is expected\nto cease for βeff(T) =α⊥\neff(T), or, in other words, the crit-\nical gradient uW, Eq. (12), diverges for βeff(T)→α⊥\neff(T).\nThe intersection point βeff(T) =α⊥\neff(T), shown in Fig. 4\n(e), is in very good agreement with the torque compensa-\ntion point shown in panels (c) and (d) which have been\ndetermined directly from wall tilting and precession, re-\nspectively. Note that by definition, this intersection point\nalso marks the temperature at which the wall velocities\nabove and below Walker breakdown, depicted in Figs. 4\n(a) and (b), coincide. For our parameters that is a steady-\nstate velocity at TT=125 K of about 90 m s−1in both\npanels.\nFurthermore, we are now able to generalize one of\nour findings, namely that the torque compensation point\nTTis found above the angular momentum compensation\ntemperature TA: if the adiabatic STTmediated by the\nthermal magnon current uacts repulsive on the DW, the\nnon-adiabaticity βeffhas to be negative, to ensure that the\nnon-adiabatic STT(9), i.e. the product βeffu, remains pos-\nitive (for∂T/∂z> 0) [8–10]. Thus, the term ( βeff−α⊥\neff)u\nin Eq. (11) can only be zero for an attractive adiabatic\nSTT, sinceα⊥\neffis strictly positive.10\nD. Domain wall motion in time domain\nFor discussing the DWmotion in the time domain it is\nhelpful to compare our results on the FI’s dynamics to the\nprevious works on FMsand AFM s[7–9,11,57,58]. For the\nFMwe can do this even in a quantitative manner by simply\nswitching the sign of the inter-sublattice coupling JAB\nto get a ferromagnetic exchange between the sublattices\nA and B. The steady-state wall velocity VDWbelow the\nWalker breakdown appears to be more or less unaffected\nby the sign change of JAB, as can be seen in the top panel\nof Fig. 6. This is indeed expected from Eq. (13) where\nonly the sum of the sublattice angular momenta enters\nand the effective exchange stiffness Aeff(T) of the system\nshould be equal in the FM,AFM, and FI.\nHowever, the time to reach this steady-state velocity\nin the FMis greatly extended with acceleration times on\nthe order of nanoseconds [Fig. 6 (j)-(l)], whereas in the FI\ntheDWcan reach its steady state velocity on time scales\nof several tens of picoseconds [Fig. 6 (a)-(c) and (g)-(i)].\nIn fact, close to the torque compensation point [Fig. 6\n(d)-(f)] the acceleration is even faster, though, the exact\ntime constant there is difficult to determine due to strong\nfluctuations of the DWmotion even for a grid cross section\nof 96×192 spins. This is especially problematic for very\nlow gradients as for instance in Fig. 6 (a), (d), and (g).\nThe steady-state below the Walker breakdown is char-\nacterized by a constant tilting angle Φ DW, where the\ntorques of the non-adiabatic STT are balanced by the\nanisotropy torques, see Eqs. (10) and(11). During the\ninitial rotation of the DWup to this angle the velocity\nincreases to its steady-state value, and hence one can\ninterpret it as an inertial mass of the wall [ 78]. As men-\ntioned before, these torques are partially compensated\nin the FIgreatly reducing the tilting angle and therefore\nalso the effective inertia of the DW. For the same rea-\nson the Walker breakdown uW, at which the wall starts\nto rotate continuously, is shifted to much higher critical\ngradients. At T=58 K we find a threshold gradient in\ntheFIof aboutkB|∂T/∂z|FI\nW/dy\nA,FI≈1.8 nm−1compared\ntokB|∂T/∂z|FM\nW/dy\nA,FM≈0.15 nm−1in the FM. At the\ntorque compensation point the FIresembles an AFM, for\nwhich there is no tilting and hence the wall can move\nquasi-inertia-free, i.e. without a relevant acceleration time\n[9]. Ultrafast DWacceleration in the FIis not only found\nat exactly the torque compensation point TT, but also\nslightly below, due to the diverging wall precession Ω DW\nclose to the angular momentum compensation point TA\n[see Fig. 4 (d)]. Thus, even though the wall has to tilt\nby a finite angle, the steady-state angle is reached on\nultrashort time scales of only few picoseconds.\nIt should be noted though that there are other effects\nin an AFM that can be attributed to a mass of the DW\n[48,66]. However, these effects are much smaller and\nproportional to the velocity of the wall, which is here\nrestricted by feasible temperature gradients.V. CONCLUSION\nTo summarize our results, we calculated the DWdy-\nnamics of a FIin a thermal gradient using both, large\nscale atomistic spin dynamics simulations based on the\nstochastic LLG-equation and analytical calculations based\nonLSWT . Our simulation results are in good agreement\nwith our theoretical findings that we derived from LSWT .\nWhereas in the thoroughly studied ferromagnetic systems\nthe adiabatic and non-adiabatic STTlead qualitatively to\nthe same result [ 7,8,10]—a motion to the hotter sample\nregion—a ferrimagnetic DWreacts differently to these two\nkinds of torques. The non-adiabatic torque leads to a\nconsistent motion towards the hotter end, as it is the\ncase for the FMand AFM and can be explained by the\nfree energy minimization via an entropic torque [ 8,9].\nOn the other hand the adiabatic STT can either push\nor pull the ferrimagnetic DWaway from or towards the\nspin-wave source, depending on whether the temperature\nis below or above the angular momentum compensation\npoint. In the FIthe copropagation of the DWwith the\nmagnon current at low temperature is not due to lin-\nearmomentum transfer resulting from magnon reflection\n[12,75–77], but due to the angular momentum transfer\nfrom the transmitted magnons. Moreover, the FIshows\nanother distinct characteristic point, besides the angular\nmomentum and magnetic compensation points, that is a\ntorque compensation point at which we find a reversal of\ntheDWrotation. Consequently, at the torque compensa-\ntion point the Walker breakdown is strongly suppressed\nwhich suggests that high DWvelocities and ultrafast DW\nacceleration should be achievable at this point.\nFinally, we want to mention that first experimental ev-\nidence on copropagation of a DWwith a thermal magnon\ncurrent, induced by ultrashort laser pulses, has been re-\nported recently by Shokr et al. [6]. In their work it is\nreported that DWsin a ferrimagnetic GdFeCo alloy will\nmove away from the laser spot center, i.e. against the\nthermal gradient and towards the cold region, corrobo-\nrating our findings for the DWmotion above the Walker\nbreakdown.\nACKNOWLEDGMENTS\nThe authors would like to thank Philipp Graus and\nJohannes Boneberg for fruitful discussions. This work was\nfinancially supported by the Deutsche Forschungsgemein-\nschaft through the SFB 767 “Controlled Nanosystems”.\nAt the FU Berlin support by the Deutsche Forschungsge-\nmeinschaft through SFB/TRR 227 “Ultrafast Spin Dy-\nnamics”, Project A08 is gratefully acknowledged. Fur-\nthermore, U.R. acknowledges funding from the Deutsche\nForschungsgemeinschaft via the project RI 2891/2-1.11\nAppendix A: Dispersion relation of the rocksalt-type\nferrimagnet\nWe start the derivation of the spin-wave dispersion by\nintroducing the complex vector S= [Sx\nA+iSy\nA,Sx\nB+iSy\nB].\nThis ansatz implies an x-y-symmetry and thus vanishing\nin-plane anisotropy K⊥=Kyy−Kxx= 0 in order to avoid\ndealing with squeezed magnon states [ 79]. We assume a\ngroundstate magnetization of mz\nA= +1 andmz\nB=−1.\nFollowing Refs. [ 55,74] one can deduce the linearized\nLLG-equation in k-space in analogy to the FMand AFM\ncase:\n∂Sk\n∂t=−Ωk·Sk (A1)\nwhere the frequency matrix on the right hand side is given\nby\nΩk=/bracketleftbigg\n(+i−αG)Ωk\nAA(+i−αG)Ωk\nAB\n(−i−αG)Ωk\nBA(−i−αG)Ωk\nBB/bracketrightbigg\n. (A2)\nThe matrix elements of Eq. (A2) are\nΩk\nAA=γA\nµA/parenleftBig\n6JAB−2dz\nA−2JAAC(2)\nk/parenrightBig\n(A3)\nΩk\nAB=γA\nµA2JABC(1)\nk(A4)\nΩk\nBB=γB\nµB/parenleftBig\n6JAB−2dz\nB−2JBBC(2)\nk/parenrightBig\n(A5)\nΩk\nBA=γB\nµB2JABC(1)\nk. (A6)\nThe structure factors C(n)\nkare related to the neighbor\npositions of shell nand can be expressed as\nC(1)\nk=/summationdisplay\nνcos(kνaν) (A7)\nC(2)\nk=/summationdisplay\nν,κ\nν/negationslash=κ[1−cos(kνaν) cos(kκaκ)] ; (A8)\n2aνare hereby the lattice constants of the face centered\northorhombic unit cell (see Fig. 1), in the following we\nassumeaν=afor simplicity.\nThe solution of Eq. (A1) is given by the eigenvalues\nof Eq. (A2) and can be computed in closed form for the\n2×2-matrix:\nΛk\n±=−αG−i\n2Ωk\nAA−αG+i\n2Ωk\nBB (A9)\n∓1\n2/bracketleftBig\n4(1 +α2\nG)/parenleftbig\nΩk\nABΩk\nBA−Ωk\nAAΩk\nBB/parenrightbig\n+/parenleftbig\n(αG−i)Ωk\nAA+ (αG+i)Ωk\nBB/parenrightbig2/bracketrightBig1/2\n.\nWe can further simplify this expression by assuming\nαG/lessmuch1, leading to\nωk\n±= +Ωk\nAA−Ωk\nBB\n2∓1\n2˜Ωk (A10)\nλk\n±\nαG=−Ωk\nAA+ Ωk\nBB\n2±(Ωk\nAA)2−(Ωk\nBB)2\n2˜Ωk(A11)\nXW L\nK XU\nk\n020406080100120140k,and k,/  G (meV)\nk, +\nk,\nk, +\nk,\nFigure 7. Dispersion relation of the FIalong the high symme-\ntry path of the magnetic BZ(ΓX=π/a). Solid lines are the\nmagnon energies and dashed lines their linewidth (normalized\ntoαG), respectively. Shown are the absolute values of the\nfrequency, neglecting the rotation sense.\nFigure 8. Sublattice-resolved magnon amplitudes, along\nthe high-symmetry lines in the BZ. From the ratio of the\namplitudes one can deduce in which direction the net-angular\nmomentum of the magnon points. The top/bottom sketch\nqualitatively show a low-/high-frequency magnon excitation\namplitude far away from the Γ-point. At the Γ-point (not\nshown as sketch) mixed excitations can occur in which both\nsublattices are excited.\nwhere the frequencies ωk\n±=/Ifractur{Λk\n±}and damping rates\nλk\n±=/Rfractur{Λk\n±}are the imaginary and real parts of the\ncomplex eigenvalues Λk\n±, respectively. The frequency ˜Ωk\nsimply reads\n˜Ωk=/radicalBig\n(Ωk\nAA+ Ωk\nBB)2−4Ωk\nABΩk\nBA. (A12)\nNote that for k= 0 Eqs. (A10) and(A11) coincide with\nthe results of Kamra et al. [68, Eq. (16) and (17) ]for the\nmagnetic resonance mode of a FI.\nNext we want to derive the “amplitude” of the magnon.\nAlthough the absolute value of a magnon is not well de-\nfined in our semiclassical picture, it is sufficient to compute12\nthe relative amplitudes between the A and B sublattices,\nas the absolute values of the amplitudes will cancel for a\nthermal magnon distribution. These relative amplitudes\nare related to the (non-normalized) eigenvectors\nSk\n±=/bracketleftbigg\n−Ωk\nAA+ Ωk\nBB∓˜Ωk\n2Ωk\nBA,1/bracketrightbigg\n, (A13)\nto the eigenvalues Λk\n±of Eq. (A2). The classical equivalent\nto the magnon amplitude µk\n±/lessmuchµA,Bfollows from simple\ngeometrical considerations as\nµk\n±=−S2\n2||Sk\n±||2/parenleftBig\nµA/parenleftbig\nSk\n±,A/parenrightbig2−µB/parenleftbig\nSk\n±,B/parenrightbig2/parenrightBig\n,(A14)\nor\nlk\n±=−S2\n2||Sk\n±||2/parenleftbiggµA\nγA/parenleftbig\nSk\n±,A/parenrightbig2−µB\nγB/parenleftbig\nSk\n±,B/parenrightbig2/parenrightbigg\n,(A15)\nwhereSis a scaling parameter which quantifies the classi-\ncal spin-wave amplitudes. The sublattice-resolved magnon\namplitudes as defined inside the brackets of Eq. (A15)\nare shown in Fig. 8. One can clearly see that the sign\nofµk\nσdoes not depend on k, but only on σ, since for a\ngiven branch σ, one sublattice is always excited much\nmore strongly. In fact, apart from the modes close to the\nΓ-point, the magnon amplitude can be approximated by\nan excitation of only one of the two sublattices: for the\nlow frequency branch (orange), that is the B-sublattice\n(top), whereas for the high frequency branch (blue) it is\nthe A-sublattice.\nAppendix B: Linear spin wave theory for thermally\ninduced domain wall motion\n1. Temperature step\nFirst we suppose a system of an extended nanostrip\nwith a temperature profile in the form of a step function\nT(z) =T0+∆TΘ(−z). The system is then isotropic along\nthex-/y-directions and we only expect a net-spin current\npropagating along z-direction. Since the thermal magnon\noccupation in the classical limit follows a Rayleigh-Jeans\ndistribution n0\nk,σ=kBT/~ωk,σ, one can drop the base\ntemperature T0as long as it is low enough that it does\nnot affect the effective magnetic parameters, i.e. as long\nas the dispersion relation (A10) and (A11) is still valid.\nNote, that unlike previous works that studied the\naction of spin-waves on DWs[43,75], an effective 1-\ndimensional model is not sufficient here, since thermally\nexcited magnons with off-axis wave vector k/negationslash=kˆ zare\nrelevant and due to the large grid cross-section included in\nthe numerical simulations. The macroscopic spin current\ndensity follows from integrating over all thermally excited\nmodes in the BZ. The two sublattices of the checkerboard\nAFM (Fig. 1) are two fcc-lattices with magnetic lattice\nconstant of 2 a, respectively. Thus, the BZis a truncated\nVDW\nm⊥/meqT/ΔTFigure 9. DWvelocity for a uniaxial FIas a function of\ndistancezfrom a temperature step of kBT=1 meV ; compari-\nson between LLG-simulations (points) and LSWT (line). The\ntheory line was obtained by integrating Eq. (B5) numerically\nvia a Monte-Carlo method with about 5×105k-points in\nthevz\nk,σ>0 half of the BZ. For the LLG simulation a grid\nof 96×96×480 and a simulation time of 192 ps was taken\n(t/greatermuchz/vz\nk,σfor the majority of the BZ). Errorbars indicate\nthe initial and final position of the DW. The DWcopropagates\nwith the magnon current, i.e. moves to the colder sample\npart. The inset shows the schmematics of the angular momen-\ntum transfer due to high (blue) and low (yellow) frequency\nmagnons.\noctahedron with qX=π/a, [80]. For a DWat a position\nz>0 away from the temperature step, we can restrict the\nk-space integral to the half space vz\nk>0 to only include\nforward propagating magnons:\nJ(z) =/summationdisplay\nσ=±1/integraldisplay\nvz\nk>0d3k\n(2π)3jz\nk,σ(z) (B1)\nEach mode k,σcontributes with\njk,σ=−σ~nk,σ∂ωk,σ\n∂kzexp/parenleftbigg−2z\nξk,σcosϑk/parenrightbigg\n(B2)\nto the net-current J. Here∂ωk,σ/∂k=vk,σis the group\nvelocity of the mode, nk,σis the magnon occupation num-\nber at the source, and the exponential factor accounts\nfor the absorption of the current with propagation length\nξk,σ=|vk,σ|τk,σ. The factor of two in the exponential\naccounts for the conversion of the spin-wave amplitude to\nthe magnon number, proportional to the squared ampli-\ntude.ϑkdenotes the angle between kand thez-direction\nand is needed to compute the actual propagated distance\nrk=z/cosϑk.\nIgnoring depletion effects at the interface, i.e. at the\nmagnon source, we can assume that the magnon occu-\npation at the source is given by the thermal population13\nnk,σ≈n0\nk,σand hence we get\njk,σ=−σ~kB∆T\n~ωk,σ∂ωk,σ\n∂kzexp/parenleftbigg−2z\nξk,σcosϑk/parenrightbigg\n(B3)\n=−σkB∆T∂lnωk,σ\n∂kzexp/parenleftbigg−2z\nξk,σcosϑk/parenrightbigg\n(B4)\nand we arrive at our preliminary result for the magnon\ncurrent density due to a temperature step\nJ=−/summationdisplay\nσ=±1kB∆Tσ/integraldisplay\nvz\nk>0d3k\n(2π)3∂lnωk,σ\n∂kzexp/parenleftbigg−2z\nξk,σcosϑk/parenrightbigg\n.\n(B5)\nWe should note that we assumed a fixed magnon am-\nplitude of ~in the derivation, which for the quantum\nmechanical case is a reasonable assumption for the FI[79],\nbut seems arbitrary for the classical case [see Eq. (A14) ].\nThis is however not further relevant, since the magnon\namplitude eventually cancels when we put in the thermal\noccupation n0\nk,σ∝1/~ωk,σ.\nIn Fig 9 we compare the DWvelocity calculated accord-\ning to Eq. (B5) with our numerical simulations results.\nSince the base temperature is set to zero, the system is\nbelow the compensation temperature. The DWvelocity\nis plotted as a function of distance zfrom a 1 meV tem-\nperature step. We find excellent quantitative agreement\nbetween numerical simulations and the LSWT . Further-\nmore, as for the thermal gradients, we also find the motion\nof the wall to be away from the magnon source.\n2. Temperature gradient\nThe solution of the previous section B 1 is easily applica-\nble to temperature gradients, by simply summing up over\nseveral temperature steps d T(z) =∂T/∂z dz. Once more\nwe will use the fact that the Rayleigh-Jeans distribution\nis linear in T(z). Therefore, in a constant temperature\ngradient, we have the same amount of magnons flowing\nfrom the right to the left (carrying spin −σ~), than we\nhave “magnon-holes” flowing from right to left (carrying\nspin +σ~). This means we can again restrict the k-space\nintegral over half of the BZand multiplying the result\nby two, such that the final result for the spin current is\nthe sum of equal contributions of magnon current and\n“magnon-hole” current.\ndJ= 2/summationdisplay\nσ=±1kBσ/integraldisplayd3k\n(2π)3∂lnωk,σ\n∂kz\n×exp/parenleftbigg\n−2z\nξk,σcosϑk/parenrightbigg∂T\n∂zdz. (B6)\nWe obtain the final result Eq. (17) by performing the\nz-integration\nJ=/summationdisplay\nσ=±1kB∂T\n∂zσ/integraldisplayd3k\n(2π)3∂lnωk,σ\n∂kzξk,σcosϑk.(B7)\n0 10 20 30 40 50 60 70 80\nz (nm)103\n102\n101\n100101VDW (m/s)BE 5 meV\nBE 10 meV\nBE 25 meV\nRJFigure 10. Comparison of the semiclassical spin current\nemitted from a kB∆T=1 meV temperature step, derived\nvia the Rayleigh-Jeans distribution (RJ; red line) and the\nquantum statistical derivation based on the Bose-Einstein\ndistribution (BE) for different base temperatures T0. At very\nlow temperature the spin current is dominated by the low-\nfrequency branch, since the high-frequency magnons are still\nfrozen out. Thus the sign of the net-spin current is reversed\nin the vicinity of the source, indicated by the dashed line.\n3. Quantum effects\nOne can further compute the spin current in the quan-\ntized form by replacing the Rayleigh-Jeans distribution\nn0\nk,σwith the Bose-Einstein distribution nBE\nk,σ. For simplic-\nity we restrict this discussion to the case of a temperature\nstep as the findings are expected to be qualitatively similar\nfor the thermal gradient.\nIn quantum statistics, the magnon occupation number\nis no longer linear in the temperature, hence, the base\ntemperature will be relevant. The resulting spin current\n(B2) emitted from our temperature step should thus be\nproportional to\nnk,σ=nBE\nk,σ[kB(T0+ ∆T)]−nBE\nk,σ[kBT0]. (B8)\nFigure 10 shows the DWvelocity corresponding to a\nthermal spin current calculated with the correct quan-\ntum statistics for a set of base temperatures kBT0(the\nCurie temperature for the given exchange constants is\nTC=616 K ). At very low temperature only the lowest\nmagnon energies will be occupied, i.e. the low-frequency\nbranch of the dispersion will dominate the spin transport,\ndespite the low propagation length. Moreover it is implied\nthat real systems can exhibit a sign change of the net-spin\ncurrentJat very low temperature. Thus, for uW→0, the\nDWvelocityVDW=Ja3/lfu[Eqs. (8)and(15)] not only\nchanges sign at the compensation point where lfuchanges\nsign, but also a second time when the temperature is suf-\nficiently high, to populate the long-range, high-frequency\nmagnons of the upper branch—the ones that carry nega-\ntive momentum (see Fig. 8). The overall magnon current14\nand magnon accumulation at low temperature is greatly\nreduced with respect to the classical case. However, for\nhigher temperatures and in particular at room tempera-\nture, we qualitatively retain the semiclassical magnon cur-\nrent derived with the Rayleigh-Jeans distribution. From\nthis we can conclude that the semiclassical treatment\nis sufficiently accurate for describing most experiments,\nwhich are usually carried out near room temperature with\nmagnetic materials of similar ordering temperature [61].\nAppendix C: Entropic torque in the ferrimagnet\nThe entropic or magnetothermal torque in the FIcan\nbe defined in analogy to the FMcase [ 8]. The effective\nexchange stiffness for our cubical FIis composed by the\nthree exchange contributions Aeff=AAA+AAB+ABB. In\nthe molecular field approximation, for a magnetic texture\nalong the (001) direction, these can be written as [71]\nAAA=2JAA\nam2\nA, ABB=2JBB\nam2\nB,\nandAAB=−JAB\n2amAmB. (C1)\nThe different numerical factors come from the symmetry\nof the shells which is fcc for the ferromagnetic exchanges\nAAAandABB(eight neighbors with ∆ z=±1), and sim-\nple cubic for the antiferromagnetic exchange AAB(two\nneighbors with ∆ z=±1), see Fig. 1. Their temperature\ndependence is hereby assumed to be well approximated\nby the mean field expressions Aij/Aij(0) =mimj. It\nshould be noted that although we chose exchange param-\neters withJAA/greatermuchJBBand thusJBBis not significantly\naffecting the magnetic ordering ( TCfor instance), it does\nadd a non-negligible contribution to the entropic torque\ndue to the faster demagnetization of mBcompared to\nmA. The temperature dependence of the equilibrium\nmagnetizations mi(T) is taken from the data in Fig.1.\nWe find∂mA/∂T≈−4.87×10−4K−1for the strongly\ncoupled sublattice A and ∂mB/∂T≈−2.15×10−3K−1\nfor the weakly coupled lattice B. The temperature deriva-\ntives which we obtained for the three exchange stiffness\ncontributions are summarized in Tb. I.\nTable I. Low-temperature exchange stiffnesses and their tem-\nperature derivatives in 10−11J/m and 10−14J/(m K), respec-\ntively.\nAA BB AB sum\nAij 2.05 0.064 0.192 2.31\n∂Aij/∂T −2.00 −0.275 −0.507 −2.78\nThe effective magnetocrystalline anisotropy density is\nKzz\neff= (dz\nA+dz\nB)/2a3=5.13×106J m−3. This value ischosen rather high in order to (i) keep the DWwidth and\nhence the required computation grid small and (ii) to\nreduce the characteristic time scale of the DWaccelera-\ntion which is proportional to the wall width ∆ DW[see\nEq.(11)]. In our simulations we observe a DWwidth\nof about 1.6 nm to 2.2 nm (depending on temperature)\nwhich is in good agreement with the theoretical predic-\ntion of ∆ DW=/radicalbig\nAeff/2Kzz\neff= 1.50 nm\nAppendix D: Computation of steady state domain\nwall dynamics\nDue to the different time scales involved in the ferrimag-\nnetic DWdynamics, determining the steady-state velocity,\nprecession, and tilting is challenging. On the one hand,\nsimulation time should be as short as possible, in order to\nminimize the thermal drift, i.e. the error margins of the\ntemperature, but at the same time one has to assure the\nsimulation time is sufficiently long for the wall to reach\nits steady-state motion.\nFor the data in Fig. 3 the steady-state dynamics were\ndetermined as follows: below Walker breakdown, we simu-\nlated a fixed amount of time of 320 ps ,128 ps ,128 ps , and\n384 ps for the panels (a)-(d), respectively. These numbers\nreflect the acceleration time scales at the different base\ntemperatures. The first 25 % of this simulation time was\nhereby discarded in order to reach the steady-state veloc-\nity (and tilting), the other 75 % were used for computing\nthe time average of VDWdisplayed in Fig. 3. The Walker\nbreakdown was defined by the wall angle tilting by more\nthan 45◦plus a five degree error margin, in order to ac-\ncount for the diverging wall precession time at exactly the\nWalker threshold. Above the torque compensation point,\npanel (d), or very close to the Walker thresholds, the DW\nprecession is slow and the precession period can be several\nhundreds of picoseconds. In this case, the steady-state\nvelocity was time-averaged over only one 180◦-rotation\nto keep thermal drift as low as possible and ensure a\nwell-defined temperature. 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A detailed magnetic phase diagram is built up, in which antiferromagnetic state dominates for  x≤0.25 and ferrimagnetic phase arises for x≥0.3. \nMeanwhile, sizeable electric polarization of spin origin is commonly observe d in all samples, no \nmatter what the magnetic state is. For the samples hosting a ferrimagnetic state, square-like magnetic hysteresis loops are reve aled, while the remnant magnetizat ion and coercive field can be \ntuned drastically by simply vary ing the Mn-content or temperatur e. Possible coexistence of the \nantiferromagnetic and ferrimagnetic phases is propos ed to be responsible for the remarkable \nmodulation of magnetic prope rties in the samples.  \n   \n \n  I. Introduction \nMagnetoelectrics has been providing a fer tile ground for studying cross-coupling among \nvarious ferroic-orders, which is fundamentally in teresting and practically important for developing \nconceptually new devices 1-4. Recently, a group of newly discovered magnetoelectric (ME) \nmaterials, i.e. the polar magnets A2Mo 3O8 (A: transition metal), have been drawing considerable \nattention, because of their concurrent spontaneous electric polarization ( P) due to the \ncrystallographic polarity, spin-driven electric P, and linear ME effect 5-11. This is rarely seen in the \nexisting materials such as other ferrites 12-13. Note that the linear magnetoelectrics usually have zero \nspontaneous P and thus are regarded as non-ferroelectric (FE) 14.  \nThe family of A2Mo 3O8 belongs to a pyroelectric space group P63mc 15-17. As illustrated in Fig. \n1(a), there are two ineq uivalent sites of the A ions, i.e. the octahedra and the tetrahedra, and the \npolar arrangement of the AO4 tetrahedra induces crystallographic polarity along the c-axis. \nMagnetism of  A2Mo 3O8 is dominated by the A-ions, while the Mo ions are trimerized. The two \nmagnetic sublattices com posed of tetragonal- A and octahedral- A are coupled in an \nantiferromagnetic (AFM) manner. In each of the sublattices, magnetic moments can be either \nparallel or antiparallel, depending on the A ions. For instance, Fe 2Mo 3O8 and Co 2Mo 3O8 have two \ninteracted AFM sublattices, and Mn 2Mo 3O8 hosts a ferrimagnetic (FRM ) ground state where two \ntypes of ferromagnetic (FM) orders  are coupled antiferromagnetically 9, 18. In Ni 2Mo 3O8, a specific \nnoncollinear magnetic configuration is stabilized due to the coupli ng of magnetic sublattices of Ni2+ \n11. The different magnetic configurations give rise  to strikingly distinct ME effect and thereby \nunderlying physics in these materials.  \nBeyond the conventional cross-control between ma gnetic and electric variables, some more fascinating properties such as the optical ME effect which is an extension of the classic ME to \ndynamic region, orbitally selective Mott feature, and giant thermal Hall effect have been revealed \ntypically in Fe 2Mo 3O8. Importantly, all these properties are found to closely relate to the magnetic \nstructure. The band gap of Fe 2Mo 3O8 exhibits notable narrow down at the AFM transition TN~60 K, \nimplying intimate coupling between the ch arge and spin degrees of freedom 19. The optical ME \ndepends on the relative arra ngement between magnetization ( M) and electric polarization P, e.g. the \ndiagonal term MꞏP leads to the gyrotropic birefri ngence and the off-diagonal term  M×P determines \nthe nonreciprocal di rectional dichroism 20-21. For the thermal Hall effect, the transverse thermal \nconductivity jumps to a unprecedentedly large value driven by the AFM to FRM transition 7. \nMotivated by these observations, it is of highly interesting and practical to study the modulation of \nmagnetic order of Fe 2Mo 3O8, which may extend the ME scenar io to a much broader scope.  \nFe2Mo 3O8 hosts a rather robust AFM ground state,  and the critical magnetic field ( Hcri) of the \nAFM-FRM transition is quite high indeed 5, 8. In this sense, chemical substitution may be a more \nfavorable route to tailor the magnetism of Fe 2Mo 3O8, considering the very different magnetic states \namong the members of A2Mo 3O8 as mentioned above. In particular , it was found that the Fe ions at \nthe octahedral (Fe O) and the tetrahedral (Fe T) sites could be selectiv ely substituted by properly \nchoosing the doping species 22. This is fundamentally important, because of the distinct electronic \nand magnetic properties between Fe O and Fe T. In comparison with non-magnetic ions, doping with \nmagnetic ions could better preserve  the fascinating properties th at essentially associate with \nmagnetic order. Especially, the effect of Mn-substitution at Fe-site on the magnetic properties deserves special attention, based on three motivations. First, Fe\n2Mo 3O8 hosts an AFM ground state, \nwhile Mn 2Mo 3O8 has a FRM ground state, wh ich facilitates the modula tion of magnetic structure efficiently. Second, although both materials have the same c-axis magnetic anisotropy, Fe2+ has \nmuch stronger spin anisotropy than Mn2+. For instance, the c-axis magnetic anisotropy remains \nnotable up to room temperature ( T) in Fe 2Mo 3O8, but disappears right ab ove the FRM temperature \nTC~41 K in Mn 2Mo 3O8 6, 8. Third, in Fe 2Mo 3O8, Fe T (~4.6 μB/Fe) has much larger magnetic moment \nthan Fe O (~4.2 μB/Fe) 22. Differently, Mn O and Mn T have the same magnetic moments of ~5 μB/Mn \nin Mn 2Mo 3O8, which are expected to fully compensate at T=0 K. Moreover, it was found that \nMn-doping in Fe 2Mo 3O8 indeed causes interesting sign-reversal  of the linear ME coefficient from \nnegative to positive, while the detailed magnetic properties of (Fe 1-xMn x)2Mo 3O8 remains largely \nunexplored 6.  \n In the present work, a series of single crystal (Fe\n1-xMn x)2Mo 3O8 (0≤x≤1) were synthesized, and \nextensive characterizations including struct ure, magnetization, a nd pyroelectric current \nmeasurements were carried out. Doping induced AFM to FRM transition is identified at x=0.25, i.e. \nthe bicritical point. It is found that the FRM pha se can be modulated remarkably, and the remnant \nmagnetization ( Mr) and coercive field ( Hc) exhibit strong dependence on the Mn-content x and T. \nInterestingly, possible coexistence of AFM and FR M phases may widely exist in the materials with \ndifferent x.      \n \nII. Experimental details \nThe (Fe 1-xMn x)2Mo 3O8 single crystals were s ynthesized using chemical vapor transport method \n23-24. High purity MnO 2, Mn, Fe 2O3, CoO, and MoO 2 powders were mixed and sealed in evacuated \nsilica tubes, which then were pl aced in a two-zone furnace for th e crystal growth. Chemical vapor transport method is more favorab le for the synthesis of (Fe 1-xMn x)2Mo 3O8 polar magnets than other \nroutes 25-26. The obtained crystals are hexagonal plate-like with typical size of 2×2×0.5 mm3. In \norder to determine the crystalline structure, the crystals were crashed thoroughly, and then powder \nX-ray diffraction (XRD) measurements we re performed at room temperature ( T). Energy Dispersive \nX-ray Spectroscopy (EDS) and X-ray Photoelect ron Spectroscopy (XPS) measurements were \nperformed to check the stoichio metry and valance state of the single crystals. The magnetization \nmeasurements as a function of T and magnetic field H were carried out using a Quantum Design \nphysical property measurement system (PPMS). For both zero field cooling (ZFC) and field cooling \n(FC) measurements, the measuring field was fixed at H=0.1 T. The M(H) measurements were \ncarried out at various temperatures for the sa mples. All the magnetization measurements were \nperformed with H//c-axis. Gold electrodes were prepared for pyroelectric current measurements. \nElectric polarization along the c-axis as a function of T were obtained by integrating the \npyroelectric current which were measured usin g Keithely 6517 electrometer during the warming \nprocess. Pulsed high magnetic field measurements  for magnetization were performed by means of a \nstandard inductive method employing a couple of coaxial pickup coils. The pulsed magnetic field \nup to ~58 T with a duration time of ~10 ms was generated by using a nondestructive long-pulse \nmagnet energized by a 1 MJ capacitor bank.   \nIII. Results and Discussions \nThe collected powder XRD spectra confirm that  all samples with different Mn-content x are \npure phase. To acquire more details of the crystalline structure, Rietveld-profile refinements of the \nXRD data were performed for all samples. The refined result for the sample with x=0.9 is shown as an example in Fig. 1(b). The difference between the measured and refined spectra is small with \nreliability parameter Rwp = 3.86%. For all other samples, the refinements are all high quality, and \nthe Rwp values are at similar levels. According to the refinements, derived lattice constants are \na=5.773 Å and c=10.054 Å for Fe 2Mo 3O8 (i.e. x=0), and a=5.799 Å and c=10.267 Å for Mn 2Mo 3O8 \n(i.e. x=1), which are in good agreem ent with previous studies 18. The obtained a, c, and cell volume \nV are plotted as a function of x in Fig. 1(c)-(e), respectiv ely. It is seen that a ll lattice parameters vary \nwith x continuously, without showing any apparent a nomaly. Since stoichiometry and valance state \nare crucial ingredients in determining the properties of oxides 27-30, further EDS and XPS \nmeasurements have been performed in the crysta ls (shown in the Supporting Information). It is \nconfirmed that no apparent deviatio n of the concentration of the orig inal cations from a given value, \nand Fe2+ and Mn2+ are quite stable in the crystals.   \n \nBecause of the ine quivalent sites, Fe 2Mo 3O8 has two AFM sublatti ces consisting of Fe O and \nFeT, respectively, which are coupled antiferromagnetically. Fe 2Mo 3O8 undergoes a paramagnetic \n(PM) to AFM transition at TN~60 K, evidenced by sharp peaks in Mc(T) curves measured under \nboth ZFC and FC sequences, as show n in Fig. 2(a). By introducing Mn2+ into the Fe2+ sites, TN is \ngradually shifted toward low- T region. For the sample with x=0.2 shown in Fig. 2(c), the peak of \nMc(T) gets evidently higher than the samples with  lower doping content, indicating the occurrence \nof ferrimagnetism in the sample. At x=0.25, the Mc(T) curves exhibit drastic enhancement ensuing \nat TC~53 K, and the subsequent peak is about ten times higher than that of the samples with x<0.2, \nevidencing development of the FRM phase. The FRM transition temperature TC is determined by \nthe minima of T derivative of Mc. Further cooling the sample causes steep decrease in Mc at TN~39 K, owing to the FRM to AFM transition.  \nAt x=0.3, the FRM phase is well stabilized below TC~53 K, and no further AFM transition can \nbe seen in the low- T region according to the FC curve. In ferrimagnets , there are generally two \nways to induce net M 31. One is due to the different magnetic moments of the interacted sublattices, \nand the uncompensated M emerges below TC and then persists down to low T. The other way is \nrelated to different T-dependence of the sublattices hosting  the same magnetic moment. In this \nsense, the M(T) curve usually first increases fast at TC, and then gradually decreases with T upon \ncooling the system. Eventually, M goes to zero as T→0, because of cancellation of the two \nsublattices. For a mixed case where the sublattices have different T-dependence and magnetic \nmoments, M is expected to be reduced to a finite value at low- T, such as the FC curve at x=0.3 \nshown in Fig. 2(e). Another phenomenon is that the ZFC curve shows tremendous difference from \nthe FC curve and becomes negative below TB~20 K. This has been wi dely observed in other \nferrimagnets such as spinels, and can be underst ood based on the classic molecular field theory 32. \nThe two FM sublattices are aligned in opposite directions, and the sign of net M depends on the \ncompetition between Zeeman energy and magnetic  anisotropy. As a consequence, negative M can \nappear. Once the Zeeman energy is beyond the anisotropy energy, the net M is thus reversed, \nleading to the sh arp variation of Mc from negative to positive around TB~20 K in the ZFC curve.   \nFurther increasing x causes continuous decrease in TC. Meanwhile, the large bifurcation \nbetween ZFC and FC curves can be seen in all samples with x>0.3, and TB is changed modestly \nwith x. Obviously, the FC curves of th e samples commonly exhibit non-zero Mc as T→0 K, \nresulting from the different magnetic  moments of the sublattices. In Mn 2Mo 3O8 (x=1), the PM-FRM \ntransition is identified at TC~41 K, consistent with previous study. Although Mn O and Mn T have the same magnetic moment of 5 μB/Mn, the different T-dependence of two FM s ublattices leads to the \nstriking Mc-enhancement at TC, a typical example of ferrima gnetism as mentioned above. \nNevertheless, a fully compensated stat e would be eventually achieved at T=0 K, revealed by the fast \napproaching between the FC and ZFC curves below TB~12 K shown in Fig. 2(m).   \n To better understand the ma gnetic properties of (Fe\n1-xMn x)2Mo 3O8, extensive M(H) \nmeasurements were carried out, and typical resu lts are shown in Fig. 3. For the non-doped sample \nFe2Mo 3O8, sharp AFM to FRM transition is trigge red by relatively large critical field Hcri, e.g. \nHcri~12 T at T=35 K. The jump at Hcri in M(H) is caused by the different magnetic moments \nbetween Fe T and Fe O, which has similar value of about ~0.5 μB/f.u. at various temperatures. \nIncreasing x causes significant reduction in Hcri. For instance, Hcri is decreased from ~12 T to ~3 T \nat T= 35 K as x is increased from 0 to 0.2, indicating stro ng suppression of the AFM phase of the \nsystem. At x=0.2, additional kink anomaly arises in M(H) above Hcri. This is getting more clear at \nx=0.25 where multiple step-like transitions accom panied by evident magnetic hysteresis can be \nidentified, shown in Fig. 3(d). In  fact, weak but unambiguous magnetic  hysteresis also exists at the \nvery low- H region. These phenomena imply coexisten ce of the AFM and FRM phases in the \nmaterial, and the steps are due to successive tran sitions from AFM to FRM domains, consistent with \nthe observation of both TC and TN from the M(T) shown in Fig. 2(d).  \nFor the sample with x=0.3 shown in Fig. 3(e), square-like M(H) hysteresis loops are obtained, \nconfirming the emergence of the FRM phase. Interestingly, the initial branch of M(H) at T=15 K \nshows step-like transition at  a reduced critical field H~0.5 T, possibly due to the H-driven \nAFM-FRM transition akin to the cases at x≤0.25. Similar phenomenon is generally seen in the samples with higher doping content. Meanwhile, the doped samples with 0.3 ≤x≤0.9 exhibit very \nlarge coercive field ( Hc) shown in Fig. 3(e)-(k), which is about  one order of magnitude larger than \nthat of Mn 2Mo 3O8 hosting a pure FRM state  (Fig. 3(l)). Therefore, strong competition between \nAFM and FRM phases may be a common feature in the (Fe 1-xMn x)2Mo 3O8 samples with \nintermediate doping levels.  \nIn addition, the M(H) curve at T=15 K of Mn 2Mo 3O8 displays another clear upturn above H~5 \nT, shown in Fig. 3(l). To understand this, high fi eld magnetization measurements were carried out, \nwhich are able to capture much more de tails of the sudden upturn in a broad H-range. As shown in \nFig. 4(a), the critical field of the upturn is moved to higher H side with increasing T. This is \nbasically different from conventio nal FRM to FM transition which is usually promoted by thermal \nfluctuation, e.g. the critical fiel d would be lowered by increasing T of the sample. Moreover, M(H) \nshows continuous linear increase after the upturn, without any trace of satu ration. This resembles \nthe typical antiferromagnetic beha vior. Therefore, the upturn-a nomaly should be assigned to a \nspin-flop transition from the c-axis FRM to the ab-plane FRM. The critical field of the upturn at \nvarious temperatures is summarized in Fig. 4(b).  \n Remnant magnetization M\nr and coercive field Hc are two key parameters of magnetic materials. \nIn fact, enhancing the macroscopic magnetization has been one of the primary goals of the ME \nresearch, which is crucial to combine the advant ages of both magnetic and ferroelectric memories. \nTypical data of Mr and Hc derived from M(H) at various temperatures are plotted as a function of \nMn-content x in Fig. 5(a) and (b), respectivel y. Similar behavior is observed for Mr(x) and also Hc(x) \nat different T. As an example, at T=15 K, Mr jumps from zero to ~0.2 μB/f.u. at x=0.3 where the doping-induced FRM phase is stabilized, and then decreases gradually for 0.4 ≤x≤1. Hc(x) displays \ndifferent evolving way that it increases quickly as the FRM phase is induced, and then reaches a \nplateau for 0.5 ≤x≤0.9. After that, Hc drops to a small value of ~0.2 T in Mn 2Mo 3O8, i.e. x=1. The \nfull data of both Mr and Hc at various T and x are summarized in Fig. 5(c) and (d), respectively. Mr \nincreases with T monotonously for all samples, and it can be as large as ~0.5 μB/f.u. for the samples \nwith 0.3≤x≤0.5. Regarding Hc(T), successive decrease can be seen for all samples hosting the FRM \nphase.  \n The above magnetization characterizations ha ve demonstrated highly tunable magnetic \nproperties in (Fe\n1-xMn x)2Mo 3O8. Further, it is confirmed that the magnetically driven electric \npolarization is maintained in all samples, no ma tter what kind of magnetic  state is hosted. Each \nsample was first cooled down far above TN or TC, and then the pyroelectric current was collected \nduring the warming process. For both the cooling and warming processes, no magnetic and electric \nfields were applied. Because of the polar crys talline structure, all samples display notable \npyroelectric current even far above TN. Importantly, additional shar p valley emerges in each \npyroelectric current curve, which matches well w ith the AFM or FRM transition of the sample, \nshown in Fig. 6(a). By integrating th e pyroelectric current versus time, T-dependence of electric \npolarization is obtained, shown in Fig. 6(b). The marked drop of ΔP(T) at TC or TN suggests a spin \norigin of the induced P, probably due to an exchange stri ction mechanism proposed by previous \nfirst principles calculations 8. Moreover, magnitude of the P-drop is about ΔP ~1000 μC/m2 for the \nsamples, which is larger than most of the multiferroics hosting a spin-origin 2, 33-37.  \n Based on the comprehensive characterizations of magnetization and pyr oelectric current, a T-x \nphase diagram is built up for (Fe 1-xMn x)2Mo 3O8 shown in Fig. 7(a). Three regions can be seen in the \ndiagram, which are dominated by PM, FRM, and AFM phases, respectively. At the phase boundary \nx~0.25, coexistence of AFM and FRM phases is identified. Interestingly, such  phase coexisting \nfeature may widely happen in the (Fe 1-xMn x)2Mo 3O8 samples, especially for the ones with middle \ndoping levels. Moreover, for the FRM phase, very large remnant magnetization Mr and coercive \nfield Hc are identified, and both parameters can be tuned remarkably by varying the Mn-content x \nand T. To understand these phenomena,  the occupation ratios of Mn2+ at the two inequivalent sites \ncould be a pivotal factor, because of the very differ ent properties of ions at the two sites in the polar \nmagnets.   \nIn the FRM phase, Mr is in principle determined by the difference of magnetic moments \nbetween Fe T (Mn T) and Fe O (Mn O), and thus can be given as:  \nMr = 2x ꞏ PO ꞏ MMn-O + (1 - 2 x ꞏ PO) ꞏ MFe-O - [2x ꞏ PT ꞏ MMn-T + (1 –2 x ꞏ PT) ꞏ MFe-T]    (1)    \nwhere P O and P T (=1- P O) are the occupation ratios of Mn2+ locating at the octahedral site and \ntetragonal site, respectively. MFe-O and MMn-O (MFe-T and MMn-T) are the magnetic moment of Fe2+ \nand Mn2+ on the octahedral (tetragonal ) site, respectively. Equation (1 ) can also be written as:   \nMr = 2x ꞏ PO ꞏ (MMn-O – MFe-O) - [2 x ꞏ PT ꞏ (MMn-T – MFe-T)] + (MFe-O – MFe-T)           ( 2 )  \nAs revealed by previous Möessbauer spectroscopy on Fe 2Mo 3O8, MFe-O and MFe-T are ~4.8 µ B and \n~4.2 µ B, respectively 22, resulting in ( MFe-O – MFe-T)~0.6 µ B. This is close to our experimental result \nof ~0.5 µ B, determined by the jump in M(H) due to the AFM-FRM transition. For Mn 2Mo 3O8, \nMMn-O and MMn-T have the same magnitude of M~5 µ B. With these parameters, it is now able to \nsimulate the Mr(x). The Mr(x) data typically measured at T=40 K close to TC (Fig. 5(a)) are used for the simulation, in order to make sure that the majo rity of the domains have  been converted to be \nFRM and uniformly aligned. The ob tained occupation ratios of Mn2+ are roughly distributed around \nPT ~ 0.6 and P O ~ 0.4 in all samples, as shown in Fig. 7(b). This is consistent with previous \nMöessbauer spectroscopy characterizations giving that Mn2+ has slight preference to occupy the \ntetragonal sites in (FeMn) 2Mo 3O8 6, 18. For Mn 2Mo 3O8 (x=1), the occupation rati os slightly deviate \nfrom 0.5, which is caused by the different T-dependence of MMn at different sites.  \nAt low- T, the difference between MMn-O and MMn-T becomes negligible, and thus Mr is \ndominated by the term ( MFe-O – MFe-T). For instance, if all doma ins are FRM, the ideal low- T Mr is \n~0.35 μB/f.u in the sample with x=0.3, by taking ( MFe-O – MFe-T)~0.5 µ B and the derived occupation \nratios of Mn2+. This is about two times larg er than the measured value  Mr~0.18 μB/f.u at T=10 K. \nThe discrepancy can be even remarkable at lower T, because the measured Mr decreases with T \nlinearly, shown in Fig. 5(c). Therefore, it is natu rally expected that AFM orders coexist with the \nFRM phase in the sample with x=0.3, responsible for the significantly reduced Mr. Difference \nbetween the ideal and measured Mr can also be found in the samples with higher doping contents, \nwhile the discrepancy is gradually descended. Taking this scenario that AFM and FRM phases \ncoexist, the giant coercive field Hc can be well understood by the strong phase competition in the \nsystem. In addition, the polar magnets (Fe 1-xMn x)2Mo 3O8 exhibit large remnant magnetization and \nelectric polarization simultaneously, which is usually accessible in composite ME systems rather than the single-phase materials \n38-39, and therefore may have interesting potentials in developing \nmicrowave materials 40-41.   \n \nIV . Conclusion A series of single crystal (Fe 1-xMn x)2Mo 3O8 (0≤x≤1) have been synthesized, and characterized \nby performing structure, magnetization, and electric  polarization measurements. It is found that \nMn-substitution of Fe can induce antiferromagne tic to ferrimagnetic phase transition at x=0.25, and \ncoexistence of AFM and FRM phases may be a comm on feature in the materials. For the samples \nwith x≥0.3 where ferrimagnetic state is stabilized, square-like magnetic  hysteresis loops with highly \ntunable remnant magnetization Mr and coercive field Hc are identified. In particular, remarkable \nelectric polarization driven by ma gnetic ordering is found in a ll samples hosting various magnetic \nstates. These findings indicate that (Fe 1-xMn x)2Mo 3O8 possessing highly tunabl e properties could be \nunique candidate for exploring em ergent physics and functions.   \n   ASSOCIATED CONTENT \nSupporting Information \nThe Supporting Information is available free of charge at *** \nAdditional characterizations such  as energy dispersive X-ray sp ectroscopy and X-ray photoelectron \nspectroscopy of the materials s upplied as Supporting Information. \n AUTHOR INFORMATION \nCorresponding Author \n* Authors to whom correspondence should be addressed, email: cllu@hust.edu.cn;  \nAuthor Contributions \n# Yuting Chang and Lei Gao contri buted equally to this work.  \nC.L.L. designed and supervised the experiments. Y.T.C. performed the measurements. L.G., B.Y., \nY.L., R.X., and J.F.W. contributed to the measurem ents and data analysis. Y.L.X. contributed to \ndata analysis. C.L.L. and J.M.L. wrote the manuscript.  \nNotes \nThe authors declare no comp eting financial interest. \n ACKNOWLEDGMENT \nThis work is supported by the National Nature Science Foundation of China (Grant Nos. 12174128, \n12074291, 92163210, and 11834002), Hubei Province Natural Science Foundation of China (Grant \nNo. 2020CFA083), and the Fundamental Research Funds  for the Central Universities (Grant No. \n2019kfyRCPY081, 2019kfyXKJC010). \n  Reference:  \n1. Lu, C.; Wu, M.; Lin, L.; Liu, J.-M., Single-Phas e Multiferroics: New Materials, Phenomena, \nand Physics. National Science Review 2019,  6 (4), 653-668. \n2. Dong, S.; Liu, J.-M.; Cheong, S. -W.; Ren, Z., Multiferroic Ma terials and Magnetoelectric \nPhysics: Symmetry, Entanglement, Excitation, and Topology. Advances in Physics 2015,  64 (5-6), \n519-626. \n3. 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Scientific Reports 2021,  11 (1), 18342. \n  Figures:  \n \n \nFigure 1. (a) Sketch of cr ystalline structure of Fe 2Mo 3O8 (top) or Mn 2Mo 3O8 (bottom). Magnetic \nmoments of Fe2+ and Mn2+ have also been labeled correspondingl y. One of the magnetic sublattices \nis indicated by dashed lines. (b) Rietveld refinement of the XRD spectra of sample \n(Fe 0.1Mn 0.9)2Mo 3O8. (c)-(e) present evalua ted lattice parameters a, c, and cell volume V as a \nfunction of Mn-content x, respectively.  \n \n   \nFigure 2. Temperature dependence of magnetization with H along the c-axis for (Fe 1-xMn x)2Mo 3O8 \n(0≤x≤1) single crystals: (a) x=0, (b) x=0.1, (c) x=0.2, (d) x=0.25, (e) x=0.3, (f) x=0.4, (g) x=0.5, (h) \nx=0.6, (i) x=0.7, (j) x=0.8, (k) x=0.9, and (l) x=1. TC and TN represent the ferrimagnetic and \nantiferromagnetic transitions, respectively. TB indicates the reversal of net magnetization.  \n \n   \nFigure 3. Magnetizati on as a function of H // c-axis measured at various temperatures for \n(Fe 1-xMn x)2Mo 3O8 (0≤x≤1) single crystals: (a) x=0, (b) x=0.1, (c) x=0.2, (d) x=0.25, (e) x=0.3, (f) \nx=0.4, (g) x=0.5, (h) x=0.6, (i) x=0.7, (j) x=0.8, (k) x=0.9, and (l) x=1.  \n  \nFigure 4. (a) High field magneti zation measurements of Mn 2Mo 3O8, in which the critical field Hcri \nof spin flop transition is indicated. (b) Summari zed magnetic phase diagram, based on the pulsed \n(black squares) and static (red do ts) magnetic fields measurements.  \n \n \nFigure 5. (a) and (b) present obtained remnant magnetization Mr and coercive field Hc at a function \nof x at various temperatures, re spectively. (c) and (d) show T-dependence of Mr and Hc, \nrespectively.    \nFigure 6. (a) Measured pyroelectric current as a function of T. (b) Integrated electric polarization as \na function of T.  \n \n  \n \nFigure 7. (a) T-x phase diagram is summarized for (Fe 1-xMn x)2Mo 3O8 (0≤x≤1), based on the data of \nmagnetization (solid squares and triangles) and pyroelectric current (stars and crosses). (b) \nCalculated Mn2+ occupation ratios P O and P T as a function of x.  \n \n  \nTable of Contents Graphic \n \n "    },    {        "title": "1703.08482v2.Electronic_structure_and_direct_observation_of_ferrimagnetism_in_multiferroic_hexagonal_YbFeO3.pdf",        "content": "1 \n Electronic structure  and direct observation of \nferrimagnetism in multiferroic hexagonal YbFeO 3 \n \nShi Cao ,1 Kishan Sinha,1 Xin Zhang,1 Xiaozhe Zhang,1,2 Xiao Wang,3 Yuewei Yin,1 Alpha T N'Diaye ,4 Jian \nWang,5 David J Keavney ,6 Tula R Paudel,1 Yaohua Liu,7 Xuemei Cheng,3 Evgeny Y Tsymbal ,1,8 Peter A \nDowben,1,8 Xiaoshan Xu1,8 \n1Depa rtment of Physics and Astronomy , University of  Nebraska, Lincoln, NE 68588, USA  \n2Department of Physics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China  \n3Department of Physics, Bryn Mawr College, Bryn Mawr, PA 19010, USA  \n4Advanced Light  Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  \n5Canadian Light Source, Saskatoon, SK S7N 2V3, Canada  \n6Advanced Photon Source, Argonne National Labora tory, Argonne, Illinois 60439, USA  \n7Quantum Condensed Matter Division, Oak Ridge National Lab, Oak Ridge, TN 37831, USA  \n8Nebraska Center for Materials and Nanoscience, University of  Nebraska, Lincoln, NE 68588, USA  \nAbstract  \nThe magnetic interaction between rare-earth and Fe ions in hexagonal rare -earth ferrites (h-REFeO 3), may \namplif y the weak ferromagnetic mo ment on Fe, making these materials more appealing as multiferroics. \nTo elucidate the interaction strength between the rare-earth and Fe  ions as well as the magnetic moment of \nthe rare -earth ions , element specific magnetic characterization is needed . Using  X-ray magnetic circular \ndichroism, we have s tudied the ferrimagnetism in h-YbFeO 3 by measuring the magnetization of Fe and Yb \nseparately.  The result s directly show  anti-alignment of magnetization of Yb and Fe  ions in h-YbFeO 3 at \nlow temperature , with an  exchange field on Yb  of about 17 kOe. The magnetic moment of Yb is about  1.6 \nµB at low -tempe rature,  significantly reduced compared with the 4.5 µ B moment of a free Yb3+. In addition, \nthe saturation magnetization of Fe in h -YbFeO 3 has a sizable enhancement compared with that in h -LuFeO 3. \nThese findings  directly demonstrate  that ferrimagnetic order exists in h -YbFeO 3; they also account for the \nenhancement of magnetization and the reduction of coercivity  in h-YbFeO 3 compared with those in h-\nLuFeO 3 at low temperature , suggesting an important role for the rare -earth ions in tuning the multiferroic  \nproperties of h-REFeO 3. \n  2 \n I. Introduction  \nThe divers e magnetic properties  of rare-earth (RE) transition -metal (TM) oxides  owe to the interplay \nbetween the distinct magnetism of rare-earth  and transition -metal  ions. For the transition -metal  ions, t he \nmagnetic moments come from d electrons  which are well exposed to the local environment . In contrast,  for \nrare-earth ions, the magnetic moments come from 4f electrons  which are close to the inner core  and have \nsignificant contributions from both spin and orbital angular moment um.1 While the stronger interaction \nbetween the transition -metal  ions determines the framework of the magnetic order  in the RE -TM oxides2–\n4, the weaker  interaction between the rare-earth  and transition -metal  ions,  on the other hand, generates \ninteresting phenomena such as spin reorientations  and moment compen sation .5–11 Despite the importance \nof the RE-TM interaction, a comprehensive understanding of its underpinnings  and implications is still \nlacking for many material systems.  \nIn this work, we study the magnetic interaction  between the rare-earth  and transition -metal  ions by \nmeasuring the magnetizati on of the rare-earth and transition -metal  ions separately using a n element -specific \nmethod.  In particular, we study hexagonal YbFeO 3, a member  of hexagonal rare -earth ferrites ( h-REFeO 3, \nRE=Ho-Lu, Y, and Sc) . Hexagonal R EFeO 3 have a layered crystal structure in which both RE and Fe atoms \nadopt a two -dimensional triangular lattice , as shown in Figure  1.12 Below about  1000 K, h -REFeO 3 crystal \nstructure undergoes a distortion corresponding to a rotation of the FeO 5 local structure and a buckling of \nthe rare -earth layer, which induces an improper ferroelectrici ty.13–17 The rotation of the FeO 5 also cants the \nmoment on Fe , via the Dzyaloshinskii -Moriya  interaction, generating weak ferr omagnetism on top of a \n120-degree antiferromagnetic order  below about 1 20 K, as illustrated  in Fig ure 1. 18–20 The spontaneous \nmagnetization is along the c axis. Recent work demonstrated that a super -lattice structure of hexagonal Lu -\nFe-O materials are promising for realizing room temperature mult iferroic materials with co -existing \nferroelectric ity and ferromagnetism,21 a property that has potential application in energy -efficient \ninformation processing and storage22. \nIn h-YbFeO 3, the Fe -Fe interaction is expected to dominate  the framework of the magnetic ordering , as \ncorroborated by the fact that  the ordering temperature of h -YbFeO 3 is almost the same as that of h -LuFeO 3 \n(noting that Lu3+ is non -magnetic)16,17,23,24. The Yb-Fe interaction is  weaker but enough to partially align \nthe moment on Yb and contribute to the total magnetization . Indeed, an enhancement of magnetization  of \nh-YbFeO 3, compared with that in h -LuFeO 3, has been ob served previously23,24, to be up to about  3 µB/f.u. \nat 3 K , in contrast to 0.0 18 µB/f.u. in h-LuFeO 3.13,16 The Yb -Fe interaction could , in principle , align or anti -\nalign the moment s of Fe and Yb. At the compensation temperature3,5, the magnetization of Fe and Yb \ncancel s, and an  indication of this was observed  previously  at about 80 K24. On the other hand , direct \nobservation of anti -alignment between the Fe and Yb magnetization is still lacking. In addition, t he \npreviously reported large magnetization (about 3 µ B/f.u.)24 at low temperature is more consistent with  a free \nYb3+, but unexpected when considering the effect of t he crystal field generated by the local environment ,25–\n29 which  could significantly change  the effective magnetic moment and the magnetic anisotropy  at low \ntemperature .28,30  \nTo elucidate the Yb-Fe interaction and the magnetic moment of  Yb, we have studied the electronic structure \n \nFigure 1  (color online) The c rystal structure of h -YbFeO 3 \nand schematic of the magnetic structure. The arrows on the \natoms indicate the atomic magnetic moments. 𝑀Fe and 𝑀Yb \nare the magnetization of Fe and Yb along the c axis \nrespectively, which are anti -aligned at low temperature.  The \nFe moments form a 120 -degree antiferromag netic order in \nthe basal plane, with only a very small component along  the \nc axis. The Yb moments are partially aligned by the Yb -Fe \nexchange field.  \n 3 \n of h-YbFeO 3 using X -ray absorption spectroscopy (XAS) and the X -ray photoemission spectroscopy (XPS)  \nand measured the magnetization of Fe and Yb separately using  X-ray magnetic circular dichroism (XMCD) . \nWe have found a large exchange field (1 7 kOe) on Yb, while the magnetic moment of Yb is significantly \nreduced from the value of a free ion.  Mixed valence of  Yb was investigated and found only at the surface \nof samples grown in reducing environment, suggesting minimal effect on the magnetism of h -YbFeO 3. \nII. Methods  \nHexagonal YbFeO 3 (001) films ( 20-50 nm) were deposited on yttrium stabilized zirconia (YSZ) (111) \nsubstrates  and on Fe 3O4 (111)/Al 2O3 (001) substrates using pu lsed laser (248 nm) deposition  in 5 mtorr \noxygen and argon environment , at 750 °C with a laser fluence of about 1 J cm−2 and a repetition rate of 2 \nHz.13,14,31 All the films studied  with X-ray absorption spectr oscopy and X -ray magnetic circu lar dichroism \nwere grown in oxygen enviro nment.  The film growth was monitored using reflection high energy electron \ndiffraction (RHEED).  The crystal structures  of the h -YbFeO 3 films were characterized by X-ray diffraction \n(XRD ) using a Rigaku D/Max -B diffractometer, with the Co K -α radiation (1.7903 Å). The linear X-ray \nabsorption spectroscopy  on the Fe L edge and O K edge was studied using X-ray photoemission electron \nmicroscope (X -PEEM) at the SM beamline of the Canadian Light Source with linearly polarized X -ray. \nThe circular X-ray absorption (fluorescence ) spectroscopy  of Yb M edge and Fe L edge  measurements were \nperformed at the bend magnet beamline 6.3.1 in the Advanced Light Source at Lawren ce Berkeley National \nLaboratory  and at the beamline 4IDC in the Advanced Photon Source at Argonne National Laboratory  \nrespectively.  The angle -resolved X -ray photoemission spectra (ARXPS) were obtained using SPECS \nPHOIBOS 150 energy analyzer. A non -monochromatized Al Kα x -ray source, with ph oton energy 1486.6 \neV, was used with various emission angles, as previously reported .32 \nIII. Results and analysis  \nA. Crystal structure and local environment of  Fe  \nTo verify  the structure and phases of the epitaxial films, we  carried out X -ray diffraction , electron \ndiffraction, and  X-ray spectroscopy measurements. Figure 2 (a) shows the X -ray diffractio n (θ-2θ scan) of \nh-YbFeO 3/YSZ films. No additional peak  other than those expected for h -YbFeO 3 and the substrate is \nvisible in this large -range scan, indicating no impurity phases.  As shown in Fig ure 2(b), RHEED images \nshow  diffraction streaks consistent with a flat surface and the structure of h -REFeO 3.13,31 \nThe X-ray absorption spectra provided further confirmation of  the local structure of Fe , from  the Fe L edge \n \nFigure 2  (color online) (a) θ -2θ X -ray \ndiffraction measurement of a n h-YbFeO 3 film \ngrown on yttrium stabilized zirconia (YSZ). (b) \nRHEED patter ns of an h-YbFeO 3 film with \nelectron beam along the <1 -10> and <100> \ndirections.  \n 4 \n spectra taken with linearly polarized X -ray. The local environment of Fe in h -YbFeO 3 is a trigonal \nbipyramid, with two apex O atoms ( top and bottom ) and three equator O atoms (in the Fe layer)  as shown \nin Fig ure 1 as well as in Fig ure 3(a) inset.  This structur e makes the out -of-plane  direction (along the c axis) \nand the in -plane direction (in the a-b plane) two distinct crystalline direction s. Using linearly polarized X -\nray, we measured the absorption spectra at the Fe L edge , as illustrated in Fig ure 3(b). As shown in Fig ure \n3(a), the spectrum with s-polarized X -ray ( E vector in the a-b plane) and that with p-polarized X -ray ( E \nvector along the c axis) show obvious contrast , consistent with the large structural anisotropy. The spectra \nand linear dichroism in Fig ure 3(a) match  those observed previously for  h-LuFeO 3,20,21,33,34 confirming that \nthe local environment of the FeO 5 moiety in the two materials are almost identical.  \nB. The Electronic structure of  Yb  \nWhile the electronic structur e of Fe in h -LuFeO 3 and h -YbFeO 3 are superficially similar, the electronic \nstructure of Yb3+ is expected to be different from that of Lu3+ by one less 4f electron. To probe the \nunoccupied states o f Yb, we measured the excitation of electrons from O 1s stat es to O 2p states (O K edge) \nusing X -ray. Nominally, O 2p states are fully occupied; the O 1s to O 2p excitation is forbidden by the Pauli \nexclusion principle. If, on the other hand, the O 2p states are hybridized with the Yb states, the O 2p states \nwill b e slightly unoccupied and give rise to observable O 1s to O 2p excitation; one can infer the energy of \nthe unoccupied Yb states using the excitation energies.20 As shown in Figure  4(a), with linearly polarized \nX-rays, several features can be observed in the absorption spectra. Previously, we carried out symmetry \nanalysis of the absorption spectra measured on h -LuFeO 3 and identified the origin of these features mainly \nas the 5d orbitals split in the crystal field: 𝑒𝜋, 𝑎1 and 𝑒𝜎 [see Figure  4(b)]20. Compared with the X -ray \nabsorption spectra of h -LuFeO 3, the spectra of h -YbFeO 3 show additional density of state s, as indicated in \nFigure 4(a), which is expected to be the unoccupied 4f state that is hybridized with the O 2p states.  \nThe 4f13 configuration  of Yb can also be probed by measuring the excitation  directly  to the  unoccupied  4f \nstates  (in the absence s -f hybridization, none exist with Lu3+). As shown in Figure  5(a), X -ray absorption \nspectra  at the Yb M edge  were measured at 18 K . Two peaks are observed in the absorption spectra at \napproximately  1513 and 1555 eV, which can be assigned to M 5 (initial state 3d 5/2) and M 4 (initial state 3d 3/2) \n \nFigure 3  (color online) (a) X -ray absorption \nspectra at the  Fe L edge measured using \nlinearly polarized X -ray. Inset: the FeO 5 \nlocal environment. (b) Schematic illustration \nof the L 2 and L 3 excitation.  \n \n  \nFigure 4  (color online) (a) X -ray \nabsorption spectra at the  O K edge of h -\nLuFeO 3 and h -YbFeO 3 measured using \nlinearly polarized X -ray. The arrow \nindicates the 4f state. (b) Schematic \nillustration of the O K edge excitation \nand the hybridization between the O and \nYb states.  \n 5 \n excitations respectively according to the photon energy35 [see Fig ure 5(b)] . The M 5 transition in  Yb, which \nis allowed by the angular -momentum selection rule,  can be described using the one-electron (hole) picture , \nwithout many -body interactions , due to the simple initial (full 3d 5/2, one hole in 4f 7/2) and final (one hole in \n3d5/2, full 4f 7/2) states, consistent with the observed  sharp, structureless  peak in Figure 5(a). The M 4 \nexcitation  (3d 3/2 to 4f 7/2), on the other hand , is not allowed by the angular -momentum selection rule. The \nnon-zero intensity of the M4 peak suggests that the crystal -field splitting and the Yb 4f -O 2p hybridization \nreduces the symmetry of the electronic states considerab ly, which is in line  with the observed contribution \nto the O K edge excitation by the Yb 4f state shown  in Fig ure 4(a). \nC. The Ferrimagnetism of h-YbFeO 3  \n1. Magnetization  of Yb and Fe   \nTo study the magnetization of Yb, we carried out X -ray magnetic circular dichroism measurements, by \ncomparing the absorption spectra using circular ly polarized X -ray in opposite  magnetic field s. As shown in \nFigure 5(a), the X -ray absorption spectra measured in 19 kOe and -19 kOe magnetic field s along the z \ndirection show a clear contrast . We define the XMCD contrast as 𝐼+−𝐼−\n(𝐼++𝐼−)/2, where 𝐼+ and 𝐼− are the M5 peak \nareas of the absorption spectra  in positive and negative magnetic fields respectively .  \nThe XMCD contrast measured at H = 19 kOe , for various temperatures  between 6.5 and 80 K, is displayed \nin Figure 6(a). The value of the XMCD signal decreases rapidly at low  temperature, inconsistent with \ntypical ferromagnetic dependence , which typically follows the Bloch’s law ( decrease slowly at low \ntemperature  but much faster close to the magnetic ordering temperature )36. Figure 6(b) shows the field \ndependence of the XMCD contrast of Yb at 18 K. A clear hysteresis is observed with a coercive field of \napproximately 3.5 kOe. The magnetization converted from the XMCD contrast (See Appendix A) is also \ndisplayed in Fig ure 6.  \n \nFigure 5  (color online) (a) X -ray \nabsorption spectra at the  Yb M edge \nmeasured using X -ray polarized \ncounter clockwise. XAS+ (XAS-) is the \nspectrum measured in magnetic field along \nthe + z (-z) direction.  (b) Schematic \nillustration of the Yb M edge excitation.  \nThe crystalline c axis of h-YbFeO 3 is along \nthe z direction.  \n \n \nFigure 6  (color online) XMCD contrast of Yb M 5 edge \nand the corresponding magnetization.  (a) Temperature \ndependence measured  in a 19 kOe magnetic field; the \nline is calculated using the parameters analyzed from \n(b). Inset: 𝐻𝑌𝑏 extracted  from the mean -field theory \n(see text in Section IV.B). (b) Magnetic field \ndependence measured at 18 K.  The magnetic field is \nalong t he c axis.  \n 6 \n Figure 7(a) shows the spectra of X -ray absorption of Fe L edge measured in circularly polarized X -ray in a \n10 kOe magnetic field at 6.5 K . A clear difference is observed between the spectra measured using  X-rays \nof different  polarization s, which can be used to estimate the magnetization of Fe .37 Figure 7(b) shows the \nmagnetic -field dependen ce of the Fe magnetization  calculate d from the XMCD contrast using the sum \nrule37–39. A hysteretic behavior is observed, with a coercive field of approximately 4 kOe, consistent with \nthe value found in  previous bulk magnetometry measurements .23,24 This coercive fields is also similar to  \nthat of Yb in Figure  6(b), indicative of the exchange field on Yb generated by Fe.  The saturation \nmagnetization of Fe is 0.05 +/ - 0.01 µB/f.u., which corresponds to a small projection of the Fe moment \nalong  the c axis. From  Figure 6 and 7 , we find that the magnetization of Fe is anti-parallel to  the magnetic \nfield and to that of the Yb magnetization  at low temperature, as also illustrated in Figure 1.  This provides  a \ndirect observation of ferrimagnetic order in h -YbFeO 3. \n2. The L ow temperature magnetic moment of Yb  \nAs shown in Fig ure 6(b),  the magnetization of Yb does not saturate in the measurement condition; instead, \nit shows a linear relation with magnetic field when the field is much larger than the coerci ve field , which is \nconsistent with a susceptibility behavior  and somewhat akin to paramagnetism for Yb. We can, nonetheless, \nfurther analyze  the magnetic moment  on Yb using  the mean -field theory36, which has been extensively \ndiscussed historically in orthorferrites and garnets10,11,40 –43. \nIn the mean -field theory , the exchange interactions are modeled using the molecular fields.  Assuming that \nthe saturation magne tization of Fe is 𝑀𝐹𝑒,𝑆 (in µB/f.u.), the magnetization of Fe is given by:  \n𝑀𝐹𝑒 =𝑀𝐹𝑒,𝑆𝐿(𝑥𝐹𝑒),     (1) \nwhere 𝐿(𝑥)=coth (𝑥)−1\n𝑥 is the Langevin function, 𝑥𝐹𝑒=(𝛾𝑌𝑏𝐹𝑒 𝑀𝑌𝑏+𝛾𝐹𝑒𝑀𝐹𝑒 +𝜇0𝐻)𝑀𝐹𝑒,𝑆\n𝑘𝐵𝑇, 𝑀𝑌𝑏 is the \nmagnetization of Yb, 𝛾𝑌𝑏𝐹𝑒 and 𝛾𝐹𝑒 are the molecular field parameters for the Yb -Fe and Fe -Fe interactions \nrespectively, 𝜇0 is the vacuum permittivity, 𝑘𝐵 is the Boltzmann constant , H is external magnetic field, and \nT is temperature . The magnetization  of Yb  is given by:  \n𝑀𝑌𝑏 =𝜇𝑌𝑏𝐿(𝑥𝑌𝑏),     (2) \nwhere 𝑥𝑌𝑏=(𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒+𝜇0𝐻)𝜇𝑌𝑏\n𝑘𝐵𝑇 and 𝜇𝑌𝑏 is the magnetic moment of Yb.  No Yb -Yb exchange interaction is \nincluded since such exchange interactions are  too weak to play a role in the temperature range investigated \nin this work.3,5 \nWhen the magnetic field is much  larger than the coer cive field and the temperature is much lower than the \n \nFigure 7 (color online) (a) Absorption spectra \nof Fe L edge measured with circularly \npolarized X -ray in a 10 kOe field at 6.5 K.  CW \nand CCW stand for clockwise and \ncounterclockwise polarization of the X -ray \nrespectively.  (b) Magnetic -field dependence of \nthe magnet ization of Fe  at 6.5 K, which \ncontains a soft and a hard component (see \ndiscussion in Section IV .D). The magnetic field \nis along the c axis.  \n 7 \n magnetic ordering temperature for the Fe (  120 K for h -YbFeO 3)24, one may treat  |𝑀𝐹𝑒|≈𝑀𝐹𝑒,𝑆 as a \nconstant. As shown in Figure  6(b), at T = 18 K , when H is between 6 and 19 kOe, the XMCD contrast \nshows a linear dependence wi th magnetic field, suggesting  that 𝑥𝑌𝑏 is small enough that  the Langevin \nfunction takes a linear form with respect to the magnetic field H: \n𝑀𝑌𝑏=𝜇𝑌𝑏2(𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒+𝜇0𝐻)\n3𝑘𝐵𝑇       (3). \nAccording to Eq. (3), the slope of the field dependence  of 𝑀𝑌𝑏 (susceptibility) is 𝜒𝑌𝑏=𝑑𝑀𝑌𝑏\n𝑑𝐻=𝜇𝑌𝑏2𝜇0\n3𝑘𝐵𝑇, \nwhich leads to  𝜇𝑌𝑏 = 1.6 +/- 0.1 µB, a value much smaller than the magnetic moment  of a free Yb  (4.5 \nµB/f.u.)44. \n3. Exchange field  on Yb  \nAccording to Eq. (3) , the remanent magnetization ( magnetization in zero H) is expected to be  \n𝑀𝑌𝑏,𝑅=𝜇𝑌𝑏2𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒\n3𝑘𝐵𝑇    (4). \nBecause  𝑀𝐹𝑒 and 𝑀𝑌𝑏 have different sign in zero H [see Figure 6(b) and Figure 7(b)], one finds  𝛾𝑌𝑏𝐹𝑒 < 0 \nfrom Eq. (4).  \nUsing the value 𝑀𝑌𝑏,𝑅 = 0.057 µB/f.u. at 18 K from Fig ure 6(b), one can calculate  the exchange field on Yb : \n𝐻𝑌𝑏=𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒\n𝜇0 = 17  kOe.  We also note that the exchange field on Yb generated by Fe in h -YbFeO 3 is \nabout an order of magnitude larger than  the value  1.6 kOe in orthorhombic YbFeO 3 and that in rare-earth \northoferrites in general3. This large difference may come from the dramatic differences between the bond \nlengths and bond angles in the hexagonal and orthorhombic YbFeO 3 structures.  \nD. The P ossible  mixed valence of Yb  \nMixed valence (Yb3+ and Yb2+) may play a role in  the magnetism of h -YbFeO 3 as well as the determination \nof the m agnetization on the Yb3+. In principle, there is a tendency to form Yb2+ due to the stability of the \n4f14 configuration. Although it will not affect the XMCD method discussed above  since Yb2+ does not \ncontribute to the Yb M 5 X-ray absorption in the first place (the excitations to the fully occupied 4f states \nare forbi dden in Yb2+), it will  be important for bulk magnetometry. We investigate d the possibility of mixed \nvalence in h -YbFeO 3 using ARXPS, by probing the core level electronic structure . \nFigure 8(a) shows  the Fe 2p X-ray photoemission spectra  for both h -LuFeO 3 and h -YbFeO 3. The good \n \nFigure 8  (Color online) (a) The X-ray \nphotoelectron spectra around Fe 2p edge of \nh-YbFeO 3 and h-LuFeO 3. (b) The X-ray \nphotoelectron spectra around Yb  5p edge of \nh-YbFeO 3 film samples grown in Ar and O 2 \nenvironment measured  at 0o and 70o take-off \nangle , corresponding to 2 nm and 0.7 nm \nprobing depth  respectively47,48. 8 \n match between the  Fe 2p 3/2 peaks  of h-LuFeO 3 and h -YbFeO 3 in Figure 8(a) indicates that Fe core level \nelectronic structure are similar in these  two ferrites . Previously, we have studied the X-ray photoemission \nspectra of Fe 2p using the Gupta and Sen (GS) multiplet fitting45,46 of Fe 2p 3/2 in h-LuFeO 3 and concluded \nthe Fe 2p and its satellite peaks are characteristic of a nominal Fe3+ valance32. The same analysis  applies \nhere in h-YbFeO 3 as well. These features also do not vary with emissio n angle  (data not shown). As a result , \nboth the surface and the bulk part of the h -YbFeO 3 are in the nominal Fe3+ valance state.   \nWe also did not find indication of Yb2+ in the film samples grown in oxygen environment (used for X-ray \nabsorption spectroscopy and X -ray magnetic circular dichroism in Figure 2 to Figure 7 ). To investigate the \npossible appearance of Yb2+, we studied ARXPS  on the h -YbFeO 3 films prepared in argon environment  A \ncomparison with two samples grown in oxygen and argon environments is displayed  in Fig ure 8(b). At the \nzero-degree  take-off angle  (perpendicular  to surface ), the XPS  spectra  of Yb are identical for both h-\nYbFeO 3 samp les. At the 70˚ take -off angle , which probes mostly the surface47,48, the XPS spectra of the \nsample grown in oxygen environment (lower panel) do not show clear  difference from that at zero degree , \nalso the surface appears to be slightly Yb rich . In contrast, for the sample grown in the argon environment, \nthe XPS spectra at the 70˚ take -off angle  exhibit additional intensity at the 5p peak, indicating a Yb2+ \nvalence49 at the surface  The correlation between the growth condition s indicates that the pr esence  of oxygen \nvacancy promotes the reduction of Yb3+. Although slightly YbO rich, t he mixed surface termination (both \niron oxide  and YbO  appear present at the surface) differs from the Fe–O termination seen for LuFeO 3.32 \nIV. Discussion  \nA. Origin of r educed moment of Yb  \nThe low -temperature magnetic moment of Yb is found to be 1.6 µB, a value significantly smaller than 4.5 \nµB for a free Yb 44. In h-YbFeO 3, Yb is surrounded by 7 oxygen atoms, a pproximately corresponding to a \nC3v symmetry. Analysis using double groups indicates that the 4f 7/2 states are split by the crystal field into \n4 levels: 3 𝐸1\n2 + 𝐸3\n2  (see Appendix B), where 𝐸1\n2 and 𝐸3\n2  are both  two dimensional44. The energy scale of the \ncrystal -field splitting is typically a few meV to a few tens meV,26–28 which cannot be resolved in the XAS \nspectra. This crystal field  splitting  means that , at low temperature , only the low -lying level (ground state) \nis populated and contribute s to the magnetization . The occupation of the  low-lying level , in turn, leads t o \nthe reduced value of 𝜇𝑌𝑏, and is the reason for the temperature -dependent magnetic momen ts and magnetic \nanisotropy observed previously in rare -earth -containing oxides28,30. \nB. Possible spin  reorientation  and magnetization compensation  \nOne can calculate the temperature dependence of Yb magnetization using Eq. (2). As shown in Fig ure 6(a), \nthe resu lt (with 𝜇𝑌𝑏=1.6 µ B, H=19 kOe,  and 𝐻𝑌𝑏= 17 kOe ) is compared with the measured values. The \nmeasured and the calculated magnetization  match well below  70 K, suggesting that mean -field theory can \ndescribe the temperature dependence of 𝑀𝑌𝑏 too.. The fact that the mean -field theory  can describe  both the \nmagnetic -field (Section III  C 2) and temperature dependence of 𝑀𝑌𝑏, indicates its validity in analyzing the \nmagnetic properties of h -YbFeO 3. \nOn the other hand,  at about 80  K, the calculated value is much larger than the measured value , suggesting \na reduction  of 𝐻𝑌𝑏 at higher temperature.  To reveal the temperature dependence of 𝐻𝑌𝑏, we calculate d 𝐻𝑌𝑏 \nfrom the measured magnetization value using Eq. (3) ( with 𝜇𝑌𝑏=1.6 µ B, H=19 kOe) ; the result is displayed \nin Fig.6(a) inset . Clearly, a sign change of 𝐻𝑌𝑏 occurs at about 80 K, indicating a possible realignment  \nbetween  the magnetization  𝑀𝑌𝑏 and 𝑀𝐹𝑒, which is discussed below.  \nIn principle, the alignment between 𝑀𝑌𝑏 and 𝑀𝐹𝑒 is determined by the minimization of total energy \n𝐸𝑡𝑜𝑡𝑎𝑙 =−1\n2𝜒𝑌𝑏(𝐻+𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒)2−𝑀𝐹𝑒𝐻, or the maximization of the total magnetization 𝑀𝑡𝑜𝑡𝑎𝑙 =\n𝑀𝐹𝑒(1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒 )+𝜒𝑌𝑏𝐻.11 Here the external field H is along the c axis and  𝑀𝐹𝑒 may point either along \nor opposite to H, corresponding to  the positive and negative  signs respectively.  9 \n Because  𝛾𝑌𝑏𝐹𝑒<0 and 𝜒𝑌𝑏=𝜇𝑌𝑏2𝜇0\n3𝑘𝐵𝑇 (see section  III C  3), the sign of 1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒 is expected to change \nwith temperature, possibly causing the reversal of  the direction of the magnetization 𝑀𝐹𝑒:  \n(1) At low temperature, 1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒 <0. In this case , 𝑀𝐹𝑒< 0 ( 𝑀𝐹𝑒 anti-parallel to H) is more \nfavorable  for maximizing 𝑀𝑡𝑜𝑡𝑎𝑙. this mean the exchange field 𝐻𝑌𝑏=𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒\n𝜇0>0 (parallel to the \nexternal field ).  \n(2) When temperature is increased and 1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒  > 0 is satisfied,  𝑀𝐹𝑒> 0 ( 𝑀𝐹𝑒 parallel to H) is \nmore favorable . In this case,  the exchange field 𝐻𝑌𝑏=𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒\n𝜇0<0 (anti parallel to the external \nfield);this could be the reason that at about 80  K 𝐻𝑌𝑏 becomes  negative  [Figure 6(a) inset ]. .  \n(3) At the compensation temperature, 1+𝜒𝑌𝑏𝛾𝑌𝑏𝐹𝑒  = 0. Therefore, 𝑀𝑡𝑜𝑡𝑎𝑙 =𝜒𝑌𝑏𝐻, as if 𝑀𝐹𝑒 is \n“screened” by the part of Yb moment induced by the exchange field 𝐻𝑌𝑏. The magnetization \ncompensation can be understood as the cancellation of 𝑀𝐹𝑒 and 𝑀𝑌𝑏 at zero field . According to \nFig. 6(a) inset, the compensation temperature appears to be between 70 and 80 K, in fair agreement \nwith the previous estimation24. \nNonetheless, the magnetization of the Yb is largely a spectator to that of the Fe. The coercivity is the same \nas that observed for iron, with the essential observation [Figure 6(b)]  that the magnetization does not easily \nsaturate indicating that much of the magnetization depends on the magnetic susceptibility and possible \nalignment of the moments with external magnetic field H and with that of Fe (Figure 7).  \nC. Exchange field on Fe  \nThe exchange field may also have an effect on the Fe, which can be understood by combining Eq. (1) and \nEq. (2) to reach  𝑥𝐹𝑒=[𝛾𝑌𝑏𝐹𝑒 𝜇𝑌𝑏𝐿(𝛾𝑌𝑏𝐹𝑒𝑀𝐹𝑒𝜇𝑌𝑏\n𝑘𝐵𝑇)+𝛾𝐹𝑒𝑀𝐹𝑒]𝑀𝐹𝑒,𝑆\n𝑘𝐵𝑇, assuming H = 0. Since Fe moments in h -\nYbFeO 3 form weak ferromagnetic order, 𝛾𝐹𝑒 must be positive. Because of the properties of the Langevin \nfunction 𝐿(𝑥), 𝛾𝑌𝑏𝐹𝑒 𝜇𝑌𝑏𝐿(𝛾𝑌𝑏𝐹𝑒 𝑀𝐹𝑒𝜇𝑌𝑏\n𝑘𝐵𝑇) is always positive regardless of the sign of 𝛾𝑌𝑏𝐹𝑒. Therefore, the \nYb always enhance s the m olecular field on the Fe. That said, because  in general 𝛾𝐹𝑒≫|𝛾𝑌𝑏𝐹𝑒 |, the effect \nmay not be significant.   \nD. Comparison between magnetic properties of h-YbFeO 3 and h -LuFeO 3 \nHexagonal LuFeO 3 (h-LuFeO 3) is the most studied hexagonal rare -earth ferrites. Because Lu3+ is non -\nmagnetic, the magnetic properties of h -LuFeO 3 is less complex. By comparing h -LuFeO 3 and h -YbFeO 3, \none may gain insight on the effect of the rare earth  on the magnetism . \nOne dr amatic difference between h -YbFeO 3 and h -LuFeO 3 is in the co ercive field of magnetization. For h -\nYbFeO 3 at 18 K, the coercive field is about 4 kOe, which is much smaller than the value 25 kOe for h -\nLuFeO 3.16 For both h -LuFeO 3 and h -YbFeO 3, the magnetization -field loop s for Fe  have  a squared shape, \nsuggesting that t he magnetic coercive field is determined by the competition between the magnetic \nanisotropy energy and the Zeeman energy . Compared with h -LuFeO 3, h-YbFeO 3 has enhanced \nmagnetization due to the contribution of Yb. Therefore, a much smaller magnetic field is needed in h-\nYbFeO 3 to overcome the magnetic anisotropy, corresponding to a much smaller coerciv e field . \nAnother  differen ce between h -YbFeO 3 and h -LuFeO 3 is in the saturation magnetization of Fe. According \nto Fig ure 7, in h-YbFeO 3, 𝑀𝐹𝑒,𝑆 = 0.05 +/ - 0.01 µB/f.u., larger than that in h -LuFeO 3 ( 0.03 µB/f.u.)16. We \nnote that previously it was observed in h -LuFeO 3 that the magnetization contains a soft component and a \nhard component, in which only the  hard component  (0.018 µB/f.u.) is bel ieved to be  intrinsic to the weak \nferromagnetic ordering  because it disappears above the magnetic ordering temperature .16 In Fig ure 7, there \nis also one soft (coercive field  1 kOe) and one hard component (coercive field  4 kOe). If we only treat \nthe hard component to be intrinsic to the  canting of the Fe moment , the weak ferromagnetic  moment of Fe \nin h-YbFeO 3 is = 0.03 +/ - 0.01 µB/Fe according to Fig. 7(b) , still larger compared with  the value  0.018 10 \n µB/f.u. in h -LuFeO 3.16 Due to  the size difference of Lu3+ and Yb3+,14 the lattice constant  of the basal plane  \nof h-LuFeO 3 is smaller than th at of h-YbFeO 3: a = 5.963 Å  for h -LuFeO 3 and a = 6.021 Å  for h -YbFeO 3.31 \nOur recent work suggests that a compressive biaxial strain may reduce the canting of the Fe moments in h -\nREFeO 3,50 which is in line wi th the correlation between the lattice constant and weak ferromagnetic moment \non Fe observed here . \nV. Conclusion  \nWe have studied the electronic structure and magnetic ordering of  h-YbFeO 3 (001) thin films on YSZ (111) \nand on Fe3O4(111)/Al 2O3(001) substrates.  The magnetism of Yb in h -YbFeO 3 was studie d using the \nelement -specific  method X-ray magnetic circular  dichroism  based on X -ray absorption spectroscopy . From \nthe temperature and magnetic -field dependence of the  Yb magnetization, we found that the low temperature \nYb magnetic moment is significantly reduced compared with the value of free  Yb3+ ions, indicating the \neffect of crystal field.  The e xchange field on Yb , generated by the Fe moments, tends to anti -align the \nmagnetization of Fe and Yb  at low temperature.  We also investigated possible  valence mixing of  Yb and \nonly found indication of Yb2+ at the surface of samples grown in an Ar environment, suggesting an \ninsignificant e ffect to the magnetism of h -YbFeO 3. We expect  that future work, such as  optical spectroscopy \non probing Yb crystal field levels  and theoretical calculation s on Yb -Fe interaction strength , may provide \nmore insight on the ferrimagnetism of  h-YbFeO 3. \nAcknowledgements  \nThis project was primarily supported by the National Science Foundation through the Nebraska Materials \nResearch Science and Engineering Center (Grant No. DMR -1420645). Additional support was provided by \nthe Semiconductor Research Corporation through the Ce nter for Nanoferroic Devices and the SRC -NRI \nCenter under Task ID 2398.001. Work of Bryn Mawr College is supported by National Science Foundation \nCareer Award (Grant No. NSF DMR -1053854 ). This research used resources of the Advanced Photon \nSource, a U.S. D epartment of Energy (DOE) Office of Science User Facility operated for the DOE Office \nof Science by Argonne National Laboratory under Contract No. DE -AC02 -06CH11357 . Use of the \nAdvanced Light Source was supported by the U.S. Department of Energy, Office of  Science, Office of \nBasic Energy Sciences  under contract no. DE -AC02 -05CH11231 . The Canadian Light Source is funded by \nthe Canada Foundation for Innovation, the Natural Sciences and Engineering Research Council of Canada, \nthe National Research Council Canada, the Canadian Institutes of Health Research, the Government of \nSaskatchewan , Western Economic Diversification Canada, and the University of Saskatchewan.  The \nresearch was performed in part in the Nebraska Nanoscale Facility: National Nanotechnology Coordinated \nInfrastructure and the Nebraska Center for Materials and Nanoscience, which are supported by the National \nScience Foundation under Award ECCS: 1542182, and the Nebraska Research Initiative.  \n  11 \n Appendix  \nA. Converting XMCD contrast to magnetization of Yb  \nThe ca lculation of the magnetization of Yb from the  XMCD contrast  at the  M edge  is different from that of \n3d metals (e.g. Fe, Co, Ni)  at the  L edge , due to the strong spin -orbit coupling  in both initial and final states . \nWe hereby present a method based on  the XMCD contrast of  excitation s from the 3d 5/2 to the individual \n4f7/2 eigen states  Jz=-7/2 to 7/2, where  Jz is the projection of total angular moment J on the z axis; all possible \n4f7/2 states are superposition of these states.   \nExcited by an X -ray polarized clockwise, the transition from one 3d 5/2 state to one 4f 7/2 state needs to satisfy \nΔJz = 1. One can calculate the transition probabilities 𝑃 between individual states; the non -zero results are \ndisplayed in Fig ure 9(a). The projection of the magnetic moment of a Jz state on the z direction is µ z = gµBJz, \nwhere g = 1.14 is the Lande g-factor and µ B is the Bohr magneton.  Therefore,  one can calculate the XMCD \ncontrast, defined as 2(𝑃𝐽𝑧−𝑃−𝐽𝑧)\n𝑃𝐽𝑧+𝑃−𝐽𝑧 , with respect to µ z, where 𝑃𝐽𝑧 (𝑃−𝐽𝑧) is the transition probability for the final \nstate represented by Jz (-Jz); the result is  shown in Fig ure 9(b). Although f or large µz, the XMCD contrast \ndoes not  distinguish the | Jz|=7/2 and | Jz|=5/2 states,  for small µz, the relation between XMCD contrast and \nµz is approximately  linear . The measured XMCD contrast in this work falls in the small µz region (all values \nare less than 0.4). Therefore, we can use the relation in Fig. 9(b) to convert XMCD contrast to magnetization  \nas a fair  approximation.  \nB. Group  theory analysis of the crystal field splitting of Yb  states  \nIn h-YbFeO 3, the local environment of Yb has a symmetry that can be described using point group C3v [see \nFigure  9(b) inset].  The degenerate electronic state s in general are split according to the symmetry of the \nlocal environment. Because of the strong spin -orbit coupling, the angular momentum of the 4f state s takes \nhalf-integer 𝐽=5\n2 or 𝐽=7\n2, the analysis of which requires the double group.  TABLE 1  shows  the character \ntable for C 3v double group,  including irreducible representations 𝐴1, 𝐴2, 𝐸, 𝐸1\n2, 𝐸3\n2  (𝐸3\n2+ and 𝐸3\n2−). The \ncharacter of the representation with angular momentum 𝐽=5\n2 and 𝐽=7\n2 are also listed. Using these \ncharacters, one can reduce the 𝐽=5\n2 and 𝐽=7\n2 representations. The results are: 𝐽=5\n2→ 2𝐸1\n2+𝐸3\n2  and 𝐽=\n7\n2→ 3𝐸1\n2+𝐸3\n2 . \n \nFigure 9 (color online) (a) Transition \nprobability between individual 3𝑑5\n2 and \n4𝑓7\n2 states excited by a clockwise polarized \nX-ray. (b) XMCD contrast as a function of \n𝜇𝑧 (see text) calculated assuming a free \nYb3+ ion. 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B 95, 94110 (2017).  \n \n "    },    {        "title": "2101.03449v1.Tuning_Dzyaloshinskii_Moriya_Interaction_in_Ferrimagnetic_GdCo__A_First_Principles_Approach.pdf",        "content": "Tuning Dzyaloshinskii-Moriya Interaction in Ferrimagnetic GdCo: A First Principles\nApproach\nMd Golam Morshed,1,\u0003Khoong Hong Khoo,2Yassine Quessab,3Jun-Wen Xu,3Robert\nLaskowski,2Prasanna V. Balachandran,4, 5Andrew D. Kent,3and Avik W. Ghosh1, 6\n1Department of Electrical and Computer Engineering,\nUniversity of Virginia, Charlottesville, VA 22904 USA\n2Institute of High Performance Computing, Agency for Science,\nTechnology and Research, 1 Fusionopolis Way, Connexis, Singapore 138632, Singapore\n3Center for Quantum Phenomena, Department of Physics,\nNew York University, New York, NY 10003 USA\n4Department of Materials Science and Engineering,\nUniversity of Virginia, Charlottesville, Virginia 22904 USA\n5Department of Mechanical and Aerospace Engineering,\nUniversity of Virginia, Charlottesville, Virginia 22904 USA\n6Department of Physics, University of Virginia, Charlottesville, VA, 22904 USA\n(Dated: January 12, 2021)\nWe present a systematic analysis of our ability to tune chiral Dzyaloshinskii-Moriya Interactions\n(DMI) in compensated ferrimagnetic Pt/GdCo/Pt 1\u0000xWxtrilayers by cap layer composition. Using\n\frst principles calculations, we show that the DMI increases rapidly for only \u001810% W and saturates\nthereafter, in agreement with experiments. The calculated DMI shows a spread in values around the\nexperimental mean, depending on the atomic con\fguration of the cap layer interface. The saturation\nis attributed to the vanishing of spin orbit coupling energy at the cap layer and the simultaneous\nconstancy at the bottom interface. Additionally, we predict the DMI in Pt/GdCo/X (X = Ta ;W;Ir)\nand \fnd that W in the cap layer favors a higher DMI than Ta and Ir that can be attributed to the\ndi\u000berence in d-band alignment around the Fermi level. Our results open up exciting combinatorial\npossibilities for controlling the DMI in ferrimagnets towards nucleating and manipulating ultrasmall\nhigh-speed skyrmions.\nIntroduction. Magnetic skyrmions are topologically\nprotected spin textures and are attractive for next-\ngeneration spintronic applications, such as racetrack\nmemory and logic devices [1{7]. The interfacial\nDzyaloshinskii-Moriya Interaction (DMI), an antisym-\nmetric exchange originating from the strong spin-orbit\ncoupling (SOC) in systems with broken inversion symme-\ntry [8, 9], is one of the key ingredients in the formation of\nskyrmions in magnetic multilayers [10{12]. Controlling\nthe DMI o\u000bers the possibility to manipulate skyrmion\nproperties, i.e., size and stability [13, 14].\nOver the past few years, the underlying DMI physics\nand overall skyrmion dynamics have been studied ex-\ntensively for ferromagnetic (FM) systems [12, 15{20].\nAlthough both heavy metal (HM)/FM bilayers and\nHM/FM/HM sandwiched structures have been explored,\nmost of the reported results are based on ideal inter-\nfaces. Indeed, very few studies focus on the role of dis-\norder on DMI [21]. Furthermore, ferrimagnetic materials\nhave drawn attention due to their low saturation magne-\ntization, low stray \felds, reduced sensitivity to external\nmagnetic \felds, and fast spin dynamics, all of which fa-\nvor ultra-fast and ultra-small skyrmions [22{26]. Very\nrecently, Quessab et al. have experimentally studied\nthe interfacial DMI in amorphous Pt/GdCo thin \flms,\nand shown a strong tunability of the DMI by varying\n\u0003mm8by@virginia.eduthe thickness of the GdCo alloy and cap layer compo-\nsition [27]. However, a detailed understanding of DMI,\nincluding the impact of two-sublattice ferrimagnetism, as\nwell as the role of an experimentally realistic, chemically\ndisordered interface are both missing.\nIn this paper, we present a systematic theoretical anal-\nysis of the DMI in a compensated ferrimagnetic alloy us-\ning \frst principles calculations. In particular, we explore\nthe variation of the DMI in Pt =GdCo =Pt1\u0000xWx(Fig. 1)\nand \fnd a strong tunability from 0 to 4 :42 mJ/m2with\nvariation in the W composition (Fig. 2). We studied the\nin\ruence of atom placement and observed that the DMI\nis sensitive to structural variations such as the GdCo con-\n\fguration in the thin magnetic \flm, and the PtW con-\n\fguration at the interface. This is important to consider\nbecause, in reality, we have an amorphous alloy and the\ninterfaces in deposited \flms are not perfect. We \fnd a\nspectrum of DMI values that show an overall saturating\ntrend, as seen in the experimental data [27]. We argue\nthat the change in SOC energy in the interfacial HM lay-\ners, especially the constancy of the SOC energy at the\nbottom layer and reduction of it in the cap layer, gen-\nerates the observed saturating trend in the DMI with\npercentage of W incorporated (Fig. 3). Additionally, we\ntheoretically predict the variation of the DMI depending\non the cap layer material, speci\fcally for Pt/GdCo/X,\nwhere X = Ta ;W;Ir (Fig. 4). We \fnd that the DMI\nis highest for W in the cap layer and lowest for Ir, a\ntrend that correlates with 3 d-5dCo-X band alignment atarXiv:2101.03449v1  [cond-mat.mtrl-sci]  10 Jan 20212\nthe cap layer interface (Fig. 5). Our results identify the\nchemical and geometric factors responsible for interfacial\nDMI, and provide a potential path forward towards the\nengineering of material properties towards next genera-\ntion skyrmion based spintronic applications.\nMethod. We use the technique of constraining the mag-\nnetic moments in a supercell to calculate the DMI within\nthe Density Functional Theory (DFT) framework [15].\nThe Vienna ab initio simulation package (VASP) is used\nfor the DFT calculations [28]. We use the projec-\ntor augmented wave (PAW) potential to describe the\ncore-electron interaction [29, 30]. The Perdew-Burke-\nErnzerhof (PBE) functional form of the generalized gra-\ndient approximation (GGA) is used for the exchange-\ncorrelation functional [31]. In order to treat the on-\nsite Coulomb interaction of Gd 4 f-electrons, we use the\nGGA+ Umethod [32] with an e\u000bective value of U= 6 eV\nfor Gd, as reported in previous studies for both bulk and\nslab calculations [33{35]. We also validate the e\u000bective\nUfor our GdCo alloy by taking a range of Uvalues from\n1\u00007 eV, and con\frming a stable ferrimagnetic ground\nstate con\fguration of GdCo at U= 6 eV. A 4\u00021\u00021 su-\npercell of Pt(2) =GdCo(2) =Pt1\u0000xWx(2) (numbers in the\nparenthesis represent the number of monolayers) is used\nin all our calculations. While creating the GdCo alloy\nby replacing Gd atoms in the hcp Co(0001) slab, a 25%\nGd composition is maintained, which is the closest to the\nexperimental proportion (22% Gd [27]) achievable within\nour structural arrangement. The trilayers are formed by\naligning fcc(111) and hcp(0001) planes. The in-plane lat-\ntice constant of the slab structure is set to 2 :81/RingA, equal\nto the calculated nearest neighbor distance of bulk Pt,\nand the supercells are separated by a vacuum layer of\n10/RingA in the [001] direction. The cuto\u000b energy is set to\n500 eV, and a 4\u000216\u00021 Monkhorst-pack k-grid is used\nfor all the calculations. We verify the convergence of our\ncalculations with cuto\u000b energy, number of k-points, and\nthe thickness of the vacuum layer.\nThe three step DMI calculation procedure starts with\nionic relaxation along the atomic z-coordinate to mimic\na thin \flm, until the forces become less than 0 :01 eV/ /RingA\nand, the energy di\u000berence between two ionic relaxation\nsteps becomes smaller than 10\u00006eV. Next, in the ab-\nsence of SOC, the non-spin polarized Kohn-Sham equa-\ntions are solved to \fnd an initial charge density. Finally,\nSOC is included, and the total energy of the system is\ncalculated self-consistently for clockwise (CW) and an-\nticlockwise (ACW) spin con\fgurations (Fig. 1) until the\nenergy di\u000berence between two consecutive steps becomes\nsmaller than 10\u00006eV.\nResults. The DMI energy ( EDMI) can be de\fned as\nEDMI =X\nhi;jidij\u0001(Si\u0002Sj) (1)\nwhere Si,Sjare the nearest neighboring normalized\natomic spins and dijis the corresponding DMI vector.\nThe total DMI strength, dtot, de\fned by the summation\nFigure 1. Schematic of Pt(2) =GdCo(2) =Pt1\u0000xWx(2) struc-\nture (number in parentheses denoting the number of mono-\nlayers) corresponding to x= 12:5% for (a) CW, and (b) ACW\nspin con\fgurations. The red arrows in the \fgure show the\nspin orientations. L1 ; :::;L6 denote the layer number while\nnumbers in circles label atomic positions.\nof the DMI coe\u000ecient of each layer, to a \frst approxima-\ntion, is calculated by the energy di\u000berence between the\nCW and ACW spin con\fgurations [15], and expressed as\ndtot= (ECW\u0000EACW )=12. The micromagnetic DMI, D\nis given by D= 3p\n2dtot=NFa2[15], where NFanda\nrepresent the number of magnetic layers and the fcc lat-\ntice constant respectively.\nBefore presenting the numerical results, it is worth\nmentioning that we can only investigate a limited sub-\nset of the structures for our calculations, as exploring all\ncombinatorial possibilities is not feasible in terms of time\nand computational resources. We consider two separate\nalloy con\fgurations: (i) Gd alloying in the magnetic lay-\ners, and (ii) W alloying in the cap layers.\nIn case (i), we \frst \fx the position of the Gd atoms\nin the GdCo alloy. We maintain 25% Gd composition\nseparately in each magnetic layer, arguing that steric re-\npulsion implies two Gd atoms are energetically unlikely\nto sit in the same layer, as assumed in previous stud-\nies [36]. The Gd atoms can thus arrange themselves in\u00004\n1\u0001\n\u0002\u00004\n1\u0001\n= 16 ways. These sixteen combinations can\nbe grouped into just four distinct sets because of their\ntranslational symmetry. In Fig. 1, looking at positions\n(1\u00008) in magnetic layers (L3 & L4), it can be seen\nthat Gd in (1 ;7);(2;8);(3;5), and (4 ;6) positions repre-\nsent equivalent structures once the unit cell is period-\nically extended. Similarly, the other three groups are\n[(1;8);(2;5);(3;6);(4;7)], [(1 ;6);(2;7);(3;8);(4;5)], and\n[(1;5);(2;6);(3;7);(4;8)]. We con\frmed this equivalence\nby calculating the energy of the Pt/GdCo stack by vary-\ning all the Gd positions and indeed \fnd equal energy\nfor the four structures within the same group. For case\n(ii), we choose one representative from each of the above\nfour groups and proceed with W positional variations in\nthe cap layer. While exploring W alloy con\fgurations,\nfor lower composition (12 :5%\u000050%), W is only incor-3\n0.00.20.40.60.81.0\nW Composition (x)0.01.02.03.04.05.0DMI, D [mJ/m2]\nW(5')\nW(6')\nW(7')\nW(8')W(5',6')\nW(5',7')W(5',8')\nW(7',8')W(6',8')\nW(6',7')W(1',3',5'-8')\nW(2',3',5'-8')W(1',4',5'-8')\nW(2',4',5'-8')\nW(3',4',5'-8')W(1',2',5'-8')W(5'-8')W(1'-8')\n(a)Gd(1,7)\n0.00.20.40.60.81.0\nW Composition (x)-1.00.01.02.03.04.05.0DMI, D [mJ/m2]\n(b)Gd(1,7)\nGd(1,8)\nGd(2,7)\nGd(3,7)\n0.00.20.40.60.81.0\nW Composition (x)-0.50.00.51.01.52.02.5Surface DMI [pJ/m]\n(c)Gd(1,7)\nGd(1,8)\nGd(2,7)\nGd(3,7)\nExp.\nFigure 2. The DMI as a function of W composition ( x) in Pt =GdCo =Pt1\u0000xWx. (a) DMI variation with respect to W positions\nwhile the Gd atoms are \fxed at (1 ;7) positions. (b) Total spectrum of the DMI as both Gd and W positions are varied. For\na speci\fc W composition, each of the di\u000berent colors represents the variation of Gd atomic positions, and the scattered points\nwithin the same color represent di\u000berent W positions for that particular Gd arrangement in the structure (Fig. 1). (c) Surface\nDMI in comparison with experimentally observed DMI [27]. The numbers followed by the symbols Gd and W represent the\npositions of the respective atoms in the structure shown in Fig. 1.\nporated in layer L6. Finally, we vary all the possible W\npositions and calculate the DMI for a total of 76 struc-\ntures.\nFigure 2(a) shows the calculated DMI, D for\nPt=GdCo =Pt1\u0000xWx, as a function of W composition. At\nx= 0%, the DMI vanishes as expected because, for a\nperfectly symmetric trilayer structure, the contributions\nfrom the bottom and top interfaces are equal and oppo-\nsite. As the W composition increases from 0% to 12 :5%,\nwe \fnd a maximum DMI of 2 :93 mJ/m2. The underly-\ning mechanism behind this non-zero DMI is the inversion\nsymmetry breaking of the Pt =GdCo =Pt structure by the\ninsertion of W atoms in the cap layer. We \fnd that a\nsmall amount of W (12 :5%) gives a large DMI change,\nand subsequent to that initial rise, with increasing W\ncontent, the DMI saturates. As the composition of W\nincreases, we \fnd a maximum DMI of 4 :42 mJ/m2cor-\nresponding to 75% W composition.\nWe \fnd that the DMI is very sensitive to the struc-\ntural details, speci\fcally the positions of the Gd and W\natoms. Figure 2(a) shows the variation of D as the posi-\ntion of the W atoms changes. In Fig. 2(a), for all cases,\nGd atoms are \fxed at the (1 ;7) positions. We show the\nvariation of W positions for the structures with 12 :5%,\n25%, and 75% compositions because for the other three\ncases there is only one combination possible in terms of\nW positions. Figure 2(b) shows the total spectrum of\nthe DMI variation while varying both the Gd and W po-\nsitions in the structure. Interestingly, for all the cases,\nthe increasing trend of the DMI is very similar. We con-\njecture that changing the position of the atoms within\nthe small unit cell will change the nature of the interface\nthat gives variations in the DMI. For example, in the\ncase of Pt/GdCo/W, when Gd atoms placed at position\n(1,7), the SOC energy change in the interfacial Pt layer\nis higher than that of position (3,7), which translates to\nthe corresponding DMI as well.\nTo validate our results against the recent experi-ment [27], we calculate the surface DMI (in units of\npJ/m) by multiplying the calculated DMI, D with the\nthickness of the magnetic layers. In our calculations,\nwe use the thickness as NFa=p\n3 = 4 :6\u0017A for the mag-\nnetic layers, while the experimental thickness is 5 nm.\nFigure 2(c) shows the surface DMI from both the DFT\ncalculation and the experiment, scaled by their respec-\ntive thickness. In the experiment, a non-zero DMI of\n0:56 pJ/m (solid black line) is found for the Pt/GdCo/Pt\nstructure because of the asymmetry in the bottom and\nthe top interfaces due to the di\u000berence of interface rough-\nness and intermixing [27]. On the contrary in our DFT\nmodel, we use a perfect crystal structure that gives a\nnear zero DMI for the symmetric cases (a small non-\nzero DMI might arise from intrinsic asymmetry within\na thin crystalline GdCo \flm modeled here). We \fnd\nan overall matching trend between the DFT and exper-\nimental data for the rest of the compositions. An exact\nquantitative agreement between the DFT results and the\nexperiment is di\u000ecult to achieve because we use a crys-\ntal structure for our model, whereas, in the experiment,\namorphous or polycrystalline materials are used. Addi-\ntionally, the magnetization also di\u000bers between our model\nand the experiment as the thickness and the dimensions\nof the structure are di\u000berent. However, we argue that\nthe structural imperfections in the experiment amount\nto an ensemble averaging over the various con\fgurations\nwe theoretically explore, so that the experimental data\nfalls in the middle of the spectrum (gray shaded area) of\nour DFT data.\nIn Fig. 2, the DMI increases non-monotonically as a\nfunction of W composition as opposed to a linear in-\ncrease one may expect. This non-monotonic trend can\nbe explained by the change of spin-orbit coupling energy,\n\u0001ESOC, between CW and ACW spin con\fgurations, in\nthe HM layers adjacent to the magnetic layers in Fig. 1.\nIn Fig. 3(a), we show the \u0001 ESOC in L2 (adjacent to the\nbottom magnetic layer) and L5 (adjacent to the top mag-4\n(a)\nL2 L5-6.0-4.0-2.00.02.04.06.0\"ESOC [meV]W(0%)\nW(12.5%)\nW(25%)\nW(50%)\nW(75%)\nW(100%)\n(b)\nL1L2L3L4L5L6-4.0-2.00.02.04.0DMI, D [mJ/m2]W(0%)\nW(12.5%)\nW(25%)\nW(50%)\nFigure 3. (a) Change in SOC energy at the interfacial HM\nlayers (L2 & L5) as a result of changing spin chirality of the\nmagnetic layers (L3 & L4) from CW to ACW. All the color\nbars on the left (right) side represents the SOC energy change\nat L2 (L5) for di\u000berent W compositions. (b) Layer resolved\nDMI for structures having W composition, x= 0%\u000050%.\nnetic layer) for all W compositions (0% \u0000100%). We \fnd\nthat \u0001 ESOC in L5 changes drastically as W composition\nchanges from 0% to 12 :5%, slowing down thereafter. On\nthe other hand, distributions of \u0001 ESOC in L2 are not\nvery sensitive to the W composition. Although we \fnd\na relatively lower \u0001 ESOC at L2 for 75% and 100% W\ncompositions, the corresponding \u0001 ESOCs at L5 are pos-\nitive. In trilayer structures, the DMIs of the bottom and\ntop interface are additive [12, 37], so that the sum arising\nfrom L2, and L5 accounts for the observed non-monotonic\nchange of DMI in Fig. 2. From our \fndings, we conjec-\nture that the inversion symmetry breaking plays a vital\nrole on the DMI while the e\u000bect of W composition is\nnot that prominent, in agreement with the recent exper-\niment [27].\nTo corroborate our analysis, we calculate the layer re-\nsolved DMI. Figure 3(b) shows the layer resolved contri-\nbution of the DMI for the structures with 0% \u000050% W\ncomposition. The results show that the DMI comes only\nfrom the interfacial magnetic layers. We can see that\nthe change in the DMI contribution from the top inter-\nfacial layer (L4) with increasing W is small, generating\n(a)\nTa W Ir0.01.02.03.04.05.0DMI, D [mJ/m2]Pt/GdCo/Ta\nPt/GdCo/W\nPt/GdCo/Ir\n(b)\nL1L2L3L4L5L6-1.00.01.02.03.04.05.0DMI, D [mJ/m2]Pt/GdCo/Ta\nPt/GdCo/W\nPt/GdCo/IrFigure 4. (a) Calculated DMI in Pt/GdCo/X, where X=Ta,\nW, Ir. (b) Layer resolved DMI.\na similar trend as \u0001 ESOC shown in Fig. 3(a). Addition-\nally, the contribution from the bottom interfacial layer\n(L3) remains almost the same throughout the range of\nW compositions. The addition of the DMI from the bot-\ntom and the top interfaces produces a saturation in the\noverall DMI curve.\nFinally, our theoretical model allows us to explore\nthe tuning of DMI in ferrimagnetic systems with dif-\nferent cap layer compositions, which could be critical\nin designing suitable materials for hosting ultrasmall\nhigh-speed skyrmions. Furthermore, for applications,\nskyrmions can be driven by current-induced spin-orbit\ntorques (SOT) [38]. Changing the cap layer HM o\u000bers\nthe ability to tune the SOT e\u000eciency and DMI simul-\ntaneously. We report the DMI of Pt/GdCo/X where\nX = Ta ;W;Ir, to demonstrate the e\u000bect of cap layer\n5dtransition HM on the DMI in Fig. 4(a). W and Ta\nare known for their giant spin-Hall angle [39, 40], and\nprevious studies have shown an additive DMI for a fer-\nromagnet sandwiched between Pt and Ir [37, 41], which\nguides us to explore these structures and see which one\nof them has the largest DMI. We \fnd that W in the\ncap layer favors higher DMI than Ta and Ir. To explain\nthe DMI trend, we calculate the layer resolved DMI con-\ntribution from bottom and top interfaces, as shown in\nFig. 4(b). From Fig. 4(b), we can observe that the DMI5\n-12.00.012.0DOS\"\n#(a)Co\nTa\n-12.00.012.0DOS\"\n#(b)Co\nW\n-6 -4 -2 0 2 4\nEnergy, E-EF [eV]-12.00.012.0DOS\"\n# (c)Co\nIr\nFigure 5. Projected density of states (p-DOS) showing 3 d-\n5dband alignment between Co (black) and X (colored) in\nPt/GdCo/X. (a) X=Ta, (b) X=W, and (c) X= Ir. The red\nup (down) arrow represents the spin-up (spin-down) channel.\ncontribution from the top interface (L4) is large when Ir\nis used as a cap layer material while the DMI contribu-\ntions are smaller for the cases of W and Ta. The ob-\nserved trend of the DMI can be explained qualitatively\nby the Co 3 d-X 5dband alignment, which controls the\ncorresponding orbital hybridization. Figure 5 shows the\nprojected density of states (p-DOS) of Co-3 dand HM-5 d\norbitals. Clearly, in Co/Ir, the band alignment around\nthe Fermi level is higher than that of Co/W and Co/Ta,\nwhich in turn produce larger DMI contributions from L4\nfor Ir over W and Ta. The band alignment of Co/W and\nCo/Ta are close to each other. However, we note that\nthe sign of the DMI contribution from the top interfaceis di\u000berent for Ir than Ta and W. By analyzing the or-\nbital projected densities of states of the cap layer HM,\nwe \fnd that Ta and W behave in a similar way i.e, dxy\nanddx2\u0000y2have major contributions near the Fermi level\nwhile for Ir, the orbitals associated with the zcharacters,\nnamely dxz,dyz, and dz2are prominent, correlating with\nthe behavior shown in Fig. 4(b). Moreover, the variation\nof the DMI sign depending on the adjacent HM has pre-\nviously been seen in both theoretical and experimental\nstudies [16, 42]. Finally, adding the DMI contribution\nfrom both the interfaces (Fig. 4(b)) gives a smaller over-\nall DMI for Pt/GdCo/Ir because of the large negative\ncontribution from the top interface.\nConclusion. In summary, we demonstrate the impact of\nW composition in the cap layer of Pt =GdCo =Pt1\u0000xWx\ntrilayer structures using \frst principles calculations. We\n\fnd excellent tunability of the DMI that shows a ten-\ndency of saturation with increasing W composition. The\nsaturating trend of the DMI is attributed to the change\nof SOC energy at the top and the bottom intefacial HM\nlayers as a function of W composition. Moreover, we\n\fnd DMI sensitivity to the structural variation. We also\ndemonstrate the DMI variation in Pt/GdCo/(Ta, W or\nIr). We \fnd W in the cap layer provides a higher DMI\nthan Ta and Ir, due to the varying degree of orbital hy-\nbridization controlled by the band alignment between 3 d-\n5dorbitals at the cap layer interface. Our results provide\ncritical insights to the control mechanism of DMI in ferri-\nmagnetic GdCo based systems, providing a path towards\nmanipulating skyrmion properties for spintronic applica-\ntions.\nAcknowledgments. We thank Shruba Gangopadhyay,\nJianhua Ma, Hamed Vakilitaleghani, and S. Joseph\nPoon for insightful discussions. This work is funded\nby the DARPA Topological Excitations in Electronics\n(TEE) program (grant D18AP00009). The calculations\nare done using the computational resources from High-\nPerformance Computing systems at the University of\nVirginia (Rivanna) and XSEDE.\n[1] A. Fert, N. Reyren, and V. Cros, Nat. Rev. Mater. 2, 1\n(2017).\n[2] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8,\n152 (2013).\n[3] W. Koshibae, Y. Kaneko, J. Iwasaki, M. Kawasaki,\nY. Tokura, and N. Nagaosa, Jpn. J. Appl. Phys. 54,\n053001 (2015).\n[4] X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 1 (2015).\n[5] H. Vakili, Y. Xie, and A. W. Ghosh, Phys. Rev. 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Lett. 120, 157204 (2018)."    },    {        "title": "2205.06579v1.Fast_scanning_nitrogen_vacancy_magnetometry_by_spectrum_demodulation.pdf",        "content": "Fast scanning nitrogen-vacancy magnetometry by spectrum demodulation\nP. Welter1;2;y, B. A. J\u0013 osteinsson1;2;y, S. Josephy2, A. Wittmann3,\nA. Morales2, G. Puebla-Hellmann2, and C. L. Degen1;4\n1Department of Physics, ETH Z urich, Otto Stern Weg 1, 8093 Z urich, Switzerland.\n2QZabre AG, Regina-K agi-Strasse 11, 8050 Z urich, Switzerland\n3Institute of Physics, Johannes Gutenberg Universit at Mainz, Staudingerweg 7, 55128 Mainz, Germany. and\n4Quantum Center, ETH Z urich, 8093 Z urich, Switzerland.\u0003\n(Dated: May 16, 2022)\nWe demonstrate a spectrum demodulation technique for greatly speeding up the data acquisition\nrate in scanning nitrogen-vacancy center magnetometry. Our method relies on a periodic excitation\nof the electron spin resonance by fast, wide-band frequency sweeps combined with a phase-locked\ndetection of the photo-luminescence signal. The method can be extended by a frequency feedback to\nrealize real-time tracking of the spin resonance. Fast scanning magnetometry is especially useful for\nsamples where the signal dynamic range is large, of order millitesla, like for ferro- or ferrimagnets.\nWe demonstrate our method by mapping stray \felds above the model antiferromagnet \u000b-Fe2O3\n(hematite) at pixel rates of up to 100 Hz and an image resolution exceeding one megapixel.\nI. INTRODUCTION\nThe scanning nitrogen-vacancy (NV) magnetometer\nis a next-generation scanning probe microscope able to\nquantitatively map surface magnetic stray \felds with\nsub-50-nm spatial resolution [1{4]. The technique re-\nlies on a single, optically-readable defect spin embedded\nin a sharp diamond tip that is scanned over the sam-\nple of interest. Scanning NV magnetometry exploits the\nprinciples of quantum metrology to reach very high sen-\nsitivities, leading to new opportunities in the imaging\nof weakly magnetic systems. In the recent past, scan-\nning NV magnetometry has been used to map the stray\n\feld of magnetic vortices and domain walls in ferromag-\nnets [5{8], antiferromagnets [9{13] and multiferroics [14{\n16], skyrmions [17{20], superconducting vortices [21{23],\nand two-dimensional ferromagnetism [24{26].\nIn the most commonly used detection scheme, the spin\nresonance frequency f0of the NV center is tracked us-\ning continuous-wave optically detected magnetic reso-\nnance (cw-ODMR) spectroscopy, and later converted to\nunits of magnetic \feld using the spin's gyromagnetic ratio\n(\re= 2\u0019\u000228 GHz=T) [27]. In this scheme, the microwave\nexcitation frequency is scanned slowly across the spin res-\nonance and the resulting absorption line-shape, detected\nusing a photo-luminescence (PL) measurement, \ftted to\nextract the resonance position. Although this scheme\nworks well for slow acquisition speeds, the non-linear\nleast squares \ftting of the spectrum is computationally\nexpensive and ill-suited for real-time performance past a\nfew Hz. On the other hand, the high PL of modern NV\ntips [28, 29] should allow measurements of magnetic \felds\nat rates of 100 Hz or faster while maintaining a high sen-\nsitivity below 10 \u0016T. The possibility of acquiring a scan\nin a matter of minutes rather than hours or days is en-\nticing, and would further bolster the versatility of the\n\u0003degenc@ethz.ch;yThese authors contributed equally.technique.\nSeveral concepts for speeding up image acquisition\nhave been presented in the past. These include quali-\ntative approaches that rely on PL quenching [3] or \fxed-\nfrequency excitation [2], semi-quantitative approaches\nusing multi-frequency excitation [30], resonance track-\ning [31], a posteriori \feld reconstruction [32], or combi-\nnations thereof [32]. The highest reported scan rates for\nsingle spin magnetometry are around 40 pixels =s in imag-\ning [32] and 100 samples =s in stationary benchmarks [31].\nFor ensemble NV sensing, real-time \feld tracking up to\nseveral hundred Hz has been reported [33].\nIn this work, we present a signal demodulation method\nthat easily scales to sample rates of 100 Hz and beyond,\nyet is directly quantitative without the need for post-\nprocessing, and that can tolerate sudden jumps of the\nmagnetic \feld. Our method is based on periodic excita-\ntion of the spin resonance by fast, wide-band frequency\nsweeps and spectral demodulation of the resulting PL\nsignal. Real-time feedback can optionally be included to\nincrease the dynamic range. We demonstrate our tech-\nnique by imaging the magnetic surface texture and do-\nmain structure of an antiferromagnetic thin \flm at pixel\nrates of up to 100 Hz.\nII. TRADITIONAL RESONANCE DETECTION\nThe canonical detection method in ODMR involves\nslowly sweeping the microwave excitation frequency\nacross the spin resonance and monitoring the optical PL\nemission [34]. By \ftting of the resonance curve with an\nappropriate line shape, most often Lorentzian,\nR(f) =R0\"\n1\u0000\u000f\u0012\n1 +[f\u0000f0]2\n\u00002\u0013\u00001#\n; (1)\nthe resonance frequency f0as well as other parameters in-\ncluding the resonance linewidth \u0000, spin contrast \u000fand PL\nemission rate R0can be extracted (Fig. 1(a)). To avoidarXiv:2205.06579v1  [quant-ph]  13 May 20222\n(a) (b)\nMicrowave frequency f\nδφ∆fwin Tmodfc\nPL signal2Γ\nϵR0R0\nf0δf0ΔRPL signal\nMicrowave frequency f Time t\nFigure 1. (a) In traditional ODMR spectroscopy, the reso-\nnance frequency f0is estimated by \ftting the spectral peak.\nChanges in magnetic \feld shift the position of the spectrum\n(dashed line). R0is the intensity of photo-luminescence (PL)\nemission,\u000fthe spin contrast, and \u0000 the linewidth parameter of\nthe absorptive line shape. (b) In the spectrum demodulation\ntechnique, the spin resonance is excited periodically by fast\nmicrowave frequency sweeps (black saw-tooth curve). The\nfrequency modulation window \u0001 fwinis chosen larger than \u0000.\nA change in the spin resonance by \u000ef0then causes a phase\nshift\u000e\u001e= 2\u0019\u000ef0=\u0001fwinof the PL signal compared to the mi-\ncrowave drive (dashed line). The phase shift is measured by\ndemodulating the PL signal, e.g., by a lock-in ampli\fer.\nthe computationally expensive \ftting, the change in reso-\nnance frequency f0can also be detected by observing the\nchange in amplitude \u0001 Rat a single frequency f(blue dots\nin Fig. 1(a)). This `amplitude detection' can be extended\nto a few discrete frequency values to increase robustness\n[35]. Alternatively, a small (sub-linewidth) modulation\nof the microwave frequency or bias \feld can be applied\nto record a di\u000berential line shape or to frequency-lock\nto the resonance [31, 36]. The above procedures work\nwell for analyzing spectra at a slow rate or for detecting\nand tracking small \u000ef0<\u0000 changes in the resonance fre-\nquency. However, they are ill-suited for real-time track-\ning of large spectral shifts.\nIII. SPECTRUM DEMODULATION\nIn our spectrum demodulation method, the microwave\ndrivef(t) is swept quickly across a wide frequency win-\ndow much larger than the resonance linewidth using a\nsaw-tooth frequency modulation. The modulation rate\nfmodis chosen much faster than the intended integration\ntime per spectrum, yet much slower than the absorp-\ntion and re-polarization rates of the spin. The recorded\nPL signal as a function of time R(t) is then a peri-\nodic concatenation of (truncated) resonance line shapes\n(Fig. 1(b)). By measuring the relative phase between thesawtooth drive f(t) and the PL signal R(t),\n\u001e=2\u0019(f0\u0000fc)\n\u0001fwin; (2)\none can directly determine the frequency o\u000bset f0\u0000fc.\nHere,fcis the center frequency and \u0006\u0001fwin=2 the fre-\nquency span of the sawtooth modulation. Importantly,\nthe phase\u001eis insensitive to the detailed line shape of the\nresonance.\nTo experimentally determine \u001e, we demodulate R(t)\nat the modulation frequency fmod(for example, using a\nlock-in ampli\fer) and compute the argument of the in-\nphase and quadrature channels,\na1:=hR(t)\u0001e2\u0019ifmodti; (3)\n\u001e= arg (a1) = arctan( Y=X ); (4)\nwhere the angled brackets represent an average or a low-\npass \flter to reject the image at 2 fmod.X= Re(a1) and\nY= Im(a1) are the in-phase and quadrature parts of the\ncomplex signal a1, respectively. The desired frequency\nf0then follows from Eq. (2),\nf0=fc+\u001e\u0001fwin\n2\u0019; (5)\nwhere the phase \u001e2[\u0000\u0019;\u0019[.\nThe demodulation can be extended to higher harmon-\nicsnfmodofR(f), yielding a series of coe\u000ecients an. A\nshift of the resonance frequency results in phase shifts of\nn\u001efor the harmonic of order n. The expected amplitudes\nof higher harmonics are generally decreasing exponen-\ntially, therefore, phase measurements of the higher har-\nmonics are increasingly noisy. Nevertheless, the higher\nharmonics can be included in the analysis to obtain an\nimproved estimate for f0(Appendices A and B). Fur-\nthermore, including harmonics up to second order allows\nextracting estimates for the resonance parameters \u0000, \u000f\nandR0. For a Lorentzian line shape, these are given by\n(see Appendix A):\n\u0000\u0019\u0001fwin\n2\u0019ln\f\f\f\fa1\na2\f\f\f\f; (6)\n\u000f\u0019\u0001fwin\n\u0019\u0000e2\u0019\u0000=\u0001fwin\f\f\f\fa1\na0\f\f\f\f; (7)\nR0\u0019a0: (8)\nA more advanced analysis (Appendix B) also exploits\nphase information from the higher order coe\u000ecients to\nimprove the estimation of both the resonance frequency\nand the other parameters. However, in practice, Eqs. (6-\n8) already provide decent estimates for the resonance pa-\nrameters.\nIV. IMPLEMENTATION\nA variety of ways may be devised to implement a spec-\ntrum demodulator. The key elements of the system are3\nSynthesizer\nAWGLaser\nfLO(t)\nfIF(t)f (t) Electronic\nspin\nPhotodetector\nLock-in\namplifier\nData processing\nunitfmod\na1 a0, a2, a3, ...\nf0 Γ ϵ R0R(t)syncfeedback (optional)fc\nSpectrum demodulatorf0 = fc +          arg(a1)Δfwin\n2\nFigure 2. Block diagram of the spectrum demodulator. The\ncomplete system consists of conventional NV magnetometer\nsetup to which a demodulator is added (dashed blue box).\nThe demodulator can be implemented in a variety of ways\nand always includes a lock-in ampli\fer and data processing\nunit. The lock-in ampli\fer computes the demodulated signals\nan, each representing the complex amplitude at the harmonic\nfrequencies nfmod. The data processing unit computes an es-\ntimate of the NV resonance frequency f0(and possibly other\nparameters including R0,\u000f, \u0000) from the signals an. Data pro-\ncessing can be a simple arithmetic operation [Eqs. (4,5)], or\ncan use advanced methods such as Kalman or particle \flters\nfor improved sensitivity. Resonance tracking may optionally\nbe implemented by feeding back the estimate of the resonance\nfrequencyf0to dynamically adjust the center frequency fcof\nthe microwave modulation. In that case the spectrum de-\nmodulator together with the microwave synthesizer form a\nphase-locked loop (PLL) that locks onto the NV resonance\nfrequency.\nshown in Fig. 2 and include: (i) the demodulator itself,\n(ii) a data processing unit for extracting the resonance\nparameters and (iii) optionally, a feedback to enable reso-\nnance tracking. In our system, the frequency modulation\nis generated on an arbitrary waveform generator, photo-\ndetection achieved via a single-photon avalanche photo-\ndiode (APD) and digital counter card, and all demodu-\nlation tasks are performed in software (see Section VII).\nMany other implementations can be considered: A\nfully analog system may combine a linear avalanche\nphoto-diode (or a Geiger-mode APD with a down-stream\nlow-pass \flter) with a lock-in ampli\fer and a PID con-\ntroller. Conversely, a fully digital version may use a\nmicro-processor to perform demodulation, signal extrac-\ntion and tracking in a single unit. Likewise, frequency\nmodulation may be realized digitally (by direct digital\nsynthesis) or fully analog (via a voltage-controlled oscil-\nlator).V. SENSITIVITY\nWe next analyze the sensitivity of the spectrum de-\nmodulation technique and compare it to the conventional\nmethods. The main source of noise in the optical detec-\ntion system is photon shot noise. For low spin contrast\n\u000f\u001c1, which is a good approximation for NV centers,\nthe noise is Poissonian and white and the power spectral\ndensity is simply given by S=R0[37]. Assuming a signal\nintegration time of tint, the equivalent noise bandwidth of\nthe \flter from Eq. (3) is 2 =tint, and the variance of the de-\nmodulated signal is \u001b2=R0=tint. This variance is evenly\ndistributed over both quadratures, \u001b2\nX=\u001b2\nY=R0=2tint.\nNext, we use Eq. (4) and Eq. (5) to convert the un-\ncertainties in XandYinto an uncertainty \u000ef0of the\nestimated frequency shift (see Appendix A for deriva-\ntion). The sensitivity \u0011, de\fned as the uncertainty \u000ef0\nnormalized to unit time, is then given by:\n\u0011=\u000ef0ptint\u00192\u0000\n\u000fpR0\u0002\u000b2e\u0019=\u000b\np\n2\u00192: (9)\nHere, we introduce the relative window size \u000b=\n\u0001fwin=(2\u0000) as the ratio between \u0001 fwinand 2\u0000, and as-\nsume that\u000b&1.\nIt is instructive to compare Eq. (9) to the optimum\nsensitivity \fgure for amplitude detection (Fig. 1(a)) [38],\n\u0011\u00192\u0000\n\u000fpR0\u00020:77 (10)\nand to that of a least squares \ft (Appendix C),\n\u0011=2\u0000\n\u000fpR0\u0002p\n2\u000b=\u0019 (11)\nClearly, for small \u000b!1, the sensitivity of all techniques\nis similar. This is not surprising, because most frequency\npoints lie in the vicinity of the resonance and contain use-\nful information. Conversely, for \u000b\u001d1, the sensitivity\nrapidly (\u0011/\u000b2) deteriorates for our spectrum demodu-\nlation technique because the signal power is increasingly\ndistributed over higher harmonic coe\u000ecients an. In prin-\nciple, the\u0011/p\u000bbehavior of least-squares \ftting can be\nrecovered by including the anin the analysis, however,\nthis comes at the cost of increased analytical complex-\nity (Appendix B). Overall, the window size \u000bis an im-\nportant parameter in our spectrum demodulation tech-\nnique, because it provides us with a knob to balance be-\ntween a large signal range (large \u000b) and a high sensitivity\n(small\u0011).\nVI. TRACKING\nTo combine a high sensitivity with a large signal range,\nit is useful to include a tracking method that dynamically\nre-centers the frequency modulation window to the reso-\nnance position f0, e\u000bectively forming a phase-locked loop4\n(a)\n(b)\n110100100110100100\nFigure 3. (a) Experimental standard deviation of the demod-\nulator frequency output f0computed from 200 data points\nwith no magnetic \feld modulation applied, plotted as a func-\ntion integration time tintand window size \u0001 fwin. The solid\nlines act as guides to the eye, illustrating that the uncertainty\nscales ast\u00001=2\nint [Eq. (9)]. Right scale gives conversion to units\nof magnetic \feld. (b) Experimental sensitivity vs. relative\nwindow size \u000b= \u0001fwin=(2\u0000). The solid gray curve is the the-\nory scaling [Eq. (9)]. The dashed gray curve is the theory\nscaling for least-squares \ftting [Eq. (11)]. The dotted gray\ncurve is the theoretical limit for amplitude detection at the\npoint of the steepest slope [Eq. (10)]. In this experiment,\nR0\u0019500 kCt=s,\u000f\u001915%, and 2\u0000 \u001910 MHz.\n(PLL). At its simplest, we adjust the center frequency fc\nof the microwave drive to the last estimate for f0after ev-\nery time step. This method is not very robust, as a single\nnoisy measurement can throw o\u000b the entire tracking, but\nwe \fnd it to be satisfactory in most of our experiments.\nBetter approaches, not implemented here, would consider\nmore than just the latest measurement. One way is to\nimplement a carefully tuned, higher-order PLL loop \fl-\nter. Another, digital approach is to make this \flter itself\nadaptive, by use of recursive maximum-likelihood esti-\nmators that optimally consider previous time steps [39].\nUltimately, the choice of window size \u0001 fwinis a trade-\no\u000b between signal-to-noise ratio (SNR) and tracking\nspeed. The maximum frequency step allowed between\ntwo samples is given by \u0001 fwin=2. This implies a maxi-\n(a)\n(b)Figure 4. Real-time monitoring of magnetic \feld gener-\nated by passing a current through a coil underneath the NV\nprobe. (a) With resonance tracking disabled. (b) With reso-\nnance tracking enabled. The waveform starts at t= 0 s, and\nconsists of two superimposed tones (0 :8 Hz and 4 Hz) with\nequal amplitude. Detection parameters are R0\u0019950 kCt=s,\n\u000f\u001918%, 2\u0000 \u001912 MHz, and \u0001 fwin= 30 MHz.\nmum tracking rate (slew rate) for frequency jumps of:\nSR =\u0001fwin\n2tint: (12)\nUsing parameters typical for the experiments presented\nbelow (\u0001fwin= 30 MHz, tint= 10 ms), the slew rate is\napproximately SR = 1 :5 GHz=s corresponding to a mag-\nnetic slew rate of2\u0019\n\reSR\u001850 mT=s.\nVII. EXPERIMENTAL RESULTS\nWe experimentally demonstrate our spectrum demod-\nulation technique using a commercial scanning magne-\ntometer instrument (QSM, QZabre). The scanning mag-\nnetometer is equipped with an arbitrary waveform gen-\nerator and local oscillator (LO) for microwave control,\nand a single photon counting module for optical detec-\ntion. We implement the frequency modulation by gen-\nerating microwave chirp pulses at fmod= 1 kHz around\na 100 MHz baseband frequency, and mix it with the LO\nto the desired 2\u00004 GHz \fnal frequency centered at fc.\nWe use an avalanche photo diode and a data acquisition\ncard to count and bin the photons at a 20 \u0016s dwell, cor-\nresponding to a data rate of 50 ;000 points per second.\nWe demodulate the time trace in software by computing\na fast Fourier transform of segments of duration tintand\nretaining the coe\u000ecients corresponding to nfmod. We\nthen extract the resonance frequency f0from the phase5\n(a)\n5 μm\nMagnetic field (mT)2.52.62.72.82.93.0\n(b) (c) (d) (e) (f)2Hz, 1h 28min 10Hz, 18min 20Hz, 8min 45s 50Hz,  4min 11 s\n 100Hz, 1min 47s\nFigure 5. Magnetometry scans on \u000b-Fe2O3\flms using the spectrum demodulation technique. (a) Large-area scan (27 \u0016m\u0002\n15:2\u0016m) containing 1.03 million pixels. Pixel rate is 100 Hz and spacing is 20 nm. Total scan time is 3 h 5 min. The diagonal\nstripe pattern is likely due to steps in the surface topography [40] while the large-scale structure re\rects the magnetic domains.\n(b-f) 100 \u0002100 pixel scans recorded at varying rates, 2 to 100 Hz, in the region corresponding to the dashed rectangle in\npanel (a). Titles indicate the scan rate and the total acquisition time. Pixel spacing is 50 nm. Experimental parameters are\nR0\u0019400 KCt=s,\u000f\u001920%, 2\u0000 \u001911 MHz, and \u0001 fwin= 30 MHz. A small bias \feld of ca. 2 :75 mT is applied along the NV\nsymmetry axis.\nof the \frst Fourier coe\u000ecient as per Eq. (5). Frequency\ntracking, when enabled, is implemented by updating the\nLO frequency according to the previously measured reso-\nnance frequency f0. In our current implementation, there\nis substantial latency associated with this process (up to\n10 ms), and in tracking mode we limit the maximum sam-\nple rate to 50 Hz. A future implementation will reduce\nthis bottleneck by using an LO with lower latency.\nWe start experiments by assessing the sensitivity of\nthe method. For this purpose, we record the resonance\nfrequencyf0at a constant rate given by tintfor a total\nof 200 samples without applying a magnetic signal. We\nthen plot the standard deviation of the data record as a\nfunction of tint. Fig. 3(a) shows the measured standard\ndeviation for four window sizes \u0001 fwin. In all cases, thestandard deviation scales with the inverse square root of\ntint. This is expected from Eq. (A13) and con\frms that\nthe measurement is limited by shot noise. In Fig. 3(b),\nwe plot the sensitivity \u0011as a function of window size\n\u000b= \u0001fwin=(2\u0000). The experimental data matches the sen-\nsitivity model from Eq. (9) exceptionally well. Fig. 3(b)\nalso shows theory curves for least-squares \ftting given by\nEq. (11) as well as the lower bound imposed by Eq. (10).\nTo test the dynamic performance, we hover the scan-\nning probe above a small copper coil ( \u001f5 mm) and feed a\ncurrent waveform through the coil. Fig. 4 presents traces\nof the resulting coil magnetic \feld recorded at a rate of\n50 Hz. For reference, we also show the applied wave-\nform on a matching scale (orange trace). Fig. 4(a) shows\na trace recorded without tracking. Here, the dynamic6\nrange is limited to \u00061\n2\re\u0001fwin=\u00060:53 mT by the chosen\nwindow size of \u0001 fwin= 30 MHz. Signals exceeding this\nrange cannot be detected (not shown). Fig. 4(b) shows\na corresponding trace with the tracking enabled. The\nsignal range is now much larger while the SNR is only\nmarginally reduced (due to feedback latency). In both\n\fgures, gray bars indicates the instantaneous tracking\nwindow \u0001fwin.\nWhether or not the tracking should be enabled de-\npends on the expected signal magnitude. If the expected\nsignal dynamic range is small, less than approximately\n\u000630 MHz (equal to approximately \u00061 mT, i.e. 2 mT\npeak-to-peak), tracking is not necessary. The linewidth\ncan be arti\fcially broadened (or narrowed) by increas-\ning (decreasing) the microwave power, such as to remain\nclose to an optimum \u000b\u00193. By contrast, if the expected\nsignal is strong ( &\u000630 MHz), tracking is recommended.\nFinally, Fig. 5 shows images of the stray \feld above a\nmagnetic thin \flm obtained by scanning magnetometry.\nThe sample is a 10 nm \flm of \u000b-Fe2O3(hematite) grown\nepitaxially on an Al 2O3(001) substrate by o\u000b-axis mag-\nnetron sputtering capped with a 5 nm layer of Pt and a\n2 nm layer of amorphous carbon. At room temperature,\n\u000b-Fe2O3exhibits weak ferromagnetism due to the canting\nof the antiferromagnetically coupled magnetic sublattices\nwithin the easy plane. The average domain size is on the\norder of 1\u0016m making it a suitable materials system for\nour demonstration [41]. The large image (Fig. 5(a)) is\n1350\u0002760 pixels, recorded at rate of 100 Hz resulting in\na total measurement time of 3 h 5 min. Despite the fast\nacquisition rate and the fairly weak signal ( \u0018500\u0016T\npeak-to-peak), the image shows exceptional detail and a\nhigh SNR. Figs. 5(b-f) show a 100 \u0002100 pixel sub-section\nof the image recorded at di\u000berent rates 2 to 100 Hz. Al-\nthough some loss in SNR becomes visible at high rates,\nfeatures are well resolved in all images while the total ac-\nquisition time is dramatically reduced from 1 h 28 min to\nbelow 2 min. For comparison, recording the large image\nat 2 Hz would have resulted in a scan time of approxi-\nmately one week.\nVIII. OUTLOOK\nIn summary, we have introduced a technique for quan-\ntitative spin resonance frequency estimation that scales\nto at least 100 measurements per second. Our method\nrelies on a rapid, large-bandwidth frequency modulation\nof the microwave excitation and demodulation of the re-\nsulting PL signal. Various trade-o\u000bs between accuracy\n(sensitivity) and speed (simplicity) are discussed. We\ndemonstrate fast scanning magnetometry by imaging the\nsurface stray \felds of a thin-\flm antiferromagnet at rates\nof up to 100 Hz and sizes of up to one megapixel.\nLooking forward, the spectrum demodulation tech-\nnique can be further improved in several directions. A\nsimple extrapolation (Appendix D) indicates that the up-per limit to the pixel rate is above 1 kHz for our experi-\nmental parameters, based on the available SNR. In non-\ntracking mode, this speed is in principle accessible with\nour instrumentation. In tracking mode, communication\noverheads currently limit the maximum rate to approx-\nimately 50 Hz. New hardware with reduced latency will\nlikely improve this limit to well beyond 100 Hz. More\nadvanced data processing and feedback techniques, such\nas the Kalman or particle \flters, should further increase\nsensitivity and robustness and allow for even faster rates.\nAlso, data can be post-processed to optimize the SNR af-\nter a scan has completed.\nAnother interesting future avenue is the implementa-\ntion of gradiometry imaging. Recent work has demon-\nstrated spectacular improvements to sensitivity and im-\nage quality in scanning experiments by detecting the\nmagnetic \feld gradient [40]. Gradiometry relies on a me-\nchanical oscillation of the sensor above the sample sur-\nface, which up-converts the local gradient into a time-\nvarying \feld set by the oscillation frequency. While the\noriginal implementation relied on pulsed AC quantum\nsensing techniques, the concept can also be exploited in\nthe context of the spectrum demodulation. Assuming the\nmechanical oscillation frequency is fTF, magnetic \feld\ngradients will lead to signal sidebands at fTF\u0006nfmod\nin addition to the original signal at nfmod. For typical\ntuning fork oscillators, fTF\u001832 kHz is much larger than\nfmod= 1 kHz, therefore, signals are spectrally well sep-\narated. The gradient signal (and if desired, higher-order\nderivatives around multiples of fTF) can be demodulated\nin exactly the same way as the standard demodulation\ntechnique (see Appendix E for details). Measuring the\ngradient in addition to the direct \feld takes no extra\nmeasurement time, the only resource consumed is addi-\ntional computation time. In the particular case where\nthe scanning probe oscillates in the direction of the fast\nscanning axis, an estimate of the gradient (even if noisy)\nneatly integrates with a recursive estimator, improving\nthe dynamic prediction of the \feld at the next pixel.\nACKNOWLEDGMENTS\nThe authors thank Marius Palm, Nils Prumbaum and\nGeo\u000brey Beach for discussions and support, and Larry\nScipioni, Adam Shepard, Ty Newhouse-Illige, and James\nA Greer at PVD Products, Wilmington, Massachusetts\n01887, USA for growing the \u000b-Fe2O3thin \flm. This\nwork was supported by the Swiss National Science Foun-\ndation (SNSF), Grant No. 200020 175600, by the Na-\ntional Center of Competence in Research in Quantum\nScience and Technology (NCCR QSIT) of the SNSF,\nGrant No. 51NF40-185902, by Innosuisse Grant 43106.1\nIP-ENG, and by the Advancing Science and TEchnology\nthRough dIamond Quantum Sensing (ASTERIQS) pro-\ngram, Grant No. 820394, of the European Commission.7\n[1] C. L. 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Jacques, Avoiding power broadening\nin optically detected magnetic resonance of single NV\ndefects for enhanced dc magnetic \feld sensitivity, Phys.\nRev. B 84, 195204 (2011).\n[39] N. Moriya, Primer to Kalman \fltering : a physicist's per-\nspective , Engineering Tools, Techniques and Tables (Nova\nScience Publishers, Hauppauge, New York, 2010).\n[40] W. S. Huxter, M. L. Palm, M. L. Davis, P. Welter,\nC. H. Lambert, M. Trassin, and C. L. Degen, Scanning\ngradiometry with a single spin quantum magnetometer,\narXiv:2202.09130 (2022).\n[41] F. P. Chmiel, N. W. Price, R. D. Johnson, A. D. Lami-\nrand, J. Schad, G. V. der laan, D. T. Harris, J. Irwin,\nM. S. Rzchowski, C. Eom, and P. G. Radaelli, Obser-\nvation of magnetic vortex pairs at room temperature in\na planar\u000b-Fe2O3/Co heterostructure, Nature Materials\n17, 581 (2018).Appendix A: Derivation of sensitivity\nWe aim to compute the sensitivity to frequency shifts\nbased on a single coe\u000ecient a1. We start by deriving an\napproximate analytical expression for a1. Assume zero\nfrequency shift ( fc=f0). We de\fne the sweep rate\nv=fmod\u0001fwinand the sweep period T= 1=fmod. Over\na single modulation period jtj< T= 2, the luminescence\ntime trace is simply the Lorentzian line shape\nL(t) :=R(vt+fc) =R0(1\u0000\u000f[1 + (vt)2=\u00002]\u00001);(A1)\nas per Eq. (1). We can use this expression to describe\nthe luminescence time trace over multiple periods,\nR(t) = 1X\nn=\u00001\u000e(t\u0000nT)!\n\u0003\u0010\nL(t) rect(t=T)\u0011\n;(A2)\nwhere rect( t=T) is the rectangular function which trun-\ncates the Lorentzian. The convolution ( \u0003) with a delta\ncomb then creates the periodic PL signal.\nWe next compute the Fourier transform of Eq. (A2),\nF[R(t)] (f) = \n1\nT1X\nk=\u00001\u000e\u0012\nf\u0000k\nT\u0013!\n\u0001\u0010\nF[L] (f)\u0003(Tsinc(Tf))\u0011\n(A3)\nHere, we apply that multiplication and convolution are\ndual operations under a Fourier transform, that the\nFourier transform of a comb is another comb, and\nthat the Fourier transform of rect( t) is a sinc function,\nsinc(x) = sin(\u0019x)=(\u0019x). The Fourier transform of the\nLorentzian is a double-sided exponential,\nF[L] (f) =R0\u000e(f)\u0000\u0019R0\u000f\u0000\nve\u00002\u0019\u0000jfj=v: (A4)\nNote that we use the following convention for de\fning\nthe Fourier transform:\nF[g](f) :=Z1\n\u00001g(t)e\u00002\u0019iftdt: (A5)\nIn Eq. (A3), for frequencies no larger than a small mul-\ntiple offmod, the sinc function is much sharper than the\ndecaying exponential in Eq. (A4), and we approximate\nTsinc(Tf)\u0019\u000e(f): (A6)\nThis approximation is equivalent to neglecting the trun-\ncation, noting that (i) the Lorentzian has largely decayed\ntowards the side of the window, and (ii) the truncation\ncreates a discontinuity that mostly contains higher fre-\nquencies (\u001dfmod) that are rejected by the demodula-\ntion.\nCombining Eqs. (A3), (A4) and (A6), we \fnd\nF[R(t)] (f) = 1X\nk=\u00001\u000e\u0012\nf\u0000k\nT\u0013!\n\u0001\u0010R0\nT\u000e(f)\u0000\u0019R0\u000f\u0000\n\u0001fwine\u00002\u0019\u0000jfj=v\u0011\n;(A7)9\nHere, the multiplication with the comb discretizes the\nspectrum of R(t), as expected from its periodicity. In-\ndeed, we can represent F[R(t)] (f) as a Fourier series:\nF[R(t)] (f)!=1X\nk=\u00001ak\u000e(f\u0000k=T): (A8)\nBy comparison of coe\u000ecients, we \fnd\na0=R0\u0000\u0019R0\u000f\u0000\n\u0001fwin\u0019R0 (A9)\na1\u0019\u0019R0\u000f\u0000\n\u0001fwine\u00002\u0019\u0000fmod\nv=\u0019R0\u000f\u0000\n\u0001fwine\u00002\u0019\u0000=\u0001fwin(A10)\na2\u0019\u0019R0\u000f\u0000\n\u0001fwine\u00004\u0019\u0000=\u0001fwin: (A11)\nAs expected, a1is purely real in the absence of a shift\noff0. Otherwise, by Eq. (2), a1is rotated in the com-\nplex plane by an angle \u001e. Since we know the magnitude\nja1jfrom Eq. (A10), as well as its variance from shot\nnoise (\u001b2\nX=\u001b2\nY=R0=2tint, see main text), it is straight-\nforward to compute the uncertainty in its phase, using\nstandard error propagation,\n\u000e\u001e=p\nR0=2tint\nja1j: (A12)\nEquivalently, the uncertainty in estimated resonance fre-\nquency ( cf.Eq. (5)) is given by\n\u000ef0=\u000e\u001e\u0001fwin\n2\u0019=2\u0000\n\u000fpR0\u000b2e\u0019=\u000b\np\n2\u00192(tint)\u00001=2:(A13)\nIn the last step we have introduced the relative window\nsize\u000b= \u0001fwin=(2\u0000). The sensitivity \u0011, de\fned as the\nuncertainty in f0normalized to unit time, is given by\n\u0011=\u000ef0\u0001ptint; (A14)\nimmediately yielding Eq. (9).\nSolving Eqs. (A9) to (A11) for \u000f, \u0000 andR0yields\nEqs. (6) to (8):\na0\u0019R0; (A15)\nja1=a2j= exp (2\u0019\u0000=\u0001fwin)\n)\u0000 =\u0001fwin\n2\u0019lnja1=a2j; (A16)\nja1=a0j\u0019\u0019\u000f\u0000\n\u0001fwine\u00002\u0019\u0000=\u0001fwin=\u0019\u000f\n2\u000be\u0000\u0019=\u000b\n)\u000f=2\u000b\n\u0019e\u0019=\u000b\f\f\f\fa1\na0\f\f\f\f: (A17)Appendix B: Higher-order coe\u000ecients\nThe phase of a1is a straightforward way to estimate\nthe resonance frequency. But we can extend our analysis\nto include higher harmonics ( nfmod), allowing us to po-\ntentially improve sensitivity and extract information on\n\u0000,\u000fandR0at the same time. In the following, we will\nmove from a continuous-time picture to discrete time,\nand assume we have sampled R(t) on a regular grid tk,\nwith a sampling time \u0001 t=tint=Nand number of sam-\nplesN. The harmonic coe\u000ecients up to n= 2 are given\nby:\na0=1\nNX\nkR(tk) (B1)\na1=1\nNX\nkR(tk)e2\u0019ifmodtk(B2)\na2=1\nNX\nkR(tk)e2\u0019i2fmodtk(B3)\n:::\nNote how we also include the DC value a0. Eq. (B2) is\nthe discrete-time analogue of Eq. (3).\nA shift of the resonance frequency that results in a\nphase shift of a1by\u001ewill shifta2by 2\u001e, and so on.\nThe expected amplitudes of higher harmonics are gen-\nerally decreasing exponentially (Appendix A), so mea-\nsurements of the phase of the higher harmonics are in-\ncreasingly noisy. Indeed, all coe\u000ecients have the same\nvariance\u001b2=R0=tint, and, by the central limit theorem,\ntheir distribution is closely normal.\nAll of the coe\u000ecients ajcarry usable information about\nall of the parameters. For instance, Eqs. (6){(8) clearly\ndiscard the phase information in a2. The challenge is to\n\fnd a suitable way to combine all of these measurements\n(real and imaginary part of each aj) into a single estimate\nof all the parameters. Because all ajsatisfy the condition\nof being normally distributed with equal variances, the\nmaximum-likelihood estimate is in fact the one produced\nby least-squares optimization. That is, the optimum es-\ntimate is:\n^f0;^\u000f;^\u0000;^R0= argminX\njj~aj\u0000ajj2; (B4)\nwhere ~aj= ~aj(f0;\u000f;\u0000;R0) represent the expected coe\u000e-\ncients based on a model of the line shape, cf.Eqs. (A9){\n(A11).\nWith the only di\u000berence that the raw data are \frst\nFourier transformed, Eq. (B4) is equivalent to directly\n\ftting the Lorentzian spectrum, a method we earlier dis-\nmissed as too computationally expensive. The key ad-\nvantage here is that we can truncate the sum in Eq. (B4)\nat perhaps n= 3, as the remaining ajare small and\nthe information they carry is minimal. But indeed there\nis a trade-o\u000b to be made here, between computational\ncomplexity and sensitivity.10\nAppendix C: Derivation of sensitivity of\nleast-squares \ftting\nNext, we aim to compare the uncertainty above in\nEq. (A13) to the standard method of least-squares \ft-\nting the Lorentzian spectrum directly. A complete calcu-\nlation of the error propagation through a full nonlinear\nleast-squares \ftting operation is intractable and unlikely\nto yield any tangible insights. Instead, to simplify the\ncalculations, we relax the requirements by making the\nfollowing assumptions:\n•We only \ft the resonance frequency f0, and assume\nthat the linewidth \u0000, contrast \u000fand count rate R0\nare known exactly.\n•We further assume that we already have a decent\nestimate of f0, and we perform a single linear least-\nsquares step starting from the true value f0.\n•Likewise, the window shall be well-centered on the\nresonance, i.e.fc\u0019f0.\nThe model function which we \ft to our data hence is\nRf0(f) =R0(1\u0000\u000f[1 + (f\u0000f0)2=\u00002]\u00001): (C1)\nOur measurement is a noisy sample of this function:\nyk=Rf0(fk)tint\nN+wk; k = 1;:::;N: (C2)\nHere, we sample the spectrum at the frequencies fk. The\ntotal integration time tintis spread across all Npoints.\nThe random variable wk\u0018N(0;\u001b2=R0tint=N) captures\nthe shot noise.\nThe Jacobian of the least-squares problem reads:\nJk=@Rf0(fk)\n@f0tint\nN: (C3)\nNote that because we are only \ftting a single parameter\nf0, the Jacobian is simply a row vector. Let ^f0be the\nleast-squares estimate of the resonance frequency. Its\nvariance is given by:\nVh\n^f0i\n=\u001b2\u0000\nJTJ\u0001\u00001(C4)\n=\u001b2 NX\nk=1J2\nk!\u00001\n(C5)\n=\u001b2 NX\nk=1\u0012@Rf0(fk)\n@f0tint\nN\u00132!\u00001\n(C6)\n\u0019\u001b2 Zfc+\u000b\u0000\nfc\u0000\u000b\u0000\u0012@Rf0(f)\n@f0tint\nN\u00132Ndf\n\u0001fwin!\u00001\n:\n(C7)\nThe integral runs over the frequency window. The ap-\nproximation in the last step is valid when the frequency\npoints are many ( N\u001d1) and equally spaced.Inserting Eq. (C1) into (C7), the integral can be solved\nanalytically, \fnally yielding\n\u000ef0=r\nVh\n^f0i\n=2\u0000\n\u000fpR0tint\u0001vuut\u000b\n\u000b4+8\n3\u0001\u000b3\u0000\u000b\n(1+\u000b2)3+ arctan\u000b:\n(C8)\nwhere\u000bis the relative window size. This expression is\nminimized for \u000b\u00191. In that case, the square root eval-\nuates to approximately unity as well. The ultimate sen-\nsitivity with least-squares \ftting is thus:\n\u0011opt\nlstsq=\u000ef0\u0001ptint\u00192\u0000\n\u000fpR0; (C9)\nsimilar to the optimum single-point sensitivity of ampli-\ntude detection, see Eq. (10) in the main text. Note that in\npractice, this sensitivity is out of reach by a small factor.\nThe reason is that resonance frequency is notthe only\nquantity that must be estimated from the data. Robust\nestimation of also the contrast and the linewidth requires\nthat the window is large enough to also capture the tails\nof the resonance ( \u000b&2).\nIf the window is much larger than the linewidth ( \u000b\u001d\n1, as required by wide-dynamic-range measurements),\nthe square root in Eq. (C8) is \u0018p\n2\u000b=\u0019, meaning that\nthe sensitivity is given by\n\u0011lstsq\u00192\u0000\n\u000fpR0\u0001p\n2\u000b=\u0019: (C10)\nThis expression scales likep\u000b, which is intuitively plau-\nsible as only a fraction of the total integration time, of\nordertint=\u000b, is spent sampling the spectrum near the\nactual resonance line.\nAppendix D: Maximum rate\nWe estimate an upper bound for the maximum track-\ning rate solely limited by the SNR. To get such a bound,\nwe determine the integration tintwhere the uncertainty in\n\u000ef0(according to Eq. A13) becomes equal to the estimate\n\u000efrand\n0one would obtain if the phase were completely ran-\ndom over [\u0000\u0019;\u0019[. De\fning \u000efrand\n0 by the square root of\nits variance,\n\u000efrand\n0=\u0001fwin\n2\u0019\u00141\n2\u0019Z\u0019\n\u0000\u0019d\u001e\u001e2\u00151=2\n=\u0001fwinp\n3;(D1)\nsetting\u000ef0=\u000efrand\n0, and solving for t\u00001\nintwe \fnd\nt\u00001\nint=\u000f2R0\u00194\n6\u000b2exp\u00002\u0019\n\u000b\u0001: (D2)\nAssuming numbers typical for our experiments ( \u000f= 0:2,\nR0= 5\u0001105s\u00001, \u0001fwin= 30 MHz, \u0000 = 5 MHz), we \fnd\nt\u00001\nint= 4:4 kHz.11\nOn the other hand, the NV center has a response time\nof\u00181\u0016s [31]. For accurate sensing, the sweep period\nmust be much larger than this, e.g.tint&100\u0016s. This\nequally imposes a limit on the maximum sampling of the\nsame order of magnitude.\nAppendix E: Demodulation of magnetic \feld\ngradient\nIn the following, we show how demodulating the lumi-\nnescence signal at fTF\u0006nfmodgives access to the mag-\nnetic \feld gradient along the cantilever oscillation axis.\nIn the presence of a magnetic gradient, the probe expe-\nriences an additional AC \feld, B(t) =B1cos(2\u0019fTFt),\nwithB1=x0@B\n@x,x0andfTFbeing the cantilever os-\ncillation amplitude and frequency, respectively,@B\n@xthe\n\feld gradient along the oscillation axis, and Bthe vector\ncomponent of the magnetic \feld along the NV anisotropy\naxis.\nIn the absence of a magnetic \feld gradient, the lumi-\nnescence signal is R(t), as per Eq. (A2). With a non-zero\n\feld gradient, R(t) incurs an additional phase modula-\ntion,\nR0(t) =R\u0012\nt+\u0001\u001e\n2\u0019fmodcos (2\u0019fTFt)\u0013\n: (E1)\nHere the phase modulation depth is \u0001 \u001e=\reB1=\u0001fwin.\nIn the limit of a small gradient (\u0001 \u001e\u001c1), we can expand\nEq. (E1) to \frst order,\nR0(t)\u0019R(t) +dR(t)\ndt\u0001\u001e\n2\u0019fmodcos (2\u0019fTFt): (E2)\nThe \frst term is simply the zero-gradient signal, from\nwhich we extract the static \feld as by demodulation atfmod. The second term is an amplitude modulation of\ndR(t)=dtat the frequency fTF. We thus observe an up-\nconverted version of d R(t)=dtcentered on fTF.\nNext, we express R(t) as a Fourier series,\nR(t) =1X\nk=\u00001akexp (2\u0019ikf modt); (E3)\nwhereak\u0018exp(\u0000\u0019jkj=\u000b), see Eqs. (A4) to (A11). The\ntime derivative is given by\n_R(t) = 2\u0019fmod1X\nk=\u00001ikakexp (2\u0019ikf modt): (E4)\nWe next rewrite the second term in Eq. (E2) as\n_R(t)\u0001\u001e\n2\u0019fmodcos (2\u0019fTFt) (E5)\n=_R(t)\u0001\u001e\n2\u0019fmodRe [exp (2\u0019ifTFt)] (E6)\n=\u0001\u001e\n2\u0019fmod\u0001Reh\n_R(t) exp (2\u0019ifTFt)i\n(E7)\n=\u0001\u001e\u0001Re\"1X\nk=\u00001ikakexp\u0000\n2\u0019i(fTF+kfmod)\u0001#\n:(E8)\nTherefore, we expect spectral components at fTF+kfmod.\nSinceakare known (by demodulation at kfmod), measur-\ning the amplitude of the fTF+kfmodcomponents will en-\nable the inference of \u0001 \u001e, and thus the gradient strength.\nAlternatively, the phase of the gradient signal itself also\ncontains useful information about the static \feld, so the\nmeasurements of kfmodandfTF+kfmodharmonics may\nbe combined into a single estimate of both the \feld and\nthe gradient."    },    {        "title": "2001.00707v1.Neutron_diffraction_and_ab_initio_studies_on_the_fully_compensated_ferrimagnetic_characteristics_of_Mn2V1_xCoxGa_Heusler_alloys.pdf",        "content": "1 \n Neutron diffraction and ab initio  studies  on the      \nfully compensated f errimagnetic characteristics of  \nMn 2V1-xCo xGa Heusler alloys  \nP V Midhunlal1, C Venkatesh2, J Arout  Chel vane3, P D Babu4 and N Harish Kumar1 \n  \n1. Department of Physics, Indian Institute of Technology Madras, Chennai -600036, India.  \n2. Saha Institute of nuclear physics,  Bidhannaga r, Kolkata, West Bengal -700064 . \n3. Defense  Metallurgical  Research  Laboratory,  Kanchanbagh  (PO),  Hyderabad -500058,  India.  \n4. UGC-DAE Consortium for Scientific Research, Mumbai Center, R -5 shed, BARC, Trombay,   \n    Mumbai -400085, India . \n \n \nKeywords :  Compensated ferrimagnets, high Curie temperature, Neutron diffraction,  \n                    ab initio  studies . \n \nAbstract:  Neut ron diffraction and ab initio  studies were  carried out on Mn 2V1-xCoxGa        \n(x=0, 0.25, 0.5, 0.75, 1) Heusler alloys which exhibits high T C fully compensated ferrimagnet ic \ncharacteristic  for x=0.5 . A combined analysis of neutron diffraction and ab initio  calculations \nrevealed the crystal structure and magnetic configuration  which could not be determined f rom \nthe X-ray diffraction and magnetic measurements . As reported earlier, R ietveld refinement of \nneutron diffraction data confirmed L21 structure for Mn 2VGa and Xa structure  for Mn 2CoGa . The \nalloy s with x=0.25  and 0.5  possess L21 structure with Mn (C)-Co disorder.  As the Co \nconcentration reaches 0. 75, a structural transition  has been observed from disordered L21 to \ndisord ered Xa. Detailed ab initio  studies also confirm ed this structural transition. The reason for \nthe magnetic moment compensation in Mn 2(V1-xCox)Ga was identified  to be different from that \nof the earlier reported fully compensated ferrimagnet (MnCo )VGa . With the help of neutron \ndiffraction and ab initio  studies, it is identified that t he disordered L21 structure with antiparallel  \ncoupling between the ferromagnetically  align ed magnetic moments of  (Mn (A)-Mn (C)) and  (V-Co) \natom pairs enables the compensation in  Mn 2V1-xCoxGa. 2 \n 1. Introduction  \n \n     Half-metallic fully compensated ferrimagnets  (HMFCF i) with high Curie temperatures are  \npotential candidates for spintronic application s especially for the spin-transfer torque based \nrandom access memory (STT -RAM)  [1]. They posse ss 100 % spin polarization at the Fermi \nlevel along with zero net magnetization and produces no  stray magnetic field which  increases  the \nefficiency of spintronic devices . Even though HMFCF i posses ses zero magnetic moment  \nmacroscopically , they significantly differ from conventional antiferromagnets . Unlike the \nconventional antiferromagnets , zero magnetization occurs in HMFCFi due to the magnetic \nmome nt compensation  arising from the different  sub-lattices which are chemically or \ncrystallographically inequivalent [2]. There are reports which discuss  the possibility of having \nzero moment half -metallicity in  different kinds of materials such as Heusler alloys  [3], dilute \nmagnetic semiconductors  [4], double perovskites  [5] and transition metal pnictides  [6]. Among \nthese, the most favourable  candidate is Heusler alloys  which are ternary intermetallics having \ngeneral formula X 2YZ  for full -Heusler alloys and XYZ for half -Heusler alloys  (where  X, Y are \ntransition metal and Z is an sp valent atom) . The  band structure and thereby the physical \nproperties of these alloys can be tuned by modifying the chemical composition through the \nsubstitution of  certain elements at specific sites.  The Slater -Pauling rule (S-P rule) , i.e.,  Mt=Zt-24 \n(for full -Heusler  alloys) , which relates the total magnetic moment per formula unit (M t) to the \ntotal number of valence electrons  (Zt) of half -metallic Heusler alloys can be utilized as an \nimportant tool to engineer new materials with improved physical properties like zero moment  \nhalf-metallicity.  The rule predicts hal f-metallic zero magnetic moment state  in full-Heusler \nalloys having 24 valence electrons  [7]. Galanakis I et al . have shown that half-metallic fully \ncompe nsated fe rrimagnetism  is achievable in  well-known  half-metallic Mn 2VZ (Z=Al ,Si) \nHeusler alloys  by substituting  Co at the Mn site  [8]. The alloys (MnCo) VAl and (Mn 1.5Co0.5)VSi \nhaving 24 valence electrons are predicted to be HMFCFi . Based on the ir theoretical prediction , \nexperimental investigations have been reported on (Mn 2-xCox)VZ (Z=Ga ,Al) alloy s [9,10] . This \nis interesting since the parent alloys Mn 2VZ (Z=Ga ,Al) are reported to have half -metallic \ncharacteristics  [11,12] . Even though the results show ed that the magnetic moment is \ncompensating  in these Co substituted systems , the Curie temperature decrease s below room \ntemperature with increase in Co concentration for both the alloy  series  which is not favourable  3 \n for devise applications. On the o ther hand , our recent  experimental  investigation show ed that the \nCo substitution at the V site has resulted in total magnetic moment compensation without \nsignificant change in Curie temperatures in the ferrimagnetic Mn 2VZ (Z=Ga ,Al) Heusler alloys  \n[13]. The observed Curie temperature s were  well above the room temperature  (> 650 K)  for \nMn 2V1-xCoxZ (Z=Ga ,Al; x=0,0.25,0.5,0.75,1)  alloys which were in contrast to the earlier reports \non Co substitution at the Mn site. The observed magnetic moment  and T C were 0.10 f.u.and \n706 K for Mn 2V0.5Co0.5Ga and  0.06 f.u.and 659 K for Mn 2V0.5Co0.5Al respectively . The \npossible reason for the magnetic moment compensation  was attributed to the  antiparallel  \ncoupling between the ferromagnetically  aligned  moments of  (Mn (A)-Mn (C)) and  (V-Co) atom \npairs . But there was no experimental or theoretical evidence for t his proposed magnetic \nconfiguratio n. Thus in order to determine the crystal  structure  and magnetic configuration  of the \ninteresting alloys Mn 2V1-xCoxGa (x=0,0.25,0.5,0.75,1) , neutron diffraction  measurement and \ndensi ty functional theory (DFT) based ab initio calcu lations  were carried out . The c rystal  \nstructure and the magnetic  configuration  were  identified  by comparing the neutron diffraction  \n(ND)  results with that of the ab initio  calculation s.  \n2. Experimental  & Computational  details  \n     The details on  sample preparation, discussions on X-ray diffraction and magnetic \nmeasurements  can be found in our earlier report  [13]. Room temperature  neutron diffraction \n(ND) measurements were carried out at a wavelength of 1.48 Å using the position -sensitive \ndetector based focusing crystal diffractometer PD -3 installed by the UGC -DAE CSR Mumbai \nCentre at the Dhruva reactor, Trombay  [14]. Spin-polarized  band structure calculations were \ncarried out for the Mn 2V1-xCoxGa (x= 0, 0.25, 0.5, 0.75, 1)  Heusler  alloys using full -potential \nlinearized augmented plane wave method (FP -LAPW) implemented in WIEN2k code [15]. \nPedrew -Burke -Ernzerhof  parameterization scheme was used for the  generalized gradient \napproximation (GGA)  exchange -correlation potential  [16]. The c alculation s were  carried out for \nthe 16 atom s supercell . The k -space integration was carried out by the tetrahedron method [17] \nby taking  150 k points in the irreducible Brillouin zone. The plane wave cut off parameter \nRMT×Kmax was taken as 9. RMT×Kmax is the product of the  smallest  atomic sphere radius  times the \nlargest K -vector of the plane wave expansion of the wave function which determines the size of \nthe basis set and thereby the accuracy  of the calculation.  The angular momentum truncation 4 \n parameter lmax was set to 9.  The energy convergence ( -ec) and charge convergence ( -cc) criterion \nwas set to 10-4 Ry and 10-4 e respectively.  \n       X2YZ Heusler alloys  (X, Y are  transition metal atom and Z is sp valent atom)  crystallize in \nthe cubic  Cu2MnAl type structure (Space group: 225, L21, Fm𝟑̅m) having four interpenetrating \nfcc sub -lattices  namely A, B, C and D  with coordinates (0,0,0), (1\n4 ,1\n4,1\n4), (2\n4,2\n4 ,2\n4) and (3\n4 ,3\n4 ,3\n4) \nrespectively . X atoms  are equally occupied in the A and C sub-lattices . The  Y and Z atoms \noccupy the B and D sub-lattices respectively.  Apart from the usual Cu2MnAl structure , Heusler \nalloys are found to crystalli ze in Hg 2CuTi type structure also, commonly known as inverse \nHeusler structure (space group: 216, Xa, F4̅3m). Unlike the  L21 alloys, the valance of X \ntransition -metal atom is smaller than that of the Y transition -metal atom for the Xa alloys and the \natoms in the B and C sub-lattices will be swapped as shown  in figure 1(a)&( b). In the case of  \nMn 2V1-xCoxGa, the atomic ordering along cube diagonal would be  Mn (A)- (V/Co )(B)-Mn (C)-Ga(D) \nin the L21 structure and Mn (A)-Mn (B)-(V/Co )(C)-Ga(D) in the Xa structure . So t he structural \nrefinement of ND pattern and ab initio  calculations  were carried out by considering both  L21 and \nXa crystal structure s.  \n \n \n \n \n \n \n \n \n X Y Z \nL21 Xa (a) (b) \nA B C D A \nFigure 1 : (a) L21 crystal structure with X (A)-Y(B)-X(C)-Z(D)  atomic configuration along \nthe cube diagonal (b) Xa crystal structure with X (A)-X(B)-Y(C)-Z(D)  atomic \nconfiguration. A, B, C and D represents the four interpenetrating fcc lattices.  5 \n 3. Results & discussion  \n3.1 Neutron diffraction studies on Mn 2V1-xCo xGa \n       Earlier neutron diffraction studies showed that Mn 2VGa alloy  were  in the cubic L21 structure  \n[18]. So i n our previous report on the structural and magnetic properties of the Mn 2V1-xCoxGa \n(x=0,0.25,0.5,0.75,1)  alloys, Rietveld refinement of X-ray diffraction  (XRD)  patterns was carried \nout by assuming the L21 structure . At the same time, the report by Liu et al. show s that the other \nend member  Mn 2CoGa crystallize in the inverse  (Xa) crystal structure  [19]. So the crystal \nstructure of substituted alloys could be either L21 or Xa. This was not taken into consideration in \nour previous report.  So in order to remove the uncertainty in the crystal structure , we have \ncarried out the refinement by considering the Xa structure also. The Rietveld refinement  carried \nout on Mn 2V1-xCoxGa alloys  showed that the X-ray diffraction patterns  were  fitting well for both \nL21 and Xa structural models  resulting in the similar  lattice parameter and χ2 value s. This makes it \ndifficult to distinguish the crystal structure of the se Co substituted alloys  through  XRD analysis . \nTo illustrate this problem  in a  better  manner , XRD  patterns  of Mn 2V1-xCoxGa \n(x=0,0.25,0.5,0.75,1)  alloys in both L21 and Xa crystal structures have been simulated using \nFullProf software [20] as displayed in figure 2 (a)&(b). It is clear  that there are no noticeable \ndifferences in the simulated L21 and Xa X-ray diffraction pattern s. Our earlier experimental report  \nshowed similar X -ray diffraction pattern s without having any order dependent  superlattice  \nreflections  (111) and (200). The absence of superlattice  reflections was  attributed to the similar \natomic scattering factors of X-ray for the elements Mn , V, Ga and Co . The presence of atomic \nanti-site disorder was also unavoidable . The ambiguity in the crystal structure  could be resolved \nby doing neutron diffraction  (ND)  measurements  since the coherent scattering length for \nindividual elements differs  significantly  (Mn=  -3.73 fm; V=  -0.38 fm; Ga=  7.288 fm; Co=  2.49 \nfm). In order to check this, the neutron diffraction patterns of the alloys were simulated as shown \nin figure 3 (a)&(b). For simulating the X -ray and neutron diffraction patterns, earlier reported \nexperimental lattice parameter was used.  Unlike the X-ray diffraction patterns, the simulated  ND \npatterns show  that e xcept the (333) peak , all other peaks are paired in both L21 and Xa crystal \nstructure . L21 and Xa crystal structures are easily distinguishable  by looking at  the relative \nintensities of these paired peaks  which were not the case  with the X-ray diffraction pattern s. For \nthe L21 structure, the second peak in each pair is more intense than the first peak and for the Xa 6 \n structure , the intensity of the first peak is higher than that of the second one.  Another noticeable \ndifference  between the two simulated ND pattern  is that the (333) peak in Xa structure is more \nintense than that in the L21 structure . Here a striking difference between the simulated XRD and \nthe ND patterns has to be mentioned. The visible peaks such as (220), (400), (422), (440) and \n(620) in the XRD patterns (indexed in green)  are having negligible intensity in the ND patterns. \nAt the same time, the low intense peak s such as (111), (200), (311), (222), (331), (420) etc…  in \nthe XRD patterns (indexed in black ) are pronounced in the ND patterns.  Thus the simulated ND \npatterns clear ly show the difference between L21 and Xa crystal structure .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 2 : Simulated XRD patterns of Mn 2V1-xCoxGa alloys in the (a) L21 and (b) Xa \ncrystal structure . \n15 30 45 60 75 90105 12002040600204060021426302346690214263\n(333)\n(620)(440)(422)(400)(220)x=0 \n \n (111)\n(200)\n(311)\n(222)\n(331)\n(420)\n(531)\n(442)x=0.25 \n x=0.5 \n x=0.75 \n x=1  \n Intensity (103 counts )\n2 (degree)\n(a)L21\n15 30 45 60 75 90105 1200275402754029580306003060\n  \n (333)\n(620)(440)(422)(400)(220)(111)\n(200)\n(311)\n(222)\n(331)\n(420)\n(531)\n(442) \n  \n  \n Intensity (103 counts )  \n x=1\nx=0.75\nx=0.5\nx=0.25\nx=0\n2 (degree)\n(b)Xa7 \n  \n \n \n \n \n \n \n \n \n \n \n \nAs the ND patterns would distinguish the L21 and Xa crystal structure , we have recorded the ND \npatterns of Mn 2V1-xCoxGa alloys at room temperature  which is shown in figure 4 (a). Before \ndiscussing the Rietveld refined patters as shown in figure 4 (b), some distinguishable features  can \nbe observed from the unrefined ND patterns.  Comparison of  the simulated L21 and Xa ND \npatterns  with experimental data reveals that Mn 2VGa  crystallizes  in L21 structure  and Mn 2CoGa \ncrystallizes in  Xa structure . The relative intensity of the nearby peaks is in good agreement with \nthe simulated patterns.  It is evident that the order dependent superlattice reflections (111) and \n(200) are present with a huge  intensity which was absent in the XRD patterns. So the reported \nabsence of superlattice reflections in the XRD patterns of Mn 2V1-xCoxGa alloys can be attributed Figure 3 : Simulated neutron diffraction patterns of Mn 2V1-xCoxGa alloys in the (a) \nL21 and (b) Xa crystal structure . \n15 30 45 60 75 90105 120081609170101901122091726\n(620)(440)(422)(400)(220)\n(622)(533)(442)(531)(333)(420)(331)(222)(311)(200) \n \n2 (degree)x=0Intensity (count)\nL21(111) x=0.25\n  x=0.5\n  x=0.75\n \n(a) x=1 \n \n15 30 45 60 75 90105 120091701019032640102001122 \n Intensity (count)x=0.25\nx=0\n2 (degree) Xa\n(b)(111)\n(200)\n(311)\n(222)\n(331)\n(420)\n(333)\n(531)\n(442)\n(533)\n(622)(220)\n(400)\n(422)\n(440)\n(620) \n  x=0.5\n  x=0.75\n  x=1 \n 8 \n to the similar atomic scattering factors of X-ray for the elements Mn , V, Ga and Co  which are in \nthe same period and not due to any atomic antisite disorder .  \n \n \n \n \n \n \n \nFor the Co substitution with x=0.25, some changes in the ND pattern can be observed  compared \nto the parent Mn 2VGa . The relative intensity of the (111) peak has been increased and the \nintensity of the (311) peak is more compared to the (222) peak.  Apart from these intensity \nchanges the overall pattern resembles that of the L21. The changes in intensity suggest the \npossibility of having an atomic disorder which will be discus sed later.  When the Co \nconcentration is increased to x=0.5, more changes in  the peak intensities were observed . The \nrelative intensity of peak s in the  pairs (111) & (200),  (311) & (222), (531) & (442)  and the \nintensity of the single peak (333) resembles the Xa structure and the pairs (331) & (420), ( 533) & \n(622)  resembles the L21 structure . Another interesting point to note is that the intensity mismatch  Figure 4 : (a) Neutron diffraction patterns of Mn 2V1-xCoxGa alloys recorded at room \ntemperature. (b) Nuclear refinement of the data for 2θ>40.  \n20 40 60 80 100061301020303690112201428\nx=0 \n (111)\n(200)\n(311)\n(222)\n(331)\n(420)\n(333)\n(531)\n(442)\n(533)\n(622) \n x=1\nx=0.75\nx=0.5\nx=0.25 \n  \n   \n Intensity (103 Counts )\n2 (degree)\n(a)\n0918\n01020\n047\n0918\n40 60 80 1000611\n  \n  \n  \n  \n  \n  \n  \n  \n(b) Yobs\n Ycalc\n Yobs-Ycalc\n Bragg position\n  \n  \n(311)\n(222)\n(331)\n(420)\n(333)\n(531)\n(442)\n(533)\n(622)\nIntensity (103 Counts )\n2 (degree)x=0x=0.25x=0.5x=0.75x=19 \n between the peaks in each pair is less compare to  the simulated pattern s. The above points \nindicate  that the crystal structure of Mn 2V0.5Co0.5Ga alloy could be  either  Xa or L21 with atomic \nantisite disorder.  For the Co concentration with x=0.75, the ND pattern is similar to that of \nMn 2CoGa which is Xa. This implies that there could be a structural transition from L21 to Xa as \nthe Co concentration increases  which sho uld be confirmed by doing the R ietveld refinement . By \nkeeping these things in mind, we tried to do the Rietveld refinement for  the ND pattern  by using \nthe FullPro f software . Since Mn 2V1-xCoxGa alloys are  ferrimagnetic at room temperature         \n(TC > 650 K) , nuclear refinement was carried out by considering the scattering angle  (2θ) above \n40˚at the beginning of the refinement . The refined ND patterns are shown in figure 4 (b). The \nrefinement showed that the Mn 2VGa alloy cry stallizes in the L21 structure with 6 % atomic \nantisite disorder between V and Ga . For x=0.25 and 0.5,  different structural models  such as  L21, \nXa, L21 + (various disorders), Xa + (various disorders)  have been considered.  For x=0.25, the best \nfit was obtained only for L21 + (Mn (C)-Co disorder).  For the alloy with x=0.5 , the structural \nmodels  L21 + (Mn (C)-Co disorder) and  Xa + (Mn (B)-Co disorder)  have resulted in identical fit to \nthe pattern . The ambiguity in crystal structure for this alloy  has been removed with the help of  \nab initio  calculations  which is di scussed in the next section. This calculation  identified the \ncorrect structure to be L21+(Mn (C)-Co) disorder . For the alloy with x=0.75, Xa + (Mn (B)-Co \ndisorder)  structural model has given the best fit.  Mn 2CoGa was found to crystallize in the Xa \nstructure with 5 % atomic antisite disorder between Mn (B) and Co . The considered structural \nmodels, the observed disorders, lattice parameter and the respective χ2 values are shown in Table \n1. It was evident from the  refinement that  the majority of the substituted Co has go ne to the Mn \nsite and the corresponding Mn atoms have gone to the V site  for the alloys with x=0.25 and 0.5 . \nThe antisite Co content was reduced from 89 % to 5 % as the Co concentration increases from \nx=0.25  to 1. The obtained crystal structure after ND pattern refinement is in agreement with the \npreliminary observations. Interesting point is that there is a gradual structural transition from  L21 \nto Xa as the Co concentration increases from x=0 to 1 which was not evident from the XRD \nanalysis.  \n  10 \n  \n \n \n \n \n \n \n x Fitted structural models  Co d isorder  \n(%)  \nLattice \nparameter  \n(Å) χ2 \n0 L21 + (V-Ga disorder)  6  5.895  11 \n0.25 L21+ (Mn (C)-Co disorder)  89  5.878  4.6 \n0.5 L21 + (Mn (C)-Co disorder)  \nor \nXa + (Mn (B)-Co disorder)  86  \n \n71  5.862  \n \n5.866  5.7 \n \n4.6 \n0.75 Xa + (Mn (B)-Co disorder)  25  5.844  8.1 \n1 Xa + (Mn (B)-Co disorder)  5  5.862  4.1 \nTable 1 : The considered structural models, the observed disorders, lattice parameter and the \nrespective χ2 values for neutron diffraction refinement of Mn 2V1-xCoxGa alloys.                                 \n \n \n 11 \n Having identified the crystal structure of Mn 2V1-xCoxGa, the next step is to do the refinement by \nconsidering the magnetic contribution to the neutron diffraction pattern by taking the whole \nscattering  angle 5 -120˚. The magnetic refinement showed that the ferromagnetically aligned Mn \natoms in the A and  C sublattices couples ferrimagnetically with V atoms in Mn 2VGa. The \nobserved magnetic moments were -1.28 μB for Mn (A) & Mn (C) and 0.71  μB for V which sums up \nto -1.85 μB/f.u. For Mn 2CoGa, refinement showed that the Mn atoms in the A and B sublattices  \nare aligned ferrimagnetically with a moment of -1.82 μB for Mn (A), 3.20  μB for Mn (B) and 0.52 μB \nfor Co which sums up to 1.9 μB/f.u. For the substituted alloys (x=0.25, 0.5, 0.75), the magnetic \nphase refinement of ND data was not possible d ue to certain constraints. Since the disordered  L21 \nand Xa structure s have 6 magnetic atoms per formula unit (Mn (A), Mn (C/B), V (B/C), Co (B/C), \ninterchanged Mn and Co),  fixing the individual moment became difficult. For example, in the \ncase of x=0.25, As the ‘ C’ sublattice is occupied by Mn and Co atoms and the ‘ B’ sublattice is \noccupied by V, Co and Mn atoms, it is difficult to figure out the individual atomic contribution \nto the total neutron scattering amplitude from these sub -lattices. Another difficulty is th at, due to \nthe ferrimagnetic  nature, magnetic reflections were overlapped with the nuclear reflection in the  \ndiffraction patterns and very small magnetic contribution was observed in the low angle (111) \nand (200) superlattice reflections only. This adds to the difficulty of magnetic phase  refinement \nas the goodness of the fit has to be monitored by observing these two peaks alone.  So in order to \nidentify the magnetic configurations and to confirm the crystal structure, density functional \ntheory based (DFT) ab initio  calculations were carried out.  The following section discusses  the \nab initio  calculations on  Mn 2V1-xCoxGa alloys.  Conclusions on the crystal structure and the \nmagnetic configurations  are given on the basis of a  combined analysis of ND and ab initio  \ncalculations .                                             \n3.2 Ab initio  studies  on Mn 2V1-xCo xGa (x=0, 0.25, 0.5 , 0.75 and 1)  \n          Spin-polarized  ab initio  calculations were carried out for L21 and Xa crystal structures for \neach x value. For each structure , energy minimization  with respect to lattice parameter  (volume) \nhas been carried  out and the lattice parameter corresponding to the minimum energy was used \nfor the self -consistent field (scf) calculations. Experimental lattice parameter  for Mn 2V1-xCoxGa \nwas used for the energy minimization.  The Figure 5(a)-(e) shows the energy minimization with \nrespect to the volume for Mn2V1-xCoxGa alloys in L21, Xa and disordered structures.   12 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 5 (a) -(d): Energy minimization with respect to the volume for Mn2V1-xCoxGa \nalloys in L21, Xa and disordered structures.  \n \n \n \n \n1.20 1.28 1.36 1.44 1.52 1.60-0.4-0.3-0.2-0.1\n(a) Volume (1000 ) [a.u]3\n  Energy (- 41686 ) (Ry)\n L21\n Xa x=0\n1.20 1.28 1.36 1.44 1.52-0.6-0.5-0.4\nEnergy (- 42574 ) (Ry) x=0.25\n Xa\n L21\n L21+ disorder\n  \n(b) Volume (1000 ) [a.u]3\n1.20 1.28 1.36 1.44 1.52-0.90-0.84-0.78-0.72-0.66 x=0.5\n  Energy (- 43462 ) (Ry)\nVolume (1000 ) [a.u]3 (c) L21\n L21+disorder\n Xa\n Xa+disorder\n1.26 1.33 1.40 1.47-0.18-0.16-0.14-0.12-0.10-0.08 \nEnergy (- 44351 ) (Ry) \n x=0.75\n L21\n Xa\nVolume (1000 ) [a.u]3 (d)\n1.20 1.28 1.36 1.44 1.52 1.60-0.52-0.48-0.44-0.40-0.36\n L21\n Xa\n  Energy (- 44351 ) (Ry) x=1\n(e) Volume (1000 ) [a.u]313 \n From the energy minimization , it is clear that the alloy with x=0 ha s minimum energy in the L21 \ncrystal structure . ND analysis indicate d that for the alloy with x=0.25, most of the substituted Co \natoms have gone  to the Mn (C) site and equivalent Mn (C) atoms have gone to the V site  in the L21 \nstructure . So along with  the L21 and Xa structures , L21+(Mn (C)-Co) disordered structure  is also \nconsidered  for the energy minimization . The stoichiometry used for the calculation is \n(M1.75Co0.25)(V 0.75Mn 0.25)Ga which still ha s the total valence electron number of 23 and thereby a \nmoment of -1 μB/ f.u. as per the S-P rule.  As shown in figure 5(b) , L21+(Mn (C)-Co) disordered \nstructure has exhibited low er energy  which is in agreement with the ND result . Similarly , ND \nrefinement suggested either a L21+(Mn (C)-Co) disordered structure or Xa+(Mn (B)-Co) disordered \nstructure for x=0.5 . So the  energy minimization was carried out by considering these disordered \nstructure s also. The minimization curve s clearly show that the L21+(Mn (C)-Co) disordered \nstructure has the minimum energy compared to other structures.  The stoichiometry used for the \ncalculation is (M 1.5Co0.5)(V 0.5Mn 0.5)Ga which still has the total valence electron number of 2 4 \nand thereby a moment of 0 μB/ f.u. as per the S-P rule . For x=0.75 and 1, Xa crystal structur e has \nthe minimum energy compared to  the L21 structure. Thi s clearly shows a crossover of crystal \nstructure from L21 to Xa as the Co concentration increases  which are in agreement with the ND \nresults . Table 2 shows the  calculated lattice parameters which are compared to that of the values \nreported in other experimental and ab initio  studies.  Table 3  shows the total and atom -resolved \nmagnetic moments  and spin polarization values of Mn 2V1-xCoxGa. The expected magnetic \nmoments from S -P rule are -2, -1, 0, 1, and 2 μ B/f.u. and the reported experimental values in our \nearlier report  are 1 .80, 0.97, 0.10, 1.11 and 2.05 μB/f.u for x=0, 0.25, 0.5, 0.75 and 1 respectively. \nThe calculated total moments also follows the experimental trend of magnetic moment \ncompensation as the Co concentration reaches x=0.5. For the x=0  alloy  (Mn 2VGa), the \ncalculated total magnetic moment  in the low er energy L21 structure is  -1.98 μB/f.u. which is close \nto the S -P value.  \n \n \n \n 14 \n  \n \n \n \n \n \n \n  \nx Crystal structure  Lattice parameter (Å)  \nPresent \nCalculation  Other Reports  \nCal. Exp. \n0 L21 5.824 5.820 [21] \n5.685 [22] \n5.808 [23] 5.914 [13] \n5.905 [11] \n6.095 [24] \n0.25 L21+disorder  5.805  - 5.874 [13] \n0.5 L21+disorder  5.796  - 5.878 [13] \n0.75 Xa 5.824 - 5.881 [13] \n1 Xa 5.789 5.78 [21] \n5.61 [25] 5.877 [13] \n5.862 [19] \n5.869 [26] \nTable 2 : Crystal structure and the corresponding calculated lattice parameters of Mn 2V1-\nxCoxGa alloys. Calculated (Cal) and experimental (exp) lattice parameters reported by other \nresearchers are also shown for the comparison.  \n                                \n \n \n 15 \n  \n \nThe obtained moment value is in agreement with the earlier report  [18]. The Mn moments in the \nA and C sublattices are found to be in ferromagnetic alignment which is common in the case of \nMn-based Heusler  alloys in the L21 structure.  For Mn 2CoGa ( x=1), Mn moments in the A and B \nsublattices are found to be in antiparallel  alignment which is observed  in the case of Mn -based \nHeusler  alloys in the Xa structure . This is also in agreement with the earlier report [27]. For \nx=0.25  and 0.5 , L21 structure  with Mn (C)-Co disorder is the minimum energy state  having  a total \nmoment of -0.99 and 0 μB/f.u. respectively. This is close to the experimental and S -P value  \n(shown in  Table 3 ). For x=0.75 also, the calculated moment  is close to the experimental value.  \nThe obtained magnetic configuration from the ab initio  calculation for the alloy with x=0.5  is \nsimilar  to the assumption made for explain ing the magnetic moment compensation  in our \nprevious report.  L21 structure with dduu magnetic configuration was assumed for Mn (A), Mn (C),  \nx \n Crystal \nstructure  Mag.  \nConfig.  atom resolved magnetic moments  \n(μB) Total \nmoment \n(Cal)  \nμB/f.u. Total \nmoment  \n(S-P rule)  \nμB/f.u. Total \nmoment    \n(Exp)[13] \nμB/f.u. P \n(%) \nMn(A) Mn(B/C) V Co \n L21 ddu -1.45 -1.45 0.82 - -1.98 -2 1.80 94 \n0.25 L21+disorder  dduu  -1.32 -1.53 0.73 \n2.90 (Mn)  0.48 -0.99 -1 0.97 99 \n0.5 L21+disorder  dduu  -1.35 -1.31 0.65 \n2.82 (Mn)  0.66 0 0 0.10 99 \n0.75 Xa dudu  -1.91 2.69 -1.43 0.94 0.99 1 1.11 98 \n1 Xa duu -1.82 2.85 - 0.99 2.00 2 2.05 99 \nTable 3: Identified crystal structure, calculated (cal) atom -resolved and total magnetic moments \nand spin polarization values of Mn 2V1-xCoxGa. Experimental (exp) magnetic moments are also \nshown for the camparison. The antisite Mn moment is shown near to the V mo ment in the case \nof x=0.25 and 0.5.  For representing the magnetic configuration  of Mn(A) Mn(B/C)V Co , upward \nand downward directions are denoted as ‘u’ and ‘d’ respectively.  \n                                \n \n \n 16 \n V(B), Co (B) atoms (here u implies upward moment direction and d implies downward moment \ndirection).  The present calculation also showed dduu  configuration  irrespective of the disorder . \nIn addition to this, the disordered Mn atom at the V site has aligned  in the upward direction.  This \nimplies t hat the  L21 structure and ferromagnetic coupling  of Mn atoms in the A and C sublattices  \nof Mn 2VGa ha ve not altered with Co substitution up to x=0.5 . Also , the substituted Co and V \natoms couples  ferro magnetically for x up to 0.5 . In short , the disordered L21 structure with the \nferrimagnetic  coupling between the ferromagnetically  aligned (Mn (A)-Mn (C)) and  (V-Co) atom \npairs enables the magnetic moment compensation in  Mn 2V1-xCoxGa. Figure  6 shows the \nvariation of atomic moments with the Co concentration.  For the  Co-substituted alloys, t he \nMn (B/C) and Co  moment increases and the Mn (A) and V moment decrease gradually  as x  increases \nfrom 0.25  to 1.  \n \n \n \n \n \n \n \n \n \n \n \nL.Wollm an et al . have carried out a detailed theoretical investigation on the structural and \nmagnetic properties of the cubic Mn 2Y(3d/4d)Z (Z=Ga,  Al) alloys  [21]. The results show  that for \nthese Mn -based alloys with Z t ≤ 24, L21 structure is favoured  (eg: Mn 2VGa) and when Z t >24, Xa \nstructure is favoured  (eg: Mn 2CoGa).  Another interesting observation is that due to the \ndifferences in the nearest neighbour  atoms, L21 Mn-based alloys have ferromagnetic coupling for Figure 6 : Variation of atomic moments with the Co concentration \nfor Mn 2V1-xCoxGa.                                \n \n \n \n0.00 0.25 0.50 0.75 1.00-2-10123\n  \n   Mn(A)\n Mn(B/C)\n V\n CoAtomic moment (B)\nCo concentration x17 \n the Mn atoms at the A and C sub -lattices  and in the case of Xa Mn-based alloys, the coupling \nbetween the Mn atoms in the A and B sub -lattices is ferrimagnetic . This is one of the \ncharacteristic features of Xa Mn-based alloys.  This observation  is comparable to our present \ninvestigation. In our investigation, the valence electron number (Z t) of Mn 2VGa  is varied from \n22 to 26 by substituting Co at the V site. Our calculation also shows a ferro magnetic  coupling \nbetween the Mn atoms at different sub -lattices for the alloys with x= 0, 0.25 and 0.5 which \ncrystallizes in  the disordered L21 structure .  The alloys with x=0.75 and 1 crystallize in the Xa \nstructure  and the Mn moments are coupled ferrimagnetically . \n       Since  the Mn 2V0.5Co0.5Ga alloy exhibits a fully compensated ferrimagnetic state with high \nTC, it is interesting to investigate the half -metallic characteristic of Mn 2V1-xCoxGa. Figure  7(a)-\n(e) shows the spin-resolved  total density of states (TDOS) and partial density of states versus \nenergy diagram for the Mn 2V1-xCoxGa alloys.  Only  the dominant d band  contribution of Mn, V \nand Co is considered for the partial density of states.  As shown in figure 7 (a), Mn 2VGa  (x=0) \nexhibit  a gap at the Fermi level in the spin-down  subband . The non -zero density of states at the \nFermi level in the spin -up subband  is mostly contributed by the Mn (A,C) - 3d states . For Mn 2VGa,  \nthe calculated  spin polarization is 92 %  which indicates that it is nearly a half -metal . This is in \nagreement with the earlier reported values which  is around 94 %  [21,23] . For the Co substituted \nalloys also, the spin polarization values are more than 90 %. The calculated spin polarization \nvalues for the alloys with x=0.25, 0.5, 0.75 and 1 are 99, 99, 98 and 99 % respectively.  As \nexpected the V density of states decreases ( shown in pink colour) and the Co density of states \nincreases ( shown in green colour) with the increase in Co concentration. The earlier ab initio  \nresults on the Mn 2-xCoxVGa  alloys  (Co substitution at the Mn site) showed that the spin \npolarization  decreases to 34 %  for the compensation point x= 1 [23]. The experimental \ninvestigation carried out by Ramesh Kumar et al. also indicates the absence of ha lf-metallicity \nwhen Co is substituted at the Mn site  [9]. In contrast to this, our present investigation shows that \nthe spin polarization for Mn 2V0.5Co0.5Ga alloy is 99 %.  The difference in the density of states \ncurve for the spin up and spin down  subbands  indicates that the exchange potential for the \nmagnetic atoms are having opposite signs resulting in the ferrimagnetic  state which is evident \nfrom the atomic moments shown in Table 3  [28].  18 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure 7(a) -(e): Spin-resolved  total and partial density of states versus \nenergy diagram for the Mn 2V1-xCoxGa (x=0, 0.25, 0.5, 0.75, 1)  alloys.  \n-6 -4 -2 0 2 4 6-6-30369\nx=0\n(a) Total DOS\n Mn(A,C) - d\n V(B) - d\n   \n E (eV)DOS  (states/eV)\nF\n-6 -4 -2 0 2 4 6-6-4-2024 x=0.25\n Total DOS\n Mn(A) - d\n Mn(C) - d\n Co(C)  - d\n V(B)    - d\n Mn(B) - d\nF\n   \n \nDOS  (states/eV)\n(b) E (eV)\n-6 -4 -2 0 2 4 6-4-2024 Total DOS\n Mn(A) - d\n Mn(C) - d\n Co(C)  - d\n V(B)    - d\n Mn(B) - d\nx=0.5\n   \n DOS  (states/eV)\nE (eV) (c)F\n-6 -4 -2 0 2 4 6-6-4-20246\nx=0.75\n(d)  Total DOS\n  Mn(A) - d\n  Mn(B) - d\n  V(C) - d\n  Co(C) - d\n   \n \nDOS  (states/eV)\nE (eV)F\n-6 -4 -2 0 2 4 6-6-4-20246\n(e) Total DOS\n Mn(A) - d\n Mn(B) - d\n Co(C) - d\n   \n DOS  (states/eV)\nE (eV)Fx=119 \n 4. Conclusions  \n     The c rystal structure and the magnetic configuration of Mn 2V1-xCoxGa alloys which exhibit \nfully compensated ferrimagnetic property  are identified through combined  neutron diffraction \nand ab initio  studies . Our investigation showed that the L21 and Xa structure of the Mn 2V1-\nxCoxGa alloys are well distinguishable from the neutron diffraction pattern which could not be \nidentifiable  from the X -ray diffraction  studies . Neutron diffraction studies on Mn 2V1-xCoxGa \nclearly indicated a structural transition from L21 to Xa as the Co concentration increases. Alloy \nwith x=0 crystallize s in L21, x=0.25 and 0.5 crystallize s in disordered L21, x=0.75  crystallizes in \ndisordered Xa and  x=1  alloy crystallize s in Xa structure. When  a slight  V-Ga disorder was present \nin Mn 2VGa, Mn -Co atomic antisite disorder was observed in the Co -substituted alloys. Ab initio  \nstudies  have confirmed the structural transition observed in the neutron diffraction patterns.  The \nidentified magnetic configuration from the ab initio  studies for x=0, 0.25, 0.5. 0.75 and 1 are \nddu, dduu , dduu , dudu  and duu respectively . It was evident that the disordered L21 structure \nwith the antiparallel  coupling between the ferromagnetically  aligned (Mn (A)-Mn (C)) and  (V-Co) \natom pairs enables the magnetic moment compensation in  Mn 2V1-xCoxGa. A high spin \npolarization value of more  than 90 %  was observed for all the alloys including the high T C fully \ncompensated ferrimagnet Mn 2V0.5Co0.5Ga.  \n \nAcknowledgement  \nP V Midhunlal acknowledge IIT Madras for all the funding  and providing VIRGO supercluster \nfor the High -Performance  Computing Environment (HPCE) . 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B  60 \n13006 –10 \n \n "    },    {        "title": "2003.04608v1.Room_temperature_ferrimagnetism_of_anti_site_disordered_Ca2MnOsO6.pdf",        "content": "1 Room-temperature ferrimagnetism of anti-site-disordered Ca 2MnOsO 6 \nHai L. Feng ( 冯海),1,2,3,*# Madhav Prasad Ghimire,4,5,6 Zhiwei Hu,2 Sheng-Chieh Liao,2 Stefano \nAgrestini,2,7 Jie Chen,1,8 Yahua Yuan,9,1 Yoshitaka Matsushita,10 Yoshihiro Tsujimoto,1,8 Yoshio \nKatsuya,11 Masahiko Tanaka,11 Hong-Ji Lin,12 Chien-Te Chen,12 Shih-Chang Weng,12 Manuel \nValvidares,7 Kai Chen,13 Francois Baudelet,13 Arata Tanaka,14 Martha Greenblatt,3 Liu Hao Tjeng,2 \nKazunari Yamaura 1,8* \n1. Research Center for Functional Material, National Institute for Materials Science, 1-1 Namiki, \nTsukuba, Ibaraki 305-0044, Japan  \n2. Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany \n3. Department of Chemistry and Chemical Biology, Rutgers, the State University of New Jersey, 610 \nTaylor Road, Piscataway, New Jersey 08854, United States \n4. Central Department of Physics, Tribhuvan University, Kirtipur, 44613 Kathmandu, Nepal  \n5. Leibniz Institute for Solid State and Materials Research, IFW Dresden, P.O. Box 270116, \nD-01171 Dresden, Germany \n6. Condensed Matter Physics Research Center, Butwal-11, Rupandehi, Nepal \n7. ALBA Synchrotron Light Source, E-08290 Cerdanyola del Vall`es, Barcelona, Spain \n8. Graduate School of Chemical Sciences and Engineering, Hokkaido University, North 10 West 8, \nKita-ku, Sapporo, Hokkaido 060-0810, Japan  \n9. School of Physics and Electronics, Central South University, Changsha 410083, China \n10. Materials Analysis Station, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, \nIbaraki 305-0047, Japan  \n11. Synchrotron X-ray Station at SPring-8, National Institute for Materials Science, Kouto 1-1-1, \nSayo-cho, Hyogo 679-5148, Japan  \n12. National Synchrotron Radiation Research Center, 101 Hsin-Ann Road, Hsinchu 30076, Taiwan, \nROC \n13. Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, 91192 Gif-sur-Yvette Cedex, France \n14. Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima 739-8526, \nJapan \n* Corresponding authors E-mail: hai.feng@iphy.ac.cn, Hai.Feng_nims@hotmail.com (HLF); \nYamaura.Kazunari@nims.go.jp (KY)  \n# Current address: The Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China \n  2 Abstract  \nRoom-temperature ferrimagnetism was discovered for the anti-site-disordered perovskite \nCa2MnOsO 6 with Tc = 305 K. Ca 2MnOsO 6 crystallizes into an orthorhombic structure with a space \ngroup of Pnma, in which Mn and Os share the oxygen-coordinated-octahedral site at an equal ratio \nwithout a noticeable ordered arrangement. The material is electrically semiconducting with \nvariable-range-hopping behavior. X-ray absorption spectroscopy confirmed the trivalent state of the \nMn and the pentavalent state of the Os. X-ray magnetic circular dichroism spectroscopy reveals that \nthe Mn and Os magnetic moments are aligned antiferromagnetically, thereby classifying the material \nas a ferrimagnet which is in accordance with band structure calculations. It is intriguing that the \nmagnetic signal of the Os is very weak, and that the observed total magnetic moment is primarily due \nto the Mn. The Tc = 305 K is the second highest in the material category of so-called disordered \nferromagnets such as CaRu 1-xMnxO3, SrRu 1-xCrxO3, and CaIr 1-xMnxO3, and hence, may support the \ndevelopment of spintronic oxides with relaxed requirements concerning the anti-site disorder of the \nmagnetic ions.  \n  3 1. INTRODUCTION \n Double perovskite oxides containing 3d and 4d/5d elements are currently much in focus \nsince promising properties for spintronic applications have been reported. For example, Sr 2FeMoO 6 \nshows a low-field magnetoresistance at room temperature[1], and Sr 2CrReO 6 half-metallic (HM) \ntransport with a remarkably high Curie temperature ( Tc) of 635 K[2,3]. Analogous closely related \nferrimagnetic (FIM) oxides have been synthesized with a wide variety of 3d and 4d/5d elements, \nsuch as A2FeMoO 6 (A = Ca, Ba)[1,4], A2FeReO 6 (A = Ca, Sr, Ba)[5,6], Ca 2CrOsO 6[2,3], and \nCa2FeOsO 6[7,8]. In addition, a ferromagnetic (FM) Dirac–Mott insulating state ( Tc ~100 K) has been \nfound in Ba 2NiOsO 6 [9] and an exchange bias effect in Ba 2Fe1.12Os0.88O6 [10], which may also be \nuseful for further development of spintronic oxides.  \n In the magnetic ground state of Sr 2FeMoO 6 and Sr 2CrReO 6, the 3d magnetic moments are \nordered parallel to each other and antiparallel to those of the 4d/5d[1-3]. Anti-site disorder between \nthe 3d and 4d/5d elements has, however, a significant negative impact on the magnetic properties, \nbecause strongly antiferromagnetic (AFM) Fe–O–Fe and Cr–O–Cr bonds are formed which interfere \nwith the long-range magnetic order[11,12]. Indeed, even a small degree of anti-site disorder \ndramatically decreases Tc, and the spin-polarization is also strongly reduced in Sr 2FeMoO 6[13,14], \naccompanied by a linearly decreasing saturation magnetization[12]. Therefore, accurate control of \nthe anti-site disorder has been a significant issue for the fabrication of practical devices since \ngrowing anti-site-disorder-free materials is highly challenging[11,12]. Alternatively, \nanti-site-disorder tolerant materials with promising properties are in demand.  \n A so-called disordered FM has been reported in a substitutional study of the paramagnetic \nperovskite CaRuO 3, where Ru is partially replaced by a variety of magnetic or nonmagnetic elements \nsuch as Sn, Ti, Mn, Fe, Ni, or Rh[15,16]. Among these, Ca 2MnRuO 6 is particularly of high interest, \nbecause its magnetic moment is relatively large (~1.6 μ B/f.u.) and its Tc is high (~230 K)[17,18]. In 4 the structure of Ca 2MnRuO 6, Mn and Ru are distributed over the perovskite B site at an equal ratio \nwithout an ordered arrangement. This material was revealed to be FIM by neutron diffraction owing \nto a balance between FM (Mn3+ to Mn4+) and AFM (Ru to Mn) interactions[17-19], and the \nexperimental magnetic state was consistent with that obtained by first-principles calculations[19]. \nAnother example is SrRu 0.6Cr0.4O3 which was found to have a high transition temperature of 400 K \nbut a very small saturated moment of 0.15 μ B/f.u.[20]. \n In this study, an anti-site-disordered compound, Ca 2MnOsO 6 was synthesized for the first \ntime by a high-pressure and high-temperature method at 6 GP and 1500 ºC. Ca 2MnOsO 6 shows a \nFIM transition at Tc = 305 K, which is the second highest Tc among the disordered FMs. Here we \nreport the refined crystal structure and bulk magnetic properties of Ca 2MnOsO 6. The results suggest \nthat the compound is useful for further development of anti-site-disorder-tolerant spintronic oxides.  \n \n2. EXPERIMENTAL \n Polycrystalline Ca 2MnOsO 6 was synthesized via a solid-state reaction from powders of \nCaO2 (lab-made from CaCl 2∙2H2O, 99% Wako Pure Chem.), Os (99.95%, Heraeus Materials), and \nMnO2 (99.997%, Alfa-Aesar). The powders were thoroughly mixed at the stoichiometric ratio, \nfollowed by sealing in a Pt capsule. The preparation was conducted in an Ar-filled glove box. The Pt \ncapsule was statically and isotropically compressed in a belt-type high-pressure apparatus (Kobe \nSteel, Ltd., Japan [21]), and a pressure of 6 GPa was continuously applied while the capsule was \nheated at 1500 °C for 1 h, followed by quenching to room temperature in less than a minute. The \npressure was then gradually released over several hours.  \n A dense, black polycrystalline pellet was obtained, and several pieces were cut out from it. \nA selected piece was finely ground for a synchrotron X-ray diffraction (SXRD) study, which was 5 conducted in a large Debye–Scherrer camera in the BL15XU beam line, SPring–8, Japan[22,23]. The \nSXRD pattern was collected at room temperature and the wavelength was confirmed to be 0.65298 Å \nby measurement of a standard material, CeO 2. The absorption coefficient was measured in the same \nline. The Rietveld method was used to analyze the SXRD pattern with the RIETAN–VENUS \nsoftware [24,25].  \n X-ray absorption spectroscopy (XAS) at the Mn- L2,3 and Os-L3 edges of Ca 2MnOsO 6 was \ncarried out at the BL11A and BL07C beamlines using the total electron yield and transmission \nmethod, respectively, in the National Synchrotron Radiation Research Center, Taiwan. The Mn- L2,3 \nspectrum of MnO and the Os- L3 spectrum of Sr 2FeOsO 6 were also measured for energy calibration \npurposes. X-ray magnetic circular dichroism (XMCD) spectra at the Mn- L2,3 and the Os- L2,3 edges \nwere obtained at the BL29 BOREAS beamline of the ALBA synchrotron radiation facility in \nBarcelona and at the ODE beamline of Soleil France, respectively. The degree of circular \npolarization in BOREAS and ODE beamline was close to 100% and 90%, respectively.  The XMCD \nspectra were measured in a magnetic field of 60 kOe at a temperature of 20 K for Mn- L2,3 and in 13 \nkOe at 4 K for the Os- L2,3. The Os L3/L2 edge-jump intensity ratio I(L3)/I(L2) was normalized to 2.08 \n[26], which accounts for the difference in the radial matrix elements of the 2p 1/2–to–5d(L2) and 2p 3/2–\nto–5d(L3) transitions.  \n The electrical resistivity ( ρ) of polycrystalline Ca 2MnOsO 6 was measured by a four-point \nmethod at a gauge current of 0.1 mA in a physical properties measurement system (Quantum Design, \nInc.). Electrical contacts on a piece of Ca 2MnOsO 6 were prepared using Pt wires and Ag paste in the \nlongitudinal direction. The temperature dependence of the specific heat capacity ( Cp) was measured \nin the same apparatus by a thermal relaxation method at temperatures between 2 K and 300 K using \nApiezon N grease to thermally connect the material to the holder stage.  \n The magnetic susceptibility ( χ) of a loosely gathered Ca 2MnOsO 6 powder was measured in a 6 magnetic properties measurement system (Quantum Design, Inc.). The measurement was conducted \nin field cooling (FC) and zero-field cooling (ZFC) conditions in a temperature range between 2 K \nand 390 K. The applied magnetic field was 10 kOe. The magnetic field dependence of the \nmagnetization ( M) was measured between −50 kOe and +50 kOe at fixed temperatures of 2, 200, \n300, and 350 K. The alternative current (ac) χ was measured in the same apparatus between 5 K and \n320 K; the amplitude and frequencies of the ac-magnetic field were 5 Oe and 0.5 –500 Hz, \nrespectively.  \n The density functional theory (DFT) calculation was performed on Ca 2MnOsO 6 with the \nall-electron full-potential local-orbital code[27] using the standard generalized-gradient \napproximation (GGA) [28]. In this study, a linear tetrahedron method was employed for all k space \nintegration with a 12 × 12 × 12 subdivision in the full Brillouin zone for an ordered phase of \nCa2MnOsO 6. The magnetic ground state was obtained by computing the total energy of possible \nmagnetic configurations. Our calculation was double-checked in selected cases using the \nfull-potential linearized augmented plane wave method, as implemented in the WIEN2k code[29]. In \norder to estimate the correlation effects, the GGA+ U (U is the Coulomb interaction) function with \natomic-limit (AL) double-counting correction was used[30,31]. The values of U were chosen to be 5 \neV for Mn-3d and 1.5 eV for Os-5d, which were comparable to the values in the literature [9,32-36]. \nThe self-consistent full-relativistic calculations were carried out with spin-orbit coupling (SOC) \nincluded. This is necessary to determine the magnetic anisotropy energy (MAE) and to check its \ninfluence on the electronic and magnetic properties. The calculation was first performed with the \nexperimental lattice parameters obtained by SXRD, and then geometry optimization was conducted. \nThe energy and charge convergence for the self-consistent calculation was set to 10-8 Hartree and \n10-6 of electron, respectively.  \n3. RESULTS 7 A. Crystal structure \n The double perovskite oxide A2BB'O6 [A: alkali earth, B: 3d, B': 4d(5d) elements] normally \ncrystallizes into a monoclinic ( P21/n), or tetragonal ( I4/m), or cubic ( Fm-3m) structure, in which two \ndistinct octahedra  BO6 and B'O6 align in a rock-salt fashion [37]. The degrees of the octahedral \ndistortion and rotation seem to have a major impact on the lattice symmetry[38]. Based on the \ngeneral trend, we attempted to characterize the crystal structure of Ca 2MnOsO 6 by applying these \nmodels. In particular, the monoclinic model ( P21/n) was very effective to refine the SXRD pattern as \nreported for the analogous compounds such as Ca 2FeOsO 6 [7], Ca 2CrOsO 6 [2], and Ca 2MnReO 6 [39]. \nThe refinement quality was, however, not fully satisfactory: a significant failure was found for the \npeaks marked by an arrow in the inset of Fig. 1. Although the major peaks were well refined by the \nP21/n model (if Mn and Os are fully ordered), several expected peaks such as (0-11), (-101), and \n(101) were not observed above the background level. This failure suggested that the rock-salt-type \nordered arrangement of MnO 6 and OsO 6 octahedra is not formed in Ca 2MnOsO 6[13], namely, Os and \nMn atoms are distributed randomly over the perovskite B-site (i.e., anti-site disorder), as was found \nin Ca2MnRuO 6 (Pnma [17]).  \n \nFig. 1 Rietveld refinement of the powder synchrotron XRD pattern collected at room temperature. \nThe crosses and solid lines show the observed and calculated patterns, respectively, with their \ndifference shown at the bottom. The Os was analyzed simultaneously as the secondary phase. The \nestimated mass proportion was Ca 2MnOsO 6: Os = 0.986: 0.014. (Inset) Analysis by an ordered Mn \n8 and Os atoms model; the arrow indicates an obvious discrepancy between the model and the \nexperiment in the analysis.  \n \n We therefore tested the Pnma model of the SXRD data, and as shown in the main panel of \nFig. 1, the Pnma model successfully characterized the structure of Ca 2MnOsO 6. The final solutions \nof the lattice parameters, atomic coordinates, and temperature factors are listed in Table 1. Note that \nthe orthorhombic model ( Pnma) was proposed for both end compounds, CaOsO 3[40] and \nCaMnO 3[41] as well, which indicates  that Ca 2MnOsO 6 is not a double perovskite, but a solid \nsolution of them.  \nTable 1 Atomic coordinates and temperature factors ( B) for Ca 2MnOsO 6 at room temperature \nobtained from synchrotron XRD  \nAtom Site Occp. x y z B  (Å2) \nCa 4 c 1 0.9523(3) 0.25 0.0102(9) 1.12(3) \nOs/Mn 4b 0.5/0.5 0 0 0.5 0.22(1) \nO1 4 c 1 0.0340(11) 0.25 0.5718(10) 1.03(8) \nO2 8 d 1 0.2106(7) 0.4578(5) 0.1850(7) 1.03 \nSpace group: Pnma; lattice constants a = 5.50669(4) Å, b = 7.60567(6) Å, and c = 5.39322(4) Å; Z = \n2; dcal = 6.1946 g/cm3; and the final R indices are Rp = 2.029% and Rwp = 3.597%.  \n \n The average of the M‒O (M = Mn or Os) bond lengths of Ca 2MnOsO 6 is compared with that \nof related compounds; it is shorter than that of CaOs4+O3[40] and LaMn3+O3[42] and longer than that \nof NaOs5+O3[43] and CaMn4+O3[41] (see Table 2). The comparison suggests that the valence state of \nM is intermediate between the two groups. In addition, we examined the impact of the Jahn–Teller \n(JT) distortion on the average structure through comparison of the distortion factors (defined as the \nratio of the longest to the shortest M–O length of an octahedron) of the compounds. It is 1.084 for 9 Ca2MnOsO 6, which is much smaller than 1.142 for LaMn3+O3 and much greater than 1.004 for \nCaMn4+O3, 1.030 for CaOs4+O3, and 1.004 for NaOs5+O3. Although the comparison indicates that the \nJT distortion could contribute to the average structure of Ca 2MnOsO 6, it is extremely difficult to \nuniquely identify the exact charge distribution on Mn and Os of Ca 2MnOsO 6 from only an analysis \nof the powder SXRD pattern, because of the anti-site disorder.  \nTable 2 Comparison of structural parameters of Ca 2MnOsO 6 and related compounds  \n LaMn3+O3  \n[42]  CaMn4+O3  \n[41]  Ca2MnOsO 6  \n[this work]  CaOs4+O3  \n[40]  NaOs5+O3  \n[43]  \nSpace group Pnma Pnma Pnma Pnma Pnma \na (Å) 5.5367(1) 5.279(1) 5.50666(4) 5.57439(3) 5.38420(1) \nb (Å) 5.7473(1) 7.448(1) 7.60562(6) 7.77067(4) 7.58038(1) \nc (Å) 7.6929(2) 5.264(1) 5.39317(4) 5.44525(3) 5.32817(1) \nV (Å3)  244.8 207.0 225.9 235.9 217.5 \nM–O1 (Å) 1.9680(3) 2 1.895 2 1.945(1) 2 2.003(1) 2 1.946(1) 2 \nM–O2 (Å) 1.907(1) 2 1.900 2 1.917(4) 2 1.978(4) 2 1.939(3) 2 \nM–O2 (Å) 2.178(1) 2 1.903 2 2.077(4) 2 2.037(4) 2 1.940(3) 2 \nDistortion a 1.142 1.004 1.084 1.030 1.004 \n<M–O> (Å) 2.018 1.899 1.980 2.006 1.942 \nM–O1–M (º) 155.48(2) 158.6 149.6(2) 151.7(1) 153.9(2) \nM–O2–M (º) 155.11(5) 157.2 155.7(1) 152.0(2) 155.2(2) \na Defined as the ratio of the longest to the shortest M–O length of the MO6 octahedron  \n \nB. X-ray absorption spectroscopy  \n To investigate the valence states of Mn and Os, we conducted the Mn- L2,3 and Os-L3 XAS \non Ca 2MnOsO 6 and compared the spectra to those of MnO[44], LaMnO 3[44], SrMnO 3[45], and \nSr2FeOsO 6[46] which serve as Mn2+, Mn3+, and Mn4+, and Os5+ references, respectively (see Fig. 2). \nThe spectral weight of the L 3 white line shifts to higher energies by 1 eV or more as the Mn valence 10 state increases by one: from Mn2+ (MnO) to Mn3+ (LaMnO 3) and further to Mn4+(SrMnO 3)[44,45]. \nThe similar energy position and multiplet spectral featur es of the Mn-L2,3 of Ca 2MnOsO 6 and \nLaMnO 3 demonstrate  the trivalent state of Mn in Ca 2MnOsO 6. Analogously, the Os- L3 spectrum of \nCa2MnOsO 6 locates at the same energy as that of Ca 2FeOsO 6, indicating the pentavalent state of the \nOs[7,47], thereby fulfilling the charge balance requirement for the Mn3+/Os5+ state. Consequently, \nthe valence state of Ca 2MnOsO 6 (Mn3+, Os5+) is different from that of the analogous compound \nCa2MnRuO 6, in which the mixed valent states of (Mn3+, Ru5+) and (Mn4+, Ru4+) are degenerate in \nenergy[17].  \n \nFig. 2 (a) Room temperature Mn- L2,3 XAS spectra of Ca 2MnOsO 6 and of MnO[44], LaMnO 3[44], \nSrMnO 3[45] as reference compounds for Mn2+, Mn3+, and Mn4+ valence states. (b) Os- L3 XAS \nspectra of Ca 2MnOsO 6 and of Sr 2FeOsO 6[46]as an Os5+ reference.  \n \nC. Electrical transport \n11  Fig. 3 shows an increasing resistivity ρ(T) of Ca2MnOsO 6 with lowering the temperature, \nexperimentally demonstrating a semiconducting-like behavior. Interestingly, it is qualitatively and \nquantitatively different from the half-metallic behavior observed for the related compound \nCa2MnRuO 6[17,19]. The temperature dependence of ρ(T) is poorly characterized by the Arrhenius \nmodel at low temperatures (< 300 K, Inset of Fig. 3); however, a variable-range-hopping conduction \nmodel better explains the data (inset of Fig. 3)[48]. Although the Arrhenius model does not exactly \nfit the data, we can roughly estimate the lower limit of the thermal activation energy in the \nhigh-temperature region (> 300 K) as ~0.11 eV from the linear fit as shown in the inset of Fig. 3.  \n \nFig. 3 Temperature dependence of ρ for polycrystalline Ca 2MnOsO 6. The inset shows an \nalternative plot of the data. The blue line indicates a fitting to the Arrhenius law at high temperatures \n(> 300 K).  \n \nD. Magnetic properties  \n The magnetic susceptibility χ(T) curves measured in the zero field cooled (ZFC) and field \ncooled (FC) conditions for Ca 2MnOsO 6 are shown in Fig. 4. Upon cooling, there is an increasing \nvalue of χ at around 305 K, indicating the establishment of a FM-like transition. The divergence \nbetween the ZFC and FC curves at Tc is negligible, but it becomes prominent at approximately 200 K, \n12 suggesting a possible formation of magnetic domains, or the like in the measurement process. We \nwould like to note that the range of the high-temperature part (> Tc) of the χ(T) curve is too narrow \nfor a meaningful Curie–Weiss analysis (see the right side of Fig. 4).  \n \nFig. 4 Temperature dependence of χ of polycrystalline Ca 2MnOsO 6 measured in a field of 10 kOe. \nAn alternative plot ( T vs. χ-1) is shown on the right side.  \n \n The AC χ (= χ’ + iχ’’) of Ca 2MnOsO 6 was measured at temperatures between 5 K and 320 K, \nand the χ’ and χ’’ vs. T curves are shown in Fig. 5. A sharp peak is observed at 305 K, which is \nconsistent with the onset temperature of the χ(T) measurement. In contrast to multiple transitions \nfound for Ca 2MnRuO 6, no additional anomaly is detected, indicating a single magnetic transition \nover the temperature range. Moreover, it substantiates that the divergence between the ZFC and FC \ncurves in the χ(T) measurements is not caused by a magnetic transition. In Fig. 6, the temperature \ndependence of the heat capacity Cp shows an anomaly evolving at 305 K in zero magnetic field, and \nit becomes broader when a magnetic field of 70 kOe is applied. This observation supports that this \nanomaly is caused by a magnetic transition. \n13   \nFig. 5 (a) Real and (b) imaginary components of ac-magnetic susceptibility for polycrystalline \nCa2MnOsO 6 measured in an ac-magnetic field ( Hac) of 5 Oe at various frequencies. The inset shows \na horizontal expansion around the magnetic transition temperature.  \n \nFig. 6 Temperature dependence of specific heat capacity of Ca 2MnOsO 6.  \n \n The field-dependence of the magnetization was measured at several temperatures between 2 \n14 K and 350 K (see Fig. 7). The linear feature at 350 K confirms the paramagnetic behavior above Tc. \nIn contrast, magnetic hysteresis appears at temperatures below Tc, suggesting that a FM component is \ninvolved in the magnetically ordered state. The magnetization at 2 K and 50 kOe is 1.40 μ B/f.u. \nwhich indicates the partial cancellation of the Mn and Os magnetic moments.  The separate Mn and \nOs contributions to the total magnetic moment will be disentangled using the element specificity of \nX-ray magnetic circular dichroism ( vide infra ).  \n \n \nFig. 7 Isothermal magnetization of polycrystalline Ca 2MnOsO 6 measured at various temperatures.  \n \nE. X-ray magnetic circular dichroism \n Figure 8a shows the Mn- L2,3 XAS spectrum (green) and the XMCD spectrum (blue) which \nis the difference between circularly polarized light with positive and negative helicities in an applied \nmagnetic field of 60 kOe at 20 K. The XMCD signal can be clearly seen in Fig.8a. In order to extract \nthe Mn moment, we performed the well-established configuration-interaction cluster calculations \nusing the XTLS code[49]. The method uses a MnO 6 cluster, which includes explicitly the full atomic \nmultiplet interaction, the hybridization of the Mn with the oxygen ligands, and the crystal field acting \non the Mn ions. The hybridization strengths and the crystal field parameters were taken from Ref.[50]. \n15 Fig. 8b shows the calculated XAS (green) and XMCD (blue) for the Mn cluster in an exchange field \n(Hex) of 30 meV. With this field (the energy scale of which reflects the experimentally determined Tc of \n305 K) the Mn is fully magnetized having a moment of 3.9 μ B. We observe that the line shape of the \ncalculated XMCD spectrum is very similar to the experimental one, but we also notice that its \nmagnitude is larger than measured. To reproduce the experimental size of XMCD signal in Fig.8a, \nwe need to rescale the calculated XMCD spectrum by a factor of 1/3 as shown in Fig. 8b. This \nimplies that the XMCD experiment presents the net Mn moment of 3.9/3 = 1.3 μ B.   \n \nFig. 8 (a) Mn- L2,3 XAS (green) and XMCD (blue) spectra of Ca 2MnOsO 6 measured at 20 K under \na magnetic field of 60 kOe and (b) the calculated XAS (green) and XMCD (blue) rescaled by a factor \n1/3. \n \n Figure 9 shows the Os- L2,3 XAS spectrum (green curve) together with the XMCD spectrum \n(blue curve) measured below Tc in an applied magnetic field of 13 kOe at 4 K. The XMCD signal of \n16 the Os has an opposite sign as that of the Mn, which indicates an antiparallel alignment of Os \nmagnetic moment with that of the Mn. We have also performed configuration-interaction cluster \ncalculations using an OsO 6 cluster with parameters taken from Ref.[51]. Fig. 9b shows the calculated \nXAS (green curve) and XMCD (blue curve) spectra. With an exchange field of 30 meV ( Tc of 305 K) \nthe Os is fully magnetized having a moment of 2.05 μ B. The line shapes of the calculated spectra match \nvery well those of the experiment. We notice, however, that the magnitude of the experimental XMCD \nis about 400 times smaller than that of the calculated, which suggests that most of the Os ions are \nmagnetically disordered and/or antiferromagnetically aligned. \n \nFig. 9(a) Os- L2,3 XAS (green) and XMCD (blue multiplied by 400) spectra of Ca 2MnOsO 6 measured \nat 4 K under a magnetic field of 13 kOe. (b) the calculated Os- L2,3 XAS (green) and XMCD (blue) \nspectra. \n \nWe would like to note that our finding from XMCD that the net Mn moment is 1.3 μ B and that \n17 the net Os moment is very small (antiparallel aligned) is in excellent agreement with the \nmagnetization measurement (see the previous section) which yielded a total net moment of 1.40 \nμB/f.u.  \nF. Density functional theory (DFT) calculation   \n We investigated the electronic and magnetic ground state of Ca 2MnOsO 6 by DFT \ncalculations. In this theoretical study, we investigated a hypothetically ordered phase ( P21/n; see \nTable S1, Figs. S1, and S2 in supplementary materials [52]). The symmetry of the P21/n structure has \nbeen reduced further to lower symmetry P-1 (space group: 2) which gives rise to two in-equivalent \natoms each of Os and Mn, respectively. The magnetic ground state of the ordered phase was studied \nby calculating the total energy for each of the five different magnetic states, namely FM (FM- ↑↑↑↑; \nOs1: Os2: Mn1: Mn2), two AFM (AFM-1-↑↓↑↓ and AFM-2-↑↓↓↑), and two FIM (FIM-1-↑↑↓↓ and \nFIM-2-↑↑↑↓) states (see Fig. 10). From the total energy calculations, FIM-1 is found to have the \nlowest energy, with an energy difference of 198 meV/unit cell to that of the next lowest order \n(AFM-1). The spin-polarized DFT calculations for the hypothetically ordered Ca 2MnOsO 6 suggest \nthat the magnetic ground state is certainly FIM-1 with the anti-parallel alignment of Mn moments to \nOs. The electronic ground state is HM (the HM gap is approximately 0.87 eV in the spin-up channel) \nwithin the GGA functional. Additionally, SOC effects has been considered where the spins are \nallowed to ordered along the [100] and [001] direction to check its influence on the electronic and \nrelated properties. Significant difference was not found for the net moments for Mn and Os (see \nTable 3), while a reasonable change has been observed in the electronic behavior. 18  \nFig. 10  Possible spin arrangements considered in the theoretical calculation for Ca 2MnOsO 6 with \nordered Mn and Os atoms; ferromagnetic (FM), antiferromagnetic (AFM), and ferrimagnetic (FIM) \nconfigurations.   \n \nTable 3 Spin < ms> and orbital < ml> moments measured by XMCD and two sets of theoretical \nvalues for ordered Ca 2MnOsO 6  \n XMCD a GGA+SO b \nOrdered structure \n[100] Ordered structure \n[001] \nMn Os Mn Os Mn Os \nms (μB/atom) 1.3 -0.005 3.13 -1.38 3.14 -1.41 \nml (μB/atom) 0.0 0.000 0.02 0.02 0.01 0.08 \nms+ml \n(μB/atom) 1.3 -0.005 3.15 -1.36 3.16 -1.33 \na T = 4 K and H = 13 kOe for Os and 20 K and 60 kOe for Mn  \nb Standard generalized-gradient approximation with spin-orbit interaction  \n The electronic band structure and the spin-resolved DOS (total and partial) within GGA \nfunctional are shown in Fig. 11 for the ordered phase, which is HM with an insulating band gap of \n~0.87 eV in the spin-up channel and metallic state in the spin-down channel. The major contributions \n19 to the total DOS around the EF are attributed to the Os-5d orbitals in both spin channels, which \nhybridize strongly with the O-2p orbitals. From the partial DOS and band structure (see Fig. 11 and \nFig. S3a [52]), the Os-5d orbitals in the spin-up channel are found to be fully occupied by 5d- t2g \norbitals, which shows the Os5+ (5d3) state. The six bands at and around the Fermi level ranging from \n-1.5 eV upto ~ +0.7 eV are from the Os-5d -t2g orbitals which hybridizes strongly with the O-2p \norbitals (see Fig. 11 and Fig. S3a [52]) in spin-up and spin down channel. On the other hand, Mn-3d \nstates are found to dominate mostly above +0.7 eV in the conduction region in spin-up channel, \nwhile in spin-down channel three d-orbitals (d- t2g) are fully occupied lying below -1.5 eV while two \nof the d-states (d- eg) cross the Fermi level signaling the partial occupancy in spin-down channel. This \nfeature is close to the ionic picture Mn3+ which should have four d-states occupied. With SOC taken \ninto account (within GGA), band splitting was observed (see Fig. S3a [52]), however, the overall \nband gap did not open. This suggests the necessity of using GGA+U. \n \nFig. 11  (a) Band dispersion and (b) DOS of Ca 2MnOsO 6 with fully ordered Mn and Os atoms \nwithin GGA (optimized).  \n20  \n To further investigate the experimentally observed semiconducting transport, we performed \na GGA+U calculation with U values ranging from 0–5 eV for Mn and 0–2.5 eV for Os[32-35]. The \nHM state changed to a semiconducting state at U(Mn) = 4 eV and U(Os) = 1.25 eV or higher[19]. \nWith U = 5 eV for Mn and 1.5 eV for Os, the calculated DOS and band structure are shown in Fig. \n12. A band gap of 0.11 eV was achieved, which agrees well with the experiment (~0.11 eV). With \nSOC, an indirect band gap of 0.14 eV is found between the high symmetry point Γ and M, and a \ndirect band gap of 0.21 eV at Γ point in the band dispersion shown in Fig. S3b [52]. Splitting of the \norbitals for the easy axis [001] is large with a gap size of 0.14 eV. From the scalar relativistic DOS \nand band structure shown in Fig. 12, and relativistic fat bands in Fig. S3b [52], the Os-5d states are \nfound to fully occupy with three t2g orbitals in the spin-up channel and the remaining eg orbitals are \nunoccupied, while in the spin-down channel, all the d-bands lie above EF. Most of the bands from \nMn-3d around EF hybridize with the O-2p and Os-5d states. The broad band lying just above EF is \nmainly contributed by the Mn-3d orbital; the exchange energy splitting is of the order of ~1.75 eV. \nBesides, MAE has been considered for the FIM-1 ground state. The calculated MAE is 29.6 \nmeV/unit cell with its easy axis along the cubic [100] direction[53] within GGA.  In contrast, within \nGGA+U, the easy axis is found along the out-of-plane with MAE of 3.46 meV/unit cell with all the \nspins aligned along the [001] direction. 21  \nFig. 12 (a) Band dispersion and (b) DOS of Ca 2MnOsO 6 with fully ordered Mn and Os atoms \nwithin GGA+ U (U = 5 eV for Mn and 1.5 eV for Os atoms, respectively). \n \nTo evaluate the impact from anti-site disorder on the electronic ground state, five different \nanti-site-disordered configurations (AS1, AS2, AS3, AS4, and AS5) were created as shown in the \nFig. S4 [52], where a 1 x 1 x 2 supercell has been generated which corresponds to a total of 40 atoms \nin a unit cell. DFT calculations were first carried out on these five different anti-site-disorder cases \nwith FIM spin state. However, in comparison with the ordered FIM-1 case these anti-site-disordered \ncases showed higher energies within ~0.6 to 1.37 meV/unit-cell (see Table S2 [52]), with energies \nthat AS1 < AS2 < AS3 < AS4 < AS5. We further calculated these anti-site-disordered cases with FM \nspin state. Energy differences (∆E) between FM and FIM states in the respective anti-site-disordered \ncases are summarized in Table S3 [52].  FIM states showed lower energies, indicating that AFM \nalignment is favorable between Mn-Os, while FM coupling may be favorable among the Mn-Mn or \nOs-Os, giving rise to long-range FIM order in the anti-site-disordered Ca 2MnOsO 6. The electronic \n22 band gap is noted for AS1 and AS2 cases, while others remain metallic within GGA+U (see table S2 \n[52]). \n \nDISCUSSION \nWe synthesized an anti-site-disordered double perovskite Ca 2MnOsO 6 (equivalent to \nCaMn0.5Os0.5O3) with Mn3+ and Os5+ ions randomly distributed at the B site of the perovskite. It is \nnoteworthy that neighbor compounds Ca 2FeOsO 6 and Ca 2CrOsO 6 all crystallize in ordered double \nperovskite structure with rocksalt arrangement of FeO 6/CrO6 and OsO 6 octahedra. The B and B' site \nordering in double perovskite is generally determined by the charge difference and effective ionic \nsize difference of B and B' ions[54].  Given that Mn3+ has the same effective ionic radii (0.645  Å) as \nthat of Fe3+ in octahedral coordination[55], the reason for the absence of Mn3+ and Os5+ ordering in \nCa2MnOsO 6 is unclear, although perhaps the JT distortions induced around the Mn3+ ions may play \nan important role. Despite the absence of Mn3+ and Os5+ ordering, Ca 2MnOsO 6 is electrically \nsemiconducting and features a remarkably high Tc above room-temperature (305 K). \nTo rationalize the magnetic properties, we first consider the double perovskite Ca 2MnOsO 6 \nwith perfect Mn3+ and Os5+ ordering. Our band structure calculations suggest that this would lead to \na FIM ground state, where the moments of Mn3+ (t2g3eg1) and Os5+ (t2g3) are arranged antiparallel. In \norder to interpret this result, we notice that the nearest-neighbor Mn3+–O–Os5+ super-exchange \ncoupling can either be FM (the Mn3+ eg electrons hop on to the empty Os5+ eg states) or AFM (the \ndown spin Os5+ t2g electrons hop on to the Mn3+ t2g states). In comparison to 3d ions, the 5d ions have \nmuch enhanced crystal field splitting, therefore the FM coupling involving the virtual hopping \nbetween e g orbitals becomes weak. In particular when the crystal structure is distorted this FM \ncoupling can become even weaker[47,56], with the result that the AFM Mn3+–O–Os5+ coupling is the \ndominant interaction and generate the FIM state in the hypothetically ordered Ca 2MnOsO 6.   23 Introducing now anti-site disorder in Ca 2MnOsO 6, we will have also NN Os5+–O–Os5+ and \nMn3+–O–Mn3+ interactions. The Os5+–O–Os5+ coupling is AFM as shown in AFM NaOsO 3[43,57]. \nThe Os5+ moments are therefore expected to compensate each other in such a situation. This would \nthen be consistent with the experimental observation that the Os XMCD effect is very small. The \ncase concerning the Mn3+–O–Mn3+ interactions is more complex. Depending on the orbital \norientation due to the JT distortion of Mn3+, the interaction can be either FM or AFM which has been \nwell discussed in LaMnO 3[42,58-61]. In our crystal structure analysis, the contribution from the JT \neffect of Mn3+ was clearly noticed (see crystal structure part). The moment, 1.3 μ B/Mn3+, obtained \nfrom Mn XMCD is not negligible, but smaller than the theoretically calculated 3.15 μ B/Mn3+ for \nordered Ca 2MnOsO 6 phase (see Table 3) and also smaller than the 3.9 μ B/Mn3+ from the cluster \ncalculations. This indicates that there are competing FM and AFM Mn3+–O–Mn3+ interactions. \nStudies of highly anti-site disordered double perovskite A2Mn3+B'O6 (A = Ca, Sr; B' = Ta, Sb), in \nwhich only Mn3+ is magnetic, found that the FM interactions are dominant in these compounds with \npositive Weiss temperatures varied from 64–107 K [62]. A large FM component with a \nmagnetization of about 1.47 μ B/Mn3+ (at 5 K and 50 kOe) was observed in Sr 2MnTaO 6 despite its \nglassy state at low temperatures [62]. These results are thus similar to our Ca 2MnOsO 6 case. \n  Recent theoretical and experimental studies of the anti-site disorder on the magnetic \nproperties of Sr 2FeMoO 6, Sr2FeReO 6, Sr2CrReO 6, Sr2CrOsO 6, and Sr 2FeRuO 6 suggested that the \nanti-site disorder suppresses Tc or destroys the magnetically ordered state, because the anti-site \ndisorder introduces strong AFM Fe3+–O–Fe3+ and Cr3+–O–Cr3+ interactions to hinder the long-range \nmagnetic order[11-14,63-65]. Different from Fe3+ and Cr3+, the Mn3+–O–Mn3+ super-exchange \ninteractions can be both FM and AFM[42,58-61] depending on the orbital orientation, which may \nhelp to maintain net partial Mn3+ moments and also a high Tc in this anti-site-disordered Ca 2MnOsO 6. \nThis scenario is likely to be different from the ferrimagnetism of anti-site-disordered \nCa2MnRuO 6[17,18], where the presence of mixed valence states of Mn3+/Mn4+–Ru4+/Ru5+ would 24 also lead to a ferromagnetic double exchange mechanism for the Mn3+–Mn4+ and a HM state for \nCa2MnRuO 6[17-19]. \n4. CONCLUSION \n Anti-site-disordered Ca 2MnOsO 6 was synthesized for the first time under high-pressure (6 \nGPa). It crystallizes into an orthorhombic structure (space group: Pnma), in which trivalent Mn and \npentavalent Os share the Wycoff 4 b position without an ordered arrangement. Ca 2MnOsO 6 is \nelectrically semiconducting. XAS measurement confirmed the trivalent Mn and pentavalent Os \noxidation states. The XMCD reveals the antiparallel alignment of the net Mn and Os magnetic \nmoments. Remarkable is that the net Mn moment is only about 1/3 of its full Mn3+ value and that the \nnet Os moment is very small. We have discussed the strength and sign of various inter-site exchange \ninteractions in this material using data from band structure calculations, taking into account also the \npresence of anti-site disorder and JT distortions around the Mn3+ ions.  The Tc = 305 K is the second \nhighest in the material category of so-called disordered ferromagnets and could therefore be useful in \nthe development of oxide spintronic devices that are less sensitive for anti-site disorder during \nfabrication.  \nACKNOWLEDGMENTS  \nMPG thanks the Alexander von Humboldt Foundation for the financial support through HERMES \nprogram and Ulrike Nitzsche for the technical support. This study was supported in part by JSPS \nKAKENHI Grant Number JP16H04501, a research grant from Nippon Sheet Glass Foundation for \nMaterials and Engineering (#40-37), and Innovative Science and Technology Initiative for Security, \nATLA, Japan. We acknowledge support from the Max Planck-POSTECH-Hsinchu Center for \nComplex Phase Materials. The work in Dresden was partially supported by the Deutsche \nForschungsgemeinschaft through SFB 1143 (project-id 247310070). The synchrotron radiation \nexperiments were performed at the NIMS synchrotron X-ray station at SPring-8 with the approval of 25 the Japan Synchrotron Radiation Research Institute (Proposal Numbers 2017A4503, 2017B4502, \n2018A4501, and 2018B4502). MG acknowledges support of NSF-DMR-1507252 grant of USA. \n  26 REFERENCES \n[1] K. I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature 395, 677 (1998). \n[2] R. Morrow, J.R. Soliz, A.J. Hauser, J.L. Gallagher, M.A. Susner, M.D. Sumption, A.A. 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Solid State Chem. 78, 281 (1989). \n \n "    },    {        "title": "2001.06918v5.Dephasing_of_Transverse_Spin_Current_in_Ferrimagnetic_Alloys.pdf",        "content": "1 \n Dephasing of Transverse Spin Current in Ferrimagnetic Alloys  \nYoungmin Lim1, Behrouz Khodadadi1, Jie-Fang Li2, Dwight Viehland2, Aurelien Manchon3,4, \nSatoru Emori1,* \n1. Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA  \n2. Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, \nUSA \n3. Physical Science and Engineering Division (PSE), King Abdullah University of Science and \nTechnology (KAUST), T huwal 23955 -6900, Saudi Arabia  \n4. Aix -Marseille Univ, CNRS, CINaM, Marseille, France  \n* email: semori@vt.edu  \n \nIt has been predicted that transverse spin current can propagate coherently (without \ndephasing) over a long distance in antiferromagnetically orde red metals. Here, we \nestimate  the dephasing length of transverse spin current in ferrimagnetic CoGd alloys by \nspin pumping measurements  across the compensation point . A modified drift -diffusion \nmodel, which accounts for spin -current transmission through the ferrimagnet, reveals \nthat the  dephasing length is about 4 -5 times longer in nearly compensated CoGd than in \nferromagnetic metals. This finding suggest s that antife rromagnetic order can mitigate \nspin dephasing – in a manner analogous to spin echo rephasing for nuclear and qubit \nspin systems – even in structurally disordered alloys at room temperature. We also  find \nevidence that  transverse spin current interacts  more strongly with the Co sublattice than \nthe Gd sublattice.   Our result s provide fundamental insight s into the interplay between \nspin current and antiferromagnetic order, which are crucial for engineering spin torque \neffects in ferrimagnetic and antiferromagnetic metals.  \n  2 \n I. INTRODUCTION  \nA spin current is said to be coherent when the spin polarization of its carriers, such as \nelectrons, is locked in a uniform orientation or precessional phase.  How far a spin current \npropagate s before decohering underpins various phenomena in solids  [1,2]. Spin decoherence \ncan generally arise from spin-flip scattering , where the carrier spin polarization is randomized \nvia momentum scattering  [3,4]. In magnetic materials, electronic spin current polarized \ntransverse to  the magnetization can also decohere by dephasing , where the total carrier spin \npolarization vanishes  due to the destructive interferences of precessing spins (i.e., upon \naveraging over the Fermi surface ) [5–9]. In typical ferromagnetic metals ( FMs), the dephas ing \nlength 𝜆𝑑𝑝 is only ≈1 nm  [5–7,10]  whereas the spin -flip (diffusion) length 𝜆𝑠𝑓 may be \nconsiderably longer (e.g., ≈10 nm)  [3,4], such that dephasing dominates the decoherence of \ntransverse spin current.   \nFigure 1(a) qualitatively illustrates  the dephasing of a coherent electronic spin current  in \na FM spin sink . In this particular illustration, a coherent ac transverse spin current is excited by \nferromagnetic resonance (FMR) spin pumping  [11], although in general a coherent transverse \nspin current may be generated by other means (e.g., dc electric current  spin-polarized \ntransverse to the magnetization  [5,6,12] ). This  spin current , carried by electrons,  then \npropagates coherently through the normal metal spacer  (e.g., Cu, where 𝜆𝑠𝑓~100 nm is much \ngreater than the typical spacer thickness ) [13]. However, this spin current enters  the FM spin \nsink with a wide distribution of incident wavevectors1, spa nned by the Fermi surface of the FM. \nElectronic spins with different wavevectors require different times  to reach a certain depth in the \nFM, thereby spending different times in the exchange field. Thus, even though these electronic \nspins enter the FM with the same phase, they  precess about the exchange field in the FM by \ndifferent amounts . Within a few atomic monolayers in the FM, the transverse spin  polarization \naverages to zero; the spin current dephases  within a short length scale 𝜆𝑑𝑝 ≈ 1 nm .   \n                                                           \n1 An insulating tunnel barrier is known to filter the incident wavevectors to a narrow distribution  [94]. This \nfiltering effect can reduce dephasing and thus extend 𝜆𝑑𝑝.  3 \n  \nFIG. 1.  Dephasing of a coherent transverse spin current excited by ferromagnetic resonance (FMR) in the \nspin source. The spin current carried by electrons is coherent in the normal metal (NM) spacer layer \n(indicated by the aligned black arrows), but ente rs the spin sink with different incident wavevectors \n(dashed gray lines). (a) In the ferromagnetic metal ( FM) spin sink, the propagating spins accumulate \ndifferent precessional phases in the ferromagnetic exchange field (red vertical arrows) and completely  \ndephase within a short distance. (b) In the ideal antiferromagnetic metal ( AFM ) or ferrimagnetic metal \n(FIM), the spin current does not dephase completely in the alternating antiferromagnetic exchange field \n(blue and green vertical arrows), as any precess ion at one sublattice is compensated by the opposite \nprecession at the other sublattice.  In the case of a FIM that is an alloy of a transition metal (TM , such as \nCo) and rare -earth metal (RE , such as Gd ), the TM constitutes one sublattice (e.g., blue arrow s) and the \nRE constitutes the other  sublattice  (e.g., green arrow s).  \n \nTransverse spin currents in antiferromagnetically ordered metals have been predicted to \nexhibit longer 𝜆𝑑𝑝 [14–17]. This prediction may apply not only to intrinsic antiferromagnetic \nmetals (AFMs) but also compensated ferrimagnetic metals (FIMs) , which  consist of transition -\nmetal (TM) and rare -earth -metal (RE) magnetic sublattices  that are antiferromagnetically \ncoupled to each other  [18]. In the ideal case as illustrated in Fig. 1(b), the spin current interacts \nwith the staggered antiferromagnetic exchange field whose direction alternates at the atomic \nlength scale. The propagating spins precess in alternating directions as they move from one \nmagnetic sublattice to the next, such that spin dephasing is suppressed over multiple \nmonolayers. This c ance llation of dephasing in AFMs and FIMs is analogous to spin rephasing \nNM spacer FM sink\nAFM/FIM sink\nsource\nNM spacer source(a)\n(b)4 \n by π-pulses (Hahn spin echo method) in nuclear magnetic resonance  [19], which has recently \ninspired several approaches of mitigating decoherence of qubit spin systems  [20–22].  \nThe above idealized picture for extended coherence in antiferromagnetically ordered \nmetals (Fig. 1(b)) assumes a spin current without any sca ttering and simple layer -by-layer \nalternating collinear magnetic order. Finite scattering, spin -orbit coupling, and complex \nmagnetization states in real materials  may disrupt transverse spin coherence  [23–25]. The \ntransverse spin coherence length 𝜆𝑐 accounting for both spin -flip scattering and spin dephasing \nis given by  [10,26] , \n1\n𝜆𝑐=Re[√1\n𝜆𝑠𝑓2−𝑖\n𝜆𝑑𝑝2].                          (1)      \nThus, in real AFMs and FIMs, a  shorter coherence length results from reduced 𝜆𝑠𝑓 due to \nincreased spin -flip rates, or reduced 𝜆𝑑𝑝 due to momentum scattering and non -collinear \nmagnetic order that prevents perfect cancellation of dephasing  [23,24] . Most experiments on \nAFMs (e.g., polycrystalline IrMn) indeed show short coherence lengths of  ≈1 nm  [7,27 –30].  \nNevertheless, a recent experim ental study utilizing a spin -galvanic detection \nmethod  [31–35] has reported a long coherence length of >10 nm  at room temperature in FIM \nCoTb  [18]. The report in Ref.  [18] is quite surprising considering the strong spin -orbit coupling of \nCoTb, primarily from RE Tb with a large orbital angular momentum, which can result in \nincreased spin -flip scattering  [36–38] and noncollinear sperimagnetic order  [39–41]. TM and RE \nelements also tend to form amorphous alloys  [39–42], whose structural disorder may result in \nfurther scattering and deviation from layer -by-layer antiferromagnetic order. It therefore remains \na critical issue to confirm whether the cancellation of  dephasing (as depicted in Fig. 1(b)) \nactually extends transverse spin coherence in antiferromagnetically  ordered metals, particularly  \nstructurally disordered FIMs.  \nHere, we test for the suppressed dephasing of transverse spin current in ferrimagnetic \nalloys. Our experimental test consists of FMR  spin pumping measurements  [7,28]  on a series of \namorphous FIM CoGd spin sinks, which exhibit significantly weaker spin -orbit coupling than \nCoTb due to the nominally zero orbital angular momentum of RE Gd.  Our experimental results \ncombined with  a modified drift -diffusion model  [9,10,43,44]  reveal that spin dephasing is indeed \npartially  cancelled  in nearly compensated CoGd , with 𝜆𝑑𝑝 extended by a factor of 4 -5 compared \nto that for FMs . Moreover, we find evidence  that transverse spin current interacts more strongly \nwith the TM Co sublattice  than the RE Gd sublattice . Our results  suggest  that, even in the \npresence of substantial structural disorder, the antiferromagnetic ally coupled sublattices  in FIMs 5 \n can mitigate the  decoherence of transverse spin current. On the other hand, the maximum 𝜆𝑑𝑝 \nof ≈5 nm in FIM CoGd found here is  quantitatively at odds with the report of 𝜆𝑑𝑝> 10 nm in FIM \nCoTb  [18]. Our study  sheds light on the interaction of  transverse spin current with \nantiferromagnetic order, which underpins spin  torque control of FIMs and AFMs for fast \nspintronic devices  [45–47].  \n \nII. FILM GROWTH AND STATIC MAGNETIC PROPERTIES  \nWe deposited spin -valve -like stacks of Ti(3)/Cu(3)/Ni 80Fe20(7)/Cu(4)/Co 100-xGd x(d)/Ti(3) \n(unit: nm) with x = 0, 20, 22, 23, 25, 28, and 30 by dc magnetron sputtering on Si/SiO 2 \nsubstrates. The base pressure in the deposition chamber was better than 8 10-8 Torr. The  Ar \nsputtering gas pressure was 3 mTorr. The Ti(3)/Cu(3) seed layer promotes the growth of \nNi80Fe20 with low Gilbert damping and minimal inhomogeneous linewidth broadening, whereas \nthe Ti(3) capping layer protects the stack from oxidation. FIM Co 100-xGd x films with various Gd \nconcentrations (x in atomic %) were deposited by co -sputtering Co and Gd targets at different \nGd sputtering powers (resulting in an uncertainty in composition of ≈±0.5 at.% Gd), except for \nCo80Gd 20 and Co 70Gd 30 films that were deposite d by sputtering compositional alloy targets.  The \ndeposition rate of each target  was calibrated by x -ray reflectivity and was set at 0.020 nm/s  for \nTi, 0.144 nm/s for Cu, 0.054 nm/s for NiFe, 0.011 or  0.020 nm/s for Co, 0.008 – 0.020 nm/s for \nGd, 0.01 4 nm/s for Co 80Gd 20, and 0.01 2 nm/s for Co 70Gd 30. \nWe performed vibrating sample magnetometry (with a Microsense EZ9 VSM) to identify \nthe magnetic compensation composition at room temperature for Co 100-xGd x films. We measured \nCo100-xGd x as single -layer films and as part of the spin -valve -like stacks (NiFe/Cu/CoGd), both \nseeded by Ti(3)/Cu(3) and capped by Ti(3). CoGd in both types of samples exhibited identical \nresults within experimental uncertainty. In the spin -valve -like stacks, the C u spacer layer \nsuppresses static exchange coupling between the NiFe and CoGd layers. This interlayer \nexchange decoupling is corroborated by magnetometry results (e.g., Fig. 2 (a)) that  indicate \nseparate switching for the NiFe and CoGd layers.  \nFigure 2(b -g) summarizes the composition dependence of the static magnetic properties \nof Co 100-xGd x. Our results  corroborate the well -known trend in ferrimagnetic alloys: the saturation \nmagnetization converges toward zero and coercivity diverges near the composition at  which the \nmagnetic moments of the Co and Gd sublattices compensate. Comparing the results for \ndifferent CoGd thicknesses, we find that the magnetization compensation composition shifts \ntoward higher Gd content with decreasing CoGd thickness. A similar thi ckness dependence of \nthe compensation composition in TM -RE FIM films has been seen in prior studies  [18]. Since 6 \n precise determination of the mag netic compensation composition wa s difficult for smaller Co 100-\nxGd x thicknesses, we tentatively identify t he magnetic compensation composition window to be \nx ≈ 22-25, which is what we find for 5 -nm-thick CoGd . We also remark that the angular \nmomentum compensation composition is expected to be ≈1 Gd at. % below the magnetic \ncompensation composition, considering  the g-factors of Co and Gd ( gCo ≈ 2.15, gGd = 2.0)  [48]. \nCoGd layers in our stack structures do not show perpendicular magnetic anisotropy  [18,49 –58], \ni.e., CoGd films here are in -plane magnetized  [59–62].  \n \n-50 -25 0 25 50\n0200400\n02040\n0100200\n04080\n0100200\n15 20 25 300816M (a.u.)\nm0H (mT)m0Hc (mT)    Ms (kA/m)\n    Ms (kA/m)CoGd 40 nm(b)\n(c)\n(d)\n(e)\n(f)\n(g)   Ms (kA/m) m0Hc (mT)(a)\n m0Hc (mT)\n  CoGd 10 nm\n   CoGd 5 nm\n \nGd at.%\n \nFIG. 2.   (a) Hysteresis loop of Ni 80Fe20(7)/Cu(4)/Co 75Gd25(13) (unit: nm) . The Ni80Fe20 and Co 75Gd25 \nmagnetization s switch separately at in -plane fields ≈ 0.2 mT and ≈13 mT, respectively.  (b,d,f) Saturation 7 \n magnetization Ms and (c,e,g) coercivity μ0Hc  of Co 100-xGdx with thicknesses of 40 nm (b,c), 10 nm (d,e), \nand 5 nm (f,g).  \n \nIII. CHARACTERIZATION OF SPIN TRANSPORT BY SPIN PUMPING  \nA. FMR Spin Pumping Experiment  \nThe multilayer stacks (described in Sec. II) in our study consist of a NiFe spin source \nand a Co100-xGd x spin sink, separated by a diamagnetic Cu spacer.  A coherent spin current \ngenerated by FMR  [11,13]  in NiFe propagates through the Cu spacer and decoheres in the \nCo100-xGd x spin sink, yielding nonlocal Gilbert damping  [7,28] . The Cu  spacer layer suppresses \nexchange coupling  – and hence direct magnon coupling – between the NiFe and CoGd \nlayers  [63]. The diamagnetic Cu spacer also accommodates spin transport mediated solely by \nconduction electrons, such that direct interlayer magnon coupling  [17,64 –66] does not play a \nrole here2.  \nIn our FMR spin pumping experiments performed at room temperature , the half -width -at-\nhalf-maximum linewidth ∆𝐻 of the NiFe spin source  is measured at microwave frequencies f = \n2-20 GHz . The d etails of the FMR measurement method are in Appendix A. The FMR response \nof the NiFe layer is readily separated  from that of pure Co (x = 0), and CoGd did not yield FMR \nsignals above our instrumental background  (see Fig. 10 in Appendix A) . Thus, as shown in Fig. \n3, the Gilbert damping parameter α for the NiFe layer is quantified from the f dependence of  ∆𝐻 \nthrough  the linear fit,  \n 𝜇0∆𝐻=𝜇0∆𝐻0+ℎ\n𝑔𝜇𝐵𝛼𝑓,  (2)  \nwhere 𝑔≈2.1 is the Landé g-factor of Ni 80Fe20, 𝜇0 is the permeability of free space, ℎ is \nPlanck’s constant, 𝜇𝐵 is the Bohr magneton, 𝜇0∆𝐻0 (< 0.2 mT) is the zero -frequency linewidth \nattributed to m agnetic inhomogeneity  [67]. For NiFe without a spin sink, we obtain αno-sink ≈ \n0.0067, similar to typically reported values for Ni 80Fe20 [68,69] .   \nA finite thickness 𝑑 of spin sink results in a damping parameter αw/sink that is greater than \nαno-sink. For examp le, the damping increases significantly with just 𝑑=1 nm of Co (Fig. 3(a)), \nsuggesting substantial spin absorption by the spin sink. By contrast, a stack structure that \nincludes an insulating layer of Ti -oxide before the spin sink does not s how the enhanc ed \ndamping (Fig. 3 (a)). This observation is consistent with the Ti -oxide layer blocking the spin \ncurrent  [70,71]  between the spin source a nd spin sink layers. Thus, the enhanced damping Δ α = \n                                                           \n2 We note that some of the electronic spin current might be converted into a magnonic spin current  [95] at \nthe Cu/CoGd interface. However, for the sake of simplicity, we assume the dominance of electronic spin \ntransport in CoGd here.  8 \n αw/sink – αno-sink is nonlocal in origin, i.e., due to the spin current propagating through the Cu \nspacer and decohering in the magnetic spin sink  [7,10,11,13,28,43] . The decoherence of \ntransverse spi n current in the spin sink is then directly related to Δ α. This spin pumping method \nbased on nonlocal damping enhancement provides an alternative to the spin-galvanic \nmethod  [18] that is known to contain parasitic voltage signals  unrelated to spin transport  [31–35], \nas discussed further in Sec. IV -C.  \n0 5 10 15 200246810\n0 5 10 15 20 0 5 10 15 20 NiFe/Cu/Co(3)\n NiFe/Cu/Co(1)\n NiFe/Cu/Co(0.6)\n NiFe/Cu/TiOx(3)/Co(1)\n NiFe/CuHWHM Linewidth (mT)\nFrequency (GHz)(a)\nFrequency (GHz)0 5 10 15 20\nFrequency (GHz) NiFe/Cu/Co75Gd25(9)\n NiFe/Cu/Co75Gd25(5)\n NiFe/Cu/Co75Gd25(2.5)\n NiFe/Cu/Co75Gd25(1.6)\n NiFe/Cu(c)\nFrequency (GHz) NiFe/Cu/Co77Gd23(9)\n NiFe/Cu/Co77Gd23(5)\n NiFe/Cu/Co77Gd23(2)\n NiFe/Cu/Co77Gd23(1)\n NiFe/Cu(b)  NiFe/Cu/Co72Gd28(8)\n NiFe/Cu/Co72Gd28(5)\n NiFe/Cu/Co72Gd28(2.1)\n NiFe/Cu/Co72Gd28(1.5)\n NiFe/Cu(d)\n \nFIG. 3.  (a) Half -width -at-half-maximum (HWHM) FMR linewidth versus frequency for stacks with a spin \nsink (NiFe/Cu/Co( d: nm)), a stack without a spin sink (NiFe/Cu), and a stack with an insulating Ti -oxide \nspin blocker before the spin sink (NiFe/Cu/TiO x/Co). (b -d) FMR linewidth versus frequency for stacks with \ndifferent thicknesses (d: nm) of Co 100-xGdx spin sink s, where x = 23 (b), 25 (c), and 28 (d).  The slope \n(proportional to the damping parameter  𝛼, see Eq. (2)) saturates at d ≈ 1 nm for the FM Co spin sink (a), \nwhereas the slope saturates at a much larger d for the FIM Co100-xGdx spin sinks (b -d).  \n \nIn contrast to the large Δ α with an ultrathin FM Co spin sink, the damping enhancement \nwith 𝑑 is more grad ual for FIM CoGd sinks. Figure 3 shows e xemplary linewidth versus \nfrequency results for the spin sink compositions of Co 77Gd 23 (Fig. 3(b)), Co 75Gd 25 (Fig. 3(c)), and \nCo72Gd 28 (Fig. 3(d)) . A damping enhancement similar in magnitude to that of the 1 -nm-thick FM \nCo spin sink is reached only when the CoGd  thickness is several nm. This suggests that \ntransverse spin -current decoherence takes place over a greater length scale in FIM spin sinks \nthan in FM spin sinks .  \nIn Fig. 4, we summarize our experimental results of transverse spin decoherence (i.e., \nΔα) as a function of spin sink thickness 𝑑. It can be seen that Δ α for each spin sink composition \nsatur ates above a sufficiently large  𝑑. This apparent saturation thickness – related to how far \nthe transverse spin current remains coherent  [7,10]  – changes markedly with the spin sink \ncomposition. With FM  Co as the spin sink , the saturation of Δ α occurs at 𝑑≈1 nm, in \nagreement with 𝜆𝑑𝑝 reported before for FMs  [7,10] . By contrast, Δ α saturates at 𝑑≫1 nm for 9 \n FIM CoGd sinks. This observation again implies that transverse spin current remains coherent \ndeeper within FIM sinks than within FM sinks.  \n \n   \n012\n012\n012\n012\n012\n012\n0 5 10012Da (10-3)\nPure Co\nCo80Gd20Da (10-3) Da (10-3)\nCo78Gd22\nCo77Gd23Da (10-3) Da (10-3)\nCo75Gd25\nCo72Gd28Da (10-3) Da (10-3)\nSpin sink thickness, d (nm)Co70Gd30  \nFIG. 4.  Nonlocal damping enhancement Δ α versus spin sink thickness d for CoGd spin sinks with \ndifferent compositions.  \n \nWe now  consider possible mechanisms  of the longer decoherence length s in FIM sinks. \nIncreasing the Gd concentration dilutes  the magnitude of the  exchange field in Co 100-xGd x alloys, \nas evidenced by the reduction of the Curie temperature  from ≈1400 K for pure Co to ≈700 K for  \nCo70Gd 30 [72]. The diluted  exchange field would lead to slower spin dephasing (i.e., longer 𝜆𝑑𝑝), \nthereby requiring a thicker spin sink  for complete spin-current decoherence. This mechanism  \nwould yield 𝜆𝑑𝑝 that is inversely proportional to the Curie temperature  [73], such  that the spin -\nsink thickness 𝑑 at which  Δα saturates would increase monotonically with Gd content. However, \nwe do not observe such a monotonic  trend in Fig. 4 . Rather , the saturation thickness appears to 10 \n plateau or peak at a Gd content of x ≈ 25, which is close to the magnetic compensation \ncomposition window.  \nWe therefore consider an alternative mechanism , where the antiferromagnetically \ncoupled Co and Gd sublattices in the FIM alloys mitigate dephasing, as qualitatively illustrated \nin Fig. 1(b). This mechanism would be expected to maximize  𝜆𝑑𝑝 near the compensation \ncomposition . In the following subsection, we describe and apply a modified spin drift -diffusion \nmodel to estimate 𝜆𝑑𝑝 and examine the possible role of the antiferromagnetic order in the \nmitigation of dephasing  in FIM CoGd .  \n \nB. Modified Drift -Diffusion Model  \nWe wish to model  our experimental results and quantify 𝜆𝑑𝑝 for the series of CoGd sinks. \nThe conventional drift -diffusion model captures spin -flip scattering in nonmagnetic  \nmetals  [11,74,75] , but not spin dephasing that is expected to be significant in magnetic metals. \nThis conventional model also predicts a monotonic increase of Δ α with 𝑑 [11,74,75] , whereas  \nwe observe in Fig. 4 a non-monotonic behavior where Δ α overshoot s before approaching \nsaturation  for some compositions of CoGd . For spin pumping studies with magnetic  spin sinks, \ntypical models assume  that the transverse spin current decoheres by dephasing as soon as it \nenters the FM spin sink, i.e., 𝜆𝑑𝑝=0 [13,75,76] . Others fit a linear increase of Δα with 𝑑 up to \napparent saturation , deriving  𝜆𝑑𝑝=1.2±0.1 nm for FMs  [7,28,73] . However,  it is questionable \nthat this linear cut -off model applies  in a physically meaningful way to our experimental results  \n(Fig. 4 ), in which the increase of Δα to saturation is not generally linear.   \nWe therefore apply  an alternative model that captures  the dephasing (i.e., precession \nand decay) of transmitted transverse spin current in the magnetic spin sinks  by invoking the \ntransmitted spin -mixing conductance  𝑔𝑡↑↓ [5,9,10,43,44,77,78] . As illustrated in Fig. 5, 𝑔𝑡↑↓ \naccounts for the spin current transmitted thro ugh the spin sink, in contrast to the conventional \nreflected  spin-mixing conductance that accounts for the spin current reflected at the spin sink \ninterface  [7,79] . 𝑔𝑡↑↓ is a function of the magnetic spin sink thickness 𝑑 that must vanishes in the \nlimit of 𝑑≫𝜆𝑑𝑝, i.e., when the transverse spin current completely dephases in  a sufficiently thick \nspin sink  [5,9,77,78] . In conventional FM spin sinks, it is often  assumed  that 𝑔𝑡↑↓=0, which is \nequivalent to assuming  𝜆𝑑𝑝=0 [79].  11 \n  \nFIG. 5.  (a) Cartoon schematic of transverse spin current transport  in the spin -source/spacer/spin -sink \nstack. The FMR -driven spin source pumps a coherent transverse spin current (polarized along the x-axis) \nthrough the Cu spacer. The spin current reflected at the Cu (spacer)/CoGd (spin sink) interface is \nparameterized by the reflected spin -mixing conductance 𝑔̃2↑↓, whereas the spin current transmitted through \nthe CoGd spin sink is parameterized by the transmi tted spin -mixing conductance 𝑔𝑡↑↓. (b) Illustration of the \ndephasing of the transverse spin current propagating in the CoGd spin sink. The shrinking circle s \nrepresent the decay of the transverse spin polarization due to dephasing while it precesses about the \neffective net exchange field 𝐻𝑒𝑥𝑛𝑒𝑡. (c) Illustration of the oscillatory decay of the transverse spin current. \nThe complex transmitted spin -mixing conductance 𝑔𝑡↑↓ captures the transmitted transverse spin \npolarization components sx and sy.  \n \nFurthermore, 𝑔𝑡↑↓(𝑑) is a complex value where the real and imaginary parts are \ncomparable in magnitude  [78]. Re[𝑔𝑡↑↓(𝑑)] and Im[𝑔𝑡↑↓(𝑑)] are related to the two orthogonal \ncomponents  of the spin current transmitted through the magnetic spin sink  [9]. As illustrated in \nFig. 5 (a,b), the incident spin polarization is along the x-axis, whereas  the y-axis is normal to \nboth the incident spin polarization and the magnetic order of the spin sink. Re[𝑔𝑡↑↓(𝑑)] and \nreflected \nxyspin source Cu spacer CoGd spin sink Ticapd\nxy\nsxsy\nxytransmitted(a)\n(b)\n(c)\n0 1 2 3 4\nd/ldp\nHexnetHexnet12 \n Im[𝑔𝑡↑↓(𝑑)] represent the x- and y-components, respectively, of the transverse  spin polarization  \n(Fig. 5) ; these components  oscillate and decay while the spin current dephases.  \nWe approximate 𝑔𝑡↑↓(𝑑) with an oscillatory decay function  [9],  \n𝑔𝑡↑↓(𝑑)=𝑔𝑡,0↑↓(𝜆𝑑𝑝\n𝜋𝑑sin𝜋𝑑\n𝜆𝑑𝑝±𝑖[(𝜆𝑑𝑝\n𝜋𝑑)2\nsin𝜋𝑑\n𝜆𝑑𝑝−𝜆𝑑𝑝\n𝜋𝑑cos𝜋𝑑\n𝜆𝑑𝑝])exp (−𝑑\n𝜆𝑠𝑓), (3) \nwhere  𝑔𝑡,0↑↓ represents the interfacial contribution to 𝑔𝑡↑↓. Equation (3)  is identical to the function \nproposed by Kim  [9], except that we incorporate an exponential decay factor (with 𝜆𝑠𝑓 = 10 nm \nas explained  in Appendix  B) that approximates incoherent spin scattering  as an additional \nsource of spin -current decoherence . The sign between the real and imaginary terms in Eq. (3) \nrepresents the net precession direction of the transverse spin polarization: the positive sign \nindicates precession about an  exchange field along  the net magnetization, whereas the \nnegative sign indicates precession about a n exchange  field opposing  the net magnetization.   \nAlthough  the effective exchange field is along the net magnetization in most cases, we discuss \nin Sec. IV -B a case where the exchange field opposes the net magnetization.  \nThe oscillatory decay of 𝑔𝑡↑↓(𝑑) modeled by Eq. (3) is illustrate d in Fig. 5(c).  Although  Fig. \n5(c) shows a case with 𝑔𝑡,0↑↓ as a positive real  quantity , 𝑔𝑡,0↑↓ is generally complex. In particular, \nRe[𝑔𝑡,0↑↓] and Im[𝑔𝑡,0↑↓] represent  the filtering and rotation  [5,6], respectively, of spin current at the \ninterface  of the Cu spacer and the Co 100-xGd x sink.  \nWe incorporate Eq. (3 ) into the drift -diffusion model  by Tanig uchi et al.  [10] that uses  the \nboundary conditions applicable to our multilayer systems. Accordingly, the nonlocal Gilbert \ndamping enhancement ∆𝛼 due to spin decoherence in the spin  sink is given by  \n∆𝛼=𝑔𝜇𝐵\n4𝜋𝑀𝑠𝑡𝐹(1\n𝑔̃1↑↓+1\n𝑔̃2↑↓)−1\n,     (4) \nwhere 𝜇𝐵 is the Bohr magneton; 𝑔=2.1 is the gyromagnetic ratio, 𝑀𝑠=800 kA/m is the \nsaturation magnetization, and 𝑡𝐹=7 nm is the thickness of the NiFe spin source. 𝑔̃1↑↓=16 nm-2 \nis the renormalized reflected spin -mixing conductance at the NiFe/Cu interface ( see Appendix  \nB). 𝑔̃2↑↓ is the renormalized reflected spin -mixing conductance at the Cu/Co 100-xGd x interface, \nwhich depends on the Co 100-xGd x spin sink thickness 𝑑 as \n𝑔̃2↑↓=(1+Re[𝑔𝑡↑↓]Re[𝜂]+Im[𝑔𝑡↑↓]Im[𝜂])\n(1+Re[𝑔𝑡↑↓]Re[𝜂]+Im[𝑔𝑡↑↓]Im[𝜂])2+(Im[𝑔𝑡↑↓]Re[𝜂]−Re[𝑔𝑡↑↓]Im[𝜂])2𝑔̃𝑟↑↓,     (5) 13 \n where 𝜂=(2𝑒2𝜌𝑙+/ℎ)coth  (𝑑/𝑙+) with 𝑙+=√1/𝜆𝑠𝑓2−𝑖/𝜆𝑑𝑝2, 𝜌 is the resistivity of the spin sink, \nand 𝑔̃𝑟↑↓ is the renormalized reflected mixing conductance a t the Cu/ Co 100-xGd x interface with \n𝑑≫𝜆𝑑𝑝.  \nGiven the rather large number of parameters in the model, some assumptions to \nconstrain the fitting are required , as outlined in Appendix  B. These assumptions leave us with \nthree  free fit parameters: 𝜆𝑑𝑝, Re[𝑔𝑡,0↑↓], and Im[𝑔𝑡,0↑↓]. It is also possible to further reduce the \nnumber of free parameters by fixing Re[𝑔𝑡,0↑↓], particularly near the compensation compositio n as \nexplained in  Appendix  C. Qualitatively similar results are obtained with Re[𝑔𝑡,0↑↓] free or fixed.   \n012\n-8-404\n-4-202\n012\n012\n-202\n012\n-202\n-101\n012\n012\n-101\n0 5 10 15-101\n0 5 10012Da (10-3)\nPure Co\nCo80Gd20 Co80Gd20Pure Co\ng↑↓\nt(nm-2)\nRe[g↑↓\nt]\nIm[g↑↓\nt]\ng↑↓\nt(nm-2) g↑↓\nt(nm-2)Da (10-3) Da (10-3)\nCo78Gd22 Co78Gd22Da (10-3)\nCo77Gd23 Co77Gd23\ng↑↓\nt(nm-2) g↑↓\nt(nm-2)Da (10-3)\nCo75Gd25\nCo72Gd28Co75Gd25\nCo72Gd28Da (10-3)\ng↑↓\nt(nm-2) g↑↓\nt(nm-2)Da (10-3)\nSpin sink thickness, d (nm)Co70Gd30 Co70Gd30\n \nFIG. 6.  Left column: spin sink thickness dependence of nonlocal damping enhancement Δ α. The solid \ncurve indicates the fit using the modified drift -diffusion model. Right column: the real and imaginary parts \nof the transmitted spin -mixing conductance 𝑔𝑡↑↓ derived from the fit. The vertical dashed line indicates the \nspin dephasing length 𝜆𝑑𝑝.  14 \n  \nThe fit results using the modified drift -diffusion model are  shown as solid curves in the \nleft column of Fig. 6. These curves adequately reproduce the 𝑑 dependence of Δα for all spin \nsink compositi ons. We also show in the right column of Fig. 6  the 𝑑 dependence of 𝑔𝑡↑↓, \nillustrating the net precession and decay of the transverse spin current.  \nWith the modeled results in Fig. 6 , the overshoot in Δα versus 𝑑 can now be attributed to \nthe precession of the transverse spin current  [9,44,77] . For a certain magnetic spin sink \nthickness, the polarization of the spin current leaving the sink is opposite to that of the spin \ncurrent entering the spin sink . Since the difference between the leaving and entering spin \ncurrents is related to the spin angular momentum transferred to the magnetic order, the spin \ntransfer – manifesting as Δα here – can be enhanced  [9,44,77]  compared to  when the spin \ncurrent is completely dephased for 𝑑≫𝜆𝑑𝑝.  \nWe acknowledge that our assumptions in the  modified drift -diffusion model  do not  \nnecessarily capture  all phenomena that could impact spin transport  in a quantitative manner . \nFor example , there remain unresolved questions regarding the relationship between  resistivity 𝜌 \nand spin -flip length 𝜆𝑠𝑓, as well as the role of the thickness dependent compensation \ncomposition on dephasing, which warrant further studies in the future . Nevertheless, as \ndiscussed in Sec. IV, our approach reveals sa lient features of spin dephasing  in compensated \nFIMs , as well as the upper  bound of the transverse spin coherence length in FIM CoGd . \n \nIV. DISCUSSION  \nA. Compositional Dependence of Spin Transport  \nFigure 7  summarizes the main finding  of our work, namely  the relationship between the \nmagnetic compensation (Fig. 7 (a)) and the spin transport parameters (Fig. 7 (b-d)) of  FIM Co 100-\nxGd x. We a lso refer the readers to Fig. 16  in Appendix C, which shows that qualitatively similar \nresults are obtained when Re[𝑔𝑡,0↑↓] is treated as a fixed parameter.  \nWe first discuss the composition dependence of 𝜆𝑑𝑝, summarized in Fig. 7(b), which is \nderived from the modified drift -diffusion  model (vertical dashed lines in Fig. 6).  The results in \nFig. 7 (b) show  a peak value of 𝜆𝑑𝑝 ≈ 5 nm at x ≈ 25-28, which is close to  the magnetic \ncompensation composition window  (x ≈ 22-25, the shaded region in Fi g. 7). It is worth noting \nthat 𝜆𝑑𝑝 is maximized on the somewhat Gd -rich side of the compensation composition  window . \nWe attribute this observation to the stronger contribution to spin dephasing  from the Co \nsublattice, as discussed more in detail in Sec. IV-B.  15 \n  \nFIG. 7.  (a) Static magnetic properties (saturation magnetization Ms and coercive field Hc) reproduced from \nFig. 2, (b) spin dephasing length 𝜆𝑑𝑝, and the (c) real and (d) imaginary parts of the interfacial contribution \nof the transmitted spin -mixing conductance 𝑔𝑡,0↑↓ versus Gd content in the spin sink. The shaded region \nindicates the window of composition corresponding to magnetic compensation. The error bars in (b -d) \nshow the 95% confidence interval.  \n \n Our results  (Fig. 7(b)) indicate  up to a factor of ≈5 enhancemen t in 𝜆𝑑𝑝 for nearly \ncomp ensated FIMs compared to FM Co. This observation is  qualitatively  consistent with the \ntheoretical prediction that antiferromagnetic order mitigates the decoherence of transverse spin \ncurrent  [14–18]. In a nearly compensated FIM CoGd spin sink, the alternating Co and Gd \n-12-8-404Re[gt,0] (nm-2)\n0246ldp (nm) \n0 20 25 30-2024Im[gt,0] (nm-2)\nGd at. %(a)\n(b)\n(c)\n(d)\n050100150Ms (kA/m)\n01020\nm0Hc (mT)\n16 \n moments of approximately equal magnitude partially cancel the dephasing of the propagating \nspins . This scenario, illustrated in Fig. 1(b), is corroborated by our tight-binding calculations \nassuming coherent ballistic transport  (see Appendix  D). Transverse spin current in \ncompensated CoGd is therefore able to remain coherent ove r a longer distance than in FMs. \nWe note , however,  that the spin current decohere s within a finite length scale in any real \nmaterials,  due to the imperfect suppression of dephasing and the presence of incoherent  \nscattering.  \nWe now comment on the compositional dependence of Re[𝑔𝑡,0↑↓] and Im[𝑔𝑡,0↑↓], particularly \nin the vicinity of magnetic compensation . Our calculations  in Appendix  C predict  Re[𝑔𝑡,0↑↓] to be \nonly weakly dependent on the net ex change splitting  𝑘Δ of the magnetic spin sink. Our \nexperimental results (Fig. 7(c)) are in qualitative agr eement with this prediction, as  Re[𝑔𝑡,0↑↓] \ntakes a roughly constant value near magnetic compensation.  \nBy contrast, the same calculations in Appendix C predict a quadrati c dependence of \nIm[𝑔𝑡,0↑↓] on 𝑘Δ. Specifically,  Im[𝑔𝑡,0↑↓] converges  to zero as the net exchange 𝑘Δ approaches zero \n(i.e., magnetic sublattices approaching compensation). Our experimental results indeed show a \nminimum in Im[𝑔𝑡,0↑↓] when  𝜆𝑑𝑝 is maximized . We remark that Im[𝑔𝑡,0↑↓] is related to  how much the \npolarization of the spin current rotates upon entering the magnetic spin sink  [5,6]. When the \nmagnetic sublattices are nearly compensated, the s pin current sees a nearly canceled net \nexchange field such that the spin rotation (precession) is suppressed. Therefore, both t he \nreduction of Im[𝑔𝑡,0↑↓] and the enhancement of 𝜆𝑑𝑝 arise naturally from the cancellation of the net \nexchange field. Our results in Figs. 6 and 7  consistently point to the suppression of spin -current \ndephasing enabled by antiferromagnetic order.  \nIt is important to note that FIM TM -RE alloys in general are amorphous with no long -\nrange structural order. Instead of the sim ple layer -by-layer alternating order illustrated in Fig. \n1(b), the TM and RE atoms are expected to be arranged in a rather disordered fashion. \nConsidering that disorder and electronic scattering tend to quench transverse spin \ncoherence  [18,23,24] , it is remarkable that such amorphous FIMs permit extended 𝜆𝑑𝑝 at all. We \nspeculate the observed enhancement of transverse spin coherence is enabled by short -range \nordering of Co and Gd atoms, e.g., finite TM -TM and RE -RE pair correlations in the film plane \n(and TM -RE pair correlation out of the film plane) as sugges ted by prior reports  [18,80] .  \n \n  17 \n B. Distinct Influence of the Sublattices on Spin Dephasing  \nOur results (Fig. 7 ) indicate  that magnetic compensation is achieved in the Co 100-xGd x \ncomposition range  of x ≈ 22-25, while 𝜆𝑑𝑝 is maximized (and Im[𝑔𝑡,0↑↓] is minimized) at x ≈ 25-28. \nThis observation deviates from  the simple expectation  (Fig. 1(b) ) that spin dephasing is \nsuppressed when the two sublattices are compensated. Here, w e discuss why 𝜆𝑑𝑝 should be  \nmaximized  at a more Gd -rich spin sink composition than the magnetic compensation \ncomposition.  \nFirst, it should  be recalled  that the magnetic compensation composition is dependent on \nthe FIM thickness . Our magnetometry data (Fig.  2(b-g)) indicate  that the magnetic \ncompensat ion composition becomes more Gd -rich with decreasing CoGd thickness  𝑑. However , \nthe CoGd thickness 𝑑= 5 nm shown in Fig. 2(f,g) , from which the compensation composition is \ndeduced,  is close to  the estimated maximum 𝜆𝑑𝑝 of ≈ 5 nm . Therefore, t he thickness \ndepende nce of the compensation composition alone does not  explain the maximum  𝜆𝑑𝑝 on the \nGd-rich side of magnetic compensation.  We consider an alternative  explanation below.   \nGenerally, it  might be expected that transverse spin current interacts more strongly with \nthe TM Co magnetization (from the spin -split itinerant 3 d bands near the Fermi level)  [50,81]  \nthan the RE Gd magnetization (primarily from the localized 4 f levels ≈7 -8 eV below the Fermi \nlevel [82,83] ). This is  analogous to magnetotransport effects dominated by itinerant 3 d band \nmagnetism in TM -RE FIMs  [60,84] . If the interaction of transverse spin current with the Co \nsublattice is stronger  [81], more Gd would be needed to compensate spin depha sing. Thus, t he \ngreater contribution  of the Co sublattice to dephasing could explain why 𝜆𝑑𝑝 is maximized at a \nmore Gd -rich composition than the magnetic compensation composition.  \n We observe additional evidence for the stronger interaction  with the TM Co sublattice  by \ninspecting the dependence of 𝑔𝑡↑↓ on spin sink thickness 𝑑. The relevant results can be seen in \nthe right column of Fig. 6, but for the sake of clarity, we highlight  𝑔𝑡↑↓ vs 𝑑 for a select ed few \nCoGd compositions near magnetic compensation in Fig. 8. For most spin sink compositions  \n(e.g., Fig. 8(a),(c)) , the results are consistent with the straightforward scenario : the net  \nexchange field , felt by the transverse spin  current,  points along the net magnetization . However, \nthe Co 75Gd 25 spin sink (Fig. 8(b)) , which is on the Gd -rich side of the magnetic compensation \ncomposition,  exhibits a qualitatively different  phase shift in Im[𝑔𝑡↑↓]. This noteworthy result is \nconsistent with the transverse spin  precessing about a  net exchange field  that opposes  the net 18 \n magnetization3. In other words , the net magnetization in Co 75Gd 25 is dominated by the Gd \nmagnetization  (from 4 f orbitals far below the Fermi level ), but the net exchange field points \nalong t he Co magnetization  (from 3d bands near the Fermi level ). The retrograde spin \nprecession in Co 75Gd 25 suggests that transverse electronic spin current interacts preferentially \nwith the itinerant 3 d TM magnetism.   \n \nFIG. 8.  Transmitted spin -mixing conductance 𝑔𝑡↑↓ versus spin sink thickness 𝑑 near the magnetic \ncompensation com position , derived from the modified drift -diffusion model fit (Fig. 6) . Note the phase shift \nin Im[𝑔𝑡↑↓] for Co 75Gd25 (b). The cartoons on the right illustrate the net precession direction of the \ntransverse spin polarization in each  CoGd spin sink. Co 75Gd25 exhibits retrograde precession, opposite to \nthe precession direction in the other spin sink compositions.   \n                                                           \n3 The modeled curves for Re[𝑔𝑡↑↓] and Im[𝑔𝑡↑↓] in Fig s. 6 and 8 for Co 75Gd25 showing retrograde \nprecession are obtained by taking the negative sign between the real and imaginary terms in Eq. (3). \nAdequate fits could also be obtained by fixing the sign between the real and imaginary terms to be \npositive, but th is would require the signs of  Re[𝑔𝑡,0↑↓] and Im[𝑔𝑡,0↑↓] for Co75Gd25 to be opposite to those of \nthe other CoGd compositions. Since a sign flip in Re[𝑔𝑡,0↑↓] or Im[𝑔𝑡,0↑↓] with respect to CoGd composition is \nnot expected (according to our calculations  in Appendix C), we  conclude that retrograde precession is the \nphysically reasonable scenario for the Co 75Gd25 sink.  \n-2-1012\n Re[gt ]\n Im[gt ]Co77Gd23gt (nm-2)\n0 5 10-1.0-0.50.00.5gt (nm-2)Co72Gd28\nspin sink thickness, d (nm)\n-2-101gt (nm-2)Co75Gd25CoGd\nCoGd\nCo\nGd\n  \n(a)\n(b)\n(c)19 \n We also comm ent on the possible role of  angular momentum compensation in FIM \nCoGd. As noted in Sec. II , angular momentum compensation composition is slightly more Co-\nrich than the magnetic compensation composition. While angular momentum compensation is \nkey to  fast antiferromagnetic -like dynamics in FIMs  [53,55] , it is evidently  unrelated to the \nmaximum  𝜆𝑑𝑝 on the slightly Gd -rich side of the magnetic compensation composition.  Rather, \nwe conclude that 𝜆𝑑𝑝 in FIMs is governed by the effective net exchange field , where the TM \nsublattice can play a greater role than the RE sublattice.   \n \nC. Comparison with a Prior Report of Long Transverse Spin Coherence in Ferrimagnetic \nMetals  \nOur results (e.g., Fig s. 6-8) point to  mitigation of spin decoherence  in nearly \ncompensated FIM CoGd. However,  we do not observe  evidence for  a transverse spin \ncoherence length  in excess of 10 nm , which was  recently reported for CoTb  [18]. Owing to the \nweaker spin -orbit coupling in CoGd than in CoTb, o ne might expect longer length scales for spin \ndephasing (more collinear antiferromagnetic order) and spin diffusion (less spin -flip scattering) \nin CoGd than in CoTb. We discuss possible reasons as to why the maximum coherence length \nin CoGd in our present study is significantly shorter than that reported in CoTb by Ref.  [18].  \nA plausible  factor  is the difference in experimental method for deducing the coherence \nlength . Yu et al.  in Ref. [18] utilize spin -galvanic measurements on Co/Cu/CoTb/Pt stacks: FMR \nin the Co layer pumps a spin current presumably through the CoTb spacer and generates a \nlateral dc voltage from the inver se spin -Hall effect in the Pt detector  [85]. A finite dc voltage is \ndetected for a range of CoTb alloy spacer thicknesses up to 12 nm, interpreted as evidence that \nthe spin current propagates from Co to Pt even with >10 nm of CoTb in between. However, \nthere could be coexisting voltage contributions besides the spin -to-charge conver sion in the Pt \ndetector . For example, spin scattering in the CoTb layer could yield an additional inverse spin -\nHall effect  [86], i.e., the  reciprocal of the strong spin -orbit torque reported in CoTb  [87]. \nFurthermore, the FMR -driven spin -galvanic measurement could  pick up spin rectification  [31–\n33] and thermoelectr ic voltages  [34,35]  from the dynamics of the FM Co layer, which might be \nchallenging to disentangle from the inver se spin -Hall effect in Pt. While  a number of control \nexperiments are performed in Ref.  [18] to rule out artifacts, it is possible that some spurious \neffects are not completely suppressed. Figures 4(d) and 5(c) in Ref.  [18] show that the spin -\ngalvanic signal drops abruptly  with the inclusion of a finite thickness of CoTb space r and \nremains constant up to CoTb thickness >10 nm . This trend , at odds with  the expected gradual 20 \n attenuation of spin current with CoTb  thickness , may arise from other mechanisms unrelated to \nspin transmission  through CoTb.  \nIn our present study , the spin pumping method measures the nonlocal damping Da that \nis attributed to spin-current decoherence in the spin sink ( see Sec. III -A). This method  does not \ninvolve any complicat ions from spin -galvanic signals. It still  might be argued , however,  that our \nmethod does not necessarily allow for precise, straightforward quantification of spin transport  \ndue to the large number of  parameters involved in modeling ( see Appendix B). Nevertheless, \neven if there are quantitative errors in our modeling, it is incontrovertible that Da saturates (i.e., \nthe transverse spin current pumped int o CoGd decoheres)  within a length scale well below 10 \nnm. Thus, our results indicate that the spin coherence length in CoGd does not exceed 10 nm.  \nThere may be other factors leading to the discrepancy  between our study and Ref.  [18]. \nThe CoTb films in Ref.  [18] may have a higher degree of layer -by-layer ordering than our CoGd \nfilms. This appears unlikely, since  CoTb “multilayers” (grown by alternately depositing Co and \nTb) and  CoTb “alloys” (grown by simultaneously depositing Co and Tb) exhibit essentially the \nsame CoTb thickness dependence of spin -galvanic signal  [18]. Another possibility is that \nmagnons, rather than spin -polarized conduction electrons, are responsible for the remarkably \nlong spin coherence through CoTb. If this were the case, it is yet unclear how CoTb might \nexhibit a longer magnon coherence length than CoGd, or how the spin -galvanic method in \nRef. [18] might be more sensitive to magnon spin transport than the nonlocal damping method \nin our present study. A future study  that directly compares  CoTb and CoGd spin sinks (e.g., with \nthe nonlocal damping method ) may verify  whether transverse spin current s survive over > 10 nm \nin ferrimagnet ic alloys with strong spin -orbit coupling .  \n \nD. Possible Implications for Spintronic Device Applications  \nThe dephasing length  𝜆𝑑𝑝 of transverse electronic spin current fundamentally impacts \nspin torque effects  in a magnetic metal  [5,6]. Specifically, the transverse spin angular \nmomentum lost by  the spin current is transferred to the magnetization , thereby giving rise to a \nspin torque within a depth  of order  𝜆𝑑𝑝 from the  surface of the magnetic metal.  Due to the  short \n𝜆𝑑𝑝 ≈ 1 nm, the spin torque is more efficient  in a thinner  FM layer, which comes  at the expense \nof reduced thermal stability of the stored magnetic information. The longer 𝜆𝑑𝑝 in compensated \nFIMs (or AFMs) may enable efficient spin torques in thicker, more thermally stable magnetic \nlayers  for high -density nonvola tile memory devices  [18,52] . Another practical benefit of \nextended 𝜆𝑑𝑝 is that  it facilitates  tuning the magnetic layer thickness to enhanc e spin 21 \n torques  [9,44,77] .  We further emphasize that the increase of 𝜆𝑑𝑝 is evident even in amorp hous  \nFIMs. This suggests that optimally engineered  FIMs or AFMs  (e.g., with high crystallinity)  could \nexhibit much longer 𝜆𝑑𝑝, potentially yielding spin torque effects that are qualitatively distinct from \nthose in FMs  [14].  \n In addition to estimating  the dephasing length scale in FIMs, our study highlights the \nroles of the chemically distinct sublattices on spin dephasing . From conventional spin  torque \nmeasurements, i t is generally difficult to deduce whether a spin current in a TM -RE FIM \ninteracts more strongly with a particular sublattice  [54]. Our results (as discussed in Sec. IV -B) \nimply that the TM sublattice contributes more strongly to the dephasing of transverse spin \ncurrent  – and hence to sp in torque s. The greater role of the TM sublattice over the RE sublattice \ncould be crucial for engineering spin torque effects in compensated TM -RE FIMs  – e.g., for \nenabling ultrafast (sub-THz-range ) spin torque oscillators  [88].  \n \nV. CONCLUSION  \nIn summary, we have utilized broadband FMR spin pumping to estimate  the dephasing \nlength 𝜆𝑑𝑝 of transverse spin current in ferrimagnetic CoGd alloys  across the compensation \npoint . We obtain a maximum of 𝜆𝑑𝑝≈5 nm in nearly compensated CoGd, consistent with the \nantiferromagnetic order mitigating the decoherence (dephasing) of transverse spin current. The \nobserved maximum 𝜆𝑑𝑝 constitutes a factor of ≈4 -5 enhancement compared to that for \nferromagnetic metals. On the other hand, we do not find evidence for 𝜆𝑑𝑝 in excess of 10 nm in \nferrimagnetic alloys reported  in a recent study  [18]. Despite this  quantitative difference, our \nresults suggest that partial spin rephasing by antiferromagnetic order – i.e., analogous to  the \nspin-echo  scheme to counter spin decoherence  – is indeed operative even in disordered \nferrimagnetic alloys at room temperature.  Moreover, o ur results suggest that transverse spin \ncurrent interacts more strongly with the itinerant Co sublattice than the localized Gd sublatt ice in \nnearly compensated CoGd. The spin rephasing effect and the sublattice dependent interaction  \ncould impact  spin torques in ferrimagnetic alloys , with possible applications in  fast spintronic \ndevices.  Our finding also points to the possibility of further extending transverse spin coherence \nin structurally pristine antiferromagnetic metals, thus opening a new avenue for fundamental \nstudies of spin transport in magnetic media.  \n \n  22 \n Acknowledgements  \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the \nCommonwealth of Virginia, as well as by the ICTAS Junior Faculty Award  and the NSF Grant \nNo. DMR -2003914 . We thank Jean J. Heremans  for helpful discussions .  \n  23 \n APPENDIX A: FMR MEASUREMENT METHOD  \n25 30 35 40 45 50 55 60dIFMR/dH\nm0H (mT)f = 6 GHz\nNiFe (7 nm)\n \nFIG. 9.  FMR spectrum  of Ni 80Fe20(7)/Cu(4) at 6 GHz. The red curve in the bottom panel represents the fit \nusing Eq. (A1).  \n \nFMR spectra are acquired using a broadband spectrometer, with each sample placed on \na coplanar waveguide (film side down) and magnetized in -plane (with a quasi -static magnetic \nfield, maximum value ~1 T, from an electromagnet). The quasi -static field is swe pt while fixing \nthe frequency of the microwave field (transverse to the quasi -static field) to acquire the \nresonance spectrum. A radio -frequency diode and lock -in amplifier (with 700 Hz modulation field \nas the reference) is used to detect the signal, which  is recorded as the derivative of the \nmicrowave power absorption with respect to the applied field, as shown in Fig.  9. To obtain  the \nhalf-width -at-half-maximum FMR linewidth Δ H, the measured signal is fit with the derivative of \nthe sum of symmetric and an tisymmetric Lorentzian functions  [89],  \n \n \nwhere Hres is the resonance field and the coefficients A and S are the proportionality factors for \nthe antisymmetric and symmetric terms, respectively.  𝑑𝐼𝐹𝑀𝑅\n𝑑𝐻=𝐴2(𝐻−𝐻𝑟𝑒𝑠)∆𝐻\n[(𝐻−𝐻𝑟𝑒𝑠)2+(∆𝐻)2]2+𝑆(𝐻−𝐻𝑟𝑒𝑠)2−(∆𝐻)2\n[(𝐻−𝐻𝑟𝑒𝑠)2+(∆𝐻)2]2, (A1)  24 \n  \n60 80 100 120 140dIFMR/dH (a.u.)\nm0H (mT)f = 10 GHz\nNiFe (7 nm)\nCo (4 nm)Ni80Fe20(7)/Cu(4)/Co100-xGdx(d) (Unit: nm)\nNo spin sink\n(x=0, d=4)\n(x=25, d=1.2)\n(x=25, d=9)\n(x=28, d=0.6)\n(x=28, d=8)  \nFIG. 10. FMR spectra for Ni80Fe20 (7 nm)/Cu (4 nm)/Co 100-xGdx (d nm) with no spin sink (magenta) and \nwith spin sink layer of pure Co (black), Gd 25% (red and green) and Gd 28% (blue and light blue). NiFe \nand Co exhibit well -separated FMR. The different thicknesses for Gd 25 and 28% was selected to show \ntwo different transp ort regimes, i.e., below/above the 𝜆𝑑𝑝 found from  Fig. 6 in the main text. No FMR \nsignal attributable to Co 100-xGdx in our samples (20 ≤ x ≤ 30) is detected.  \n \nAPPENDIX B: ASSUMPTIONS IN THE MODIFIED DRIFT -DIFFUSION MODEL  \nGiven the rather large number of  parameters in the model  (𝑔̃1↑↓, 𝑔̃𝑟↑↓, 𝜌, 𝜆𝑠𝑓, 𝜆𝑑𝑝, Re[𝑔𝑡,0↑↓], \nand Im[𝑔𝑡,0↑↓]), some assumptions are required to constrain the modified drift -diffusion model \n(Sec. III -B). Our assumptions are as follow:  \n(1)  The renormalized reflected spin-mixing  conductance (accounting for the Sharvin \nconductance of Cu  [79]) is set equal for the Ni 80Fe20/Cu and Cu/Co interfaces, i.e., \n𝑔̃1↑↓=𝑔̃𝑟↑↓. Here, 𝑔̃𝑟↑↓ is the renormalized reflected spin -mixing conductance for the \nCu/Co interface, 𝑔̃2↑↓, in the limit of 𝑑≫𝜆𝑑𝑝. We compute 𝑔̃1↑↓ from a modified form of Eq. \n(4),  25 \n ∆𝛼𝑠𝑎𝑡𝐶𝑜=𝑔𝜇𝐵\n4𝜋𝑀𝑠𝑡𝐹(2\n𝑔̃1↑↓)−1\n,     (B1) \nwhere ∆𝛼𝑠𝑎𝑡𝐶𝑜=0.0022  is the average of ∆𝛼 for Co spin sink thicknesses 𝑑≥3 nm (large \nenough that the transverse spin current is essentially  completely dephased). The \nrenormalized spin -mixing conductance 𝑔̃1↑↓ at the Ni 80Fe20/Cu interface is found to be 16 \nnm-2.  \n(2)  For each ferrimagnetic Co 100-xGd x spin sink composition, we compute 𝑔̃𝑟↑↓, shown in Eq. \n(5)) via \n∆𝛼𝑠𝑎𝑡𝐶𝑜𝐺𝑑=𝑔𝜇𝐵\n4𝜋𝑀𝑠𝑡𝐹(1\n𝑔̃1↑↓+1\n𝑔̃𝑟↑↓)−1\n,     (B2) \nwhere ∆𝛼𝑠𝑎𝑡𝐶𝑜𝐺𝑑 is the average of ∆𝛼 for the sa mples with the three largest spin sink \nthicknesses , where ∆𝛼 is essentially saturated . The values of 𝑔̃𝑟↑↓ are summarized in Fig. \n11.  \n \n0 20 25 30051015g\nr   \nGd at. %~(nm-2)  \nFIG. 1 1. Composition dependence of the renormalized reflected mixing conductance 𝑔̃𝑟↑↓ at the Cu/ Co 100-\nxGdx interface with the Co 100-xGdx spin sink thickness 𝑑≫𝜆𝑑𝑝.  \n \n(3) We set 𝜆𝑠𝑓=10 nm, which is of the same order as  the reported spin diffusion length in \nCo [4]. We further note that if the claim of 𝜆𝑐>10 nm for ferrimagnets in Ref.  [18] were \ncorrect, 𝜆𝑠𝑓 would necessarily need to be >10 nm. This relatively long 𝜆𝑠𝑓 is equivalent \nto assuming quasi -ballistic spin transport in the spin sink, such that spin dephasing \n(rather  than scattering) governs the transverse spin coherence length, 𝜆𝑑𝑝<𝜆𝑠𝑓. Our \nquantitative results of 𝜆𝑑𝑝 are essentially unaffected  if 𝜆𝑠𝑓 ≳ 4 nm (e.g., Fig. 12), while \nsome variation can be seen in  Im[𝑔𝑡,0↑↓] with 𝜆𝑠𝑓.  26 \n  \nFIG. 12.  Dependence of the spin dephasing length 𝜆𝑑𝑝 and interfacial transmitted spin -mixing \nconductance 𝑔𝑡,0↑↓ of Co 72Gd28 on the spin -flip length 𝜆𝑠𝑓 in the modified drift -diffusion model.  \n \n(4) For the sake of simplicity, we assume constant values of 𝜆𝑠𝑓 and 𝜌 for a given spin -sink \ncomposition (although in general, 𝜆𝑠𝑓 may depend on the resistivity 𝜌 of the spin sink, \nwhich in turn can depend on the spin -sink thickness). In particular, we set 𝜌 at the \nvalues obtained from four -point resistivity measurements of 10 -nm-thick Co 100-xGd x films \n(Fig. 13) ; the results are summarized in the figure below. (The exception was pure Co \nwhere 𝜌 was set at 1.0x10-6 m, since using the measured resistivity 𝜌 = 0.3x10-6 m \nyielded an unphysically large |𝑔𝑡,0↑↓| of ≫10 nm-2. The higher effective resistivity for Co is \npossibly justified considering that ∆𝛼 saturates at 𝑑≈1 nm, where the resistivity is likely \nmuch higher than 0.3x10-6 m due to surface scattering.) We remark that the \nuncertainty in the spin -sink thickness dependence of 𝜌 and 𝜆𝑠𝑓 may result in  a \nsystematic  error in quantifying 𝜆𝑑𝑝; further detailed studies are warranted to elucidate \nthe relationship between electronic and spin transport in ferrimagnets. Nevertheless, \nour approach is sufficie nt for semi -quantitative examination of 𝜆𝑑𝑝 as a function of CoGd \ncomposition. I t is incontrovertible that FIM CoGd has a significantly longer transverse \ncoherence length than FM Co and that this coherence length (albeit well below 10 nm)  \nshows a maximum  near the magnetic compensation .  \n0246ldp (nm)\n4 6 8 10-2-10123Im[gt,0]\nRe[gt,0]gt,0 (nm-2) \nlsf (nm)\n\n27 \n \n0 10 20 30 400.00.51.01.52.02.5Resistivity, r (10-6 m)\nGd at.% \nFIG. 1 3. Resistivity 𝜌 with varying Gd at.% for SiO x(thermally oxidized)/Co 100-xGd x(10)/TiO x(3) (unit: nm) . \nThe resistivi ties of intermediate compositions between x = 20 and 30 are derived via linear interpolation.  \n \n(5) We assume 𝜆𝑑𝑝 to be a constant parameter for each spin -sink composition, i.e., \nindependent of the spin sink thickness. We note that there could be a deviation from \nthis assumption, considering that the magnetic compensation composition (hence net \nexchange splitting) ev idently depends on the thickness of the ferrimagnetic spin sink, as \nseen in our magnetometry results (Fig. S3).  \n \nWe also add a few remarks regarding the details of our fitting protocol. In fitting the \nexperimental data ∆𝛼 versus 𝑑 using the modified dri ft-diffusion model  (Fig. 6, left column) , we \nassign a weight to each data point that is inversely proportional to the square of the error bar for \n∆𝛼 (from the linear fit of linewidth versus frequency). We fix Im[𝑔𝑡,0↑↓]=0.2𝑔̃𝑟↑↓ for pure Co, similar \nto what is suggested by Zwierzycki et al. [78]; for Co 80Gd 20, we impose the constraint 0.2𝑔̃𝑟↑↓<\n|𝑔𝑡,0↑↓|<0.3𝑔̃𝑟↑↓, since having the magnitudes of Re[𝑔𝑡,0↑↓] and Im[𝑔𝑡,0↑↓] as free parameters resulted \nin unphysically large |𝑔𝑡,0↑↓| with large error bars.   \n \nAPPENDIX C.CONSTANT 𝐑𝐞[𝒈𝒕,𝟎↑↓] NEAR MAGNETIC COMPENSATION  \nWhile the assumptions outlined in Appendix B results in three free fit parameters in the \ndrift-diffusion mode l (𝜆𝑑𝑝, Re[𝑔𝑡,0↑↓], and Im[𝑔𝑡,0↑↓]), it is possible to reduce the number of free \nparameters further by fixing Re[𝑔𝑡,0↑↓]. Here, we provide a physical justification for setting Re[𝑔𝑡,0↑↓] \nconstant, particularly near the magnetic compensation composition.  \nWe consider the simplest non -magnet and ferromagnet interface. In the nonmagnetic \nmetal, 𝑘𝑥=√𝑘𝐹2−𝜅2, and in the ferromagnet, 𝑘𝑥𝜎=√𝑘𝐹2+𝜎𝑘𝛥2−𝜅2. Here 𝜅 is the momentum 28 \n lying in  the plane of the interface and 𝑘𝛥=√2𝑚𝛥\nℏ quantifies the s -d exchange. The transmission \ncoefficient  𝑡𝜎 for spin 𝜎 reads  \n𝑡𝜎=2𝑘𝑥\n𝑘𝑥𝜎+𝑘𝑥 ,  (C1) \nand consequently, the transmitted mixing conductance at the interface  is \n𝑔𝑡,0↑↓=∫𝑑2𝜿\n4𝜋2𝑡↑(𝑡↓)∗ . (C2) \nFigure 14 displays the dependence of 𝑡↑(𝑡↓)∗ as a function of the in -plane momentum 𝜅2. \nThe real part is given in blue and the imaginary part is given in red. The first interesting feature \nis that all incident states contribute to Re[𝑔𝑡,0↑↓] whereas only states with incidence larger than \n𝜅2≥𝑘𝐹2−𝑘𝛥2 (rather close to grazing incidence) contribute to Im[𝑔𝑡,0↑↓].  \n \nFIG. 1 4. Momentum -resolved 𝑔𝑡,0↑↓ versus the in -plane momentum 𝜅2. Re[𝑔𝑡,0↑↓] in blue and Im[𝑔𝑡,0↑↓] in red.  \n \nUpon integration over the Fermi surface, the real and imaginary parts of the interfacial \ntransmitted mixing conductance reduce to Re[𝑔𝑡,0↑↓]≈𝑘𝐹2\n2𝜋 and Im[𝑔𝑡,0↑↓]=𝑘𝛥2\n4𝜋. In other words, the \nreal part is mostly independent of  the exchange  splitting , whereas the imaginary part is directly \nproportional to it (in fact, it is quadratic). This behavior is reported in Fig. 15 . When varying the \ncontent of Gd in the vicinity of  the magnetic compensation composition , it is therefore \nreasonable to assume that  Re[𝑔𝑡,0↑↓] is constant whereas Im[𝑔𝑡,0↑↓] varies.  \n29 \n  \nFIG. 15 . Interfacial transmitted mixing conductance versus  the exchange splitting  𝑘Δ. Re[𝑔𝑡,0↑↓] in blue and \nIm[𝑔𝑡,0↑↓] in red.  \n \nFigure 16  compares the fitting results of our experimental data with free Re[𝑔𝑡,0↑↓] (i.e., \nsame as Fig. 7) and with fixed Re[𝑔𝑡,0↑↓].  In Fig. 16(b), Re[𝑔𝑡,0↑↓] is fixed at a constant magnitude of \n1 nm-2 for all samples, except for Co and Co 80Gd 20 where larger Re[𝑔𝑡,0↑↓] (e.g., 8 and 4 nm-2, \nrespectively) is needed to obtain adequate fit curves. We observe that the qualitative conclusion \nis unaffected by whether or not Re[𝑔𝑡,0↑↓] is treated as a free parameter: 𝜆𝑑𝑝 is maximized (and \nIm[𝑔𝑡,0↑↓] is minimized) close to – or, more specificall y, on the slightly Gd -rich side of – the \nmagnetic compensation composition.   \n \n30 \n  \nFIG. 1 6. Comparison of modeling results with (a) free Re[𝑔𝑡,0↑↓] and (b) fixed Re[𝑔𝑡,0↑↓]. The shaded region \nindicates the window of composition corresponding to magnetic compensation.  \n \n \n  \n0246ldp (nm) \n-12-8-404Re[gt,0] (nm-2)\n0246ldp (nm) \n0 20 25 30-2024Im[gt,0] (nm-2)\nGd at. %(a) (b) \n-12-8-404Re[gt,0] (nm-2)\n0 20 25 30-2024Im[gt,0] (nm-2)\nGd at. %\n 31 \n APPENDIX D.  MODELING SPIN DEPHASING IN A FERRIMAGNETIC HETEROSTRUCTURE  \nTo understand the influence of the (collinear) magnetic order on the reflected and \ntransmitted mixing conductance s, we consider a magnetic trilayer composed of two equ ivalent \nnonmagnetic leads and a ferrimagnetic spacer, as illustrated in Fig. 17 . We compute the mixing \nconductances of this system assuming coherent ballistic transport.  \n \n \nFIG. 17.  Schematic of a ferrimagnetic trilayer as modeled below. The two magnetic sublattices are \ndenoted by red and blue arrows pointing in opposite directions.  \n \nSystem definition and boundary conditions:   Each layer is made of a three -dimensional \nsquare lattice with equal lattice parameter a for simplicity. The ferrimagnet is composed  of a \ntwo-atomic unit cell with sublattices A and B. In the {A,B} basis, the Hamiltonian for spin 𝜎 reads  \nℋ=(𝜀𝐴+𝜎𝛥𝐴−4𝑡𝐴𝜒𝑘 −2𝑡𝐴𝐵𝛾𝑘\n−2𝑡𝐴𝐵𝛾𝑘 𝜀𝐵+𝜎𝛥𝐵−4𝑡𝐵𝜒𝑘) \nwith \n𝜒𝑘=cos𝑘𝑥𝑎cos𝑘𝑦𝑎+cos𝑘𝑥𝑎cos𝑘𝑧𝑎+cos𝑘𝑧𝑎cos𝑘𝑦𝑎 \n𝛾𝑘=cos𝑘𝑥𝑎+cos𝑘𝑦𝑎+cos𝑘𝑧𝑎 \nIn this expression,  𝜀𝑖, 𝑡𝑖, 𝛥𝑖 are the onsite energy, hopping integral and magnetic exchange on \nsublattice i, and 𝑡𝐴𝐵 is the inter -sublattice hopping integral. The energy dispersion for spin 𝜎 and \nband 𝜂 is \n𝜀𝜂,𝜎=𝜀̅+𝜎𝛥̅−4𝑡̅𝜒𝑘+𝜂√(𝛿𝜀+𝜎𝛿𝛥 −4𝛿𝑡𝜒𝑘)2+4𝑡𝐴𝐵2𝛾𝑘2 \nand the associated eigenstate function reads  \n𝜙̂𝜂,𝜎=1\n√2√1+𝜂𝛽𝑘,𝜎|𝐴〉⊗|𝜎〉−𝜂\n√2√1−𝜂𝛽𝑘,𝜎|𝐵〉⊗|𝜎〉 \n𝛽𝑘,𝜎=𝛿𝜀+𝜎𝛿𝛥 −4𝛿𝑡𝜒𝑘\n√(𝛿𝜀+𝜎𝛿𝛥 −4𝛿𝑡𝜒𝑘)2+4𝑡𝐴𝐵2𝛾𝑘2 \n32 \n In the above  expressions, 𝛿𝜀=𝜀𝐴−𝜀𝐵\n2,𝜀̅=𝜀𝐴+𝜀𝐵\n2,𝛿𝛥=𝛥𝐴−𝛥𝐵\n2,𝛥̅=𝛥𝐴+𝛥𝐵\n2,𝛿𝑡=𝑡𝐴−𝑡𝐵\n2,𝑡̅=𝑡𝐴+𝑡𝐵\n2.  \nLet us now build the scattering wave function across the trilayer. The electron wave functions in \nthe left, central and right layers read  \n𝜓𝜎𝐿(𝒌)=(𝑒𝑖[𝑘𝑥𝐿(𝒌⊥)𝑥+𝒌⊥⋅𝝆]+∫𝑑2𝜿\n4𝜋2𝑟𝜎(𝜿)𝑒−𝑖[𝑘𝑥𝐿(𝜿)𝑥+𝜿⋅𝝆])|𝐶〉 \n𝜓𝜎𝐹(𝒌)=∫𝑑2𝜿\n4𝜋2∑𝑒𝑖𝜿⋅𝝆[𝐴𝜂,𝜎cos𝑘𝑥𝜂,𝜎(𝜿)𝑥+𝐵𝜂,𝜎cos𝑘𝑥𝜂,𝜎(𝜿)𝑥]𝜙̂𝜂,𝜎\n𝜂 \n𝜓𝜎𝑅(𝒌)=∫𝑑2𝜿\n4𝜋2𝑡𝜎(𝜿)𝑒𝑖[𝑘𝑥𝑅(𝜿)(𝑥−𝑑)+𝜿⋅𝝆]|𝐶〉 \nHere, 𝑘𝑥𝐿,𝑅(𝒌⊥) is a solution of the dispersion relation in the leads, 𝜀(𝒌)=𝜀𝑁−2𝑡𝑁𝛾𝑘=𝜀𝐹, 𝜀𝐹 \nbeing the Fermi energy, and 𝒌⊥is the in -plane component of the incoming wave vector. Similarly, \n𝑘𝑥𝜂,𝜎(𝜿) is determined by  the condition  𝜀𝜂,𝜎(𝒌)=𝜀𝐹. Now, let us determine the matching \nconditions at  𝑥=0 and  𝑥=𝑑. Because the leads and the ferr imagnetic layer possess a different \nfirst Brillouin zone, there will be Umklapp scattering  [90] that must be taken into account in the \nmatching procedure. In fact, the f irst Brillouin zone of the ferrimagnet is twice smaller than the \nfirst Brillouin zone of the leads, introducing an Umklapp momentum  𝑸=𝜋\n𝑎(𝒚+𝒛) [90–92]. We \nthen project the wave function of the ferrimagnetic layer on the normal metal orbital |𝐶〉 and use \nthe fact that  〈𝐶|𝐵〉=〈𝐶|𝐴〉𝑒𝑖(𝜅𝑦−𝑘⊥,𝑦)𝑎. In summary, the boundary conditions for normal \nscattering are  \n1+𝑟𝜎=∑[𝜂\n√2√1−𝜂𝜎𝛽 +1\n√2√1+𝜂𝜎𝛽 ]𝐴𝜂,𝜎\n𝜂 \n𝑖𝑘𝑥𝐿(1−𝑟𝜎)=∑[𝜂\n√2√1−𝜂𝜎𝛽 +1\n√2√1+𝜂𝜎𝛽 ]𝑘𝑥𝜂𝐵𝜂,𝜎\n𝜂 \n∑[𝜂\n√2√1−𝜂𝜎𝛽 +1\n√2√1+𝜂𝜎𝛽 ](𝐴𝜂,𝜎cos𝑘𝑥𝜂𝑑+𝐵𝜂,𝜎sin𝑘𝑥𝜂𝑑)\n𝜂 =𝑡𝜎 \n∑[𝜂\n√2√1−𝜂𝜎𝛽 +1\n√2√1+𝜂𝜎𝛽 ]𝑘𝑥𝜂(−𝐴𝜂,𝜎sin𝑘𝑥𝜂𝑑+𝐵𝜂,𝜎sin𝑘𝑥𝜂𝑑)\n𝜂 =𝑖𝑘𝑥𝑅𝑡𝜎\n  \nand for the Umklapp scattering, we obtain  \n𝑑𝜎=∑[𝜂\n√2√1−𝜂𝜎𝛽𝑄−1\n√2√1+𝜂𝜎𝛽𝑄]𝐴𝜂,𝜎\n𝜂 \n−𝑖𝑘𝑥,𝑄𝐿𝑑𝜎=∑[𝜂\n√2√1−𝜂𝜎𝛽𝑄−1\n√2√1+𝜂𝜎𝛽𝑄]𝑘𝑥,𝑄𝜂𝐵𝜂,𝜎\n𝜂 33 \n ∑[𝜂\n√2√1−𝜂𝜎𝛽𝑄−1\n√2√1+𝜂𝜎𝛽𝑄](𝐴𝜂,𝜎cos𝑘𝑥,𝑄𝜂𝑑+𝐵𝜂,𝜎sin𝑘𝑥,𝑄𝜂𝑑)\n𝜂 =𝑢𝜎 \n∑[𝜂\n√2√1−𝜂𝜎𝛽𝑄−1\n√2√1+𝜂𝜎𝛽𝑄]𝑘𝑥,𝑄𝜂(−𝐴𝜂,𝜎sin𝑘𝑥,𝑄𝜂𝑑+𝐵𝜂,𝜎sin𝑘𝑥,𝑄𝜂𝑑)\n𝜂 =𝑖𝑘𝑥,𝑄𝑅𝑢𝜎 \n \nIn order to make the above expressions easier to track, we  have  set \n𝑟𝜎=𝑟𝜎(𝒌⊥),𝑡𝜎=𝑡𝜎(𝒌⊥),𝛽=𝛽𝒌⊥,𝑘𝑥𝐿,𝑅=𝑘𝑥𝐿,𝑅(𝒌⊥),𝑘𝑥𝜂=𝑘𝑥𝜂(𝒌⊥) \n𝑑𝜎=𝑟𝜎(𝒌⊥+𝑸),𝑢𝜎=𝑡𝜎(𝒌⊥+𝑸),𝛽𝑄=𝛽𝒌⊥+𝑸,𝑘𝑥,𝑄𝐿,𝑅=𝑘𝑥𝐿,𝑅(𝒌⊥+𝑸),𝑘𝑥,𝑄𝜂=𝑘𝑥𝜂(𝒌⊥+𝑸) \n \nMixing Conductance s: The reflected and transmitted mixing conductances are \ndefined  [93] \n𝑔𝑟↑↓=(𝑒2\nℎ)∫𝑑2𝜿\n4𝜋2(1−𝑟↑𝑟↓∗−𝑑↑𝑑↓∗) \n𝑔𝑡↑↓=(𝑒2\nℎ)∫𝑑2𝜿\n4𝜋2(𝑡↑𝑡↓∗+𝑢↑𝑢↓∗) \nWe now compute these two quantities (both real and imaginary parts) as a function of the \nferrimagnetic layer thickness upon varying the magnetic exchange. For these calculations, we \nset 𝜀𝑁=𝜀̅=2.5 eV, 𝛿𝜀=0,𝑡𝑁=𝑡𝐴𝐵=1 eV, 𝑡̅=0. We also set 𝛥𝐴=1 eV and 𝛥𝐵 is varied \nbetween +1 eV (ferromagnet) and -1 eV (antiferromagnet). The results a re reported i n Fig . 18. \nThe reflected mixing conductance is weakly affected by the magnetic order, which is expected \nbased on simple free -electron arguments  [93], whereas the transmitted mixing conductance is \ndramatically modified when tuning the magnetic exchange. In the ferromagnetic limit ( 𝛥𝐵=\n+1 eV), it displays the expected damped osc illatory behavior already observed  [93] and \nattributed to the destructive interferences between precessing spins with different incidence (in \nother words, spin dephasing). Redu cing and then inverting the exchange leads to an overall \nreduction of the average exchange field, leading to an increase of the dephasing length, which \ndiverges in the antiferromagnetic limit ( 𝛥𝐵=−1 eV). 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Butler, \"Tunneling magnetoresistance from a symmetry filtering effect,\" Sci. \nTechnol. Adv. Mater. 9, 14106 (2008).  \n95.  Y. Wen, F. Zhuo, Y. Zhao, P. Li, Q. Zhang, A. Manchon, and X. Zhang, \"Competition \nbetween Electronic and Magnonic Spin Currents in Metallic Antiferromagnets,\" Phys. Rev. \nAppl. 12, 54030 (2019).  \n "    },    {        "title": "1703.10310v2.Current_driven_dynamics_and_inhibition_of_the_skyrmion_Hall_effect_of_ferrimagnetic_skyrmions_in_GdFeCo_films.pdf",        "content": "Current-driven dynamics and inhibition of the skyrmion Hall effect of ferrimagnetic skyrmions in GdFeCo films   Authors: Seonghoon Woo,1†* Kyung Mee Song,1,2† Xichao Zhang,3 Yan Zhou,3 Motohiko Ezawa,4 Xiaoxi Liu,5 S. Finizio,6 J. Raabe,6 Nyun Jong Lee,7 Sang-Il Kim,7 Seung-Young Park,7 Younghak Kim,8 Jae-Young Kim,8 Dongjoon Lee,1,9 OukJae Lee,1 Jun Woo Choi,1,10 Byoung-Chul Min,1,10 Hyun Cheol Koo,1,9 and Joonyeon Chang1,10  Affiliations: 1Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea 2Department of Physics, Sookmyung Women’s University, Seoul 04130, Korea 3School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China 4Department of Applied Physics, University of Tokyo, Hongo 7-3-1, Tokyo 113-8656, Japan 5Department of Electrical and Computer Engineering, Shinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan 6Swiss Light Source, Paul Scherrer Institut, 5232 Villigen, Switzerland 7Spin Engineering Physics Team, Division of Scientific Instrumentation, Korea Basic Science Institute, Daejeon, 305-806, Korea 8Pohang Accelerator Laboratory, Pohang University of Science and Technology, Pohang 37673, Korea 9KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02792, Korea 10Department of Nanomaterials Science and Engineering, Korea University of Science and Technology, Daejeon 34113, Korea  † These authors contributed equally to this work. * Authors to whom correspondence should be addressed: shwoo_@kist.re.kr    Page 2 of 24 Magnetic skyrmions are swirling magnetic textures with novel characteristics suitable for future spintronic and topological applications. Recent studies confirmed the room-temperature stabilization of skyrmions in ultrathin ferromagnets. However, such ferromagnetic skyrmions show undesirable topological effect, the skyrmion Hall effect, which leads to their current-driven motion towards device edges, where skyrmions could easily be annihilated by topographic defects. Recent theoretical studies have predicted enhanced current-driven behaviour for antiferromagnetically exchange-coupled skyrmions. Here we present the stabilization of these skyrmions and their current-driven dynamics in ferrimagnetic GdFeCo films. By utilizing element-specific X-ray imaging, we find that the skyrmions in the Gd and FeCo sublayers are antiferromagnetically exchange-coupled. We further confirm that ferrimagnetic skyrmions can move at a velocity of ~50 m s-1 with reduced skyrmion Hall angle, θSkHE ~20°. Our findings open the door to ferrimagnetic and antiferromagnetic skyrmionics while providing key experimental evidences of recent theoretical studies.   Page 3 of 24 Introduction Magnetic skyrmions are non-trivial topological objects1,2 that have been greatly highlighted recently, mainly due to their unique and fascinating topological characteristics suitable for future spintronic applications such as skyrmion-based racetrack memory.3–5 Magnetic skyrmions can be stabilized in the presence of a strong Dzyaloshinskii-Moriya interaction (DMI),6,7 which prefers non-collinear spin orientation between neighbouring magnetic moments. Recent investigations have revealed that, in structures where ultrathin ferromagnets are interfaced with large spin-orbit coupling materials, such as Ta/CoFeB/MgO,8 Pt/Co/Ir,9 Pt/Co/MgO,10 Pt/Co/Ta and Pt/CoFeB/MgO,11,12 the interfacial DMI can be strong enough to stabilize the chiral skyrmion structure even at room temperature. It has also been shown that skyrmions can move along magnetic tracks upon the injection of an electrical current,8,11,13,14 which indicates that the skyrmions can be adopted in practical devices.5 However, contrary to theoretical predictions, ferromagnetic skyrmions have shown relatively slow and pinning-dominated current-driven dynamic behaviours at room temperature.8,11,13–15 More seriously, ferromagnetic skyrmions exhibit an inevitable topology-dependent effect, namely, the skyrmion Hall effect,16–19 where the magnetic skyrmions do not move collinear to the current flow direction but acquire a transverse motion due to the appearance of a topological Magnus force acting upon the non-zero topological charge. The Magnus force is opposite for a ferromagnetic skyrmion whose topological charge (Q) is either Q = 1 or -1. To avoid this issue, theoretical studies have suggested that the skyrmion Hall effect can be suppressed by utilizing antiferromagnetically exchange-coupled skyrmions with Q = 1 and -1 in antiferromagnetic materials.20–23 Along with the recent interest in antiferromagnets resulting from their intrinsic ultrafast dynamics and insensitivity to disturbing magnetic fields,24 these advantages in skyrmion dynamics have generated an intense interest in antiferromagnetic skyrmion textures. However, since such an excitation behaves as if it were a simple bubble with Q = 0, it is quite difficult to experimentally verify if it is really a topological object. In this work we investigate a similar object in ferrimagnet multilayers. Namely, we consider skyrmion excitations in one magnetic layer and in the magnetic other layer. Since they are antiferromagnetically coupled, their topological charges are opposite. Let us refer to such a pair of skyrmions as a ferrimagnetic skyrmion. The distinctive nature is the topological Magnus forces are not balanced exactly because the magnetization is different between two layers. We predict that ferrimagnetic skyrmions show a small but nonzero skyrmion Hall effect, which we may use as a verification of the topological excitation. We Page 4 of 24 note a recent study reporting the stabilization of ferrimagnetic skyrmions in Fe/Gd multilayers,25 in which Bloch-type skyrmions are stabilized by long-range dipolar interactions. However, current-driven dynamics of these skyrmions cannot be used for actual applications, mainly due to the random chirality of each individual skyrmion that results in a complex and random current-driven motion.26,27 Thus, the actual observation of deterministic and efficient current-driven dynamics of chiral ferrimagnetic skyrmions remains elusive so far and needs to be pursued for the realization of skyrmion-based memory and logic computing devices.  Results X-ray microscopy observation of magnetic domains In our multilayer stack, the Pt heavy metal underlayer is used to induce a strong DMI that stabilizes chiral magnetic textures. Note that the [Pt(3 nm)/Gd25Fe65.6Co9.4(5 nm)/MgO(1 nm)]n structure with a large repetition number, n = 20, is used in this study, due to: i) the sizable internal demagnetization field that effectively drives as-grown the magnetization state into the multi-domain state (see Supplementary Fig. 1 and Supplementary Note 1 for its hysteresis behaviours and the top left panel in Fig. 1a for its as-grown multi-domain state), ii) a large amount of magnetic material that resulted in the enhanced magnetic contrast in our X-ray transmission measurement, and iii) promising skyrmion characteristics observed in the structure, which will be presented throughout this Article. Using vibrating sample magnetometry (VSM) measurements, we estimate the magnetization compensation point, TM, of our GdFeCo film to be > 450 K (see Supplementary Fig. 2 and Supplementary Note 2 for details). Hence, our magnetic devices remain at an uncompensated ferrimagnetic state throughout all the room-temperature X-ray measurements. Moreover, to correctly extract the magnetic parameters of our ferrimagnetic films, we have performed several experiments including ferromagnetic resonance (FMR), spin-torque FMR (ST-FMR), and X-ray magnetic circular dichroism (XMCD) spectroscopy measurements on a companion GdFeCo film grown on SiOx/Si substrate (see Supplementary Figs. 3-6, Supplementary Note 3 and Methods for detailed measurement descriptions and acquired parameter values). To reveal the nature of ferrimagnetic skyrmions in our system, we first performed element-specific scanning transmission X-ray microscopy (STXM) in the presence of an external perpendicular magnetic field, Bz. Figure 1a shows the STXM images of the domain structure in a patterned 2.5-µm-wide and 5-µm-long Pt/GdFeCo/MgO film with decreasing external magnetic field from Bz = 0 mT to Bz = -130 mT. The upper and lower panels show Page 5 of 24 corresponding STXM images taken at the absorption edges of Fe (L3-edge) and Gd (M5-edge), respectively. In these STXM images, dark and bright contrasts correspond to upward (+Mz) and downward (-Mz) magnetization directions, respectively. At Bz = 0 mT, a labyrinth stripe domain state with the average domain width of ~220 nm is achieved. Moreover, it is immediately obvious that STXM images at Fe- and Gd-absorption edges show opposite contrast, revealing their expected antiferromagnetic spin ordering within the GdFeCo alloy. Note that, since the measurements are conducted at room temperature, which is lower than the compensation point TM, the magnetic moment of Gd aligns parallel with the external magnetic field while the moment of Fe aligns in an anti-parallel fashion.28 As the magnetic field increases, fewer domains remain and magnetic configurations become less complex. Eventually, by reaching a magnetic field of Bz = -130 mT, we observe multiple isolated skyrmions, and it is evident that the Gd and Fe magnetic moments are still antiferromagnetically exchange-coupled within these skyrmions, therefore confirming that we observed ferrimagnetic skyrmions. The high spatial resolution (~25 nm) of STXM allows us to measure the diameter of the observed skyrmions as discussed in Supplementary Figs. 7-8 and Supplementary Note 4, and we find that the average skyrmion diameter is roughly ~180 nm, which is as small as the skyrmions found in ferromagnetic multilayers with a large DMI value, 1.5 ~ 2 mJ m-2, studied in Refs. [9,11,12]. We later confirm that our skyrmions exhibit a left-handed Néel-type chirality by observing their current-driven behaviours. Figure 1b schematically illustrates the orientation of the antiferromagnetically exchange-coupled internal magnetic moments within the observed ferrimagnetic skyrmion structure.  Current-driven behaviours of ferrimagnetic skyrmions Having established that ferrimagnetic skyrmions can form at a finite external field in this material, we next study their current-induced dynamics in the magnetic track. Figure 2a shows a schematic drawing of our ferrimagnetic track and electric contacts patterned on a 100-nm-thick Si3N4 membrane for transmission X-ray measurements. The actual scanning electron microscopy (SEM) micrograph of our device is also included, and two indicated areas within the image, (i) and (ii), are used to analyse current-driven skyrmion behaviours shown in Fig. 2b and 2c, respectively. In Fig. 2b and 2c, each STXM image, taken at Fe-edge, was acquired after injecting a single current pulse, with the various pulse amplitudes of between 4.90×1010 A m-2 ≤ |ja| ≤ 3.55×1011 A m-2 and the pulse duration of 5 ns (see Supplementary Fig. 9 and Supplementary Note 5 for details on the electronic connections and the actual pulse shape). Note that each skyrmion is colour-circled, and a single colour is used Page 6 of 24 for the same skyrmion throughout each sequence. The pulse polarity, defined by the electric current flow direction, is indicated in the figure as red and blue pulse-shaped arrows. It is also noteworthy that, while skyrmion core points +z and –z directions in STXM images in Fig. 2b and Fig. 2c, respectively, the effective core magnetization points -z and +z directions in Fig. 2b and Fig. 2c, respectively, because Gd moments are dominant in our material at room temperature as discussed above. Fig. 2b first shows a sequence of STXM images of skyrmions stabilized by a magnetic field of Bz = 145 mT near the left Au electrode on a magnetic track. In our field of view, there are initially two skyrmions, and as we inject leftward current pulses, additional two skyrmions appear while all of the skyrmions show homogeneous propagations. Moreover, it is noticeable that, when a train of the skyrmions propagates along the track, their alignment/trajectory shows a finite angle with respect to the current flow direction, which is the hallmark of the skyrmion Hall effect. We then reversed the magnetic field to Bz = -145 mT and investigated another region near the right Au electrode on the same track, as shown in Fig. 2c. While no skyrmion is observed in the field of view of the first image, up to four skyrmions appear with pulse injections, and all of them show pinning-free homogeneous displacements, as was observed for the other polarity of skyrmions in Fig. 2b. Note that a train of these bright skyrmions also shows transverse velocity component, and surprisingly, the sign of slope is opposite to the case of dark skyrmions. This is due to the opposite topological polarity of skyrmions for two cases, which experience the opposite sign of the topological Magnus force that consequently provides opposite transverse propagation directions. This observation of finite skyrmion Hall effect and the symmetry of skyrmion Hall angle in our ferrimagnetic material agrees well with the cases of ferromagnetic skyrmions in Ta/CoFeB/TaOx17 and Pt/CoFeB/MgO.18 Moreover, in the last image of each sequence, we observe the repulsion between skyrmions and the edges of the sample, which locates right underneath the field of views as shown in Fig. 2a. The repulsion occurs due to the DMI boundary condition,29 which results in the skyrmion motion i) back toward the sample center (blue-circled skyrmion in Fig. 2b) or ii) straight along the sample edge (yellow-circled skyrmion in Fig. 2b and blue-circled skyrmion in Fig. 2c), which also agrees with previously observed ferromagnetic skyrmion motion along the edge in Ref. [17]. With these observations shown in Fig. 2b and 2c, three important qualitative conclusions on the skyrmion physics within ferrimagnetic material can be drawn. First and most importantly, our investigation reveals that ferrimagnetic skyrmions can also be displaced by electric currents at room temperature just as the ferromagnetic skyrmions.8,11–Page 7 of 24 14,17,18 Moreover, we show that the skyrmion propagation direction is along the current flow direction (against the electron flow) for both +Mz-core and -Mz-core skyrmions, and this same directionality agrees well with the spin Hall current-driven motion of homochiral left-handed Néel-type hedgehog skyrmions stabilized by interfacial DMI in Pt/ferromagnet thin films.11–14,18 This implies that the interfacial DMI at the Pt/GdFeCo interface plays a crucial role in stabilizing skyrmions and also driving them on the track in our ferrimagnetic structure. Furthermore, the skyrmion pinning, which was often observed in many of ferromagnetic systems,11,13,14,17 is significantly reduced in our ferrimagnetic material. This low pinning may originate from the amorphous nature of GdFeCo because the absence of grain boundaries leads to lower skyrmion pinning, as was observed in amorphous CoFeB ferromagnetic films.11 Overall, it is noteworthy that our observation serves as the first experimental observation of current-driven excitation of nanoscale magnetic skyrmions in ferrimagnets. The pulse amplitude-dependent skyrmion velocity and its skyrmion Hall angle are plotted in Fig. 2d and 2e, respectively. To correctly calculate the distance and angle between two images, we have performed image-displacement correction using the edge between our magnetic track and Au electrode. It is first noticeable that skyrmion velocity increases linearly with pulse amplitudes, and the maximum velocity approaches ~50 m s-1 at |ja| = 3.55×1011 A m-2, which is comparable to the current state-of-the-art skyrmions observed in a few ferromagnetic heterostructures.11,14,18 Moreover, as shown in Fig. 2e, we observe a very small skyrmion Hall angle, |θSkHE| up to ~20°, which is far lower than the skyrmion Hall angles, |θSkHE| > 30°, observed for ferromagnetic skyrmions in Ta/CoFeB/MgO and Pt/CoFeB/MgO structures.17,18 Antiferromagnetic coupling between two sublayers and the corresponding largely-reduced net magnetization within GdFeCo films have led to the effective inhibition of the skyrmion Hall effect. It is noteworthy that relatively large skyrmion Hall effect in conventional ferromagnets may lead skyrmions toward device edges, where they could easily be annihilated by topographic defects.20,30 Moreover, the skyrmion Hall effect-driven strong transverse motion may pose a limitation for the maximum skyrmion speed due to the finite edge repulse. Therefore, we believe that our ferrimagnetic multilayers can serve as an important magnetic material for such future skyrmionic devices that could replace conventional ferromagnets with enhanced reliability and mobility. Note that skyrmion Hall angle increases monotonically at low current densities, |ja| < ~2×1011 A m-2, and saturates at high current densities, because the skyrmion dynamics is dominated by pinning sites at low driving forces, which is similar to the creep motion of ferromagnetic skyrmions in low-current-density regime caused by the pinning potential.15,17 The remnant finite Page 8 of 24 skyrmion Hall angle, θSkHE ~ 20°, results from the uncompensated magnetic moments between Gd and FeCo sublayers at 300 K < TM, where MS_Gd ≠ MS_FeCo. (see Methods). Therefore, by adjusting material compositions, it will be possible to further reduce the effective skyrmion Hall angle in ferrimagnetic GdFeCo films.  Micromagnetic simulation on ferrimagnetic skyrmion dynamics For more comparison, we simulated the current-driven dynamics of a ferrimagnetic skyrmion (see Methods for more modelling details) in a checkerboard-like two-sublayer spin system based on the G-type antiferromagnetic structure with simple square lattices,20,30,31 where the two sublayers, corresponding to Gd and FeCo, are coupled in a ferrimagnetic manner with a net saturation magnetization, while each sublayer is ferromagnetically ordered. We have also examined simulations using the two-sublattice model with classic J1-J2-J2’ Heisenberg exchange interactions as shown in Supplementary Figs. 10-12 and Supplementary Note 6. While we find that the intra-sublattice exchange interactions indeed affect on the ferrimagnetic skyrmion size and dynamics, however, the influence of these effects on the overall dynamics, especially on the skyrmion Hall effect, turns out not to be significant in both qualitative and quantitative results. Simulations were performed with both models: with and without pinning defects, using experimentally measured materials parameters given in Methods. Moreover, in simulations, we considered the error range of damping coefficient measurement shown in Supplementary Fig. 3 and Supplementary Note 3, because the ferrimagnetic skyrmion dynamics can be strongly influenced by small changes in damping coefficient (see Supplementary Figs. 13-14 and Supplementary Note 7 for details). Note that the error ranges of simulations are shown as shaded areas in Fig. 2d and 2e. The simulated skyrmion velocity as a function of the current density is first shown in Fig. 2d. Simulation results show qualitative and quantitative agreement with experimental observations, revealing that the ferrimagnetic skyrmion velocity is linearly proportional to the driving current density. We also calculated the skyrmion Hall angle as a function of the current density as shown in Fig. 2e. It is first noticeable that the skyrmion Hall angle in the model without pinning defects is independent of the current density, while the skyrmion Hall angle in the model with certain pinning defects increases with increasing current density and approaches a constant value calculated with the pinning-absent model. This linear increase is qualitatively consistent with our experimental observations and also with previous report17, indicating the existence of certain pinning effects due to impurities or defects in the real material. Moreover, the Page 9 of 24 calculated skyrmion Hall angle considering the damping errors shows a fair quantitative agreement with experimental observation. Although the averaged skyrmion Hall angle for simulation is still slightly larger than experiments, we speculate that the small difference may originate from the larger effective damping associated with skyrmion dynamics. Weindler et al. recently reported that local FMR (α = 0.0072) and domain wall dynamics measurements (α = 0.023) yield very different damping parameters for the same material, Permalloy, and magnetic texture-induced nonlocal damping may be responsible for the increase in effective damping.32 Gerrits et al. also reported that, unlike small-angle magnetization dynamics such as conventional FMR, large-angle magnetization dynamic could induce an increase in the apparent damping,33 and we believe this scenario could also be used to explain our case, where skyrmion motion involves the large-angle magnetization dynamics. Moreover, because a patterned 2.5-µm-wide and 5-µm-long nanowire structure was used for skyrmion study while continuous films were employed in material parameter analysis, roughness-induced extrinsic damping enhancement could be another source of damping increase.34 Nevertheless, by considering above possible scenarios, our quantitative results on the suppression of the skyrmion Hall effect can be reasonably understood.  Discussion We have so far observed that the current-driven behaviours of ferrimagnetic skyrmions are indeed attractive compared with their ferromagnetic counterparts. However, it may also be possible that the small value of skyrmion Hall angle in a ferrimagnetic material is only from the small saturation magnetization in a ferrimagnet and not from its antiferromagnetic characteristic. Thus, for a fair comparison between ferrimagnetic and ferromagnetic skyrmions, we have performed simulations on the current-driven dynamics of skyrmions with the same low (net) saturation magnetization, as shown in Fig. 3. It should first be pointed out that, by using the same material parameters, the sizes of the ferrimagnetic and ferromagnetic skyrmions are measured to be identical at certain out-of-plane magnetic fields (Fig. 3a, inset). Figure 3a shows the velocities of the current-driven ferrimagnetic and ferromagnetic skyrmions as a function of the driving current density. It can be seen that both the ferrimagnetic and ferromagnetic skyrmion velocities increase with increasing driving current density, where the mobility of ferrimagnetic skyrmion is slightly larger than that of ferromagnetic skyrmion. More significantly, as shown in Fig. 3b, the skyrmion Hall angle of ferrimagnetic skyrmion is significantly smaller than that of the ferromagnetic skyrmion by a Page 10 of 24 factor of 2, even the net saturation magnetizations are the same for both skyrmions. Note that skyrmion Hall angles are independent of the current densities for both cases, as the pinning effect is not considered in this comparison. To compare the effect of pinning on ferrimagnetic and ferromagnetic skyrmions more carefully, we have simulated the current-driven dynamics of both ferrimagnetic and ferromagnetic skyrmions in the same disorder model, which is shown in Supplementary Fig. 15 and Supplementary Note 8. It is found that the given pinning effect on the current-driven skyrmion dynamics is not significant for both ferrimagnetic and ferromagnetic cases, especially at a large driving current density (e.g. 5×1011 A m-2), however, at a small driving current density (e.g. 1×1011 A m-2), the ferrimagnetic skyrmion motion is more influenced by the pinning effect. The reason could be that the ferromagnetic skyrmion experiences a stronger Magnus force, which helps it in overcoming obstacles.35,36 Nevertheless, larger skyrmion velocity and smaller skyrmion Hall angle for ferrimagnetic case were maintained over the whole range of examined current densities. Overall, simulation results indicate that, even when the ferrimagnetic and ferromagnetic skyrmions have the same low (net) saturation magnetization, the current-driven dynamics of the ferrimagnetic skyrmion is much more reliable for the transport in narrow channels as information carriers. In conclusion, we have observed and studied the stabilization and current-driven dynamics of antiferromagnetically exchange-coupled skyrmions in ferrimagnetic GdFeCo films. By utilizing the element-specific X-ray imaging, we have identified that the ferrimagnetic skyrmion in the GdFeCo films consists of two antiferromagnetically exchange-coupled skyrmions in the Gd and FeCo sublayers. We further confirm that current-driven ferrimagnetic skyrmions can move at a velocity of ~50 m s-1 with reduced skyrmion Hall angle, |θSkHE| ~20°. With micromagnetic simulations, we reveal that ferrimagnetic skyrmions are much more attractive than their ferromagnetic counterparts in many technological-relevant aspects, such as larger skyrmion mobility and strongly suppressed skyrmion Hall effect, mainly due to their antiferromagnetic nature. Our findings reveal the promising dynamic properties of ferrimagnetic skyrmions, and highlight the possibility to build more reliable skyrmionic devices using ferrimagnetic and antiferromagnetic materials.   Page 11 of 24 Methods Sample preparation and experimental method The [Pt(3 nm)/Gd25Fe65.6Co9.4(5 nm)/MgO(1 nm)]20 films were grown by DC magnetron sputtering at room temperature under 1 mTorr Ar for Pt and GdFeCo and 4 mTorr Ar for MgO at a base pressure of roughly ~2×10-8 Torr. Samples were grown on a 100-nm-thick SiN substrate and then patterned using electron beam lithography and lift-off technique. Nominally sample films were grown on SiOx/Si substrate for vibrating-sample magnetometry (VSM), ferromagnetic resonance (FMR), spin-torque FMR (ST-FMR), asymmetric bubble expansion measurements. The series of measurements yielded material constants: anisotropic field µ0Hk = 0.15 T, net saturation magnetization MS = 2×105 A m-1, uniaxial anisotropy, Ku = 4.01×104 J m-3, damping coefficient, α = 0.205±0.035, spin Hall angle, θSH = 0.055, magnetization ccompensation ratio, 𝑛=𝑀!!\"#$𝑀!!\"=0.78 and DMI magnitude, D = -0.98 mJ m-2 (see Supplementary Note 3 for details).  All microscopy images were acquired using the STXM installed at the PolLux (X07DA) beamline of the Swiss Light Source (SLS) at the Paul Scherrer Institute in Villigen, Switzerland. The device used for the experiments was 2.5-µm-wide and 5-µm-long, which yielded an electrical resistance of ~57 Ohms measured in 2-point. This resistance reduced the impedance mismatch of the device, and allowed an almost complete transmission of 5-ns-long short electrical pulses across the device. This was verified by simultaneously measuring the injected (through a -20 dB pickoff T) and transmitted samples with a real-time oscilloscope. Pulse current densities above ~4×1011 A m-2 led to a damage of the Au contact, which eventually limited the maximum current applied in Fig. 2. Skyrmion velocities were determined using the total displacements, measured by acquiring XMCD-STXM images before and after the injection of the pulses, and the integrated pulse time. Three to ten displacements were recorded for each pulse amplitude, and the average value and standard deviations of the individual velocity measurements are plotted in Fig. 2d. Current densities were calculated by dividing the injected current with stripe width and effective total thickness of Pt and GdFeCo.  Simulation method The spin dynamics simulation is carried out by using the Object Oriented MicroMagnetic Framework (OOMMF) with the home-made extension modules for the periodic boundary condition.37 The model is treated as a checkerboard-like two-sublattice Page 12 of 24 spin system based on the G-type antiferromagnetic structure with simple square lattices, where the two sublattices are coupled in a ferrimagnetic manner with a net spontaneous magnetization, while each sublattice is ferromagnetically ordered. The Hamiltonian is based on the classical Heisenberg model, given as ℋ=−𝐽!\"𝐒!∙𝐒!!!,!!−𝐽!\"!𝐒!∙𝐒!≪!,!≫!−𝐽!\"!𝐒!∙𝐒!≪!,!≫!+𝐷𝐮!\"×𝑧∙𝐒!×𝐒!!!,!! −𝐾𝐒!!!!−𝜇!𝐒!∙𝑯!+𝐻!!\"                                           (1) where 𝐒! represents the local spin vector reduced as 𝐒!=𝑴!𝑀!! at the site i, and 𝐒! represents the local spin vector reduced as 𝐒!=𝑴!𝑀!! at the site j. 𝑴! and 𝑴! are the magnetization at the site i and j, respectively. 𝑀!! denotes the saturation magnetization of the sublattice i, while the saturation magnetization of sublattice j is defined as 𝑀!!=𝑛𝑀!! with the compensation ratio n. <i, j> runs over all the nearest-neighbor sites in the two-sublattice spin system. <<i, j>>A and <<i, j>>B run over all the nearest-neighbor sites in the sublattice A and sublattice B, respectively (see Supplementary Fig. 10). 𝐽!\" is the exchange coupling energy constant between the two adjacent spin vector 𝐒! and 𝐒!, which has a negative value (𝐽!\"<0) representing the antiferromagnetic spin ordering of the two sublattices. 𝐽!\"! and 𝐽!\"! are the exchange coupling energy constants for the sublattice A and sublattice B, respectively, which are positive numbers (𝐽!\"!>0, 𝐽!\"!>0) representing the ferromagnetic intra-sublattice coupling. 𝐷 is the interface-induced DMI constant, 𝐮!\" is the unit vector between spins 𝐒! and 𝐒!, and 𝑧 is the interface normal, oriented from the heavy-metal layer to the ferrimagnetic layer. 𝐾 is the perpendicular magnetic anisotropy (PMA) constant, H is the applied magnetic field, and 𝐻!!\" stands for the dipole-dipole interaction, i.e., the demagnetization effect. The time-dependent dynamics of the spin system is controlled by the Landau-Lifshitz-Gilbert (LLG) equation augmented with the damping-like spin Hall torque, which is expressed as !𝐒!!\"=−𝛾!𝐒!×𝐇!\"\"+𝛼𝐒!×!𝐒!!\"+𝜏𝐒!×𝑝×𝐒!                (2) where 𝐇!\"\"=−1𝜇!𝑀!!∙𝛿ℋ𝛿𝐒! is the effective field on a lattice site, 𝛾! is the Gilbert gyromagnetic ratio, and α is the phenomenological damping coefficient. The coefficient for the spin Hall torque is given as 𝜏=𝛾!ℏ𝑗𝜃!\"2𝜇!𝑒𝑀!!𝑏, where 𝑗 is the applied charge current density, 𝜃!\" is the spin Hall angle, and 𝑏 is the thickness of the ferrimagnetic layer. 𝑝=𝒋×𝒛 denotes the spin polarization direction. Page 13 of 24 For the simulation on the multilayer structure, we employed an effective medium approach11 with the lattice constant of 5 Å, which improves the computational speed by converting the multilayer into a two-dimensional effective model with reduced parameters. The intrinsic magnetic parameters used in the simulation are measured from our experimental samples as well as adopted from Refs. [11,18,38]: the damping coefficient α = 0.205 ± 0.035, the inter-sublattice exchange stiffness AGd-Fe = -10 pJ m-1, AGd-Gd = 5 pJ m-1, AFe-Fe = 5 pJ m-1, the spin Hall angle θSH = 0.055, the DMI constant D = -0.96 mJ m-2, the PMA constant Ku = 4.01×104 J m-3, and the net saturation magnetization 𝑀!=𝑀!!\"−𝑀!!\"=200 kA m!!. The compensation ratio is measured as 𝑛=𝑀!!\"𝑀!!\"=0.78. In the experimental multilayer system, the thickness of one ferrimagnetic layer is tm = 5 nm, the thickness of one repetition is tr = 9 nm, and the number of repetitions is nrep = 20. For the simulation on the model with pinning defects, the defects with the size of 10 Å × 10 Å and a higher PMA (Kp = 5Ku) are randomly distributed in the whole ferrimagnetic layer. The density of the defects in the whole model equals 5 %.  Data availability Data supporting the findings of this study are available within the article and its Supplementary Information files and from the corresponding author upon request. The micromagnetic simulator OOMMF used in this work is publicly accessible at http://math.nist.gov/oommf.   Page 14 of 24 References 1. Rößler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442, 797–801 (2006). 2. Mühlbauer, S. et al. Skyrmion Lattice in a Chiral Magnet. Science 323, 915–919 (2009). 3. Jonietz, F. et al. Spin Transfer Torques in MnSi at Ultralow Current Densities. Science 330, 1648–1651 (2010). 4. Yu, X. Z. et al. Skyrmion flow near room temperature in an ultralow current density. Nat. 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Phys. Rev. Lett. 109, 96602 (2012).    Page 18 of 24 Acknowledgements This work was primarily supported by Samsung Research Funding Center of Samsung Electronics under Project Number SRFC-MA1602-01. Part of this work was performed at the PolLux (X07DA) beamline of the Swiss Light Source. S.W. and J.W.C. acknowledge the support from KIST Institutional Program. K.M.S acknowledges the support from the Sookmyung Women's University BK21 Plus Scholarship. X.Z. was supported by JSPS RONPAKU (Dissertation Ph.D.) Program. Y.Z. acknowledges the support by the President's Fund of CUHKSZ, the National Natural Science Foundation of China (Grant No. 11574137), and Shenzhen Fundamental Research Fund (Grant Nos. JCYJ20160331164412545 and JCYJ20170410171958839). M.E. acknowledges the support by the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grant Nos. JP17K05490, 25400317 and JP15H05854), and also the support by CREST, JST (Grant No. JPMJCR16F1). S.F acknowledges the support by the EU Horizon 2020 MAGicSky project (Grant No. 665095). K.M.S. and J.W.C. acknowledge the travel fund supported by the National Research Foundation of Korea (NRF) funded by the MSIP (2016K1A3A7A09005418). S.-Y.P. and B.-C.M. acknowledge the support from the National Research Council of Science & Technology (NST) grant (No. CAP-16-01-KIST) by the Korea government (MSIP). S.-Y.P. also acknowledges the support from KBSI Grant (D37614). S.W. also acknowledges S. Emori for his helpful comments on the manuscript.  Author Contributions S.W. designed and initiated the study. K.M.S. optimized structure, fabricated devices and performed the film characterization. S.W., K.M.S., S.F. and J.R. performed X-ray imaging experiments using STXM at Swiss Light Source in Villigen, Switzerland. X.Z. and Y.Z. performed the numerical simulations. M.E. carried out the theoretical analysis. During the revision of this article, N.J.L., S.-I.K. and S.-Y.P. performed ferromagnetic resonance (FMR), D.L. and O.L. performed spin-torque FMR, and S.W., K.M.S., Y.H.K., J.-Y.K. and J.W.C. performed XMCD spectroscopy at 2A beamline at Pohang Accelerator Laboratory in Pohang, Korea. S.W., X.Z. and M.E. drafted the manuscript and revised it with assistance from X.L., D.L., O.L., S.-Y.P., J.W.C., B.-C.M., H.C.K. and J.C.. All authors commented on the manuscript.  Competing Financial Interest The authors declare no competing financial interests. Page 19 of 24  Author Information S.W. and K.M.S. contributed equally to this work. Correspondence and requests for materials should be addressed to S.W. (shwoo_@kist.re.kr). Page 20 of 24 Figure Legends  Figure 1. Scanning transmission X-ray microscopy (STXM) imaging of domain structure upon magnetic field application. a, STXM images acquired by sweeping the external perpendicular magnetic field from Bz = 0 mT to Bz = -130 mT. Dark and bright contrasts correspond to magnetization oriented up (along +z) and down (along -z), respectively. Upper panel and lower panel show corresponding images acquired at the L3 and M5 absorption edges of Fe and Gd, respectively. Note that, due to longer penetration depth associated with the higher energy used for Gd, ~1189 eV, compared with that of Fe, ~709 eV, magnetic contrast under Au electrodes is visible for Gd magnetic moment imaging. b, Schematic of antiferromagnetically exchange-coupled ferrimagnetic skyrmion on a magnetic track as observed in our GdFeCo films as indicated in the red dashed-square boxes in the last image of a. Scale bar, 1 µm.  Figure 2. Current-driven behaviour of ferrimagnetic skyrmions and their velocity and skyrmion Hall effect. a, Schematic of scanning transmission X-ray microscopy (STXM) geometry, and a scanning electron microscopy (SEM) image of the actual device used for experiments. Scale bar, 2 µm. Sequential STXM images taken at Fe-edge showing the responses of multiple skyrmions after injecting unipolar current pulses along the track at b, Bz = 145 mT and c, Bz = -145 mT, respectively. With a fixed pulse-length of single pulse, 5 ns, the pulse amplitude is changed between 4.90×1010 A m-2 ≤ |ja| ≤ 3.55×1011 A m-2. Pulse polarities are indicated as red- and blue-coloured arrows inside each image. Within STXM images, the same skyrmion is indicated with the same colour. Scale bars, 500 nm. d, Experimental and simulated average skyrmion velocity of Pt/GdFeCo/MgO versus current density. e, Experimental and simulated average skyrmion Hall angle of Pt/GdFeCo/MgO versus current density. In d-e, The shaded areas in plots represent the simulation results considering the damping coefficient error ranges of α = 0.205 ± 0.035, which was measured experimentally as described in Supplementary Fig. 3 and Supplementary Note 3. Note that pulse current densities above ~4×1011 A m-2 led the damage of the Au contact, which eventually limited the maximum applicable current to our sample. Error bars denote the standard deviation of multiple measurements.    Page 21 of 24 Figure 3. Comparison between the current-driven dynamics of ferrimagnetic and ferromagnetic skyrmions of the same net saturation magnetization. (a) The ferrimagnetic and ferromagnetic skyrmion velocities as a function of the driving current density. Inset shows the close-up top view of the ferrimagnetic skyrmion in a square film with periodic boundary conditions in both x and y directions. The lattice constant is set as 5 Å. (b) The ferrimagnetic and ferromagnetic skyrmion Hall angle as a function of the driving current density. The net saturation magnetization was set to be MS = 2×105 A m-1 for both cases. Scale bar, 5 nm.    Page 22 of 24    \n  Figure 1 Seonghoon Woo et al.   \nPage 23 of 24    \n Figure 2 Seonghoon Woo et al.   \nPage 24 of 24     \n  Figure 3 Seonghoon Woo et al. \n"    },    {        "title": "2301.08300v1.Ab_initio_comparison_of_spin_transport_properties_in_MgO_spaced_ferrimagnetic_tunnel_junctions_based_on_Mn__3_Ga_and_Mn__3_Al.pdf",        "content": "Ab initio comparison of spin-transport properties in MgO-spaced\nferrimagnetic tunnel junctions based on Mn 3Ga and Mn 3Al\nM. Stamenova,1,\u0003P. Stamenov,1and N. Baadji2\n1School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland.\n2Laboratoire de Physique des Mat\u0013 eriaux et ses applications & D\u0013 epartement de Physique,\nFacult\u0013 e des Sciences, Universit\u0013 e Mohamed Boudiaf, M'sila, 28000, Alg\u0013 erie\nWe report on \frst-principles spin-polarised quantum transport calculations (from NEGF+DFT)\nin MgO-spaced magnetic tunnel junctions (MTJs) based on two di\u000berent Mn-based Heusler fer-\nrimagnetic metals, namely Mn 3Al and Mn 3Ga in their tetragonal DO 22phase. The former is a\nfully compensated half-metallic ferrimagnet, while the latter is a low-moment high-spin-polarisation\nferrimagnet, both with a small lattice mismatch from MgO. In identical symmetric and asymmetric\ninterface reconstructions across a 3-monolayer thick MgO barrier for both ferrimagets, the linear\nresponse (low-voltage) spin-transfer torque (STT) and tunneling magneto-resistance (TMR) e\u000bects\nare evaluated. A larger staggered in-plane STT is found in the Mn 3Ga case, while the STT in\nMn3Al vanishes quickly away from the interface (similarly to STT in ferromagnetic MTJs). The\nroles are reversed for the TMR, which is practically 100% in the half-metallic Mn 3Al-based MTJs\n(using the conservative de\fnition) as opposed to 60% in the Mn 3Ga case. The weak dependence on\nthe exact interface reconstruction would suggest Mn 3Ga-Mn 3Al solid solutions as a possible route\ntowards optimal trade-o\u000b of STT and TMR in the low-bias, low-temperature transport regime.\nI. INTRODUCTION\nAmong the Mn-based Heusler ferrimagnets are\na number of topical binary materials, combining\nlow moments with high Curie temperature and\nspin-polarisation [1{4], as well as high anisotropy\nwith low Gilbert damping [5, 6] { hence holding\npromise for the emerging \feld of Antiferromagnetic\nSpintronics[7, 8]. Ferrimagnetic electrodes, and es-\npecially, the compensated ones, in magnetic-tunnel\njunction (MTJ) devices lead to the reduction of\nthe demagnetizing \feld and, more importantly, the\nreduction of the critical current needed to switch\nthe magnetization together with the fast dynam-\nics of such switching, involving inter-sublattice ex-\nchange interactions. The possible exploitation of\nthis class of materials, however, depends on the\nability to produce high-frequency oscillations (in\nthe 100s of GHz range) and the depth of resis-\ntance modulation that junctions using these could\nsupport. Predictive computational guidance for\nthe magnitude and resilience of both the spin-\ntransfer torque (STT) and the tunneling magneto-\nresistance (TMR) e\u000bects, towards the di\u000ecult-to-\ncontrol experimentally barrier quality and inter-\nface reconstruction, can help to speed-up the de-\nvelopment of functional prototype devices.\nOne such candidate is Mn 3Ga, which in its\ntetragonal DO 22phase is a low-moment ferrimag-\nnet with a high spin-polarisation and anisotropy [1,\n2, 9], but also a low Gilbert damping and an estab-\nlished epitaxial relationship with MgO(001) [10].\nA large staggered long-ranged STT e\u000bect has been\nfound theoretically in MTJs based on DO 22-phased\nMn3Ga [11], present both in Fe/MgO/Mn 3Ga and\nin Mn 3Ga/MgO/Mn 3Ga tri-layers, and related to\n\u0003Contact email address: stamenom@tcd.iethe mismatch of the Fermi wavevectors of the ma-\njority and minority \u0001 1symmetry band in Mn 3Ga\nin the direction of transport. Theoretically, the\nTMR e\u000bect in these stacks reaches a few tens or\npercent and exhibits a sign change below 1V, which\nis in accordance with experimental observations for\nsimilar ferrimagnetic MTJs [12].\nAnother Mn-based Heusler Mn 3Al has been\nshown to exhibit half-metallicity and almost ide-\nally fully compensated moment in its cubic DO 3\nphase [13], similarly to Mn 3Ga in this phase[3],\nand proposed MTJs with GaAs have shown large\ntheoretical TMR ratios [14]. A GGA-PBE geome-\ntry optimisation of Mn 3Al reveals a stable tetrag-\nonal DO 22solution with almost fully compensated\nmoment and an in-plane lattice constant commen-\nsurate with MgO. As geometrically the Mn 3Ga and\nMn3Al stacks with MgO are very similar, but o\u000ber\ndi\u000berent placement of the Fermi level with respect\nto the main band dispersions, a comparison of the\nspin-transport in analogous MTJs could unveil fur-\nther insights about the Spintronic capacity of the\ntwo Heuslers. Here we \frst examine the electronic\nstructure properties of both materials in bulk (Sec-\ntion II) and then we compare the spin-dependent\ntransport properties of two pairs of MTJs, all with\n3-monolayers (ML) thick MgO spacers, but featur-\ning two di\u000berent terminations at one of the inter-\nfaces (Section III).\nII. BULK PROPERTIES\nIntermetallic Heusler alloys X 2YZ crystallize\nusually in the cubic L2 1structure, especially at\nhigh temperature, and at low temperature they\ncan develop a DO 22tetragonal structure with a\nratioc=aaroundp\n2. The larger departure of this\nratio fromp\n2 is a prelude for a higher magneto-arXiv:2301.08300v1  [cond-mat.mes-hall]  19 Jan 20232\nFIG. 1. Band structures and spin-polarised density of states of (a,c) Mn 3Al and (b,d) Mn 3Ga, respectively\nin their depicted unit cell, calculated using LDA-PW92 (for Mn 3Al) and LDA-CA (for Mn 3Ga) on the MgO-\nmatching (strained) geometries (black or red/blue \flled curves, see text and insets). In all panels a comparison\nis shown with a corresponding GGA-PBE calculation for the relaxed unit cell (green curves, see text/legends for\ndetails). In (a,b) the DO 22unit cells of both materials with the local spins of the Mn atoms shown as arrows.\ncrystalline anisotropy [1]. Our calculated optimum\nlattice parameters, using the GGA-PBE, of both\ncompounds in their antiferromagnetic con\fgura-\ntion area= 4:057,c= 5:911\u0017A anda= 3:78,\nc= 7:1\u0017A for Mn 3Al and Mn 3Ga, respectively.\nThe in-plane lattice constants are close to that\nof bulk MgO ( aMgO = 4:21\u0017A), which makes\ntheir integration in conventional magnetic tunnel\njunctions feasible. The DO 22Mn3Al(Ga) struc-\ntures are constructed by alternating planes of Mn-\nAl(Ga) (Mn in 2 bWycko\u000b position: Mn I) and Mn-\nMn (Mn in 4 dWycko\u000b position: Mn II) coupled\nanti-ferromagnetically, along the z-axis (see unit\ncell schematics in Fig. 1). In such a lattice struc-\nture, Mn 3Ga is metallic and both spins contribute\nto the conductivity, while Mn 3Al is a half-metal\nand only spin-up states contribute to the conduc-\ntion. Consequently, Mn 3Al has a 100% spin po-\nlarization at the Fermi level and therefore one can\nexpect a high TMR for junctions based on Mn 3Al.\nMn3Ga also shows a high spin polarization of 88%\nin other GGA-based calculations [2]. Similar re-\nsults for the Fermi level spin-polarisation are ob-\ntained also when the transverse lattice constants\narea=b= 4:1\u0017A in both materials (approaching\nthat of MgO thin \flms), as shown in the density\nof states presented in Fig. 1(c,d).\nThe corresponding band structure of both com-\npounds is presented in Fig. 1(a,b) and one can see\nthat for Mn 3Al the spin-down channel exhibits a\ngap of the order of 0.4 eV when computed with\nGGA-PBE on the relaxed unit cell (green points)\nand similarly about 0.25 eV, when calculated with\nLDA-PW92 (the black points), as implementedin the Siesta code [15]. This gap can be tuned\nby changing the lattice parameters. In the LDA\ncase, we apply a longitudinal tensile strain of 4%\n(c= 6:027\u0017A) to open the gap and approximate\nthe GGA result (where c= 5:795\u0017A). The impact\nof that on the layer-resolved magnetic moments\n(calculated by Mulliken population analysis) is a\nsmall decrease by about 7 % for both Mn sublat-\ntices compared to GGA. We have additionally es-\ntablished that Mn 3Al keeps its fully-compensated\nferrimagnetic character for applied strains ranging\nbetween -4% to 8% (so that the 100% polarization\nis kept for such a strain). Note that the orbitals\nbelow the Fermi level are mainly a hybridisation\nbetween d-t2gorbitals of the Mn atoms, while the\nempty bands above the Fermi level are egbands of\nthe Mn occupying site 4 d.\nFor Mn 3Ga we are using a longitudinal lat-\ntice constant c= 6:6\u0017A, which has been found\nto reproduce more reasonably the experimen-\ntal values of the Mn spins (as extracted from\nneutron-di\u000braction results [1]), within the LDA\n[see Fig. 2(b,c) for the computed layer-resolved\nmagnetic moments], compared to the GGA-relaxed\nor the experimental lattice parameter values c=\n7:1\u0017A (larger by about 7 %), for which the LDA-\ncomputed magnetic moments are larger by over\n15 % with respect to the c= 6:6\u0017A case and\noutside the experimental range. We assume that,\nwith these structural amendments (consistent also\nwith Ref. 11 for Mn 3Ga-based MTJs), the LDA,\nwhich is not currently replaceable by GGA in our\nnon-collinear-spin method for spin-transfer torque\n(STT)[11], captures the essential Fermi-surface3\nFIG. 2. (a) Schematic of the four MTJs considered, including the directions of the spin quantisation axes in the\ntwo leads at the 90 °alignment for the STT calculations. Note, that the corresponding 'asymmetric' MTJs have\none layer of Mn removed from the right interface. (b,c) Self-consistently calculated layer-resolved spin-components\n(xandz) for the four MTJs at equilibrium (see legend for the color code). (d,e) Corresponding layer-resolved in-\nplane STT components calculated at the Fermi level in linear response regime (so-called, torkance \u001c=dT(V)=dV\nat the limit V= 0, whereT(V) is the STT, see Ref. 16 for precise de\fnition) in the four di\u000berent MTJs (same\ncolour code). A=a2= 16:81\u0017A2is the cross-sectional area of the junction (it is the same for all; note the periodic\nboundary conditions in the x\u0000yplane).\nproperties of the two materials relevant for the\nlinear-response regime investigated here and we\ncontinue exclusively with the LDA and the de-\nscribed above lattice parameters of both materials\nin Section III.\nConsequently, for the just described geometries,\nboth compounds are ferrimagnetic with Mn 3Al be-\ning fully compensated (total magnetic moment '0\n), while Mn 3Ga having a 2.6 \u0016Bper cell (1.3 \u0016B\nper formula unit). The advantage of having a very\nlow-spin-moment lead is to reduce the demagnetiz-\ning \feld, but more importantly, to reduce the crit-\nical current for a spin-transfer torque switching,\nwhich we will discuss it in the next Section III. We\nshould mention here that the magneto-crystalline\nanisotropy calculated for Mn 3Ga is much bigger\nthan that of Mn 3Al and this is because of the much\nstronger spin-orbit interaction at the Ga site com-\npared to Al. The local spins (extracted by Mul-\nliken population analysis) on the two Mn sublat-\ntices in Mn 3Al are nearly fully compensated with\n3.31\u0016Bon the Mn Isite and -1.58 \u0016Bon the Mn II\nsite, respectively. In Mn 3Ga the corresponding val-\nues are 3.53 and -2.46 \u0016Bfor Mn Iand Mn II, re-\nspectively, which are consistent with the measured\nmoments[1].III. SPIN-TRANSPORT IN Mn 3Al AND\nMn3Ga JUNCTIONS WITH MgO\nBARRIERS\nFour di\u000berent junctions based on Mn 3Al and\nMn3Ga all sandwiching 3 MLs of MgO have been\ninvestigated [Fig. 2a]. In the junctions, which we\nrefer to as 'symmetric', the two interfaces are the\nsame, that is, in both cases the interface is be-\ntween the MgO and the Mn II-plane of the DO 22\nlattice and overall the junction is mirror-symmetric\nwith respect to the central plane in the MgO. In\nthe 'asymmetric' junctions we have removed one\nmonolayer of Mn from the right interface, but all\ndistances, including the interface spacing, are pre-\nserved. Note that these geometries have not been\nrelaxed { the interface distance we have chosen in\nthe Mn 3Al case is 2 \u0017A, while in the Mn 3Ga junc-\ntions we have chosen 2.2 \u0017A, as motivated in Ref.\n11.\nIn Fig. 2 we compare the linear-response STT\n[11, 16] for a 90 °misalignment of the spin po-\nlarisations in the two leads, computed within the\nnon-equilibrium Green's function (NEGF) open-\nboundaries method implemented in the Smeagol\ncode [17]. Panels (b) and (c) show the layer-\nresolved local moments in the scattering region {\nin (b) the mirror symmetry is readily observed be-4\ntween thexandzcomponents of the spins on both\nsides of the junction. In comparison, in (c) the\nasymmetry after the removal of one Mn plane at\nthe right interface is evident. It is worth noting\nthat the spins in the left lead across the MgO ap-\npear practically una\u000bected by this local structural\ndisturbance on the right interface, in both junc-\ntions.\nThe calculated layer-resolved in-plane STT for\nthe symmetric junctions [Fig.2(d)] displays a per-\nfect left-to-right symmetry as well { this time an in-\nversion symmetry with respect to the central layer\nof MgO. For both ferrimagnets the STT at the op-\nposite interfaces has an opposite sign. There is,\nhowever, a qualitative di\u000berence between the two\nferrimagnets { if the STT in the Mn 3Al case is lo-\ncalised at the interface, in Mn 3Ga it shows the fa-\nmiliar long-range oscillatory decay with periodicity\ndetermined by the di\u000berence of majority and mi-\nnority spin wave-vectors of the \u0001 1-symmetry band\nin Mn 3Ga, as described in Ref. 11. This ideal\ninversion symmetry of the STT across the bar-\nrier, however, will hinder the switching of the such\nideally symmetric junctions. For instance, a posi-\ntivex-component of STT will rotate anti-clockwise\nthe Mn IIspins at the left interface (aligned along\n\u0000z), and also anti-clockwise the Mn IIspins aligned\nalong\u0000xat the right-hand-side interface. Hence\nsuch an alignment of the STT cannot drive switch-\ning of one lead with respect to the other. It is\nworth noting, however, that the STT changes sign\nin the next bi-layer { this will act against the sub-\nlattice exchange coupling. This is would enable\nthe excitation of high frequency anti-phase modes\nand potentially lead to fasted switching dynamics.\nIt is likely that the ideal symmetry in our calcu-\nlations will be broken in real structures and there\nwill be an STT imbalance leading to the switch-\ning of one of the layers. In the Mn 3Al junction the\nSTT has analogous symmetry, which does not lead\nto switching. It is, however, much more localised\nat the interfacial layer and signi\fcantly lower in\namplitude.\nTo illustrate the e\u000bect of possible structural im-\nperfection we consider the 'asymmetric' versions of\nboth junctions { with removed Mn-Mn monolayer\nfrom the right interface, while keeping the inter-\nface spacing unchanged [Fig. 2(a)]. In this case\nthe STT pro\fles change substantially [Fig.2(d,e)].\nNow we see both in the Mn 3Al and Mn 3Ga junc-\ntions the STT becomes asymmetric on both sides\nof the barrier. This time it shows a tendency to ro-\ntate spins in opposite directions, especially in the\nMn3Al case, thus driving a switching of one of the\nlayers with respect to the other. In the asymmetric\nMn3Al junction the STT is signi\fcantly increased\nin magnitude from the symmetric case, especially\non the right-hand side, where we see a large STT\nalso beyond the interfacial layer.\nWe then examine the energy dependence of the\ntotal transmissions in the four junctions in their\nFIG. 3. Energy-resolved properties at 0 V equilibrium\nfor both the P and AP states of the junctions (see text).\n(a,b) transmission coe\u000ecients and (c,d) TMR ratio,\nde\fned as TMR = ( TP\u0000TAP)=(TP+TAP) for the two\nsymmetric and the two asymmetric MTJs, respectively.\ncollinear spin states { parallel (P) state in which\nthe Mn spin in each sublattice (Mn Ior Mn IItype)\non the two sides of the barrier are parallel, and\nAP state, when they are antiparallel. We clearly\nsee the half-metallicity of Mn 3Al manifesting it-\nself in the vanishing transmission in the AP state\naround the Fermi level. This in turn drives a large\nTMR (practically 100% from the conservative de\f-\nnition with the sum of the transmission coe\u000ecients\nin the denominator) for a wide range of energies\naroundEFin both the symmetric and the asym-\nmetric junctions [green curve in Fig. 3 (c,d)]. In\nthe case of Mn 3Ga we \fnd a TMR of about 60 %\nfor the symmetric junction, which drops to 30 %\nfor the asymmetric case. We \fnd a signi\fcant en-\nhancement of the TMR for energies higher than\n0.2 eV above EF, which is due to the \u0001 1symmetry\nband edge and the band gap opening above that\nenergy for majority spin in the \u0000 \u0000Z(transport)\ndirection [11] (note that the position of this band\nedge above EFis consistent between the LDA and\nGGA calculations in Fig.1). The role of the in-\nterfacial asymmetry appears to be in reducing the\ntransmission in the P state, where in the asymmet-\nric junctions the interfacial spins are pointing in\nopposite directions, giving rise to additional scat-\ntering at the interfaces.\nThis can also be seen in the 2D portraits of the\nFermi-level transmission coe\u000ecients in the trans-\nverse 2D Brillouin zone (2D-BZ) (Fig. 4). In the\nMn3Ga case the transmission in all spin-states and5\nFIG. 4. Contour portraits of the total transmission coe\u000ecient (both spin species) at the Fermi level decomposed\nover the transverse 2D-BZ [as a function of ( kx;ky) on each panel, where \u0000\u0019=a\u0014kx;y\u0014\u0019=a]. From left to right,\nthe four di\u000berent MTJs are depicted (as indicated above), while the two rows correspond to the parallel (P) and\nanti-parallel (AP) spin states of the junctions, respectively (see text). Color-code bars for each panel are for the\nT(kx;ky;E=EF), i.e. dimensionless transmission probabilities (see, e.g. Ref.16).\njunction geometries is predominantly around the\n\u0000 point. The P states show stronger transmis-\nsion in both junctions, while in the asymmetric\ncase both spin-polarised transmissions are some-\nwhat suppressed with respect to the correspond-\ning ones in the symmetric MTJ. In contrast to\nMn3Ga, the transmission around the \u0000 point is\nsuppressed in the two Mn 3Al junctions in the P\nstate, as in this material there is a band gap along\n\u0000\u0000Zdirection for majority (up) spins [see Fig.\n1(a)]. Interestingly, despite the small quantita-\ntive changes, which the asymmetric interface in-\ntroduces in some regions of the transverse 2D-BZ,\nfor the small range of wavevector angles relevant\nto realistic specular transport around the \u0000 point,\nthe transmission appears relatively una\u000bected by\nthe interfacial detail. This e\u000bect is expected to be\nenhanced further for thicker MgO barriers, where\nthe transmission is more strongly \fltered in the\nvicinity of \u0000 point and we expect the overall trans-\nmission in Mn 3Al-based MTJs to be even less sen-\nsitive to features of the interfaces. We also note\nthat in the AP state of the Mn 3Al junctions the\ntransmission is extremely small and at the verge\nof computational accuracy, as expected from its\nhalfmetallicity (see Fig. 1).\nIV. CONCLUSIONS\nWe have compared linear-response spin-\npolarised transport properties in ferrimagnetic\ntunnel junctions made from two similar Mn-\nbased binary Heuslers in their tetragonal phase\nsandwiching a MgO barrier. Junctions featuring\nMn3Al o\u000ber a di\u000berent compromise between the\nmagnitude of STT and TMR, when compared\nto otherwise equivalent Mn 3Ga-based ones. The\noverall transmission, and therefore the expecteddi\u000berential conductivity, is higher in the ferrimag-\nnetic Mn 3Ga case, together with the low-bias STT.\nThe absence of one type of spin states for Mn 3Al-\nbased junctions, leads to smaller overall absorbed\nmomentum (with STT acting only within a couple\nof lattice spacings away from the interfaces) and\na signi\fcantly suppressed low-bias di\u000berential\nconductance. It can be argued, therefore that at\nleast at low temperature and applied bias, when\ncompared on the basis of STT per unit dissipated\npower, the Mn 3Ga-based junctions would o\u000ber\nan edge over the otherwise better in their TMR\nperformance Mn 3Al counterparts. The relatively\nlow sensitivity of the calculated parallel-state\ntransmission (normal to the interface) and also\nof the STT, although to a lesser extent, on the\nexact surface reconstruction in both Mn 3Al and\nMn3Ga-based junctions, may further motivate the\nexperimental veri\fcation of spectroscopic TMR\nand STT features (at low temperatures), hence,\n\fnite-bias calculations of the same would be well-\nwarrantied. In view of the above, aluminium-rich\nsolid solutions of Mn 3Al-Mn 3Ga, with or without\nSn doping, could o\u000ber enhanced low-bias TMR\nperformance, while preserving tetragonality, at\nmanageable growth-induced strain, in practical\ndevice stacks.\nACKNOWLEDGMENTS\nWe gratefully acknowledge funding from the\nScience Foundation Ireland (SFI Grant No.\n18/SIRG/5515 and 18/NSFC/MANIAC). 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Sanvito, Physical Review B\n70, 094406 (2004), 0403351."    },    {        "title": "0910.2377v1.Monte_Carlo_Study_of_Mixed_Spin_S__1_2_1__Ising_Ferrimagnets.pdf",        "content": "arXiv:0910.2377v1  [cond-mat.mtrl-sci]  13 Oct 2009Monte Carlo Study of Mixed-Spin S=(1/2,1) Ising\nFerrimagnets\nW Selke1,2and J Oitmaa2\n1Institut f¨ ur Theoretische Physik, RWTH Aachen, 52056 Aachen, Germany\n2School of Physics, The University of New South Wales, Sydney, NSW 2052,\nAustralia\nAbstract. WeinvestigateIsingferrimagnetsonsquareandsimple–cubiclattice swith\nexchange couplings between spins of values S=1/2 and S=1 on neighb ouring sites and\nan additional single–site anisotropy term on the S=1 sites. Based ma inly on a careful\nand comprehensive Monte Carlo study, we conclude that there is no tricritical point in\nthe two–dimensional case, in contradiction to mean-field prediction s and recent series\nresults. However, evidence for a tricritical point is found in the thr ee–dimensionalcase.\nIn addition, a line of compensation points is found for the simple–cubic , but not for\nthe square lattice.\nPACS numbers: 75.10.-b, 75.10.Hk, 75.40.Mg, 75.50.Gg\nSubmitted to: Institute of Physics Publishing\nJ. Phys.: Condens. Matter\n1. Introduction\nMixed-spin Ising models have been studied for some time as simple mode ls of\nferrimagnets, and there has been renewed interest recently in co nnection with\n’compensation points’. These are temperatures, below the critical temperature, at\nwhich the sublattice magnetizations cancel exactly, giving zero tot al moment. As the\ntemperature is tuned through such a point the total magnetizatio n changes sign, which\nmay be used in technological applications. In this context, Ising mod els may be exactly\nsolvable in special cases [1, 2, 3, 4] or they may be studied by a variet y of powerful\napproaches, including Monte Carlo [5, 6, 7, 8, 9, 10] or other [11, 1 2, 13, 14, 15] methods.\nIn the present work we revisit oneof the simplest such models, a mixe d–spin Ising model\nwith spins S=1/2 and 1 occupying the sites of a bipartite square or sim ple cubic lattice\nwith the Hamiltonian\nH=−J/summationdisplay\n/angbracketlefti,j/angbracketrightσiSj+D/summationdisplay\nj∈BSj2(1)\nwith couplings Jbetween spins σi=±1 on the sites of sublattice ’A’, and\nneighbouring spins Sj= 1,0,−1 on sites forming the sublattice ’B’. D denotes theMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 2\nstrength of a single–ion term acting only on the S=1 spins of sublattic e B. Following\nprevious convention, we choose σi=±1 rather than ±1/2, which has to be taken into\naccountwhencalculatingsublatticemagnetizationsandwhendefinin gthecompensation\npoint. Theconventionsimplyamountstoarescalingoftheexchange coupling. Notethat\nthe nearest neighbour coupling Jmay be either antiferromagnetic, J <0, as assumed\noften for ferrimagnets, or ferromagnetic, J >0. Both cases are completely equivalent\nby a simple spin reversal on either sublattice. We shall use in this artic le ferromagnetic\ncouplings. As a consequence, in our case the magnetizations of bot h sublattices are\nidentical at the compensation point, while in the antiferromagnetic c ase, at the same\ncompensation point, the sublattice magnetizations have equal mag nitude but different\nsign leading to the above mentioned vanishing of the total magnetiza tion.\nThe model on the square lattice has been studied by several autho rs. Kaneyoshi\nand Chen [13], via a mean-field treatment, found a line of compensatio n points in a\nnarrow region 4 > D/J≥2 ln6 (= 3.583..) and a tricritical point at Dt/J= 3.72,\ni.e. a first-order transition for D > D t. Buendia and Novotny [8], using transfer matrix\nmethods, supplemented by Monte Carlo simulations, found no eviden ce of either a\ncompensation point or a tricritical point, although a compensation p oint was observed\nin an extended model with additional ferromagnetic interactions be tweenσspins. More\nrecently, Oitmaa and Enting [16] studied the same model using a comb ination of high-\nand low-temperature series. No compensation point was found, bu t evidence for a\nfirst-order transition, and hence a tricritical point was observed from an apparent\ncrossing of the high- and low-temperature branches of the free e nergy with different\nslopes, for D/J≥3.2. Thus the phase diagram of this simple model remained\nuncertain, motivating partly the present extensive Monte Carlo st udy, improving\nprevious simulations substantially. In fact, our study provides clea r evidence that the\nmodel in two dimensions has no compensation point or tricritical point . Moreover, the\nmodel is found to exhibit very interesting thermal behaviour, both for the specific heat\nand the magnetization, especially in the low–temperature region nea rD= 4, which\nhas not been discussed in detail before. This behaviour is the likely ex planation for the\napparent ’first-order’ behaviour observed in Ref. 16.\nFor the simple cubic lattice, to our knowledge, no detailed analyses ha ve been done\nso far. Of course, mean–field theory may be easily applied, leading ag ain to a tricritical\npoint and a line of compensation points.\nThe outline of the article is as follows. In Section 2 we present and disc uss our\nresults for the square lattice. In Section 3 we consider the simple–c ubic lattice. Here, in\ncontrasttothetwo–dimensional case, wefindaclearoccurrence ofalineofcompensation\npoints. Furthermore, we obtain clear evidence of transitions of fir st-order, and thence of\na tricritical point, which we locate approximately. In the final sectio n, a brief summary\nwill be given.Monte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 3\n2. The model on the square lattice\nLet us first consider the ferrimagnet, eq. (1), in the case of a squ are lattice. We have\nperformed mainly standard Monte Carlo simulations, using the Metro polis algorithm\nwith single–spin flips, providing, indeed, the required accuracy, so t hat there was no\nneed to apply other techniques like cluster–updates or the Wang–L andau approach [17].\nWe studied lattices with LxLsites, employing full periodic boundary conditions. L\nranged from 4 to 80, to study finite–size effects. Typically, runs of 107Monte Carlo\nsteps per spin have been done, with averages and error bars obta ined from evaluating a\nnumber of such runs, at least three, using different random numbe rs. These rather long\nruns lead to very good statistics, improving appreciably results of p revious simulations\n[7, 8]. The estimated errors, unless shown otherwise, are smaller th an the symbols\ndepicted in the figures.\n33.13.2 3.3 3.4 3.53.63.7 3.8 3.9 4\nD/J00.20.40.60.811.21.41.6kBT/J\nFigure 1. Phase diagram of the mixed–spin model on a square lattice.\nWe recorded the energy per site, E, the specific heat, C, both from the energy\nfluctuationsandfromdifferentiating Ewithrespecttothetemperature, andtheabsolute\nvalues of the sublattice magnetizations of the two sublattices\n|mA|=<|/summationdisplay\nAσi|> /(2(L2/2)) (2)\nand\n|mB|=<|/summationdisplay\nBSj|> /(L2/2) (3)\nas well as the absolute value of the total magnetization,\n|m|=<|/summationdisplay\nAσi+/summationdisplay\nBSj|> /L2(4)\nwhere the brackets <>denote the thermal average. Note the factor of 1/2 in\nthe definition of |mA|, taking into account the correct length of the S=1/2 spins, soMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 4\nthat|mA(T= 0)|= 1/2, while |mB(T= 0)|= 1 for the ferromagnetic ground state.\nIn addition, the corresponding susceptibilities, χA,χB, andχ, have been computed\nfrom the fluctuations of the magnetizations. We also analysed histo grams for the\ntotal magnetization, p(m), i.e. the probability to encounter a configuration with the\nmagnetization m, as well as the fourth–order cumulant of the order parameter, t he\nBinder cumulant [18], defined by\nU= 1−< m4> /(3< m2>2) (5)\nwith< m2>and< m4>being the second and fourth moment of the total\nmagnetization. Finally, we monitored typical equilibrium Monte Carlo co nfigurations,\nillustrating the microscopic behaviour of the system.\nTo test the accuracy of the simulations, we computed numerically ex act results\nfor various quantities by enumerating all possible configurations fo r small lattices with\nL= 4.\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8 2 2.2\nkBT/J00.20.40.60.811.2 C\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8 2 2.2 2.4 2.6\nkBT/J00.050.10.150.20.250.3C\nFigure 2. (a) Left: Specific heat at D/J= 3.0, showing numerically exact, for\nL= 4(solid line), and Monte Carlo data for sizes L= 10 (circles), 20 (squares),\n40 (diamonds) and 60 (triangles). (b) Right: Specific heat at D/J=3.8, showing\nnumerically exact, for L= 4 (solid line), and Monte Carlo data for sizes L= 20\n(circles), 40 (squares) and 80 (diamonds).\nIn agreement with previous work, the model is observed to display a ferromagnetic\nground state and low–temperature phase for D/J <4. The energy to flip a B spin\nfrom its ferromagnetic orientation, ’+’ or ’ −’, surrounded by four A spins of the same\norientation, to the state 0 is obviously ∆ E= 4J−Dwhich vanishes at D= 4J.\nHence the ground state at D/J= 4 will comprise configurations with ’0’ states on B\nsites and arbitrarily oriented spins on the neighbouring A sites, as we ll as ferromagnetic\nplaquettes (of either sign) on B sites and neighbouring A sites. Due t o the resulting\nhigh degeneracy, one may call ( D/J= 4,T= 0) the ’degeneracy point’. For D >4J\nat zero temperature, all B spins will be in the state 0, with the A spins being randomlyMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 5\noriented. This leads to a lower, but still macroscopic degeneracy. A tD/J≥4, there is\nno ordered phase even at zero temperature.\nMost of our Monte Carlo work deals with the interesting range 3 ≤D/J <4,\nwhich had been discussed controversially before, augmented by so me simulations at\nlower values of D/J. The resulting phase diagram is depicted in Fig. 1, based on\nmonitoring the size–dependence of the position of the (critical) max ima in the specific\nheat and susceptibility, and the intersection points of the Binder cu mulant, see below.\nOur findings are in accordance with a continuous transition in the Isin g universality\nclass for all values of D/Jwe studied, D/J≤3.98. There is no compensation point.\nIn the following, we shall discuss main properties of the physical qua ntities\nmentioned above.\nThe specific heat C, for negative or relatively small positive D/J, is observed to\nresemble qualitatively that of the nearest–neighbour Ising model o n a square lattice.\nThere is a unique maximum in C(T), for finite L, turning into a logarithmic singularity\nin the thermodynamic limit. Indeed, in the limit D/J→ −∞, one recovers the simple\nIsing model. Increasing D/J, as displayed in Fig. 2a for D/J= 3.0, an additional\nshoulder or maximum evolves at a lower temperature, Tl, being largely independent of\nlattice size and being non–critical. Its origin becomes clear by furthe r increasing D/J,\nas shown in Fig. 2bfor D/J= 3.8. In fact, one finds kBTl/J≈0.42(4−D/J), reflecting\nthe thermally activated flipping of B spins from the ferromagnetic st ate ’1’ (or –’1’) to\nthe state zero, requiring, as stated above, an energy proportio nal to 4 −D/J. It is\ninteresting to note that the height of the pronounced non–critica l peak, signalling the\npartial disordering of the B sublattice, depends only very weakly on D/J. In the range\n3.5≤D/J <4, one has C(Tl)≈0.22.\nAs illustrated in Fig. 3b for D/J= 3.8, the critical peak, located at Tm, may\nseparate from the upper maximum, at Tu, when increasing the strength of the single–ion\nterm. Thus, the specific heat may display a three–peak structure , with two non–critical\nmaxima and a critical peak in between. The origin of the maximum at Tuis due to\nthe fact that at the critical point, the σspins on the A sublattice form rather large\nclusters of different orientations, leading to the vanishing of the or der parameter. That\nbehaviour may be seen by monitoring typical equilibrium configuration s. These clusters\nshrink quickly near Tu, due to thermally activated flipping of σspins, determined by the\ncoupling constant J. Indeed, Tuis essentially independent of D. As seen in Fig. 3b, the\nmaximum in CatTudepends rather weakly on the size of the lattice, L, demonstrating\nits non–critical character.\nThe height of the critical maximum at Tmis expected, for Ising universality,\nto increase logarithmically with Lfor sufficiently large values of L. Our results are\nconsistent with this expectation. However, on approach to the de generacy point, the\nbackground contribution to the specific heat becomes more and mo re relevant. Then\nlarger and larger lattices, with L > L 0, are needed to see the anticipated logarithmic\nbehaviour. For example, at D/J= 3.6, one gets L0≈40, andL0≈60 atD/J= 3.95.\nIn fact, in that range, the Ising–like character of the transition m ay be inferred moreMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 6\nclearly from other quantities, as discussed below.\n0 0.1 0.2 0.3 0.4 0.5 0.6\nkBT/J00.10.20.30.40.50.60.70.80.91|mA,B|\nFigure 3. Sublattice magnetizations |mA|and|mB|atD/J= 3.8 (circles) and 3.95\n(diamonds), with lattices of size L= 40 (dashed lines) and 60 (solid lines).\nThe partial disordering of the B sublattice, near Tl, leads to a rapid decrease\nof the magnetization |mB|, as illustrated in Fig. 3. Actually, the anomaly in |mB|\nbecomes more and more dramatic on approach to the degeneracy p oint. In contrast,\nthe magnetization of the A sublattice, |mA|, is hardly affected by the disordering of the\nB sublattice. Indeed, this behaviour may open the possibilty of a com pensation point,\nat which the two sublattice magnetizations, |mA|and|mB|, would coincide. However, as\ndepicted in Fig. 3, we find no evidence for such a compensation point in two dimensions\nfor all cases we studied, with D/Jgoing up to 3.95.\nThe susceptibility χis found to show, in all cases we studied, only one maximum,\nclose to the critical temperature. The background term is much we aker than for the\nspecific heat, allowing an analysis of critical properties for smaller lat tices. In fact, as\nillustrated in Fig. 4, the size dependence of the height of the maximum inχ,χmax(L),\nis observed to be nicely compatible with the asymptotic form χmax∝L7/4, expected for\nthe Ising universality class, for all cases studied and sufficiently larg e lattices. Note that\nthe susceptibility shows a rather mild anomaly near Tl, where the specific heat shows a\npronounced maximum, close to the degeneracy point. At that anom aly,χ(T) exhibits\na maximal slope, as may be easily identified using exact enumeration fo r small lattices.\nThe shrinking of the A clusters, as indicated by the broad maximum in CatTu, leads\nto no obviously unusual features in the susceptibility.\nAs usual, one may estimate the bulk transition temperature, Tc, from the size\ndependent position of the corresponding peaks in χandC. We obtain consistent\nestimates, shown in Fig. 1, with the location of the maxima varying, fo r largeL,\nproportionally to 1 /L, as expected for Ising–like transitions. Of course, one gets distin ct\nproportionality factors for the two quantities.Monte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 7\n2 2.5 3 3.5 4 4.5 5\nln (L)02468ln (χmax(L))\nFigure 4. Log–log plot of the susceptibility χmaxversus system size Lfor the square\nlattice at D/J= 3.6 (circles), 3.8 (squares) and 3.95 (diamonds). For comparison, the\ndashed line shows χmax∝L7/4.\nThe transition temperature may be also conveniently estimated fro m the Binder\ncumulant, U. Indeed, the estimates follow from the location of the intersection\ntemperatures of the cumulants for different lattice sizes [18]. Finite size corrections\noften turn out to be rather small. Actually, this is also true for the p resent model, as\nshown in Fig. 5 for D/J= 3.95. We find very good agreement with the estimates\nofTcbased on the susceptibilty and the specific heat. Note that the valu e ofUat the\nintersection temperature is, already for fairly small systems sizes , close to the accurately\nknown [19] critical Binder cumulant U∗=U(Tc,L=∞) for isotropic Ising models,\nU∗= 0.6069.... One may emphasize that anisotropic interactions and correlations may\nlead to non–trivial dependences of U∗on such interactions [20, 21]. However, here we\nare dealing with an isotropic system, and excellent agreement with th e known critical\nvalue is observed, demonstrating that the the transition belongs t o the Ising universality\nclass.\nAdditional insight into the phase transition is provided by the histogr ams for the\ntotal magnetization, p(m). An example is displayed in Fig. 6. As expected for a\ncontinuous transition, p(m) shows, in the ferromagnetic low–temperature phase, two\nsymmetric peaks, at ±m0, moving closer and closer to each other on approach to Tcand\nwhen increasing the lattice size. Above Tc,p(m) tends to acquire a Gaussian shape [18].\nWe emphasize that Fig.6 refers to the case D/J= 3.98, i.e. very close to the degeneracy\npoint. There is no indication of a transition of first order, which might be signalled by a\ncentral peak, in addition to the two peaks at ±m0, as would be the case for coexistence\nof the disordered and ordered phases. Accordingly, we may safely conclude, based on\nthe analysis of several quantities, that we have clear evidence for continuous transitions\nof Ising type along the boundary of the ferromagnetic phase, at le ast for the regionMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 8\n0.11 0.115 0.12 0.125 0.13 0.135 0.14\nkBT/J0.50.520.540.560.580.60.620.640.66 U\nFigure 5. Binder cumulant U(L,T) at D/J=3.95 for L=20 (circles), 40 (squares),\n60 (diamonds), and 80 (triangles). The horizontal line indicates the critical Binder\ncumulant of an isotropic Ising model in the thermodynamic limit [19].\nD/J≤3.98.\n-400 -300 -200 -100 0100200 300 400m00.0010.0020.0030.0040.0050.0060.0070.0080.009p(m)\nFigure 6. Histogram of the total magnetization, p(m), for the square lattice\nwithL= 20 at D/J= 3.98 and temperatures below and above the transition,\nkBT/J= 0.04,0.05,0.06, and 0.07 (peak positions moving towards the center), with\nkBTc/J≈0.051.\n3. The model on the simple–cubic lattice\nLet us now turn to the analysis of the mixed–spin model, eq. (1), on a simple cubic\nlattice. In complete analogy to the two–dimensional case, we did sta ndard Monte CarloMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 9\nsimulations, applying the Metropolis algorithm. We studied lattices with L3sites, with\nLranging from 4 to 32. Full periodic boundary conditions were employe d. Typically,\nruns of 2 ×106to 5×106Monte Carlo steps per spin were performed, averaging over a\nfew, at least three, such runs to estimate thermal averages and error bars.\n3.63.8 4 4.2 4.4 4.64.85 5.2 5.4 5.6 5.8 6\nD/J00.511.522.533.54kBT/J\nFigure 7. Phasediagramofthe mixed–spinmodelonasimple–cubiclattice. The s olid\nline denotes the boundary of the ferromagnetic phase, while the da shed line denotes\ncompensation points.\nAs for the square lattice, the energy E, the specific heat C, magnetizations\n|mA|,|mB|, and|m|, as well as corresponding susceptibilities, the Binder cumulant U,\nand histograms for the total magnetization, p(m), were recorded. Typical Monte Carlo\nequilibrium configurations were generated to illustrate the microsco pic behaviour.\nFor the cubic lattice, one has a ferromagnetic ground state at D/J <6. The\ndegeneracy point occurs now at D/J= 6, with ground states comprising local\nferromagnetic plaquettes of neighbouring A and B spins as well as B s pins in the state\n0 with surrounding A spins being randomly oriented. For D/J >6, a high, but reduced\ndegeneracy prevails, with all B spins being zero, and the A spins point ing randomly ’up’\nor ’down’.\nForD/Jsmall or negative, a continuous transition of Ising type is expected to\noccur, as we confirm in simulations with moderate efforts. Most of ou r work has been\ndone for 3 .5≤D/J <6, to identify possible deviations from that kind of transition.\nIndeed, significant deviations from Ising universality have been obs erved for D/J≥5.9,\nwhile for smaller values of D/Jthe simulational data are consistent with an Ising–like\ntransition. In addition, we identified and located a line of compensatio n points in the\nrange 5.5< D/J < 6. The main features of the phase diagram are summarized in Fig.\n7. The phase transition line is based on analyzing various quantities an d taking into\naccount finite–size effects, as for the square lattice. Details of ou r Monte Carlo findings\nwill be discussed in the following.Monte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 10\nThe specific heat C(T) shows for small and negative values of D/Ja single\nmaximum, giving rise to critical behaviour in the thermodynamic limit. In case of an\nIsing–like transition, itsheight isexpected [22] togrowlike Cmax∝Lα/νwith thecritical\nexponents of the Ising universality class, α≈0.11 andν≈0.63 [23]. Our simulational\nfindings confirm this scenario. As in the case of the square lattice, u pon increasing D/J,\none encounters, eventually, three maxima in C(T), see Fig. 8. In complete analogy to\nthe two–dimensional case, the peak at the lower temperature, Tl, is rather sharp and\ndepends only very weakly on lattice size. It signals the partial disord ering of the B\nsublattice, with B spins being flipped thermally from the ferromagnet ic (’+’ or ’ −’)\nstate to 0. The maximum occurs at kBTl/J≈0.6(6−D/J). The upper, rather broad\nmaximum, at Tu, is non–critical as well, stemming from dissolving the, at criticality\nstill quite large spin clusters on the A sublattice. Tuis only very weakly affected by\nthe strength of D, being determined by the ferromagnetic coupling J. In between the\ntwo non–critical maxima in C(T), a critical peak shows up. It signals the transition, at\nwhich both sublattice magnetizations vanish, with quite pronounced local spin order on\nthe A sublattice.\nThe type of the transition may be inferred from the size dependenc e of the critical\npeak,Cmax(L). Indeed, for single–ion terms up to D/J= 5.8, we find agreement with\nan Ising–type transition, α/ν≈0.17. On further approach to the degeneracy point,\naccurate Monte Carlo data with a fine temperature resolution are r equired, due to the\nrather large nonanalytic background term in Cand the sharpness of the peak. In\nfact, other quantities may provide more easily and clearly reliable clue s on the type of\ntransition for that part of the transition line of the ferromagnetic phase.\n0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2\nkBT/J00.050.10.150.20.250.3C\nFigure 8. Specific heat versus temperature for the model on the simple–cub ic lattice\natD/J= 5.9for systems with L= 4(circles), 10 (squares), 16(diamonds), 20(triangles\nup) and 32 (triangles left).\nBefore discussing further the type of the phase transition close t o the degeneracyMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 11\npoint, we shall deal with the compensation points. Indeed, we ident ified such points\nin the range 5 .5< D/J < 6. The resulting line is depicted in Fig. 7. Two concrete\nexamples are shown in Fig. 9, for D/J= 5.7 and 5.9. As may be inferred from that\nfigure, the sublattice magnetization at the compensation point dec reases monotonically\nwith decreasing single–ion term. Therefore, when the compensatio n occurs at low\nmagnetizations, the accurate location of the compensation point is difficult, because\nof strong finite–size effects in the critical region. On the other han d, with increasing\nD/J, the compensation point moves towards lower temperatures, and finite size effects\nplayusuallynosignificant role. Inanyevent, incontrasttothetwo– dimensional case, we\nfindalineofcompensationpointsforthesimple–cubic lattice. Obvious ly, thedecreasein\nthe magnetization of the B sublattice, |mB|, occurs in three dimensions more drastically\nthan for the square lattice, while |mA|changes there rather mildly in both cases.\n0 0.2 0.4 0.60.81 1.2 1.4 1.6 1.8\nkBT/J00.10.20.30.40.50.60.70.80.91|mA,B|\nFigure 9. Sublattice magnetizations |mA|and|mB|for the simple–cubic lattice with\nL= 20 atD/J= 5.7 (circles) and 5.9 (squares).\nLet us now turn back to the discussion on the type of phase transit ion. For\nD/J≤5.8, the data on the susceptibilty χconfirm the Ising–like character of the\ntransition. In particular, the size dependence of the height of the maximum in χ,\nχmax(L), isfoundtobeconsistent withIsingcriticality, χmax∝Lγ/ν, whereγ≈1.24and\nν≈0.63, thusγ/ν≈1.97. Indeed, from our simulational data we obtain characteristic\nexponents close to 2. However, at D/J= 5.9, we observe, for systems sizes ranging from\nL= 8 toL= 32, a substantially lower (effective) exponent, of about 1.7. Beca use\nthe peak in χgets extremely sharp, very accurate simulational data with a very fine\ntemperature mesh are needed to arrive at safe conclusions. A mor e convenient way to\nmonitor the possible change in the type of the transition will be discus sed below.\nInterestingly, our analysis of the Binder cumulant Useems to indicate substantial\ndeviations from an Ising–like transition at about D/J≈5.9 as well. For smaller values\nofD/Jthe intersection values of the cumulant curves for different syste m sizes, alreadyMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 12\n1.9 1.92 1.94 1.96 1.98 2 2.02 2.04\nkBT/J0.320.360.40.440.480.520.560.6U\n1.2 1.24 1.28 1.32 1.36 1.4\nkBT/J0.320.360.40.440.480.520.560.6U\nFigure 10. (a) Left: Binder cumulant Uversus temperature at D/J= 5.5 for lattices\nwithL= 8 (circles), 12 (squares), 16 (diamonds) and 20 (triangles). (b) Right:U\nversus temperature at D/J= 5.9 for lattices with L= 10 (circles), 16 (squares),\n20 (diamonds), and 32 (triangles). The horizontal lines indicate the critical Binder\ncumulant of an isotropic three–dimensional Ising model in the therm odynamic limit\n[24].\nfor fairly small systems, seem to agree with the expected asympto tic value of the critical\nBinder cumulant for isotropic Ising systems [24], U∗≈0.465. An example is depicted in\nFig. 10a, for D/J= 5.5, with the intersection points, for the simulated finite lattices,\napproachingtheasymptoticvaluefrombelow, whenincreasingthes ystemsize. Atlarger\nsingle–ion anisotropy, D/J≥5.9, the intersection points of the curves are appreciably\nlower than U∗, as shown in Fig. 10b for D/J= 5.9. However, it is not completely clear,\nwhether the tendency reflects stronger finite–size effects or a c hange in the type of the\nphase transition.\nTo get more evidence for a possible change of the nature of the tra nsition, the\nhistograms for the magnetization, p(m), turned out to be most instructive. Already\nfor small lattices, L= 4, one sees, close to the transition, a qualitative change of the\nhistograms. We did simulations close to the transitions in the range 5 .85≥D/J≥5.98,\nusing an increment of 0.01. We observe a dramatic change in the form of the histograms\naroundD/J≈5.91. Below that value, there is no central peak and thus no indication\nof phase coexistence when crossing the transition, in contrast to the situation closer to\nthe degeneracy point, where a central peak, in addition to the sym metric peaks at ±m0,\nindicates coexistence of the ordered and disordered phases and, accordingly, a transition\nof first order. That distinction persists for larger system sizes. E xamples are displayed\nin Figs. 11 a, for D/J= 5.85, and 11 b, for D/J= 5.975. Based on these observations,\nwe may tentatively locate the tricritical point at D/J= 5.91±0.03. Note that such\na change in the form of the histograms does not occur in two dimensio ns, as has been\ndiscussed above, see also Fig. 6.\nInsummary, thepresent analysis onthemixed–spin model onasimple –cubic latticeMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 13\n-500-400 -300 -200 -1000100200 300 400 500m00.0010.0020.003p(m)\n-500-400 -300 -200 -1000100200 300 400 500m00.0010.0020.0030.0040.005p(m)\nFigure 11. (a) Left: Histogram of the total magnetization, p(m), forL=8 and\nD/J= 5.85, at temperatures crossing the transition, kBT/J= 1.32, 1.36, 1.40, 1.44,\nand 1.48, where the maxima move towards the center with increasing temperature.\n(b) Right: p(m) forL= 8 and D/J= 5.975, at temperatures crossing the transition,\nkBT/J= 0.47, 0.51, 0.55, 0.59, and 0.63, where the central peak grows in he ight with\nincreasing temperature.\nshows clearly a line of compensation points, and allows to locate appro ximately the\ntricritical point.\n4. Summary\nWe have studied a mixed–spin Ising model with ferromagnetic coupling s,J, between\nspins 1/2 and 1 on neighbouring sites of square and simple–cubic lattic es, the two types\nof spins forming a bipartite lattice. An additional quadratic single–ion term,D, acts\nupon the S=1 spins. We mainly used standard Monte Carlo simulations t o compute\nvarious thermodynamic properties as well as the Binder cumulants a nd histograms of\nthe total magnetization.\nThe model on the square lattice has been shown to display a continuo us phase\ntransition of Ising–type, presumably up to the degeneracy point a tD/J=4. No\ncompensation point has been found. Close to the degeneracy point , the model displays\nan intriguing three–peak structure in the specific heat as a functio n of temperature.\nThe sharp, but non–critical anomaly at low temperatures arises fr om flipping S=1 spins\ninto the state 0, while the broad non–critical maximum at high temper atures stems\nfrom thermal activation of spins in fairly large clusters of S=1/2 spin s persisting above\nthe phase transition. At temperatures in between, the critical pe ak shows up. Both\nanomalies may cause difficulties in low– and high–temperature expansio ns, which have\npredicted, incorrectly, the existence of a tricritical point. The su ggestion on the absence\nof a compensation point has been confirmed, albeit the magnetizatio n on the S=1\nsublattice decreases rapidly near the anomaly of the specific heat a t low temperatures.\nInthecaseofthesimple–cubic lattice, thespecificheatdisplaysasim ilarthree–peakMonte Carlo Study of Mixed-Spin S=(1/2,1) Ising Ferrimagne ts 14\nstructure, with two non–critical maxima and the critical peak in bet ween. Sufficiently\nfar away from the degeneracy point, the ferromagnetic phase dis orders via a continuous,\nIsing–like transition. In the vicinity of the degeneracy point, D/J= 6, this transition\nseems to beof first order. The evidence forthat kind of transition ismainly based onthe\ntype of the histograms of the magnetization, showing phase coexis tence. We tentatively\nlocate the tricritical point at D/J= 5.91±0.03. In addition, we determined a line of\ncompensation points, arising from the degeneracy point. Thus, in t hree dimensions, the\nmean–field theory appears to give at least qualitatively correct pre dictions. However,\nin two dimensions the mean–field theory is found to be incorrect, eve n qualitatively.\n5. Acknowledgement\nW.S. thanks the School of Physics at the University of New South Wa les for the very\nkind hospitality during his stay there, which was supported by the Go rdon Godfrey\nFund.\nReferences\n[1] Goncalves L L 1985 Phys. Scripta 32248.\n[2] Lipowski A and Horiguchi T 1995 J. Phys. A: Math. Gen. 28L261.\n[3] Jascur M 1998 Physica A 252217.\n[4] Dakhama A 1998 Physica A 252225.\n[5] Oitmaa J and Zheng W-H 2003 Physica A 328185.\n[6] Jascur M and Strecka J 2005 Condens. Mat. Phys. 8869.\n[7] Zhang G-M and Yang C-Z 1993 Phys. Rev. B489452.\n[8] Buendia G M and Novotny M 1997 J. Phys.C: Cond Mat. C95951.\n[9] Nakamura Y 2000 J. Phys.C: Cond. Mat. 124067.\n[10] Godoy M and Figueiredo W 2004 PhysicaA339392.\n[11] Godoy M, Leite V S, and Figueiredo W 2004 Phys. Rev. B69054428.\n[12] Oitmaa J 2005 Phys. Rev. B72224404.\n[13] Kaneyoshi T and Chen J C 1991 J. Mag. Mag. Mat. 98201.\n[14] Boechat B, Filgueiras R, Marins L, Cordeiro C, and Branco NS 200 0Mod.Phys. Lett. B 14749.\n[15] Sarmento E F, Cressoni J C, and dos Santos R J V 2000 Int. J. Mod. Phys. B 14521.\n[16] Oitmaa J and Enting I G 2006 J. Phys. C: Cond. Mat 1810931.\n[17] Landau DP and Binder K 2005 A Guide to Monte Carlo Simulations in Statistical Physics\n(Cambridge: University Press)\n[18] Binder K 1981 Z. Physik B43119.\n[19] Kamieniarz G and Bl¨ ote HWJ 1993 J. Phys. A: Math. Gen 26201.\n[20] Chen XS and Dohm V 2004 Phys. Rev. E70056136; Dohm V 2008 Phys. Rev. E77061128.\n[21] Selke W and Shchur LN 2005 J. Phys. A: Math. Gen. 38L739; Selke W 2006 Eur. Phys. J. B\n51223.\n[22] Fisher ME and Barber MN 1972 Phys. Rev. Lett 281516.\n[23] Pelissetto A and Vicari E 2002 Phys. Rep. 368549.\n[24] Hasenbusch M, Pinn K, and Vinti S 1999 Phys. Rev. B5911471."    },    {        "title": "1402.4634v1.Theory_of_Ferrimagnetism_in_the_Hubbard_Model_on_Bipartite_Lattices_with_Spectrum_Symmetry.pdf",        "content": "arXiv:1402.4634v1  [cond-mat.str-el]  19 Feb 2014Theory of Ferrimagnetism in the Hubbard Model on Bipartite L attices with Spectrum\nSymmetry\nYang Xue,1Jing He,2Xing-Hai Zhang,1and Su-Peng Kou1,∗\n1Department of Physics, Beijing Normal University, Beijing , 100875, P. R. China\n2Department of Physics, Hebei Normal University, HeBei, 050 024, P. R. China\nInthis paperwe developedtheoryoftheferrimagnetism inth eHubbardmodel on bipartitelattices\nwith spectrum symmetry. We then study the defect-induced fe rrimagnetic orders in three models\nand explored the universal features.\nThe magnetic orders are important magnetic proper-\nties for two dimensional (2D) Hubbard model, to which\nresearchers also have payed much attention. At half fill-\ning case, the ground state is known to be the long-range\n(LR)antiferromagnetic(AF) order. Nagaokamadeasur-\nprising discovery about the induced ferromagnetic (FM)\nby a single hole for infinite coupling limit U→ ∞[1].\nAt heavily doping region a possible FM order may exist.\nThe starting point on the issue of ferrimagnetism of the\nHubbard model is a theorem by Lieb[2]. In this theo-\nrem, it is pointed out that the total spin Sof the ground\nstate for the Hubbard model on bipartite lattice with\nparticle-hole (PH) symmetry and real hopping parame-\nters is givenby S=|NA−NB|/2 (we call it Lieb spin mo-\nment) whereNAandNBare the numbers of lattice sites\non A and B sublattice, respectively. Then, in Ref.[3], it\nis pointed out that the ground state of the 2D sublattice-\nunbalanced Hubbard model on bipartite lattice with the\nsame conditions (PH symmetry and real hopping param-\neters) is really the ferrimagnetic (FR) order that pos-\nsesses a finite total magnetic moment and the LR AF\norder and obeys m(0)≤m(Q) (we call it Shen-Qiu-Tian\n(SQT) inequality[4]) where m(0) =/summationdisplay\nij/angbracketleftbig\nˆs+\niˆs−\nj/angbracketrightbig\n/Nand\nm(Q) =/summationdisplay\nij(−1)i+j/angbracketleftbig\nˆs+\niˆs−\nj/angbracketrightbig\n/N(ˆsi=1\n2ˆc†\niσˆci, Nis the\ntotal number of the lattice sites). However, the FR order\nwas seldom studied [3, 5, 7, 8] and the detailed proper-\nties of the FR order in 2D Hubbard model have not been\nexplored.\nThe 2D Hubbard Model : Our starting point is the fol-\nlowing Hamiltonian\nH=−/summationdisplay\ni,jtij/parenleftBig\nˆc†\niˆcj+h.c./parenrightBig\n+U/summationdisplay\niˆni↑ˆni↓−µ/summationdisplay\ni,σˆc†\niσˆciσ,\n(1)\nwhere ˆci= (ˆci,↑,ˆci,↓)Twith ˆci,σrepresenting the\nfermion annihilation operator at site i.σ=↑,↓denote\nspin index. For this bipartite lattice, we have two sub-\nlattices, A and B. tijis the hopping amplitude. Uis the\nstrength of the repulsive interaction. µis the chemical\npotential which is set to µ= 0 for the half filling case.\n∗Corresponding author; Electronic address: spkou@bnu.edu .cnThe spectrum symmetry : Firstly, we defined the spec-\ntrum symmetry . For the Hamiltonian in Eq.(1) with the\nspectrum symmetry (S symmetry), the denisty of state\n(DOS)ρ(E) is alwayssymmetricvia Easρ(E) =ρ(−E).\nAsaresult, eachenergylevelwithpositiveenergy Emust\nbe paired with an energy level with negative energy −E.\nTo make it clearer, we define an operator of the S sym-\nmetry as S=P · DwherePis the PH transformation\noperator P=R · Kintroduced in Ref.[9, 10] and Ris\nan operator that leads to ˆ ci↔(−1)iˆci,Kis the com-\nplex conjugate operator, Dis a discrete transformation\noperator that commutes with lattice translation opera-\ntorsTx,yas, [D,Tx] = [D,Ty] = 0. For a Hamiltonian\nwith the S symmetry, we have H=−S†HS. Thus, each\nenergy level |ψ/angbracketrightwith positive energy Eis paired with an\nenergy level S|ψ/angbracketrightwith negative energy −E. So, the PH\nsymmetry is a special case of the S symmetry and for the\ncase ofD= 1,the S symmetry is reduced into the PH\nsymmetry.\nVacancy-induced zero-modes : We then consider the\ncase of free Hamiltonian ( U= 0) with lattice-defect -\nthe vacancy by adding a potential on given lattice site\nR,HR=−/summationtext\ni,jtij/parenleftBig\nˆc†\niˆcj+h.c./parenrightBig\n+VRˆc†\nRˆcR. In the uni-\ntary limit, the lattice-defect becomes a missing lattice\nsite that is just a vacancy, on which we have an infinite\non-site potential, i.e., VR→ ∞. It is pointed out that the\nvacancydoesn’tbreaktheSsymmetry( HR=−S†HRS).\nDue to the S symmetry, there exists a zero energy state\n(the so-called zero-mode) |ψ0/angbracketrightwhen we add a vacancy\non A sublattice. For the zero-mode (ZM) |ψ0/angbracketright,we have\n|ψ0/angbracketright=S|ψ0/angbracketright. We denote the wave-function of the\nZM byψ0(ri−R) whereRis the position of the va-\ncancy. When there exists n-vacancy on A sublattice,\nn=|NA−NB|zero energy modes will necessarily ap-\npear. In addition, for the case with nearest neighbor\nhoppingtij=t/angbracketleftij/angbracketright(t/angbracketleftij/angbracketrightis the real or complex near-\nest neighbor (NN) hopping parameter), these vacancy-\ninduced (VI) zero-modes (ZMs) localize only on the B\nsublattice and are orthotropic each other.\nFor some models with S symmetry, the DOS may be fi-\nnite atthe Fermi level. Exceptforthe VI ZMs, theremay\nexist additional zero energy states |ψ/angbracketright /negationslash=S|ψ/angbracketright. However,\nthe additional zero energy states are not protected by\nsymmetry and are fragile against perturbations. On the\ncontrary, since the VI ZMs are protected by S symme-\ntry (|ψ0/angbracketright=S |ψ0/angbracketright), they are fixed precisely at the Fermi2\nlevel, the interaction term is highly relevant. In particu-\nlar, arbitrary small (repulsive) interaction will drive the\nspin moments of the VI ZMs into an FM ordered state.\nConsequently, the ground state of the original Hamilto-\nnian turns into a LR FR order.\nFerrimagnetism : Let us show the universal features of\nthe FR order in the Hubbard model in the small Ulimit.\nFor a system with Lx×Lyvacancy-lattice ( Lxalong\nxdirection and Lyalongydirection), in a unit cell\n(UC) there are Lx×Ly/a2−1 lattice sites. In gen-\neral, we have Lx×Ly/a2−1 magnetic order parame-\ntersMi=/angbracketleftˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketright/2 to denote the local mag-\nnetizations on the lattice sites in a UC. To simply\ncharacterize the FR order, we introduce two order pa-\nrameters, the total FM moment in a unit cell M=/summationtext\ni∈unit-cell/angbracketleftˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketrightand the total AF moment in\na unit cell N=/summationtext\ni∈unit-cell(−1)i/angbracketleftˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketright, re-\nspectively. One can see that m(0) =/summationdisplay\nij/angbracketleftbig\nˆs+\niˆs−\nj/angbracketrightbig\n/N=\n/summationdisplay\ni/angbracketleftBig\nˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketrightBig\n/N=M/(Lx×Ly/a2−1) and\nm(Q) =N/(Lx×Ly/a2−1). We then define the spin\noperatorofthe VI ZM as ˆSR(i) =1\n2ˆψ†\n0(ri−R)σˆψ0(ri−R)\nwhereˆψ0= (ˆc0,↑,ˆc0,↓)Tψ0and ˆc0,σis the particle anni-\nhilation operator of the zero-mode with spin σ. When/angbracketleftBig\nˆSz\nR/angbracketrightBig\n/negationslash= 0,the FR order is a direct physics consequence\nof the FM order of the spin moments of the different\nZMs. The total FM moment in a UC is given by M=\n2/summationtext\ni∈unit-cellMi= 2/summationdisplay\nR/angbracketleftBig\nˆSz\nR/angbracketrightBig\n,and the total AF moment\ninaUCis N= 2/summationtext\ni∈unit-cell(−1)iMi= 2/summationdisplay\nR/angbracketleftBig\n(−1)iˆSz\nR/angbracketrightBig\n.\nFrom above discussion, we already know that there ex-\nists a ZM for each vacancy. In the small Ulimit,U→0,\nthe low energy physics is dominated by these ZMs. Be-\ncause the ZMs induced by two different vacancies at R\nandR′are orthotropic each other, when considering the\non-site interaction there exists the Hund rule’s coupling\nas−Jeff(R,R′)ˆSR·ˆSR′whereJeff(R,R′)>0 is the effec-\ntive FM spin coupling constant. Thus, the low energy ef-\nfective Hamiltonian becomes Heff=−/summationdisplay\nR,R′Jeff(R,R′)ˆSR·\nˆSR′,of which the ground state is a long range FM order\ndenoted by/angbracketleftBig\nˆSz\nR/angbracketrightBig\n=1\n2/angbracketleftBig\nˆψ†\n0,R↑ˆψ0,R↑−ˆψ†\n0,R↓ˆψ0,R↓/angbracketrightBig\n=1\n2or\nM= 1. As a result, the total spin moment of the ground\nstate must be the total number of spin moments of the\nVI ZMsS=1\n2/summationdisplay\nMthat is just the Lieb spin moment\nS=|NA−NB|/2. The local magnetizations are given by\nMi→/summationdisplay\nR/angbracketleftBig\nˆSz\nR(i)/angbracketrightBig\n.\nFor the case with only NN hopping tij=t/angbracketleftij/angbracketrightin small\nUlimit, it is obvious that Mi∈A= 0.Thus, we have\nN=M= 1.M=Nmeans the ground state is an FM-\nAF-balanced FR order. On the contrary, in the strong/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s52/s54/s56/s49/s48\n/s85/s47/s116/s69/s47/s116\nFIG. 1: (Color online) (a) Part of particle density distribu tion\nof VI ZM on a 60 a×60asquare lattice (a vacancy at the white\nspot); (b) The illustration of the uniform FM-AF-balanced\nferrimagnetic order for the Hubbard model on square lattice\nandU= 0.01t; (c) The two order parameters; (d) The density\nof state for case of Hubbard model on square lattice and U=\nt.\ncoupling limit, U→ ∞,the low energy effective Hamil-\ntonian turns into the (un-frustrated) Heisenberg model\nwith vacancy-lattice as Heff=J/summationdisplay\n/angbracketleftij/angbracketrightˆ si·ˆ sj(J→4t2\nU). The\nground state is characterized by the AF ordered stag-\ngered magnetization, M= 2(−1)iMi→1/2. So, we\nhaveN= 2(Lx×Ly/a2−1)M→Lx×Ly/a2−1 and\nM= 1. The LR FR order is really a defect-diluted AF\norder[5, 17].\nFor case with both NN hopping tij=t/angbracketleftij/angbracketrightand next\nnearest neighbor (NNN) hopping, tij=t/angbracketleft/angbracketleftij/angbracketright/angbracketright/negationslash= 0, the\nsituation becomes complex. The VI ZM still exists due\nto S symmetry. However, the wave-functions of the ZM\nmay distribute on both sublattices. As a result, in small\nUlimit, we have M= 1 and the total FM moment is also\nLieb spin moment |NA−NB|/2. In the large Ulimit, the\nlowenergyeffectiveHamiltonianturnsintothefrustrated\nHeisenberg model, of which the ground state may be not\nan AF order.\nExample 1 - the Hubbard model on square lattice : For\nthe Hubbard model on square lattice, the hopping pa-\nrameters are t/angbracketleftij/angbracketright=t. For this model, the operator of\nthe S symmetry is S=P. The S symmetry protected\nVI ZM is an extended state, of which the wave-function\nΨ0can be naturally be ψ0,i∈A= 0, ψ0,i∈B=1\nNB. See\npart of the particle density distribution of this ZM on a\n60a×60asquare lattice in Fig.1(a).\nTo check the validity of above discussion, we use the\nmean field approach to study the Hubbard model on\nsquare lattice with a vacancy-lattice. The lattice con-\nstant of the vacancy-lattice is set to be d= 6a.We\nchoose 6a×6asites to be a UC (6 aalongxdirection\nand 6aalongydirection). To search the ground state3\nwith the lowest energy, we need to solve 6 ×6−1 = 35\norder parameters that denote the local magnetizations\nMi=/angbracketleftˆc†\ni↑ˆci↑−ˆc†\ni↓ˆci↓/angbracketright/2 on 35 lattice sites in a UC.\nFrom the mean field calculation, we find that for the\nweakcouplinglimit, U→0,thegroundstateisa uniform\nFerrimagnetic order, Mi∈A= 0, Mi∈B=1\n6×6−1=1\n35.\nThe FR ordered state is illustrated in Fig.1(b). From\nFig.1(c) one can see that the total FM moment in a\nUC is indeed a constant, M ≡1, which is consistent\nto the prediction of Lieb spin moment |NA−NB|/2. On\nthe other hand, the total AF moment in a UC is also\nN= 1. The FM-AF-balance character ( M=Nor\nm(0) =m(Q)) comes from the fact that the VI ZMs\nonlydistributeononesublattice. With theincreasingthe\ninteraction strength, the average magnetizations on the\nsitesof Bsublatticebecomefinite valueswithanopposite\npolarization to those on the sites of A sublattice Mi∈A:\nMi∈A\n|Mi∈A|=−Mi∈B\n|Mi∈B|/negationslash= 0. Because the amplitudes of Mion\nall lattice sites increase, we have an AF-dominated FR\norder with M<N(orm(0)< m(Q)). Now, the SQT\ninequalitym(0)≤m(Q) is satisfied. In the large Ulimit,\nwe have a saturated value, N →35 butM ≡1. Fig.1(d)\nshows the DOS of the FR order, of which there exists an\nenergy gap. Near the gap, the DOS is enhanced due to\nthe VI ZMs.\nExample 2 - the staggered-flux Hubbard model on\nsquare lattice : Recently, people had realized the photon-\nassisted tunneling on optical lattice and then generated\na large effective (staggered) magnetic flux on optical\nlattice[11–13]. When two-component fermions with re-\npulsive interaction are put into such optical lattice, one\ncangetaneffectivestaggered-fluxHubbardmodel. These\nprogresses may provide new research platform to learn\ntheferrimagnetism. Itiseasytochangethepotentialbar-\nrier by varying the laser intensities to tune the Hamilto-\nnian parametersincluding the hopping strength ( t-term),\nthe staggered flux ( φ) and the particle interaction ( U-\nterm). For this reason, we take the Hubbard model on\nsquare lattice with staggered-flux (SF) as the second ex-\nample, where ti,i±δy=ei±φtfori∈A,ti,i±δx=tfor\ni∈A andti,i±δy=ti,i±δx=tfori∈B. See the illustra-\ntion in Fig.2(a). In particular, we only consider the NN\nhoppings. For this model, the operator of the S symme-\ntry isST=P ·TwhereTis the time-reversal transfor-\nmation operatorwhich commutes with lattice translation\noperators [ T,Tx] = [T,Ty] = 0.\nAfter diagonalization of the Hamiltonian in\nmomentum space, the energy spectra are ob-\ntained as E±=−t[cos(ky−φ)+cosky]±/radicalBig\nt2[cos(ky−φ)−cosky]2+4t2cos2kx.For the case\nofφ= 0, the SF disappears and we get a uni-\nform Hubbard model on square lattice. In the BZ\nkx∈(−π,π),ky∈(−π,π), the energy S turns into\nE=−2t(coskx+ cosky).Now the Fermi surface at\nhalf filling has perfect nesting condition. Away from\nthis case,φ/negationslash= 0, the BZ is reduced into a half one.\nThe system becomes a semi-metal and also has the\nFIG. 2: (Color online) (a) The illustration of the staggered -\nflux lattice; (b) The dispersion of the π/2-flux case: there are\na hole pocket and an electron pocket in a reduced BZ. The\nred plane denotes the position of chemical potential.\nperfect nesting condition. For the + band, there exists\na hole pocket; For the −band, there exists an electron\npocket. See the illustration in Fig.2(b). The density\nof state (DOS) ρ(E) near Fermi surface is reduced\nwith increasing φ. For the case of π-flux, the energy S\nturns intoE±=±2t/radicalbig\ncos2ky+cos2kxand the electron\npocket and hole pocket shrinks into two Dirac nodes at\nK1= (π/2,π/2) andK2= (π/2,−π/2).\nBecause the SF Hubbard model on bipartite lattices\nat half-filling is unstable against antiferromagnetic (AF)\ninstability, the ground state becomes an insulator with\nAF order for the case of finite U. Such AF order is de-\nscribed by the following mean field ansatz /angbracketleftˆc†\ni,σˆci,σ/angbracketright=\n1\n2/parenleftbig\n1+(−1)iσM/parenrightbig\nwhereMis the staggered magnetiza-\ntion. For the cases of spin up and spin down, we have\nσ= +1 and σ=−1, respectively. Then in the mean\nfield theory, by minimizing the ground state energy in\nthe reduced Brillouin zone, we could solve the staggered\nmagnetization. Due to the perfect nesting condition, the\narbitrary small interaction term leads to an AF spin-\ndensity-wave (SDW) order and then the BZ of φ/negationslash= 0,\nπcase is reduced into a quarter one. For the π-flux\ncase, the DOS near Fermi surface is zero and the crit-\nical point between the semi-metal and AF insulator is\naboutUc= 3.11t[14, 15].\nFor the case of φ/negationslash= 0,the VI ZM is a quasi-localized\nstate. See the particle density distribution of this ZM\nfor the case of φ=π/2 in Fig.3(a). The VI ZMs are\nanisotropicdue to the rotation-symmetrybreaking of the\noriginal Hamiltonian. In the continuum limit, for the\ncase ofφ=π,the wave function of VI ZM distributes\non B sublattice that has a simple form of ψ0(x,y)≃\neiK1.r\nx+iy+eiK2.r\nx−iy[16]. The amplitude of this state decays\nwith the distance to the vacancy as 1 /r. It is needed to\npoint out that these VI ZMs are all protected by the S4\n/s85/s47/s116\n/s69/s47/s116\nFIG. 3: (Color online) (a) Part of particle density distribu tion\nof VI ZM on a 60 a×60asquare lattice for φ=π/2 andU= 0;\n(b) The illustration of the cluster AF-balanced ferrimagne tic\norder for φ=π/2 andU= 0.001t; (c) The two order param-\neters for φ=π/2; (d) The density of state for φ=π/2 and\nU=t.\nsymmetry.\nWe use the mean field approach to study the Hubbard\nmodel onSF squarelattice with a 6 a×6avacancy-lattice.\nWe focus on the case of φ=π/2. Now the ground state\nis aclusterFerrimagnetic order for weak coupling case.\nThe word ”cluster” means that the local magnetic order\nparameters Miis larger near the vacancy but smaller\nfar from it. See the illustration Fig.3(b). This ”cluster”\nbehaviorobviouslyis aphysicalconsequenceofthe quasi-\nlocalized VI ZMs. From Fig.3(c) one can also see that\nN →1,M ≡1 (U→0) andN →35,M ≡1 (U→\n∞). From the DOS of the system (Fig.3(d)), one can see\nthat there exists energy gap of the mid-gap states (zero-\nmodes) that dominate the low energy physics. All these\nfeatures indicate an AF-dominated FR order from the\nFM order of spin moments of the quasi-localized ZMs.\nFrom the mean field calculation, for the case of φ=π,\nwe find that the quantum phase transition between the\nmetallic(orsemi-metallic)statesandthemagneticorders\nshiftsfromU=UctoU= 0. Arbitraryinteractiondrives\nthe systeminto a longrangeFR order, which is obviously\ninduced by the vacancy-lattice. At U∼Uc, there is no\ntrue phase transition, instead, a crossover occurs. For\nU < U c,the ground state can be regarded as an FR\norder from the FM ordered spin moments of ZMs. Now,\natinyenergygapopenswhichisduetothe spinpolarized\neffect of the ZMs. On the other hand, for U > U c,the\nground state can be regarded as the defect-diluted AF\norder, and a big energy gap opens which is just AF-Mott\ngap for the double occupied particles on one site.\nExample 3 - the spinful Haldane model on square lat-\ntice: An interesting issue is the FR order for the case\nfrom a topological insulator with NNN hoppings. Now\nwe consider the spinful Haldane model on square lat-\n/s85/s47/s116/s69/s47/s116/s85/s47/s116\nFIG. 4: (Color online) (a) Part of particle density distribu tion\nof VI ZM for the interacting spinful Haldane model on 60 a×\n60asquare lattice at φ=π, t′= 0.1,andU= 0; (b) The\nillustration of the cluster FM-balanced ferrimagnetic ord er\nfor the interacting spinful Haldane model on square lattice at\nφ=π, t′= 0.1 andU= 0.001t; (c) The two order parameters\nof the case of φ=π, t′= 0.1; (d) The density of state of the\ninteracting spinful Haldane model on square lattice at φ=π,\nt′= 0.1 andU=t.\ntice, of which the Hamiltonian is given by HT=H(φ=\nπ) +H(t′) whereH(φ=π) is the Hamiltonian of\nthe SF Hubbard model on square lattice and t′is the\nNNN hopping. The Hamiltonian has PH symmetry from\nHT=−P†HTP. In addition, the free Hamiltonian is a\ntopological Chern insulator.\nWe numerically calculated the free Hamiltonian with\na vacancy, and found a (S or PH symmetry protected)\nVI ZM on a 60 a×60alattice[10]. Fig.4(a) shows the\nparticle density of the ZMs, localized around the defect\ncenter within a length-scale of ∼(∆E)−1, where ∆Eis\nthe energy gap of the vacancy-free case. In particular,\nthe wave-function of the ZM distribute not only on B\nsublattice but also on A sublattice.\nWe then use the mean field approach to study the FR\norder. From the results given in Fig.4(b), we find that\nthegroundstateisalsoaclusterFR.Inthe small Ulimit,\nthe average magnetizations on the sites of A sublattice\nbecomefiniteasMi∈A\n|Mi∈A|=Mi∈B\n|Mi∈B|/negationslash= 0. Now, the magnetic\norder in A sublattice has the same polarized direction to\nthat of B sublattice but the value of it is much smaller\nto that of B sublattice. The total spin Sof the ground\nstate still obeys the prediction of Lieb spin moment as\n|NA−NB|/2. However, we have M>Nthat mean\nm(0)> m(Q). The ground state is an FM-dominated\nFR orderand the SQT inequality is violent. We conclude\nthat the violence of SQT inequality for this case is due\nto the NNN hoppings.\nWhen we increase the interaction strength, m(0) =\nm(Q) atU0. When we further increase the interaction\nstrength, the ground state turns into an AF-dominated5\nFR order with m(0)< m(Q). For larger interaction\nstrength, a topological quantum phase transition occurs.\nThe energy gap closes and opens again. The system then\nhas no nontrivial topological properties and becomes a\ndefect-diluted AF order.\nConclusion : In this paper, we developed a universal\nformula of the ferrimagnetism in the Hubbard model be-\nyond Lieb’s theorem by taking into account for the S\nsymmetry with larger universality than traditional PH\nsymmetry. Then, by taking three models as exam-\nples, we study the defect-induced FR orders that emerge\nfrom three typical fermionic systems - metal, semi-metal,\n(Chern) insulator. We found that there may exist vari-\nous FR orders (uniform FM-AF-balanced FR order, uni-\nform AF-dominated FR order, cluster FM-AF-balanced\nFR order, cluster AF-dominated FR order, cluster FM-\ndominated FR order...). The total spin of all these FR\norders is equal to the Lieb spin moment ( |NA−NB|/2).From the common feature of these FR orders, we con-\njecture that it is the S symmetry protected VI ZMs\n(|ψ0/angbracketright=S|ψ0/angbracketright) that dominate the low energy physics of\nthe system in small Ulimit. However, we found that the\nSQT inequality ( m(0)≤m(Q)) is valid for the model\nwith only the NN hoppings and can be violent by the\nNNN hoppings. In addition, we may point out that the\nformula can be straightforwardly applied to other Hub-\nbard models on bipartite lattices with S symmetry.\n* * *\nThis work is supported by National Basic Research\nProgram of China (973 Program) under the grant No.\n2011CB921803, 2012CB921704 and NSFC Grant No.\n11174035.\n[1] Y. Nagaoka, Phys. Rev. 147, 392-405 (1966).\n[2] E. H. Lieb, Phys. Rev. Lett. 62, 1201-1204 (1989), [Er-\nrata62, 1927 (1989)].\n[3] S. Q. Shen, Z. M. Qiu and G. S. Tian, Phys. Rev. Lett.\n72, 1280 (1994).\n[4] After a (partial) particle-hole transformation, the re pul-\nsive Hubbard model can be transformed into an attrac-\ntive Hubbard model, of which the ground state always\nhas singlet superconducting (SC) pairing. This singlet\nSC just corresponds to the ground state with a total spin\n|NA−NB|/2[2, 18, 19]. Then, the SQT inequality corre-\nsponds to non-negative off-diagonal long-range SC pair-\ning of the attractive Hubbard model in earlier paper[18].\n[5] G. S. Tian, J. Phys. A: Math. Gen. 272305 (1994).\n[6] G. S. Tian, Phys. Rev. B 50,6246 (1994).\n[7] G. S. Tian and T. H. Lin, Phys. Rev. B 53, 8196 (1996).\n[8] S. Q. Shen, Int. J. Mod. Phys. B 12, 709 (1998).[9] C. N. Yang and S. C. Zhang, Mod. Phys. Lett. B 4,759\n(1990).\n[10] J. He, et al, Phys. Rev. B 87, 075126 (2013).\n[11] M. Aidelsburger, et al, Phys. Rev. Lett. 107,255301\n(2011).\n[12] M. Aidelsburger, et al, Phys. Rev. Lett. 111, 185301\n(2013).\n[13] H. Miyake, et al, Phys. Rev. Lett. 111, 185302 (2013).\n[14] T. C. Hsu, Phys. Rev. B. 41, 11379 (1990).\n[15] G. Y. Sun and S. P. Kou, Europhys. Lett. 87,67002\n(2009).\n[16] V. M. Pereira, et.al, Phys. Rev. Lett. 96,036801 (2006).\n[17] E. H. Lieb, D. C. Mattis, J. Math. Phys. 3,749 (1962).\n[18] G. S. Tian, Phys. Rev. B 45, 3145 (1992).\n[19] S. Q. Shen and Z. M. Qiu, Phys. Rev. Lett. 71, 4238\n(1993)."    },    {        "title": "1704.08896v3.Thermal_Control_of_the_Magnon_Photon_Coupling_in_a_Notch_Filter_coupled_to_a_Yttrium_Iron_Garnet_Platinum_System.pdf",        "content": "Thermal Control of the Magnon-Photon Coupling in a Notch Filter coupled to a\nYttrium-Iron-Garnet/Platinum System\nVincent Castel\u0003\nLab-STICC (UMR 6285), CNRS, IMT Atlantique, Technopole Brest Iroise, 29200 Brest, France\nRodolphe Jeunehomme\nIMT Atlantique, Technopole Brest Iroise, 29200 Brest, France\nJamal Ben Youssef\nLab-STICC (UMR 6285), CNRS, Universit\u0013 e de Bretagne Occidentale,\n6 Avenue Victor le Gorgeu, 29200 Brest, France\nNicolas Vukadinovic\nDassault Aviation, 78 quai Marcel Dassault, 92552 St-Cloud, France\nAlexandre Manchec\nElliptika (GTID), 29200 Brest, France\nFasil Kidane Dejene\nMax Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany\nGerrit E. W. Bauer\nInstitute for Materials Research, WPI-AIMR, and Center for Spintronics\nResearch Network, Tohoku University, Sendai 980-8577, Japan and\nZernike Institute for Advanced Materials, University of Groningen,\nNijenborgh 4, 9747 AG Groningen, The Netherlands\n(Dated: June 25, 2021)\nWe report thermal control of mode hybridization between the ferromagnetic resonance (FMR)\nand a planar resonator (notch \flter) working at 4.74 GHz. The chosen magnetic material is a\nferrimagnetic insulator (Yttrium Iron Garnet: YIG) covered by 6 nm of platinum (Pt). A current-\ninduced heating method has been used in order to enhance the temperature of the YIG/Pt system.\nThe device permits us to control the transmission spectra and the magnon-photon coupling strength\nat room temperature. These experimental \fndings reveal potentially applicable tunable microwave\n\fltering function.\nI. INTRODUCTION\nMagnon-photon coupling has been investigated in mi-\ncrowave resonators (three-dimensional cavity1{10and\nplanar con\fguration11{13) loaded with a ferrimagnetic\ninsulator such as Yttrium Iron Garnet (YIG, thin \flm\nand bulk). More recently, research groups6{8have de-\nveloped an electrical method to detect magnons coupled\nwith photons. This method has been established by plac-\ning a hybrid YIG/platinum (Pt) system into a microwave\ncavity, showing distinct features not seen in any previous\nspin pumping experiment but already predicted by Cao\net al.14. These later studies have been realized in a 3D\ncavity (with high Qfactor), but insertion of an hybrid\nstack including a highly electrical conductor such as plat-\ninum has been reported to induce a drastic enhancement\nof the intrinsic loss rate.\nHere, we demonstrate thermal control of the magnon-\nphoton coupling at room temperature in a compact, de-\nsign based on a notch \flter resonating at 4.74 GHz and a\nhybrid YIG/Pt system. The thermal control of frequen-\ncies hybridization at the resonant condition is realizedby current-induced Joule heating in the Pt \flm, which\ncauses an out-of-plane temperature gradient. The planar\ncon\fguration of our rf device has a limited Qfactor but\navoids the negative impact of the YIG/Pt stack.\nII. COMPACT DESIGN DESCRIPTION\nThe used insulating material consists of a single-crystal\n(111) Y 3Fe5O12(YIG) \flm grown on a (111) Gd 3Ga5O12\n(GGG) substrate by liquid phase epitaxy (LPE). A Pt\nlayer of 6-nm grown by dc sputtering has been used as a\nspin (charge) current detector (actuator)15. The hybrid\nYIG/Pt system has been cut into a rectangular shape (1\nmm\u00027 mm) by using a Nd-YAG laser working at 8 W\nand placed on the stub line as shown in Fig. 1 (a) with\nthe crystallographic axis [1,1, \u00162] parallel to the planar mi-\ncrowave \feld generated by the stub line. The distance\nbetween electrodes (+ and -) is \fxed at 4 mm. Figure\n1 (a) illustrates the rf design of the present study which\nis based on a notch \flter (supplementary information in\nRef.16) coupled to a hybrid YIG/Pt system placed atarXiv:1704.08896v3  [cond-mat.mtrl-sci]  17 Jul 20172\nFIG. 1. (a) Experimental setup: A vector network analyzer\n(VNA) is connected to a 50 \n microstrip line which is capac-\nitively coupled to the resonator. The hybrid YIG/Pt system\nis placed at 0.25 \u0015. (+) and (-) are the electrical contacts\nwhich permit to detect (inject) a dc charge current from (at)\nthe edges of the Pt layer. (b) Frequency dependence of Spa-\nrameters with and without the YIG/Pt system. (c) Magnetic\n\feld dependence of the frequency: Observation of the strong\ncoupling regime via the anti-crossing \fngerprint. The color\nmap illustrates the magnitude of S21 from 0 to -12 dB. (d)\nFrequency dependence of S21 at for di\u000berent applied magnetic\n\felds using the analytic solution from Ref.9.\n\u0015=4. The in-plane static magnetic \feld ( H) is applied\nperpendicularly to the YIG/Pt bar with H?hrf, where\nhrfcorresponds to the generated microwave magnetic\n\feld. This latter con\fguration permits to maximize (i)\nthe precession of the magnetization at the resonant con-\ndition and (ii) the dc voltage detection at the edges of\nthe Pt layer. P1 and P2 illustrate the 2 ports of the\nvector network analyzer (VNA) that were calibrated, in-\ncluding both cables. The frequency range is \fxed from\n4 to 5 GHz at a microwave power of P= 0 dBm. In\nthe notch \flter con\fguration, the maximum of magnetic\n\feld is located at \u0015=4 (corresponding to a short circuit,\nSC) and a maximum of electric \feld at \u0015=2 (open circuit,\nOC). Insertion of the YIG/Pt device at the SC does not\nimpact the resonator features which is well illustrated by\nthe frequency response of Sparameters (with and with-\nout the device) as shown in Fig. 1 (b). Even if the hybrid\nstack includes a highly electrical conductor, no shielding\ne\u000bect has been observed. The S21 resonance peak of the\nempty resonator (loaded) has a half width at half max-\nimum (HWHM) \u0001 FHWHM of 32.75 MHz (34.45) indi-\ncating that the damping of the resonator (working at thefrequencyF0) is\f= \u0001FHWHM=F0= 1=2Q=6.92\u000210\u00003\n(7.27\u000210\u00003). This leads to a quality factor Qof 72.3\n(68.7). It should be noted that the latter de\fnition of Q\ndoes not re\rect the electrical performance of the notch \fl-\nter which is de\fned by Q0=F0=\u0002\n\u0001FS21\n\u00003dB(1\u0000S11F0)\u0003\n,\nwhere \u0001FS21\n\u00003dBandS11F0correspond respectively to the\nbandwidth of S21 at -3 dB and the magnitude (linear)\nofS11 atF0.Q0is reduced from 148 to 140.5 for the\nempty and loaded con\fguration, respectively. The pla-\nnar rf design used here o\u000bers many opportunities for inte-\ngrated spin-based microwave applications and is not sig-\nni\fcantly a\u000bected by the hybrid YIG/Pt device. In con-\ntrast, insertion of such device (including electrical con-\nnections for the detection or actuation of the magnetiza-\ntion) enhanced \fby a factor of 56to 127.\nIII. RESULTS AND DISCUSSION\nA. Strong coupling regime\nWe \frst studied the frequency dependence of Sparam-\neters atP= 0 dBm with respect to the applied magnetic\n\feld,H. Figure 1 (c) demonstrates the strong coupling\nregime via the anti-crossing \fngerprint. The FMR and\nthe notch \flter interact by mutual microwave \felds, gen-\nerated by the oscillating currents in the stub and the\nFMR magnetization precession which led to the follow-\ning features: (i) hybridization of resonances, (ii) anni-\nhilation of the resonance at F0, and (iii) generation of\ntwo resonances at F1andF2. At the resonant condition\nH=HRES, the frequency gap, Fgap, betweenF1and\nF2is directly linked to the coupling strength of the sys-\ntem (Fgap=2 =geff=2\u0019). Here, the analysis is based on\nthe harmonic coupling model9according to which we can\nde\fne the upper ( F1) and lower ( F2) branches by:\nF1;2=1\n2\u0014\n(F0+Fr)\u0006q\n(F0\u0000Fr)2+k4F2\n0\u0015\n(1)\nThe FMR frequency, Fr, is modelled by the Kittel equa-\ntion,Fr= (\r=2\u0019)p\nH(H+ 4\u0019Ms), which is the pre-\ncession frequency of the uniform mode in an in-plane\nmagnetized ferromagnetic \flm. The parameter kcor-\nresponds to the coupling strength which is linked to the\nexperimental data geff=2\u0019by the following equation9:\nFgap=F2\u0000F1=k2F0. A good agreement of F1;2(solid\nred and blue lines) with experimental data is obtained\nwithk=0.142, the saturation magnetization 4 \u0019Ms=1775\nG, and the gyromagnetic ratio \r=1.8 107rad Oe\u00001s\u00001.\nThe color plot is associated with the S21 parameter for\nwhich the dark area corresponds to a magnitude of -12\ndB. The same feature has been observed for negative\nmagnetic \feld (not shown). Figure 1 (d) represents the\nfrequency dependence of the transmission spectra for the\nnotch/YIG/Pt system for di\u000berent magnetic \feld, lower\nand higher than HRES. We \fnd good agreement be-\ntween experimental data (solid lines) and the calculated3\nresponse (dash dot lines) based on the analytic solution\nproposed in Ref.9, leading to an e\u000bective damping param-\neter of\u000beff=1.75 10\u00003for the YIG/Pt system. It should\nbe noted that \u000beffdoes not re\rect the intrinsic magnetic\nloss but includes the inhomogeneous broadening due to\nthe excitation of several modes. We studied the YIG/Pt\nsample by standard FMR using a highly sensitive wide-\nband resonance spectrometer within a range of 4 to 20\nGHz (at room temperature with a static magnetic \feld\napplied perpendicular to the sample). These characteri-\nzations lead to the intrinsic damping parameter \u000b\u001910\u00004.\nB. Thermal control\nSending an electric current through the structure re-\nduces or enhances the magnetic losses, i.e. the Gilbert\ndamping parameter \u000b, of the YIG \flm (depending on the\nmagnetic/current setting). Furthermore, the current in-\nduced heating gives rise to a temperature di\u000berence, \u0001 T,\nover the YIG/Pt system. As discussed below, the strong\ndependence of the YIG/Pt device to \u0001 Tovershadows\nthe anticipated modulation of \u000bby the STT (without\n\u0001Tcontribution). The absence of strong coupling in the\nISHE signal (supplementary information in Appendix A)\nis another possible reason for the failure to control \u000bof\nthe bulk magnetization.\nFigure 2 illustrates the charge current dependence of\nthe frequency response at the particular magnetic \feld,\nH=HRES at which the S21 splitting is maximized.\nRelatively small dc currents in the range of -40 to 40 mA\n(corresponding to a current density between -6.7 to 6.7\n109Am\u00002) were sent through the Pt contact shown in\nthe inset from Fig. 2 (c), which are one order of mag-\nnitude smaller than Ref.17. The Joule heat produces a\ntemperature gradient. Figure 2 (a) represents the fre-\nquency dependence of the transmission spectra ( S21 in\ndB) at the resonant condition ( H=HRES) for di\u000berent\napplied charge currents. Labels [1] to [7] correspond to\nthe magnitude of the applied charge current which is in-\ndicated in Fig. 2 (b). The reference measurement was\ncarried out for zero current density ([4]). By sending a\ncurrent density of 3.3 ([5]), 5.0 ([6]), and 6.7 109Am\u00002\n([7]) into the Pt strip, the resonant frequency F2de-\n\fned by Eq.(1) can be signi\fcantly tuned and drastically\nchange the transmission spectra. By reversing the sign\nof the charge current, the response of [1]&[7] (as well as\n[2]&[6] and [3]&[5]) present the same behaviour. Figure 2\n(b) gives an overview of the latter dependence where the\ncolor map is associated to the magnitude of S21 from 0\nto -12 dB. In this \fgure, we can clearly observe the rela-\ntively small changes of F1which approaches F0(resonant\nfrequency of the notch \flter) for larger current densities.\nThis behaviour cannot be attribute to current-induced\nmagnetic \feld because the response does not depend on\nthe sign of j(and the sign of H, as shown in Fig. 1\n(c)). Indeed, magnitude of the in-plane Oersted \feld can\nbeen estimated18to not exceed\u00060.3 Oe (extracted in\nFIG. 2. Injection: (a) Frequency dependence of the Spa-\nrameters at the resonant condition ( H=HRES) for di\u000berent\napplied charge current. Labels [1] to [7] correspond to the\nmagnitude of the applied charge current as indicated in (b).\n(b) Charge current dependence of the frequency measured at\nthe resonant condition. The color map is associated to the\nmagnitude of S21 from 0 to -12 dB. Solid cyan curves corre-\nspond to the current dependence of the hybridized frequencies\nbased on Eq.(1), including the temperature dependence of Fr.\nRed and blue triangles indicate the position of F1andF2. (c)\nMagnetic \feld dependence of the frequency: the correlation\nbetween magnetic \feld and charge current dependence of F1\nandF2. Inset shows a schematic of the measurement setup.\nDependence of Pt strip resistance (d), the temperature in-\ncrease of the Pt strip (e), and \u0001 Fr(f) with respect to the\ninjected current ( I2).\nthe middle of the YIG section) when I=\u000650 mA is sent\nthrough the Pt section of 6.10\u000012m2. Figure 2 (c) illus-\ntrates the correlation between the e\u000bects of magnetic \feld\nand charge current on the frequency. Open blue and red\ntriangles correspond to the experimental values of F1and\nF2atH=HRES extracted from the measurement pre-4\nsented in Fig. 2 (b). A perfect equivalence of the e\u000bects\nof magnetic \feld and charge current on the resonance fre-\nquencies is evident. For example, the response of S21 at\nH=\u0006890 Oe (with j= 0 Am\u00002) is equivalent to S21\nj=\u00066:7109Am\u00002(withH=HRES) which corresponds\nto a shift of\u0006100 Oe. As shown in Fig. 2 (d), the mea-\nsured resistance of the Pt layer increases quadratically\nwith the applied current de\fned by R(I)=R0+R2I2,\nwhereR2=21.019 k\nA\u00002and the total resistance of the\nPt at room temperature is R0=280.35 \n corresponding\nto a Pt conductivity of 2.38 106\n\u00001m\u00001in agreement\nwith previous work15.\nIn order to obtain quantitative information we carry\nout three-dimensional \fnite element modeling19of our\ndevice which takes into account material and tempera-\nture dependent transport coe\u000ecients for the Pt layer20.\nThe temperature dependence of the electrical conductiv-\nity can be \ftted by \u001b(T) =\u001b=(1+a(T\u0000T0)), with\u001bequal\nto the resistance at vanishing currents, a= 4\u000210\u00003K\u00001\nis the temperature coe\u000ecient of resistance (TCR) of Pt\nwhich is well tabulated in the literature, and T\u0000T0is\nthe overall temperature increase (\u0001 T). The thermal con-\nductivity of the YIG substrate is kept at 5 W/m/K21\nwhereas that of Pt is calculated using the Wiedemann-\nFranz relation \u0014=\u001bL0T0whereL0=2.44 10\u00008V2=K2is\nthe Lorenz number and T0is the reference temperature.\nFigure 2 (e) presents the dependence \u0001 Tas a function of\nI2extracted from our model (green triangles). It should\nbe noted that the latter dependence agreements with the\nanalytic solution17: \u0001T=\u0014Pt(R(I)\u0000R0)=R0, where\n\u0014Pt=254 K. We see that the temperature increase of the\nPt layer is estimated at 47 \u00060.6 K for a current density\nof 8.33 109Am\u00002.\nNext, we measured the magnetic \feld dependence of\nthe frequency response as shown in Fig. 1 (c) as func-\ntion of charge current as illustrated in Fig. 3 (d) to (f).\nFigure 2 (f) represents the dependence of \u0001 Fas func-\ntion ofI2de\fned asFr(T) =FT0=RT\nr\u0000\u0001F(T). The\nobserved reduction is caused by the temperature depen-\ndence of the magnetization22,23. Substituting this equa-\ntion into Eq.(1) reproduces the current dependence of\nthe hybridized frequencies, F1andF2in Fig. 2 (c) (solid\ncyan curves). It should be noted that a good agreement\n(onF1andF2) has been found as well by using the same\nprocedure for an applied magnetic \feld larger than HRES\n(supplementary information in Appendix B).\nA map of the coupling regime3,9can achieved through\nthe representation of the dependence of K=\u000b as function\nofK=\f whereK=k2p\nF0=2Fmcorresponds to the e\u000bec-\ntive coupling strength with Fm= (\r\n2\u0019)4\u0019Ms. Together,\nthese latter parameters determine the coupling features\nof the system that can be con\fgured from weak ( K=\u000b< 1\nandK=\f < 1) to strong coupling regime ( K=\u000b > 1 and\nk=\f > 1).Kcan be tuned with (i) the volume of YIG14,16,\n(ii) the volume of the cavity24, and (iii) the magnitude of\nthe microwave magnetic \feld3,16. Control of the coupling\nregime has been already demonstrated7,25by tuning the\nFIG. 3. (a)-(c) Calculated magnetic \feld dependence of\nthe microwave transmission spectrum based on Ref.9for dif-\nferent values of the damping parameter \u000b. (d)-(f) Measured\nmagnetic \feld dependence of the frequency as function of the\ninjected current magnitude. The color map of (a)-(f) is as-\nsociated to the magnitude of the transmission parameter S21\nfrom 0 to -3 dB. (g) Calculated dependence of S21 (color\nmap from 0 to -12 dB) as function of the frequency and the\ndamping parameter. Representation of F1,F2, andF0. (h)\nThermal annihilation of the coupling strength measured (blue\nstars) and extracted from (g) (dash dot red line).\ncavity losses \f. However, the charge current dependence\nof the transmission spectra of our system suggests a con-\ntrol of the magnetic losses ( \u000beffin our case). It should\nbe noted that our notch \flter features remain unchanged\nas function of the current density sent into the Pt strip.\nThe e\u000bect of the Pt heating is not only a shift of the\nresonance (tuning) but also an increased broadening (de-\ncoupling) caused by inhomogeneous heating (top part of\nthe YIG/Pt device is hotter than the bottom part, while\nthe microwaves see the whole sample). On the other\nhand, the temperature dependence of the damping pa-\nrameter cannot be attributed to the Spin Seebeck E\u000bect\n(SSE) because of the thickness of our sample (6 \u0016m).\nFigures 3 (a) to (c) represent the calculated magnetic\n\feld dependence of the frequency based on Ref.9for dif-\nferent values of \u000beff(de\fned above). Here we show by\nmodel calculations how to control the coupling strength\nby increasing the magnetic losses26,27. Enhancement of\n\u000befffrom 1.75 10\u00003to 1 10\u00001suppresses the frequency\ngap between F1andF2and thus the coupling strength.\nThe same behaviour has been observed experimentally by\nchanging the current density from 0 to 8.3 109Am\u00002as\nshown Fig. 3 (d) to (f). By following the same procedure,\nthe experimental determination of the coupling strength,\ngeff=2\u0019=1\n2(F2\u0000F1), at the resonant condition (by ad-\njustingHin order to keep Fr=F0) has been done for\nseveral current density. The experimental (blue star) and\ncalculated (red dash-dot line) dependences of geff=2\u0019are5\nrepresented in Fig. 3 (h) as function of the charge current\ndensity. Between 0 and 6 109Am\u00002,geff=2\u0019is closed to\n40 MHz and abruptly reduced to zero beyond this value.\nThe calculated dependence, which permits to well repro-\nduced the experimental behaviour, is based on the color\nplot from Fig. 3 (g). It represents the dependence of\nthe transmission spectra S21 (color map form 0 to -12\ndB) as function of \u000beffand the frequency. It should be\nnoted that this \fgure represents the decoupling of the\nsystem, whereas Fig. 2 (c) illustrates the detuning of\nthe transmission spectra at a \fxed value of the magnetic\n\feld.\nIV. CONCLUSION\nWe reported thermal control of mode hybridization\nbetween the ferromagnetic resonance and a planar res-\nonator by using a current-induced heating method. Our\nset-up allows simultaneous detection of the ferromagnetic\nresponse and the dc voltage generation in the Pt layer\nof the system, which reveals an absence of the strong\ncoupling signature in the ISHE signal. We demonstrate\nthe potential for tunable \flter application by electrically\ncontrol the transmission spectra and the magnon-photon\ncoupling strength at room temperature by sending a\ncharge current through the Pt layer.\nACKNOWLEDGMENTS\nThis work is part of the research program supported\nby the European Union through the European Regional\nDevelopment Fund (ERDF), by Ministry of Higher Edu-\ncation and Research, Brittany and Rennes M\u0013 etropole,\nthrough the CPER Project SOPHIE/STIC & Ondes,\nand by Grant-in-Aid for Scienti\fc Research of the JSPS\n(Grant Nos. 25247056, 25220910, and 26103006).\nAPPENDIX A: SPIN CURRENT DETECTION\nOur set-up allows us to simultaneously detect the dc\nvoltage generated in the Pt layer ( VISHE ) and the ferro-\nmagnetic response (microwave absorption) of the system\nas shown in Fig. 4. Keeping the previous calibration of\nthe VNA, we carried out a step-by-step measurement (by\nshrinking the trigger mode) in order to improve the qual-\nity of the measured dc voltage by the nanovoltmeter. Fig-\nures 4 (a) and (b) correspond respectively to the response\nofS21 and the measured voltage, VISHE , as function of\nthe frequency for speci\fc values of the applied magnetic\n\feld. A nonzero VISHE comes from the fact that at (and\nclose to)Fra spin current ( js) is pumped into the Pt\nlayer, which results in a dc voltage by the inverse spin\nHall e\u000bect (ISHE). The sign of the electric voltage signalis changed by reversing the magnetic \feld and no sizable\nvoltage is measured when His equal to zero, as expected.\nThe sign reversal of Vshows that the measured signal is\nnot produced by a possible thermoelectric e\u000bect, induced\nby the microwave absorption. The strong coupling in the\nspin pumping signal (predicted by Cao et al.14) has been\nexperimentally observed in a YIG(2.6-2.8 \u0016m)/Pt system\nfor which the spin sink layer presents a thickness of 57\nand 10 nm6,8. Even though the S21 parameter illustrates\nin Fig. 1 (b) is strongly coupled, no such a signature has\nbeen observed in the dc voltage generation. We therefore\nconclude that in contrast to the bulk magnetization the\ninterface sensed by the Pt layer is not strongly coupled\nto microwaves. However, from the comparison between\nFig. 4 (a) and (b), we are able to identify a correlation in\nthe peak positions (vertical dashed lines) of VISHE and\nfeatures in S21.\n/s52/s46/s52 /s52/s46/s53 /s52/s46/s54 /s52/s46/s55 /s52/s46/s56 /s52/s46/s57 /s53/s46/s48/s45/s49/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s56/s49/s48\n/s45/s49/s50/s45/s49/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48\n/s52/s46/s52 /s52/s46/s53 /s52/s46/s54 /s52/s46/s55 /s52/s46/s56 /s52/s46/s57 /s53/s46/s48/s45/s56/s45/s52/s48/s52/s56\n/s86\n/s73/s83/s72/s69/s32/s91 /s86/s93\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s91/s71/s72/s122/s93/s72/s32/s105/s110/s99/s114/s101/s97/s115/s105/s110/s103\n/s32/s72/s61/s48/s32/s79/s101\n/s32/s72/s61/s43/s57/s57/s48/s32/s79/s101\n/s32/s72/s61/s45/s57/s57/s48/s32/s79/s101/s83/s50/s49/s32/s91/s100/s66/s93/s40/s99/s41\n/s40/s98/s41/s40/s97/s41/s86\n/s73/s83/s72/s69/s32/s91 /s86/s93\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s91/s71/s72/s122/s93\nFIG. 4. Detection: Simultaneous detection of (a) S21 and\n(b) the dc voltage generated by the conversion of a spin cur-\nrent into a charge current via Inverse Spin Hall E\u000bect (ISHE)\nat resonant condition for the positive and negative con\fgura-\ntions of the magnetic \feld. (c) Measured voltage, VISHE , as\nfunction of the frequency for di\u000berent applied magnetic \feld.\nAPPENDIX B: HYBRIDIZED FREQUENCIES\nFigure 5 illustrates the charge current dependence of\nthe frequency response at an applied magnetic \feld where\nno mode hybridization is observed (H=1128 Oe). The\nsolid black lines correspond to the calculated dependence\nofF1andF2based on Eq.(1) in which the tempera-\nture dependence of Fris included. It should be noted\nthat the temperature enhancement reduces the coupling\nstrength of our system from k=0.142 (geff=2\u0019\u001945 MHz\natj= 0 Am\u00002) tok=0.110 (geff=2\u0019\u001928.5 MHz at\nj= 7:1109Am\u00002), in agreement with current density de-\npendence of geff=2\u0019represented in Fig. 3 (g).6\n-8.3-7.5-6.7-5.8-5.05.05.86.77.58.34.44.54.64.74.84.95.0S\n21 [dB]geff/2π=28.45 MHzk\n=0.110H\n=1128 OeFrequency [GHz]j\n [109Am-2]\n 0-\n2-\n4-\n6-\n8-\n10-\n12\nFIG. 5. Charge current dependence of the frequency mea-\nsured at H=1128 Oe (o\u000b resonance). The color map is asso-\nciated to the magnitude of S21 from 0 to -12 dB. Solid black\ncurves correspond to the current dependence of the hybridized\nfrequencies based on Eq.(1), including the temperature depen-\ndence ofFrwithk=0.110.\n\u0003vincent.castel@imt-atlantique.fr\n1L. Kang, Q. Zhao, H. Zhao, and J. Zhou, Opt. Express\n16, 8825 (2008).\n2J. N. Gollub, J. Y. Chin, T. J. Cui, and D. R. 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Khartsev, and J. kerman, Journal of Applied Physics\n117, 17D119 (2015)."    },    {        "title": "2112.10036v1.Pressure_induced_charge_orders_and_their_coupling_to_magnetism_in_hexagonal_multiferroic_LuFe2O4.pdf",        "content": " \n \n1 \n Pressure -induced charge orders and their coupling to magnetism in \nhexagonal multiferroic LuFe 2O4 \nFengliang  Liu1,2,3#, Yiqing Hao1#, Jinyang Ni1#, Yongsheng Zhao2, Dongzhou Zhang4, Gilberto Fabbris5, \nDaniel Haskel5, Shaobo Cheng6, Xiaoshan Xu7, Lifeng  Yin1,8,9 ,10,11, Hongjun Xiang1,9, Jun Zhao1,9,10, Xujie \nLü2, Wenbin Wang8,9,10,11,*, Jian Shen1,8,9 ,10,11,* and Wenge  Yang2,* \n1State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, \nChina  \n2Center for High Pressure Science and Technology Advanced Research (HPSTAR), Shanghai 201203, China  \n3Department of Physics, Nanchang University, Na nchang, 330031, China  \n4Hawaii Institute of Geophysics & Planetology, University of Hawaii Manoa, Honolulu, HI, USA.  \n5Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA  \n6Department of Condensed Matter Physics and Materials Sci ence, Brookhaven National Laboratory, Upton, \nNY 11973, USA  \n7Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588, USA  \n8Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China  \n9Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China  \n10Shanghai Qi Zhi Institute, Shanghai 200232, China  \n11Shanghai Research Center for Quantum Sciences, Shanghai 201315, China  \n \n#These authors  contributed equally.  \n* Corresponding authors. E-mail addresses: wangwb@fudan.edu.cn (W.B.W.) , \nshenjian5494@fudan.edu.cn (J.S.), yangwg@hpstar.ac.cn (W.G.Y.) . \n \nHexagonal LuFe 2O4 is a promising charge -order (CO) driven multiferroic material with high charge and spin \nordering temperatures. The coexisting charge and spin orders on Fe3+/Fe2+ sites result in novel \nmagnetoelectric behaviors, but the coupling mechanism between the charge and spin orders remains elusive. \nHere, by tuning external pressure, we reveal three correlated spin -charge ordered phases in LuFe 2O4: i) a \ncentrosymmetric incom mensurate three -dimensional CO with ferrimagnetism, ii) a non -centrosymmetric \nincommensurate quasi -two-dimensional CO with ferrimagnetism , and iii) a centrosymmetric commensurate \nCO with antiferromagnetism. Experimental in -situ single -crystal X-ray diffrac tion and X-ray magnetic \ncircular dichroism measurements combined with density functional theory calculations suggest that the  \n \n2 \n charge density redistribution caused by pressure -induced compression in the frustrated double -layer [Fe 2O4] \ncluster is responsible  for the correlated spin-charge phase transitions. The pressure -enhanced effective \nCoulomb interactions among Fe -Fe bonds drive the frustrated (1/3, 1/3) CO to a less frustrated (1/4, 1/4) CO, \nwhich induces the ferrimagnetic to antiferromagnetic transition . Our results not only elucidate the coupling \nmechanism among charge, spin and lattice degrees of freedom in LuFe 2O4 but also provide a new way to \ntune the spin -charge orders in a highly controlled manner.  \nINTRODUCTION  \nMultiferroric  materials have attracted lots of research interests during the last decades because of their great \npotential applications in electronic devices and spintronics1,2. Hexagonal LuFe 2O4 is a promising candidate \nmaterial for charge -order (CO) driven multiferro icity, which has been intensively investigated both from \nfundamental and applied perspectives3-10. Previous X-ray diffraction (XRD)4 and transmission electron \nmicroscopy (TEM) measurements11,12 have suggested that LuFe 2O4 exhibits three -dimensional (3D) CO at \nambient pressure, manifesting a periodic arrangement of low valence (Fe2+) and high valence (Fe3+) ions. \nIndication of the CO -driven ferroelectricity of LuFe 2O4 was also revealed in the observ ation of the \nspontaneous electronic polarization above room temperature3,13,14. In addition to the 3D order of Fe2+-Fe3+ \nions observed below the CO transition temperature TCO (~ 320 K), a quasi -2D (Q2D) ordering of Fe2+-Fe3+ \nwas also observed above TCO persisting up to ~ 525 K12. Furthermore, neutron diffraction measurements have \nrevealed ferrimagnetic order of Fe moments below the Neel transition temperature TN (~ 240 K)15. It is \ntherefore suggested that the correlations between charge and magnetic order associated with Fe2+ and Fe3+ \nions may play a crucial role in the multiferroicity in LuFe 2O44,16-19. \nApplication of external pressure is a powerful and clean tool to gain deep insights into the interplay \nbetween the charge, magnetic and structural degrees of freedom, because the strong frustration involved in \nthe spin and charge orders in the triangular lattice of LuFe 2O4 may result in highly tunable ground states16,19. \nHowever, high pressure diffraction measurements are often difficult  owing to the reduced beam flux and \nincreased  background  caused by  pressure cell s. Previous n eutron diffraction  measurement on powder \nsamples of LuFe 2O4 revealed ~ 30% reduction of the ferrimagnetic ordered moments  up to ~3 GPa20, and X-\nray powder diffraction showed  indications  for pressure -induced structur al phase  transitions21. However,  the \npositions and intensities of superlattice reflections observed in powder diffraction  show ed non-systematic \nevolution  with increasing pressure probably due poor powder averaging , thus preventing accurate description \nof the pressure -induced phases21. Moreove r, neutron and X-ray powder diffraction measurements  have \nrevealed a high -pressure polymorph phase LuFe 2O4-hp22,23 above 12 GPa. The LuFe 2O4-hp phase retained its \nstructure after pressure release, thus allowing  ex-situ measurement at ambient pressure22,23. This LuFe 2O4-hp \nphase adopts a rectangular Fe lattice23 and is not directly relevant to  the frustrated triangular lattice of \nLuFe 2O4 at ambient condition s. Therefore, despite the intensive  efforts, the pressure dependent evolution of \nthe frustrated charge and magnetic interactions in the multiferroic  LuFe 2O4 remains unclear .  \nHere, we investigated the evolution of charge and spin orders under pressure in LuFe 2O4 using the in -situ \nhigh-pressure single -crystal X-ray diffraction (HP -SXD) and high -pressure X-ray magnetic circular \ndichroism (HP -XMCD) spectroscopy. A series of pressure -induced charge -order phases, including a Q2D \nCO phase below 5.5 GPa and a 3D CO phase at higher pressur e (6.0 ~ 12.6 GPa), were identified by HP -\nSXD measurements. In addition, the HP -XMCD measurements suggest that the Q2D and 3D CO phases are \nassociated with ferrimagnetic and antiferromagnetic order, respectively. The pressure induced charge - and \nmagnetic -order phase transitions were further confirmed by density functional theory (DFT) calculations. \nThese results suggest that the CO phases are intimately coupled with magnetism, both of which can be \nmanipulated by external pressure in a highly controlled mann er in hexagonal layered multiferroic LuFe 2O4. \n \n \nRESULTS  \nLuFe 2O4 adopts a bi -layered triangular lattice structure with space group R-3m (No.166) at ambient \npressure. Considering the CO and the atom ic displacement it induces, the trigonal lattice splits into three 120o  \n \n3 \n twinned monoclinic lattices, each of which adopts the space g roup C2/m (No.12). The transformation \nbetween the trigonal and monoclinic lattices is shown in Fig. 1 a. The optimal stoichiometry of our sample \nwas confirmed by magnetic susceptibility and transmission electron microscopes measurements at ambient \npressure (Fig. S1) . Our high -pressure synchrotron X-ray diffraction measurement of LuFe 2O4 (Fig. S2) was \ncarried out on a high quality single -crystalline sample. Within  the representation of space group C2/m (Fig. \nS3), the lattice parameters are a = 5.957(2) Å, b = 3.434(4) Å, c = 8.642(3) Å, and = 103.28(2) ° at 300 K \nand 0.8 GPa. The lattice parameters a, b, and c shrink with increasing pressure. At 300 K and 12.6 GPa, the \nlattice parameters become a = 5.809(5) Å, b = 3.339(16) Å, c = 8.374(11) Å, and =103.39(9) ° with a volume \nreduction of 8.2(6)%.  \n \n \nFig. 1 Crystal structure and charge order evolution of LuFe 2O4 under high pressure. (a) Crystal \nstructure of LuFe 2O4 at ambient pressure. Dotted lines represent the trigonal lattices. Red, blue and green \nsolid lines represent three equivalent monoclinic lattices with different twinned directions. ( b-e) Single \ncrystal  X-ray diffraction (SXD) intensities in the ( HHL ) scattering plane measured near the charge ordering \n(CO) superlattice peaks at room temperature at 0.8, 3.1, 5.0 and 6.0 GPa, respectively. ( f-h) SXD intensity \nalong the Q = (1/3, 1/3, L), (1/3, 1/3, L) and (1/4, 1/4, L) direction measured at 0.8 GPa, 5.0 GPa and 6.0 \nGPa, respectively. (I) Summary of the charge order peak positions in HK-plane measured from 0.8 to 12.6 \nGPa. The black hexagon in center indicates ( H, H) = (1/3, 1/3).  \n \n \n \n4 \n Apart from the lattice shrinkage, a series of superlattice peaks were also observed with a wave vector kAP \n= (1/3, 1/3, 3/2) T in trigonal lattice or (0, 2/3, 1/2) M in monoclinic lattice at 0.8 GPa, as illustrated in Fig. 1 b \nand Fig. 1f. This wave vector is  close to that observed at ambient pressure where the charge order of low -\nvalence (Fe2+) and high -valence (Fe3+) Fe ions form s a √3×√3×2 super -lattice4 (Fig. 2 b). As pressure \nfurther increases, the superlattice peaks exhibit drastic broadening along the L-direction (Fig. 1( b-d)), \nindicating that the charge order becomes quasi -two-dimensional. Meanwhile, the maximum of peak intensity \nin L-direction moves from half  integer ( L = n+1/2) at ambient pressure to integer ( L = n) at 5.0 GPa (Fig. 1( d, \ng) and Fig. S4), indicating that the inter -plane polarization emerges under pressure (Fig. 2 d). Interestingly, \nsimilar quasi -two-dimensional charge order was also observed above the 3D charge ordering temperature of \n320 K in LuFe 2O4 at ambient pressure4. \n \n \nFig. 2 Charge order models of LuFe 2O4 corresponding to observed  SXD pattern from 0.8 GPa to 12.6 \nGPa. Red and blue spheres represent Fe2+ and Fe3+ ions, respectively. Sphere size in ( a, c, e) indicate the Fe \nions in different planes. Thick and thin solid lines represent nearest neighbored (NN) and next nearest \nneighbored (NNN) Fe -Fe pairs, respectively. Dashed lines represent lattice boundaries.  (a, b) \nCentrosymmetric CO -AP model with k ~ (1/3, 1/3, 3/2). (c, d) Non-centrosymmetric CO -2D model with k ~ \n(1/3, 1/3, 0) at P = 5.0 GPa. Noted the inter -layer arrangements in ( b) and ( d) are different due to change in \nL-component of the wave vector. (e, f) CO-HP model for k = (1/4, 1/4, 0), corresponding SXD patterns \nobserved between 6.0 and 12.6 GPa.  \n \nA closer inspection on the diffraction pattern actually reveals a tiny incommensurability of the charge order \nwave vector [ kAP = (1/3+ δ, 1/3+ δ, 3/2) T and k2D = (1/3 -δ, 1/3-δ, 0) T] at 0.8 GPa. The incommensurability \nincreases dramatically with increasing pressure, and eventually reaches a commensurate position kHP = (1/4, \n1/4, 0) T at pressures above 6.0 GPa. In contrast to the broadened diffraction patterns alo ng the L-direction for \nthe superstructure below 5.0 GPa (Fig. 1 d), the (1/4, 1/4, 0) T phase shows sharp peak features both along the \nin-plane ( HK) and out -of-plane ( L) directions (Fig. 1 (e, h)), indicating restoration of  robust 3D order . This \ncommensurate CO -HP phase persists up to 12.6 GPa without a major change in the primary structure.  \nThe evolution of the incommensurability of the superlattice peaks in the HK -plane under various pressures \nis summarized in Fig. 1 i. There are three 120 -degree twinned charge orders in LuFe 2O4, each with a unique \nk-vector, therefore the superlattice peak position differs for each charge order twin, forming a spiral -like \ndiffraction pattern in reciprocal space. As pressure increases, the centers of charge order peaks move away \nfrom (1/3, 1/3) along the 120 -degree directions in the HK-plane and eventually reach (1/4, 1/4), (1/2, 1/4), \n(1/4, 1/2) at 6.0 GPa. Therefore, the quasi -2D charge order phase observed in the 3.1 GPa to 5.0 GPa range \ncan be regarded as  an intermediate phase between the commensurate (1/3, 1/3) and (1/4, 1/4) charge -order \nphases.  \n \n \n5 \n We now discuss the possible charge order models for the high -pressure phases. We consider two valence \nstates of Fe ions (Fe2+ and Fe3+). Based on diffraction data, we found that the charge order of (1/4, 1/4, 0) T \nphase can be best described with the Pb2/c (BNS 13.71) black -white space group  (Fig. 2( e,f) and Fig. S5). \nThe CO phase models of CO -AP [kAP = (1/3, 1/3, 3/2) T], CO -2D [k2D = (1/3, 1/3, 0) T] and CO -HP [kHP = \n(1/4, 1/4, 0) T] are illustrated in Fig. 2( a,b), (c,d) and ( e,f), respectively. These CO models are further \nsupported by DFT calculation (Fig. 4d). The evolution from (1/3, 1/3) T to (1/4, 1/4) T can be understood \nthrough the Coulomb interactions on different types of Fe -Fe bonds. In CO -AP and CO -2D phases, 5/9 of \nthe nearest neighbor (NN), 2/3 of the 2nd NN and 5/9 of the 3rd NN Fe bonds are Fe2+-Fe3+ bonds. On the \ncontrary, in CO -HP phase, 2/3 o f the NN, 2/3 of the 2nd NN and 1/3 of the 3rd NN Fe bonds are Fe2+-Fe3+ \nbonds. Above observation provides a natural understanding, that the effective Coulomb interactions of NN \nand 3rd NN Fe -Fe pairs  are tuned by the compression in frustrated triangular d ouble -layer Fe 2O4 structure \ndue to increased pressure. We also calculated the interlayer next -nearest -neighbor ( VcNNN) and intralayer \nnearest -neighbor ( VabNN) Coulomb interactions at each pressure point. Our result shows that VcNNN/VabNN \nremains almost unchanged in the region of 1 to 5 GPa, and drops sharply above 6 GPa where the (1/4, 1/4) T \nCO phase is favored24 (Fig. S6).  \n \n \nFig. 3 Magnetic properties of LuFe 2O4 under pressure.  (a-c) Fe K -edge X -ray absorption near edge \nstructure (XAS) and XMCD spectroscopy data measured at T = 100 K, H = 5 T and P = 1.9 GPa ( a), 4.3 GPa \n(b), 9.5 GPa ( c), respectively. The pressures correspond to the three observed charge order phases (CO -AP, \nCO-2D, CO-HP) in LuFe 2O4. Insets of ( a-c) show the XAS and XMCD data of pre -edge region at 1.9 GPa, \n4.3 GPa and 9.5 GPa, respectively. ( d) Ferrimagnetic model for the centrosymmetric CO -AP phase ( e) \nFerrimagnetic model for the non -centrosymmetric CO -2D phase ( f) Antiferromagnetic model for the \ncentrosymmetric CO -HP phase for DFT calculations in pressurized LuFe 2O4. (Ferri: ferrimagnetic. AFM, \nanti-ferromagnetic).  \n \nIn order to inform on the magnetic ground states associated with these charge -ordered phases, we util ized \nHP-XMCD spectroscopy to monitor the evolution of net magnetization of LuFe 2O4 under pressure. Fig. 3a \nshows Fe K -edge isotropic absorption (XAS) and dichroic (XMCD) signals in LuFe 2O4 at 100 K and 5 T. \nThe measurements were performed at the Fe K-edge instead of the more commonly used Fe L-edge because \nsoft X-ray MCD is incompatible with the highly absorbing diamond anvil cell environment25-27. The dichroi c \nsignal at 1.9 GPa consists of a positive peak in the pre -edge region (7114.0 eV) and anothe r positive peak \nnear the main rising edge peak (7129.0 eV) with larger intensity  (Fig. 3a). With increasing pressure, these \ntwo dichroic peaks become much weaker and vanish at 9.5 GPa  (Fig. 3c). This result indicates non -zero net \nmagnetization at 1.9 GPa  and 4.3 GPa, and zero net magnetization at 9.5 GPa. The peaks of Fe K-edge XAS \nand XMCD signals observed at 1.9 GPa are in agreement with those observed at ambient pressure ( Fig. S7)28. \nTherefore, the magnetic order at 1.9 GPa  is best interpreted as a ferrimagnetic structure which was unveiled \n \n \n6 \n by the previous L -edge XMCD18 and neutron diffraction15 measurements at ambient pressure. Meanwhile, \nthe peak positions of dichroism peaks are unchanged at 1.9 GPa and 4.3 GPa (Fig. 3 (a-c)). This result suggests \nthat the ferrimagnetic arrangement of Fe2+ and Fe3+ moments is preserved at 4.3 GPa, but the net \nmagnetization is gradually reduced by pressure. Insets of Fig. 3(a-c) illustrate the pre -edge region of XAS \nand XMCD signals. The posi tions of XAS pre -edge peak and leading edge do not shift by pressure, which \nindicates that the valence ratio of Fe2+ and Fe3+ remains unchanged ( Fig. S8), in agreement with the CO \nmodels illustrated in Fig. 2.  \n \n \nFig. 4 Phase diagram of LuFe 2O4 under pressure.  (a) Relative reduction of lattice parameters a and c by \nincreased pressure measured between 0.8 and 12.6 GPa. ( b) Evolution of in -plane incommensurate CO wave \nvector δ by pressure. δ is defined by (1/3+ δ, 1/3+ δ) in CO -AP and (1/3 -δ, 1/3 -δ) in CO -2D. The \ncommensurate CO wave vector (1/4, 1/4) of CO -HP is represented by δ=1/12. ( c) The pressure dependence \nof XMCD spectral weight. ( d) The pressure dependence of relative enthalpies between the charge order \nphases mentioned in Fig. 2 and Fig. 3d-f. (ferri: ferrimagnetic. AFM:anti -ferromagnetic).  \n \nMore insight into the nature of the magnetic ground states associated with the CO phases in pressurized \nLuFe 2O4 can be obtained by DFT calculations. For CO -AP phase and CO -2D phase, DFT calculations show \nthat LuFe 2O4 adopts the “2:1 ferrimagnetic state” as ground state. In this state, as shown in Fig 3( d, e), \nmajority spin orientation consists of all Fe2+ ions plus  one-third of total Fe3+ ions while the minority spin \norientation consists of remaining Fe3+ ions. For CO -HP phase, DFT calculation shows that the \nantiferromagnetic structure (Fig. 3f) is the magnetic ground state ( Fig. S9). In this state, both Fe2+ and Fe3+ \nare evenly separated in opposite spin orientations, resulting in the centrosymmetric electromagnetic order. \nThis result is consistent with the XMCD measurement, where the net magnetization  is completely suppressed \nby pressure.  \n \nDISCUSSION S \nCombining the  HP-SXD, HP -XMCD and DFT calculations, we have found a series of 3D -2D-3D charge \norder transitions in the hexagonal LuFe 2O4. The emergen ce of the CO-HP phase at high pressure is \naccompanied by a ferri -magnetic to antiferromagnetic transition (Fig. 3( d-f)). We calculate the pressure \ndependence of the enthalpy for each spin -charge ordered phase using GGA+ U method29,30, as illustrated in \nFig. 4d. According to the calculation, at ambient pressure, the 3D centrosymmetric CO phase (CO -AP) \ncombined with 2:1 ferrimagnetic order (Fig. 3d) is most favored. Then, as the lattice shrinks with applying \nhydrostatic pressure (Fig. 4a), a 2D non -centrosymmetri c CO phase (CO -2D) with 2:1 ferrimagnetic order \n(Fig. 3e) becomes the ground state.  At higher pressure, a 3D centro -symmetric CO phase (CO -HP) with \nantiferro -magnetic order (Fig. 3f) becomes the ground state. During the CO -AP to CO -2D phase transition, \nthe Fe-Fe bonds within [Fe 2O4] bilayer structure are compressed by pressure. Such compression allow s \ncharge transfer among low valence and high valence  Fe sites  in the frustrated lattice , which could result in \n \n \n7 \n damping of out -of-plane correlations and incommensurability in ab -plane (Fig. 1( b-d)). As the pressure \nfurther increases, the effective Coulomb interaction among Fe -Fe bonds reaches a critical point where the Q \n= (1/3, 1/3) charge orders cannot be sustained. Thus the system evolves into a less fr ustrated Q = (1/4, 1/4) \nCO-HP charge order where all Fe layers are charge neutral. Due to the strong coupling between charge and \nspin correlations, such drastic redistribution of charge density in turns causes substantial change in the \nmagnitude of spin -spin interactions. As a result, the magnetic moments on Fe2+ and Fe3+ are re -arranged and \nthe magnetic structure evolves from ferrimagnetism to anti -ferromagnetic order. The DFT calculation s and \nexperimental evidence indicate  that the observed charge orderin g and magnetic transitions are coupled and  \nstrongly correlated with the compression of  the crystalline lattice. Our calculation reveals that the enthalpies \nof the three phases (CO -AP, CO -2D, CO -HP) are close to each other (Fig. 4d). This is not surprising as the \ndouble -layer triangular structure of LuFe 2O4 induces strong frustration in both charge and spin exchange \ninteractions.  \nIn summary, we have conducted a systematic study on the evolution of the crystal structure, charge orders \nand spin orders in hexagonal LuFe 2O4 under pressure using the in -situ HP -SXD and HP -XMCD \nmeasurements. With increasing pressure, the system exhibits three correlated charge ordered ground states: \n1) the centrosymmetric 3D CO -AP phase with ferrimagnetic order, 2) the non -centrosymmetric CO -2D phase \nwith ferrimagnetic order, and 3) the centrosymmetric 3D CO -HP phase with antiferromagnetic order. These \nresults suggest strong coupling between charge and magnetic orders in hexagonal LuFe 2O4. The evolution of \nthe charge and magneti c order under pressure is further supported by the DFT calculations, which show that \nthe phase transitions are the result of tuned frustrated charge and spin interactions induced by the compression \nof triangular Fe 2O4 double layer structures. Our study pro ves that hydrostatic pressure is a powerful tool to \nunveil novel charge and magnetic states in hexagonal LuFe 2O4 and other candidate multiferroics , and also \nprovides new insights on realiz ing the potential charge -order induced multiferroicity in LuFe 2O4. \n \nMETHODS  \nSample Preparation  \nSingle crystals of LuFe 2O4 were grown by the floating -zone method using a CO/CO 2 (~ 2.7:1) mixed \natmosphere to control oxygen stoichiometry. Our electron probe micro analyzer (EPMA) measurement on \nthe single crystalline sample shows almost optimal stoichiometry of Lu 1.01(1) Fe2O3.97(4) . \nHigh -pressure measurement  \nSingle -crystal X -ray diffraction (SXD) experiment was conducted at synchrotron radiation beamline 13 -BM-\nC, Advanced Photon Source (APS), Argonne National Lab (ANL), using  monochromatic X-rays with 0.434 \nÅ wavelength, with the pressure increasing from 0.8 GPa up to 14.5 GPa. The 1 -deg step scan, wide -step \nscan and whole -range scan were performed with a scanning angle range of ± 35 degrees at each pressure \npoint31. Symmetry a nalysis was conducted using Sarah software32 and Bilbao Crystallographic Server33, and \nstructure refinements were conducted using FULLPROF program suite34. High -pressure synchrotron X -ray \nMagnetic Circular Dichroism (XMCD) spectroscopy measurements with th e energy scanning across the Fe \nK-edge were conducted at beamline 4 -ID-D, APS, ANL. Circularly polarized X -rays were generated with a \ndiamond phase retarder. To obtain the XMCD signal, the X -ray helicity was modulated at 13.1 Hz, and \nXMCD signals were dete cted with a phase lock -in amplifier. In addition, XMCD scans were repeated with \nopposite applied field direction to ensure artifact -free XMCD signals35. Corresponding to the three observed \nCO phases in pressurized LuFe 2O4, the XMCD data were collected at 1 .9 GPa, 4.3 GPa, and 9.5 GPa. In \nthese measurements, magnetic field was set to +5T/ -5T and temperature to 100 K. X -ray Absorption \nSpectroscopy (XAS) data is collected simultaneously during the XMCD measurement and obtained by \naveraging X -ray absorption for  opposite X -ray helicities. The pressure was determined by the in -situ ruby \nfluorescence measurement system36 with the pressure uncertainty less than 5%. An offline ruby system was \nused in the SXD measurement, before and after each manual pressure change a t ambient temperature. For  \n \n8 \n the low temperature XMCD measurements, online membrane and ruby fluorescence systems were used to \napply and measure pressure, respectively (See details  in supplementary Section S 1). \n \nDensity Functional Theory (DFT) calculations  \nThe first -principle density functional theory calculations are based on the projector augmented wave (PAW) \nmethod37 encoded in the Vienna ab initio simulation package (VASP)38. The exchange -correlation functional \nof the Perdew -Becke -Erzenh (PBE)39 form is adopted and the plane -wave cutoff energy is set to 500 eV. To \nproperly describe the strong electron correlation in the Fe 3 d states, the GGA plus on -site repulsion U method \n(GGA+ U)29 was employed with the effective U value ( Ueff = U - J) of 4 eV. Calculati ons with various Ueff \nshow that the main results remain valid when Ueff is varied between ~3.1 and 5.5 eV. The structural \noptimizations are carried out until the forces acting on atoms are smaller than 0.01 eVÅ-1. To obtain the \nenergy of each charge order structure, GGA+ U calculations were carried out in two steps30. For each charge \norder structure, we first optimize the chosen charge order structure using a large U (e.g., Ueff = 7.5 eV). Then \nwe re -optimize the charge order structure with a smaller U (i.e., Ueff = 4 eV) using the converged charge \ndensities obtained with the large U. It is noted that a larger Ueff was employed only to generate an initial \ncharge density with the desired charge order structure. The enthalpies of each charge order phase under \nvarious pressure are calculated by follow expression: H = U – PV, where H is enthalpy, U is total energy and \nPV is product of pressure and volume. 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Forman, Calibration of the pressure dependence of \nthe R1 ruby fluorescence line to 195 kbar. J. Appl . Phys . 46, 2774 -2780  (1975 ). \n37. P. E. Blochl, Projector augmented -wave method. Phys . Rev. B: Condens . Matter . 50, 17953 -17979  \n(1994 ). \n38. G. Kresse, J. Furthmuller, Efficient iterative schemes for ab initio total -energy calculations using a \nplane -wave basis set. Phys . Rev. B: Condens .  Matter .  54, 11169 -11186  (1996 ). \n39. J. P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple. Phys . \nRev. Lett. 77, 3865 -3868  (1996 ). \n  \n \n10 \n ACKNOWLEDGMENTS  \nThis work was supported by National Natural Science Foundation of China (Grant No U1930401, 12074071, \n51772184 and 11991060), and National Key Research Program of China (2016YFA0300702). We \nacknowledge Dr. Lili Zhang, Dr. Sheng Jiang and Dr. Aiguo Li of 15 U1 at SSRF, and Dr. Changyong Park \nat HPCAT, APS, ANL for the technical supports on the high pressure XRD experiment. Gas loading by \nSergey N. Tkachev is also acknowledged. HPCAT operations are supported by DOE -NNSA under Award \nNo. DE -NA0001974 and DOE -BES under Award No. DE -FG02 -99ER45775, with partial instrumentation \nfunding by NSF. The SXD measurement was conducted at 13BM -C, APS, ANL. 13BM -C operation is \nsupported by COMPRES through the Partnership for Extreme Crystallography (PX2) project, under NSF \nCooperative Agreement EAR -1661511 and by GSECARS under NSF EAR -1634415. The XMCD \nexperiment was conducted at beamline 4ID -D, APS, ANL. This research used resources of the Advanced \nPhoton Source, a U.S. Department of Energy (DOE) Office of Science User Facili ty operated for the DOE \nOffice of Science by Argonne National Laboratory under Contract No. DE -AC02 -06CH11357.  \n \nAUTHOR CONTRIBUTIONS  \nWenbin Wang, Jian Shen and Wenge Yang proposed and conceived this project. Fengliang Liu conducted \nthe high-pressure XRD an d XMCD experiments. Yiqing Hao, Fengliang Liu, Wenbin Wang, Jun Zhao, \nLifeng  Yin, and Wenge Yang analyzed the data. Jinyang Ni and Hongjun Xiang performed the DFT \ncalculations. Dr. Dongzhou Zhang supported the SXD measurement, Drs. Gilberto Fabbris and Dan iel Haskel \nsupported the XMCD measurement, Dr. Yongsheng Zhao offered help in the XMCD measurements. Drs. \nShaobo Cheng, Xiaoshan Xu and Xujie Lü  offers valuable discussion during this project. Fengliang Liu, \nYiqing Hao, Wenbin Wang, Wenge Yang and Jian She n wrote the original version of the manuscript, and \nreceived feedbacks from all authors.  Fengliang  Liu, Yiqing Hao  and Jinyang Ni  contributed equally  to this \nwork.  \n \nCONFLICT OF INTEREST  \nThe authors declare no conflict of interests.  \n \nADDITIONAL INFORMATION  \nSupplementary Information  \nSupplementary data associated with this article can be found, in the  supplementary files.  \n \n \n \n \n "    },    {        "title": "2401.04678v1.Microwave_magnetic_excitations_in_U_type_hexaferrite_Sr__4_CoZnFe___36__O___60___ceramics.pdf",        "content": "1 \n Microwave magnetic  excitations in  U-type hexaferrite  \nSr4CoZn Fe36O60 ceramics  \nM. Kempa ,1 V. Bovtun,1 D. Repček,1,2 J. Buršík,3 C. Kadlec ,1 S Kamba1 \n1Institute of Physics of the Czech Academy of Sciences , Na Slovance 2, Prague 182 00, Czech \nRepublic  \n2Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, \nBřehová 7, 115 19 Prague, Czech Republic  \n3Institute of Inorganic Chemistry, Czech Academy of Sciences, 250 68 Řež near Prague, Czech \nRepublic  \n \nAbstract.  \n \nMicrowave  (MW) spectra of ferrimagnetic U-hexaferrite Sr4CoZnFe 36O60 ceramics were \nstudied  using several  experimental techniques from 1 00 MHz to 50 GHz at temperatures \nbetween  10 and 390  K. They revealed 9 excitations , which exhibit remarkable temperature \ndependences  near the magnetic phase transition s at 14 5 K and 305 K . Some of them also change \nunder the application of  a weak bias magnetic  field at room temperature . Three lowest -\nfrequency modes seen between 200 MHz and 3 GHz were assigned to  the dynamics of the \nmagnetic domain s. The mode attributed to the natural ferromagnetic resonance exhibits a \ndramatic critical slowing -down from 25 GHz at 390 K to 5 GHz near the 305 K  phase transition , \nand again a hardening to ~11 GHz on further cooling.  The higher -frequency  excitations  are \nmost likely  spin wave (magnon) modes arising from  the complex ferrimagnetic structure of \nSr4CoZnFe 36O60. The high sensitivity of the MW spectra to the weak magnetic bias field H < \n700 Oe at room temperature is  shown to be caused  by the transformation of the polydomain \nmagnetic structure  in randomly  oriented ceramic grains  to a monodomain one . The MW \nresponse measured above 2 GHz using coplanar and microstrip line s with different \nelectromagnetic field distribution and sample coupling revealed the same excitations  with \nsimilar temperature and bias field dependences . It confirms the reliability of the results  and \nproves the effectiveness of the used MW techniques . \n \nCorresponding author.  \nE-mail address: kamba@fzu.cz  (S. Kamba)  2 \n 1. Introduction  \n \nFerrite materials , called hexaferrites , exhibit  a complicated ferrimagnetic arrangement with a \nhexagonal crystal structure that can be described as  stacking sequences of three basic blocks: \nS Me2Fe4O8, (spinel block),  where Me denotes divalent metal ion ; R [(Ba,Sr)Fe 6O11]2-, \n(hexagonal block ) and T (Ba,Sr) 2Fe8O14, (hexagonal block ). Depending on the stacking \nsequences of these blocks, hexaferrites are divided into six main types  [1,2]. Typical examples \nare M -type hexaferrite BaFe 12O19 (stacking sequence SR),  Y-type BaSrCoZnFe 11AlO 22 (ST), \nW-type SrZnCoFe 16O27 (SSR) , Z-type (Ba,Sr) 3Co2Fe24O41 (STSR) , X-type Ba 2Co2Fe28O46 \n(SRS*S*R) and U -type Sr4Co2Fe36O60 (SRS*R*S*T), where the (*) symbol means that the \ncorresponding block has been turned 180  around the hexagonal  c-axis. Due to their unique \nmagnetic properties and economic availability, M -hexaferrites in particular are often used in \nengineering practice , e.g., in refrigerator seal gaskets, microphones, loudspeakers , small \nmotors, clocks, magnetic separators and in microwave generators, filters and absorbers [ 3,4].  \n Hexaferrites are also very interesting from the scientific point of view. Nearly twenty \nyears ago, Kimura found that the conical magnetic structure in Y -hexaferrite induces a \nferroelectric polarization, and that such a multiferroic system exhibits strong magnetoelectric \ncoupling  [5]. Similar spin -induced  ferroelectricity has been uncovered in a variety of materials \nwith different hexaferrite crystal structures, which often exhibit  strong magnetoelectric \ncoupling not only at heliu m temperatures , but also at  and above  room temperature  (RT)  [6,7,8]. \nDepending on the chemical composition, these materials are ferrimagnetic up to relatively high \ntemperatures of 600 -750 K and generally exhibit a set of magnetic phase transitions upon \ncooling from collinear structures to various conical ones, among  which the transverse and \nalternat ing longitudinal conical structures are multiferroic  [1]. Static magnetoelectric coupling \nhas so far been studied mainly in hexaferrites with Y - and Z -crystal structures , but \nmulti ferroicity has been  reported also in some hexaferrites with M -, W-and U -hexaferrite \nstructures  [9,10,11,12]. Interestingly, despite the chemical and structural affinities of different \nhexaferrites, the magnetoelectric coupling in Y - and Z -hexaferrite arises from  different origins : \ninverse Dzyaloshinskii -Moriya interaction (  Si  Sj) is responsible for the static \nmagneto electric  coupling in the Y -type hexaferrite , while additional p-d hybridization [  (ei  \nSj)2ei ; here Si and ei denote i-th spin and the bond direction, respectively] becomes dominant \nin the Z -type hexaferrite  [13]. 3 \n  A dynamic magnetoelectric coupling  can be responsible for the activation of spin \nexcitations is microwave (MW) and terahertz (THz)  dielectric spectra. Such excitations are \ncalled electromagnons and in hexaferrites they are activated, regardless of the type of static \nmagnetoelectric coupling, by exchange striction  ( Si ∙ Sj) [14,15,16,17,18,19]. One of the \ncharacteristic features of an electromagn on is a directional dichroism : it means  that the reversal \nof the light incidence direction gives  a different electromagnon absorption  [20]. Such an effect \nhas mainly been observed in the THz region but has recently been discovered also in the MW \nregion for Y hexaferrite  [21]. In this case the sign of nonreciprocal MW response could be \ncontrolled by the poling electric field , which opens a new avenue for practical applications in \nmagnonic s and other future wireless communication technologies.  This motivated us to study \nthe microwave properties of Sr4CoZnFe 36O60 (CoZnU)  with U-type hexaferrite  structure . \nThe microwave dielectric  permittivity  and magnetic permeability  of various materials \nwith U -hexaferrite structure have been published more frequently, but  practically only at RT \nand in a limited frequency range up to 6, 12 or maximum 18 GHz  [22,23,24,25]. In our paper, \nwe focus on CoZnU , which is chemically and structurally related to multiferroic Sr4Co2Fe36O60 \n(trigonal space group 𝑅3̅𝑚) [26]. Very recently , we reported that it exhibits  a transition from \na paramagnetic to a collinear ferrimagnetic structure  at Tc1 = 635 K  [27]. At Tc2 = 305 K , it \ntrans forms  into a first conical magnetic structure and at Tc3 = 145 K  most probably in to a second  \none, yet not clearly identified [27]. Dielectric studies did not reveal any anomalies near \nmagnetic phase transitions  and magnetoelectric measurement did not find any intrinsic \nferroelectric polarization at 10 K  [27]. It means that CoZnU  is not multiferroic.  Here we report \nthe results of  our MW investigations  performed up to 50 GHz  and down to 100 K,  which reveal \n9 magnetic excitations, including a ferromagnetic resonance. Most of these excitations show \nstrong temperature dependences near the magnetic phase transitions  and we will discuss their  \norigin . \n \n \n2. Experimental  details  \n \nA detailed description of the CoZnU ceramic s preparation , structure  and characterization of its \nstatic and dynamic dielectric and magnetic properties from 10 to 800 K (measured up to 3 GHz ) \ncan be found  in ref.  [27]. In this report, t he MW spectroscopy study was performed using two \nkinds of planar lines : a microstrip line (MSL) and a grounded coplanar waveguide (CPW) with 4 \n different electromagnetic field distribution and coupling between the sample and the line. In \nboth the MSL and the CPW, the propagation of the only quasi -TEM mode with transversal \nelectric and magnetic components of the MW electromagnetic field was ensured [28]. The full \nset of the scattering  S-parameters  (S11, S22, S21, S12, both magnitude and phase) [29] spectra  of \nthe Southwest Microwave MSL  and CPW  50 GHz test boards [30] loaded with the parallel \nplate samples ( ~ 1 mm thick, ~ 100 mm2 surface ) at their top were recorded  in the transmission  \nsetup between 100 MHz  and 50 GHz on heating from  10 K to 390 K in a Janis closed -cycle He \ncryostat with a temperature rate of 0.5 K /min using the Agilent E8364B network analyzer.  The \ninfluence of t he bias magnetic field H (up to  700 Oe) on the  S-parameters spectra , reflectio n \nloss, magnetic nonlinearity and magneto -impedance effect were studied at RT in the MSL  and \nCPW setups . Reflectio n loss (S11 parameter in the reflect ion setup [ 29]) spectra were measured \nbetween 10 0 MHz and 20 GHz using the “short” calibration standard.  \nThe resonance MW measurements were performed using a) the thin cylindrical sample \n(~0.22 mm thick, ~12 mm in diameter) as a dielectric resonator  (DR)  with a few resonance \nfrequencies  [31] between 8 and 16 GHz and b) a composite dielectric resonator  (CDR)  with \nthe same sample  [31,32,33] as a part of the CDR at TE01δ resonance  near 5.8 GHz . The \nresonators were measured  in a cylindrical shielding cavity using the transmission setup  with a \nweak coupling by an Agilent E8364B network analyzer  during heating  from 10 to 390 K with \na temperature rate of 0.5 K/min in a Janis  closed cycle He cryostat.  In the case (a) , the S21 \n(transmission coefficient) was recorded. In the case (b), the TE01δ resonance frequency,  quality \nfactor, and insertion loss of the base cylindrical dielectric  resonator with and without sample \nwere recorded . The effective refractive index and loss tangent  of the samples were calculated \nfrom the measured parameters of the DR and the CDR  using the MATLAB -based software \n[31], and calculations accounting for volume fractions of the CDR parts  were performed  \n[32,33]. \n A custom -made time -domain THz -transmission spectrometer was used to determine \nthe complex refractive index  of a 0.22 mm thick ceramics from 0.3 to 1.7 THz . The d etailed \ndescription  of the THz spectrometer and the principle of  the time-domain THz measurement \ncan be found  elsewhere  [34]. \n \n \n  5 \n 3. Broadband m icrowave  transmission and absorption . \n \nThe recorded full set of S-parameters  of the MSL  and the CPW  loaded with CoZnU  ceramics \nwas analyzed (see  Fig. S1 and  details in Supplement ary Information ). As the most sensitive \nand informative, the following parameters were selected and calculated  [29,35,36]: S21 \nmagnitude  in dB (transmission coefficient Tr), input impedance Zin and absorption coefficient \nA. Tr is the most reliable  directly measured S-parameter.  Its recorded  spectra  with multiple \nresonances  are similar for the CPW and MSL experiments and  show qualitatively similar \ntemperature changes.  This similarity is well demonstrated by the comparison of  the Tr(f,T) \ntemperature -frequency  maps ( Fig. 1 ). In more detail,  the temperature evoluti on of b roadband \nTr(f) spectra  is presented  in Fig. 2  for the CPW , and in Fig. S2 of Supplementary Information  \nfor the MSL.  \n \n \nFig. 1. Temperature -frequency  maps  of the transmission coefficient  Tr (S21 magnitude)  of \nCoZnU  ceramic s obtained using MSL (a) and CPW  (b). Points correspond to the main \nTr(f) minima shown in Fig. 2 for CPW , and in Fig. S2 of Supplementary Information  \nfor MSL.  Temperatures of the phase transitions are marked  by dash ed lines . Apparent \njumps of the Tr at several temperatures are instrumental artifacts.  \n   \n6 \n  \nFig. 2. Broadband s pectra  of the transmission coefficient Tr (S21 magnitude)  of the CPW loaded \nwith a CoZnU  sample  at selected temperatures below Tc3 (a), between Tc3 and Tc2 (b), \nand above Tc2 temperature (c, d). The main Tr(f) minima are marked  T3÷T9. \n We limited our analysis to the frequency range 1÷35 GHz because of a too high density \nof the observed resonances and noise level of the Tr(f) spectra above 35 GHz  (see Fig. S1 in \nSupplementary Information ). Below 35 GHz, 7 well-defined resonance -like or diffuse Tr(f) \nminima were selected and numbered from T3 to T9 with increasing frequency  (here we \ncontinue in the numbering of the excitations seen below 3 GHz in the complex magnetic \npermeabilit y 𝜇∗= 𝜇′+𝑖𝜇\" published in [27]). These minima correspond to the MW \nexcitations ( transmission modes, T -modes ) of the MW line loaded with a CoZnU  sample . The \nsame set of T-modes is observed in both the CPW and MSL measurement of  the same CoZnU \nsample. Moreover, the temperature evolution of the CPW and MSL T -modes is similar, with a \nremarkable c hange of their behavior near the  CoZnU magnetic  phase transitions . The T7, T8 \nand T 9 modes are well separated below Tc2, but bec ome overlapped above Tc2, forming one \nT(7-9) mode. The T4 and the T5 modes bec ome overlapped near Tc2. Also, the T4 and the T6 \nmodes become overlapped above Tc2. In spite of the overlapping, the F4 mode dominates above \nTc2. Two modes, the T4 and the T7, are characterized by a very pronounced temperature \ndependence.  \n-16-14-12-10-8-6-4-20\n-8-6-4-20\n(c) T9T3\nT5 T4T6\nT(7+8)T4T4\n 311 K\n 314 K\n 317 K\n 320 K\n 323 K\n0 10 20 30-25-20-15-10-50\n(d)T(7−9)T3\nT5\nFrequency (GHz)Tr = S21 magnitude (dB)\nT4T4 325 K\n 350 K\n 370 K\n 390 KT9(a)T8T5Tr = S21 magnitude (dB)T3\nT6\nT7 10 K\n 50 K\n 100 K\n 125 K\n0 10 20 30-14-12-10-8-6-4-20\n(b)T8T9T3\nFrequency (GHz)T5\nT7T6\nT4 150 K\n 175 K\n 200 K\n 225 K\n 250 K\n 275 K\n 300 KT47 \n  The a nalysis of the  transmission coefficient  spectra  Tr(f) allowed us to observe, reliably \nseparate and study 7 MW excitations in the CoZnU hexaferrite. For a better understanding of \ntheir nature, the analysis of the absorption coefficient spectra A(f) could be  very useful. The \nA(f) spectrum  characterizes the absorption of the electromagnetic energy in a sample. \nConsequently, A(f) maxima characterize the frequency ranges of the increased absorption of \nelectromagnetic energy due to the interaction with the excitations in the studied material. While  \nthe transmission coefficient  Tr characterizes the electromagnetic energy which was not \nabsorbed in a sample and not reflected from the sample, the frequencies of the Tr(f) minima \nare defined by the superposition of the reflection and absorption  maxima. Only in the case \nwhere the absorption prevails, the frequencies of the Tr(f) minima correspond to  those of  the \nA(f) maxima , both being related  to frequencies of the material loss maxima  and could be \nreasonably used for the physical interpretation of the MW excitation , for the study of their \nmicroscopic origin . The c alculated  (see Supplementary Information)  A(f) spectra are shown for \nthe CPW experiment in Fig . 3 and for the MSL experiment in  Fig. S3 in Supplementary \nInformation . \n \nFig. 3. Broadband s pectra  of the absorption coefficient A of the CPW loaded with a CoZnU  \nceramics  at selected temperatures below Tc3 (a), between Tc3 and Tc2 (b), and above Tc2 \n(c, d). The main A(f) maxima are marked  A3÷A9.  \n-60-50-40-30-20-10\n-60-50-40-30-20-10(c)\nA9\nA4\nA5A(7+8)\nA6 A4\nA3\n 311 K\n 314 K\n 317 K\n 320 K\n 323 KA5\n10 20 30-30-25-20-15-10-50\n(d)A6A4A9A5A(7+8)A4A4\nFrequency (GHz)A (dB)\n 325 K\n 350 K\n 370 K\n 390 KA9\n(a)A8A (dB)A3A6A7  10 K\n 50 K\n 100 K\n 125 K\n10 20 30-60-50-40-30-20-100\n(b)A8A9A3\nFrequency (GHz)A7\nA6\nA4 150 K\n 175 K\n 200 K\n 225 K\n 250 K\n 275 K\n 300 KA48 \n  \nThe absorption spectra are noisier than the transmission ones, nevertheless 7 diffuse \nA(f) maxima can be recognized below 35 GHz and numbered from A3 to A9 with increasing \nfrequency. These maxima evidence 7 MW excitations ( absorption modes , A-modes ) of the \nCoZnU  hexaferrite.  The same set of the A -modes and their similar temperature evolution were \nderived from both CPW and MSL experiments. The A -mode system corresponds to the T -\nmode one. The f requencies of the A -modes remarkably c hange near the  magnetic  phase \ntransitions , like th ose of the T-modes. Therefore, we consider all observed modes (T3÷T 9, \nA3÷A 9) to be related to the electromagnetic properties of the CoZnU hexaferrite, and their \nfrequencies  characterize the  mean frequencies of the MW excitations (F3÷F9) of the CoZnU \nhexaferrite . We also analyzed the input impedance spectra Zin(f), see Fig. S4  in Supplementary \nInformation . The main Zin(f) maxima  also correspond to  the frequencies of  the F3÷F9 \nexcitations . The temperature evolution of the F3 ÷F9 excitations is presented in Fig. 4.  \nThe transmission spectra of the thin cylindrical CoZnU sample, measured as a \ndielectric resonator  (DR) in a shielding cavity [ 29,31], show a few temperature -dependent \nresonances with an anomalous behavior near Tc2 = 305 K (see t emperature -frequency  maps in \nFig.S5 of Supplementary Information ). This confirms critical dynamics of the MW excitations \nin the CoZnU ceramics.  \nAn essential fact is that independently of the used measurement setup (CPW, MSL or \nDR) and of the type of coupling, the main resonance modes in transmission, absorption or \nimpedance MW spectra exhibit  a similar temperature behavior with anomalies near the phase \ntransitions.  This shows that the MW excitations seen in Fig. 4 are indeed genuine  in the CoZnU \nceramics  and we will explain their origin in the next section.  \n 9 \n  \nFig. 4. Temperature  evolution of the mean frequencies of the MW excitations F3 ÷F9 \ncorresponding to Tr(f) minima and A(f) maxima of the CPW (line s with filled symbols \nand crosses, respectively ) and MSL (empty symbols) loaded with a CoZnU  sample . The \nmean frequencies of the magnetic excitations F1 and F2, corresponding to the µ”(f) \nmaxima  from coaxial measurements below 3 GHz and published in [27], are also added.  \n \n \n0 100 200 300 40005101520253035\nF9\nF8\n CPW-T\n CPW-A\n MSL-T\n \"\n \"Tc3Tc2\nF1F2F3F3F5F5F6\nTemperature  (K)Frequency of Tr(f) minima, A(f) and \"(f) maxima (GHz)\nF4F4F7F(7-9)10 \n 4. Temperature evolution of the microwave magnetic  excitations.  \n \nNo anomalies of the complex dielectric permittivity 𝜀∗= 𝜀′−𝑖𝜀\" (measured between 1 Hz \nand 1 GHz)  were observed near the ferrimagnetic  phase transition s at Tc3 = 145 K  and Tc2 = \n305 K [27]. It is true that at low frequencies a colossal permittivity was observed  in CoZnU  \n[27], which could mask the dielectric anomaly at a possible spin-induced ferroelectric phase \ntransition, but this  colossal  permittivity is due to the Maxwell -Wagner relaxation s from \ninhomogeneous conductivity in the CoZnU ceramic . Therefore,  the permittivity decreases with \nincreasing frequency , and at 100 MHz or 1 GHz it already takes intrinsic values of 20 -30 (see \nFig. 5a). Thus, if a spin -induced ferroelectric phase were to form at Tc2 or Tc3, the associated \ndielectric anomaly should be observed. The absence of such an anomaly , together with the fact \nthat no intrinsic ferroelectric polarization is observed in [27], suggests that CoZnU is not a \nmultiferroic but only a paraelectric ferrimagnet  at all temperatures below Tc1. \nOn the other hand, temperature dependences of the permeability μ’(T) and loss μ’’(T) \nat 100 MHz and 1 GHz are characterized by the remarkable maxima at Tc3 and Tc2. Below 3 \nGHz, we separately measured the permittivity and the permeability  using coaxial techniques \n[27]. At higher frequencies , the dielectric and magnetic parameters cannot simply be \ndistinguished [31]. Our MW measurements, using the MSL, CPW and DR techniques, as well \nas the THz one, characterize the  full response of the studied material on both electric and \nmagnetic components of the applied electromagnetic field, which is dependent on both \nmaterial’s permittivity and permeability . Our MSL and CPW techniques revealed the \ntemperature evolution of the main MW excitation  in the material, but they  do not allow the \nquantitative estimati on of the dielectric and magnetic parameters. The e xperimental data of  the \nDR and THz techniques can be analyzed  [32,33] using the following mixed electromagnetic \nparameters:  the product  𝜀′𝜇′ as a parameter characterizing the reduction of the wave length, and \n𝜀′′𝜇′′/𝜀′𝜇′ as a parameter characterizing  electromagnetic loss es (loss  tangent ) in the material. \nThese parameters have the meaning of the square  of the refracti ve index and of the index of \nabsorption  [37]. We also calculated 𝜀′𝜇′ and 𝜀′′𝜇′′/𝜀′𝜇′ parameters for 1 GHz (using \npermittivity and permeability values from coaxial measurements [ 27]) and show all of them in \nFig. 5b,c. \n 11 \n  \nFig. 5. Temperature dependenc es of the  (a) 𝜀′ (lines) and permeability 𝜇′ (symbols) at 100 MHz \nand 1 GHz; (b) and (c)  𝜀′𝜇′ and 𝜀′′𝜇′′/𝜀′𝜇′ electromagnetic parameters at 1, 5.8 and 300 \nGHz  respectively.  \n \nThe c omparison of the 1 GHz  𝜀′𝜇′ parameter (Fig. 5b) with 𝜀′ and  𝜇′ (Fig. 5a) proves \nthat the 𝜀′𝜇′ maxima at Tc3 = 145 K  and Tc2 = 305 K are caused by the μ’(T) anomalies.  The \nabsence of the dielectric loss 𝜀′′ maxima in the 1 GHz temperature dependence  [27] evidences  \nan exclusive magnetic nature of the 𝜀′′𝜇′′/𝜀′𝜇′ loss maxima in the 1 GHz curve (Fig. 5c).  Since \nthe 300 GHz  𝜀′𝜇′ value is nearly equal to the 1 GHz  𝜀′, we can conclude that  𝜇′ ≈ 1 at 300 GHz \nand no essential dielectric dispersion takes place in the MW range between 1 and 300 GHz , at \nleast below 300 K . So, the frequency change of the electromagnetic parameters (Fig. 5b,c) is \ncaused by  the only MW magnetic dispersion, which is also observed between 1 MHz and 1 \nGHz [ 27]. \nNote that 𝜀′𝜇′(𝑇) and 𝜀′′𝜇′′/𝜀′𝜇′(𝑇) dependences at 5.8 GHz show sharp anomalies \nnear Tc2. The 𝜀′𝜇′(𝑇) minimum (in fact, the 𝜇′(𝑇) minimum ) could be explained by the critical \nslowing down of the mean F4 frequency (Fig. 4) below the frequency of experiment , like that \nof the soft  mode in  ferroelectrics [ 38]. In the collinear phase above Tc2, at temperatures where \nF4 > 5.8 GHz, 𝜀′𝜇′ values  at 1 and 5.8 GHz are nearly equal, so the MW magnetic dispersion \ntakes place mainly above 5.8 GHz. In the conical phases below Tc2, the higher level of the 5.8 \n110   '\n 100 MHz\n    1 GHz   '\n 100 MHz\n 1 GHz(a)\n', '\n0 100 200 300 4000.00.40.81.2(c)\nTemperature  (K)\"\"/''204060 (b)Tc2 Tc3\n  1 GHz\n  5.8 GHz\n  300 GHz''12 \n GHz parameters in comparison with  the 300 GHz ones also evidences MW magnetic dispersion \neven above 5.8 GHz . It correlates with  the presence of the magnetic excitations (magnons) in \nthe THz range  (see Fig. S6 in Supplementary Information ). \nWe assume that all MW excitations  in Fig. 4 are related to the spin dynamics in CoZnU \nhexaferrites . MW excitations F4  and F7 with a very pronounce d temperature dependence are \nwell observed in both MSL and CPW (the A -modes are shown in Fig. 4 only for CPW to \npreserve the figure clarity). Above Tc2, the F 7, F8  and F 9 excitations join together into  the F(7-\n9) one. The F 6 excitation is well resolved only below and near Tc2, and absent above 315 K . It \nis characterized by the we ak temperature dependence. The F 5 excitation is better seen as a \ntransmission mode below Tc2 and as an absorption mode above Tc2. It is characterized by a \nweak temperature dependence  below Tc2, but its frequency essentially increases with \ntemperature above Tc2, similar to that of F4. The F3 excitation is well resolved only above 120 \nK, and its  mean frequency increases from the 2 GHz level up to 5 GHz at temperatures near \nboth phase transitions ( Tc2 and Tc3). Let us note, that the F3 mean frequency is close to the \nfrequency of the F 3 excitation observed at RT in the coaxial experiment [27], so we consider \nthem as the same excitation.  \nThe F4 mean frequency critically decreases towards Tc2. We compare its temperature \ndependence with that of MW inverse permeability at 1 GHz which obeys the Curie -Weiss law \nabove Tc2: \n1\n𝜇′(𝑇)⁄ =1\n(𝜇𝐿+𝐶\n𝑇−𝜃)⁄ ,    (1) \nwith parameters: μL = 1.07, C = 8 K and θ = 305 K [ 27]. The temperature dependences of the \nF4 mean frequency from the CPW experiment (CPW -T in Fig. 4) and of the inverse 1 GHz \npermeability are shown together in Fig. 6. The F4 mean frequency is proportional  to the inverse \n1 GHz permeability with a constant factor of 25 GHz at temperatures above Tc2. It means, that \ndespite  the permeability dispersion above 1 GHz, the μ’(T) Curie -Weiss law remains valid up \nto ~22 GHz  at high temperatures . For that reason,  we can claim that the temperature \ndependence of the 1 GHz permeability is mainly caused by the temperature change of  the F4 \nfrequency. We can therefore assign the F4 excitation to the  critical magnetic excitation , which \nshould  correspond to the  ferromagnetic resonance  (FMR) in zero bias magnetic field  (i. e. \nnatural ferromagnetic  resonance) . Other two ferrimagnets with U -hexaferrite structure, Ba 4Co2-\n3xCr2x Fe36O60 and Ba 4 Zn2-xCoxFe36O60, have at RT FMR frequency near 8 GHz,  [25] and 9.4 \nGHz, respectively ( the latter  one in external magnetic field of 0.3 T)  [39]. It supports our \nsuggestion  that FMR in CoZnU corresponds to  the F4 mode.  13 \n  \n \nFig. 6. Temperature dependenc es of the  inverse  1 GHz magnetic permeability with its Curie -\nWeiss  fit (Eq. 1) above 305 K and  eigenfrequency  of the  F4 magnetic excitation  \ncorresponding to FMR . \n \nBased on the  analysis  described above, we can propose an explanation for the origin  \nof the observed MW magnetic modes . The FMR mode ( F4) is the lowest frequency mode  of \nthe complex ferrimagnetic excitation spectrum  of individual grains . The higher frequency \nmodes (F 5 – F9) are other spin wave  (magnon) excitations  of the exchange origin.  [40,41,42]. \nThe h igh number of the  magnon excitations is related to  the complex magnetic structure of \nCoZnU  consisting of (SRS*R*S*T )3 magnetic blocks  (note  that CoZnU  contains  three formula \nunits with a total of 108 Fe and 3 Co magnetic cations ) [26]. It is responsible for the formation \nof many magnon branches in the Brillouin zone [ 42] and some of the magnons from the \nBrillouin zone them can be activated in MW and THz spectra . The c hange of the magnetic \nstructure at the transition from the conical to the collinear phase modifies the evolution of the \nmagnon s frequencies  with temperature  and in some cases even induces their seeming  splitting. \nThis can explain the observation of the F7, F8 and F9 modes in the conical phase  below Tc2, \nwhich overlay in the collinear  phase  above Tc2 (Figs. 1 and 4 ). \n0 100 200 300 4000.20.40.60.81.0\nTc2     experiment\n 1/', 1 GHz\n F4, CPW\n    Curie-Weiss fit\n   1 GHz, 305 K1/'\nTemperature  (K)Tc3510152025\nF4 (GHz)14 \n The lower frequency modes (F1 – F3) are attributed to dynamics of the \ninhomogeneous magnetic structure of the non -magnetized hexaferrite ceramics with various \norientation of the local magnetic moments of grains and domains, with the presence of grain \nboundaries and domain walls  and related magnetization mechanisms  [27]. \n \n \n5. Magnetic nonlinearity of the microwave excitations.  \n \nEssential influence of the weak magnetic bias field H on the RT permeability dispersion , \nmeasured in the coaxial setup below 3 GHz, and the bias field dependences of the F1, F2 and \nF3 mean frequencies responsible for this dispersion were reported in Ref. [27]. It motivated  us \nto study the  bias field influence on the MW magnetic excitations above 3 GHz, i.e., on the  \nmagnon  modes F4 – F9. The b roadband s pectra  of the RT transmission of the CPW and MSL \nloaded with a CoZnU  sample  at different magnetic bias fields  (Fig. 7) evidence the main \nevolution of the F4 – F9 modes with increasing H. The  F4 Tr(f) minimum near 6 GHz,  \ncorresponding to the FMR, changes non -monotonically. With increasing H, the minimum first \nbecomes sharper,  and its depth achieves a maximum of ~10 dB at ~300 Oe , then it broadens \nand its depth decreases (see also Fig. 8). The F4 Tr(f) minimum  is asymmetric at low H and \nbecomes symmetric at H > 400 Oe. It can be related to the presence of the F3 contribution at \nlower frequencies , which reduces with increasing H. The F4 frequency non -monotonically \ndecreases with increasing H, with a local maximum at ~300 Oe corresponding to the depth \nminimum (Fig. 8).  Non-monotonic behavior of the FMR minimum could be related to the \nevolution of the polydomain structure of the CoZnU ceramics under application of the bias \nfield.  \nThe field H > 400 Oe  essentially suppresses all MW modes above 8 GHz, i.e., F5 – F9, \nboth in CPW and MSL (see Fig. 7). The F5÷F9 modes in MSL are much more pronounced \nthan in CPW (due to stronger coupling of the CoZnU  sample with MSL), nevertheless, \nrelatively we ak H = 400 Oe reduces the depth of their minima down to the 4÷5 dB level, similar \nto that in CPW. Moreover, the peak system of the F 7, F8 and F 9 modes becomes more  damped  \nand finally these peaks  merge above 300 Oe. This behavior reminds the ir temperature evolution \non heating above the phase transition at Tc2 = 305 K (see Fig. 4 ). Such a high sensitivity of the \nMW modes to low bias could be related to the low (hundreds of Oe) in -plane anisotropy field \ninducing rotation of the magnetization in the c -plane of hexaferrites [ 43]. \n 15 \n   \nFig. 7. RT transmission coefficient Tr spectra  of the CPW (a) and MSL (b) loaded with a \nCoZnU  sample  at different magnetic bias fields. The main Tr(f) minima correspond to \nthe MW excitations F3 -F9. Note the log frequency scale.  \nFig. 8. RT bias magnetic field dependences of frequencies (a) and depth s (b) of the Tr(f) minima \ncorresponding to the MW excitations F4, F 7 and F 9 in CPW or MSL  loaded with the \nCoZnU  sample . \n \nThe RT reflection loss  (RL) spectra of the CPW loaded with a CoZnU  sample  evidence \nthe splitting of the F4 mode  into two components  (F4_1 and F4_2)  under application of the \nweak bias H field (Fig. 9), which is seen both in RL magnitude and phase. The 15 dB deep \ndiffuse F4 RL(f) minimum becomes sharp and 37 dB deep at H = 140 Oe, then splits into the \n35 dB deep F4_1 and 28 dB deep F4_2 sharp minima at H = 280 Oe. At H = 420 Oe, the F4_1 \nminimum reduces to 20 dB, while the F4_2 minimum sharpens, and its depth increases to 35 \ndB. A subsequent increase of the bias field leads to the depth decreas ing and widening of both \nminima. The e volution of the F4 minimum frequency and depth is shown in Fig. 10.  The \nsplitting and the frequency change of the F4 mode is related to the evolution of the polydomain \nmagnetic structure of the polycrystall ine CoZnU ceramics  into the magnetic monodomain s. \n \n-8-6-4-20\n5 10 15202530-30-20-100F7F(7-9)\n(a)CPWF3 F6F5\nF9F8\nF4Tr = S21 magnitude (dB)\n(b)F7F(7-9)F6\nFrequency (GHz)F9F8\nF6F5\nF4F3\n 0 Oe\n 140 Oe\n 280 Oe\n 420 Oe\n 560 Oe\n 700 OeMSL\n6.16.36.5283032\n0100 200 300 400 500 600 700-30-25-20-10-8-6-4(a)\n F4_MSL\n F7_CPW\n F9_MSLFrequency (GHz)\n(b)Depth of Tr(f) minimum (dB)\nBias magnetic field H (Oe)16 \n  \nFig. 9. RT  reflectio n loss RL magnitude (a)  and phase (b) spectra  below 10 GHz of the CPW \nloaded with the CoZnU  sample  at different magnetic bias fields. The RL magnitude  \nminima correspond to the MW modes F3 , F4 and F5 . \nFig. 10. RT bias magnetic field dependences of the frequencies (a) and depth (b) of the RL \nmagnitude  minima corresponding to the MW modes F4 and F3 in the CPW loaded with \na CoZnU  sample . \n \nThe RL spectra also confirm the presence of the F3 excitation near 4 GHz (Fig. 9 ) and \nits dependence on the applied bias field ( Fig. 10 ). The F3 is seen in the Tr(f) spectra (Fig. 7) , \nbut only as a contribution to the asymmetry of the diffuse F4 minimum. In the RL spectra, the \nF3 minimum is well separated from the F4 one. The a pplication of the bias field suppresses F3 \nand reduces the depth of its RL minimum from 6 dB to 2.5 dB ( Fig 10b ).  \nLet us note, that  the RL magnitude is used for the characterization of the electromagnetic \nMW absorption ( MA) of materials [ 29,35,36]: MA = -RL (dB). So,  the RL(f) minima correspond \nto the MA(f) maxima and, similar to the absorption coefficient  A(f) maxima in the transmission \nexperiment ( Fig. 3 ), they are related  to the loss maxima  in studied materials.  The c omparison \nof the room -temperature RL(f) minima ( Fig. 9a ) at zero-bias field and of 275 - 300 K A(f) \nmaxima ( Fig. 3b ) allows to explain the very asymmetric shape of the A(f) maxima by the loss \n-30-20-100\n2 4 6 8 10-180-120-60060120180F5\n(a)F3\nF4_1F4\nF4_2     0 Oe\n 140 Oe\n 280 Oe\n 420 Oe\n 560 Oe\n 700 OeRL magnitude (dB)\nF5\n(b)\nFrequency (GHz)RL phase (deg)F3\nF4_1F4_2\nFrequency (GHz)F4\n3.803.855.56.06.57.0\n0100 200 300 400 500 600 700-35-30-25-20-15-6-4(a)\n F3\n F4_1\n F4_2Frequency (GHz)\n(b)Depth of RL(f) minimum (dB)\nBias magnetic field H (Oe)17 \n contributions from both F4 and F3 excitations. The loss d ependence on the applied bias field \nproves the magnetic nature of the F3 and F4 excitations.  \nFor quantitative analysis  of the CoZnU magnetic nonlinearity , we use a simple method  \ndeveloped for the description of the g iant magnetoimpedance  effect  [44], i.e. comparison of \nthe H-dependent  parameters ( Zin) with th ose under the maximal bias field Hmax applied in our \nexperiment:  \n  ∆𝑍𝑖𝑛\n𝑍𝑖𝑛(%)=100% ∗[𝑍𝑖𝑛(𝐻)−𝑍𝑖𝑛(𝐻𝑚𝑎𝑥 )]\n𝑍𝑖𝑛(𝐻𝑚𝑎𝑥 )                                       (2) \n \nFig. 11. Broadband s pectra  of the input impedance Zin of the CPW loaded with the CoZnU  \nsample  at different magnetic bias fields  (a) and magnetoimpedance  effect  (b) measured \nat RT. The main Zin(f) maxima Z3÷Z9 correspond to the MW excitations F3  – F9. \n \nThe b roadband s pectra  of the input impedance Zin of the CPW loaded with the CoZnU  \nsample  are shown in Fig. 11a.  They present remarkable dependence on the bias magnetic field \nin the whole studied frequency ra nge, from 100 MHz to 35 GHz, especially near Zin(f) maxima \ncorresponding to the MW modes F3-F9. The i ncreas e of the  bias magnetic  field suppresses the \nF3 and F6 maxima, nonmonotonically influences the F4 and F7 maxima , and results in increase \n6080100120 (a)\nF7F(8+9)\nF9F8F6\nF5\nF4     0 Oe\n 140 Oe\n 280 Oe\n 420 Oe\n 560 Oe\n 700 OeZin  (Ohm)F3\n0 10 20 30-80-60-40-2002040 F9F4\n(b)F8\nF6F7Zin/Zin (%)\nFrequency (GHz)18 \n and emergence of the F8 and F9 maxima. The magnetoimpedance  effect (Fig. 11b) is the most \npronounced near F4 (> 20 %), F6 (> 70 %) and F8 -F9 maxima  (± 50 %) . Generally, the \nobserved magnetoimpedance  effect proves the magnetic nature of the main MW modes \nbetween 2 and 35 GHz.  \nIn order to understand  why the  MW response of the CoZnU ceramics  (transmission \nspectra, reflection loss spectra and input impedance ) essentially change s with increasing \ntemperature or under application of a weak magnetic bias field H < 700 Oe  at RT, let us analyze \nmagnetization  (discussed in detail  in [27]) in a weak  field. The temperature dependence of the \nweak -field magnetization MW(T) at H = 100 Oe ( Fig. 1 2a) shows a sharp and substantial \nanomaly at the phase transition  from the conical to the collinear phase [ 27] at Tc2 = 305 K , i.e., \nclose to RT. The main change s of the magnetization M(H) curves at RT and below  it, i.e., in \nthe conical magnetic phase s, take place in a weak field H < 700 Oe and have similar step -like \nshape  and value  (Fig. 1 2b). This step corresponds  to the weak -field magnetization MW [27]. \nThe s hape of the  M(H) curve at H < 700 Oe in the collinear phase, at 348 K, is not step -like; \nthe MW(H) contribution is smaller. Such MW(H) temperature evolution corresponds to the \nMW(T) curve in Fig. 1 2a. The weak -field magnetization MW and its temperature or field \nevolution is attributed to the transformation of the polydomain magnetic structure to the \nmonodomain one [ 27]. At RT, i.e., near Tc2, the transformation from the polydomain conical \nstructure  to the polydomain collinear one with increasing weak bias field can also contribute \nto MW. Beside MW, the high -field MH contribution is seen at H = 2-7 kOe in the conical phase \nbelow Tc2 = 305 K ( Fig. 12b ), which is attributed to the field -induced transition from the conical \nto the collinear phase [ 27]. \n 19 \n  \nFig. 1 2. Temperat ure dependence of the weak -field magnetization  Mw (a). Magnetic field \ndependences of the magnetization  M (b) at selected temperatures . (Data are adopted \nfrom [27].) \n \nOverall , the high sensitivity of the MW response and of the main magnetic modes of \nthe CoZnU ceramics  to the weak magnetic bias field H < 700 Oe at RT is defined by the weak -\nfield magnetization Mw caused by the transformation of the polydomain magnetic structure to \nthe monodomain one . The temperature evolution of the MW transmission  spectra and main \nmagnetic modes without bias field (part s 3 and 4 ) is affected by the temperature change of the \npolydomain magnetic structure and is defined by the magnetic properties of the polydomain \nand polycrystalline sample.  The a pplication of a high bias magnetic field up to 10 kOe  could \nreveal the T- and H-evolution of the monodomain sample  in a broad temperature range . \n \n \n9. Conclusions  \n100 200 300 400 500 600 7000246810\nTc1\nTc2 Tc3 MW(T), 100 Oe\nTemperature  (K)Magnetization M (B/f.u.)(a)\n0 2 4 6 8 10010203040\n(b) 700 Oe\nMagnetic field H (kOe)MH MHMH\nMW 153 K\n 223 K\n 298 K\n 348 K20 \n Using three different experimental methods (CPW, MSL and DR), we studied the microwave \nresponse of Sr4CoZnFe 36O60 ferrimagnetic ceramics in the frequency range from 100 MHz to \n35 GHz at temperatures from 10 to 400 K. In total, accounting for results of the coaxial \nmeasurements  (below 2 GHz)  published in [27], we observed 9 excitations. Since the \nfrequencies of all modes exhibit anomalies at the magnetic phase transitions ( Tc2 = 305 K  and \nTc3 = 140 K) and are also sensitive to external magnetic fields, we interpret all modes as \nmagnetic excitations. The lowest frequency modes (F1 – F3) seen below 3 GHz  are attributed \nto the dynamics  of the magnetic domain walls  in randomly  oriented ceramic grains [27]. The \nmost pronounced temperature dependence is exhibited by the F4 mode, which softens from 23 \nGHz (at 400 K) to 5 GHz near Tc2 and hardens to 11 GHz with further cooling. The behavior \nof this mode is responsible for the  Curie-Weiss behavior of the permeability above  Tc2. Since \nit is also the strongest mode, we assign  it as the natural ferromagnetic resonance. The frequency \nof the same mode exhibits, together with the F6 mode, a shallow minimum at Tc3, and this is \nagain responsible for the maximum permeability at Tc3 measured at 100 MHz and 1 GHz . The \nother modes are also interpreted as spin modes (magnons).  The u nit cell of the CoZnU \nhexaferrite contains  three formular units with a total of 108 Fe and 3 Co magnetic cations  \narranged in 18 spin blocks  (SRS*R*S*T) 3 [26], so there can be expected many magnetic \nexcitations  in the microwave and in the THz region s, and we see 9 of them .  \nThe MW spectra measured using coplanar and microstrip lines with different \nelectromagnetic field distribution and sample coupling revealed the same frequencies of \nexcitations with similar temperature and bias field dependences. It confirms the reliability of \nthe results and proves the effectivity of the used MW techni ques. The high number of magnetic \nexcitations observed in the MW region confirms the suitability of using our U -hexaferrite as a \nmicrowave absorber s and  highly  tunable filters . \n \nAcknowledgments  \nThis work has been supported by the Czech Science Foundation (Project No. 21 -06802S)  and \nGrant Agency of the Czech Technical University in Prague (Project No. \nSGS22/182/OHK4/3T/14) , and by the Research Infrastructure NanoEnviCz (funded by MEYS \nCR, Projects No. LM2018124) . \n \nDeclaration of Competing Interest  \nThe authors declare that they have no known competing financial interests or personal \nrelationships that could have appeared to influence the work reported in this paper.  21 \n  \nData availability  \nData will be made available on request.  \n \n \n1 T. Kimura. Magnetoelectric Hexaferrites. Ann. Rev. Condens. Matter Phys.  3, 93-110 (2012).  \n2 R. C. Pullar. Hexagonal ferrites: A review of the synthesis, properties and applications of hexaferrite \nceramics. 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Microwave magnetic excitations in Sr 4CoZnFe 36O60 U-type \nhexaferrite ceramics  \nM. Kempa,1 V. Bovtun,1 D. Repček,1,2 J. Buršík,3 C. Kadlec ,1 S. Kamba1 \n1Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, Prague 182 00, \nCzech Republic  \n2Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in \nPrague, Břehová 7, 115 19 Prague, Czech Republic  \n3Institute of Inorganic Chemistry, Czech Academy of Sciences, 250 68 Řež near \nPrague, Czech Republic  \n \nS1. Broadband microwave transmission and absorption.  \n \nThe full set of S-parameters ( S11, S22, S21, S12, both magnitude and phase) of t he \nmicrostrip line (MSL) and coplanar waveguide (CPW) loaded with CoZnU  ceramics \nrecorded in the broad frequency range  10 MHz - 50 GHz from  10 K to 390 K shows a \nreciprocal agreement of the S11 and S22, S21 and S12 spectra. It proves the high quality of \nthe calibration procedure and the reliability of the experiment. For further analysis, the \nS21 and S11 spectra were mainly used. As the most sensitive and informative, the \nfollowing parameters were selected and calculated  [1,2,3] : \na) S21 magnitude in dB (transmission coefficient Tr corresponding to the \nshielding efficiency SE = - Tr); \nb) input impedance Zin of the MSL or CPW loaded with CoZnU  samples, which \ncan be calculated from the S11 magnitude in dB (reflection coefficient R) as \n                                    𝑍𝑖𝑛=𝑍01+10𝑆1120⁄\n1−10𝑆1120⁄ ,                                                    ( s1)    \nwhere Z0 = 50 Ohm is an input impedance of the unloaded, empty MSL or \nCPW;  \nc) absorption coefficient A which can be calculated as  \n                           𝐴=20log(1−|2𝑍𝑖𝑛\n𝑍𝑖𝑛+𝑍0|−|𝑍𝑖𝑛−𝑍0\n𝑍𝑖𝑛+𝑍0|)  ,                         \n(s2)     The experimentally recorded S11 (S11 magnitude in dB is a reflect ion coefficient  \nR) and S21 (S21 magnitude in dB is a transmission  coefficient  Tr) spectra  of the CPW \nloaded with a CoZnU  ceramic s (see Fig. S1 ) show multiple resonances with the phase \nchange between –180 and +180 degrees, R(f) maxima above -10 dB and Tr(f) minima \nbelow -10÷20 dB.  The unloaded CPW is characterized (after the calibration) by the \nparameter values R < -30 dB, Tr ≈ 0 dB and S21 phase ≈ 0 degree s in the whole \nfrequency range of the experiment. So, all observed resonances are caused by the \nCoZnU  sample coupled with the CPW transmission line. The c hange and the frequency \nshift of the observed resonances with temperature are much larger than those caused by \nthe temperature change of the sample and the line sizes, and  should be related to the \ntemperature evolution of the sample’s electromagnetic properties. The most \ntemperature dependent S21 resonance (transmission mode) is marked as T4 in Fig. S1c. \nS-parameters spectra of the MSL loaded with a CoZnU  sample slightly differ from those \nof CPW (due to the different electromagnetic field distribution and sample coupling ) \nbut show qualitatively similar temperature changes. This similarity also supports  the \nattribution of the S-parameters temperature change to the temperature evolution of the \nCoZnU  electromagnetic properties.  S-parameters  spectra are noisy above 35 GHz, \ntherefore we evaluate data mainly below 35 GHz (see Fig. 1 – Fig. 4 of the main text).  \n  \nFig. S1. Recorded  broadband s pectra  of the S11 (a, b) and S21 (c, d) parameters of the \nCPW loaded with a CoZnU  sample at selected temperatures.   \n \nThe same set of the transmission modes is observed in both CPW and MSL \nexperiments, with the same CoZnU sample. Moreover, the temperature evolution of the \nCPW (Fig. 2 in the main text) and MSL (Fig. S2 ) T-modes is similar, with a remarkable \nchange of their behavior near the  CoZnU magnetic  phase transitions . \n \n-20-100\n-20-100R = S11 magnitude (dB)\n(a) (c) T4T4Tr = S21 magnitude (dB)\n0 10 20 30 40 50-150-100-50050100150\n(b) 10 K\n 100 K\n 150 K\n 300 K\n 390 K\nFrequency (GHz)0 10 20 30 40 50-150-100-50050100150\n(d)S21 phase (degree)S11 phase (degree)\nFrequency  (GHz) \nFig. S2. Broadband s pectra  of the transmission coefficient Tr (S21 magnitude)  of the \nMSL loaded with a CoZnU  sample  at selected temperatures below Tc3 (a), \nbetween Tc3 and Tc2 (b), and above Tc2 temperature (c, d). The main Tr(f) minima \nare mark ed T3  - T9. \n \nIn the MSL absorption spectra  (Fig. S3 ), 6 diffuse A(f) maxima can be \nrecognized below 35 GHz,  and they are numbered from A 4 to A9 with increasing \nfrequency . These maxima evidence microwave excitations (absorption modes, A -\nmodes) . The same system of the A -modes and their similar temperature evolution was \nderived from both CPW and MSL  experiments  (except for the A3 mode, which is weak \nand absent in the MSL spectra) . Temperature behavior of the A -mode s remarkably \nchanges near the magnetic phase transitions, similar to those of T-modes.  \n \n-40-30-20-100\n-30-25-20-15-10-50\n(c)T6T3\nT5T4\nT(7+8)\nT7 T8T4 311K\n 314K\n 317K\n 320K\n 323K\n0 10 20 30-30-20-100\n(d)T3\nT5\nT5\nT4\nT(7+8)\nFrequency (GHz)Tr = S21 magnitude (dB)\n 325 K\n 350 K\n 370 K\n 390 K(a)T5 T3\nT6T7\nT8\nT9T4T3Tr = S21 magnitude (dB) 10 K\n 50 K\n 100 K\n 125 K\n0 10 20 30-40-30-20-100\n(b)T5\nT6T4\nT7\nT7\nT8T8T4T3\nFrequency (GHz) 150 K\n 175 K\n 200 K\n 225 K\n 250 K\n 275 K\n 300 K \nFig. S3. Broadband spectra of the absorption coefficient A of the MSL loaded with a \nCoZnU  sample  at selected temperatures below Tc3 (a), between Tc3 and Tc2 (b), \nand above Tc2 (c, d). The main A(f) maxima are marked  A4 - A9. \n \nWe also analyzed  the input impedance spectra Zin(f), calculated using Eq. s1 and \nshown for CPW in Fig. S4. The spectra for MSL look very similar. The impedance \nspectra are noisy, similar to the absorption ones  (Fig. S3), but less diffuse d. 7 main \nZin(f) maxima seen below 35 GHz are marked  Z3 - Z9. These maxima correspond to \nthe transmission minima T3÷T 9 (Fig. 2  in the main paper) . Sometimes,  the Zin(f) \nmaxima are more pronounced, less diffuse,  and better separated than the Tr(f) minima.  \nThe impedance spectra  also confirm a remarkable c hange of the  T-modes  behavior near \nthe magnetic phase transition  at Tc2 in the  CoZnU  ceramics . \n-20-15-10-50\n-30-20-100\n(c)A4\nA5A6 A7\nA4A(8+9)\nA4\n 311 K\n 314 K\n 317 K\n 320 K\n 323 KA5\n15 20 25 30-20-15-10-50\n(d)A(7−9)A6\nA6A4A5 \nA4\nFrequency (GHz)A (dB)\n 325 K\n 350 K\n 370 K\n 390 K(a)A9\nA5A8A (dB)A6A7\n 10 K\n 50 K\n 100 K\n 125 K\n10 20 30-40-30-20-100\n(b)\nA5\nA4A(8+9)\nFrequency (GHz)A7\nA6\nA4A4\n 150 K\n 175 K\n 200 K\n 225 K\n 250 K\n 275 K\n 300 K \n \nFig. S 4. Broadband s pectra  of input impedance Zin of the CPW loaded with a CoZnU  \nsample  at selected temperatures below Tc3 (a), between Tc3 and Tc2 (b), and \nabove Tc2 (c). The main Zin(f) maxima are marke d Z3 - Z7.  \n \nThe transmission spectra  of the thin cylindrical CoZnU sample, measured as a \ndielectric resonator (DR) in the shielding cavity  [4,5], show a few temperature -\ndependent resonances ( S21 maxima) between 8 and 16 GHz with an anomalous behavior \n50100150200250\n(a)Z(8+9)\nZ7Z6\nZ5\nZ4\nZ3 10 K\n 30 K\n 50 K\n 75 K\n 100 K\n 125 KZin (Ohm)\n50100150200250\n(b)Z(8+9)\nZ7 150 K\n 175 K\n 200 K\n 225 KZ6\nZ5\nZ4 Z3 250 K\n 275 K\n 300 K\n0 10 20 3050100150\n(c)Z(8+9)\nZ7Z5\nZ4Z3 325 K\n 350 K\n 370 K\n 390 K\nFrequency  (GHz)near Tc2 = 305 K (see t emperature -frequency  maps in Fig. S5). Opposite to the \ntransmission spectra of CPW or MSL ( Fig. S2), the DR resonance frequencies (DR -T \nmodes) correspond to the S21 maxima, not minima, because the effective transmission \nis possible only in the case of sufficient coupling at the resonance frequency. The DR-\nT modes with the most pronounced S21 maxima (≥ -40 dB, dark red to brown colors) at \n~10 GHz, ~11 GHz, ~13.5 GHz and ~15.5 GHz, and with weak S21 maxima (≥ -60 dB, \ngreen to yellow colors) are shown also as black points in Fig. S5. Of course, the mean \nfrequencies of the CPW -T and DR -T modes are different, but their temperature \nbehavior is similar.  All DR -T modes below 14 GHz slow down toward Tc2, similarly to \nthe CPW and MSL T -modes below 22 GHz (Fig.  4 in the main paper ). The F4 excitation \n(CPW -T4) mode, which is characterized by the high absorption and the strongest \ntemperature dependence above Tc2, is also shown as a solid line in Fig. S5. It crosses \nall DR -T modes near Tc2 with increasing temperature, and the crossing points \ncorrespond well to the DR -T mode temperature anomalies.  \n \n \nFig. S5. Temperature -frequency  map of the transmission coefficient  (S21 magnitude)  of \nthe CoZnU  dielectric resonator . Points correspond to the main S21(f) maxima, \nsolid line near Tc2 corresponds  to the softening of the F4 frequency towards Tc2 \n(CPW -T curve from Fig.  4 in the main paper ). \n \nS2. THz spectra . \n \nThe time-domain THz transmission spectra were measured down to 4 K. The amplitude \nand phase of the passing wave were  determined,  and this allow ed the complex refractive \nindex  𝑁= (𝜀∗𝜇∗)12⁄ to be calculated  directly  [6]. Seven excitations are visible in the \nTHz spectrum, but since we are not sure which are magnetic and which are polar \n(phonon)  excitations , we present in Fig. S6 only the components of the complex \nrefractive index  N and not the complex dielectric permittivity  𝜀∗ and magnetic \npermeability  𝜇∗ spectra . However, we estimate from the shape of the excitations which \nones are magnons and which are phonons. We label their frequencies in Fig. S6 with \narrows.  \n 4.64.74.84.95.05.15.2\n0.4 0.6 0.8 1.0 1.2 1.4 1.60.00.10.20.30.4     4 K\n   50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 Kn\nFrequency (THz)kmagnonsmagnon or phonon\nphonon \nFig. S6. Temperature dependence of the complex refractive index in the THz region. \nThe f requencies of the magnetic and of the lattice excitations are marked.   \n \nReferences.  \n \n[1] H. W. Ott “Electromagnetic Compatibility Engineering”, I. Wiley & Sons, New \nYork, 2009 . \n[2] V. Bovtun, M. Kempa, D. Nuzhnyy, J. Petzelt, O. Borisova, O. Machulianskyi , Y. \nYakymenko \"Microwave absorbing and shielding properties of inhomogeneous \nconductors and high -loss dielectrics \", Ferroelectrics 532, (2018)  57-66.  \n[3] V. Bovtun, M. Kempa, D. Nuzhnyy, J. Petzelt, O. Borisova, O. Machulianskyi , Y. \nYakymenko “Composition dependent microwave properties of dielectric -\nconductor nanocomposites”, Phase Transitions  91, (2018)  1027 -1035 .  \n[4] J. Krupka \"Frequency domain complex permittivity measurements at microwave \nfrequencies \", Meas. Sci. Technol.  17, (2006)  R55–R70.  [5] A. Maia, C . Kadlec , M. Savinov, R. Vilarinho , J. A. Moreira, V . Bovtun, M . Kempa, \nM. Míšek, J . Kaštil, A . Prokhorov, J . Maňák, A. A. Belik, S . Kamba , Can the \nferroelectric soft mode trigger an antiferromagnetic phase transition?  J. Eur. \nCeram . Soc. 43, (2023) 2479 –2487 .  \n[6] P. Kužel, H. Němec, F. Kadlec, and C. Kadlec. Gouy  shift correction for highly \naccurate refractive index retrieval in time -domain terahertz spectroscopy. Opt. \nExpress 18 (2010) 15338 -15348.  "    },    {        "title": "2103.06165v1.Experimental_Demonstration_of_a_Rowland_Spectrometer_for_Spin_Waves.pdf",        "content": "Experimental Demonstration of a Rowland\nSpectrometer for Spin Waves\n´Ad´am Papp1,2,+, Martina Kiechle1,+, Simon Mendisch1, Valentin Ahrens1, Levent Sahin1,\nLukas Seitner1, Wolfgang Porod3, Gyorgy Csaba1,2, and Markus Becherer1,*\n1Technical University of Munich (TUM), Department of Electrical and Computer Engineering, Munich, Germany\n2P´azm ´any P ´eter Catholic University, Faculty of Information Technology and Bionics, Budapest, Hungary\n3University of Notre Dame, Department of Electrical Engineering, Notre Dame, IN, 46556\n+These authors contributed equally to this work.\n*markus.becherer@tum.de\nABSTRACT\nWe experimentally demonstrate the operation of a spin-wave Rowland spectrometer. In the proposed device geometry,\nspin waves are coherently excited on a diffraction grating and form an interference pattern that spatially separates spectral\ncomponents of the incoming signal. The diffraction grating was created by focused-ion-beam irradiation, which was found\nto locally eliminate the ferrimagnetic properties of YIG, without removing the material. We found that in our experiments\nspin waves were created by an indirect mechanism, by exploiting nonlinear resonance between the grating and the coplanar\nwaveguide. Our work paves the way for complex spin-wave optic devices – chips that replicate the functionality of integrated\noptical devices on a chip-scale.\nIntroduction\nInformation processing in today’s computers is done almost exclusively by charges (electric currents). Photons, albeit they\nare ideal for information transmission, never became a mainstream technology for computing. Despite their numerous\nadvantages, photonic devices have practical limitations: they are challenging to integrate on-chip and optical wavelengths\n(about a micrometer) are huge compared to nanoscale devices, limiting the scalability of any photonic interference-based\ndevice.\nSpin waves (aka magnons) are wave-like excitations in ferromagnetic (and ferrimagnetic) materials that travel via cou-\npling between precessing magnetic moments. Their wavelength can be adjusted in a wide range (from several micrometers\ndown to potentially nanometer scale), and they have an electronics-friendly frequency range (1-100 GHz)1. This makes\nspin waves attractive for on-chip applications, especially in wave-based microwave signal processing. They also interact\nwith each other (scatter), and the resulting nonlinearity may enable general-purpose computation. Spin waves exist only\nin magnetic media and one needs high-quality materials to achieve ideal conditions for propagation and also carefully\ndesigned waveguide structures to launch (and pick up) the waves. It has only recently become possible to demonstrate\nshort-wavelength, long-distance propagation2, 3, spin-wave equivalents of optical laws4or larger-scale refractive devices\nusing local heating5. Spin-wave variants of complex optical devices are within reach of experimental demonstrations6.\nA key element of such a spin-wave optics device is a source which launches coherent spin waves. In the simplest\ncase, the source of spin waves can be a simple coplanar waveguide that is placed atop the magnetic film. For many\ndevice constructions, such a simple construction is insufficient. Microwave waveguides alone are fairly inefficient at short\nspin-wave wavelengths8and they are limited to generation of plane wavefronts or curved wavefronts with relatively small\ncurvature. For generation of short-wavelength spin waves or non-planar wavefronts, lithographically patterning the edge of\nthe magnetic film is desirable. A periodically patterned edge can serve both as a wave source and a diffracting element.\nThe focus of the present paper is the experimental study of such a patterned edge as a spin-wave launcher. We use\nFocused Ion Beam (FIB) irradiation to write a high-resolution concave grating pattern in an yttrium iron garnet (YIG)\nthin film. The grating is used in the Rowland spectrometer arrangement, which is frequently used in optical and X-ray\nspectroscopy9. This arrangement does not require a separate lens component, and thus it is ideal for waves with limited\npropagation length. We experimentally demonstrate that the edge of a FIB-irradiated pattern in YIG generates a coherent\nspin-wave wavefront. Using time-resolved Magneto Optical Kerr Effect (trMOKE) imaging we found that the diffraction\npatterns closely match those expected from theory and micromagnetic simulations. An unexpected discovery was that in\nour experiments spin waves were not primarily excited directly by the field of the waveguide. Instead, spin waves werearXiv:2103.06165v1  [physics.app-ph]  10 Mar 2021generated indirectly by the dipole fields of high-amplitude, nonlinear, and long-wavelength standing waves that developed\nbehind the grating in the unirradiated area. This process has a significantly higher efficiency of spin-wave generation at a\ndistance from the waveguide. However, this quasi-homogeneous oscillation behind the grating only forms at a sufficiently\nhigh amplitude and only if a single frequency component is applied.\nSpectral Decomposition with a Concave Diffraction Grating\nThe main component of the proposed device is a concave diffraction grating that acts both as a wave source and as a\ndiffractor. The fabrication of such a grating requires sub-wavelength patterning resolution as the pitch of the grating has to\nbe comparable to the spin-wave wavelength. We use the so-called Rowland arrangement, a detailed description is given in7.\nThe device generates a spectral decomposition of a time-domain signal by converting temporal frequency components\nto spatially separated spin-wave intensity peaks. The layout of the fabricated device is shown in Fig. 1. The microwave\nsignal is converted to spin waves by a waveguide antenna. Each frequency component of the signal generates spin waves\nwith corresponding wavelengths. Along the edge of the grating (FIB-irradiated region) the time-varying magnetic field is\nalmost homogeneous, but due to the abrupt parameter change in YIG, every point along the edge acts as a wave source\n(also described in10). The curved grating not only diffracts different wavelengths to different directions, but also focuses the\nwavefronts. The drawing of Fig. 1 bshows the geometry to determine the diffraction pattern. With a concave grating of\nradius R and ridge pitch d, the diffraction peaks (i.e. wavelength-dependent focal points) will form on a circle with radius\nR/2 drawn tangentially to the grating (Rowland circle). The nth-order diffraction angle can be calculated as ®Æarcsin³\nn¸\nd´\n,\nwhere ¸is the wavelength of the spin wave.\nFigure 1. a) Sketch of the experimental setup, indicating the coplanar waveguide, the FIB irradiated grating, the trMOKE\nprobing laser, and the spin-wave interference pattern. b) Geometry of the curved diffraction grating in the Rowland\narrangement. The angle under which the first-order diffraction peak is seen from center of the grating is denoted by ®.\nIn traditional optical or X-ray Rowland spectrometers the wave source is placed opposing the curved grating, which\nreflects the waves, acting as a secondary source. Such arrangement would be rather impractical for spin waves: the relatively\nlong path between the source and the diffraction grating will cause much higher attenuation of spin waves. Thus, it is\ndesirable that the grating and the source are in the same structure, i.e. the coherent spin-wave source itself is shaped as a\ncurved diffraction grating. In this geometry, diffraction is caused by the phase difference between waves that originate\nfrom the bottom and the top of the ridges. This phase difference will also depend on the ratio of the ridge depth and the\nwavelength, which does not influence the diffraction angle, but it changes the relative amplitude between diffraction orders.\nIn the designed structure the ridge depth also introduces an amplitude difference between waves that are generated on the\ntop and the bottom of the ridge. The device is thus a combination of an amplitude grating and a phase grating.\nWe fabricated the designed device and recorded spin-wave interference patterns with trMOKE. The trMOKE images\nat three different excitation frequencies are shown in Fig. 2. Diffraction peaks are clearly observed close to the expected\ndiffraction angles, as indicated by black lines. In case of the smallest wavelength the second-order peaks can also be\nobserved (2¸\ndÇ1). We found that the peaks are not perfectly focused along the Rowland circle, but an arc can be fitted\nto the peaks that works well for all three wavelengths. We attribute this to the fact that our grating is much wider than\n2/7conventional concave gratings (180±instead of a few degrees), and thus the approximations used in the derivation of the\nRowland circle do not hold perfectly. This is confirmed by micromagnetic simulations, which also show that focal points are\nlocated on an arc with a slightly smaller curvature (see Fig. 4 c). Another possible cause of deviations is the slight anisotropy\nof spin waves introduced by a tilt in the external bias field, which can not be fully eliminated in our current experimental\nsetup.\nFigure 2. Spin-wave interference patterns recorded by trMOKE at multiple excitation frequencies. In a-c) the FIB\nirradiated region (grating) is indicated by a semi-transparent red overlay (radius R=30 ¹m, pitch d=12 ¹m), the gray stripe\nrepresents the ground line of the CPW. The green circles are the theoretical Rowland circles with a radius R/2, while the\nradius of red arcs are fitted to the data. Black lines indicate expected diffraction angles, and white arrows point to focal\npoints in the measured data. d-f) Spin-wave intensities extracted along the red arcs in a-c).\nGeneration Mechanism of Spin Waves by Nonlinear Resonance\nIn most spin-wave devices, the magnetic field of the waveguide is directly responsible for launching the spin waves, as\ndescribed in7. The relatively delocalized magnetic field of the waveguide and the localized demagnetizing field of the film\nedge jointly create a high, periodically changing torque and launch the spin waves10.\nWe found that in our device an indirect, nonlinear mechanism is dominant for spin-wave excitation. In the design of7\nall material in the YIG film is assumed to be removed behind the grating. However, our FIB method is performed as the final\nfabrication step, after the CPW is already in place. We did not irradiate the total area between the grating and the CPW,\nonly a narrow region, which is wide enough to block spin waves at the designed wavelength to significantly couple through\nvia dipole fields. However, at sufficiently large excitation amplitudes nonlinear effects cause the spin-wave wavelength\nto increase. This is due to the lowered OOP demagnetization-field component14. The wavelength at sufficiently large\namplitudes becomes much larger than the distance between the CPW and the grating. The spin waves that are generated\nunder the CPW reflect back from the back side of the grating, creating a standing-wave pattern. Since the wavelength is\nmuch larger than the size of the region between the CPW and the grating, this resembles a homogeneous resonance in that\nregion. The trMOKE images of Fig. 2 a-calready show the high-amplitude region on the left of the grating: it is observable\nthat in this region the colormap is saturated and without apparent pattern, indicating large-amplitude, uniform precession.\nThe dipole field of the quasi-homogeneous resonant excitation reaches significantly farther than that of the short-\nwavelength spin waves, thus it can excite coherent spin waves on the ridges of the grating. This field is in fact much stronger\nthan the magnetic field of the CPW, becoming the dominant effect for spin-wave generation on the grating edge.\n3/7Figure 3 shows the spin-wave-generation process in more detail. At sufficiently small excitation power ( PrfÆ0 dBm),\nlinear spin waves are excited under the CPW. These small-amplitude, short-wavelength spin waves cannot significantly\ncouple through the FIB-irradiated region. However, the Oersted field of the CPW is not sufficient to create spin waves at the\ngrating edge, that could be detected by our trMOKE apparatus. In Fig. 3 b(PrfÆ5 dBm) nonlinear behavior is observable\nbehind the grating, but the nonlinear wavelength is not yet long enough to create uniform precession. One can observe a\npartial interference pattern on the right possibly due to larger uniform standing waves on the top. At PrfÆ10 dBm excitation\npower (Fig. 3 c) the resonance almost uniform (apart from the top region being out-of phase), and the interference pattern\nis complete.\nFigure 3. TrMOKE images of miscellaneous gratings expressing the difference in performance with respect to small a),\nhigher b)and high c)microwave current applied at the input waveguide. The right edge of the CPW ground line is 3 ¹m\naway from grating posterior in each case. The semi-transparent red overlay indicates the FIB-irradiated region, black lines\nshow expected diffraction angles. The grating operation visibly enhances with the level of uniformity in the nonlinear spin\nwave excitation behind the FIB area.\nTo gain more insight into the excitation process, we performed both one-dimensional and two-dimensional micro-\nmagnetic simulations using mumax312. These simulations confirmed the behavior that we observed in the experiments\n(Fig. 4). At small excitation fields, spin waves beyond the grating are very small in amplitude (Fig. 4 a). At a higher (nonlinear)\nexcitation field, however, a uniform standing wave is observed between the grating and the CPW, and coupling through\nthe FIB region is strong (Fig. 4 b). Left from the CPW the wavelength change of the spin waves can be observed as they\ndecay due to magnetic damping. Beyond about 50 &m propagation self-modulational instability is also causing spikes in\nthe waveform13. 2D simulations also confirmed the operation of the device (Fig. 4 c). Here the diffraction angles match\nperfectly the theory, but the position of the focal points are also somewhat behind the Rowland circle. This is the same\neffect we observe in the experiments. Gratings with larger radius and shorter width would probably not suffer from this\ndeviation, especially at small diffraction angles. However, the position of the peaks is predictable, so this does not affect the\nusability of the device.\nAn additional, very unusual aspect of this nonlinear excitation can be observed in Fig. 2 c. In case of the direct excitation\nmechanism one would expect that segments of the grating that are closest to the CPW will excite the highest amplitude\nwaves, since the field of the CPW decays with the lateral distance. However, here we observe exactly the opposite: the\nstrongest \"beams\" seem to form on the farthest parts of the grating, and amplitudes are less strong in the middle part,\nwhich is very close to the CPW (Fig. 3 c). This is because the largest area where homogeneous oscillations can occur are on\nthe sides, where there is enough distance between the CPW and the grating. The larger the area, the higher are the dipole\nfields, and the higher the excitation on the opposite side of the grating. Thus, the discovered indirect excitation mechanism\nis very efficient at exciting short-wavelength spin waves on a finely patterned edge at a distance from a CPW. This can be\nadvantageous in applications where a complex wavefront has to be launched.\nA significant limitation of the nonlinear excitation method is that it only works for single-frequency excitations. If\nmultiple frequency components are excited, the uniform standing waves cannot form, moreover nonlinear mixing creates\nunwanted spectral pollution. If multiple frequencies are present in the excitation signal (as it is often desirable in a spectrum\n4/7Figure 4. Micromagnetic simulations of nonlinear indirect excitation of spin waves. Yellow rectangles indicate the CPW\nposition and width. The red stripe represents the FIB irradiated region, modelled by zero M s.a)is a 1D example of linear\nexcitation, while bis a strongly nonlinear case. c)represents a 2D simulation of the experimental scenario in Fig. 2 b.\nanalyzer), the nonlinear method cannot be used, but, as we demonstrated, the proposed method is very effective at creating\ndevices with complex interference patterns at a single frequency.\nMethods\n1 Sample Fabrication and Characterization\nYIG thin films were deposited on a GGG substrate using rf-magnetron sputtering (we used 100 nm thick films in the grating\nexperiments). Their magnetic properties were evaluated by means of ferromagnetic resonance (FMR), revealing a saturation\nmagnetization of Ms=120 kA/m and a damping constant of ®YIGÆ4.4£10¡4. For the excitation of spin waves shorted\naluminum CPW antennas were fabricated on top of the YIG film. The antennas were wire bonded to a PCB based CPW with\nconnections to an RF signal generator (Stanford Research SG 386).\nThe gratings were fabricated in YIG next to the CPW via FIB irradiation. We used 50 keV GaÅions with a relatively low\ndose (1015ions/cm2), which is high enough to almost completely destroy the magnetic properties of YIG, but no material\nis removed. Much higher doses (in case of ion milling) would likely deteriorate the YIG film around the patterned region\ndue to ion scattering, which makes this method more suitable. Sub-micron resolution patterning can easily be achieved\n(possibly down to 100 nm in our facility). A similar method was recently described in11, where comparably lower doses\nwere used to change magnetic properties of YIG on film level. Here, we deliberately used higher doses to drastically reduce\nthe saturation magnetization of YIG locally, to create a region which inhibits spin-wave transmission. We found that this\nmethod is in effect very similar to actually removing material, as it was proposed in7.\nThe effect of FIB irradiation was also investigated using transmission electron microscopy (TEM). In Fig. 5 b,cTEM\nimages indicate that the crystalline structure of YIG is completely destroyed down to a depth of approximately 25 nm, and\nfurther significant damage is observable at even higher depths. These results are in good agreement with the previously\nperformed SRIM simulations of our system in Fig. 5 a, suggesting a peak implantation depth of 24 nm.\nGaÅions are relatively large compared to other frequently used ions such as HeÅ, which explains their low penetration\ndepth and the resulting bilayer formation in YIG. We only had access to GaÅFIB, but SRIM simulations suggest that higher\nimplantation depths could be achieved with HeÅions, and, with higher doses compared to GaÅ, similar modification of YIG\nproperties could be achieved with better uniformity across the film thickness.\n2 Imaging of 2D Spin-Wave Patterns\nTo image spin-wave interference patterns we built a time-resolved magneto-optical Kerr microscope (trMOKE). Since\nthe Rowland spectrometer requires isotropic spin-wave propagation, we had to use out-of-plane bias. Thus our trMOKE\nmeasures the longitudinal Kerr effect, i.e. it is sensitive to changes in the in-plane magnetization component that lies in\nthe incidence plane of the laser15. We used a ps-laser with 50 ps pulse width and 405 nm wavelength (PicoQuant Taiko\nPDL M1 with LDH-IB-405 laser head). With this, we can measure spin waves up to approximately 5 GHz frequency (with\n10 MHz steps) and down to 2 &m wavelength. We scan through the sample with an XYZ stage with 0.4 &m resolution. Larger\narea scans (such as the ones presented in this paper) take about a few hours of measurement time. We use a stroboscopic\ntechnique in which the excitation signal is phase locked to the lock-in amplifier and the ps-laser, thus we can extract phase\n5/7Figure 5. Crystal investigation of the FIB impact in YIG by means of TEM. a)Shows the simulated ion implantation depth\nfor 50 keV GaÅions in Y 3Fe5O12. A cross-sectional image of a 80 nm thick irradiated YIG film is depicted in b).c)The\nmagnification of the orange square in b)exposes an amorphous toplayer of the thickness expected from the SRIM\nsimulations in a).\ninformation as well. The amplitude scale is not calibrated, but we estimate that the setup is sensitive to a few percent\nchange in in-plane magnetization.\nCurrently our setup uses a permanent magnet under the sample for biasing. This makes calibration challenging due\nto the inhomogeneous field profile. The bias field values in the measurements are approximate values with a few mT\nuncertainty, and perfect out-of-plane biasing is difficult to achieve.\n3 Micromagnetic simulations\nMicromagnetic simulations were performed in mumax312. We used experimental values for parameters where they were\navailable ( Ms=120 kA/m and ®YIGÆ4.4£10¡4, 100 nm thickness), and values from literature where we could not directly\nmeasure parameters ( AexÆ3.65£10¡12J/m). For discretization we used 30 nm £30 nm £100 nm cells, i.e. approximating the\nfilm by a single layer. The lateral cell size is somewhat larger than the exchange length lexÆq\n2Aex/(¹0M2s)¼20 nm, but it\nis still at least a hundred times smaller than the wavelength, and no discrepancies could be observed compared to smaller\ncell sizes, while the simulation can be completed in a reasonable time. To avoid reflections, in the left and right hand side of\nthe simulation 3¹mwide artificial absorbing layers were created using a quadratically increasing damping constant. On\nthe lateral boundaries periodic boundary conditions (single repetition) were used to simulate a long waveguide and avoid\nenergy loss on the sides. The external field was chosen to be 221 mT, using an analytical dispersion formula to match the\nmeasurement wavelength at the given frequency (the external field at the exact position of the sample cannot be measured\nwith sufficient precision in our setup). The FIB irradiation was modeled as a region with ( Ms=0 A/m. The field of the CPW\nwas calculated by HFSS, assuming 1 mA current (peak) in the waveguide. The simulation was run for 180 ns, which was\nlong enough to form a steady interference pattern.\nConclusions\nOptically inspired magnonic devices represent a promising route to wave-based computing, which are themselves sought\nafter for post von-Neumann computing. To our knowledge, the Rowland spectrometer we demonstrate here is the most\ncomplex spin-wave interference device on the micrometer scale. The spin-wave patterns we observe behave remarkably\nsimilar to expectations and to the behavior of ideal isotropic waves.\nWe used FIB irradiation to draw patterns in YIG with nanoscale precision but without removing material. This minimizes\nthe damage to the adjacent YIG areas. The irradiated patterns can influence spin-wave propagation and may also act as wave\nsources nearby a waveguide. The manipulation of magnetic properties via FIB in other material systems is well-established,\nbut we are not aware of fabricated spin-wave elements in YIG using a similar approach.\nBesides demonstrating complex spin-wave patterns in YIG films, we also described a newfound way of creating spin\nwaves via nonlinear resonance, a method that exploits high-amplitude standing-wave oscillations to indirectly excite\nshort-wavelength spin waves with complex wavefronts.\n6/7Acknowledgements\nThe authors are grateful for fruitful discussions and encouragement from Gary Bernstein, Hadrian Aquino (University of\nNotre Dame), Andrii Chumak (University of Vienna) and Philipp Pirro (TU Kaiserslautern). Many thanks to all staff members\nat TUM, especially Anika Kwiatkowski. Furthermore, we would like to acknowledge the support of the Central Electronics\nand Information Technology Laboratory – ZEITlab. This work was partially funded by the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) – Projektnummer (429656450), and by the US National Science Foundation (NSF)\nwith a grant from the SpecEES (Spectrum Efficiency, Energy Efficiency, and Security) program. Adam Papp received funding\nfrom the TUM TUFF postdoctoral grant, and from the postdoctoral grant of the Hungarian Academy of Sciences/Eötvös\nLoránd Research Network (PPD 2019). We are grateful for support from PicoQuant.\nAdditional information\nCompeting financial interests: The authors declare no competing financial interests.\nReferences\n1.Csaba, G., Papp, Á. and Porod, W., 2017. Perspectives of using spin waves for computing and signal processing. Physics\nLetters A, 381(17), pp.1471-1476.\n2.Chang, Houchen, Peng Li, Wei Zhang, Tao Liu, Axel Hoffmann, Longjiang Deng, and Mingzhong Wu. \"Nanometer-thick\nyttrium iron garnet films with extremely low damping.\" IEEE Magnetics Letters 5 (2014): 1-4.\n3.Maendl, Stefan, Ioannis Stasinopoulos, and Dirk Grundler. \"Spin waves with large decay length and few 100 nm\nwavelengths in thin yttrium iron garnet grown at the wafer scale.\" Applied Physics Letters 111, no. 1 (2017): 012403.\n4.Stigloher, Johannes, Martin Decker, Helmut S. Körner, Kenji Tanabe, Takahiro Moriyama, Takuya Taniguchi, Hiroshi\nHata et al. \"Snell’s law for spin waves.\" Physical review letters 117, no. 3 (2016): 037204.\n5.Vogel, Marc, Burkard Hillebrands, and Georg von Freymann. \"Spin-Wave Optical Elements: Towards Spin-Wave Fourier\nOptics.\" arXiv preprint arXiv:1906.02301 (2019).\n6.Albisetti, Edoardo, Silvia Tacchi, Raffaele Silvani, Giuseppe Scaramuzzi, Simone Finizio, Sebastian Wintz, Christian\nRinaldi et al. \"Optically Inspired Nanomagnonics with Nonreciprocal Spin Waves in Synthetic Antiferromagnets.\"\nAdvanced Materials (2020): 1906439.\n7.Papp, Á., Porod, W., Csurgay, Á.I. and Csaba, G., 2017. Nanoscale spectrum analyzer based on spin-wave interference.\nScientific reports, 7(1), p.9245.\n8.Papp, A., Csaba, G., Dey, H., Madami, M., Porod, W. and Carlotti, G., 2018. Waveguides as sources of short-wavelength\nspin waves for low-energy ICT applications. The European Physical Journal B, 91(6), p.107.\n9.James, J. Spectrograph design fundamentals (Cambridge University Press, 2007\n10. Davies, C. S., and V . V . Kruglyak. \"Generation of propagating spin waves from edges of magnetic nanostructures pumped\nby uniform microwave magnetic field.\" IEEE Transactions on Magnetics 52, no. 7 (2016): 1-4.\n11. Ruane, W.T., White, S.P ., Brangham, J.T., Meng, K.Y., Pelekhov, D.V ., Yang, F .Y. and Hammel, P .C., 2018. Controlling and\npatterning the effective magnetization in Y3Fe5O12 thin films using ion irradiation. AIP Advances, 8(5), p.056007.\n12. Vansteenkiste, Arne, Jonathan Leliaert, Mykola Dvornik, Mathias Helsen, Felipe Garcia-Sanchez, and Bartel Van\nWaeyenberge. \"The design and verification of MuMax3.\" AIP advances 4, no. 10 (2014): 107133 see also M.J. Donahue\nand D.G. Porter Interagency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD\n(Sept 1999)\n13. Wu, Mingzhong, Boris A. Kalinikos, and Carl E. Patton. \"Generation of dark and bright spin wave envelope soliton trains\nthrough self-modulational instability in magnetic films.\" Physical review letters 93.15 (2004): 157207.\n14. Wu, Mingzhong. \"Nonlinear spin waves in magnetic film feedback rings.\" Solid State Physics 62 (2010): 163-224.\n15. Arregi, Jon Ander, Patricia Riego, and Andreas Berger. \"What is the longitudinal magneto-optical Kerr effect?.\" Journal\nof Physics D: Applied Physics 50.3 (2016): 03LT01.\n7/7"    },    {        "title": "1812.00455v1.Direct_detection_of_induced_magnetic_moment_and_efficient_spin_to_charge_conversion_in_graphene_ferromagnetic_structures.pdf",        "content": "*Corresponding author: Joaquim B. S. Mendes, joaquim .mendes @ufv.br  Direct d etection of induced magnetic moment and efficient spin -to-charge conversion in \ngraphene/ferromagnetic structures  \nJ. B. S. Mendes1,*, O. Alves Santos1,2, T. Chagas3, R. Magalhães -Paniago3, T. J. A. Mori4,  \nJ. Holanda2, L. M. Meireles3, R. G. Lacerda3, A. Azevedo2, and S. M. Rezende2 \n1Departamento de Física, Universidade Federal de Vi çosa, 36570 -900, Viçosa, MG, Bra zil \n2Departamento de Física, Universidade Federal de Pernambuco, 50670 -901, Recife, PE, Bra zil. \n3Departamento de Física, Universidade Federal de Minas Gerais, 31270 -901, Belo Horizonte, MG, Bra zil. \n4Laboratório Nacional de Luz Síncrotron , Centro Nacional de Pesquisa em Energia e Materiais, 13083 -970 Campinas, SP, Brazil  \nThis article  shows th at the spin-to-charge  current conversion in single -layer graphene (SLG) by means of  the \ninverse Rashba -Edelstein effect  (IREE) is made possible with t he integration of this remarkable 2D -material with the \nunique ferrimagnetic insulator yttrium iron garnet  (YIG  = Y 3Fe5O12) as well as  with the ferromagnetic metal permalloy  \n(Py = Ni 81Fe19). By means of X -ray absorption spectroscopy (XAS) and magnetic circular dichroism (XMCD) \ntechniques, we show that the carbon atoms of the SLG  acquires a n induced magnetic moment due to the proximity \neffect with the magnetic layer. The spin currents are gene rated in the magnetic layer by spin pumping from microwave \ndriven ferromagnetic resonance an d are detected by a dc voltage along the graphene layer , at room temperature . The \nspin-to-charge  current conversion , occurring at the graphene layer , is explained by the extrinsic spin -orbit  interaction \n(SOI)  induced by the proximity effect with the ferromagnetic layer.  The r esults obtained for the SLG/YIG and SLG /Py \nsystems confirm very similar values for the I REE parameter, whic h are larger than the v alues report ed in previous \nstudies for SLG.  We also report systematic investigations of the electronic and magnetic properties of the SLG /YIG by \nmeans of scann ing tunneling microscopy (STM).  \nKEYWORDS:  graphene, magnetic proximity, magnetic circular dichroism, spin-pumping,  spin-to-charge conversion,  \ninverse Rashba -Edelstein effect, extrinsic SOI in graphene .  \n \nI - INTRODUCTION  \nNew phenomena discovered in  condensed matter \nphysics and materials science in the last decades have \nshown that S pintronic s has many advantages over \nconventional electronics and several spin based devices \nare being used in  consumer  electronic s1,2. Nowadays, \ngeneration, transport and manipulation of spin current s - \ni.e., net flows of spin angular momentum  - are amo ng \nthe main challenging  topics  in this  research  area. In turn, \namong the main phenomena to generate and manipulate \nspin currents, one can mention the spin pumping effect \n(SPE),  the spin Hall effect (SHE) and its Onsager \nreciproc al, the inverse SHE (ISHE)3–11. In the SPE,  the \nprecessing magnetization in a ferromagnetic material \n(FM) generates a pure spin cu rrent in an adjacent normal \nmetal (NM) , and in the SHE, an unpolarized charge \ncurrent is partially converted i nto a transv erse spin \ncurrent by the spin -orbit coupling (SOC). Spin curre nt \nhas the advantage of flowing in both metallic and \ninsulating materials that can be magnetic or non -\nmagnetic. This feature s increase  the range of spintronics \nphenomena that are of interest for basic and applied \nresearch , such as:  (i) spin transport in magnetic \ninsulators and organics materials12–17; (ii) spin -to-charge \ncurrent -conversion in paramagnetic, ferromagnetic and \nantiferromagnetic  meta ls, as well as in \nsemiconductors18–28; (iii) magnetization manipulation b y spin transfer torque (STT)29–33; and spin-charge  interpl ay \nin two dimension al systems  based on the Rashba -\nEdelstein effect34–40. \nFrom the applications point of view  the creatio n of \nspin-logic devices, which allow the control and transport \nof the spin current over long distances, is one of the \nmajor  research challenges in spintronics. In this regard, \ngraphene - a single atomic layer of carbon atoms in a \nhoneycomb lattice (see Figure 1(c)) - has attracted great \nattention as a p romising material f or spin based  devices \ndue to its exceptional electronic transport properties, \nexcellent charge carrier mobility, quantum  transport, \nlong spin diffusion lengths and  spin relaxation tim es41–\n45. However, due to the low atomic number of carbon, \nthe Dirac electrons in intrinsic graphene ha ve a weak \nSOC and no magnetic moments, thus, reducing the \npromises of applications  in spintronics2. In order to \novercome this limitation, several mechanisms and \nstructures have been proposed mainly focused on the \nimprovement of local magnetic moments and spin -orbit \ncoupling in graphene . Among them we may cite : (i) \nfunctionaliz ation of  graphene with small doses of \nadatoms or nanoparticles44,45; (ii) development of \nnanostructures with gate electrode s of ferromagnetic \nmaterials46,47; (iii) and by directly attaching  graphene in \ncontact with a ferromagnetic insulator (FMI)  or a \nferromagnetic metal (FM), to induce magnetic proximity \neffects38,48 –51. More recently, it has been shown a large enhancement of the SOC in graphene when  it is directly \nfabricated on top of semiconducting transition -metal \ndichalcogenides52–54. \nRecent experimenta l studies have shown that YIG \nturned out to be  an ideal ferromagnetic insulating (FMI)  \nmaterial to induce s pin properties in hybrid structures \nwith single -layer graphene (SLG)38,55 –59. Among other \nreasons for the use of YIG, we can mention its excellent \nmecha nical and structural properties , as well as its \nextraordinary  low magnetic damping. At the same time , \nthe metallic ferromagnet ic Py (Ni 81Fe19) can used instead \nof YIG if a conducting material is required for device \ndevelopment . In this manuscript, we report a n extensive \ninvestigation  of spin -curre nt to charge -current \nconversion by inverse Rashba -Edelstein effect (I REE) in \nlarge area SLG grown by chemical vapor  deposition \n(CVD) on YIG and Py films.  Thus , we show that the \nSLG in atomic con tact with both magnetic materials  \nexhibi ts an appreciable extrinsic SOC that enhances the \nIREE . Besides that, as the unequivocal proof of the \nproximity -induced magnetism  in graphene , we \nperform ed X-ray magnetic circular dichroism (XMCD)  \nmeasurements in YIG/SLG  and Py/SLG sample s. The \nXMCD results confirmed the existence of an induced \nmagnetic moment at the carbon atoms of graphene.  \n II - SAMPLE PREPARATION  AND \nCHARACTERIZATION  \nIn the experiments reported here , we investigated  two \ntypes of heterostructures. In the first structure , the \ngraphene is interfaced with a ferrimagnetic insulator \nmaterial (sample A: YIG/SLG), and in the second \nheterostructure,  the graphene is interfaced with a \nconductive ferromagnetic material (sample B : Py/SLG). \nAs ferrimagnetic insulator material we have used a \nsingle -crystal YIG film with thickness 6.0 µm grown by \nliquid -phase epitaxy on a 0.5 mm thick substrate of (111) \ngadolinium gallium garnet (GGG). The YIG samples \nwere cut from the same GGG/YIG w afer in the form of \nrectangles with lateral dimensions 2.5 x 8.0 mm2. The \nsamples were prepared in the following way. The \ngraphene  layers were  grown on Cu foils inside a CVD \nchamber at 1000 °C with a mixture of CH 4 (33% volume) \nand H 2 (66% volume) at 330 m Torr for 2.5 hours60,61. A \n120 nm thick layer of PMMA is spun on top of the \ngraphene/Cu surface and then the Cu is  etched away. \nThe PMMA/graphene st ructure  is now transferred to the \nYIG film. Finally, the PMMA is removed using solvents \nand the sample receives two silver paint contacts as \nillustrated in Figure 1(a) [See Sup. Mat., Section I and \nFigure S1, for further details on the growth conditions as \nwell as the  trans ference  of graphene]. For sample B \nFigure 1.  (color online) (a) Sketch showing the YIG/graphene structures and the electrodes used to measure the dc \nvoltage due to th e IREE charge cur rent in the graphene layer resulting from the spin currents generated by microwave \nFMR spin pumping. (b) STM image sh owing the surface details from an arbitrary region in the SLG/YIG sample. (c) \nHigh -resolution image showing the typical hexagons of the graphene deposited on YIG. (d) Raman spectroscopy \nspectra of GGG/YIG, SiO 2/graphene and GGG/YIG/graphene. (e) I-V curve of the YIG/SLG structure demonstrating \nthe formation of Ohmic contacts between the electrodes and the SLG. (f) Low-magnetic -field dependence of the Hall \nresistance at room temperature.  preparation, we followed the same procedure , but in this  \ncase the graphene is transferred directly to the substrate \nof SiO 2 (300nm)/Si(100) . As schematically shown in  \nFigure 5(a), the SLG surface is partially covered by a Py \nlayer leaving the ends free to coat two Ag electrodes.  \nWith this configuration, the effect of the Py conductivity \ndoes not affect the electrical detection of the spin -current \nto charge -current that occurs on the SLG. The scanning \ntunneling microscopy (STM) image  of Figure 1(b) \nshow s surface details  for the SLG/YIG sample, where it \nallows us to observe the flat and homogeneous  graphene \nsurface. The honeycomb hexagon lattice of the SLG , \ndeposited onto YIG,  is clearly seen by the high -\nresolution  STM  image shown in  Figure 1(c) . Additional \ncharacterizations of the  high-quality  YIG layer  used in \nthis work can be verified in Fig. S2 of the Sup. Mat., \nsuch as X -ray diffraction measurements, surface \nroughness, magnetic hysteresis, a s well as the magnetic \ndomain structures measured by  means of the magnetic \nforce microscopy technique . \nFigure 1(d) shows  the Raman spectroscopy spectra of \nGGG/YIG, SiO 2/SLG and GGG/YIG/SLG. The \nspectrum at the top of Fig.  1(d), shows the lines of \nGGG/YIG /SLG. The  characteristic peaks from the G \n(1580 cm−1) and 2D (2700 cm−1) bands of the SLG  \nconfirm the  successful transfer  of graphene  to the \nsurface of the  YIG layer . Figure 1(e) shows the \nmeasured linear I×V characteristics, indicating Ohmic \ncontacts between the Ag -electrodes and the graphene. \nThe magnetic -field dependence of the Hall resistance \ncan be seen checked in Figure 1(f), in the field range \nwhere the magnetization of YIG rotates out of plane.  \nIII - STM/STS EXPERIMENTS  \nScanning tunneling microscopy/spectroscopy \n(STM/STS) measurements were carried out using a \nNanosurf microscope (in constant -current mode) \noperating at room temperature (300 K) using freshly \ncleaved Pt -Ir tips. Spectroscopy data were obtained \nthrough numerical differentiation of measurements of \nthe tunneling current as a function of the s ample bias at \nselected points (averaged over ten individual STS \nmeasurements). The STM images were obtained \nemploying a voltage of 40.0 mV – 70.0 mV and a \ntunneling current of 1.0 nA – 1.5 nA.  In order to \ninvest igate the electronic response of the sample under \nthe application of a magnetic field ( H), STM/STS  \nmeasurements were carried out in atomically flat regions \nas well as in the vicinity of steps and edges. These are \nregions where the spin orientation of the electrons is \nchanged due to the presence of a  magnetic field62. A \nSTM image of the sample,  where one observes a region of low roughness with the presence of some steps,  is \nshown in Fig. 2 (a). STS  spectra, Fig. 2 (b) were measured \nat the points marked with the green circles in the STM \nimage and labeled as 1 (left plot), i.e., at a flat region and  \n2 (right plot), i.e., at a step edge. Both measurements \nwere repeated with and without the presence of a \nmagnetic field parallel to the sample surface. In this \nfigure the black and the red curves, in both panels, are \nrelated to STS spectra measured for H = 0.0  Oe and H = \n10.0 Oe , respectively. In both conditions a V -shaped \nspectrum, compatible with the typical electronic \nsignature of a monolayer graphene43, is observed.  In this \nmeasurement one also observes a natural broadening of \n4K BT ~ 100 meV, near the Dirac point, which is a \nconsequence of temperature of the experiment (300 K)63. \nThis result evidences that the graphene layer is \ndecoupled from the substrate for this particular region \nsince its local density of states (LDOS) is not \nsignificant ly affected by the substrate.  \nOn the contrary, for Figs.2(c) and 2 (d) a different \nscenario is observed. In Fig. 2(c) an atomically flat \nregion with the presence of some steps is presented. \nDifferently from the previous case, for this area of the \nFigure 2. (color online) (a) STM topography image of the \ngraphene/YIG sample showing a region of low roughness. (b) \nSTS spectra acquired at the points marked with the green \ncircles (shown in a) and labeled as 1 (left panel) and 2 (right \npanel). The black curves were measured in the absence of a \nmagnetic field and the red curves were measured for a \nmagnetic field of 1.0 0 mT parallel to the sample surface. (c) \nSTM image showing an atomically flat region with some steps \nin the left lower corner. (d) STS spectra acquired at the points \nmarked with the green circles (shown in c) and labeled as 1 \n(left panel) and 2 (right pane l). The black curves were \nmeasured in the absence of a magnetic field and the red curves \nwere measured for a magnetic field of 1.00 mT perpendicular \nto the sample surface. Scale bars are: (a) 50 nm and (c) 40 nm.  system a ma gnetic field perpendicular to the sample \nsurface was applied. STS spectra for points 1 (flat \nregion) and 2 (step region) indicated in this figure are \nshown in  Fig. 2 (d), in the left and right panels, \nrespectively. For the atomically flat region marked as 1, \nthe STS spectra slightly differ from the ones shown in \nthe left panel of Fig. 2 (b). In this last case the STS \nspectra observed do not present a completely linear \nrelationship between LDOS and energy, which indicates \na higher degree of coupling of the graphene sheet to the \nsubstrate. In addition, no significant changes are \nobserved due to the application of H. For the STS spectra \nacquired at point 2 (shown in the rig ht panel of  Fig. 2 (d)) \na distinct electronic behavior is noticed. Analyzing the \nLDOS measured at the step without the application of a \nmagnetic field one observes a peak at -0.25 V, which is \nprobably related to an edge state. Interestingly, this peak \nis highly suppressed in the presence of a magnetic field \nof 10.0 Oe showing that the LDOS is influenced by H \nunder this condition. This effect may be directly related \nto the change of the magnetoresistance for different \nvalues of magnetic fields reported in ref erence  [38]. \nThus, this result indicates that the changes observed in \nthe magnetoresistance for H = 0.0  Oe and H  = 10.0 Oe \nmust be are related to the local electronic density of edge \nstates.  \nIV - XAS AND XMCD EXPERIMENTS  \nElectronic orbital orientation and element -specific \nmagnetic characterization were performed b y X-ray \nabsorption spectroscopy , i.e., X -ray linear (XLD) and \nmagnetic circular dichroism, respectively. The measurements were carried out at the U11A -PGM \nbeamline of the Brazilian Synchrotron Ligh t Laboratory \n(LNLS) near both the carbon K and the iron L 3,2 \nabsorption edges, at room temperature by total electron \nyield (TEY) mode. In order to determine the average \nspatial orientation of the molecular orbitals (π or σ) at \nthe YIG/SLG interface, severa l XAS spectra were \nacquired by keeping the electric field of the linearly \npolarized light pointin g along the vertical direction \nwhilst the sample was tilted so that the angle between the \nelectric field and the surface normal was varied from 10° \nto 90° (see  inset of Figure 3 (a) for details on the \ngeometry). To check the magnetism at both carbon and \niron absorption edges, the XAS spectra were collected \nunder four configurations of applied magnetic field and \npolarization of light (+/ -300 Oe and right/left -handed \ncircular polarization), with a grazing incidence angle of \n15°. Although it is faster to change polarization than \nmagnetic field at the end station/beamline utilized for the \nexperimen t, the strong X -ray natural circular dichroism \n(XNCD) due to the chira lity of graphene requires the \ninversion of the magnetic field to get rid of the \ncontribution of the XNCD to the circular dichroism. \nTherefore, in order to get the best quality data, the \nXMCD asymmetry spectra were obtained by taking the \naverage of the diff erences (modulus) between the spectra \nacquired for two different polarizations for each \nmagnetic field direction ( XMCD=\n[|TEY_(H_+,μ_+ )−TEY_(H_+,μ_− ) |+TEYH−,μ+−\nTEY_(H_−,μ_− ) |]⁄2 where H+H−⁄ and μ+μ−⁄stand \nfor the direction of the magnetic field and for t he hand \nof the circu lar polarization, respectively).  \nFigure 3.  (color online) (a) Spectra acquired around the carbon K edge of the YIG/graphene, with the el ectric field \nvector of the linearly polarized beam pointing along the vertical direction, for several incidence angles α. The dash -\nlined spectrum is acquired from a HOPG reference sample with an incidence angle of 15°. (b) XAS spectra acquired \naround the F e L3,2 (left) and C K (right) edges, for both right and left -handed circular polarizations of the light, under \nan applied magnetic field of 300 Oe and at an incidence angle of 15°. The bottom of the figures presents the XMCD \nasymmetries.  The average spatial orientation of the molecular \norbitals at the YIG/SLG  interface can be  addressed by \nacquiring several X-ray absorption  spectra for different \norientations of the sample out -of-plane direction with \nrespect to the electric field direction of the linearly \npolarized X -ray beam. In this experiment, the absorption \nintensity of a specific orbital final state presents a \nmaximum if the electric field is parallel to the direction \nof maximum density of states of the molecular orbital. \nOn the other hand, the absorption intensity vanishes if \nthe electric field is perpendicular to the molecular orbital \naxis. Figure 3(a) presents spectra a cquired near the \ncarbon K edge, with the electric field vector of the \nlinearly polarized beam pointing along the vertical \ndirection, for several incidence angl es 𝛼. In th e geometry \nillustrated in the inset of the figure, the electric field of \nthe light is  along  the surface normal at  𝛼=0°, and \nparallel to the surface at  𝛼=90°. The changes of the \nXAS lineshape show that the spectral feature around 285 \neV, which is assigned to the C 1s→π∗ transition of 1s \ncore electrons into unoccupied π∗ states, is remarka bly \nstrong for grazing incidence angles and tends to vanish \nas the incidence angle is increased towards normal \nincidence. This strong angular dependence of the XAS \nlineshape is due to different orbital orientations and \ndemonstrates that the π∗ unoccupied s tates are aligned \nout-of-plane , as it is expected for the C p z orbitals in the \ngraphene -type layered sp2 coordination.  This agrees with \nthe structural characterization by STM and also proves \nthe good quality of the SLG.  \nIt is worth noting the spectral features around 286 -\n290 eV, which is ascribed to the PMMA presence on the \nsample surface (residue of the graphene transferring \nprocess) and jeopardizes the analysis of the spectra as it \nis in the very middle region between the spectral features \ndue to C 1s →π∗ (283-289 eV) and C 1s→ σ∗ (289-315 \neV) transitions. A comparison with a spectrum around \nthe C K edge of highly oriented pyrolytic graphite (dash -\nlined spectrum in Figure 3(a), acquired from a HOPG \nsample with an incidence angle of 15°) allows one to \nnotice changes in the lineshape of the YIG/SLG \nspectrum. These changes were attributed to \nchemisorption as suggested in references64 and65. Small \nshifts in energy and remarkable broadening of the π∗ and \nσ∗ resonances suggest orbital hybridization and electron \nsharing along the interface , with delocalization of the \ncorresponding core -excited state. Such a covalent \ninterfacial bonding between carbon atoms in the SLG \nand iron atoms in the YIG could be the origin of the \nmagnetic proximity effect detected in carbon atoms. In a \nvery similar wa y as it has been demonstrated  for the \nsystem Ni(111)/graphene64,65, we suggest that a hybridization between graphene π and valence -band \nstates at the YIG/graphene interface leads to partial \ncharge transfer of spin -polarized electrons from the \nsubstrate to C atoms in graphene. In this sense, an \ninduced magnetic moment of carbon atoms can be \ndetected by XMCD, as shown in Figure 3(b). \nFigure 3(b) shows XAS spectra acquired around the \nFe L 3,2 (left) and C K (right) edges, for both right and left \nhand circular polarizations of light, under an applied \nmagnetic field of 300 Oe  and at an incidence angle of \n15°. The bottom of the figures present the XMCD \nasymmetries, which were evaluated by taking the \naveraged difference between the difference of the \nspectra taken with two different polarizations under 300 \nOe (shown in the figure ) and the difference of the spectra \ntaken under -300 Oe (not shown), as explained in the \nmethods section . The Fe L 3,2 XAS/XMCD spectrum is \nin agreement with published spectroscopic data of \niron(III) coordinated as it is in YIG66. The most \nimportant result of the XAS experiments is the clear \nobservation of a dichroic signal at the C K absorption \nedge. Despite the presence of the PMMA feature in the \nXAS spectra, it is clear in the XMCD spec trum that the \nasymmetry has its maximum at the 1s→π∗ resonance. In \nother words, the major magnetic response of the carbon \natoms arises from transitions of the 1s electrons into the \nunoccupied π∗ states, the transitions to the σ∗ states \nyielding a very weak  magnetic signal. This result, also \nin accordance with references64 and65 for \nNi(111)/graphene, suggests that mostly the C 2p z \norbitals of the SLG are magnetically polarized due to the \nhybridization with the valence -band states of the YIG \nsubstrate. In reference67the authors report on magnetic \nsignals of carbon observed by the X -ray resonant \nmagnetic reflectivity technique, the ferromagnetism of \ncarbon in the Fe/C multilayered system being related to \nthe hybridizati on between the Fe 3d band and the C p z \norbitals. Besides, there are other reports on magnetism \nin the Fe/C interface investigated by XMCD and on the \nmagnetism  induced in carbon nanotubes on \nferromagnetic Co substrate due to spin -polarized charge \ntransfer a t the interface68,69. \nThe XAS and XMCD spectra of a sample Py/SLG \nwere also acquired (see Figure S3 in the supplementary \nmaterial s). Similar to the results obtained for the sample \nYIG/SLG, the XAS spectra for the Py/SLG , measured \nwith linear polarization and varying the incide nce angle , \nconfirm the good quality of that sam ple, with  the π∗ \nunoccupied states of carbon pointing  out-of-plane. In its \nturn, the circular dichroism measured at carbon K edge \nalso present a remarkable asymmetry around the feature \nrelated to the π∗ unoccupied states. Combined with the XMCD spectra at the Ni L3,2 the results also suggest that \nthe C 2p z orbitals of the SLG are magnetically polarized \ndue to the hybridization with the valence -band states of \nthe adjacent Py layer.  \nV - SPIN -PUMPING EXPERIMENTS  \nIn the spin pumping ex periments, the FM /SLG  \nbilayer sample is  under action of both a small radio -\nfrequency (rf)  magnetic field applied perpendicular to a \ndc magnetic field in the ferromagnetic resonance (FMR) \nconfiguration. The precessing magnetization  𝑀⃗⃗  in the \nFM layer generates a spin current density at the FM/ SLG  \ninterface,  𝐽 𝑆=(ℏ𝑔𝑒𝑓𝑓↑↓4𝜋𝑀2⁄)(𝑀⃗⃗ ×𝜕𝑀⃗⃗ \n𝜕𝑡), where  𝑔𝑒𝑓𝑓↑↓ \nis the real part of the effective spin mixing conductance \nat the interface that takes into account the spin -pumpe d \nand back -flow spin currents6. The  net spin current that \nflows into the adjacent conducting layer produces two \neffects: (i) increase s the  damping of the magneti zation  \ndue to the flow of spin  angular momentum  out to the \nadjacent material ; (ii) due to the inverse Rashba \nEdelstein effect  (IREE) , the spin current that flows out \nof the FM layer is converted  into a perpendicular charge \ncurrent that produces a dc-voltage at the ends of the \ngraphene layer , measured  between the two Ag \nelectrodes. By investigating, t hese two independent  \nphenomena it is possible to extract material parameters \nfrom the measurements of the FMR absorption and the spin-pumping voltage.  \nIn the microwave FMR absorption and spin -charge \nconversion experiments the sample with electrodes , \nshow  in Fig. 1(a) and 5(a), is mounted on the tip of a \nPVC rod and inserted into a small hole in the back wall \nof a rectangular microwave cavity operating in the TE 102 \nmode , in a nodal position of maximum rf magnetic field \nand zero electric field.  For sample A, we did not use a \nmicrowave cavity because the detuning of the resonance \nat the strong FMR absorption  of YIG introduce s \ndistortions in the lineshapes . The waveguide is placed \nbetween the poles of an electromagnet so that the sample \ncan be rotated while maintaining the static and rf fields \nin the sample plane and perpendicular to each other. \nWith this configuration we can investigate the angular \ndependenc e of the FMR absorption spectra. Field scan \nspectra of the derivative  𝑑𝑃𝑑𝐻⁄  of the microwave \nabsorption are obtained by modulating the dc field with \na weak ac field at 1.2 kHz  that is used as the reference \nfor lock-in detection. Figure s 4(b) and 5(b) show the \nderivative of the FMR spectra of the two samples \ninvestigated in this work . Both spectra were  obtained \nwith H in-plane and normal to the long sample  \ndimension, frequency  𝑓=9.4 GHz and input \nmicrowave power  of 80 for sample A  and 24 mW  for \nsample  B. In Fig. 4(a), for a bare YIG film, one clearly \nsees the various lines corresponding to standing spin -\nwave modes with quantized wave numbers k due to the \nFigure 4.  (color online) (a) and (b) shows the field scan FMR microwave absorption derivative spectra for YIG and \nYIG/SLG respectively, with the magnetic field is applied in the film plane, normal to the long dimension. (c) Driving \nfrequency versus FMR resonance field, where the solid red li ne corresponds to best fit of Kittel equation. (d) Field \nscan of dc spin pumping voltage measured with 9.4 GHz microwave driving with power of 80 mW in YIG/SLG for \nthree different directions between in plane magnetic field and detection electrodes. (e) Field scan dc SPE voltage for \ndifferent input microwave power. The results shows the linear variation of the peak voltage with microwave power in \nYIG/SLG. (f) SPE voltage for four different driving microwave frequencies for 𝜙=0°, 180° . boundary conditions at the edges of the film. The \nstrongest line corresponds to the uniform (FMR) mode \nthat has frequency close to the spin -wave mode with  𝑘=\n0, the lines to the left of the FMR correspond to \nhybridized standing spin -wave surface modes whereas \nthose to the right are volume modes. All modes have \nsimilar peak -to-peak linewidth of 0.39 Oe,  \ncorresponding to a half -width -at-half-maximum \n(HWHM) of Δ𝐻≈0.34 Oe of direct microwave \nabsorption . Fig 4(b) show the FMR absorption for \nYIG/SLG samp le. Due to the spin injection  into \ngraphene the linewidth of YIG/SLG increases up to \nΔ𝐻𝑌𝐼𝐺𝐿𝑆𝐺⁄=0.90 Oe. As can be seen in Fig. 4(b), the \ncontact of the YIG film with the graphene  layer increases \nthe FMR linewidth of YIG. From the line broadening we \ncan obtain a good estimate for the spin mixing \nconductance 𝑔𝑒𝑓𝑓↑↓, using the relation 𝑔𝑒𝑓𝑓↑↓=\n4𝜋𝑀0𝑡𝐹𝑀\nℏ𝜔(Δ𝐻𝐹𝑀−Δ𝐻𝐹𝑀𝑆𝐿𝐺⁄), from which we obtained \n𝑔𝑒𝑓𝑓↑↓≈1014cm−2 from the YIG/SLG interface.  \nFigure 4(c) show s the FMR measurements carried at \nseveral microwave frequencies employing a  FMR  \nbroadband spectrometer. The microwave excitation was \ncarried out  by means of a microstrip transmission line of \ncopper, 0.5 mm wide and characteristic impedance Z0 = 50 W that was fabricated on a Duroid substrate, where \nthe return is the ground plane on the back side of the \nboard.  The YIG film placed on top o f the microstrip line \nseparated by a 60 μm thick Mylar sheet is excited by the \nrf magnetic field perpendicular to the static field H \napplied in the film plane. By keeping the frequency \nvalue fixed and sweeping the static  magnetic  field, the \nvariation of th e transmitted power due to the FMR \nabsorption is  detected by a Schottky diode . The applied \nmagnetic field is modulated with frequency 8.7 kHz and \namplitude 0.45 Oe so as to allow lock -in detection of the \nfield derivative of the power absorption. The solid red \nline in the Fig. 4 (c) was obtained from the fit with Kittel \nequation,  𝑓=𝛾[(𝐻𝑅−𝐻𝐴)(𝐻𝑅+𝐻𝐴+4𝜋𝑀𝑆)]12⁄, \nwhere 𝛾=𝑔𝜇𝐵ℏ⁄ (≈2.8 𝐺𝐻𝑧/𝑘𝑂𝑒) is the \ngyromagnetic ratio, 𝑔 is the spectroscopic splitting \nfactor, 𝜇𝐵 the Bohr magneton,  ℏ the reduced Plank \nconstant,  𝐻𝐴 the in -plane anisotropy field, and  4𝜋𝑀𝑆=\n1760 G the saturation magnetization.  With the  fit shown \nin Fig. 4(c) we obtained 𝐻𝐴=28 𝑂𝑒. \nThe net spin current that reach es the YIG/SLG \ninterface creates a spin accumulation on SLG, this \naccumulation can be converted into charge current , we \ncan write the dc spin pumping spin -current density at \nYIG/SLG interface generated by magnetization \nprecession as  \n𝐽𝑆(0)=ℏ𝜔𝑝𝑔𝑒𝑓𝑓↑↓\n16𝜋(ℎ𝑟𝑓\nΔ𝐻)2\n𝐿(𝐻−𝐻𝑅),     (1) \nwere Δ𝐻 and 𝐻𝑅 are the linewidth and fi eld for re sonance \nof YIG/SLG bilayer, 𝐿(𝐻−𝐻𝑅) represents  the \nLorentzian function , 𝑝 is the pre cession ellipticity  and \n𝜔=2𝜋𝑓 and ℎ𝑟𝑓 are the frequency and amplitude of the \ndriving microwave magnetic field, respectively18. This \nspin current is converted into a transversal charge \ncurrent, at the SLG by means of the IREE, which can be \ndetected by measuring a dc voltage that sets up between \nthe two Ag electrodes. The dc voltage is measured by  a \nnanovoltmeter directl y connected to the edges of sample \nas sketched  in Figures 1(a) and 5(a). The direct \nmeasurement of the spin -pumping voltage  𝑉𝑆𝑃(𝐻) \nspectr a are obtained by sweeping the  dc magnetic  field \nwith no AC field modulation. By rotating the sample in \nthe plane,  we measured the  angular  dependence of the \nvoltage as a function of the angle  𝜙 (see inset Fig. 4(d)). \nFigure 4(d) shows the spectra of 𝑉𝑆𝑃(𝐻) obtained with \nmicrowave frequency  𝑓=9.4 GHz and incident power \nof 80 mW, for t hree field directions,  𝜙=0°,90°,180° . \nThe spectra exhibit a large peak at the FMR field position \nand lateral small peaks corresponding to the standing \nspin-wave mode s. Moreover, this intensity is \nFigure 5.  (color  online) (a) Sketch showing the \nSi/SiO2/SLG/Py(20 nm) structure. (b) Field scan of derivative \npower absorption of Si/SiO2/SLG/Py(20 nm). The best fit was \nadjusted with linewidth of 39 Oe, and the inset show FMR \nabsorption of Py on top of Si with 𝛥𝐻=23 Oe. (c) dc spin \npumping voltage in SLG/Py. (d) The blue line represents a \nsum of a Lorentzian (symmetric red line) and a Lorentzian \nderivate (antisymmetric green line) best fit form data of \n𝑉𝑆𝑃 (𝜙=180°). proportional to pumped spin current, the signal reverses \nat 180°  and fall to noi se level in 90°. It is important to \nmention th at the voltage measured in Fig 4 (d) with \nmicro -wave power of 80 mW is almost one order of \nmagnitude greater than the values reported in Ref [38], \nwhere  it was applied a rf power of 150 mW. The \nasymmetry between  the positive and negative peaks is \nsimilar to that observed in othe r bilayer systems and can \nbe attributed to a thermoelectric effect70,71. Fig 4(e) \nshows the field scan of spin pumping  voltage spectra for \ndifferent  input microwave power obtained for  the 𝜙=0° \nconfiguration . Note that the amplitude of the peak \nvoltage in YIG/SLG has a linear dependence with the \nmicrowave power, showing that nonlinear effects are not \nbeing excited  in the power range of the experiments . \nVI - SPIN -TO-CHARGE CURRENT \nCONVERSION  RESULTS  \nFigure 6(a) show s the schematic illustration of a spin \ncurrent generation drive n by the spin pumping effect,  \nwhere  the charge current  𝐽 𝐶 genera ted at the  SLG is \nsimultaneous ly orthogonal to the dc magnetic field  𝐻⃗⃗  and \nto the spin current  𝐽 𝑆, for either YIG or Py.  In systems as \nYIG/SLG38, Ag/Bi34 and LAO/STO35, the presence of \ntwo-dimensional SOC driven by  the Ras hba effect72, as \nillustrated in  Fig 6(b). This effect creates a spin -split in \nthe energy dispersion surfaces of the Rashba states, thus, \ncausing  a locking between  the momentum 𝑘 and the spin \nangular  of the electron. In this case, a 3D spin current \noverpopulates states on one side of the Fermi contour and \ndepopulates state s on the other, thereby, creating a shift \nΔ𝑘 on the Fermi contours which is equivalent to  a charge \ncurrent. This effect is called inverse Rashba -Edelstein \neffect34, as illustrated in Figure 6 (c), where a spin current \n(−𝑧 direction) with spin polarization ( −𝑥 direction) \ncreates a charge current ( −𝑦 direction). Using the relation34 𝑗𝐶=(2𝑒ℏ⁄)𝜆𝐼𝑅𝐸𝐸𝐽𝑆, between the 3D spin \ncurrent in Eq. (1) and 2D charge current in SLG we can \nestimate the 𝜆𝐼𝑅𝐸𝐸 which is a coefficient characterizing \nthe IREE, with dimension of length and proportional to \nthe Rashba coefficient, and consequently to the \nmagnitude of the SOC in graphene. The charge current \ndensity produces a measured voltage described as \n𝑉𝐼𝑅𝐸𝐸=𝑅𝑆𝑤𝑗𝐶. Using Eq. (1)  one can express the IREE \ncoefficient in terms of the measured voltage peak value \nby, \n𝜆𝐼𝐸𝐸=4𝑉𝐼𝑅𝐸𝐸𝑝𝑒𝑎𝑘\n𝑅𝑆𝑤𝑒𝑓𝑝𝑔𝑒𝑓𝑓↑↓(ℎ𝑟𝑓Δ𝐻⁄)2,     (2) \nwhere 𝑅𝑆 is the  shunt  resistance and 𝑤 is the width of \nSLG . Since YIG is insulating, here 𝑅𝑆  is the resistance \nof the graphene layer. Computing the equation (2) with \nthe parameters: 𝑉𝐼𝑅𝐸𝐸𝑝𝑒𝑎𝑘=40 𝜇V, 𝑅𝑆=9 kΩ, 𝑤=3 mm, \nℎ𝑟𝑓=3.5×10−2Oe and 𝑝=0.3, we obtain 𝜆𝐼𝑅𝐸𝐸≈\n0.002 nm, which is about twice the value measured for \nthe graphene reported  in a previous paper38. On the other \nhand, this value is about 2 orders of magnitude smaller \nthan the one obtained for the Ag/Bi interface34, and about \none order of magnitude smaller than the measured for the \ntopological insulators39,40. Figure 4(f) shows the \nmeasur ement  of the spin -pumping voltage with scanning \nH for several frequencies. Here amplitude of the spin -\npumping voltage falls abruptly as the frequency \ndecreases below 4 GHz. The reason is that in the YIG \nfilms with thickness in the μm range73 the three -magnon \nnonlinear process have the coincidence of frequencies at \npower levels high, as 100 mW74. \nIn a further investigation,  we replac ed the injector of \nspin current by a metallic ferromagnetic material. I n this \ncase, a  layer of Py with 20 nm of thickness was deposited \non top of a SLG transferred on Si/SiO( 300 nm) substrate. \nFig 5(a) shows the sketch of the heterostructure \nFigure 6.  (color online) (a) Schematic of the Spin -to-charge conversion by the inverse Edelstein effect in FMR spin -\npumping experiments on the graphene. (b) The spin -polarized band structure (electron energy E as a function of in -\nplane momentum k) of 2D electron states at a Rashba interface. (c) The unbalanced spin states resulting from the 3D \nspin current creates a 2D charge current in the contour of Rashba interface, i.e inverse Edelstein effect.  Si/SiO/SLG/Py  with lateral dimensions of 3×1.5 mm. \nOnce again, the sample was mounted on the tip of a PVC \nrod and inserted through a hole in the center of a back \nwall of a rectangular micr owave cavity operating in the \n𝑇𝐸102 mode, at a frequency of 9.4 GHz  with a Q factor \nof 2000.  Fig. 5(b) show s the FMR absorption , at 24 mW,  \nfor SiO 2/SLG/Py(20 nm) . The FMR line shape  was fit \nwith the derivative of a Lorentzian function and the \nextracted FMR  linewidth was 39 Oe. The inset  of Fig. \n5(b) shows the FMR absorption spectrum for an \nidentical Py layer deposited on  Si substrate,  with a \nlinewidth of Δ𝐻𝑃𝑦=23 Oe. As occu rred in YIG/SLG, \nthe broadening of  the FMR  linewidth of Py in contact \nwith graphene layer produces an additional damping due \nthe spin pumping . We obtain ed 𝑔𝑒𝑓𝑓↑↓≈1015cm−2 for \nthe SLG /Py interface , which is a value similar to the one  \nobserved in  Py/Pt interface19,20,75. The spin pumping \nvoltage shown if Fig 5(c) was measured by exciting the \nFMR with the same microwave power. The scan field \nspectra exhibit the same feature as  the VSP measured in  \nYIG/SLG,  so that the voltage peak occurs at the FMR  \nfield value . However, the direction of spin injection in to \nSLG/Py it’ s opposite in comparison  of YIG/SLG, once \nthat the ferromagnetic material  is now on top of  \ngraphene, therefore  was measured an opposite spin \npumping voltage for the same direction , i.e.,  the \nmaximum  positive  value  occurs  for 𝜙=180° , negative \nfor 𝜙=0°, and null value for 𝜙=90°. The field \ndependence voltage 𝑉(𝐻) in the case of Py, can be \nadjusted by the sum of two components,  as shown in Fig. \n5(d), where the solid blue line of Fig 5(c), was adjusted \nby 𝑉(𝐻)=𝑉𝑠𝑦𝑚𝐿(𝐻−𝐻𝑅)+𝑉𝑎𝑠𝑦𝑚𝐷(𝐻−𝐻𝑅). Here  \n𝐿(𝐻−𝐻𝑅) is the (symmetric - solid red line) Lorentzian \nfunction and 𝐷(𝐻−𝐻𝑅) is the (antisymmetric – solid \ngreen line) Lorentzian derivative centered about the \nFMR resonance field . Taking  into account,  the \namplitude of the microwave field in Eq (2) in O e is \nassociated to the incident power, in W, by ℎ𝑟𝑓=\n1.776(𝑃𝑖)12⁄, 𝑅𝑆=2.5kΩ, 𝑤=1.5mm, and 𝑉𝑠𝑦𝑚=\n𝑉𝐼𝐸𝐸𝑝𝑒𝑎𝑘=1.14 𝜇V which correspond s to the amplitude of \nsymmetric component of 𝑉(𝐻), we obtain   𝜆𝐼𝑅𝐸𝐸≈\n0.003 nm, very similar  with the previous value for  \nYIG/SLG . The antisymmetric component 𝑉𝑎𝑠𝑦𝑚=𝑉𝐴𝑀𝑅=0.52 𝜇V is related to the contribution of the \nanisotropic magnetoresistance (AMR) that occurs on the \nPy and reflects on the edges of graphene layer18,76. \nVII - CONCLUSIONS  \nIn this investigation we have shown that the carbon \natoms of single layers of graphene acquires an induced \nmagnetic moment  due to the proximity effect with \nferromagnetic surfaces. This astonishing result was \nconfirmed by means of X -ray absorption spectroscopy \n(XAS) and magnetic circular dichroism (XMCD) \ntechniques, which shows  that the C 2p z orbitals of the \nSLG  are magnetically polarized due to the hybridization \nwith the valence -band states of the YIG  and Py  \nsubstrate s. We have also demonstrated the optimal \nconversion of a spin current into a charge current in large \narea of single layers of graphene deposited on  YIG and \nPy films, which is attributed to the inverse Rashba -\nEdelstein effect (I REE). The results obtained for the \nSLG/YIG and SLG/Py systems revealed  very similar \nvalues for the I REE parameter, which are larger than the \nreported values in the previous stu dies for SLG38. We \nbelieve that the spin -to-charge conversion efficiency can \nbe attributed to the excellent structural and interface \nquality of SLG/FM heterostructures produced, which \noptimizes the transfer of spins. 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Quantifying Spin Hall Angles from Spin \nPumping: Experiments and Theory. Phys. Rev. Lett.  104, \n046601 (2010).  \n \nSUPPLEMENTAL MATERIAL  \nI. Conditions for growth and transferring of \ngraphene  \nThe sample A (YIG/SLG) was prepared in the \nfollowing way. Graphene is grown on Cu foils inside a \nCVD chamber at 1000 °C with a mixture of CH 4 (33% \nvolume) and H 2 (66% volume) at 330mTorr for 2.5 \nhours. A 120 nm thick layer of PMMA is spun on top \nof the graphene/Cu surface and Cu is then etched away60,61. The PMMA/graphene structure is then \ntransferred onto the YIG film, with lateral dimensions \nof 2.5 x 8.0 mm2. Finally, the PMMA is removed and \nthe sample receives two silver paint contacts as \nillustrated in Fig. 1 (a). Figure S1 shows all the steps \ninvolving the transfer of graphene. In the case of \nSample B (Si/SiO 2/SLG/Py) the graphene is transferred \nonto Si/SiO 2 substrate with lateral dimensions of 1.5 x \n3.0 mm2, by the same steps of the previous systematics.  \nFigure S1 . (color online) The sequence of steps made in order to insure the best transference of the single -layer -\ngraphene to YIG. Using a similar process, the SLG is transferred to Si/SiO 2.  \nAfter this process is deposited a rectangle of Py film \nwith a thickness of 20 nm at mid part of graphene, using \na 1.5 x 2.0 mm2 shadow mask, as seen in the Fig 5(a). \nOnce again two silver paint contacts are made on the \nedges of graphene.  \nII. Additional information about the sample \npreparation  \nThe YIG samples consist in a single -crystal yttrium \niron garnet (111) films, with thickne ss of in a 6 𝜇m, \ngrown by liquid phase epitaxy (LPE) onto a 0.5mm thick \n(111) -oriented Gd 3Ga5O12 (GGG) substrates. It is \nimportant to emphasize that the YIG films were grown \nby standard LPE technique from a supersaturated melt in \nwhich Y 2O3 and Fe 3O3 are added to a flux of PbO+B 2O3. \nThe quality of the YIG samples is attested by the very \nsmall FMR linewidth, which is 0.3 Oe (See Fig. 4(a)). \nAny composition different from Y 3Fe5O12 or phase \ndifferent from the garnet phase would be impossible to \nprovide such n ice small linewidth. In turn, the Py films \nwere deposited by DC sputtering on SiO 2/SLG with base \npressure of \n7100.2  Torr, and  the deposition pressure \ncorresponds to 3 mTorr argon atmosphere in the sputter -\nup configuration, with the substra te at a distance of 9 cm \nfrom the target. A 10 min pre -sputtering procedure was \nused for cleaning of the targets.   \nIII. The X -ray diffraction, magnetic hysteresis, \nand AFM/MFM characterizations in the YIG films  \nThe crystallographic structure of the YIG was \nassessed by X -ray diffraction (XRD) measurements. \nThe out -of plane XRD pattern is shown in Fig. S2(a). \nThe diffraction patterns were obtained at angles \nbetween 10° and 90° (2θ). Only the line associated with \nthe (444) crystal plane of YIG appears in the out -of-\nplane XRD pattern, indicating that no impurity phases \nprecipitated. The XRD spectrum at high resolution \ndetailing the position of the peaks of the YIG film and \nthe GGG substrate is shown in the insert. The  results of \nXRD measurements indicate that the present YIG film \nis epitaxially grown on the GGG substrate. The XRD \npatterns were recorded using the Bruker D8 Discover \nDiffractometer equipped with the Cu Kα radiation \n(λ=1.5418 Å).  \nFigure S2(b) shows an atom ic force microscopy \n(AFM) image of the YIG film surface. The line trace in \nFig. S2(b) confirms the uniformity of the YIG film \nsurface with very small roughness. Figure S2(c) shows \na magnetization hysteresis curve for a representative \nYIG sample measured by  VSM (Vibrating Sample \nMagnetometer). The data show a small hysteresis, with \na coercive field 1.8 Oe and remanent magnetization \n0.12 of the saturation value. Figure S2(d) shows an \nFigure S2 . (color online) (a) Out -of-plane XRD patterns (ω -2θ scans) of YIG film grown on GGG substrate. The \nXRD spectrum detailing the position of the peaks of the YIG film and the GGG substrate is shown in the insert. (b) \nAFM image of the YIG film surfa ce. The line trace in (b) confirms that the YIG surface has a very small roughness. \nFigure (c) shows a VSM hysteresis curve for the YIG samples used in this work. (d) MFM image of the as -grown 6 -\n𝜇m-thick YIG film showing up - and down -oriented domains as r egions of light and dark contrast.  magnetic force microscopy  (MFM) image for the \ndemagnetized state in the 6 -𝜇m thick YIG film, which \nexhibits a stripes domain structure typical. All the \nmeasurements were obtained at room temperature.  \nIV. X -ray absorption and magnetic circular \ndichroism  of the Py/SLG sample  \nFigure S3(a) shows carbon K edge x -ray absorption \nspectra acquired with linear vertical polarization of the \nelectric field for sample Py/SLG. Similarly to the resul ts \nobtained for sample YIG/SLG (main text), the angular \ndependence of the XAS lineshape confirms that the π∗ \nunoccupied states (feature around 285 eV, C 1s→π∗ \ntransition) are aligned out -of-plane. This result also \ncorroborates the good quality of the SLG t ransferred to \nPy thin film.  \nFigure S3(b) shows XMCD asymmetry spectra \nmeasured at both Ni L 3,2 and carbon K edges. The Ni spectrum is representative of the Py  layer and \ncharacteristic of the metallic environment of this \nmaterial, whilst the C spectrum is somehow similar to \nthat of the YIG/SLG, presenting a remarkable \nasymmetry around the feature related to the π∗ \nunoccupied states. This result is also in accord ance with \nreferences62 and63 for Ni(111)/graphene, suggesting that \nthe C 2p z orbitals of the SLG are magnetically polarized \ndue to the hybridization with the valence -band states of \nthe adjacent Py layer.  \nV. The EDX spectrum of graphene grown by CVD  \nThe energy dispersive X -ray (EDX) spectrum of SLG \nby CVD onto the YIG film can be seen in Figure S4(a). \nThe EDX spectrum taken from an arbitrary region of the \nsample shows the presence only of yttrium (Y), iron \n(Fe), oxygen (O) of the YIG film, and carbon (C) of \nSLG.  Different samples were prepared, in order to \nFigure S3 . (color online) ( a) XAS spectra of Py/graphe ne measured at α=15°. ( b) XAS spectra acquired around the \nNi L 3,2 (left) and C K (right) edges, for both right and left -handed circular polarizations of the light, under an applied \nmagnetic field of 300 Oe  and at an incidence angle of 15°. The bottom of the figure s presents the XMCD \nasymmetries . \nFigure S4 . (color online) ( a) EDX spectrum from an arbitrary region in the sample of GGG/YIG/graphene . (b) The \nEDX spectrum of SLG grown by CVD onto the nickel grids. The additional peaks of the Ni, Al and Mg in the EDX \nspect rum are due to the presence of Ni TEM grids and the stub (compound of Al and Mg).  confirm the results EDX measurements.  In Figure \nS4(b), the samples for EDX measurements are prepared \non a standard transmission electron microscope (TEM) \nnickel grids and are hence suspended graphene samples \n(as shown at the top of the insert). The image of the Ni -\ngrid/SLG sample analyzed by electron microscopy is \nshown on the bottom of the insert of Figure S4(b). The \nEDX spectrum shows the presence only of oxygen (O) and carbon (C) of SLG.   The additional peak s of the Ni, \nAl and Mg in the EDX spectrum are due to the presence \nof Ni TEM grids and the stub (compound of Al and Mg) \nthat served as support on which the graphene samples \nare prepared for analysis. This result confirms that there \nare no visible traces of  Cu adatoms on the graphene \nlayer.  \n "    },    {        "title": "1911.00607v1.Tuning_interfacial_Dzyaloshinskii_Moriya_interactions_in_thin_amorphous_ferrimagnetic_alloys.pdf",        "content": "*yassine.quessab@nyu.edu Tuning interfacial Dzyaloshinskii-Moriya interactions in thin amorphous ferrimagnetic alloys Y. Quessab,1,∗ J.-W. Xu,1 C. T. Ma,2 W. Zhou,2 G. A. Riley,3,4 J. M. Shaw,3 H. T. Nembach,3,5 S. J. Poon,2 and A. D. Kent1  1Center for Quantum Phenomena, Department of Physics, New York University, New York, New York 10003, USA 2Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA 3Quantum Electromagnetics Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA 4Center for Memory and Recording Research, University of California San Diego, La Jolla, CA92093, USA 5JILA, University of Colorado, Boulder, Colorado 80305, USA Abstract  Skyrmions can be stabilized in magnetic systems with broken inversion symmetry and chiral interactions, such as Dzyaloshinskii-Moriya interactions (DMI). Further, compensation of magnetic moments in ferrimagnetic materials can significantly reduce magnetic dipolar interactions, which tend to favor large skyrmions. Tuning DMI is essential to control skyrmion properties, with symmetry breaking at interfaces offering the greatest flexibility. However, in contrast to the ferromagnet case, few studies have investigated interfacial DMI in ferrimagnets. Here we present a systematic study of DMI in ferrimagnetic CoGd films by Brillouin light scattering. We demonstrate the ability to control DMI by the CoGd cap layer composition, the stack symmetry and the ferrimagnetic layer thickness. The DMI thickness dependence confirms its interfacial nature. In addition, magnetic force microscopy reveals the ability to tune DMI in a range that stabilizes sub-100 nm skyrmions at room temperature in zero field. Our work opens new paths for controlling interfacial DMI in ferrimagnets to nucleate and manipulate skyrmions.    Introduction: Magnetic skyrmions due to their non-trivial topology have interesting properties1-3 that make them attractive for spintronic applications, such as racetrack memory and logic devices4-6. A magnetic skyrmion designates a chiral spin texture with a whirling spin configuration7. Skyrmions can be stabilized by broken inversion symmetry and chiral interactions, such as the Dzyaloshinskii-Moriya interactions (DMI)8,9, which is an antisymmetric exchange interaction that favors non-collinear neighboring spins. Ultrathin magnetic materials with interfaces to heavy non-magnetic metals with large spin-orbit coupling exhibit interfacial DMI that stabilizes skyrmions and chiral domain walls10-13. The interfacial DMI and the nucleation of skyrmions have been extensively investigated in ferromagnetic materials10,14-18. Very recently, magnetic skyrmions and chiral domains were reported in ferrimagnetic systems19-21. Nearly compensated thin ferrimagnetic films with interfacial DMI are interesting materials due to their low stray fields, reduced sensitivity to external magnetic fields, and fast spin dynamics, which are predicted to lead to ultrasmall and ultrafast skyrmions19,22. Unlike in ferromagnets where fast current-induced motion of chiral textures is impeded by the Walker breakdown and domain wall pinning13,23-25, high domain wall velocities––reaching 1000 m s-1––have been observed in ferrimagnetic CoGd films near the angular momentum compensation temperature19. Hence, ferrimagnetic thin films are promising candidates for ultrafast skyrmion-based spintronics.  Recently, bulk DMI was reported in an amorphous ferrimagnetic GdFeCo alloy26. However, the significant advantages of interfacial DMI are that it can be controlled by the nature of the interfaces and widely tuned to stabilize skyrmions. Yet, interfacial DMI has not been studied in this class of materials. The DMI competes with the perpendicular magnetic anisotropy and Heisenberg exchange interaction and tuning the DMI in a range that favors small skyrmions can be challenging. Asymmetric domain wall nucleation and motion is commonly used to measure DMI12,27. However, these methods require advanced models of the domain wall dynamics28,29. A more direct method is Brillouin light scattering (BLS) in the Damon-Eshbach geometry, which relies on the asymmetric spin-wave frequency dispersion in the presence of DMI30; the asymmetry is directly related to the strength of the DMI.  Here we present a systematic study of the interfacial DMI in CoGd thin films by BLS as a function of the capping layer composition and magnetic layer thickness. We aim to understand how the DMI in CoGd films depends on the structural symmetry and magnetic properties. The interfaces are studied by cross-sectional transmission electron microscopy (TEM). We found that as little as 10% of W in the cap layer in Pt/CoGd/Pt1-xWx thin films is sufficient to induce a DMI of about 0.15 mJ m-2, larger than the bulk DMI found in much thicker films26. We also observed that the DMI is inversely proportional to the magnetic thickness in asymmetric CoGd stacks, confirming the interfacial nature of the DMI. In addition, we were able to tune the DMI in a range that stabilizes sub-100 nm skyrmions at room temperature in zero field, as observed by magnetic force microscopy (MFM). Our findings provide insight into the key parameters that control the DMI in ferrimagnetic films toward achieving ultrasmall and ultrafast skyrmions.   Results: Tuning the DMI with the capping layer composition: The ferrimagnetic CoGd thin films were grown by RF magnetron co-sputtering on oxidized silicon wafers in the following sequence: W(3)/Pt(3)/Co78Gd22(t)/Pt1-xWx(3)/Pt(3) (thicknesses in nanometers) [Methods]. The W/Pt seed layer provides good adhesion to the substrate and texture to ensure perpendicular magnetic anisotropy (PMA). The top Pt layer prevents sample oxidation. The DMI of a 5-nm thick CoGd film was studied as a function of the W composition (x) of the cap layer Pt1-xWx. The magnetic properties of the films were measured by vibrating sample magnetometry (VSM) and are summarized in Table I (Methods). Figures 1(a) and 1(b) show an out-of-plane field room-temperature magnetization hysteresis loop and the temperature (T) dependence of the saturation magnetization (MS) for Pt/CoGd(5 nm)/W, respectively. MS (T) is greatly reduced around 150 K, corresponding to the magnetization compensation temperature (TM).  Spin wave spectroscopy using BLS was performed to measure the DMI in the CoGd films (Methods). The DMI leads to an asymmetric frequency dispersion of the counterpropagating spin waves30. The DMI energy D (mJ m-2) is proportional to the frequency shift (∆f DMI) and given by: |#|=\t&'\t()*+,(-./∆12-3,         (1) where g is the spectroscopic splitting factor (we take g = 2), μB the Bohr magneton and h Planck’s constant and k = 16.7 µm-1 is the spin wave vector. Notably, the DMI energy given by BLS measurements is an effective DMI averaged over the film thickness, i.e., a sum of the bottom and top interfacial contributions. An example of BLS spectra is displayed in Fig. 1(c) for Pt/CoGd(5 nm)/W. We fitted the spectra for positive and negative field polarity. The frequency shift was determined for the Stokes and the anti-Stokes peaks separately and then averaged.  The diameter of a skyrmion results from the competition between different energies such as the Heisenberg exchange energy, the magnetic anisotropy energy and the DMI strength. Ultrasmall skyrmions can be nucleated at room temperature only in a narrow range of D31-33. For DMI strength larger than a scale set by the magnetic anisotropy, the formation of stripe domains become energetically favorable32,33. Conversely, a weak DMI cannot stabilize a skyrmion. Theoretical work has predicted that ferrimagnetic materials are better candidates than ferromagnets to host ultrasmall and ultrafast skyrmions due to their low saturation magnetization, which causes only small stray fields22. Indeed, in ferrimagnetic materials, the interfacial DMI can dominate over the dipolar interactions and enable the formation of ultrasmall DMI skyrmions, which is difficult to achieve in ferromagnets. Hence, our goal was to provide a new method for controlling the interfacial DMI in thin ferrimagnetic CoGd films, which could allow one to precisely tune the DMI in a range that would enable skyrmion nucleation. Changing the nature of the CoGd interfaces can be used to engineer the DMI strength. Therefore, the idea is to leave the Pt underlayer at the bottom interface of the CoGd film unchanged and insert a Pt1-xWx alloy at the top interface. Thus, by changing the composition of the Pt1-xWx alloy, the structural symmetry of the film can be gradually broken to induce DMI. Pt is chosen for its strong spin-orbit coupling that gives a large interfacial DMI on Co spins34,35. Theoretical calculations based on Hund’s first rule have shown that, on the contrary, a weaker  DMI arises from interactions between W and Co34 and with the same chirality as Pt and Co. In addition, W, due to its giant spin-Hall angle36,37, would serve as a spin current source to enable skyrmion motion induced by spin-orbit torque (SOT). Figure 2(a) shows the DMI energy as a function of the W composition (x) in Pt/CoGd(5 nm)/Pt1-xWx measured by BLS. A maximum DMI of 0.23 ± 0.02 mJ m-2 is obtained for the asymmetric stack (x = 1). In comparison, a bulk DMI of up to 0.10 mJ m-2 was reported in ferrimagnetic GdFeCo films26. Conversely, the DMI is almost zero for the symmetric film (x = 0). Indeed, in the Pt/CoGd/Pt film, the top and bottom interfaces induce an interfacial DMI of similar amplitude but opposite sign (as the DMI is a chiral interaction), thus, resulting in a near vanishing effective DMI. In Pt/CoGd/W, since W gives rise to a weaker interfacial DMI, the contributions of the two interfaces are not compensated, leading to a larger effective DMI. As seen in Fig. 2(a), as little as 10 % of W introduced in the top layer is enough to significantly break the symmetry and induce a DMI of 0.15 mJ m-2. Yet, for x > 0.1, the DMI is less sensitive to the W content in the top interface. This would indicate that for 0.1 < x < 0.9, the W rather greatly reduces the DMI between the Co spins and the Pt. Indeed, if the W were actively contributing to the interfacial DMI, a stronger dependence of the DMI energy with the alloy composition would have been expected. Additionally, the quality of the interfaces, which has a great impact on the DMI38,39, was assessed by cross-sectional TEM. Figure 2(b) shows a cross-section of the asymmetric Pt/CoGd(5 nm)/W film, while a closer view of the top and bottom interfaces of the CoGd layer is displayed in Fig. 2(c). The CoGd alloy and the W layers are amorphous and the Pt is polycrystalline. Figure 2(c) shows that the Pt/CoGd and the CoGd/W interfaces are smooth.  Thickness-dependence of the DMI: It is necessary to study the dependence of the DMI on CoGd thickness to establish its nature, i.e. to know whether the DMI is arising from interfacial effects. In our Pt/Co78Gd22(t)/Pt1-xWx films, the W composition (x) was fixed either to 0 or 1 to investigate the DMI in a symmetric (x=0) and asymmetric (x=1) stack as a function of the magnetic thickness t. t was increased from 5 nm to 15 nm. The magnetic properties were systematically measured by VSM as a function of thickness. The results are presented in Fig. 3(a) and the DMI is plotted versus the inverse magnetic thickness. In the asymmetric Pt/CoGd/W stack, the DMI is inversely proportional to the magnetic thickness and reaches a minimum of 0.09 ± 0.01 mJ m-2 for 1/t = 0.067 nm-1 (t = 15 nm). The DMI linearly increases with the inverse thickness at a rate of ~ 1 mJ nm-1. This confirms that the strength of the DMI at the interface remains unchanged and underlines its interfacial nature. In Fig. 3(b), the saturation magnetization times the magnetic thickness is plotted versus the CoGd thickness. It has a linear dependence on thickness with an x-axis intercept near zero thickness, which indicates that there is no measurable dead layer in the CoGd film. Notably, in Fig. 3(a), the intercept of the linear fit is non-zero for 1/t = 0 (i.e. an infinitely thick film). This indicates that there is a residual DMI of 0.03 mJ m-2, which may result from a change of the magnetization compensation temperature throughout the thickness as evidenced in another rare-earth transition-metal alloy40 that could induce inversion symmetry breaking. Yet, as the thickness decreases, the interfacial effects become more important and the DMI increases as seen in Fig. 3(a). Thus, the interfacial DMI dominates in the entire thickness range we have studied. On the other hand, for the symmetric Pt/CoGd/Pt film, the DMI increases with the thickness to a value of 0.09 ± 0.02 mJ m-2 as seen in Fig. 3(a) (red data points). This behavior is surprising as the interfacial DMI is expected to be almost zero in symmetric layer structures. This result shows there is a difference in the nature of the top and bottom CoGd interfaces. In order to verify the latter, we performed TEM imaging in the Pt/CoGd(15 nm)/Pt film. The full stack is shown in Fig. 4(a) and a closer view of the top and bottom interfaces in Figs. 4(b) and 4(c), respectively. In Fig. 4(b), a thin layer of intermediate gray contrast (indicated by the white arrows) can be seen at the top CoGd interface and not in the bottom interface. It appears that the Pt from the capping layer has diffused into the amorphous CoGd film. As a result, the bottom and top interfaces have different roughness and intermixing. Hence, the DMI contributions of the top and bottom interfaces are not equal. Thus, due to the chirality of the interaction, they do not cancel out, leading to an increase of the net DMI. Intermixing and roughness effects appear to be more predominant in thicker films as the DMI increases with thickness in Pt/CoGd/Pt as seen in Fig. 3(a). Interestingly, the residual DMI observed for the asymmetric stack is of the same order of magnitude as that for the symmetric Pt/CoGd(5 nm)/Pt structure. Therefore, this suggests that this residual DMI is independent of the CoGd interfaces. Finally, as the skyrmion size depends on the magnetic film thickness22,33, it is thus important to understand how the interfacial DMI scales with the thickness. Evidence of magnetic skyrmions by MFM: Finally, we aimed to verify whether these thin ferrimagnetic alloy films would indeed host skyrmions. We focused on the asymmetric Pt/CoGd/W stacks as they are more promising for skyrmion motion via spin-orbit torque because of the giant spin-Hall angle of W36,37. In fact, in Pt/CoGd/Pt, the spin-orbit torques from the top and bottom interfaces would tend to cancel each other out. The Pt/CoGd/W films were subject to AC in-plane magnetic field demagnetization and imaged by atomic and magnetic force microscopy (AFM and MFM) at room temperature in zero field. Figure 5 shows images for Pt/CoGd(10 nm)/W. The left column is AFM data (Figs. 5(a) and (c)) and right column MFM images (Figs. 5(b) and (d)). The surface roughness was measured and is on the order of 0.2 nm (rms). Magnetic contrast is indicated by dark areas in the MFM images. By comparing the AFM and MFM images, it is clear that this contrast comes from magnetic textures and is not due to topography. Several skyrmion-like textures can be seen in Fig. 5(b). Figure 5(d) corresponds to a smaller MFM scan performed around of one of them marked by a square box in Fig. 5(b). This skyrmion-like texture is on the order of 100 nm. 50 nm skyrmions were observed in Pt/CoGd(8 nm)/W (see supplemental materials41). Arguably, considering the size of these textures, the DMI energy values [see Fig. 3(a)], and the fact that the CoGd films are weakly magnetized (MS ~ 140 – 150 kA m-1, see supplemental materials41), it is unlikely that these textures be magnetic bubbles4,7 stabilized by dipolar interactions. Thus, MFM images would rather indicate the presence of skyrmions. However, accurate estimation of the skyrmion size is difficult. Indeed, the MFM tip is sensitive to the dipolar field emerging from the magnetic texture which is spatially spread out. Furthermore, smaller magnetic features may be present in Fig. 5(b), yet they cannot be clearly distinguished due to the background noise and small magnetic contrast.  Discussion: To summarize, we have demonstrated that by capping the ferrimagnetic CoGd layer with a PtW alloy we could tune the DMI energy over a large range, from almost no DMI to an interfacial DMI energy of 0.23 mJ m-2. The DMI thickness dependence reveals the interfacial nature of the DMI in CoGd thin films. Thus, the DMI strength can be controlled by the interfaces in the thickness range we studied, which is also the range relevant for skyrmion nucleation. Moreover, the DMI was found to be non-zero in thicker symmetric structures emphasizing the role of interface roughness and intermixing. Lastly, we showed evidence that films can have a DMI in a range that allows sub-100 nm skyrmion nucleation at room temperature in zero field. Our experimental results provide insight into the key parameters that control the DMI in ferrimagnetic films toward achieving ultrasmall and ultrafast skyrmion motion for spintronic applications.  Methods: Thin film deposition:  The thin films were prepared by RF magnetron sputtering and deposited onto Si-SiO2 substrates at room temperature with a base pressure of 2.7´10-5 Pa. The Ar deposition pressures of W, Pt, CoGd, and Pt1-xWx were 0.93 Pa, 0.1 Pa, 0.16 Pa, and 0.16 Pa, respectively. CoGd films were obtained by co-sputtering from the Co and Gd targets. The powers of the Co and Gd sources were tuned to obtain CoGd films with approximately 78 at. % of Co. The deposition rates were calibrated using x-ray reflectometry.  Magnetometry: The magnetic properties of the samples were measured by vibrating sample magnetometry. Magnetic hysteresis loops were measured by varying the temperature from 100 K to 300 K with steps of 25 K in order to extract the temperature dependence of the saturation magnetization and the coercive field. Magnetometry was systematically performed prior to BLS experiments.  Brillouin light scattering: Spin wave spectroscopy using BLS is sensitive to interfacial effects and can be used to measure the DMI strength. The spin waves (SWs) inelastically scatter the monochromatic laser beam that is focused onto the sample surface. The frequency of the scattered photons is shifted by the SWs frequency. The SWs frequency is determined by analyzing the backscattered light with a (3 + 3)-pass tandem Fabry-Pérot interferometer. The counterpropagating Damon-Eshbach SWs have a non-reciprocal frequency dispersion characterized by a frequency shift (noted ∆fDMI). The frequency shift is considered here in absolute value. An in-plane bias magnetic field was applied to allow the SW propagate in-plane (Damon- Eshbach geometry). For a λ = 532 nm laser beam with an incidence of θi = π/4, the SW vector, k defined as k = 4πsin(θi)/λ was set to 16.7 μm-1.  Acknowledgments: This work was supported by DARPA grants No. D18AP00009 and R186870004 and by the DOE grant No DE-SC0018237.  Author contributions: A.D.K. and S.J.P. conceived of and supervised the project. Y.Q, H.T N., S.J.P. and A.D.K. planned the experiments. BLS measurements were conducted by G.A.R., J.M.S. and H.T.N. BLS data was analyzed by Y.Q., J.-W.X. A.D.K. and H.T.N. Sample fabrication and magnetometry measurements were performed by C.T M. and W.Z. MFM imaging was done by Y.Q. and J.-W.X. All of the authors participated in the discussion and interpretation of the experimental results. Y.Q. and A.D.K. drafted the manuscript and all of the authors contributed to and commented on it.   x MS (kA m-1) µ0HC (mT) TM (K) 0 160 12.5 125 – 150 0.1 166 11.0 125 0.25 160 10.0 125 0.5 180 12.0 125 – 150 0.75 160 12.0 125 – 150 1 145 11.0 125 – 150 TABLE I. Summary of the magnetic properties of Pt/Co78Gd22(5 nm)/Pt1-xWx films as a function of W composition (x). The room temperature saturation magnetization (MS) and coercive field (µ0HC) are indicated. \n  FIG. 1. (a) Out-of-plane magnetization hysteresis loop (b) and temperature dependence of the saturation magnetization measured by VSM for the Pt/CoGd(5 nm)/W sample. Magnetic compensation of this CoGd composition occurs around 150 K. (c) Spin wave spectroscopy obtained by BLS in Pt/CoGd(5 nm)/W. The shift in the frequency dispersion, ∆fDMI, is proportional to the DMI. The applied in-plane field was 0.460 T. The solid lines are fit to the BLS data obtained for positive (blue curve) and negative (red curve) field polarity.    StokesAnti-Stokes2ΔfDMI = 2.92 GHz-1000100-200-1000100200\n100150200250300050100150\n-30-20-101020300.00.51.0Magnetization (kA m-1)\nMagnetic Field (mT) Pt/CoGd(5 nm)/W(a)(b)\n(c)\nMagnetization (kA m-1)\nTemperature (K) Pt/CoGd(5 nm)/W\n  Positive field polarity  Negative field polarityPhoton counts (a.u.)\nFrequency (GHz) \n  FIG. 2. (a) DMI energy measured by BLS in Pt/CoGd(5 nm)/Pt1-xWx as a function of the W composition (x). The solid black line is a guide to the eye. (b) and (c) Cross-sectional TEM images of the Pt/CoGd(5 nm)/W film. (b) The full stack and (c) a magnified view of the top and bottom interface of the CoGd layer.   \n FIG. 3. (a) Magnetic thickness (t) dependence of the DMI in Pt/CoGd(t)/(W or Pt) with the DMI energy plotted against 1/t. In Pt/CoGd/W, the increase indicates the interfacial nature of the DMI interactions. (b) Room temperature magnetization thickness product versus thickness. The solid blue lines are linear fits to the Pt/CoGd/W data.   \n10 nm\n(b)(c)(a)SiO2WCoGdPtPtW\nWCoGdPtPtW0.00.20.40.60.81.00.000.050.100.150.200.25DMI (mJ m-2)\nTungsten composition (x) Pt/CoGd(5 nm)/Pt1-xWx\n0.00.10.20.000.050.100.150.200.25\n0510150.00.51.01.52.02.5Pt/CoGd(t)/WPt/CoGd(t)/PtDMI (mJ m-2)\n1/CoGd (t) thickness (nm-1)Pt/CoGd(t)/WPt/CoGd(t)/PtMS t (x10-6A)\nCoGd (t) thickness (nm)(b)(a) FIG. 4. Cross-sectional TEM images of the symmetric Pt/CoGd(15 nm)/Pt structure. The full stack is shown in (a) with a magnified view of the top (b) and bottom (c) interface of the CoGd layer.  \n FIG. 5. AFM (a,c) and MFM (b,d) images showing skyrmion-like magnetic textures nucleated in Pt/CoGd(10 nm)/W at room temperature in zero-field. The skyrmion imaged in (d) is indicated by a square box in (b).       \n(a)(b)\n(c)SiO2PtCoGdPtCoGdPt\nCoGdPt\nMFMAFM(a)(b)\n(c)(d)1 µm1 µm\n100 nm100 nmReferences:   1 Mühlbauer, S. et al. Skyrmion Lattice in a Chiral Magnet. Science 323, 915 (2009). 2 Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nature Nanotechnology 8, 899–911, doi:10.1038/nnano.2013.243; (2013). 3 Rohart, S., Miltat, J. & Thiaville, A. Path to collapse for an isolated Neel skyrmion. Physical Review B 93, 214412 (2016). 4 Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nature Nanotechnology 8, 152–156, doi:10.1038/nnano.2013.29; https://www.nature.com/articles/nnano.2013.29#supplementary-information (2013). 5 Koshibae, W. et al. Memory functions of magnetic skyrmions. 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Ivanov,5, 6Kathrin Ganzhorn,4, 7Francesco Della Coletta,4\nMatthias Althammer,4, 7Hans Huebl,4, 7, 8Rudolf Gross,4, 7, 8J¨ urgen\nKosel,5Mathias Kl¨ aui,1, 2,∗and Sebastian T. B. Goennenwein4, 7, 8, 9,†\n1Institut f¨ ur Physik, Johannes Gutenberg-Universit¨ at Mainz, 55099 Mainz, Germany\n2Graduate School of Excellence Materials Science in Mainz, 55128 Mainz, Germany\n3Quantum Condensed Matter Division,\nOak Ridge National Laboratory, Oak Ridge, 37830 TN, United States\n4Walther-Meißner-Institut, Bayerische Akademie\nder Wissenschaften, 85748 Garching, Germany\n5Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE),\nKing Abdullah University of Science and Technology (KAUST), Thuwal, 23955, Saudi Arabia\n6now at: Erich Schmid Institute of Materials Science,\nAustrian Academy of Sciences, A-8700 Leoben, Austria\n7Physik-Department, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany\n8Nanosystems Initiative Munich (NIM),\nSchellingstraße 4, 80799 M¨ unchen, Germany\n9Institut f¨ ur Festk¨ orperphysik, Technische Universit¨ at Dresden, 01062 Dresden, Germany\n1arXiv:1703.03218v2  [cond-mat.mtrl-sci]  10 Mar 2017Abstract\nWe investigate the generation of magnonic thermal spin currents and their mode selective spin\ntransport across interfaces in insulating, compensated ferrimagnet/normal metal bilayer systems.\nThe spin Seebeck effect signal exhibits a non-monotonic temperature dependence with two sign\nchanges of the detected voltage signals. Using different ferrimagnetic garnets, we demonstrate\nthe universality of the observed complex temperature dependence of the spin Seebeck effect. To\nunderstand its origin, we systematically vary the interface between the ferrimagnetic garnet and\nthe metallic layer, and by using different metal layers we establish that interface effects play a\ndominating role. They do not only modify the magnitude of the spin Seebeck effect signal but\nin particular also alter its temperature dependence. By varying the temperature, we can select\nthe dominating magnon mode and we analyze our results to reveal the mode selective interface\ntransmission probabilities for different magnon modes and interfaces. The comparison of selected\nsystems reveals semi-quantitative details of the interfacial coupling depending on the materials\ninvolved, supported by the obtained field dependence of the signal.\n2In recent years, the generation and flow of spin currents and thus the transport of spin\ninformation has been studied intensively as it is an essential prerequisite for the realization\nof spintronic devices. In spintronics not only the electron charge but also its spin state is\nused for information transmission and manipulation1,2, which promises reduced power con-\nsumption compared to conventional charge-based electronics that entails ohmic losses. The\nviability of this concept has recently been corroborated for instance by the demonstration\nof multi-terminal logic majority gates based on the superposition of spin currents3–6.\nIn general, spin currents can be generated in magnetic materials in different ways as, for\ninstance, ferromagnetic resonance (FMR) based spin pumping7–9, electrical injection by the\nspin Hall effect (SHE)10,11or the excitation of thermal spin currents via the spin Seebeck\neffect (SSE)12–14. The induced spin currents are then detected in an adjacent normal non-\nmagnetic metal (NM) layer by the inverse spin Hall effect (ISHE)15. SSE measurements on\nY3Fe5O12/Pt (YIG/Pt) bilayers with different YIG thicknesses revealed a strong thickness\ndependence of the effect, confirming that in ferromagnetic insulators (FMI) thermally gen-\nerated spin currents are carried by magnonic excitations of the spin system14. Beyond the\nkey influence of bulk properties for the spin current generation and transport in insulat-\ning ferrimagnets, indications were found that the SSE signal also depends on the YIG/NM\ninterface16–18. Guo et al.18have demonstrated that in different YIG/NM hybrids both the\nSSE amplitude and its temperature dependence are influenced by the NM material. This\nsuggests a more complex behavior of the SSE signal strength and sign with a possible expla-\nnation being based on different frequency and momentum dependencies of spin transmission\nprobabilities across different interfaces to different NMs. While different materials are shown\nto yield different SSE characteristics, it is not clear to what extent material properties or the\nmorphology of the interface dominate spin transmission properties. Previous findings have\nnot allowed for a quantitative understanding, as the magnetic structure of YIG prohibits\na distinction and separation between the different magnon modes that make up the total\nspin current. Furthermore, given indications by spin pumping measurements that the signal\namplitude measured only weakly depends on the excitation frequency19, it is perceived as\ncommon wisdom that the spin transmissivity across interfaces is magnon mode independent.\nThe possible mechanism of different transmission amplitudes across the interface is therefore\ndifficult to resolve as there is no means to distinguish between the modes excited.\nRecent SSE experiments performed using the compensated ferrimagnet Gd 3Fe5O12\n3(GdIG), which comprises a more complex magnetic sublattice configuration, have revealed\na more complicated temperature dependence of the SSE signals20. When measured as a\nfunction of temperature, two sign changes of the signal amplitude appear. The sign change\noccurring near the so-called magnetic compensation point, where the magnetizations of\nthe sublattices cancel each other out, is explained by the reorientation of the magnetic\nsublattices20. A second sign change at lower temperatures, however, is the result of a com-\npetition between two bulk magnon modes with opposite precession yielding spin currents of\nopposite spin polarity. The first mode is the fundamental ferrimagnetic magnon mode ex-\nhibiting a gapless dispersion with a narrow bandwidth, while the second mode is gapped by\nthe highly temperature dependent exchange coupling between the Gd and the Fe sublattice.\nThis demonstrates that the SSE is a probe of magnon properties of the respective material.\nWhile established for GdIG, it is unclear if this complex behavior is a universal phenomenon\nthat occurs for other ferrimagnetic materials as well. Given this particular temperature\ndependence, the SSE experiments in different compensated ferrimagnets provide the unique\nopportunity to infer the importance of bulk vs interface effects for the spin transport across\nthe interface. The evolution of the SSE signal at temperatures around the low temperature\nSSE sign change allows one to select the dominating magnon mode and thus to probe the\nmode-selective spin transmission across the interface to the NM, which has not been possible\nso far. Furthermore, intentional modifications of the interface as well as using different NM\nlayers for the spin current detection yield access to the magnon selective interface-related\nspin transmissivity properties.\nIn this Letter we compare the SSE response of different rare-earth iron garnets in order\nto extract the mode- and interface-dependent spin transmission. We start by checking\nwhether the complex SSE behavior found in Gd based iron garnets indeed is a universal\nphenomenon for rare-earth iron garnets with varying thickness. To isolate the interface effect,\nwe systematically vary the interfaces of the FMI/NM bilayers by either changing the NM\ndetection layer or using different surface treatments of the FMI before the NM deposition.\nFinally, we combine this with the high magnetic field influence on the temperature dependent\nspin transmission efficiency and identify the spin current contribution of distinct magnon\nmodes.\nFor our investigation of the temperature dependent excitation of thermal magnons we\nhave selected ferrimagnetic garnet thin films of the stoichiometry Gd 3Fe5O12, Dy 3Fe5O12and\n4(Gd 2Y)(Fe 4In)O 12(GdYInIG). We chose GdYInIG to study the effect of combining mag-\nnetic (Gd) and non-magnetic (YIn) rare earth ions while Dy 3Fe5O12is used to investigate\nthe effect of other magnetic rare-earth ions (Dy) compared to the previously used Gd (in\ncontrast to Gd3+ions, Dy3+ions feature a finite orbital moment ( L= 5)). All thin (20 nm\nto 200 nm) films are grown epitaxially on either Gd 3Ga5O12(GGG) or Y 3Al5O12(YAG)\nsubstrates by pulsed laser deposition in an oxygen atmosphere. Afterwards, the Pt or Pd\ndetection layer is deposited by magnetron-sputtering (ex-situ) or electron beam evaporation\n(in-situ) on top of the garnet thin film. Similarly to our previous work18, high resolution\nx-ray diffraction as well as transmission electron microscopy have been used in order to check\nfor the quality of the deposited films and the interfaces between them. X-ray reflectivity and\natomic force microscopy reveal smooth surfaces of the grown garnet films (see supplemental\ninformation21). The respective high resolution scanning TEM (HR-STEM) images and elec-\ntron energy loss spectroscopy (EELS) elemental maps clearly reveal the crystalline structure\nof the grown garnet films as well as well-defined interfaces between epitaxial ferrimagnetic\ngarnet thin film and the normal metal layer. Additionally, superconducting quantum in-\nterference device (SQUID) magnetometry measurements were carried out for each sample\nto determine the magnetization compensation point. For garnet films grown on YAG sub-\nstrates, the temperature dependence of the magnetization M has been measured by sweeping\nthe system temperature while applying an external magnetic field of µ0H = 1 T. For thin\nfilms on GGG substrates, however, the large paramagnetic moment of GGG prohibits this\nprocedure. M thus has been determined via magnetic field sweeps at constant temperatures.\nSSE measurements are performed in different setup geometries and measurement meth-\nods. Samples produced at the WMI are patterned into a Hall bar mesa structure, as shown in\nFig. 1 a), which allows for simultaneous heating and signal detection20,22. To detect the SSE\nsignal, either measurements as a function of the magnetic field magnitude at a fixed in-plane\nfield orientation (hysteresis measurements) or in-plane angular-dependent measurements at\na fixed magnetic field strength are performed. The latter enable to quantify the SSE signal\nby the characteristic sinusoidal angular dependence, while in the hysteresis measurements\nthe signal is extracted from the difference between positive and negative magnetic field, as\nshown in Fig. 1 b) and d) for different temperatures. A second type of SSE setup has\nbeen used for samples fabricated and measured in Mainz. An additional heater is placed\non top of the measurement-stripe structure electrically isolated from the detection layer20,23,\n5as shown in Fig. 1 c). For this setup, hysteresis measurements14and temperature sweep\nmeasurements23are performed. In the latter the SSE signal is obtained by the difference of\ntemperature dependent measurements at a fixed positive and negative magnetic field. All\nmeasurement methods yield a quantitatively identical SSE signal and show the character-\nistic linear dependence of the signal strength on the applied heating power. Simultaneous\nmeasurements of the detection (metal) layer and bottom layer resistance ( RtandRb) are\nused to determine the temperature gradient applied to the sample stack and the actual tem-\nperature of the sample surface. This temperature is used in the following as the reference\ntemperature for the sample. Note that we show here the measured spin Seebeck voltage\nand not the scaled data in µV K−1. This is to obtain robust information as the flattening of\ntheRvs.Tdependence at very low temperatures complicates and partially prohibits the\naccurate determination of the temperature gradient. Since we have measured all samples\nthat have similar thermal properties in nominally identical conditions, a robust comparison\nof the data is possible. In the following measurement data are thus plotted in units of µV.\nAll SSE measurement methods consistently yield two sign changes of the SSE voltage\namplitude as a function of temperature for the garnets studied here (Fig. 2). At 300 K,\nthe signal has the same sign as the SSE voltage commonly observed in YIG/Pt24,25. Upon\ndecreasing the temperature, the signal amplitude decreases, resulting in a first SSE voltage\nsign change. At this first sign change the SSE-hysteresis flips orientation as shown in Fig.\n1 d). The temperature of the first sign change T sign,1 matches with the temperature of the\nmagnetic compensation point T compwithin the error bar for all investigated garnets, as can\nbe seen in Fig. 2 a)-c).\nWe further investigate the GdIG thickness dependence of this sign change of the SSE\nvoltage. We find that the temperature for magnetic compensation, T comp, changes with\nthe thickness of our GdIG films grown epitaxially on GGG as shown in Fig. 2 d)-f). For\nincreasing film thickness we observe a shift of the compensation point temperature towards\nthe reported bulk value of 286 K26. The shift of the compensation point T comp to higher\ntemperatures for increasing the GdIG film thickness is reflected in the shift of the first sign\nchange of the SSE signal as T sign,1 coincides with T compfor all investigated samples. As the\ntemperature is decreased further, a second sign change of the SSE voltage is observed in\nall investigated samples at T sign,2. Around T sign,2 a continuous increase of the net magnetic\nmoment is observed, indicating that no qualitative change of the sublattice magnetizations\n6takes place in this temperature range. This is further strengthened by recent observations\nmade by Ganzhorn et al.27, where the spin Hall magnetoresistance (SMR) has been inves-\ntigated in GdYInIG/Pt bilayers as a function of temperature. The SMR reveals a distinct\ndependence on the orientation of the magnetic sublattices with the result that in the vicinity\nof T comp, where a canted spin state appears, the SMR changes sign. In the region of T sign,2,\nhowever, the SMR does not exhibit unexpected behavior. Generally, the temperature of\nthe second sign change T sign,2 increases slightly as the temperature of the first sign change\nTsign,1 increases, however, there is no direct proportionality of T sign,1 and T sign,2 (cf. Fig.\n2 d)-f)). Two possible explanations were put forward to explain the low temperature sign\nchange in GdIG20. The first explanation is based on a competition of the two dominating\nbulk magnon modes with temperature and the second one on different interfacial magnon\nmode transmission probabilities at the GdIG/Pt interface. To check the relevance of these\ntwo different models, we investigate the influence of different detection layer materials on the\nSSE signal. For this purpose we cleave a GdIG/GGG (100) sample into two parts, which\nallows us to ensure identical GdIG thin film properties. The two equal GdIG thin films\nare then covered by either Pt or Pd layers. The HR-STEM analysis indicates a very good\nepitaxial quality of the GdIG film as well as a high surface flatness of both metallic films.\nIn addition to that, EELS elemental maps reveal similar sub-nanometer interfaces between\nPt or Pd and GdIG. Fig. 3 a) and b) show the SSE signal as a function of temperature\nfor both samples. While the first sign change occurs at the same temperature T sign,1, as\nexpected for one and the same GdIG thin film if the bulk magnetic properties are decisive,\nthe temperature of the second low temperature sign change at T sign,2 depends strongly on\nthe NM layer. Compared to the Pt capped sample, in the sample with Pd as detection layer\nTsign,2 is shifted to lower temperatures and the overall signal amplitude is reduced.\nTo probe the influence of the interface quality on the SSE signal, a second GdIG/GGG\nsample is cleaved into two parts, whose surfaces are treated differently before the deposition\nof nominally identical Pt detection layers. One sample is only treated by acetone and\nisopropyl alcohol cleaning prior the deposition (marked as “untreated”), while the other\nsample is etched by an O-ion bombardment in-situ prior to the Pt deposition (marked as\n“etched”). As shown in Figs. 3 c) and d), again, T sign,1 remains unaltered while T sign,2\nchanges. Compared to the untreated sample, T sign,2 of the etched sample is shifted to a\nhigher temperature accompanied by an overall reduced signal amplitude. These results\n7demonstrate the strong influence of the interface quality on the temperature of the second\nsign change.\nAs shown by Kikkawa and coworkers28, the magnetic field strength has a significant influ-\nence on the SSE amplitude. To further investigate this matter in compensated garnet/NM\nheterostructures, we probe the dependence of the magnetic field on the SSE signal. Fig. 4\nshows the SSE voltage as a function of temperature for different magnetic field strengths in\nGdIG/Pt and DyIG/Pt heterostructures. T sign,1 slightly shifts towards lower temperatures\nas the magnetic field strength is increased. While for GdIG the temperature shift of T sign,2\nwith magnetic field strength is similar to the temperature change of change T sign,1, a more\npronounced temperature shift of the T sign,2 can be observed for DyIG. The sign change even\nvanishes at low magnetic field strengths, indicating a more complex SSE in DyIG as com-\npared to GdIG. Above the compensation point, both GdIG and DyIG reveal an increase of\nthe SSE signal with increasing external field.\nWe now analyze the experimental findings presented above, considering that the SSE volt-\nage signal can be decomposed into a contribution arising from (i) the temperature dependent\ndominating magnon modes within the ferrimagnet20and (ii) the interfacial spin transmission\nacross the garnet/metal interface. In particular, the comprehensive investigation with dif-\nferent magnetic garnet materials, different metal layers and different interface preparation\nprocedures enables us to determine how the observed interface and magnetic field effects\nreveal the influence of the interfacial boundary between ferrimagnet and metal. As shown\nabove, the temperature of the first sign change T sign,1 coincides with the temperature of the\nmagnetic compensation point determined by SQUID magnetometry measurements within\nthe experimental error for all investigated garnets, independent of the metal detection layer\nor interface quality. Especially the interface quality and metal detection layer insensitivity\nof T sign,1 confirm the interpretation of the origin of this sign change put forward in Ref.20,\nbased on the theoretical proposal of Ohnuma et al.29. All investigated garnets consist of at\nleast three coupled magnetic sublattices. Two of these sublattices (a and d lattice site) are\noccupied by Fe ions, which are antiferromagnetically coupled. This sublattice configuration\nis also present in the classical ferrimagnetic insulator YIG, which shows no magnetic com-\npensation point. By adding a further magnetic material on the c lattice site (e.g., Gd in\ncase of GdIG), a third magnetic sublattice is introduced. For Gd or Dy on the c lattice site,\nthis c sublattice couples ferromagnetically to the Fe on the a lattice site20,30. Furthermore,\n8this third sublattice possesses a strong temperature dependent magnetic moment, leading to\na temperature below the N´ eel point (the magnetic ordering temperature) at which the net\nmagnetic moment of the material is zero. The temperature T compof this so-called magnetic\ncompensation point depends on the magnetic ions on the c sublattice as well as on their\ntemperature dependence26. In a finite magnetic field the magnetic moments invert their ori-\nentation when the temperature is swept across T comp. As a result, the sign of the SSE voltage\nis inverted if measured above and below the compensation point, as the polarization of the\nemitted magnonic spin current follows the orientation of the magnetic lattices20. Any influ-\nence of the metal detection layer or the quality of the interface on T sign,1 would be evidence\nthat interface - and not bulk magnetic properties – are relevant for T sign,1. The fact that we\nobserve T comp≈Tsign,1 irrespective of the thicknesses of the garnet layer, its composition,\nand its interface properties is a strong evidence that the sign change of the SSE voltage at\nTsign,1 indeed simply reflects the change in the ‘bulk’ sublattice moment orientations viz.\nbulk magnon properties. It must be noted that besides the magnetic compensation point,\ncompensated ferrimagnets can also exhibit an angular momentum compensation point. The\ntheoretical model by Ohnuma et al.29however does not predict any change of the thermal\nspin current due to the latter. Indeed, we do not observe any features in our experimental\ndata which would point to a modification of the thermal spin currents around the angular\nmomentum compensation point. While in GdIG the angular momentum of Gd is quenched\nsuch that magnetic compensation and angular momentum compensation temperature coin-\ncide, there would be a distinct difference between the two in DyIG.\nA more complex behavior must be invoked to understand the second SSE voltage sign\nchange at lower temperatures (T sign,2), as we discuss next. As mentioned above, no changes\nof the net magnetization are observed in the vicinity of T sign,2 that could explain the observed\nSSE behaviour. However, the shift of T sign,2 with both the material of the detection layer\n(Fig. 3 a) and b)) and the garnet surface treatment (Fig. 3 c) and d)) demonstrates the\ninfluence of the interface, which was previously neglected. Expanding the original theory\ninvoked by Ohnuma et al.29, Tsign,2 is explained by a temperature dependent competition\nbetween two bulk magnonic modes in the garnet. While at temperatures below T sign,2\nthe spin current is mainly given by the low-energy mode (acoustic like mode), the second\nmode (optical like mode) becomes dominating above T sign,2. Since the two modes are of\nopposite precession, their contributions to the spin current are of opposite spin polarity\n9which results in the sign change of the SSE signal at T sign,2. A more detailed description\nof this process is provided by Gepr¨ ags et al.20. The efficiency of the spin transfer from the\ngarnet into the metal detection layer, which is described by the element specific spin mixing\nconductance G ↑ ↓31, depends on the detection layer material. Due to a generally lower spin\nmixing conductance as well as spin Hall angle of Pd as compared to Pt32, the overall signal\nis reduced as seen in Fig. 3 c)-d). Nevertheless, the observed shift of the second sign change\nto lower temperatures as compared to Pt indicates that magnons of the high energy mode\ncan be transferred more efficiently across the interface into Pd. This is either attributable\nto intrinsic properties of the metal detection layer or the chemical surface termination of the\ngarnet film. While in case of GdIG/Pt a Gd/Fe rich interface is observed, GdIG/Pd exhibits\na Gd/O rich interface (see supplemental information21). One thus might speculate that\nthe decreased Fe concentration at the GdIG/Pd interface results in an increased interface\ndamping of the low energy mode. Besides changing the metal detection layer the interface\ntransparency is manipulated via O-ion bombardment, yielding an universal reduction of the\nsignal strength. This points to a reduction of G ↑ ↓for both modes into the detection layer.\nHowever, as compared to the unetched interface the second sign change is found at higher\ntemperatures, suggesting that the high energy mode is more affected by the reduced interface\ntransparency.\nFinally we corroborate the magnon mode sensitivity of the second sign change by ana-\nlyzing the magnetic field dependence shown in Fig. 4. For YIG it has been shown that high\nmagnetic fields cause a reduction of the magnonic propagation length, which is relevant for\nthe whole temperature range33. An energy gap (Zeeman effect) is opened by the magnetic\nfield, which suppresses the excitation of magnons with energies lower than the gap28,33,34.\nThe field gap is especially important at low temperatures, as in YIG the field induced energy\ngap is in the order of a few Kelvin34. It therefore causes a shift of the whole dispersion re-\nlation to higher energies, thus mainly introducing a constant offset with respect to the zero\nfield state33. The resulting shift of the dispersion relation towards higher energies moves\nTsign,2 to higher temperatures, as it now requires more thermal energy to sufficiently occupy\nthe high energy mode. The different sensitivities of T sign,2 on the magnetic field for GdIG\nand DyIG might originate from the different electronic states of the Gd3+and Dy3+ions.\nIn contrast to Gd3+, Dy3+has a finite orbital moment ( L= 5), which may lead to differ-\nences in the magnetic field dependences of magnon modes due to the different multiplett\n10states occupied. The comparison to field dependent magnetization data (see supplemental\ninformation21) shows that the increase of VSSEat high magnetic fields above the compen-\nsation point cannot be explained by an increase of the magnetization. Instead, one has to\nconsider that in contrast to YIG, where the thermal signal is governed by the fundamental\nferrimagnetic magnon mode, in compensated iron garnets the spin current reflects the field\ndependence of a complex magnon mode structure. The quantification of the latter as well\nas the magnon mode dependence on different electronic states of the magnetic ions require\nfurther theoretical input, which is beyond the scope of this work.\nIn conclusion, we have shown that the ferrimagnetic nature and magnon mode structure\nof compensated garnets can be accessed by temperature dependent SSE measurements. Our\nstudy of different compensated iron garnets, detection metal layer materials and interface\nqualities allows us to clearly show the universal correlation between the magnetic compen-\nsation point and the sign change of the SSE voltage at high temperatures, T sign,1. We find\nthat the second sign change at the lower temperature T sign,2 is strongly influenced by the\ndetection layer material as well as the interface nature, which can be rationalized in terms\nof a modified coupling strength of the two magnon modes. We study these individually by\nvarying the temperature to identify the mode selective magnon transmissivity. The mag-\nnetic field dependence of T sign,2 is consistent with our developed magnonic mode selective\npicture of the SSE. T sign,2 is shifted towards higher temperatures for higher magnetic fields.\nOur results reveal that the SSE provides an elegant solution to gain insight into the magnon\nmode structure of a ferrimagnet and, on the other hand, show for the first time evidence for\nan interface sensitive magnon mode selective coupling towards normal metal layers.\nI. ACKNOWLEDGMENTS\nThe authors would like to express their gratitude to Prof. Gerrit E.W. Bauer and Dr. Joe\nBarker for valuable discussions and the Institute for Materials Research at Tohoku University\nfor the hospitality during a visiting professor stay (MK). Furthermore, they would like to\nthank Prof. Kathrin D¨ orr for the help in sample preparation. This work was supported by\nDeutsche Forschungsgemeinschaft (DFG) SPP 1538 “Spin Caloric Transport,” the Graduate\nSchool of Excellence Materials Science in Mainz (MAINZ), and the EU projects (IFOX,\n11NMP3-LA-2012246102, INSPIN FP7-ICT-2013-X 612759).\n∗Klaeui@uni-mainz.de\n†Sebastian.Goennenwein@tu-dresden.de\n1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln´ ar, M. L. Roukes,\nA. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001).\n2I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Reviews of Modern Physics 76, 323 (2004).\n3M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Applied Physics\nLetters 87, 153501 (2005).\n4S. Klingler, P. Pirro, T. Br¨ acher, B. Leven, B. 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Du, W. Zhang, and H. F. Ding,\nJournal of Applied Physics 115, 17C504 (2014).\n33U. Ritzmann, D. Hinzke, A. Kehlberger, E.-J. Guo, M. Kl¨ aui, and U. Nowak, Physical Review\nB92, 174411 (2015).\n34H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Physical Review B 92, 054436\n(2015).\n35S. Geller, G. P. Espinosa, and P. B. Crandall, J. Appl. Crystallogr. 2, 86 (1969).\n36G. P. Espinosa, J. Chem. Phys. 37, 2344 (1962).\n14−10 0 10−4−2024VSSE(µV)\n−10 0 10\n−10 0 10\n0 180−2024VSSE(µV)\n0 180\n0 180 360\nYAG (111) / GdIG (26 nm)YAG (111) / GdIG (26 nm) a) b)\nc) d)25 K 166 K 297 K\n20 K 158 K 272 KFigure 1. Schematics of the setups used to measure the spin Seebeck effect. At WMI, a) the\nout-of-plane temperature gradient is generated by driving a large current I dthrough a Hall bar\nmesa structure while the thermo-voltage is recorded as b) a function of the in-plane magnetic\nfield orientation as well as d) a function of the magnetic field strength at a fixed magnetic field\norientation. In Mainz c) the out-of-plane temperature gradient is generated by an external heater\nattached on top of the sample. The SSE amplitude is obtained by sweeping a perpendicular\nmagnetic field.\n150100 200 300\nT(K)−4−202VSSE(µV)\n101102103\n0100 200 300\nT(K)−4−2024VSSE(µV)\n200400600\n0100 200 300\nT(K)−10−505VSSE(µV)\n100101102103YAG (111) / GdIG (26 nm) YAG (111) / DyIG (30 nm) YAG (111) / GdYInIG (61 nm)\na) b) c)\nGGG (001) / GdIG (41 nm) GGG (001) / GdIG (100 nm) GGG (001) / GdIG (155 nm) \nd) e) f)Figure 2. In each panel, the upper plot shows the magnetic moment of the garnet film as a\nfunction of temperature determined by SQUID magnetometry measurements. The SSE voltage as\na function of temperature is shown in the bottom plot for the same sample. The upper three figures\na), b), c) show the data of REIG/Pt heterostructures with REIG = GdIG, DyIG and GdYInIG\ngrown on YAG. The lower three figures d), e) and f) contain the data for GdIG/Pt heterostructures\ngrown on GGG, featuring different GdIG layer thicknesses. The dashed lines mark the temperature\nof the magnetic compensation point T comp determined from SQUID magnetometry data and the\ntemperatures at which the SSE voltage changes sign. For all measurements a magnetic field of µ0H\n= 1 T is applied.\n16−1.0−0.50.00.51.0VSSE(µV)\n0 100 200 300\nT(K)−0.50.00.5VSSE(µV)GGG (001) / GdIG (100 nm)\nPt\nPd179 K\n179 K72 K\n58 K\n0 100 200 300\nT(K)−0.50.00.5VSSE(µV)\n−1.0−0.50.00.51.0VSSE(µV)GGG (001) / GdIG (41 nm)\nuntreated\netched153 K\n153 K44 K\n71 Ka) c)\nb) d)Figure 3. SSE voltage as function of temperature measured in GdIG/NM heterostructures grown\non GGG (001). (a), (b) V SSEof GdIG/NM heterostructures using Pt as well as Pd as detection\nlayer. The same GdIG thin film sample was used for both heterostructures. (c), (d) V SSEof two\nGdIG/Pt heterostructures. The Pt thin films were deposited on both an untreated and etched\nsurface of the same GdIG thin film sample. The dashed lines mark the temperatures at which the\nSSE voltage changes sign.\n0 100 200 300\nT(K)−6−4−2024VSSE(µV)\n0 100 200 300\nT(K)−6−4−2024VSSE(µV)YAG (111) / DyIG (30 nm) YAG (111) / GdIG (26 nm)\n1T\n15T1T\n15T\nFigure 4. SSE voltage as function of temperature for (a) GdIG/Pt and (b) DyIG/Pt heterostruc-\ntures for different magnetic field strengths.\n17Magnon mode selective spin transport in compensated\nferrimagnets\nSupplemental Information\nS2. SAMPLE CHARACTERIZATION\nS2.1. X-ray diffraction & Reciprocal space mapping\nThe crystalline quality of the investigated rare earth iron garnet films has been exam-\nined via X-ray diffractometry (XRD) and reciprocal space mapping (RSM). In Fig. S1,\n2Θ−ωscans of the garnet films grown at the WMI are depicted, each showing the YAG\n(444) reflection as well as the respective REIG (444) reflection ( REIG = GdIG, DyIG, and\nGdYInIG). The presence of the latter indicates crystalline growth of the films, reinforced\nby the appearance of Laue fringes for GdIG and DyIG signifying high crystalline quality.\nFor GdIG and DyIG the (444) reflection appears at a scattering angle which is below the\ntheoretical value obtained for the lattice constants aGdIG = 12.471˚A,aDyIG = 12.405˚A35,36\nand the Bragg equation\n2dhklsin (Θ) =λ\nwithdhkl=a√\nh2+k2+l2,\nassuming a cubic geometry. The shift to lower scattering angles signifies an enhance-\nment of the out-of-plane lattice constant, which in turn can be explained by an in-plane\ncompressive strain considering the lattice constant aYAG= 12.008˚A of the substrate35. To\nthe knowledge of the authors there are no reported values of afor GdYInIG, but regarding\nits close chemical relation to GdIG a similar lattice constant and thus a strained state is\nexpected.\nXRD data of the GdIG films investigated at JGU Mainz are shown in Fig. S2. Since\nin this case the GdIG films of varying thickness are grown on GGG susbtrates with (001)\norientation, the 2Θ −ωscans cover the GGG (004) and GdIG (004) reflection. Again,\nthe presence of the reflection peaks and the clear appearance of Laue fringes for GdIG with\n18( ( (\n49 50 51 52 53 54100101102103104105106Intensity (a.u.)\n49 50 51 52 53 54100101102103104105106\n49 50 51 52 53 54100101102103104105106\n2Θ◦) 2Θ ) 2Θ )YAG (111) / GdIG (26 nm) YAG (111) / DyIG (30 nm) YAG (111) / GdYInIG (61 nm)\n◦ ◦\nYAG (444)YAG (444)YAG (444)GdIG (444)\nDyIG (444)\nGdYInIG (444)(a) (b) (c)\nBulk value Bulk valueFigure S1. X-Ray diffraction data of the rare earth iron garnets Gd 3Fe5O12(GdIG), Dy 3Fe5O12\n(DyIG), and (Gd 2Y1)(Fe 4In1)O12(GdYInIG). The films are grown epitaxially via pulsed laser\ndeposition (PLD) in an oxygen atmosphere on Y 3Al5O12(YAG) substrates with (111) surface\norientation. The vertical dashed lines indicate the (444) reflection calculated from the respective\nbulk lattice constant.\nthickness d= 41 nm and 100 nm) indicates high crystalline quality of the films. Furthermore,\nas in the case of YAG, the lattice constant of GGG aGGG= 12.375˚A is smaller than the bulk\nvalue ofafor GdIG, leading to in-plane compressive stress and a shift of the scattering angle\nof GdIG to lower values. The coherent growth of high quality GdIG films up to 155 nm is\nfurther emphasized by the reciprocal space maps around the GGG (228) reflection presented\nin Fig. S3.\n1926 27 28 29101102103104105106107\n26 27 28 29101102103104105106107\n26 27 28 29101102103104105106107Intensity (a.u.)\n2Θ(◦) 2Θ( ) 2Θ( ) ◦ ◦GGG (001) / GdIG (41 nm) GGG (001) / GdIG (100 nm) GGG (001) / GdIG (155 nm)GGG (004)\nGGG (004)GdIG (004)\nGdIG (004)\nGdIG (004)\nGGG (004)Bulk value Bulk value Bulk valueFigure S2. X-Ray diffraction data of Gd 3Fe5O12(GdIG) thin films of various thickness ( d=\n41 nm, 100 nm and 155 nm). The films are grown epitaxially via pulsed laser deposition (PLD)\nin an oxygen atmosphere on Gd 3Ga5O12(GGG) susbtrates with (001) surface orientation. The\nvertical dashed lines indicate the GdIG (004) reflection calculated from the respective bulk lattice\nconstant.\n0.225\n0.230\n0.235\n0.63\n0.64\n0.65\nGdIG (41 nm)\nRSM 228\nl (rlu)\nh (rlu)\nGGG\n(a)GGG/GdIG(41 nm)\n0.230\n0.235\n0.63\n0.64\n0.65\nl (rlu)\nh (rlu)\nGdIG (155 nm)\nRSM 228\nGGG (b)GGG/GdIG(155 nm)\nFigure S3. Reciprocal space maps (RSM) around the GGG (228) reflection for GGG/GdIG(41 nm)\nand GGG/GdIG(155 nm).\nS2.2. X-ray reflectometry & Atomic force microscopy\nTo obtain information about the thickness and surface roughness of the grown garnet\nfilms X-ray reflectometry (XRR) as well as atomic force microscopy (AFM) measurements\nare performed. From the XRR data (Fig. S4 and S5) the film thickness as well as the root\nmean square roughness σrmsof the sample surface are extracted by fitting a simulation curve\n200.05 0.10 0.15 0.20 0.25\nQz(Å−1)10−410−310−210−1100Intensity (a.u.)Data\nSim\n0.05 0.10 0.15 0.20 0.25\nQz(Å−1)10−410−310−210−1100\nData\nSim\n0.05 0.10 0.15 0.20\nQz(Å−1)10−410−310−210−1100\nData\nSimYAG (111) / GdIG (26 nm) YAG (111) / DyIG (30 nm) YAG (111) / GdYInIG (61 nm)\n(a) (b) (c)Figure S4. X-Ray reflectometry data of rare earth iron garnets Gd 3Fe5O12(GdIG), Dy 3Fe5O12\n(DyIG), and (Gd 2Y1)(Fe 4In1)O12(GdYInIG) investigated at the WMI.\n0.05 0.10 0.15 0.20 0.25\nQz(Å−1)101102103104105106107108Intensity (a.u.)Data\nSim\n0.05 0.10 0.15 0.20 0.25\nQz(Å−1)101102103104105106107108\nData\nSim\n0.05 0.10 0.15 0.20 0.25\nQz(Å−1)101102103104105106107\nData\nSimGGG (001) / GdIG (41 nm) GGG (001) / GdIG (100 nm) GGG (001) / GdIG (155 nm)\nFigure S5. X-Ray reflectometry data of Gd 3Fe5O12(GdIG) thin films of various thickness ( d=\n41 nm, 100 nm and 155 nm) investigated at JGU Mainz.\nto the raw data. In case of large thickness ()GGG/GdIG(155 nm, Fig. S5 (c)) no Kiessig\nfringes can be observed so that the film thickness is estimated by the growth time.\nThe AFM measurements (Fig. S6) allow to extract σrmsdirectly from the topographical\ninformation yielded by the data. As seen in Fig. S6 (a)-(c), the sample surfaces are partially\nor significantly contanimated by dirt particles resulting from sample handling, leading to an\nincrease of σrmsas compared to the values obtained by XRR.\nTheσrmsvalues obtained by the different methods for the investigated films are summa-\nrized in Tab. S1.\n212.89 nm\n0.171.502.00\n2 µm\n1.59 nm\n0.200.801.001.20\n2 µm\n3.07 nm\n0.411.502.002.50\n2 µm(a) GdIG\n(b) DyIG\n(c) GdYInIGFigure S6. Atomic force microscopy data of rare earth iron garnets (a) Gd 3Fe5O12(GdIG), (b)\nDy3Fe5O12(DyIG), and (c) (Gd 2Y1)(Fe 4In1)O12(GdYInIG).\nTable S1. Thickness and roughness data for the investigated garnet films obtained by XRR and\nAFM.\nFilm d(nm)σrms,XRR (nm)σrms,AFM (nm)\nYAG (111) / GdIG 26 0.60 0.54\nYAG (111) / DyIG 30 0.40 0.80\nYAG (111) / GdYInIG 61 0.35 0.43\nGGG (001) / GdIG 41 1.37 -\nGGG (001) / GdIG 100 1.71 -\nGGG (001) / GdIG 155 2.27 -\n22S2.3. Transmission electron microscopy characterization\nA direct insight into the crystalline structure of part of the investigated samples is ob-\ntained via high resolution scanning transmission microscopy (HR-STEM). For HR-STEM\nthe samples have been prepared using the standard focused ion beam (FIB) protocol. Re-\nspective HR-STEM images and electron energy loss spectroscopy (EELS) spectra have been\ntaken using an aberration-corrected scanning transmission electron microscope (FEI Probe\nCorrected Titan3TM80-300 S/TEM) operating at 200 kV and equipped with a Gatan Quan-\ntum ERS spectrometer.\nFig. S7 (a),(b) show two examples of high-angle annular dark field (HAADF) cross\nsections of the GGG(001)/GdIG/Pt and GGG(001)/GdIG/Pd samples, whose spin Seebeck\ndata are presented in Fig. 3(a),(b) in the main text. The images already indicate the\nepitaxial growth of the garnet films as well as reveal the overall flatness of the metal films.\nIn Fig. S7 (c) we show an image example of the interface between substrate and grown\ngarnet film of the GGG(001)/GdIG/Pd, combining the information obtained by annular dark\nfield (ADF) and electron energy loss spectroscopy (EELS) imaging. The clear crystalline\nstructure of substrate and grown film are pictured with atomic resolution, revealing the high\nepitaxial quality of the latter and a well defined interface. The elemental maps given by the\nEELS data allow us to assign or rather identify elements at the lattice sites, which show\nlittle interdiffusion of Gd and Ga atoms between GGG and GdIG.\nIn Fig. S8 we show an example of a detailed HR-STEM images of the GdIG/Pt interface,\nincluding ADF information as well as information about the distribution of single elements\nvia EELS. On the one hand, the images reveal that the garnet film retains its crystalline\nstructure up to at least 100 nm thickness. On the other hand, a sub-nanometer, Gd/Fe\nrich interface between Pt and GdIG is observed via the EELS elemental map. In case of\nGdIG/Pd (not shown here) a Gd/O rich interface is observed.\n23GGGGdIGPt\nGGGGdIGPd(a)\n(b)\n(c)\nFigure S7. (a),(b) Annular dark field (ADF) images of (a) GGG(001)/GdIG/Pt and (b)\nGGG(001)/GdIG/Pd samples investigated at JGU Mainz. (c) ADF (left) and electron energy\nloss spectroscopy (EELS, right) image of the GGG/GdIG interface of the sample shown in (b).\nThe images are taken alongside the [110] crystal orientation using a scanning transmission electron\nmicroscope. For comparison, the expected crystal structure is shown schematically showing good\nagreement.\n0 1 2 3 4 5 6\nPosition (nm)0.00.20.40.60.81.0Nomralized counts (a.u.)\nPt\nGd\nFe\nOGGG (001) / GdIG (100 nm) / Pt (5 nm)\n(a) (b)\nPt\nGdIGPosition\nFigure S8. (a) High resolution scanning electron microscopy ADF image of the GdIG/Pt interface\nof the GGG(001)/GdIG/Pt sample investigated at JGU Mainz. Respective spin Seebeck data are\nshown in Fig. 3 in the main text. (b) Distribution of single elements across the interface obtained\nby an electron energy loss spectroscopy elemental map. The direction of the scan is indicated by\nthe arrow in (a).\n24S3. SQUID MAGNETOMETRY OF GDIG AND DYIG\nIn Fig. S9, SQUID magnetometry data obtained for GGG(001)/GdIG(100 nm) and\nYAG(111)/DyIG(30 nm) are shown, depicting the sample magnetization as a function of\nexternal magnetic field at different temperatures. While measuring the respective data, in\naddition to the magnetic moment of the grown iron garnets, the paramagnetic (diamag-\nnetic) moment of GGG (YAG) is recorded as well. To remove this contribution a linear fit is\napplied to the raw magnetiation data in the field range where the sample is assumed to be\nfully magnetized. Subsequently the raw data are rectified by means of the obtained linear\nslope. In case of YAG/DyIG the diamagnetic background is determined once for 2 K and\nthereafter, assuming a constant diamagnetic contribution of YAG, all temperature datasets\nare corrected using the determined parameters. For GGG/GdIG the paramagnetic back-\nground of GGG is calculated for each temperature. Since the latter is large as compared to\nthe GdIG magnetization, this procedure may lead to an error in the slope of M(H) in the\nsaturated state.\n20 K\n40 K\n60 K\n80 K\n100 K\n120 K\n140 K\n160 K\n180 K\n200 K\n220 K\n240 K\n260 K\n280 K\n300 K2 K\n5 K\n10 K\n20 K\n40 K\n60 K\n100 K\n140 K\n180 K\n190 K\n195 K\n200 K\n205 K\n215 K\n225 K\n250 K\n300 K −100 −50 0 50 100\nµ0H(mT)−400−2000200400M(kA/m)\n−8−6−4−20 2 8\nµ0H(T)−800−600−400−2000200400600800M(kA/m)\n4 6GGG (001) / GdIG (100 nm) YAG (111) / DyIG (30 nm)\n(a) (b)\nFigure S9. Magnetization of (a) GdIG and (b) DyIG as a function of external magnetic field at\ndifferent temperatures.\n25S4. TYPICAL SSE TEMPERATURE DEPENDENCE MEASUREMENTS\nFor a comprehensive characterization of the spin Seebeck (SSE) temperature dependence\ndifferent measurements can be performed, here shown exemplarily for the GGG(001)/GdIG(100 nm)/Pt(5 nm)\nsample measured at JGU Mainz.\nFig. S10 (a) shows the temperature dependence of the recorded SSE voltage VLSSE for\ndifferent heating currents. As expected, VLSSE increases with increasing heating current.\nUsing the temperature calibration method described in the main text one can extract the\ndependence of VLSSE on the temperature difference ∆ Tapplied across the sample, revealing\na linear dependence (Fig. S10 (b)). This corresponds to the behavior expected for the SSE\nand is observed for both regimes above and below the magnetization compensation point\nwhereVLSSE changes its sign. Generally, the position of the first and second sign change of\nVLSSE is independent on the applied heating power, emphasizing their physical significance\nand their dependence on intrinsic material properties.\nUsing the evaluated temperature difference ∆ Tacross the sample (plotted in Fig. S10 (c),\nagain for the different applied heating currents) one can deduce the temperature dependence\nof the spin Seebeck coefficient σLSSE, see Fig. S10 (d). Within the error the different curves\nobtained for different heating currents superimpose, demonstrating the robustness of the\nmeasurement method.\n26Figure S10. Typical data obtained from the GGG/GdIG (100 nm) sample during measurements\nusing the setup at JGU Mainz. (a) Spin Seebeck voltage VLSSE as a function of temperature\nfor different applied heating currents. (b) VLSSE as a function of temperature difference between\nsample top and bottom at different temperatures, revealing a linear dependence. (c) Temperature\ndifference between sample top and bottom as a function of temperature for different applied heating\ncurrents. (d) Spin Seebeck coefficient σLSSE =VLSSE/∆Tas a function of temperature.\n27"    },    {        "title": "1705.06398v1.Magnetic_vortex_nucleation_annihilation_in_artificial_ferrimagnet_microdisks.pdf",        "content": " \n1 \n Magnetic vortex nuclea tion/annihilation in artificial -ferrimagnet micro disks \n \nPavel N. Lapa ,1,2 Junjia  Ding,1 Charudatta Phatak,1 John  E. Pearson,1 J. S. Jiang,1 Axel \nHoffmann1 and Valentine  Novosad1 \n1Material Science Division, Argonne National Laboratory, Argonne, IL 60439, USA  \n2Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843 -4242, USA  \nThe topological nature of magnet ic-vortex state gives rise to peculiar magnetization reversal observed in magnetic microdisks. \nInterestingly, magnetostatic and exchange energies which drive  this reversal can be effe ctively controlled in artificial \nferrimagnet heterostructures composed of rare -earth and transition met als. [Py( t)/Gd( t)]25 (t=1 or  2 nm) superlattices \ndemonstrate a pronounced change of the magnetization and exchange stiffness in a 10–300 K temperature range as well as very \nsmall magnetic anisotropy . Due to these properties , the magnetization of  cylindrical micro disks composed of these artificial \nferrimagnets  can be transformed from the vortex to uniform ly-magnetized  states in a permanent magnetic field by changing \nthe temperature . We explore d the behavior of magnetization  in 1.5 -µm [Py( t)/Gd( t)]25 (t=1 or 2 nm) disk s at different \ntemperatures and magnetic fields  and observed  that due to the energy barrier separating vortex and uniform ly-magnetized  \nstates , the vortex  nucleation and annihilation occur  at different temperatures. T his causes  the temperature dependences  of the \nPy/Gd disks  magnetization to demonstrate unique hysteretic behavior  in a narrow temperature range . It was discovered that for \nthe [Py(2 nm)/Gd(2 nm)] 25 microdisks the vortex can be metastable at a certain temperature rang e. \n \nI. INTRODUCTION  \nIn order to minimize magnetic stray field  and magnetostatic energy associated  with it, the magnetization of  a macroscopic  \nferromagnetic object  typically develops  an inhomogeneous domain structure . For  nano -sized  objects, the  cost of  exchange \nenergy due to  the domain wall s makes multi -domain state energ etically unfavorable. H ence, for these objects, a uniformly -\nmagnetized single -domain s tate is stable , even in a zero magnetic field . For cylindrical micro disks, which have sizes in between \nmacro - and nano -size regimes , another mechanism of a magnetic -flux elimination  can be realized. Namely, in a zero magnetic \nfield, magnetization curls  azimuthally  around the disk geometrical center , forming a so-called magnetic vortex.1 To avoid \ndiscontinuity  of magnetization  in the geometrical center of the vortex , the magnetization smoothly points perpendicular to  the \ndisk plane within a narrow  area containing the vortex center. The area with a nonzero out-of-plane component of magnetization \nis usually called a vortex core.2 An increasing magnetic  field caus es the vortex  to shift away  from the geometrical center of the \ndisk in the  direction perpendicular to the magneti c field.  When the magnetic field exceeds a critical value, which is defin ed by \nmicroscopic  parameters of the material (exchange stiffness and magnetization)  and geometrical parameters of the disks \n(diameter and height ), the vortex annihilates.  Vice versa, a decreasing magnetic field drives the vortex nucleation and it \nsubsequently moves  towards the geometrical center of the disk.  \n2 \n It is argued that  the drastically reduced dipole -dipole interaction  between the disks and the topological nature  of the vortex \nstate can be utilized  for storing binary information .3, 4 For this, t he static5 and dynamic6-8 behavior of magnetic vortices  have  \nbeen studied extensively. More fundamental aspects , like a magnetic structure of a vortex core or microscopic  mechanisms of \nvortex  nucleation/ annihilation , have also attracted  a lot of attention and have been investigate d using a variety of tools .9-15 Thus, \nfabrication of multilayered microdisks in which different layers are either coupled to each other16, 17 or exchange biased  with \nan antiferromagnet18 has proved to be an efficient way to control magnetization reversal.  \n One of the main requirement for an observation of a vortex state in a magnetic micro disk is that the magnetic material s \nhave  very low magnetic anisotropy.19 A typical  choice of materials which satisfy this requirement includes  magnetically soft  \nPermalloy (Py =Ni 0.81Fe0.19),20 polycrystalline Ni21 and Fe .15 For micro disks compos ed of these materials, a conventional \napproach to switch the magnetization from the vortex to uniformly -magnetized states is to change  the amplitude of the \nexternally -applied magnetic field. Since the  Curie temperature s for these materials  are high, their  magnetic parameters \n(magnetization and exchange stiffness ) are almost constant  below room temperature. This  means t hat for the microdisk s \ncomposed of these materials,  the nucleation and annih ilation fields demonstrate an insignificant change while  temperature is \nvaried within a  10–300 K range.14 \nDue to an antiferromagnetic coupling between rare -earth Gd and transition metal s, the heterostructures composed of these \nmaterials are widely used for artificial ferrimagnet application s.22-27 As we observed previously,26, 27 if an artificial ferrimagnet  \nis composed of thin layers of Py and Gd, the heterostructure has very low anisotropy which is comparable with that of Py. \nHence, it is possible that the magnetic vortex may  be a stable magnetization configuration for the micro disks composed of \nthese artificial ferrimagnets . Importantly, the Curie temperature  of Gd (293  K for bulk ) is much lower than that for Py which \nmeans that the Gd/Py superlattice s have significant  change s of magnetization between 10 and 300 K. This suggests  that the \nvortex nucle ation and annihilation fields for  the Py/Gd superlattice micro disks may also var y significantly  within  this \ntemperature range . Our motivation  for this work  was to detect and study vortex nucleation /annihilation  transitions that occur \nin a permanent magnetic field  due to  the change  of the microscopic  parameters of the material  for different temperatures . \nII. EXPERIMENTAL DETAILS  \nWe prepared two groups of  disk array s for the study. T he disks from the first and second group s have \n[Py(1  nm)/Gd(1  nm)]25 and [Py(2  nm)/Gd(2  nm)] 25 structures along the height , respectively. The nominal  diameter of each disk \nis 1.5  µm. In addition, two different approaches  were used fo r fabrication  of each group: etching and lift-off. For  fabrication  of \nthe lift -off technique,  Si/SiO 2 wafers were spin -coated with photoresist, and an array of 1.5  µm holes were patterned by means \nof optical lithography. Then, after magnetron sputtering of the multilayers , the lift -off process yielded array s of 1.5  µm disks.  \n3 \n For fabrication of the etched disks , the artificial ferrimagnet films were sputtered on top of the Si/SiO 2 wafer . After that,  the \narrays of holes  were patterned  on top of the films. T his pattern was transferred into arrays of photoresist disks using an image \nreversal technique (baking the wafer at 100ºC in NH 3 environment for 30 minutes followed by flood exposure and \ndevelopment).  The final step  of this process  was Ar-ion milling. The deposition rates  used for the sputtering of Gd and Py  are \n1.4 Å/sec  and 0.7  Å/sec, respectively. For seeding and a capping , 5-nm thick  Ta layer s were deposited before and after \nsputtering each multilayer . In addition to the disk array s, two control films with the same  structure s were sputtered for the \nstudy. The wafer s containing fabricated disk arrays were cut into 6  mm×6  mm pieces , and their magnetic behavior was studied \nusing a superconducting quantum interference device (SQUID) magnetometer.  We also fabricated the  [Py(1  nm)/Gd(1  nm)] 25 \ndisk array on top of a Si3N4 membrane window  for a Lorentz transmission  electron microscopy ( TEM ) study  using the lift -off \ntechnique . TEM images at different temperatures were taken in Fresnel imaging mode .28 \nIII. MATERIAL  PROPERTIES AND SIMULATION S DETAILS  \nMagnetic and structural properties of the Py/Gd  multilayers have been studied by us26, 27 and other groups25, 29, 30 previously. \nImportantly for this work,  a polycrystalline Gd film has comparative ly high anisotropy,31 and only the Py/Gd multilayers with \nGd-layer  thickness  of 2 nm or less are magnetically soft. At the same time, it was  demonstrated that the influence  of material \nintermixing becomes significant  for multilayers w ith thin Py and Gd  layers .26, 27 The temperature dependences  of magnetization \nfor the [Py(t)/Gd( t)]25 (t=1 or 2 nm) control films  are shown in Fig.  1(a). It is seen that only the magnetization for the film with \nt=2 nm demonstrates ferrimagnet -like behavior. Its magnetization is small at room temperature . Then when the Gd layers  \nmagnetization s begin  to rise  the total magnetization decrease s; at around 172  K, the magnetic moments  of the Py and Gd layers \ncompensate each other. At low temperatures, the total magnetization in the multilayer is Gd -aligned. In contrast, th e \n[Py(1  nm)/Gd(1  nm)] 25 film becomes ferromagnetic only at around 275 K, which  is below  the Curie temperature s of bulk Gd \nand Py, and then  its magnetization rises monotonically with decreasing temperature.  Due to the strong intermixing of Py and \nGd, the [Py(1  nm)/Gd(1  nm)] 25 superlattice  can be considered as a homogeneous  film composed of a PyGd alloy . We adopt ed \nthis model for micromagnetic simulation s of magnetization behavior for the [Py(1  nm)/Gd(1  nm)] 25 disks. The analysis26, 27 \nshowed that the magnetic and at omic structure s of the [Py(2  nm)/Gd(2  nm)] 25 superlattice  is more complicated . According to \nour estimates ,26, 27 only the 0.5-nm thick core parts of  the Py and Gd lay ers are  not subjected to intermixing, while the rest of \nthe film is  the PyGd alloy.  Taking into account this magnetization profile over the [Py(2  nm)/Gd(2  nm)] 25 film thickness would \nlead to a very complicated  micromagnetic model which would require  a very small mesh , and hence, long calculation time. To \navoid this , similar ly to the microm agnetic model used for the [Py(1  nm)/Gd(1  nm)] 25 disks, we assume d that the  \n4 \n [Py(2  nm)/Gd(2  nm)] 25 disks are composed of a homogeneous material. Microscopically, the effective magnetization  of the \nPy/Gd  disks , Meff, is expected  to be equal to the magnetization s of the corresponding control films at a given temperature.   \n \nFIG. 1. a) Magnetization and b) effective exchange stiffness, Aeff, as a function of temperature for the [Py(1  nm)/Gd(1  nm)] 25 (black dots) \nand [Py(2  nm)/Gd(2  nm)] 25 (red dots) control films. Magnetization was measured in the 100 -Oe magnetic field applied in -plane. The lines \nrepresents dependences used for the micromagnetic simulations.  \n \nOnce  it is assumed  that the disks are composed of a homogene ous material, the effective exchange stiffness of this material  \nat different temperatures must  be estimated . For this, we utilize  a technique used by us previously.26, 27 Shortly, the  films with \nstructures Py(50  nm)/ [Py(t)/Gd( t)]25 (t=1 or  2 nm) were sputtered and their magnetization curve s were measured  at different \ntemperatures. The total magnetic moment of the [Py(t)/Gd( t)]25 stack is  coupled antife rromagnetically  with the magnetic \nmoment  of the 50 -nm thick Py layer . An applied magnetic field results in  an alignment of  these magnetic moments along the \nfield, and since  the effective exchange stiffness of the stack s is much smalle r than  that of Py, the process is realized through an \nexchange -spring -like magnetization twist  in the Py/Gd stacks.  Modeling the observed non -linear dependences  of the \nmagnetization on magn etic field, and assuming that the exchange stiffness ( Aeff) is constant across the Py/Gd stacks, allowed  \nus to estimate  Aeff in the [Py(t)/Gd( t)]25 (t=1 or  2 nm) multilayers for  different temperatures  [Fig.  1(b)] .  \nSummarizing, for a micromagnetic simulation  at a given temperature, T, it was assumed that a disk is composed of an \nisotropic homogeneous material with magnetization Meff(T) and exchange stiffness Aeff(T). The simulation s were  conducted \nusing a GPU -accelerated micromagnetic simulation program , mumax3.32 The size of the mesh  cell is 2.9  nm×2.9  nm in plane \nof a disk and 12.5  nm over  its thickness.  \nIV. EXPERIMENTAL RESULTS  \nFor both superlattice s composition s, Meff is at a maximum at low temperatures. Magnetization curves for the etched \n[Py(t)/Gd( t)]25 (t=1 or  2 nm) disks measure d at 10  K are shown in Fig.  2(a). Both curves correspond  to vortex -like behavior5 \n \n5 \n in low magnetic fields, i.e. the magnetization  changes linearly with  the field and the coercivity is negligibly small (the coercive \nfields are  0.5 and 2.5  Oe for t=1 and 2  nm, respectively ). The micromagnetically -simulated curves shown in  Fig. 2(b), \nquantitatively and qualitatively resemble the experimental ones. According to the simulations, the vortex  configuration is stable \nin the magnetic field s of up to 340  Oe for  the [Py(1  nm)/Gd(1  nm)] 25 disks, and up  to 580 Oe for the  [Py(2  nm)/Gd(2  nm)] 25 \ndisks. Due to an energy barrier separating  the vortex and uniformly -magnetized state s, the simulated and experimental loops \nhave  hysteretic  regions  in higher  magnetic  fields [Fig.  2(a, b)]. Because the disks diameter is 1.5  µm and Aeff is low, the vortex \nnucleation and annihilation are very different fr om those  observed in submicron Py disks.33 First, i ncreasing magnetic field \ncauses the magnetic vortex  to deform  in addition to shifting . In magnetic fields below the annihilation  fields, the vortices are \nmoon -shaped [schematics in Fig.  2(b)]. Second, upon magnetic field decrease,  the magnetization does not transform  from  a \nuniform state to a vortex state  directly, but instead , two vortices appear which move  toward each other and merge into a single \nvortex in 210 and 360 -Oe magnetic fields for t=1 and 2  nm, respectively [schematics in Fig.  2(b)] . Only the vortex annihilatio n \nlooks like a sharp transition . \n \nFIG. 2. a) Experimental b) micromagnetically -simulated hysteresis loops (10  K) for the etched [Py(1  nm)/Gd(1  nm)] 25 (black line) and \n[Py(2 nm)/Gd(2  nm)] 25 (red line) disks; b) contains the schematics of magnetization configurations realized in the disks at different \nmagnetic fields.  \n \nTo determine which magnetizatio n configurations are realized at  different temperatures , magnetization  of the etched  arrays \nwas measured in a 100 -Oe magnetic field while the temperature was slowly (1  K/min) swept from 300 to 10  K and back to \n300 K [Fig.  3(a)].  At the initial stage of the cooling down, the curves demonstrate the behavior almost identical to  that \n \n6 \n demonstrated by  the control films. T hen, at 112  K for t=1 nm and 66  K for t=2 nm, a phase  transition occurs  and the \nmagnetization begin s to decrease when  the temperature  is increased . At low temperatures, the magnetization is almost constant.  \nWhen the temperature is ramped up , the magnetization  decrease s but not monotonically. For the [Py(1  nm)/Gd(1  nm)] 25 disks, \nthe magnetization falls slowly until 143 K, and then the curve merge s with the one obtained while cooling down  the sample . A \nvery similar decrease  of the magnetizati on is demonstrated by the [Py(2  nm)/Gd(2  nm)] 25 disks in the initial stage  of the \nwarm -up, but right before the merging with the cool-down curve , the magnetization  kinks up. Interestingly , the magne tic phase \ntransition s yield  the temperature dependences  of the magnetization  to demonstrate  hysteretic behavior within the  40-160 K and \n40-110 K ranges, for t=1 and 2  nm, respectively.  \n \nFIG. 3. a) Experimental b) micromagnetically -simulated temperature dependences of magnetization in the 100 -Oe magnetic field for the \netched [Py(1  nm)/Gd(1  nm)] 25 (black line) and [Py(2  nm)/Gd(2  nm)] 25 (red line) disks; b) contains the schematics of magnetization \nconfigurations realized in the disks at  different temperatures.  \n \nThe tem perature dependences  of the disks magnetization in the 100-Oe magnetic field  were si mulated micromagnetically \n[Fig.  3(b)].  Importantly,  it is a zero-temperature  simulation s without  thermal field fluctuations  and any dependence on \ntemperature is only introduced by temperature -dependent parameters,  Meff(T) and Aeff(T). This simplification is reasonable \nbecause any fluctuation effect s, such as a thermal excitation over an energy barrier ,34 are expected to be insignificant for 1.5  µm \ndisks. The s imulated curves demonstrate hysteretic behavior  similar to the one  observed in the experiment  [Fig.  3(a, b)]. The \nsimulation s reveal that the phase transition s observed in the experiment are  due to the vortex nucleation and annihilation.  At \n \n7 \n the high temperatures, the disks magnetization s are  low and the disks are in the uniformly -magnetized state. W hen the \ntemperature  is decreased  the disks magnetization increase s, and at a critical temperature , it becomes energet ically favorable to \nnucleate vortices,  thus minimizing  the magnetic  flux. Again, similar ly to the nucleation processes observed at 10  K, the disks \nmagnetization does not switch  from uniform ly-magnetized  to single -vortex states  directly ; instead, the magnetization  transfer s \nthrough a series of intermediate states in which the doub le vortices configurations are  stable.  The transition s between each \nintermediate state cause s an abrupt decrease  of magnetization. Since the arrays consist of disks with slightly differe nt diameters, \nthe abrupt transitions observed in the simulation s are smeared out for the experimental curves. According to the simulations , \nincreasing the temperature causes the vortex core  shifting  from the center of the disk and its deformation, eventually the disks \nswitch  to the quasi -uniformly -magnetized state  [schematics in Fig.  3(b)] .  \nIt was of particular  interest to investigate the stability of the vortex state  at the temperatures which are within the thermal \nhysteresis regions , e.g. at 115 K for the [Py(1  nm)/Gd(1  nm)] 25 disk array and 75 K for the [Py(2  nm)/Gd(2  nm)] 25 disk array.  \nFor this, the disks were coole d down to 10  K, and the magnetic field was set to 0 Oe. T his procedure allowed to ensure  that the \nvortex  is the  initial magnetic state for the disks . Then , the temperature was increased to 115  K and 75  K for the \n[Py(1  nm)/Gd(1  nm)] 25 and for [Py(2  nm)/Gd(2  nm)] 25 disk array s, respectively, and  the arrays  magnetization  was measured \nwhile  the magnetic field was ramped  up from 0  Oe to 500  Oe and then cycle d between 500 and -500 Oe [Fig.  4(a)]. The \nexperiment s were  simulated microma gnetically [Fig.  4(b)].  For the [Py(1  nm)/Gd(1  nm)] 25 disk array , the magnetization  shows  \nthe behavi or identical to that observed for the array at  10 K, i.e. the magnetization rises l inearly in low magnetic field s and the \nhysteresis is  present only i n the high  fields. T he initial part of the magnetiza tion curve measured between 0 and  500 Oe coincides \nwith its  main part (500 Oe → -500 Oe → 500  Oe) indicat ing that the v ortex state is stable for these  disks in low magnetic field . \nThe simulated curve is  in qualitative agreement with the experimental one. In contrast, the main  part of the magnetization curve \nfor the [Py(2  nm)/Gd(2  nm)] 25 disk array has a double -waist ed shape, while its initial  curve demonstrates typical vortex -like \nbehavior.  Moreover, the initial  magnetization curve passes outside the area under the main magnetization curve. This unusual \nbehavior indicate s that the vortex state  is stable  at the initial stage of the measurements , but after the saturation of the disk \nmagnetization, the magnetization  does not switch back to the vortex state while the main hysteresis loop is measured. It was \nalso observed that, if the disk is demagnetized after measuring the main hysteresis loop, and after that, the magnetization i s \nmeasured while the magnetic field is  increased, the resulting curve passes within the main hysteresis loop and does not c oincide \nwith the i nitial magnetization curve.  Thus,  the (0,0) point at the M -H plane can correspond to two different magnetization \nstates, one of which is the vortex  state. This means  that the vortex is a metastable  state for the [Py(2  nm)/Gd(2  nm)] 25 disk at \n75 K. By a metastable state , it is implied that the state cannot be accessed by isothermal changing of an external magnetic field.   \n8 \n Interesting, the magnetic field depe ndence of magnetization for  the [Py(2  nm)/Gd(2  nm)] 25 disk array was not completely \nreproduced in the s imulation [Fig.  4(b) red line]. It is possible that  the fine magn etic structure within the [Py(2  nm)/Gd(2  nm)] 25 \nsuperlattice  as well as its very low magnetic anisotropy  must be taken into account  for an adequate modeling of the \nmagnetization reversal for  the corresponding disks. \n \nFIG. 4. a) Experimental b) micromagnetically -simulated hysteresis loops for the etched [Py(1  nm)/ Gd(1  nm)] 25 disks at 115  K (black lines) \nand the etched [Py(2  nm)/Gd(2  nm)] 25 disks at 75  K (red lines); the initial parts of the magnetization curves (from 0 to 500  Oe) are shown \nwith thin lines while the main parts of hysteresis curves (500  Oe → -500 Oe → 500  Oe) with thick lines; b) contains the schematics of \nmagnetization configurations realized in the disks at different magnetic fields. The inset in the upper -left corner of (b) shows the hysteretic \npart of the loop for the [Py(2  nm)/Gd(2  nm)] 25 disks.  \n \nAn important factor that strongly affects magnetic properties of the Py/Gd superlattice disks is the intermixing of Py and \nGd. Due to a shadow effect, t he intermixing is much more pronounced for the lift -off disks than that for the etched  disks. Since \nthe intermixing is almost c omplete for the [Py(1  nm)/Gd(1  nm)] 25 superlattice, t he lift -off and etched  [Py(1  nm)/Gd(1  nm)] 25 \ndisks demonstrate almost identical temperature and field dependences . At the same time, due to the intermixing, the \ncompen sation tempe rature of the lift -off [Py(2  nm)/Gd(2  nm)] 25 disks is about 50  K higher than that for the etched  disks and \nthe corresponding  control films. Although  the same physical phenomena are observed in the  lift-off [Py(2  nm)/Gd(2  nm)] 25 \ndisks as that  observed in the etched  ones ( vortex nucleation and annihilation ), some quantitative parameters, like  the \nnucleation/annih ilation fields and temperatures, are different for the  [Py(2  nm)/Gd(2  nm)] 25 disks prepared using di fferent \nfabrication techniques.  \n \n9 \n  \nFIG. 5. Under -focused Lorentz transition electron microscopy images of a lift-off [Py(1  nm)/Gd(1  nm)] 25 disk on a Si 3N4 membrane \nwindow acquired at 153  K (a) and 98  K (b). A white dot at the center of the disk at 98  K indicates the presence of magnetic vor tex; c) \nmagnetization of the etched [Py(1  nm)/Gd(1  nm)] 25 disk array as a function of temperature measured in the 10 -Oe magnetic field.  \n \nTo get a direct confirm ation that vortices nucleate  in the Py/Gd superl attice disks at low temperatures, we acquired a  series \nof under -focused Lorentz TEM images of a lift -off [Py(1  nm)/Gd(1  nm)] 25 disk on a Si 3N4 membrane at different temperatures. \nFig. 5(a) and (b) shows  the Lorentz T EM obtained at 152 and 98  K, respectively. A white dot at the center of the disk in \nFig. 5(b), which appears when the temperature drops below 129  K, indicates the presence of a  magnetic vortex . Fig. 5(c) shows  \na temperature dependence of  the magnetization measured in very low magnetic field (10 Oe), accordi ng to  which the \nmagnetization of  the [Py(1  nm)/Gd(1  nm)] 25 disks switches to the vortex state below 180  K in the low magnetic field.  \nImportantly, during the TEM imaging the temperature was s wept continuously. T he real temperature of the disk on the Si 3N4 \nmembrane  can be very different from the readings of  the microscope’s stage thermo couple . This yields slightly different  vortex \nnucleation temperatures detected using  Lorentz TEM and  using  magnetometry . \nV. CONCLUSION  \nWe observed that the 1.5 -µm disks composed of the [Py( t)/Gd( t)]25 (t=1 or  2 nm) artificial ferrimagnets experience phase \ntransitions.  For these disks, t he resulting temperature dependences  of magnetization measured in the 100 -Oe magnetic field are  \nhysteretic in narrow temperature regions . Micromagnetic  simulation s revealed that these phase transitions are due to nucleation \nand annihilation of magnetic vortices which happen because the magnetic properties of the artificial ferrimagnet change  \nsignificantly within  the 10 –300 K temperature range. It was shown that  at 75 K,  the vortex  in the [Py(2  nm)/Gd(2  nm)] 25 disk \nis in a meta stable state. I t is impossible to nucleate the vortices  in the [Py(2 nm)/Gd(2  nm)] 25 disks by changing the external \n \n10 \n magnetic  field isothermally , at the same time,  if the disk  with the magnetic vortex is brought to 75  K, the vortex does not \nannihilate until the disk magnetization is saturated.  \nREFERENCES  \n1. K. Y. Guslienko, J. Nanosci. Nanotechnol. 8, 2745 -2760 (2008).  \n2. T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T. Ono, Science 289 (5481), 930 -932 (2000).  \n3. K. Bussmann, G. A. Prinz, S. -F. Cheng and D. Wang, Applied Physics Letters 75 (16), 2476 -2478 (1999).  \n4. R. 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Waeyenberge, AIP Advance s 4 (10), \n107133 (2014).  \n33. G. Gubbiotti, G. Carlotti, F. Nizzoli, R. Zivieri, T. Okuno and T. Shinjo, IEEE Transactions on Magnetics 38 (5), 2532 -\n2534 (2002).  \n34. K. Liu, J. Nogués, C. Leighton, H. Masuda, K. Nishio, I. V. Roshchin and I. K. Schuller, Ap plied Physics Letters 81 \n(23), 4434 -4436 (2002).  "    },    {        "title": "1504.04547v1.Model_for_the_FC_and_ZFC_Ferrimagnetic_Spinel.pdf",        "content": "arXiv:1504.04547v1  [cond-mat.str-el]  17 Apr 2015Model for the FC and ZFC Ferrimagnetic Spinel\nN. Karchev\nDepartment of Physics, Sofia University, James Bourchier 5 b lvd., 1164 Sofia, Bulgaria\n(Dated: October 20, 2021)\nThere are two methods of preparation of ferrimagnetic spine l. If, during the preparation, an\nexternal magnetic field as high as 300 O¨ e is applied upon cool ing the material is named field-cooled\n(FC).Iftheappliedfieldisabout1O¨ ethematerial iszero-fi eldcooled (ZFC).Toexplorethemagnetic\nand thermodynamic properties of these materials we conside r two-sublattice spin system, defined\non the bcc lattice, with spin- sAoperators SA\niat the sublattice Asite and spin- sBoperators SB\niat\nthe sublattice Bsite, where sA> sB. The subtle point is the exchange between sublattice A and\nB spins, which is antiferromanetic. Applying magnetic field along the sublattice A magnetization,\nduring preparation of the material, one compensates the Zee man splitting, due to the exchange, of\nsublattice B electrons. This effectively leads to a decrease of thesBspin. We consider a model with\nsBvarying parameter which accounts for the applied, during th e preparation, magnetic field.\nIt is shown that the model agrees well with the observed magne tization-temperature curves of\nzero field cooled (ZFC) and non-zero field cooled (FC) spinel f errimagnetic spinel and explains the\nanomalous temperature dependence of the specific heat.\nPACS numbers: 75.50.Gg,71.70.Ej,75.10.Dg,75.10.Lp\nI. INTRODUCTION\nThe magnetization-temperature and magnetic suscep-\ntibility curves for zero field cooled (ZFC) and non-zero\nfield cooled (FC) ferrimagnetic spinel display a notable\ndifference below N´ eel TNtemperature [1–13]. The (ZFC)\ncurve exhibits a maximum and then a monotonic de-\ncrease upon cooling from TN, while the (FC) curve in-\ncreases steeply, shows a dip near the temperature at\nwhich the (ZFC) curve has a maximum and finally in-\ncreases monotonically. There is also difference between\nso-called field-cooled-cooling and field-cooled-warming\nprocedures [14].\nThe specific heat curves for spinel ferrimagnetics show\nsharp peak at N´ eel temperature, which indicates fer-\nrimagnetic to paramagnetic transition, and tiny peak\nat temperature below N´ eel’s where the magnetization-\ntemperature curve has a maximum [3, 6, 9].\nAlthough the FC and ZFC spinel ferrimagnetics have\nintensively been studied, their magnetic and thermody-\nnamic properties have not been understood. There is\nnot an effective model which in unified way to explain\nthe experimental results.\nIn the present paper we consider two-sublattice spin\nsystem, defined on the bcc lattice, with spin- sAopera-\ntorsSA\niat the sublattice Asite and spin- sBoperators\nSB\niat the sublattice Bsite, where sA> sB. The subtle\npoint is the exchange between sublattice A and B spins,\nwhich is antiferromanetic. Applying magnetic field along\nthesublatticeAmagnetization, duringpreparationofthe\nmaterial, one compensates the Zeeman splitting of sub-\nlattice B electrons due to the exchange. This effectively\nleads to decrease of the sBspin. One can obtain an intu-\nition for this from spin-fermion model of spinel ferrimag-\nnetic with spin-1 /2itinerantelectronsat the sublattice B\nsite and spin- slocalizedelectronsat the sublattice Asite.\nAn applied, alongthe magnetizationof the localized elec-trons, external magnetic field compensates the Zeeman\nsplitting due to the spin-fermion exchange. Integrating\nout the fermions we obtain an effective spin model with\neffective spin sBwhich depends on the external magnetic\nfield (see Appendix A).\nWe consider a model with sBvarying parameter which\naccounts for the applied, during the preparation, mag-\nnetic field. First we present the method of calcula-\ntion exploring ZFC spinel with sA= 1.5 andsB= 1.\nWe obtained that the system has two phases. At low\ntemperature (0 ,T∗) the magnetic orders of the Aand\nBspins contribute to the magnetization of the sys-\ntem, while at the high temperature ( T∗,TN), the mag-\nnetic order of the sublattice B with smaller spin sBand\nwith a weaker intra-sublattice exchange is suppressed by\nmagnon fluctuations. Only the sublattice A spins, with\nstronger intra-sublattice exchange, have non-zero spon-\ntaneous magnetization. There is no additional symmetry\nbreaking, and the Goldstone boson has a ferromagnetic\ndispersion in both phases, but partial-order transition\ndemonstrates itself through tiny peak of specific heat as\na function of temperature.\nPartial order is well known phenomenon and has been\nsubject to extensive studies. Frustrated antiferromag-\nnetic systems has been studied by means of Green func-\ntion formalism. Partial order and anomalous temper-\nature dependence of specific heat have been predicted\n[15]. Experimentally the partial order has been observed\ninGd2Ti2O7[16]. Monte Carlo method has been uti-\nlized to study the nature of partial order in Ising model\nonkagom´elattice [17]. There are exact results for the\npartially ordered systems which precede the above stud-\nies [17–19]. The advantage of the present method of cal-\nculation is that it permits to consider the sublattice B\nspin as a varying parameter sB<1. We calculate the\nmagnetization-temperature curves and the specific heat\nas a function of temperature for sB= 0.7,sB= 0.4,2\nsB= 0.2 and fixed sA= 1.5, thus accounting for the in-\ncreasing of the applied, during preparation of FC spinel,\nmagnetic field.\nII. METHOD OF CALCULATION\nThe Hamiltonian of the ZFC ferrimagnetic spinel is\nH=−JA/summationdisplay\n≪ij≫ASA\ni·SA\nj−JB/summationdisplay\n≪ij≫BSB\ni·SB\nj\n+J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj (1)\nwhere the sums are over all sites of a body-centered cu-\nbic (bcc) lattice: ∝angbracketlefti,j∝angbracketrightdenotes the sum over the nearest\nneighbors, ≪i,j≫Adenotes the sum over the sites of\nthe A sublattice, ≪i,j≫Bdenotes the sum over the\nsites of the B sublattice. The first two terms describe\nthe ferromagnetic Heisenberg intra-sublattice exchange\nJA>0,JB>0, while the third term describes the inter-\nsublattice exchange which is antiferromagnetic J >0.\nTo study a theory with the Hamiltonian Eq.(1) it\nis convenient to introduce Holstein-Primakoff represen-\ntation for the spin operators SA\ni(a+,a) andSB\ni(b+,b).\nRewriting the effective Hamiltonian in terms of the Bose\noperators ( a+,a,b+,b) we keep only the quadratic and\nquartic terms. The next step is to represent the Hamil-\ntonian in the Hartree-Fock approximation:\nH≈HHF=Hcl+Hq (2)\nwith\nHcl= 6NJA(sA)2(uA−1)2+6NJB(sB)2(uB−1)2\n+ 8NJsAsB(u−1)2, (3)\nand\nHq=/summationdisplay\nk∈Br/bracketleftbig\nεa\nka+\nkak+εb\nkb+\nkbk−γk/parenleftbig\na+\nkb+\nk+bkak/parenrightbig/bracketrightbig\n,\n(4)\nwhereN=NA=NBis the number of sites on a sublat-\ntice. The two equivalent sublattices A and B of the bcc\nlattice are simple cubic lattices. The wave vector kruns\nover the reduced first Brillouin zone Brof a bcc lattice\nwhich is the first Brillouin zone of a simple cubic lattice.\nThe dispersions are given by equalities\nεa\nk= 4sAJAuA(3−coskx−cosky−coskz) + 8sBJu\nεb\nk= 4sBJBuB(3−coskx−cosky−coskz) + 8sAJu\nγk= 8J u√\nsAsBcoskx\n2cosky\n2coskz\n2(5)\nThe equations (5) show that Hartree-Fock parameters\n(uA,uB,u) renormalize the intra and inter-sublattice ex-\nchange constants ( JA,JB,J) respectively.To diagonalize the Hamiltonian one introduces new\nBose fields αk, α+\nk, βk, β+\nkby means of the transforma-\ntion\nak=ukαk+vkβ+\nka+\nk=ukα+\nk+vkβk\n(6)\nbk=ukβk+vkα+\nkb+\nk=ukβ+\nk+vkαk,\nwherethe coefficients ofthe transformation ukandvkare\nrealfunctionsofthewavevector k(B7). Thetransformed\nHamiltonian adopts the form\nHq=/summationdisplay\nk∈Br/parenleftBig\nEα\nkα+\nkαk+Eβ\nkβ+\nkβk+E0\nk/parenrightBig\n,(7)\nwith new dispersions\nEα\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk+εa\nk/bracketrightbigg\n(8)\nEβ\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+εb\nk−εa\nk/bracketrightbigg\nand vacuum energy\nE0\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk−εa\nk/bracketrightbigg\n(9)\nFor positive values of the Hartree-Fock parameters and\nall values of k∈Br, the dispersions are nonnegative\nEα\nk≥0, Eβ\nk≥0. When sA> sBtheαkboson is\nthe long-range (magnon) excitation in the system with\nEα\nk∝ρk2, near the zero wavevector, while the βkbo-\nson is a gapped excitation, with gap proportional to the\ninter-sublattice exchange constant Eβ\n0= 8Ju(sA−sB) .\nThe free energy of a system with Hamiltonian HHF\nequations (2), (3) and (4) is\nF= 6NJA(sA)2(uA−1)2+6NJB(sB)2(uB−1)2\n+ 8NJsAsB(u−1)2+1\nN/summationdisplay\nk∈BrE0\nk (10)\n+1\nβN/summationdisplay\nk∈Br/bracketleftBig\nln/parenleftBig\n1−e−βEα\nk/parenrightBig\n+ ln/parenleftBig\n1−e−βEβ\nk/parenrightBig/bracketrightBig\n,\nwhereβ= 1/Tis the inverse temperature. Then, the\nsystem of equations for the Hartree-Fock parameters is\n∂F/∂uA= 0, ∂F/∂uB= 0, ∂F/∂u= 0.(11)\nThe Hartree-Fock parameters are positive functions\nofT/J, solution of the system of equations (11) (see\nEqs.(B8)). Utilizing these functions, one can calculate\nthe spontaneous magnetization on the two sublattices\nMA=< S3\n1j> j is from sublattice A\n(12)\nMB=< S3\n2j> j is from sublattice B3\nandM=MA+MB, the spontaneous magnetization of\nthe system. In terms of the Bose functions of the αand\nβexcitations they adopt the form\nMA=sA−1\nN/summationdisplay\nk∈Br/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\n(13)\nMB=−sB+1\nN/summationdisplay\nk∈Br/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\n,\nwhereukandvkare functions of the wavevector k, coef-\nficients in the transformation (6).\nThemagnonexcitation- αkisacomplicatedmixtureof\nthe transversal fluctuations of the AandBspins (6). As\na result the magnons’ fluctuations suppress in a different\nwaythemagnetizationonsublattices AandB. Quantita-\ntivelythis dependsonthe coefficients ukandvk. At char-\nacteristic temperature T∗spontaneous magnetization on\nsublattice Bbecomes equal to zero, while spontaneous\nmagnetization on sublattice Ais still nonzero (see Fig.1).\nTo study the magnetic properties of the system above\nT∗we first consider the paramagnetic phase. To this end\nwe make use of the Takahashi modified spin-wave theory\n[20] and introduce two parameters λAandλBto enforce\nthe magnetization on the two sublattices to be equal to\nzero. The new Hamiltonian is obtained from the old one\nequation (1) by adding two new terms:\nˆH=H−/summationdisplay\ni∈AλASA3\ni+/summationdisplay\ni∈BλBSB3\ni (14)\nIn momentum space the new Hamiltonian ˆHadopts the\nform Eq.(4) with new dispersions\nˆεa\nk=εa\nk+λA,ˆεb\nk=εb\nk+λB.(15)\nUtilizing the same transformation (6) one obtains the\nHamiltonian ˆHin diagonal form (7) with dispersions ˆEα\nandˆEβobtained from Eqs.(8) replacing εa\nkandεb\nkwith\nˆεa\nkand ˆεb\nkrespectively. It is convenient to represent the\nparameters λAandλBin the form\nλA= 6JusB(µA−1), λB= 6JusA(µB−1).(16)\nThe dispersions ˆ εa\nkand ˆεb\nkarepositivefor all valuesofthe\nwavevector k, if the parameters µAandµBare positive.\nThe dispersions ˆEα\nkandˆEβ\nkare well defined if\nµAµB≥1. (17)\nTheβkexcitation is gapped ( ˆEβ\nk>0) for all values of\nparameters µAandµBwhich satisfy equation (17). The\nαexcitation is gapped if µAµB>1, but in the particular\ncase\nµAµB= 1 (18)\nˆEα\n0= 0, and near the zero wavevector ˆEα\nk≈ˆρk2. There-\nfor, in the particular case Eq. (18) αkboson is the long-\nrange excitation (magnon) in the system./s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48\n/s45/s49/s44/s48/s45/s48/s44/s53/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s83/s80/s79/s78/s84/s65/s78/s69/s79/s85/s83/s32/s32/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78/s84/s47/s74/s77/s65 \n/s77/s65 \n/s43/s77/s66\n/s77/s66/s115/s65 \n/s61/s49/s46/s53\n/s115/s66\n/s61/s49\nFIG. 1: (Color online) Spontaneous magnetization MA,MB\nandMA+MBas a function of the temperature in units of\nexchange constant T/Jfor ferrimagnetic spinel on bcc lattice\nwithsA= 1.5,sB= 1,JA/J= 2 and JB/J= 0.002. The\npartial order transition temperature T∗is the temperature\nabove which MB= 0.\nAboveN´ eeltemperature we introduced the parameters\nµAandµB(λA,λB) to enforce the sublattice AandB\nspontaneous magnetizations to be equal to zero. We find\nout these and Hartree-Fock parameters, as functions of\ntemperature, solving the system of five equations, equa-\ntions (11) and the equations MA=MB= 0, where the\nspontaneous magnetization has the same representation\nasequations(13)butwithcoefficients ˆ uk,ˆvk, anddisper-\nsionsˆEα\nk,ˆEβ\nkin the expressions for the Bose functions.\nThe numerical calculations show that above N´ eel tem-\nperature µAµB>1. When the temperature decreases\nthe product µAµBdecreases, remaining larger than one.\nThe temperature at which the product becomes equal to\none (µAµB= 1) is the N´ eel temperature Fig.(6).\nBelowTN, the spectrum contains long-range(magnon)\nexcitations. Thereupon, µAµB= 1 and one can use the\nrepresentation µB= 1/µA. Then,µAand Hartree-Fock\nparameters are solution of a system of four equations,\nequations (11) and the equation MB= 0.\nWe utilize the obtained functions µA(T),µB(T)uA(T),\nuB(T),u(T) (see Appendix C) to calculate the sponta-\nneous magnetization as a function of the temperature.\nFor a system with sA= 1.5,sB= 1,JA/J= 2 and\nJB/J= 0.002 the functions MA(T/J),MB(T/J) and\nMA(T/J) +MB(T/J) are depicted in figure (1). The\npartial order transition temperature T∗is the tempera-\nture above which MB= 0.\nThe customary formula for the entropy of a Bose sys-\ntem with Hamiltonian (7) is\nS=1\nN/summationdisplay\nk,δ/bracketleftbig\n(1+nδ\nk)ln(1+nδ\nk)−nδ\nklnnδ\nk/bracketrightbig\n,(19)\nwhereδstays for αandβ. The dispersions Eα\nkandEβ\nk\n(8) areused to define the Bosefunctions nα\nkandnβ\nkbelow\nT∗, dispersions ˆEα\nkandˆEβ\nkwithµB= 1/µAare used for4\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48 /s50/s50 /s50/s52 /s50/s54 /s50/s56 /s51/s48 /s51/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s67\n/s84/s47/s74/s115/s65\n/s61/s49/s46/s53\n/s115/s66\n/s61/s49\nFIG. 2: (Color online) Specific heat Cvs temperature, in\nunits of the exchange constant T/J, forJA/J= 2,JB/J=\n0.002,sA= 1.5 andsB= 1. The high temperature (green)\nvertical line marks N´ eel TNtemperature, while the low tem-\nperature (red) line marks partial order transition tempera ture\nT∗.\npartial order phase T∗< T < T N, and with µAµB>1\nfor paramagnetic phase above N´ eel temperature. With\nentropy, as a function of temperature in mind, one can\ncalculatethecontributionofmagnonstothespecificheat:\nC=TdS\ndT(20)\nThe resultant curve C(T/J), for a system with the same\nparameters, as above, is depicted in figure (4).\nThefigures(1)and(2)showthatthepresentmethodof\ncalculation describes correctlythe features of the system,\npartial order transition and anomalous behavior of the\nspecific heat at the temperature of this transition T∗.\nIII. FC SPINEL FERRIMAGNETICS\nFC spinel ferrimagnetics are prepared applying mag-\nnetic field along the sublattice A magnetization upon\ncoolingthematerial. ThiscompensatestheZeemansplit-\nting of sublattice B electrons which effectively leads to\ndecrease of the sBspin. The Hamiltonian of the FC\nspinel ferrimagnet is given by Eq.(1) with sBa varying\nparameter. The advantage of the method of calculation,\npresented above, is that we can use the same systems of\nequations for the three phases, 0 < T < T∗,T∗< T <\nTN,T > T N, and different values of the parameter sB.\nWe calculate the magnetization-temperature curves and\nthespecificheatasafunctionoftemperaturefor sA= 1.5\nand three different values of sB(sB= 0.7,sB= 0.4 and\nsB= 0.2), thus accounting for the increasing of the ap-\nplied, during preparation of FC spinel, magnetic field.\nThe resultant magnetization-temperature curves are\ndepicted in figure (3). The curve ”a” corresponds to\nZFC spinel (see figure (1)). The increasing of the ap-\nplied, during the preparation, magnetic field is mod-\neled by decreasing of sB. The curves ”b”, ”c” and ”d”/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52/s83/s80/s79/s78/s84/s65/s78/s69/s79/s85/s83/s32/s32/s77 /s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78\n/s84/s47/s74/s97/s98/s99/s100\nFIG. 3: Color online) Spontaneous magnetization MA+MB\nas a function of the temperature in units of the exchange\nconstant T/Jfor ferrimagnetic spinel on bcc lattice with\nJA/J= 2,JB= 0.002,sA= 1.5 and :a)(black) sB= 1,\nb)(red)sB= 0.7, c)(blue) sB= 0.4 and d)(green) sB= 0.2.\nThe dash (magenta) line shows the spontaneous magnetiza-\ntionMAof sublattice AforsA= 1.5 andsB= 0.2.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s67/s32/s115/s66\n/s61/s49\n/s32/s115/s66\n/s61/s48/s46/s55\n/s32/s115/s66\n/s61/s48/s46/s50/s115/s65\n/s61/s49/s46/s53\n/s84/s47/s74\nFIG. 4: (Color online) Specific heat Cvs temperature, in\nunits of the exchange constant T/J, forsA= 1.5 and different\nvalues of the effective spin sB: black curve sB= 1, red curve\nsB= 0.7 and blue curve sB= 0.2 .\nare magnetization-temperature curves for FC spinel fer-\nrimagnetics with sB= 0.7,sB= 0.4 andsB= 0.2. The\ncurve ”c” agrees well with the observed magnetization-\ntemperature curve of FC spinel MnV2O4with an exter-\nnal magnetic field as high as 300 O¨ e applied upon cool-\ning the material [4, 5]. The dash line shows the spon-\ntaneous magnetization MAof sublattice AforsA= 1.5\nandsB= 0.2. Comparing with line ”d”, which shows\nthe spontaneous magnetization MA+MBof the same\nsystem one can conclude that contribution of the sublat-\ntice B magnetization is small and Zeeman splitting, of\nsublattice B electrons, is approximately compensated.\nThe dependence of specific heat on temperature for\ndifferent values of sBis shown in figure (4). The curve\nwith higher maximumcorrespondsto ZFC spinel (see fig-\nure (2)) With increasing the applied, during the prepara-\ntion, magnetic field (with decreasing sB) the anomalous\ntemperature behavior of specific heat, at temperature of\npartialordertransition, remainswell defined but the pick5\ndecreases. The anomalous pick is well observed experi-\nmentally [3, 6, 9], and one can use it to determine the T∗\ntemperature of the partial order transition.\nIV. SUMMARY\nIn this paper is pointed out that a model which agrees\nwell with the observed magnetic and thermodynamic\nproperties of ZFC and FC spinel ferrimagnets is a two-\nsublattice Heisenberg model with spin- sAoperators at\nthe sublattice Asite, spin- sBoperators at the sublat-\nticeBsite, ferromagnetic intra-sublattice exchange and\nantiferromagnetic inter-sublattice exchange. The ap-\nplied magnetic field, along the sublattice A magnetiza-\ntion upon cooling the material, is accounted for varying\nonly one parameter sB.\nThe subtle point is that MnV2O4spinel hasan obvious\nanomalousmagnetic behavior,but the appearanceofthis\nanomaly is attributed to the strong orbital-spin coupling\n[4] which is not discussed in the model (1). The spinel\nMnV2O4is a two-sublattice ferrimagnet, with site Aoc-\ncupied by the Mn2+ion, which is in the 3 d5high-spin\nconfiguration with quenched orbital angular momentum,\nwhich can be regarded as a s= 5/2 spin. The B site is\noccupied by the V3+ion, which takes the 3 d2high-spin\nconfiguration in the triply degenerate t2gorbital and has\norbital degrees of freedom. Because of the strong spin-\norbitalinteractionit isconvenientto consider jjcoupling\nwithJA=SAandJB=LB+SB. The sublattice A\ntotal angular momentum is jA=sA= 5/2, while the\nsublattice Btotal angular momentum is jB=lB+sB,\nwithlB= 3, and sB= 1 [1]. Then the g-factor for\nthe sublattice AisgA= 2, and for the sublattice B\ngB=5\n4. The sublattice Amagnetic order is antiparallel\nto the sublattice Bone and the saturated magnetization\nisσ= 25\n2−5\n44 = 0, in agreement with the experimen-\ntal finding for ZFC spinel that the magnetization goes to\nzero when the temperature approaches zero. The Hamil-\ntonian of the system is\nH=−κA/summationdisplay\n≪ij≫AJA\ni·JA\nj−κB/summationdisplay\n≪ij≫BJB\ni·JB\nj\n+κ/summationdisplay\n/angbracketleftij/angbracketrightJA\ni·JB\nj (21)\nThe first two terms describe the ferromagnetic Heisen-\nberg intra-sublattice exchange κA>0,κB>0, while the\nthird term describes the inter-sublattice exchange which\nisantiferromagnetic κ >0. Theoperators JA\njandJB\njsat-\nisfy theSU(2) algebra, therefor we can use the Holstein-\nPrimakoff representation of the total angular momen-\ntum vectors JA\nj(a+\nj,aj)andJB\nj(b+\nj, bj), wherea+\nj, ajand\nb+\nj, bjare Bose fields. Farther on, we repeat the calcula-\ntions from sections II and III. The only difference is the\nexpression for the magnetization gAMA+gBMB. The\njjmodel shows that the anomalous magnetic and ther-\nmodynamic behavior of MnV2O4is a consequence of theferrimagnetic nature of the system.\nAppendix A: Spin-fermion model of spinel\nferrimagnet\nThe Hamiltonian of the spin-fermion model of ferri-\nmagnetic spinel defined on a body centered cubic lattice\nis\nH=−t/summationdisplay\n≪ij≫B/parenleftbig\nc+\niσcjσ+h.c./parenrightbig\n−µ/summationdisplay\ni∈Bni\n−JB1/summationdisplay\n≪ij≫BSB\ni·SB\nj+J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj(A1)\n−JA/summationdisplay\n≪ij≫ASA\ni·SA\nj−H/summationdisplay\ni∈ASzA\ni−H/summationdisplay\ni∈BSzB\ni,\nwhereSνB\ni=1\n2/summationtext\nσσ′c+\niστν\nσσ′ciσ′, with the Pauli matrices\n(τx,τy,τz), is the spin of the itinerant electrons at the\nsublattice Bsite ,SA\niis the spin ofthe localizedelectrons\nat the sublattice Asite,µis the chemical potential, and\nni=c+\niσciσ. The sums are over all sites of a body cen-\nteredcubic lattice, ∝angbracketlefti,j∝angbracketrightdenotesthe sum overthe nearest\nneighbors, while ≪ij≫Aand≪ij≫Bare sums over\nall sites of sublattice AandBrespectively. The Heisen-\nberg term ( JA>0) describes ferromagnetic Heisenberg\nexchangebetweenlocalizedelectronsand J >0is the an-\ntiferromagnetic exchange constant between localized and\nitinerant electrons. H >0 is the Zeeman splitting en-\nergy due to the external magnetic field (magnetic field\nin units of energy). We represent the Fermi operators,\nthe spin of the itinerant electrons and the density oper-\natorsniσin terms of the Schwinger bosons ( ϕi,σ,ϕ+\ni,σ)\nand slave fermions ( hi,h+\ni,di,d+\ni). The Bose fields are\ndoublets ( σ= 1,2) without charge, while fermions are\nspinless with charges 1 ( di) and -1 ( hi):\nci↑=h+\niϕi1+ϕ+\ni2di, c i↓=h+\niϕi2−ϕ+\ni1di,\nni= 1−h+\nihi+d+\nidi, sν\ni=1\n2/summationdisplay\nσσ′ϕ+\niστν\nσσ′ϕiσ′,\nc+\ni↑ci↑c+\ni↓ci↓=d+\nidi (A2)\nϕ+\ni1ϕi1+ϕ+\ni2ϕi2+d+\nidi+h+\nihi= 1 (A3)\nTo solve the constraint (Eq.A3), one makes a change of\nvariables, introducing Bose doublets ζiσandζ+\niσ[21]\nζiσ=ϕiσ/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig−1\n2,\nζ+\niσ=ϕ+\niσ/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig−1\n2,(A4)\nwhere the new fields satisfy the constraint ζ+\niσζiσ= 1. In\nterms of the new fields the spin vectors of the itinerant\nelectrons have the form\nSνB\ni=1\n2/summationdisplay\nσσ′ζ+\niστν\nσσ′ζiσ′/bracketleftbig\n1−h+\nihi−d+\nidi/bracketrightbig\n(A5)6\nWhen, in the ground state, the lattice site is empty, the\noperator identity h+\nihi= 1 is true. When the lattice\nsite is doubly occupied, d+\nidi= 1. Hence, when the lat-\ntice site is empty or doubly occupied the spin on this\nsite is zero. When the lattice site is neither empty nor\ndoubly occupied ( h+\nihi=d+\nidi= 0), the spin equals\nsi= 1/2ni,where the unit vector\nnν\ni=/summationdisplay\nσσ′ζ+\niστν\nσσ′ζiσ′(n2\ni= 1) (A6)\nidentifies the local orientation of the spin of the itinerant\nelectron.\nThe part of Hamiltonian Eq.(A1) with itinerant\nfermions can be rewritten in terms of Bose fields Eq.(A4)\nand slave fermions\nHB=−t/summationdisplay\n≪ij≫B/bracketleftbig/parenleftbig\nd+\njdi−h+\njhi/parenrightbig\nζ+\niσζjσ\n+/parenleftbig\nd+\njh+\ni−d+\nih+\nj/parenrightbig\n(ζi1ζj2−ζi2ζj1)+h.c./bracketrightbig\n×/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig1\n2/parenleftbig\n1−h+\njhj−d+\njdj/parenrightbig1\n2\n−JB1/summationdisplay\n≪ij≫Bmimjni·nj\n+U/summationdisplay\ni∈Bd+\nidi−µ/summationdisplay\ni∈B/parenleftbig\n1−h+\nihi+d+\nidi/parenrightbig\n(A7)\n−H/summationdisplay\ni∈B1\n2nz\ni[1−h+\nihi−d+\nidi].\nwhere\nmi=1\n2[1−h+\nihi−d+\nidi]. (A8)\nThe mixed, spin-fermion term, adopts the form\nHAB=J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·njmj (A9)\nThe Hamiltonian (A1) can be rewritten in the form\nH=HA+HB+HAB(A10)\nwhereHAis the contribution of sublattice A spins SA\ni.\nAn important advantage of working with Schwinger\nbosons and slave fermions is the fact that Hubbard term\nis in a diagonal form. The fermion-fermion and fermion-\nboson interactions are included in the hopping term.\nOne treats them as a perturbation. To proceed we ap-\nproximate the hopping term of the Hamiltonian Eq.(A7)\nsetting/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig1\n2∼1 and keeping only the\nquadratic, with respect to fermions, terms. This means\nthat the averagingin the subspace of the fermions is per-\nformed in one fermion-loop approximation. Further, we\nrepresenttheresulting hBandhABHamiltonianasasum\nof two terms\nHf=Hf\n0+Hfb\nint, (A11)where\nHf\n0=−t/summationdisplay\n≪ij≫B/parenleftbig\nd+\njdi−h+\njhi+h.c./parenrightbig\n+U/summationdisplay\ni∈Bd+\nidi\n−µ/summationdisplay\ni∈B/parenleftbig\n1−h+\nihi+d+\nidi/parenrightbig\n(A12)\n+4sJ−H\n2/summationdisplay\ni∈B/parenleftbig\nh+\nihi+d+\nidi/parenrightbig\n,\nis the Hamiltonian of the free dandhfermions, and\nHfb\nint=−t/summationdisplay\n≪ij≫B/bracketleftbig/parenleftbig\nd+\njdi−h+\njhi/parenrightbig/parenleftbig\nζ+\niσζjσ−1/parenrightbig\n(A13)\n+/parenleftbig\nd+\njh+\ni−d+\nih+\nj/parenrightbig\n(ζi1ζj2−ζi2ζj1)+h.c./bracketrightbig\n−JB1/summationdisplay\n≪ij≫Bmimjni·nj\nis the Hamiltonian of boson-fermion interaction.\nThegroundstateofthe system, without accountingfor\nthe spin fluctuations, is determined by the free-fermion\nHamiltonian h0and is labeled by the density of electrons\nn= 1−< h+\nihi>+< d+\nidi> (A14)\n(see equation (A2)) and the ”effective spin” of the sub-\nlattice B electron\nsB=1\n2/parenleftbig\n1−< h+\nihi>−< d+\nidi>/parenrightbig\n.(A15)\nAt half-filling\n< h+\nihi>=< d+\nidi> . (A16)\nTo solve this equation one sets the chemical potential\nµ=U/2. Utilizing this representation of µwe calculate\nthe effective spin sBas a function of applied magnetic\nfieldhfor parameters 4 t/U= 2.2 and 4sAJ/U= 4. The\nresult is depicted in figure (5).\nLet us introduce the vector,\nMν\ni=sB/summationdisplay\nσσ′ζ+\niστν\nσσ′ζiσ′M2\ni= (sB)2.(A17)\nThen, the spin-vector of itinerant electrons Eq.(A5) can\nbe written in the form\nSB\ni=1\n2mMi/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig\n,(A18)\nwhere the vector Miidentifies the localorientationof the\nspin of the sublattice B itinerant electrons.\nThe Hamiltonian is quadratic with respect to the\nfermions di,d+\niandhi,h+\ni, and one can average in the\nsubspace of these fermions (to integrate them out in the\npath integral approach). As a result, accounting for the\ndefinition of sB(see Eq.A15), one obtains SBi=Miand\nan effective model for spin vectors SBiwith Hamiltonian\nHeff=−JB2/summationdisplay\n≪ij≫BSB\ni·SB\nj,(A19)7\n/s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52/s48/s44/s53\n/s52/s116/s47/s85/s61/s50/s46/s50\n/s52/s115/s65\n/s74/s47/s85/s61/s52/s115/s66\n/s104\nFIG. 5: (Color online) Sublattice B effective spin sBas a\nfunctionofapplied magnetic field hfor parameters 4 t/U= 2.2\nand 4sAJ/U= 4.\nwhere the effective exchange constant JB2is calculated\nin the one loop approximation. The term with JB1ex-\nchange constant in equation (A7) and the mixed, spin-\nfermion term (A9), adopt the form\n−JB1/summationdisplay\n≪ij≫BSBi·SBj (A20)\nand\nJ/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj (A21)\nrespectively. Collecting all terms one obtains the Hamil-\ntonian of the FC ferrimagnetic spinel\nH=−JA/summationdisplay\n≪ij≫ASA\ni·SA\nj−JB/summationdisplay\n≪ij≫BSB\ni·SB\nj\n+J/summationdisplay\n/angbracketleftij/angbracketrightSA\ni·SB\nj,\nwhereJB=JB1+JB2andsBdepending on the field\napplied during preparation Fig.(5). After the process of\npreparation the external magnetic field is set equal to\nzero. For systems with sB= 1, discussed in the pa-\nper, one has to consider two-band model for sublattice\nB fermions. The result remains the same even for this\nmore complicate system: the effective spin sBdecreases\nwhen the applied, under the preparation, magnetic field\nincreases.\nAppendix B: Hartree-Fock approximation\nLet us consider a theory with Hamiltonian (1). We\nintroduce Holstein-Primakoff representation for the spinoperators\nSA+\nj=SA1\nj+iSA2\nj=/radicalBig\n2sA−a+\njajaj\nSA−\nj=SA1\nj−iSA2\nj=a+\nj/radicalBig\n2sA−a+\njaj(B1)\nSA3\nj=sA−a+\njaj\nwhen the sites jare from sublattice Aand\nSB+\nj=SB1\nj+iSB2\nj=−b+\nj/radicalBig\n2sB−b+\njbj\nSB−\nj=SB1\nj−iSB2\nj=−/radicalBig\n2sB−b+\njbjbj(B2)\nSB3\nj=−sB+b+\njbj\nwhen the sites jare from sublattice B. The operators\na+\nj, ajandb+\nj, bjsatisfythe Bosecommutationrelations.\nIn terms of the Bose operators and keeping only the\nquadratic and quartic terms, the effective Hamiltonian\nEq.(1) adopts the form\nH=H2+H4 (B3)\nwhere\nH2=sAJA/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+sBJB/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(B4)\n+J/summationdisplay\n/angbracketleftij/angbracketright/bracketleftBig\nsAb+\njbj+sBa+\niai−√\nsAsB/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightBig\nH4=1\n4JA/summationdisplay\n≪ij≫A/bracketleftbig\na+\nia+\nj(ai−aj)2+(a+\ni−a+\nj)2aiaj/bracketrightbig\n+1\n4JB/summationdisplay\n≪ij≫B/bracketleftbig\nb+\nib+\nj(bi−bj)2+(b+\ni−b+\nj)2bibj/bracketrightbig\n+1\n4J/summationdisplay\n/angbracketleftij/angbracketright/bracketleftBigg/radicalbigg\nsA\nsB/parenleftbig\naib+\njbjbj+a+\nib+\njb+\njbj/parenrightbig\n(B5)\n+/radicalbigg\nsB\nsA/parenleftbig\na+\niaiaibj+a+\nia+\niaib+\nj/parenrightbig\n−4a+\niaib+\njbj/bracketrightBigg\nThe next step is to represent the Hamiltonian in the\nHartree-Fock approximation (2,3) and\nHq=sAJAuA/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+sBJBuB/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(B6)\n+Ju/summationdisplay\n/angbracketleftij/angbracketright/bracketleftBig\nsAb+\njbj+sBa+\niai−√\nsAsB/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightBig\nIt is convenient to rewrite the Hamiltonian in momen-\ntum space representation Eq.(4) The two equivalent sub-\nlattices A and B of the bcc lattice are simple cubic lat-\ntices. The wave vector kruns over the reduced first Bril-\nlouin zone Brof a bcc lattice which is the first Brillouin8\nzone of a simple cubic lattice. The dispersions are given\nby equalities Eqs. (5)\nTo diagonalize the Hamiltonian one introduces new\nBose fields αk, α+\nk, βk, β+\nkby means of the transforma-\ntion (6) where the coefficients of the transformation uk\nandvkare real function of the wave vector k\nuk=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+ 1\n\n(B7)\nvk=sign(γk)/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−1\n.\nThe transformed Hamiltonian adopts the form Eqs.(7)\nwith dispersionsEqs.(8) The free energy ofa system with\nHamiltonian HHF(7,8) is Eq.(10) The system of equa-\ntions for the Hartree-Fock parameters Eqs.(11), in terms\nof the Bose functions of the αandβexcitations, adopt\nthe form\nuA= 1−1\n3sA1\nN/summationdisplay\nk∈Brεk/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\nuB= 1−1\n3sB1\nN/summationdisplay\nk∈Brεk/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\nu= 1−1\nN/summationdisplay\nk∈Br/bracketleftbigg1\n2sA/parenleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/parenrightBig\n(B8)\n+1\n2sB/parenleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/parenrightBig\n−8Ju/parenleftBig\n1+nα\nk+nβ\nk/parenrightBig/parenleftBig\ncoskx\n2cosky\n2coskz\n2/parenrightBig2\n/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk\n\nwherenα\nkandnβ\nkare the Bose functions of αandβex-\ncitations.\nAppendix C: Takahashi modified spin-wave theory\n[20]\nTo study the paramagnetic phase of the system we\nmakeuseofthe Takahashimodified spin-wavetheory[20]\nand introduce two parameters λAandλBto enforce the\nmagnetization on the two sublattices to be equal to zero.\nIt is convenientto representthe parameters λAandλBin\nthe form Eqs.(16). We find out these and Hartree-Fock\nparameters, as functions of temperature, solving the sys-\ntem of five equations, equations (B8) and the equations\nMA=MB= 0. The numerical calculations show thatabove N´ eel temperature µAµB>1 (blue line in Fig.(6)).\nWhen the temperature decreases the product µAµBde-\ncreases, remaining larger than one. The temperature at\nwhich the product becomes equal to one ( µAµB= 1) is\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s49/s50/s51/s52/s53/s54\n/s84/s47/s74/s32\n/s32\n/s32\nFIG. 6: (Color online) Parameters µA,µBandµAµBas a\nfunction of the temperature, in units of the exchange con-\nstantT/J, for ferrimagnet with sublattice A spin sA= 1.5\nand sublattice B spin sB= 1. The high temperature vertical\n(cyan) line corresponds to the N´ eel temperature of ferrima g-\nnet to paramagnet transition. The low temperature vertical\n(green) line corresponds to partial order transition tempe ra-\nture.\nthe N´ eel temperature, marked by vertical cyan line in\nFig.(6).\nAt low temperature µA= 1 and µB= 1 (λA=λB=\n0). The Hartree-Fock parameters are positive functions\nofT/J, solution of the system of equations (B8). Uti-\nlizing these functions, one can calculate the spontaneous\nmagnetizationon the twosublattices at low temperature.\nAt characteristic temperature T∗spontaneous magneti-\nzationon sublattice Bbecomes equalto zero, while spon-\ntaneous magnetization on sublattice Ais still nonzero\n(see Fig.1 in the paper).\nAboveT∗the spectrum contains long-range (magnon)\nexcitations. Thereupon, µAµB= 1 and one can use the\nrepresentation µB= 1/µA. Then,µAand Hartree-Fock\nparameters are solution of a system of four equations,\nequations (B8) and the equation MB= 0.\nThe functions µA(T/J),µB(T/J) and\nµA(T/J)µB(T/J) are depicted in figure (6) for a\nsystem with sA= 1.5,sB= 1,JA/J= 2 and\nJB/J= 0.002.\nFor the same system the Hartree-Fock parameters\nuA(T/J),uB(T/J)u(T/J) are depicted in figure (7).\n[1] K. Adachi, T. Suzuki, K. Kato, K. Osaka, M. Takata and\nT. Katsufuji, Phys. Rev. Lett. 95, 197202 (2005).[2] Zhaorong Yang, Shun Tan, Zhiwen Chen, and Yuheng9\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52/s49/s44/s54/s49/s44/s56/s50/s44/s48/s50/s44/s50\n/s84/s47/s74/s117\n/s117/s65\n/s117/s66/s72/s65/s82/s84/s82/s69/s69/s45/s70/s79/s67/s75/s32/s80/s65/s82/s65/s77 /s69/s84/s69/s82/s83\nFIG. 7: (Color online) Hartree-Fock parameters as a functio n\nof the temperature, in units of the exchange constant T/J, for\nferrimagnet with sublattice A spin sA= 1.5 and sublattice\nB spinsB= 1. The high temperature vertical (cyan) line\nmarks the N´ eel temperature of ferrimagnet to paramagnet\ntransition. The low temperature vertical (blue) line marks\nthe partial order transition temperature.\nZhang, Phys. Rev. B 62, 13872 (2000).\n[3] H. D. Zhou, J. Lu, and C. R. Wiebe, Phys. Rev. B 76,\n174403 (2007).\n[4] V. O. Garlea, R. Jin, D. Mandrus, B. Roessli, Q. Huang,\nM. Miller, A. J. Schultz, and S. E. Nagler, Phys. Rev.\nLett.100, 066404 (2008).\n[5] S-H. Baek, K-Y. Choi, A. P. Reyes, P. L. Kuhns, N.\nJ. Curro, V. Ramanchandran, N. S. Dalal, H. D. Zhou,\nand C. R. Wiebe, J. Phys.: Condens. Matter 20, 135218\n(2008).[6] Kim Myung-Whun, J. S. Kim, T. Katsufuji, and R. K.\nKremer, Phys. Rev. B 83, 024403 (20011).\n[7] A. Kiswandhi, J. S. Brooks, J. Lu, J. Whalen, T. Siegrist,\nand H. D. Zhou, Phys. Rev. B 84, 205138 (2011).\n[8] A. Kismarahardja, J. S. Brooks, A. Kiswandhi, K. Mat-\nsubayashi, R. Yamanaka, Y. Uwatoko, J. Whalen, T.\nSiegrist, and H. D. Zhou, Phys. Rev. Lett. 106, 056602\n(2011).\n[9] Q. Zhang, K. Singh, F. Guillou, C. Simon, Y. Breard,\nV. Caignaert, and V. Hardy, Phys. Rev. B 85, 054405\n(2012).\n[10] Y. Nii, H. Sagayama, T. Arima, S. Aoyagi, R. Sakai, S.\nMaki, E. Nishibori, H. Sawa, K. Sugimoto, H. Ohsumi,\nand M. Takata, Phys. Rev. B 86, 125142 (2012).\n[11] Z. H. Huang, X. Luo, S. Lin, Y. N. Huang, L. Hu, L.\nZhang, Y. P.Sun, Solid State Communications 159, 88\n(2013).\n[12] Z. H. Huang, X. Luo, L. Hu, S. G. Tan, Y. Liu, B. Yuan,\nJ. Chen, W. H. Song, and Y. P. Sun, Journal of Applied\nPhysics115, 034903 (2014).\n[13] Dina Tobia, Juli´ an Milano, Maria Teresa Causa and\nElin L. Winkler, J. Phys.: Condens. Matter 27, 016003\n(2015).\n[14] Vincent Hardy, Yohann Br´ eard, and Christine Martin,\nPhys. Rev. B 78, 024406 (2008).\n[15] R. Quartuand H. T. Diep, Phys.Rev. B 55, 2975 (1997).\n[16] J. R. Stewart, G. Ehlers, A. S. Wills, S. T. Bramwell,\nand J. S. Gardner, J. Phys.: Condens. Matter 16, L321\n(2004).\n[17] P. Azaria, H. T. Diep, and H. Giacomini, Phys. Rev.\nLett.59, 1629 (1987).\n[18] V. G. Vaks, A. I. Larkin, and Y. N. Ovchinnikov, JETP\nLetters. 22, 820 (1966).\n[19] H. T. Diep, Ed., Frustrated Spin Systems , World Scien-\ntific (2004)\n[20] M. Takahashi, Phys. Rev. Lett. 58, 168 (1987).\n[21] D. Schmeltzer, Phys. Rev. B 43, 8650 (1991).\n[*] Electronic address: naoum@phys.uni-sofia.bg"    },    {        "title": "1706.01925v1.Magnetic_order_and_interactions_in_ferrimagnetic_Mn3Si2Te6.pdf",        "content": "Magnetic order and interactions in ferrimagnetic Mn 3Si2Te6\nAndrew F. May,1,\u0003Yaohua Liu,2Stuart Calder,2David S. Parker,1Tribhuwan\nPandey,1Ercan Cakmak,1Huibo Cao,2Jiaqiang Yan,1and Michael A. McGuire1\n1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831\n2Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831\n(Dated: June 8, 2017)\nThe magnetism in Mn 3Si2Te6has been investigated using thermodynamic measurements, \frst\nprinciples calculations, neutron di\u000braction and di\u000buse neutron scattering on single crystals. These\ndata con\frm that Mn 3Si2Te6is a ferrimagnet below TC\u001978 K. The magnetism is anisotropic, with\nmagnetization and neutron di\u000braction demonstrating that the moments lie within the basal plane\nof the trigonal structure. The saturation magnetization of \u00191.6\u0016B/Mn at 5 K originates from the\ndi\u000berent multiplicities of the two antiferromagnetically-aligned Mn sites. First principles calculations\nreveal antiferromagnetic exchange for the three nearest Mn-Mn pairs, which leads to a competition\nbetween the ferrimagnetic ground state and three other magnetic con\fgurations. The ferrimagnetic\nstate results from the energy associated with the third-nearest neighbor interaction, and thus long-\nrange interactions are essential for the observed behavior. Di\u000buse magnetic scattering is observed\naround the 002 Bragg re\rection at 120 K, which indicates the presence of strong spin correlations well\naboveTC. These are promoted by the competing ground states that result in a relative suppression\nofTC, and may be associated with a small ferromagnetic component that produces anisotropic\nmagnetism below \u0019330 K.\nKeywords: Mn3Si2Te6, ferrimagnetism, competing interactions, magnetic frustration, di\u000buse scattering\nI. INTRODUCTION\nhttps://doi.org/10.1103/PhysRevB.95.174440\nUnderstanding and manipulating magnetic anisotropy\nis important in both basic and applied physics research.\nFor instance, anisotropic magnetism is essential for the\ndevelopment of permanent magnets, and it appears to\nbe a fundamental component in many systems that dis-\nplay unconventional superconductivity. Recently, re-\nsearch on materials that are crystallographically lay-\nered (held together by van der Waals bonds) has gained\nprominence in condensed matter physics. This is largely\ndriven by the now-realized prospect of building van der\nWaals heterostructures from materials with complemen-\ntary properties.1Two-dimensional and quasi-2D materi-\nals are of fundamental interest from a bulk perspective\nas well, in part because they exist in the limit of very\nanisotropic interactions (strong in-plane interactions and\nweak cross-plane interactions). For instance, the com-\npounds CrSiTe 3, MnPS 3, and CrI 3have layers connected\nby van der Waals bonds, and demonstrate bulk mag-\nnetic ordering with anisotropic interactions manifesting\nthemselves in anisotropic properties, suppressed 3D or-\ndering temperatures, two-dimensional order, and persis-\ntent short-range correlations above TC.2{9In this work,\nwe probe the behavior of Mn 3Si2Te6, which can be con-\nsidered as a three-dimensional analogue of CrSiTe 3.\nWhile its crystal structure is interesting, very little re-\nsearch has been reported on Mn 3Si2Te6.10,11The initial\nreport of its existence and physical properties discussed\nthe magnetism within the framework of an incorrect sto-\nichiometry and structure,10and thus the initial hypoth-\nesis of ferrimagnetism could not be fully evaluated. De-\nspite containing nominally Mn2+with no orbital mo-ment (3d5with S=5/2, L=0), the magnetism displays\na large anistropy \feld on the order of 10 T at 5 K.10\nThe crystal structure was reported several years after\nthe initial characterization,11but the magnetic proper-\nties were not revisited. Mn 3Si2Te6has a trigonal crystal\nstructure (space group No. 163) that is shown in Fig. 1.11\nFig. 1 also contains the magnetic structure obtained from\nour neutron di\u000braction data. Mn 3Si2Te6is composed of\nMnTe 6octahedra that are edge-sharing within the ab\nplane (Mn1 site), and along with Si-Si dimers this creates\nlayers of Mn 2Si2Te6. The layered framework is analogous\nto that of CrSiTe 3, which is hexagonal and has a van der\nWaals gap between the layers. In Mn 3Si2Te6, however,\nthe layers are linked by the \flling of one-third of the octa-\nhedral holes within the van der Waals gap by Mn atoms\nat the Mn2 site, yielding a composition of Mn 3Si2Te6.11\nImportantly, the multiplicity of Mn1 is twice that of Mn2.\nIn this work, Mn 3Si2Te6was characterized using mag-\nnetization, speci\fc heat and electrical resistivity mea-\nsurements, as well as via powder x-ray and single crystal\nneutron di\u000braction. We \fnd Mn 3Si2Te6to be a ferri-\nmagnet due to antiparallel alignment of moments on the\nMn1 and Mn2 atomic positions, the di\u000berent multiplicity\nof which yields a net magnetization with moments pre-\nferring to lie in the trigonal plane. The data, including\ndi\u000buse neutron scattering, reveal the existence of strong\nspin correlations well-above TC, which may be associated\nwith short range order or the persistence of correlated\nexcitations in the paramagnetic region. The experimen-\ntal probes were complemented by \frst principles calcula-\ntions that revealed a competition between antiferromag-\nnetic (AFM) exchange interactions that are frustrated\nwith respect to each other. Interestingly, the longer-\nrange, third-nearest neighbor coupling (Mn1-Mn2 pairs\nat 5.41 \u0017A) dominates over the in-plane coupling of second-arXiv:1706.01925v1  [cond-mat.str-el]  6 Jun 20172\nnearest neighbors at 4.06 \u0017A (Mn1-Mn1 pairs). The net\nresult is an antiferromagnetic con\fguration close in en-\nergy to the ferrimagnetic one, and this competition for\nthe ground state leads to a relative suppression of the\nordering temperature.\nFIG. 1: (a) Crystal structure of Mn 3Si2Te6viewed down [110].\n(b,c) Single layers of the di\u000berent MnTe 6octahedra viewed\ndown [001], where (b) is an image of Mn1 layers and (c) a\nMn2 layer. In all panels, one unit cell is outlined, MnTe 6\noctahedra are shown, and arrows indicate relative alignments\nof Mn moments within the trigonal plane.\nII. EXPERIMENTAL DETAILS\nMn3Si2Te6single crystals were obtained by melting a\nstoichiometric mixture of the elements in a vacuum sealed\nquartz ampoule at T=1125 K. High-purity Si lump (Alfa\nAesar 99.9999%), and Te shot (Alfa Aesar, 99.999%)\nwere combined with Mn granules (99.98% Alfa Aesar)\nthat were arc-melted prior to use. Single crystals of\nMn3Si2Te6were mechanically isolated from the as-grown\ningot.\nField-cooled magnetization measurements were per-\nformed in a Quantum Design Magnetic Property Mea-\nsurement System. AC magnetic susceptibility, electri-\ncal transport, and speci\fc heat capacity measurements\nwere performed in a Quantum Design Physical Property\nMeasurement System. Powder x-ray di\u000braction mea-\nsurements were performed using a PANalytical X'Pert\nPro MPD with a Cu K \u000b;1incident-beam monochroma-\ntor. An Oxford PheniX Cryostat was utilized to obtain\ndata between 20 K and 300 K. Additional powder x-ray\ndi\u000braction measurements were performed between 300\nand 400 K on a PANalytical X'Pert di\u000bractometer with\nCu K\u000bradiation and an XRK900 oven-type furnace. The\nfurnace chamber was purged with He for 12 hours prior\nto initiating the high-temperature measurements and gas\nwas continuously \rowed throughout the measurements.\nThe measurements were delayed at each temperature for\n30 minutes to allow the sample to equilibrate. At 300 K,\nRietveld re\fnement of data from the low Tstage yieldeda=7.0321(6) \u0017A andc=14.249(1) \u0017A, while re\fnement of the\ndata from the high Tstage yielded a=7.0343(2) \u0017A and\nc=14.249(2) \u0017A.\nSingle crystal neutron di\u000braction data were measured\nusing the HB3A di\u000bractometer at the High Flux Isotope\nReactor at Oak Ridge National Laboratory (ORNL). The\nneutron wavelength of 1.546 \u0017A from the bent perfect Si-\n220 monochromator was used for the data collections,12\nand 380 re\rections were measured at 4, 100, and 380 K.\nThe nuclear-only re\fnement at 380 K yielded occupan-\ncies equal to unity within the error bars, and thus the\noccupancies were \fxed at unity for all T. The magnetic\nre\fnement at 4 K required three equivalent magnetic do-\nmains. All re\fnements were performed using the pro-\ngram FullProf.13\nDi\u000buse neutron scattering measurements were per-\nformed on the CORELLI di\u000buse scattering spectrome-\nter at the Spallation Neutron Source located at ORNL.\nCORELLI is a time-of-\right spectrometer with a pseudo-\nstatistical chopper, which generates both an elastic scat-\ntering signal (with an average energy resolution \u00191 meV)\nand a total scattering signal from a single measurement.\nAn approximately 7 mg single crystal was attached to\nan Al plate in a manner to facilitate inspection of the\nHHL scattering plane. The measurements were taken\nat temperatures of 6, 120, and 350 K, where the sam-\nple was rotated in steps of 3\u000eover ranges of 120\u000e(6 K),\n120\u000e(120 K) and 90\u000e(350 K). A signi\fcant amount of\ntime was spent at 120 K measuring 30 degrees centered\naround the 002 Bragg re\rection. Data around 002 were\nalso collected on warming from 6 K to 120 K. Mantid was\nutilized to perform the Lorentz and spectrum corrections\nand merge the full volume of the scattering data.14\nFirst principles calculations were performed us-\ning the linearized augmented plane-wave (LAPW)\ncode WIEN2K,15within the generalized gradient\napproximation.16Spin-orbit coupling was only included\nduring the calculation of magnetic anisotropy within the\nobserved ground state. Sphere radii of 2.04, 2.5 and\n2.5 Bohr were chosen for Si, Mn and Te, respectively,\nwith an RK maxof 7.0 (here RK maxis the product of the\nsmallest sphere radius and the largest planewave expan-\nsion wavevector). All calculations used the experimen-\ntal, room-temperature lattice parameters of a=7.029 \u0017A\nandc=14.255 \u0017A taken from Ref. 11. For each mag-\nnetic con\fguration, the internal coordinates were relaxed\nwithin the magnetic ordering pattern until the forces\nwere less than 2 mRyd/Bohr. Optimization within the\nmagnetic state yields signi\fcantly di\u000berent atomic coor-\ndinates than optimization within the non-magnetic state\n(an energy gain of some 250 meV/Mn results), suggest-\ning a strong coupling of magnetism to the lattice. The\nstructure theoretically optimized within the ferrimag-\nnetic ground state is much closer to the experimental\nstructure than that optimized within the non-magnetic\nstate. For example, the value for Mn1 of \u0001 z \u0011zExp: -\nzCalc is -8.7\u000210\u00004for the ferrimagnetic ground state\nbut nearly 5 times as large in magnitude at 4.02 \u000210\u000033\nfor the non-magnetic state. Similarly large ratios apply\nfor the Si z and Te y and z coordinates, with the Te x co-\nordinate \u0001x roughly twice as large for the non-magnetic\nstate as for the ferrimagnetic state. Additional details\nare provided in the Supplemental Information.\nIII. RESULTS AND DISCUSSION\nA. Magnetization, neutron di\u000braction and\ntheoretical calculations\nThe magnetization Mdata for single crystalline\nMn3Si2Te6are shown in Fig. 2a. Mn 3Si2Te6is observed\nto have a Curie temperature of TC=78 K. The ordering\ntemperature was also con\frmed by AC magnetic suscep-\ntibility measurements, the results of which are shown in\nFig. 3. The anisotropy of MbelowTCsuggests the or-\ndered moments lie primarily within the ab-plane. This is\ndemonstrated by a signi\fcantly larger Mwhen the ap-\nplied \feldHlies within the ab-plane as compared to when\nHjjc; note the di\u000berent vertical axes in the main panel\nof Fig. 2a. Consistent with this, below TCthe magnetiza-\ntion saturates rapidly for Hjjaband reaches\u00191.6\u0016B/Mn\natT=5 K (inset Fig. 2a). The data imply an anisotropy\n\feld of approximately 9 T (90 kOe), and a small ferro-\nmagnetic component is observed for Hjjc. These data\nare consistent with the initial report on Mn 3Si2Te6.10\nWe note there is no remanant moment for either orien-\ntation, which is di\u000berent from the prior report and, to\nsome extent, suggests the present single crystals are of\nhigh quality.\nFig. 2b plots the magnetization data as H=M , which is\nequivalent to 1 =\u001fwhen the susceptibility can be de\fned\nas\u001f=M=H (whenMis linear inH). The data demon-\nstrate Curie-Weiss behavior between approximately 350\nand 750 K, the region where 1/ \u001fis linear in T. The\ndata above 400 K were \ft to a simple Curie-Weiss law,\n\u001f= C/(T\u0000\u0002), where C is the Curie constant and \u0002\nthe Weiss temperature. This \ftting produced an e\u000bec-\ntive moment of 5.6 \u0016B/Mn and a Weiss temperature of\n-277 K. The e\u000bective moment is consistent with the pres-\nence of Mn2+ions and the negative Weiss temperature\nindicates antiferromagnetic correlations. The saturation\nmagnetization (inset, Fig. 2a) is about one-third of that\nexpected on the basis of this e\u000bective moment, which sug-\ngests the presence of one uncompensated Mn2+local mo-\nment per formula unit in the ordered phase. In Fig. 2b,\nthe low-temperature data were collected on a large mass\nof ground crystals, while high-temperature data were col-\nlected using six small crystals sealed under vacuum in a\nthin quartz tube (the crystals are free to rotate in an\napplied \feld).\nThe AC susceptibility \u001fAC= dM/dHcan be described\nby in-phase (real) \u001f' and out-of-phase (imaginary) \u001f\"\ncomponents, which are shown in Fig. 3. The out-of-\nphase component \u001f\" relates to dissipative losses, for in-\nstance the movement of domain walls in ferromagnets;\n/s48 /s53 /s48 /s49 /s48 /s48 /s49 /s53 /s48 /s50 /s48 /s48 /s50 /s53 /s48 /s51 /s48 /s48 /s51 /s53 /s48 /s52 /s48 /s48 /s48 /s50 /s48 /s52 /s48 /s54 /s48 \n/s48 /s49 /s48 /s50 /s48 /s51 /s48 /s52 /s48 /s53 /s48 /s54 /s48 /s48 /s46 /s48 /s48 /s46 /s53 /s49 /s46 /s48 /s49 /s46 /s53 \n/s48 /s49 /s48 /s48 /s50 /s48 /s48 /s51 /s48 /s48 /s52 /s48 /s48 /s53 /s48 /s48 /s54 /s48 /s48 /s55 /s48 /s48 /s56 /s48 /s48 /s48 /s53 /s48 /s49 /s48 /s48 /s49 /s53 /s48 /s50 /s48 /s48 /s50 /s53 /s48 /s40/s98 /s41/s32/s77 /s47/s72 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s84 /s32/s40/s75 /s41/s72 /s32 /s61 /s32 /s50 /s48 /s32 /s79 /s101 /s40/s97 /s41\n/s48 /s46 /s48 /s48 /s46 /s50 /s48 /s46 /s52 /s48 /s46 /s54 \n/s32\n/s72 /s32 /s32 /s99 \n/s72 /s32 /s124/s124/s32 /s99 /s77 /s32 /s40\n/s66 /s47 /s77 /s110 /s41\n/s72 /s32 /s40/s107 /s79 /s101 /s41/s84 /s32 /s61 /s32 /s53 /s75 \n/s32\n/s72 /s32 /s61 /s32 /s49 /s32 /s107 /s79 /s101 \n/s72 /s32 /s61 /s32 /s49 /s48 /s32 /s107 /s79 /s101 /s72 /s47/s77 /s32/s40/s109 /s111 /s108/s45/s77 /s110 /s47/s99 /s109 /s32/s51 \n/s41\n/s84 /s32/s40/s75 /s41/s84 /s32 /s60 /s32 /s51 /s56 /s48 /s75 /s32 /s112 /s117 /s108 /s118 /s101 /s114/s105 /s122 /s101 /s100 \n/s99 /s114/s121 /s115/s116 /s97 /s108 /s115/s84 /s32 /s62 /s51 /s52 /s48 /s75 /s32 /s99 /s114/s121 /s115/s116 /s97 /s108 /s115\n/s114/s97 /s110 /s100 /s111 /s109 /s32 /s111 /s114/s105 /s101 /s110 /s116 /s97 /s116 /s105 /s111 /s110 FIG. 2: (a) Anisotropic magnetization data for\nMn3Si2Te6upon cooling in an applied \feld of 20 Oe;\nthe left axis ( H?c) and right axis ( Hkc) have the same\nunits. The inset shows isothermal magnetization data at\nT=5 K, and together these results reveal easy-plane mag-\nnetization. (b) Inverse susceptibility data combining high\nand lowTmeasurements. The dashed curve represents a\nCurie-Weiss \ft extended to lower T.\n\u001f\" is typically zero for simple antiferromagnets and non-\nzero in ferromagnets and sometimes in metals.17The AC\ndata shown in Fig. 3 were collected in zero applied DC\n\feld, with a small amplitude A=2 Oe and a frequency\nf=997 Hz. These data reveal a sharp onset at the Curie\ntemperature, with \u001f' and\u001f\" increasing below TCfor both\norientations. In addition, the data reveal anisotropy\naboveTC, which was also observed with the DC mea-\nsurements. A comparison of \u001f' and the DC analog M=H\n(H=20 Oe) is shown in the Supplemental Information.\nAs shown in Fig. 2b, the low-\feld magnetization data\non ground crystals deviate from the high TCurie-Weiss\nbehavior below\u0019330 K. Single crystal magnetization\ndata reported in Fig. 4 demonstrate that this behav-\nior is associated with a relatively abrupt onset of mag-\nnetic anisotropy below \u0019330 K, which is observed in\nboth the low-\feld DC data (Fig. 4a) and the AC data\n(Fig. 4b). The anisotropy is the same as that in the ferri-\nmagnetic phase below TC(easy-plane anisotropy). Upon4\ncooling below 330 K, \u001f\" becomes non-zero for in-plane\ndata, and it reaches a maximum near 110 K before ris-\ning sharply at TC. This may suggest the anisotropy is\nassociated with, or derived from, a ferromagnetic compo-\nnent (uncompensated moment) that is present for H?c.\nIndeed, isothermal magnetization data reveal a small\n(soft) ferromagnetic contribution for H?cin the re-\ngionTC<T < 330 K, which increases upon cooling. The\nlow-\feldM(H) data atT=110 K are shown in Fig. 5, and\nthe corresponding AC data are shown in the Supplemen-\ntal along with M(H) curves at di\u000berent T. Interestingly,\n\u001f' plateaus near 85-110 K, which would suggest that this\nferromagnetic component has saturated. That is, this\nplateau in\u001f' seems to suggest that the anisotropy does\nnot originate in precursory short-range order of the ferri-\nmagnetic phase, which would be expected to continue in-\ncreasing all the way down to TCas the correlation length\nbegins to diverge. However, the ferromagnetic-like con-\ntribution suggests the anisotropy originates from some\nform of magnetic order, as opposed to a crystal-\feld in-\nduced anisotropy of a paramagnet. As discussed below,\nour di\u000braction measurements did not reveal any signi\f-\ncant structural change near 330 K, but a structural con-\ntribution and/or origin cannot be ruled out at this time.\nThe true nature of the transition near 330 K is thus un-\nclear and warrants further investigation. As discussed in\nthe Supplemental Information, this behavior seems to be\nintrinsic to our single crystals.\nSingle crystal neutron di\u000braction and spectroscopy\nwere utilized to examine the nuclear and magnetic struc-\ntures of Mn 3Si2Te6. As shown in Fig. 6, the intensity of\nthe 002 Bragg re\rection increases upon cooling through\nTC. This suggests a strong in-plane component of the\nmoment, consistent with the magnetization data. The\nlack of additional scattering at the 110 re\rections at low\nTis consistent with the moment being in the ab-plane,\nbut this observation alone is not de\fnitive given the pres-\nence of three magnetic domains. The square root of\nthe integrated magnetic intensity is proportional to the\nordered moment, and can thus be used as a magnetic\norder parameter. Order parameter data employing in-\ntensity from the 002 re\rection were obtained using both\nHB3A and CORELLI, as shown in Fig. 6b where \fts to\npower law behavior are included. The data in Fig. 6b\nfrom CORELLI are the integrated intensity of 002, while\nthat from HB3A were constant Q data collections; nei-\nther data set are corrected for background. Additional\ntemperature-dependent data from HB3A are shown in\nthe Supplemental Information.\nThe order parameter \fts shown as solid curves in\nFig. 6b yield a critical exponent \f\u00190.25 andTCvalues\nconsistent with that obtained from magnetization mea-\nsurements, though some minor discrepancies are observed\nbecause the data were collected while ramping temper-\nature in di\u000berent conditions. The \fts employ a sim-\nple power law, ( I\u0000I0)0:5=A(TC-T)\f, whereI0repre-\nsents the non-magnetic intensity (background and nu-\nclear, assumed temperature independent). Data from\n/s48 /s46 /s48 /s49 /s46 /s48 /s50 /s46 /s48 /s51 /s46 /s48 /s52 /s46 /s48 /s53 /s46 /s48 /s54 /s46 /s48 \n/s56 /s48 /s49 /s48 /s48 /s49 /s50 /s48 /s49 /s52 /s48 /s49 /s48 /s45/s53 /s49 /s48 /s45/s52 /s49 /s48 /s45/s51 /s49 /s48 /s45/s50 /s49 /s48 /s45/s49 /s49 /s48 /s48 \n/s55 /s48 /s56 /s48 /s57 /s48 /s49 /s48 /s48 /s49 /s49 /s48 /s49 /s50 /s48 /s49 /s51 /s48 /s49 /s52 /s48 /s49 /s53 /s48 /s48 /s46 /s48 /s48 /s48 /s46 /s48 /s49 /s48 /s46 /s48 /s50 /s48 /s46 /s48 /s51 /s48 /s46 /s48 /s52 /s56 /s48 /s49 /s48 /s48 /s49 /s50 /s48 /s49 /s52 /s48 /s49 /s48 /s45/s50 /s49 /s48 /s45/s49 /s49 /s48 /s48 /s49 /s48 /s49 /s72 \n/s68 /s67 /s61 /s48 \n/s40/s98 /s41/s32\n/s72 \n/s65 /s67 /s32/s99 \n/s72 \n/s65 /s67 /s32/s124/s124/s32 /s99 /s39 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41/s40/s97 /s41\n/s65 /s67 /s32/s109 /s97 /s103 /s110 /s101 /s116/s105/s99 /s32/s115 /s117 /s115 /s99 /s101 /s112 /s116/s105/s98 /s105/s108/s105/s116/s121 \n/s39/s39\n/s84 /s32/s40/s75 /s41/s32/s39 /s39 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s84 /s32/s40/s75 /s41/s39\n/s84 /s32/s40/s75 /s41FIG. 3: (a) The real component \u001f' and (b) the imaginary\ncomponent \u001f\" of the AC magnetic susceptibility data near\nthe ferrimagnetic ordering temperature. The insets show the\ndata on log-scale using the same units as the primary panels.\nData collected upon cooling using A=2 Oe and f=997 Hz.\njust below TCdown to\u001940 K are \ft, and the result-\ning curves are extended to lower Tin Fig. 6b for com-\nparison. For the CORELLI data we obtain \f=0.26(1)\nandTC=81.7(2) K while for the HB3A data we obtain\n\f=0.23(1) and TC=78.5(1) K. The standard deviations\noriginate from the \ftting procedures, but the absolute\nerrors on critical exponents estimated in this way are\nexpected to be larger. The inset contains these same\nresults in log-log representation to emphasize the simi-\nlar temperature-dependence for the two data sets, and\nto highlight the good quality of the \fts. The criti-\ncal exponent of \f\u00190.25 for Mn 3Si2Te6can be com-\npared to those obtained for simple mean \feld mod-\nels. For instance, a 2D Ising system is expected to\nhave\f=0.125, while the 3D Ising and Heisenberg mod-\nels have\f=0.326 and 0.367, respectively.18In CrSiTe 3,\na critical exponent of \f=0.12 was reported, suggesting\n2D character of the magnetism in that van der Waals\nbonded ferromagnet.2,4Mn3Si2Te6is a crystallographi-\ncally three-dimensional material. The deviation of \ffrom\nthe simple 3D models is not surprising, however, given\nthe existence of multiple Mn sites and ferrimagnetic or-5\n/s48 /s53 /s48 /s49 /s48 /s48 /s49 /s53 /s48 /s50 /s48 /s48 /s50 /s53 /s48 /s51 /s48 /s48 /s51 /s53 /s48 /s52 /s48 /s48 /s49 /s48 /s45/s52 /s49 /s48 /s45/s51 /s49 /s48 /s45/s50 /s49 /s48 /s45/s49 /s49 /s48 /s48 \n/s53 /s46 /s48 /s54 /s46 /s48 /s55 /s46 /s48 /s56 /s46 /s48 \n/s51 /s48 /s48 /s51 /s49 /s48 /s51 /s50 /s48 /s51 /s51 /s48 /s51 /s52 /s48 /s51 /s53 /s48 /s45/s48 /s46 /s49 /s48 /s46 /s48 /s48 /s46 /s49 /s48 /s46 /s50 /s48 /s46 /s51 /s40/s97 /s41\n/s40/s98 /s41\n/s32/s77 \n/s68 /s67 /s32/s40/s101 /s109 /s117 /s47/s103 /s41\n/s84 /s32/s40/s75 /s41/s72 \n/s68 /s67 /s32 /s61 /s32 /s50 /s48 /s32 /s79 /s101 /s72 \n/s68 /s67 /s32 /s32 /s99 \n/s72 \n/s68 /s67 /s32 /s124/s124/s32 /s99 \n/s72 \n/s68 /s67 /s32 /s61 /s32 /s48 /s72 \n/s65 /s67 /s32 /s32 /s99 \n/s72 \n/s65 /s67 /s32 /s124/s124/s32 /s99 \n/s39 /s39 /s39 \n/s32/s32/s39 /s32/s97 /s110 /s100 /s32 /s39 /s39 /s32/s40/s49 /s48 /s45 /s51 \n/s32/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s84 /s32/s40/s75 /s41\nFIG. 4: (a) Anisotropic DC magnetization data collected\nupon cooling in an applied \feld of 20 Oe. (b) Anisotropic\nAC susceptibility data showing the real \u001f' and imaginary \u001f\"\ncomponents. The AC data were collected using A=14 Oe and\nf=997 Hz with zero applied DC \feld.\n/s48 /s50 /s48 /s48 /s52 /s48 /s48 /s54 /s48 /s48 /s56 /s48 /s48 /s49 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s52 /s48 /s46 /s48 /s48 /s54 \n/s72 \n/s68 /s67 /s32 /s32/s99 \n/s72 \n/s68 /s67 /s32 /s32/s99 \n/s72 /s32/s40/s79 /s101 /s41\n/s32/s77 \n/s68 /s67 /s32/s40\n/s66 /s47/s77 /s110 /s41/s84 /s32 /s61 /s32 /s49 /s49 /s48 /s75 \nFIG. 5: Isothermal DC magnetization data at 110 K demon-\nstrating the ferromagnetic contribution for H?cin\nMn3Si2Te6.\nder.\nRe\fnement of the 4 K neutron di\u000braction data from\n/s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50 /s49/s51 /s49/s52 /s49/s53/s48/s49/s50/s51/s52/s53/s54\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48 /s52/s48/s48/s48/s49/s50/s51\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s48/s48/s50/s72/s66/s51/s65/s48/s48/s50\n/s49/s49/s48/s84 /s32/s61/s32/s52/s32/s75\n/s49/s48/s48/s32/s75/s49/s49/s50\n/s49/s49/s50/s99/s111/s117/s110/s116/s115/s47/s116/s105/s109/s101/s32/s40/s49/s48/s51\n/s47/s51/s115/s41\n/s32/s40/s100/s101/s103/s114/s101/s101/s115/s41\n/s40/s98/s41/s32/s73/s32/s48/s46/s53\n/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s84 /s32/s40/s75/s41/s40/s97/s41\n/s32\n/s32/s72/s66/s51/s65\n/s32/s67/s79/s82/s69/s76/s76/s73/s76/s111/s103/s91/s40 /s73 /s45/s73\n/s48/s41/s48/s46/s53\n/s93\n/s76/s111/s103/s91 /s84/s45/s84\n/s67/s93FIG. 6: (a) Rocking curves of various Bragg re\rections at 4\nand 100 K. The increase of intensity upon cooling into the\nmagnetically-ordered state is strongest for re\rections with\nnon-zero L. (b) Order parameter for the ferrimagnetic struc-\nture (002 intensity), with data from the single crystal neutron\ndi\u000bractometer (HB3A) and the elastic di\u000buse neutron scatter-\ning spectrometer (CORELLI). The solid curves are \fts to a\nsimple power law, and the inset shows the data and \fts in a\nlog-log plot.\nHB3A yielded the magnetic structure shown in Fig. 1\n(Rf= 4:6 and\u001f2= 3:6). This spin structure is de-\n\fned by parallel alignment of the moments on Mn1 com-\nbined with anti-parallel alignment to the moments on\nMn2. Within the limits of these data, the re\fned mo-\nments are the same on both Mn sites, consistent with\nthe expectation of similar oxidation states (both posi-\ntions are octahedrally-coordinated by Te). At 4 K, the\nre\fned moment is 4.2(1) \u0016B/Mn when the moments are\nconstrained to be equal and to lie within the trigonal\nplane. When permitted, a small moment of 0.2(2) \u0016B/Mn\nalong thecaxis is observed, but the quality of the re\fne-\nment does not improve. However, the isothermal mag-\nnetization data M(H) at 5 K suggest a soft ferromag-\nnetic component for Hjjc(inset Fig. 2a), and the low-\n\feldM(T) data are anisotropic below TC(Fig. 2a). It is\npossible that an applied \feld causes alignment of canted\nmoments or domains, and the lack of an applied \feld dur-\ning neutron di\u000braction leads to an average canting that is6\nnegligible. Note that in Fig. 1, the moments are shown to\nbe along the aaxis, but their speci\fc orientation within\ntheab-plane cannot be determined due to symmetry.\nIn trying to understand this magnetic con\fguration,\nwe \frst consider the crystal structure and apparent ex-\nchange interactions. Mn1-Mn2 have an interatomic dis-\ntance of 3.55 \u0017A at room temperature,11and are linked\nthrough face-sharing octahedra along the caxis (Fig.1),\nwhich would lead to a direct exchange interaction that\nwe expect to be antiferromagnetic (AFM) for these Mn2+\nions. The Mn1-Mn1 interaction is more complicated be-\ncause it occurs via edge sharing octahedra, which re-\nsults in a competition of direct interaction (AFM) and\n\u001990\u000eMn1-Te-Mn1 interactions that can be either ferro-\nmagnetic (FM) or AFM depending on the panddor-\nbitals involved.19Since the ground state is ferrimagnetic\nwith parallel alignment of Mn1 moments, it is tempt-\ning to suggest that FM superexchange is the dominant\nin-plane interaction. Such an explanation would seem\nconsistent with the observation of ferromagnetism in the\nstructurally-related material CrSiTe 3, in which the anal-\nogous edge-sharing octahedra have ferromagnetic cou-\npling and orovide the shortest exchange pathway (there is\nno inter-plane occupancy).4Theoretical calculations were\nperformed to get a better understanding of the coupling\nmechanisms.\nFirst principles calculations were performed to under-\nstand the ferrimagnetic ground state and the associated\nTC. Energies of the following six states were calculated:\nnon-magnetic, ferromagnetic (FM), ferrimagnetic states\nFI1 and FI2, and antiferromagnetic states AF1 an AF2.\nFI1 is the ferrimagnetic ground state, as also observed\nexperimentally, and its magnetic symmetry is shown in\nFig. 1. The \frst competing state is AF1, in which all\nMn1-Mn1 planar neighbors are anti-aligned, while all\nMn1-Mn2 nearest neighbors are aligned. Additional de-\ntails regarding the various magnetic con\fgurations are in\nthe Supplemental Information. The energies associated\nwith the magnetic states studied fall more than an eV per\nMn below the non-magnetic state, and the Mn moments\ncan accurately be considered local moments.\nTABLE I: The calculated relative energies for several mag-\nnetic ordering patterns, along with net magnetic moments\nreported as \u0016B/Mn. In all cases, each Mn is calculated to\nhave a moment consistent with Mn2+.\nOrdering \u0001E (meV/Mn) M (\u0016B/Mn)\nFI1 (ground state) 0 1.67\nAF1 19.1 0\nFI2 31.5 1.67\nAF2 32.2 0\nFM 105.4 4.82\nThe calculations predict that the ferrimagnetic con-\n\fguration is the ground state, being substantially lower\nin energy than the paramagnetic or ferromagnetic states\n(105 meV/Mn below the ferromagnetic state). As sum-marized in Table I, AF1 is only 21 meV/Mn above the\nground state, while AF2 and FI2 are slightly more than\n30 meV/Mn above FI1. The energy of the ground state\nrelative to competing states determines TC, which is to\nsay that the existence of competing magnetic con\fgura-\ntions acts to suppress TC. Using a mean \feld approach,\nthe Curie temperature is one-third of the energy di\u000ber-\nence between the ground state and the competing state,\nwhich in this case gives 19.1 meV/3k B= 73.9 K, in good\nagreement with the experimentally observed TC=78 K.\nUltimately, the similar energies of the competing states\nreveal a competition between di\u000berent coupling mecha-\nnisms and/or pathways.\nThe theoretical analysis was extended by mapping the\nobserved energetics to a Heisenberg model assuming an\nIsing spin S=\u00061, which yielded the \frst three e\u000bective\nexchange constants. The de\fnitions and relative values\nof these J iare given in Table II. Compared to the third-\nnearest neighbor at 5.42 \u0017A, the other Mn-Mn distances\nare\u00197\u0017A or greater and have been neglected. The mean\n\feld approach to a Weiss temperature is one-third of the\naverage interaction energy per magnetic ion, which in\nthis case is \u0002\u0019-757 K/3 = -252 K (accounting for the\nmultiplicities), which is in good agreement with the ex-\nperimental value of -277 K. To give a sense for the energy\nscales, the calculations yielded J 1=-35(4) meV/Mn.\nThe calculations revealed that the \frst three exchange\nconstants are all antiferromagnetic in nature, which re-\nsults in a frustrated system of competing interactions.\nAs shown in Table II, J 1provides the strongest exchange,\nand its AFM nature is consistent with our expectations\nbased on the direct exchange pathway. Interestingly, the\nsecond-nearest neighbor interaction (J 2) is AFM and is\nweaker than the third-nearest neighbor interaction (J 3).\nThus, the longer-range J 3dominates the in-plane J 2and\ntogether with the strong J 1produces FM alignment of the\nMn1 moments within the ab-plane despite the energy cost\ndue to an AFM J 2. The occupancy of the Mn2 position is\ntherefore essential for generating FM alignment of Mn1\nmoments in the ground state of Mn 3Si2Te6. Thus, this\nseems to contrast the behavior observed in ferromagnetic\nCrSiTe 3where the hypothetical Cr2 position is vacant\nbut the in-plane coupling appears to be ferromagnetic.4\nThe multiplicity of each exchange interaction is criti-\ncal in determining the order of competing ground states.\nFor instance, in ferromagnetic MnBi the nearest-neighbor\nexchanges are AFM in nature, but a larger number of\nlonger-range, ferromagnetic interactions results in a high\nCurie temperature of 630 K.20In Mn 3Si2Te6, the multi-\nplicity of J 2and J 3are equal despite a di\u000berent number\nof pertinent neighbors (due to crystallographic site mul-\ntiplicities), and are a factor of three larger than that of\nJ1(see Table II). The energy minimization in the ground\nstate comes from satisfying both J 1and J 3. However, due\nto the multiplicity di\u000berence, the \frst competing state\n(AF1) is realized by satisfying both J 3and J 2. Thus, due\nto the Mn-Mn coordinations, several competing states\nexist in Mn 3Si2Te6.7\nSince under some circumstances strong correlations\ncan be present in 3 d-based chalcogenides, we have also\nperformed GGA+U calculations of the ferrimagnetic\nground state and the ferromagnetic state, applying a\nHubbard U of 3 eV to the Mn dorbitals. There are two\nmain e\u000bects of adding this U: the ferrimagnetic band\ngap is increased slightly, and the energetics of the sys-\ntem change signi\fcantly. In particular, the ferromagnetic\nstate is now just 43 meV/Mn above the ferrimagnetic\nstate, to be compared with 105 meV/Mn in the GGA-\nonly calculations. It is likely that similar e\u000bects would\nbe obtained for the other excited states. Since the GGA\nitself provides excellent agreement with both the mag-\nnitude of the Curie temperature and the Curie-Weiss \u0002\nobtained from the exchange constants, it appears that the\nregular GGA is indeed an appropriate tool for studying\nthis system.\nSince Mn 3Si2Te6shows a surprisingly large anisotropy\n\feld - approximately 9 T - we have also performed \frst\nprinciples calculations21{23of the magnetic anisotropy in\nthe ferrimagnetic state in this system, which necessitates\nthe use of the spin-orbit interaction. Here the moments\nare constrained to lie either along the caxis, or within the\nabplane. We \fnd that the energy is some 0.65 meV/Mn\nlower in the planar con\fguration, leading to a calcu-\nlated anisotropy \feld HAof 13 T, in reasonable agree-\nment with the experimental observations. It is interest-\ning to note that this anisotropy appears to arise from an\nanisotropy in the Mn orbital moments, which are of mag-\nnitude 0.023 \u0016B/Mn in the axial con\fguration (all orbital\nmoments are parallel to the corresponding spin moments\nand hence are of opposite sign for Mn1 and Mn2), but\nare found to be 0.037 \u0016B/Mn1 and -0.048 \u0016B/Mn2 in the\nplanar con\fguration. The existence of an orbital moment\nimplies a deviation from the 3 d5, S=5/2 electronic con-\n\fguration, though it appears that only a small orbital\ncontribution exists.\nIt is interesting to compare the magnetic exchange\nconstants in Mn 3Si2Te6to those in MnTe. MnTe has\nthe hexagonal NiAs structure-type in its bulk form, and\nit contains the same exchange pathways as de\fned in\nTable II, but with slightly di\u000berent bond angles, in-\nteratomic distances, and multiplicities.24MnTe has an\nantiferromagnetic structure composed of ferromagnetic\nplanes that stack antiferromagnetially along the caxis.25\nThe planar J 2is ferromagnetic in MnTe,24and thus\nthe interactions in MnTe are not frustrated. There-\nfore, while the magnetic structures of Mn 3Si2Te6and\nMnTe are similar (ferromagnetic planes stacked antifer-\nromagnetically), the origins are rather di\u000berent. In both\ncompounds, J 2is associated with edge-sharing octahe-\ndra (see Table II). The octahedra are nearly ideal in\nMnTe (bond angles of 90.0 \u00060.1\u000e), while they are rather\ndistorted in Mn 3Si2Te6(for instance, one bond angle\ndrops to 85.1\u000e). In relation to J 2, the distortions in\nMn3Si2Te6lead to a signi\fcant deviation from the ideal\nangle of 90\u000e(87.2\u000e) associated with Mn-Te-Mn superex-\nchange, as well as a shorter interatomic distance for thecoupling (4.06 \u0017A compared to 4.13 \u0017A in MnTe). This\ncomparison further highlights the importance of frus-\ntrated interactions in Mn 3Si2Te6, which produce com-\npeting magnetic ground states. As discussed above, the\nenergy di\u000berence between the lowest energy (magnetic)\ncon\fgurations determines the magnetic ordering tem-\nperature, and thus the existence of competing interac-\ntions/states leads to a suppressed Curie temperature in\nferrimagnetic Mn 3Si2Te6. However, since the dominant\nexchange interactions in MnTe are not frustrated, they\ndo not result in a competing ground state that lies close\nin energy. The bulk ordering temperature of MnTe is\nthus rather large, TN=310 K.25\nTABLE II: Schematic de\fning the three nearest neighbor\nexchange constants. All are found to be antiferromagnetic\n(J<0), and the values shown were obtained for an Ising spin\nof\u00061 in the Hamiltonian. The relevant multiplicities and\ndistances in the relaxed cell are provided; see Fig. 1 for more\ncrystallographic details. The interaction multiplicity takes\ninto account the number of neighbors at a given site and the\nassociated crystallographic multiplicity (four Mn1 and two\nMn2 per unit cell).\nExchange Multiplicity Distance\n(K/Mn) (per unit cell) ( \u0017A)\nJ1= -402(50) 4 3.54\nJ2= -73(9) 12 4.06\nJ3= -172(22) 12 5.42\nB. Lattice behavior and electrical resistivity\nAs shown in Fig. 7, the evolution of the aandclattice\nparameters appears smooth across TCand near 330 K.\nThe thermal expansion along the aaxis is fairly typical\nfor allT. Thecparameter, however, shows a change in\ntemperature dependence across TCand an unusual curva-\nture up to near room temperature. This seems to bolster\nthe idea that an ordering transition occurs near 330 K\nand has some coupling to the lattice, though additional\nmeasurements would be useful. The neutron di\u000braction\ndata at 100 and 380 K re\fned well within the published\ncrystal structure.\nThe speci\fc heat capacity of Mn 3Si2Te6is shown in\nFig. 8. An anomaly can be observed near TC, and as\nshown in the inset an applied \feld ( Hjjc) causes a broad-\nening of this feature to higher temperature. This behav-\nior is consistent with ferromagnetic-like ordering, where\nan applied \feld suppresses \ructuations and broadens the\nspeci\fc heat anomaly by removing more of the magnetic\nentropy above TC. While we do not have a good lattice\n(phonon) standard for Mn 3Si2Te6, an overestimation of\nthe entropy change that occurs near TCcan be made by\nintegrating CP=Twith a straight line for a background\n(see Supplemental Information). When \ftting between\n60 and 95 K, this procedure yields an entropy change of8\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48 /s52/s48/s48/s48/s46/s57/s57/s48/s48/s46/s57/s57/s50/s48/s46/s57/s57/s52/s48/s46/s57/s57/s54/s48/s46/s57/s57/s56/s49/s46/s48/s48/s48/s49/s46/s48/s48/s50\n/s97\n/s32/s84 /s32/s40/s75/s41/s99/s97 /s47/s97\n/s51/s48/s48/s75/s32/s38/s32 /s99 /s47/s99\n/s51/s48/s48/s75\nFIG. 7: Temperature dependence of the lattice parameters\nnormalized to values at 300 K, with data for co\u000bset for clarity.\nError bars are smaller than the size of the data markers, and\nthe dashed line indicates TC.\nonly 8% of that expected for ordering of S=5/2 local\nmoments on Mn2+. The magnetic entropy per mole of\nMn in the paramagnetic state is Smag= RLn[2S+1] =\nR Ln(6) = 14.9 J/mol-Mn/K, and the integration from\n60 to 95 K only yields \u0001 Smag\u00191.2 J/mol-Mn/K. Again,\nwe emphasize that this is likely to be an overestimation\nof \u0001Smagdue to the use of a linear background; this\napproach is utilized to emphasize how much entropy is\nreleased away from TC(both above and below). A sim-\nilarly small \u0001 Smag is observed in MnPS 3, where two-\ndimensional order is observed above TC,5,26as well as\nin CrSiTe 3where short range correlations are observed\naboveTCand the magnetism is strongly coupled to the\nlattice.3,4\nAs shown in Fig. 8b, there is a strong contribution to\nthe speci\fc heat capacity at low temperature, and this\ndecreases with increasing applied \feld. At the lowest\ntemperatures, the data can be well-described by a Debye\nterm (\fT3) plus a linear term ( \rT), the latter of which is\nthe typical way to include an electronic contribution. In\nthis case, however, Mn 3Si2Te6is a semiconductor and the\nstrong magnetic-\feld dependence also points to a mag-\nnetic origin; \rshould not be associated with the typical\nelectronic Sommerfeld coe\u000ecient. As shown in the Sup-\nplemental Information, the low Tmagnetic contribution\nin Mn 3Si2Te6(and CrSiTe 3) can also be described using\na term proportional to T1:5. Data below 2 K would likely\nhelp in considering the nature of the low Tmagnetic con-\ntribution in Mn 3Si2Te6.\nThe in-plane electrical resistivity \u001aabof Mn 3Si2Te6is\nshown in Fig. 9 for two crystals. The value at 300 K is\nconsistent with that reported by Ref. 10, and similar to\ntheir data we observe activated-conduction and a strong\n/s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s49/s46/s56/s50/s46/s48/s50/s46/s50/s50/s46/s52\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s50 /s52 /s54 /s56 /s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s40/s98/s41/s32\n/s48/s72 /s61/s48/s84\n/s52/s84\n/s56/s84/s67\n/s80/s32/s40/s82/s47/s109/s111/s108/s45/s97/s116/s41\n/s84 /s32/s40/s75/s41/s40/s97/s41\n/s84 /s32/s40/s75/s41\n/s32/s67\n/s80/s32/s40/s82/s47/s109/s111/s108/s45/s97/s116/s41\n/s56/s84/s52/s84\n/s32/s32/s67\n/s80/s47/s84 /s32/s40/s109/s74/s47/s109/s111/s108/s45/s77/s110/s47/s75/s50\n/s41\n/s84/s50\n/s32/s40/s75/s50\n/s41/s48/s72 /s61/s48FIG. 8: (a) Speci\fc heat capacity of Mn 3Si2Te6, with inset\nshowing the \feld dependence of the anomaly observed near\nTC. (b) Field dependence of the the low-temperature speci\fc\nheat data demonstrating a strong magnetic contribution.\nanomaly at the ferrimagnetic ordering temperature. An\nincrease in the carrier mobility is expected below TCdue\nto the suppression of spin \ructuations, and this may ex-\nplain the sharp dip in \u001aabbelowTC. The feature at TCis\nsomewhat dramatic, though, especially when considering\nthe implications of a small \u0001 SmagnearTC. The present\ndata, which extends to higher temperatures than pre-\nviously reported, also shows a broad feature near 330 K.\nThis appears to demonstrate the bulk-nature of magnetic\nordering associated with the susceptibility anomaly ob-\nserved at the same temperature in Fig. 2, and the asso-\nciated behavior in Fig. 9 could be related to carrier scat-\ntering or electronic structure e\u000bects.\nTo inspect the changes in \u001a(T) above and below\nthe magnetic transitions, we \ft the data to \u001a=\n\u001a0Exp[EA=kBT] whereEAandkBare the activation en-\nergy and Boltzmann constant, respectively. The \ftted\ncurves are shown in red in Fig. 9. Firstly, we note that\nthese \fts su\u000ber from inadequacy of the simple model\n(particularly at the lowest T), as well as the limited\namount of data utilized. However, the trends and magni-\ntudes are worth noting. The activation energy obtained\nby \ftting\u001afrom 350-380 K is roughly 0.3 eV; this is the9\n/s48 /s49 /s48 /s48 /s50 /s48 /s48 /s51 /s48 /s48 /s52 /s48 /s48 /s49 /s69 /s45/s51 /s48 /s46 /s48 /s49 /s48 /s46 /s49 /s49 /s49 /s48 \n/s32/s32/s97 /s98 /s32/s40 /s45/s109 /s41\n/s84 /s32/s40/s75 /s41\nFIG. 9: The in-plane electrical resistivity of\nMn3Si2Te6generally increases with decreasing tempera-\nture, with a notable feature near TC. The decrease of \u001a\nbecomes more rapid above \u0019330 K. The black and green\ncurves are data collected on di\u000berent crystals, and the red\nlines are \ftted curves using \u001a=\u001a0Exp[EA/kBT], which only\n\fts the data over small temperature ranges but does reveal\nan apparent increase in activation energy upon warming\nacross the two magnetic anomalies.\nregion where all of the material appears to be in a para-\nmagnetic state. Below the magnetic anomaly observed\nat 330 K, the activation energy decreases substantially,\nEA=0.04 eV for 160-300 K. Below TC, the simple Arrhe-\nnius behavior is not a good model for the data. For\ncomparison sake, a small temperature range allows for a\nreasonable \ft to be obtained, with EA=0.007 eV for 25-\n40 K. Interestingly, these results show the decrease of the\nactivation energy with increasing magnetic order, and if\nEais related to an energy gap then this is rather unusual\nbehavior. Note that the reduction of the activation en-\nergy at low Tcould be due to the activation of low-level\ndefects.\nC. Di\u000buse neutron scattering\nMotivated by the bulk measurements that revealed\nanomalous behavior below 330 K, neutron scattering was\nperformed on CORELLI to search for di\u000buse scattering.\nThese measurements revealed di\u000buse scattering near the\n002 re\rection at 120 K (Fig. 10), and di\u000buse scattering\nalong L in the (H-HL) plane at 6 K (Fig. 11). The scatter-\ning data were binned into 3D bins with three orthogonal\naxes along [H H 0], [H -H 0] and [0 0 L] directions, respec-\ntively. The bin sizes are 1%, 2% and 5% of the reciprocal\nvectors [1 1 0], [1 -1 0] and [0 0 1], respectively.\nFigure 10a is a 2D map of the elastic neutron scat-\ntering intensity for Mn 3Si2Te6at 120 K, showing di\u000busescattering around the 002 re\rection. The 2D map shows\nthe integrated intensity from [-0.035 -0.035 0] to [0.035\n0.035 0], i.e., 7 bins, along the [H H 0] out-of-plane direc-\ntion. The strong scattering intensity reaching out from\n000 in Fig. 10 was strongly angle dependent and did not\nappear consistent with the usual instrument background,\nbut it was essentially independent of temperature. We\nthus speculate that this feature may have been caused\nby the glue utilized to orient the small crystal.\nThe di\u000buse scattering is emphasized in Fig. 10c, which\nis a map of the intensity at 120 K with the intensity at\n350 K subtracted. By contrast to the 002 re\rection, the\nstrong nuclear-only 004 re\rection is sharp and is not sur-\nrounded by di\u000buse scattering. This di\u000buse cloud around\n002 is not observed at 6 or 350 K, suggesting it is from\nshort-range magnetic scattering. It is interesting to spec-\nulate whether this di\u000buse scattering may be related to the\nweak ferromagnetic component of the magnetization seen\nbelow 330 K. This hypothesis could be tested with addi-\ntional temperature-dependent INS measurements. The\nabsence of this di\u000buse scattering at 6 K suggests that it\ncoalesces into the ferrimagnetic order. This is consistent\nwith the observation of di\u000buse scattering around the 002\nre\rection, which has a strong magnetic contribution in\nthe ferrimagnetic phase. The strong contribution from\nthe long-range magnetic order to the intensity of the 002\nre\rection would also hinder detection of di\u000buse scattering\nat 6 K.\nIn Fig. 10b, line scans along L are shown around the\n002 and 004 re\rections using di\u000berent integration areas\ndH (resolution is inversely proportional to dH). The data\nfor 004 demonstrate the behavior expected for a nuclear-\nonly Bragg peak; the 004 re\rection has a sharp peak with\nlittle dependence on the temperature or integration area.\nBy contrast, the nuclear component of the 002 re\rection\nis very weak (see Fig. 10d) and the data at 120 K demon-\nstrate a very broad region of increased intensity centered\naround 002. The sharpening of the intensity at 002 with\ndecreasing dH (higher resolution) suggests the presence\nof a small, sharper Bragg re\rection underneath a di\u000buse\ncloud of scattering.\nThe di\u000buse scattering in Fig. 10 demonstrate the ex-\nistence of short-range spin correlations well-above TCin\nMn3Si2Te6, though the nature of these correlations is\nunclear. The scattering could be from short range order,\npotentially relating to the magnetization anomaly near\n330 K. Alternatively, the di\u000buse scattering could be asso-\nciated with excited states that persist above TCand cause\ninelastic scattering that cannot be entirely excluded due\nto the nature of the experiment. Above TC, the param-\nagnons could be broad, un-gapped and without a de\fned\ndispersion, which would promote isotropic, di\u000buse scat-\ntering in these nearly-elastic measurements. Both short-\nrange order and excited states will exist above a second\norder magnetic transition, but in some cases these short-\nrange correlations persist to much higher temperatures\nthan is typically observed (alternatively, their correla-\ntions lengths are relatively longer above TC).10\nElastic scattering 120K \nTotal Scattering 120K Total Scattering 350K (a) \n(b) (d) (c) (120K minus 350K) Total Scattering, difference map \nFIG. 10: Data from the neutron elastic di\u000buse scattering spec-\ntrometer CORELLI. (a) Estimate of the elastic scattering at\n120 K showing di\u000buse scattering near the 002 Bragg re\rec-\ntions. Line scans of the total scattering at (b) 120 K and (d)\n350 K across 002 and 004 re\rections with di\u000berent integration\nwidths along H indicated in the legend. (c) Di\u000berence map of\ntotal scattering (120 K with 350 K baseline) emphasizing the\ndi\u000buse scattering near the 002 re\rection.\nThe existence of short-range correlations above TCis\ngenerally enhanced when TCis suppressed relative to the\nstrongest exchange interaction. Geometric frustration,\ncompeting interactions, or strong anisotropy in the ex-\nchange constants can all lead to such behavior. In these\ncases, strong magnetic exchange interactions can pro-\nmote short-range correlations (of the ground or excited\nstates) well-above TCbecause thermal energy has less in-\n\ruence near TC. As noted in Ref. 27, the relevant ratio\nto consider when thinking about short-range correlations\nnearTCis notT=T Cbut ratherkBT/JS. In many quasi-\n2D systems, anisotropic interactions lead to a relative\nsuppression of kBTC, and a strong JS can enhance cor-\nrelation lengths above TC. For Mn 3Si2Te6, competing\nantiferromagnetic interactions appear to provide the un-\nderlying mechanism that ultimately promotes the persis-\ntent, short-range correlations above TC. A counter ex-\nample is MnF 2, where the thermal energy is large relative\nto the exchange energy at the antiferromagnetic ordering\nTN,27and as a result the majority of the magnetic en-\ntropy in MnF 2is released below TN\u001967 K.27,28\nAt 6 K, di\u000buse scattering was detected along L in the\n(H-HL) plane, as shown in Fig. 11. The anisotropy of\nthis low-temperature di\u000buse scattering suggests it is as-\nsociated with a short correlation length along the caxis.\nThe 6 K di\u000buse scattering is centered around L=2, and\n(a) \n(b) 6K \n350K FIG. 11: Total neutron scattering in the (H-HL) plane at\n(a) 6 K and (b) 350 K. Di\u000buse scattering is observed along L,\nnotably at 6 K near 1-12. These 2 D maps show the intensity\nfrom [-0.005 -0.005, 0] to [0.005, 0.005 0], i.e., a single bin,\nalong the [H H 0] out-of-plane direction.\nis only observed around particular re\rections (H-K =\n3n+2) that are only contributed to by Mn scattering\n(both sites contribute). This enhances the relative contri-\nbution of the magnetic intensity in these re\rections and\nfacilitates detection of this di\u000buse scattering. Despite ap-\npearing to have a magnetic origin, there is perhaps a mi-\nnor indication of this di\u000buse scattering at 350 K (though\nthis is within the noise of the background scattering).\nTherefore, this feature may be associated with defects\non the Mn positions. From a magnetic perspective, the\ndi\u000buse scattering seems to suggest that there is a short\ncorrelation length of spin canting along the caxis. While\nthis seems plausible and consistent with our other mea-\nsurements, the current data cannot determine the origin\nof this di\u000buse scattering. An additional scattering ex-\nperiment with an applied \feld would be the next step in\ndetermining the origin of this anisotropic di\u000buse scatter-\ning. We note that the tails extending toward L = 0 (the\nasymmetry of 004 for instance) are an artifact of the time\nof \right measurement technique and are not observed in\nthe elastic data (see Supplemental Information).\nIV. SUMMARY\nThis work has characterized the magnetism of\nMn3Si2Te6and found it to be a ferrimagnet due to anti-\nparallel alignment of moments on di\u000berent Mn atomic po-11\nsitions. If the potential canting of the moments along the\ncaxis is ignored, then the magnetic structure is fairly sim-\nple. By contrast, the mechanisms that lead to this ground\nstate are relatively complex. A competition between an-\ntiferromagnetic exchange interactions exists, and it is\na third-nearest neighbor interaction that ultimately de-\ntermines the ground state. This frustration suppresses\nTCand likely enhances the importance of \ructuations,\nleading to persistent short-range correlations above TC.\nMagnetization measurements also suggest that some ad-\nditional ordering may exist below \u0019330 K. Di\u000buse scat-\ntering was observed at 6 K that may indicate short-range\ncorrelations of spin canting along the caxis. Due to the\ncompeting antiferromagnetic interactions, the manipula-\ntion of bond distances or the targeted substitution of theMn2 atoms would likely yield new ground states with\npotentially higher ordering temperatures.\nV. ACKNOWLEDGMENTS\nThis work was supported by the U. S. Department of\nEnergy, O\u000ece of Science, Basic Energy Sciences, Materi-\nals Sciences and Engineering Division. Work at ORNLs\nHigh Flux Isotope reactor and Spallation Neutron Source\nwere supported by the Scienti\fc User Facilities Division,\nO\u000ece of Basic Energy Sciences, U.S. Department of En-\nergy (DOE).\n\u0003Electronic address: mayaf@ornl.gov\n1A. K. Geim and I. V. Grigorieva, Nature 499, 419 (2013).\n2V. Carteaux, F. Moussa, and M. Spiesser, EPL (Euro-\nphysics Letters) 29, 251 (1995).\n3L. D. Casto, A. J. Clune, M. O. Yokosuk, J. L. Musfeldt,\nT. J. Williams, H. L. Zhuang, M.-W. Lin, K. Xiao, R. G.\nHennig, B. C. Sales, et al., APL Materials 3, 041515 (2015),\nURL http://scitation.aip.org/content/aip/journal/\naplmater/3/4/10.1063/1.4914134 .\n4T. J. Williams, A. A. Aczel, M. D. Lumsden, S. E. Nagler,\nM. B. Stone, J.-Q. Yan, and D. Mandrus, Phys. Rev. B 92,\n144404 (2015), URL http://link.aps.org/doi/10.1103/\nPhysRevB.92.144404 .\n5P. A. Joy and S. Vasudevan, Phys. Rev. B 46,\n5425 (1992), URL http://link.aps.org/doi/10.1103/\nPhysRevB.46.5425 .\n6A. Wildes, S. Kennedy, and T. 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Hen-\nnion, Phys. Rev. B 73, 104403 (2006).\n25K. Adachi, J. Phys. Soc. Jpn. 16, 2187 (1961).\n26Y. Takano, N. Arai, A. Arai, Y. Takahashi, K. Takase, and\nK. Sekizawa, J. Magn. Magn. Mater. 272, E593 (2004).\n27L. J. de Jongh and A. R. Miedema, Adv. Phys. 23, 1\n(1974).\n28W. O. J. Boo and J. W. Stout, J. Chem. Phys. 65, 3929\n(1976).12\nSupplementary Information\nThis Supplemental Materials contains additional infor-\nmation regarding magnetization measurements and \frst\nprinciples calculations, as well as neutron di\u000braction and\nscattering data for Mn 3Si2Te6. It also includes addi-\ntional details for the analysis of speci\fc heat capacity\ndata, and a comparison of the low-temperature speci\fc\nheat for Mn 3Si2Te6, CrSiTe 3and CrI 3.\nMagnetization measurements\nThe bulk ordering temperature of TC=78 K was char-\nacterized using AC and DC magnetization measure-\nments. These measurements also revealed that the mag-\nnetization is anisotropic below 330 K, which is well-above\nthe bulk ordering temperature. The anisotropy of the\nmagnetization in this intermediate temperature region is\nconsistent with that in the ordered region - the moment\nis largest for \felds applied perpendicular to the c-axis.\nThis is emphasized in Fig. S1, where isothermal magne-\ntization data are shown for a variety of temperatures. A\nsmall, ferromagnetic-like contribution for H ?ccan be\nseen above TC=78 K. This is best observed at 110 and\n150 K, though anisotropy is easily observed in the M(H)\ndata at higher T.\nA small ferromagnetic component can sometimes\nbe attributed to an impurity, but the observed\nanisotropy strongly suggests that this is intrinsic to our\nMn3Si2Te6single crystals. Also, there was a change\nin the in-plane resistivity at approximately this same\ntemperature. None of the known binaries in the Mn-\nSi and Mn-Te phase diagrams order ferromagnetically\nas bulk materials and no other Mn-Si-Te ternary com-\npounds are known. Furthermore, we do not detect ob-\nvious signs of these binary impurities in powder x-ray\ndi\u000braction or the 3D full volume single crystal scattering\ndata from CORELLI. We note that the previous work on\nMn3Si2Te6did not report data above 300 K, nor did they\nreport anisotropic susceptibility measurements, and the\nCurie Weiss \ftting was performed using data collected in\na large applied \feld.1It is worth mentioning that some of\nthe potential impurities, such as MnSi 1:7,2have complex\nmagnetism that can be tuned to ferromagnetism using\n\fnite size e\u000bects.\nThe AC magnetization measurement naturally probes\nthe ferromagnetic-like contribution observed at low \felds.\nTo demonstrate this, data at T=110 K are examined in\nmore detail in Fig. S2. The (soft) ferromagnetic con-\ntribution for H?cis manifested as a larger \u001f' than\nforHjjc(especially at HDC=0). When the DC \feld\nis increased from zero, the ferromagnetic component is\nrapidly quenched (or saturated), and thus \u001f' decreases\nwith increasing applied \feld for H?c. Similarly, the\nout-of-phase component of the AC susceptibility ( \u001f\") de-\ncreases with applied \feld as the ferromagnetic portion is\nquenched (likely due to the loss of domain movement). In\n/s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s52 /s48 /s46 /s48 /s48 /s46 /s50 /s48 /s46 /s52 /s48 /s46 /s54 \n/s48 /s50 /s48 /s48 /s52 /s48 /s48 /s54 /s48 /s48 /s56 /s48 /s48 /s49 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s49 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s49 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s51 /s84 /s32 /s61 /s32 /s49 /s49 /s48 /s32 /s75 /s72 /s32 /s32 /s99 \n/s72 /s32 /s32 /s124/s124/s32 /s99 \n/s32/s84 /s32 /s61 /s32 /s55 /s48 /s32 /s75 \n/s72 \n/s68 /s67 /s32/s40/s79 /s101 /s41\n/s32\n/s84 /s32 /s61 /s32 /s50 /s48 /s48 /s32 /s75 \n/s32/s77 \n/s68 /s67 /s32/s40\n/s66 /s47/s77 /s110 /s41\n/s84 /s32 /s61 /s32 /s49 /s53 /s48 /s32 /s75 FIG. S1: Isothermal magnetization curves showing the\nanisotropy of the induced moment as a function of applied\n\feld, with these plots highlighting the data at small applied\n\felds.\nthe other orientation, Hjjc,\u001f' is essentially independent\nof applied \feld because MDCis very linear with Hfor this\norientation (the ferromagnetic contribution is negligible\naboveTC=78 K forHjjc, see Fig. S1). The anisotropy\nof\u001f' and\u001f\" are greatly reduced by H=1000 Oe.\nThe data in Figures S1 and S2 thus reveal a ferromag-\nnetic contribution that is related to the anisotropy in the\nmagnetization below 330 K. This is consistent with the\nanisotropy observed in the magnetization measured dur-\ning DC measurements, which is greatest at small applied\n\felds.\nThe small ferromagnetic contribution leads to a dis-\ncrepancy between \u001fACand the DC analog MDC=HDC.\nMDC=HDC=\u001fwhenMis linear inH(as for a paramag-\nnet at high T). For comparison sake, we plot these two\nquantities together in Fig. S3. The di\u000berence between\nthese two quantities starts to grow below the 330 K fea-\nture, and then increases again below TC. Note that for\nthis orientation H?c, the demagnetization e\u000bects are13\n/s48 /s46 /s48 /s48 /s48 /s46 /s48 /s50 /s48 /s46 /s48 /s52 /s48 /s46 /s48 /s54 /s48 /s46 /s48 /s56 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s50 /s48 /s46 /s48 /s48 /s52 /s48 /s46 /s48 /s48 /s54 \n/s48 /s50 /s48 /s48 /s52 /s48 /s48 /s54 /s48 /s48 /s56 /s48 /s48 /s49 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s48 /s48 /s46 /s48 /s48 /s52 /s48 /s46 /s48 /s48 /s56 /s48 /s46 /s48 /s49 /s50 \n/s32/s32/s72 \n/s68 /s67 /s32 /s32 /s99 \n/s72 \n/s68 /s67 /s32 /s124/s124/s32 /s99 \n/s40/s99 /s41/s40/s98 /s41\n/s32/s77 \n/s68 /s67 /s32/s40\n/s66 /s47/s77 /s110 /s41/s84 /s32 /s61 /s32 /s49 /s49 /s48 /s75 /s40/s97 /s41\n/s39 /s39 /s39 \n/s32/s32/s72 \n/s65 /s67 /s32 /s32 /s99 \n/s72 \n/s65 /s67 /s32 /s124/s124/s32 /s99 /s39 /s32/s97 /s110 /s100 /s32 /s39 /s39 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s72 \n/s68 /s67 /s32/s40/s79 /s101 /s41\nFIG. S2: Isothermal magnetization for T=110 K showing (a)\nDC magnetization and (b,c) AC susceptibility data. The AC\ndata were collected using an amplitude of A=14 Oe and a\nfrequency of f=997 Hz.\nexpected to be small (\feld applied in the plane of a thin\nplate).\nFirst principles calculations\nFig. S4 shows the magnetic con\fgurations explored\nwith \frst principles calculations (the non-magnetic and\nferromagnetic states are not shown). Our \frst princi-\nples calculations included a non-magnetic state, a ferro-\nmagnetic state, the ferrimagnetic states FI1 (Mn1 and\nMn2 spins opposed, the calculated ground state) and\nFI2 (Mn1-Mn1 planar neighbors anti-aligned and half of\nMn1-Mn2 neighbors aligned and half antialigned), and\ntwo antiferromagnetic states AF1 and AF2 (both these\nstates areq= 0 states). In state AF1, all Mn1-Mn1 pla-\nnar neighbors are anti-aligned, while all Mn1-Mn2 near-\nest neighbors are ferromagnetically aligned. In AF2 the\nMn1-Mn1 planar neighbors are antialigned and Mn1-Mn2\nnearest neighbors are anti-aligned.\nThe calculated energies were mapped to a Heisenberg\nmodel with J 1the nearest-neighbor exchange (between\nMn1-Mn2), J 2the second-nearest neighbor exchange that\nis between planar Mn1-Mn1, and J 3that couples Mn1\nand a third-nearest spin at Mn2. These calculations as-\nsumed an e\u000bective spin S of \u00061 in the Hamiltonian. The\n/s48 /s53 /s48 /s49 /s48 /s48 /s49 /s53 /s48 /s50 /s48 /s48 /s50 /s53 /s48 /s51 /s48 /s48 /s51 /s53 /s48 /s52 /s48 /s48 /s49 /s48 /s45/s51 /s49 /s48 /s45/s50 /s49 /s48 /s45/s49 /s49 /s48 /s48 /s49 /s48 /s49 /s49 /s48 /s50 \n/s77 \n/s68 /s67 /s47/s72 \n/s68 /s67 \n/s39 \n/s72 /s32 /s32/s99 \n/s32/s32\n/s65 \n/s65 /s67 /s32/s61 /s32/s49 /s52 /s32/s79 /s101 \n/s72 \n/s68 /s67 /s32/s61 /s32/s50 /s48 /s32/s79 /s101 /s39 /s32/s32/s97 /s110 /s100 /s32 /s77 \n/s68 /s67 /s47/s72 \n/s68 /s67 /s32/s40/s99 /s109 /s51 \n/s47/s109 /s111 /s108/s45/s77 /s110 /s41\n/s84 /s32/s40/s75 /s41FIG. S3: Comparison of AC susceptibility and quantity M=H\nfrom DC measurements. The DC measurements were con-\nducted upon cooling in an applied \feld of 20 Oe. The AC\ndata were obtained using HDC=0,A=14 Oe and a frequency\noff=997 Hz. The data are for magnetization with \felds lying\nin the basal plane of the trigonal crystal.\nvalues obtained are all antiferromagnetic: -34.6 , -6.3,\nand -14.8 meV/Mn for J 1, J2, and J 3, respectively. Ef-\nfective uncertainties of approximately 10-15 percent are\nobtained for each of these constants. The uncertainty\narises due to an over-determined set of equations deter-\nmining the exchange constants (one more equation than\nvariable, so that the constants are \\best-\ft\" constants).\nRemoval of one of the equations allows an exact set, for\nthis equation subset, to be determined and thereby the\nuncertainty within the \\best-\ft\" constants assessed.\nAs mentioned, the \frst principles calculations depict\na strong relationship of physical structure to the mag-\nnetism (speci\fcally, the non-symmetry-dictated internal\npositions). In Table S3, we compare the atomic coor-\ndinates extracted from the relaxations to the published\nstructure. Note that all calculations used the experimen-\ntal lattice parameters of a= 7.029 \u0017A andc= 14.255 \u0017A,\nwhich were determined at room-temperature and re-\nported in Ref. 3. LAPW sphere radii of 2.04 Bohr for\nSi, and 2.5 for Mn and Te were employed and su\u000ecient\nnumbers of k-points - at least 600 in the full Brillouin\nzone - were used to determine energy di\u000berences. An\nRKmaxof 7.0, where RK maxis the product of the small-\nest LAPW sphere radius (in this case Si) and the largest\nplanewave expansion wavevector, was employed.\nIn a certain sense, it is natural that signatures of the\nferrimagnetic ground state would remain well above the\nCurie temperature of 78 K. The DFT results \fnd the en-\nergy of the ferrimagnetic state to be well more than an\neV per Mn below that of the non-magnetic state. This\nis more than two orders of magnitude larger than the\n7 meV/Mn scale of TC, indicating a strong likelihood of14\nFIG. S4: The magnetic con\fgurations examined by \frst prin-\nciples calculations in Mn 3Si2Te6are shown (the non-magnetic\nand ferromagnetic states are not shown). FI1 is the ground\nstate, and relative to that energy we obtained AF1=20.8,\nFI2=32.6, and AF2=33.8 meV/Mn.\n\\disordered local moments\"4well aboveTC. Supporting\nthis interpretation is the behavior of the resistivity, which\nshows semiconducting behavior well above TC. This is\nconsistent with the semiconducting gap we \fnd theoret-\nically in the ferrimagnetic state, while the non-magnetic\nstate from theory has a metallic behavior.\nTABLE S3: The internal coordinates of the non-symmetry\ndictated atomic positions. We include the symmetry-dictated\nplanar coordinates of Mn1 and Si for reference. \\Exp.\" refers\nto the experimental published structure,3NM to the non-\nmagnetic calculations, and FI to the ferrimagnetic calcula-\ntions.\nAtom Site Symmetry Origin x y z\nMn1 4f Exp. 1/3 2/3 0.00068\nFI-Calc. 1/3 2/3 0.00155\nNM-Calc. 1/3 2/3 0.00470\nSi 4e Exp 0 0 0.08153\nFI-Calc. 0 0 0.08186\nNM-Calc. 0 0 0.08298\nTe 12i Exp. 0.65869 0.00361 0.12788\nFI-Calc. 0.65040 0.00081 0.12697\nNM-Calc. 0.64213 0.02033 0.11615Single crystal neutron di\u000braction data - HB3A data\nRocking curves from the four-circle di\u000bractometer\nHB3A at the High Flux Isotope Reactor (ORNL) are\nshown in Fig. S5. These demonstrate a strong temper-\nature dependence for the 002 re\rection and not the 004\nre\rection. The re\fned magnetic structure prohibits mag-\nnetic scattering contributions to (004), which is shown\n(Fig. S6) to increase only slightly upon cooling across the\nmagnetic ordering temperature of TC\u001978 K. The inten-\nsities of other nuclear-dominated re\rections are shown in\nFig. S6.\n/s53 /s54 /s55/s45/s49/s48/s49/s50/s51/s52/s53/s54/s55\n/s49/s50 /s49/s51/s48/s48/s52/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s49/s48/s51\n/s32/s99/s111/s117/s110/s116/s115/s41\n/s32/s40/s100/s101/s103/s46/s41/s32/s48/s48/s50\n/s52/s75\n/s49/s48/s48/s75\n/s51/s56/s48/s75\n/s32/s40/s100/s101/s103/s46/s41\nFIG. S5: Rocking curves for single crystal neutron di\u000braction\ndata of (a) the 002 Bragg re\rection and (b) the 004 Bragg\nre\rection at 4, 100, and 350 K.\nDi\u000buse Scattering - CORELLI data\nDi\u000buse magnetic scattering was investigated using the\nsame crystal as the neutron di\u000braction measurements.\nThe crystal is approximately 7 mg in mass, which is small\nwhen searching for di\u000buse scattering intensity. As such,\nlong counting times were employed. In addition, the mea-\nsurement is facilitated by the large moment on the Mn\natoms. The main text discussed di\u000buse scattering along\nL in the (H-HL) plane, which was strongest at 6 K. This\ndi\u000buse scattering is centered around L=2, and is essen-\ntially absent for L <1 and L>3. As discussed in the\nmain text, the source for this scattering is not entirely\nclear. Given that this occurs for non-zero H, and is not\npresent for large L, it potentially relates to a canting of\nthe moments out of the abplane. Additional data for this\nscattering plane, including the estimated elastic contri-\nbution, are shown in Fig. S7\nThe di\u000buse scattering at 6 K is very anisotropic, occur-15\n/s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s46/s56/s53/s48/s46/s57/s48/s48/s46/s57/s53/s49/s46/s48/s48/s49/s46/s48/s53/s49/s46/s49/s48/s49/s46/s49/s53\n/s32\n/s49/s48/s48 /s49/s49/s48\n/s48/s48/s52 /s50/s50/s48/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121\n/s84 /s32/s40/s75/s41\nFIG. S6: Temperature dependence of di\u000braction intensity at\nseveral Bragg re\rections as obtained from the single crys-\ntal neutron di\u000braction beamline HB3A. Based on the re\fned\nmagnetic structure, the 004 re\rection should not possess any\nmagnetic contribution and thus the relative change with tem-\nperature is just a re\rection of the evolution of the lattice or\npeak position (constant Q measurements). The other re\rec-\ntions, with H6= 0, could contain some magnetic contribution\nif the moments have a component along the caxis. However,\nthe relative change with Tfor these peaks is similar to that\nfor 004, suggesting that magnetic scattering is minor for the\npeaks shown here. The data have been normalized and then\no\u000bset from one another.\n350K  \n120K  \n6K Total scattering  Elastic estimate  \n(a) (d) \n(b) \n(c) (e) \n(f) \n120K  \n6K 350K  \n120K  \n6K 350K  \nFIG. S7: Neutron scattering intensity maps from the 2D de-\ntectors of CORELLI showing scattering in the H-HL plane\n(a,d) 6 K, (b,e) 120 K and (c,f) 350 K. The two columns are the\ntotal scattering and the estimated elastic scattering. These\n2D maps show the intensity from [-0.005 -0.005, 0] to [0.005,\n0.005 0], i.e., a single bin, along the [H H 0] out-of-plane di-\nrection.ring only along L and not along H. Such one-dimensional\ndi\u000buse scattering is sometimes observed in quasi-2D crys-\ntal structures due to the presence of stacking faults. An\nimportant feature observed here is the lack of di\u000buse scat-\ntering for H = 0. From a magnetic scattering perspective,\nonly non-zero H re\rections in this plane would possess\nscattering from a component of the moment along the c\naxis. Therefore, these results would be consistent with\nthe presence of some canting along the caxis that only\nhas short coherence lengths. This would be consistent\nwith the average structure being that of a non-canted\nmoment, and we recall that only a very small canted mo-\nment is observed when allowed during re\fnement of the\nsingle crystal neutron di\u000braction data at 4 K. Short co-\nherence lengths of a canted moment could perhaps be\ncaused by the competing exchange interactions. The ex-\nistence of a large J 1promotes ferrimagnetic Mn1-Mn2-\nMn1 units along the caxis, and the coupling of these\nvia competing interactions produces the long-range fer-\nrimagnetic structure. Thus, it may be possible that each\nof these units (or clusters of these units) has a certain\ncanting angle (or lack thereof), but the coherence be-\ntween them is somewhat weak. It is also possible that\na crystallographic defect on the Mn positions causes the\ndi\u000buse scattering, which becomes observable through a\nstrong magnetic contribution at low T.\nAnalysis of speci\fc heat capacity data\nAs mentioned in the main text, we performed a \ft of\nthe speci\fc heat capacity data to estimate the magnetic\nentropy change at TC. Importantly, this was done in a\nmanner that would generally maximize any contribution\n(overestimation). The reason for this is to show that even\nwhen overestimated, the entropy change across TCis very\nsmall (about 8% of anticipated, for the data and baseline\nshown). The data utilized are shown in Figure S8, and\nthe inset shows the data after subtracting the baseline\n(same units as primary y-axis).\nTo further examine the magnetic contribution to the\nspeci\fc heat, we have compared data for Mn 3Si2Te6with\nthat of CrSiTe 3and CrI 3in Fig. S9. Both of these\nCr-based compounds are ferromagnetic semiconductors\nand are quasi-2D with van der Waals gaps; CrSiTe 3\nhasTC=33 K and CrI 3hasTC=61 K.5,6While the mag-\nnetic order of these materials is di\u000berent from that of\nMn3Si2Te6, they all possess relatively large magnetic\ncontributions to the speci\fc heat capacity at low T. In\nFig.S9a, the data are plotted in the conventional CP=T\nversusT2manner. This appears valid for CrI 3and does\nan adequate job of describing the data for Mn 3Si2Te6,\nthough the data for Mn 3Si2Te6have some curvature over\nthe wide temperature range considered. A more extreme\ncurvature is found for CrSiTe 3, where non-linearity of\nCP=TversusT2is clearly evident. Instead, CrSiTe 3is\nbetter modeled using a magnetic contribution that is pro-\nportional to T1:5(Fig.S9b). This treatment also seems16\n/s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s46/s55/s48/s48/s46/s55/s53/s48/s46/s56/s48/s48/s46/s56/s53/s48/s46/s57/s48/s48/s46/s57/s53\n/s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50\n/s32/s32/s100/s97/s116/s97\n/s32/s98/s97/s115/s101/s108/s105/s110/s101/s67\n/s80/s47/s84 /s32/s40/s74/s47/s109/s111/s108/s45/s77/s110/s47/s75/s50\n/s41\n/s84 /s32/s40/s75/s41/s67\n/s80/s47/s84/s32/s45/s32/s98/s97/s115/s101/s108/s105/s110/s101\n/s84 /s32/s40/s75/s41\nFIG. S8: Speci\fc heat data as CP=TversusT, the integra-\ntion of which provides an entropy change. To integrate over\nthe anomaly near TCwithout a phonon reference material, we\nhave taken a straight line for the baseline and this will typ-\nically cause an overestimation of the entropy change across\nTC. The inset shows the data after removal of this baseline,\nwhich would essentially correspond to the magnetic contribu-\ntion if an appropriate baseline were utilizedto work for Mn 3Si2Te6, but clearly fails for CrI 3. At the\nlowestT, the data for Mn 3Si2Te6are well-described by a\nlinear term, as shown in the main text.\n\u0003Electronic address: mayaf@ornl.gov\n1R. Rimet, C. Schlenker, and H. Vincent, J. Magn. Magn.\nMater. 25, 7 (1981), URL http://dx.doi.org/10.1016/\n0304-8853(81)90141-4 .\n2S. Zhou, K. Potzger, G. Zhang, A. M ucklich, F. Eichhorn,\nN. Schell, R. Gr otzschel, B. Schmidt, W. Skorupa, M. Helm,\net al., Phys. Rev. B 75, 085203 (2007), URL http://link.\naps.org/doi/10.1103/PhysRevB.75.085203 .\n3H. Vincent, D. Leroux, D. Bijaoui, R. Rimet, and\nC. Schlenker, J. Solid State Chem. 63, 349 (1986).\n4A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter, J.Phys. F 13, 979 (1983).\n5J. Dillon Jr and C. Olson, J. Appl. Phys. 36, 1259 (1965).\n6M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales,\nChem. Mater. 27, 612 (2015), URL http://pubs.acs.org/\ndoi/abs/10.1021/cm504242t .\n7L. D. Casto, A. J. Clune, M. O. Yokosuk, J. L. Musfeldt,\nT. J. Williams, H. L. Zhuang, M.-W. Lin, K. Xiao, R. G.\nHennig, B. C. Sales, et al., APL Materials 3, 041515 (2015),\nURL http://scitation.aip.org/content/aip/journal/\naplmater/3/4/10.1063/1.4914134 .17\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s77/s110\n/s51/s83/s105\n/s50/s84/s101\n/s54\n/s67/s114/s83/s105/s84/s101\n/s51\n/s67/s114/s73\n/s51\n/s32/s77/s110\n/s51/s83/s105\n/s50/s84/s101\n/s54\n/s67/s114/s83/s105/s84/s101\n/s51\n/s67/s114/s73\n/s51/s67\n/s80/s47/s84 /s32/s40/s109/s74/s47/s109/s111/s108/s45/s84/s77/s47/s75/s50\n/s41\n/s84/s50\n/s32/s40/s75/s50\n/s41\n/s40/s98/s41\n/s32/s67\n/s80/s47/s84/s49/s46/s53\n/s32/s40/s109/s74/s47/s109/s111/s108/s45/s84/s77/s47/s75/s50/s46/s53\n/s41\n/s84/s49/s46/s53\n/s32/s40/s75/s49/s46/s53\n/s41/s40/s97/s41\nFIG. S9: Low temperature speci\fc heat capacity of\nMn3Si2Te6, CrSiTe 3and CrI 3are plotted assuming (a) CP\n=\rT+\fT3formulation and (b) using a CP=\r0T1:5+\fT3\nrepresentation. In the axis labels, the term TM = transition\nmetal. Here, \fis assumed to be related to the Debye tem-\nperature as usual, but \ris not meant to represent the usual\nSommerfeld coe\u000ecient because these materials are semicon-\nducting and should not possess a signi\fcant electronic contri-\nbution toCPat lowT. The data for CrSiTe 3are taken from\nRef. 7 and data for CrI 3are from Ref. 6."    }]