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+[    {        "title": "2202.02461v2.Gyromagnetic_bifurcation_in_a_levitated_ferromagnetic_particle.pdf",        "content": "Gyromagnetic bifurcation in a levitated ferromagnetic particle\nT. Sato,1T. Kato,1Daigo Oue,2, 3, 4and M. Matsuo2, 5, 6, 7\n1Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n3The Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom\n4Instituto Superior T´ ecnico, University of Lisbon, 1049-001 Lisboa, Portugal\n5CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n7RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: September 20, 2023)\nWe examine mechanical rotation of a levitated magnetic particle that is induced by ferromagnetic\nresonance under microwave irradiation. We show that two stable solutions appear in a certain range\nof parameters by bifurcation when the rotation frequency is comparable to the microwave frequency.\nThis phenomenon originates from the coexistence of the Barnett and the Einstein-de Haas effects.\nWe also reveal that this measurement is sensitive to the strength of the spin-rotation coupling. Our\nwork provides a platform for accessing a microscopic relaxation process from spin to macroscopic\nrotation.\nIntroduction.— The discovery of gyromagnetism [1, 2],\ni.e., the interconversion between spin and mechanical ro-\ntation, was a milestone in magnetism research, because\nit revealed the origin of magnetism to be the intrinsic\nangular momentum of electrons, which classical physics\ncannot explain, even before the establishment of quan-\ntum mechanics. Originally, gyromagnetic effects were in-\nvestigated in bulk magnetic materials with aim of deter-\nmining their gyromagnetic ratios [3]. Recently, gyromag-\nnetic effects have been recognized as universal phenom-\nena, as demonstrated in various systems at scales ranging\nfrom those of condensed matter [4–16] to those of particle\nphysics [17–21]. They have also provided powerful tools\nwith which to measure and control both the mechanical\nand magnetic degrees of freedom [4, 5, 22–27].\nIndeed, the Barnett effect (conversion of angular mo-\nmentum of mechanical rotation into spin [1]) has been\nutilized to identify the angular momentum compensa-\ntion temperature of ferrimagnets [24, 25] and to gener-\nate spin current from rigid-body rotation [6–8], surface\nacoustic waves [13–15], and vorticity in fluids [9–12]. On\nthe other hand, the Einstein–de Haas effect (mechanical\ntorque generated from spin polarization [2]), which is ab-\nbreviated as the EdH effect hereafter, has been utilized to\nmeasure the faint torque caused by a single electron spin\nflip [23], identify the gyromagnetic ratio of a nanomag-\nnetic thin film [22], and reveal the demagnetization pro-\ncess in ferromagnets on sub-picosecond time scales [26].\nVery recently, it has been demonstrated, in a solid-\nstate device, that the Barnett and EdH effects can coexist\nin the GHz-frequency regime [16], although the Barnett\nand EdH effects have been treated independently so far.\nThrough these two routes, angular momenta can bounce\nback and forth between magnetic and mechanical degrees\nof freedom in a single system; hence, their coexistence\nmay bring out rich physics and possibly highlight themicroscopic features of the spin-rotation coupling, the\nfundamental coupling between spin and mechanical an-\ngular momentum of a rotating body [28, 29]. However,\nsolid-state platforms involve various excitations; hence,\nthe gyromagnetic phenomena are damped by and could\neven be buried in, e.g., spin and charge transport and im-\npurity scattering. In this sense, it is crucial to construct\na platform that is detached from those relaxation paths\nand enables high-frequency, stable rotation.\nLevitated optomechanics, inspired by laser tweezers,\nlevitation, and cooling [30–34], can provide such a plat-\nform. Stable levitation of small particles has been demon-\nstrated under a high vacuum through the use of, e.g.,\noptical and radio-frequency forces [35]. In particular,\nwith optical forces controlled by parametric feedback,\nsub-kelvin cooling of center-of-mass motion of small par-\nticles has been demonstrated [36–38]. By combining the\nfeedback cooling scheme with the cavity cooling tech-\nnique [39, 40], even zero-point fluctuation of the center of\nmass has been revealed [41]. Since motion of the center\nof mass can be significantly suppressed, high-frequency\nrotation ( ∼GHz) of small particles in a high vacuum\nhas been studied recently [42, 43]. The levitation of fer-\nromagnets, which is necessary for our setup, has been\nstudied theoretically in previous works [44–52] and has\nachieved also experimentally [53, 54]. Though the levi-\ntated optomechanical measurements are thus best suited\nfor investigating the interplay between the Barnett and\nEdH effects, such an application has been overlooked so\nfar.\nIn this Letter, we propose a levitated optomechanical\nsetup to directly probe angular-momentum transfer be-\ntween a spin system and mechanical rotation via Gilbert\ndamping with high precision. We consider a small levi-\ntated particle and study its uniaxial rotation under mi-\ncrowave irradiation in a ferromagnetic resonance (FMR)arXiv:2202.02461v2  [cond-mat.mes-hall]  19 Sep 20232\nFIG. 1. (a) Schematic picture of the model. The magnetic\nnanosphere is optically levitated. The static magnetic field\nH0is applied in the zdirection and the circularly-polarized\nmicrowave field in the xyplane. They induce precession of\nthe magnetization Mand macroscopic rotation Ωabout the\nzdirection via the spin-rotation coupling. (b) Net angular\nmomentum gain f(Ωz). There are three steady-state solu-\ntions, Ωz,i(i= 1,2,3). For Ωz,1andΩz,3(red points), an\ninfinitesimal change in Ωzproduces a restoring torque, and\nthus, the solutions become stable. In contrast, an infinites-\nimal deviation subsequently grows for Ωz,2, and thus, that\nsolution is unstable.\nexperiment (see Fig. 1 (a)). We calculate the steady-\nstate rotation frequency by balancing energy injection\nfrom microwaves and energy loss from air resistance and\nconclude that the particle can be rotated with a high ro-\ntation frequency up to GHz order in a vacuum. We show\nthat the steady-state solution exhibits bifurcation when\nthe rotation frequency is comparable to the microwave\nfrequency, which is the requirement for the strong Bar-\nnett effect, in other words, coexistence of the Barnett and\nthe EdH effects. The present setup enables us to sensi-\ntively measure the g-factor in the spin-rotation coupling.\nOur result illustrates the usefulness of levitated optome-\nchanical techniques in the study of gyromagnetism.\nDynamics of gyromagnetic systems.— We consider a\nspherical ferromagnetic levitated particle of radius r, in\nwhich the motion of the center of mass is suppressed.\nFor simplicity, the particle is regarded as a rigid body\nwith a moment of inertia, I= 2mptcr2/5, where mptc\nis the particle mass. We assume that the magnetization\nof the particle, M, is initially directed in the z-direction\nby an external static magnetic field. We consider a ferro-\nmagnetic resonance (FMR) experiment in which external\nmicrowaves irradiate a sample particle [55]. In this ex-\nperiment, the angular momentum of the excited spins is\ntransferred via the Gilbert damping to the rigid-body ro-\ntation of the particle and therefore the particle starts to\nrotate around the z-axis. Its rotation frequency vector is\ndenoted as Ω= (0,0, Ωz).\nFor the steady state, the Hamiltonian in the rotating\nframe fixed to the particle is written as [28, 29, 56]\nH=−(µ0γH+gSRΩ)·ℏStot. (1)\nThe first term describes the Zeeman energy, where Stot\nis the total spin of the particle, µ0is the vacuum perme-ability, γ(<0) is the gyromagnetic ratio, and His the\nmagnetic field in the rotating frame,\nH= (hcos(ω−Ωz)t, hsin(ω−Ωz)t, H 0). (2)\nHere, H0is a static magnetic field and h(≪H0) and\nωare the amplitude and frequency of the microwaves,\nrespectively. The second term of the Hamiltonian (1)\ndescribes the spin-rotation coupling that explains the\nEinstein-de Haas effect [2] and the Barnett effect [1].\nNote that the g-factor for the spin-rotation coupling, gSR,\ngenerally deviates from one [54, 56].\nThe magnetization of the particle is given as M=\nℏγStot/V, where V= 4πr3/3 is the volume of the par-\nticle. The Landau–Lifshitz–Gilbert (LLG) equation is\ngiven as\n˙M=M×(µ0γH+gSRΩ) +α\nM0M×˙M,(3)\nwhere M0=|M|,˙M= dM/dt, and αis the Gilbert\ndamping constant. The steady-state solution of the LLG\nequation can be obtained by assuming ˙Mz= 0 and\n(Mx, My) =M(cos [( ω−Ωz)t+ϕ],sin [(ω−Ωz)t+ϕ])\nand using M2\n0=M2+M2\nz. For this steady state, the\nzcomponent of the gain rate of the angular momentum\nfrom the spin system is given as\n[M×µ0γH]z=−α\nM0(ω−Ωz)M2. (4)\nThis coincides with the Gilbert damping term\nα/bracketleftig\nM×˙M/bracketrightig\nz/M0, indicating that the gain in an-\ngular momentum from the microwaves is lost through\nGilbert damping. This spin relaxation causes continuous\ntransfer of the angular momentum to the rigid-body\nangular momentum.\nMagnetically driven rigid-body rotation.— The equa-\ntion for the time evolution of the angular momentum in\nthe present system is given as [54]\nd\ndt/bracketleftbig\nIΩz+gSRℏStot\nz/bracketrightbig\n=Γin+Γair, (5)\nwhere Γin=µ0γℏ[Stot×H]zis the torque supplied from\nthe microwaves and Γairis the torque due to the air re-\nsistance (discussed later). When the Gilbert damping\nterm is included phenomenologically as in the last term\nof Eq. (3), a steady state, ˙Sz= 0, arises from the balance\nbetween the Gilbert damping and the gain in angular\nmomentum from the microwaves, Eq. (4). Then, Eq. (5)\nleads to the equation of motion for the steady-state ro-\ntation frequency Ωz.\nThe steady-state rotation of the particle receives\ntorque from the surrounding air. In this paper, we fo-\ncus on the molecular flow region in which λ≫rholds,\nwhere λis the mean free path, in order to realize high-\nfrequency rotation [43, 57–59]. We assume diffuse reflec-\ntion at the surface. Accordingly, the torque induced by3\nthe air resistance is calculated as [54]\nΓair=−8r4\n3/radicalbiggπmair\n2kBTΩzp, (6)\nwhere mairis the mass of the air molecules, kBis the\nBoltzmann constant, Tis temperature, and pis pressure.\nIn the following estimate, we assume that the particle is\nin the atmosphere, for which the average molecular mass\nismair= 4.78×10−26kg and take r= 1×10−6m and\nT= 273 K [60]. The mean free path satisfies p·λ=\nkBT/(√\n2πξ2), where ξis the diameter of an air molecule.\nSubstituting ξ= 3.76×10−10m, we have p·λ≃6.0×\n10−3N/m, and taking p≲100 Pa is sufficient to enter\nthe molecular flow region λ≫r.\nGyromagnetic bifurcation.— First, we set gSR= 1 in\norder to see the features of the steady-state rotation.\nThe Euler equation of a spherical particle is presented\nin Eq. (5). Defining the net angular momentum gain as\nf(Ωz)≡Γin(Ωz)+Γair(Ωz), the rotation frequency of the\nsteady state can be obtained by solving f(Ωz) = 0. Our\nestimate employs the parameters, M0= 1.557×105A/m,\nα= 6.7×10−5,H0= 2.6×105A/m, of a spin pumping\nexperiment for YIG [61].\nBefore showing our results, we should explain that\nboth stable and unstable solutions may appear in gen-\neral. Figure 1 (b) is a schematic graph of f(Ωz) and the\nsolutions of f(Ωz) = 0 when f(Ωz) has three solutions,\nΩz,i(i= 1,2,3). Two solutions, Ωz,1andΩz,3, are stable\nbecause an infinitesimal change in Ωzinduces a restoring\ntorque. On the other hand, Ωz,2is an unstable solution\nbecause no restoring torque works there. The emergence\nof three solutions highly depends on the microwave am-\nplitude hand the Gilbert damping parameter α, as will\nbe discussed later. Hereafter, we calculate the steady-\nstate rotation frequency as a function of the microwave\nfrequency ωand study how it changes as hvaries.\nFigures 2 (a)-(d) show the rotation frequency of the\nsteady state as a function of the microwave frequency for\nh= 4 A /m. The red and blue curves indicate the stable\nand unstable solutions, respectively. For a sufficiently\nhigh pressure, only one solution can be realized for ar-\nbitrary microwave frequencies (not shown in Fig. 2). As\nthe pressure is lowered, two stable solutions and one un-\nstable solution appear (Fig. 2 (a)). These new solutions\nproduced by bifurcation appear only near the resonant\nfrequency ω0=−µ0γH0≃57.52 GHz, while one stable\nsolution exists away from the resonant frequency. As the\npressure is lowered, the lower stable branch approaches\nthe upper one and they become connected to each other\nat a certain pressure, p1(Fig. 2 (b)). Below this crit-\nical pressure, the topology of the graph of Ωzchanges\n(Fig. 2 (c)), and a transition from the lower to the upper\nbranch becomes possible when the microwave frequency\nωapproaches ω0. For a sufficiently low pressure, the\nbifurcation disappears and only one stable solution ex-\nists for any microwave frequency (Fig. 2 (d)). In orderto see the characteristics of the branches, we show the\nmagnetization of the particle at the critical pressure in\nFig. 2 (e). The particle has a small (large) magnetization\nin the upper (lower) branch with a large (small) steady-\nstate rotation frequency. This means that the distribu-\ntion of the angular momentum between magnetization\nand rigid-body rotation is different in these two branches.\nFigure 2 (f) shows the steady-state solutions at ω=ω0\ninp-Ωzspace. The green area in the figure is the forbid-\nden region in which Mzbecomes an imaginary number.\nThe critical pressure is defined as the lowest pressure in\nthe lower branch in Fig. 2 (f) and is calculated as [54]\np1=4|γ|µ2\n0M0h2\nrαω2\n0/radicalbigg\nπkBT\n2mair. (7)\nIn the present estimate, the critical pressure at h=\n4 A/m isp1≃4.4×10−3Pa.\nFigures 2 (g)-(j) show the steady-state rotation fre-\nquency for h= 12 A /m. In this case, there is no bifur-\ncation and only one stable solution at any frequency and\nany pressure. Note that the graph of Ωzhas a cusp at\nω=ω0at a specific pressure p=p2. At this pressure, the\nmagnetization reaches zero at ω=ω0(Fig. 2 (k)). The\nbifurcation disappears because the unstable solutions are\ninside the forbidden region, as shown in Fig. 2 (l) which\nplots the steady-state solutions at ω=ω0on the p-Ωz\nplane. The pressure p=p2corresponds to the crossing\npoint of the two solutions, which is always on the bound-\nary of the forbidden region [54]. Finally, we should note\nthat for gSR= 1 the ferromagnetic resonance frequency\nis not affected by the rotation frequency. This is be-\ncause the decrease in microwave frequency in the rotat-\ning frame (see Eq. (2)) is completely compensated by the\ndecrease in the effective magnetic field due to the spin-\nrotation coupling (the Bernett effect) through the LLG\nequation (3).\nThus, whether the bifurcation occurs or not depends\non the pressure. The condition for a bifurcation to appear\nis obtained by solving p1< p 2atω=ω0ash/H 0<\nα/2 [54]. This indicates the bifurcation disappears at\nhigh microwave amplitudes, which is consistent with the\nresults shown in Fig. 2.\nNow, let us consider the case when gSRdeviates from\n1. Figure 3 shows the rotation frequency as a function of\nthe microwave frequency ωfor different values of gSRfor\np=p1andh= 4 A /m. The curves are strongly tilted\neven for a small deviation in gSRfrom 1. Furthermore,\neven when gSR−1 is small (see Fig. 3 (c),(d)), the two\nstable branches exist in a finite range of ω; therefore,\ntransitions from the lower branch to the upper branch\nalways occur as ωchanges, for example ω≃57.51 GHz\nin Fig. 3 (c). These changes originate from incomplete\ncompensation between the microwave frequency shift and\nthe spin-rotation coupling in the rotating frame. This\nbehavior can be utilized for measuring the effective g-\nfactor of the spin-rotation coupling. Note that if gSR−14\n(a) (b)\n(c) (d)(e)\n(g) (h)\n(i) (j)(k)\n(l)(f)\nFIG. 2. Steady-state rotation frequency Ωzfor various pres-\nsures with microwave amplitudes (a)–(d) 4 A /m and (g)–(j)\n12 A/m. (e)(k) The steady-state solutions of the magnetiza-\ntionMzthat correspond to Ωzin (b) and (h), respectively.\n(f)(l) Red and blue curves express the pdependence of the\nsteady-state solutions for the rotation frequency. The green\narea is the region in which the zcomponent of the magnetiza-\ntionMzbecomes imaginary. Thus, the physical solutions are\nthe red curves, and the solutions depicted by the blue curves\nare unstable.)\nis negative, the graph of Ωz(ω) is tilted in the opposite\ndirection.\nConclusion.— We examined angular momentum trans-\nfer between the spins and the mechanical angular mo-\nmentum of a levitated ferromagnetic particle driven by\nmicrowave irradiation in a vacuum. This setup is suit-\nable not only for making precise measurements but also\nfor very fast mechanical rotation, due to the absence of\na restoring torque, as shown in recent experiments on\nlevitated optomechanics [35]. We formulated the steady-\nstate rotation frequency using the LLG equation in com-\nbination with angular momentum conservation and es-\ntimated it using realistic experimental parameters. We\nfound a bifurcation phenomenon in the solutions for the\nsteady-state rotation frequency when the rotation fre-\nquency is very fast and comparable to the microwave\nfrequency. Our setup is a great candidate for satisfying\nthese conditions, which are essential for the strong Bar-\nnett effect, in other words, coexistence of the Barnett\nand the EdH effects. When the g-factor for the spin-\n(a) (b)\n(c) (d)\nFIG. 3. Steady-state solutions for the rotation frequency Ωz\nfor various gSR: (a) gSR= 1; (b) gSR= 1.0001; (c) gSR=\n1.001; (d) gSR= 1.01. The pressure and microwave amplitude\nare fixed: p=p1andh= 4 A /m. Red (blue) curves are stable\n(unstable). The scale of the horizontal axis is different in each\nfigure.\nrotation coupling, gSR, is unity, a transition in rotation\nfrequency is observed near the resonant frequency due\nto transitions between the two branches. Even a slight\ndeviation in the g-factor gSRfrom unity separates the\ntwo branches more significantly and the transition in ro-\ntation frequency appears in a wider region of pressure.\nThis feature of a bifurcation sensitive to the value of gSR\ncan be used for accurate measurement for the effective g-\nfactor. Our proposal will provide a powerful way to inves-\ntigate angular momentum conversion from magnetism to\nmacroscopic motion in ferromagnets and will promote in-\nterdisciplinary research between two developing research\nfields, spintronics and optomechanics.\nWe thank Y. Ominato, H. Taya, and M. Hongo for\nhelpful comments. 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Mulholland, and Michael R.\nZachariah, “Calculating the rotational friction coefficient\nof fractal aerosol particles in the transition regime using\nextended kirkwood-riseman theory,” Phys. Rev. E 96,\n013110 (2017).\n[60] In our estimate, a fluctuation of rotation velocity, which\nis dominantly induced by thermal fluctuation of air re-\nsistance torque, becomes much smaller than its average.\nTherefore, the noise of rotation frequency does not affect\nits precious measurement.\n[61] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. Saitoh, “Transmis-\nsion of electrical signals by spin-wave interconversion in\na magnetic insulator,” Nature 464, 262–266 (2010).Supplementary Information:\nGyromagnetic bifurcation in a levitated ferromagnetic particle\nT. Sato,1T. Kato,1Daigo Oue,2, 3, 4and M. Matsuo2, 5, 6, 7\n1Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n3The Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom\n4Instituto Superior T´ ecnico, University of Lisbon, 1049-001 Lisboa, Portugal\n5CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n7RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: September 20, 2023)\nMEASUREMENT OF THE ROTATION\nFREQUENCY\nThe fast rotation of an optically levitated particle has\nbeen observed in a number of previous works [1–4]. For\nexample, Ref. [2] reported a GHz rotation of a single\n100 nm silica particle in vacuum in pressure range from\n10−3Pa to 10 Pa. In Ref. [2], the authors analyzed the\nfrequency shift of the detection beam which goes through\nthe particle. Because this shift is independent of the ro-\ntation detection mechanism, it has been utilized for accu-\nrate measurement of the rotation frequency of the parti-\ncle. Though these experiments are mainly performed for\nnon-magnetic particles, levitation of ferromagnetic parti-\ncles has also been realized recently. For example, Ref. [5]\nachieved the levitation of a spherical iron particle having\na diameter around 1 µm by a Paul trap method. There-\nfore, we expect that both levitation of ferromagnetic par-\nticles and measurement of their rotation frequency that\nare assumed in our work are possible within the present\nexperimental techniques. We note that our theoretical\ndescription based on the LLG equation and the angular-\nmomentum conservation does not depends on the kind\nof magnetic materials and is applicable to various ferro-\nmagnetic particles.\nAIR RESISTANCE IN VACUUM\nWe formulate the torque by the air resistance in vac-\nuum referring to Refs. [3, 6–8]. We assume the condition\nofλ≫r, where λis the mean free path of the air and\nris the radius of the particle. We further assume the\ndiffuse reflection at the surface. Then, the total torque\nof the air resistance for the particle is given as [9]\nΓair=−2πr4Ωz\n3ρ¯c, (1)\nwhere ρand ¯care the density and the mean velocity of\nmolecules, respectively. From the Maxwell-Boltzmann\ndistribution for an ideal gas, these quantities are givenp mean free path λ\n10−3Pa 6 m\n10−2Pa 60 cm\n0.1 Pa 6 cm\n1 Pa 6 mm\n10 Pa 0 .6 mm\nTABLE I. The pressure dependence of the mean free path for\nthe air molecules.\nas\nρ=nmair=mairp\nkBT,¯c=2√π/radicalbigg\n2kBT\nmair, (2)\nwhere mairis the molecule mass, nis the number of\nmolecules per unit volume, Tis the temperature, pis\nthe pressure, and kBis the Boltzmann constant. Substi-\ntuting Eq. (2) into Eq. (1), we obtain\nΓair=−8r4\n3/radicalbiggπmair\n2kBTpΩz≡ −βΩz. (3)\nWe assume the mass of air molecules, m= 4.78×\n10−26kg. We also take r= 10−6m and T= 273 K.\nTo check the condition λ≫r, we need to evaluate the\nmean free path, which is given in Ref. 7 as\nλ=kBT√\n2πξ2p(4)\nwhere ξis the diameter of an air molecule. Taking ξ=\n3.76×10−10m [10] and T= 273 K, the mean free path\nof the air satisfies\np·λ≃6.0×10−3N/m. (5)\nThe mean free path for several typical pressures is given\nin Table I. For r= 10−6m, the condition λ≫ris well\nsatisfied when p≲100 Pa.arXiv:2202.02461v2  [cond-mat.mes-hall]  19 Sep 20232\nDERIVATION OF THE ANGULAR-MOMENTUM\nCONSERVATION LAW\nWe derive the angular-momentum conservation law of\nthe system. For this purpose, it is convenient to start\nfrom the Lagrangian in the rotating frame given as\nL=1\n2I˙φ2+ℏStot˙ϕs(cosθs−1)− H\n=1\n2I˙φ2+ℏStot˙ϕs(cosθs−1) +µ0γH·S+gSRΩ·S\n=1\n2I˙φ2+ℏStot˙ϕs(cosθs−1)\n+ℏStotµ0γ{hsinθs[cos(ωt−φ) cosϕs\n+ sin( ωt−φ) sinϕs] +H0cosθs}\n+gSRℏStotcosθs˙φ, (6)\nwhere Stot=Stot(sinθscosϕs,sinθssinϕs,cosθs),φis\na rotation angle whose time derivative corresponds to\nΩ= (0,0,˙φ), and we have imposed φ(t= 0) = 0 as\nthe initial condition. From the Euler-Lagrange equation\nwith respect to φ,d\ndt∂L\n∂˙φ=∂L\n∂φ, we obtain\nd\ndt/bracketleftbig\nI˙φ+gSRℏStot\nz/bracketrightbig\n=ℏStotµ0γh[sin(ωt−φ) cosϕs−cos(ωt−φ) sinϕs]\n=µ0γℏ/bracketleftbig\nStot×H/bracketrightbig\nz≡Γin, (7)\nwhere Γin=µ0γℏ[Stot×H]zis the torque supplied from\nthe microwave. By introducing the dissipative function\nto describe the air resistance, an additional torque Γair\nis added to the right-hand side of the above equation.\nBy considering the steady state, ˙ φ=Ωz, and using φ=/integraltextt\n0˙φdt=Ωzt, we obtain Eq. (7) in the main text.\nSTEADY-STATE EULER EQUATION\nThe dynamics of the system is determined by the LLG\nequation\n˙M=M×(µ0γH+gSRΩ) +α\nMM×˙M,(8)\nand the Euler equation\nI˙Ωz+gSRℏ˙Stot\nz=Γin+Γair, (9)\nwhere we have assumed Ω= (0,0, Ωz). We further as-\nsume the xandycomponents of Mas\nMx=Mcos [(ω−Ωz)t+ϕ], (10)\nMy=Msin [(ω−Ωz)t+ϕ],\nwhere M2=M2\n0−M2\nz. By taking\n˙Mz=˙Ωz= 0, (11)the steady-state solution is obtained as\ntanϕ=αMz\nM0(ω−Ωz)\nω−ω0+∆gΩ z, (12)\nM=µ0γMzh/radicalbig\n(ω−ω0+∆gΩ z)2+ (α(ω−Ωz)Mz/M0)2.\n(13)\nwhere ω0=−µ0γH0and we defined ∆g=gSR−1. Using\nM2+M2\nz=M2\n0, the equation for M2\nzcan be derived. By\nsolving this equation, we obtain\nM2\nz=1\n2α2(ω−Ωz)2/M2\n0/bracketleftig\n−A+/radicalbig\nA2+B/bracketrightig\n, (14)\nA=µ2\n0γ2h2+ (ω−ω0+∆gΩ z)2−α2(ω−Ωz)2,\n(15)\nB= 4α2(ω−Ωz)2(ω−ω0+∆gΩ z)2. (16)\nHere, we have chosen the solution in which the sign in\nfront of the square root is plus, because Mzhas to be\nreal. Since Γinis written as\nΓin=µ0γℏ/bracketleftbig\nStot×H/bracketrightbig\nz\n=−αV\nM0γ(ω−Ωz)(M2\n0−M2\nz), (17)\nthe steady-state Euler equation (9) becomes\n−αV\nM0γ(ω−Ωz)(M2\n0−M2\nz)−βΩz= 0, (18)\nwhere βis the constant defined in Eq. (3). Substituting\nEq. (14) into Eq. (18), we obtain the cubic equation for\nΩzas\na3Ω3\nz+a2Ω2\nz+a1Ωz+a0= 0, (19)\nwhere\na3=−/parenleftbigg2βα\nM0/parenrightbigg2\n+4βα\nM0V\nγ(∆g2+α2), (20)\na2=/parenleftbigg2βα\nM0/parenrightbigg2\nω+8βα\nM0V\nγ(∆g(ω−ω0)−α2ω),(21)\na1=4βα\nM0V\nγ(µ2\n0γ2h2+ (ω−ω0)2+α2ω2)\n−4V2\nγ2µ2\n0γ2h2α2, (22)\na0= 4V2\nγ2µ2\n0γ2h2α2ω. (23)\nThe solution of Eq. (19) has to satisfy the condition\nh(Ωz) =/radicalbig\nA(Ωz)2+B(Ωz)\n=µ2\n0γ2h2+ (ω−ω0+∆gΩ z)2+α2(ω−Ωz)2\n+2αβγ\nM0VΩz(ω−Ωz)≥0, (24)\nwhich appears when squaring Eq. (18). The solutions\nof Eq. (19) which don’t satisfy Eq. (24) correspond to\nnegative M2\nz, in other words imaginary Mz.3\nD⒩\nD⒪D⒩D⒪(a) (b)\nFIG. 1. The steady-state rotation frequency Ωzas a function\nof the pressure for ∆g= 0 and ω=ω0at (a) h= 4 A /m and\n(b)h= 12 A /m. Red (blue) solid curves express the steady-\nstate stable (unstable) solutions of rotation frequency. Green\narea is the region that the zcomponent of the magnetization\nMzbecomes imaginary. When solving Eq. (19), not only the\nred and blue solid curves but also red dotted curves appear.\nHowever, red dotted curves are prohibited since its zcompo-\nnent of the magnetization becomes imaginary.\nBIFURCATION AND CRITICAL PRESSURE\nIn this section, we describe the condition for bifurca-\ntion and detailed calculation on the critical pressure for\ngSR= 1 and ω=ω0given in the main text. We show the\nreal solutions of Ωzas a function of the pressure pby the\nsolid lines in Fig. 1. The red (blue) part of the solid line\nindicates the stable (unstable) solution. Fig. 1 (a) and (b)\ncorrespond to the cases of h= 4 A /m and h= 12 A /m,\nrespectively. We also show the forbidden region, in which\nthe condition (24) does not holds, by the green hatch in\nFig. 1. The dashed lines in the hatch indicate the un-\nphysical solutions, which give imaginary values for Mz.\nBy comparison between Fig. 1 (a) and (b), we can\nsee that the behavior of the steady-state solutions is dis-\ntinct for h= 4 A /m and h= 12 A /m; one unstable so-\nlution exists for weak amplitude of microwave, while it\ndisappears for large amplitude of microwave. This dif-\nference is analyzed by the relative position between the\nbifurcation point D1and the crossing point D2shown in\nFig. 1. These two points correspond to the multiple roots\nof Eq. (19) for ∆g= 0 and ω=ω0. We define x,y, and\nzas\nx=2βα\nM0, y=−V\nγα2, z=−V\nγµ2\n0γ2h21\nω2\n0.(25)\nWe note that x,y, and zare all positive since γ < 0.\nThen, the cubic equation given in Eq. (19) is rewritten\nfor∆g= 0 and ω=ω0as\nΩ3\nz(x+ 2y)x−Ω2\nz(x+ 4y)xω0 (26)\n+Ωz(4yz+ 2xz+ 2xy)ω2\n0−4yzω3\n0= 0.\nThe discriminant of Eq.(26) is given as\nD= 4ω4\n0x(x−8z)(x2y−x2z−4xyz−4y2z)2.(27)By solving D= 0, the condition for the multiple roots is\ngiven as x=x1, x2where\nx1= 8z, x 2=2y(z+√yz)\ny−z. (28)\nWe note that the sign in front of the square root is chosen\nas a plus since xhave to be positive. The solutions, x=\nx1andx=x2, correspond to D1andD2, respectively.\nThe steady-state rotational frequencies are calculated for\nthese two solutions as\nΩz(D1) =1\n2ω0, (29)\nΩz(D2) =y−z\ny+√yzω0. (30)\nNext, we consider the boundary of the forbidden area\n(the green hatched area in Fig. 1). It is written with x,\ny, and zas\nh(Ωz) =−γ\nV/bracketleftig\n(x+y)Ω2\nz+ (−x−2y)ω0Ωz\n+(y+z)ω2\n0/bracketrightig\n= 0. (31)\nWe can check that the solution for D2given by x=\nx2and Eq. (30) satisfies this condition. This indicates\nthat the crossing point D2of the steady-state solution is\nalways on the boundary of the forbidden region in Fig. 1.\nWe note that the D2point disappears when y < z ; in this\ncase only one branch is observed just like the usual FMR.\nFory > z , the unstable solution (indicated by the blue\nline in Fig. 1 (a)) exists only when Ωz(D1)< Ω z(D2).\nThis condition is written as\nh\nH0<α\n2. (32)\nWhen the unstable solution exists, the D1point cor-\nresponds to the critical pressure p1at which the two\nbranches connect each other:\np1=−2M0\nγαr/radicalbigg\n2πkBT\nmair/parenleftbiggh\nH0/parenrightbigg2\n. (33)\nForh= 4 A /m, which correspond to the situation given\nin Fig. 3 (b) in main text, the critical pressure is evalu-\nated as p1≃4.4×10−3Pa.\nEFFECT OF THE GILBERT DAMPING\nWe discuss the effect of the Gilbert damping for the\nsteady-state rotation. We first show the rotation fre-\nquency Ωzas a function of the microwave frequency ω\nforgSR= 1,p= 5.0×10−3Pa, and h= 4 A /m in Fig. 2.\nAs the Gilbert damping αis increased from Fig. 2 (a) to\n(c), the bifurcation first occurs in the lower branch and\nsubsequently the upper branch is disconnected from the4\n(a)\n(c) (d)\n(b)\nFIG. 2. Steady real solutions of rotation frequency for each α\natgSR= 1, p= 5.0×10−3Pa and h= 4 A /m are described.\nαcr≃5.89×10−5determined from Eq. (33).\nlower one. Finally, the pair of the stable and unstable\nsolutions disappears after the bifurcation in the upper\nbranch and only one stable solution is left as shown in\n(d). The increase of αgives a similar effect as the de-\ncrease of the amplitude of the microwave, h.\nNext, we show the rotation frequency for gSR= 1.001,\np= 5.0×10−3Pa, and h= 4 A /m. If αis sufficiently\nsmall, there are two stable solutions (red lines) and one\nunstable one (a blue line) as shown in Fig. 3 (a) and\n(b), and the shape of Ωz(ω) is tilted as pointed out in\nthe main text. As αincreases, the critical pressure p1\nbecomes lower than the present pressure and the upper\nbranch is disconnected from the lower branch as shown\nin Fig. 3 (c). If αfurther increases, the upper branch\ndisappears and only one solution is realized as shown\nin Fig. 3 (d), for which the tilt of the graph of Ωz(ω)\nbecomes obscure because of large broadening of the reso-\nnant peak. We note that if the pressure pis much lower,\ntwo separated branches exist for any value of the Gilbert\ndamping αsince the pressure at the D2point approaches\nto∼4.7×10−3Pa.\nIn the main text, we chose α= 6.7×10−5from a\ntypical value for yttrium iron garnet (YIG). The tilting of\nthe ferromagnetic resonance (FMR) peak for Ωzcannot\nbe observed if αbecomes large as shown in Fig. 3 (d).\nHowever, the tilting of the FMR peak can be recovered\nby increasing the microwave amplitude hfor such a large\nvalue of the Gilbert damping.\n(a)\n(c) (d)\n(b)\nFIG. 3. Steady real solutions of rotational frequency Ωzfor\ndifferent αare described. gSR,p, and his fixed to 1 .001 and\n5.0×10−3Pa, and 4 A /m respectively.\ng-FACTOR FOR THE SPIN-ROTATION\nCOUPLING\nIn the main text, we have pointed out the possibility\nthat the g-factor for the spin-rotation coupling, gSR, de-\nviates from 1. Here, we briefly outline how gSRis shifted\nfrom 1 for ferromagnetism induced by localized electrons\nin 3dtransition-metal ions by considering the crystal field\nand the spin-orbit coupling following Ref. 11.\nWe consider a localized spin in an ion of 3 dtransition\nmetals whose Hamiltonian is given as\nH=Hintra+Hcry+HSO+HZ+HSR. (34)\nHere, HintraandHcrydescribe the intraatomic Coulomb\nand the crystal field, respectively. The last three terms\nare the Hamiltonians of the spin-orbit coupling, the Zee-\nman energy, and the spin-rotation coupling and are given\nas\nHSO=λL·S, (35)\nHZ=µBH·(L+ 2S), (36)\nHSR=Ω·(L+S), (37)\nrespectively. We assume H0=Hintra +Hcry≫\nHSO, HZ, HSR, which is satisfied for 3 dtransition-metal\nions. We assume that the ground state of H0is nonde-\ngenerate and express it as |G, M s⟩, where Msis the spin\nquantum number. Since H0does not mix spin and orbit\nstates, this eigenfunctions are written as the products,\n|G, M s⟩=|G⟩|Ms⟩. Using the perturbation theory, the\nfirst- and the second-order corrections to the energy for\nan orbitally nondegenerate ground state, E(1), E(2), are5\nderived as\nE(1)= 2µBH·S+Ω·S, (38)\nE(2)=−/summationdisplay\ni̸=G|⟨G|L·(λS+µBH+Ω)|i⟩|2\nE(0)\ni−E(0)\nG(39)\nwhere we have used the fact that ⟨G|L|G⟩vanish. When\nwe define\nΛµν=/summationdisplay\ni̸=G⟨G|Lµ|i⟩⟨i|Lν|G⟩\nE(0)\ni−E(0)\nG, (40)\nthe energy correction reduces to\nE(2)=−Λµν/bracketleftbig\nλ2SµSν+ (µBHµ+Ωµ)(µBHν+Ων)\n+2λSµ(µBHν+Ων)] (41)\nwhere the Einstein summation convention has been used.\nBy combining these results, the effective Hamiltonian be-\ncomes\nHeff=H0+gZ\nµνµBHµSν+gSR\nµνΩµSν−λ2ΛµνSµSν\n−Λµν(µBHµ+Ωµ)(µBHν+Ων), (42)\ngZ\nµν= 2δµν−2λΛµν, (43)\ngSR\nµν=δµν−2λΛµν. (44)\nwhere gZ\nµνandgSR\nµνareg-factor tensors for the Zeeman\nterm and the spin-rotation coupling, respectively. This\nresult indicates that the spin-orbit interaction changes\nnot only the g-factor of the Zeeman term but also that of\nthe spin-rotation coupling in the same way through Λµν.\nIn general, Λµνcan be diagonalized by taking a certain\nprinciple axis. If the crystal field has cubic symmetry,\nΛµνbecomes\nΛµν= 0 (µ̸=ν), Λxx=Λyy=Λzz. (45)\nThis means an isotropic constant shift of the g-factor. In\nthe main text, we have assumed an isotropic change of\ntheg-factor for simplicity.\nSTABILITY ANALYSIS OF THE ROTATION\nThrough our article, we have assumed that the in-plane\nrotation velocities, ΩxandΩy, are zero. We have checked\nnumerically that the in-plane rotation always decays even\nwhen the initial condition includes the in-plane rotation.\nThis indicates that the uniaxial rotation around the z\naxis is stable against perturbation to incline the rotation\naxis. In this section, we show the stability of the uniaxial\nrotation by solving Eqs. (48) and (56) within a simple\napproximation, which leads to the same result obtained\nby numerical calculation.\nIn order to discuss a general formalism of our model,\nwe have to ride on the body frame in which the in-plane\nrotation is also allowed. The coordinate transformationfrom the laboratory frame, ex,ey,ez, to the body frame,\nex′,ey′,ez′, can be implemented by using Euler angles,\n(φ, θ, ψ ). The transformation matrix R ( Ab= RA,\nwhere Ais a vector in the lab frame, and Abis that\nin the body frame) is given as\nR =\ncosψsinψ0\n−sinψcosψ0\n0 0 1\n·\n1 0 0\n0 cos θsinθ\n0−sinθcosθ\n (46)\n·\ncosφsinφ0\n−sinφcosφ0\n0 0 1\n.\nThe rotation frequency Ωis expressed by Euler angles\nas\n\n\nΩx′= ˙φsinθsinψ+˙θcosψ,\nΩy′= ˙φsinθcosψ−˙θsinψ,\nΩz′= ˙φcosθ+˙ψ.(47)\nWhen θ=φ= 0,˙θ= ˙φ= 0, the body frame rotates only\naround zaxis, and it becomes the same as the model of\nour article. For stability analysis, it is sufficient to con-\nsider states of θ≪1. The dynamics of the magnetization\nin the body frame, Mb, is determined from a modified\nversion of Eq. (8):\n˙Mb=Mb×(µ0γHb+Ωb) +α\nMMb×˙Mb, (48)\nwhere gSRis set as unity for simplicity and Hbis defined\nas\nHb= RH= R\nhcosωt\nhsinωt\nH0\n (49)\n≃\nhcos(ωt−φ−ψ)\nhsin(ωt−φ−ψ)\nH0\n+θ\nH0sinψ\nH0cosψ\n−hsin(ωt−φ)\n.\nBy expanding Mbwith respect to θas\nMb=M+θM1+O(θ2), (50)\nwe can derive equations from the zeroth- and first-order\nofθ. The zeroth-order equation leads to the equation of\nMgiven in the main text and therefore the steady-state\nsolution of Mis given as\nM≡\nMx′\nMy′\nMz′\n=\nMcos[(ω−Ωz′)t+ϕ]\nMsin[(ω−Ωz′)t+ϕ]\nMz\n. (51)\nHere, we used Ωz′= ˙φcosθ+˙ψ≃˙φ+˙ψand as-\nsumed that ˙ φand ˙ψare almost constant. On the\nother hand, from the equation of the first order of θ,\nM1≡(M1x′, M1y′, M1z′) obeys6\n\n∂tα\nM0Mz′∂t−µ0γH0−Ωz′−α\nM0My′∂t+µ0γHy′+α\nM0˙My′\n−α\nM0Mz′∂t+µ0γH0+Ωz′ ∂tα\nM0∂t−µ0γHx′−α\nM0˙Mx′\nα\nM0My′∂t−µ0γHy′−α\nM0˙My′−α\nM0Mx′∂t+µ0γHx′−α\nM0˙Mx′ ∂t\n\nM1x′\nM1y′\nM1z′\n\n=−µ0γ\nH0Mz′cosψ+hsin(ωt−φ)My′\n−H0Mz′sinψ−hsin(ωt−φ)Mx′\nH0(sinψMy′−cosψMy′)\n, (52)\nwhere ( Hx′, Hy′)≡h(cos(ω−Ωz′)t,sin(ω−Ωz′)t). Ex- tracting upper two equations, and assuming h≪H0,\nthese equations are reduced to\n/parenleftig\n∂t−i˙ψ+i/parenleftig\n−α\nM0Mz′(∂t−i˙ψ) +µ0γH0+Ωz′/parenrightig\niα\nM0M+∂t−iµ0γH++α\nM0(ω−˙φ−˙ψ)M+/parenrightig/parenleftigg\nM1\nM1z′/parenrightigg\n=−µ0γH0Mz′, (53)\nwhere M1≡(M1x′+iM1y′)eiψ,M+≡Mei(ωt−φ+ϕ)and\nH+≡(Hx′+iHy′)eiψ=hei(ωt−φ).M+andH+oscil-\nlate with the microwave frequency ωbecause ˙ φis much\nsmaller than ω. Since the dynamics of M1is roughly\ndominated by slowly oscillating terms, we can drop these\nfast oscillating terms. Technically speaking, M+and\nH+becomes zero by being integrated out from t= 0\ntot= 2π/ω. Then, Eq. (53) becomes\n/parenleftbigg\n1−iα\nM0Mz′/parenrightbigg\n∂tM1 (54)\n+/bracketleftbigg\ni(µ0γH0+Ωz′−˙ψ)−α\nM0Mz′˙ψ/bracketrightbigg\nM1=−µ0γMz′,\nand the steady-state solution of this is obtained as\nM1=−µ0γH0Mz′\ni(µ0γH0+Ωz′−˙ψ)−α\nM0Mz′˙ψ. (55)\nEuler equation of the particle is written as\n\n\nI1˙Ωx′+∆IΩ z′Ωy′= [Γg+Γoptical +Γair]x′,\nI1˙Ωy′−∆IΩ z′Ωx′= [Γg+Γoptical +Γair]y′,\nI2˙Ωz′= [Γg+Γoptical +Γair]z′,(56)\nwhere we assumed that the particle is symmetrical top\nandI1(I2) is a principle moment of inertia around x′, y′\n(z′) axis and ∆I≡I2−I1. Here, Γgis a torque generated\nfrom Gilbert damping given as\nΓg=−V\nγα\nM0Mb×˙Mb, (57)Γoptical is a torque from optical trap, and Γair=−βΩis\na torque from air resistance. Using Eq. (47), equation of\nmotion for θ,φandψis written as\nI1¨θ+I1˙φ˙ψsinθ+∆IΩ z′˙φsinθ\n= [Γg+Γoptical +Γair]x′cosψ\n−[Γg+Γoptical +Γair]y′sinψ, (58)\nI1( ¨φsinθ+ ˙φ˙θcosθ−˙θ˙ψ)−∆IΩ z′˙θ\n= [Γg+Γoptical +Γair]x′sinψ\n+ [Γg+Γoptical +Γair]y′cosψ, (59)\nI2(¨ψ+ ¨φcosθ−˙φ˙θsinθ) = [Γg+Γoptical +Γair]z′.\n(60)\nNeglecting the fast oscillating term, the first term in the\nbracket of the right hand side is approximated as\nΓg,x′cosψ−Γg,y′sinψ≃ −V\nγα\nM0θ˙ψMz′Re(M1),(61)\nΓg,x′sinψ+Γg,y′cosψ≃ −V\nγα\nM0θ˙ψMz′Im(M1),(62)\nΓg,z′≃ −V\nγα\nM0(ω−Ωz′)(M2\n0−M2\nz′). (63)\nThe torque from optical trap, Γoptical , which usually\nworks to restore the axis of inertia to zaxis, is written\nas\nΓoptical ,x′cosψ−Γoptical ,y′sinψ=−ξθ, (64)\nΓoptical ,x′sinψ+Γoptical ,y′cosψ= 0, (65)\nΓoptical ,z′= 0, (66)7\nwhere ξis a coefficient of the optical torque. In addition,\nwe take the spherical particle limit ∆I=I2−I1→0\nas considered in the main text. By these assumptions,\nEq. (58) and (59) are simplified as\nI1¨θ+β˙θ+/parenleftbigg\nI1˙φ˙ψ+ξ+V\nγα\nM0˙ψMz′Re(M1)/parenrightbigg\nθ≃0,\n(67)\nI1θ¨φ+ (I1˙θ+βθ) ˙φ−I1˙θ˙ψ+V\nγα\nM0θ˙ψMz′Im(M1)≃0,\n(68)\nI1(¨ψ+ ¨φ)≃ −βΩz′−V\nγα\nM0(ω−Ωz′)(M2\n0−M2\nz′).\n(69)\nIn our setup, we estimate I1≃9×10−27N·m·s2,β≃\n5×10−29N·m·s and\nV\nγα\nM0Mz′Im(M1)≃V\nγαM0ω0−˙φ\nω0\n∼(ω0−˙φ)×(−4.3×10−39) N·m·s,(70)\nV\nγα\nM0Mz′Re(M1)≃ −V\nγα2M0˙ψ\nω0\n∼˙ψ×2.9×10−43N·m·s. (71)\nIn addition, since the optical force is typically order of\nsub-pico Newton, the torque ξon a particle with radius\n1µm from optical trap is estimated to be ξ∼10−15N·m.\nConsidering that ˙ψis order of 10 GHz at most, the third\nterm in the bracket of Eq. (67) can be neglected.\nThe right hand side of Eq. (69) is same as the total\ntorque f(Ωz) defined in the main text, and vanishes in\nthe stable solutions of Ωz, which indicates\nΩz′≃˙ψ+ ˙φ= const . (72)\nSubstituting ˙ψ=Ωz′−˙φ, and treating Ωz′as constant\nvalue, Eq. (67)(68) are transformed to\nI1¨θ+β˙θ+/parenleftig\nI1˙φ˙ψ+ξ/parenrightig\nθ≃0, (73)\nI1θ¨φ+/parenleftbigg\n2I1˙θ+βθ−V\nγαM0θ/parenleftbigg\n1 +Ωz′\nω0/parenrightbigg/parenrightbigg\n˙φ\n≃/parenleftbigg\nI1˙θ−V\nγαM0θ/parenrightbigg\nΩz′, (74)\nup to the first order of ˙ φ. From Eq. (73), ˙ φcan be\nexpressed as\n˙φ=−1\nI1Ωz′θ/parenleftig\nI1¨θ+β˙θ+ξθ/parenrightig\n. (75)By substituting Eq. (75) into Eq. (74), we obtain\nI1...\nθ+ (2β+A)¨θ\n+/parenleftigg\nβ(β+A)\nI1+ 2ξ+I1Ω2\nz′+β˙θ\nθ+I1¨θ\nθ/parenrightigg\n˙θ\n+/parenleftbiggξ(β+A)\nI1+BΩ2\nz′/parenrightbigg\nθ= 0, (76)\nA=−V\nγαM0/parenleftbigg\n1 +Ωz′\nω0/parenrightbigg\n(>0), (77)\nB=−V\nγαM0(>0). (78)\nWe assume that the non-linear terms proportional to\n˙θ2/θand¨θ˙θ/θin Eq. (76) can be neglected (the validity\nof this assumption will be checked later). Then, Eq. (76)\nis reduced to a linear homogeneous differential equation\nof third order;\n...\nθ+a2¨θ+a1˙θ+a0θ= 0, (79)\na2= (2β+A)/I1(>0),\na1=/parenleftbiggβ(β+A)\nI1+ 2ξ+I1Ω2\nz′/parenrightbigg\n/I1(>0),\na0=/parenleftbiggξ(β+A)\nI1+BΩ2\nz′/parenrightbigg\n/I1(>0).\nAccording to Routh–Hurwitz stability criterion, the gen-\neral solution of Eq. (79) decays in time when and only\nwhen a2, a1, a0>0 and a2a1> a 0. Using I1≃9×\n10−27N·m·s2,β≃5×10−29N·m·s,ξ≃10−15N·m,\nB≃2.5×10−28N·m·s, and Ωz′∼1010Hz, we can\nconfirm a2a1> a0. Therefore, we can conclude that the\ndeclination angle θalways decays towards zero.\nFinally, we check the validity of the assumption of ig-\nnoring the non-linear terms in Eq. (76). Using the same\nparameter estimate shown above, Eq. (79) is approxi-\nmately rewritten as\n...\nθ+B\nI1/parenleftbigg\n1 +Ωz′\nω0/parenrightbigg\n¨θ+Ω2\nz′˙θ+B\nI1Ω2\nz′θ≃0. (80)\nThe general solution of this differential equation is writ-\nten as\nθ≃C1e−B\nI1t+ (C2sinΩz′t+C3cosΩz′t)e−BΩz′\n2I1ω0t,\n(81)\nwhere C1,C2, and C3are determined from the initial\nconditions for θ,˙θ,¨θ. If the initial condition is chosen to\nsatisfy ˙θ(t= 0)≪Ωz′θ(t= 0) and ¨θ≪Ω2\nz′θ(t= 0), C2\nandC3become much smaller than C1, since C1,C2Ωz′\nandC3Ω2\nz′are of the same order. Then, the amplitude of\nβ˙θ/θandI1¨θ/θbecome much smaller than a1≃I1Ω2\nz′,\nand neglecting these non-linear terms is justified.\nIn summary, we conclude that θdecays into zero and\nnever shows complex dynamics. This result indicates8\nthat the solution, Ω= (0,0, Ωz), which we derived in\nthe main article, is stabilized with respect to small per-\nturbation tilting the rotation axis.\n[1] Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced\nrotation and cooling of a trapped microgyroscope in vac-\nuum,” Nat. Commun. 4(2013).\n[2] Ren´ e Reimann, Michael Doderer, Erik Hebestreit,\nRozenn Diehl, Martin Frimmer, Dominik Windey, Fe-\nlix Tebbenjohanns, and Lukas Novotny, “Ghz rotation\nof an optically trapped nanoparticle in vacuum,” Phys.\nRev. Lett. 121, 033602 (2018).\n[3] Jonghoon Ahn, Zhujing Xu, Jaehoon Bang, Yu-Hao\nDeng, Thai M. Hoang, Qinkai Han, Ren-Min Ma, and\nTongcang Li, “Optically levitated nanodumbbell torsion\nbalance and ghz nanomechanical rotor,” Phys. Rev. Lett.\n121, 033603 (2018).[4] Fernando Monteiro, Sumita Ghosh, Elizabeth C. van As-\nsendelft, and David C. Moore, “Optical rotation of levi-\ntated spheres in high vacuum,” Phys. Rev. A 97, 051802\n(2018).\n[5] P. Huillery, T. Delord, L. Nicolas, M. Van Den Bossche,\nM. Perdriat, and G. H´ etet, “Spin mechanics with levitat-\ning ferromagnetic particles,” Phys. Rev. B 101, 134415\n(2020).\n[6] P. S. Epstein, “On the resistance experienced by spheres\nin their motion through gases,” Phys. Rev. 23, 710–733\n(1924).\n[7] A. Roth, Vacuum Technology (North Holland, 1990).\n[8] J. Corson, G. W. Mulholland, and M. R. Zachariah,\n“Calculating the rotational friction coefficient of fractal\naerosol particles in the transition regime using extended\nkirkwood-riseman theory,” Phys. Rev. E 96, 013110\n(2017).\n[9] This equation corresponds to Eq. (64) in Ref. 6.\n[10] This value for the molecular diameter ξis taken from\nFig. 2.9 of Ref. 7.\n[11] R. M. White, Quantum Theory of Magnetism (Springer,\nBerlin, Heidelberg, 2007)."    },    {        "title": "1609.07863v1.Ferromagnetic_resonance_study_of_composite_Co_Ni___FeCoB_free_layers_with_perpendicular_anisotropy.pdf",        "content": "Ferromagnetic resonance study of composite Co/Ni - FeCoB free layers\nwith perpendicular anisotropy\nT. Devolder,1,a)E. Liu,2, 3J. Swerts,2S. Couet,2T. Lin,2S. Mertens,2A. Furnemont,2G. Kar,2and J. De\nBoeck2, 3\n1)Centre for Nanoscience and Nanotechnology, CNRS, Univ. Paris-Sud, Universit ´e Paris-Saclay, 91405 Orsay,\nFrance\n2)imec, Kapeldreef 75, 3001 Heverlee, Belgium\n3)Department of Electrical Engineering (ESAT), KU Leuven, Leuven 3001, Belgium\n(Dated: 18 June 2021)\nWe study the properties of composite free layers with perpendicular anisotropy. The free layers are made of a soft\nFeCoB layer ferromagnetically coupled by a variable spacer (Ta, W, Mo) to a very anisotropic [Co/Ni] multilayer\nembodied in a magnetic tunnel junction meant for spin torque memory applications. For this we use broadband ferro-\nmagnetic resonance to follow the field dependence of the acoustical and optical excitation of the composite free layer in\nboth in-plane and out-of-plane applied fields. The modeling provides the interlayer exchange coupling, the anisotropies\nand the damping factors. The popular Ta spacer are outperformed by W and even more by Mo, which combines the\nstrongest interlayer exchange coupling without sacrificing anisotropies, damping factors and transport properties.\nMagnetic tunnel junctions (MTJ) based on perpendicular\nmagnetic anisotropy (PMA) systems are under active devel-\nopment for the next generations of spin transfer torque (STT)\nmagnetic random access memories (MRAM). In this tech-\nnology, the information is stored in the magnetization state\nof a free layer composed typically of Fe-rich FeCoB alloys\nsandwiched between Ta and MgO (so-called ”single” MgO\nfree layer1) or between two MgO layers (”dual” MgO free\nlayers2). Each MgO interface provides an interface energy\nthat promotes PMA. The dual MgO option is gradually be-\ncoming more frequent for memory applications as it provides\nup to double anisotropy, hence more resilience to thermal\nfluctuations. This strategy ensures scalability3,4for junctions\ndown to 20-30 nm of diameter but material solutions have to\nbe found to scale further. To match with STT-MRAM ob-\njectives, one possible configuration comprises (i) an Fe-rich,\nbcc-structured, low damping FeCoB at the MgO interface (ii)\ncoupled ferromagnetically with a material supplying a strong\nanisotropy energy while maintaining damping and thickness\nas low as possible.\nAmong the material systems providing large PMA, the\nCo/Ni multilayers belong to the few ones that have a reason-\nably low damping, from50.033 down to6,70.021 and even8\n0.014 depending on compositions and thicknesses. The un-\nresolved questions are whether these low damping values can\nbe maintained in ultrathin Co/Ni multilayers and whether the\nfcc-based Co/Ni multilayer can be coupled strongly to the bcc-\nbased FeCoB layers across a texture transition.\nIn this paper, we compare three different spacers and\nevaluate their impact on the anisotropies, the interlayer\nexchange coupling and the damping in each layer of an\nMgO/FeCoB/spacer/[Co/Ni] \u00024/Pt system embodied in an\nMTJ. The studied spacers include the popular9Tantalum, as\nwell as Molybdenum and Tungsten spacers. The choice of bcc\nmetals (Ta, W and Mo) is to promote the fcc to bcc texture\na)Electronic mail: thibaut.devolder@u-psud.frtransition. The refractory character of W, Ta and Mo is also\na foreseen advantage since large resilience to atomic diffu-\nsion upon annealing is desirable for CMOS back-end-of-line\ncompatibility. W/CoFeB and Mo/CoFeB systems have indeed\nproven large resistance to annealing10,11which correlate with\nthe very slow diffusion of refractory metals in Fe12–14. To as-\nsess the performance of these spacers, we use broadband fer-\nromagnetic resonance (FMR) to follow the field dependence\nof the acoustical and the optical excitations of the compos-\nite free layer. The modeling provides the interlayer exchange\ncoupling, the anisotropies and the damping factors. The pop-\nular Ta spacer is outperformed by Mo, which combines the\nstrongest interlayer exchange coupling with no detrimental ef-\nfect on the anisotropies and the damping within the magnetic\nmaterials of the stack. This qualifies Molybdenum as a mate-\nrial of choice for the spacer layer in composite free layers.\nOur objective is to study hybrid free layers. By ”hybrid”\nwe mean comprising a bcc FeCoB layer which ensures op-\ntimal transport properties, coupled ferromagnetically to an\nfcc [Co/Ni] multilayer whose anisotropy strengthens thermal\nstability of the whole free layer. The multilayer is grown\nfirst on a Pt buffer that provides low coercivity and high\nanisotropy15. We then grow top-pinned MTJs of the follow-\ning configuration15: Pt / [Co(3 ˚A)/Ni(6 ˚A)]\u00024(t2= 3:5nm)\n/ spacer / Fe 60Co20B20(t1= 1 nm) / MgO / reference layer\n/ cap. The [Co/Ni] multilayer is terminated by a nickel layer\nin contact with the spacer. The studied spacers are Ta(3 ˚A),\nMo(3 ˚A) and W(3 ˚A). In addition, thicker (i.e. 5 ˚A) Mo and\nW spacers were used to decouple the two parts of the free\nlayer and thereby measure of their respective moments and\neasy axes. W(5 ˚A) leads to in-plane magnetization of the Fe-\nCoB layer (not shown). We aim to optimize the spacer within\nhybrid free layers in realistic MTJs, i.e. comprising reference\nlayers that might influence the crystallization within the free\nlayer. Our reference system15is a standard Co/Pt based syn-\nthetic ferrimagnet (SAF). All samples were annealed at 300\u000eC\nfor 30 min. in a field of 1 T.\nFor Mo(5 ˚A), the magnetizations are found to be Ms1=\n1:21\u0002106A=m,Ms2= 0:763\u0002106A=mfor the FeCoBarXiv:1609.07863v1  [cond-mat.mtrl-sci]  26 Sep 20162\n/s48/s50/s48/s52/s48/s54/s48\n/s48/s50/s48/s52/s48/s54/s48\n/s45/s50 /s45/s49 /s48 /s49/s48/s50/s48/s52/s48/s54/s48\n/s45/s49 /s48 /s49/s45/s50 /s45/s49 /s48 /s49 /s50/s32/s32/s69/s105/s103/s101/s110/s109/s111/s100/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s105/s101/s115/s32/s40/s71/s72/s122/s41/s78/s111/s32/s99/s111/s117/s112/s108/s105/s110/s103/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s97/s110/s105/s115/s111/s116/s114/s111/s112/s105/s101/s115\n/s70/s101/s67/s111/s66/s58/s32 /s51/s50/s49/s32/s38/s32 /s67/s111/s47/s78/s105/s58/s49/s32/s50/s56/s55/s32/s109/s84\n/s32/s32\n/s74/s61/s48/s46/s57/s32/s109/s74/s47/s109/s50/s32/s32\n/s74/s61/s48/s46/s53/s32/s109/s74/s47/s109/s50/s45/s54/s32/s38/s32 /s49/s48/s49/s56/s32/s109/s84\n/s32/s32/s32/s32\n/s79/s117/s116/s32/s111/s102/s32/s112/s108/s97/s110/s101/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s69/s97/s115/s121 /s32/s97/s120/s105/s115/s32/s73/s110/s32/s112/s108/s97/s110/s101/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s72/s97/s114/s100/s32/s97/s120/s105/s115/s45/s50/s51/s32/s38/s32 /s52/s52/s48/s32/s109/s84/s32/s76/s111/s111/s112/s115 /s69/s105/s103/s101/s110/s109/s111/s100/s101/s115\nFIG. 1. (Color online). Eigenmode frequencies and hard axis loops for 3 hybrid free layers that share the same calculated eigenexcitation\nfrequencies at remanence. The anisotropies and the interlayer couplings are chosen to get the eigenmode frequencies of 9 GHz (blue dot)\nand 36 GHz (red dot) at remanence like in the case of Mo(3 ˚A) spacer. Left column: eigenmode frequencies versus out-of-plane fields for an\nincreasing (black) or decreasing (green) field sweep. In practice the layers’ magnetizations switch near zero field and thus visit mainly the\ncolored branches that correspond to the most stable states. Middle column: idem versus in-plane fields. Right column: normalized hard axis\nloops (i.e.MxversusHx) of the FeCoB layer (black lines) and the [Co/Ni] multilayer (red lines).\nlayer and the [Co/Ni] multilayer, respectively. For the thinner\nspacers, the two parts of the free layer are sufficiently coupled\nto switch as a single block during easy axis loops, while the\nSAF reference system switches in two steps at much larger\npositive and negative fields [Fig. 2(a)]. Hard axis loops (not\nshown) comprise also the signals of the SAF of the MTJ such\nthat the informations are too intertwined for a separate identi-\nfication of the contribution of each magnetic sub-block.\nTo see to what extent the two parts of the free layer ful-\nfill their roles, we measured the system’s eigenexcitations, in\naddition its ground state determined by magnetometry. We\nillustrate our method in Fig. 1. We are concerned by hybrid\nfree layers with PMA and full remanence, i.e. in which the\nmagnetizations of the two layers at remanence are collinear to\nthe out-of-plane axis (z). In this case the acoustical and opti-\ncal eigenmode frequencies of the free layer under out of plane\napplied fields17arefexp\n1;2=\r0\n2\u0019(Hsoft\n1;2\u0006Hz)where the fields\nadd for magnetizations up [i.e. along +(z)] and subtract for\nmagnetizations down. The softening fields read17:\nHsoft\n1;2=He\u000b\nk1+He\u000b\nk2\n2+J\n2Ms2t2+J\n2Ms1t1\u0006p\n\u0001\n2Ms1Ms2t1t2(1)\nwhere \u0001 = 2 JMs1Ms2t1t2(He\u000b\nk2\u0000He\u000b\nk1)(Ms1t1\u0000\nMs2t2) +Ms12Ms22t12t22(He\u000b\nk2\u0000He\u000b\nk1)2+J2(Ms1t1+Ms2t2)2. We have used the notation He\u000b\nk1;2for the effective\nanisotropy fields of FeCoB (layer 1) and Co/Ni (layer 2), and\nJfor their interlayer exchange coupling.\nWhen later conducting FMR versus Hzexperiments on the\nfree layer (Fig. 2), we will confirm the finding of two V-shaped\nlines, with the apex of the V at fexp\n1;2and slopes\u0006\r0\n2\u0019. Un-\nfortunately, the two data ( fexp\n1andfexp\n2) are insuffisient to\ndetermine the three unknown parameters fJ; He\u000b\nk1; He\u000b\nk2g. We\nhave illustrated this problem in Fig. 1 (left panels) by choos-\ning three different triplets fJ; He\u000b\nk1; He\u000b\nk2gthat yield the same\ncalculated eigenexcitation frequencies in the Hzfield config-\nuration (see the red and blue lines common to all left panels).\nTo get an overdetermined problem, we performed in addi-\ntion FMR vs in-plane field. Indeed as the two parts of the free\nlayer have distinct anisotropies and thickness, their magneti-\nzations tilt at different angles and the dynamics gets very sen-\nsitive to the exchange coupling J. The fitting procedure can\nbe summarized this way: we first apply Eq. 1 on the two re-\nmanent frequencies fexp\n1;2to get all the triplets fJ; He\u000b\nk1; He\u000b\nk2g\ncompatible with the easy axis field data. To determine which\ntriplet is the correct one, we numerically calculate the mag-\nnetic configuration and eigenexcitations in in-plane applied\nfield for all possible triplets, (Fig. 1, right panels) and select\nthe triplet that best match with the experiments.3\nTABLE I. Properties of the free layer subsystems. The magnetizations were fixed at Ms1= 1:21\u0002106A=m,Ms2= 0:763\u0002106A=m. The\nysymbol emphasizes the number that come from high field extrapolation when the dispersion curve departs from linear behavior at low fields.\nThe linewidths are calculated from out-of-plane measurements of the lowest and highest frequency eigenmodes. In the absence of coupling,\nthey would be indicative of the damping of the FeCoB and Co/Ni systems, respectively.\nfexp\n1fexp\n2 Fe60Co20B20[Co3 ˚A/Ni6 ˚A]\u00024Coupling1\n2@\u0001f\n@f1\n2@\u0001f\n@fTMR RA\n(GHz) (GHz)\u00160He\u000b\nk1\u00160He\u000b\nk2J(mJ/m2)FeCoB Co/Ni\nSpacer \u000630 mT\u000650 mT\u00060.01\u00060:001 \n:\u0016m2\nTa 3 ˚A 1.0 24.5 -20 850 0.07 0.009 0.03\u00060:004 137% 7.0\nMo 3 ˚A8.6 35 -50 920 0.58 0.010 0.03\u00060:006 137% 8.7\nW 3 ˚A-0.7y27.6 -190 890 0.22 0.014 0.030\u00060:003 137% 8.7\nxxxxRef. \nRef. Ref. layersFree. layer\nFIG. 2. (Color online). Properties of the MTJ with Mo 3 ˚A spacer. (a)\nEasy axis loop of the MTJ. (b) Permeability in the ffrequency-fieldg\nplanes for (b) out-of-plane field and (c) in-plane applied fields. The\ncontrast has been strengthened in the green box. The violet arrows\npoint at an eigenmode of the MTJ’s reference layer. (d) Eigenmode\nfrequencies along a hard axis loops (symbols) and modeling thereof\n(lines) with the parameters of Table 1.\nIn the absence of coupling [Fig. 1, top middle panel], the\nhard-axis field eigenmode lines are W-shaped with two dis-\ntinct softening fields at Hx=\u0006He\u000b\nk1for the FeCoB layer and\natHx=\u0006He\u000b\nk2for the Co/Ni multilayer. For finite coupling\nJ, an anticrossing appears in the dispersion curves and there\nremains only one softening field per applied field sign. The\nlayers still tilt at a different pace with the field. For large cou-\npling (bottom panels, J= 0:9 mJ=m2), the layers essentially\ntilt together and behave like a single unit.\nIn practice the shape of the in-plane field FMR curves is a\nway to select the set of material parameters that best describe\na sample. Besides, the linewidth of each mode in the out-of-plane field configuration can be used to estimate the damping\nof the two parts of the free layer. Indeed we will see that the\ncoupling is small enough so that the lowest mode linewidth re-\nflects the value of the FeCoB damping, while the linewidth of\nthe highest frequency mode reflects the damping of the Co/Ni\npart of the free layer (for a detailed justification see for in-\nstance Figs. 6 and 7 in ref. 17 and the related analysis).\nThe free layer eigenmodes were determined experimen-\ntally using Vector Network Analyzer FerroMagnetic Reso-\nnance (VNA-FMR18). We applied fields either in the plane of\nthe sample or perpendicularly to it. Thanks to the very differ-\nent anisotropies of the magnetic subsystems within the MTJ,\nthe eigenmodes have well separated frequencies [Fig. 2(b) and\n(c)]. There is one eigenmode undergoing sudden frequency\njumps at exactly the two switching fields of the SAF reference\nlayers of the MTJ, but not undergoing any change at the free\nlayer coercivity. The frequency of this mode is independent\nof the free layer inner spacer (i.e. Mo, W or Ta). This mode\ncan thus be assigned to the other part of the MTJ (i.e. to the\nSAF reference system), and we shall thus not consider here-\nafter. Following the methods described earlier (Fig. 1), a fit of\nthe other modes to coupled macrospins was used to determine\neach layer’s properties with conclusions gathered in Table I.\nfexp\n1describes the strength of the effective fields that hold\nthe softest part of the free layer. It is thus a relevant indica-\ntion of its non volatility. The hybrid free layer with Mo spacer\nclearly outperforms substantially the other spacers in term of\nnon volatility. Mo also ensures a low damping of the FeCoB\nlayer at no expense of the overall anisotropy and of the trans-\nport properties. The main influence of the chemical nature of\nthe spacer is the coupling strength (Table 1). The magneto-\ntransport properties are insensitive to the spacer layer. The\ncouplings through the Ta spacer is low, and our experience\nis that it is weaker in our present Ni-terminated multilayers\nthan when Co-terminated19,20. Comparatively to Ta, W could\nbe considered as better owing to its larger interlayer exchange\ncoupling. However, the W spacer has a clear detrimental ef-\nfect on both the damping and the anisotropy of the FeCoB\nlayer, which is consistent with other studies21.\nIn summary we have studied hybrid perpendicular\nanisotropy free layers that couple a soft FeCoB layer with a\nvery anisotropic [Co/Ni] multilayer through various spacers\nmade of refractory metals. The formerly used Ta(3 ˚A) spacer\nis outperformed by W(3 ˚A) and even more by Mo(3 ˚A) spac-\ners, which combine the strongest interlayer exchange coupling\n(0.58 mJ/m2) without sacrificing the anisotropies, the damp-4\ning factors and the magneto-transport properties within the\nstack.\n1S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo,\nS. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nature Materials 9,\n721 (2010).\n2H. Sato, M. Yamanouchi, S. Ikeda, S. Fukami, F. Matsukura, and H. Ohno,\nApplied Physics Letters 101, 022414 (2012).\n3H. Sato, E. C. I. Enobio, M. Yamanouchi, S. Ikeda, S. Fukami, S. Kanai,\nF. Matsukura, and H. Ohno, Applied Physics Letters 105, 062403 (2014).\n4J.-H. Kim, J.-B. Lee, G.-G. An, S.-M. Yang, W.-S. Chung, H.-S. Park, and\nJ.-P. Hong, Scientific Reports 5, 16903 (2015).\n5W. Chen, J.-M. L. Beaujour, G. d. Loubens, A. D. Kent, and J. Z. Sun,\nApplied Physics Letters 92, 012507 (2008).\n6H.-S. Song, K.-D. Lee, J.-W. Sohn, S.-H. Yang, S. S. P. Parkin, C.-Y . You,\nand S.-C. Shin, Applied Physics Letters 103, 022406 (2013).\n7H.-S. Song, K.-D. Lee, J.-W. Sohn, S.-H. Yang, S. S. P. Parkin, C.-Y . You,\nand S.-C. Shin, Applied Physics Letters 102, 102401 (2013).\n8M. Haertinger, C. H. Back, S.-H. Yang, S. S. P. Parkin, and G. Woltersdorf,\nJournal of Physics D: Applied Physics 46, 175001 (2013).\n9L. Cuchet, B. Rodmacq, S. Auffret, R. C. Sousa, and B. Dieny, Applied\nPhysics Letters 105, 052408 (2014).\n10T. Liu, Y . Zhang, J. W. Cai, and H. Y . Pan, Scientific Reports 4, 5895\n(2014).11Y . Liu, T. Yu, Z. Zhu, H. Zhong, K. M. Khamis, and K. Zhu, Journal of\nMagnetism and Magnetic Materials 410, 123 (2016).\n12R. W. Powers and M. V . Doyle, Journal of Applied Physics 30, 514 (1959).\n13S. Takemoto, H. Nitta, Y . Iijima, and Y . Yamazaki, Philosophical Magazine\n87, 1619 (2007).\n14H. Nitta, T. Yamamoto, R. Kanno, K. Takasawa, T. Iida, Y . Yamazaki,\nS. Ogu, and Y . Iijima, Acta Materialia 50, 4117 (2002).\n15E. Liu, J. Swerts, S. Couet, S. Mertens, Y . Tomczak, T. Lin, V . Spamp-\ninato, A. Franquet, S. V . Elshocht, G. Kar, A. Furnemont, and J. D. Boeck,\nApplied Physics Letters 108, 132405 (2016).\n16M. Gajek, J. J. Nowak, J. Z. Sun, P. L. Trouilloud, E. J. OSullivan, D. W.\nAbraham, M. C. Gaidis, G. Hu, S. Brown, Y . Zhu, R. P. Robertazzi, W. J.\nGallagher, and D. C. Worledge, Applied Physics Letters 100, 132408\n(2012).\n17T. Devolder, Journal of Applied Physics 119, 153905 (2016).\n18C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. P. Freitas,\nJournal of Applied Physics 101, 074505 (2007).\n19T. Devolder, S. Couet, J. Swerts, and A. Furnemont, Applied Physics Let-\nters108, 172409 (2016).\n20A. Le Goff, R. Soucaille, T. Tahmasebi, J. Swerts, A. Furnemont, and\nT. Devolder, Japanese Journal of Applied Physics 54, 090302 (2015).\n21R. Soucaille, M. Belmeguenai, J. Torrejon, J.-V . Kim, T. Devolder, Y . Rous-\nsign, S.-M. Chrif, A. A. Stashkevich, M. Hayashi, and J.-P. Adam,\narXiv:1604.05475 [cond-mat] (2016), arXiv: 1604.05475."    },    {        "title": "1501.02938v1.Strain_Induced_Extrinsic_High_Temperature_Ferromagnetism_in_the_Fe_Doped_Hexagonal_Barium_Titanate.pdf",        "content": "  \n1 \n Correspondence and requests for materials should be addressed to  A.Z. (andrej.zorko@ijs.si)  \n \nStrain -Induced  Extrinsic High-\nTemperature Ferromagnetism in the Fe -\nDoped Hexagonal Barium Titanate  \nA. Zorko1,*, M. Pregelj1, M. Gomilšek1, Z. Jagličić2,3, D. Pajić4, M. Telling5,6, I. Arčon1,7, I. \nMikulska7, and M. Valant7 \n1Jožef Stefan Institute, Jamova c. 39, SI -1000 Ljubljana, Slovenia , 2Institute of Mathematics, \nPhysics and Mechanics, Jadranska c. 19, 1000 Ljubljana,  Slovenia , 3Faculty of Civil and Geodetic \nEngineering, University of Ljubljana, Jamova c. 2, SI -1000 Ljubljana, Slovenia , 4Department of \nPhysics, Faculty of Science,  University of Zagreb,  Bijenička  c. 32, HR-10000 Zagreb, Cr oatia, \n5ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX,  United \nKingdom , 6Department of Materials, University of Oxford, Parks Road, Oxon, UK , 7University of \nNova Gorica, Vipavska 13, SI -5000 Nova Gorica, Slovenia . \n \nDiluted magnetic semiconductors possessing intrinsic static magnetism  at high \ntemperatures  represent a promising class of multifunctional materials with high \napplication potential in spintronics  and magneto -optics . In the hexagonal Fe -doped \ndiluted magneti c oxide , 6H-BaTiO 3-, room -temperature ferromagnetism has been \npreviously reported . Ferromagnetism  is broadly  accepted as  an intrinsic property  of \nthis material , despite  its unusual  dependence on doping concentration and \nprocessing conditions . However, t he here reported combination of bulk \nmagnetization and complementary in-depth local -probe electron spin resonance and \nmuon spin relaxation measurements , challenges  this conjecture . While a \nferromagnetic transition occurs around 700 K, it does so only in ad ditionally \nannealed samples  and is accompanied by an extremely small average value  of the \nordered magnetic moment . Furthermore, several additional magnetic instabilities \nare detected at lower temperatures. These coincide with electronic instabilities of \nthe Fe-doped 3C-BaTiO 3- pseudocubic p olymorph . Moreover , the distribution of iron \ndopants  with frozen magnetic moments  is found to be non-uniform.  Our results \ndemonstrate that the intricate  static magnetism of the hexagonal phase  is not   \n2 \n intrinsic, but rather  stems from  sparse  strain -induced pseudocubic regions . We \npoint out the vital role of internal strain in establishing defect ferromagnetism in \nsystems with competing structural phases.   \nThe search  for dilute magnetic oxides (DMOs) is at the forefront  of spintronic s and magneto -\noptics research and application1–6. The great interest has been triggered by the possibility of \ncombining diverse functionalities of semiconductor  electronic s and magneti sm in a single \nmaterial , which  would boost its application  potential . Existence of magn etoelectric coupling in \nwide-band-gap semiconducting oxides at  room temperature would be of great technological \nimportance , given that the magnetic ordering is indisputably  intrinsic. Indeed, a  number of \nstudies have claimed  such intrinsic ordering has been observed in non-magnetic oxide \nsemiconductors lightly doped with paramagnetic transition -metal ions. These findings are, \nhowever, a subject of controversy7–9. Very often the  corresponding studies  are focused  only \non routine  characterization and modelling  of the material's properties  rather than  critical \nanalysis of the origin of the observed static magnetism . Recently, more detailed experiments  \nhave proven  wrong  a significant part  of the claims  about  intrinsic magnetism in D MOs10–14. \nMoreover, a ggregation of magnetic ions that leads to  chemical phase separation on a nano -\nscale, also known as spinodal decomposition into regions with high/low concentration of \ndopants and a crystal structure imposed  by the semiconducting host, has been  lately  \nwitnessed  in various cases2,3,15-17. Thus, the need for careful analysi s of the magnetic \nproperties  of these systems at the microscopic level  and their critical assessment are of \nparamount importance .  \n \nSince the discovery18 of room -temperature ferromagnetism in the bulk Fe-doped hexagonal19 \n6H-BaTiO 3- (Figure 1)  that can be stabilized from the pseudocubic  (3C) perovskite struct ure \n(Figure 1)  by transition -metal doping20, this system has  been in the focus  of investigation s. \nAlthough simultaneous mag netic and polar orders (multiferroicity), highly desired  for device \napplications , cannot be achieved in this ma terial (1% Fe doping destroys ferroelectric \norder18), the magnetoelectric coupling has recently  been demonstrated21. Moreover, \narguments speaking in  favour  of its intrinsic ferromagnetism would  make Fe -doped  6H-  \n3 \n BaTiO 3- rather special in the  field of ferromagnetic  (FM) DMOs.  In contrast to many other \nmaterials  where dopant -ion segregation and formation of precipitates a re regularly \nencountered, the very high solubility of 6 H-BaTiO 3 for Fe (and other transition metals ), the \npossibility of growing single -crystal DMOs, and the apparent absence of any spurious \nprecipitates all seem  as strong evidences for its intrinsic FM  character18,22,23. \n \nFigure 1 : Schematic interface of the two crystallographic phases in BaTiO 3-. \nPseudocubic (3 C) and hexagonal (6 H) crystallographic polymorph of BaTiO 3-Fe3+ cations \nsubstitute for Ti4+ ions (spheres) within FeO 6 ochatedra (polyhe dra). Two crystallographically \ndifferent  Ti sites are found in the 6 H structure, while only one exists in the 3 C structure.  High-\ntemperature ferromagnetism  (FM) is ascribed to sparse regions of the 3 C phase, while the \nmajority 6 H phase remains paramagnetic  (PM). \nThe FM  state of the Fe -doped 6 H-BaTiO 3- is, however, highly unusual, if not even \ncontroversial . While t he amount of the room -temperature FM signal significantly decreases \nwith increasing dopant concentration24, the average value of the static magnetic moment per \nFe3+ ion never exceeds a few hundredth s of a Bohr magneton ( B); compared to the full \nmoment of 5 B. Different hypotheses  about such a drastic moment reduction have been put \nforward in the literature . These include  competition between s uperexchange interactions \nleading to intrinsic magnetic inhomogeneity in the system18 or, alternatively, the formation  of \ndispersed Fe3+ clusters around oxygen vacancies24. Furthermore, the size of the average FM \nmoment substantially  depend s on various factors, including temperature, atmosphere21,25, and \nduration  of the synthesis26. Also  the inevitable  oxygen vacancies in the Fe-doped 6 H-BaTiO 3- \n  \n4 \n material, which compensate for  the charge mismatch between Fe3+ and Ti4+, are \nsuggested21,23 to play an intrinsic role in establishing ferromagnetism  through a dynamical \nexchange of  trapped e lectrons among bound polarons27. However,  the corresponding reports  \nare contradictory , since the FM response can apparently be enhanced  in either reduced \nsamples22 or in samples annealed in oxygen  atmosphere25. Moreover, it has recently been \nsuggested  that the FM behaviour  of the Fe -doped 6 H-BaTiO 3- should rather be related to \nthermally activated di ffusion , leading to  a cation order -disorder transition , and that  the \nsegregation of oxygen vacancies plays no major role in  ferromagnetism28. The result ing \nclustering of  Fe3+ ions at  the two neighbouring  pentahedral Ti(2) sites (Figure 1) and the \nsegregation of oxygen vacancies at the connecting oxygen  site29, lead to the formation of  \nlattice -matched Fe 2O8 dimers . These have been predicted by ab -initio calculations18, and are \nregularly observed in experiment s28,30,31, further revealing  that a major portion of the Fe \ndopants are involved in their formation28,31. This fact thus evidently  questions the  cation \nordering  as the  intrinsic origin  of ferromagnetism in 6 H-BaTiO 3-, given the extremely small \nsize of the average value of the magnetic  moment per Fe dopant.  \n \nThe FM behaviour  of the Fe-doped  6H-BaTiO 3- has not yet been explored above room \ntemperature  or examined by local -probe magnetic spectroscopy techniques , which could \npotentially help in identifying any spurious effects. Therefore, we performed a comprehensive \nmagnetic investigation, combin ing both bulk magnetic measurements as well as the \ncomplementary local -probe measurements of electron spin resonance (ESR) and muon spin \nrelaxation ( SR), in the broad temperature range between 2 and 850 K. Here, we report a \nplethora  of magnetic anomalies  that appear in the Fe -doped 6 H-BaTiO 3- below  ~700 K. These \ncoincide with the electronic instabilities of the Fe -doped pseudocubic 3 C-BaTiO 3 polymorph. \nWe provide firm experimental evidence that the  previously reported  static magnetism is not \nan intrinsic property of the doped 6 H-BaTiO 3 hexagonal phase, but is instead  related  to sparse \nregions of strain -induced pseudocubic phase , which arise  as a consequence of locally \ncompeting structural phases . \nResults    \n5 \n In order to pr ovide clear evidence for either intrinsic  or extrinsic origin of magnetism in  the \nFe-doped 6 H-BaTiO 3-, an in-depth experimental study , which  combin es bulk and local -probe \nmagnetic characterization techniques, is necessary. All our measurements were performed on \nhigh-quality polycrystalline samples  (see Methods ); some  of which were additionally \nannealed . This  annealing  prove s crucial for the existence  of FM behaviour . In what follows, we \nlabel the non -annealed and annealed samples with F cBTO and F cBTOa, respectively . Here c \ndenotes the doping percent age of Fe3+ ions at the Ti sites. \n \nDete ction  of magnetic instabilities by b ulk magnetic measurements  \nBulk magnetization  (M) measurements display a clear difference between the non-annealed \nand annealed samples (Figure 2a).  While t he latter exhibit complicated zero -field cooled \n(ZFC)/field -cooled (FC) split  M(T) curves , a paramagnetic (PM) phase is found in the former . \nThe PM phase  is characterized by  a monotonic  temperature dependence  and absence of the  \nZFC/FC  bifurcation . The corresponding  molar  susceptibility (per mole Fe) for the non -annealed \nsamples,  \nmol MH  (H denotes the applied  magnetic field ), agreeswell with the Curie -Weiss \nmodel  \n   2\nmol A B CW 3 N k T . Here NA and kB denote the  Avogadro and the Boltzman \nconstant,  respectively,  CW is the Weiss temperat ure and   = 4.6(1) B is the average magnetic \nmoment per Fe dopant.  The derived magnetic moment is common to both F10BTO and \nF20BTO samples and is somewhat reduced from th e value of  \n BB 1 5.92 g S S  expected \nfor Fe3+ (S = 5/2) moments. However, the Fe3+ valence was clearly revealed by X -ray \nabsorption near -edge structure (XANES) spectroscopy28,29 as being by far dominant in 6 H-\nBaTiO 3-. For F10BTO and F20BTO samples we  find antiferromagnetic  (AFM)  Weiss \ntemperature of -3(1) K and -7(2) K, respectively. This parameter  scales  linearly with the \ndoping concentration  and provides the energy scale of the average excha nge coupling  J \nbetween  the iron dopants . We find J to be rather  small  compared to room temperature , where \nthe FM state has been reported . Secondly, it is AFM and thus questions the intrinsic origin  of \nthe FM order.  \n   \n6 \n At high temperatures  (T > Tc1 ~ 700 K)  the M(T) curves of the anneal ed samples match those \nof the non -annealed ones, proving the same PM state. This  behaviour , however,  drastically \nchanges  below Tc1, especially in the low applied field  of 10 mT  (Figure 2a). On decreasing \ntemperature a ZFC/FC splitting  is observed  in F20BTOa at Tc1, with the FM component  being  \nenhanced  below Tc2 ~ 570 K, and the ZFC/FC di fference increased below Tc3 ~ 450 K . Finally, \nan additional  magnetic anomaly , reflected in a local magnetization maximum , is found  at Tc4 \n~ 110 K. The M(H) magnetization curves  of the same sample  (Figure 2b) clearly display  the \nsame  high-temperature magnetic  transitions , as a hysteresi s appears below Tc1 and is \nenhanced below Tc2. The deduced average value of the FM ordered  magnetic moment (FM = \n1.5∙10-2B per Fe at 300 K; see Methods ) is dramatically reduced from  the expected  full 5B \nFe3+ value . Moreover, the M(H) curves change linearly with H at larger  fields; a response \nbeing typical for  PM and not FM systems . Therefore, below Tc1 one can think of the system as \nbeing predominantly PM  with an additional small FM component. Such a simple picture is \nconfirmed by M(T) data in higher magnetic fields, where the high -temperature magnetic \nanomalies, although appearing at the same critical temperatures, become drastically \nsuppressed  (Figure 2a)  due to saturation of the FM component . Similarly, as regularly \nreported before18,23, the PM component that shows a Curie -Weiss temperature dependence \ndominates the magnetic response at low -temperature s (see the inset  in Figure 2b),  as the \nsplitting of the FM hysteresis is only moderately enhanced at 2 K compared to room \ntemperature (Figure 2b).  The PM component is far from being saturated at low temperatures \neven at 5 T , similarly as found before18,22,23. The average value of the PM magnetic moment \nper Fe3+ ion can be modelled32 with a modified Brillouin function for spin 5/2 , \n5/2 B effB M f B k T  \n, by renormalizing the moment  (\nB 52g ; g denotes the Landé g-\nfactor)  with a factor f and introducing an  effective temperature  \neff 0T T T . The fit to the \nFBTO20a data yields the empirical temperature  T0 = 2.1 K (inset in Figure 2b) , confirming  \nthat AFM interactions are pertinent to this phase32. As the actual concentration of iron dopants \nin all our samples is very close to the nominal one (see Methods), t he obtained \nrenormalization factor f = 0.25 reveals that the majority of spins do not contribute to the \nmagnetic signal at low temperatures. Such renor malization factor  that has been found to   \n7 \n decrease steadily with the doping concentration18 can then be accounted for by formation of  \nnon-magnetic  AFM entangled  iron pairs at low temperatures , which  are correlated with the \nformation of  the Fe2O8 structural dimers28,31. The bulk-magnetization results thus  imply the \npresence of FM and PM magnetic phases in the annealed 6H -BaTiO3 - samples  below Tc1 and \ncorroborate impurity -based ferromagnetism, as further verified below by local -probe  magnetic \ncharacterization s. \n \nFigure 2: Bulk magnetization measurements.  (a) The temperature dependence of zero -\nfield cooled (full symbols) and field -cooled (open symbols) field -normalized bulk magnetization \nin the 20% Fe -doped 6 H-BaTiO 3-. The solid line is the fit to the Curie -Weiss model  \n2\n0 A B CW 3 M H N k T  \n (see text  for details ), while the vertical dashed lines display the \ntemperatures of the magnetic transitions in the annealed sample. (b) Magnetization curves of \nF20BTOa at different temperatures  displaying the FM hysteresis . Inset shows the 2 K data \n(circles) and the fit with the modified Brillouin function  \n5/2 B effB M f B k T    (see text for \ndetails) . (c) The time -dependent magnetization in the F20BTOa sample at 300 K after applying \nand rem oving the magnetic field of 1 T . \nPeculiarities of FM resonance modes  \nFurther insight to the unusual  magnetism of the Fe -doped BaTiO 3- samples on a local scale is \nprovided by ESR.  Again, dra stically di fferent ESR spectra are found for the non -annealed and \nannealed samples . The former are featureless  single -line ESR spectra , while the latter possess \nseveral lines that significantly evolve with temperature  below Tc3 (Figure 3a). We note that \n0 200 400 600 8000.00.10.20.30.40.50.6\n-0.10 -0.05 0.00 0.05 0.10-0.4-0.20.00.20.4\n-1000100-303\n0 10 20 30 400.010.111\nTc1Tc2Tc3Tc4\n  M/0H (Am2 T -1 per mol Fe)\nT (K)a\n   F20BTO\n  10 mT   F20BTOa\n 10 mT\n 0.34 T,\n,F20BTOa\n2 K400 K\n850 K800 K700 K600 K500 Kb\n M (10-2B per Fe)\nBH (T)data\n fit \n \n  \n2 KF20BTOac\n0 T0 T1 T M (10-2B per Fe)\nt (min)  \n8 \n the ESR spectra of the PM non-annealed samples  are different from the fine-structured ESR \nspect ra reported in previous studies22,33 –35. However, it should be stressed  that all previous  \nstudies were limited to doping concentrations below 5%. The disappearance of the fine \nstructure in our non -annealed samples suggests  enhanced magnetic interactions  (dipolar \nand/or exchange)  between  iron dopants .  \n \nFigure 3: ESR characterization of 6 H-BaTiO 3-. (a) The temperature dependence of the ESR \nspectra in 20% Fe -doped 6 H-BaTiO 3- displaying a  single line in the non -annealed sample and \na multi -feature line in the annealed sample . Spectra are displaced vertically for clarity.  The g = \n2 position is marked with the vertical line.  (b) The increase of the  ESR line width  with doping \nconcentration  in the non -annealed samples  at 300 K . The peak -to-peak line width  B of both \nspectr a is indicated by the horizontal arrow s. (c) Comparison of the 300 -K spectra of F20BTOa \nat 9.4 GHz and at 24.3 GHz . Side -band shifts that are frequency independent  are marked with \nvertical lines. ( d) Resonance field of various resonance modes observed in the annealed Fe -\ndoped 6 H-BaTiO 3- samples  below Tc3. The temperature independent position (horizontal mode) \ncorresponds to the g = 2 mode.  (e) Comparison of the ESR  intensity decrease of the single -line \n0 200 400 600-6061218243036\n-200 0 200 400-0.50.00.5\n0 200 400 600-0.9-0.6-0.30.00.30.6\n0 200 400 6000.00.20.40.6\n450 500 550 6000.00.20.40.60.81.0g = 2600 K\n500 K (x 1/2)\n400 K\n300 K\n200 K\n100 K\n5 K (x 1/2)\nF20BTOa\nF20BTOa\n ESR signal (arb. units)\n0H (mT)c\n0(H - H0) (mT)   F20BTOa\n9.4 GHz\n24.3 GHzF10BTO\nF20BTO\n0H (mT) \n b\nB\ng = 2e\nT (K)T 20%\nc3T 10%\nc3Tc4d\n  Bres (T)F10BTOa\nF20BTOa\n F20BTOa\n F20BTOESR int. (arb. units)\nT (K) \nTc3Tc2  \n9 \n FM mode observed above Tc3 in F20BTOa and the PM signal of the F20BTO sample with Curie -\nlike dependence. Both ESR intensities are normalized at 450 K.  \n \nIn order to support the claim of sizable magnetic interactions we  analyse the line width of the \nPM ESR spectra in the non -annealed samples. The peak -to-peak line width B (Figure 3b) \nscales proportionally with the doping level (\n10%45 2  mT B ,\n20%91 2  mT B ). Such a \nlinear increase is in accordance with the linearly increasing magnetic anisotropy and exchange \ninteractions (see Methods), as deduced from the Weiss temperatures of both non -annealed \nsamples. On the contrary, the dipolar interactions in the absence of isotropic exchange \ncouplin g36 would yield much smaller line widths (see Methods). ESR thus verifies that the \naverage exchange interaction J among dopants is not negligible. Furthermore, the line width, \nbeing proportional to c, is a fingerprint of a uniform PM phase. Aggregation of magnetic ions \ninto nano -scale clusters, on the other hand, would result in an inverse dependence of the ESR \nline width on c, as was recently observed17 in Mn -doped SrTiO 3. \n \nIn sharp contrast to the simple ESR line shape of the PM non -annealed samples, the ESR \nspectra of the annealed samples are much more complex below Tc3. Here, they exhibit several \nwell-displaced lines  (Figure 3c)  that shift significantly with temperature (F igure 3d). The field \nshift of the side bands from the g = 2 position  is the same at 9.4 GHz and at 24.3 GHz  (Figure \n3c), which is indicative of static internal magnetic fields characteristic of the FM phase. Very \nimportantly, above Tc3 a single -line spectrum is observed at  \n2g  (Figure 3a) in all the \nannealed samples despite the fact that the FM state persists to much higher temperature, Tc1 \n~ 700 K. With its peak-to-peak line width \n9 1  mTB , this spectrum is markedly narrower  \nthan the PM spectrum of the non -annealed samples . Moreover,  its intensity strongly \ndecreases between Tc3 and Tc2 (Figure 3e). The latter behaviour is in disagreement with the \nCurie-like decrease, characteristic of the PM non -annealed samples  (Figure 3e) , and is rather \nreminiscent of the FM magnetization decrease in the same temperature range (Figure 2a). \nTherefore, this \n2g  mode is still a FM mode. We stress that its single -component line shape   \n10 \n reveals that the local magnetic anisotropy, which dictates the shape of the ESR spectrum in \nthe FM state37 and depends on local symmetry, drastically ch anges at Tc3. \n \nClustering of FM moments as evidenced by SR \nThe ESR measurements  provide  convincing evidence of FM moments in the annealed samples , \nas well as  their absence in the non -annealed samples . In addition , they reveal  a novel and \nimportant (vide infra) discovery about  the local -symmetry reduction below  Tc3. However,  ESR \ndoes not yield any information about  the spatial distribution of the FM moments. T o answer \nthis question  we turn to the complementary SR method  (see Methods  for introduction to th is \nmethod ). This local-probe  technique  is renown ed in magnetic spectroscopy for its ability to \ndetect even the smallest local magnetic fields at  the muon stopping site and for the ease in \nwhich the experimentalist can discriminate  between static and dynamical internal fields38. We \nfind that i n zero magnetic field (ZF) , the time -dependent muon spin polarization P(t) is \nmonotonic at all temperatures in the experimental time window, independe nt of the PM/FM \ncharacter of a particular sample ( Figure 4 a and 4b). It can be modelled with the stretched -\nexponential  (SE) time dependence  \n \nSE etPt  , (1) \nwell suited for  dynamical local fields in  diluted spin systems39. Here  denotes the muon \nrelaxation rate and  the stretch exponent.  For each sample the relaxation rate is found to \nincrease notably with increasing temperature  (P(t) curves in Figure 4 a and 4b decrease faster  \nat higher temperature ). This increase of the relaxation rate is very similar in the annealed and \nnon-annealed sample s for a given concentration of dopants . Therefore, the muon relaxation in \nthe Fe -doped BaTiO 3- system is attributed to the fluctuating paramagnetic moments . \n \nDirect confirmation  of this statement is provided by comparing the ZF data and the data \nobtained in the weak longitudinal field (LF) of 1 mT. Such field comparison shows a very \nsimilar relaxation response (Figure 4c). We note that in addition to the dynamical model of \nequati on (1), the Lorentzian Kubo -Toyabe (KT) model38   \n11 \n  \n\n   KT121e33tP T t , (2)  \nalso provides a satisfactory fit to the ZF data. The KT model is applicable to disordered static \nlocal fields with the distribution width = 7.0∙10-2 mT = 2∙135.5 MHz/T is the muon \ngyromagnetic ratio  However, in this static case the 1 mT longitudinal field would almost \ncompletely remove the relaxation of P(t), which clearly contradicts the experiment (Figure \n4c). The muon relaxation due to local fields fluctuating  with the exchange frequency \n\neB ~1 THz kJ\n on the other hand, is not affected by such a small LF field38, because\ne 1 mT=0.85 MHz\n. \n \n \nFigure 4: SR characterization of 6 H-BaTiO 3-. The temperature dependence of muon \npolarization in (a) 10% and (b) 20% Fe -doped 6 H-BaTiO 3- samples in zero applied field  \nshowing increased relaxation rates with increasing temperature . The upper panels correspond \nto non -annealed samples and the lower panels to annealed samples. The solid lines are fits to \nthe stretched expon ential (SE) model of e quation (1 ). (c) The room -temperature muon \npolarization of the F20BTOa sample in ZF and in  a weak longitudinal applied field. The dashed \nlines correspond to the static Kubo -Toyabe (KT) model of e quation (2 ), while the solid lines \nare fits to the model encompassing both static KT and dynamical SE relaxation (see text  for \ndetails ). The error bars of muon polarization data are defined as a square root of the total \nnumber of detected positrons.  \n \n0.10.40.71.0\n0 2 4 6 810 120.20.40.60.81.0\n0 2 4 6 810 120.10.40.71.00.40.71.0\n0 2 4 6 810 120.40.71.0323 K\n423 K\n523 K623 K\n723 K\n823 Ka Muon polarizationF10BTOc\nF20BTOa\nt (s)0 mT\n data\n KT\n KT * SE\n1 mT\ndata\n KT\n KT * SE \n293 K\nF10BTOa 323 K\n423 K\n523 K623 K\n723 K\n823 K \nt (s)323 K\n423 K\n523 K623 K\n723 K\n823 Kb \nF20BTO\nF20BTOa 323 K\n423 K\n523 K623 K\n723 K\n823 K \nt (s)  \n12 \n A close r look at the ZF and LF data of the F20BTOa sample at room temperature, however, \nreveals a somewhat  decreased muon relaxation  in the LF experiment  (Figure 4 c). This can be \nattributed to the presence of some weaker  static relaxation in addition to the dominant \ndynamical one. A simultaneous fit of both datasets to the corresponding model  \n SE KT P t P t P t\n yields the static -field distribution width  of = 6.1 (3)∙10-3 mT. This value  \nrepresents  the upper bound  of the frozen -FM-moments ’ contribution  in the annealed samples , \nsince the static -field distribution may also partially arise from nuclear magnetic moments . \nHowever,  for homogeneously distributed FM moments the calculated field -distribution widths  \nat both oxygen sites , in the vicinity of which the muons are highly likely to reside  in oxides38, \nare almost an order of magnitude larger , = 3.4 (2)∙10-2 mT (see Methods) , even for the \nvery small average  value of the  FM moments in F20BTOa (1.5∙10-3B at 300 K).  As the \ninteraction  strength  between the muon and the FM moment decreases with the cube of their \ndistance, t he much smaller experimental value  of  can be reconciled by considering \nclustering of the FM moments . This thus provides microscopic evidence that , in contrast to the \nhomogeneous distribution of PM moments evidenced by ESR , the FM moments are not \nuniformly distributed throughout the sample . \nDiscussion  \nThe small size of the average  FM moments and their non -uniform distribution in the annealed \nFe-doped  6H-BaTiO 3- samples  cast doubt on the intrinsic origin of these moments . It is \ntherefore  crucial to  critically assess th e origin of ferromagnetism in this system . It could well \nbe that the  FM behaviour  either correspond s to some kind of “impurity” phase or that the \nspinodal decomposition occurs and very small dispersed lattice -matched  FM clusters of Fe3+ \nions form during the annealing process. The latter scenario has been recently demonstrated17 \nin Mn -doped SrTiO 3. In ord er to check for the presence of such small clusters that would \nminimally deform  the local crystal structure and would be practically undetectable even by high -\nresolution imaging techniques, we performed a magnetization relaxation experiment on the FM \nF20BT Oa sample. After being exposed to the magnetic field of 1 T for 20 minutes, the \nsubsequent  remnant relaxation in zero field was measured. No magnetization relaxation   \n13 \n towards zero has been detected on the time scale  of several minutes (Figure 2c).  This implies \nan unexpectedly slow relaxation process within the Stoner -Wohlfarth model  of single -domain \nFM (nano)particles40 and thus excludes the presence of small FM clusters of Fe3+ ions. \nTherefore, the alternative scenario  of “impurity” ferromagnetism is much more plausible and \nshould be carefully considered. The Curie temperature s of the most common iron oxides41 that \nmay develop as a by-product  during the synthesis are much higher than Tc1 ~ 700 K;  ~950 K \n(Fe2O3) and ~850 K (Fe 3O4). On the other hand,  the Curie temperature of  ~680 K has been \nrecently  reported for the 3C bulk BaTi0.95Fe0.05O3- sample, where a much larger  magnetic \nmoment of 0 .75B per Fe has been  found42. The high -temperature ferromagnetism has regularly \nbeen encountered in this pseudocubic polymorph, even at much  higher Fe -doping \nconcentrations  (up to 75%), where the 3 C phase is stabilized  only by the confined geometry of \nthin films and nanoparticles43–46. We note though that the Curie temperature of an impurity \nphase is usually size dependent for nano -particles, as well as it may depend on the level of \ndoping. Therefore, the fair agreement of Tc1 with the Curie temperature of the 3C bulk \nBaTi 0.95Fe0.05O3- sample cannot be taken as a solid proof that the  FM cha racter of the Fe -doped \n6H-BaTiO 3- is due to the impurity 3 C phase . However, an unambiguous evidence that this is \nindeed the case  comes from the slightly doping  dependent Tc3 transition  temperature  (\n10%\nc3 430 K T\n,\n20%\nc3 450 K T ; see Figure 3d), which exactly coincides with the doping dependent \ncubic-to-tetragonal structural transition of the 3 C phase43. Moreover, t his structural transition \nis clearly reflect ed in the FM ESR  line-shape change from the  high-temperature single line to a \nmulti-feature  spectrum  below  Tc3, which  corroborates the  change  of the local sym metry  from \nbeing highly symmetric above Tc3 to being less symmetric below Tc3. It is also worth noting that \nin epit axially grown thin films, the 3C phase grows  at the initial stages, while later on a \ndisordered phase composed of  3C and 6 H intergrowths forms43. Since increasing thickness \ndecreases the FM response of thin -film samples47, their ferromagnetism is most naturally \nattributed to the  3C phase, for which it is also theoretically pre dicted as the ground state42. \nAll the above -presented arguments  give clear evidence that the FM behaviour  of the annealed  \nFe-doped BaTiO 3- samples is related to internal  strains  that locally destabilize the 6 H phase in   \n14 \n favour  of the 3 C phase , much as it happens globally in the confined geometries of thin films \nand nanoparticles . The SR results yielding only minor differences between the FM annealed  \nand PM non-annealed  samples , as well as  the non-uniform distrib ution of  FM moments , \ncorroborate  this scenario. Furtherm ore, ferromagnetism  arising from the sparse strain -\nstabilized  defect regions with  the 3C crystal structure  explain s both the extremely small \nmagnitude  of the  average ordered Fe moment  and its unusual decreas e with increasing  doping \nconcentration;  the increasing doping level being  known to destabilize the 3C phase35,48. Finally, \nin the non -annealed samples that were hea t-treated at lower temperature  (see Methods  for \ndetails ) the crystallites are under significantly less strain than in the densely sintered annealed \nsamples after additional annealing at higher temperature . Therefore, i n the latter  samples \nstrong  strain fields along crystallographically mismatch ed grain boundaries , or around plane \ndefect s, are likely regions for nucleation of stable pseudocubic domains with their own intrinsic \nferromagnetism.  These are not single -domain nano -sized FM defects, as m agnetization \nrelaxation is absent.  \nOur comprehensive magnetic investigation of various  Fe-doped 6 H-BaTiO 3- samples  has thus \nrevealed that the FM response  observed in the annealed samples  is not intrinsic to the \nhexagonal crystallographic phase,  as broadly  speculated before. This conclusion is based  on our \ncomplementary bulk and local -probe magnetic  investigation s, providing new microscopic insight \nand extending the experimental temperature  range  far beyond  all previous reports. Bulk \nmagnetization measureme nts have demonstrated that extremely small average static magnetic \nmoments develop in the annealed samples below  Tc1 ~ 700 K and several additional  magnetic \ninstabilities occur a s the temperature is lowered further . Below the Tc3 transition  (at aroun d 430 \nK and 450 K in 10% - and 20% -doped  samples, respectively ) a signi ficant symmetry reduction  \nof the local structure is detected by ESR. The Tc1 and Tc3 transitions coincide with the \nferromagnetic and  ferroelectric transitions of the Fe -doped 3 C-BaTiO 3 polymorph, respectively, \nwhile the microscopic characteristics  determined by  SR imply  non-uniformly distributed  FM \nregions in the samples . This demonstrates  that the  FM response of the  Fe-doped 6 H-BaTiO 3- \nsystem originates from  sparse  regions  where the  pseudocubic structural poly morph is stabilized  \nby strain fields. The strain -induced  local competition between the two structural  (and magnetic)    \n15 \n phases  is inherent only to den sely sintered annealed samples.  Such scenario of ferromagnetism \nmay turn to be important  for other DMO materials, where the competi tion between  different \nstructural phases is intrinsically present , in particular in confined geometries where  strain \neffects  are enhanced . \nMethods  \nSamples  \nHigh quality 6 H-BaTiO 3- polycrystalline samples with c = 10, 20% Fe3+ ions substituted for Ti4+ \nwere synthesized according to the procedure thoroughly explained  in Ref.  28. For each \ncomposition, after heat treatment  at 1250 °C, a part of the sample was additionally annealed \nin oxygen atmosphere at 1500 °C, typically for 5 – 10 hours.  We label the non -annealed and \nannealed samples with F cBTO and F cBTOa, respectively. The variations of the annealing time \nshowed  no significant influence on the magnetic properties28. X-ray powder diffraction  (XRD)  \nwas used to verify the single -phase hexagonal struct ure of all our samples.  The elemental \nanalysis was performed by  Energy Dispersion X -Ray Spectroscopy (EDX) on polished ceramic \nsurfaces with JSM -7100 F (Jeol) field -emission scanning ele ctron microscope equipped with an \nx-ray detector (X -Max 80, Oxford Instrument). Ten EDX characterizations  were performed on \neach sample and were statistically treated to obtain average values and standard deviations. \nThe iron concentration in the nominally  10%-doped samples was found at 10.3±0.5%, while in \n20%-doped samples it amounted to 19.9±0.6%. For all our samples t he analytically determined \ncompositions thus correspond to the nominal compositions within the small error bars.  \nBulk magnetization  \nBulk ma gnetization measurements were performed on a couple of Quantum Design SQUID \nmagnetometers and a vibrating sample magnetometer (VSM) as a function of temperature \n(between 2 and 850 K) in various magnetic fields and at various fixed temperatures as a \nfunctio n of varying magnetic field. Zero -field-cooled (ZFC) and field -cooled (FC) temperature -\ndependent data were collected. A high -temperature insert was used for measurements above \n400 K. The SQUID measurements were performed on samples sealed in quartz capilla ries, while   \n16 \n boronitride sample holders were used with VSM. The signal from the sample holders was \ncarefully evaluated. The size of the ordered magnetic moment was estimated from the \nsaturation value of the FM component in the hysteresis, by sub tracting  the linear ly increasing \nPM part.    \nESR \nElectron spin resonance was measured with home -built resonator -cavity based X -band (9.4 \nGHz) and K -band (24.3 GHz)  spectrometer s. In the X band, a continuous -flow cryostat was \nused in the temperature range 5 – 300 K , while  a preheated -nitrogen -flow heating system was \nused above room temperature up to 620 K. The samples were sealed in ESR silent quartz tubes.  \nThe ESR intensity was calibrated at room temperature with a reference sample (CuSO 4∙5H2O). \nThe ESR susceptibil ity of the non -annealed PM samples ESR = 0.06(3) Am2/ T per mole Fe was \nfound compatible  with the bulk molar susceptibility b = 0.086 Am2/T, proving that ESR  was \ndetect ing the  intrinsic signal of Fe dopants in BaTiO 3. \nThe second moment36 of the ESR line M2 arising from dipolar interactions is given by  \n \n22 2\n40\n2B 63cos 1 3144jk\nk jkM S S gr       (3) \nwhere 0 is the vacuum permeability. The sum runs over all neighbours k of a given site j, \nconnected by the vector rjk. jk is the angle between rjk and the applied magnetic field . M2 was \ncalculated by powder -averaging e quation (3) for a lattice fully occupied with Fe3+ ions on either \nthe Ti(1) or Ti(2) site. In the limit of negligible exchange interactions M2 yields the ESR  peak-\nto-peak line width  \n 2B3 B M g  \n  for the fully occupied lattice and the line width  \n  B c c B\n in the dilute limit17,36. We find \n20%\nTi(1)23 mT B  and \n20%\nTi(2)31 mT B , which is far \nbelow the experimental values . In the exchange narrowing limit  (\n\nBBk J g B ), on the other \nhand,  the ESR line width is given by  \n \n2 B B B M g k J . The second moment M2 is quadratic in the \nanisotropic interaction (i.e., in the doping level c), therefore B also scales linearly wi th c for a \nlinearly dependent J.   \n17 \n  \nSR \nIn a SR experiment almost 100% spin -polarized muons are implanted into a sample. As \nparticles possessing a magnetic moment, they interact with local magnetic fields, which drive \nthe time dependence of muon polarization P(t). At the time of the muon  decay a positron is \nemitted preferentially in the direction of  the muon ’s spin direction . The detection of positrons \ntherefore allows  for the reconstruction of P(t) and thus for the determination of a magnitude, \ndistribution, and fluctuation  rate of local internal magnetic fields .  \nMuon spin relaxation measurement were performed on the EMU instrument at the ISIS facility, \nRutherford Appleton Laboratory, UK. A flat -plate furnace was used to cover the temperature \nrange between 290 and 820 K. The background signal from a titanium sample holder was \nestimated with a hematite reference sample. It was found to typically amount to ~30%  of the \ntotal signal  and was subtracted from the SR data shown above. 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Appl. Phys.  98, 063907 (2005).    \n22 \n 46. Yang, L. et al. Magnetic properties of BaTiO 3 and BaTi 1−xMxO3 (M=Co, Fe) nanocrystals  \nby hydrothermal method. J. Magn. Magn. Mater.  350, 1–5 (2014).  \n47. Maier, R. & Cohn, J. L. Ferroelectric and ferrimagnetic iron -doped thin -film BaTiO 3: \nInfluence of iron on physical properties. J. Appl. Phys.  92, 5429 (2002).  \n48. Nguyen, H. M. et al.  Tetragonal and hexagonal polymorphs of BaTi 1−xFexO3−δ \nmultiferroics using x -ray and Raman analyses. Appl. Phys. Lett.  99, 202501 (2011).  \n49. Zorko, A., Adamopoulos, O., Komelj, M., Arčon, D. & Lappas, A. Frustration -induced \nnanometre -scale inhomogeneity in a t riangular antiferromagnet. Nat. Commun.  5, 3222 (2014).  \nAcknowledgements  \nWe acknowledge the financial support of the Slovenian Research Agency Programs P1 -0125, \nP2-0377, P1 -0112 and Projects N2 -0005, Z1 -5443, as well as the Research Support \nProgramme of University of Zagreb . The SR study has been supported by the European \nCommission under the 7th Framework Programme through the 'Research Infrastructures' \naction of the 'Capacities' Programme, Contract No. 283883 -NMI3 -II. \nAuthor contributions  \nA.Z. and M.V.  designed and supervised the project. I.A., I.M. and M.V. synthesized and \nstructurally characterized the samples. Z.J. and D.P. performed bulk magnetization \nmeasurements. M.P. and  M.G. conducted ESR measurements. A.Z., M.G. and M.T. carried out \nthe SR study. All the experimental data were analysed by A.Z., who also wrote the paper.  All \nauthors discussed the results and reviewed the manuscript.  \nAdditional information  \nCompeting financial interests  \nThe authors declare no competing financial interests.  "    },    {        "title": "1809.08644v1.Ferromagnetic_resonance_in_thin_ferromagnetic_film_with_surface_anisotropy.pdf",        "content": "Ferromagnetic resonance in thin ferromagn etic film with surface anisotropy \n \nN. A. Usov1,2\n \n1National University of Science and Tec hnology «MISiS», 119049, Moscow, Russia  \n2Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, \nRussian Academy of Sciences, IZMIRAN, 108480, Troitsk, Moscow, Russia \n \nAbstract The ferromagnetic resonance frequencies are obtaine d for a thin ferromagnetic film with surface \nanisotropy for the cases when th e external magnetic field is applied perpe ndicularly or parallel to the film surface, \nand for various combinations of boundary conditions on the film surface. It is shown that in the presence of surface \nanisotropy the ferromagnetic resonance frequency essentially depends both on th e film thickness and on the value of \nthe surface anisotropy constant. The results obtained provide  a basis for the correct interpretation of experimental \ndata obtained by means of broadband ferroma gnetic resonance in thin film structures. \n \nPACS:    75.50.Kj, 75.80.+q, 75.60.Ej, 75.60.Jk \nKeywords:   Thin ferromagnetic film, surface magnetic anisotropy, ferromagnetic resonance frequency \n \nI. Introduction \nThin ferromagnetic films with su rface magnetic anisotropy are cu rrently of great technological \ninterest [1-4]. In particular, the ultra-thin CoFe B/MgO bilayers [2-4] exhibit a perpendicular magnetic \nanisotropy. The latter provides a high thermal stability and is highly de sirable for reducing the critical \ncurrent for spin-torque switching [2]. Therefore, thin  CoFeB ferromagnetic films are considered currently \nas promising electrode material in magnetic rando m access memory and spin-transfer torque memory \ndevices. \n From physical considerations, it is obvious that only the surface magnetic anisotropy [5-8] is \ncapable of providing an out-of-plane magnetization in a thin ferromagnetic  film made of a soft magnetic \nmaterial. Actually, if a surface anisotropy constant is negative and sufficiently large in absolute value, the \norientation of the unit magnetization v ector perpendicular to the film pl ane is energetically favorable, in \nspite of a significant increase in the magnetostatic en ergy of the film. Experimentally, a rotation of the \nunit magnetization vector as a function of the film thickness was observed in a number of experiments \nwith thin ferromagnetic films of iron, cobalt, and other ferromagnets [9-12]. \nThe existence of surface magnetic anisotropy may be related to various physical origins, such as \nthe specifics of the spin-orbit interaction on the ferromagnetic surface [13], the difference in atomic periods of various layers of a multilayer sample, or  a film substrate [14], th e presence of non-uniform \nmechanical stresses near the inte rface [15], etc. However, in terms of phenomenological description of \nmagnetic phenomena [5], it is important that the energy density of surface anisotropy is concentrated in a \nvery narrow region near the sample surface. Therefor e, it is a surface contri bution to the total energy \nwhich is proportional to the surface area of the film , and not to its volume. The general variational \napproach [5] shows that the influence of surface ma gnetic anisotropy on the behavior of a ferromagnet is \nmanifested only under special boundary condition. The latt er acts on the sample surface or the interface \nbetween different materials [5,16].  \nSince the theoretical calculation of  the surface anisotropy constant Ks is rather complicated [13-\n15], it is desirable to have reliable methods to dete rmine this phenomenological constant experimentally. \nOne of such methods can be the ferromagnetic res onance (FMR) [17,18], in whic h the film magnetization \nundergoes a rapid precession near a certain equilib rium position of the unit magnetization vector. The \nbroadband FMR [19-23] can provide valuable information on the presence or absence of the energy \ninteractions on the surface of a ferromagnetic film, or on the interfaces of ferromagnetic multilayers. The \ninfluence of surface anisotropy on the ferromagnetic resonance in a thin ferromagnetic film was previously considered in seminal papers by Rado and Weertman [24,25]. Un fortunately, no explicit \n 1formulas for the FMR frequency were actually obt ained. Whereas in the present paper, the FMR \nfrequencies have been derived for ferromagnetic f ilm with surface anisotropy for the cases when the \nexternal magnetic field is applied perpendicularly or parallel to the film surface. Besides, various \ncombinations of boundary conditions on the film surface s have been studied. Thes e results seem helpful \nfor correct interpretation of the experimental data obt ained by means of the broadband FMR in thin film \nstructures.  \n \nII. Out-of-plane external magnetic field \n Consider a thin ferromagnetic film of thickness L parallel to the XY plane and located in the \ndomain 0 < z  < L. The unit magnetization vector of the film ()t r,rrα  satisfies the Landau-Lifshitz-Gilbert \n(LLG) equation \n[]⎥⎦⎤\n⎢⎣⎡\n∂∂× + × − =∂∂\ntHtefαα κ α γαrrrrr\n,       ( 1 )  \nwhere γ is the gyromagnetic ratio, κ being the phenomenological dampi ng constant. The total effective \nmagnetic field of the film efHr takes into account the contributions due to the exchange and dipolar \ninteractions as well as from th e magnetic anisotropy energy. \n \n0H HMw\nMCH\nsa\nsefr r\nrr r+′+∂∂− Δ =αα .       ( 2 )  \nHere C = 2A is the exchange constant, Ms is the saturation magnetization,  is the vector of the \nhomogeneous applied magnetic field, 0Hr\nH′r is the vector of the demagnetiz ing field which is created by the \nvolume and surface magnetic charges distributed in  the volume and on the surface of the film, \nrespectively. Finally, ()αrw  is the energy density of the magne tic anisotropy in the film volume. \n The presence of surface magnetic anisotropy can be described [5-8] by introducing a surface \ninteraction with the ener gy density per unit area  \n \n     ()2n K ws sarrα= .       ( 3 )  \nHere Ks is the phenomenological surface anisotropy constant, and nr is the unit vector of the external \nnormal to the film surface. In th e presence of the surface magnetic  anisotropy, Eq. (3), the boundary \ncondition for the unit magnetiza tion vector is given by [5] \n \n    () ()(α α α)α rrrr rrr\nn n n KnCs− =∂∂2 .      ( 4 )  \nIt works on the upper and lower surfaces of the ferro magnetic film. In the absence of surface anisotropy \non one or both surfaces of the film, Ks = 0, the usual boundary condition, 0= ∂ ∂nαr, acts on the \ncorresponding surface.  \n Suppose that the ferromagnetic film is placed in  a sufficiently strong external magnetic field \nperpendicular to the film surface, ()0 0 , 0 , 0H H=r. In what follows we neglect for simplicity the effect of a \nsmall volume magnetic anisotropy of the film, setting ()0=αrw . In equilibrium, in the absence of \nmagnetization perturbations, the unit magnetization vector is perpe ndicular to the film surface, \n. Indeed, in the case considered the v ector of effective ma gnetic field has only z component, ()(1 , 0 , 00=αr)\n) (s ef M H H H H π4 , 0 , 00) 0 (\n0) 0 (− = ′+ =rrr. Therefore, the equilibrium condition for the unperturbed unit \nmagnetization vector  \n     .  () ()[]00 0= ×efHrrα .      ( 5 )  \n 2as well as the boundary condi tion (4) are satisfied. \n Let us now consider small deviations of th e unit magnetization vector from the equilibrium \nposition. The first-order correction of  the perturbation theory [5] to the unit magnetization vector is given \nby () () ()()0 , ,1 1 1\ny xα α α=r. The components of the first-order co rrection satisfy the equation of motion \n \n   ()\n() ()[]() ()[]()()\n⎥⎦⎤\n⎢⎣⎡\n∂∂× + × − × − =∂∂\ntH Htef ef1\n0 1 0 0 11αα κ α γ α γαrrrrrrr\n   (6) \nSince the magnetization in the film pl ane is homogeneous, it is  reasonable to assume that the perturbation \nof the magnetization depends only on the z coordinate. Then the first order correction to the effective \nmagnetic field vector is given only by the exchange interaction, so that \n \n    () ()\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂\n∂∂= 0 , ,21 2\n21 2\n) 1 (\nz MC\nz MCHy\nsx\nsefα α r.      ( 7 )  \nTaking into account Eqs.  (6), (7) and neglec ting the attenuation, κ → 0, one obtains the following \nequations of motion for the component s of the unit magnetization vector \n \n    () ()\n()()1\n21 2 1\ny H my\nsx\nz MC\ntα ω ωαγα− +∂∂=∂∂;     ( 8 a )  \n    () ()\n()()1\n21 2 1\nx H mx\nsy\nz MC\ntα ω ωαγα− −∂∂− =∂∂,     ( 8 b )  \nwhere we denote ωH = γH0 and ωm = 4πγMs. The boundary conditions to Eq. (8) follow from Eq. (4)  \n \n  ()()()()t L Kzt LCx sx, 2,1\n11\nαα− =∂∂;  ()()()(t L Kzt LCy sy, 2,1\n11\nαα− =∂∂);  (9a) \n  ()()()()t KztCx sx, 0 2, 01\n21\nαα=∂∂;  ()()()(t KztCy sy, 0 2, 01\n21\nαα=∂∂)\n).  (9b) \nHere it is assumed that the surface anisotropy constants at the upper, K1s, and lower, K2s, film surfaces can \nbe different.  \n If one of the surface anisotropy constants is zero, for example, K2s = 0, the solution of Eq. (8), (9) \nhas the form of the unit ma gnetization vector precession \n \n  ;  ()() ( )( t kz A t zx ω α cos cos ,1=()()()()t kz B t zy ω α sin cos ,1= .    (10) \nFrom the equations of motion (8) one obtains for the precession frequency \n \n    2kMC\nsm Hγ ω ω ω+ − = ,       ( 1 1 )  \nwhereas the boundary conditions (9) lead to the dispersion equation \n \n     ()CKkL ks12tan= .       ( 1 2 )  \nIn the limit kL << 1 for the lowest quasi-homogeneous pre cession mode one obtains from Eq. (12) \n 3    CLKks12= ;   L MK\nss\nm H12γ ω ω ω+ − = .   (13) \nFor inhomogeneous FMR modes in the limit 11<<C L Ks  one has  \n \n     n CK\nLnks\nnππ 12+ ≈ ;  n = 1,2,  … \n However, if the surface anisotropy is pres ent on both surfaces of the ferromagnetic film, the \ndispersion relation (1 2) takes the form \n \n    ()( )\nCK KkLkCK Kks s s s 2 1\n22 1 2tan4 +=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− .     (14) \nAs a result, in the limit kL << 1 the resonance frequency of the quasi-homogeneous mode equals  \n \n   ( )\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+++ − =C MK K\nL MK K\nss s\nss s\nm H2 1 2 1 4 2γ ω ω ω .    (15) \n Eqs. (11-15) for FMR frequencies are valid in a sufficiently strong perpe ndicular magnetic field, \nunder the condition that the unperturbed film magnetizati on is perpendicular to the film surface. They can \nbe used to determine experimentally the values of the film surface anisotropy constants. \n \nIII. In-plane external magnetic field \nSuppose now that a sufficiently strong external ma gnetic field is applied parallel to the film \nsurface, for definiteness, along the Y axis, so that ()0 , , 00 0 H H=r. Then in equilibrium, in the absence of \nmagnetization perturbation, the unit magnetization vector is parallel to the external magnetic field, \n. Indeed, in the case under considera tion the demagnetizing field is absent, , and the \neffective magnetic field vector is given by ()(0 , 1 , 00=αr) 0) 0 (=′Hr\n()0 , , 00) 0 (\n0) 0 (H H H Hef =′+ =rrr. Therefore, both the equilibrium \nequation (5) and the boundary condition (4) are satisfied.  \nNext, the first-order correction of the unit magnetization vector has the form () ()()()1 1 1, 0 ,z xα α α=r. \nThe first order correction to the eff ective magnetic field vector contains the contributions of exchange and \nmagneto- dipole interactions associated with the devi ation of the unit magnetization vector perpendicular \nto the film surface \n    () ()\n()\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−∂∂\n∂∂=1\n21 2\n21 2\n) 1 (4 , 0 ,z sz\nsx\nsef Mz MC\nz MCH α πα α r.   (16) \nAs a result, neglecting attenuation, the equation of motion (6) takes the form \n \n   () ()\n()()1\n21 2 1\nz H mz\nsx\nz MC\ntα ω ωαγα+ +∂∂− =∂∂;      ( 1 7 a )  \n   () ()\n()1\n21 2 1\nx Hx\nsz\nz MC\ntα ωαγα−∂∂=∂∂.      (17b) \nThe boundary conditions for the components of the unit magnetization vector are given by \n \n   ()()0,1\n=∂∂\nzt Lxα;  ()() ()(t L Kzt LCz sz, 2,1\n11\nαα=∂∂);   (18a) \n 4   ()()0, 01\n=∂∂\nztxα;  ()() ()(t Kzt\nz sz, 0 2, 01\n21\nαα− =∂∂).    (18b) \nIn the absence of the surface magnetic anisotropy, K1s = K2s = 0, the solution of Eqs. (17), (18) is \nsimilar to Eq. (10). For the FMR frequency one obtains the Kittel’s relation [18] \n \n ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+ + =2 2\nn\nsH n\nsm H n kMCkMCγ ω γ ω ω ω ; kn = πn/L;   n = 0,1 … \nFor the homogeneous precession mode, n = 0, it reduces to well known result [17,18] ()m H Hω ω ω ω+ = . \nIf the surface magnetic anisotropy is present on ly on the upper surface of the ferromagnetic film, \nK1s < 0, K2s = 0, the solution of Eqs. (17), (18) is of the form \n \n   ()() ()() () ()t z k A z k A t zx ω α cos cosh cosh ,2 2 1 11+ = ;  \n   ()() ()() ( )()t z k B z k B t zz ω α sin cosh cosh ,2 2 1 11+ = .     (19) \nHere the wave vectors k1,2 and the coefficients B B1,2 are determined by the equations \n \n  ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+ ± + =4 22\n2 2\n2 , 1m m\nHs\nCMkωωωωγ;   2 , 12\n2 , 1\n2 , 1 AkMC\nBsH\nωγω−\n= . \nIn the given case, K2s = 0, the boundary conditions (18b) for the so lutions (19) are auto matically satisfied, \nwhereas the boundary conditions (18a) lead to the dispersion equation for the FMR frequency ω \n \n () ()\n22 2\n21 2\n2\n11 2\n11 2\n1coth 2 coth 2\nkL kkMC\nСKkMC\nkL kkMC\nСKkMC\nsHs\ns sHs\ns⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− + =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− +γωγ γωγ.  (20) \nIn the limit k1L, k2L << 1, using the series expansion coth( x) = 1/ x + x/3 + … for the lowest precession \nmode one obtains from Eq. (20) the relation \n \n    ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− − + =CL K\nL MKs\nss\nm H H1 1\n3212γ ω ω ω ω .    (21) \nEq. (21) is valid for sufficiently small film thicknesses L, when dimensionless parameter 11<C L Ks . \nWith increasing film thickness, the value of the FMR frequency ω should be found numerically from Eq. \n(20). \nEq. (21) for the lowest FMR frequency is used in  the literature [19-21,23] by analogy with Kittel's \nformulas [17], but without any ju stification. In fact, Eq. (21) is valid only in the limit k1L, k2L << 1, and \nfor the case when the surface anisotropy is present only on one of the film surfaces.  \n If the surface anisotropy is present both on th e upper and lower surfaces of the ferromagnetic film, \nK1s, K2s < 0, the solution of Eqs. ( 17) has a more general form \n \n ()() ()()()() () t z k A z k A z k A z k A t zx ω α cos sinh cosh sinh cosh ,2 4 2 3 1 2 1 11+ + + = ();  \n ()() () ()()() () ()t z k B z k B z k B z k B t zz ω α sin sinh cosh sinh cosh ,2 4 2 3 1 2 1 11+ + + = ,   (22) \nwhere the coefficients B Bi are expressed in terms of Ai as follows \n 5   2 , 12\n1\n2 , 1 AkMC\nBsH\nωγω−\n= ;   4 , 32\n2\n4 , 3 AkMC\nBsH\nωγω−\n= . \nUsing the solutions (22), it is pos sible to satisfy the general boundary  conditions (18). Then, from the \ncorresponding dispersion equation in the limit k1L, k2L << 1 one can obtain an approximate expression for \nthe FMR frequency \n \n   ( )\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−+− + =C MK K\nL MK K\nss s\nss s\nm H H2 1 2 1 4 2γ ω ω ω ω .   (23) \nAs an example, Fig. 1 shows the FMR frequencies, f = ω/2π, in the thin-film structure [2-4] \nTa/CoFeB/MgO in the case when the external magnetic field H0 is applied parallel to the film plane, and \nunder the assumption that the surface anisotropy c onstant is nonzero only at the CoFeB/MgO interface. \nThe solid curves in Fig. 1 are drawn according to Eq. (21). \n For sufficiently small CoFeB film thicknesses this equation approximates the numerical solution of \nEq. (20) with reasonable accuracy. The values of the saturation magnetization Ms = 1250 emu/cm3, and \nthe exchange constant A = C/2 = 1.5×10-6 erg/cm, used in the calculation were determined experimentally \n[22] for ferromagnetic fi lm of the composition Co 40Fe40B B20.  \n \n02468 1 0 1 2 1 4010203040\n432f  (GHz)\nH0  (kOe)Ks = - 1.0 erg/cm2\nMs = 1250 emu/cm3\nA = 1.5*10-6 erg/cmCo40Fe40B20 film\n1\n(a)\n   0 2 4 6 8 1 01 21 4010203040\n32Co40Fe40B20 filmL = 1.0 nm\nMs = 1250 emu/cm3\nA = 1.5*10-6 erg/cmf  (GHz)\nH0  (kOe) (b)1\n \nFig. 1. a) The frequency of ferromagne tic resonance in a ferromagnetic film Co 40Fe40B B20 as a function of \nin-plane magnetic field for different film thicknesses: 1) L = 1.0 nm; 2) L = 0.8 nm; 3) L = 0.7 nm; 4) L = \n0.6 nm. b) The same for different values of the surface anisotropy constant at fixed thickness L = 1.0 nm: \n1) K 1s = - 1.0 erg/cm , 2) K2\n1s = - 1.25 erg/cm , 3) K2\n1s = - 1.5 erg/cm . 2\n \nIt was experimentally shown [3] that for thin -film structure Ta/CoFeB /MgO the out-of-plane \nmagnetization exists in the film thickness range 0.6 - 1.2 nm. It ca n be shown also [26] that the \nmagnetization vector becomes parallel to the film plane above the saturation field Hs = 2|K1s|/MsL - 4πMs. \nThis value closely coincides with the magnetic fiel d at which the FMR frequencies, Eq. (21), become \nnonzero. \nAs Fig. 1 shows, in the presence of surface an isotropy the FMR frequenc ies essentially depend \nboth on the ferromagnetic film thickness and on the valu e of the surface anisotropy constant. Note that \nEqs. (21), (23) are valid al so at film thicknesses greater than the cr itical value, when in the absence of \napplied magnetic field there is a canted or inhomoge neous micromagnetic state in the film [11,16]. The \n 6only condition is that there is a su fficiently strong in-plane magnetic field in which the unperturbed film \nmagnetization is parallel  to the film surface. \n \nIV. Conclusion \nIt this paper, it is shown that the peculiarit ies of the FMR spectra observed in the ultra-thin \nCoFeB/MgO bilayers [2-4] and similar ferromagnetic structures may be related with the presence of the \nsurface magnetic anisotropy of appreciable value at various surfaces of the f ilm. A proper theoretical \nmethod to describe the influence of the surface magnetic anisotropy on the film behavior is the use of the \nnatural boundary condition that can be  obtained on the basis of the genera l variational approach [5]. Using \nthis technique, the FMR frequencies are obtained in th is paper for a thin ferromagnetic film with surface \nanisotropy for the cases when the external magnetic field is applied perpendicularly or parallel to the film \nplane, and for various combinations of boundary conditio ns on the film surface. These results can be used \nfor determination of the surface anisotropy constant of the film by means of the broadband FMR \nexperiment. \n \nAcknowledgement \n This work was supported by the Ministry of E ducation and Science of the Russian Federation in \nthe framework of Increase Competitiveness Program of NUST «MISiS», contract № K2-2017-008. \n \nReferences \n[1] S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, E. E. Fullerton, Nature Mater. 5 (2006) 210. \n[2] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, H. \nOhno, Nature Mater . 9 (2010) 721. \n[3] W. X. Wang, Y. Yang, H. Naganuma, Y. Ando, R. C. Yu, X. F. Han. Appl. Phys. Lett. 99 (2011) 012502. \n[4] Q. L. Ma, S. Iihama, T. Kubota, X. M. Zhang, S. Mi zukami, Y. Ando, T. Miyazaki, Appl. Phys. Lett. 101 (2012) \n122414. \n[5] W. F. Brown, Jr., Micromagnetics, Wiley-Interscience, New York - London, 1963. \n[6] M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broe der, J. J. de Vries, Rep. Prog. Phys. 59 (1996) 1409.  \n[7] H. N. Bertram, D. I. Paul, J. Appl. Phys. 82 (1997) 2439. \n[8] C. A. F. Vaz, J. A. C. Bland, G. Lauhoff, Rep. Prog. Phys. 71 (2008) 056501. \n[9] R. Allenspach, M. Stampanoni, A. Bischof, Phys. Rev. Lett. 65 (1990) 3344. \n[10] H. Fritzsche, J. Kohlepp, H. J. Elmers , U. Gradmann, Phys. Rev. B 49 (1994) 15665.  \n[11] M. Speckmann, H. P. Oepen, H. Ibach, Phys. Rev. Lett. 75 (1995) 2035.  \n[12] M. Hehn, S. Padovani, K. Ounadjela, J. P. Bucher, Phys. Rev. B 54 (1996) 3428. \n[13] H. X. Yang, M. Chshiev, B. Dieny, J. H. Lee, A. Manchon, K. H. Shin, Phys. Rev. B 84 (2011) 054401.  \n[14] K. H. He, S. J. Chen, J. Appl. Phys. 111 (2012) 07C109.  \n[15] J. I. Hong, S. Sankar, A. E. Berkowitz, W. F. Egelhoff Jr., J. Magn. Magn. Mater. 285 (2005) 359. \n[16] N. A. Usov, O. N. Serebryakova, J. Appl. Phys. 121 (2017) 133905. \n[17] C. Kittel,  Phys. Rev. 73 (1948) 155.  \n[18] C. Kittel, Phys. Rev. 110 (1958) 1295. \n[19] J.-M. L. Beaujour, W. Chen, A. D. Kent, J. Z. Sun, J. Appl. Phys. 99 (2006) 08N503. \n[20] J-M. L. Beaujour, J. H. Lee, A. D. Kent, K. Krycka, C-C. Kao, Phys. Rev. B 74 (2006) 214405. \n[21] X. Liu, W. Zhang, M. J. Carter, G. Xiao, J. Appl. Phys. 110 (2011) 033910. \n[22] A. Conca, J. Greser, T. Sebastian, S. Klingler, B. Obry, B. Leven, B. Hillebra nds, J. Appl. Phys. 113 (2013) \n213909. \n[23] A. Conca, A. Niesen, G. Reiss, B. Hillebra nds, J. Phys. D.: Appl. Ph ys. (2018), DOI: 10.1088/1361-\n6463/aab5d8. \n[24] G. T. Rado, J. R. Weertman, J. Phys. Chem. Solids 11 (1959) 315. \n[25] G. T. Rado, Phys. Rev. B 26 (1992) 295. \n[26] N. A. Usov, O. N. Serebryakova, J. Magn. Magn. Mater. 142 (2018) 453. \n 7"    },    {        "title": "2108.10581v1.Higher_order_ferromagnetic_resonances_in_periodic_arrays_of_synthetic_antiferromagnet_nanodiscs.pdf",        "content": "Higher-order ferromagnetic resonances in periodic arrays of synthetic-antiferromagnet nanodiscs\nV . Yu. Borynskyi,1D. M. Polishchuk,1, 2A. K. Melnyk,3A. F. Kravets,1, 2A. I. Tovstolytkin,1and V . Korenivski2\n1Institute of Magnetism of NAS of Ukraine and MES of Ukraine, 36-b Akad. Vernadsky blvd., Kyiv 03142, Ukraine\n2Nanostructure Physics, Royal Institute of Technology, Stockholm 10691, Sweden\n3Institute for Sorption and Problems of Endoecology of NAS of Ukraine, 13 Naumov str., Kyiv 03164, Ukraine\nWe investigate spin dynamics in nanodisc arrays of synthetic-antiferromagnets (SAF) made of Py/NiCu/Py\ntrilayers, where the NiCu spacer undergoes a Curie transition at about 200 K. The observed ferromagnetic\nresonance spectra have three distinct resonance modes at room temperature, which are fully recreated in our\nmicromagnetic simulations showing also how the intra-SAF asymmetry can be used to create and control the\nhigher-order resonances in the structure. Below the Curie temperature of the spacer, the system effectively tran-\nsitions into a single-layer nanodisc array with only two resonance modes. Our results show how multi-layering\nof nano-arrays can add tunable GHz functionality relevant for such rapidly developing fields as magnetic meta-\nmaterials, magnonic crystals, arrays of spin-torque oscillators and neuromorphic junctions.\nThe last decade has seen large research efforts focused on\ncollective spin dynamics of magnonic crystals and logic cir-\ncuits aimed for various high-speed applications [1–4]. Con-\ntrollable size of the individual elements used in such meta-\nmaterials, down to the 10 nm range, enables rich higher-order\nFMR dynamics such as standing spin waves and edge-mode\nresonances. The typical geometry of a periodic array offers\nhigh flexibility in engineering novel properties. At the same\ntime, the increased complexity requires systematic investiga-\ntions of various effects and parameters involved [5–8].\nRelatively simple periodic arrays of single-layer ferromag-\nnetic nanodiscs exhibit anisotropic spin wave modes (com-\nmonly, with four-fold in-plane anisotropy) affected by the\ninter-element dipolar interactions [9–11]. The properties be-\ncome more intriguing for arrays of multilayered elements such\nas SAF’s, which are of interest in antiferromagnetic spintron-\nics [12] as well as magnonics [13]. Their collective spin dy-\nnamics is dictated by the inter-element interactions, the intra-\nparticle dipolar coupling, and the interplay of the two [14–16].\nArrays of multilayered nanomagnets can exhibit distinct\nspin dynamic modes not found in single-layer systems. The\nintra-particle dipolar interactions lead to complex demagne-\ntizing field distributions, altering the collective spin dynam-\nics in such systems, resulting in mode-frequency shifts [17]\nor even splitting of the resonant modes [18]. For example,\na systematic micromagnetic study performed by Pauselli et\nal.[14] for a sub-40-nm magnetic tunnel junction read head\nexplained additional higher-order modes as originating from a\nstrong localization of spin excitations to the junction’s edges,\nwhere an excitation in one of the junction’s magnetic lay-\ners drives a localized resonance in the neighboring, dipole-\ncoupled magnetic layer. Multi-layering is thus a capable tool\nfor varying spin wave modes and therefore spectral modifi-\ncation [19, 20], additional to element grouping or sequencing\nin arrays of single-layer nanoparticles [18, 21]. Furthermore,\nparticle shape imperfections unavoidable on the experiment,\nsuch as edge deviations [22] or a sidewall angle [23], can\nhave a significant effect on the spin dynamics of the system.\nStrongly localized edge-mode excitations are highly sensitive\nto this kind of distortions, which can lead to lifting of their\ndegeneracy for both single-layer [24] and multilayered [25]\nsystems. While arrays of singe-layer nanoparticles are wellstudied [26–28], a systematic investigation of arrays of multi-\nlayered nano-elements is still pending.\nIn this work, we investigate higher-order FMR excitations\nin periodic arrays of multilayered SAF nanodiscs, designed\nwith an in-situ thermal control of the interlayer coupling ca-\npable of making the nanodiscs effectively single-layered. This\nin-situ control of the SAF versus single-layer character of the\narray elements allows us to identify the SAF-unique proper-\nties from a direct comparison of the FMR spectra measured\nabove and below the Curie point of the spacer. Using detailed\nmicromagnetic simulations, we explain the occurrence, po-\nsitions, and anisotropy of the measured resonances and how\nthey are affected by strong localizations of the spin excitations\nand a complex interplay between the inter- and intra-particle\ninteractions, pronounced in SAF-based nanostructures.\nWe focus our investigation on arrays of SAF nanodiscs\nPy(7 nm)/NiCu(10 nm)/Py(7 nm) with periodicity 250 nm and\nnominal diameter d=150 nm. Here, Py and NiCu stand\nfor Ni 80Fe20(Permalloy) and Ni 60Cu40, respectively. The\nspacer is paramagnetic at room temperature (RT) and medi-\nates no direct exchange between the outer Py layers, which\ninteract through magnetostatic coupling only, thereby form-\ning a SAF element. At low temperature, the spacer is fer-\nromagnetic, which allows in-situ switching between the SAF\nand the single-layer states in the system [29, 30]. Multilay-\ners Py/NiCu/Py were grown by magnetron sputtering (Orion,\nAJA Int.) onto thermally oxidized Si (001) wafers at room\ntemperature. The multilayers were then used to fabricate ar-\nrays of SAF nanodiscs using electron-beam lithography (V oy-\nager, Raith Inc.). The arrays were etched using Ar-plasma\nvia a double-layer hard mask, TaN(60 nm)/Al(20 nm). Here,\nthin Al was used as a lift-off layer to form a mask (with the\nnominal diameter d0=150 nm) for selective reactive plasma\netching with SF6 of the thick TaN layer underneath. Addi-\ntional array geometries were fabricated and tested for calibra-\ntion and verification purposes, in the process of optimizing the\nSAF arrays, with most interesting properties presented herein.\nSEM images of the fabricated arrays in Fig. 1(a–b) show\ngood uniformity in the element size (the spread is less than\n5 %) and good edge morphology. Angled SEM in Fig. 1(b)\nshows a side-wall angle of 20–30\u000e, formed in the process of\nplasma etching, making the effective diameter of the top PyarXiv:2108.10581v1  [cond-mat.mes-hall]  24 Aug 20212\nFIG. 1. (a) Top-view SEM image of select region in planar array of\n150-nm SAF nanodiscs (Py/NiCu/Py). Inset indicates particle peri-\nodicity and in-plane orientation of external magnetic field. (b) 45\u000e\nSEM view of same sample. (c) Typical M-H loops, measured using\nMOKE at 300 and 170 K; inset illustrates vertical cross-section of\nsingle SAF nanodisc. TaN is used as etching mask and for capping.\ndisc about 15 % smaller than that of the bottom disc. The\ntop dome-like cap, visible as the central concentric circles in\nthe top-view and contrasted-top in the side-view SEM images,\nFig. 1(a–b), is the remaining TaN mask that protects the fab-\nricated particles from oxidization.\nCavity-FMR measurements were carried out using an X-\nband ELEXSYS E500 spectrometer (Bruker Inc.) equipped\nwith an automatic goniometer for precise angle control. The\nmeasurements were performed at a constant operating fre-\nquency of 9.36 GHz. The magnetic volume of the ar-\nrays of area 1 :4\u00021:4 mm2was large enough for a high\nsignal-to-noise ratio. Additionally, magnetostatic properties\nof the arrays were measured using a custom-made variable-\ntemperature magneto-optical Kerr-effect (MOKE) setup.\nFigure 1(c) shows two characteristic MOKE loops mea-\nsured in-plane ( jH=0\u000e). Zero remnant magnetization at\nH=0 and relatively high saturation field Hs\u0019300 Oe for RT\nloop indicate a pronounced antiferromagnetic-type interlayer\ncoupling of the top and bottom magnetic moments of the indi-\nvidual SAF nanodiscs. The double-step shape and small hys-\nteresis can arise from the presence of magnetic anisotropy [31]\nas well as a non-ideal circular shape of the nanodiscs [32, 33].\nFigure 2(a) shows the FMR spectra measured at RT for\nthe array of 150-nm SAF nanodiscs with different in-plane\norientations of the external magnetic field, jH=0;45, and\n90\u000e. Each curve shows three well-defined resonance lines\nmarked as L m, Le1, and L e2. We associate the L mline with\nthe quasi-uniform (hereafter main ) FMR mode since its po-\nsition coincides with that of the resonance line for a contin-\nuous Py film. With respect to the position of L m, lines L e1\nand L e2are observed at higher resonance fields. A similarhigher-field mode, along with the main mode, was reported\npreviously [11] for arrays of single-layer nanodiscs and was\nattributed to the edge-mode resonance. For our arrays of SAF\nnanodiscs, we observe two higher-field modes, both of which\nwe associate with higher-order edge-mode resonances in the\nSAF nanostructure – the interpretation supported by our mi-\ncromagnetic simulations, detailed below. We note that any\nFMR modes related to potential domains or vortices forming\nin the SAF particles are excluded from consideration since the\nresonance fields of L m, Le1, and L e2(Hr>1 kOe) are much\nhigher than the SAF saturation field [ Hs\u0019300 Oe; Fig. 1(c)].\nThe resonance field, Hr, of each resonance line show a pro-\nnounced angular dependence, as seen in Fig. 2(b). All lines\nshow bi-axial anisotropy with, however, opposite hard and\neasy axes for L mand L e1;e2. While L mexhibits easy axes\n(minimum Hr) at 45\u000ewith respect to the main xandyaxes\nof the array [Fig. 1(a)], the edge-modes, L e1and L e2, have\neasy axes along xandy. Such anisotropic behavior was ex-\nplained in Refs. 9 and 34 by the inter-element dipole coupling\nleading to a non-uniform effective field distribution within the\nindividual discs, even at rather high external fields.\nIn order to explain the origin and behavior of the three FMR\nmodes observed, we performed detailed micromagnetic sim-\nulations [35], using MuMax3 [36, 37]. The layout of each\nelement as well as the base diameter ( d0) used were those\nobserved experimentally [Fig. 1(c)]. Figure 2(c) displays the\ncalculated spectra for two isolated single-layer Py discs (7 nm\nthick) with diameters d0=150 nm (blue line) and d=0:85d0\n(red dashed line). This difference in size corresponds to the\nSEM-measured difference in the diameters of the bottom and\ntop layers of our SAF nanodiscs; cf. Fig. 1(b). Both simu-\nlated spectra show two resonance lines corresponding to the\nmain and edge-resonance modes, illustrated by the insets to\nFig. 2(c). Of note is that despite line L mis associated with the\nuniform mode of a continuous Py film, the related excitation\nwithin a nanodisc is not fully spatially uniform, which is why\nwe call it main rather than uniform FMR mode. Importantly,\nneither spectrum shows any trace of a third line observed on\nthe experiment, nor a related additional line can be obtained\nwhen the two spectra are superposed.\nThe simulated spectra are substantially different when the\ntwo discs of d0=150 nm and d=0:85d0form a SAF ele-\nment, as shown in Fig. 2(d). Three resonance lines are visible,\nwith the positions and intensities resembling the experimen-\ntal data. The insets to Fig. 2(d) clarify why we associate the\nhigh-field lines with edge-mode resonances. Whereas the L e1\nmode is excited mostly in the bottom layer, the L e2mode is\nmore intensive in the top layer. Importantly, when the top and\nbottom layers are of equal diameter, only two modes, L mand\nLe1, are excited; the third mode appears only when the top and\nbottom diameters become noticeably different (by 5–10 %).\nFigure 2(e) shows the simulated angle-dependence of the\nresonance fields of the L m, Le1, and L e2modes obtained for\nan array of SAF nanodiscs. The simulations are in excellent\nagreement with the experimental resonance-field versus angle\ndata of Fig. 2(b), which further validates our interpretation of\nthe observed modes and their properties.\nFigure 3(a) compares the SAF FMR spectra measured as3\nFIG. 2. (a) FMR spectra measured at RT for 150-nm SAF array at 0, 45, and 90\u000e-oriented external field. L m, Le1and L e2mark resonance\nlines attributed to main and two edge-modes, respectively. (b) Resonance fields of individual modes versus in-plane angle. (c) FMR spectra\ncalculated for individual single-layer discs with d0=150 nm and d\u00190:85d0, corresponding to bottom and top layers of SAF nanodiscs. Insets\nshow excitation regions for two eigen-modes. (d) Calculated FMR spectrum for individual SAF particle with d0andd\u00190:85d0. Insets display\nexcitation regions in both layers of SAF nanodisc. All spectra were calculated with in-plane field at jH=0\u000e(black arrow in left inset to c). (e)\nIn-plane angular dependence of resonance fields simulated for SAF disc array with d0=150 nm, d\u00190:85d0, and array periodicity 200 nm.\nthe temperature is lowered from room to below the Curie\npoint of the NiCu spacer. Our SAF design is rather unique\nas it effectively transforms the nano-elements from trilayers\nto single-layers when the spacer transitions from para- to fer-\nromagnetic state at about 220 K [29, 30, 38, 39]. The as-\nsociated transition in the FMR spectrum is very illuminat-\ning – the three-mode resonance is reduced to effectively two\nmodes, which is a known characteristic of single-layer nano-\narrays. Figure 3(b) shows the fits to the extracted high-order\nresonances (after subtracting the main peak): the pronounced\ndouble-resonance at room temperature with dominating L e1\n(red) and essentially a single broad peak at 170 K where the\nLe1peak is <1% of its RT value. The transition from two high-\norder peaks to one takes place precisely at the Curie temper-\nature of the spacer, deduced independently from our MOKE\nmagnetization data shown in Fig. 3(c). We thus confirm, in a\ndirect experiment, that the origin of the double spin-wave res-\nonance is unique to SAF nano-arrays. Specifically, the high-\norder (non-uniform) double resonance is a result of hybridiza-\ntion of predominantly acoustic and optical (in-phase and out-\nphase) oscillations in the two ferromagnetic layers comprising\nthe SAF. These modes collapse into one on direct exchange\ncoupling the outer layers below the TCof the spacer and, gen-\nerally, can be tuned by changing the SAF geometry and the\nspacer properties (via magnetic dilution and/or temperature).\nFIG. 3. (a) FMR spectra of SAF arrays measured at select tempera-\ntures. (b) Fits of high-order resonances (L e1,Le2) at above and below\nspacer’s Curie point. (c) Normalized remnant magnetization versus\ntemperature obtained from MOKE loops.\nThe above results demonstrate that the magnetization dy-\nnamics of periodic arrays of SAF nanomagnets have key dis-\ntinctions from the known behavior of single-layered nano-4\narrays. Our variable field-angle and temperature FMR and\nMOKE studies, supported by morphological characterization\nand micromagnetic simulations, enable us to identify the ori-\ngins of the distinct SAF properties observed. The three FMR\nmodes and their pronounced in-plane anisotropy are two such\ndistinct properties that, however, are governed by different\nfactors. Our micromagnetic simulations indicate that the split-\nting of the higher-order edge mode into two is due to the\ndipolar coupling within the individual, asymmetric SAF par-\nticles. Here, the inter-particle interaction within the array is\na secondary effect, which on the other hand dominates the\nanisotropic properties of the array. We find that the high-\norder spin excitations are hybridized acoustic (predominantly\nin-phase for L e1) and optical (predominantly out-of-phase for\nLe2) oscillations in the two layers comprising the SAF.\nThe occurrence of similar edge modes in single-layer nan-\nodiscs is usually explained by a non-uniform internal field dis-\ntribution, with its minima located at the edges orthogonal to\nthe static magnetization of the discs [14]. Following similar\nlogic for our asymmetric SAFs, we associate the two edge res-\nonances with non-uniform field distributions that are different\nin each of the two nanodiscs owing to their different internal\nand external (inter-layer) demagnetization field profiles. The\ndemagnetization is stronger in the smaller-diameter top layer,\nwhich results in a higher external field needed to excite the\nedge mode in this layer. In a symmetric SAF, with symmetric\ndemagnetization fields, only one edge mode is excited.\nThe four-fold anisotropy of the three FMR modes is due to\nthe bi-axial symmetry of our periodic arrays forming a square\nlattice. For our circular 150-nm SAF nanodisc arrays, this\nanisotropy is clearly caused by the inter-particle dipolar inter-actions. The observed 45\u000e-angle between the easy axes of the\nmain mode and the edge modes can be explained by the dif-\nference in the spacing of the respective excitation nodes for\nsingle-layer nanodot arrays [34]. We are able to recreate this\ntype of anisotropy in our micromagnetic simulations of SAF\nnano-arrays (to be discussed elsewhere).\nIn summary, we have investigated experimentally and mi-\ncromagnetically the spin dynamics of arrays of three-layer\nSAF nanoparticles where the intra-SAF coupling can be in-\nsitu controlled by varying temperature. The results show\nhow the intra- and inter-particle dipolar interactions com-\nbine to produce rather unique GHz properties of the system,\nsuch as additional spin-wave modes that can be tuned by the\nSAF geometry and/or temperature, which are relevant for the\nemerging field of nanostructured magnetic metamaterials and\nmagnonic devices.\nACKNOWLEDGMENTS\nWe thank Dr. Roman Verba for fruitful discussions. Sup-\nport from the Swedish Research Council (VR 2018-03526),\nthe Olle Engkvist Foundation (2020-207-0460), the V olkswa-\ngen Foundation (90418), and the National Academy of Sci-\nences of Ukraine (Projects 0119U100469, and 0120U100457)\nare gratefully acknowledged.\nDATA A VAILABILITY\nThe data supporting the findings of this study are available\nfrom the corresponding author upon request.\n[1] A. V . Chumak, V . I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, “Magnon spintronics,” Nature Physics 11, 453–461\n(2015).\n[2] X. M. Liu, J. Ding, G. N. Kakazei, and A. O. 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Dzhezherya, A. I. Tovs-\ntolytkin, V . O. Golub, and V . Korenivski, “Anisotropic mag-\nnetization relaxation in ferromagnetic multilayers with variable\ninterlayer exchange coupling,” Physical Review B 94, 064429\n(2016)."    },    {        "title": "1207.4522v1.Interface_Ferromagnetism_in_a_SrMnO3_LaMnO3_Superlattice.pdf",        "content": "arXiv:1207.4522v1  [cond-mat.str-el]  18 Jul 2012INTERFACE FERROMAGNETISM IN A SrMnO 3/LaMnO 3SUPERLATTICE\nS. Smadici,1B.B. Nelson-Cheeseman,2A. Bhattacharya,2,3and P. Abbamonte1\n1Frederick Seitz Materials Research Laboratory, Universit y of Illinois, Urbana, Illinois 61801\n2Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439\n3Center for Nanoscale Materials, Argonne National Laborato ry, Argonne, Illinois 60439\nResonantsoft x-rayabsorption measurementsat theO Kedge onaSrMnO 3/LaMnO 3superlattice\nshow a shoulder at the energy of doped holes, which correspon ds to the main peak of resonant\nscattering from the modulation in the doped hole density. Sc attering line shape at the Mn L3,2\nedges has a strong variation below the ferromagnetic transi tion temperature. This variation has a\nperiod equal to half the superlattice superperiod and follo ws the development of the ferromagnetic\nmoment, pointing to a ferromagnetic phase developing at the interfaces. It occurs at the resonant\nenergies for Mn3+and Mn4+valences. A model for these observations is presented, whic h includes\na double-exchange two-site orbital and the variation with t emperature of the hopping frequency tij\nbetween the two sites.\nI. INTRODUCTION\nDoped La 1−xSrxMnO3(LSMO) has multiple FM,\nAFM and canted magnetic orders as a function of dop-\ningandtemperature1–3fromsuperexchange6anddouble-\nexchange4,5interactions, which favor an antiferromag-\nnetic (AFM) insulating phase and a ferromagnetic (FM)\nmetallic phase, respectively. The FM phase at low tem-\nperatures is near the x= 0.33 doping, at which the\nclosely related manganite La 1−xCaxMnO3shows very\nlarge (“colossal”) magnetoresistance (CMR)7.\nThe wave vectors of AFM and orbital orders in bulk\nmanganites can be accessed with soft x-ray scattering\nat the Mn L3,2edges. For instance, studies were made\nfor Pr 1−xCaxMnO3(Ref. 8), La 2−xSrxMnO4(Refs. 9–\n11)andLa 2−2xSr1+2xMn2O7(Ref.12)manganites. More\nrecent measurements showed that magnetic and orbital\nscattering amplitudes are similar13, studied the dop-\ning dependence14, confirmed the separation in energy\nof Ref. 13 between the magnetic and orbital scattering\nresonances,15and studied their evolution after photoex-\ncitation16. The broader features of the measurements\nare obtained in calculations of line shapes at the Mn\nL3,2edges with atomic multiplet models of magnetic17\nand orbital18scattering, and more recently, with a finite-\ndifference method19. However, the investigation of the\nbulk FM phase near the x= 0.33 doping is not possible\nwith soft x-ray scattering at the Mn Ledges, due to lack\nof contrast for this order.\nThe FM phase can be studied with soft x-ray scat-\ntering in (SMO)n/(LMO)2nsuperlattices (SL), in which\nthe Sr and La atoms are ordered in SrMnO 3(SMO) and\nLaMnO 3(LMO) layers. External magnetic fields are not\nnecessary. The SL growth sequence can be used to define\nthe period and symmetry of a reflection along the c-axis.\nThis was demonstrated for a n= 4 SL, in which the\nscattering wave vector was decreased to a range accessi-\nble at the O Kedge resonance and a higher symmetry of\na reflection used to probe interface scattering.20\nIn this work, we have applied these ideas to SL reflec-\ntions at the O Kand MnL3,2edges and studied thedevelopment of the FM moment in a shorter superpe-\nriodn= 2 SL with soft x-ray absorption and scatter-\ning. Measurements at the O Kedge showed modulated\nhole doping at oxygen sites. We have observed scatter-\ning at the Mn Ledges from the SL interfaces at SL re-\nflectionL= 2, following the temperature dependence\nof the FM moment. The symmetry of the SL reflec-\ntion allowed us to probe all Mn valences in the inter-\nface layers. In addition to Mn3+valence resonances of\nbulk AFM order measurements8–16, a peak in the reso-\nnant line shape, which has not been observed before, is\naligned with the fluorescence yield edge for the Mn4+va-\nlence. We present a model of the x-ray scattering from\nthe SL interfaces, which includes the temperature depen-\ndence of the double-exchange hopping frequency tijand\nthe change in the configurations of the Mn ions in the\nFM state.\nII. EXPERIMENTS\nA. Structure\nThen= 2 SL was grown on the (001) surface of\nSrTiO 3(STO)atArgonneNationalLaboratoryat700◦C\nin a 2×10−6Torr ozone pressure, followed by cool-\ning to 100◦C and pump down. The structure was\n{r×[2(SMO) + 4(LMO)] + SMO }withr= 13 [Fig.\n1(a)]. SMO ( aSMO= 3.805˚A, Ref. 21) and LMO\n(aLMO= 3.99˚A, Ref. 22) layers on the STO substrate\n(as= 3.905˚A) are under +2 .6 % tensile and −2.2 %\ncompressive strain. The surface RMS roughness, mea-\nsured with an atomic force microscope, was σs= 2.85˚A.\nFrom hard and soft x-ray reflectivity measurements, the\nSL superperiod was cSL= 22.5±0.5˚A and the average\nc-axisparameterfor1ML(acoverageofoneformulaunit\nof SMO or LMO over a as×bsarea) was 3 .86±0.05˚A.2\n(a) \n0.9 1.0 1.1 1.8 2.0 2.2 0510 15  Mn edge (643.6 eV) \n O edge (531.8 eV) \n 8048 eV Intensity (arb.units) \nL (r.l.u.) \n (c) I\nII L\nL\nS\nS\nL\nL\nL\n3+ \n3.5+ \n4+ \n3.5+ \n3+ \n3+ Sf\nifnif\nac\nnifif\nLf\n(b) L=1 \nL=2 \nπ−pol θ2θ b\naφbay\nx\nπσ\nFIG. 1: (Color online) (a) Sketch of a 6 ML superperiod of\nthe (SMO) 2/(LMO) 4superlattice with LaO ( L) and SrO ( S)\nplanes, layer form factors f, Mn valences in the MnO 2planes\nestimated from the neighboring LorSplanes, a magnetic\norder in the FM state and periods of the L= 1 andL= 2 SL\nreflections. (b) Scattering geometry for azimuth φ= 0◦. The\nscattering vector is along the c-axis, normal to the surface.\n(c) SL momentum scans at the Mn L3, OKedges and with\nhard x-rays (8048 eV).\nB. X-ray absorption\nX-ray absorption spectroscopy (XAS) measurements\nin fluorescence (FY) and electron yield (EY) modes were\nmade at undulator beamline X1B at the National Syn-\nchrotronLightSource. Theincidentlightwas π-polarized\nand the incidence and detector angles were θ= 80◦and\n2θ= 110◦[Fig. 1(b)]. The calculated energy resolution\nwas 0.39 eV and 0 .59 eV at O K(520 eV) and Mn L3\n(640 eV) edges, respectively.\nSL FY and EY measurements at the O Kedge show\ndoped holes on the oxygen sites (Fig. 2). Because the\nprobing depth exceeds the total SL thickness, FY has\ncontributions from both the SL and the substrate. In\ncontrast, because of the short electron escape depth, EY\nmeasurementsarefromtheSLtoplayersonly. Theshoul-der in FY measurements at 530.3 eV is aligned with the\nfirst peak in EY and is not present in FY measurements\nof the bare STO substrate. This shoulder corresponds to\ndoped holes in LSMO (Ref. 24) and to the L= 1 scatter-\ning peak at 529.6 eV. The peak at 531.8 eV is from the\nSTO substrate. The SL FYand EY measurementsat the\nOKedgeshownodiscerniblevariationwith temperature\nbetween 300 K and 255 K.\nSL FY and EYmeasurements at the Mn L3,2edges are\ncompared to FY measurements on bulk samples with dif-\nferent Mn valences (Refs. 25–27) in Fig. 3. SL FY does\nnot have the sharp peak at the lower energies character-\nistic of the Mn2+valence, which shows that the SL has\nonly the Mn3+and Mn4+valences. The SL FY measure-\nment was aligned in energy with the average of bulk FY\nfor the Mn3+and Mn4+valences, according to the num-\nber of SMO (Mn4+valence) and LMO (Mn3+valence)\nlayers in one superperiod. No discernible variation was\nobserved in FY or EY between 300 K and 255 K.\nXAS measurementsprobethe averagevalenceofOand\nMn atoms in the SL. To discern a variation with tem-\nperature in different SL layers it is necessary to turn to\nscattering.\nC. Resonant soft x-ray scattering\nResonant soft x-ray scattering (RSXS) measurements\nwere made at the same beamline with an ultra-high\nvacuum diffractometer. For other RSXS experiments\non bulk and SL at this endstation see Refs. 20,23,28–\n31. Detection was for both π−πandπ−σchannels.\nThe scattering momentum in the reflectivity geometry\nQ= (0,0,2πL/cSL) is indexed with respect to the SL\nsuperperiod cSL. The energy resolution was 0 .20 eV and\n0.34 eV at O K(520 eV) and Mn L3(640 eV) edges, re-\nspectively. The sample was cooled in zero magnetic field\nand scattering measurements for L= 1 at the O Kedge\nand forL= 1,2 at the Mn L3,2edges [Fig. 1(c)] were\nmade at different temperatures (Figs. 2 and 4).\nFM order in metallic films has been studied with x-\nray resonant magnetic scattering of linearly32and cir-\ncularly33polarized light at the Fe and Co Ledges. The\nFMorderinaAg/NiSLhasbeeninvestigatedwithcircu-\nlarly polarized light34. However, unlike previous studies,\nwhere an external magnetic field was applied to separate\nthe magnetic and charge scattering, the SL FM order is\naccessed here at SL reflections with no applied magnetic\nfields.\n1. OKedge line shape\nThe spatial modulation in the density of holes doped\non the oxygen sites can be observed with RSXS. Specifi-\ncally, the O Kedge line shape for L= 1 scattering shows\na peak closeto the energyof the shoulderin FY measure-\nments (Fig. 2).3\n525 530 535 540 545 0510 15 020 40 \n330 K 310 K 290 K 270 K 255 K 185 K 160 K 135 K 110 K 90 K RSXS intensity \n(arb. units) \nEnergy (eV)  FY and EY  \n(arb. units)  FY (SL+STO at 300 K) \n FY (SL+STO at 255 K) \n FY (STO only) \n EY (SL only) \n \n \n  \nSTO \nSL \neL,S \ng,\neL\ng,eL,S \ng,\n2 ,gt↑,ge↑2 ,gt↓,ge↓\n2-3 eV 1-2 eV \nFIG. 2: (Color online) Top: FY at the O Kedge for the\nSL on STO at different temperatures and the bare STO\nsubstrate (black line), compared to EY measurements. FY\nmeasurements were aligned with the linear transformation\nFYplotted=aFYmeasured +b, whereaandbare constants.\nBottom: Temperature dependence of resonant scattering at\nL= 1. The scans have been normalized tothe pre-edge values\nand shifted vertically for clarity. The inset shows the orde r\nof the energy levels for SMO in the ground state, with the\nJahn-Teller splitting neglected. The upper level eg,↓is split\nin LMO.\nSince the order of levels in the RSXS line shape at the\nOKedge follows that of the ground state, the RSXS line\nshape canbe analyzedusingthe hybridizationbetween O\npand Mneglevels in the ground state. XAS experiments\nand calculationsat the Mn Ledge(Ref. 35) givea crystal\nfield splitting 10 Dqbetween the egandt2glevels of 1.5\neV for bulk LMO and 2.4 eV for SMO. Fig. 2 (inset)\nshows the unoccupied eg,↑,t2g,↓andeg,↓levels in SMO.\nThe scattering line shape at the O edge is described well\nby Opstates hybrids with the Mn eglevels in the SMO\nand LMO layers, shown with arrows in Fig. 2, followed\nat higher energy by hybrids with La and Sr states. Two\neg,↓levels (3.5 eV and 5 eV above eg,↑) are present in\nLMO and only one for SMO (3.5 eV above eg,↑) (Ref. 36)\nbecause the electron in the eg,↑level in LMO splits the\nunoccupied eg,↓levels by Coulomb interaction, even in\nthe absence of any Jahn-Teller distortion.\nHowever, no variation across the FM transition is ob-\nserved within the error bars. To access T-dependent in-635 640 645 650 655 0246FE D\nEY 205 K EY 300 K  Mn 2+ \n Mn 3+ \n Mn 4+ L=2 (T=90 K) CBA\n  Intensity (arb. units) \nEnergy (eV)  FY \n(T=300 K )\n FY \n(T=255 K )\nFIG. 3: (Color online) SL EY and FY (lower curves) and\nRSXS (top) at L= 2 at the Mn L3,2edges compared to\nFY measurements for different Mn valences (middle, from\nRef. 26).\nterface states (Sec. III A), it is necessary to reach the\nL= 2 reflection, for which the O Kedge energy is too\nlow. In contrast, L= 2isaccessibleatthe Mn L3,2edges.\n2. MnL3,2edge valences\nThere is no discernible variation in the line shape at\nL= 1 at the Mn L3,2edges across the FM transition\ntemperature (Fig. 4). However, a pronounced variation\nis visible for L= 2. From under the relatively broad\nXAS at the Mn Ledge, T-dependent RSXS at L= 2\nselects those states that are sensitive to the temperature\nvariation. Specifically, an increased intensity of the A,\nBandCpeaks at lower T, and a decreased intensity of\ntheαpeak is observed at the L3edge (Fig. 4). Parallel\nvariations occur at the L2edge for peaks E,Fandβ.\nSimilar results were obtained for azimuthal angle φ=\n45◦(Fig. 5), consistent with the azimuthal dependence\nof magnetic scattering. The temperature dependence of\nheight and width of peak Care shown in Fig. 6.\nCoulomb and exchange interactions for ground and\nRSXS excited states are different at the Mn L3,2edges.\nAn analysis of the RSXS line shapes based on ground\nstatecalculations,similartothatattheO Kedge,cannot\nbe made. However, both FY and RSXS measurements4\nprobe excited states and FY measurements on bulk sam-\nples for different Mn valences will be used to identify the\nvalence of scatterers for different resonances in the RSXS\nline shapes. This approach is supported by the relatively\nsmall difference in energy between RSXS resonances for\nscatterers of the same valence and different specific scat-\ntering contrasts ( /lessorsimilar1 eV between magnetic and orbital\nscattering for bulk measurements13,15) compared to the\n∼2 eV separation in energy between the main FY fea-\ntures for different valences (Fig. 3).\nFY measurements in Sec. II B showed that the Mn2+\nvalence is absent in the SL. The AandBresonances\nin the RSXS line shape for L= 2 correspond to the\nMn3+valence, while resonance Clines up at the energy\nof FY edge for a Mn4+valence. More T-dependent mea-\nsurements on SL with different superperiods are needed\nbefore a quantitative discussion of peak α.\nIII. DISCUSSION\nThe model of Sec. III A relates the variation with\ntemperature of the scattering contrast for L= 2 and the\nabsence of variation for L= 1 to changes in the form fac-\ntorδfiof interface layers and interface ferromagnetism.\nThe variation of δfi=fi,FM−fi,PMwith the transi-\ntion from the PM to the FM state for specific interface\nmagnetic and orbital x-rayscattering models is discussed\nin Sec. III B.\nA. Interface ferromagnetism\nThe line shape and structure factor at the Mn edge at\nL= 2(Fig. 4)aremadeoftwoparts. TheT-independent\nstructurefactor Sis givenbythe SLstructure, to whicha\nT-dependentcontribution δSisaddedwiththetransition\nto the FM state [Fig. 6(a)].\nThe scattering intensity follows the evolution with\ntemperature of the FM moment [Fig. 6(a)]. The pos-\nsibility that the transition to the FM state gives exclu-\nsively non-magnetic scattering contrast is not supported\nby an analysis of the line shapes (Sec. III B). The T-\ndependent scattering δSis, at least partially, magnetic in\norigin and, for simplicity, we discuss only x-ray resonant\nmagnetic scatteringin this section. Orbitalcontributions\nto temperature variations in line shapes are addressed in\nSec. III B.\nThe scattering intensity is I∝/vextendsingle/vextendsingleS+δS/vextendsingle/vextendsingle2. The scat-\ntering form factors ( f)mnand (δf)mnare tensors, which\nare multiplied with the final (ˆ ǫf) and initial (ˆ ǫi) light po-\nlarization vectors. This gives an overall factor which, for\ncharge (f) and magnetic ( δf) scattering, is\nS∝ˆǫ∗\nf,m(f)mnˆǫi,n∝(ˆǫ∗\nfˆǫi)f(ω) (1)\nδS∝ˆǫ∗\nf,m(δf)mnˆǫi,n∝i[(ˆǫ∗\nf׈ǫi)ˆzl]δf(ω) (2)635 640 645 650 655 660 0510 15 20 0510 \n330 K 310 K β\nL=1 \nL=2 α Intensity (arb. units) \n295 K 275 K 260 K 240 K 220 K 205 K 185 K 165 K 145 K 125 K 105 K 90 K \nEY 300 K \nEnergy (eV)  \n  330 K 310 K 280 K 255 K 90 K \n Mn 3+ Mn 4+ \nFIG. 4: (Color online) Temperature dependence of RSXS at\nthe MnL3,2edges atL= 1 andL= 2 for azimuthal angle\nφ= 0◦.\nwhere ˆzlis the direction of the local moment at site l\n(Refs. 32,34). The sum over the in-plane sites lfor each\nlayeris proportionalto the magnetization− →Mofthe layer.\nf(ω) andδf(ω) are scalar functions.\nRSXS peaks that persist above the FM transition tem-\nperature are due to scattering contrast defined by the SL\nstructure, which is held constant by the internal field\nbetween Sr2+and La3+ions arranged in the SL layers.\nWith the form factors shown Fig. 1(a) for “interface”\n(fi), “near-interface” ( fni), “middle SMO” ( fS) and\n“middle LMO” ( fL) layers, and neglecting inter-diffusion\nroughnessand structuraldifferences between SMO/LMO\nand LMO/SMO interfaces, the T-independent structure\nfactorsS(Q) =/summationtext\nlfleiQzlatL= 1,2 are\nS(L= 1) =−fL+fS+fi−fni (3)\nS(L= 2) =fL+fS−fi−fni (4)\nThe origin has been chosen so that an arbitrary phase\nfactor between S(L= 1) andS(L= 2) is zero.\nHigher momenta Lare not accessible at the Mn L5\nedges. However, the absence of a variation with temper-\nature in the scatteringintensity at L= 1 [δS(L= 1) = 0]\nand the variation at L= 2 [δS(L= 2)∝ne}ationslash= 0] strongly sug-\ngests that the unit cell of the T-dependent contribution\nto the structure factor ( δS) is half the SL superperiod or\n3 ML. The middle of the SMO and LMO layers (sepa-\nrated by 3 ML) are the most dissimilar parts of the SL\nstructure, while the SMO/LMO and LMO/SMO inter-\nfaces (with two interfaces every superperiod, also sepa-\nrated by 3 ML) are similar. In the following, we consider\na scattering component δSthat develops at the SL inter-\nfaces.\nThese conditions are contrary to those expected for\nscattering from crystal field effects or structural differ-\nences in a SL, from either differences in the c-axis lattice\nconstant or Jahn-Teller distortions, which do not have a\n3 ML unit cell. In addition, no discernible variation was\nobserved for scattering at the La M5,4edges forL= 1\norL= 2 between 300 K and 225 K (data not shown).\nThis shows that the change in the line shape is due to a\nvariation with temperature in the resonant form factors\nof the SL layers ( δfl), not of structural factors ( δzl).\nIn addition, these symmetry conditions on δSset\nmore stringent constraints on the variation of δflin the\nSL, beyond the experimental observation δS(L= 1) =\n−δfL+δfS+δfi−δfni= 0. To obtain a 3 ML unit cell,\nthe variation in the middle of the LMO and SMO layers\nmust be equal, δfL=δfS. In addition, the variation\nin the interface and near-interface layers must be equal,\nδfi=δfni[Fig. 1(a)].\nFM order in the iandnilayers is consistent with esti-\nmates of the average Mn valence in a MnO 2plane based\non the type ( LorS) of neighboring planes [Fig. 1(a)].\nSpecifically, a comparison to magnetic orders of equiv-\nalent bulk LSMO doping shows that Mn3+and Mn3.5+\nvalencesareneartheFMdomeforbulkLSMO.Themag-\nnitude of the FM moment depends on the Mn valence\nand implicitly on the SL interface roughness, with struc-\ntural imperfections in a (SMO) 4.4/(LMO) 11.8SL corre-\nlatingwiththeaverageinterfaceFMmoment.38However,\nthe FM moment distribution is more symmetrical in the\nsmaller superperiod n= 3 SL (Ref. 39), consistent with\nthe symmetric FM moment distribution in Fig. 1(a).\nThe 3 ML unit cell of T-dependent scattering shows\nthat there must be two regions within a superperiod\nwhich are different from the FM iandnilayers. With\none the middle of the SMO layers ( fS), the other must\nbe the middle of the LMO layers( fL). The x-ray scatter-\ning measurements imply that the magnetic scattering in\nthesetwolayersis thesame( δfS)mag= (δfL)mag. There-\nfore, these layers have either the same magnetization− →M\nor no magnetization at low temperatures. The different\nhole doping of these layers does not support the possibil-\nity of an equal magnetization. The remaining possibility\nis that, as the SL is cooled and becomes FM in zero ap-\nplied field, there is no magnetization in both these layers,\nor (δfS)mag= (δfL)mag= 0. Therefore, for the n= 2\nSL, the variation in magnetic scattering ( δf)magand FM(a) \n(b) Imin Imax \nT=255 K \nT=300 K ABCEF\nD\nαβ\nFIG. 5: (Color online) Two-dimensional resonance profiles a t\nthe MnL3,2edges at 255 K and 300 K for azimuth φ= 45◦.\nphase are localized at the SL interfaces.\nA model of the magnetic state for the FM SL is shown\nin Fig. 1(a), where IandIIrepresent magnetic phases\nof thefSandfLlayers in no applied field (in con-\ntrast, the polarized neutron reflectivity measurements in\nRefs. 38,41 were made in applied fields). Since the aver-\nage magnetization is zero for both IandIIphases, the\nmagnetic scattering has a 3 ML period. There are sev-\neral different possible IandIIphases: a PM phase, an\nordered AFM phase (for instance, a C-type or a G-type),\nor an irregular phase with canted moments41pointing in\ndifferent directions in the sample regions with slight vari-\nations in local doping3, even though the moments in the\nfiandfnilayers are always parallel.\nThe valences of the fLandfSlayers are close to\nMn3+and Mn4+, which correspond to A-type and G-\ntype AFM magnetic orders in bulk LMO and SMO, with\ntransition temperatures of T N,LMO= 135 K (Ref. 37)\nand T N,SMO= 235 K (Ref. 21), respectively. This sug-\ngests PMIorIIphases, at least for the higher tem-\nperature range, below the SL FM transition tempera-\nture of 305 K [Fig. 6(a)]. However, the SL saturation6\nFM moment of ∼2.5µBat 5 K (Ref. 40) gives the ex-\ntent along the c-axis of the FM region in high fields of\n/greaterorsimilar(2.5/3.22)×6 ML∼4.65 ML for each superperiod,\nwhere∼3.22µBis the maximum FM moment of the\nx= 0.33 alloy39. This value is too high for both IandII\nphases to remain PM at the lowest temperatures. There-\nfore, at least one magnetic transition, other than the FM\ntransition, occurs in the SL.\nThe RSXS measurements are consistent with transi-\ntions in the fSandfLlayers from a PM phase at higher\ntemperatures to either a G-type or C-type AFM phase\n(near the Mn4+doping of bulk LSMO) and to an irreg-\nular canted phase (near the the Mn3+doping3) at lower\ntemperatures, respectively. However, since there is no\naverage layer magnetization in all these cases, the RSXS\nintensity does not vary at these transitions, in contrast\nto the FM transition in the iandnilayers.\nWith these constraints on δfl, the change in the struc-\nture factor δS(Q) =/summationtext\nlδfleiQzlat the FM transition\nandL= 1,2 from Eqs. (3)-(4) is\nδS(L= 1) = 0 (5)\nδS(L= 2) =−2δfi (6)\nEq. (6) relates the changes with temperature in the line\nshape atL= 2 to variations of form factor of interface\nδfiand near-interface δfni=δfilayers in the SL.\nTheδS(L= 2) reflectionis allowedin this n= 2 SL for\nall Mn sites in the FM layers. However, similar to AFM\norders in bulk LSMO, scattering from a Mn4+valence\nwasnotobservedat L= 3foran= 4SL (Ref. 20) [it was\nobserved in a n= 3 SL atL= 3 (data not shown)]. The\nsymmetry that very effectively forbids reflections from\nthe Mn4+ions atL= 3 for the n= 4 SL is not known\nandsurprising,giveninherentsmallimperfectionsofaSL\nstructure. More measurements are needed for different\nSL to answer this question.\nB. Interface x-ray scattering\nWe now discussthe temperaturevariationofthe RSXS\nline shape at L= 2 and Mn L3,2edges.\nThe width of resonance C, corresponding to the Mn4+\nvalence and to interface and near-interface layers, has an\nsharp increase at the FM transition temperature [Fig.\n6(b)]. The increase in the scattering intensity in the FM\nstate is also taking place ∼0.2 eV below the chargeorder\nresonance that corresponds to the Mn4+valence in the\nPM state (at 644 .65 eV in Fig. 4).\nIn general, the line shape of resonantmagnetic scatter-\ning is related to variations in the occupation of orbitals\ninduced by a magnetic field42near the FY edges for Mn\nions of different (Mn3+and Mn4+) valences. However,\nthe magnetic scattering is slightly shifted to lower en-\nergies compared to orbital scattering for AFM bulk or-\nders.13,15We cannot resolve two peaks at Cin the SL\nline shapes at low T, but this suggests that, with the(a) \n0 100 200 300 400 10 -4 10 -3 \n0.00 0.75 1.50 2.25 \n R(B=0 T) \n R(B=9 T)  \n R (ohm-cm) \nT (K) \n Moment ( µB/f.u.) \n FC (200 Oe) \n ZFC \n0.00 0.25 0.50 0.75 \n Mn 4+  scattering \nheight (arb. units) \n \n(b) \n50 100 150 200 250 300 350 0.7 0.8 0.9 1.0 1.1 1.2 Mn 4+  scattering FWHM (eV) \nT (K) 1ψ\n2ψ\n()()Mn O Mn i j− − \nFIG. 6: (Color online) (a) Temperature dependence of res-\nonanceCheight forφ= 0◦compared to the SL FM mo-\nment measured with SQUID for ZFC and in-plane FC=200\nOe (Ref. 40). The SL has a 305 ±5 K FM transition tempera-\nture, which is lower than the ∼355 K transition temperature\nof thex= 0.33 LSMO alloy (Ref. 39). Hysteresis loops show\nthat the easy axis is in-plane (data not shown). SL resistanc e\nbecomes metallic-like at low T (Ref. 40). (b) Temperature\ndependence of resonance Cwidth. The line is a guide to the\neye. Inset shows a sketch of the double-exchange configura-\ntions|ψ1/angbracketrightand|ψ2/angbracketrightfor Mn sites iandj(Ref. 4).\nincrease of the FM moment at lower T, a T-dependent\nmagneticscatteringcontributionisadded ∼0.2eV below\nthe charge scattering resonance. This addition to fiof\na temperature dependent ( δfi)magexplains the observed\nvariation in line shape at L= 2. The charge scatter-\ning resonance might also increase at lower T, concomi-\ntantly with magnetic scattering and variations in orbital\nscattering with T are discussed briefly at the end of this\nSection.\nA more gradual increase in width is observed at lower\nT [Fig. 6(b)]. For x-ray scattering in the FM phase, it is\nnecessary to consider a double-exchange two-site orbital,\nwhichsuggeststhat thiswidth increaseisrelatedtotheT\ndependenceofthedouble-exchangefrequency tijbetween\nthe two Mn sites. Both resonant magnetic and orbital\nscattering are ultimately scattering off orbitals, and the\nconsideration of two-site orbitals in the FM state applies\nto both cases.\nThe double-exchange process involves two coordinated\njumps from the Mn to the O atoms [Fig. 6(b), inset].7\nIt is useful to consider the simpler process of one jump\nfirst, which is sometimes included in XAS calculations of\ncomplex oxides. In this case, inter-site charge transfer\nbetweend-states of a transition metal and a neighboring\n(ligand,L) O ion43and consideration of multiple con-\nfigurations (for instance, d8andd9Lfor Cu1+and Cu2+\nvalences) change the scattering form factor fat the tran-\nsition metal edge. In particular, satellite peaks develop\nin XAS (and, implicitly, in RSXS) at additional Cu va-\nlences.44\nIn the double-exchange process, specific to FM com-\nplex oxides, charge transfer takes place between transi-\ntionmetalsites,beyondtheneighboringOatoms. Specif-\nically, the |ψ1∝an}bracketri}htand|ψ2∝an}bracketri}htconfigurations are coupled in a\ntwo-site ground state wave function [Fig. 6(b), inset],\nwhich for this FM manganite is\n|ψ±∝an}bracketri}ht=1√\n2(|ψ1∝an}bracketri}ht±|ψ2∝an}bracketri}ht) (7)\n|ψ1∝an}bracketri}ht=|Mn3+,O2−,Mn4+∝an}bracketri}ht\n|ψ2∝an}bracketri}ht=|Mn4+,O2−,Mn3+∝an}bracketri}ht\nwith Mn valences in FM layers in a superposition of\nMn3+and Mn4+. In the ground state (without an x-\nray photon absorbed), the charge transfer splits the two\nlevels|ψ±∝an}bracketri}htby the exchange energy 2 tij(Ref. 4), where\nthe double-exchange hopping between sites iandjis\ntij=tDEcos[(θi−θj)/2], withtDEa constant and θi,j\nthet2g,↑spins orientations on the two sites (Ref. 5).\nTo account for the double-exchange process in x-ray\nscattering, the orbitals |ψ1,2∝an}bracketri}htare replaced with the two-\nsite orbitals |ψ±∝an}bracketri}ht. Similar to the case of ligand holes\non oxygen atoms, the charge transfer between Mn sites\nbeyond the neighboring O atoms changes the scattering\nfactorfat satellite peaks in RSXS, which correspond to\nthe Mn3+and Mn4+valences.\nIn addition, the splitting by 2 tijof the|ψ±∝an}bracketri}htstates\nresults, for our relatively large energy resolution and core\nhole width, in an increase of the measured width by 2 tij.\nMoreprecisely,thethebandwidthof egelectronsdepends\non the hopping frequency between the iandjsites as37\nW∝cos[(θi−θj)/2]cosφ∝tijcosφ (8)\nwhere (π−φ) is the angle between the Mn-O-Mn bonds.\nA width increase at lower T in the ground state is trans-\nferred to a increase of the RSXS line width.\nThe hoppingfrequency tijincreaseswith increasedFM\norder of spins θi,jat lower T, and broadens the scatter-\ning form factors fand the line width. In this model,\nthe XAS and therefore, the RSXS peaks, should become\nbroader at lower temperatures. The width increase at\nlower temperatures of peak C[Fig. 6(b)] is consistent\nwith this model and tband∼0.2−0.5 eV for each of the\negstates, 2tDE∼2TCurie∼0.05 eV and contributions\nfrom experimental resolution (0 .34 eV at the Mn Ledge)\nand core-hole width ( wFWHM∼0.3−0.5 eV, Ref. 8).\nInadditiontothedouble-exchangeprocessesintheFM\nstate, lattice distortions are also relevant to the CMRtransition45,46. In bulk manganites, they may depend on\nT,changingthebondalignment φandbandwidth W[Eq.\n(8)]. However, the average angle between the Mn-O-Mn\nbonds for SL samples is fixed by the substrate.\nOrbital scattering at the Mn Ledges has a compa-\nrable amplitude to magnetic scattering for bulk AFM\norders.13,15It can come from occupation contrast or po-\nlarization contrast from different atomic orbital orien-\ntations in the anomalous scattering tensor. The anal-\nogous occupation contrast in SL FM is a T-dependent\ncharge transfer across SL interfaces (which includes the\nelectronic reconstruction of Ref. 20), in addition to the\nT-independent part defined by the SL structure. The T-\ndependent polarization contrast in the SL may also be\nsubstantial; for instance, on closely related SL (Ref. 47),\nin-planeeg(x2−y2) occupation and FM near LMO in-\nterfaces and out-of-plane eg(3z2−r2) orbital occupation\nand AFM in the middle of LMO layers was inferred from\nXMLD and XMCD measurements. Polarization-resolved\nscattering measurements in a magnetic field with πand\nσincident light and scattered beam polarization analysis\nare necessary to separate different magnetic and orbital\ncontributions to scattering at the Mn L3,2edges.\nWe discuss the O Kedge briefly. Oxygen doping is\nconsistent with our observations (Fig. 2), other measure-\nments24and certain models14. The interface FM state of\nthisn= 2 SL is the metallic state observed in a n= 4\nSL (Ref. 20). The reflection at L= 1 (Fig. 2) is not sen-\nsitive to T-dependent scattering because, as for the Mn\nLedges,δS(L= 1) = 0. The intensity of the L= 3 re-\nflection in Ref. 20 corresponds to the L= 2 reflection for\nthis SL. Inparticular, to determinewhether T-dependent\nscattering occurs at the O Kedge in this SL, it would\nbe necessary to measure at L= 2, a momentum which is\nnot accessible at the O Kedge.\nIV. CONCLUSION\nX-ray absorption measurements at the O Kedge in a\nSrMnO 3/LaMnO 3superlattice showed a shoulder, corre-\nsponding to holes doped on oxygen sites. The shoulder is\nalignedwith the main resonantpeak ofsoft x-rayscatter-\ning from the spatial modulation in the density of doped\nholes.\nA large variation in the Mn L3,2line shapes at L= 2,\nbut not atL= 1, was observed across the FM transition,\npointing to scattering from ferromagnetic interfaces.\nComparison to fluorescence yield edge energies for dif-\nferentMn valencesshowedthe presenceofscatteringcon-\ntrast at both Mn3+and Mn4+valences. An x-ray scat-\ntering model, which includes double-exchange orbitals in\nthe FM state, explains the observed line broadening at\nlower temperatures.\nMeasurements on bulk 113 (Ref. 8), 214 (Ref. 9–11)\nand 327 (Ref. 12) manganites in the AFM state observe\ntwo main resonances only at the Mn L3edge, atAand\nB. Scattering from Mn4+ions (corresponding to res-8\nonanceC) has the symmetry of a forbidden reflection.\nSpecifically, it has a spatial periodicity of 2 u.c. and is\nnot allowed at the bulk in-plane orbital order reflection\nwavevector of 4 u.c. along the tetragonal axes (Ref. 8,9).\nHaving to rely on measurements of the Mn3+resonances\nonly, different methods to determine the charge dispro-\nportionation for bulk AFM orders are controversial, with\nboth small and large chargedisproportionation obtained.\nOur RSXS line shapes, for a SL structure with a large\nintrinsic charge disproportionation, add an experimental\nconstraint on these competing models.\nThe development of the SL FM order was accessed\nwith x-ray resonant magnetic scattering and no applied\nmagnetic fields. An open question is the trace [FC or\nZFC in Fig. 6(a)] that the height of resonance Cwould\nfollow on further cooling.\nWe would like to contrast our measurements to polar-\nized neutron reflectivity (PNR) data on SMO/LMO su-\nperlattices (Refs. 38,41), where a magnetic modulation\nwas measured with a period equal to the SL superpe-\nriod (magnetization strongly suppressed in SMO, high in\nLMO). In contrast, the RSXS measurements presented\nhereshowanorderingofmagneticmomentswithaperiod\nequal to half the SL superperiod. Several factors may be\nat the origin of this difference. First, the experimentalconditions of the PNR and RSXS measurements were\ndifferent. Specifically, PNR measurements were made in\nrelativelyhighfields (0 .55T and0.82T inRef. 38and41,\nrespectively), while the RSXS measurements were made\nwith no applied fields. 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The antiferromagnetic resonance is clearly reflected in both the transmission\nthrough the waveguide as well as the DC voltage in the Pt strip. The DC voltage remains largely unaffected\nby field reversal and thus presumably stems from microwave rectification and/or heating effects. However, we\nidentify a small magnetic field orientation dependent contribution, compatible with antiferromagnetic spin\npumping theory.\nI. INTRODUCTION\nFerromagnetic resonance1describes the collective, res-\nonant excitation of magnetic moments in a ferromag-\nnet and is a powerful tool to investigate a specimen’s\nmagnetic properties. In ferromagnet/normal metal het-\nerostructures, ferromagnetic resonance furthermore rep-\nresents an effective source of pure spin currents2, i.e. an-\ngular momentum transport without an associated charge\nflow. In the so-called spin pumping process, the nor-\nmal metal acts as a spin sink, accepting non-equilibrium\nangular momentum from the resonantly driven magne-\ntization in the adjacent ferromagnet. If heavy metals\nwith large spin-orbit coupling are used as spin sinks, the\nspin current can be converted into an electrical current\nby means of the inverse spin Hall effect3,4and thus de-\ntected electrically. The electrically detected spin pump-\ning process has been investigated in a large variety of\nmagnetic materials5–7, including magnetic insulators8,9\nsuch as yttrium iron garnet. The magnetic materials\nhereby are either ferro- or ferrimagnetic which brings\nalong some limitations for experiments and potential ap-\nplications. At moderate (few Tesla) magnetic fields ferro-\nand ferrimagnets are limited to magnetic resonance fre-\nquencies of a few to a few ten GHz and their finite mag-\nnetization makes them susceptible to e.g. unintended\nstray fields. While some of these features can be useful\nfor certain applications, faster dynamics and resilience\nto external factors are often priority design goals. An-\ntiferromagnets offer magnetization dynamics at several\nhundred GHz even at zero external field, have negligi-\nble stray fields and are virtually immune to unintended\nchanges in their magnetization state. These qualities at\nthe same time, however, make it challenging to investi-\ngate antiferromagnetic dynamics. One option is to apply\nlarge magnetic fields to tune the antiferromagnetic reso-\na)Electronic mail: philipp.ross.13@ucl.ac.uk\nb)Electronic mail: michael.schreier@wmi.badw.denance frequency to a frequency range covered by conven-\ntional magnetic resonance spectrometers. Furthermore,\nthe properties of antiferromagnets with respect to spin\ncurrents are largely unexplored. However, recent exper-\niments demonstrate spin current transmission through\nantiferromagnets10and theory predicts11that antiferro-\nmagnets also could be useful as spin current source.\nHere we present antiferromagnetic resonance experi-\nments on the antiferromagnetic insulator MnF 212close\nto the spin-flop transition. We simultaneously record the\nmicrowave absorption signal and the DC voltages aris-\ning along a thin Pt strip deposited on top of the MnF 2\ncrystal. While the antiferromagnetic resonance is clearly\nreflected in the DC voltage, the voltage does not invert\nsign under reversal of the external field as one would\nexpect for conventional spin pumping. This indicates\nthe presence of additional DC voltage generation mecha-\nnisms, potentially rectification by means of the spin Hall\nmagnetoresistance, and/or thermal effects. A small but\nfinite change in the amplitude of the DC voltage under\nfield reversal may be attributed to spin pumping from\nMnF 2into Pt.\nII. THEORY\nWe consider a simple antiferromagnet with two mag-\nnetic sublattices and a parallel aligned external magnetic\nfieldH0. As detailed in Refs. 13 and 14, the fundamen-\ntal (exchange) magnetic resonance mode of such a system\nallows for the two solutions\nω±\nres=γµ0/bracketleftBig\n±H0+/radicalbig\nHA(2HE+HA)/bracketrightBig\n. (1)\nThe above, however, only holds for external fields H0be-\nlow a critical field HC, which marks the so-called spin-flop\ntransition. When H0> H cboth sublattice magnetiza-\ntions rotate to a new configuration nearly perpendicular\nto the magnetic field and bend towards H0with increas-\ning field strength. As the two sublattices are degenerate\nin this configuration, the magnetic resonance responsearXiv:1507.04490v1  [cond-mat.mtrl-sci]  16 Jul 20152\n0 3 6 9 120200400600\nμ0H0(T)f(GHz)ωres+\nωres-\nωresμ0HC\nFIG. 1. Antiferromagnetic resonance in MnF 2for external\nfields along the magnetic easy-axis. Below the spin-flop field\nµ0HC≈9.5 T magnetic resonance can occur for two different\nfrequencies, one increasing ( ω+\nres) and one decreasing ( ω−\nres) as\nH0increases.ω−\nreseventually approaches zero as H0→HC.\nAboveHCthe two collapse to a single resonance line.\nreduces to a single line at the frequency15\nωres=γµ0/radicalBig\nH2\n0−HA(2HE+HA). (2)\nIn Eqs. (1) and (2) above γis the gyromagnetic ratio,\nµ0is the vacuum permeability, HEis the exchange field,\nHAis an uniaxial anisotropy field and HHH0is assumed\nparallel toHHHA. MnF 2is well approximated in this pic-\nture usingµ0HE≈53 T,µ0HA≈0.85 T,µ0HC≈9.5 T\n(Refs 12, 16, and 17) and γ= 0.92γe. Whileω+\nresis typi-\ncally too large to be accessible using standard microwave\nequipment, ω−\nresapproaches zero frequency when H0is\nclose toHC(cf.Fig. 1). Even a small misalignment\nθH(cf.Fig. 2) between HHH0andHHHA, however, will shift\nω−\nres(H0≈HC) upwards by approximately 4 GHz/deg\nagain.\nIn (ferro-)magnetic resonance the magnetic material\n“pumps” a spin current into an adjacent normal metal,\nwhich thereby provides an additional damping channel\nfor magnetization excitation in the magnetic material2.\nIn electrically detected spin pumping this spin current\ngenerates an electric field EISHalong the normal metal\ndue to the inverse spin Hall effect3,4. AsEISHdepends\non the relative orientation between voltage-taps and spin\ncurrent spin polarization (which itself is linked to the\nmagnetization orientation of the ferromagnet), the volt-\nageVISHmeasured in experiment is antisymmetric with\nrespect to a reversal of the external magnetic field, i.e.\nVISH(+H0) =−VISH(−H0)6,18. Theory suggests11that\nthe same should also hold for antiferromagnetic reso-\nnance experiments, with the spin current spin polariza-\ntion determined by the antiferromagnet’s N´ eel-vector.\nhrf\ngold wir e\ncontact padsCPW with\nSU-8 isolationlayerMnF 2 with Pt strip  \nMini-SMP\nconnectorsFIG. 2. The MnF 2/Pt sample is placed on a coplanar waveg-\nuide structure capped with an insulating resist layer. The Pt\nstrip is contacted via Au wires to enable detection of DC volt-\nages. The external magnetic field is applied along the mag-\nnetic easy-axis of MnF 2, however, the mechanical mounting\nlimits alignment precision to a few degrees.\nIII. SAMPLE AND EXPERIMENTAL SETUP\nFor the experiments we use a commercial MnF 2single\ncrystals in the shape of a 3 mm ×1 mm×0.5 mm cuboid\nwhose magnetic easy-axis points along the 0 .5 mm side.\nAfter polishing the sample a 3 mm ×250µm, 7 nm thick\nPt film is deposited on the 3 mm ×0.5 mm side of the sam-\nple by electron beam evaporation using a shadow mask.\nThe sample is then placed on a coplanar waveguide such\nthat the microwave field is perpendicular to the magnetic\neasy-axis (Fig. 2) and thereby able to effectively excite\nthe antiferromagnetic resonance12. The waveguide in-\ncluding the sample mounted on top is placed in a liquid\nHelium magnet cryostat which provides a magnetic field\nalong the magnetic easy-axis of the MnF 2crystal. All ex-\nperiments are performed in He exchange gas at pressures\nof few mbar and a temperature of 4 K /lessmuchTN´ eel\nMnF 2≈67 K19.\nIV. RESULTS\nFigure 3b shows a false color plot of the transmission\n|˜S21|2through the coplanar waveguide for microwave fre-\nquencies 15 .5 GHz≤f≤19.5 GHz and external field\nstrengths of 9 T≤µ0H0≤9.9 T. The data were recorded\nusing a vector network analyzer by performing magnetic\nfield sweeps at fixed frequencies ( Pmw= 11 dBm). For\nclarity, the transmission at an off-resonant offset value\n|S21(H0= 9 T)|2(Fig. 3b) [|S21(H0=−9 T)|2, Fig. 3a]\nhas been subtracted from each field sweep. The ap-\nproximately parabolic feature with reduced transmis-\nsion corresponds to the excitation of the antiferromag-\nnetic resonance by the microwave field. As discussed in\nSec. II the resonance frequency does not drop to zero at\nH0=HC≈9.5 T but remains finite due to a small mis-\nalignment of about θH≈4.21◦, inadvertently introduced3\n|S\n212\n0\n-0.04-0.03-0.02-0.01\n-9.9 -9.6 -9.3 -9.016171819\nμ0H0(T)f(GHz)a\n|S\n212\n0\n-0.04-0.03-0.02-0.01\n9.0 9.3 9.6 9.916171819\nμ0H0(T)f(GHz)b\nVDC(µV)\n-10+1\n-9.9 -9.6 -9.3 -9.016171819\nμ0H0(T)f(GHz)c\nVDC(µV)\n-10+1\n9.0 9.3 9.6 9.916171819\nμ0H0(T)f(GHz)d\nFIG. 3. Transmission change through the coplanar waveguide in vicinity of the critical field ( a,b). The antiferromagnetic\nresonance along with some spin wave modes are well reflected as a drop in the transmitted power. Only minor differences exist\nbetween the data for positive ( b) and negative ( a) applied external magnetic fields. The dashed line is a simulation of the\nantiferromagnetic resonance frequency for a misalignment between uniaxial anisotropy axis and the external magnetic field of\n4.21◦. The simultaneously recorded DC voltage along the Pt film ( c,d) closely resembles the transmission data. Differences\nbetween positive and negative applied external fields are, again, small but a slight change in amplitude is visible.\nwhen mounting the sample. The dashed line indicat-\ning the antiferromagnetic resonance frequency is calcu-\nlated based on an extension of the model by Skrotskii\nand Kurbatov20to two sublattices. When using the mis-\nalignment given above and the parameter set introduced\nin Sec. II the experimental data are well reproduced.\nUpon closer inspection at least two spin wave modes21\ncan be observed. Figure 3a shows data recoded in the\nsame fashion for fields applied antiparallel to HHHA, i.e.\n−9 T≥H0≥−9.9 T. As expected this spectrum closely\nmirrors the one for positive external magnetic fields, with\nonly minor differences which we will discuss in Sec. V.\nFigures 3c and 3d show the DC voltage across the\nPt strip recorded simultaneously with the transmission\ndata. As with the transmission data an offset voltage\nVDC(H0=−9 T) (Fig. 3c) [ VDC(H0= 9 T), Fig. 3d] was\nsubtracted from the raw data for clarity. Clearly, the an-\ntiferromagnetic resonance is also visible in VDC, as are theindividual spinwaves. In contrast to |˜S21|2, however, the\ndifferences between VDC(H0>0) andVDC(H0<0) are\nmore pronounced. Although the sign of the DC voltage\nremains unchanged under field reversal the magnitude of\ntheVDC(H0<0) data is slightly less than that of the\nVDC(H0>0) data. It is worth noting that, in the imme-\ndiate vicinity of HC, neither transmitted power nor DC\nvoltage seem to be significantly affected by the spin-flop\ntransition.\nV. DISCUSSION\nThe character of the data prevent a clear identification\nof the mechanism(s) responsible for the DC voltage\ngeneration. It is apparent from Figs. 3c and 3d that VDC\nis largely unaffected by the field reversal. Since these\nvoltages are not compatible with the characteristic sign4\nΔ|S\n212\n0\n-0.04-0.03-0.02-0.01\n9.0 9.3 9.6 9.916171819\n|μ0H0| (T)f(GHz)a\nΔVDC(µV)\n-10+1\n9.0 9.3 9.6 9.916171819\n|μ0H0| (T)f(GHz)b\nFIG. 4. ( a) Transmitted power difference between negative\nand positive applied external magnetic fields. Both positive\nand negative values are recorded, however, the average value\nis less than zero. This indicates more efficient microwave ab-\nsorption at negative fields. ( b) DC voltage difference between\nnegative and positive applied external magnetic fields. A clear\nbias towards positive values is observed, signifying the larger\nmagnitude of the DC voltage for positive external magnetic\nfields.\ninversion upon field reversal expected from electrically\ndetected spin pumping in ferromagnets, a different\nmechanism must be dominating.\nMicrowave absorption by the antiferromagnet is\ntied to the antiferromagnetic resonance ( cf. Figs. 3a\nand 3b) and can heat the sample substantially22,23. If\nthe heating is nonuniform across the sample a thermal\nvoltage can be generated in the Pt strip24. At the\nincident microwave power of 11 dBm and given the\nlarge thermal conductivity of the sample25,26, however,\nit appears unrealistic to expect nonuniform heating of\nmore than few ten millikelvins between the two contacts\nseparated by a distance of 300 µm. On the other hand,\nwith the Seebeck coefficient for the Pt-Au-wire-junction\nwell below 1 µV/K27,28atT= 4 K, it would take a\ntemperature difference between the two voltage contacts\nof more than 1 K to account for all of the magnetic fieldorientation independent offset.\nThe spin Seebeck effect29is another possible spurious\neffect and may occur if the temperature profile has a\nfinite slope along the MnF 2/Pt interface normal. The\ndetection of the effect, however, relies on the same\nmechanism as spin pumping and should therefore also\ndepend on the sign of the external magnetic field. Also,\nas supported by first experiments30, the spin Seebeck\neffect is predicted31to vanish in antiferromagnets.\nThus, the spin Seebeck effect is an unlikely candidate to\naccount for the dominant features detected in the DC\nvoltage.\nFilms in which the electrical resistivity is tied to the\nmagnetization orientation can show a so-called mi-\ncrowave rectification effect32. In these materials, under\nmagnetic resonance, the resistivity oscillates at the same\nfrequency as the external driving field. Simultaneously\nthe microwave driving field can induce an oscillating\ncharge current in the material. The product of the\nhigh frequency resistance and the charge current results\nin a DC voltage. While MnF 2itself is insulating, the\nresistivity of the Pt film may be linked to the sublattice\nmagnetization orientations in the MnF 2by means of the\nspin Hall magnetoresistance33. Unfortunately spin Hall\nmagnetoresistance theory34can not readily be applied to\nantiferromagnets. The field dependence of the resulting\nantiferromagnetic spin Hall magnetoresistance mediated\nrectification effect35(if any) is thus unclear as of writing\nthis manuscript.\nTo analyze the difference between positive and\nnegative applied external field in more detail we\ncompute the microwave transmission difference\n∆|˜S21|2=|˜S21(H0<0)|2−|˜S21(H0>0)|2(Fig. 4a).\nWe find that/angbracketleft∆|˜S21|2/angbracketright<0, where/angbracketleft···/angbracketright denotes the\naverage over all data points. In contrast, the analogously\ncomputed ∆ VDC=VDC(H0<0)−VDC(H0>0)\n(Fig. 4b) yields/angbracketleft∆VDC/angbracketright>0. Hence, although slightly\nmore power was absorbed at negative fields, the DC\nvoltage magnitude was notably larger for positive fields.\nFrom its symmetry properties under field inversion, this\nfield dependent voltage ∆ VDCis consistent with the\nspin pumping mechanism outlined in Sec. II. If spin\npumping indeed is at the origin of the observed voltage\ndifference, this would correspond to a contribution of\naboutVSP(±H0)≈∓100 nV which is about a fifth to\na tenth of the total DC voltage observed. Note also\nthat electrically detected spin pumping experiments\ntypically yield DC voltages in the range of a few 10 nV\nto a few 10 µV, such that the magnitude of the voltage\nalso appears reasonable.\nVI. CONCLUSION\nIn summary we investigated DC voltages arising when\nexciting antiferomagnetic resonance in MnF 2/Pt bilayer\nsamples. Placing the sample on a coplanar waveguide,5\nthe antiferromagnetic resonance below and above MnF 2’s\nspin-flop transition is observed in both microwave trans-\nmission through the waveguide as well as in the DC volt-\nage detected along the Pt strip. Since the DC voltage\ndoes not invert its sign under field reversal, and thus does\nnot exhibit the fingerprint of electrically detected spin\npumping, the DC signal must be dominated by other\nprocesses. We envisage rectification or, somewhat less\nlikely, thermal effects potentially result in the DC volt-\nage signature. In detail, we note the presence of a small\ndifference in VDCbetween positive and negative applied\nexternal magnetic fields, which may indicate antiferro-\nmagnetic spin pumping. However, further studies are\nrequired to substantiate this conjecture.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge financial support\nby the DFG via SPP 1538 “Spin Caloric Transport”\n(project GO 944/4-2).\nREFERENCES\n1C. Kittel, Phys. Rev. 73, 155 (1948).\n2Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n3M. I. D’yakonov and V. I. Perel’, JETP Lett. 13, 467 (1971).\n4J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n5M. V. Costache, M. 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Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France\n4CNRS, SPINTEC, F-38000 Grenoble, France\n5CEA, INAC-SPINTEC, F-38000 Grenoble, France\n∗Correspondence to: web54@columbia.edu\n(Dated: July 9, 2021)\nAbstract\nInertial magnetization dynamics have been predicted at ult rahigh speeds, or frequencies ap-\nproachingtheenergy relaxation scaleof electrons, inferr omagnetic metals. Hereweidentifyinertial\nterms to magnetization dynamics in thin Ni 79Fe21and Co films near room temperature. Effective\nmagnetic fields measured in high-frequency ferromagnetic r esonance (115-345 GHz) show an addi-\ntional stiffening term which is quadratic in frequency and ∼80 mT at the high frequency limit of\nour experiment. Our results extend understanding of magnet ization dynamics at sub-picosecond\ntime scales.\n1The magnetization M(t) in ferromagnetic materials is generally understood to evolve\nwithout memory of its prior motion. As described by the Landau-Lifs hitz (LL) equation[1],\nmagnetization dynamics dM(t)/dtcan be written in terms of the magnetization M(t) alone,\nexcluding any temporal derivatives dnM(t)/dtn. Magnetization is then ’inertialess’: it re-\nsponds to a step magnetic field H(t) with an instantaneous change in speed and with infinite\nacceleration.\nAs pointed out by Ciornai et al.and F¨ ahnle et al.[2–4], followed by other theoreticians[5,\n6], the absence of inertia is questionable for magnetization dynamics at very high frequen-\ncies. The high frequency behavior, >100 GHz, becomes relevant in ultrafast ( <ps-scale)\nmagnetization switching schemes envisaged for magnetic informatio n storage technology[7–\n11]. In metallic ferromagnets, the electrons which comprise the mag netization themselves\npossess inertia, expressed through a finite (Bloch-state) lifetime τand cannot change their\nmomenta infinitely quickly. These authors[2–4] have proposed that the LL equation should\nbe amended to include an inertial term:\ndm\ndt=−µ0γ[m×Heff+αm×(m×Heff)]±αm×τd2m\ndt2(1)\nHeremis the unit magnetization vector, γis the gyromagnetic ratio, Heffis the ef-\nfective field, and αis the Gilbert damping coefficient. The additional nonlinear term\n±αm×τd2m/d2tis second-order in time. It is straightforward to show that this iner tial\nterm leads to an effective field H′\neff=∓αω2τ/γ, quadratic in frequency, which acts on\nmagnetization dynamics. Increasing the frequency limit for ferrom agnetic resonance (FMR)\nexperiments by a factor of 10 will therefore increase the magnitud e of the inertial effective\nfield by a factor of 100.\nThe sign of the inertial term in Eq. (1) refers to two different origins for the proposed\ninertial dynamics. Ciornai et al.[2, 12] argue that additional energy terms for the precession,\ntypically neglected, come about from transverse magnetization mo tion. Finite rotational\nmoments of inertia about the two transverse axes were introduce d semiclassically to account\nfor this energy in Ref. [2]. On the other hand, F¨ ahnle et al.[3, 4] have pointed out that\nsimilar dynamics arise from a more complete consideration of damping. Here the inertia\ncan be associated with the linear momentum of electrons: electrons cannot change their\nmomenta kmore rapidly during precession than their Bloch state relaxation time τB, and\ntend to remain at higher energy states. In this sense, inertia is pro portional to τB. The\n2rotational moment of inertia (”+” in Eq. 1) softens the resonance frequency while the linear\ninertia due to relaxation (”-” in Eq. 1) stiffens the resonance frequ ency.\nIn this work, we test for the existence of inertial effective fields wh ich act on the mag-\nnetization of the device ferromagnets permalloy (Ni 79Fe21, or Py) and Co. To do so, we\nmeasure FMR up to 345 GHz, roughly an order of magnitude higher th an is accessible\nusing conventional sources. Through measurements of the freq uency-dependent resonance\nfieldµ0Hres(ω), we find in all samples a negative nonlinear shift quadratic in frequen cy\nwhich reflects a positive effective field added by the inertial term. Th e sign of the shift\nshows the dominance of inertia due to the finite Bloch state relaxatio n time[3, 4] over the\ninertia due to finite rotational moment of inertia of electrons[2]. In a ll samples the extracted\nvalues of τare comparable with low-fluence limits of remagnetization times τEmeasured in\nultrafast demagnetization experiments. Our results show that hig h-frequency FMR (HF-\nFMR) provides a near-ground-state ( ∼1 meV) view of the processes involved in ultrafast\nmagnetization dynamics, heretofore accessible through far-fro m-equilibrium techniques.\nPy and Co thin films were deposited on Si/SiO 2substrates using ultra-high vacuum mag-\nnetron sputtering. The capping layers are Cu(5 nm)/SiO 2(5 nm) for Py and Cu(5 nm)/Ta(3\nnm) for Co; the seed layers are Ta(5 nm)/Cu(5 nm) in both series. T he thicknesses tFM=\n6, 10 and 30 nm for Py and 6, 10, 15, 30 nm for Co, respectively. Two series of field-swept,\nvariable-frequency FMR experiments were carried out, both with a pplied field and static\nmagnetization perpendicular to film plane. At low fields and frequencie s (3-26 GHz, 1-2.5\nT, 298 K), a coplanar waveguide was used to measure the power tra nsmission through the\nthin films. These measurements were carried out at Columbia Univers ity using an Fe-core\nelectromagnet[13]. At high fields and frequencies (115, 230 and 345 GHz, 4-13 T, 273-280\nK) continuous-wave high-field electron paramagnetic resonance w as used to measure the\npower reflection from the samples. These measurements were car ried out at the LNCMI-G\nusing a superconducting magnet[14]. Six field sweeps were taken for HF-FMR to reduce\nmeasurement and hysteresis errors.\nSeveral steps have been taken to ensure the consistency of low- and high-frequency mea-\nsurements. These are essential to detect the nonlinear term in ma gnetization dynamics.\n3(a)Fi/g427ng residual \nlow-f\nHF-FMRextrapola/g415onsignal x3\n230 GHz\n115 GHz345 GHz\nCo 15 nm (b)\nFIG. 1. (a) Low-frequency resonance fields for Py and Co <26 GHz. Lines and dashed lines are\nfits toω/γ=µ0(Hres−Meff). The upper panel shows the fitting residuals of the lower pan el.\n(b) High-frequency FMR absorption spectra for Co 15 nm. The ( horizontal) field axis shows the\ndifference of measured high-frequency resonance fields with e xpected values based on extrapolation\nof the low-frequency resonance fields without inertial effect s. The solid data points indicate the\nresonance field µ0Hres, with fits shown in the dashed curve.\nFirst, the magnetic fields of both low- and high-frequency FMR were calibrated in-situ\nusing paramagnetic markers[15]. An accuracy of 0.01% is obtained, c orresponding to ∼1\nmT at 345 GHz ( ∼13 T). Second, for low-frequency FMR, each sample normal has be en\nindependently aligned along the applied field by rotating the waveguide with respect to two\naxes (with <0.2◦precision) to maximize the value of µ0Hresat 3 GHz. The samples were\naligned optically, to lower precision, in HF-FMR. Third, we have correc ted for the small ( ∼\n20K)temperaturedifferences between thetwo apparati, throug hvariable-temperatureFMR\nmeasurements at Columbia University, which allow us to correct for t emperature-dependent\nmagnetization µ0Meff(T). See the Supplemental Information for details.\nFig. 1(a) shows the low-frequency µ0Hresas a function of ω/2πfor Py and Co. Contin-\nuous and dashed lines are fits to the linear Kittel equation ω/γ=µ0(Hres−Meff). The\ntop panel shows the fitting residuals, which are randomly distribute d within ±1 mT. In\norder to examine whether resonance field offset exists at high freq uencies, we plot in Fig.\n1(b) the HF-FMR lineshapes (average of six measurements) for Co 15 nm, as a function of\nµ0(H−Hextrap\nres), withµ0Hextrap\nresthe resonance field extrapolated from low-frequency FMR.\n4The solid circles indicate the resonance field µ0Hresextracted from the FMR lineshapes for\nthe three frequencies. The dashed lines are a fit of the resonance positions to a quadratic\nnonlinear term to frequency. A negative curvature is found in the d ashed lines. This implies\na negative sign in Eq. (1) with positive effective inertial field. Similar effe cts are found in\nother Py and Co films.\nFi/g427ng residual \nFi/g427ng residual \n(a) (b)\nFIG. 2. Deviation of µ0Hresfrom the linear Kittel equation for (a) Py and (b) Co. Main pan els:\nµ0Hres|exp−(µ0Meff+ω/γ)|fitin solid data points; α0ω2τ/γ|fitin curved lines. Meff,γandτ\nare the fit parameters from Eq. (2). Upper panels: fitting resi duals of main panels as a function\nof frequency. Insets: full plots of µ0Hresand fits to Eq. (2).\nTo quantify the nonlinear term from Eq. (1), we fit the resonance fi elds of both low- and\nhigh-frequency FMRmeasurements together into the linear Kittel equation, plus a quadratic\nterm:\nµ0Hres=µ0Meff+ω/γ−α0ω2τ/γ (2)\ntakingMeff,γandτas fit parameters. The parameter τgoverns the magnitude of the\ninertial term. The Gilbert damping coefficients α0are extracted by fitting the low-frequency\nFMR linewidths into µ0∆H1/2=µ0∆H0+ 2α0ω/γ, whereµ0∆H0is the inhomogeneous\nlinewidth. To estimate errorbars for HF-FMR, we take the scatter of all six measurements\n5into the data sequence.\nFigure 2 contains the central result of this manuscript. In the main panels we show\nexperimental resonance fields µ0Hresand fits to Eq. (2) as a function of frequency ω/2π\nfor (a) Py and (b) Co. The conventional effective field terms µ0Meff+ω/γhave been\nsubtracted away; plotting the data as µ0Hres|exp−(µ0Meff+ω/γ)|fit(solid data points)\nandα0ω2τ/γ|fit(curves) isolates the inertial contribution. The fit values are tabu lated in\nTable 1; error bars denote 3 σof uncertainty. We find a reasonable fit to the data through\nthe introduction of the quadratic term. For both Py and Co, the va lues ofτstay roughly\nat the same level between 10-30 nm: ∼0.1 ps for Py and ∼0.3 ps for Co. For Py 06 nm\nnegligible shifts in µ0Hresis found, as will be discussed.\nSample µ0Meff geff α0 τ(µ0Hres)τ(µ0∆H1/2)\nPy 06 nm 0.915 T 2.106 0.0075∗0.28±0.06 ps 0.09 ±0.09 ps\nPy 10 nm 0.935 T 2.100 0.0072 0.13 ±0.03 ps 0.11 ±0.04 ps\nPy 30 nm 0.993 T 2.101 0.0069 0.12 ±0.05 ps 0.06 ±0.08 ps\nCo 06 nm 1.409 T 2.147 0.0067 0.47 ±0.08 ps 0.19 ±0.07 ps\nCo 10 nm 1.547 T 2.153 0.0053 0.26 ±0.09 ps 0.14 ±0.07 ps\nCo 15 nm 1.596 T 2.154 0.0051 0.32 ±0.03 ps 0.15 ±0.05 ps\nCo 30 nm 1.672 T 2.155 0.0056 0.26 ±0.05 ps 0.21 ±0.09 ps\nTABLE I. Fit parameters of experimental resonance fields µ0Hres(ω) to the inertial term Eq. (2)\nand linewidths µ0∆H1/2to Eq. (3) for Py and Co. Values of low-frequency Gilbert damp ing\nα0are found from linear fits of low-frequency linewidths. Erro rbars of τrepresent 3 σdeviations.\n∗Misaligment analysis has been applied; see text for details .\nA primary source of uncertainty may come from sample misalignment in HF-FMR measure-\nments. Given the experimental geometry, the samples cannot be a ligned independently of\nmounting. Canting the sample off normal alignment in HF-FMR will also lea d to reduced\nresonance fields as in Fig. 1(b). Misalignment will be manifested as a lineardeviation of\nµ0Hres|exp−(µ0Meff+ω/γ)|fitfrom being zero at low frequencies ( ω/2π <30 GHz) in\n6Fig. 2. For Co samples, the deviations at low frequencies are negligible , indicating that\nCo samples are well aligned perpendicular to biasing field. For Py the sa me is true, with\nthe exception of the 6 nm sample. This sample shows a significant linear deviation at low\nfrequencies (green circles), indicating that the sample plane may no t be perfectly perpendic-\nular to the biasing field. According to simulations, shown in the Supplem ental Information,\nsuch a deviation would be produced by a misalignment of ∼6◦for Py 06 nm, which will\nadd an additional negative µ0Hresshift of∼20 mT. With this correction, the estimated τ\nis increased from 0.01 ps to 0.28 ps in Py 06 nm, consistent with the larg er value for Co 6\nnm. The corrections are much smaller for other samples and have be en neglected.\nThe influence of the inertial dynamics which we observe is a stiffening, rather than\na softening, of the resonance. We conclude therefore that the in ertial term from linear\nmomentum[3, 4, 6] dominates over the inertial term from angularmomentum proposed in\nRef. [2]. The former term is a direct product of the breathing Fermi surface model causing\nconductivity-like Gilbert damping[16, 17]. In the conductivity-like mod el the value of ob-\nservedτ, 0.1-0.5 ps, represent the average of Bloch state lifetime τBnear the Fermi surface\nfor Py and Co.\nLow-frequency \nextrapola/g415on Data \n& fitsLow-frequency \nextrapola/g415on Data \n& fits(a) (b)\nFIG. 3. Ferromagnetic resonance half-power linewidths µ0∆H1/2as a function of ω/2πfor (a) Py\nand (b) Co, between 4 to 345 GHz. Dashed lines are linear extra polations of the low-frequency\nFMR linewidths; solid curves are fits to Eq. (3). Each point wi th error bar represents a summary\nof six measurements.\n7Note that Gilbert damping coefficient α, calculated by the breathing Fermi surface model,\nshould also be frequency-dependent. A relation of α=α0/(1+ω2τ2\nB) can be obtained[16–\n18] with α0the zero-frequency damping. Fig. 3 shows the linewidths of both low -frequency\nand high-frequency ranges for Py and Co. Here points with errorb ars (one σ) are used for\nfrequencies >100 GHz to reflect the scattering of six repeats. The dashed lines a re low-\nfrequency linear extrapolations from α0. Compared with extrapolated linewidths from the\nlow-frequency ∆ H1/2(ω), we observe reduced linewidths at high frequencies (115-345 GHz ).\nNo explicit prediction has been made for the effect of rotational iner tia on the linewidth in\nRef. [2, 12], but the observed behavior matches well with the predic tion of linear inertial\nterms in Ref. [16–18]. The solid curves are fits to the form:\nµ0∆H1/2=µ0∆H0+2α0ω\nγ1\n1+ω2τ2(3)\ntaking the α0from the low-frequency linewidths. The fitted τare listed in Table 1. The\nvalues of τextracted from µ0∆H1/2are close but slightly smaller than from µ0Hres. We\nnote here that the linewidth measurements are less precise than µ0Hresand more sensitive\nthan resonance fields to various sources of inhomogeneities in the s amples.\nAt room temperature, the relaxation rate of Bloch states (1 /τB) is determined by the\nelectronscatteringwithphononsandimpurities. Inthissense, τBissimilar totheremagneti-\nzation time τEin the ultrafast demagnetization experiments[19–21], where optica lly excited\nelectrons also relax through electron-phonon and electron-impur ity interactions. In the limit\nof zero laser fluence, nonlinear effects due to high occupation numb er of excited states are\nreduced[21, 22]. The zero-fluence τEhas been reported to be 0.2-0.25 ps for Py 10-30 nm[22]\nand∼0.4 ps for Co 15 nm[21], close to the value of τin this work.\nAt high frequencies the (nonmagnetic) skin depths δsin the ferromagnetic ���lms become\nsmaller and the enhanced eddy current effect may influence the res onance field[23]. The res-\nonance field will be enhanced by ∼µ0Ms/δ4\nsk4\n0, wherek0is the lowest-energy wavenumber\ndetermined by the surface anisotropy. Because δ2\ns∝1/ω, the resonance field enhancement\nis proportional to ω2and may influence the quadratic term in Eq. (2). Our calculations\nshow that this term is negligible, about 0.4 mT for Py 30 nm and 0.09 mT fo r Co 30 nm,\ncompared with the observed effects up to 80 mT (See the Supplemen tal Information for\ndetails).\nWe do not believe that interfacial effects, including spin pumping, play an important role\n8in the observed high-frequency behavior. Both Gilbert damping α0and the two inertial\ndynamics lifetimes τ(µ0Hres) andτ(µ0∆H1/2) in Table 1 show little thickness dependence\nfor either Py or Co, indicating that bulk relaxation is dominant. The we ak thickness de-\npendence of α0is consistent with the very low spin pumping effect of Py/Ta identified in\nRef. [24], in any case negligible for 30 nm films and without quadratic fre quency depen-\ndence. Theoretical predictions of resonance shifts from imaginar y spin mixing conductance\nare three orders of magnitude lower than observed here[25]. Only τ(µ0Hres) measured from\nresonance shifts for 6 nm films, not matched in τ(µ0∆H1/2) measured from linewidths, differ\nsignificantly. This enhancement may be structural in origin.\nA technological implication of our results is that effective field require ments for preces-\nsional switching will be reducedas switching times in magnetic storage decrease into the\nfew-picosecond range. In this sense, the inertial dynamics ease u ltrafast switching, if the\nbehaviors of Py and Co are representative of other metallic ferrom agnets. The effective field\nreduction, here up to 80 mT in Co, is not small in an absolute sense, an d might according\nto Eq. (2) be enhanced significantly in ferromagnets with higher Gilbe rt damping. On the\nother hand, prior switching experiments with high-field relativistic ele ctron bunches seem\nto indicate that nonlinear damping increases effective field requireme nts by a rather larger\namount for large-angle dynamics in CoCrPt[8], underscoring the utilit y of HF-FMR to iden-\ntify the inertial effect.\nIn summary, we identify a novel term to magnetization dynamics in th e ferromagnetic\nmetal films Py and Co which is quadratic in frequency and becomes sign ificant above 100\nGHz. Thetermstiffensthefrequency, aidingappliedfieldsindriving ult rafastmagnetization\nmotion. The behavior is best explained by dynamics retarded throug h a finite Bloch-state\nrelaxation time τBas proposed in Refs. [3, 4, 6]. Extracted relaxation times are 0.1-0.2 ps\nfor Py and 0.2-0.4 ps for Co, close to the remagnetization times meas ured in optical pump-\nprobe demagnetization experiments. Our findings extend underst anding of magnetization\ndynamics at picosecond time scales and may open up new possibilities fo r high-speed inertial\nswitching in ferromagnetic materials, previously demonstrated only in antiferromagnets[10].\nWe thank M. F¨ ahnle, J. E. Wegrowe and J. Fransson for discussion s and X. Yang for\nsuggestions on statistical analysis. We acknowledge fundings by th e EC through CRONOS\n(N◦280829), NSF-DMR-1411160, and the Chair of Excellence Program of the Nanosciences\nFoundation in Grenoble France for support.\n9[1] Eq. (1) is equivalent to the Landau-Lifshitz-Gilbert (L LG) equation with renormalized gyro-\nmagnetic ratio γ. See: T. L. Gilbert, IEEE Trans. Magn. 6, 3443 (2004).\n[2] M.-C. Ciornei, J. M. Rubi and J.-E. Wegrowe, Phys. Rev. B 83, 020410(R) (2011).\n[3] M. F¨ ahnle, D. Steiauf and C. Illg, Phys. Rev. B 84, 172403 (2011).\n[4] M. F¨ ahnle, D. Steiauf and C. Illg, Phys. Rev. B 88, 219905(E) (2013).\n[5] D. B¨ ottcher and J. Henk, Phys. Rev. B 86, 020404(R) (2012).\n[6] S. Bhattacharjee, L. Nordstr¨ om and J. Fransson, Phys. Rev. Lett. 108, 057204 (2012).\n[7] C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Wel ler, E. L. Garwin and H. C.\nSiegmann, Science285, 864 (1999).\n[8] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann , J. St¨ ohr, G. Ju, B. Lu and\nD. Weller, Nature428, 831 (2004).\n[9] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Ts ukamoto, A. Itoh and Th. Rasing,\nPhys. Rev. Lett. 99, 047601 (2007).\n[10] A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev, A. Kirilyuk and Th. Rasing, Nature\nPhys.5, 727 (2009).\n[11] S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V. Uhl´ ıˇ r, L. Pang, M. Hehn, S. Alebrand,\nM. Cinchetti, G. Malinowski, Y. Fainman, M. Aeschlimann and E. E. Fullerton, Nature Mater.\n13, 286 (2014).\n[12] J.-E Wegrowe and M.-C. Ciornei, Am. J. Phys. 80, 607 (2012).\n[13] Y. Li and W. E. Bailey, arXiv1401.6467\n[14] A.-L. Barra, A. K. Hassan, A. Janoschka, C. L. Schmidt an d V. Sch¨ unemann, Appl. Magn.\nReson.30, 385 (2006).\n[15] Paramagnetic markers used: BDPA complex with benzene( 1:1) free radical for low-frequency\nFMR,geff=2.0025; MnO diluted in the diamagnetic host MgO for high-fr equency FMR,\ngeff=2.00101\n[16] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[17] V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007).\n[18] A. M. Clogston, Bell Syst. Tech. J. 34, 739 (1955).\n[19] E. Beaurepaire, J.-C. Merle, A. Daunois and J.-Y. Bigot ,Phys. Rev. Lett. 76, 4250 (1996).\n10[20] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa and W. J. M. d e Jonge, Phys. Rev. Lett. 95,\n267207 (2005).\n[21] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. F¨ ahnle, T. Roth, M. Cinchetti\nand M. Aeschlimann, Nature Mater. 9, 259 (2009).\n[22] C. A. C Bosco, A. Azevedo and L. H. Acioli, Appl. Phys. Lett. 83, 1767 (2003).\n[23] P. Pincus, Phys. Rev. 118, 658 (1960).\n[24] S. Mizukami, Y. Ando and T. Miyazaki, J. Magn. Magn. Mater. 226, 1640 (2001).\n[25] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas a nd G. E. W. Bauer, Phys. Rev. B\n71, 064420 (2005).\n11"    },    {        "title": "1502.04128v1.Antiferromagnet_Mediated_Spin_Transfer_Between_Metal_and_Ferromagnet.pdf",        "content": "Antiferromagnet-Mediated Spin Transfer Between Metal and Ferromagnet\nSo Takei,1Takahiro Moriyama,2Teruo Ono,2and Yaroslav Tserkovnyak1\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Institute for Chemical Research, Kyoto University, 611-0011 Uji, Kyoto, Japan\n(Dated: June 8, 2022)\nWe develop a theory for spin transported by coherent N ´eel dynamics through an antiferromagnetic insulator\ncoupled to a ferromagnetic insulator on one side and a current-carrying normal metal with strong spin-orbit cou-\npling on the other. The ferromagnet is considered within the mono-domain limit and we assume its coupling to\nthe local antiferromagnet N ´eel order at the ferromagnet jantiferromagnet interface through exchange coupling.\nCoupling between the charge current and the local N ´eel order at the other interface is described using spin\nHall phenomenology. Spin transport through the antiferromagnet, assumed to possess an easy-axis magnetic\nanisotropy, is solved within the adiabatic approximation and the e \u000bect of spin current flowing into the ferromag-\nnet on its resonance linewidth is evaluated. Onsager reciprocity is used to evaluate the inverse spin Hall voltage\ngenerated across the metal by a dynamic ferromagnet as a function the antiferromagnet thickness.\nPACS numbers: 72.25.Mk, 75.47.-m, 75.50.Ee, 76.50. +g\nSpintronics of antiferromagnets (AFs), where AFs take on\nthe role of the central active component, is identified as one of\nthe most important emerging topics in the field of magnetism\ntoday [1]. Robustness to magnetic perturbations due to their\ntotal magnetic compensation, as well as characteristic dynam-\nical scale in the THz range may render AFs advantageous over\nferromagnets (Fs) for spintronics device applications. In ad-\ndition, recent works on AFs have shown that the important\nphenomena responsible for the success of F-based spintron-\nics also have AF counterparts, giving added impetus for AF-\nbased spintronics research. Indeed, giant magnetoresistance\nand current-induced torques [2], anisotropic magnetoresis-\ntance [3] and spin superfluidity [4], as well as current-induced\ndomain wall motion [5] and coupled dynamics between con-\nduction electrons and background magnetic texture [6], are all\nshown to be possible in AFs as well.\nAn important aspect of AF-based spintronics is the use of\nAFs as a medium to transport spin angular momentum. Spin\ntransfer through AFs has been the focus of several recent ex-\nperimental endeavors. Both Hahn et al . [7] and Wang et\nal. [8] demonstrated spin transport through an AF insulator,\nNiO, using an YIG jNiOjPt heterostructure (YIG standing for\nthe insulating ferrimagnet yttrium iron garnet). Inverse spin\nHall signal showed robust spin pumping from YIG into Pt\neven in the presence of the intervening NiO, suggesting e \u000e-\ncient spin transport through the AF. More recently, Moriyama\net al . used spin-torque ferromagnetic resonance (ST-FMR)\nto demonstrate the propagation of spin excitations through a\nmetallic AF, IrMn, using a Pt jIrMnjCoFeB trilayer [9] as well\nas NiO using a PtjNiOjFeNi trilayer [10]. Spin current injected\nfrom the Pt was shown to change the FMR linewidth, also sug-\ngesting the transfer of spin angular momentum through the\ncentral AF. Given the rising interest in AF spintronics and\nthe recent experimental focus, a theoretical account of spin\ntransport through an experimentally relevant normal metal\n(N)jAFjF trilayer is highly desirable.\nIn this Letter, we develop a general phenomenology for spin\ntransport through an AF by collective N ´eel order parameter\nx=0x=LxzNAFFS(t)✓⌧\"n(x)j(j,\")(S,⌧)JhJssttihJsGiNAFF(#,↵\"#)#⇤effective junctionVISHE`\n↵0hJsspihJsexiFIG. 1. Normal-metal (N) jantiferromagnet (AF) jferromagnet (F) tri-\nlayer considered in this work. N sustains a dc charge current jand\nF is described by a time-dependent macrospin S(t). Spin transfer\nhJs\nexioccurs via the exchange coupling Jat the AFjF interface, while\nspin transfer across the AF jN interface has a spin transfer torque con-\ntributionhJs\nstti(proportional to the e \u000bective interfacial spin Hall an-\ngle#) and a spin pumping contribution hJs\nspi(proportional to the\ninterfacial spin-mixing conductance \u000b\"#). The AF Gilbert damping,\nparametrized by \u000b0, leads to the loss of spin current hJs\nGiin the AF\nbulk. The central AF can be thought of as an e \u000bective interface that\ncouples jandSwith an e \u000bective spin Hall angle #\u0003.\ndynamics, focusing on an N jAFjF trilayer relevant for both the\nspin-pumping /inverse spin Hall as well as the ST-FMR exper-\niments mentioned above (see Fig. 1). Spin Hall phenomenol-\nogy, applicable to a wide range of di \u000berent AFjN interfaces\nobeying certain structural /crystalline symmetries, is utilized\nto model the spin transfer at the AF jN interface, while the ex-\nchange coupling is assumed at the AF jF interface. As one of\nthe main achievements of this work we develop a simple “cir-\ncuit” model, a pictorial visualization of spin flow, that allows\none to keep track of spin transfer through various parts of the\nheterostructure (see bottom half of Fig. 1). From the circuit\nmodel, we see that spin is both injected into (i.e., hJs\nstti) andarXiv:1502.04128v1  [cond-mat.mes-hall]  13 Feb 20152\nejected (i.e.,hJs\nspi) out of the AF at the AF jN interface due to\nspin Hall /spin-torque e \u000bects and spin pumping, respectively.\nThe collective N ´eel dynamics leads to Gilbert damping and to\nthe loss of spin current (i.e., hJs\nGi) within the AF bulk, and the\nexchange coupling at the AF jF interface leads to spin transfer\n(i.e.,hJs\nexi) across the interface. We first study how spin trans-\nport through the AF modifies the linewidth at FMR, akin to the\nST-FMR [9, 10]. The FMR linewidth is quantified in terms of\nthe e\u000bective spin Hall angle and spin-mixing conductance at\nthe AFjN interface, the exchange coupling at the AF jF inter-\nface, as well as AF Gilbert damping. We show that linewidths,\nmeasured for di \u000berent electrical currents in N and AF thick-\nnesses, can be used to extract the e \u000bective spin Hall angle\nand spin-mixing conductance at the N jAF interface, as well as\nthe bulk Gilbert damping. By invoking Onsager reciprocity,\nwe also make connections with the inverse spin Hall experi-\nments [7, 8] and compute the inverse spin Hall voltage gener-\nated across N arising as a result of a dynamic F macrospin.\nAs shown in Fig. 1, an insulating AF is attached on one side\nto a mono-domain F and on the other to a paramagnetic N with\nstrong spin-orbit coupling. The N and F sustain dc charge\ncurrent density jand a time-dependent macrospin S(t), re-\nspectively. We consider a bipartite AF, which can be charac-\nterized by two hydrodynamic variables, n(x;t) andm(x;t),\nparametrizing the staggered (N ´eel) and smooth (magnetic)\ncomponents of the spins, respectively [11]. We assume easy-\naxis magnetic anisotropy along the zaxis in the AF, as well\nas full translational and rotational symmetries in the yzplanes\nso that our treatment essentially reduces to a one-dimensional\nproblem that depends only on the coordinate x. The free en-\nergyFfor the AF and its coupling to the F reads\nF=ZL\n0dx(A\n2[@xn(x)]2+m(x)2\n2\u001f\u0000\u0014\n2nz(x)2)\n\u0000JS\u0001n(L);(1)\nwhere Aand\u001fare the N ´eel order sti \u000bness and spin suscepti-\nbility, respectively, \u0014the uniaxial anisotropy parameter, and J\nthe exchange coupling between AF and F.\nThe Landau-Lifshitz-Gilbert dynamics in the bulk AF cor-\nresponding to Eq. (1) can be written as\ns( ˙n+\u000bn\u0002˙m)=\u001f\u00001m\u0002n; (2)\ns( ˙m+\u000bm\u0002˙m+\u000b0n\u0002˙n)=n\u0002(A@2\nxn+\u0014nzez);(3)\nwhere\u000band\u000b0are (independent) Gilbert damping param-\neters and sis the roughly saturated spin density (per unit\nlength) [12]. In the limit of slow dynamics (i.e., \u001c\u001d~=Eex,\nwhere\u001cis the time scale for AF dynamics and Eexis the\nAF exchange energy scale) and strong local N ´eel order (i.e.,\njm(x;t)j\u001c1), one may first use Eq. (2) to solve for m, insert\nthe solution into Eq. (3), and arrive at the following dynamics\nfor the AF N ´eel vector,\n\u001fs2n\u0002¨n+s\u000b0n\u0002˙n\u0019An\u0002@2\nxn+\u0014n\u0002nzez:(4)The AF spins are excited by the current and the dynamic\nmacrospin. The e \u000bects of these external perturbations are lo-\ncalized at the interfaces and thus enter the AF dynamics as\nboundary conditions. At the AF jN interface, spin currents\narise via spin transfer torque and spin-pumping, both of which\ncan be accounted for using spin Hall phenomenology [13].\nBased on structural symmetries at the interface, spin Hall phe-\nnomenology allows us to write down a general expression for\nthe spin transfer torque that applies to a variety of F- and AF-\nbased heterostructures with di \u000berent microscopic details. In\nthe presence of full translational and rotational symmetries\nin the yzplane and with the breaking of reflection symmetry\nalong the xaxis, there are two contributions to the spin current\n(integrated over the interface area) flowing into AF [13]\nJs\nl=[#n\u0002(ex\u0002j)\u0002n\u0000~\u000b\"#n\u0002˙n]jx=0\u0011Js\nstt\u0000Js\nsp;(5)\nwhere the first term is the so-called spin Hall-like (dissipative)\ncontribution and the second term describes spin-pumping.\nThe coe \u000ecient#is proportional to the (tangent of the) ef-\nfective spin Hall angle at the AF jN interface [13]; although #\ncan, in general, depend on the orientation of n, we will treat\nit as a constant here. We will also disregard any anisotropies\nof\u000b\"#with respect to the orientations of nand ˙n, assuming\nthat the exchange energy scale at the interface dominates over\nthe energy scale of spin-orbit interactions.\nWhile the coe \u000ecient\u000b\"#is proportional to the real part of\nthe (generally complex) spin-mixing conductance g\"#for the\nAFjN interface [13], its imaginary part gives rise to a term\n/˙nin Eq. (5). The assumed structural symmetry, in prin-\nciple, also allows for the so-called field-like (reactive) con-\ntribution given by \u0011(ex\u0002j)\u0002n. Assuming the magnitudes\nof all four terms are small (as compared top\nA\u0014to be pre-\ncise) we will compute the FMR linewidth to linear-order in\nthe coe \u000ecients#,\u0011andg\"#. Since the terms that do not ap-\npear in Eq. (5) are non-dissipative terms, they do not enter the\nlinewidth at linear-order in these coe \u000ecients. We will, there-\nfore, drop these contributions from the following discussion\nand only consider the two terms in Eq. (5).\nThe exchange coupling J[given by the last term in Eq.(1)]\nleads to the following expression for spin current flowing\nacross the AFjF interface,\nJs\nr\u0011Js\nex=@F\n@S\u0002S=JS\u0002njx=L: (6)\nThe spin current in the AF bulk can be read o \u000bfrom Eq. (3)\n(dropping the Gilbert damping and anisotropy terms) and the\nresulting continuity equation [i.e., s˙m=\u0000@x(\u0000An\u0002@xn)], so\nthatJs\nAF(x)=\u0000An(x)\u0002@xn(x). The continuity of spin current\nacross each interface then leads to the boundary conditions\nJs\nl=Js\nAF(x=0);Js\nr=Js\nAF(x=L): (7)\nThe dynamic N ´eel texturen(x;t) can be obtained using the\nlow-frequency (adiabatic) approximation, valid in the regime\n\n\u001c\n0, where \nand\n0are the FMR and the AF resonance3\n(a)(b)L/\u0000\u0000↵F\u0000↵F\nFIG. 2. The interfacial contribution \u000e\u000b(i)\nF(dashed lines), the bulk\ncontribution \u000e\u000b(b)\nF(dotted lines) and the total contribution \u000e\u000bF(solid\nlines) to the extrinsic FMR linewidth ( Sset to unity) are plotted as a\nfunction of the (normalized) system size L=\u0015. We fix the following\nparameters: \u0011=1,#jy=b0=0:01 and\u000b\"#=0:01. Two regimes\nare considered for the AF Gilbert damping ˜ \u000b: (a) the strong damping\nregime ˜\u000b=0:2; and (b) the weak damping regime ˜ \u000b=0:01 (see text\nfor more details).\nfrequencies, respectively. Within this approximation, the AF\nN´eel texture is first solved for an arbitrary static S, the result\ndenoted byn(0)(x;S). SinceS(t) varies su \u000eciently slowly in\ntime compared to the characteristic AF time scale, the N ´eel\ntexture in the adiabatic limit will arrange itself into the static\nconfiguration corresponding to S(t) at every moment in time\nand is given by n(x;t)\u0019n(0)[x;S(t)]\u0011n(0)(x;t). The above\ncalculation does not account for spin current losses due to the\nAF dynamics (i.e., spin-pumping at the AF jN interface and\nGilbert damping in the AF bulk). Taking these losses into ac-\ncount up to linear-order corrections to the adiabatic result, the\nspin currenthJs\nexientering F, time-averaged over a cycle of\nFMR precession (the angle brackets h\u0001\u0001\u0001ihereafter represent-\ning time-average over a cycle of FMR precession), is given\nbyhJs\nexi=hJs\nstti\u0000hJs\nspi\u0000hJs\nGi(c.f. Fig. 1), where the spin-\ntransfer torque contribution is given by inserting the adiabatic\nresult for the N ´eel texture into Eq. (6)\nhJs\nstti=JhS(t)\u0002n(0)(L;t)i; (8)\nand the loss terms read\nhJs\nspi=~\u000b\"#hn(0)(0;t)\u0002˙n(0)(0;t)i;\nhJs\nGi=s\u000b0ZL\n0dxhn(0)(x;t)\u0002˙n(0)(x;t)i:(9)\nThe first term in Eq. (9) describes (time-averaged) spin current\nlost due to spin pumping at the AF jN interface and the second\nterm corresponds to Gilbert damping in the AF bulk.\nAn analytical result for the FMR linewidth can be ob-tained if we consider small deviations of S(t) away from\nthezaxis (parallel to the static FMR field and the AF easy-\naxis); we take j=jyey+jzezand assumejjjto be weak\nsuch that a linear-response treatment is su \u000ecient. In this\ncase, the N ´eel unit vector nshould not deviate far from the\nzaxis and we may evaluate the above results with respect\nto small transverse fluctuations, i.e., S(t)\u0019S[sx(t);sy(t);1]\nandn(x;t)\u0019[nx(x;t);ny(x;t);1] withjsx(t)j;jsy(t)j\u001c 1 and\njnx(x;t)j;jny(x;t)j \u001c 1. Within this treatment, the trans-\nverse components n?=(nx;ny)Tobey\u001fs2¨n?+s\u000b˙n?=\nA@2\nxn?\u0000\u0014n?[c.f. Eq. (4)], and n(0)(x;t) has the form\nn(0)\u0019ez+f(x)[ez\u0002S(t)]\u0002ez+g(x)ez\u0002S(t)+h(x)ex;(10)\nwhere the functions f(x) and g(x) (to linear-order in the cur-\nrent) are given by\nf(x)=1\nScoshx\n\u0015\ncoshL\n\u0015+1\n\u0011sinhL\n\u0015; (11)\ng(x)=1\nSsinhL\u0000x\n\u0015+1\n\u0011coshL\u0000x\n\u0015\u0010\ncoshL\n\u0015+1\n\u0011sinhL\n\u0015\u00112#jyp\nA\u0014; (12)\nandh(x)/#jzis not explicitly shown here since this term will\nnot contribute to the linewidth within the current theoretical\ntreatment. Here, \u0011\u0011JS=p\nA\u0014, and\u0015\u0011pA=\u0014is the AF\nhealing length.\nThe spin current Js\nexentering F modifies the F dynamics as\n~˙S=b\u0002S\u0000~\u000bF\nSS\u0002˙S+Js\nex; (13)\nwhere\u000bFis the intrinsic Gilbert damping parameter in F and\nb=\u0000b0ezis the static FMR field (in units of energy). Insert-\ning Eq. (10) into Eqs. (8) and (9) and performing the time-\naverage over the last two terms in Eq. (13), the full FMR\nlinewidth can be read o \u000bdirectly by summing the coe \u000ecients\nappearing in front of hS\u0002˙Si. The total Gilbert damping pa-\nrameter is then given by \u000b0\nF=\u000bF+\u000e\u000b(i)\nF+\u000e\u000b(b)\nF\u0011\u000bF+\u000e\u000bF,\nwhere the extrinsic contribution \u000e\u000bFhas the interfacial contri-\nbution\u000e\u000b(i)\nFand the AF bulk contribution \u000e\u000b(b)\nF:\n\u000e\u000b(i)\nF=1\nS\u0010#jy\nb0+\u000b\"#\u0011\n\u0010\ncoshL\n\u0015+1\n\u0011sinhL\n\u0015\u00112; (14)\n\u000e\u000b(b)\nF=˜\u000b\nSL\n\u0015+1\n2sinh2L\n\u0015\u0010\ncoshL\n\u0015+1\n\u0011sinhL\n\u0015\u00112; (15)\nwhere ˜\u000b=s\u000b0\u0015=2~. The former originates from spin injection\nand spin-pumping at the AF jN interface while the latter from\nGilbert damping in the AF bulk. Eqs. (14) and (15) constitute\nthe main result of this work.\nAs seen from Eqs. (14) and (15), the healing length \u0015sets\nthe distance over which spin propagation decays inside the\nAF. The healing length is determined from the slope of the\nlinewidth vs. jycurves for various thicknesses Land by ex-4\ntracting the decay length. It is important to note that the cur-\nrent theory only considers spin transport mediated by coherent\nN´eel dynamics, and does not take account of spin transported\nby incoherent thermal magnons. The latter contribution is\nsuppressed at su \u000eciently low temperatures by some power of\nthe ratio T=TN, where TNis the N ´eel ordering temperature\nof the AF, and since magnon-mediated transport is expected\nto decay over the spin di \u000busion length \u0015sd, it is strongly sup-\npressed for \u0015sd\u001cL.\nOnce the AF healing length is known, the parameters #,\u000b\"#\nand ˜\u000bcan be extracted by measuring the FMR linewidth for\nvarious jyandL. While the e \u000bective spin Hall angle #can be\nobtained from the slope of a linewidth vs. jycurve, the Gilbert\ndamping parameter ˜ \u000bcan be extracted in the regime L\u001d\u0015,\nin which the linewidth depends only on ˜ \u000b(see Fig. 2),\n\u000e\u000bFL\n\u0015!1\n!˜\u000b\nS \u0011\n1+\u0011!2\n\u0011\u000e\u000b1\nF: (16)\nForL\u001c\u0015, we expand \u000e\u000bFto linear order in L=\u0015,\n\u000e\u000bF\u00191\nS #jy\nb0+\u000b\"#!\n+1\nS\"\n˜\u000b\u00002\n\u0011 #jy\nb0+\u000b\"#!#L\n\u0015\n\u0011c0+c1L\n\u0015;(17)\nfrom which we see that \u000b\"#can be extracted at zero current\n(i.e., jy=0) and measuring the linewidth for L\u001c\u0015.\nThe extrinsic linewidth exhibits qualitatively di \u000berent be-\nhavior depending on the relative magnitudes of the bulk and\nthe interfacial contributions (see Fig. 2). For strong Gilbert\ndamping ( c1>0) the bulk damping in the AF dominates over\nthe interface e \u000bects and the linewidth grows initially as Lin-\ncreases, saturating eventually as L=\u0015!1 [see Fig. 2(a)]. In\nthe limit of weak Gilbert damping [see Fig. 2(b)], i.e., c1<0\n(and\u000e\u000b1\nF<c0),\u000e\u000bFexponentially decays as Lincreases. In\nRef. 8, the FMR linewidth was measured for various AF thick-\nnesses in the absence of the electrical current. The gradual\nincrease in the linewidth obtained there as a function of the\nthickness is more consistent with our strong Gilbert damping\nregime [c.f. Fig. 2(a)].\nMoriyama et al . [10] has recently reported an ST-FMR\nmeasurement for single crystal NiO using a Pt jNiOjFeNi tri-\nlayer, observed a linear dependence of the FMR linewidth\non the Pt electrical current, and extracted the slope [c.f.\nEq. (14)]. The reported ratio \f\u00190:82 between the slopes\nfor the PtjNiOjFeNi and PtjFeNi samples indicated relatively\ne\u000ecient spin transfer through NiO compared to previous re-\nports [7, 8], and was attributed to the single crystal nature of\nthe AF layer. Applying Eq. (14) directly to the experiment\n(see also Ref. 10), the theoretical ratio \fth(ignoring a pos-\nsible di \u000berence in#between the two samples) is given by\n\fth=[cosh( L=\u0015)+sinh( L=\u0015)=\u0011]\u00002. Using\u0015\u0018100 nm,\u0011\u00181\n(appropriate for NiO and NiO jFeNi interface) and L\u001810 nm,\nwe obtain\fth\u00190:8, which is quantitatively consistent with\nthe experiment [10].Our results can be used to make a connection with the recip-\nrocal experiments [7, 8], in which spin transfer through the AF\nis quantified by measuring the inverse spin Hall voltage VISHE\ngenerated across N by a dynamic F (see Fig. 1). From spin\nHall phenomenology and Onsager reciprocity, the electromo-\ntive force generated in N is given by \"=#(n\u0002˙n)\u0002exjx=0[13].\nUtilizing the adiabatic result n(0)(x;t) [i.e., Eq. (10)] with\nj=0, the (time-averaged) motive force becomes h\"i=\n\u0000(#\u0003\u00122b0=~)ey, where\u0012\u0019(s2\nx+s2\ny)1=2is the cone angle,\n#\u0003=#=fVF[cosh( L=\u0015)+sinh( L=\u0015)=\u0011]2g, andVFis the vol-\nume of F. This leads to an inverse spin Hall voltage\nVISHE=\u0000#\u00122b0`\n~VF\u0010\ncoshL\n\u0015+1\n\u0011sinhL\n\u0015\u00112; (18)\nwhere`is the length of N in the ydirection. We can arrive at\nthe same result by treating the central AF as an e \u000bective junc-\ntion between the N and F subsystems (see Fig. 1). Namely,\nfrom Eq. (14), the macroscopic coupling between current j\nand the macrospin dynamics in F is given through the torque\n\u001c=#\u0003\u00122jyez+(term/jz) acting on the latter, where #\u0003is\nthe overall torque coe \u000ecient for the e \u000bective junction. By On-\nsager reciprocity, this torque gives rise to the inverse spin Hall\nvoltage Eq. (18). Within the current theory, Eq. (18) indicates\nthat the decay length for VISHE as the AF thickness increases\nis set by\u0015.\nWe note in conclusion that the current work considers spin\ntransfer purely mediated by the coherent N ´eel dynamics, cor-\nresponding to the so-called superfluid contribution to spin\ntransport. As the relevant experiments are performed at room\ntemperature [7, 8, 10] reconsidering AF spin transport by ac-\ncounting for the incoherent thermal magnons and studying\ntheir e \u000bect on the FMR linewidth would be valuable, and\nwill contribute to the general understanding of the “two-fluid”\n(condensate and thermal cloud with mutual interactions be-\ntween them) nature of spin transport via collective excitations\nin an AF.\nThe authors would like to thank P. Chris Hammel and\nFengyuan Yang for useful discussions. This work was sup-\nported by FAME (an SRC STARnet center sponsored by\nMARCO and DARPA) (S.T. and Y .T.) and in part by Grants-\nin-Aid for Scientific Research (S) and Grant-in-Aid for Young\nScientists (B) from the Japan Society for the Promotion of Sci-\nence (T.M. and T.O.).\n[1] A. H. MacDonald and M. Tsoi, Philosophical Transactions of\nthe Royal Society A 369, 3098 (2011); R. Duine, Nat Mater 10,\n344 (2011); A. Brataas, A. D. Kent, and H. 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Jungwirth, Nat Mater 10, 347 (2011).\n[4] J. K ¨onig, M. C. Bønsager, and A. H. MacDonald, Phys.\nRev. Lett. 87, 187202 (2001); E. B. Sonin, Adv. Phys. 59,\n181 (2010); S. Takei, B. I. Halperin, A. Yacoby, and\nY . Tserkovnyak, Phys. Rev. B 90, 094408 (2014).\n[5] R. Wieser, E. Y . Vedmedenko, and R. Wiesendanger, Phys. Rev.\nLett. 106, 067204 (2011); S. K. Kim, Y . Tserkovnyak, and\nO. Tchernyshyov, Phys. Rev. B 90, 104406 (2014).\n[6] K. M. D. Hals, Y . Tserkovnyak, and A. Brataas, Phys. Rev.\nLett. 106, 107206 (2011); A. C. Swaving and R. A. Duine,\nPhys. Rev. B 83, 054428 (2011); R. Cheng and Q. Niu, Phys.\nRev. B 86, 245118 (2012); E. G. Tveten, A. Qaiumzadeh, O. A.Tretiakov, and A. Brataas, Phys. Rev. Lett. 110, 127208 (2013).\n[7] C. Hahn, G. de Loubens, V . V . Naletov, J. B. Youssef, O. Klein,\nand M. Viret, EPL (Europhysics Letters) 108, 57005 (2014).\n[8] H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev. Lett.\n113, 097202 (2014).\n[9] T. Moriyama, M. Nagata, K. Tanaka, K. Kim, H. Almasi,\nW. Wang, and T. Ono, ArXiv e-prints (2014), arXiv:1411.4100\n[cond-mat.mtrl-sci].\n[10] T. Moriyama, S. Takei, M. Nagata, Y . Yoshimura, N. Mat-\nsuzaki, T. Terashima, Y . Tserkovnyak, and T. Ono, in prepa-\nration.\n[11] V . G. Bar’yakhtar and B. A. Ivanov, Sov. J. Low. Temp. Phys. 5,\n361 (1979); F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983);\nA. Auerbach, Interacting Electrons and Quantum Magnetism\n(Springer-Verlag, New York, 1994).\n[12] The saturated spin density is defined by s=~S0A=V, where\nS0is the magnitude of each AF spin, Ais the cross-sectional\narea (along the yzplane) of the AF, and Vis the volume occu-\npied by each spin.\n[13] Y . Tserkovnyak and S. A. Bender, Phys. Rev. B 90, 014428\n(2014)."    },    {        "title": "2106.11127v1.Resonant_Measurement_of_Non_Reorientable_Spin_Orbit_Torque_from_a_Ferromagnetic_Source_Layer_Accounting_for_Dynamic_Spin_Pumping.pdf",        "content": "Resonant Measurement of Non-Reorientable Spin-Orbit Torque from a\nFerromagnetic Source Layer Accounting for Dynamic Spin Pumping\nJoseph A. Mittelstaedt1and Daniel C. Ralph1, 2\n1Cornell University, Ithaca, NY 14850, USA\n2Kavli Institute at Cornell, Ithaca, NY 14853, USA\n(Dated: June 22, 2021)\nUsing a multilayer structure containing (cobalt detector layer)/(copper spacer)/(Permalloy source\nlayer), we show experimentally how the non-reorientable spin-orbit torque generated by the Permal-\nloy source layer (the component of spin-orbit torque that does not change when the Permalloy\nmagnetization is rotated) can be measured using spin-torque ferromagnetic resonance (ST-FMR)\nwith lineshape analysis. We \fnd that dynamic spin pumping between the magnetic layers exerts\ntorques on the magnetic layers as large or larger than the spin-orbit torques, so that if dynamic spin\npumping is neglected the result would be a large overestimate of the spin-orbit torque. Nevertheless,\nthe two e\u000bects can be separated by performing ST-FMR as a function of frequency. We measure a\nnon-reorientable spin torque ratio \u0018Py= 0:04\u00060:01 for spin current \row from Permalloy through an\n8 nm Cu spacer to the Co, and a strength of dynamic spin pumping that is consistent with previous\nmeasurements by conventional ferromagnetic resonance.\nI. INTRODUCTION\nCurrent-induced spin-orbit torques o\u000ber the potential\nfor e\u000ecient manipulation of nanoscale magnets for mem-\nory applications [1{3]. Among the families of materi-\nals that are of interest as potential sources of spin-orbit\ntorque are metallic ferromagnets [4{19]. They are pre-\ndicted to generate current-induced spin currents with\nboth a reorientable component in which the spin is al-\nways parallel to the source-layer magnetization, plus a\nnon-reorientable component (with a \fxed spin direction\nin-plane and perpendicular to the charge current) that,\nremarkably, is not dephased within the source magnet\neven though the spin is not aligned with the magnetiza-\ntion [14]. The non-reorientable component has the same\nsymmetries as the torque generated by the spin Hall ef-\nfect from a non-magnetic source material.\nPreviously, spin-orbit torques from metallic ferromag-\nnets have been probed indirectly by exciting nonlocal\nspin currents through magnetic insulators [10, 20{23],\nand the reciprocal process of spin-to-charge conversion\nhas been measured for spin currents injected from an in-\nsulating ferromagnetic into a metallic ferromagnet [24{\n29]. However, it is more challenging to make direct mea-\nsurements of spin-orbit torques generated by ferromag-\nnets in magnetic trilayer structures that would be nec-\nessary for memory applications (i.e., ferromagnetic spin-\nsource layer / non-magnetic spacer / ferromagnetic free\nlayer to be manipulated). Ideally, one would like to apply\na current and make quantitative measurements of just\nthe de\rection of the free-layer magnetization. However,\nin structures with two magnetic layers an applied cur-\nrent will exert torques on both layers and their dynamics\nwill also be coupled by interlayer interactions, making it\nchallenging to isolate just the strength of the spin-orbit\ntorque acting on the free layer. For the reorientable com-\nponent of the spin torque, one approach to address this\nchallenge is to perform spin-torque ferromagnetic reso-\nnance (ST-FMR) experiments and analyze the depen-\ndence on dc current of the separate linewidths for eachmagnetic layer [7, 11, 15, 17{19], although care should\nbe taken in avoiding artifacts that can a\u000bect this method\n[30, 31]. Here, we demonstrate that ST-FMR can also be\nused to achieve quantitative measurements of the non-\nreorientable (independent of the source-layer magneti-\nzation) component of spin-orbit torque generated by a\nferromagnetic metal layer in a magnetic trilayer, by an-\nalyzing the resonant amplitudes and lineshapes. For a\ncorrect analysis of trilayers containing two metallic mag-\nnets (e.g., Co and Permalloy (Py)), it is critical to take\ninto account that dynamic spin pumping [32, 33] cou-\nples the two magnetic layers and alters their resonance\namplitudes. Still, the e\u000bects of spin currents associated\nwith spin-orbit torques and those arising from dynamic\nspin pumping can be separated by analyzing the fre-\nquency dependence of the ST-FMR signals. Therefore,\nST-FMR performed on magnetic trilayers can provide\nquantitative measurements of both the non-reorientable\nspin-orbit torque produced by a magnetic source layer\nand dynamic spin pumping. Our approach is similar to\nthat of Yang et al. [34], except that their method does\nnot account for dynamic spin pumping.\nII. SAMPLE FABRICATION AND\nMEASUREMENTS\nWe used DC magnetron sputtering onto thermally ox-\nidized high-resistivity silicon wafers to grow heterostruc-\ntures of containing both a Co \\detector\" layer, whose\nresonance dynamics we analyze, and a Ni 81Fe19(Py)\n\\source\" layer that generates a spin current to act on the\nCo. Speci\fcally, the sample layer structure is SiO x/Ta\n(1)/Cu (8)/Co (8)/Cu (8)/Py ( tPy)/Ta (1) with num-\nbers in parentheses indicating thickness in nm. The two\nCu spacer layers have the same thickness, and are de-\nsigned so that the in-plane component of the Oersted\n\feld that they generate will cancel within the Co de-\ntector, and consequently the only net in-plane Oersted\n\feld acting on the Co layer will be due to current \rowingarXiv:2106.11127v1  [cond-mat.mes-hall]  21 Jun 20212\nwithin the Py layer. This will simplify our analysis of the\nspin-orbit torque acting on the Co layer. Cu was chosen\nfor the spacer layers because of its long spin di\u000busion\nlength and negligible spin Hall e\u000bect [35]. The Ta lay-\ners provide layer-smoothing and protection against ox-\nidation, and have su\u000eciently large resistivity that they\ncarry negligible current.\nDuring the measurement, we apply an RF current to-\ngether with an in-plane external magnetic \feld at an\nangle\u001erelative to the axis of current \row, and sweep\nthe external \feld strength to tune through the magnetic\nresonance (Figure 1(b)). Each sample is studied using\nseveral di\u000berent RF frequencies and \feld angles \u001e, and\nall measurements are performed at room temperature.\nWe detect a DC signal that arises from mixing between\nthe applied RF current and the resistance variations due\nto the angle-dependent magnetoresistance in both mag-\nnetic layers, originating from both anisotropic magne-\ntoresistance within the layers and spin Hall magnetore-\nsistance [36]. (We work at su\u000eciently large applied mag-\nnetic \felds that the magnetizations of the Co and Py\nlayers are saturated parallel in equilibrium, so contri-\nbutions from giant magnetoresistance are second order\nin the precession angles, and negligible for the measure-\nment.) The circuitry of the measurement is described\nelsewhere [37{39]. The two magnetic layers produce two\nresonances, with the lower resonant \feld being primarily\nfrom the Co and the higher resonant \feld primarily from\nthe Py. We \ft these resonances to two sets of symmet-\nric and antisymmetric lorentzians to capture the signal\nvoltages from each of the magnetic layers.\nThe dependence on the angle \u001efor the symmetric and\nantisymmetric resonance amplitudes of the Co detector\nlayer are shown in Figure 1(c) and (d): both follow a\nsin(2\u001e) cos(\u001e) dependence. This is the form expected\nfrom \\conventional\" current-induced torques { from the\nin-plane component of the Oersted \feld and from spin-\norbit torques arising from a spin current with polariza-\ntion in-plane and perpendicular to the current. It is also\nthe form expected in our geometry for torques arising\nfrom interfacial spin rotation as discussed in refs. [6, 8],\nand based on the analysis below the same angular de-\npendence will be produced by the torques from dynamic\nspin pumping. We observe no signi\fcant contribution\nfrom unconventional current-induced torques that would\nlead to an altered angular dependence [40]. We note,\nhowever, that since both magnetizations have the same\nequilibrium orientation for our geometry, our measure-\nments are not sensitive to the reorientable torques arising\nfrom the spin anomalous Hall e\u000bect or planar Hall e\u000bect\n[4, 11] which create spin currents whose polarization is\nparallel to the magnetization.\nIII. ANALYSIS: EFFECT OF DYNAMIC SPIN\nPUMPING\nWe can account for the resonant dynamics of the\nmagnetic layers, including the e\u000bects of spin pumpingbetween the layers, using a modi\fed Landau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation [32, 33]\n_mi=\u000bimi\u0002_mi+\u001cneq;i+\u001ceq;i\u0000\u000b0\nimj\u0002_mj(1)\nwhereiandjidentify the magnetic layers, \u001ceq;iare\nthe equilibrium torques acting on layer icoming from\nthe applied \feld and anisotropy, and \u001cneq;iare the non-\nequilibrium torques acting on layer icoming from spin-\norbit torques and current-generated Oersted \felds. \u000bi\nis the e\u000bective Gilbert damping parameter for layer\ni, including both the intrinsic damping and the time-\naveraged e\u000bect of the dynamically-pumped spins emerg-\ning from layer i, while the term containing \u000b0\nidescribes\nthe e\u000bect of the dynamically-pumped spins from layer j\nimpinging on layer ito exert a time-dependent torque.\nDipole coupling of the form mi\u0001mjis generally also\nrelevant in systems with more than one ferromagnetic\nlayer, but we designed our Cu spacers to be su\u000eciently\nthick to minimize this e\u000bect, and found by measuring the\nshift of the minor loop in hysteresis measurements that\nit is indeed negligible in our system (see Supplementary\nInformation).\nOur treatment of the torques due to dynamic spin\npumping follow references [32, 33, 41]. Compared to the\nmore-familiar time-averaged DC component of the spin-\npumped spin current that generates DC voltage signals\nin inverse spin Hall experiments [42{44] and modi\fes the\nmagnetic damping [45, 46], the time-varying spin current\nassociated with dynamic spin pumping is much larger. In\nsystems with just a single ferromagnetic layer the e\u000bects\nfrom the dynamic portion of the pumped spins are gener-\nally time-averaged to zero, but with a second ferromag-\nnetic layer present the pumped spins will induce dynamic\ncoupling between the magnetic layers [32, 33, 47, 48].\nBecause the spins pumped by layer jare oscillating and\nare generally out-of-phase with the precession in layer i,\nthey exert a nontrivial torque on layer iwhich depends\non the phase di\u000berence between the oscillations of the\ntwo layers, introducing \feld and frequency dependence\nto the interaction. Based on this \feld and frequency de-\npendence, we will show that it is possible to separate the\ne\u000bects of the spin-pumping-induced interaction between\nthe layers from the direct e\u000bects of the non-reorientable\nspin-orbit torque from the source layer.\nTo model the resonance lineshapes, we consider the\ncase that the resonance \feld is much larger than the co-\nercive \felds of both magnetic layers so that the equilib-\nrium magnetizations of both layers are saturated in the\nsame direction. We then solve for in-plane ( x) and out-\nof-plane (z) de\rections of each layer, and only keep terms\nto \frst order in these de\rections from equilibrium. We\ncan \fnd the de\rections of each layer by solving Eq. (1)\nfollowing the same procedure as [49], noting that we can\nthink of the dynamic spin pumping as e\u000bectively mod-\nifying the torques on each layer. After decoupling the\nin-plane and out-of-plane de\rections of each layer, the3\nFIG. 1. (a) Schematic of our device near the Co resonance. The Oersted \feld from the symmetric spacer layers around the\nCo will cancel inside of the Co layer. Even at the Co resonance condition the Py precession is still signi\fcant. (b) example\nresonance signal of the tPy= 5:7 nm sample showing the Co (left, lower resonant \feld) and Py (right, higher resonant \feld)\nresonances and their symmetric and antisymmetric components for an applied microwave frequency f= 8 GHz. The dashed\nvertical line represents the Co resonance \feld. The antisymmetric part of the Py resonance is still substantial there. (c,d) The\nangular variation in the Co symmetric (c) and antisymmetric (d) resonance components, showing the sin(2 \u001e) cos(\u001e) angular\ndependence expected from SHE-like torques and in-plane Oersted torques, respectively.\nde\rections of the detector layer are\nmdx=Ld(\u0000!d2[\u001cdz\u0000i!\u000b0\ndmsx]\n+i![\u001cdx+i!\u000b0\ndmsz])(2)\nmdz=Ld(!d1[\u001cdx+i!\u000b0\ndmsz]\n+i![\u001cdz+i!\u000b0\ndmsx])(3)\nwhere the d subscript denotes parameters\nassociated with the detection layer, Ld =\u0002\u0000\n!2\u0000!2\nd0\u0001\n+i!!+\nd\u000bd\u0003\u00001,\u001cdxand\u001cdzthe in-plane and out-of-plane non-equilibrium torques,\n!d1=\rB,!d2=\r(B+\u00160Me\u000b;d),!2\nd0=!d1!d2,\nand\u00160Me\u000b;d= 1:82\u00060:09 T for the Co layer based\non the precession frequency. The expressions for the\nsource-layer de\rections are the same as the detec-\ntor layer's but with sanddinterchanged and with\n\u00160Me\u000b;s= 0:93\u00060:05 T for the Py.\nWe can then completely decouple this system by ig-\nnoring any terms which are above \frst order in the \u000b's\nand\u000b0's. The result for the in-plane de\rection of the de-\ntector layer, which generates the signal we are primarily\nconcerned with, is4\nmdx=Ld\u0000\n\u0000!d2\u001cdz+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)(!s1+!d2)!2\u001csx\u0000!2!+\ns\u000bs(!s2!d2+!2)\u001csz\u0003\n+i!\u0000\n\u001cdx+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)(!s2!d2+!2)\u001csz+!2!+\ns\u000bs(!s1+!d2)\u001csx\u0003\u0001\u0001 (4)\nwithLs=\u0002\n(!2\u0000!2\ns0)2+!2!+2\ns\u000b2\ns\u0003\u00001and!+\ni=!i1+\n!i2. The real part of the expression within the large\nparentheses corresponds to the antisymmetric compo-\nnent of the detector-layer resonance and the imaginary\npart to the symmetric component. The terms pro-\nportional to Ls\u000b0\ndrepresent modi\fcations of the usual\nexpression for the ST-FMR resonance caused by spin\npumping from the source layer. Through these terms,\ntorques acting on the source layer ( \u001csx,\u001csz) produce a\nmodi\fcation of the torques acting on the detector layer,\nwith the strength of the e\u000bect depending on the am-\nplitude of the source-layer resonance through Ls. The\nterms in Eq. (4) inside the square brackets containing\n(!2\ns0\u0000!2) correspond to the antisymmetric component\nof the source-layer resonance while the other terms cor-\nrespond to the symmetric component.\nThis expression can be simpli\fed signi\fcantly for ap-\nplication to our measurements. First, the symmetric\npart of our source (Py) resonance is essentially zero near\nthe detector (Co) resonance \feld as seen in Figure 1(b),\nso we can ignore all terms in Eq. (4) which correspond\nto the symmetric part of the source resonance. Second,\nthe dampinglike torque acting on the source layer, \u001csx,\nis quite small compared to the Oersted torque on the\ndetector layer, \u001cdz, as illustrated in Figure 1(b) by the\nfact that the symmetric part of the Py resonance is tiny\ncompared to the antisymmetric part of the Co resonance.\nThis allows us to ignore all terms in Eq. (4) which in-\nclude\u001csxas well. After these simpli\fcations, the in-plane\noscillation of the detector layer takes the form\nmdx=Ld(\u0000!d2\u001cdz\n+i!\u0000\n\u001cdx+Ls\u000b0\nd(!2\ns0\u0000!2)(!s2!d2+!2)\u001csz\u0001\u0001\n:\n(5)\nThe amplitude of the DC mixing signal we detect has\nthe form\nVmix=I\n2\u0000\nRAMR\nd Re [mdx] +RAMR\ns Re [msx]\u0001\nsin(2\u001e);\n(6)\nwhereIis the total current \rowing in the multilayerandRAMR\ni sin(2\u001e) is the\u001e-derivative of the resistance\nchange of the whole heterostructure due to the angle-\ndependent magnetoresistance of layer i. In general, it is\nalso necessary to include an additional voltage signal due\nthe DC component of the pumped spin current and the\ninverse spin Hall e\u000bect [42{44], but we calculate that this\ncontribution is negligible for our samples (see Appendix\nA).\nEven at the position of the detector-layer resonance\nit is not correct to ignore the contribution proportional\nto the source-layer magnetoresistance (the second term\nin Eq. (6)), because precession of the detector layer will\nproduce dynamic spin pumping that drives precession of\nthe source layer. Since this precession of the source layer\nis driven by spin pumping from the detector layer, near\nthe detector layer's resonance condition the source-layer\nprecession has the same resonant \feld and linewidth as\nthe detector-layer resonance. This means that when we\nmeasure the lineshape which we attribute to the detector\nlayer, some of this signal is actually due to the source-\nlayer precession and the mixing voltage which results.\nWe can take into account the source-layer oscillations\nby writing their in-plane component in the form\nmsx=Ls(\u0000!s2\u001csz+i!\u001csx)\n+LdLs(!2\ns0\u0000!2)\u0002\n!2(!d1+!s2)\u001cdx\u000b0\ns\n+i!(!d2!s2+!2)\u001cdz\u000b0\ns\u0003(7)\nwhere the \frst term is the source-layer resonance in the\nabsence of any dynamic spin pumping coupling and the\nLdterm the one producing a resonance in the source\nlayer of the same shape as the detector layer. Near the\ndetector-layer resonance condition, the \frst term is es-\nsentially constant and hence will not play a signi\fcant\nrole in our analysis.\nCombining Eqs. (5), (6) and (7), the voltage am-\nplitudes we measure upon \ftting the detector-layer\nresonance as a sum of symmetric and antisymmetric\nLorentzians is\nVdS=IRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd\u0002\n\u001cdx+LsA(!s2!d2+!2)\u001csz\u000b0\nd\u0003\n+IRAMR\ns sin(2\u001e)\n21\n\u000bd!+\ndLsA(!s2!d2+!2)\u001cdz\u000b0\ns(8)\nVdA=IRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd!d2\n!\u001cdz (9)\nwhereLsA=Ls(!2\ns0\u0000!2). The \frst part of each mixing\nvoltage comes from the detector layer and the secondpart is due to the source layer. There is also a small\ncontribution of the source-layer resonance to VdAthat is5\nFIG. 2. (a,b) Measured values of \u0018FMR for a sample with 5.7 nm Py with a \ft to Eq. (12). In (a) we plot the data versus\nfrequency and in (b) we plot the same data versus calculated values of R(!) to show that the data vary linearly with R(!).\nThe horizontal line is the extracted value of \u0018DL, which is the asymptote of (a)/the intercept of (b). (c) The e\u000bect of Py\nthickness on \u0018DL. (d) The dependence of the dynamic spin pumping parameter on the source Py layer thickness.\nnot included in Eq. (9) because it contributes no more\nthan 1 % of signal from the detector layer for our system,\nso we can ignore it in our analysis.\nWe are left with two signi\fcant modi\fcations to the\nST-FMR signals caused by dynamic spin pumping, both\nof which modify the measured symmetric lineshape am-\nplitude of the detector layer. The \frst is due to spins\npumped from the source layer to the detector layer,\nthereby altering the detector layer resonance, an ef-\nfect which is proportional to the Oersted torque on the\nsource layer, \u001csz, with an amplitude proportional to\nthe antisymmetric component of the source-layer reso-\nnance. This contribution adds directly to the damp-\ninglike torque on the detector layer, \u001cdx. The second\nis due to spins pumped from the detector layer to the\nsource layer, causing the source layer to oscillate with\nthe same general lineshape as the detector-layer reso-\nnance. The size of this term proportional to the Oersted\ntorque on the detector layer \u001csz, and it also has an am-plitude proportional to the antisymmetric component of\nthe source-layer resonance. From Eqs. (8) and (9) it is\nclear that the spin-pumping terms are not expected to al-\nter the usual angular dependence of the ST-FMR signals\n(/sin(2\u001e) cos(\u001e)) as long as the current-induced torques\nare proportional to cos( \u001e), the dependence expected for\nboth conventional-symmetry spin-orbit torques and Oer-\nsted torques [40].\nThe e\u000eciency of the antidamping spin-orbit torque\ngenerated by the Py source layer acting on the Co detec-\ntor layer can be characterized by the antidamping torque\ne\u000eciency, de\fned as\n\u0018DL=\u001cdxeMsat,dtd\n\u0016BJc;s; (10)\nwhereeis the electron charge, Msat;d= 1:19\u00060:08\u0002106\nA/m for the Co layer, td= 8 nm for the Co layer, and\nJc;sthe charge current density within the source mag-\nnetic layer. This e\u000eciency, and also the in\ruence of the6\ndynamic spin pumping, can be determined by analyzing\nhow the ratio of the symmetric and antisymmetric res-\nonance amplitudes depends on measurement frequency.\nFollowing ref. [50], we \frst de\fne an intermediate pa-\nrameter\n\u0018FMR =e\u00160Msat;dtstd\n~r!d2\n!d1VdS\nVdA(11)\nwherets=tPy. The values of \u0018FMR extracted for a\nsample with ts= 5:7 nm as a function of di\u000berent applied\nRF frequencies are shown by the circles in Figure 2(a).\nWe \fnd that \u0018FMR increases by more than a factor of 3 as\nthe frequency is decreased from 14 GHz to 6 GHz, a much\nstronger dependence than is found in samples with non-\nmagnetic source layers [37]. We will show that this can\nbe explained by dynamic spin pumping, in that for lower\nRF frequencies the resonance \felds of the two magnetic\nlayers move closer together, increasing the source layer\nresonance amplitude, and hence the value of LsA, at the\ndetector-layer resonance condition which enhances the\ne\u000bects of the dynamic spin pumping.\nFor magnetic layers as thick as those we employ,\n\u00155 nm, and since the detector layer's interfaces are\nsymmetric and are with the light metal Cu, Oersted\ntorques should dominate over any weak interfacial \feld-\nlike torques for both the source and detector layers [50{\n52]. We therefore assume that \u001cdz=\r\u00160Jc;sts=2 (only\nthe current density within the source layer contributes to\nthe Oersted \feld acting on the detector layer because the\ncontributions from the Cu spacer layers cancel and the\nTa layers have high resistivity and negligible current den-\nsities) and \u001csz=\r\u00160(Jc;dtd+Jc;CutCu)=2 (the current\ncreating the Oersted \feld in the source layer comes from\nthe detector layer and the Cu layers, which comprise the\nrest of the conductive layers in the device). The ratio\nof these torques can be written \u001csz=\u001cdz=\u0000(1\u0000xs)=xs\nwherexs, the fraction of the total RF current \rowing\nwithin the source layer, is in the range of 0.06 to 0.12\ndepending on the Py thickness and is calculated from\nthe layer resistivities using a parallel-conduction model\n(see Supplementary Information). It then follows that\n\u0018FMR =\u0018DL+R(!)\u000b0\ne\u000b (12)\nwith\nR(!) =\u0000e\u00160Msat;dtstd\n~LsA(!s2!d2+!2)1\u0000xs\nxs(13)\n\u000b0\ne\u000b=\u000b0\nd\u0000xs\n1\u0000xsRAMR\ns\nRAMR\nd\u000b0\ns: (14)\nAll of the parameters that are frequency dependent are\ncontained within R(!). Furthermore, all of the param-\neters that enter R(!) are independently measurable, so\nwe can use Eq. (12) to \ft to the data in Figure 2(a),\nusing just two adjustable \ft parameters, \u0018DLand\u000b0\ne\u000b.\nThe \ft is shown both in Figure 2(a) and in Figure 2(b),\nwhich display the same data points plotted as a func-\ntion of measurement frequency in (a) and as a function\nofR(!) in (b). For the tPy= 5:7 nm sample, we \fnd\u0018DL= 0:05\u00060:01 and\u000b0\ne\u000b= 4:0\u00060:4\u000210\u00003. The dom-\ninant source of uncertainty comes from the determina-\ntion ofMe\u000b;dwhich we measure by \ftting the frequency\ndependence of the detector-layer resonant \feld to the\nKittel equation. We note in samples without dynamic\nspin pumping, and in which like our samples the \feld-\nlike spin-orbit torque is negligible relative to the Oersted\ntorque,\u0018DL=\u0018FMR. Therefore, Figure 2(a) illustrates\nthat the neglect of spin-pumping in our analysis would\nlead to a large overestimate of \u0018DL.\nWe have performed the same analysis on samples in\nwhich the Py layers vary in thickness from 5 nm to 12\nnm. The results for \u0018DLand\u000b0\ne\u000bare shown in Fig-\nure 2(c) and (d). The overall dampinglike torque e\u000e-\nciency\u0018DLof the magnetization-independent spin-orbit\ntorque from Py through the 8 nm Cu spacer to the Py\nlayer is 0:04\u00060:01, and it does not have a signi\fcant\ndependence on the Py thickness within our experimen-\ntal uncertainty. The value we determine for \u000b0\ne\u000bdoes\ndepend on the Py thickness, changing by more than a\nfactor of 2 between Py thicknesses of 5 and 12 nm. The\nparameter\u000b0\ndshould not depend on the Py thickness, so\nthe changes in \u000b0\ne\u000b(tPy) indicate that the second term in\nEq. (14) is important as well as the \frst { that the sig-\nnal at the detector resonance is a\u000bected by spin pumping\nfrom the detector layer to the source layer that excites\nprecession in the source (Py) layer. We are not able to\nmake a separate determination of the parameters \u000b0\ndand\n\u000b0\nsin Eq. (14), however, because we are not able to make\nan accurate calibration of the ratio RAMR\ns=RAMR\nd. We\ncan calibrate the anisotropic magnetoresistance of indi-\nvidual Co and Py layers, but we \fnd that when they are\ncombined within a trilayer there is also a signi\fcant ad-\nditional contribution from spin Hall magnetoresistance\nthat cannot be calibrated without separate control over\nthe magnetization angles in the two layers.\nIV. DISCUSSION\nWe can compare our results to previous measure-\nments. E\u000borts to determine the non-reorientable spin\nHall e\u000bect in ferromagnetic layers via spin Seebeck and\nspin pumping experiments are complicated by di\u000ecul-\nties in determining the interface spin mixing conduc-\ntance [53] and the spin di\u000busion length. Furthermore,\nmany of these experiments do not provide su\u000ecient in-\nformation to distinguish between the reorientable and\nnon-reorientable components of the spin current gener-\nated by Py. The two measurements with direct quan-\ntitative claims we have found in this class of experi-\nments give a value for the non-reorientable spin Hall\nratio within Py of \u0012SH(Py) = 0.02 by spin pumping in\nYIG/Cu/Py [25, 26]. The quantity we measure, the spin-\norbit torque e\u000eciency, \u0018DL, is related to the spin Hall\nratio as\u0018DL=\u0012SH(Py)TintwhereTintis an interfacial\nspin transfer coe\u000ecient less than or equal to 1. There-\nfore our measurements establish a lower bound for the\nnon-reorientable spin Hall e\u000bect for the Py in our sam-7\nples,\u0012SH(Py)\u00150:04\u00060:01. We suspect the di\u000berence\nis that the spin mixing conductance may be overesti-\nmated [53] in refs. [25, 26], leading to an underestimate\nof\u0012SH(Py). Miao et al. [24] and Wu et al. [27] studied\nYIG/Py samples and quoted values of \u0012SH(Py) relative\nto values for similar samples made with Pt instead of\nPy:\u0012SH(Py)=\u0012SH(Pt) = 0.38 [24] and 0.98 [27]. These\nestimates rely on assumptions that the interfacial spin\ntransparency and spin di\u000busion lengths are similar for\nYIG/Py and YIG/Pt.\nDas et al. [10, 20] detected both reorientable and non-\nreorientable components of spin-orbit torque generated\nby Py using nonlocal measurement scheme where a spin\ncurrent generated in Py was converted to magnons in\nYIG which then were converted back into a spin cur-\nrent in Pt or Py and transduced into a voltage through\nthe inverse spin Hall e\u000bect. Their value of \u0012SH(Py)\nis only given in relation to that of Pt, they quote\n[GPy\u0012SH(Py)]=[GPt\u0012SH(Pt)] = 0:09 whereGiis a param-\neter describing the spin current to magnon conversion in\neach of the materials. Yang et al. [34] attempted to mea-\nsure the non-reorientable component of the spin curernt\nalone from Py using lineshape analysis of ST-FMR sig-\nnals for Py/YIG and CoFeB/YIG samples (both with no\nspacer layer), but without accounting for dynamics spin\npumping. They claimed values of the non-reorientable\nspin-orbit torque e\u000eciencies \u0018DL= 0:009 for Py acting\non YIG and\u00000:0014 for CoFeB acting on YIG; values\nmuch lower in magnitude than our result. We suspect\nthat these low values may be due primarily to poor inter-\nfacial spin transmission from the Py into the YIG. Low\nspin transparency has also been a general feature found\nin spin-orbit torque measurements from heavy metals\nacting on iron garnets (see the comparisons by Gupta et\nal. [54]).\nBased on measurements of conventional FMR in mag-\nnet/spacer/magnet samples, the parameter \u000b0\ne\u000bis ex-\npected to have a scale comparable to the intrinsic Gilbert\ndamping in magnetic layers, \u000b[32]. This is true for our\nsamples. The values of \u000b0\ne\u000bplotted in Fig. 2(d) are about\nhalf the damping parameters in our Co and Py layers de-\ntermined by the ST-FMR \fts.V. CONCLUSION\nWe have measured and analyzed spin-torque ferromag-\nnetic resonance (ST-FMR) signals for samples contain-\ning the layer structure Py source layer/Cu spacer/Co\ndetector layer. We \fnd that the resonance amplitudes\nand lineshapes are determined by a combination of di-\nrect current-induced torques and dynamic spin pumping\nbetween the magnetic layers. In fact, the strength of the\ntorque created by dynamic spin pumping can be substan-\ntially larger than the direct current-induced spin-orbit\ntorque, so the contribution from dynamics spin pump-\ning should be considered whenever analyzing resonant\nmeasurements in samples with more than one ferromag-\nnetic layer [13, 34]. The frequency dependence of the\nST-FMR signals allows these two e\u000bects to be separated,\nand therefore our experiment provides independent mea-\nsurements of the non-reorientable spin-orbit torque gen-\nerated by the Py layer (the portion of the spin-orbit\ntorque that does not depend on the orientation of the\nPy magnetization) and the strength of dynamical spin\npumping. We \fnd an e\u000eciency for the non-reorientable\nspin-orbit torque generated by Py (and acting through\na 8 nm Cu spacer) of 0 :04\u00060:01. The strength of the\ndynamic spin pumping, as characterized by the parame-\nter\u000b0\ne\u000b, appears consistent with previous measurements\nusing conventional ferromagnetic resonance [32].\nVI. ACKNOWLEDGEMENTS\nWe thank Bob Buhrman for discussions. This research\nwas supported in part by Task 2776.047 of ASCENT,\none of six centers in JUMP, a Semiconductor Research\nCorporation program sponsored by DARPA, and by the\nNSF (DMR-1708499). The devices were fabricated us-\ning the shared facilities of the Cornell NanoScale Facility,\na member of the National Nanotechnology Coordinated\nInfrastructure (NNCI), and the Cornell Center for Mate-\nrials Research, both of which are supported by the NSF\n(Grants No. NNCI-1542081 and No. DMR-1719875).\nAppendix A: DC Spin Pumping Contribution\nThe DC component of the pumped spin current from\nthe Co is converted into a DC charge current through\nthe inverse spin Hall e\u000bect in the Py layer, as described\nin ref. [49]. The generated DC spin pumping voltage\nfollows the form\nVDC\nSP=\u00002e\n~\u0012SHRtotWsin\u001e~\n4\u0019g\"#\ne\u000b\u0015sd,s\u0014!d1\u001c2\ndx+!d2\u001c2\ndz\n(\u000bd!+\nd)2LdS(B)\u0015\ntanh\u0012ts\n2\u0015sd;s\u0013\n(A1)\nwhereRtotis the total resistance of the heterostructure, Wis the width of the device, \u0015sd;sis the spin di\u000busion8\nFIG. 3. Comparison of the DC spin pumping contribution to\n\u0018FMR to the total signal for 5.7 nm Py. The contribution is\nso small as to be negligible.\nlength of the source, and LdS(B) is the symmetric line-\nshape of the detector layer. We have either measured\nor can estimate with reasonable certainty all quantities\nin this expression. For the ts= 5:7 nm Py sample,\nRtot= 12:1 \n,W= 20\u0016m,\u000bd= 0:014,\u0012SH\u00190:05,we approximate g\"#\ne\u000b\u00198 nm\u00002[53] and\u0015sd;s\u001910 nm\n[55] as reasonable estimates compared to other systems,\nwe approximate 1.5 mA of RF current is \rowing in the\nPy layer using the input microwave power after account-\ning for loss in the cabling and current shunting, which\nleads to\u001cdx\u00194\u0002106s\u00001and\u001cdz\u00198\u0002106s\u00001, and\n!d1; !d2and!+\nd, which are evaluated at the Co reso-\nnant \feld and hence depend on frequency, are on the\norder of 10 GHz, 300 GHz and 300 GHz, respectively.\nTo compare this more directly to what we have mea-\nsured before, we utilize the fact that this will only add\nto the symmetric component of the resonance. Since we\nlook at the ratio of the symmetric and antisymmetric\nresonance amplitudes, we compute VDC\nSP=VAwhereVA\nis the amplitude of the antisymmetric component of the\nLorentzians. 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The \frst covers a \feld range\nlarger than the coercive \feld for both magnetic layers and allows us to read the full hyseresis curve of the bilayer.\nThe other two are minor loops which only switch the magnetization of one layer while keeping the magnetization\nof the other layer constant. The results of these hysteresis sweeps on a device with a 6.7 nm Py layer are shown in\nSupplementary Fig. 4.\nThe di\u000berence in the center of the minor hysteresis loops should be equal to the bilinear coupling e\u000bective \feld\nbetween the two layers, since the magnet with the larger coercive \feld has a di\u000berent orientation and hence would\ninduce opposite directions of the bilinear exchange \feld in the two minor loops. The di\u000berence in the minor loop\ncenters is only 2 Oe, and is similar for all of the other Py thicknesses. This is su\u000eciently weak coupling that it can\nbe neglected in our analysis relative to the coupling produced by dynamic spin pumping.\nFIG. 4. (a) The full hysteresis curve with the two minor loops superimposed for the 6.7 nm Py sample. The vertical lines\nindicate the center of the minor hysteresis loops. (b) The minor hysteresis loops superimposed. The centers of the minor loops\nare 2 Oe apart, indicating that there is not signi\fcant bilinear coupling.11\nFIG. 5. Conductance of (a) Co and (b) Py devices with a \ft to the Fuchs-Sondheimer model to determine the resistivity of\neach layer.\nAppendix C: Resistivities of the Ferromagnetic Layers\nTo measure the resistance of the ferromagnetic layers in our devices, we use a separate series of samples with the\nstructure Ta (1) / Cu (3) / FM ( tFM) / Ta (1), which were created so that there would be less shunting from the\nferromagnetic layers leading to a better signal to noise ratio. By investigating the dependence of the heterostructure's\nresistance on the ferromagnetic layer's thickness, we are able to estimate the resistivity of the ferromagnetic layer.\nIn a simple parallel conductor model, we would expect the heterostructure resistance to have a linear dependence\non the inverse ferromagnetic layer thickness, but we found that since the thickness of our ferromagnetic layers are\ngenerally comparable to the mean free path of the ferromagnets, a full Fuchs-Sondheimer model [56] was needed to\nget reasonable results. Using electron mean free paths from the literature [57] for Co and Ni (which we used as an\napproximation for the mean free path of Py), we were able to \fnd the bulk resistivities of the Co and Py layers using\nthe Fuchs-Sondheimer model to \ft the thickness dependence of the device conductivity as shown in Supplementary\nFigure 5. We found the bulk resistivity of Co to be 16 :2\u00060:2\u0016\n-cm and the bulk resistivity of Py to be 24 :9\u00060:4\n\u0016\n-cm. When calculating the shunting factors for the main text, we calculated the resistances of the various layers\nusing the full Fuchs-Sondheimer model for that particular thickness of the ferromagnetic layer.\nAppendix D: Demonstration that Both Anisotropic Magnetoresistance and Spin Hall Magnetoresistance\nContribute to the Angle-Dependent Magnetoresistance of the Magnetic Trilayers\nTo measure the anisotropic magnetoresistance (AMR) in our devices, we use a separate series of samples with\nthe structure Ta (1) / Cu (3) / FM ( tFM) / Ta (1), which were created so that there would be less shunting from\nthe ferromagnetic layers leading to a better signal-to-noise ratio. We measure the resistance of our device with a\nnanovoltmeter while rotating a magnetic \feld in the plane. We measure the resistance change of the whole device,\nbut we would like to measure the resistance change of just the ferromagnetic layer. To get this quantity, we use a\nparallel resistor model and assume that the AMR resistance change is small compared to the resistance of the device.\nWe de\fne\u001eas the angle between the current and external magnetic \feld, Rtot(\u001e) the total device resistance, RNM\nthe resistance of the non-magnetic layer, and RFM(\u001e) =R0+RAMR ;0cos2\u001ethe ferromagnet resistance with R0the12\nFIG. 6. Device conductance as a function of applied \feld angle for (a) the 10 nm Py device and (b) the 8 nm Co device.\nbase resistance and RAMR ;0the AMR resistance. Expanding, we \fnd that\n1\nRtot(\u001e)=1\nRNM+1\nRFM(\u001e)(D1)\n=1\nRNM+1\nR01\n1 +RAMR ;0=R0cos2\u001e(D2)\n1\nRtot(\u001e)\u00191\nRNM+1\nR0\u0000RAMR ;0\nR2\n0cos2\u001e (D3)\nwhich provides a way for us to extract the desired quantity from \ftting the total conductance as a function of applied\n\feld angle. Fits for the devices with 10 nm Py and 8 nm Co are shown in Supplementary Fig. 6.\nWe have also measured the AMR resistance change of a full trilayer structure, Cu (8) / Co (8) / Cu (8) / Py\n(tPy=10) from the main text. Using the values we have measured above for the 8 nm Co and 10 nm Py single layers,\nwe would expect the conductance of this device to vary by 75 \u0016S due to the AMR by adding the contributions\nfrom each of the ferromagnetic layers. However, we actually measure a conductance variation of 23 \u0016S, which is\nover three times smaller than what we expect from the single-layer measurements. We attribute this discrepancy to\nthe presence of spin Hall magnetoresistance in the full trilayer samples, which only exists when we have layers with\nnon-negligible spin Hall e\u000bect. In the single layers, the only other conductive layer was the Cu which has a negligible\nspin Hall e\u000bect. This has precluded a simple measurement of the appropriate RAMR\ni to use in the expressions in the\nmain text.\nAppendix E: Derivation of the E\u000bects of Dynamic Spin Pumping\nTo derive the e\u000bect which the dynamic spin pumping has on the resonances of our layers, we begin with the\nmodi\fed LLGS equation proposed by Heinrich [32]\n_mi=\u000b0\nimi\u0002_mi+\u001cneq;i+\u001ceq;i+\u000b0\ni(mi\u0002_mi\u0000mj\u0002_mj) (E1)\nwhere \u001cneq;iare the nonequilibrium torques which oscillate at the RF frequency such as the Oersted torque or the\nspin-orbit torque, \u001ceq;iare the equilibrium torques which do not oscillate such as those from the static external \feld\nor from anisotropy, \u000b0\niis the intrinsic damping parameter and \u000b0\niis a parameter describing the e\u000bect of spin pumping\nthat both layers have on layer i. We can combine the e\u000bect of damping with the spin pumping out of layer ito arrive\nat a modi\fed damping parameter \u000bi=\u000b0\ni+\u000b0\ni\n_mi=\u000bimi\u0002_mi+\u001cneq;i+\u001ceq;i\u0000\u000b0\nimj\u0002_mj: (E2)\nFirst we assume that both layers are saturated in the same direction and have the same equilibrium magnetization\ndirection. We use a coordinate system where ^ yis along the equilibrium direction of the magnetization and ^ zis normal13\nto the \flm. The magnetization is essentially constant along the ^ yaxis, so our problem reduces to a 2D problem in\nthex\u0000zplane. The torques we need to consider are\n\u001ceq;i=\u0012\nmiz!i2\n\u0000mix!i1\u0013\n;\u001cneq;i=\u0012\n\u001cix\n\u001ciz\u0013\n;mi\u0002_mi=\u0012\n_miz\n\u0000_mix\u0013\n(E3)\nwith!i1=\rB,!i2=\r(B+\u00160Me\u000b;i),\rthe gyromagnetic ratio and Me\u000b;ithe e\u000bective magnetization of layer i.\nPutting this all into the modi\fed LLGS, arrive at\n0\nB@_msx\n_msz\n_mdx\n_mdz1\nCA=0\nB@\u000bs_msz+\u001csx+!s2msz\u0000\u000b0\ns_mdz\n\u0000\u000bs_msx+\u001csz\u0000!s1msx+\u000b0\ns_mdx\n\u000bd_mdz+\u001cdx+!d2mdz\u0000\u000b0\nd_msz\n\u0000\u000bd_mdx+\u001cdz\u0000!d1mdx+\u000b0\nd_msx1\nCA: (E4)\nTo solve this, we \frst consider the system for the source and detector layers independently, treating the coupling\nterms between the source and detector layers as modi\fcations to the nonequilibrium torques\n\u0012\n_msx\n_msz\u0013\n=\u0012\n\u000bs_msz+!s2msz+\u001csx\u0000\u000b0\ns_mdz\n\u0000\u000bs_msx\u0000!s1msx+\u001csz+\u000b0\ns_mdx\u0013\n(E5)\n\u0012\n_mdx\n_mdz\u0013\n=\u0012\n\u000bd_mdz+!d2mdz+\u001cdx\u0000\u000b0\nd_msz\n\u0000\u000bd_mdx\u0000!d1mdx+\u001cdz+\u000b0\nd_msx\u0013\n(E6)\nand use the ansatz\nmi(t) =mie\u0000i!t(E7)\nto arrive at the linearized form of the LLGS\n\u0012\n\u0000i!m sx\n\u0000i!m sz\u0013\n=\u0012\n\u0000msz(i!\u000bs\u0000!s2) +\u001csx+i!\u000b0\nsmdz\nmsx(i!\u000bs\u0000!s1) +\u001csz\u0000i!\u000b0\nsmdx\u0013\n(E8)\n\u0012\n\u0000i!m dx\n\u0000i!m dz\u0013\n=\u0012\n\u0000mdz(i!\u000bd\u0000!d2) +\u001cdx+i!\u000b0\ndmsz\nmdx(i!\u000bd\u0000!d1) +\u001cdz\u0000i!\u000b0\ndmsx\u0013\n(E9)\nWe can solve for the dynamics of each subsystem without decoupling as usual [49], which results in\nmsx=Ls(\u0000!s2[\u001csz\u0000i!\u000b0\nsmdx] +i![\u001csx+i!\u000b0\nsmdz])\nmsz=Ls(!s1[\u001csx+i!\u000b0\nsmdz] +i![\u001csz+i!\u000b0\nsmdx])\nmdx=Ld(\u0000!d2[\u001cdz\u0000i!\u000b0\ndmsx] +i![\u001cdx+i!\u000b0\ndmsz])\nmdz=Ld(!d1[\u001cdx+i!\u000b0\ndmsz] +i![\u001cdz+i!\u000b0\ndmsx])\nwithLi=\u0002\u0000\n!2\u0000!2\ni0\u0001\n+i!!+\ni\u000bi\u0003\u00001,!2\ni0=!i1!i2and!+\ni=!i1+!i2. We now need to decouple the source and\ndetector layer resonances. If we ignore any terms greater than \frst order in the damping parameters, this is relatively\neasy to do via substitution. We will only solve for the detector layer magnetizations as this system is symmetric\nunder interchanging s$d. The in-plane de\rection becomes\nmdx=Ld\u0000\n\u0000!d2\u001cdz+i!\u001cdx+\u000b0\ndLs\u0002\n\u0000!2\u001csx(!s1+!d2)\u0000i!\u001csz(!s2!d2+!2)\u0003\u0001\n(E10)\nand the out-of-plane de\rection becomes\nmdz=Ld\u0000\n\u0000!d1\u001cdx+i!\u001cdz+\u000b0\ndLs\u0002\n!2\u001csz(!s2\u0000!d1) +i!\u001csx(!d1!s1\u0000!2)\u0003\u0001\n: (E11)\nWe now separate the dynamic spin pumping term into real and imaginary parts. It will be convenient to de\fne\nLi=(!2\u0000!2\ni0)\u0000i!!+\ni\u000bi\n(!2\u0000!2\ni0)2+!2!+2\ni\u000b2\ni\u0011Li\u0000\n(!2\u0000!2\ni0)\u0000i!!+\ni\u000bi\u0001\n(E12)\nso that the in-plane de\rections become\nmdx=Ld\u0000\n\u0000!d2\u001cdz+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)!2\u001csx(!s1+!d2)\u0000!2!+\ns\u000bs\u001csz(!s2!d2+!2)\u0003\n+i!\u0000\n\u001cdx+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)\u001csz(!s2!d2+!2) +!2!+\ns\u000bs\u001csx(!s1+!d2)\u0003\u0001\u0001 (E13)14\nand the out-of-plane de\rections become\nmdz=Ld\u0000\n!d1\u001cdx+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)!2\u001csz(!d1\u0000!s2) +!2!+\ns\u000bs\u001csx(!d1!s1\u0000!2)\u0003\n+i!\u0000\n\u001cdz+Ls\u000b0\nd\u0002\n(!2\ns0\u0000!2)\u001csx(!2\u0000!d1!s1) +!2!+\ns\u000bs\u001csz(!d1\u0000!s2)\u0003\u0001\u0001\n:(E14)\nBefore converting to \feld units, we make two approximations. First, the latter part of the dynamic spin pumping\ncontributions to Eq. (E13) and Eq. (E14) correspond to the symmetric part of the source layer Lorentzian. This\ncomponent falls o\u000b quickly away from the source resonant \feld and will be nearly zero near the detector layer\nresonance \feld, and are thus insigni\fcant for our analysis. Also for our system, the dampinglike torque on the source\nlayer\u001csxis quite small, as is evidenced by the symmetric part of the source layer resonances being small compared\nwith all other resonance components, including \u001cdzin particular. This justi\fes ignoring the occurrences of this torque\nappearing in the antisymmetric source layer Lorentzian in Eq. (E13) and Eq. (E14). This leaves only one part of the\ndynamic spin pumping having a signi\fcant e\u000bect, and the in-plane and out-of-plane de\rections can be written as\nmdx=Ld\u0000\n\u0000!d2\u001cdz+i!\u0000\n\u001cdx+Ls(!2\ns0\u0000!2)\u000b0\nd\u001csz(!s2!d2+!2)\u0001\u0001\n(E15)\nand\nmdz=Ld\u0000\n!d1\u001cdx+Ls(!2\ns0\u0000!2)\u000b0\nd!2\u001csz(!d1\u0000!s2) +i!\u001cdz\u0001\n: (E16)\nWe now wish to write the source resonance in \feld coordinates. To do so we expand the resonant frequency about\nthe source resonant \feld:\n!2\ns0(B) =!2\ns0jB=Bs0+(B\u0000Bs0)d!2\ns0\ndB\f\f\f\f\nB=Bs0(E17)\n=!2+ (B\u0000Bs0) \n!s1d!s2\ndB\f\f\f\f\nB=Bs0+!s2d!s1\ndB\f\f\f\f\nB=Bs0!\n(E18)\n=!2+\r(B\u0000Bs0)!+\nsjBs0 (E19)\n!2\ns0\u0000!2=\r(B\u0000Bs0)!+\nsjBs0 (E20)\nso in \feld coordinates\nLs(!2\ns0\u0000!2) =\r!+\nsjBs0(B\u0000Bs0)\n\r2!+2sjBs0(B\u0000Bs0)2+!2!+2s\u000b2s=1\n\u000bs!+sjBs01\n!(B\u0000Bs0)\u0001s\n(B\u0000Bs0)2+ \u00012s\u0011LsA (E21)\nwhere we have de\fned \u0001 i\u0011!\u000bi=\rwhich is the linewidth of the resonance associated with layer i.\nWe can extract the symmetric and antisymmetric detector-layer resonance amplitudes by taking the real part of\nEq. (E15) and Eq. (E16) combined with Eq. (E21) and identifying the prefactors on the symmetric and antisymmetric\nLorentzian shapes, as done in more detail in [49]. Doing so, we \fnd that\nAdx=1\n\u000bd!+\nd1\n!!d2\u001cdz (E22)\nSdx=1\n\u000bd!+\nd\u0000\n\u001cdx+LsA(!s2!d2+!2)\u001csz\u000b0\nd\u0001\n(E23)\nAdz=\u00001\n\u000bd!+\nd1\n!\u0000\n!d1\u001cdx+LsA!2(!d1\u0000!s2)\u001csz\u000b0\nd\u0001\n(E24)\nSdz=1\n\u000bd!+\nd\u001cdz (E25)\nwithSandArepresenting the symmetric and antisymmetric amplitudes and the subscript xandzrepresenting that\nthose quantities deal with the in-plane and out-of-plane magnetization de\rection amplitudes, respectively. Everything\nwith a \feld dependence is evaluated at Bd0, except where noted in LsA, since all of these resonance amplitudes are\nmeasured close to the detector layer resonance \feld. The e\u000bect of dynamic spin pumping captured in Eqs. (E23)\nand (E24) comes from spins pumped by the source layer which travel to and exert torques on the detector layer.\nIn the next section, we deal with another slightly more subtle e\u000bect which the dynamic spin pumping has on our\nmeasurements.15\n1. Dynamic Spin Pumping from Detector Layer to Source Layer\nHere we show spins pumped from the detector layer to the source layer also end up having an e\u000bect on our\nmeasurement of the detector-layer resonance lineshape. To see that, we will start at the equivalent of Eq. (E10) for\nthe source layer magnetization (simply replacing d$s) and rearrange some terms\nmsx=Ls\u0000\n\u0000!s2\u001csz+i!\u001csx+\u000b0\nsLd\u0002\n\u0000!2\u001cdx(!d1+!s2)\u0000i!\u001cdz(!d2!s2+!2)\u0003\u0001\n(E26)\n=Ls(\u0000!s2\u001csz+i!\u001csx) +\u000b0\nsLdLs\u0002\n\u0000!2\u001cdx(!d1+!s2)\u0000i!\u001cdz(!d2!s2+!2)\u0003\n(E27)\n=Ls(\u0000!s2\u001csz+i!\u001csx) +\u000b0\nsLdLs\u0002\n(!2\ns0\u0000!2)!2(!d1+!s2)\u001cdx\u0000!2!+\ns(!d2!s2+!2)\u000bs\u001cdz\n+i!\u0000\n(!2\ns0\u0000!2)(!d2!s2+!2)\u001cdz+!2!+\ns(!d1+!s2)\u000bs\u001cdx\u0001\u0003\n:(E28)\nThe \frst part of Eq. (E28) is the usual equation for the source layer resonance, with the second part being due to\nthe dynamic spin pumping. Near the detector resonance, the \frst term in Eq. (E28) is essentially a constant o\u000bset,\nbut theLdterm means that the source layer magnetization will resonate at the deteciton layer's resonance \feld and\nwith the detector layer's linewidth. The voltage we measure is still due to recti\fcation from the soure layer AMR\nmixing with the applied current, but since we measure the voltage across the whole heterostructure this signal will\nadd to the resonance we measure at the detector-layer's resonant \feld and linewidth.\nTo see how this a\u000bects things, we can make a few simpli\fcations again. If we want to look near the detector layer\nresonance, the terms related to the symmetric part of the source layer resonance (on the right in the square brackets\nin Eq. (E28) will be negligible (same argument as above), so we can ignore them, letting the source layer oscillation\nnear the detector layer resonance look like\nmsx=Ls(\u0000!s2\u001csz+i!\u001csx) +Ld\u0002\nLsA!2(!d1+!s2)\u001cdx\u000b0\ns+i!LsA(!d2!s2+!2)\u001cdz\u000b0\ns\u0003\n: (E29)\nSince we are only concerned with this signal near the detector-layer resonance, and the \frst term in Eq. (E29) is\nessentially a constant o\u000bset, we ignore it in the following analysis. We can decompose the signal from the second\npart into symmetric and antisymmetric parts as above. We \fnd that\n~Ad=\u00001\n\u000bd!+\nd1\n!LsA!2(!d1+!s2)\u001cdx\u000b0\ns (E30)\n~Sd=1\n\u000bd!+\ndLsA(!d2!s2+!2)\u001cdz\u000b0\ns (E31)\nwhere the\u0018signi\fes that this appears as additions to the detector-layer resonance since these lorentzians have the\nsame resonant \feld and lineshape, but actually come from the source layer.\n2. Overall E\u000bect on Measured Detector-Layer Resonance\nCombining Eqs. (E22) and(E30), and Eqs. (E23) and (E31), keeping note of which layer gives rise to the recti\fcation\nproducing the voltage we measure, we \fnd the expression for the voltage we measure when \ftting the resonance\nwhich has the resonant \feld and linewidth of the detector layer:\nVdS=IRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd\u0002\n\u001cdx+LsA(!s2!d2+!2)\u001csz\u000b0\nd\u0003\n(E32)\n+IRAMR\ns sin(2\u001e)\n21\n\u000bd!+\ndLsA(!s2!d2+!2)\u001cdz\u000b0\ns (E33)\nVdA=IRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd1\n!!d2\u001cdz\u0000IRAMR\ns sin(2\u001e)\n21\n\u000bd!+\nd1\n!\u000b0\nsLsA!2(!d1+!s2)\u001cdx: (E34)\nwithIthe total current in the heterostructure and and RAMR\ni the resistance change of the whole stack due to the\nAMR of layer i.\nIn the conventional ST-FMR setting with only one ferromagnetic layer, we could look at the ratio of VdSandVdA\nto \fnd the charge-to-spin conversion e\u000eciency, but with the added terms this ratio is now harder to interpret, so\nwe will work to simplify this expression a bit and see what information we can get from this ratio. We \frst try to\nsimplify our expression for VdA. We can write it as\nVdA=IdRAMR\nd sin(2\u001e)\n21\n\u000bd!+\nd1\n!!d2\u001cdz\u0012\n1\u0000RAMR\ns\nRAMR\ndLsA!2(!d1+!s2)\n!d2\u001cdx\n\u001cdz\u000b0\ns\u0013\n(E35)16\nFIG. 7. Source layer thickness dependence of the fraction of current \rowing in the Py layer, xs.\nand will now show that the term subtracting from 1 is small. For the Co/Py system, all of the terms subtracting\n1 are known or can be reasonably approximated. First, LsA!2(!d1+!s2)=!d2\u0019\u00001 which we can \fnd from \ftting\nthe resonances. We estimate the torque ratio to be in the range 0 :2\u00000:5. The AMR ratio is close to 1 from our\nAMR measurements in Section D. The spin pumping coupling term \u000b0\nsshould be on the same order as damping in\nPy, so estimating 0 :01 should be reasonable. Putting all of this together, the whole term to the right of the 1 inside\nthe parentheses comes out to no more than 0 :01, so this should be no more than a 1% correction to VdA. Therefore,\nwe ignore the e\u000bect of dynamic spin pumping on VdA, which will simplify our analysis.\nAfter simplifying VdA, theVdS=VdAratio becomes\nVdS\nVdA=!\n!d2\u0014\u001cdx\n\u001cdz+LsA(!s2!d2+!2)\u001csz\n\u001cdz\u0012\n\u000b0\nd+\u001cdz\n\u001cszRAMR\ns\nRAMR\nd\u000b0\ns\u0013\u0015\n(E36)\nand we can then de\fne\n\u0018FMR =\u0018DL+\u0011LsA(!s2!d2+!2)\u001csz\n\u001cdz\u0012\n\u000b0\nd+\u001cdz\n\u001cszRAMR\ns\nRAMR\nd\u000b0\ns\u0013\n(E37)\nwhich is the expression that appears in the main text.\nAs discussed in the main text, the torque ratio can be expressed as \u001csz=\u001cdz=\u0000(1\u0000xs)=xswhere the shunt factor\nxsis the fraction of the total RF current \rowing within the source layer. Based on a parallel-conduction model\nusing the layer resistances determined as described above, the value of xsas a function of Py thickness is shown in\nSupplementary Fig. 7."    },    {        "title": "1901.01207v1.How_to_accurately_determine_a_saturation_magnetization_of_the_sample_in_a_ferromagnetic_resonance_experiment_.pdf",        "content": "How to accurately determine a saturation magnetization of the sample\nin a ferromagnetic resonance experiment?\nP. Tomczak\u0003\nQuantum Physics Division\nFaculty of Physics, Adam Mickiewicz University ul. Umultowska 85,\n61-614 Pozna\u0013 n, Poland\nH. Puszkarski\nSurface Physics Division\nFaculty of Physics, Adam Mickiewicz University ul. Umultowska 85,\n61-614 Pozna\u0013 n, Poland\n(Dated: March 11, 2022)\nThe phenomenon of ferromagnetic resonance (FMR) is still being exploited for determining the\nmagnetocrystalline anisotropy constants of magnetic materials. We show that one can also deter-\nmine accurately the saturation magnetization of the sample using results of FMR experiments after\ntaking into account the relationship between resonance frequency and curvature of the spatial dis-\ntribution of free energy at resonance. Speci\fcally, three examples are given of calculating saturation\nmagnetization from FMR data: we use historical Bickford's measurements from 1950 for bulk mag-\nnetite, Liu's measurements from 2007 for a 500 mn thin \flm of a weak ferromagnet (Ga,Mn)As, and\nWang's measurements from 2014 for an ultrathin \flm of YIG. In all three cases, the magnetization\nvalues we have determined are consistent with the results of measurements.\nIntroduction | For a very long time, the ferromag-\nnetic resonance has been used successfully to \fnd mag-\nnetocrystalline anisotropy constants and spectroscopic\nsplitting factors of ferromagnets, see e.g., Ref. [1]. Re-\ncently, we have shown [2] that using this classic experi-\nmental technique completed with a cross-validation of the\nnumerical solutions of Smit-Beljers equation [3{6], not\nonly makes it possible to determine very precisely the\nspectroscopic splitting factor gand magnetocrystalline\nanisotropy constants, e.g., up to fourth order for the di-\nluted magnetic semiconductor (Ga,Mn)As, but also to\nstate which anisotropy constants are necessary to prop-\nerly describe the spatial distribution of the energy which\nis related to magnetocrystalline anisotropy. In what fol-\nlows we present that one can interpret the results of FMR\nexperiment in such a way that it is possible to determine\nthe saturation magnetization of the sample under inves-\ntigation, i.e, that it is possible to \fnd the spatial depen-\ndence of magnetocrystalline energy stored in a ferromag-\nnet using data collected in a single FMR experiment.\nWe will show how to determine saturation magneti-\nzation for three cases: for bulk magnetite, Fe 3O4, we\nwill use the historical Bickford's measurements [7] from\n1950, for a 500 mn \flm of a weak diluted ferromagnet,\n(Ga,Mn)As, we will use Liu's measurements [8] from 2007\nand for an ultrathin \flm of YIG we will use Wang's mea-\nsurements [9] from 2014.\nWhat does one measure while performing FMR\nexperiment? | The most simple answer is that the\nspatial distribution of resonance \feld Hris measured,\ni.e., its dependence on angles #Hand'H, see Fig. 1.\nHowever, let us consider two possible ways of performing\nthe FMR experiment. First, while \fxing a frequency !of\nan alternating microwave \feld one measures the depen-\nFIG. 1: The geometry of the FMR experiment. The\norientation of the applied magnetic \feld His usually\ndescribed in the coordinate system attached to the\ncrystallographical axes of the sample. The \feld\ndirection is represented by angles #Hand'Hand\nequilibrium direction of the sample magnetization Mis\nrepresented by angles #and'.\ndence of the static magnetic \feld Hron angles#Hand\n'Hat resonance, and the second | a static magnetic\n\feld is \fxed and one changes the frequency of microwave\n\feld to \fnd a frequency at which the precession of the\nmagnetic moment of the sample occurs. The resonance\ncondition, for both measurement methods, is given by\nthe following Smit-Beljers equation [3{6]\n\u0012~!\ng\u0016B\u00132\n=1\nsin2#(f##f\u001e\u001e\u0000f2\n#\u001e); (1)\nwherefis free energy of the sample divided by its satu-\nration magnetization M, i.e., free energy is expressed in\nterms of real (Zeeman term) and \fctitious (demagnetiz-\ning and anisotropy terms) \felds. f#\u001e=@f\n@#@f\n@',gis the\nspectroscopic splitting factor, \u0016B{ the Bohr magneton\nand~{ the Planck constant.\nLet us recall, that the Gaussian curvature of the sur-\nfaceS(#;') at one of its points, being the product of twoarXiv:1901.01207v1  [cond-mat.str-el]  4 Jan 20192\nprincipal curvatures \u00141and\u00142, is the determinant of the\nHessian matrix calculated at this point. One obtains in\nspherical coordinates\n\u00141\u00142= det\"\n@2S\n@#21\nsin#@2S\n@S#@'\u0000cot#\nsin#@S\n@'\n1\nsin#@2S\n@'@#\u0000cot#\nsin#@S\n@'1\nsin2#@2S\n@'2+ cot#@S\n@##\n:\n(2)\nLet us now assume that we are interested in the curva-\nture of the the free energy landscape of the sample at\nresonance and put S=fin Eq. (2). Remembering that\nMprecesses around its eqilibrium position, that is, the\nequations\n@f\n@#= 0;@f\n@'= 0 (3)\nare satis\fed, we arrive at Eq. (1) if we interpret\u0000~!\ng\u0016B\u00012\nas a Gaussian curvature of the spatial distribution of free\nenergy.\nReturning to the two ways of measuring the resonance\n\feld we see, that by \fxing a static \feld and subsequent\nmeasuring the frequency !of a microwave \feld for di\u000ber-\nent angles at resonance, we see directly a spatial distri-\nbution of the curvature of the free energy landscape. If,\nhowever, we set the microwave \feld, and then by chang-\ning the static \feld we hit the resonance ( Hr) for di\u000ber-\nent angles#Hand'Hthen we will maintain a constant\nGaussian curvature of the energy landscape during the\nmeasurement.\nThe answer to the question posed at the beginning\nof this section is, that while performing a typical FMR\nexperiment by \fxed !, we measure such a \feld Hrat\nwhich the Gaussian curvature \u00141\u00142of the free energy\nlandscape is constant, i.e., does not depend on #Hand\n'H. As we shall see, this observation will not only allow\n\fnding contributions from various types of anisotropy to\nthe free energy but also will enable the determining of\nsaturation magnetization of the sample.\nWhat determines the spatial distribution of the\nGaussian curvature of the free energy at res-\nonance? | Assume that the free energy density\nF(#H;'H) of a ferromagnet placed in a magnetic \feld,\nexpressed in terms of magnetic \felds, is given by\nF(#H;'H)\nM=f(#H;'H) =\u0000HX\n\u000bn\u000bnH\n\u000b\n+MX\n\u000b;\fN\u000b\fn\u000bn\f+1\n2HcX\n\u000b6=\fn2\n\u000bn2\n\f:(4)\n\u000b;\f =x;y;z . The \frst term on the right-hand side of\nEq. (4) is the Zeeman energy, the second term is the\ndemagnetizing energy, with N\u000b\fbeing the demagnetiza-\ntion tensor. The last term describes the energy related\nto cubic magnetocrystalline anisotropy. The components\nof vectorsnH\n\u000b=H=Handn\u000b=M=Mare de\fned as\nusuall, see e.g., Eqs. (3a) and (3b) in Ref. [2] and Fig. 1.Note that with ful\flled relations (3), the free energy of a\nsample with a speci\fc geometry and saturation magneti-\nzation in a magnetic \feld does not depend on angles #;'\nexplicitly.\n(a)\n (b)\n (c)\n (d)\nFIG. 2: Terms in Eq. (4) representing \fctitious\nmagnetic \felds and their Gaussian curvatures\n(a)MP\n\u000b;\fN\u000b\fn\u000bn\f(ellipsoid) (b) its curvature\n(c)1\n2HcP\n\u000b6=\fn2\n\u000bn2\n\f(d) its curvature.\nEach component of Eq. (4) describes a spatially vary-\ning magnetic \feld with a certain curvature. For exam-\nple the spatial dependencies of the \frst two terms and\ntheir curvatures are shown in Fig. 2. Surface (a) - el-\nlipsoid - represents the \fctitious demagnetizing \feld for\nthe sample oriented along axes [100], [010] and [001]. Its\ncurvature is shown in Fig. 2b. Note that it is non-zero\nin the (001) plane, although this is not visible due to\nthe used scale. The surface (c) represents the \frst order\ncubic magnetocrystalline \fctitious \feld and in Fig. 2d\nis shown its curvature. The curvature of the Zeeman\nterm in Eq. (4) depends on the static magnetic \feld and\ncan be changed during the measurement while the cur-\nvatures associated with the \fctitious demagnetizing and\ncubic anisotropy \felds remain constant during the mea-\nsurement.\nHow to \fnd saturation magnetization and\nanisotropy \felds numerically? | During the FMR\nexperiment the total Gaussian curvature of the free en-\nergy remains constant: contributions of curvatures from\nvarious \fctitious magnetic \felds (anisotropy, demagne-\ntizing), related to the sample itself, are compensated by\nthe curvature of the Zeeman term. To \fnd numerically\nvalues of those \felds, g-factor and saturation magnetiza-\ntion we apply a procedure to some extent reverse to the\nmeasurement: for a given set of measured Hr(#H;'H)\ndata, collected in a single experiment, the constant g,M\nand anisotropy and demagnetizing \felds should be cho-\nsen so that Eq. (1) should be met for all measured static\nresonance \felds Hr(#H;'H). It means, that minimizing\nan appropriate objective function, which measures the\ndeviation from the constant curvature with respect to un-\nknown anisotropy \felds, gandMlets us \fnd them. This\nprocedure is described in detail in Ref. [2]. The key point\nfor \fnding saturation magnetization is that its change\nleads to the change of the curvature of the demagnetizing\n\feld. But to ensure that the determined magnetization\nvalue is unambiguous, the constraint Nx+Ny+Nz= 1\nshould be met during the minimization procedure.\nTo carry out the minimization procedure, it is neces-3\nsary to know the functional form of free energy. Usu-\nally one assumes its speci\fc form taking into account the\nsymmetry of the system | free energy should be invari-\nant under relevant symmetry transformations. Then it is\npossible to expand it into basis functions (orthogonal or\nnon-orthogonal) with the same symmetry. Typically this\nexpansion is limited to some low-order terms of system-\natically decreasing basis functions. It should be remem-\nbered, however, that in the case of examination of cur-\nvature, the higher-order basis functions may give greater\ncontributions to total curvature of free energy than the\nlower-order ones.\nNumerical example I: Saturation magnetization\nof bulk magnetite | Bickford [7] measured the reso-\nnance \felds for disk-shaped samples of magnetite cut in\n(100) and (110) planes. The results of his measurements\nare shown as blue squares in Figs 3a and 3b, respectively.\nEach sample was oriented di\u000berently with respect to the\ncrystallographic planes and thus it has di\u000berent demag-\nnetization tensor. We will assume, however, that both\ntypes of samples have the same g-factor and the same\nsaturation magnetization. The demagnetizing tensor is\nassumed to be diagonal for samples cut in (100) plane\nN(100) =2\n4Nx0 0\n0Ny0\n0 0Nz3\n5: (5)\nThe curvature of free energy distribution, given by\nEq. (1), depends now on six parameters which we denote\ncollectively by the vector hB= (g;M;Nx;Ny;Nz;Hc):\nUsing the procedure described in Ref. [2], we get the\nfollowing coordinates of the vector hB\nhB= (2:220(88);5:906(150);0:2216(267);0:2212(268);\n0:5572(535);\u00000:2392(95)):\n(6)\nBoth,MandHcare given in kOe. The errors (in brack-\nets) refer to the last signi\fcant digits and were calculated\nassuming that the measurement errors are subject to nor-\nmal distribution. Having determined the vector hB, we\nsolve Eq. (4) numerically with respect to Hrand get the\nresonance \feld dependence on the #Hangle (black line in\nFig. 3a). To consider measurements of samples cut in the\nplane (110) let us rotate the coordinate system in such a\nway, that the zaxis is perpendicular to the (110) plane.\nThe demagnetizing tensor is given by\nN(110)=2\n641\n4(Nxy+ 2Nz)1\n4(Nxy\u00002Nz)p\n2\n4(Ny\u0000Nx)\n1\n4(Nxy\u00002Nz)1\n4(Nxy+ 2Nz)p\n2\n4(Ny\u0000Nx)p\n2\n4(Ny\u0000Nx)p\n2\n4(Ny\u0000Nx)1\n2Nxy3\n75;\n(7)\nNxystands forNx+Ny. Solving again Eq. (4) for such a\ntensorN(110), we reach the dependence of Hr(#H) shown\nin Fig. 3b. In Table I the relevant anisotropy constants\nfound by Bickford are presented compared with those\nobtained in the way as described in the text.\nFIG. 3: Resonance \feld versus out-of-plane angle #H\nfor bulk magnetite for samples cut in (100) { (a) and\n(110) planes { (b). Squares - Bickford's measurements\n[7], black line - solution of Smit-Beljers equation for the\nvector hBgiven by Eq. (6). Figs (a) and (b) correspond\nto Figs 3 and 4 in Ref. [7], respectively.\nTABLE I: Comparison of g-factor, saturation\nmagnetization Mand cubic anisotropy constant Kc\nvalues found by Bickford [7]with those obtained by the\nmethod described in the text. The errors (in brackets)\nrefer to the last signi\fcant digits.\ng M [kA/m] Kc[kJ/m3]\n2.220(88) 470(12) -14.12(92)\nRef. [7] 2.07 472a-11\naMeasurement result, [7].\nTo summarize this example, let us stress two things.\nFirst, we observe a fairly large statistical errors that are\nthe result of the scattering of original measurements re-\nported in Ref. [7]. Second, in fact, the Bickford's samples\nhad the shapes of a circular \rattened disks, since in (100)\ngeometryNx\u0019Ny6=Nz.\nNumerical example II: Saturation magnetization\nof the (Ga,Mn)As thin \flm | In Ref. [2] we showed,\nanalyzing Liu's measurements [8] of resonance \felds for\nthe weak ferromagnet (Ga,Mn)As, that in order to prop-\nerly describe the free energy spatial distribution at reso-\nnance, one should expand it up to the fourth order with\nrespect to cubic anisotropy \felds. Here we add to this ex-\npansion demagnetizing \felds, and, since the sample was\noriented along crystallographical axes of (Ga,Mn)As we\nuse a diagonal form od the demagnetizing tensor. The\nvector hLdepends now on eleven parameters\nhL\u0011(g;M;Nx;Ny;Nz;Hc1;:::;Hc4;H[001];H[110]):(8)\nAnisotropy \felds are denoted like in Ref. [2]. The mini-\nmization procedure leads to (\felds are given in Oe)\nhL= (1:984(3);383:5(2:2);0:08104(90);0:06796(95);\n0:8510(20);76:86(0:42);\u0000539:5(10:0);42:86(0:92);\n1412(70);4213(24);68:08(0:64)):\n(9)\nThe numerical solutions of Eq. (1) for such an hLvector\nis shown in Fig. 4. They are equivalent to those obtained4\nin Ref. [2] but now the use of the demagnetizing \felds\nmade it possible to determine saturation magnetization\nM= 30:52(0:18) [emu/cm3].\nFIG. 4: Resonance \feld versus out-of-plane angle #H\n(a) and versus in-plane angle 'H(b) for (Ga,Mn)As\n\flm. Squares - Liu measurements [8], black line -\nsolution of Smit-Beljers equation for the vector hL\n(Eq. 9). Figs (a) ad (b) correspond to Figs 5 and 6 in\nRef. [8], respectively.\nNumerical example III: Saturation magnetization\nof the YIG ultrathin thin \flm | The measurements\nof the resonance \feld of the 9.8 nm thin \flm are taken\nfrom Ref. [9] and shown in Figs 5a and 5b, respectively, as\nblue squares. To determine the anisotropy \felds let us as-\nsume that the free energy is given by Eq. (1) from Ref. [9]\nand that the demagnetizing \felds, as in the (Ga,Mn)As\ncase, are calculated using a diagonal demagnetizing ten-\nsor. Then the vector hHdepends on nine parameters\nhH= (g;M;Nx;Ny;Nz;H2?;H4?;H2k;H4k):(10)\nFactorg= 2 for YIG [10], while saturation magnetization\nM= 1851 [Oe] was determined for this sample using\na vibrating sample magnetometer [9].\nThe components of the vector hHcorresponding to the\nconstant curvature of free energy, i.e., for the sample at\nresonance, are collected in Table II for four cases. In the\n\frst case, we take gandMvalues as measured experi-\nmentally and \fnd remaining ones. In the second case we\n\fx only experimental value of M, in the third case - only\nvalue ofg, and \fnally, we treat both gandMas unknown\n(last row of in Table II). In the last column of Table II the\nerror function E1\nRMS(hH) is given, as de\fned in Ref. [2].\nIts value informs us about the quality of the prediction\nof new experimental results of resonance \felds calculated\nfrom a given free energy formula with speci\fc values (row\nof Table II) of anisotropy \felds. We see that analysis\nof FMR measurements of the YIG ultrathin \flm givesworse results when we treat measured gandMvalues\nas known. This points out that the form of free energy\nused for this analysis is not chosen optimally, because it\nwas not possible to reproduce the experimental values g\nandMwith satisfactory accuracy. Thus the presented\nmethod may also serve as a test for the correctness of\nthe assumed free energy form. Note also, that although\nthe investigated \flm was ultrathin, we obtained non-zero\nvalues of the demagnetization \felds in the (001) plane.\nThe dependencies of resonance \felds on angles #Hand\n'Hare shown, for hHfrom the \frst row of Table II, in\nFig. 5.\nFIG. 5: Resonance \feld for 9.8 nm ultrathin YIG \flm\nversus out-of-plane angle #H(a) and in-plane angle \u001eH\n(b). Squares - Wang's measurements [9], black line -\nsolution of Smit-Beljers equation for the vector hH\ngiven by Eq. (10). Figs (a) and (b) correspond to\nFigs 2(b) and 2(c) in Ref. [9], respectively .\nConclusion | We presented in this article one more ap-\nproach to the interpretation of the classical FMR experi-\nmental data. It is based on the analysis of the curvature\nof the spatial dependence of free energy of the sample at\nresonance and makes it possible, after assuming the func-\ntional form of free energy, to determine (with an accu-\nracy of 0.5-4% in the presented numerical examples) the\nsample saturation magnetization and, consequently, the\nspatial dependence of magnetocrystalline energy stored\nin the sample in a single FMR experiment. The method\npresented here can also be a test for the correctness of the\nassumed form of free energy of the sample at resonance.\nThis may be important for people working in the \feld of\nmagnetic resonance.\nAcknowledgment | This study is a part of the project\n\fnanced by Narodowe Centrum Nauki (National Science\nCentre of Poland) No. DEC-2013/08/M/ST3/00967. Nu-\nmerical calculations were performed at Pozna\u0013 n Super-\ncomputing and Networking Center under Grant No. 284.\n\u0003e-mail: ptomczak@amu.edu.pl\n1M. Farle, Reports on Progress in Physics 61, 755 (1998).\n2P. Tomczak and H. Puszkarski, Phys. Rev. B 98, 144415(2018).\n3J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955).\n4J. O. Artman, Phys. Rev. 105, 74 (1957).5\nTABLE II: Comparison of g-factor, saturation magnetization Mand and anisotropy \feld values found in Ref. [9]\nwith those obtained by the method described in the text. The errors (in brackets) refer to the last signi\fcant digits.\ng M [Oe] Nx Ny Nz H2?[Oe] H4?[Oe] H2k[Oe] H4k[Oe] E1\nRMS(hH)\n2.000a1851b0.08429(10) 0.08721(11) 0.8285(10) -497.7(2.0) 372.3(1.1) 62.23(0.27) 49.02(0.22) 0.815\n2.014(1) 1851b0.08795(17) 0.08985(10) 0.8222(15) -466.5(1.5) 276.9(0.9) 61.83(0.23) 47.27(0.18) 0.691\n2.000a1800(11) 0.08102(15) 0.08408(11) 0.8349(16) -538.9(1.7) 372.6(1.0) 62.16(0.11) 49.05(0.16) 0.815\n2.013(1) 1772(12) 0.08263(8) 0.08467(13) 0.8327(13) -526.2(1.6) 277.5(0.9) 61.82(0.09) 47.42(0.18) 0.691\nRef. [9] 1851(37) -1253(25) 60.4(1.2) 42.0(1.2)\naMeasurement result, see Apendix A of Ref. [10].\nbMeasurement result, Ref. [9].\n5L. Baselgia, M. Warden, F. Waldner, S. L. Hutton, J. E.\nDrumheller, Y. Q. He, P. E. Wigen, and M. Mary\u0014 sko,\nPhys. Rev. B 38, 2237 (1988).\n6A. Morrish, The Physical Principles of Magnetism (Wiley-\nIEEE Press, 2001).\n7L. R. Bickford, Phys. Rev. 78, 449 (1950).8X. Liu, Y. Y. Zhou, and J. K. Furdyna, Phys. Rev. B 75,\n195220 (2007).\n9H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev.\nB89, 134404 (2014).\n10D. Stancil, A. Prabhakar, Spin Waves. Theory and Appli-\ncations (Springer, 2009)."    },    {        "title": "1907.03097v1.Angle_resolved_broadband_ferromagnetic_resonance_apparatus_enabled_through_a_spring_loaded_sample_mounting_manipulator.pdf",        "content": "Angle-resolved broadband ferromagn etic resonance \napparatus enabled through  a spring -loaded sample \nmounting manipulator  \n \nShikun He ,1,2,a Qing Qin1, Tiejun Zhou1 and Christos \nPanagopoulos2,b\n \n1Data Storage Institute, A*STAR (Agency for Science Technology and Research),  2 Fusionopolis Way \n08-01 Innovis, Singapore 138634  \n2Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, \nNanyang  Technological University, Singapore 637371  \n \n \nAbstract  \nBroadband ferromagnetic resonance is a useful technique to determi ne the \nmagnetic anisotropy and study the magnetization dynamics of magnetic \nthin films. We report  a spring -loaded sample loading manipu lator for \nreliable sample mounting and rotati on. The manipulator enables maxim um \nsignal, enhances system stability and is particular ly useful for fully \nautomated in-plane -field angle-resolved measurement s. Th is angle -\nresolved broadband ferromagnetic resonance apparatus pro vides a viable \nmethod to study anisotropic damping  and weak magnetic anisotropies , both \nvital for funda mental research and application s.   \n                                                             \na heshikun@gmail.com  \nb christos@n tu.edu.sg  Manipulating magnetization through electrical current s via spin -transf er \ntorque and spin -orbit torque  is superior to conventional schemes such as \nthe usage of magnetic field s due to its sc alability and energy efficiency1. \nConsequently, e merging applications based on spin torques have attracted \nconsiderable attention. In particular, it has been demonstrated that spin-\ntransfer -torque mag netic random access memory ha s the potential to \ncompete with mainstream memory technologies2,3. Magnetic thin films and \nmultilayers  with low damping constant , and high anisotropy are essential \nfor reducing the  switching current density without scarifying  data retention  \nperformance .  \n \nBroadband ferromagnetic resonance  (FMR)  is widely used for studying the \nmechanism of magnetization relaxation in ultrathin films4,5. It has been \nshown that the damping parameter can be determined accurately in a field \nperpendicular to the plane geometry6,7, assuming that damping is isotropic . \nHowever, both phenomenological theory and recent experiments indicate \nthat the Gilbert damping constant can depend on the orientation of the \nmagnetization vector8,9. Crucially, p rior to the calculation o f anisotropy in \ndamping, it is important to determine the extrinsic contributions to the \nFMR linewidth such as inhomoge neous broadening and magnetic \nanisotropy7,10 therefore , it is desirable to develop an angle -resolved FMR \napparatus to measure spectr a at various magnetization orientations and frequencies.   \n \nHere, we present a fully automated angle -resolved broadband FMR design  \nwhich meets those requirements. T he key component of th e apparatus is an \nin-plane sample mounting manipulator , which incorporates  a spring -\nloaded sample holder and  is attached to a 2-axis motorized stage  for \nautomation .  \n \nIn the broadband FMR technique, the thin film sample is flipped and placed \non top of  a coplanar waveguide  (CPW) . The gap between the film and the \nCPW  is minimized to ensure that the microwave field  (hrf) generated by \nthe RF -current -flow in the cent ral conductor of the CPW  is applied \neffectively on the sample  under investigation . Consequently,  the Figure 1: Comparison of two different schemes for achieving field - in-plane \nangle -resolved FMR measurements. (a), (b) Field rotation scheme. (c) -(f) \nSample rotation scheme . First, lift the sample up to create a gap between \nsample and CPW. Then, rotate the sample to  the desired orientation . Finally, \nreload the sample.  \nmicrowave field  will always  be in the sample plane.  Figure 1 depicts the \ntwo possible schemes to vary the orientation of the magnetization of the \nsample  to enable  field-in-plane angle -resolved FMR . The first scheme  \nallows a  chang e in the orientation of the external field ( Hdc) as shown by \nFigs. 1a and 1b. This method is  straightforward to implement by using  a \nvector -magnet  (if rot ating the magnet base is cumbersome ). Because the \nprecession of magnetization is caused by the component of h rf \nperpendicular to the external field Hdc, there will be no FMR signal for \nHdc||hrf (Fig. 1b). Furthermore, there will be an unavoidable and significant  \nreduction of signal when  Hdc is tilted away from the x-axis. Therefore , such \nconfiguration  would hinder the  investigation of ultra -thin films.  In the  \nsecond scheme one rotates the sample , keeping the waveguide and the \ndirection of Hdc fixed . This scheme is preferred  from a sensitivity \nperspective  because  Hdc and hrf remain perpendicular  to each other  (Figs. \n1c and 1f). However,  it is necessary to decouple the contact between the \nsample and  the CPW before each sample rotation  in order to avoid  \nscratching the sample and  the CPW . Hence, one would follow the \nprocedure s illustrated in Fig. 1c to 1f to rotate the sample. It is well know n \nthat the sample mounting method ha s a significant impact on the signal \nstrength and noise level of FMR spectrum. Hence, a reliable loading and \nunloading method is crucial to the  performance  of this technique.  To meet \nthese stringent requirements , we developed a n in-plane sample mounting manipulator.    \n \nAs shown in Fig.  2a, the axis of the in -plane sample mounting manipulator \nis aligned to the center of the FMR probe head where a CPW  is mounted . \nThe FMR probe head (Fig.  2b) is attached to a separate supporting rod , the \naxis of which is perpendicular to the in -plane sample mounting manipulator. \nPrior to activat ing the in -plane manipulator, the orientation of the FMR \nprobe head is se t to the default value  stored in the configuration data file  \nby the associated rotary positioner. The manipulator is attached to a rotary \npositioner , which controls  the sample  orientation. The rotary positioner , in \nturn, is put into  a line ar position er to control the height of the manipulator \nand hence the location of the sample.  \nThe in -plane rotation procedure is as follows: First, attach  the sample (item \n11 of Fig . 2d) to the bottom of the sample holder using glue or double -\nsided tape. Then, mount the sample holder to the manipulator just as mount \na screw. Next, the software will follow the procedure listed in Fig. 1c to \nFig. 1f  to measure FMR spectr a at specified i n-plane field orientations.  To \navoid damag ing the CPW and the sample, the sample holder height is \ncontrolled by a Cu spring  (item 5  of Fig. 2d ) and two slots (item 3  of Fig. \n2d). The usage of the spring minimizes the gap between the sample and the \nCPW with a controllable  force  (Fig. 2a) , which is essential for the \nreliability of this manipulator . The tolerance of the travel distance is about 5 mm, which is orders magnitude larger than  the control precision of  \ncommercially available linear positioners.  The difference between the  \nwidth of the guiding slots and the outer diameter of the set screw is 0.05  \nmm. Thus , the maximum  error of the  sample orientation is 0.7 degree s, as \nestimated by the outer diameter of th e housing  tube (25  mm, item 4  of Fig. \n2d).    \nThe procedure for  mounting the sample in  the out-of-plane rotation \nmeasurement  using the probe head only (Fig. 2b)  is as follows: F irst, the \nin-plane sample mounting manipulator is deactivated and pulled up \nautomatically by the linear positioner. Next, we place the flipped sample at \nthe center of the CPW (Fig.  2c). Third,  we press the clip base (item 6), Figure 2: (a) A schematic of the FMR head and sample location during in -plane \nrotation. (b) The FMR probe head. 1 -Copper housing; 2 -Endlaunch connector; 3 -\ncoplanar wave guide (CPW); 4 -2.4mm cable connection port; 5 -BeCu spring clip; \n6-clip base; 7 -Cu spring (c) The angle -resolved FMR system based on an \nelectromagnet. The in -plane sample mounting manipulator is shifted back for \nclarity. ( d) The in -plane sample mounting manipulator. 1 -sample holder; 2 -spring \nholder; 3 -set screw; 4 -housing tube; 5 -Cu spring; 6 -shaft; 7 -adapter to positioner; \n8-screw hole for item 3; 9 -guiding slot; 10 -mounting screw hole ; 11-sample.  \nrotate the spring clip (item 5) to a location on top of the sample and release \nthe clip base. The out -of-plane measurement through rotating the FMR \nprobe head is trivial so therefore needn’t be discussed . \n \n \nFigure 3: Results of a Si/Py  10nm/Ta 5nm sample. (a) Schematic of the coordinate \nsystem used for the in -plane rotation. (b) FMR raw data at various field orientations for \nf=10  GHz. (c) 2D mapping of the FMR spectra at 10 GHz. (c) The field orientation 𝜑𝐻 \ndependence of the resonance field. (e) The FMR linewidth versus field orientation 𝜑𝐻. \n(f) The effective saturation magnetization and anisotropy field value calculated at \nvarious frequencies.       \n \nShown in Fig.  3 is the  in-plane FMR  data obtained f or a 10 nm Py  sample \nprepared by magneto -sputtering.  Resonance lineshape is observed for all \nof the field/magnetization  orientations.  The nearly isotropic FMR response \nis shown in Fig. 3c.  Solving the Landau –Lifshitz –Gilbert equation in polar \ncoordinates11,12, we derive the general formula of dynamic susceptibility \nfor the in -plane  angle-resolved FMR : \n  \n \nM\n2 M4 cos( - )[P i ]cos( - ) 2(2 / ) i (P Q)2s H\nHM H\nHPQ f     (1) \nHere,\n2\n21\nMEPM , \n2\n221\nsinMMEQM  are the second partial derivatives of total \nfree energy density E. However, fitting the real and imaginary part of the \nspectrum to Eq.1 simultaneously  requires  prior knowledge of E, which is \ninconvenient . Alternatively, to determine the FMR linewidth accurately \nwithout prior information on the actual anisotropy energies, we fit the data \nto a Lorentz lineshape:  \n𝑆21=𝐴⋅(∆𝐻/2)2\n[(𝐻−𝐻𝑅𝑒𝑠)2+(∆𝐻/2)2]+𝐵⋅∆𝐻/2⋅(𝐻−𝐻𝑅𝑒𝑠)\n[(𝐻−𝐻𝑅𝑒𝑠)2+(∆𝐻/2)2]+𝐶 (2) \nwhere A and B are the amplitude of symmetric and antisymmetric  lineshape,  \nrespectively,  C is the background of the spectrum, HRes is the resonance \nfield and ΔH is the full  width at half maximum. We find a weak two -fold \nsymmetry in the magnetization dependence of Hres (Fig. 3d) and therefore, \nwe can determine the resonance condition by adding a uniaxial anisotropy \nterm in the total energy and  solving the Smit -Beljers equation11: \n2\nres eff uni res uni2=\n( cos( ) 4 cos )( cos( ) cos 2 )M H M M H Mf PQ\nH M H H H\n            \n (3) \nHere, \nM is calculated numerically by searching for the energy minimum \nFigure 4: (a) Frequency dependence of FMR linewidth of a Si/Py  10nm/Ta 5nm \nfilm. (b) The extracted damping constant for the film at various field orientations.  \nof E in a single domain approximation. Notably, \neff 4M and \nuniH are \nmaterial related parameter s, and do  not change with measurement \nfrequency.  As shown in Fig. 3f, the values  determined  are nearly identical  \nfor all the frequencies  used. Remarkably, the small anisotropy, which is \nless than 3 Oe , possibly due to steps in the substrate , can be determined \nwith a standard deviation of less than 5%. The FMR linewidth at a given \nfrequency  is also essentially the same for all magnetization orientations , as \nexpected by the isotropic nature of  the Py film (Fig. 3e) .  \n \nFigure 4a depicts the frequency dependence of the FMR linewdith. The \nobserved linearity  for any field orientation allows the calculation of \ndamping along each field direction  \n04H f H   . Here, w e find the \ndamping constant for Py to be  isotropic (Fig.  4b). Notably,  besides angle -\ndependent damping  one can use the nonlinear linewidth versus frequency \ncurves , which are typical for ultra -thin films, to extract the extrinsic \ncontributions to magnetization relaxation and the associated angle -\ndependence.   \n \nIn summary, we developed a spring -loaded sample loading manipulator for \nreliable and fast angle -resolved  broadband  FMR measurements . We \ndemonstrated  the suitability of the apparatus for determining the magnetic anisotropy with high precision and measuring the damping constant at  all \nin-plane magnetization orientation s. Importantly, s uch a manipulator can \nbe designed in a variety of configurations to meet the geometrical \nconstraint s of a particular system.  \n \nAcknowledgements  \nCP acknowledges support from the National Research Foundation (NRF), \nNRF -Investigatorship (No. NRFNRFI2015 -04). \n \nReferences  \n1 N. Locatelli, V. Cros, and J. Grollier, Nat. Mater. 13, 11 (2014).  \n2 S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H.D. Gan, M. Endo, S. \nKanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat. Mater. 9, 721 (2010).  \n3 K.L.W. and J.G.A. and P.K. Amiri, J. Phys.  D. Appl. Phys. 46, 74003 \n(2013).  \n4 I. Harward, T. O’Keevan,  a Hutchison, V. Zagorodnii, and Z. Celinski, \nRev. Sci. Instrum. 82, 095115 (2011).  \n5 S. He and C. Panagopoulos, Rev. Sci. Instrum. 87, 43110 (2016).  \n6 M.A.W. Schoen, D. Thonig, M.L. Schneider, T .J. Silva, H.T. Nembach, \nO. Eriksson, O. Karis, and J.M. Shaw, Nat Phys 12, 839 (2016).  \n7 A. Okada, S. He, B. Gu, S. Kanai, A. Soumyanarayanan, S. Ter Lim, M. \nTran, M. Mori, S. Maekawa, F. Matsukura, and others, Proc. Natl. Acad. Sci. 114, 3815 (2017).  \n8 K. Gilmore, M.D. Stiles, J. Seib, D. Steiauf, and M. Fä hnle, Phys. Rev. B \n81, 174414 (2010).  \n9 L. Chen, S. Mankovsky, S. Wimmer, M.A.W. Schoen, H.S. Kö rner, M. \nKronseder, D. Schuh, D. Bougeard, H. Ebert, D. Weiss, and C.H. Back, \nNat. Phys. 14, 490 (2018).  \n10 S. He, Y. Liu, Y. Zheng, Q. Qin, Z. Wen, Q. Wu, Y. Yang, Y. Wang, Y. \nFeng, K.L. Teo, and C. Panagopoulos, Phys. Rev. Mater. 1, 64401 (2017).  \n11 J. Smit and H.G. Beljers, Philips Res. Rep 10, 31 (1955).  \n12 M. Farle, Reports Prog. Phys. 61, 755 (1998).  \n "    },    {        "title": "0704.2442v1.Ferromagnetic_resonance_force_microscopy_on_a_thin_permalloy_film.pdf",        "content": "arXiv:0704.2442v1  [cond-mat.mtrl-sci]  19 Apr 2007Ferromagnetic resonance force microscopy on a thin permall oy\nfilm\nE. Nazaretski, I. Martin, and R. Movshovich\nLos Alamos National Laboratory, Los Alamos, NM 87545\nD. V. Pelekhov and P. C. Hammel\nDepartment of Physics, Ohio State University, Columbus OH 4 3210,\nM. Zalalutdinov\nSFA Inc., Crofton, MD 21114,\nJ. W. Baldwin and B. Houston\nNaval Research Laboratory, Washington DC 20375\nT. Mewes\nDepartment of Physics and Astronomy,\nUniversity of Alabama, Tuscaloosa, AL 35487\nAbstract\nFerromagnetic Resonance Force Microscopy (FMRFM) offers a me ans of performing local ferro-\nmagnetic resonance. We have studied the evolution of the FMR FM force spectra in a continuous 50\nnm thick permalloy film as a function of probe-film distance an d performed numerical simulations\nof the intensity of the FMRFM probe-film interaction force, a ccounting for the presence of the\nlocalized strongly nonuniform magnetic field of the FMRFM pr obe magnet. Excellent agreement\nbetween the experimental data and the simulation results pr ovides insight into the mechanism of\nFMR mode excitation in an FMRFM experiment.\nPACS numbers: 07.79.Pk, 07.55.-w, 76.50.+g, 75.70.-i\n1Magnetic resonance force microscopy (MRFM) offers a very high se nsitivity approach\nto detection of magnetic resonance and has demonstrated three dimensional imaging with\nexcellent spatial resolution. Proposed by J.A. Sidles [1], it has been us ed for the detection\nof electron spin resonance (ESR) [2], nuclear magnetic resonance (NMR) [3]; recently Rugar\nand co-workers reported detection of a force signal originating f rom a single electron spin\n[8], emphatically demonstrating MRFM sensitivity. Incorporating bas ic elements of MRI,\nMRFM can provide much higher spatial resolution than conventional MRI. Electron spin\ndensity images with micrometer scale resolution in an arbitrarily shape d sample can be\ndeconvolved from MRFM spatial force maps [9]. MRFM image deconvolu tion requires a\nthorough understanding of the underlying interaction between th e MRFM probe and the\nobject imaged. This deconvolution process, analyzed for the case of noninteracting spins in\na paramagnetic sample is given in Ref. 10. Recently, ferromagnetic r esonance (FMR) has\nbeen detected by MRFM in YIG bar [4], YIG dot [5, 6], YIG film [7] and perm alloy dots\n[17]. The role of the probe magnet in FMRFM is dual: it both perturbs th e FMR modes,\nand detects the force signal. However, in Ref. [4, 5, 6] the FMR mod es were only weakly\nmodified by the tip field.\nIn this letter we focus on the regime when the effect of the tip is non- perturbative: the\nfield inhomogeneity due to the tip field strongly modifies the resonanc e modes as well as\nleads to the formation of the localresonance under the tip. We report FMRFM spectra from\nacontinuous, 50nmthick permalloyfilm, performedwithacantilever w ithanearlyspherical\nmicron-size magnetic tip. We report the evolution of the FMRFM spec tra as a function of\nthe tip-sample spacing, and propose a model which describes the ob served behavior.\nThe cantilever is mounted on top of a double scanning stage, compris ed of a 3D Attocube\nscanner [11] for coarse scanning and a piezotube for fine scanning . The optical feedback\ncontrol of the Attocube allows to position and move the cantilever s tage with an accuracy\nbetter than 250 nm. The microwave power is used to manipulate the s ample magnetization\nand is generated by the Giga-tronics 12000A synthesizer at a freq uencyωrf/2π= 9.55 GHz,\n79 mW of power and amplitude modulated with a modulation depth of 70% . It is fed into\na strip line resonator with the broad resonant characteristics and allows to record FMRFM\nspectra in the frequency range between 9 and 11.5 GHz. A more det ailed description of\nthe microscope can be found elsewhere [12]. We use a commercially ava ilable silicon nitride\ncantilever with a fundamental resonant frequency ωc/2π≈8.06 kHz and a spring constant\n2k∼10 mN/m. The magnetic tip is a 2.4 µm diameter spherical Nd 2Fe14B particle [13]\nshown in the inset to Fig. 2. We removed the original tip of the cantilev er by focused ion\nmilling, and manually glued the magnetic sphere to the cantilever with St ycast 1266 epoxy\nin the presence of an aligning magnetic field of a few kOe. The 50 nm thic k permalloy\nfilm was deposited on a 20 nm thick Ti adhesion layer on a 100 µm thick silicon wafer. The\npermalloywascappedwithaprotective20nmTilayer. Anapproximat ely2×2mm2sample\nwas glued to the strip line resonator and the film plane was oriented pe rpendicular to the\ndirection of the external magnetic field Hext. In Fig. 1. we show FMRFM spectra recorded\nas a function of the probe-sample distance at a constant tempera tureT= 11.000±0.005 K.\nEach spectrum displays two distinctive features: the main resonan ce signal, which occurs at\napproximately Hext≈14.54 kOe, and the secondary resonance structure at lower fields . The\nstructureoftheFMRFMsignalisreminiscent ofthoseobservedinp ermalloymicrostructures\n[17]. The intensities of both features decrease as the probe magne t is moved away from the\nsurface of the sample. It is important to note that retraction of t he probe does not change\nthe position of the main resonance peak significantly, and at the sam e time the width of the\nsecondary feature changes substantially. The quality factor Qof the cantilever decreases\nfrom∼11,000 to∼6,000 as the probe approaches the sample. The change in Qis due\nto tip-sample interactions (other than magnetic resonance) and is consistent with previous\nreports [15, 16]. We measured Qby two methods (ring-down technique and swept frequency\nthrough resonance) at each probe-film spacing , and subsequent ly calculated the magnitude\nof the force signal acting on the cantilever, Fig. 1.\nIn general, the force Fdetected in an MRFM experiment is a convolution of δm(r,t) (the\nchange in sample magnetization due to rfmanipulation) with the field gradient ∇Htip(r)\nof the magnetic tip. The force is given by the following volume integral: F=/integraltext\nVs(δm(r,t)·\n∇)Htip(r)dr.In our experiments, the spherical shape of the probe magnet allow s analytical\ncalculation of its magnetic field profile Htip(r) [19], and is used to provide precise knowledge\nof the magnetization term δm(r,t) needed to interpret FMRFM spectra correctly.\nThe total magnetic field inside the sample is Htot=Hext+Htip+Hd, whereHextis a\nuniform external magnetic field, Htipis a nonuniform magnetic field of the probe magnet\nandHdis the demagnetizing field. The exact spatial profile of Htotdepends on the total\nmagnetic moment of the probe magnet, the probe-film spacing and t he relative orientation\nofHextwith respect to the orientation of probe magnet magnetization (in o ur case they\n3are parallel). The well defined shape of the magnetic tip allows us to sc hematically divide\nthe sample into two regions according to the magnitude of the ˆzcomponent of the total\nmagnetic field Hz\ntot. The first, region I, is a circular region directly under the magnetic\ntip where its field is significant and positive. The area of this region is de termined by the\ndistance between the center of the probe tip and the sample. In re gion II the field of the\nprobe is much weaker and negative and its area encompasses the re maining sample area of\n∼2×2mm2. The schematics of two regions is shown in the inset to Fig. 2.\nFor a conventional FMR experiment the expected resonance field f or the uniform FMR\nmode is Hu\nres=ωrf\nγ+ 4πMs, where we neglect the anisotropy contribution, [16, 18]. For\nour experimental parameters γ/2π= 2.89±0.05 GHz/kOe [16] and ωrf/2π= 9.55 GHz,\nwe obtain the value of Hu\nres= 14.6 kOe, which agrees within the error with the observed\nresonance field for the main peak (dotted line in Fig. 1); we thereby a ttribute it to the\nresonance originating from region II of the sample and representin g its large area. The main\nresonance peak in Fig. 1 can be understood as the fundamental FM R mode observed in\nconventional FMR experiments [20, 21], modified by the tip field. Ana lytical derivation of\nthe exact profile of such a modified mode is difficult, so we have perfor med micromagnetic\nsimulations based on the numerical solution of the Landau-Lifshitz- Gilbert equation [22].\nFor simulation we used a damping constant α= 0.01, an exchange constant A = 1.4 ×10−6\nerg·cm−1and values of 4 πMs= 13.2 kG for the probe magnet and 4 πMs= 11.3 kG for the\npermalloy film were measured independently by the SQUID magnetome try [14]. Simulations\nindicate that the FMR mode excited in region II of the sample at the re sonant field Hresdoes\nnot penetrate significantly into spatial region I, where Hz\ntot>ωrf\nγ. We will present details of\nthe analysis elsewhere [23].\nWe assign the lower field feature in the FMRFM spectra shown in Fig. 1 t o the reso-\nnance contributions originating from the localized FMR excitations spatially confined ap-\nproximately to region I of the sample. In this region, the resonance occurs at lower values\nofHextthan that of the main peak (Fig. 1, dotted line). The frequency shif t oflocalized\nFMR is determined by two factors: (1) the strength of the tip field Htipat the sample\nsurface, and (2) the effect of mode confinement to the spatial re gion I with characteristic\ndimensions defined by the tip-sample distance, which further increa ses thelocalmode fre-\nquency relative to the bulk resonance by a value ∆ ωconf(r′). Both effects cause the local\nresonance to occur at the external field value that is lowerthan that of the bulk resonance\n4by the amount ∆ Hext≈ −Htip(r′)−∆ωconf(r′)/γ. From numerical simulations [23], for a\nconfinement within a disc of radius r′∼10µm, ∆ωconf(r′)/γ≈30 Oe, which combined with\nthe estimated value of Htip(r′)≈20 Oe results in a total shift consistent with experimental\nfindings. Both the tip field Htipand ∆ωconf(r′) decrease as the probe magnet is retracted\naway from the film surface, and local modes merge into the main reso nance (Fig. 1, dotted\nline).\nThe non-lorentzian and broad shape of the signal possibly indicates the presence of mul-\ntiple modes contributing to the resonance. While numerical simulation s are required to\ndetermine the possibility of such modes in particular geometry/mate rials, we believe that\ntheir appearance is generic in thin-film soft magnets and is induced by the local field inho-\nmogeneity. We will present the detailed theoretical analysis elsewhe re[23].\nThe normalized FMRFM spatial force map obtained from the uniform F MR mode mod-\nified by the tip field is shown in Fig. 2. The semicircular region of the plot w here the\ntip-sample interaction force is set to zero corresponds to region I . In the lower panel of\nFig. 2 we show the intensity of the main resonance signal as a functio n of the tip-sample\ndistance and compare simulations with the experiment. The dashed lin e shows the expected\nforce signal for the uniform FMR mode, not modified by the tip field. I n this case there is\nno force exerted by a uniformly magnetized infinite film on a spherical probe tip.\nIn conclusion, we conducted FMRFM experiments in a thin permalloy film . We per-\nformed quantitative analysis of the force exerted by the fundame ntal mode and observed\nlocally excited FMR. We conducted simulations and determined two dist inctive regions of\nthe sample contributing to the FMRFM spectra. We find clear evidenc e for local modifica-\ntion of the FMR mode structure by the probe tip providing insight into the interaction of\nthe probe tip with ferromagnetic samples in FMRFM.\nThe work performed at Los Alamos National Laboratory was suppo rted by the US De-\npartment of Energy, Center for Integrated Nanotechnologies, contract W-7405-ENG-36 at\nLos Alamos National Laboratory and contract DE-AC04-94AL850 00 at Sandia National\nLaboratories. The work at Ohio State University was supported by the US Department\nof Energy through grant DE-FG02-03ER46054. The work at Nava l Research Laboratory\n(NRL) was supported by the Office of Naval Research through the Institute for Nanoscience\nat NRL.\n5[1] J. A. Sidles, Appl. Phys. Lett. 58, 2854 (1991)\n[2] D. Rugar, C. S. Yannoni, and J. A. Sidles, Nature 360, 563 (1993)\n[3] D. Rugar and O. Z¨ uger and S. Hoen and C. S. Yannoni and H. M. Vieth and R. D. Kendrick,\nScience264, 1560 (1994)\n[4] Z. Zhang, P. C. Hammel, and P. E. Wigen, Appl. Phys. Lett. 68, 2005 (1996)\n[5] V. V. Naletov, V. Charbois, O. Klein, and C. Fermon Appl. P hys. Lett. 83, 3132 (2003)\n[6] V. Charbois, V. V. Naletov, J. Ben Youssef, and O. Klein, A ppl. Phys. Lett. 80, 4795 (2002)\n[7] R. Urban, A. Putilin, P. E. Wigen, S.-H. Liou, M. C. Cross, P. C. Hammel, and M. L. Roukes,\nPhys. Rev. B 73, 212410 (2006)\n[8] D. Rugar, R. Budakian, H. J. Mamin, and W. Chui, Nature 430, 329 (2004)\n[9] O. Z¨ uger, and D. Rugar, Appl. Phys. Lett. 63, 2496 (1993)\n[10] A. Suter, D. V. Pelekhov, M. L. Roukes, and P. C. Hammel, J . Magn. Res. 154, 210 (2002)\n[11] www.attocube.com, models ANPx(z) 100/LIN\n[12] E. Nazaretski, T. Mewes, D. Pelekhov, P. C. Hammel, and R . Movshovich, AIP Conf. Proc.\n850, 1641 (2006)\n[13] www.magnequench.com, partciles type: MQP-S-11-9\n[14] E. Nazaretski, J. D. Thompson, D. V. Pelekhov, T. Mewes, P. E. Wigen, J. Kim, M. Zalalut-\ndinov, J. W. Baldwin, B. Houston, P. C. Hammel, and R. Movshov ich, J. Magn. Magn. Mat.\n(2006), doi:10.1016/j.jmmm.2006.10.994\n[15] I. Dorofeyev, H. Fuchs, G. Wenning, and B. Gotsmann, Phy s. Rev. Lett. 83, 2402 (1999)\n[16] E. Nazaretski, J. D. Thompson, M. Zalalutdinov, J. W. Ba ldwin, B. Houston, T. Mewes, D.\nV. Pelekhov, P. Wigen, P. C. Hammel, and R. Movshovich, J. App l. Phys. MS JR06-3902R\n(accepted)\n[17] T. Mewes, J. Kim, D. V. Pelekhov, G. N. Kakazei, P. E. Wige n, S. Batra, and P. C. Hammel,\nPhys. Rev. B 74, 144424 (2006)\n[18] Z. Frait, Physica 86-88B, 1241 (1977)\n[19] J. D. Jackson Classical Electrodynamics 3rdedition, Wiley, New York, 1999\n[20] C. Herring, and C. Kittel, Phys. Rev. 81, 869 (1951)\n[21] M. Sparks B. R. Tittmann, J. E. Mee, and C. Newkirk, J. App l. Phys.40, 1518 (1969)\n6[22] T. L. Gilbert, Phys. Rev. 100, 1243 (1955); L. D. Landau, E. M. Lifshitz, and L. P. Pitaevsk i,\nStatistical Physics, Part 2 (Pergamon, Oxford, 1980), 3rd ed.\n[23] D.V. Pelekhov, unpublished\n7Figure Caption\nFigure 1: Evolution of the FMRFM signals as a function of the probe-fi lm spacing.\nDotted line indicates the position of the main resonance peak, indepe ndent of the probe-\nfilm distance. Arrows mark the onset of the lower field resonance fe ature. Experimental\nparameters: ωrf/2π=9.55 GHz, T = 11 K\nFigure2: Upper panel: forcemapoftheFMRFMprobe-filminteractio n. Theforceacting\non a cantilever due to an elementary ring-shaped area is calculated a s a function of radius of\nthe ring andthe cantilever-film spacing. The forcemap is normalized t o a maximum positive\nforce value at each probe-film distance. Force contribution from r egion I of the sample is set\nto zero because the fundamental resonant mode which occurs at Hresdoes not significantly\npenetrate into region I where Hz\ntot>ωrf\nγ. The inset shows schematically the probe-sample\narrangement andtwo regionsofthesamplecontributing totheFMR FMsignal. Lower panel:\nintegrated probe-film interaction force as a function of the cantile ver-film spacing for the\nmain resonance peak at Hext∼14.54 kOe. Solid symbols represent experimental points and\nsolid line is the result of calculations based on the exclusion of region I. The dashed line\nshows the force if region I were included in integration. Two insets sh ow SEM micrographs\nof the cantilever tip.\n8102\n101\n102\n101\nExternal□field□[kOe]Signal□force□[10 N]-16\n100\n14 15 14.8 14.6 14.4 14.214.0 m /c1091009.0 m/c109103\n102\n101\n1005.5 m/c109103\n102\n101\n1002.5 m/c1092.5 m/c109\nFIG. 1:\nFIG. 2:\n9"    },    {        "title": "1605.07850v2.Exploring_the_ferromagnetic_behaviour_of_a_repulsive_Fermi_gas_via_spin_dynamics.pdf",        "content": "Exploring the ferromagnetic behaviour of a repulsive Fermi gas via spin dynamics\nG. Valtolina,1;2;3F. Scazza,1;2A. Amico,1;2A. Burchianti,1;2\nA. Recati,4;5T. Enss,6M. Inguscio,1;2M. Zaccanti1;2and G. Roati1;2\n1INO-CNR, Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy\n2LENS and Dipartimento di Fisica e Astronomia, Universit `a di Firenze, Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy\n3Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy\n4INO-CNR BEC Center and Dipartimento di Fisica, Universit `a di Trento, 38123 Povo, Italy\n5Technische Universit ¨at M ¨unchen, James-Franck-Straße 1, 85748 Garching, Germany and\n6Universit ¨at Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany\nFerromagnetism is a manifestation of strong repulsive interactions between itinerant fermions in condensed\nmatter. Whether short-ranged repulsion alone is sufficient to stabilize ferromagnetic correlations in the absence\nof other effects, like peculiar band dispersions or orbital couplings, is however unclear. Here, we investigate\nferromagnetism in the minimal framework of an ultracold Fermi gas with short-range repulsive interactions\ntuned via a Feshbach resonance. While fermion pairing characterises the ground state, our experiments pro-\nvide signatures suggestive of a metastable Stoner-like ferromagnetic phase supported by strong repulsion in\nexcited scattering states. We probe the collective spin response of a two-spin mixture engineered in a magnetic\ndomain-wall-like configuration, and reveal a substantial increase of spin susceptibility while approaching a criti-\ncal repulsion strength. Beyond this value, we observe the emergence of a time-window of domain immiscibility,\nindicating the metastability of the initial ferromagnetic state. Our findings establish an important connection\nbetween dynamical and equilibrium properties of strongly-correlated Fermi gases, pointing to the existence of\na ferromagnetic instability.\nThe magnetic properties of a variety of quantum systems,\nranging from electrons in transition metals [1, 2] and normal\n3He liquids [3] to neutron and quark matter within the crust\nof neutron stars [4, 5], emerge from strong interactions be-\ntween itinerant fermions, i.e. not localized into a crystal lat-\ntice. Itinerant ferromagnetism can be qualitatively captured\nby an intuitively simple mean-field framework, as first en-\nvisioned by Stoner [6]: a free electron gas can become fer-\nromagnetic once a short-ranged screened Coulomb repulsion\nbetween oppositely oriented electron spins overcomes the ef-\nfect of Fermi pressure, which would favour a paramagnetic\nstate with no spin ordering. A sufficiently strong repulsion\npromotes the parallel alignment of magnetic moments, at the\nprice of an increased kinetic energy, making the paramagnetic\nstate unstable towards a ferromagnetic one. Stoner’s picture\nhas proven successful to qualitatively describe the phases of a\nwealth of electron systems. However, electrons in solids are\nsubject to various effects beyond short-range repulsion [1, 2],\nwhich are thought to play a role in promoting or suppress-\ning ferromagnetism. In particular, there exist materials where\na Stoner-type instability of the repulsive Fermi liquid state\nis altered by competing mechanisms. Notable examples are\nthe emergence of unconventional superconductivity adjoining\nferromagnetism in intermetallic compounds [7] and the non-\nFermi liquid behaviour of the paramagnetic state in itinerant-\nelectron ferromagnets [8].\nQuite paradoxically, even the paradigmatic scenario of an\nultracold atomic Fermi gas with short-ranged repulsive inter-\nactions entails a superfluid ground state of paired fermions,\nrather than a ferromagnetic one. This stems from the fact\nthat genuine zero-range repulsion, encoded in the s-wave scat-\ntering length athat can be adjusted via magnetic Feshbach\nresonances, necessarily requires an underlying attractive po-\ntential with a weakly bound molecular state [9]. Hence, the\nrepulsive Fermi gas corresponds to a metastable excited (up-per) energy branch of the many-body problem [10–12] (see\nFig. 1a). It is intrinsically unstable against pair formation via\ninelastic decay processes [11, 13, 14] that were found to be\nrapid with respect to the development of magnetic domains\n[13–15]. Nonetheless, theoretical approaches intentionally\nneglecting pairing, based on Landau’s Fermi liquid theory\nand quantum Monte Carlo (QMC) calculations [12, 16–21],\nconfirm Stoner’s ferromagnetic instability of a homogenous\nFermi gas driven only by short-range repulsion. In light of\nthe metastable nature of the repulsive branch, it is debated\nwhether such a system may exhibit ferromagnetic correlations\nat all, at least temporarily, and how the magnetic and transport\nproperties of the repulsive gas are affected by the large cou-\npling to lower-lying states.\nPrevious experiments with ultracold spin mixtures prepared\nin the paramagnetic phase and quenched to strong repul-\nsion [13, 22] found the Stoner instability to be precluded\nby the pairing one [15], concluding that a Fermi gas with\nstrong short-range repulsion does not undergo a ferromag-\nnetic phase transition [13]. On the other hand, a recent spec-\ntroscopic study [23] of highly-imbalanced repulsive mixtures\nin the polaronic regime [11, 12] revealed the Fermi liquid\nphase to become energetically and thermodynamically unsta-\nble beyond a critical repulsion strength, in quantitative agree-\nment with theoretical predictions for polarized repulsive gases\n[12, 19, 20, 24], while still featuring an unexpectedly long\nlifetime before decaying onto lower-lying states. This sup-\nports a scenario where the repulsive Fermi gas may exhibit\na transient time during which repulsion and ferromagnetic\ncorrelations dominate the evolution of the many-body system\nalso in the balanced case, for which a ferromagnetic instability\nis maximally favoured [20].\nIn this work we investigate the spin response and the sta-\nbility of a balanced spin \"–#mixture prepared in an ar-\ntificial magnetic domain-wall structure (see Fig. 1b). SucharXiv:1605.07850v2  [cond-mat.quant-gas]  22 Feb 20172\nz\nyx\nba\n-2 -1 0 1-2-1012ε/atom( a.u. )\n-1/kF a\n3\n03\nOptical density\nLR\nFIG. 1. Engineering a ferromagnetic state with an ultracold\natomic Fermi gas. a , A Fermi mixture with resonant short-range\ninteractions, parametrised by 1=kFa, features two distinct energy\nbranches, the lower (upper) being associated to a net interspecies\nattraction (repulsion). Ferromagnetism may develop at strong repul-\nsion along the upper branch, whose stability is limited due to en-\nhanced decay processes onto the lower branch. Depending on the\nspin imbalance, the lower branch corresponds either to a paired phase\nor to an attractive Fermi liquid. b, In our experiment we circumvent\nthe pairing instability by preparing an atomic Fermi gas in a fer-\nromagnetic domain-wall structure, probed via spin-selective in situ\nimaging (see lower image). The initial state is obtained by segregat-\ning the two spin components into two initially disconnected reser-\nvoirs at equilibrium by means of a thin optical barrier with a waist of\nabout 2\u0016m (sketched in green).\na fully-ferromagnetic initial configuration features a vanish-\ningly small\"–#density overlap in contrast to a paramagnetic\nstate [13, 22]. This greatly suppresses the effect of pairing\nprocesses, ensuring that the overall relaxation rate remains\nmuch slower than that given by the Fermi energy, thereby\nmaintaining the system on the upper branch for a compara-\ntively long time. We probe the ferromagnetic properties of the\nrepulsive Fermi gas through two distinct but interconnected\nmeasurements of spin dynamics at varying interaction. On the\none hand, by probing the spin-dipole mode [25], i.e. the out-\nof-phase relative oscillation of the approaching spin \"–#do-\nmains and observing its softening, we demonstrate an increase\nof the spin susceptibility as a function of repulsion strength,\nup to a sharp discontinuity in the spin response. On the other\nhand, by studying spin diffusion [28–30] at short and long\nevolution times we reveal the metastability of the initial ferro-\nmagnetic state and gain new insights on its relaxation mech-\nanisms [12, 13, 15]. In particular, our work shows that the\nshort-time collective dynamics of an artificially-created ferro-\nmagnetic state are governed by strong repulsion, before in-elastic decay to the attractive branch eventually leads to de-\nmagnetization.\nWe initially prepare a weakly interacting mixture of ultra-\ncold6Li atoms [31], equally populating the two lowest Zee-\nman states, hereafter denoted as j\"iandj#i. The atoms are\nheld in a cylindrical optical dipole trap with axial and radial\nfrequencies \u0017z'21Hz and\u0017?'265Hz, respectively. By\nadjusting the evaporation procedure we can tune the degree of\ndegeneracy from T=TF<0:1up to\u00181. HereTis the gas\ntemperature, while TFis the Fermi temperature of a single-\ncomponent Fermi gas of Natoms in a harmonic trap, given\nbykBTF=EF=h(6N\u0017z\u00172\n?)1=3, withhandkBdenot-\ning the Planck’s and Boltzmann’s constants. At a magnetic\nfield of about 1 G, where the magnetic moments of j\"iandj#i\nstates are opposite, the application of a magnetic field gradient\nallows us to spatially separate the two spin components along\nthe weak axis of the trap. Once the overlap between the two\nclouds is perfectly zero, we superimpose a 2 \u0016m thin optical\nrepulsive barrier as high as V0\u001810EFonto the center of\nthe harmonic potential, in order to split the trap into two inde-\npendent reservoirs [32], as sketched in Fig. 1b. We then adia-\nbatically turn off the magnetic field gradient and end up with\nall\"(#) fermions in the left (right) reservoir (see Methods\nand Supplementary Information for details). This creates two\nmacroscopic spin domains at rest, separated by a distance only\na few times wider than the mean interparticle spacing of the\ngas. Such configuration resembles the density distribution ex-\npected for a spin-mixture at full magnetization in an elongated\nharmonic trap, with a central domain wall of thickness around\nthe interparticle spacing, additionally surrounded by an unpo-\nlarized low-density shell on the cloud surface. From here, we\nlet the two spin components start interacting by removing the\noptical barrier. Before switching off the barrier, the interac-\ntion strength is adjusted by setting the magnetic field close to\nthe center of a broad \"–#Feshbach resonance located around\n832 Gauss [33].\nSpin response of a repulsive Fermi gas\nIn a first experiment the spin dynamics is triggered by\nabruptly switching off the barrier from its initial value on a\n\u0016s timescale. Owing to the small initial separation of about\n5\u0016m, the two spin clouds approach each other with small\nrelative momentum ~k\u001c~kF=p2mLiEF. We follow\nthe clouds dynamics by resonant in-situ absorption imaging,\nmonitoring the evolution of the two spin domains (see Meth-\nods and Supplementary Information). On top of an over-\nall slow drift, the relative distance between their centres of\nmassd(t) =z\"(t)\u0000z#(t)presents a small-amplitude out-of-\nphase oscillation, signalling the excitation of the spin-dipole\nmode (see Fig. 2a-c). The measurement of the spin-dipole\nfrequency[25, 26], next to the one of spin fluctuations [13, 27],\nis a powerful probe for disclosing the magnetic properties of\nharmonically trapped gases, connecting the local behaviour of\nthe spin susceptibility \u001fto a directly accessible quantity. Fol-\nlowing a sum-rule approach [25], the spin-dipole frequency\n\u0017SDis found to be:\n\u00172\nSD=N\"+N#\n4\u00192mLiR\ndrz2\u001f(n(~ r)); (1)3\nTime (ms)-1000100∆d(μm)\n0 20 40 60 80 100\nκF a = 0.04(μm)∆d05\n-5\n0 20 40 60 80 100κF a = 3.65\n0 20 40 60 80 10005\n-510(μm)∆dκF a = 1.02a\nb\nc\n d\nκF aνSD /νz\n0\n0 0.5 1 2 2.5 1.51.5\n0.52\n1\nνSD /νz\n123-1 0 1-1/(κF a)\ne\nFIG. 2. Spin response of a repulsive Fermi gas. a-c , After subtracting a slow exponential drift from the centre-of-mass distance d(t)between\nthe two spin clouds, the residual out-of-phase dynamics \u0001d(t)after sudden barrier removal is fitted to a damped sinusoidal function (dashed\nlines), from which \u0017SDis extracted for several interaction strengths. The bare trap oscillation is shown for comparison (dotted red line in\na). Shaded areas denote the standard confidence bands of the fits. Data points result from at least 5 independent measurements with error\nbars given by the standard error of the mean (s.e.m.) combined with the uncertainty on the subtracted exponential drift. d, The normalized\nspin-dipole frequency \u0017SD=\u0017zis plotted versus \u0014FaforT=TF= 0:12(2) (blue circles) and T=TF= 0:25(4) (purple squares), with error bars\nbeing 95 %confidence intervals of sinusoidal fits. A decrease of \u0017SDfollowed by a clear discontinuity is visible, suggesting the occurrence of a\nferromagnetic instability. Here, the shaded blue interval denotes the range of interaction strengths where the spin-dipole mode does not exhibit\na distinct single-frequency oscillatory behaviour. The dashed blue (violet) lines are the average \u0017SD=\u0017zmeasured beyond the discontinuity\npoint atT=TF'0:12(T=TF'0:25) up to unitarity. The solid (dashed) green lines are the T= 0 predictions from a sum-rule approach\nassuming 25 %(100%)\"–#spatial overlap (see Methods). e,\u0017SD=\u0017zmeasured atT=TF'0:12across the Feshbach resonance is shown for\nthe gas initially prepared in the upper branch (blue circles) or in the lower branch (red diamonds). The solid (dashed) red line represents the\nspin-dipole frequency prediction using the attractive gas spin susceptibility from Ref. 37 and setting a 20 %(100%)\"–#spatial overlap (see\nSupplementary Information). The dashed blue line coincides with the one shown in panel d.\nwhere\u001fdepends on the local density n(~ r)in the trap (see\nSupplementary Information for details). In particular, for a\nnon interacting gas \u0017SD=\u0017z. An increase of \u001fin the re-\npulsive Fermi liquid phase can therefore be distinctly iden-\ntified by\u0017SD< \u0017z, i.e. by a softening of the spin-dipole\nmode. Although \u001fis locally divergent if a ferromagnetic in-\nstability is reached at the trap center, \u0017SDcan only reach a\nnon-zero minimum [25] owing to the inhomogeneous density\nprofile of the trapped gas, whose outer low-density paramag-\nnetic region contributes with a small spin susceptibility. The\nmeasurement of this collective oscillation is extremely chal-\nlenging when starting from a paramagnetic configuration due\nto strong damping [34] and inelastic processes [11, 13, 23].\nHere, instead, where the two spin domains just partially over-\nlap over the timescale of the measurement, we are able to trace\na few oscillation periods of the spin-dipole mode from which\nwe extract\u0017SDthrough a fit to a damped sinusoidal function\n(see Supplementary Information). Performing such a mea-\nsurement at several magnetic field values, we obtain the trend\nof the spin-dipole frequency as a function of the repulsive in-\nteraction strength, displayed in Fig. 2d for two distinct tem-\nperatures. The interaction strength is described by the dimen-\nsionless parameter \u0014Fa, where\u0014F(and correspondingly \u000fF)\nis the average Fermi wave number (energy) weighted over the\ninitial density distribution close to the interface between the\ntwo domains (see Methods).Let us discuss here the results for the colder samples at\nT=TF= 0:12(2) (blue circles in Fig. 2d). By starting from\nthe weakly interacting regime, where \u0017SD'\u0017z(see Fig. 2a),\nan increase of the interspecies repulsion leads to a progres-\nsive reduction of the spin-dipole frequency, down to values as\nlow as\u0017SD'0:6\u0017zat about\u0014Fa'1(Fig. 2b). We find\nthe decrease of \u0017SDto be accompanied by a strong increase\nof the damping of the oscillations [34]. By further increasing\nthe interspecies repulsion, an abrupt change occurs in the spin\ndynamics: for \u0014Fa&1:1, the spin-dipole frequency sharply\njumps above the bare trap frequency, \u0017SD'1:70(4)\u0017z(see\nFig. 2c), while the damping of the oscillations is strongly re-\nduced. Once this narrow interaction region is crossed, a fur-\nther increase of \u0014Fadoes not produce any significant change,\nneither in the damping rate nor in \u0017SD. Higher tempera-\nture data exhibit a qualitatively similar trend, with the abrupt\nchange in\u0017SDoccurring at a higher \u0014Favalue.\nThe observed trend of \u0017SDup to\u0014Fa'1agrees with the\nmode-softening predicted by Eq. (1), reflecting a substantial\nincrease of spin susceptibility. This is further supported by\nthe good agreement of our low-temperature data with the spin-\ndipole frequency trend calculated in linear response [25] (see\ngreen lines in Fig. 2d), inserting into Eq. (1) the susceptibility\n\u001f(\u0014Fa)from QMC calculations for a T= 0homogeneous re-\npulsive Fermi gas [20] (see Methods). Furthermore, the value\nof\u0017SDfor\u0014Fa > 1.1, compatible with previous measure-4\nments at unitarity [28], compares well with the one expected\nfor two immiscible clouds of unpaired fermions bouncing off\neach other [35], as if one spin component acted like an im-\npenetrable potential barrier for the other. The increase of spin\nsusceptibility detected up to a critical repulsion strength to-\ngether with the discontinuity of the spin response frequency\nhint at a ferromagnetic instability located at \u0014Fa\u00191for\nT=TF'0:12.\nThe measured behaviour of \u0017SDcould not be explained if\npairing processes dominated the system dynamics. On the\nlower branch, owing to effective attraction, the spin suscep-\ntibility decreases monotonically while crossing the Feshbach\nresonance from a < 0toa > 0, vanishing completely for a\nBose-Einstein condensate (BEC) of pairs. This trend equally\noccurs in the attractive Fermi liquid state [36] as well as in\nthe superfluid or in the elusive pseudo-gap state [27, 37].\nTherefore, Eq. (1) implies a corresponding monotonic in-\ncrease of\u0017SDmoving towards the BEC limit. We experi-\nmentally demonstrate this behaviour by intentionally initial-\nizing the system in the lower branch (see Supplementary In-\nformation), observing an increase of \u0017SDeven above 2 \u0017zfor\n\u0014Fa > 0(see Fig. 2e). The spin response measured for the\nattractive gas is consistent with predictions from Eq. (1), ob-\ntained by inserting the spin susceptibility \u001frecently calcu-\nlated for a BEC-BCS crossover Fermi gas above the critical\ntemperature for superfluidity [37], in agreement with previous\nmeasurements [27]. In particular, it qualitatively differs from\nthe spin response at \u0014Fa>0shown in Fig. 2d, leading us to\nexclude that the latter arises from paired fermions in the inter-\nface region. On the other hand, to confirm that during the spin\ndynamics shown in Fig. 2a-c the upper branch is indeed pre-\ndominantly occupied, we estimate the molecular fraction un-\nder the same experimental conditions by an independent mea-\nsurement (see Supplementary Fig. S6). We detect a molecu-\nlar fraction not exceeding 15% after the first 50 ms of evolu-\ntion for any \u0014Favalue herein investigated, and below 5% for\n\u0014Fa\u001d1. Additionally, our findings seem incompatible with\na purely collisional picture [38] (see Supplementary Informa-\ntion), also considering the markedly distinct behaviour of \u0017SD\non the attractive branch shown in Fig. 2e.\nObservation of spin domain immiscibility\nIf the initially prepared fully ferromagnetic state was indefi-\nnitely stable above a critical \u0014Fa, the two spin domains would\nremain immiscible, i.e. spin diffusion would be impeded [34].\nTo investigate this aspect, we study spin diffusion in a sec-\nond set of measurements, exploring various interaction and\ntemperature regimes. We initialize the dynamics by adiabati-\ncally lowering the barrier height through a 30 ms linear ramp\nfromV0\u001810EFdown to 2EF, letting the two clouds slowly\napproach each other. Their relative distance is reduced from\nabout 5\u0016m down to 1 \u0016m, yet each spin domain remains con-\nfined within its own reservoir. At this point, we remove the\nbarrier in 5 ms and we monitor the subsequent evolution of\nthe relative population of the i=\";#component in the left and\nright reservoirs, Mi= (Ni;L\u0000Ni;R)=(Ni;L+Ni;R), from\nwhich we obtain the magnetization \u0001M= (M\"\u0000M#)=2.\nSince the distance between the two nearby cloud edges is ap-\nproximately equal to the local interparticle spacing at the in-terface between the two spin domains, this procedure does not\nexcite any detectable spin-dipole oscillation.\nFigure 3a shows the short-time evolution \u0001M(t)for differ-\nent interaction strengths: above a critical value of repulsion,\nafter an initial slight decrease of \u0001Mfrom 1 to about 0.92, we\nindeed observe a time window during which spin diffusion is\ncompletely arrested. We attribute the initial drop in magne-\ntization to spin transport within the low-density outer shells\nof the two spin clouds. The duration of the magnetization\n“plateau” is however finite, since the stability of our ferro-\nmagnetic state is limited by the intrinsic tendency of the sys-\ntem to relax from the excited upper branch onto lower-lying\nenergy states [12, 15] (see Fig. 1a). A thorough character-\nization of this interesting feature is summarized in Fig. 3b,\nwhere we plot the measured plateau duration \u001cpof constant\n\u0001Mas a function of 1=(\u0014Fa), for various temperatures. The\nvalue of\u001cpis determined through a piecewise linear fit to the\ndata, yielding zero when no noticeable halt of spin dynam-\nics is detected. A non-zero \u001cpis detected only above a crit-\nical\u0014Favalue and below a certain T=TF. Notably, finite\nplateaus appear above an interaction strength nearly coinci-\ndent with the one at which the spin-dipole mode frequency in\nFig. 2d reaches its minimum and exhibits an abrupt change\n(see Fig. 3c). Furthermore, by increasing \u0014Fa(increasing\nT=TF)\u001cpincreases (decreases), reaching its maximum at the\nunitary point 1=(\u0014Fa) = 0 . On the other hand, no dynamical\narrest is observed for T=TF\u00150:7. In addition, at low tem-\nperatures the trends for \u001cpand\u0017SD(see Fig. 3c and Fig. 2e)\nsuggest that we access the upper branch even within a narrow\n1=(\u0014Fa)<0region beyond unitarity, in accord with recent\npredictions [15].\nWe find the behaviour of the plateau duration for\n1=(\u0014Fa)>0to be captured (see curves in Fig. 3b) by a\nphenomenological model based only on the knowledge of the\nlifetime and energy spectrum of the upper and lower branches\nof the many-body system [24], calculated in the extremely po-\nlarized limit of one single \"(#) impurity embedded in a #(\")\nFermi gas [12, 24]. Based on such a description, the ferro-\nmagnetic state is destroyed by inelastic processes occurring\nat the interface between the two macroscopic spin domains:\nfermions of one kind, overcoming the surface tension associ-\nated with a domain wall, can deposit an overall excess energy\nthrough decay from the upper to the lower branch. Only af-\nter some time, once a sufficient energy has been released into\nthe system, the domain wall is melted and spin diffusion is\nestablished.\nUpon identifying for each temperature the lowest \u0014Fa\nvalue at which a non-zero \u001cpof steady magnetization is ob-\nserved, we delimit a region in the interaction-temperature\nplane where the ferromagnetic domains remain temporarily\nimmiscible, as displayed Fig. 3d. The critical interaction\nstrength for the onset of a non-zero \u001cpdisplays a non-linear\ndependence upon temperature, and by fitting the T=TF<0:3\ndata points with T=TF/((\u0014Fa)(T)\u0000(\u0014Fa)(0))\u000b, we ob-\ntain\u000b= 0:52(5) and(\u0014Fa)(0) = 0:80(9) . The fitted expo-\nnent matches within its uncertainty the value \u000b= 1=2ex-\npected from the low-temperature behaviour of a Fermi liq-\nuid exhibiting a magnetic instability (see Supplementary In-5\nτ  P\n0 10 20 30 400.750.80.850.90.951\n  \n \n \n \n \n∆M\nHold time (ms)1/κFa\n0.05\n0.45\n0.75\n1.3a\n2.5 2 1.5 1 0.5 0- 0.5 -105101520\nτP(ms)\n1/(κFa)0.06(2)\n0.12(2)\n0.31(4)\n0.53(6)T/TFb\n0 1 2 3 4 5 600.20.40.60.8T/T F\nκFaτ  > 0Pτ  = 0P\nd c\n3 2 1 0 -10.40.81.21.62 \n051015\n \nτP(ms)νSD /νz\n1/(κFa)\nFIG. 3. Metastability of a fully magnetized ultracold Fermi gas. a , Evolution of \u0001M(t)for different interactions at T=TF= 0:12(2) . For\n\u0014Fa\u00151:1a time window \u001cpof vanishing spin diffusion is detected. Error bars are the s.e.m. of 4-5 independent measurements. Data sets\nare artificially shifted by 0.05 along the y-axis from one another for clarity. b,\u001cpfrom piecewise fits to the data (dashed lines in a) is plotted\nat varying 1=\u0014Fafor various temperatures, with error bars being the fit standard uncertainty. Shaded curves are derived from the proposed\nmodel for domain-wall melting (see Methods), adjusting E+cwithin a 20% variation. c, Connection between \u001cp(red diamonds) and \u0017SD\n(blue circles) at T=TF= 0:12(2) . The grey region depicts the regime of \u001cp>0, which coincides within the experimental accuracy to the\nregion where \u0017SD'1:7. The dotted blue line denotes the prediction on \u0017SDfor the repulsive Fermi liquid (see Fig 2d). d, Metastability\n(\u001cp>0) region of the ferromagnetic state in the temperature-interaction plane. y-error bars denote the experimental uncertainty on T=TF,\nwhilex-error bars account for the uncertainty on estimating the critical \u0014Faat which a\u001cp>0is observed. The solid line is a power-law fit\n(see text) toT=TF<0:3points, extended over all values of \u0014Faas a dashed line. The black arrow marks the temperature for which \u001cp= 0\nat any\u0014Fa.\nformation). The extracted zero-temperature value (\u0014Fa)(0)\nis interestingly found in good agreement with the critical one\nobtained from repulsive QMC calculations [20, 21], and is sig-\nnificantly lower than Stoner’s mean-field criterion for an un-\npolarized gas [6, 12, 16–18, 20] \u0014Fa=\u0019=2. Notwithstand-\ning the metastable nature of the ferromagnetic state and the\ndynamical character of our study, our findings agree with the-\noretical expectations for a repulsive Fermi gas at equilibrium\nundergoing a ferromagnetic instability in the absence of pair-\ning. Moreover, the close correspondence between the trends\nof\u0017SDand\u001cp(see Fig. 3c) further suggests that the critical in-\nteraction strength for \u001cp>0corresponds to the one required\nfor the fully ferromagnetic state to be favoured.\nLong-time diffusive dynamics and spin drag coefficient\nOnce spin diffusion is established [28–30], the analysis of the\nlong-time evolution \u0001M(t)(or equivalently d(t)) within a\nsimple kinetic model (see Methods) allows us to determine\nalso the spin drag coefficient \u0000Sas a function of temperature\nand interaction. The results are displayed in Fig. 4. These are\ncompared with theoretical predictions for \u0000S, calculated fora single impurity diffusing in a homogeneous ideal Fermi gas\nwithin kinetic theory, accounting for scattering in all available\nstates within T-matrix approximation for the scattering cross\nsection (see Supplementary Information). The model is able\nto quantitatively reproduce the measured maximum of \u0000Sfor\nT=TF\u00150:3, that is the expected range of validity of the T-\nmatrix approximation, as well as the position of the maximum\nof\u0000Sat all temperatures (see Fig. 4). At low temperatures the\ndata sets exhibit a small but appreciable asymmetry around\nthe unitary point towards \u0014Fa>0. Such a feature, which dis-\nappears progressively as temperature is increased, highlights\nthe significant effect of collisions within the medium of sur-\nrounding particles on the dynamical properties of the diffusing\nquasi-particles (see Supplementary Information). A similar\nasymmetry in transport coefficients has already been reported\nfor the shear viscosity [39] and transverse spin diffusion [40],\nbut not in previous measurements of longitudinal spin diffu-\nsion [28].\nIn conclusion, we have probed the ferromagnetic behaviour\nof the repulsive Fermi gas by investigating spin dynamics in a\nresonantly interacting ultracold6Li spin mixture. Our findings6\n-1.0 -0.5 0 0.5 1.0 1.500.10.20.3 \n ћ     ΓS  / εF\n \n1/κFa-1.0 -0.5 0 0.5 1.0 1.50.00.10.20.30.4\n \n ћ    ΓS  / εFa\nbT/TF\n \n0.06(2) 0.12(2) 0.31(4) \n 0.53(6)\n 0.70(5)\nFIG. 4. Spin drag coefficient of a strongly interacting Fermi gas.\n~\u0000S=\u000fFis plotted as a function of 1=\u0014Fafora,0:31\u0014T=TF\u0014\n0:7, and b,T=TF\u00140:31. Experimental points are obtained by fit-\nting the dynamics at t >50ms to the solution of a diffusion model\n(see Methods). Error bars combine uncertainties of the fit and of\nour determination of \u000fF. Lines are predictions from a T-matrix ki-\nnetic theory (see Supplementary Information), assuming the nominal\ninitialT=TFand allowing a\u000620%temperature variation (shaded ar-\neas).\nindicate that the short-time evolution of the system is gov-\nerned by the effect of strong interparticle repulsion, and point\nto the existence of a Stoner-like ferromagnetic instability of\nthe zero-range repulsive Fermi gas, before relaxation onto at-\ntractive states alters irrevocably the nature of the many-body\nstate and its correlations. Furthermore, our study provides an\nimportant link between the dynamic response and the static\nproperties of a strongly correlated fermionic state. In the fu-\nture, our techniques may be exploited to probe spin transport\nand collective modes in different systems, such as repulsive\nand attractive Fermi gases with reduced dimensionality [41],\nand in the presence of weak optical lattices [42] and controlled\ndisorder [43].\nNote added – After submission of the manuscript, we became\naware of related theoretical work[44] by He et al. , report-\ning finite-temperature predictions qualitatively consistent with\nour findings.\nMETHODS\nExperimental protocols\nOur procedure to create weakly-interacting two-component Fermi\nmixtures of6Li atoms has been already described elsewhere[31]\n(see also Supplementary Information). The gas degeneracy parame-\nterT=TFis adjusted by exploiting the tunability of the collisional\nproperties of6Li mixtures[33] during evaporation. We typically\nend up with N\";#'5\u0002104atoms, confined in a cigar-shaped\nharmonic potential characterised by axial (radial) trap frequency of\n\u0017z= 21:0(1) Hz (\u0017?= 265(5) Hz). To spatially separate the twospin components, at a magnetic field of about 1 G, where the \"and\n#states possess equal but opposite magnetic moments, we turn on a\nmagnetic quadrupole gradient of about 1 G/cm along the weak trap\naxis, that pushes the two spin clouds towards opposite directions.\nOnce the overlap between the two components is zeroed, a repulsive\noptical potential centred at z= 0, and characterised by a short (long)\n1=e2waist ofwz= 2:0(2)\u0016m (wy= 840(30)\u0016m) is employed to\nconfine the two clouds into two disconnected reservoirs[32]. These\nare characterised by an axial oscillation frequency \u0017R= 1:78(5)\u0017z,\nand no appreciable particle tunnelling is detected over more than 2 s\nwith a barrier height of about 10 EF.\nTo excite the spin-dipole mode at a fixed Feshbach field, we\nabruptly switch off the barrier potential within less than 1 \u0016s. The\ntwo spin clouds, initially separated by a distance of about 5 \u0016m, start\nmoving one towards the other at a small relative velocity. From here\non, we monitor the dynamics of the relative distance between the\ncentres of mass of the two components. For each value of evolution\ntime, in two independent and successive experimental runs, we ac-\nquire two in-situ absorption images with two high-intensity optical\npulses, each of which is resonant only with one spin state. For each\npoint of interaction, temperature, and evolution time herein investi-\ngated, the relative distance between the two clouds is determined by\naveraging over at least five independent measurements. To investi-\ngate the spin diffusion of the two domains into each other, and in\nparticular to reveal the dynamical arrest of spin diffusion, we em-\nploy a different protocol turning off the repulsive barrier slowly in a\ntwo-stage sequence (see Supplementary Information for details).\nThe centre of mass of each cloud is equivalently determined either\nvia a Gaussian fit to the imaged density distribution (used for data\ndisplayed in Fig. 2a-c) or via the evaluation of the centre of mass zi\nof each sample i=\";#obtained via direct integration of the bare im-\nages. Once the spin dynamics is recorded, we fit d(t) = (z\"\u0000z#)(t)\nto an exponential decay. By subtracting the fitted exponential drift\nfromd(t), we extract the spin-dipole collective dynamics \u0001d(t),\nwhose frequency is then derived by means of a damped sinusoidal\nfunction (see Supplementary Information). Similarly, direct integra-\ntion of the images at each evolution time allows to determine the\nrelative population of the i=\";#component in the left and right\nreservoirs,Mi= (Ni;L\u0000Ni;R)=(Ni;L+Ni;R), from which we\nobtain the magnetization \u0001M= (M\"\u0000M#)=2. The behaviour of\nd(t)closely reflects the one of \u0001M(t).\nEffective Fermi energy and wavevector\nWe have defined \u0014Fand\u000fFas relevant length and energy scales.\nThese are evaluated by approximating the thin barrier at z= 0 as\na delta-like potential. This is reasonable considering that the bar-\nrier thickness of 2 \u0016m is about 70 times smaller than the typical\nThomas-Fermi radius of the cloud along the z-axis. Namely, we\napproximate the density distribution of a Fermi gas of Natoms\nconfined in a “half” of the harmonic trap bisected by the thin\nbarrier as half of the distribution of a Fermi gas of 2Natoms\nin the whole harmonic trap. The latter is evaluated using the\nfinite-temperature Fermi-Dirac distribution of an ideal Fermi gas\nnF(r;T=T F), given the measured T=TF, trap frequencies and atom\nnumber. From nF(r;T=T F)the local wavevector and Fermi en-\nergy are given by kF(r;T=T F) = (6\u00192nF(r;T=T F))1=3and\nEF(r;T=T F) = ~2k2\nF(r;T=T F)=(2mLi), respectively. The two\nparameters\u0014Fand\u000fFare derived by averaging kF(r;T=T F)and\nEF(r;T=T F)over a region as thick as the local interparticle spac-\ning(6\u00192)1=3=kF(x;y;0;T=T F)aroundz= 0 . Integration of\nnF(r;T=T F)within such region yields the total number Nintof\"\nand#fermions at the interface between the two spin domains (see\nSupplementary Information for details).7\nSum-rule approach for the spin-dipole mode frequency\nGiven a perturbation operator D, the moments of the strength\ndistribution function, or sum rules, are given by mk=P\nnjh0jDjnij2(En\u0000E0)k. In particular, the spin-dipole mode is\nexcited by the operator D=P\ni\"zi\u0000P\ni#zi, wherezis the lon-\ngitudinal coordinate and i\";#= 1;:::;N\";#. The frequency !SDof\nthe spin-dipole oscillation is estimated using the ratio (see Supple-\nmentary Information and Ref. 25):\n~2!2\nSD=m1\nm\u00001: (2)\nWithin the local density approximation (LDA) one can obtain (see\nSupplementary Information)\n!2\nSD=N\nmR\ndrz2\u001f(n); (3)\nwhere\u001f(n)is the zero-temperature magnetic susceptibility of a uni-\nform gas obtained at the density nfrom QMC calculations[20]. The\ndashed green line plotted in Fig. 2 is calculated through Eq. (3)\nby inserting the density profile n(r)of two fully overlapped spin\ncomponents, determined in LDA using the equilibrium condition\n\u0016(n)\u0000Vtrap=\u00160, where\u0016(n)is the chemical potential as a function\nof local density nfrom QMC calculations and \u00160is fixed by the nor-\nmalisation condition. The solid green line in Fig. 2 accounts instead\nfor a reduced overlap of 25 %around the trap centre, in closer anal-\nogy to the experimental condition. In this case, the integral in Eq. (3)\nover the outer spin-polarised regions contributes thus only with the\nspin susceptibility \u001f0of an ideal Fermi gas, yielding a reduced de-\nviation of the spin-dipole frequency from the bare trap frequency.\nTherefore, we expect the measured frequencies to be higher than the\nresults of Ref. 25, derived at full overlap. Consequently, the solid and\ndashed lines in Fig. 2 delimit a confidence region in which most ex-\nperimental data are found. Most importantly, the critical interaction\nstrength at which the abrupt change in !SDoccurs does not depend\non the initial overlap configuration. Moreover, the spin diffusion dy-\nnamics happens on a longer timescale ( &200ms), as displayed in\nthe Supplementary Information, compared to the spin-dipole period\n(\u001850\u0000100ms). Hence, while we cannot assume that the system\noscillates near an equilibrium configuration as in linear-response the-\nory, the (slow) timescale for diffusion is sufficiently separated from\nthe (faster) timescale for the spin oscillations.\nPolaron model for the domain-wall melting\nThe proposed modelling of the plateau data shown in Fig. 3 proceeds\nas follows. In the case of purely repulsive interaction, the ferromag-\nnetic state, if energetically allowed, would be indefinitely stable andthe miscibility of the two components would be prevented by the ex-\nistence of a domain wall. In particular, a #fermion at the interface\nwould need to pay a finite amount of energy \u001b > 0in order to ac-\ncess the other spin domain forming a repulsive polaron at energy E+.\nIn our metastable system, however, if a repulsive polaron is created,\nit can subsequently decay onto the lower branch with a rate \u0000, re-\nleasing an energy equal to the mismatch between the two branches,\nE+\u0000E\u0000. Hence, this two-step process will cause a net increase of\nenergy \u0001E=E+\u0000E\u0000\u0000\u001bat a rate \u0000. Importantly, the behaviour\nof\u0000,E+andE\u0000as a function of the interaction strength can be\nderived from recent non-perturbative theory approaches[12, 24]. We\nassume that at the beginning of the dynamics, the energy associated\nto the domain wall is given by \u001bN int,Nintbeing the total number of\nfermions within a slice around z= 0 of total thickness equal to one\ninterparticle spacing. The duration of the plateau \u001cpis then set by the\ncondition:\n\u001bN int= (E+\u0000E\u0000\u0000\u001b)\u001cp\u0000: (4)\nWe assume that \u001b=E+\u0000E+c, whereE+is the energy of a re-\npulsive polaron, while E+cis the energy of one free fermion at the\ninterface, left as a phenomenological parameter, and is independently\nadjusted for each T=TFherein investigated. From Eq. (S.19) we fit\nthe experimental data by optimising the only free parameter E+c(see\nalso Supplementary Information).\nDiffusion model for extracting the spin drag coefficient\nThe equation for the dynamics of the relative centre of mass d=\nz\"\u0000z#can be easily obtained from the Boltzmann equation and is\nwritten as (see Supplementary Information)\nd+ \u0000s_d+!2\nzd= 0; (5)\nwhere!zis the longitudinal trap frequency, and \u0000sthe spin drag co-\nefficient due to collisions[28, 29]. We obtain the experimental spin\ndrag coefficient by fitting the solution of Eq. (5) to the data, consid-\nering as initial condition d(0) =d0and_d(0) = 0 .\nIn Fig. 4 we compare the experimental results with a theoreti-\ncal prediction based on a T-matrix approximation for the scattering\ncross-section corrected by the available scattering states in order to\nhave a well defined scattering amplitude in the collisional integral\n(see Supplementary Information). The agreement is especially good\ndown to temperatures as low as T= 0:3TF. At lower tempera-\ntures, both the shape and the magnitude of the spin drag coefficient\nas a function of the interaction compare more poorly. This is ex-\npected since at very low temperature the T-matrix approximation is\nnot quantitatively correct, and moreover the gas may suffer some\nheating during the dynamics due to decay processes, making its tem-\nperature higher than the one measured at the start of the dynamics,\nwhich is used for the comparison with the theory model.\n[1] V ollhardt, D., Blumer, N. & Kollar, M. Metallic ferromagnetism –\nAn electronic correlation phenomenon . V ol. 580 of Lecture Notes in\nPhysics (Springer, 2001).\n[2] Brando, M., Belitz, D., Grosche, F. M. & Kirkpatrick, T. R. Metallic\nquantum ferromagnets. Rev. Mod. Phys. 88, 025006 (2016).\n[3] V ollhardt, D. & W ¨olfle, P. The superfluid phases of helium-3 (Taylor\nand Francis, 1990).\n[4] Silverstein, S. D. 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Ferromagnetism of a repulsive\natomic Fermi gas in an optical lattice: a quantum Monte Carlo study.\nPhys. Rev. Lett. 112, 015301 (2014).\n[43] Pilati, S. & Fratini, E. Ferromagnetism in a repulsive atomic Fermi gas\nwith correlated disorder. Phys. Rev. A 93, 051604(R) (2016).\n[44] He, L., Liu, X.-J., Huang, X.-G. & Hu, H. Stoner ferromagnetism of a\nstrongly interacting Fermi gas in the quasirepulsive regime. Phys. Rev.\nA93, 063629 (2016).\nAcknowledgments – We thank A. Morales and J. Seman for contributions\nin the early stage of the experiment, and G. Bertaina, G. M. Bruun, C. Di\nCastro, C. Fort, S. Giorgini, R. Grimm, W. Ketterle, P. Massignan, S. Pilati,\nR. Schmidt, W. Zwerger, M. Zwierlein and the LENS Quantum Gases group\nfor many stimulating discussions. We thank H. Tajima and Y . Ohashi for pro-\nviding us recent data of the lower branch spin susceptibility. This work was\nsupported under European Research Council grants no. 307032 QuFerm2D,\nand no. 637738 PoLiChroM. A.R. acknowledges support from the Alexan-\nder von Humboldt foundation. T.E. acknowledges the Physics Department,\nSapienza University of Rome, for hospitality, and the Humboldt foundation\nfor financial support during part of this work.\nAdditional information – Correspondence and requests for materials should\nbe addressed to M.Z. (e-mail: zaccanti@lens.unifi.it).1\nSUPPLEMENTARY INFORMATION\nExploring the ferromagnetic behaviour of a repulsive Fermi gas via spin dynamics\nG. Valtolina,1;2;3F. Scazza,1;2A. Amico,1;2A. Burchianti,1;2\nA. Recati,4;5T. Enss,6M. Inguscio,1;2M. Zaccanti1;2;\u0003and G. Roati1;2\n1INO-CNR, Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy\n2LENS and Dipartimento di Fisica e Astronomia, Universit `a di Firenze, Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy\n3Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy\n4INO-CNR BEC Center and Dipartimento di Fisica, Universit `a di Trento, 38123 Povo, Italy\n5Technische Universit ¨at M ¨unchen, James-Franck-Straße 1, 85748 Garching, Germany\n6Universit ¨at Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany\n\u0003E-mail: zaccanti@lens.unifi.it\nS.1. EXPERIMENTAL METHODS\nFermi gas preparation procedure\nOur procedure to create weakly interacting two-component Fermi mixtures of6Li atoms has been already described in detail\nelsewhere [31]. To realize highly degenerate samples at T=TF<0:2, a balanced mixture of atoms equally populating the lowest\nand third-to-lowest Zeeman states (hereafter denoted j1iandj3i, see Fig. S1a) is evaporated in a crossed optical dipole trap\n(ODT) at a bias magnetic field of 300 Gauss. At such a field the j1i\u0000j3imixture features a large [33], though off-resonant,\nscattering length value of about -900 a0which makes the forced evaporation process extremely efficient [31]. Here a0is the\nBohr’s radius. We typically end up with N1'N3'5\u0002104atoms per spin component, confined in a cigar-shaped harmonic\npotential with axial and radial trap frequencies \u0017z= 21:0(1) Hz and\u0017?= 265(5) Hz, respectively.\nFor the preparation of less degenerate samples, we either heat up in a controlled way the cloud by quickly turning off and\non the ODT for an excitation time up to 1 ms, or we perform evaporation at 300 Gauss with a j1i\u0000j2imixture. This features\nan interspecies scattering length of about -300 a0, which makes the evaporation process less efficient than for the j1i\u0000j3ispin\ncombination, leading to similar atom number and trap frequencies, though at higher temperature. The degeneracy parameter\nT=TFis determined by fitting the cloud density profiles in situ or after 5ms of time of flight to a finite temperature Fermi-Dirac\ndistribution.\nAt the end of evaporation in the j1i\u0000j3imixture, we transfer all atoms from the j3ito thej2istate via a resonant 80 \u0016s\nradio-frequency (RF) \u0019-pulse. To avoid detrimental finite-state interaction effects, that would limit the transfer efficiency, we\nperform the transfer at a bias field of 584.5 G, where the scattering lengths of the j1i\u0000j3iandj1i\u0000j2ispin mixtures are equal\nand non-resonant [33].\nSpin separation procedure\nThe procedure for creating two separate spin domains is described here and summarized in Fig. S1c. In order to spatially\nseparate thej1iandj2icomponents, we adiabatically lower the magnetic field down to \u00181G, where the two states are essen-\ntially non-interacting and possess equal but opposite magnetic moments (see Fig. S1b). Subsequently, we turn on a magnetic\nquadrupole gradient of about 1 G/cm along the weak trap axis through a 40 ms linear ramp, shifting the two spin clouds in\nopposite directions. After 180 ms, once the overlap between the two components is completely zeroed, we turn on through\na 40 ms linear ramp a strongly anisotropic 532 nm optical beam [32] centered at z= 0, and characterized by a short (long)\n1=e2waist ofwz=2.0(2)\u0016m (wx=840(30)\u0016m). The beam, blue-detuned with respect to the 671 nm lithium main transition,\nresults in a repulsive potential which bisects the cloud along the weak axis into two reservoirs. The typical maximum barrier\nheight after the 40 ms ramp is about 10 EF, a value that impedes any appreciable particle tunnelling between the two reservoirs\nover more than 2 s. Once the two spin clouds are separated and disconnected, we adiabatically turn off the magnetic gradient\nand increase the bias Feshbach field with a 500 ms linear ramp up to the target value. To prevent atom losses due to the mag-\nnetic field gradient, after the evaporation we turn on two additional elliptic plug beams at 532 nm with short (long) waist of\nwz= 50\u0016m (wy= 120\u0016m). The short waist is aligned along the weak trap axis and the plugs enter perpendicularly to this,\ncreating two effective repulsive walls that reduce as much as possible spilling of atoms due to the magnetic gradient. The plugs\nare turned on through a 200 ms linear ramp after evaporation and switched off through a 450 ms linear ramp when the Feshbach2\nFeshbac h\nB-fieldGradient\noff300  G584.5  Gtarget\nPlugsBarrier\noff off\n200 150 450 1030  5 450 5050305\ntime(ms)\n~2εF~10εF\n4030 1501 G\noffoff\n-180-170-160-150\n10 20 30 40100 200 300-400-2000200400\nMagn eticfield(G)Magn eticfield(G)Energylevels(MHz) Energylevels(MHz)a\nbc\nFIG. S1. a, Hyperfine and Zeeman levels structure of6Li in theF= 1=2andF= 3=2manifolds as a function of the magnetic field. b,\nEnergy dependence of the two lowest Zeeman states ( j1iandj2i) at low magnetic field. Below 5 G, the two states have magnetic moments\nwith nearly equal amplitude but opposite sign. c, The experimental sequence used for spin separation is sketched: the Feshbach field (light\nblue), the magnetic gradient (purple), the side plugs (dark green) and the central barrier (light green) are plotted as a function of time.\nfield is increased up to the final target value. The overall spin-separation procedure causes a decrease of about 25 %of the atom\npopulation, whereas no change of T=TFis measured. The degeneracy parameter after spin separation is estimated by a finite\ntemperature Fermi-Dirac distribution fit of the density distribution of spin polarized clouds recorded after a 5 ms time-of-flight\nexpansion at a bias field of 300 G.\nIt is important to stress that the final configuration is perfectly symmetric, both in total spin population and density distribution,\nand each component does not present any appreciable shape nor center-of-mass oscillation within the separated reservoirs.\nWe note here that a measurement of the oscillation frequency of a single spin component within one isolated reservoir yields\n\u0017R= 1:78(5)\u0017z. This is only about 10 %lower than the value 2 \u0017zexpected for the case of an infinitely thin barrier. This justifies\nthe approximation of the barrier potential as delta-like for modelling the cloud density, as discussed in detail in Section S.2.\nExcitation and measurement of spin response\nIn order to excite the spin-dipole mode at a fixed Feshbach field, we abruptly turn off the barrier potential from its maximum\nheight of about 10EFdown to zero, within less than one \u0016s. The two spin clouds, whose edges are initially separated by\na distance of about 5 \u0016m, start moving one towards the other at a small, though non-zero, relative velocity. From here on,\nwe monitor the dynamics of the relative distance between the cloud centers of mass. For each value of the hold time, in two\nindependent and successive experimental runs we acquire two in situ absorption images, each of which is resonant only with one\nspin state. For each value of interaction, temperature, and evolution time herein investigated, the relative distance between the\ntwo clouds is determined by averaging over at least five independent measurements.\n0 20 40 60 80 100 12080100120140\nHold time (ms)d(t)(µm)\nFIG. S2. Evolution of the center-of-mass distance between the two spin clouds, d(t) = (z\"\u0000z#)(t), at1=(\u0014Fa)'0:05andT=TF'0:12.\nThe dashed curve is obtained by fitting the data with the sum of an exponential decay and a damped sinusoidal oscillation.3\nThe center of mass ziof thei=\";#cloud is equivalently determined either via a Gaussian fit to the imaged density distribution\n(used for data displayed in Fig. 2a-c) or by directly evaluating zivia integration of the bare images. Once the spin dynamics is\nrecorded (see e.g. Fig. S2), we fit d(t) = (z\"\u0000z#)(t)to an exponential decay. By subtracting the fitted exponential decrease\nofd(t)due to diffusion from the data, we can isolate the spin-dipole mode, whose frequency is extracted from the data by\nmeans of a damped sinusoidal fitting function f(t) =Acos(2\u0019\u0017st)e\u0000t=\u001c. Similarly, direct integration of the images at each\nevolution time allows to determine the relative population of the i=\";#component in the left and right reservoirs, Mi=\n(Ni;L\u0000Ni;R)=(Ni;L+Ni;R), from which we obtain the total magnetization \u0001M= (M\"\u0000M#)=2. The behavior of d(t)\nclosely reflects the one of \u0001M(t).\nMeasurement of magnetization evolution and spin diffusion\nAs already mentioned in the main text, to investigate the spin diffusion of the two domains into each other, and in particular to\nreveal the dynamical arrest of spin diffusion, we employ a different protocol which does not excite the spin-dipole mode. For this\npurpose, we turn off the repulsive barrier slowly in a two-stage sequence. First, we lower the barrier from 10 EFdown to about\n\u00142EFthrough a linear 30 ms ramp (see Fig. S1c). The ramp speed and final barrier height are chosen to ensure an adiabatic re-\nadjustment of the density distributions of the two clouds in the barrier region, while preventing a detectable tunnelling of atoms\nacross the barrier and not inducing any shape excitations. In this way, the magnetization in each reservoir does not change, but\nthe relative distance between the edges of the two spin domains near z= 0 is drastically reduced, from 5 \u0016m down to about\n1\u0016m, a length-scale comparable with the mean interparticle spacing of our gas at the interface.\nIn a second stage, we ramp the intensity of the repulsive barrier down to zero. We have investigated different durations of the\nsecond ramp, spanning from 0 up to 30 ms. For ramps durations between 10 and 30 ms, we detect an appreciable flow of atoms\nacross the barrier region already during the ramp. In this case, once the barrier is off, the spin dynamics is well described by\na continuous, single exponential decay. For ramps shorter than 10 ms, in the strongly interacting regime, we observe a \u001cp>0\nwindow of vanishing spin diffusion. For ramp times below 5 ms, the duration of the plateaus is maximized and, for each target\nfield explored in this work, it does not show any dependence on the ramp speed. For the measurements presented in Fig. 3 and\n4 of the main text, the barrier is turned off in 5 ms. Examples of the short-time evolution of the cloud density profiles and of\nthe longitudinal magnetization M(z) = (n\"\u0000n#)(z)=(n\"+n#)(z)are displayed in Fig. S3. Consistently with the trend of the\ntotal magnetization \u0001Mshown in Fig. 3 of the main text, the evolution of the longitudinal magnetization at strong interactions\nexhibits a halt after a short evolution time (see Fig. S3b). After spin diffusion has been established at t >50ms, the evolution\nof the magnetization at each temperature and interaction strength is well described by Eq. (S.23) (see e.g. Fig. S4).\nt = 0 ms\nt = 10  ms\nt = 15  mst = 20  ms\nt = 70  ms\nt = 350  msb at = 0 ms\nt = 15  ms\nt = 50  ms\n[1 px = 1.04  µm]\nFIG. S3. a, Radially-integrated density profiles n\"(z)(blue) andn#(z)(red) of the two spin domains at different hold times after slowly\nremoving the optical barrier, with 1=(\u0014Fa)'0:05andT=TF'0:12.b, Longitudinal magnetization M(z)at different hold times for\n1=(\u0014Fa)'0:05(purple) and 1=(\u0014Fa)'0:9(green) atT=TF'0:12. It is visible here that M(z)remains substantially unchanged for\n1=(\u0014Fa)'0:05at hold times between 10 and 20 ms.4\n200 400 600 8000.050.10.51\nHold time (ms)∆M1/κF a = 0.05\n1/κF a = 1.35\n1/κF a = 2.401/κF a = -0.25\nFIG. S4. Spin diffusion dynamics at hold times t >50ms forT=TF= 0:12(2) . The evolution of the magnetization \u0001M(t)is plotted for\nvarious interaction strengths. Data points averaged over 4-5 indipendent experimental realizations (with error bars representing the s.e.m. of\nthe data) are shown together with fits to the solution of the simple kinetic model from which we extract \u0000S(see Eq. (S.23)).\nS.2. EFFECTIVE FERMI ENERGY AND FERMI WA VEVECTOR\nIn order to account for the inhomogeneity, the finite temperature, and the initial density of the two spin components in\ndisconnected reservoirs, in the main text we have defined \u0014Fand\u000fFas relevant length and energy scales. These are evaluated\nby approximating the thin barrier at z= 0 as a delta-like potential. As already discussed in Section S.1, this is a reasonable\nassumption, considering that the barrier thickness of 2 \u0016m is about 70 times smaller than the typical Thomas-Fermi radius of\nthe cloud along the zaxis. This is also confirmed by the measurement of the oscillation frequency within each reservoir, that is\nfound to be\u001810% lower than the value 2\u0017zexpected for the case of a delta-like potential. Namely, we approximate the density\ndistribution of a Fermi gas of Natoms confined in “half” a harmonic trap as half of the distribution of a Fermi gas of 2 Natoms\noccupying the whole dipole trap. The latter is evaluated using the finite-temperature Fermi-Dirac distribution of an ideal Fermi\ngasnF(r;T=TF), given the measured T=TF, trap frequencies and atom number (see Fig. S5a). The obtained profile excellently\nreproduces the two \"and#clouds, imaged in situ independently. From nF(r;T=TF)the local Fermi energy and wavevector are\ngiven respectively by kF(r;T=TF) = (6\u00192nF(r;T=TF))1=3andEF(r;T=TF) =~2kF(r;T=TF)2=(2mLi).\nThe two parameters \u0014Fand\u000fFare derived by averaging the local kF(r;T=TF)andEF(r;T=TF)over a region as thick as the\nlocal interparticle spacing (6\u00192)1=3=kF(x;y;0;T=TF)aroundz= 0(see Fig. S5b). Parallel to this, integration of nF(r;T=TF)\nwithin this region yields the total number Nintof\"and#fermions at the interface between the two spin domains. The definition\nof such an energy and length scale are inspired by a recent study on the spatial distribution of a fully ferromagnetic two-fermion\nmixture in a harmonic trap [45], which predicts that the domain wall at the interface between the two magnetic domains has a\ntotal thickness of one interparticle spacing.\n0050\n-5025\n-25\n5 -5 10 -10\nkF zkF y\n01\n00\n-100100\n-800 800 400 -400\nkF zkF y\narb. unitsa b\nFIG. S5. a, Model for total density distribution of the initial state: the optical barrier is approximated as infinitely thin, and the total profile of\nthe two spin components is given by the one of a Fermi gas of N\"+N#atoms at the given temperature, occupying the whole trap. b, The\nvalues of the effective Fermi energy \u000fFand wave vector \u0014Fare obtained by averaging over a slice (shown) around z= 0 of total thickness\nequal to one local interparticle spacing.5\nS.3. SUM-RULE APPROACH FOR THE PREDICTION OF THE SPIN-DIPOLE FREQUENCY\nThe spin-dipole mode is an oscillation of the relative position of the center of mass (COM) of each cloud of j\"iandj#i-atoms,\nbut not of their total COM position, i.e. a motion where atoms of one species move in the opposite direction with respect to the\natoms of the other species. For this reason, it is also referred to as an out-of-phase (spin) mode. As discussed in the main text,\nthe measurement of the spin-dipole mode frequency provides a spectroscopic tool for disclosing the spin susceptibility of the\ngas. An intuitive reasoning can clarify the relation between the mode frequency and the susceptibility, expressed in Eq. (1) of the\nmain text. A finite frequency of the spin-dipole mode is due to the presence of a restoring force when one tries to pull apart j\"i\nandj#i-particles in oppositedirections. With no interactions, the only restoring force is provided by the trapping potential and\nthe COM of each cloud oscillates at the trap frequency. A repulsive (attractive) interaction between the two atomic states leads to\na smaller (larger) energy cost to pull them apart with respect to the non-interacting case. This leads to a smaller (larger) restoring\nforce in the equation of motion, eventually resulting in a frequency smaller (larger) than the trap frequency. We indeed observe\na reduced(increased) frequency of the spin dipole mode when the system is prepared on the repulsive (attractive) branch (see\nFig. S7). For similar reasons, the spin susceptibility of a Fermi gas increases (decreases) if the interspecies particle interaction\nis repulsive (attractive). Therefore, a frequency decrease of the spin dipole corresponds to an increase of the spin susceptibility,\nand vice versa. The previous argument can be made more formal using a sum-rule approach [25, 46], which provides an upper\nbound of the spin-dipole mode frequency in terms of the system susceptibility. When the response is characterised by a single\ndelta-function (in the dynamic structure factor), the upper bound becomes an exact result. As emphasized in the main text, the\nmeasured spin-dipole oscillation frequency compares well with the sum-rule estimate supplemented with quantum Monte Carlo\n(QMC) data and employing a local density approximation [25].\nThe sum-rule approach is based on the knowledge of different momenta of the linear response of the system to the perturbation\nexciting the mode of interest. Given a perturbation operator D, the moments of the strength distribution function, or sum rules,\nare given by mk=P\nnjh0jDjnij2(En\u0000E0)k. An upper bound to the mode frequency is provided by the ratios of the sum\nrules. In particular, the spin-dipole mode is excited by the operator D=P\ni\"zi\u0000P\ni#zi, wherezis the longitudinal coordinate\nandi\";#= 1;:::;N\";#. We estimate the frequency of this mode using the ratio [25]:\n~2!2\nSD=m1\nm\u00001(S.1)\nThe reason is two-fold: (i) the inverse energy-weighted sum rule m\u00001is sensitive to low frequencies, which dominate since the\nstronger the repulsive interaction the softer the spin-dipole mode; (ii) m\u00001is related to the susceptibility of the system.\nThe so-called energy weighted sum rule m1is easily calculated in terms of a double commutator:\nm1=1\n2h0j[D;[H;D ]]j0i=N~2\n2m: (S.2)\nThe inverse energy-weighted sum rule is proportional to the spin dipole susceptibility. The latter can be determined by minimiz-\ning the total energy of the system in the presence of an external static coupling of the form \u0000\u0015D.\nSince our system is inhomogeneous, we can write the energy within the local density approximation (LDA):\nE=Z\ndr[\u000f(n\"(r);n#(r))\u0000\u0015z(n\"(r)\u0000n#(r))] (S.3)\nwhere\u000f(n\";n#)is the energy density of the uniform gas. By expanding \u000f(n\";n#)up to quadratic terms in n\"\u0000n#, minimization\nofEyields the result n\"\u0000n#=\u0015z\u001f(n)for the polarization induced by the external field. Here \u001f\u00001(n) =@2\u000f=@(n\"\u0000n#)2is\nthe zero-temperature inverse magnetic susceptibility of a uniform gas.\nThe calculation of the induced spin-dipole moment then yields m\u00001= 1=2R\ndrz2\u001f(n)and hence the result (Eq. (1) in the main\ntext)\n!2\nSD=N\"+N#\nmR\ndrz2\u001f(n)(S.4)\nfor the spin dipole frequency. Eq. (S.4) is exact within LDA and it shows that an increase of the magnetic susceptibility will\nresult in a decrease of !SD. If a ferromagnetic transition occurs, the spin-dipole mode frequency exhibits a minimum associated\nwith a diverging spin susceptibility within the central high-density region of the trap [25]. Once a ferromagnetic domain is\nformed, Eq. (S.4) is no longer valid and the dynamical behaviour of the system strongly depends on the geometry of the domain\nwall itself.\nFor an ideal Fermi gas trapped in a harmonic potential one gets the simple result !SD=!zwhere!zis the trap frequency\nalong thez-axis. For a balanced interacting Fermi gas on the repulsive branch, both for the energy density and the bulk magnetic6\nsusceptibility, we use the QMC results by Pilati et al. [20]. Both quantities can be written easily as an expansion in kFaabove\nthe mean-field results:\n\u000f\n\u000f0= 1 +kFa+C\u000f(kFa)2(S.5)\n\u001f0\n\u001f= 1\u00002\n\u0019kFa\u0000C\u001f(kFa)2; (S.6)\nwhere\u000f0and\u001f0are the chemical potential and the susceptibility of a spin- 1=2free Fermi gas, respectively. The constants\nC\u000f= 0:28andC\u001f= 0:62are obtained by fitting to the QMC results [20]. The previous expressions are suitable to determine\nan upper bound to the spin-dipole mode frequency for an unpolarized gas at equilibrium in the trap [25].\nProviding a rigorous theoretical description of the experimental results shown in Fig. 2 of the main text is challenging since\nthe gas has a time- and space-dependent local polarization. The dashed green line in Fig. 2 is calculated through Eq. (S.4) by\nassuming a cylindrically symmetric density profile n(r)with fully overlapped spin components, determined in LDA using the\nequilibrium condition \u0016(n)\u0000Vtrap=\u00160, where\u0016(n)is the chemical potential as a function of local density nfrom QMC\ncalculations and \u00160is fixed by the normalization condition. The solid green line in Fig. 2 accounts instead for a reduced overlap\nof 25 %around the trap center, in closer analogy to the experimental condition. In this case, the density profile used to compute\nEq. (S.4) is comprised of three longitudinal regions: two external non-interacting polarized tails and a central part where the\ntwo components fully overlap. For small oscillations, the overlap region can still be described through \u0016(n), i.e. the equation of\nstate of the repulsive branch. The size of the overlap region is chosen according to the typical experimental one. In particular,\na value of 25% denotes a ratio of 0.25 between the axial extension of the overlap region and the total axial size of the system.\nThe integral in Eq. (S.4) over the outer spin-polarized regions contributes thus only with the spin susceptibility \u001f0of an ideal\nFermi gas, yielding a reduced deviation of the spin-dipole frequency from the bare trap frequency. Therefore, we expect the\nmeasured frequencies to be higher than the results of Ref. 25, derived at full overlap. The solid and dashed lines in Fig. 2 delimit\na confidence region in which most experimental data are found. Most importantly, the critical interaction strength at which\nthe abrupt change in \u0017SDoccurs does not depend on the initial overlap configuration. Moreover, the spin diffusion dynamics\nhappens on a longer timescale ( &200ms), as displayed in Fig. S4, compared to the spin-dipole period ( \u001850\u0000100ms). Hence,\nwhile we cannot assume that the system oscillates near an equilibrium configuration as in linear-response theory, the (slow)\ntimescale for diffusion is sufficiently separated from the (faster) time scale of the spin oscillations.\nS.4. TEMPERATURE SHIFT OF THE CRITICAL INTERACTION STRENGTH\nWithin Landau-Fermi liquid theory the inverse susceptibility at T= 0can then be written as [47]\n\u001f\u00001(T= 0) =2\ng(eF)(1 +Fa\n0) (S.7)\nwhereFa\n0is thel= 0 antisymmetric (magnetic) Landau parameter, g(eF) =m\u0003kF=\u00192is the density of states with m\u0003=\nm(1 +Fs\n1=3)> m andFs\n1is thel= 1 symmetric (density) Landau parameter. To the first order in the interaction one has\nm\u0003=mandFa\n0=\u00002kFa=\u0019.\nAt finite temperature but T\u001cTFwe can consider the corrections to previous expression only due to free quasiparticles which\nis proportional to n, and therefore to T2. The interaction term depending on n2will contribute to higher order O(T4):\n\u001f\u00001(T) =2\ng(eF)\u0012\n1 +Fa\n0+\u00192\n12T2\nT2\nF\u0013\n(S.8)\nThe paramagnetic state becomes unstable when \u001f\u00001= 0, which at zero temperature can occur if there exists a critical value\n(kFa)csuch thatFa\n0=\u00001. At lowTwe can expand the Landau parameters around their values at the critical point (kFa)cat\nzero temperature\nFa\n0=Fa\n0;c+\u0012@Fa\n0\n@(kFa)\u0013\nc(kFa\u0000(kFa)c) (S.9)\nFs\n1=Fs\n1;c+\u0012@Fs\n1\n@(kFa)\u0013\nc(kFa\u0000(kFa)c): (S.10)\nTherefore we find that the critical temperature for the paramagnetic state to be unstable can be simply written as\n\u0012T\nTF\u0013\nc=2p\n3\n\u0019s\n\u0000\u0012@Fa\n0\n@(kFa)\u0013\ncp\nkFa\u0000(kFa)c (S.11)7\nFor instance using the expression for Fa\n0at the second order in kFa[16] one gets the first correction to the usual expression. In\nparticular (kFa)c'1:05and the critical temperature reads\n\u0012T\nTF\u0013\nc'23=2p\n3\n\u00193=2r\n1 + (kFa)c8\n3\u0019(1\u0000ln 2)p\nkFa\u0000(kFa)c (S.12)\nIn principle, application of the aforementioned power law as a fit to the experimental data in Fig. 3d is not justified: in fact,\n\u001cp>0is interpreted as the region of metastability of the fully ferromagnetic state, whereas Eq. (S.11) marks the boundary\nbetween paramagnetic and partially ferromagnetic phases. On the other hand, however, all QMC results show that the partially\nferromagnetic phase for a homogeneous system occupies a very narrow region of the phase diagram, and we expect that in\nour trapped system its presence is essentially washed out by the inhomogeneity of the clouds density distribution and finite\nexperimental resolution.\nS.5. PAIRING INSTABILITY AND MOLECULE FORMATION DURING SPIN DYNAMICS\nOne major issue that has hindered the study of strongly repulsive Fermi gases in previous experiments [13, 22] is represented\nby the pairing instability [15]. As discussed in the main text, ferromagnetic behaviour develops along the upper branch of\nthe many-body system: however, this state features an additional instability, represented by the tendency of the paramagnetic\nphase to turn via inelastic processes into a gas of pairs, which represents the true ground state of the balanced system at low\ntemperatures. At least for homonuclear mixtures and broad resonances [12, 13, 15], the pairing instability always overcomes the\nStoner’s one.\nThanks to our preparation scheme, that artificially initialises the system into a fully ferromagnetic configuration, we are able\nto contain the system tendency towards pairing, allowing for the investigation of the metastable upper branch. Furthermore, as\ndiscussed below in Section S.7, attractive polarons, rather than pairs, seem to be the preferential decay products in our system, at\nleast in the strongly interacting regime. Nonetheless, molecule formation has represented a major issue in previous experiments,\nand ruling out pairing effects for explaining the dynamics observed in our studies is fundamental for further supporting our\ninterpretation.\nIn previous experiments [13], the population of molecules and atoms has been identified via a rapid magnetic field ramp\ntechnique. After some evolution time at a target field close to the Feshbach resonance center, where molecules could be formed\nvia recombination processes, two successive absorption images were acquired. A first, taken at high field, allowed to monitor\nthe total population of atoms and molecules, since the molecule binding energy close to the Feshbach resonance is two to three\norders of magnitude smaller than the natural linewidth of the imaging transition. This allows to take a picture of the molecules\nwith the same imaging light employed for the atom, being the latter in the \"or#state. A second imaging pulse was taken after\na rapid sweep of the magnetic field to zero. The ramp converts weakly bound pairs into deeply bound molecules, which become\ntransparent to the atom imaging light. Hence, the second imaging selectively monitors the atom population, i.e. the population\nof the upper branch.\nOur setup does not allow to perform fast ( \u0018102G=\u0016s) ramps to low fields, hence we employed a different protocol to monitor\nthe presence of molecules in the system. This is based on acquiring, within a single experimental run, two subsequent absorption\nimages, by means of 4 \u0016s long pulses resonant with the \"and#states, respectively, and separated by 300 \u0016s. If no molecules\nare present, the effect of the first imaging pulse resonant with the \"state on#atoms is found to be negligible, since the \"\nimaging light is off-resonant to the #component. Furthermore, the short delay between the two pulses greatly limits the effect of\nheating of the#cloud associated to collisions with escaping \"imaged atoms. The effect of the first imaging pulse is completely\ndifferent if molecules are present. Since the \"\u0000# dimers are only weakly bound, the first imaging photon, resonant with the \"\noptical transition, dissociates the bound state into two atomic products [48], each of which symmetrically acquires a significant\nmomentum. The latter is associated to the density of states of the two outgoing atoms, to the binding energy of the dimer\n(negligible in this case), and to the photon momentum ~kL. The increase of the cloud size detected by the second imaging pulse\nis thus directly related to the amount of molecules in the system. We therefore monitor the increase of the radial width after the\nfirst pulse for different interaction strengths and different evolution times during the spin diffusion.\nA simple model allows to link the molecular fraction to the increase of the cloud width after the first pulse, following the\nexperimental protocol described above. In general, the size of a trapped cloud can be written as:\nhx2\n0i=2hUi\nm!2(S.13)\nWherehUiis the potential energy of one atom weighted over the density distribution of the cloud, the latter being eventually\nmodified by interaction effects. In the case of a pure gas of dimers, application of an imaging pulse resonant with the \"-\ncomponent causes the dimers to dissociate with a certain transfer of energy E1to the#-component. Since the photon energy is8\nalways larger than the binding one, we assume E1to be independent from the molecular binding, and hence independent from\n\u0014Fa. According to the same argument of Eq. (S.13), the width measured through the second pulse, following the first, can be\nwritten as:\nhx2\n1i=2hU+E1i\nm!2(S.14)\nIf the gas is a mixture of Nafree atoms and Nmmolecules, the mean size is set by:\nhx2i=Nahx2\n0i+Nmhx2\n1i\nNa+Nm(S.15)\nDefining the molecular fraction of the gas at time tduring the spin dynamics, fm\u0011fm(\u0014Fa;t) =Nm=N, we get:\nhx2i=2\nm!2hUi+2\nm!2hE1ifm=hx2\n0i+2\nm!2hE1ifm (S.16)\nWhen starting from a pure molecular sample, fm= 1, we would have:\n2\nm!2hE1i=hx2\n1mi\u0000hx2\n0mi (S.17)\nWe can therefore express the molecular fraction as:\nfm=hx2\n1i\u0000hx2\n0i\nhx2\n1mi\u0000hx2\n0mi(S.18)\nFor every investigated interaction strength, the denominator of Eq. (S.18) is experimentally determined by applying the double-\npulse imaging technique on a superfluid gas with the same interaction parameter \u0014Faand a temperature T=TF<0:1, ensuring\na molecular fraction fm'1on the BEC side of the resonance. The change of the density distribution when moving from\nthe unitary limit to the BEC one is accounted by renormalizing the measured radial width by the average density of the gas,\nevaluated with the conventional single-pulse absorption imaging.\nThe numerator of Eq. (S.18) is evaluated by measuring the second-pulse radial size of the cloud hx2\n1iat different evolution times\nduring the spin dynamics, initialized by the same experimental procedure discussed in Section S.1 and in the main text. hx2\n0i\nis in turn the size measured through the first imaging pulse at the corresponding times. Results of this analysis are reported in\nFig. S6, for various interaction strengths and different evolution times after abruptly removing the barrier, for a repulsive Fermi\ngas mixture initially prepared at T=TF= 0:12(2) .\nThe general trend for these measurements is interpreted as follows. In the weakly repulsive regime, the upper branch is very\nlong-lived, and despite rapid mixing of the two spin clouds only a small number of molecules is formed. Increasing interactions,\nMol.fraction50ms\n100ms\n1000ms\n00.10.20.30.40.50.60.7\n0 0.5 1 1.5 2 2.5\n1/κFa\nFIG. S6. Measured molecular fraction at T=TF= 0:12(2) for different evolution times after the barrier removal, obtained through the\ndouble-pulse imaging technique. The peak of molecule formation appears below the measured critical interactions, at 1=(\u0014Fa)'1:3.9\nthe decay rate of the upper branch monotonically increases [11, 12, 24, 49]: hence, despite an increase of the spin drag coefficient\n[28, 34], which tends to slow down the diffusion and reduces the spatial overlap of the \"and#clouds, the molecule formation\nbecomes more sizable, reaching a maximum near 1=(\u0014Fa)'1:3, in good agreement with Ref. 45. However, as one accesses the\nstrongly interacting regime where ferromagnetism is promoted, the molecule formation is again strongly reduced, highlighting\nthe tendency of the system to suppress the overlap between the two spin components. The trend at large \u0014Favalues, which\npersists also after long evolution times when the two clouds have come together, suggests that also at small values of local\npopulation imbalance the system may be a Fermi liquid state of attractive polarons, rather than a Bose gas of dimers. The Fermi\nliquid state might be favored by our way of initializing the system dynamics, as well as by the temperature increase associated\nwith the exothermic decay process from the upper to the lower branch. Importantly, for timescales below 100 ms, over which\nboth the magnetization plateau and the spin-dipole oscillations were measured, the observed heating is relatively small and the\nderived molecule fraction remains below 10 %for interaction strengths exceeding the critical value for the arrest of spin diffusion\nto occur. We therefore conclude that neither the behavior of the spin-dipole frequency nor the appearance of plateaus in the spin\ndiffusion can arise from dimer formation.\nThis conclusion is further supported by the measured temperature dependence of the critical repulsion strength (see Fig. 3d and\nEq. (S.11)). Such a trend is antithetic to the one that would result if the observed spin dynamics was governed by the molecular\npopulation within the overlap region. Assuming that such a molecular layer is in local chemical equilibrium with the atoms owing\nto fast thermalization, the molecular fraction is set by the Boltzmann factor Eb=(kBT)/1=(Ta2). Therefore, an increase of\nthe sample temperature would cause the same molecular fraction to form at smaller interaction strengths, corresponding to\nlarger molecule binding energies Eb, and leading to (T=TF)c/1=(\u0014Fa)2. This is qualitatively different from the observed\ntemperature dependence, displayed in Fig. 3d. While we cannot rule out a beneficial effect associated with the presence of\nmolecules at the interface, which may partially lower the tendency of the atoms to recombine into dimers, we can rule out that\ndimer formation lies at the origin of the observed dynamical behaviour.\nS.6. SPIN RESPONSE ON THE LOWER (ATTRACTIVE) BRANCH\nTo validate further our interpretation of the spin-dipole oscillation data presented in Fig. 2 and 3 of the main text, we have\nperformed additional spin response measurements, employing a system intentionally initialized in the lower energy branch. The\nresults are displayed in Fig. S7. The importance of such additional measurements is two-fold: on the one hand, we obtain novel\nexperimental hints on the trend of the lower-branch spin susceptibility via the same protocol employed for investigating the spin\nresponse in the upper branch; on the other hand, we can rule out that the trend of \u0017SD=\u0017zmeasured at strong coupling and\nshown in Fig. 2 of the main text could be ascribed to spin dynamics in the lower branch. From both experimental data and\ntheoretical calculations [27, 36, 37], the spin susceptibility of a Fermi gas on the lower branch of the Feshbach resonance is\n-3 -2 -1 0 10123\nνSD /νz\n-1/(κFa)\nFIG. S7. Spin-dipole frequency of an attractive Fermi gas at the BEC-BCS crossover. Red squares: \u0017SD=\u00170obtained after intentionally\ninitializing the system in the lower branch. Blue circles: spin-dipole frequency data shown in Fig. 2-3 of the main text, initializing the system\nto a repulsive Fermi gas. Green shaded region: theory prediction from Eq. (1) for a repulsive Fermi liquid, as shown in Fig. 2d of the main text.\nRed shaded region: theory prediction based on Eq. (S.4) for an attractive Fermi gas at T=TF= 0:2, obtained from the spin susceptibility and\nchemical potential given in Ref. [37]. The upper (lower) curve is obtained assuming full (20 %) spatial overlap between the two spin clouds.\nThe dashed red line denotes the theoretical expectation based on first-order perturbation theory [25].10\nknown to exhibit a monotonic decrease when passing from the BCS to the BEC side of the crossover. This is equally true for\nan attractive Fermi liquid and for a paired state, and for the latter both in the superfluid phase or in a pseudogap regime. In\nparticular, while the spin susceptibility of a weakly attractive system in the BCS limit tends to the value \u001f0of a non-interacting\nFermi gas, it strongly decreases when crossing the resonance and moving towards the BEC limit. Similarly, for all \u0014Favalues,\n\u001fis expected to monotonically decrease with decreasing temperature, especially if this drops below the critical temperature Tc\nfor superfluidity [37, 50]. Correspondingly, Eq. (1) of the main text (Eq. (S.4) here) predicts \u0017SD=\u0017z>1for an attractive Fermi\ngas at all temperatures and interaction strengths, and \u0017SDshould present a sharp increase when crossing the resonance from\na<0to thea>0(or once the temperature drops below Tc).\nThis supplementary characterization of the system dynamics is carried out by means of a slightly different experimental\nprotocol with respect to the one described in Section 1 and in the main text. We initially prepare a deeply degenerate system\natT=TF'0:12in the usual domain-wall configuration, at a bias field B0= 910 G, corresponding to \u0014Fa'\u00001. Here, the\nmeasured spin-dipole frequency, though still exceeding \u0017z, lies well below the value of \u0017SD= 1:70(4) obtained at\u0014Fa >1,\nand no arrest in the spin diffusion is ever observed. Consequently, we can assume that at this field, after the barrier removal, the\ngas immediately occupies the lower branch of the many-body system, as the upper one is neither well-defined nor energetically\naccessible.\nOnce the barrier is removed, we hold the field at B0for 20 ms, allowing a small mixing of the two spin clouds, and we\nsubsequently ramp it to a target value Btarget< B 0through a 25 ms ramp. We have checked that such a ramp is sufficiently\nadiabatic, so that the subsequent spin dynamics depends only weakly on the ramp duration. Once Btarget is reached, we\nmonitor the spin dynamics measuring the evolution of the relative distance between the centers-of-mass of the two spin clouds.\nThrough the same analysis procedure already described in the main text and Section 1 of the Supplementary Information, we\nextract\u0017SD=\u00170as a function the target interaction strength, as shown in Fig. S7 (red squares). It is evident how the spin-dipole\nfrequency measured after the adiabatic ramp from the BCS side, while being consistent with data shown in the main text for\n\u00001=\u0014Fa>0:5, presents a strong deviation further towards the BEC side. In this region, \u0017SDsharply increases to values above\n2\u00170already at unitarity, exceeding 3\u00170at1=\u0014Fa= 0:25. Further decreasing Btarget , hence\u0014Fa, no clear frequency can be\nextracted, owing to strong damping of the oscillations. Comparing the lower-branch data with the ones obtained by initializing\nthe system in the repulsive state (blue circles in Fig. S7) seems to indicate that the upper branch becomes completely unstable\ntowards pairing beyond unitarity at \u00001=\u0014Fa >0:5, such that the associated spin response is entirely compatible with the gas\noccupying the lower branch already at the beginning of the evolution.\nThe trend of our lower-branch data is well captured by Eq. (S.4), provided that we insert the spin susceptibility \u001fand chemical\npotential\u0016of a BEC-BCS crossover Fermi gas above the critical temperature for superfluidity. For our theory prediction (shown\nin Fig. S7 as the red shaded region), we have used recent data obtained within an extended T-matrix approximation and including\npairing fluctuations [37], which match both QMC results at unitarity [50] and experimental measurements of the lower-branch\nspin susceptibility [27].\nThe validity of the theoretical treatment based on Eq. (S.4) is therefore successfully confirmed in the well-established case\nof a BEC-BCS crossover attractive Fermi gas and our Supplementary Data of \u0017SDmatch previous experimental measurements\nof\u001fby Sanner et al. [27]. Finally, the reasonable agreement we find between our theoretical prediction and experimental data\nsuggests that our system is in a pseudo-gap regime of pre-formed pairs, rather than in an attractive Fermi liquid phase. For\nthe latter case, in fact, both experimental investigation and QMC studies at zero temperature [36] reported a value for the spin\nsusceptibility at unitarity that would lead to a spin-dipole frequency significantly lower than the one we measure. Such an\nintriguing aspect definitely will be the subject of future investigations.\nS.7. THEORETICAL MODEL FOR THE DOMAIN-WALL MELTING\nIn this Section, we describe the phenomenological model developed to explain the finite duration of the spin diffusion plateaus,\nas a function of interaction and temperature. Such a model is based on the precise knowledge of the spectral properties and\nlifetime of#impurities embedded in a Fermi sea of \"particles [24]. As discussed in the main text, the spectral function of\nsuch system is characterized by an upper and a lower branch. Depending on the interaction strength, associated to the lower\nbranch there exist in the extremely imbalanced case two kinds of Landau’s quasi-particles, coined attractive polarons and dressed\nmolecules [12]. For interaction parameters 1=(\u0014Fa)>0:9(1=(\u0014Fa)<0:9) dressed molecules (attractive polarons) represent\nthe absolute ground state of the many-body system. The upper branch, in turn, features a third kind of quasi-particle, termed\nrepulsive polaron, whose existence has been experimentally demonstrated both in a three-dimensional mass-imbalanced Fermi\nmixture of40K and6Li atoms [11], and also in a two-dimensional Fermi mixture of40K atoms [51]. While the lower branch is\nassociated to a net attractive interaction between the impurity and the particles of the medium, the upper one requires repulsion\nbetween the two species in order to develop. For increasing repulsive interaction ( 1=(\u0014Fa)!0+), the energy of the repulsive\npolaron progressively increases. However, parallel to this, the repulsive polaron acquires a progressively shorter lifetime, set by\nthe tendency of the system to decay onto the lower-lying states of the attractive branch. The decay rate \u0000associated with such11\ninelastic processes, together with the energies E+andE\u0000of repulsive and attractive polarons, has been determined through\nnon-perturbative theory approaches [12, 24].\nAlthough our system is a 50-50 balanced mixture of \"and#fermions, the results obtained in the impurity limit are extremely\nrelevant for understanding the existence of a plateau of zero diffusivity in the spin dynamics. Since our studies start by preparing\na phase-separated state, rather than a mixed paramagnetic phase, the initial mixing processes at the interface between the two spin\ndomains can be viewed as events in which fermions of one kind enter in the Fermi sea of the other component, and vice-versa.\nThose can be described in terms of dressed quasiparticle properties derived in the impurity limit.\nOur proposed explanation of the conductance plateau shown in Fig. 3 of the paper proceeds along the following line of thought.\nIn the case of purely repulsive interactions, the ferromagnetic state, if energetically allowed, would be indefinitely stable, and\nin a system with separately fixed spin populations would correspond to a phase-separated state. In fact, the miscibility of the\ntwo components would be prevented by the existence of a domain wall: namely, a #fermion at the interface would need to\npay a finite amount of energy \u001b > 0in order to access the other spin domain, forming a repulsive polaron. In our system,\nhowever, if a repulsive polaron is created, it can subsequently decay onto the lower branch with a rate \u0000, releasing an energy\nequal to the mismatch between the two branches, E+\u0000E\u0000. Hence, this two-step process will cause a net increase of energy\n\u0001E=E+\u0000E\u0000\u0000\u001bat a rate \u0000. We assume that at the beginning of the dynamics, the energy associated to the domain wall\nis given by \u001b\u0002Nint,Nintbeing the total number of fermions within a slice around z= 0 of total thickness equal to one\ninterparticle spacing (see Section S.2). The duration of the plateau \u001cpis then set by the condition:\n\u001bNint= (E+\u0000E\u0000\u0000\u001b)\u001cp\u0000 (S.19)\nWe write\u001b=E+\u0000E+c, whereE+cdenotes the energy of one free fermion at the interface. In homogeneous systems and\nin the zero-momentum impurity limit [12], E+c=EF. In our inhomogeneous sample at finite temperature, and in the vicinity\nof the interface between the two spin clouds, we introduce E+cas a phenomenological fitting parameter. Consequently, from\nEq. (S.19) we obtain the following expression for \u001cp:\n\u001cp=Nint(E+\u0000E+c)\n\u0000(E+c\u0000E\u0000)1\n2\u0019\u0017F(S.20)\nwhereh\u0017F=\u000fF. To compare the prediction of Eq. (S.20) to our data we employ the values of E+,E\u0000and\u0000given as a function\nof interactions by Schmidt et al. [24], and the values of \u0014F,\u000fFandNintobtained after radial averaging over finite temperature\ndensity profiles as described in Section S.2.\nFor each temperature regime herein investigated, the only free parameter of the model, E+c, is fixed by optimizing the\nagreement between experimental data and the prediction of Eq. (S.20). E+cnon-linearly increases upon increasing temperature:\nthis is expected since E+croughly corresponds to the energy of the repulsive polaron branch at the \u0014Favalue at which a first\nnon-zero\u001cpis detected. It is worth stressing how such a simple theory model, which accounts for finite temperature effects only\nviaE+cand a renormalization of \u0014F,\u000fFandNint, is able to provide a quantitative description of the experimentally observed\ntrend.\nThe model prediction stops at the unitary point and does not extend on the a <0side of the Feshbach resonance, because\nthe theory becomes unreliable in this region, the decay rate of the upper branch reaching the order of the Fermi energy, and\nhence making the repulsive polaron an ill-defined quasiparticle [24]. As a matter of fact, we are at present unable to provide a\ndescription for such a feature (and for its disappearance) on the BCS side of the resonance, which we cannot exclude to arise\nfrom a combination of collisional and many-body effects. Certainly, the measured \u001cpand\u0017SDat low temperatures are consistent\nwith the system temporarily accessing the upper branch even for \u0014Fa<0[15]. This extremely exotic many-body state, which\nis a three-dimensional analogue of the super-Tonks regime in one dimension [52], is thus far poorly explored and definitively\ndeserves deeper theoretical and experimental investigation.\nS.8. SPIN DRAG COEFFICIENT AND THEORY MODEL FOR THE DIFFUSION OF AN IMPURITY IN A FERMI SEA\nThe equation for the diffusion dynamics of the relative center of mass d(t) =z\"\u0000z#can be easily obtained starting from the\nBoltzmann equation and using the method of the averages [53]. One obtains the following coupled equations:\n@t(z\"\u0000z#)\u0000(v\"\u0000v#) = 0\n@t(v\"\u0000v#) +!2\nz(z\"\u0000z#) = (@t(v\"\u0000v#))coll; (S.21)\nwhere!zis the trapping frequency in the direction of the motion, v\"(v#) is the average velocity of the \"(#) spin component\nand(@t(v\"\u0000v#))collis the collisional term. The effect of the medium is included in this approach only in the collisional\nterm. Assuming that the distribution function changes in time only through the change of velocity, the collisional term is simply\nproportional to the relative velocity and can be written as\n(@t(v\"\u0000v#))coll=\u0000\u0000S(v\"\u0000v#); (S.22)12\nwhere \u0000Sis the so-called spin drag coefficient due to collisions [28, 29]. Therefore, substituting Eq. (S.22) in Eq. (S.21) the\nequation of motion for dis simply given by the one of a damped harmonic oscillator:\nd+ \u0000S_d+!2\nzd= 0 (S.23)\nWe obtain the experimental spin drag coefficient by fitting the solution of Eq. (S.23) to the data, with initial conditions d(0) =d0\nand_d(0) = 0 , or equivalently \u0001M(0) = \u0001M0and_\u0001M(0) = 0 (see Fig. S4).\nA main theoretical task is to determine the spin drag coefficient. For #impurities moving through a fully polarized noninter-\nacting\"Fermi sea, the drag rate can be computed as\n~\u0000S=2\u0019\nkBTn#Zd3p1\n(2\u0019~)3d3p2\n(2\u0019~)3d3p3\n(2\u0019~)3\u000e(\"1+\"2\u0000\"3\u0000\"4)\n\u0002(4\u0019~2)2\nm2d\u001b\nd\np1j(v1j\u0000v3j)n1#n2\"(1\u0000n3#)(1\u0000n4\"):(S.24)\nThis collision integral describes the scattering of an impurity atom with momentum ~ p1and a medium atom with momentum ~ p2\ninto outgoing states ~ p3and~ p4=~ p1+~ p2\u0000~ p3, conserving the total momentum and the total kinetic energy \"1+\"2of both\nparticles. Each particle has kinetic energy \"~ p=p2=2mand massm. The drag rate is proportional to the change in impurity\nvelocity,v1j\u0000v3j, wherejdenotes the spatial component in the direction of the initial impurity velocity. The scattering process\noccurs with probability n1#n2\"that the initial states are occupied, and probability (1\u0000n3#)(1\u0000n4\")that the final states are\nunoccupied, where n~ p\u001b= [exp(\f(\"~ p\u001b\u0000\u0016\u001b)) + 1]\u00001is the Fermi-Dirac distribution at chemical potential \u0016\u001b. The drag rate\nEq. (S.24) is derived from spin diffusion theory for a polarized Fermi gas [54–56] in the limit of vanishing minority density n#\n(the integral over the impurity distribution n1#cancels the factor n#in the denominator of Eq. (S.24) to yield a finite drag rate).\nEquivalently, Eq. (S.24) is obtained from the impurity drag rate [34, 57] in the limit of vanishing impurity velocity.\nIn order to compute the cross section d\u001b=d \nfor a dilute Fermi gas we use the T matrix in the ladder approximation [58]. In\nan ultracold Fermi gas the bare interaction is a s-wave contact interaction between unequal spins, thus also the T matrix only has\nans-wave component `= 0. One can then express the differential cross section\nd\u001b\nd\n=jf`=0(~ q;!)j2(S.25)\nin terms of the s-wave scattering amplitude f`=0(~ q;!)of two incoming particles with total momentum ~ q=~ p1+~ p2and total\nkinetic energy ~!=\"~ p1\"+\"~ p2#. The scattering amplitude, in turn, is given in terms of the T matrix T`as [58]\nf`=0(~ q;!) =\u0000mQ\n4\u0019~2T`=0(~ q;!): (S.26)\nFor two particles scattering in vacuum ( Q= 1), the two-body T matrix reads\nT(0)\n`=0(~ q;!) =4\u0019~2a=m\n1 +iak(S.27)\nwhereadenotes the s-wave scattering length, and ~~k= (~ p1\u0000~ p2)=2is the relative momentum of incoming particles. This\nresults in a vacuum scattering amplitude f(0)\n`=0=\u0000a=(1 +iak)and vacuum scattering cross section d\u001b(0)=d\n =jf(0)\n`=0j2=\na2=(1 +a2k2). At weak coupling jkF\"aj\u001c 1, the drag rate is proportional to the scattering cross section, \u0000S/a2. Note\nthat with the vacuum cross section, the drag rate depends only on the modulus jajof the scattering length and is symmetric\ninaaround unitarity ( a\u00001= 0) in the BCS-BEC crossover. The experimental data for the drag rate, however, exhibit a small\nasymmetry in a. A similar asymmetry in transport coefficients has been observed in the shear viscosity [39] and transverse spin\ndiffusion [40].\nIn order to explain the asymmetry in the drag rate it is necessary to include medium scattering, where the Fermi sea is Pauli\nblocked for intermediate states, and which entails a tendency toward molecule formation on the BEC side [59–62]. Medium\nscattering is described by the many-body T matrix T`,\nT\u00001\n`=0(~ q;!) =T(0)\u00001\n`=0(~ q;!) +Zd3p\n(2\u0019~)3n~ p#+n~ q\u0000~ p\"\n!\u0000\"~ p#\u0000\"~ q\u0000~ p\"+i0: (S.28)\nThe medium T matrix includes the effect of quantum degeneracy, which leads to a large increase in the partial-wave scattering\namplitudes (S.26) and would by itself violate the unitarity bound\njkf`j\u00141; (S.29)13\nwhich is the prerequisite for expressing the scattering amplitude kf`=ei\u000e`sin\u000e`in terms of real phase shifts \u000e`(~ q;!). The\ndefinition of the scattering amplitudes (S.26) in the presence of the medium therefore includes a phenomenological Qfactor\nwhich accounts for the fact that only unoccupied states are available for outgoing waves [58] (Pauli blocking),\nQ=Z1\n\u00001dcos\u0012\n2\u0010\n1\u0000n~ q=2+~~k#\u0000n~ q=2\u0000~~k\"\u0011\n: (S.30)\nThe integral averages over the angle \u0012between total momentum ~ qand relative momentum ~~k, while the modulus of ~kis fixed\nby the condition ~!=\"~ q=2+~~k#+\"~ q=2\u0000~~k\"=~2k2=m+q2=4m. In vacuum Q= 1, and also at high temperatures T\u001dTF\"\none hasQ\u00191. However, in a degenerate Fermi gas Q < 1lowers the scattering amplitude sufficiently to always satisfy the\nunitarity bound (S.29) also in the case of medium scattering.\nThe drag rate Eq. (S.24) is known analytically in the whole BCS-BEC crossover in the high-temperature limit T\u001dTF\"\nwhere the majority Fermi gas is non-degenerate [56],\n\u0010~\u0000S\nEF\"\u0011\nhightemp=16p\n2\n9\u00193=2\u0010TF\"\nT\u00111=2\u0002\n1\u0000x\u0000x2exEi(\u0000x)\u0003\nx=~2=(ma2kBT)(S.31)\nwhere Ei denotes the exponential integral. In this regime, the medium factor Q= 1. Conversely, medium scattering becomes\nimportant in a degenerate Fermi gas ( T.TF\"). In general, the collision term in Eq. (S.24) can be reduced to a three-dimensional\nintegral which is readily evaluated numerically. The resulting drag rate exhibits a maximum slightly on the BEC side of the\nresonance, in agreement with our experimental results. As the temperature is increased, the maximum shifts toward unitarity\nin agreement with the high-temperature expression (S.31) which is symmetric in a. At very low temperatures the agreement\nbetween this theory prediction and the experimental data is poorer, since there the T-matrix approximation is known to be\nquantitatively incorrect. Moreover the gas may suffer some heating during the dynamics due to decay processes, making its\ntemperature higher than the one measured at the start of the dynamics, which is used for the theory comparison.\nS.9. BREATHING MODE CHARACTERIZATION AND COLLISIONAL EFFECTS\nIn this Section we elaborate on the role of collisional effects, which could in principle dominate the system collective dynamics\nleading to misinterpretation of the observed spin-dipole mode behaviour. In strongly interacting many-body systems, collective\nmodes can be crucially affected by collisional hydrodynamics [60]. Within a purely collisional framework, the amplitude of\nspin-dipole oscillations is governed by the differential equation (S.23), which we use to analyze the long-time spin diffusion\ndynamics. Neglecting in-medium corrections, the collisional approach is based on the two-body scattering cross-section \u001b(k) =\n4\u0019a2=(1 + (ka)2)for relative momentum k. This leads to predictions independent of the sign of aand irrespective of whether\nthe system is prepared on the upper or lower branch, in clear contrast with our observations summarized in Fig. S7. This would\nnot change if one takes in-medium corrections into account (see Section S.8): at low temperatures these would lead only to a\nsmall asymmetry in the collisional integral [60], letting for instance many transport coefficients reach their maxima or minima\nslightly on the BEC side of resonance (see e.g. Fig 4 of the main text).\nLet us now consider the upper branch data for 0< \u0014Fa <1: the collisional hydrodynamics prediction (S.23) for the spin-\ndipole frequency might appear qualitatively compatible with the observed softening, given that \u0000Sincreases with interactions\n[38] (see Fig. 4 of the main text). As long as \u0000S<2!z, Eq. (S.23) would indeed give a spin-dipole frequency !SD=\n(!2\nz\u0000\u00002\nS=4)1=2. However, the damping rates \u0000Sexperimentally extracted by fitting the long-time evolution with the full\nsolution of Eq. (S.23) greatly exceed !z, yielding imaginary frequencies. On the other hand, we can consider Eq. (S.23) as a\nphenomenological model for the spin-dipole oscillations. In this case, the damping rate is the value 1=\u001cas extracted from the\nfitting procedure described in Section S.1, and therefore the measured spin-dipole frequency should be !SD= (!2\nz\u00001=\u001c2)1=2.\nOur fitted parameters do not satisfy the previous relation and more generally a collisional picture. Indeed the fitted \u001cvalues (i)\nwould give a very small correction to the bare frequency for any interaction strength and temperature and (ii) do not decrease by\nincreasing the value of \u0014Fa. Against a simple collisional interpretation speaks also the larger \u0017SDfound at higher temperature\nfor fixed\u0014Fa <1(purple squares in Fig. 2d). In the collisional dynamics framework the oscillation frequency should instead\ndecrease with temperature in the degenerate regime, where the damping rate increases because Pauli blocking becomes gradually\nless effective, as shown in Fig. 4.\nBreathing dynamics\nBesides exciting the spin-dipole mode, our experimental protocol also excites weakly the axial breathing mode, which we\nseparately characterize by studying the evolution of the cloud axial width \u001bz. Similarly to what we have done for the COM\nseparation dynamics, we isolate the breathing oscillation by subtracting from \u001bz(t)an overall exponential drift and fitting\nthe residual modulation \u0001\u001bz(t)with a damped sinusoidal function, as in Fig. S8a. The resulting frequency and damping14\nκF a0.1 1 101.21.41.61.82.0\nνB / νz\n√12/5\nκF a0.1 1 10Damping time (ms)\n50100150250\n200\n0\n-4-2024∆σz (µm)\n0 20 40 60 80 100\n0 20 40 60 80 100-202\nTime (ms)∆σz (µm)\n-505∆σz (µm)\n0 20 40 60 80 100\n-20-1001020∆σz (µm)\n0 20 40 60 80 100κF a = 0.04\nκF a = 0.29\nκF a = 0.77\nκF a ~ 50a b\nc\nFIG. S8. Characterization of the axial breathing mode. a, Evolution of the axial width \u0001\u001bz(t)for different interaction strengths at T=TF=\n0:12(2) after a sudden barrier switch-off. b, Normalized breathing frequency as a function of \u0014Fa. The breathing mode frequency \u0017B=\u0017z\ndecreases from the collisionless value 2, reaching a value betweenp\n12=5and2already at\u0014Fa'0:2.c, Damping time of the breathing\nmode versus \u0014Fa. The transition from a collisionless to a collisional regime is additionally signalled by a minimum of the damping time at\n\u0014Fa'0:2.\ntime are displayed in Fig. S8b-c. For weak interactions, the axial breathing mode oscillates at twice the axial trap frequency\n\u0017z, as expected in the collisionless regime. Increasing the interaction strength, we observe a decrease of the breathing mode\nfrequency\u0017B, which for\u0014Fa&0:2reaches an approximately constant value slightly larger than the hydrodynamic expectation\n\u0017B=p\n12=5\u0017z'1:55\u0017z[63] (see Fig. S8b).\nIn contrast to Ref. [28], we find the breathing mode amplitude to be substantially smaller than the one of the out-of-phase\nCOM oscillation at all interactions, owing to the small initial relative momentum imparted to the clouds by our experimental\nprotocol. For this reason, the spin-dipole mode is largely decoupled from the breathing one as long as the two clouds do not\nstart bouncing off each other, as if they became impenetrable. In this latter regime, observed at strong repulsion, the two modes\nare coupled and possibly lock to a common frequency within the experimental uncertainty. Such a frequency \u0017SD'1:70(4)\u0017z\nis compatible to \u0017R= 1:78(5)\u0017zthat we measured for a spin-polarized cloud oscillating in “half” the trap when the optical\nbarrier is left on; it is also similar to the experimental value 1:63\u0017z[28], and especially to the theory prediction 1:78\u0017z[35]. The\nvalue is also close to the frequencyp\n12=5\u0017zof the breathing mode in the hydrodynamic regime. In Ref. 38, where a purely\ncollisional approach is developed for two colliding clouds, it is found that the breathing mode remains essentially collisionless\nas long as the two clouds can diffuse into each other rapidly. Conversely, when the diffusion is slow, the breathing and the\nspin-dipole modes become coupled and the two clouds are predicted to enter into a bouncing regime. 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Transverse spin diffusion in strongly interacting Fermi gases. Phys. Rev. A 88, 033630 (2013).\n[57] Bruun, G. M., Recati, A., Pethick, C. J., Smith, H. & Stringari, S. Collisional properties of a polarized Fermi gas with resonant\ninteractions. Phys. Rev. Lett 100, 240406 (2011).\n[58] Bishop, R. F., Ghassib, H. B. & Strayer, M. R. Composite pairs and effective two-body scattering in a many-body medium. Phys. Rev. A\n13, 1570 (1976).\n[59] Bruun, G. M. & Smith, H., Viscosity and thermal relaxation for a resonantly interacting Fermi gas. Phys. Rev. A 72, 043605 (2005).17\n[60] Chiacchiera, S., Lepers, T., Davesne, D. & Urban, M. Collective modes of trapped Fermi gases with in-medium interaction. Phys. Rev.\nA79, 033613 (2009).\n[61] Enss, T. Quantum critical transport in the unitary Fermi gas. Phys. Rev. A 86, 013616 (2012).\n[62] Bluhm, M. & Sch ¨afer, T. Medium effects and the shear viscosity of the dilute Fermi gas away from the conformal limit. Phys. Rev. A 90,\n063615 (2014).\n[63] Gensemer, S.D. & Jin, D.S. Transition from collisionless to hydrodynamic behavior in an ultracold Fermi gas. Phys. Rev. Lett. 87, 173201\n(2001)."    },    {        "title": "1502.04934v1.Direct_Observation_of_Ferromagnetic_State_in_Gold_Nanorods_Probed_using_Electron_Spin_Resonance_Spectroscopy.pdf",        "content": "arXiv:1502.04934v1  [cond-mat.mes-hall]  17 Feb 2015Direct Observation of Ferromagnetic State in Gold Nanorods\nProbed using Electron Spin Resonance Spectroscopy\nYuji Inagaki∗and Tatsuya Kawae\nDepartment of Applied Quantum Physics, Faculty of Engineer ing,\nKyushu University, Fukuoka 819-0395, Japan\nNatsuko Sakai and Yuta Makihara\nDepartment of Material Physics and Chemistry,\nFaculty of Engineering, Kyushu University, Fukuoka 819-03 95, Japan\nHiroaki Yonemura and Sunao Yamada\nDepartment of Applied Chemistry, Faculty of Engineering,\nKyushu University, Fukuoka 819-0395, Japan\nAbstract\nX-band electron spin resonance (ESR) spectroscopy has been performed for gold nanorods\n(AuNRs) of four different sizes covered with a diamagnetic sta bilizing component, cetyltrimethy-\nlammmonium bromide. The ESR spectra show ferromagnetic fea tures such as hysteresis and reso-\nnance field shift, depending on the size of the AuNRs. In addit ion, the ferromagnetic transition is\nindicated by an abrupt change in the spectra of the two smalle st AuNRs studied. A large g-value\nin the paramagnetic region suggests that the ferromagnetis m in the AuNRs originates from strong\nspin-orbit interaction.\n1I. INTRODUCTION\nIn the past decade, rod-shaped gold nanoparticles, called Au nano rods (AuNRs), have\nattracted much attention because of their potential applications in sensing, imaging, and in\nvivo photothermal cancer therapy1–4. These applications are based on the inherent tunable\noptical properties of the AuNRs by changing the aspect ratio ardefined by the ratio of the\nlength to the diameter5. In contrast, there have been few reports thus far concerning the\nmagnetic properties of AuNRs6, although there have been many studies on the magnetism\nof spherical Au nanoparticles (AuNPs)7–12.\nRecently, Yonemura et al.examined the effects of magnetic processing and observed\nthe magnetic orientation and the side-by-side aggregation of AuNR s/poly(styrenesulfonate)\ncomposites, AuNRs with three different aspect ratios ( ar= 8.3, 5.1, and 2.5), and Au\nnanowires under a magnetic field13–15, which implies that the magnetic response of AuNRs\nis positive, although bulk Au has been believed to be diamagnetic for a lo ng time. However,\nthe detailed properties and origin of magnetism in AuNRs is not yet well understood.\nGenerally, magnetic properties have been investigated through ma gnetization measure-\nments using highly sensitive magnetometers, such as a commercial S QUID magnetometer.\nIn nano-sized particles coated with a stabilizing polymer, however, it is difficult to extract\nthe net magnetization precisely owing to the superposition of a large amount of diamagnetic\ncomponents16. We focus on the electron spin resonance (ESR) technique to elimina te the\nsignal from the stabilizing polymer because ESR is insensitive to diamag netism. Moreover,\nthe skin depth of Au at the X-band microwave region ( ∼9 GHz) is about 0.8 µm, which is\nmuch larger than the size of AuNRs in the present study. These fea tures indicate that ESR\nis a powerful tool for investigating the magnetic properties of AuN Rs.\nIn the present study, X-band ESR measurements were performe d in detail for AuNRs\nof four different sizes, where arwas varied from 2.5 to 8.3. Ferromagnetic features were\nobserved in all AuNRs. In addition, ferromagnetic transitions were clearly detected in the\ntwo smallest AuNRs studied. From the ESR parameters deduced fro m the conduction\nelectron spin resonance (CESR) at high temperatures, strong sp in-orbit interaction and\nresultant large effective mass of conduction electrons are pointed out for the origin of the\nmagnetism in the AuNRs on the basis of Elliott-Yafet theory.\n2TABLE I. Dimensions, aspect ratio, estimated content of Au, CTAB and other metals for each\nAuNR\nsize (nm) arAu(%) CTAB(%) Ag(%)\ns-AuNR 4.6 φ×11.6 2.5 3.6 Br:22.6, C:59.3, H:10.8, N:3.7 -\ns′-AuNR 5.0 φ×20.0 4.0 41.3 Br:16.7, C:31.4, H:5.8, N:1.9 2.9\nm-AuNR 7.2 φ×36.6 5.1 3.1 Br:22.9, C:59.4, H;11.0, N:3.6 -\nl-AuNR 7.7 φ×63.8 8.3 1.9 Br:22.7, C:60.6, H:11.2, N:3.7 -\nII. EXPERIMENTAL\nAuNRs studied here are named as s-, s′-, m-, and l-AuNR from small to large values\nofar. All AuNRs are prepared by the soft template method using cetyltr imethylamm-\nmonium bromide (CTAB)17. The s- and s′-AuNR are prepared using a reducing agent,\ntriethylamine, with or without acetone. The m-AuNR is prepared usin g a combination of\nchemical reduction and photoreduction. The l-AuNR is prepared us ing two kinds of re-\nducing agents, sodium borohydride and triethylamine. Typical TEM im ages of AuNRs are\ngiven in ref.13. X-ray fluorescence and CHN elemental analyses confirm that AuN Rs contain\nno contamination of other magnetic metals, as listed in Table I, which e nsures that the\nmagnetic response in the present study originates from the Au ato ms in AuNRs. ESR mea-\nsurements are performed using an X-band microwave system (JEO L ES-SCEX) equipped\nwith a continuous-He-flow-type cryostat (Oxford ESR910) oper ating down to T∼5 K.\nIII. RESULTS AND DISCUSSION\nFigures 1(a) and 1(b) show the temperature dependence of ESR s pectra observed for\ns-AuNR and s′-AuNR, respectively. Two spectra plotted with gray lines in Fig. 1(a) corre-\nspond to the blank signals recorded at 7 K and 295 K, which originate f rom the background\nfrom instruments including the sample holder (quartz tube) and the CTAB. This indicates\nthat the absorptions at around 320 mT are not intrinsic signals from the AuNR. At a glance,\nit seems that the spectra from the two AuNRs show quite similar temp erature dependences.\nHowever, on closely observing both spectra, several differences can be discerned. At room\ntemperature, the ESR spectrum of s-AuNR consists of two compo nents: a sharp absorption\n30 100200300400500600700 \nH (mT) 286 K \n200 K \n100 K \n6 K \n(d) 0 100 200 300 400 500 600 \nH (mT) (b) 297 K \n251 K \n202 K \n150 K \n120 K \n100 K \n80 K \n62 K \n49 K \n41 K \n30 K \n21 K \n12.5 K \n5.5 K \n0 100 200 300 400 500 Derivative Intensity (arb. units) \nH (mT) 295 K \n200 K \n140 K \n103 K \n75 K \n60 K \n51 K \n40 K \n32 K \n16.5 K \n9.7 K \n6.0 K \nblank 7 K blank 295 K \n(a) \n0 100 200 300 Derivative intensity (arb.units) \nH (mT) 300 K \n250 K \n200 K \n100 K \n88 K \n75 K \n62 K \n45 K \n32 K \n15 K \n10 K \n6.8 K 150 K \n(c) \nFIG. 1. (Color online) Temperature dependence of ESR spectr a in (a) s-AuNR, (b) s′-AuNR (c)\nm-AuNR and (d) l-AuNR. The absorption around 320 mT is due to i nstrumental background, as\nobserved from the blank spectra recorded at 295 K and 7 K shown by the gray lines in (a).\ncentered at s 1= 275 mT and a broad one around s 2= 230 mT, which is strongly suppressed\nwhen the temperature is decreased. In contrast, the spectra o f s′-AuNR have no broad\ncomponent and show only a sharp absorption centered at s′\n1= 289 mT. Thus, the sharp ab-\nsorptions at s 1and s′\n1are regarded as intrinsic properties of AuNRs, and their temperat ure\nvariations are examined below.\nThe temperature dependences of g-values estimated from resonance fields and line widths\n∆Hfor s1and s′\n1are summarized in Fig. 2. The resonance field and line width are nearly\nindependent of temperature for both samples above ∼80 K, while the spectra show a drastic\nshift of the resonance field and broadening of the width below 55 K fo r s-AuNR and 75\n41.5 2.0 2.5 3.0 \n020 40 60 80 100 \n0 50 100 150 200 250 300 g-value ΔH (mT) \nT (K) TcsTcs' \nFIG. 2. Temperature dependence of g-value (circle) and line width ∆ H(square). Open and solid\nsymbols represent the results for s-AuNR and s′-AuNR, respectively.\nK for s′-AuNR. The g-value of s-AuNR (s′-AuNR) increases with decreasing temperature\nand reaches 3.19 (3.07) at the lowest temperature T∼6 K. For s-AuNR (s′-AuNR), the\nline width shows a maximum at approximately 40 K (60 K), which is followed by a gradual\nnarrowing with decreasing temperature. This series ofbehaviors is atypical featureobserved\nin the magnetic ordering process. Hence, we conclude that the res onance field shift and\nbroadening of the width in s- and s′-AuNRs are caused by magnetic ordering with transition\ntemperatures of Ts\nc∼55 K and Ts′\nc∼75 K, respectively.\nThe results for m- and l-AuNRs are depicted in Figs. 1(c) and 1(d), r espectively. In\nm-AuNR, the ESR spectrum cannot be observed in the high-temper ature region. When the\ntemperature is decreased below T∼100 K, a narrow absorption with ∆ H= 8 mT appears\nat around 150 mT. The intensity of the spectrum grows with decrea sing temperature, while\nboth the resonance center and line width are almost independent of temperature down to T\n= 6.8 K. This temperature dependence will be discussed later.\nIn contrast, l-AuNR shows broad ESR spectra in the entire temper ature range, as shown\nin Fig. 1(d). The resonance field at ∼70mT in l-AuNR is the lowest among all AuNRs, while\nthe line width of ∼130 mT is the broadest. It is significant that a hysteresis emerges in all\nthe spectra of l-AuNR below ∼60 mT in the magnetic field sweep, as indicated by arrows. A\nhysteresis between magnetizing and demagnetizing processes is a c haracteristic feature of a\n50 50 100 150 200 250 300 350 400 Derivative intensity (arb. units) \nH (mT) g = 2 \nFIG. 3. (Color online) ESR spectra of all AuNRs at T ∼6K. Red, blue, green and orange colors\ncorrespond to s-, s′-, m- and l-AuNR, respectively. The vertical dotted line ind icates the resonance\nfield forg=2.\nferromagnet with domain structures and/or magnetic anisotropy18,19. Note that the domain\nstructure and anisotropy give rise to the line broadening in ESR spec tra. In other words,\nthese features observed in l-AuNR are well explained by assuming a la rge scale ferromagnet.\nNext, we discuss the systematic resonance shift in all AuNRs. The s pectra at the lowest\ntemperature T∼6 K are plotted together in Fig. 3 to make a comparison between all\nAuNRs. In a ferromagnet with a cylindrical shape like the present Au NRs, ferromagnetic\nresonance (FMR) occurs at a resonance field that depends on the demagnetizing form factor\nowing to the aspect ratio ar, as well as on the magnitude of moment Msand anisotropy K.\nIn fact, the magnitude of shift increases with ar in the present systems; the shift is estimated\nto be approximately 248, 176, 114, and 118 mT for l-, m-, s′-, and s-AuNR, respectively,\nwhich corresponds to the order of arexcept for s- and s′-AuNR. The reversal between them\nmay be caused by the resonance shift dominated by an effective anis otropy field represented\nbyK/Ms. Thus, a small anisotropy Kcan give rise to large effective field if Msis small. As\nfor the anisotropy, the following qualitative discussion can be made. In randomly oriented\nferromagnetic species, a powder pattern is expected in the FMR sp ectrum. Although a\nclear powder pattern was not recorded for all AuNRs, finite ESR int ensity can be seen like\na tail in a higher field range than the main absorption peak, as repres ented in l-AuNR. It\nis difficult to refer about other AuNRs because of the superposition of background signal\nat around 320 mT. However, a small intensity like shoulder is visible at a bout 270 mT in\n6s′-AuNR, as indicated by the vertical arrow in Fig. 3. Such FMR spectr a correspond to the\ncase with a negative anisotropy in the cubic symmetry20. Detailed frequency dependence of\nESR is required to obtain further information about the FMR parame tersMsandK.\nThe observed systematic shift, which is generally not expected in ot her antiferromagnets,\nparamagnets, and ferromagnets with a spherical shape, provide s further evidence to the\nexistence of ferromagnetic states at low temperatures in all the p resent AuNRs. On the\nbasis of the hysteresis and systematic shift of the resonance field , it is reasonable to consider\nthat all AuNRs are in the ferromagnetic state at low temperatures . To our knowledge, this\nis the first time that a ferromagnetic state has been found in AuNRs .\nIn the final part of this Letter, we examine the ESR spectra in the p aramagnetic region.\nThe spectra of s- and s′-AuNRs above Tccan be understood as CESR ones. In the CESR\nregion,g-values are estimated to be 2.34 ±0.09 and 2.26 ±0.06 for s- and s′-AuNRs, respec-\ntively. These values are considerably larger than the value of 2.11 re ported for bulk Au\nbut comparable to that of 2.26 ±0.02 for small particles of Au with a mean diameter of 3\nnm21,22. For CESR, the difference ∆ gbetween the g-value of real metals and that of ideal\nfree electrons (2.0023) gives an approximate value of the spin-orb it interaction through the\nequation ∆ g∼λ/∆E. Here,λis the spin-orbit coupling constant and ∆ Eis the difference in\nenergy between the 6s band and the nearest 5d band23. Thus, the large ∆ gobserved in the\npresent measurements indicates the large contribution of the orb ital moment in the 5d band\nto conduction electrons in the 6s band, which governs the magnetic properties of AuNRs\nand leads to the ferromagnetic state at low temperatures. A stro ng spin-orbit coupling was\nalso confirmed in recent studies of X-ray magnetic circular dichroism in bulk Au as well as\nAuNPs24,25.\nThe strong spin-orbit interaction also causes a significant broaden ing of line width. The\nline width of CESR in metals is closely related to the spin-lattice relaxatio n time, i.e., the\nspin flip rate by phonons. Accordingly, ∆ Hvaries linearly with temperature. Therefore,\nat high temperatures, CESR is hardly detected in not only bulk metals but also m-AuNR.\nThis is a reason why we could not observe the ESR in m-AuNR above 100 K. In contrast,\nnarrow ESR absorptions are obtained for both s- and s′-AuNRs, which show rather gentle\ntemperature dependences without a marked increase in the width. These features may be\nexplained by considering the system size of AuNRs. In small systems , there exists a lower\nlimit of phonon mode frequency given by ν=vs/2L, whereLis the largest dimension of\n710 510 610 710 810 910 10 10 11 10 12 10 13 \n10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0ΔH/ρ (mT/ Ω/m) \nΔg2Au \nPd \nNa KRb Cs \nCu Ag \nAl \ns'-AuNR \ns-AuNR \nFIG. 4. Beuneu-Monod plot, which connects ∆ g2and ∆H/ρ, for pure metals given in ref.27and\nfor two systems in the present study. The value ∆ H/ρof most metals differs from the fit, which\nis indicated by the solid and dashed lines, by less than one or der of magnitude.\nthe system and vsis the speed of sound in the crystal; this results in an increase in the\nspin-lattice relaxation time of conduction electrons26. As a result, narrow absorptions are\nobserved in s- and s′-AuNRs. This scenario is qualitative and does not account for the\nnearly equal line widths of s-AuNR and s′-AuNR; other factors should be considered for the\nquantitative explanation.\nNevertheless, owing to the narrow line widths, we could successfully observe CESR of\nAuNR, which allows identification of the characteristic feature of th e present CESR results\nthrough sorting with other metals in the so-called BeuneuMonod plot shown in Fig. 427,28.\nThe BeuneuMonod plot is an empirical plot that connects CESR param eters ∆gand ∆H\nfor pure metals via ∆ H/ρ=α∆g2, whereρis the resistivity and αis a metal-dependent\nconstant. Most metals follow a straight-line fit in the log-log plot of ∆ H/ρvs. ∆g2by less\nthan one order of magnitude, as indicated by the solid and dashed line s in Fig. 4. The values\nused in this plot are taken at a temperature of approximately TD/7, where TDis the Debye\ntemperature. In the case of bulk Au, TD/7 corresponds to 20 K. The CESR parameters\nfor the two systems in the present study (s- and s′-AuNR) are not available for such a low\ntemperature, which is less than Tc. Thus, we plot the estimate taken in the temperature\n8range between Tcand room temperature, while the resistivity is fixed at the bulk value a t 20\nK. Error bars for both axes originate from the temperature varia tion of ∆ gand ∆H. In the\nplot, the two data points corresponding to s- and s′-AuNR deviate significantly fromthose of\nother metals, but they are located near the point corresponding t o Pd. It is well known that\nPd is close to satisfying the Stoner criterion even in a bulk form. Inde ed, ferromagnetism is\nrealized by the downsizing of Pd29–31. Accordingly, it is confirmed that the two AuNRs are\nalso close to the ferromagnetic state.\nTo reproduce the results for s-AuNR, s′-AuNR, and Pd, a smaller αis needed. This\nsuggests that the low value of αis closely related to the realization of ferromagnetism.\nAccording to the Elliott-Yafet theory, the coefficient αis given by ne2/γm∗, wheren,e, and\nm∗are the density, charge, and effective mass of the conduction elec tron, respectively, and γ\nis the magnetomechanical ratio. Therefore, the origin of the devia tion caused by the narrow\n∆Hand the large ∆ gis suggested to be an enhancement of the effective mass. This is a\nreasonable conclusion because a large effective mass is realized in a ty pical ferromagnet of\nNi, as revealed recently by high-resolution angle-resolved photoem ission studies32.\nIn summary, we performed ESR measurements for AuNRs of four s izes with different\naspect ratios. The ESR spectra at low temperatures are explained in the context of ferro-\nmagnetic resonance for all AuNRs. Detailed frequency dependenc e of FMR measurements\nwill enable further quantitative discussion on the size- and shape-d ependent magnetism of\nAuNRs. In the two smallest AuNRs, we detected ferromagnetic tra nsitions in the ESR spec-\ntra atT∼60 K, which offers an opportunity to explore the critical behavior of the phase\ntransition in nano-rod systems. The CESR above Tcsuggests that the strong spin-orbit\ninteraction is responsible for ferromagnetism in the AuNR systems.\nACKNOWLEDGMENTS\nThe authors are grateful to T. Asano and T. Sakurai for the help of ESR experiments\nand useful discussion. This work was partially supported by a Grant -in-Aid for Scientific\nResearch, No.23540392, No.25220605 and No.25287076. The auth ors thank to Dr. Daigou\nMizoguchi (Dai Nippon Toryo Co. Ltd.) for providing them with four d ifferent types of\nAuNRs. 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B72, 214438 (2005).\n11"    },    {        "title": "1106.3519v1.Current_effect_on_magnetization_oscillations_in_a_ferromagnet___antiferromagnet_junction.pdf",        "content": "arXiv:1106.3519v1  [cond-mat.mtrl-sci]  17 Jun 2011Current effect on magnetization oscillations\nin a ferromagnet–antiferromagnet junction\nE. M. Epshtein∗, Yu. V. Gulyaev, P. E. Zilberman,\nV. A. Kotelnikov Institute of Radio Engineering and Electronics\nof the Russian Academy of Sciences, Fryazino, 141190, Russia\nAbstract\nSpin-polarized current effect is studied on the static and dy namic mag-\nnetization of the antiferromagnet in a ferromagnet–antife rromagnet junc-\ntion. The macrospin approximation is generalized to antife rromagnets.\nCanted antiferromagnetic configuration and resulting magn etic moment\nare induced by an external magnetic field. The resonance freq uency and\ndamping are calculated, as well as the threshold current den sity corre-\nsponding to instability appearance. A possibility is shown of generating\nlow-damping magnetization oscillations in terahertz rang e. The fluctua-\ntion effect is discussed on the canted antiferromagnetic con figuration.\n1 Introduction\nThe discovery of the spin transfer torque effect in ferromagn etic junctions\nunder spin-polarized current [1, 2] has stimulated a number of works in\nwhich such effects were observed as switching the junction ma gnetic con-\nfiguration [3], spin wave generation [4], current-driven mo tion of magnetic\ndomainwalls [5], modification offerromagnetic resonance [ 6], etc. Itiswell\nknown that spin torque transfer from spin-polarized electr ons to lattice\nleads to appearance of a negative damping. At some current de nsity, this\nnegative damping overcomes the positive (Gilbert) damping with occur-\nring instability of the original magnetic configuration. Th e corresponding\ncurrent density is high enough, of the order of 107A/cm2. This, naturally,\nstimulates attempts to lower this threshold. Various ways w ere proposed,\nsuch as using magnetic semiconductors [7], in which the thre shold cur-\nrent density can be lower down to 105–106A/cm2because of their low\nsaturation magnetization. However, using of such material s requires, as\na rule, low temperatures because of low Curie temperature. B esides, the\nferromagnetic resonance frequency is rather low in this cas e.\n∗E-mail: epshtein36@mail.ru\n1FM AFMNM\nMFH−+n\nxyz\n0 LAFMj/e\nFigure 1: Scheme of the ferromagnet (FM)–antiferromagnet (AF M) junction;\nNM being a nonmagnetic layer. The main vector directions are shown.\nIn connection with these difficulties, the other approaches w ere pro-\nposed, based on high spin injection [8] or joint action of ext ernal magnetic\nfield and spin-polarized current [9, 10]. It seems promising , also, using\nmagnetic junction of ferromagnet–antiferromagnet type, i n which the fer-\nromagnet (FM) acts as an injector of spin-polarized electro ns. The anti-\nferromagnetic (AFM) layer, in which the magnetic sublattic es are canted\nby external magnetic field, may have very low magnetization t hat pro-\nmotes low threshold [11]. The AFM resonance frequency may be both\nlow and high reaching 1012s−1, i.e. terahertz (THz) range. However,\ninvestigation and application of THz resonances is prevent ed because of\ntheir large damping. Such a damping in ferromagnetic juncti ons can be\nsuppressed, as mentioned above, by means of spin-polarized current. The\nquestion arises about possibility of such a suppression in F M–AFM junc-\ntions. Note, that this problem has been paid attention of a nu mber of\nauthors [12]–[20].\n2 The equations of motion\nLet us consider a FM–AFM junction (Fig. 1) with current flowin g per-\npendicular to layers, along xaxis. An external magnetic field is parallel\nto the FM magnetization and lies in the layer plane yz. The simplest AF\nmodel is used with two equivalent sublattices.\nThe AFM energy (per unit area), with uniform and nonuniform e x-\nchange, anisotropy, external magnetic field, demagnetizat ion and the sd\nexchange interaction of the conduction electrons with the m agnetic lattice\ntaking into account, takes the form [21]\n2W=/integraldisplayLAFM\n0dx/braceleftBigg\nΛ(M1·M2)+1\n2α/braceleftBigg/parenleftbigg∂M1\n∂x/parenrightbigg2\n+/parenleftbigg∂M2\n∂x/parenrightbigg2/bracerightBigg\n+α′/parenleftbigg∂M1\n∂x·∂M2\n∂x/parenrightbigg\n−1\n2β/braceleftbig\n(M1·n)2+(M2·n)2/bracerightbig\n−β′(M1·n)(M2·n)\n−((M1+M2)·H)−αsd((M1+M2)·m)+2π(M1+M2)2\nx/bracerightBigg\n,(1)\nwhereM1,M2are the sublattice magnetization vectors, Λ is the uni-\nform exchange constant, α, α′are the intrasublattice and intersublattice\nnonuniform exchange constants, respectively, β, β′are the corresponding\nanisotropy constants, nis the unit vector along the anisotropy axis, His\nthe external magnetic field, mis the conduction electron magnetization,\nαsdis the dimensionless sdexchange interaction constant; the last term\ndescribes demagnetization effect. The integral is taken ove r the AFM\nlayer thickness LAFM. We are interested in the spin-polarized current\neffect on the AFM layer, so we consider a case of perfect FM inje ctor with\npinned lattice magnetization and without disturbance of th e electron spin\nequilibrium, that allows to not include the FM layer energy i n Eq. (1).\nTwo mechanisms are known of the spin-polarized current effec t on\nthe magnetic lattice, namely, spin transfer torque (STT) [1 , 2] and an\nalternative mechanism [22, 23] due to the spin injection and appearance of\nnonequilibrium population of the spin subbands in the colle ctor layer (this\nis AFM layer, in our case). In the case of antiparallel relati ve orientation\nof the injector and collector magnetization vectors, such a state becomes\nenergetically unfavorable, so that the antiparallel config uration switches\nto parallel one (such a process in FM junction is considered i n detail in\nreview [24]). The latter mechanism is described with the sdexchange\nterm in Eq. (1). As to the former mechanism, it is of dissipati ve character\n(it leads to negative damping), so that it is taken into accou nt by the\nboundary conditions (see below), not the Hamiltonian.\nThe equations of the sublattice motion with damping taking i nto ac-\ncount take the form\n∂Mi\n∂t−κ\nM0/bracketleftbigg\nMi×∂Mi\n∂t/bracketrightbigg\n+γ/bracketleftBig\nMi×H(i)\neff/bracketrightBig\n= 0 (i= 1,2),(2)\nwhereM0is the sublattice magnetization, κis the damping constant,\nH(i)\neff=−δW\nδMi(i= 1,2) (3)\nare the effective fields acting on the corresponding sublatti ces.\nFrom Eqs. (1)–(3) the equations are obtained for the total ma gnetiza-\ntionM=M1+M2and antiferromagnetism vector L=M1−M2:\n3∂M\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nM×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nL×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[M×H]+γ[M×Hd]+γ[M×Hsd]\n+1\n2γ(β+β′)(M·n)[M×n]+1\n2γ(β−β′)(L·n)[L×n]\n+1\n2γ(α+α′)/bracketleftbigg\nM×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nL×∂2L\n∂x2/bracketrightbigg\n= 0,(4)\n∂L\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nL×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nM×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[L×H]+γ[L×Hd]+γ[L×Hsd]−γΛ[L×M]\n+1\n2γ(β+β′)(M·n)[L×n]+1\n2γ(β−β′)(L·n)[M×n]\n+1\n2γ(α+α′)/bracketleftbigg\nL×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nM×∂2L\n∂x2/bracketrightbigg\n= 0,(5)\nwhereHd=−4π{M1x+M2x,0,0}is the demagnetization field,\nHsd(x) =δ\nδM(x)/integraldisplayLAFM\n0dx′/parenleftbig\nM(x′)·m(x′)/parenrightbig\n(6)\nis the effective field due to sdexchange interaction. This field determines\nthe spin injection contribution to the interaction of the co nduction elec-\ntrons with the antiferromagnet lattice.\nTo findHsd(x) field, the conduction electron magnetization m(x) is\nto be calculated. The details of such calculations are prese nted in our\npreceding papers [25, 9]. Here we adduce the result for the ca se, where\nthe antiferromagnet layer thickness LAFMis small compared to the spin\ndiffusion length lwith the current flow direction corresponding to the\nelectron flux from FM to AFM:\nm= (m+∆m)ˆM,∆m=µBτQj\neLAFM/parenleftBig\nˆM(0)·ˆMF/parenrightBig\n, (7)\nwheremis the equilibrium (in absence of current) electron magneti zation,\n∆mis the nonequilibrium increment due to current, ˆM=M/|M|is the\nunit vector along the AFM magnetization, ˆMFis the similar vector for\nFM,µBis the Bohr magneton, eis the electron charge, τis the electron\nspin relaxation time, jis the current density.\nIt should have in mind in varying the integral (6), that the el ectron\nmagnetization mdepends on the vector Morientation relative to the FM\nmagnetization vector MF. From Eqs. (6)and (7) we have [9]\nHsd=αsdmˆM+αsdµBτQj\neLAFMˆM+αsdµBτQj\neˆMFδ(x−0).(8)\nBy substitution (8) into (4) and (5), we obtain\n4∂M\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nM×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nL×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[M×H]+γ[M×Hd]+γαsdµBτQj\ne/bracketleftBig\nM׈MF/bracketrightBig\nδ(x−0)\n+1\n2γ(β+β′)(M·n)[M×n]+1\n2γ(β−β′)(L·n)[L×n]\n+1\n2γ(α+α′)/bracketleftbigg\nM×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nL×∂2L\n∂x2/bracketrightbigg\n= 0,(9)\n∂L\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nL×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nM×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[L×H]+γ[L×Hd]+γαsdµBτQj\ne/bracketleftBig\nL׈MF/bracketrightBig\nδ(x−0)\n−γ/parenleftbigg\nΛ−αsdm\nM−αsdµBτQj\neLAFMM/parenrightbigg\n[L×M]\n+1\n2γ(β+β′)(M·n)[L×n]+1\n2γ(β−β′)(L·n)[M×n]\n+1\n2γ(α+α′)/bracketleftbigg\nL×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nM×∂2L\n∂x2/bracketrightbigg\n= 0.(10)\n3 The boundary conditions\nThe equations of motion (9) and (10) contain derivative over the space\ncoordinate x. Therefore, boundary conditions at the AFM layer surfaces\nx= 0 and x=LAFMare need to find solutions. The way of derivation\nwas described in Ref. [9] in detail. The conditions depend on the electron\nspin polarization and are determined by the continuity requ irement of the\nspin currents at the interfaces.\nThe terms with the space derivative in Eq. (9) may be written i n the\nform of a divergency:\n1\n2γ(α+α′)/bracketleftbigg\nM×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nL×∂2L\n∂x2/bracketrightbigg\n=∂\n∂x/braceleftbigg1\n2γ(α+α′)/bracketleftbigg\nM×∂M\n∂x/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nL×∂L\n∂x/bracketrightbigg/bracerightbigg\n≡∂JM\n∂x. (11)\nTheJMvector is the lattice magnetization flux density.\nLet us integrate Eq. (9) over xwithin narrow interval 0 < x < ε with\nsubsequent passing to ε→+0 limit. Then only the mentioned terms with\nthe space derivative and the singular term with delta functi on will con-\ntribute to the integral. As a result, we obtain an effective ma gnetization\nflux density with sdexchange contribution at the AFM boundary x= +0\ntaking into account:\nJeff(+0) =JM(+0)+γαsdµBτQj\ne/bracketleftBig\nM(+0)׈MF/bracketrightBig\n.(12)\n5The magnetization flux density coming from the FM injector is\nJ(−0) =µBQ\nejˆMF. (13)\nThe component J/bardbl=/parenleftBig\nJ(−0)·ˆM(+0)/parenrightBig\nˆM(+0) remains with the elec-\ntrons, while the rest,\nJ⊥=J(−0)−J/bardbl=µBQ\nej/braceleftBig\nˆMF−ˆM(+0)/parenleftBig\nˆMF·ˆM(+0)/parenrightBig/bracerightBig\n=−µBQ\neM2j/bracketleftBig\nM(+0)×/bracketleftBig\nM(+0)׈MF/bracketrightBig/bracketrightBig\n, (14)\nis transferred to the AFM lattice owing to conservation of th e magnetiza-\ntion fluxes [1, 2].\nBy equating the magnetization fluxes (12) and (14), we obtain\nJM=−µBQ\neM2j/bracketleftBig\nM×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig\n−γαsdµBτQ\nej/bracketleftBig\nM׈MF/bracketrightBig\n,(15)\nall theMvectors being taken at x= +0.\nSince the AFM layer thickness is small compared to the spin di ffusion\nlength and the exchange length, we may use the macrospin appr oximation\nwhich was described in detail in Ref. [9]. In this approximat ion, the\nmagnetization changes slowly within the layer thickness. T his allows to\nwrite\n∂JM\n∂x≈JM(LAFM)−JM(+0)\nLAFM=−JM(+0)\nLAFM, (16)\nbecause the magnetization flux is equal to zero at the interfa ce between\nAFM and the nonmagnetic layer closing the electric circuit, JM(LAFM) =\n0. This allows to exclude the terms with space derivative fro m Eq. (9).\nIn the rest terms, M(x, t) andL(x, t) quantities are replaced with their\nvalues at x= 0. Then Eq. (9) takes a more simple form:\n∂M\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nM×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nL×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[M×H]+γ[M×Hd]\n+1\n2γ(β+β′)(M·n)[M×n]+1\n2γ(β−β′)(L·n)[L×n]\n+K/bracketleftBig\nM×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig\n+P/bracketleftBig\nM׈MF/bracketrightBig\n= 0, (17)\nwhere\nK=µBQ\neLAFMM2j, P =γαsdµBτQ\neLAFMj. (18)\nThe term with delta function does not present here, since it i s taken into\naccount in the boundary conditions.\nNow we are to use again the macrospin approximation to exclud e the\nspace derivatives from Eq. (10), too.\nOwing to known relationships [21] between MandLvectors, namely,\nM2+L2= 4M2\n0and (M·L) = 0, we have the following conditions:\n/parenleftbigg\nM·∂M\n∂t/parenrightbigg\n+/parenleftbigg\nL·∂L\n∂t/parenrightbigg\n= 0,/parenleftbigg\nL·∂M\n∂t/parenrightbigg\n+/parenleftbigg\nM·∂L\n∂t/parenrightbigg\n= 0.(19)\n6By substituting Eqs. (10) and (17) in (19) we find that conditi ons (19)\nare fulfilled if the terms in (10)\n1\n2γ(α+α′)/bracketleftbigg\nL×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nM×∂2L\n∂x2/bracketrightbigg\n≡X(20)\nsatisfy the following equations:\n(X·M)+K/parenleftBig\nL·/bracketleftBig\nM×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig/parenrightBig\n+P/parenleftBig\nL·/bracketleftBig\nM׈MF/bracketrightBig/parenrightBig\n= 0,\n(X·L) = 0. (21)\nLet us decompose the considered Xvector on three mutually orthog-\nonal vectors:\nX=aM+bL+cγ[L×M]. (22)\nThe substitution (22) in (21) gives a=K/parenleftBig\nL·ˆMF/parenrightBig\n−P/parenleftBig\n[L×M]·ˆMF/parenrightBig\n,\nb= 0. As to ccoefficient, it is a current-induced correction to the co-\nefficient of γ[L×M] term in Eq. (10), i.e., a correction to the uniform\nexchange constant Λ. Let us estimate the correction. Multip lying (22)\nscalarly by [ L×M] with (20) taking into account gives\nc=1\nM2L2/parenleftbigg\n[L×M]·/braceleftbigg1\n2γ(α+α′)/bracketleftbigg\nL×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nM×∂2L\n∂x2/bracketrightbigg/bracerightbigg/parenrightbigg\n=1\n2/braceleftbigg\n(α+α′)1\nM2/parenleftbigg\nM·∂2M\n∂x2/parenrightbigg\n−(α−α′)1\nL2/parenleftbigg\nL·∂2L\n∂x2/parenrightbigg/bracerightbigg\n. (23)\nIt is seen that c∼α/L2\nAFM, while Λ ∼α/a2, where ais the lattice\nconstant [21]. Since LAFM≫a, the mentioned correction to Λ may be\nneglected.\nAs a result, Eq. (10) takes the form\n∂L\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nL×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nM×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[L×H]+γ[L×Hd]−/parenleftbigg\nγΛ−P\nM/parenrightbigg\n[L×M]\n+1\n2γ(β+β′)(M·n)[L×n]+1\n2γ(β−β′)(L·n)[M×n]\n+K/bracketleftBig\nL×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig\n−P1\nM2/bracketleftBig\n[L×M]×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig\n= 0. (24)\nHere, Λ constant contains also the equilibrium contributio n of the con-\nduction electrons −αsdm/M.\nEquations (17) and (24) are the result of applying the macros pin con-\ncept to AFM. It is shown that such an approximation may be just ified\nformally for AFM layer. Earlier, it was justified for FM layer s [1, 2] and\ngeneralized [9] with spin injection taking into account. Th e macrospin\napproach corresponds well to experimental conditions and s implifies cal-\nculations substantially. The terms with Kcoe���cient in Eqs. (17), (24)\ndescribe effect of STT mechanism, while the terms with Pcoefficient take\nthe spin injection effect into account.\n74 The magnetization wave spectrum and\ndamping\nWe assume that the easy anisotropy axis lies in the plane of AF M layer\nand is directed along yaxis, the FM magnetization vector is parallel to\nthe positive direction of zaxis, the external magnetic field is parallel to z\naxis too (see Fig. 1).\nWe are interesting in behavior of small fluctuations around t he steady\nstateM={0,0,Mz},L={0,Ly,0}, i.e. thesmallquantities Mx, My,/tildewiderMz=\nMz−Mz, Lx,/tildewideLy=Ly−Ly, Lz.\nLet us project Eqs. (17), (24) to the coordinate axes and take the\nterms up to the first order. The zero order terms are present on ly in the\nprojection of Eq. (24) to xaxis. They give\nMz=Hz+P\nγ\nΛ+1\n2(β−β′)≈Hz+P\nγ\nΛ,\nLy=±/radicalBig\n4M2\n0−M2\nz≈ ±2M0. (25)\nNote that the spin-polarized current takes part in creating magnetic\nmoment together with the external magnetic field dueto thesp in injection\ninduced interaction of the electron spins with the lattice [ 22, 23], which\nPparameter in Eq. (25) corresponds to. Such an interaction le ads to\nappearance of an effective magnetic field parallel to the inje ctor magneti-\nzation. As a result, a canted antiferromagnet configuration may be create\nwithout magnetic field. However, such a configuration corres ponds to\nparallel orientation of FM and AFM layers, M/bardblMF. As is shown below,\nthe instability does not occur with this orientation, so tha t an external\nmagnetic field is to be applied to reach instability.\nWith Eq. (25) taking into account, the equations for the first order\nquantities take the form\n∂Mx\n∂t−1\n2κ\nM0/braceleftbigg\n−Mz∂My\n∂t+Ly∂Lz\n∂t/bracerightbigg\n+(γHz+P)My\n−1\n2γ(β+β′)MzMy−1\n2γ(β−β′)LyLz+KMzMx= 0,(26)\n∂My\n∂t−1\n2κ\nM0Mz∂Mx\n∂t−(γHz+P+4πγMz)Mx+KMzMy= 0,(27)\n∂/tildewiderMz\n∂t+1\n2κ\nM0Ly∂Lx\n∂t+1\n2γ(β−β′)LyLx= 0, (28)\n∂Lx\n∂t−1\n2κ\nM0/braceleftBigg\nLy∂/tildewiderMz\n∂t−Mz∂/tildewideLy\n∂t/bracerightBigg\n−γHzLy\nMz/tildewiderMz= 0,(29)\n∂/tildewideLy\n∂t−1\n2κ\nM0Mz∂Lx\n∂t−1\n2γ(β−β′)MzLx= 0, (30)\n8∂Lz\n∂t+1\n2κ\nM0Ly∂Mx\n∂t+(γHz+P+4πγMz)Ly\nMzMx+KMzLz= 0.(31)\nThe set of equations (26)–(31) splits up to two mutually inde pendent\nsets with respect to ( Mx, My, Lz) and (Lx,/tildewideLy,/tildewiderMz). They describe two\nindependent spectral modes, one of them corresponds to prec ession of the\nAFMmagnetization vectoraroundthemagnetic field, while an other tope-\nriodic changes ofthevectorlength alongthemagnetic field. Webeginwith\nthe spectrum and damping of the first mode. We consider monoch romatic\noscillation with ωangular frequency and put Mx, My, Lz∼exp(−iωt).\nThen we obtain from Eqs. (26), (27), (31)\n/parenleftbig\n−iω+KMz/parenrightbig\nMx+/braceleftbigg\nγHz+P−1\n2γ(β+β′)Mz−1\n2iκω\nM0Mz/bracerightbigg\nMy\n−/braceleftbigg1\n2γ(β−β′)−1\n2iκω\nM0/bracerightbigg\nLyLz= 0, (32)\n/parenleftbig\n−iω+KMz/parenrightbig\nMy−/braceleftbigg\nγHz+P+4πγMz−1\n2iκω\nM0Mz/bracerightbigg\nMx= 0,(33)\n/parenleftbig\n−iω+KMz/parenrightbig\nLz+/braceleftbigg\nγ(Λ+4π)+1\n2γ(β−β′)−1\n2iκω\nM0/bracerightbigg\nLyMx= 0.\n(34)\nNote that aforementioned additivity (in the algebraic sens e, the sign\ntakingintoaccount) oftheexternalmagnetic field andthein jection-driven\neffective field takes place not only in the steady magnetizati on (25), but\nalso in the oscillations of the magnetization and antiferro magnetism vec-\ntors, so that both fields appear in Eqs. (32), (33) “on an equal footing”.\nUsually, Λ ≫4π, β, β′. With these inequalities and stationary so-\nlution (25) taking into account we find the dispersion relati on for the\nmagnetization oscillation\n(1+κ2)ω2+2iνω−ω2\n0= 0, (35)\nwhere\nω0=/radicalBig\n2γ2HAHE+(KMz)2+(γHz+P)2, (36)\nν=κγHE+KMz, (37)\nHE= ΛM0is the exchange field, HA= (β−β′)M0is the anisotropy field.\nFormulae (36)and(37)(without currentterms KMzandP)coincide with\nknownones [21,26]. At HE∼106–107G,HA∼103Gwehaveoscillations\nin THz range, ω0∼1012s−1. In absence of current the damping is rather\nhigh: at κ∼10−2\nν\nω0=κ/radicalbigg\nHE\n2HA∼1. (38)\n9Let us consider the contribution of spin-polarized current to the fre-\nquency and damping of AFM resonance. At first we consider STT m ech-\nanism effect [1, 2]. According to (18) and (25),\nKMz=µBQΛ\neLAFMHzj. (39)\nAtHz<0, that corresponds to direction of the magnetic field (and,\ntherefore, the AFM magnetization) opposite to the FM magnet ization,\nthis quantity is negative. The total attenuation becomes ne gative also\n(an instability occurs), if\nj >eκγM 0|Hz|LAFM\nµBQ≡j0. (40)\nAtκ∼10−2,γM0∼1010s−1,|Hz| ∼102G,LAFM∼10−6cm,Q∼1\nwe have j0∼105A/cm2. Atjnear toj0weakly damping THz oscilla-\ntion can be obtained. At j > j 0, instability occurs which may lead to\neither self-sustained oscillations, or a dynamic stationa ry state. The lat-\nter disappears with the current turning off. To answer the que stion about\nfuture of the instability it is necessary to go out the scope o f the linear\napproximation.\nThespin-polarized currentcontributesalsototheoscilla tion frequency.\nAt the mentioned parameter values, we have |KMz| ∼1012s−1that is\ncomparable with the frequency in absence of the current. Thi s allows\ntuning the frequency by the current or excite parametric res onance by\nmeans of the current modulation.\n5 Current-induced spin injection effect\nNow let us discuss the injection mechanism effect [22, 23]. As mentioned\nbefore, the role of the mechanism is reduced to addition of an effective\nfieldP/γto the external magnetic field. At reasonable parameter valu es,\nthat field is much less than the exchange field HE, so that it does not\ninfluence directly the eigenfrequency (36). Nevertheless, that field can\nmodify substantially the contribution of the STT mechanism , because\nEq. (39) with (25) taking into account now takes the form\nKMz=µBQΛ\neLAFM(Hz+P/γ)j. (41)\nSuch a modification leads to substantial consequences. At Hz<0,P <\nγ|Hz|the instability threshold (40) is lowered, since |Hz|−P/γdifference\nappears now instead of |Hz|. If, however, P > γ|Hz|then the AFM\nmagnetization steady state\nMz=Hz+P/γ\nΛ(42)\nbecomes positive that corresponds to the parallel (stable) relative orienta-\ntion of the FM and AFM layers. In this case, the turning on curr ent leads\nto switching the antiparallel configuration (stated before hand by means\n10of an external magnetic field) to parallel one. With turning o ff current,\nthe antiparallel configuration restores.\nSincethementionedinjection-drivenfielddependsonthecu rrent(see(18)),\nthe instability condition (40) is modified and takes the form\nj0\n1+η< j <j0\nη, (43)\nwhereη=αsdκγM0τ,j0being defined with Eq. (40). In absence of\nthe injection mechanism, this condition reduces to (40). Un der rising\nrole of this mechanism we have lowering the instability thre shold, on the\none hand, and the instability range narrowing, on the other h and. At\nj > j 0/ηthe antiparallel configuration switches to parallel one. Th e\nrelative contribution of the injection mechanism is determ ined with η\nparameter. At typical values, αsd∼104,κ∼10−2,γM0∼1010s−1,\nτ∼10−12s, this parameter is of the order of unity, so that the injecti on\neffect may lower noticeably the instability threshold.\nNow let us return to the set of equations (26)–(31) and consid er the\nsecond mode describing with Eqs. (28)–(30). The current infl uences this\nmode by changing steady magnetization Mzdue to the injection effective\nfield effect (see (26)), while the STT mechanism does not influe nce this\nmode. A calculation similar to previous one gives the former dispersion\nrelation (35), but now\nω2\n0= 2γ2HEHAγHz\nγHz+P, (44)\nν=κγHEγHz\nγHz+P. (45)\nAtHz<0,P >|Hz|, thatcorrespondstocurrentdensity j > j0/η, the\ntotal attenuation becomes negative, while the frequency be comes imagi-\nnary, that means switching the antiparallel configuration t o parallel one.\nThus, current does not cause instability of that mode.\n6 Easy plane type antiferromagnet\nLetusconsiderbrieflythesituation whereAFMhaseasy-plan eanisotropy.\nWe take the AFM layer yzplane as the easy plane and xaxis as the (hard)\nanisotropy axis. The magnetic field, as before, is directed a longzaxis.\nWithout repeating calculations, similar to previous ones, we present\nthe results. A formal difference appears only in Eq. (36) for t he eigenfre-\nquencyω0of the first of the modes considered above. We have for that\nfrequency\nω0=/radicalBig\n(γHz+P)2+(KMz)2. (46)\nThe damping has the former form (37), so that the instability threshold\nis determined with former formula (43).\nIn absence of the current ( K= 0, P= 0) with not too small damping\ncoefficient κ, the frequency appears to be much less than damping, so\nthat the corresponding oscillations are not observed. The c urrent effect\nincreases the frequency, on the one hand, and decreases the d amping (at\nHz<0), on the other hand, that allows to observe oscillation reg ime.\n117 Fluctuation effect\nIt follows from Eq. (43) that the threshold current density i s proportional\ntotheexternalmagnetic fieldstrength |Hz|anddecreases withthefield. A\nquestion arises about permissible lowest limit of the total field|Hz|+P/γ.\nIn accordance with Eq. (25), such a limit may be the field which create\nmagnetization |Mz|comparable with its equilibrium value due to thermal\nfluctuations. Let us estimate this magnetization and the cor responding\nfield.\nThe AFM energy change in Vvolume under canting the sublattice\nmagnetization vectors with θ <180◦angle between them is\n∆E= ΛM2\n0(1−cosθ)V=1\n2ΛVM2\nz, (47)\nthe anisotropy energy being neglected compared to the excha nge energy.\nThe equilibrium value of the squared magnetization is calcu lated using\nthe Gibbs distribution:\n/an}bracketle{tM2\nz/an}bracketri}ht=∞/integraltext\n−∞M2\nzexp/parenleftbigg\n−ΛVM2\nz\n2kT/parenrightbigg\ndMz\n∞/integraltext\n−∞exp/parenleftbigg\n−ΛVM2\nz\n2kT/parenrightbigg\ndMz=kT\nΛV(48)\n(strictlyspeaking, themagnetization maybechangedwithi n(−2M0,2M0)\ninterval, however, Λ VM2\n0≫kT, so that the integration limits may be\ntaken infinity).\nTo observe the effects described above, the magnetization Mzwhich\nappears under joint action of the external field and the curre nt (see (25))\nshould exceed in magnitude the equilibrium magnetization /an}bracketle{tM2\nz/an}bracketri}ht1/2. At\nthe current density j=j0/(1+η) corresponding to the instability thresh-\nold, this condition is fulfilled at magnetic field\n|Hz|>/radicalbigg\nΛkT\nV(1+η)≡Hmin. (49)\nAt Λ∼104,η∼1,LAFM∼10−6cm and lateral sizes of the switched\nelement 10 ×10µm2we have V∼10−12cm3andHmin≈30 G at room\ntemperature. This limit can be decreased under larger eleme nt size.\nIt should be mentioned also about other mechanisms of AFM can ting.\nThe most known and studied one is the relativistic Dzyaloshi nskii–Moria\neffect(see, e.g.[21,27]). Besides, possible mechanismsha vebeendiscussed\ndue to competition between sdexchange and direct exchange interaction\nof the magnetic ions in the lattice [28]. At the same time, the re are no\nindications, to our knowledge, about measurements of canti ng in conduc-\ntive AFM. So, present theory is related to conductive AFM, in which the\nlattice canting is determined with external magnetic field.\n8 Conclusions\nThe obtained results show a principal possibility of contro lling frequency\nand damping of AMF resonance in FM–AFM junctions by means of s pin-\npolarized current. Under low AFM magnetization induced by a n external\n12magnetic field perpendicular totheantiferromagnetism vec tor, thethresh-\nold current density corresponding to occurring instabilit y is less substan-\ntially than in the FM–FM case. Near the threshold, the AFM res onance\nfrequency increases, while damping decreases, that opens a possibility of\ngenerating oscillations in THz range.\nAcknowledgments\nThe authors are grateful to Prof. G. M. Mikhailov for useful d iscussions.\nThe work was supported by the Russian Foundation for Basic Re -\nsearch, Grant No. 10-02-00030-a.\nReferences\n[1] J.C. Slonczewski. J. Magn. Magn. Mater. 159, L1 (1996).\n[2] L. Berger. Phys. Rev. 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Co., Amsterdam, 1968.\n[22] C. Heide, P.E. Zilberman, R.J. Elliott, Phys. Rev. B 63, 064424\n(2001).\n[23] Yu.V. Gulyaev, P.E. Zilberman, E.M. Epshtein, R.J. Ell iott, JETP\nLett.76, 155 (2002).\n[24] Yu.V. Gulyaev, P.E. Zilberman, A.I. Panas, E.M. Epshte in.\nPhysics — Uspekhi 52, 335 (2009).\n[25] Yu.V. Gulyaev, P.E. Zilberman, E.M. Epshtein, R.J. Ell iott. J. Exp.\nTheor. Phys. 100, 1005 (2005).\n[26] A.G.Gurevich, G.A.Melkov.Magnetizsation Oscillati ons andWaves,\nCRC Press, Boca Raton, FL, 1996.\n[27] V.E. Dmitrienko, E.N. Ovchinnikova, J. Kokubun, K. Ish ida. JETP\nLett.92, 383 (2010).\n[28] J.M. Robinson, P. Erd¨ os. Phys. Rev. B 6, 3337 (1972).\n14"    },    {        "title": "2009.01057v1.Effect_of_the_Optical_Pumping_and_Magnetic_Field_on_the_States_of_Phase_Separation_Domains_in_Eu__0_8_Cr__0_2_Mn_2O_5.pdf",        "content": "1 \n  \n \nEffect of the Optical Pumping and Magnetic Field on the States  of Phase \nSeparation Domains in Eu 0.8Ce0.2Mn 2O5 \n \nE. I. Golovenchitsa, B. Kh. Khannanova, and V. A. Saninaa,* \n \na Ioffe Institute, St. Petersburg, 194021 Russia  \n*e-mail: sanina@mail.ioffe.ru  \n \nThe effect of optical pumping and applied magnetic field on the characteristics of ferromagnetic \nlayers in  one-dimensional superlattices is studied. At low enough temperatures, these layers \ncorrespond to phase separation  domains in RMn 2O5 and R 0.8Ce0.2Mn 2O5 multiferroics. The \nformation of such domains occurs owing  to the charge ordering of Mn3+ and Mn4+ ions and to the \nfinite probability for eg electrons to tunnel between  these pairs of ions. The volume occupied by \nsuch superlattices is rather small, and they can be treated as isolated  ferromagnetic semiconductor \nheterostructures, spontaneously formed in the host crystal. The  sequences of ferromagnetic \nresonances related to the s uperlattice layers in Eu 0.8Ce0.2Mn 2O5 are studied. The  characteristics of \nthese resonances give information on the properties of such layers. For the first time, it is  \ndemonstrated that the optical pumping gives rise to a new metastable state of superlattices, which \ncan be  recovered by the magnetic field cycling to the state existing before the optical pumping. It \nis found that the  superlattices recovered by the magnetic field exist up to temperatures higher than \nthose in as -grown crystals.  \n \n \n      The presence of equal numbers of Mn3+ and Mn4+ manganese ions is a characteristic feature of \nRMn 2O5 multiferroic compounds ( Pbam symmetry), ensuring  the formation of charge ordering. \nMn4+ ions are surrounded  by oxygen octahedra and located in the layers  having the positions z = \n0.25c and (1 − z) = 0.75 c. These ions contain three t2g electrons and have an  empty degenerate \norbital doublet corresponding to the  eg state. Mn3+ ions contain three t2g electrons in their  3d shell \nand one eg electron in the orbital doublet.  These ions are located in the layers having the positions  \nz = 0.5 c, and their local environment in the form  of pentagonal pyramids has no center of \nsymmetry.  R3+ ions with the environment similar to that of Mn3+ are located in the z = 0 layers [1]. \nCharge ordering and  the finite probability of the eg electron transfer  between neighboring Mn3+–\nMn4+ ion pairs (double  exchange [2, 3]) are key factors determining the multiferroic  characteristics \nof RMn 2O5 compounds in the  range from low to room temperatures. Magnetic  ordering with a \ncomplicated structure occurs below  the Neel temperature TN ≈ 40−45  K, whereas the ferroelectric  2 \n ordering induced by the magnetic order  arises below the Curie temperature TC ≈ 35−40 K [4].  The \npolar order is mainly related to the exchange striction  induced by neighboring Mn3+ and Mn4+ ion \npairs  located along the b axis and having alternating ferromagnetic  and antiferromagnetic spin \norienta tions [5].  The eg electron transfer between Mn3+ and Mn4+ ions located in adjacent layers \nperpendicular to the c axis leads to the formation of local polar regions of phase  separation with a \ndifferent distribution of Mn3+ and Mn4+ ions as compared to that in the initial host crystal.  The \nstates resulting from phase separation were  studied in detail in pure RMn 2O5 multiferroic \nmanganites,  as well as in the doped ones, R0.8Ce0.2Mn 2O5 (R = Eu, Gd, Bi, Tb, and Er), having the \nsame Pbam  symmetry [6 –13]. In these  materials, the comparative  studies of dielectric and \nmagnetic properties, heat  capacity, X -ray diffraction, Raman light scattering [6,7], electric \npolarization [8 –11], and μ-SR [12, 13]  were carried out.  \n      Phase separation was observed in undoped  RMn 2O5 crystals, whereas their doping with Ce4+ \nions substituting R3+ ions leads to a significant increase in  the density of neighboring Mn3+–Mn4+ \nion pairs in  the planes perpendicular to the c axis. In  R0.8Ce0.2Mn 2O5 and in the z = 0 plane, the \nR3+ = Ce4+ + e reaction provides additional electrons. These  electrons transform Mn4+ ions to Mn3+ \nones in the z = 0.25c and 1 − z = 0.75 c planes. As a result, the number  of Mn3+–Mn4+ pairs and the \nnumber of phase separation  domains increase. At the same time, the phas e separation domains \noccupy a small part of the crystals  both in RMn 2O5 and in R0.8Ce0.2Mn 2O5 [6, 7]. Phase  separation \ndomains are formed in RMn 2O5 and in  R0.8Ce0.2Mn 2O5 in much the same way as those in  \nLaAMnO 3 (A = Sr, Ca, and Ba) compounds, which  also contain Mn3+ and Mn4+ ions [3, 14 –16]. \nThese  domains are formed because of the balance between  strong interactions occurring in the \nsubsystem of Mn  ions, such as the double exchange with the characteristic  energy of about 0.3 eV, \nthe Jahn –Teller interactio n (0.7 eV), and the Coulomb repulsion (1 eV). That  is why these domains \nin RMn 2O5 and in R0.8Ce0.2Mn 2O5 exist in a wide temperature range from low temperatures  to  3 \n Fig. 1. (a) Schematic image of 1D ferromagnetic superlattices (shaded areas) located in the bulk \nof host crystal (white background);  magnetic field H = 0. (b) Schematic image of one of such \nsuperlattices consisting of L N ferromagnetic layers perpendicular  to the c axis having different \ndensities of Mn3+−Mn4+ ion pairs and of eg electrons located in potential wells of different  depths \n(shaded areas corresponding to different energies E). In Eu 0.8Ce0.2Mn 2O5, the linear size of \ndomains with superlattices is  about 700 Å [6, 7].  \n \nthose above room temperature. At T < 60 K,  phase separation domains are isolated one -\ndimensional  (1D) superlattices consisting of ferromagnetic  layers containing Mn3+ and Mn4+ ions \nwith different  relative densities, as well as eg electrons changing their  charge (Fig. 1). These \ndomains manifest themselves as  a serie s of ferromagnetic resonances (FMR) related to  the \nindividual layers of the superlattices. The characteristic  features of such resonances reveal the \nproperties  of these layers and of the superlattice as a whole  [17–20]. \n     The aim of this work is to study the effect of optical  pumping, applied magnetic field, and \ntemperature on  the properties of 1D superlattices in Eu0.8Ce0.2Mn2O5  single crystals. Optical \npumping can affect the electron  density in the layers of superlattices and, thus,  result in its \nredistribution in these layers. This, in turn,  should change the ratio of the numbers of ions with  \ndifferent valences in the layers, which can be detected  by changes in the characteristics of the FMR \nlines.   \n4 \n      Eu0.8Ce0.2Mn 2O5 single crystals were grown b y spontaneous crystallization from the solution –\nmelt [21, 22]. They were platelets 1 −3 mm thick with an  area of 3−5 mm2. For FMR measurements, \nwe used a  transmission type magnetic resonance spectrometer  with a low -amplitude magnetic \nmodulation. The  measurements were performed in the temperature  range of 13 –300 K at \nfrequencies of 30 –40 GHz in an  applied magnetic field up to 2 T generated by an electromagnet.  \nThe cryostat with optical windows was  located in the microwave channel ensuring a uniform  \ndistribution of the microwave field close to the sample.  The microwave radiation (with the wave \nvector k) was  directed along the c axis perpendicular to the platelet  plane. The static magnetic field \nH was oriented along  the a axis of the crystal and perpendicular to the direction  of the microwave \nfield h. The measured FMR signals  were amplified by an SR530 lock -in amplifier.  Naturally \nfaceted single crystals were used. The symmetry  of the crystals and their composition were  \ndetermined by the X -ray pha se analysis and X -ray fluorescence  technique, respectively. Optical \npumping  was performed by a solid -state pulsed neodymium  laser LTIPCH -8 with the simultaneous \ngeneration of  the first (1.06 μm) and second (532 nm) harmonics.  The second harmonic \ncorrespond s to the 7F0−5D1 transition in Eu3+ ions. The 5D1 state is located within  the electron –\nphonon band of Mn ions in RMn 2O5, with the edge beginning at about 485 nm [23].  \n         Earlier, the FMR studies of layers in 1D superlattices  in a series of RMn 2O5 (R = Eu, Er, Tb, \nand Gd)  and R 1 – x CexMn 2O5 (R = Eu and Gd , x = 0.2, 0.25 ) crystals demonstrated  that the magnetic \nfields at which individual  FMR lines are observed differ only slightly and are  independent of the \nR ion type [17 –20]. In this case,  Er3+ and Tb3+ (unlike Gd3+) are strongly coupled to  the lattice. \nThis means that the FMR resonance fields  are determined by the internal fields far exceeding the  \nanisotropy fields characteristic of the superlattice layers;  the latter fields should depend on specific \nR ions.  Indeed, as mentioned above, the equilibrium state of 1D layers in the superlattices is \ndetermined by the balance  of interactions just within the subsystem of Mn  ions. In this case, the \nlayers in the 1D superlattices can  be treated as isotropic ferr omagnetic layers with ferromagnetic  5 \n boundaries between them, which do not put  obstacles in the way of the eg electron transfer between  \nthe layers related to the double exchange (Fig. 1b)  [17–20]. A model describing the spin -wave \nexcitations  in superlattice  layers was developed in these works.  The model is based on the \ndispersion equation for spin  waves in isotropic ferromagnetic films forming 2D  multilayers [24]. \nIt is also demonstrated that the FMR  lines in 1D superlattices are observed in individual frequ ency  \nbands separated by the ranges in which the  FMR is not observed. This occurs because the 1D  \nsuperlattices (similar to semiconductor superlattices  [25]) have a band structure consisting of \nminibands  separated by gaps. The frequency of 34.5 GHz, at  which  the measurements were \nperformed, is located  inside one of the main minibands in Eu 0.8Ce0.2 Mn 2O5 (29–36 GHz) [17].  \n    It was reported in [17, 18] that the formation of the  equilibrium state of superlattices for \nEuMn 2O5 and Eu0.8Ce0.2 Mn 2O5, in which a set of FMR lines was  observed, required the magnetic \ncycling (subsequently  increasing and decreasing the magnetic field) of as -grown  samples. The \ndynamically equilibrium  (ground) state of the superlattices is achieved after triple  magnetic \ncycling in the r ange of 0 –20 kOe. Note  that the ground state of the superlattices was recovered  \nupon the cooling of the sample to helium temperatures  after its heating to room temperature and \nits long (several  weeks) keeping at this temperature. The repeated  cycling of the magnetic field is \nnot necessary.  \n       For the 1 -mm-thick Eu0.8Ce0.2 Mn 2O5 sample with  an area of 5 mm2, the sequence of FMR \nlines in the  equilibrium state is shown in Fig. 2a. Generally speaking,  these lines correspond to \nthose measured earlier  in [17, 18]. We can see five FMR lines originating from different layers  of \nsuperlattices  schematically  shown in Fig. 1b . The most intense central L0 line  appears in the \napplied magnetic field corresponding to  the FMR at this frequency for an isotropic fe rromagnet  \nwith the g factor g = 2. Less intense lines, to the left  of the main resonance (L1 and L2) and to the \nright of  it (R1 and R2), are approximately symmetrically  located in lower and higher magnetic \nfields with  respect to that corresponding to the L0 line. In this  case, the most intense lines (L1, L0, \nand R1) are doubled.  At somewhat lower fields, less intense wide lines  are observed  (Fig. 2a). 6 \n  \nFig. 2. Distribution of intensities for the observed FMR lines in Eu 0.8 Ce0.2 Mn 2O5 at 34.5 GHz. \nThe sensitivity of  the amplifier is 5 μV; H || a . The magnetic field was varied at a rate of  1.2 \nkOe/min.  \n(a) As -grown sample after the third cycle of  the magnetic field increase (dynamically equilibrium \nstate of a 1D superlattice); T = 20 K.  \n(b) Dynamically equilibrium state of  the superlattice (black curve) is compared to the distribution \nof intensities for the FMR lines after optical pumping for 1 min by  15 ns pulses with a power of \n0.5 MW and a repetition rate of 10 Hz. Magnetic field H decreases (red curve); T = 22 K.  \n(c) Changes  in the state of the sample after 16 h of its natural heating up to 250 K and its subsequent \ncooling down to 17.5 K, with an increase  in H (black curve) and with its subsequent decrease     \n(red curve).  \n(d) Recovery of the state similar to the initial dynamically equilibrium  one (Fig. 2a) during the \nthird cycle of the magnetic field growth; T = 17.5 K.  \n  \n \n7 \n        In [17, 18], the FMR lines were observed at temperatures  below 60 K. Their intensity \ndecreases gradually  with an increase in the temperature. The resonant  magnetic fields for a \nsequence of FMR lines exhibit a  linear frequency dependence ωn = γn(H + Hneff). Here,  ωn are the \ncircular frequencies for n lines ( n = L2, L1,  L0, R1, and R2), γn are the gyromagnetic ratios, and  \nHneff are the internal effective fields responsible for the  gaps in ωn(H). The values Hneff are positive \nfor lines L1  and L2 and increase with the number L. For lines  R1and R2, the values Hneff are \nnegative. The g factors  for the lines R1and R2 are slightly smaller than 2,  whereas the g factors \nfor L1 and L2 lines exceed 2 [17,  18]. \n       Following the approach reported in [18 –20], we  analyze the properties of superlattice layers \nusing the  model of semiconductor heterostructures c ontaining  alternating L, 0, and R layers, in \nwhich Mn4+ and Mn3+ ions are treated as acceptor and donor impurities,  respectively. Analyzing \nthe characteristics of the  observed FMR lines related to the layers of superlattices,  we revealed the \nproperties of these layers. We  assume that the L0 layers contain equal numbers of  Mn4+ and Mn3+ \nions; i.e., these layers are fully compensated  semiconductors. In such layers, the Fermi  level is \nlocated in the center of the band gap, and the  insulating state arises in th e L0 layers [26]. The  \nabsence of free electrons in the L0 layers explains the  maximum intensity of the L0 lines in the \nequilibrium  states of superlattices (Fig. 2a). The L and R layers are  partially compensated \nsemiconductors: Mn4+ ions play the dominant role in the L layers, whereas the  Jahn–Teller Mn3+ \nions responsible for local structural  distortions prevail in the R layers. The R layers exhibit  \nthe deepest potential wells with the highest electron  density (Fig. 1b). According to [17 –19], the \nR layers  are characterized by negative Hneff values and by g factors  below 2. As mentioned above, \nfor L layers the values Hneff are positive and the corresponding g factors slightly  exceed 2. The  \ndynamic equilibrium resulting from the  balance of competing interactions (double exchange  and \nthe Jahn –Teller effect, increasing the electron  density in the layers of the superlattices, and the \nCoulomb  repulsion of electrons), as well as the charge neutrality  of the superlattice as a whole, \nsuggests that the  states of the layers should be mutu ally correlated and  form a periodic (L−O−R) 8 \n array. That is why the resonance  fields of individual FMR lines turn out to be  strictly fixed and \nalmost the same in RMn 2O5 and R0.8Ce0.2Mn 2O5 compounds with different R ions  [18–20].  \n         Let us now discuss  the effect of optical pumping on  the state of layers in the superlattices. \nPumping was  carried out for 1 min at T = 22 K by 15 -ns pulses with  a peak power of about 0.5 \nMW and a repetition rate of  10 Hz. In Fig. 2b, we show a sequence of FMR lines  (red poi nts on \nthe curve) measured at the same temperature a few minutes after the end of optical pumping.  These \nlines arise with an increase in the magnetic field at a rate of 1.2 kOe/min. These data are plotted \nagainst the background of the equilibrium state of the layers in the superlattice (black points and \nthe curve). The sequence of FMR lines is observed at the same resonance fields as that before \npumping, but the intensities of the L0 and R1 lines change. The intensities of other lines remain \nalmost unchange d. Before pumping, the L0 line is twice as intense as the R1 line (Fig. 2a), whereas \nthe intensities of these lines after pumping become equal (Fig. 2b). We suppose that, upon \nrelaxation of optical excitations in the electron -phonon band related to Mn ions , electrons \nappearing in the layers L0 and R1 of superlattices turn out to be in excess as compared to the initial \nelectron density formed because of doping with Ce4+ ions. Most of these electrons are localized at \nMn4+ ions (Mn4+ + e = Mn3+), transforming them to Jahn –Teller ions, which lead to local lattice \ndistortions in the corresponding layers, thus reducing the energy of superlattices. An increase in \nthe density of Jahn –Teller Mn3+ ions in the L0 layers results in the deepening of potential wells \ninsid e these layers, bringing their states closer to the R1 states. This suggests an intense exchange \nof electrons between the L0 and R1 layers at T = 22 K, which makes electron densities in these \nlayers equal (Fig. 2b). A new state of superlattices arises, whi ch is long -lived at low temperatures, \nsince it does not disappear for at least 20 min after the end of pumping.   \n      In Fig. 2c, we demonstrate a sequence of FMR lines for the same sample after 16 h of natural \nheating up to 250 K (after optical pumping) and its subsequent cooling down to 17.5 K (without \nnew optical pumping). Black and red points and lines correspond to increasing and decreasing \nmagnetic fields, respectively.  We can see that the intensities of the L2 and L1 lines still do not 9 \n change at inc reasing and decreasing  fields. At the same time, as the field increases, the intensities \nof the R1 and R2 lines increase steeply and exceed the intensity of the L0 line. Thus, the state \nforming after optical pumping is not recovered when the sample is cool ed down to 17.5 K after its \nheating and subsequent cooling. This means that the state arising after optical pumping with the \nsame populations of the L0 and R1 layers (Fig. 2b) is long -lived.  \n      It is natural to assume that the heating of the sample  after pumping leads to a certain \nredistribution in  the L0, R1, and R2 layers of excess electrons generated  by optical pumping, which \nleads to the reduction of  the superlattice energy. The heating of the sample  should result in an \nincrease in the kinetic en ergy of  electrons, thus causing their redistribution among the  layers owing \nnot only to tunneling between the L0 and  R1 layers (Fig. 2b) but also to the hopping conductivity,  \nwhich favors overcoming higher barriers at the  boundaries of other layers of supe rlattices. The \npopulation  of the deeper potential wells with the maximum  number of Mn3+ ions is the most \nprobable. This is the  case for the R2 layer, which is nearly empty at low temperatures  (Figs. 2a, \n2b). At the same time, two other  layers, R1 and L0, a re populated but with a lower  probability. \nStructural distortions also increase in  them owing to an increase in the number of Jahn –Teller Mn3+ \nions, for which the potential wells in these  layers are deeper. In this case, all excess electrons are  \nlocalized at Mn3+ ions. As a result, intense narrow  FMR lines related to the R2, R1, and L0 layers \nare observed (Fig. 2c, black points and curves). With a  subsequent decrease in the magnetic field, \nthe intensities  of the L0, R1, and R2 lines decrease steeply  (Fig. 2 c, red curves). In this case, the \nR2 line intensity  becomes nearly equal to its value in dynamical equilibrium  before pumping, \nwhereas the R1 line intensity  exceeds that of the L0 line, although only slightly.  Only an additional \nincrease in the magnetic field recovers a state similar to that corresponding to the  initial \nequilibrium state of the superlattice before  pumping (Fig. 2d), but with slightly lower intensities  \nof the FMR lines. The cycling of the magnetic field,  recovering the ferromagnetic orient ation of \nthe layers  in superlattices, enhances the double exchange, leading  to the tunneling of electrons 10 \n between the layers in  superlattices. It turns out that the resulting state exists  up to a higher \ntemperature than the equilibrium state  before optic al pumping (Fig. 3).  \nFig. 3. (a) Sequence of FMR lines corresponding to the layers in superlattices after optical pumping \nand cycling  in the applied magnetic field at various temperatures. Actually coinciding zeros of \nintensity are displaced relative to each other.  \n(b) Temperature dependence of intensities for the L0 and R1 lines.  \n        In Fig. 3, we show the set of the FMR lines in the  recovered equilibrium state after optical \npumping at  various temperatures (Fig. 3a) and the temperature  dependence of the intensities of the \nL0 and R1 lines in  this state (Fig. 3b). We can see that the equilibrium  states of superlattices after \noptical pumping exist up to  T ≈ 115 K, whereas before pumping, they are observed  up to T ≈ 60 K \n[12].  \n        As mentioned above, more intense and narrower  L1, L0, and R1 lines in Fig. 2 are doubled: \nthe low  intensity  wide satellites are observed near these lines at  somehow lower fields. This occurs \nbecause the initial  crystal contains not only Ce4+ ions but also Ce3+ ions,  although in a smaller \nnumber, which induce local  regions with strong lattice distortions because of the  presence of alone \n6s2 electron pairs in their outer  shells. We previously observed this effect in  R0.8Ce0.2Mn 2O5 (R = \nEr and Tb) [27]. In these regions  with a ce rtain number of localized electrons, superlattices  with \nslightly different parameters are formed.  \n11 \n        To summarize, it has been demonstrated for the first  time that the states of 1D superlattices \n(ferromagnetic  semiconductor heterostructures) in Eu 0.8Ce0.2Mn 2O5 multiferroics, which at \nsufficiently low temperatures  are phase separation domains, can be controlled by  optical pumping \nand applied magnetic field. Optical  pumping leads to the formation of a new long -lived  state of \nsuperlattices with a different  distribution of  Mn3+ and Mn4+ ions and of electrons changing their  \ncharge in these layers. The cycling of the applied magnetic  field in this new state of superlattices \nafter pumping  recovers the state with the distribution of intensities  of resonance lines similar to \nthat existing before  pumping. In this case, the intensities in the sequence  of resonance lines \nrecorded after pumping are slightly  lower than those before pumping. This is due to the  increased \ndensities of electrons and Mn3+ and Mn4+ ion p airs in the layers. However, these lines exist up to  \na higher temperature, which means that the new equilibrium  state recovered by the magnetic field \nafter optical pumping is more favorable in energy than the  state before pumping.  \n \nFUNDING  \nThis work was supported by the Russian Foundation for  Basic Research (project no. 18 -32-\n00241) and by the Presidium  of the Russian Academy of Sciences (program 1.4  “Topical \nProblems of Low -Temperature Physics”).  \n \nREFERENCES  \n \n1. P. G. Radaelli and L. C. Chapon, J. Phys.:  Condens.  Matter 20, 434213 (2008).  \n2. P. G. de Gennes, Phys. Rev. 118, 141 (1960).  \n3. L. P. Gor’kov, Phys. Usp. 41, 581 (1998).  \n4. Y. Noda, H. Kimura, M. Fukunaga, S. Kobayashi,I. Kagomiya, and K. Kohn, J. Phys.: \nCondens. Matter  20, 434206 (2008).  \n5. J. v an den Brink and D. I. Khomskii, J. Phys.: Condens.  Matter 20, 434217 (2008).  \n6. V. A. Sanina, E. I. Golovenchits, V. G. Zalesskii,  S. G. Lushnikov, M. P. Scheglov, S. N. \nGvasaliya,  A. 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Phys. \nSolid State 34, 56 (1992).  \n23. A. S. Moskvin and R. V. Pisarev, J. Low Temp. Phys.  36, 489 (2010).  \n24. A. G. Gurevich and G. A. Melkov, Magnetization Oscillations  and Waves (Fizmatlit, \nMoscow, 1994; CRC,  New York, 1996).  \n25. A. P. Silin, Sov. Phys. Usp. 28, 972 (1985).  \n26. B. I. Shklovskii and A. L. Efros, Electronic Properties of  Doped Semiconductors (Springer, \nHeidelberg, 1984).  \n27. B. Kh. Khannanov, E. I. Golovenchits, and V. A. Sanina,  Phys. Solid State 62, 660 (2020).  "    },    {        "title": "2008.05268v1.Accelerating_the_laser_induced_demagnetization_of_a_ferromagnetic_film_by_antiferromagnetic_order_in_an_adjacent_layer.pdf",        "content": "Accelerating the laser-induced demagnetization of a\nferromagnetic \flm by antiferromagnetic order in an adjacent layer\nI. Kumberg,1E. Golias,1N. Pontius,2R. Hosseinifar,1K. Frischmuth,1I. Gelen,1T.\nShinwari,1S. Thakur,1C. Sch u\u0019ler-Langeheine,2P. M. Oppeneer,3and W. Kuch1\n1Institut f ur Experimentalphysik, Freie Universit at Berlin,\nArnimallee 14, 14195 Berlin, Germany\n2Helmholtz-Zentrum Berlin f ur Materialien und Energie,\nAlbert-Einstein-Stra\u0019e 15, 12489 Berlin, Germany\n3Department of Physics and Astronomy,\nUppsala University, Box 516, SE-75120, Uppsala, Sweden\n(Dated: August 13, 2020)\nAbstract\nWe study the ultrafast demagnetization of Ni/NiMn and Co/NiMn ferromag-\nnetic/antiferromagnetic bilayer systems after excitation by a laser pulse. We probe the\nferromagnetic order of Ni and Co using magnetic circular dichroism in time-resolved pump{probe\nresonant X-ray re\rectivity. Tuning the sample temperature across the antiferromagnetic ordering\ntemperature of the NiMn layer allows to investigate e\u000bects induced by the magnetic order of the\nlatter. The presence of antiferromagnetic order in NiMn speeds up the demagnetization of the\nferromagnetic layer, which is attributed to bidirectional laser-induced superdi\u000busive spin currents\nbetween the ferromagnetic and the antiferromagnetic layer.\n1arXiv:2008.05268v1  [cond-mat.mes-hall]  12 Aug 2020I. INTRODUCTION\nMagnetic recording and storage media will face major challenges in the near future re-\ngarding energy e\u000eciency and data transfer speed due to increased demand and continuing\ngrowth of data storage volume. Much could be gained by switching from \feld-induced\nmanipulation of magnetic order, where speed is limited by spin precession dynamics1{3, to\nmanipulation by ultra short light pulses4{11. To explain ultrafast magnetization dynamics\nfollowing the excitation by an ultrashort laser pulse, a number of di\u000berent approaches are\nbeing discussed. One can distinguish between local mechanisms such as scattering with\n(quasi-)particles5,6,12{14and non-local mechanisms such as transport by superdi\u000busive spin\ncurrents15,16. In heterostructures, the latter may dominate the observed dynamic behavior\nsince they allow transfer of angular momentum between di\u000berent ferromagnetic layers, which\nmay even lead to an entirely di\u000berent dynamic behavior as compared to isolated layers17,18.\nTransient superdi\u000busive spin currents are also expected to occur at the interface between\nan antiferromagnet (AFM) and a ferromagnet (FM). Although unpolarized in the interior\nof an AFM layer, superdi\u000busive currents excited by a laser pulse become spin-polarized in\nthe vicinity of the interface to an FM layer because of spin-polarized interface re\rection and\ntransmission. In this work, we present a study on the ultrafast optical demagnetization of\nan FM/AFM layered system. We show that the presence of antiferromagnetic order in an\nadjacent layer accelerates the demagnetization of the FM layer, which we attribute to the\nexchange of superdi\u000busive spin currents between the two layers.\nFM/AFM layered systems are well known in device design, as the exchange interaction\nbetween the two layers leads to an exchange bias (EB) e\u000bect, resulting in the magnetic\npinning of the FM layer19,20. NiFe/NiO as an FM/AFM system has been already studied\nby time-resolved magneto-optical Kerr e\u000bect in the pioneering works of Ju et al. where an\nunpinning of the interface spins within times \u00141 ps was reported21{23., More recently, the\ntorque on the FM spins in an FM/AFM bilayer after laser excitation was investigated24,\nsuggesting, like the works of Ju et al., that the magnetization in such a system is switchable\nby a laser pulse. In extension of these previous investigations, which were mainly targeting\nthe evolution of EB after laser excitation and the correlated precessional motion of the FM\nmacrospin, we investigate here the ultrafast processes in an FM/AFM bilayer system after an\noptical excitation pulse. Using Ni and Co as ferromagnets and NiMn as an antiferromagnet,\nwe have grown epitaxial FM/AFM bilayer samples with easily accessible N\u0013 eel temperature\n2TNof the AFM layer. We employ resonant X-ray magnetic circular dichroism in re\rectiv-\nity to probe the magnetization with elemental resolution. Comparing the demagnetization\nbehavior above and below TNallows us then to investigate the in\ruence of the magnetic\norder in the NiMn layer on the demagnetization of Ni and Co following the excitation with\na femtosecond laser pulse. We observe a signi\fcantly faster demagnetization for the same\namount of demagnetization of the FM layer if the adjacent NiMn layer is antiferromagneti-\ncally ordered. We discuss this in terms of the exchange of angular momentum through the\nFM/AFM interface by superdi\u000busive spin currents.\nII. EXPERIMENT\nThe samples are grown by molecular beam epitaxy in our home lab at the Freie Uni-\nversit at Berlin. All \flms are prepared on Cu(001) single crystals, which were cleaned by\nmultiple Ar+sputtering and annealing cycles. The surface integrity is veri\fed by LEED\nand the sample's cleanliness by Auger electron spectroscopy. The materials are thermally\nevaporated by electron bombardment from a rod of 99.998% purity for Ni and Co, and from\nMn \rakes with 99.98% purity in a Ta crucible. During evaporation the sample is kept at\nroom temperature and deposition rates of about 1 monolayer (ML) per minute are used. The\npressure is kept below 1 \u000210\u00009mbar while evaporating, with a base pressure of \u00196\u000210\u000010\nmbar. Deposition is monitored by medium-energy electron di\u000braction to ensure layer-by-\nlayer growth. Composition and thickness are additionally checked and calibrated by the\nsignal intensity in Auger electron spectroscopy. After preparation, the samples are capped\nby 20 ML Cu with purity of 99.99%, evaporated from a Ta crucible, to prevent sample oxi-\ndation during transport. We prepared a 20 ML Cu /15 ML Co /20 ML Ni 31Mn69/Cu(001)\nand a 20 ML Cu /12 ML Ni /14 ML Ni 38Mn62/Cu(001) sample. For the AFM layer we chose\nNiMn, an alloy which orders antiferromagnetically around equiatomic concentrations. Fur-\nthermore, EB can occur both in in-plane and out-of-plane geometries25,26and a vanishing of\ndirect exchange coupling between out-of-plane-magnetized FM layers across NiMn indicates\na spin reorientation, possibly between an in-plane and an out-of-plane AFM spin structure\nat temperatures about halfway between the N\u0013 eel temperature and the onset of EB26. For\nthe present investigation we chose a stoichiometry close to Ni 40Mn60, the Ni to Mn ratio for\nwhich we have a detailed investigation of the N\u0013 eel-Temperature26. Both, the thickness and\nthe concentration, were chosen to have T Naround 360 K25{27. For the FM layer we selected\n3Ni with an out-of-plane and Co with an in-plane easy axis of magnetization.\nThe experiments are carried out at the femtoslicing facility at beamline UE56/1 ZPM of\nthe synchrotron radiation source BESSY II in Berlin. This facility provides ultra-fast soft\nX-ray pulses for time-resolved experiments28. The setup allows to measure the transmitted\nor re\rected intensity of the circularly polarized X-ray probe pulse following the pump laser\npulse. The system measures at 6 kHz repetition rate, alternating between X-ray probe\npulses with and without a leading laser pump pulse with variable pump{probe delay time.\nThis measurement scheme allows for quasi-simultaneous recording of the pumped and the\nunpumped signal. The X-ray and laser spot sizes are 140 \u0016m\u000240\u0016m and 1500 \u0016m\u0002200\n\u0016m, respectively. Both are co-propagating with an angle of 1 \u00002\u000ebetween them to separate\nthem again after re\rection at the sample. By switching the magnetization of the sample by\nan external magnetic \feld, we make use of the X-ray magnetic circular dichroism (XMCD)\nin re\rection29as an element-resolved probe of the magnetization. All measurements are\nperformed in saturation conditions under applied magnetic \felds of \u0006300 mT for Co/NiMn\nand\u0006400 mT for Ni/NiMn. The laser pulse has a temporal full width at half maximum\n(FWHM) of 60 fs and the X-ray pulse of 100 fs, with a resolving power of E/\u0001E = 500.\nTo \fnd the optimum measurement conditions, a range of static re\rectivity scans are\nperformed, identifying the angle with the highest magnetic contrast for a given photon\nenergy while still providing enough photon intensity for the slicing measurements. Energies\naround the L3edges of Ni and Co were checked and chosen such that the anticipated data\nacquisition time is minimized. For Co the optimum conditions are found at 7\u000eangle of\nincidence with respect to the surface and 775.6 eV photon energy, indicated in the re\rectivity\nand energy scans presented in Fig. 1. In a similar fashion, the optimum conditions for Ni\nare identi\fed at 6 :5\u000eangle of incidence and a photon energy of 845.0 eV.\nThe range of pump \ruences investigated is chosen such that the system completely de-\nmagnetizes at the highest \ruence and shows a considerable demagnetization at the lower\nend, in order to study the in\ruence of di\u000berent excitation strengths on the system. The\ncobalt system is thus investigated with incident \ruences between 10 and 50 mJ/cm2, cov-\nering a demagnetization range from 20 to 100%, as shown in Fig. 2. For Ni/NiMn we\nrecorded data for two di\u000berent incident \ruences of 15 and 46 mJ/cm2, resulting in 60 and\n90% relative demagnetization, respectively. From a layerwise absorption calculation the\nsurface-transmitted \ruence is estimated to be about 15-20 % of the incident \ruence, see\nappendix. To investigate the e\u000bect of the magnetic order of the antiferromagnetic layer on\n4(a)\n(b)\n6 8 10 12 1401234567\n+ M\n- M=  7°\nCu (001)NiMn (3.1 nm)Co (2.6 nm)Cu (3.1 nm) \n1.6 nm\n800 nm\n765 770 775 780 785\nIntensity (arb. units)\nInt. (arb. units)\n(deg.)Ephoton (eV)\nEphoton = 775.6 eVFIG. 1. (a) Schematic representation of the Co/NiMn sample. The measurements are performed in\nre\rectivity with an 800 nm pump and 775.6 eV ( \u00191:6 nm) probe pulse. (b) Optimum measurement\nconditions are found by taking spectra at di\u000berent angles and comparing \u0012\u00002\u0012scans at di\u000berent\nphoton energies. The most e\u000ecient conditions with respect to a minimized data acquisition time\nfor Co are found at \u0012= 7\u000e, highlighted by a vertical dashed line in the \fgure. The optimum photon\nenergy for Co is identi\fed as E= 775:6 eV, highlighted by the dashed vertical line shown in the\ninset of the \fgure, where a spectrum recorded at 7\u000eincidence is shown.\n5Time delay (ps)XMCD\n-0.25      0.00       0.25      0.50      0.75       1.00      1.25      1.50      1.750.5\n0.4\n0.3\n0.2\n0.1\n0.010 mJ/cm2\n20 mJ/cm2\n30 mJ/cm2\n40 mJ/cm2\n50 mJ/cm2FIG. 2. Time delay traces of the Co demagnetization in Co/NiMn. Here the XMCD at the Co L3\nedge as a function of the delay at 390 K for di\u000berent laser \ruences is shown. With increased laser\n\ruence, it takes longer to reach the minimum magnetization, and also the remagnetization takes\nlonger. Complete demagnetization is reached with 50 mJ/cm2incident laser \ruence.\nthe demagnetization dynamics, two base temperatures of 80 and 390 K for Co/NiMn and\n80 and 360 K for Ni/NiMn were chosen, respectively. While the FM layer keeps its mag-\nnetization, the NiMn layer undergoes a phase transition from AFM at low temperatures to\nparamagnetic at the higher base temperature.\nIII. RESULTS\nFig. 2 shows delay traces of the time-dependent XMCD of the Co/NiMn sample, mea-\nsured at the Co L3edge, for 390 K sample temperature at di\u000berent incident pump \ruences.\nThe XMCD is de\fned as the asymmetry of the intensity of the re\rected beam with respect\nto the direction of the applied magnetic \feld. Upon excitation by the laser pulse, a de-\ncrease of the XMCD signal to a minimum in XMCD before 600 fs is observed in all cases.\n6With increasing \ruence the minimum shifts to longer times and the total change in XMCD\nincreases, following the expected behavior for ferromagnetic thin \flms after a higher heat\nload8. In order to compare the dynamics in Ni/NiMn and Co/NiMn, two measurements at\ndi\u000berent \ruences that show about the same relative demagnetization are presented in Fig.\n3. In the Co/NiMn sample [Fig. 3 (a)], the XMCD before time zero is virtually unchanged\nupon increase of the sample temperature due to the relatively high T Cof Co\u00181300 K at\n15 ML thickness30, decreasing only from 0.42 to 0.41. This is in contrast to the Ni/NiMn\nsample, where the XMCD decreases from 0.32 at 80 K to 0.18 at 360 K, shown in Fig. 3 (b)\nat negative delay times.\nWhile the static measurement does not reveal a di\u000berence between the two tempera-\ntures for Co/NiMn, the dynamic XMCD shows a di\u000berent time evolution. The decrease\nof magnetization of the Co layer at the lower base temperature, where the NiMn layer is\nantiferromagnetically ordered, is distinctly faster compared to the higher base temperature.\nTo evaluate this quantitatively, all delay traces recorded were \ftted using the sum of two\nexponentials describing a fast demagnetization and a subsequent slower initial remagneti-\nzation, respectively, from which the de- and remagnetization time constants are evaluated.\nThe sum of the exponentials is convoluted with a Gaussian function of FWHM = 120 fs to\naccount for the experimental resolution. The \ft function corresponds thus to\nM\nM0(t) =g(t)\n\u0010\nC\n+h\na(e\u0000t\u0000t0\n\u001cd\u00001)\u0000b(e\u0000t\u0000t0\n\u001cr\u00001)i\n\u0012(t\u0000t0)\u0011\n; (1)\nwith the initial XMCD C, the amplitudes for de- and remagnetization aandb, the de-\nand remagnetization time constants \u001cdand\u001cr, the Gaussian pro\fle g(t) of the instrumental\nresolution, and the Heaviside step function \u0012att0. The curves are \ftted with di\u000berent\namplitudes for de- and remagnetization, since the complete remagnetization dynamics may\nnot be adequately described by a single exponential. However, for the time window 2 ps\nused for the \ftting, the remagnetization, which is anyways not in the focus of the present\ninvestigation, can be well reproduced with a single exponential. As the relative timing of\npump and probe pulses shows slow drifts over the time period of this study, the curves\nplotted in this work are shifted by t0obtained from the \fts, in order to have a common zero\nfor all graphs.\nA \ruence-dependent comparison of the demagnetization times and amplitudes for dif-\nferent temperatures is presented in Fig. 4. For both base temperatures and samples, the\n7Time delay (ps)XMCD XMCD0.6\n0.4\n0.2\n(a)\n(b)Co/NiMn\nNi/NiMn80 K\n390 K\n0.4\n0.3\n0.2\n0.1\n0.080 K\n360 K\n-0.4       -0.2        0.0        0.2        0.4        0.6       0.8        1.0 FIG. 3. Selected time delay traces for the demagnetization of Co and Ni in Co/NiMn and Ni/NiMn,\nrespectively. Red circles indicate the high and blue squares the low base temperature, as indicated\nin the legends; the \flled symbols show data for the Co/NiMn sample and open symbols for the\nNi/NiMn sample. Panel (a) shows delay traces of the demagnetization of Co at 30 mJ/cm2laser\n\ruence at 80 and 390 K, panel (b) the demagnetization of Ni/NiMn after laser excitation with 15\nmJ/cm2laser \ruence.\nquench of the magnetization, depicted in Fig. 4 (a) as change of XMCD \u0001 XMCD in the same\nabsolute values as on the vertical axes in Figs. 2 and 3, increases with \ruence. Except for\nthe highest \ruence, this is true also for the demagnetization time \u001cd. For the higher sample\ntemperature, demagnetization times are longer and the relative demagnetization stronger,\nsimilar to the observations in Ref.31. In absolute values, i.e., in XMCD asymmetry \u0001 XMCD ,\nthe cold Ni/NiMn sample demagnetizes more than the warm one, but when considering the\n8∆XMCD (fs)τd\n10                20                30                 40                50200\n180\n160\n140\n120\n100\n80\n60\n400.40\n0.35\n0.30\n0.25\n0.20\n0.15\n0.10\nFluence (mJ/cm2)(a)\n(b)\nCo 80 K\nCo 390 K\nNi  80 K\nNi  360 K\nFIG. 4. (a) Demagnetization amplitude in XMCD asymmetry after laser excitation for Co (\flled\nsymbols) and Ni (open symbols), at 80 K and 360 K for Ni and 390 K for Co, respectively, as\ndenoted in the legend in panel (b). The amplitude of the demagnetization follows a linear trend at\nlower \ruences and is slightly higher for elevated sample temperatures for cobalt, considering the\nsame \ruence. (b) Magnetization decay time \u001cdversus \ruence. Generally, a higher base temperature\nleads to longer demagnetization times with the largest di\u000berence for Co/NiMn around 30 to 40\nmJ/cm2laser \ruence.\nrelative amount of magnetization quenched at the lower temperature it is actually less than\nat the higher temperature. When the Co/NiMn system fully demagnetizes, which is the\ncase for 50 mJ/cm2at 390 K, the demagnetization is again faster.\nTo obtain more information on the role of the antiferromagnetic order in the NiMn layer,\nwe compare the magnetization decay times for Co and Ni as a function of the reduction of\nthe XMCD signal in Fig. 5. For Co/NiMn, the \ruence range covers similar demagnetization\n9(fs)τd\n∆XMCD200\n180\n160\n140\n120\n100\n80\n60\n40\n0.35 0.30 0.25 0.20 0.15 0.10Co 80 K\nCo 390 K\nNi  80 K\nNi  360 K\nFIG. 5. Demagnetization times \u001cdvs. demagnetization amplitude for Co/NiMn (\flled symbols)\nand Ni/NiMn (open symbols) at 80 K (blue) and 360 K for Ni and 390 K for Co, respectively (red).\nThe demagnetization time, considering a similar quench of XMCD asymmetry, shows a di\u000berence\nfor the two temperatures measured. In Co/NiMn, for both base temperatures an XMCD of 0.42 for\n80 K and 0.41 for 390 K was measured before time zero, while for Ni/NiMn an XMCD asymmetry\nof 0.32 for 80 K and 0.18 for 360 K was observed at negative delay times.\nat both temperatures, while for Ni/NiMn, where the static di\u000berence in magnetization is\nalready quite sizeable, this is not the case. We observe a di\u000berence of \u001cdof up to 50 fs for\nCo/NiMn for the two temperatures around a demagnetization corresponding to a change\nof the XMCD of 0.3. The di\u000berence, albeit small, is still greater than the experimental\nerror and the uncertainty from \ftting. The data for Ni/NiMn suggest that the di\u000berence\nin demagnetization times for demagnetization amplitudes around 0.15{0.20 could be even\ngreater than that.\n10IV. DISCUSSION\nThe comparison of the demagnetization time constants at di\u000berent temperatures for equal\ndemagnetization amplitudes (Fig. 5) highlights the role of the AFM order in the NiMn layer\nfor the speed of demagnetization of the FM layer. The results show that the demagnetization\nrate of the FM layer is higher if the AFM layer exhibits magnetic order. It makes the\ninclusion of new dissipation channels introduced by the adjacent AFM layer necessary for the\ndescription of the demagnetization dynamics. We suggest in the following that superdi\u000busive\nspin currents15{18are responsible for the faster demagnetization of the FM layer at the lower\ntemperature, i.e., in the presence of antiferromagnetic order in the NiMn layer.\nTransport by superdi\u000busive currents plays an important role for the ultrafast demagneti-\nzation as shown, for example, in Refs.16{18, where a contribution of transport to an adjacent\nlayer to the ultrafast demagnetization has been reported. Before we discuss superdi\u000busive\nspin currents in the sample, we take a short look at the simulated di\u000berential absorbed laser\n\ruence across the Co/NiMn bilayer, presented in the Appendix, Fig. 6. The energy absorbed\nin the Co layer is about the same as the one absorbed in the NiMn layer, with a di\u000berence\nof only about 1{2%. Hot electrons will thus be excited both in the Co and in the NiMn\nlayer. Excitations in the FM Co layer give rise to a superdi\u000busive spin current that enters\nthe NiMn layer as well as the cap layer. Similarly, the excitation of the NiMn layer will lead\nto superdi\u000busive hot electrons that will be transported through the interface into the FM\nlayer and the Cu substrate. We will discuss in the following how these superdi\u000busive spin\ncurrents contribute to the observed di\u000berence in FM-layer demagnetization times below and\nabove the AFM ordering temperature.\nThe decisive aspect for the demagnetization of the FM layer is the dissipation of angular\nmomentum by the exchange of spin currents with the NiMn layer. Although spin mixing is\nenhanced after pumping, the superdi\u000busive spin current carried by hot electrons from the\nFM layer consists mainly of spin-majority species15. BelowTN, we assume the magnetization\nin the FM layer to be collinear with the magnetization axis of the sublattices of the ordered\nAFM NiMn layer. This con\fguration results in an injection of the spin-polarized electrons\ninto empty states of the AFM of the same spin character. In the high-temperature param-\nagnetic state, though, the magnetic moments in the NiMn layer are disordered, resulting in\na spin-character mismatch due to the random relative orientation of the spin moments in\nthe NiMn layer with respect to the FM-aligned spins in Co. Under such conditions, the spin\n11penetration depth becomes strongly reduced to about one nanometer32, leading to re\rection\nof carriers at the interface. Conversely, spin-polarized electron penetration into magnetic\nsystems with collinear spin orientation is much larger33. In the AFM-ordered state with a\ncollinear spin structure, the superdi\u000busive majority-type spin current from the FM layer can\nthus propagate much farther into the NiMn layer, making it a more e\u000ecient sink for angular\nmomentum. This leads to shorter demagnetization times for the FM layer adjacent to the\nordered AFM layer, as is observed experimentally.\nIn turn, the superdi\u000busive current of hot electrons originating in the NiMn layer will be\ntransported through the interface into the FM layer. Likewise, for collinear AFM spin order\nin the NiMn layer, interface re\rection is reduced compared to the case of a magnetically\ndisordered NiMn layer. Despite the absence of a macroscopic magnetization in an AFM,\na spin current representing the minority species of the FM layer will be preferentially en-\ntering the FM layer, as there are more unoccupied minority-spin states available, thus also\naccelerating its demagnetization.\nWe now argue that the change of temperature alone, without considering the ordering\ntransition in the NiMn layer, cannot explain our experimental observation. The e\u000bect of\ntemperature has not yet been discussed in detail in the model of superdi\u000busive transport\n(see15and34), where the FM layer is considered fully magnetic. In general, spin lifetimes and\nspin-dependent velocities of the excited electrons change with temperature. This could play\na role in the Ni/NiMn sample, where the asymmetry in the spin lifetimes and spin velocities\nwill be reduced at 360 K, in the same way as the static XMCD. For Co, though, a change\nof this asymmetry is negligible since the change of the experimentally measured XMCD is\nmarginal between the two temperatures.\nIn addition, the excited electron lifetime decreases at higher temperatures due to enhanced\nscattering probabilities. Furthermore, slightly di\u000berent states will be addressed at di\u000berent\ntemperatures, depending on the slope of the band structure. This could in\ruence the electron\nvelocity in either direction, but this is also considered a negligible e\u000bect here as the change\nin temperature is only about 300 K. Following this line of argumentation, we do not expect\na signi\fcant change in the demagnetization time for equal demagnetization as a result of\nthe di\u000berent base temperature in the FM.\nNext we exclude local mechanisms for angular momentum dissipation as a possible ex-\nplanation. Local demagnetization can be caused by Elliott-Yafet electron{phonon spin-\rip\nscattering8or ultrafast magnon generation14,35{37. In the work of Schellekens et al.38, lo-\n12cal demagnetization due to Elliott-Yafet spin-\rip scattering has been proposed as the main\ndriving mechanism that could be orders of magnitude stronger compared to spin transport.\nHowever, the relative contributions of the microscopic processes strongly depend on the sys-\ntem and properties under investigation14,39. While electronic-structure-based calculations of\nthe Elliott-Yafet electron{phonon spin-\rip scattering predict that this can only give a small\ncontribution to the laser-induced demagnetization40,41, others suggested that a feedback\nmechanism of the magnetization change and the spin-dependent chemical potential on the\nband structure would lead to an additional reduction of the exchange splitting42,43. However,\nsuch a collapse of the exchange splitting has not been found in femtosecond spin-polarized\nphotoemission experiments36and also not in extreme ultraviolet transversal magneto-optical\nKerr e\u000bect (TMOKE) measurements14on Co. In any case, using the expressions given in\nRef.40one can calculate that an increase in temperature from 80 to 390 K leads to an in-\ncrease in the Elliott-Yafet spin-\rip scattering by about a factor of 2.5 due to the increased\nphonon population. This corresponds to a faster electron{phonon demagnetization of Co at\nhigher temperatures, which is, however, opposite to the here-measured trend, which shows\naslower demagnetization of Co in Co/NiMn at 390 K.\nMagnetization dissipation by transversal spin excitations, i.e., ultrafast magnon generation14,36,37,\nis another mechanism we exclude, based on the timescale of our observations. Analogous to\nthe case of ultrafast magnon generation followed by spin pumping into an adjacent layer44,45,\nmagnons excited in the FM layer by ultrafast carriers could couple to magnon modes in the\nordered AFM layer, a coupling that would be absent in the disordered case. This could also\nlead to a faster demagnetization of the FM layer when in contact with the ordered AFM\nlayer. However, the timescale of the initial energy transfer process from hot electrons to hot\nmagnons in an FM has not been de\fnitely established. E\u000bects of hot magnon excitation\nhave been detected in laser-excited Co at already 100 fs36. The timescale to couple the hot\nFM magnons to magnon modes in the AFM is neither precisely known. Magnon transport,\nassuming even a quite high magnon group velocity of \u0018104m/s46, would only result in an\nangular momentum transfer over 1 nm in 100 fs, which is not enough to explain the faster\ndemagnetization timescale seen in Fig. 5. Thus, although the angular momentum transfer\nprocess, starting with hot magnons in the FM layer and their subsequent di\u000busion into the\nadjacent layer, is possible, it is expected to occur on a longer timescale than transfer by\nsuperdi\u000busive spin currents.\nWe \fnally note that, although our data from the Ni/NiMn sample do not allow the\n13comparison of demagnetization times for the two temperatures at equal demagnetization\namplitudes, we expect the same e\u000bect as in the Co/NiMn sample. This is motivated by the\nNi data presented in Fig. 5, where one pair of data points of almost equal magnetization\nquench shows a much faster rate at the lower temperature.\nV. CONCLUSIONS AND OUTLOOK\nIn conclusion, we infer from our experimental \fndings superdi\u000busive spin transport to\nplay a signi\fcant role for the demagnetization of an FM/AFM system. A general temper-\nature dependence of demagnetization as well as the contribution of Elliott-Yafet processes\ncannot account for the observed temperature dependence of the demagnetization. More-\nover, a magnon-related contribution to the demagnetization is expected to occur on a longer\ntimescale. Therefore, we attribute the accelerated demagnetization at low temperatures in\nCo/NiMn and Ni/NiMn to the presence of AFM order in the NiMn layer.\nSuperdi\u000busive spin currents can be exchanged more easily between the FM layer and the\nadjacent AFM-ordered NiMn layer, while the disordered NiMn layer acts as a barrier for\nthe penetration of superdi\u000busive currents. The AFM-ordered NiMn layer thus represents\non the one hand a sink for majority spin current from the FM layer, while on the other\nhand it injects minority spin current into the FM layer. Both accelerates the FM-layer\ndemagnetization and explains the experimental observations.\nTo accelerate the ultrafast demagnetization by AFM spin order in an adjacent layer\nmay become important for the ultrafast optical manipulation of magnetic order in magnetic\nrecording or spintronic devices. Rotating the spin axis of the AFM layer between perpen-\ndicular and parallel to the FM-layer magnetization, for example by staggered spin-orbit\ntorques47, could be a means of tuning the demagnetization speed of an FM layer. It might\n\fnally be interesting to see, possibly by time-resolved magnetic linear dichroism, how in a\nreverse experiment the presence of FM spin order in an adjacent layer in\ruences the optical\nquench of the magnetic order in an AFM layer.\nACKNOWLEDGMENTS\nThis work was supported by the Deutsche Forschungsgemeinschaft via the CRC/TRR\n227 \\Ultrafast Spin Dynamics\", projects A03 and A07, and by the Swedish Research Council\n14(VR). We further acknowledge support from the K. and A. Wallenberg Foundation (grant\nNo. 2015.0060) and the Swedish National Infrastructure for Computing (SNIC). We thank\nthe Helmholtz-Zentrum Berlin for the allocation of synchrotron radiation beamtime.\nAppendix A: Di\u000berential Absorption\nAbsorption (% nm-1)\nDistance from surface (nm)3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\n0.0       2.5       5.0       7.5      10.0      12.5      15.0      17.5\nCu\nCapCu\nSubstrateCo \nNiMn\n7.0 %\n6.9 %7.1 %\n6.7 %total absorbed\n      power\n16.6 %\n16.1 %nNiMn = nNi\nnNiMn = nMn\nFIG. 6. Di\u000berential absorption over sample depth for the Co/NiMn sample. The optical constants\nfor NiMn are estimated to lie between the ones for bulk Ni and for bulk Mn. To estimate the\nabsorbed energy in each layer, two absorption pro\fles are shown, one with the optical constants of\nNi for the NiMn alloy (red) and one with the optical constants of Mn (blue). The numbers indicate\nthe fraction of absorbed pump-pulse \ruence in each layer.\nWe simulate the layerwise absorption of the pump pulse in the Co/NiMn sample using the\nmatrix formalism described in Ref.48. 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We measure the SAW magneto-transmission at r oom temperature as a function of\nfrequency, external magnetic field magnitude, and orientat ion. Our data are well described by a\nmodified Landau-Lifshitz-Gilbert approach, in which a virt ual, strain-induced tickle field drives the\nmagnetization precession. This causes a distinct magnetic field orientation dependence of elastically\ndriven FMR that we observe in both model and experiment.\n1Inverse magnetostriction, or magnetoelasticity, enables the con trol of the magnetization in\nferromagnetic materials via elastic stress [1–6]. This spin-mechanica l interaction prevails at\nradio frequencies (RF), so that magnonic and phononic degrees of freedom become coupled, as\ndiscussed theoretically [7–10]. We here apply this concept to a surfa ce acoustic wave (SAW)\ntraversing a ferromagnetic thin film. Due to magnetoelastic coupling [11], the elastic defor-\nmation periodic in time and space results in a change of the magnetic an isotropy, which in\nturn exerts a torque on the magnetization. Since typical SAW freq uencies range from a few\nMHz to several GHz, SAW-based RF spin-mechanics should enable th e study of magnetization\ndynamics such as ferromagnetic resonance, driven only via RF elast ic deformation, not via ex-\nternally applied RF magnetic fields. The interaction of SAWs and ferro magnetic thin films has\nbeen studied experimentally by several groups [12–15]. Based on SA W magneto-transmission\nmeasurements with the static external magnetic field applied either perpendicular or parallel\nto the SAW propagation direction, these authors suggested that a magnetoelastic interaction\nmost probably was the dominant interaction mechanism. However, c onclusive evidence for the\noccurrence of an elastically driven, acoustic FMR has proven elusive , and important aspects of\nthe interaction mechanism still await explanation, as stated by Wiege rt as recently as 2002 [16].\nOur experimental study of SAW-based RF spin mechanics as a funct ion of magnetic field mag-\nnitude and orientation provides clear evidence for elastically excited , acoustic ferromagnetic\nresonance. Our findings thus extend the application and understa nding of magnetoelastic in-\nteraction phenomena in the RF regime.\nWe study the hybrid device depicted schematically in Fig. 1(a). It con sists of a 50nm\nthick polycrystalline ferromagnetic nickel film deposited onto the ce ntral part of a 5 ×6mm2\ny-cut lithium niobate substrate. 70nm thick Aluminum interdigital tra nsducers (IDTs) [17]\nwith a finger width of 5 µm are employed to launch and detect a SAW. The complex forward\ntransmission S21oftheapproximately850 µmlongdelaylineisdeterminedusingvectornetwork\nanalysis with a time domain window to cancel the contributions of elect romagnetic crosstalk\nand triple transit interference from the signal. For the description of the experimental and\ntheoretical results we employ the cartesian coordinate system sh own in Fig. 1(a), where αis\nthe angle between an external magnetic field Happlied in the Ni film plane and the SAW\npropagation direction kSAW∝ba∇dblx. We first discuss the influence of the SAW on the magnetization\nvectorMbasedonmagnetoelasticcoupling. Inthestaticlimit, themagneticfr eeenergydensity\n2FIG. 1. (color online) Principles of elastically driven fer romagnetic resonance. (a) Illustration of\nexperimental setup and coordinate system. The closeup to th e right shows the strain εin the Nickel\nthin film. (b) The color coded caps show calculations of the ma gnetic free energy density F. The\nblack arrows depict the gradient of Fwith respect to the magnetization direction m. A finite gradient\nat the equilibrium orientation of M(red arrow) and thus FMR is found only for α=−45◦andε∝ne}ationslash= 0.\nF0of the Ni film normalized to the saturation magnetization Msis given by [18]\nF0(m) =−µ0H·m+Bdm2\nz+Bu(u·m)2+const. , (1)\nwherem=M/Mswith components {mx,my,mz}.Bd=µ0Ms/2 is the shape anisotropy and\nBuis a uniaxial in-plane anisotropy field along u={ux,uy,0}. In equilibrium, the magnetiza-\ntion is oriented along a minimum of F0. Due to the shape anisotropy, and as we only consider\nHin the film plane here, the equilibrium morientation is in the film plane, at an angle θ0\nbetween xandm.\nThe SAW generates an RF strain εin the ferromagnetic thin film plane. Due to magne-\ntoelasticity [18], this strain results in an RF contribution FRFto the magnetic free energy\ndensity\nFRF(m) =B1ε(x,t)m2\nx+const. , (2)\nwhereB1is the magnetoelastic coupling constant and shear strains have bee n neglected for\nsimplicity. The effective field Heffacting on mis given by −∇mF=−∇m/parenleftbig\nF0+FRF/parenrightbig\n[19, 20],\n3FIG. 2. (color online) Experimental observation of elastic ally driven FMR in SAW transmission. (a)\n|S21|as a function of frequency at H= 0. The inset shows |S21|around the 9thharmonic, recorded\nwith the external magnetic field applied at α= 30◦, forµ0|H|= 100mT (solid line) and µ0|H|= 7mT\n(dotted line). (b) |S21|as a function of the external magnetic field applied at α= 30◦(solid lines:\nupsweep, dashed lines: downsweep). With increasing freque ncy, the damping of the SAW increases\nand the field-symmetric absorption dips shift to higher magn etic fields, as expected for FMR.\nevaluated at the equilibrium orientation θ0.\nThe qualitative effect of the SAW on the magnetization is illustrated in F ig. 1(b), where we\nexemplarilyconsider anin-planemagneticallyisotropicnickel film( Bu= 0). Thetotalmagnetic\nfree energy density Fis depicted by the color code in the caps close to the magnetization\nequilibrium position θ0=αforα∈ {−90◦,−45◦,0◦}andε <0,ε= 0 and ε >0. The ”tickle\nfield”, i.e., the components of Heffperpendicular to M, is depicted by the black arrows on top\nof the caps. One observes from Fig. 1(b) that the tickle field stron gly depends on the external\nstatic magnetic field orientation αand vanishes for α= 0◦andα= 90◦. Thus, for Bu= 0, the\nacoustic FMR signal will vanish for α=n·90◦,n∈Z, resulting in a four-fold symmetry. This\nis in stark contrast to conventional FMR, in which the RF magnetic fie ld is applied externally\nand does notdepend on α. Taken together, the above model suggests that the characte ristic\ndependence on the orientation of the external static magnetic fie ld can be used as a fingerprint\nto distinguish acoustically driven FMR from conventional FMR.\nWe next discuss our experimental results. Figure 2(a) shows the m agnitude of the trans-\nmission|S21|measured in the nickel/lithium niobate hybrid as a function of frequen cy at zero\n4external magnetic field. Several RFtransmission maxima, corresp onding to propagatingSAWs,\nare observed. The first transmission maximum occurs at the SAW de lay line fundamental fre-\nquencyof172MHz. SAWtransmissionisalsoobservedatoddharmon ics, whichallowschoosing\nthe magnetoelastic interaction frequency in a range from 172MHz t o 3.6GHz. Here, we con-\nstrain our discussion to the fundamental frequency as well as the 5th, 9thand 13thharmonic\nfrequency, as these frequencies exhibit the most intense SAW tra nsmission for the chosen IDT\nmetallization ratio of 0.5. Using appropriate IDT designs [17] it is possib le to excite SAWs\nat several almost arbitrarily chosen frequencies, or even a broad frequency range. The inset\nin Fig. 2(a) exemplarily shows the frequency dependence of |S21|around the 9thharmonic\n(1.55GHz) for two different values of the external magnetic field. The series of transmission\nlobes is characteristic of a SAW delay line passband [17]. As µ0|H|is changed from 100mT\nto 7mT the damping of the SAW increases by approximately 5dB. Figur e 2(b) depicts the\nevolution of the SAW transmission as a function of magnetic field stre ngth for a full magnetic\nfield sweep from −150mT to +150mT and back to −150mT. The data correspond to the SAW\ntransmission averaged over the FWHM of the central passband lob e|S21|. In addition to the\nhysteretic magnetization switching at |µ0H|<3mT already reported [13, 14], two absorption\nmaxima can be discerned as pronounced dips in Fig. 2(b). The latter s how no hysteresis, are\nsymmetric to zero external magnetic field, and are bothpresent in the magnetic field up- as well\nas the downsweep. Furthermore, the maximal SAW attenuation inc reases with frequency from\nless than 0 .1dB at 0 .17GHz to approximately 16dB at 2 .24GHz, and simultaneously shifts to\nlargerH. Due to the characteristic non-hysteretic behavior of the atten uation maxima together\nwith their shift as a function of frequency, we attribute the atten uation maxima to FMR. The\nslight asymmetry in the attenuation maxima in Fig. 2(b) is observed fo r allHorientations.\nFurther work will be necessary to determine its origin.\nTo identify the mechanism driving the FMR, a detailed study of the ang ular dependence\nis mandatory, as discussed in the context of Fig. 1(b). We thus per formed a series of mea-\nsurements to determine |S21|as a function of magnetic field magnitude and orientation. The\norientation αof the in-plane static magnetic field was varied in 5◦steps from −90◦to 90◦.\nFigure 3(a) shows the |S21|2data obtained as a false color plot. The color code represents\nthe normalized transmission |S21|2\nnorm=|S21|2(µ0H)/|S21|2(−150mT), with black indicating\nmaximal absorption. Figure 3(b) depicts the corresponding signal phase arg( S212), setting\narg(S212)(µ0H=−150mT) = 90◦, with black and yellow visualizing deviations of the signal\n5FIG. 3. (color online) Evolution of the SAW transmission as a function of the magnitude and orien-\ntation of the external magnetic field. Column (a) shows the ex perimentally determined |S21|2\nnorm. In\nresonance, the SAW transmission is attenuated, as evidence d by the black color. Column (b) shows\nthe experimentally determined arg( S212). Column (c) and column (d) show the transmitted SAW\npower density, simulated using Eq. (5). The simulation repr oduces all the salient features observed in\nthe experiment and demonstrates that the FMR is elastically driven.\nphase from 90◦(orange). As evident from Fig. 3, the SAW transmission exhibits a ch arac-\nteristic, approximately four-fold angular symmetry (four transm ission minima as a function of\n0◦≤α≤360◦for a given external magnetic field magnitude). Thus, in addition to t he shift of\nthe resonance field with frequency already discussed in the contex t of Fig. 2, the anisotropy of\nthe SAW magneto-transmission exhibits the characteristic fingerp rint of SAW-driven acoustic\nFMR. In particular, no attenuation occurs for α= 0◦andα=±90◦, as expected according to\nthe picture shown in Fig. 1(b). The fact that the attenuation maxim a are not located exactly\nat±45◦is due to a uniaxial in-plane magnetic anisotropy Bu∝ne}ationslash= 0, as discussed in the following.\nTo model the SAW magneto-transmission, we use a Landau-Lifshitz -Gilbert (LLG) ap-\nproach, taking into account RF magnetoelastic effects. We neglect spatial variations of the\nstrains and restrict the calculations to the collective FMR mode. The LLG equation [21, 22]\n∂tm=γm×µ0Heff+am×∂tm, (3)\ndescribes theevolution ofthemagnetizationdirection minaneffective magnetic field Heff, with\nγandabeing the gyromagnetic ratio and the Gilbert damping parameter, re spectively. We\nsolve theLLGinacartesiancoordinatesystem with thefirst axispoin ting alongtheequilibrium\ndirectionof m, whichinourcaseisinthefilmplaneatanangle θ0tothexaxisandthethirdaxis\n6pointing out-of-plane. In the following, the subscripts {1,2,3}refer to this frame of reference.\nThus,m={1,m2,m3}, wherem2andm3denote small deviations from equilibrium. The\nnon-vanishing component of the effective RF magnetic field h(t) =−∇mFRF(t)|θ=θ0[19, 20] is\ngiven by\nµ0h2(t) =−2B1ε(t)cos(θ0)sin(θ0). (4)\nIn contrast to classical FMR, where an external RF magnetic field is exploited to drive the\nresonance, Eq. 4 shows that acoustic FMR is excited by a purely inte rnal RF magnetic field\nh(t). The latter is due to RF spin mechanics, i.e., it is generated by the mag netoelastic\ninteraction of the SAW elastic strain field with the ferromagnet. This coupling between mand\nthe SAW results in a resonant attenuation and phase shift of the SA W transmission when the\ncondition for ferromagnetic resonance is met. We write the transm itted RF power density Pt\nas\nPt∝P0−1\n2µ0ωh†↔χh, (5)\nwhere the Polder susceptibility↔χis a function of the static magnetic free energy density F0,P0\nis the power density transmitted off resonance and ω= 2πνis the SAW angular frequency. The\nnon-vanishing components of the Polder susceptibility tensor read asχ22=Msγµ0ω1/D,χ33=\nMsγµ0ω2/Dandχ23=χ∗\n32=Msγµ0iω/DwithD=ω1ω2−ω2,ω1=γ(µ0H1+2Bd−2Buu2\n1)+\niωaandω2=γ(µ0H1+2Bu(u2\n2−u2\n1)) +iωa. Eq. (5) corresponds to the conventional FMR\nformula [8], but employs the purely virtual h(t) defined in Eq. (4) instead of an externally\napplied, real RF magnetic field.\nFigure 3(c) shows |Pt|norm=|Pt|(µ0H)/|Pt|(−150mT) calculated using Eq. (5), with Bd=\n+300mT, a= 0.25,ε= 10−5andBu= +4mT along the x-direction for the three frequencies\ninvestigated in the experiment. Figure 3(d) depicts the correspon ding transmission phases\narg(Pt). The simulation reproduces all the salient features observed in th e experiment, in\nparticular the angular dependence. Considering our rather simple m odel, the agreement is\nexcellent. This demonstrates that the SAW absorption is indeed cau sed by acoustically driven\nferromagneticresonance, i.e., byRFmagnetoelasticinteractions, andnotbyconventional FMR.\nMoreover, it corroborates the use of a virtual, purely internal h(t) on the same footing as a real\nexternally applied magnetic field in Eq. (5) a posteriori. We note that o ur damping constant\na= 0.25 is about a factor of ten larger than that expected from cavity F MR experiments [23].\nThis value was chosen to account for line broadening presumably due to the inhomogeneous\n7tickle field periodic with the SAW wavelength λ≈1.5µm at 2.24GHz, corresponding to a\nwavevector kSAW≈4×106m−1. In coplanar waveguide FMR such non-uniform excitation\nfields are known to lead to a line broadening by 100% already for k= 6×104m−1[24]. The\nmuch larger wavevectors in our case will therefore lead to consider ably higher linewidths, i.e.\na= 0.25. To circumvent such inhomogeneous broadening effects, one ma y miniaturize the\nferromagnetic thin film to lateral dimensions much smaller than the SA W wavelength. In such\nsamples, a determination of the influence of magnon-phonon intera ctions on magnetic damping\nshould be possible.\nIn conclusion, we have consistently found both in experiment and in s imulation that the\nmagnetoelastic interaction of a SAW with a ferromagnetic thin film allow s to excite FMR in\nthe film. The FMR is driven acoustically in the sense that no external R F magnetic field is\napplied tothe ferromagnet. Rather, a purely internal RFmagnetic field arises dueto magnetoe-\nlastic coupling between the SAW elastic strain field and the ferromagn et. Using a free energy\napproach as well as LLG calculations, we showed that the magnitude of the ”acoustic” RF\nmagnetic field characteristically depends on the orientation of the m agnetization. The angular\ndependence of the SAW transmission thus is a fingerprint of acoust ic FMR, as observed in our\nexperiments. Our experimental findings open a third alternative fo r the excitation of FMR,\nin addition to externally applied RF magnetic fields [25, 26] or the spin to rque effect [27–31].\nMore fundamentally, acoustic FMR can provide a pathway for the st udy of magnon-phonon\ninteraction phenomena studied predominantly on theoretical grou nds to date [7–10], or also\nfor mechanical spin pumping [32, 33]. The coupling between elastic and magnetic degrees of\nfreedom will open additional channels for spin dephasing, so that th e magnetization damping\nis linked to the magnon-phonon coupling strength. Furthermore, a detailed study of the in-\nteraction between magnonic and phononic dispersions, of the stre ngth of the coupling, of the\nanticrossing of these branches and of the generation of strongly coupled elastic/magnetic states\nare appealing challenges for the future.\nWe gratefully acknowledge stimulating discussions with B. Botters, R . Huber and D.\nGrundler. This work is financially supported by the DFG via project no s GO 944/3-1, SFB\n631 C3, and by the Cluster of Excellence Nanosystems Initiative Mun ich (NIM).\n∗goennenwein@wmi.badw.de\n8[1] H. Zheng et al., Science 303, 661 (2004).\n[2] N. A. Spaldin and M. Fiebig, Science 309, 391 (2005).\n[3] W. Eerenstein et al., Nat. Mater. 6, 348 (2007).\n[4] C. Bihler et al., Phys. Rev. B 78, 045203 (2008).\n[5] A. W. Rushforth et al., Phys. Rev. B 78, 085314 (2008).\n[6] M. Weiler et al., New J. Phys. 11, 013021 (2009).\n[7] H. F. Tiersten, J. Math. Phys. 5, 1298 (1964).\n[8] T. Kobayashi et al., Phys. Rev. B 7, 3273 (1973).\n[9] T. Kobayashi, R. C. Barker, and A. Yelon, Phys. Rev. B 7, 3286 (1973).\n[10] P. A. Fedders, Phys. Rev. B 9, 3835 (1974).\n[11] H. B¨ ommel and K. Dransfeld, Phys. Rev. Lett. 3, 83 (1959).\n[12] A. K. Ganguly, J. Appl. Phys. 47, 2696 (1976).\n[13] I. Feng et al., J. Appl. Phys. 53, 177 (1982).\n[14] R. F. Wiegert and M. Levy, J. Appl. Phys. 61, 4270 (1987).\n[15] R. F. Wiegert and M. Levy, IEEE Trans. Magn. 37, 2708 (2001).\n[16] R. F. Wiegert, J. Appl. Phys. 91, 8231 (2002).\n[17] S. Datta, Surface Acoustic Wave Devices (Prentice Hall, Englewood Cliffs, 1986).\n[18] S. Chikazumi, Physics of Ferromagnetism (Oxford Univ. Press, Oxford, 1997), 2nd ed.\n[19] J. Fidler and T. Schrefl, J. Phys. D: Appl. Phys. 33, R135 (2000).\n[20] M. J. Pechan et al., J. Appl. Phys. 89, 7514 (2001).\n[21] L. Landau and E. M. Lifshitz, Phys. Z. Sowjet. 8, 153 (1935).\n[22] T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[23] S. M. Bhagat, L. L. Hirst, and J. R. Anderson, J. Appl. Phy s.37, 194 (1966).\n[24] G. Counil et al., J. Appl. Phys. 95, 5646 (2004).\n[25] C. Kittel, Phys. Rev. 73, 155 (1948).\n[26] L. R. Bickford, Phys. Rev. 78, 449 (1950).\n[27] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[28] E. B. Myers et al., Science 285, 867 (1999).\n[29] J. A. Katine et al., Phys. Rev. Lett. 84, 3149 (2000).\n[30] M. D. Stiles and A. Zangwill, Phys. Rev. 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Lett. 104, 046601 (2010).\n10"    },    {        "title": "1710.05166v1.Repulsive_polarons_in_alkaline_earth__like__atoms_across_an_orbital_Feshbach_resonance.pdf",        "content": "Repulsive polarons in alkaline-earth(-like) atoms across an orbital Feshbach resonance\nTian-Shu Deng,1, 2Zhuo-Cheng Lu,3Yue-Ran Shi,3Jin-Ge Chen,3Wei Zhang,3, 4,\u0003and Wei Yi1, 2,y\n1Key Laboratory of Quantum Information, University of Science and Technology of China,\nChinese Academy of Sciences, Hefei, Anhui, 230026, China\n2Synergetic Innovation Center of Quantum Information and Quantum Physics,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n3Department of Physics, Renmin University of China, Beijing 100872, China\n4Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices,\nRenmin University of China, Beijing 100872, China\n(Dated: June 9, 2021)\nWe characterize properties of the so-called repulsive polaron across the recently discovered orbital\nFeshbach resonance in alkaline-earth(-like) atoms. Being a metastable quasiparticle excitation at\nthe positive energy, the repulsive polaron is induced by the interaction between an impurity atom\nand a Fermi sea. By analyzing in detail the energy, the polaron residue, the e\u000bective mass, and\nthe decay rate of the repulsive polaron, we reveal interesting features that are intimately related to\nthe two-channel nature of the orbital Feshbach resonance. In particular, we \fnd that the life time\nof the repulsive polaron is non-monotonic in the Zeeman-\feld detuning bewteen the two channels,\nand has a maximum on the BEC-side of the resonance. Further, by considering the stability of a\nmixture of the impurity and the majority atoms against phase separation, we show that the itin-\nerant ferromagnetism may exist near the orbital Feshbach resonance at appropriate densities. Our\nresults can be readily probed experimentally, and have interesting implications for the observation\nof itinerant ferromagnetism near an orbital Feshbach resonance.\nI. INTRODUCTION\nThe recently discovered orbital Feshbach resonance\n(OFR) in173Yb opens up the avenue of investigating\nstrongly-interacting many-body physics using alkaline-\nearth(-like) atoms [1{3]. In an OFR, the spin-exchange\ninteraction between the ground1S0and the long-lived ex-\ncited3P0hyper\fne manifolds can be tuned by an exter-\nnal magnetic \feld. It follows that the wealth of precision\nquantum control techniques, which have been developed\nfor the purpose of quantum metrology and quantum in-\nformation using the clock-state manifolds ( f1S0;3P0g),\ncan be employed to engineer highly non-trivial many-\nbody scenarios [4{31]. Recent studies in this regard range\nfrom interaction-induced topological states [32, 33], to\nimpurity problems such as the Kondo e\u000bects [34{39] and\nthe polaron to molecule transitions [40, 41]. Naturally,\nthe key properties of these phenomena are \frmly based\non the features of interactions of an OFR.\nLike the interactions of Feshbach resonance in alkali\natoms, the interactions of OFR can be understood as\nthe resonant scattering between an open and a closed\nchannel. Consider two alkaline-earth(-like) atoms respec-\ntively in the1S0(denoted asjgi) and the3P0(denoted\nasjei) manifolds, as J= 0 for these so-called clock-state\nmanifolds, the nuclear and the electronic spin degrees of\nfreedom are decoupled. Denoting a particular nuclear\nspin statemI(mI+1) in each manifold as j \"i (j #i),\nwe may associate the open channel with the jg#iand\n\u0003Electronic address: wzhangl@ruc.edu.cn\nyElectronic address: wyiz@ustc.edu.cnje\"istates, and the closed channel with the jg\"iand\nje#istates. Due to the di\u000berential Zeeman shift in the\nclock-state manifolds [42, 43], an external magnetic \feld\ncan conveniently shift the detuning between the open-\nand the closed-channel scattering thresholds. Further, as\nthe short-range interaction of the OFR occurs either in\nthe electronic spin-singlet and nuclear spin-triplet chan-\nnel, or the electronic spin-triplet and nuclear spin-singlet\nchannel, it couples the closed- and the open channels to-\ngether. The scattering resonance occurs when the en-\nergy of a bound state in the closed channel is tuned\nto the open-channel scattering threshold. As a result\nof OFR, a crossover from the Bardeen-Cooper-Schrie\u000ber\n(BCS) to the Bose-Einstein condensation (BEC) regime\ncan be realized in alkaline-earth(-like) atoms by tuning\nthe magnetic \feld, which is similar to the magnetic Fesh-\nbach resonance in alkali atom. However, the existence of\nmultiple nuclear spin states, as well as the spin-exchange\ninteractions in the OFR complicate the two-body scatter-\ning process, and lead to rich physics in the many-body\nsetting.\nAn illuminating example here is the system consist-\ning of a mobile impurity interacting with its environ-\nment. As the limiting case of a many-body system in\nthe large polarization limit, mobile impurity and its as-\nsociated quasi-particle excitations contain valuable in-\nformation of the underlying system. Whereas impurity\nproblems in the background of Bose gases or Fermi con-\ndensates have attracted considerable attention in recent\nyears [44{63], here we focus on the case of an impurity\nagainst a non-interacting Fermi sea. In alkali atoms,\nit has been shown that the impurity can either form\na tightly bound molecule with a majority atom, or in-\nduce collective particle-hole excitations in the Fermi seaarXiv:1710.05166v1  [cond-mat.quant-gas]  14 Oct 20172\nand form the so-called Fermi polaron [64{69]. A polaron\nto molecule transition has been observed experimentally,\nas the interaction is tuned. Further, at positive ener-\ngies, a so-called repulsive polaron branch exists, which\nis metastable and associated with the elusive itinerant\nferromagnetism [68{75]. In OFR, a recent theoretical\nstudy suggests that the transition between the attrac-\ntive polaron and the molecule also exists when tuning\nthe magnetic \feld [40, 41]. However, the existence and\nproperties of the repulsive polaron branch have not been\ninvestigated.\nIn this work, we characterize properties of the repulsive\npolaron across the OFR, using the parameters of173Yb\natoms as a concrete example. As illustrated in Fig. 1, we\nconsider a single impurity atom in the je\"istate, which\ninteracts with a Fermi sea of atoms in the jg#istate.\nWhile the impurity and the background atoms are ini-\ntially in the open channel, the spin-exchange interactions\nwould scatter atoms into the closed channel. Adopting\nthe T-matrix formalism [72, 73], we demonstrate the ex-\nistence of a metastable repulsive polaron branch at pos-\nitive energies across the OFR. We characterize various\nproperties of the repulsive polaron, such as the energy,\nthe polaron residue, the e\u000bective mass, and the decay\nrate. In particular, we identify unique features in all of\nthese quantities, which are intimately related to the two-\nchannel nature of the OFR. An interesting result of the\ninter-channel scattering is that the life time of the repul-\nsive polaron is non-monotonic in the e\u000bective interaction\nstrength, and has a maximum on the BEC-side of the\nresonance. We further analyze the condition for the ex-\nistence of itinerant ferromagnetism in these atoms near\nan OFR. By considering the stability of a homogeneous\nmixture of the impurity je;\"iatoms and the majority\njg;#iatoms against phase separation, we show that a\nphase-separated state, and hence the itinerant ferromag-\nnetism, can be stabilized beyond a critical Zeeman-\feld\ndetuning. Since such a conclusion is conditional on the\nstability of the repulsive polaron, we further demonstrate\nthat for appropriate atomic densities, a parameter win-\ndow exists where the system favors phase separation and\nthe repulsive polaron is long-lived and away from the\nmolecule-hole continuum. Our \fndings can be readily\nprobed experimentally, and have interesting implications\nfor the observation of itinerant ferromagnetism near an\nOFR.\nThe paper is organized as follows. In Sec. II, we present\nthe T-matrix formalism for Fermi polarons in the context\nof an OFR. We demonstrate the existence of the repulsive\npolaron, and characterize its energy by calculating the\nspectral function in Sec. III. We then study in detail the\npolaron residue and the e\u000bective mass in Sec. IV, where\nkinks in these properties are identi\fed and associated\nwith resonant scatterings in the many body background.\nWe characterize the decay rate of the repulsive polaron in\nSec. V, and discuss in detail the potential stability region\nof the itinerant ferromagnetism near an OFR in Sec. VI.\nFinally, we summarize in Sec. VII.II. T-MATRIX FORMALISM\nWe start from the non-interacting Hamiltonian corre-\nsponding to the con\fguration in Fig. 1(a)\nH0=X\nk\u000fo\nk(ay\ng;#kag;#k+ay\ne;\";kae;\";k)\n+X\nk\u000fc\nk(ay\ne;#kae;#k+ay\ng;\"kag;\"k); (1)\nwhereay\nj;\u001b;k(aj;\u001b;k) creates (annihilates) an atom in the\ncorresponding pseudo-spin state jj;\u001bi(j2fg;eg,\u001b2\nf\";#g) with momentum k. Here,\u000fo\nk=~2k2=2mand\n\u000fc\nk=~2k2=2m+\u000e=2. The detuning between the two\nchannels\u000e\u0011\u0001g\u0000\u0001e= (gg\u0000ge)\u0016BBoriginates from\nthe di\u000berential Zeeman shift of the clock states in the\npresence of a magnetic \feld B, wheregg(ge) is the Lande\ng-factor for thejgi(jei) manifold, and \u0016Bis the Bohr\nmagneton.\nThe typical inter-orbital spin-exchange interaction of\nan OFR can be written as\nHint=g+\n2X\nqAy\n+(q)A+(q) +g\u0000\n2X\nqAy\n\u0000(q)A\u0000(q);(2)\nwhere we have\nA\u0006(q) =X\nk(ae;#;kag;\";q\u0000k\u0007ae;\";kag;#;q\u0000k);(3)\nand the interaction strengths g\u0006are related to the\nphysical ones via the renormalization relation 1 =g\u0006=\n1=~g\u0006\u0000P\nk1=2\u000fo\nkwith ~g\u0006= 4\u0019~2a\u0006=m. Throughout\nthis work, we adopt the parameters of173Yb atoms, with\na+= 1900a0anda\u0000= 219:5a0[2, 3, 76, 77].\nDiagrammatically, the polaron properties can be cal-\nculated using the retarded self-energy of the impurity\natom [72], which is given by [see Fig. 1(b)]\n\u0006(Q;E) =Zdq\n(2\u0019)3Zd!\n2\u0019G0\ng#(q;!)Too(q+Q;E+!);\n(4)\nwhereG0\ng;#(q;!) = (!+i0+\u0000\u000fo\nq)\u00001is the free-fermion\npropagator of the majority atoms, and Toois the T-\nmatrix describing the open-channel scattering processes.\nHereEandQare respectively the energy and the center-\nof-mass momentum of the self-energy, and !is the Mat-\nsubara frequency. Due to the spin-exchange nature of the\ninteraction, the open- and the closed-channel scattering\nmatrices are coupled. Accordingly, there should be four\nkinds of T-matrices Too,Toc,Tco, andTcc, with the in-\ncoming and the outgoing states being in either the open\nor the closed channel, as indicated by the superscript la-\nbels. As discussed in Ref. [31], under the ladder approx-\nimation, we may write down a set of coupled equations\nfor the T-matrices, which lead to the solution\nToo(q;!) =1\n2(g++g\u0000)\u0000g+g\u0000\u001fc\n1\u00001\n2(g++g\u0000)(\u001fo+\u001fc) +g+g\u0000\u001fo\u001fc:(5)3\n|e,↓t\n|g,↓t(a)\nToo(b)\n=Gg↓G0\nGe↑0Ge↑0|e,↑t\n|g,↑t∆e\n∆g\nȞGe↑0Ge↑0\nGe↑0Ge↑0\nFIG. 1: (a) Level diagram of an OFR in alkaline-earth(-\nlike) atoms. An impurity of je;\"iis immersed in a majority\nFermi sea ofjg;#iatoms, and can be scattered to the other\ntwo atomic states forming the closed channel via interaction.\n\u0001g=gg\u0016BBand \u0001 e=ge\u0016BBare the Zeeman shifts of the\njgiandjeimanifolds, respectively. (b) The one-hole polaron\nself-energy \u0006 near an OFR. The solid lines with arrows indi-\ncate free propagator G0forjg;#iorje;\"i, and the square Too\nindicates the T-matrix with the incoming and the outgoing\nstates being both in the open channel.\nHere, the pair propagators for the closed and the open\nchannel\u001fc(q;!) and\u001fo(q;!) can be written as\n\u001fc(q;!) =X\nk1\n!+i0+\u0000\u000fc\nk\u0000\u000fc\nq\u0000k; (6)\n\u001fo(q;!) =X\njkj>kF1\n!+i0+\u0000\u000fo\nk\u0000\u000fo\nq\u0000k; (7)\nwhere the Fermi wave vector kFis related to the Fermi\nenergyEFofjg;#iatoms asEF=~2k2\nF=2m. From the\nequations above, we see that \u001fo(q;!) and\u001fc(q;!) are\nisotropic in q. For the convenience of discussion, we de-\n\fneq=jqj.\nSubstituting Eq. (5) into (4), we obtain\n\u0006(Q;E) =X\nq<kF\u00141\n2\u00121\n~g++1\n~g\u0000\u0013\n\u0000~\u001fo(q+Q;E+\u000fo\nq)\n\u00001\n4\u00121\n~g+\u00001\n~g\u0000\u001321\n1\n2(1\n~g++1\n~g\u0000)\u0000~\u001fc(q+Q;E+\u000foq)\u0015\u00001\n;\n(8)\nwhere the renormalized pair propagator ~ \u001fc=\u001fc+P\nk1=2\u000fo\nkand ~\u001fo=\u001fo+P\nk1=2\u000fo\nk. With Dyson's equa-tion, the Green's function of an impurity je;\"idressed\nby a Fermi sea of jg#iatoms can therefore be written as\nGe\"(Q;E) =1\nE+i0+\u0000\u000fo\nQ\u0000\u0006(Q;E); (9)\nfrom which we may extract various properties of the\nquasi-particle excitations.\nIII. SPECTRAL FUNCTION AND THE\nPOLARON ENERGY\nWe \frst calculate the spectral function at zero temper-\nature (T= 0)\nA(Q;E) =\u00002ImGe\"(Q;E): (10)\nIn Fig. 2, we plot A(Q= 0;E) in the\u000e{Eplane. The\nspectral function is strongly peaked at the energies of\npolaron excitations satisfying\nE\u0006= Re[\u0006( Q;E\u0006)]: (11)\nAs is apparent in Fig. 2, there exist two solutions for\nEq. (11). The lower branch with E=E\u0000<0 corre-\nsponds to the attractive polaron, and the upper branch\nwithE=E+>0 corresponds to the repulsive polaron.\nIn contrast to the attractive polaron, which is undamped\nunder the ladder approximation here, the repulsive po-\nlaron, being a mestastable quasipaticle excitation with\nE+>0, features a \fnite width in the spectral func-\ntion as illustrated in Fig. 3, which originates from the\ndecay into low-lying states. Under the inter-orbital spin-\nexchange interactions of the OFR, as we will show later,\nthe \fnite spectral width and hence the decay of the repul-\nsive polaron mainly come from the resonant coupling of\nthe quasi-particle excitation to the open- and the closed-\nchannel scattering continuum. Finally, we notice the ex-\nistence of a broad wing between the two polaron peaks as\nshown in Fig 3, which corresponds to the molecule-hole\ncontinuum.\nIV. IMPURITY RESIDUE AND THE\nEFFECTIVE MASS\nWe now characterize the impurity residue and the ef-\nfective mass of the repulsive polaron. For a polaron ex-\ncitation, the quasi-particle residue is de\fned as [72]\nZ\u0006=1\n1\u0000Reh\n@\u0006(0;!)\n@!i\f\f\f\f\f\n!=E\u0006; (12)\nand its e\u000bective mass as\nm\u0003\n\u0006=1\nZ\u00061\n1 + Reh\n@\u0006(Q;!)\n@Q2i\f\f\f\f\f\nQ=0;!=E\u0006; (13)4\n-2 02\n/E0-3-2-1012E/E0\n246810\nFIG. 2: False color plot of the spectral function A(Q= 0;E)\nof an impurityje;\"iin a Fermi sea of non-interacting jg#i\nparticles on the \u000e{Eplane. The solid red lines depict the\npolaron energies given by Eq. (11), and the dashed green\nline is the molecular energy. The light-blue area between the\ntwo polaronic branches is the molecule-hole continuum. The\nupper repulsive polaron branch merges into the molecule-hole\ncontinuum for \u000e&3:8E0. Here, we de\fne the unit of energy\nE0=~2k2\n0=2m, where the unit Fermi wave vector k3\n0= 6\u00192n0\nand the unit density n0= 5\u00021013cm\u00003. In this plot, we take\nn=n0.\nwhere the subscript + ( \u0000) labels the repulsive (attrac-\ntive) branch of polarons.\nWe have shown the quasi-particle residue as well as\nthe e\u000bective mass of the repulsive polaron in Fig. 4. For\ncomparison, we have also plotted the residue and the ef-\nfective mass of the attractive polaron. In an OFR and\nunder the setup illustrated in Fig. 1, the resonance occurs\nat\u000e0\u00183:06E0, and the system is on the BCS-side of the\nresonance for \u000e>\u000e 0. In Fig. 4, we see that as \u000eincreases\n(i.e., moves towards the BCS side of the resonance), Z+\ndecreases and m\u0003\n+increases, which are qualitatively con-\nsistent with the case of alkali atoms. A prominent dif-\nference in the current case is the existence of kinks in\nboth the residue and the e\u000bective mass at \u000e=E+and\n\u000e=E++EF=2. The occurrence of these kinks can be\nexplained by the qualitative di\u000berence, between regions\nwith di\u000berent values of \u000e, in the way that the atoms in\nthe open-channel Fermi sea are scattered into the closed-\nchannel continuum in forming the repulsive polaron.\nThe location of the kinks can be determined analyti-\ncally by considering the scattering process between the\nimpurity and the majority atoms, in which the out-going\nstates are at the closed-channel scattering threshold. In\nparticular, because \u000erepresents the closed-channel de-\ntuning of the two atoms and E+is the interaction-\ninduced energy shift of the impurity atom, at \u000e=E+an\nimpurity atom with zero momentum can interact with\na majority atom at the bottom of the Fermi sea (with\nq= 0), which are resonantly scattered to two atoms\nin the closed-channel scattering threshold. Likewise, at\n-2 02\nE/E00510A(0,E)=0(a)\n-2 02\nE/E00510A(0,E)=1(b)\n-2 02\nE/E00510A(0,E)=2(c)\n-2 02\nE/E00510A(0,E)=3(d)FIG. 3: The spectral function A(Q= 0;E) as functions of E\nwith di\u000berent detunings. We have taken the same parameters\nas those in Fig. 2.\n\u000e=E++EF=2, an impurity atom with zero momentum\ninteracts with a majority atom on the Fermi surface (with\nq=kF), which, under the momentum conservation, are\nresonantly scattered to two atoms in the closed channel\neach with a momentum q=kF=2. The process can there-\nfore be qualitatively described as resonant scatterings in\nthe many-body background.\nTo further demonstrate this point, in Fig. 5(a) and\n5(b), we explicitly show the imaginary parts of the pair\npropagators in the open- and the closed-channel, respec-\ntively, on the \u000e{qplane. As the imaginary parts of the\npair propagators are related to the removable singular-\nities in the summation of Eqs. (6) and (7), they re\rect\nthe contribution to the polaron self-energy as atoms in\nthe Fermi sea with momentum q<kFis resonantly cou-\npled to the scattering states in the open(closed)-channel\ncontinuum, forming particle-hole excitations. In the case\nof the open-channel pair propagator, for any given \u000e, a\n\fnite imaginary part exists only when the magnitude of\nthe center-of-mass momentum of the hole excitation qis\nbelow a critical value. This implies that, for any given \u000e,\npart of the Fermi sea is blocked by the energy and mo-\nmentum conservation conditions such that atoms therein\ncannot be resonantly scattered into the open-channel\ncontinuum. The case of the closed-channel pair prop-\nagator is more complicated. For \u000e<E +,\u001fc(q;E++\u000fq)\nfeatures a \fnite imaginary part for all q<kF. Therefore,\nall atoms in the Fermi sea can be resonantly scattered\ninto the closed-channel continuum to form the particle-\nhole excitations. For \u000e>E ++EF=2, on the other hand,\n\u001fc(q;E++\u000fq) can be completely real for any q < kF.\nHence, none of the atoms in the Fermi sea can be reso-\nnantly scattered into the closed-channel continuum, and\nthe particle-hole excitations in the repulsive polaron is\nopen-channel dominated. Therefore, in di\u000berent regions\nof\u000e, the closed-channel scattering continuum contribute5\n-2 02\n/E000.51Z(a)\n-2 02\n/E0-505m*/m(b)\nFIG. 4: (a) Quasi-particle residues Z\u0006for the attractive (blue\ndashed) and repulsive (red solid) polarons as functions of \u000e.\n(b) E\u000bective masses of the attractive (blue dashed) and the\nrepulsive (red solid) polarons as functions of \u000e.\nin qualitatively di\u000berent ways to the polaron self-energy,\nwhich gives rise to the appearance of kinks at the bound-\naries of these regions.\nV. DECAY RATE OF THE REPULSIVE\nPOLARON\nBeing a mestable state, the repulsive polaron can de-\ncay into low-lying states. Experimentally, it has been\nshown that for alkali atoms, the dominating decay chan-\nnel for the repulsive polaron is the coupling to the bare\nimpurity state in the attractive-polaron branch, so long\nas the interaction is not in the deep BEC regime [75].\nWe assume that the case with alkaline-earth(-like) atoms\nis similar. One should then include the corresponding\ndecay channel in the diagrams leading to the repulsive\npolaron self-energy. Such a decay rate can be calculated\nas [75]\n\u0000 =\u00002Z+[Im\u0006(0;E+)]; (14)\nwhereZ+is the residue for the repulsive polaron. Fur-\nther, we can replace the free-fermion propagator G0\ne\"with\n(1\u0000Z+)G0\ne\"in the self-energy \u0006, which implies substitut-\ning\u001fowith (1\u0000Z+)\u001fo. This leads to \u0000 =P\njqj<kF\u0000(q)\nwith\n\u0000(q) =Imn\n\u00002Z+h1\n2\u0000\n~g\u00001\n++ ~g\u00001\n\u0000\u0001\n\u0000(1\u0000Z+)~\u001fo(q;E++\u000fq)\n\u00001\n4\u0000\n~g\u00001\n+\u0000~g\u00001\n\u0000\u00012\n1\n2\u0000\n~g\u00001\n++ ~g\u00001\n\u0000\u0001\n\u0000~\u001fc(q;E++\u000fq)i\u00001o\n:(15)\nWe plot the calculated \u0000( q) and \u0000 in Fig. 5(c) and\n5(d), respectively. In Fig. 5(c), we see that the decay\nrate consists of two di\u000berent contributions, which can\nbe numerically associated with contributions from the\npair propagators ~ \u001foand ~\u001fc. At large and positive \u000e,\nthe decay of the repulsive polaron is open-channel dom-\ninated, which increases as the system is tuned further\n-2 02\n/E000.51q/k0(a)\n-3-2-10\n-2 02\n/E000.51q/k0(b)\n-3-2-10\n-2 02\n/E000.51q/k0(c)\n00.20.40.6\n-1 0 1 2 3 \n/E000.10.2/E0(d)FIG. 5: (a) Imaginary part of the closed-channel pair propa-\ngator\u001fc(q;E++\u000fo\nq) on the\u000e{qplane. (b)Imaginary part of\nthe open-channel pair propagator \u001fo(q;E++\u000fo\nq) on the\u000e{q\nplane. (c) \u0000( q) on the\u000e{qplane. (d) The decay rate \u0000 as\nfunction of \u000e. In (a-c), we take n=n0. In (d), the green\ndashed line, the red solid line, and the blue dash-dotted line\ncorrespond to cases of n= 1:2n0,n0, and 0:8n0, respectively.\ntowards the BCS side. At smaller or negative \u000e, the\nclosed-channel contribution becomes important, which\nincreases as the system is tuned towards the BEC regime.\nWhile this result is consistent with the previous analy-\nsis of the pair propagators, the competition between the\ntwo channels gives rise to the non-monotonic behavior\nof \u0000 as shown in Fig. 5(d), where the lowest decay rate\noccurs near \u000e=E++EF=2. Notably, as illustrated in\nFig. 5(d), the decay rate is density dependent, and in\nmost cases, we have \u0000 \u001cEF, which suggests that the re-\npulsive polaron is a well-de\fned quasi-particle through-\nout the OFR. We also note that further into the BCS\nside with large enough \u000e, the repulsive polaron branch\nwould eventually enter the molecule-hole continuum as\nillustrated in Fig. 2, where the repulsive polaron would\nbecome unstable. However, at large \u000ebut before the\nrepulsive polaron branch merges into the molecule-hole\ncontinuum, the decay rate appears to decrease with in-\ncreasing\u000e, which is due to the decreasing quasi-particle\nresidueZ+at large\u000e.\nVI. PHASE SEPARATION\nOne of the reasons for the recent interest in repul-\nsive polarons is the potential existence of itinerant fer-\nromagnetism in repulsively interacting two-component\nfermions. Previous theoretical studies have shown that\nitinerant ferromagnetism may be stabilized for alkali\nfermionic atoms in the repulsive branch [70{73]. How-6\n1.5 2   2.5 3   3.5 \n/E00.50.60.70.80.91P\nFIG. 6: Phase diagrams in terms of the detuning \u000eand the\npolarization P= (N1\u0000N2)=(N1+N2). Above and to the left\nof the line is the mixed phase, while below and to the right\nof the line is the phase-separated state. The green dashed,\nthe red solid, and the blue dash-dotted curves correspond to\nn= 1:2n0,n0, and 0:8n0, respectively.\never, a direct experimental con\frmation is still lack-\ning. A natural question is whether itinerant ferromag-\nnetism exists in alkaline-earth(-like) atoms under the\nspin-exchange interactions near an OFR.\nIn an e\u000bort to answer this question, here we consider a\nsystem ofN2impurity atoms of the state je;\"iimmersed\ninN1majority atoms of the state jg;#i. Following the\ntreatment in Ref. [73], we derive the free energy of a\nhomogeneous mixture and study the condition for the\noccurrence of a phase separation. For a highly polarized\nmixture with y=N2=(N1+N2)\u001c1, the energy per\nparticle for a homogeneous mixture at zero temperature\ncan be written as\nEmix=3\n5E(1)\nF(1\u0000y) +3\n5E(2)\nFy+yE+(N1;\u000e);(16)\nwhereE(1)\nFandE(2)\nFare the Fermi energies of the atoms\nin thejg;#iand theje;\"istates, respectively. The last\ntermE+(N1;\u000e) in the expression above is the energy of\na singleje;\"iimpurity atoms interacting with N1jg;#i\natoms in the repulsive branch, which can be related to the\nrepulsive polaron energy as E+(N1;\u000e) =E+(1\u0000y)2=3,\nwhereE+is the repulsive polaron energy discussed in\nthe previous sections with a total particle number N=\nN1+N2. By applying the usual Maxwell construction\ntoEmix[73], we obtain the critical polarization from\nthe minimum of Emix. In Fig. 6, we plot the resulting\nphase diagram on the plane of detuning \u000eand polariza-\ntionP\u0011(N1\u0000N2)=(N1+N2). Note that in the large\npolarization limit P\u00181, the mixture becomes unstable\ntowards phase separation beyond a critical \u000ec, which is\ndensity dependent due to the non-universal nature of the\nOFR.\nThe stability region of the phase-separated state shown\nin Fig. 6 is conditional on the stability of the repulsive\nFIG. 7: Phase diagram for the stability of the phase-separated\nstate on the \u000e-nplane. The background color shows the de-\ncay rate \u0000=E0of the repulsive polaron. The black dashed\nline indicates the phase boundary between the mixed and the\nphase-separated states. The red solid line is the boundary\nwhere the repulsive polaron merges into the molecule-hole\ncontinuum. Here we take the large polarization limit with\nP= 1.\npolaron. More speci\fcally, the repulsive polaron should\nbe a long-lived, well-de\fned quasi-particle away from the\nmolecule-hole continuum to justify the free-energy con-\nsiderations leading to the phase diagram in Fig. 6. To\nprovide further perspective on this point, we show in\nFig. 7 the phase diagram in the large polarization limit\nP= 1 on the \u000e{nplane, against the false color plot of\nthe polaron decay rate. We conclude that the phase-\nseparated state, and hence the itinerant ferromagnetism,\nmay be stabilized to the immediate left of the red-dotted\nline, where the decay rate \u0000 <0:1E0and the phase-\nseparated state is energetically favorable.\nVII. SUMMARY\nWe have characterized in detail the key properties of\nthe repulsive polaron near an OFR, using the parame-\nters of173Yb. We \fnd that the two-channel nature of\nthe OFR has signi\fcant impact on the properties of the\nrepulsive polaron. In particular, the decay rate features\na minimum at small magnetic \feld, on the BEC side of\nthe resonance point. The dressing of the repulsive po-\nlaron by the closed-channel scattering states would also\ngive rise to visible kinks in both the residue and the ef-\nfective mass of the repulsive polaron. We also estimate\nthe parameter region where the itinerant ferromagnetism\nmay be stabilized and observed. Our results can be read-\nily checked using the exiting experimental techniques in\nalkaline-earth(-like) atoms.7\nAcknowledgments\nWe thank Pietro Massignan and Zhenhua Yu for\nhelpful comments and discussions. 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Baena3\n1Department of Physics, University of California, San Diego , CA, 92037 USA\n2Department of Electrical and Computer Engineering, Duke Un iversity, Durham, NC, 27708 USA and\n3Department of Physics, National University of Colombia, Bo got´ a, Colombia\n(Dated: August 6, 2018)\nWe explore the hybridization of fundamental material reson ances with the artificial resonances\nof metamaterials. A hybrid structure is presented in the wav eguide environment that consists of\na resonant magnetic material with a characteristic tuneabl e gyromagnetic response that is inte-\ngrated into a complementary split ring resonator (CSRR) met amaterial structure. The combined\nstructure exhibits a distinct hybrid resonance in which eac h natural resonance of the CSRR is split\ninto a lower and upper resonance that straddle the frequency for which the magnetic material’s\npermeability is zero. We provide an analytical understandi ng of this hybrid resonance and define\nan effective medium theory for the combined structure that de monstrates good agreement with nu-\nmerical electromagnetic simulations. The designed struct ure demonstrates the potential for using a\nferrimagnetic or ferromagnetic material as a means of creat ing a tunable metamaterial structure.\nINTRODUCTION\nMetamaterials encompass a class of artificial electro-\nmagnetic media that provide electromagnetic properties\nbeyond those available from natural materials [1, 2, 3].\nTheir unique electromagnetic properties are obtained by\nharnessing the resonant behavior of many periodic and\nsubwavelength composite structures. Unfortunately, the\nresonant nature of metamaterials also ensures that they\nare frequency dispersive and limited to a narrow fre-\nquency band of operation. One method of bypassing\nthis constraint is to design tunable metamaterials, which\nthough still dispersive, can be tuned to have the desired\npropertiesoverthefrequencyrangeofinterest. Atunable\nmetamaterial can be made by placing a material with a\ntunable electromagnetic parameter, permittivity or per-\nmeability, in a region of the metamaterial cell where the\nlocal fields are concentrated. Tuning the local fields will\nin turn affect the bulk electromagnetic response of the\nmetamaterial. This hasbeen demonstratedbytuning the\ncapacitance in the gap regionof metamaterials composed\nof split-ring resonators (SRRs) using ferroelectric mate-\nrials [4, 5]. In this paper we investigate a tunable meta-\nmaterial structure that incorporates a resonant magnetic\nmaterial, that in contrast to ferroelectric material, is it-\nselffrequencydispersiveoverthe rangeofoperation. The\ntwo dispersive systems, metamaterial structure and mag-\nnetic material, combine to exhibit a distinct hybrid res-\nonance for which we provide an analytical model and\ndemonstrate its good agreement with numerical results.\nGyromagnetic materials have a permeability tensor of\nthe form [6],\nµ=µ0\nµ1iµ20\n−iµ2µ10\n0 0 1\n, (1)\nwhereµ1andµ2are resonant functions of frequency. InMagnetic Layer \nDielectri c Layer E\nHkzI/2 I/2 \nH\n60 70 80 90 100 110 120 \n-2 2468\nFrequency (Ghz) Relative Parameters Real( ε)        Imag( ε)\nReal( μ)        Imag( μ)\n1st Resonance 2nd Resonance yx\nz\nFIG. 1: A diagram of a unit cell of the CSRR parallel plate\nwaveguide is shown. At resonance, current flows along the\nedges of the CSRR structure and produces a magnetic field\nthat is approximately perpendicular in the gap region. The\nextracted permeability and permittivity are shown for the f re-\nquency range that includes the first two resonances of the\nCSRR waveguide (without the influence of the magnetic ma-\nterial).\nthis study we consider a simplified “generic” resonant\nmagnetic material which is isotropic and does not in-\nclude the off-diagonal permeability component, µ2, in or-\nder to elucidate the underlying physics of the hybridiza-\ntion. Never-the-less, this structure is suggestive of how\nferrimagnetic or ferromagnetic materials might be used\nto make a tunable metamaterial structure. At the end\nof this letter we briefly discuss the potential for integrat-2\ning a real gyromagnetic material into the metamaterial\nstructure.\nMetamaterials with artificial magnetic properties have\nbeen constructed using SRR structures and exhibit an\napproximate magnetic Drude-Lorentz response that has\nbeen well documented in the literature [2]. A related\nresonant structure is called the complimentary split-ring\nresonator (CSRR) and has the “complementary” meta-\nsurface of a SRR (as shown in Fig. 1). The dual CSRR\nand SRR structures obey Babinet’s principle of equiva-\nlence with equal resonant responses [7, 8], but the CSRR\nis excited by a perpendicular electric field while the SRR\nstructure is excited by a perpendicular magnetic field.\nThe CSRR structure is well suited for implementation\nof the metamaterial concept in a waveguidegeometrybe-\ncausethe CSRR structures canbe etched into the ground\nplane of a waveguide and excited by an incident TEM\nwave. Just as for bulk metamaterial structures, it is pos-\nsible to define effective bulk parameters for a waveguide\nmetamaterialstructure[9]. ByBabinet’sequivalence,the\nCSRRs permittivity is the dual of the SRR structure and\nis given by\nǫ(ω) = 1−Fω2\nω2−ω2\n0+iωΓc, (2)\nwhereFis a constant, ω0is the resonant angular fre-\nquency, Γ cis the dissipation factor, and ωis the angular\nfrequency [3]. The resonant frequency ω0is related to\nthe inductance, L, and capacitance, C, of the structure\nby\nω0=1√\nLC. (3)\nTheinclusionofathinlayerofmagneticmaterialbelow\nthe CSRR structure as shown in Fig. 1 provides a means\nto influence the the local fields of the CSRR structure.\nWe considera magnetic materialwith relativepermeabil-\nity of the form,\nµr(ω) =(ξMs+ξH0−iωΓm)2−ω2\n(ξH0−iωΓm)(ξMs+ξH0−iωΓm)−ω2(4)\nwhereMsis the magnetic saturation of the material, H0\nis the magnetic bias field, Γ mis the damping constant,\nandξ=γµ0is a constant with γdefined as the gyromag-\nnetic ratio [10]. The CSRR resonance and magnetic ma-\nterial resonance interact to produce a hybrid resonance.\nThe mechanism of interaction can be understood by con-\nsideringthedynamicsofthecurrentflowwhenthe CSRR\nis resonating. An incident TEM wave in the waveguide\ndrives current symmetrically in and out of the CSRR\nstructure along the edges of the gap region as shown in\nFig. 1. The current is concentrated on either side of the\ngap of the CSRR structure but flows in opposite direc-\ntions creating a nearly perpendicular magnetic field in\nthe gap and the regions directly above and below thegap. Consequently, determining the inductance of the\nCSRR structure is analogous to analyzing the magnetic\ncapacitormodelofparallelcurrentsheetsfilledwith some\nvolume fraction q(to be determined) of frequency depen-\ndent magnetic material. Exploiting this simple model,\nthe inductance is found to be,\nL=µ0/parenleftbiggµr(ω)\nµr(ω)(1−q)+q/parenrightbigg\nggeom, (5)\nwhereµr(ω) is the relative permeability of the magnetic\nmaterial and ggeomis a constant with units of length\nthat is determined by the geometry of the CSRR struc-\nture. If the parallel current approximation were exact\nthenggeom= (1/2)wd/hwithwthe length of the gap, d\nthe width ofthe gap, and hthe height ofthe gap(the fac-\ntor of 1/2 follows from the parallel current flow into the\nCSRR structure). In general, the geometrical function\nis more complex but it can be extracted from numerical\nsimulations of the empty CSRR waveguide as we demon-\nstrate below. The capacitance of the CSRR structure\nfollows from the capacitance across the gap (above and\nbelow the metal-surface). It is given by, C=ǫ0fgeom,\nwhere again fgeomis a constant with units of length that\nis determined by the geometry of the CSRR structure\nand can be extracted from numerical simulations.\nInserting Eq. 5 into Eq. 3 provides an equation that\ncan be solved to determine the resonant frequency of the\nhybrid CSRR/magnetic structure,\nω′\n0=1/radicalBig\nµr(ω′\n0)\nµr(ω′\n0)(1−q)+qω0. (6)\nwhere,ω0= 1//radicalbig\nǫ0µ0fgeomggeom. We note that Eq. 6 re-\nduces to the resonantfrequency ofthe empty structure in\nthe case µr(ω) is of unity. As previously mentioned, the\nmagneticfieldgeneratedinthegapregionoftheCSRRat\nresonanceispredominatelyperpendicular(seeFig.1)and\ninteracts principally with the magnetic material through\nthex-component ofthepermeability. Infact, simulations\n(not shown here) have shown that variation of any of the\nother diagonal permeability components has no effect on\nthe response of the structure. We can insert the perme-\nability Eq. 4 into Eq. 6 and solve to get the new resonant\nfrequencyofthe hybridstructure. Eq.6istranscendental\nin nature, as a result of the magnetic materials frequency\ndependence, but it can be solved straightforwardly using\nnumerical methods.\nIn order to understand the dynamics of Eq. 6 it is in-\nstructive to plot the left and right side of the equation as\na function of frequency, as shown in Fig. 2(a). Note that\nthe solution to Eq. 6 is found at the intersection of these\ntwo lines, i.e. where the function is self-consistent. The\ncharacteristic phenomenon of the hybridization is seen\nto be a splitting of the CSRR’s resonance into a lower,\n1a, and upper, 1 b, hybrid resonance which straddle the3\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 60 65 70 \nBiasing Component H 0 (kG) Frequency (Ghz) HFSS Simulations \nAnalytic Solution Empty CSRR Resonance 0 20 40 60 80 100 120 20 40 60 80 100 120 Frequency (Ghz) \nFrequency (Ghz) 1st Resonance (CSRR) 2nd Resonance (CSRR) \nLine of equality RHS for 1st Resonance RHS for 2nd Resonance \n1a1b\n2a2b(a)\n(b)\nFIG. 2: (a) The left hand side (line of equality) and right\nhand side (RHS) of Eq. 6 are plotted for the 1st/2nd CSRR\nresonance with H0= 1.5 kG and characteristic parameters\nstated in the text. (b) The dispersion curve for the 1st or-\nder hybrid mode of the CSRR/magnetic material waveguide\nis calculated both analytically and through numerical elec tro-\nmagnetic simulations as a function of the biasing field, H0.\nfrequency at which the magnetic material’s permeabil-\nity is zero (above resonance—not the zero point at reso-\nnance). The hybridization is strongest when the natural\nresonance of the CSRR structure is near the zero perme-\nability point of the magnetic material. If one increasethe\nbiasing field of the magnetic material, such that its zero\npermeability frequency advances through the CSRR’s\nnatural resonance, a characteristic growth and shift of\nthe 1aresonanceis observedand then asubsequentdecay\nand shift of the 1 bresonance is observed. Hence, tuning\nthe magnetic material effectively tunes the response of\nthe metamaterial structure. Because a CSRR structure\ninherently exhibits multiple resonance (the first two res-\nonances of the CSRR structure are shown in Fig. 1) it\nis not only the fundamental resonance that is hybridized\nbut also the higher order resonances. However, these\nhigher order resonances are highly damped if they are\nfar away from the magnetic material’s zero permeability\nfrequency. In practice this means that if the zero per-\nmeability frequency of the magnetic material is aligned\nwith the fundamental CSRR structure resonance, then\ndepending on losses in the system, residual higher order\nhybrid resonances 2 a, ..., n amight be found near the\nfundamental, 1 a, hybrid resonance as shown in Fig. 2(a).\nIn contrast, the 1 b,2b, ..., n aresonances remain largely\nseparated. This suggests that the 1 bresonance—which\nis strongly resonant, isolated from the other hybrid reso-\nnances, and strongly tunable—maybe the most useful for\napplication.In order to investigate the hybridization of the CSRR\nand magnetic material we used Ansoft’s commercially\navailable electromagnetic finite element solver (HFSS).\nHFSS calculates the scattering matrix for a simulated\nstructure via a full wave analysis. It is sufficient to sim-\nulate one CSRR unit cell, as shown in Fig. 1, and apply\nappropriate boundary conditions to reproduce the char-\nacteristic response of a parallel plate waveguide struc-\nture. The simulated structure consisted of two parallel\nperfectly conducting plates and adjoining perpendicular\nperfect magnetic boundaries. The structure was excited\nthrough waveports on either sides of the waveguide with\na fundamental TEM mode. The CSRR structure was\ncut into the top waveguide plate and an exterior volume\nabove the structure was defined in order to accommo-\ndate electromagnetic fields emanating above the struc-\nture. Perfect magnetic boundaries were defined on the\nexterior volume’s surfaces perpendicular to the propa-\ngation mode while perfect electric boundaries were de-\nfined on the other surfaces including the top of the ex-\ntra boundary volume (note that the height of the extra\nvolume is chosen such that on the boundary the electro-\nmagnetic fields decay to near zero—and hence its speci-\nfication is irrelevant, but numerical convergence is faster\nby defining an electric boundary here).\nThe geometrical parameters of the CSRR were chosen\nto have a resonance near 64 Ghz. The unit cell was 1\nmm, the ring radius was 0.4 mm, the neck was 0.15 mm\nwide, and the notch width was 0.035 mm. The spac-\ning between the plates, which determines the strength\nof the CSRR resonance, was set to 0.4 mm. The top\nand bottom waveguide plates were 1 µm thick and had\nthe conductivity of copper ( σ= 5.6×107). For sim-\nplicity, the dielectric between the plates was set to have\na dielectric constant of ǫr= 1. The structure was first\nsimulated without the magnetic layer to determine the\nDrude-Lorentz fit parameters in Eq. 2. A parameter ex-\ntraction was performed on the S-parameters [11] of the\nsimulated structure to determine the effective permit-\ntivity and then a least square fit of the Drude-Lorentz\nfunction, Eq. 2, was used to determine the resonant fre-\nquency,ω0= 1//radicalbig\nǫ0µ0ggeomfgeom; the constant, F; and\nthe loss tangent, Γ c. In fact, CSRR structures (and\nmetamaterials in general) are not perfect Drude-Lorentz\nresonators as they exhibit spatial dispersion effects near\nresonance. These spatial dispersion effects were incor-\nporated into our fitting procedure [12] to determine the\nresonant frequency more accurately but then the simpler\nDrude-lorentz model was used in our analytic analysis\nwith good accuracy.\nOnce the empty CSRR structure was characterized,\nnumerical simulations incorporating the magnetic ma-\nterial were performed. In the interest of exploring the\npotential of incorporating high frequency (40-70 Ghz)\nthin film magnetic materials, such as hexagonal fer-\nrites [13], we considered a 1 µm thick layer of magnetic4Transmission (dB) 55 60 65 70 75 Frequency(Ghz) \n-20 -15 -10 -5 \n55 60 65 70 75 \n-20 -15 -10 -5 \n55 60 65 70 75 \n-20 -15 -10 -5 \n55 60 65 70 75 \n-20 -15 -10 -5 Bias = 0 kG \nBias = 3 kG Bias = 2 kG Bias = 1 kG HFSS Simulation \nAnalytic Solution 1a\n1a\n1a\n1a1b\n1b\n1b\n1b2a\n2a\n2a\n2a\nFIG. 3: The transmission results are shown for the HFSS\nelectromagnetic simulation and the analytic prediction us ing\neffective medium theory.\nmaterial with parameters (Eq. 4): ξ= 18 GhzkG−1,\nΓm= (0.70Ghz)/ωandξMs= 380Ghz. Thepermeabil-\nity was calculated in MATLAB and imported into HFSS\nas a data table of real values (Real[ µ]) and loss tangent\nvalues (Imag[ µ]/Real[µ]). As previously mentioned, it is\nonlythex-componentof µr(ω) thatcontributesto thein-\nteraction. Usingthefitted parametersextractedfromthe\nempty structure we solved the analytic equation, Eq. 6,\nfor the first hybrid mode, 1 a/1b, using a numerical root\nfinding method implemented in MATHEMATICA. The\nmagnetic filling fraction, q, was determined to have a\nvalue of 0.013 through comparison to the HFSS simula-\ntions results. A comparison of the analytic dispersion\ncurve calculated from Eq. 6 versus the numerical simu-\nlations performed with HFSS is shown in Fig. 2(b) and\ndemonstrates excellent agreement with variation of the\nbias field.\nThe transmission through the CSRR/magnetic mate-\nrialwaveguidecanbe calculatedanalyticallybyassigning\nbulk electromagnetic parameters to the waveguide struc-\nture. The effective bulk permittivity of the waveguide\nstructure can be calculated by inputing Eq. 6 into Eq. 2\nand then solving the Fresnel equations [11] to determine\nthe transmission and reflection of the structure. This\nwas done for the CSRR/magnetic waveguide structure\nand compared to HFSS simulations as shown in Fig. 3.\nIn the HFSS simulations a splitting of the second order\nCSRR resonance can also be seen. In the analytic ex-\npression we were able to reproduce this by applying our63 64 65 66 \nFrequency (Ghz) \nFrequency (Ghz) Relative Permeability ξMs=380 Ghz \nξMs=189 Ghz \nξMs=38 Ghz \n60 62 64 66 68 70 \n-5 510 20 30 40 50 60 70 \n-100 -50 50 100 Magnetic Material \nHybrid Structure zero point (a)\n(b)\nRelative Permittivity \nFIG. 4: (a) The real part of the permeability is shown for sev-\neral magnetization values of the magnetic material with the\nbiasfieldchosentohavethesame zeropermeability frequenc y.\n(b) The resulting hybrid permeability is shown.\ntechnique to the first (electric) and second (magnetic)\nresonance of the CSRR structure. The overall agreement\nis seen to correlate well over a range of biasing values.\nIt is remarkable to note that the width and strength\nof the hybrid resonances is primarily a function of the\nCSRR’s resonant properties. The properties of the mag-\nneticmaterial’sresonance(andfillingfraction q)predom-\ninantly determines the bandwidth over which the hybrid\nmode interaction exists. In Fig. 4 the permeability of the\nmagnetic material, Eq. 4, and the associated effective\npermeability of the hybrid structure are plotted for vari-\nous value of magnetization Ms. When the biasing field is\nchosen such that the zero point of the magnetic material\ncorrelates to the empty CSRR structure resonance, we\nsee that the relative permeability strength of the hybrid\nstructure is the same but that the splitting of the res-\nonance is smaller for smaller values of Msvalues. This\nsuggest that it may be possible to harness narrowly reso-\nnant magnetic materials, that cannot be utilized directly,\nby using this metamaterial approach (though only over\na narrow bandwidth). These potential materials include\nhexagonal ferrites and even antiferromagnetic materials\nsuch as MnF 2[14].\nIn summary, the hybrid resonance that results from\ncombining a resonant magnetic material and CSRR\nstructure results in a unique hybrid resonance which can\nbe harnessed to make tunable metamaterial structures.\nGyromagnetic materials have a resonant permeability\ntensor form similar to that considered here but with the\nadded complexity of off-diagonal resonant components.\nAs a result, the resonant response of the combined struc-\nture has a more complicated dependance on the orienta-\ntion of the biasing direction of the gyromagnetic mate-\nrial with respect to the geometry of the CSRR structure\n[15, 16]. Preliminary work suggest that if we limit our-5\nselves to using thin layers we should expect to see an\nanalogous phenomenon as demonstrated here. Though\ngyromagnetic materials have been used directly to make\ntunable microwave devices [17, 18, 19], using them in-\ndirectly in metamaterial structures has the potential ad-\nvantageofincreasingthe rangeofeffective materialprop-\nerties while at the same time reducing the amount of the\nmagnetic material needed in the structure and its asso-\nciated losses.\nThis research was supported by U.S. Army Research\nOffice DOA under Grant No. W911NF-04-1-0247. We\nalsoacknowledgesupportfromtheAirForceOfficeofSci-\nentific Research through a Multiply University Research\nInitiative under Contract No. FA9550-06-1-0279.\n[1] V. G. Veselago, Soviet Physics Uspekhi 10, 509 (1964).\n[2] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-\nNasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).\n[3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J.\nStewart, IEEE MTT 47, 2075 (1999).\n[4] S. Lim, C. Caloz, and T. Itoh, IEEE Trans. on Microw.\nTheory. Tech. 52, 2678 (2004).\n[5] T. Hand and S. Cummer, J. of Appl. Phys. (2007).\n[6] C. Kittel, Introduction to Solid State Physics (Wiley &\nSons, Inc., 1996).[7] F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena,\nJ. Bonache, M. Beruete, R. Marqu´ es, F. Mart´ ın, and\nM. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).\n[8] J. D. Baena, J. Bonache, F. mart´ ın, R. M. Sillero, F. Fal-\ncone, T. Lopetegi, M. A. G. Laso, J. Garc´ ıa-Garc´ ıa,\nI. Gil, M. F. Protillo, et al., IEEE MTT-S Digest 53\n(2005).\n[9] C. Caloz, C. C. Chang, and T. Itoh, J. Appl. Phys. 90,\n5483 (2001).\n[10] V. S. Liau, W. S. T. Wong, S. Ali, and E. Schloemann,\nIEEE MTT-S Digest p. 957 (1991).\n[11] D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis,\nPhys. Rev. B 65, 195104 (2002).\n[12] R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith,\nPhys. Rev. E 76, 26606 (2007).\n[13] H. L. Glass, Proc. IEEE 76, 151 (1988).\n[14] M. Lui, J. Drucker, A. R. King, J. P. Kotthaus, P. K.\nHansma, andV.Jaccarino, Phys.Rev.B 33, 7720(1986).\n[15] R. Marques, F. Mesa, and F. Medina, IEEE Microwave\nand Guided Wave Letters 10, 225 (2000).\n[16] M. Horno, F. L. Mesa, F. Medina, and R. Marques, IEEE\nTrans. Microwave Theory Tech. 38, 1059 (1990).\n[17] N. Cramer, D. Lucic, R. E. Camley, and Z. Celinski, J.\nAppl. Phys. 87, 6911 (2000).\n[18] R. J. Astalos and R. E. Camley, J. Appl. Phys. 83, 3744\n(1998).\n[19] R. E. Camley and D. L. Mills, J. Appl. Phys. 82, 3058\n(1997)."    },    {        "title": "1207.3345v1.Surface_induced_Magnetism_Fluctuations_in_Single_Crystal_of_NiBi3_Superconductor.pdf",        "content": "arXiv:1207.3345v1  [cond-mat.supr-con]  13 Jul 2012Surface-induced Magnetism Fluctuations in Single Crystal of NiBi 3Superconductor\nXiangde Zhu,1,∗Hechang Lei,2C. Petrovic,2and Yuheng Zhang1\n1High Magnetic Laboratory, Chinese Academy of Sciences and U niversity of\nScience and Technology of China, Hefei 230026, People’s Rep ublic of China\n2Condensed Matter Physics and Materials Science Department ,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n(Dated: September 26, 2018)\nWe report anistropy in superconducting and normal state of N iBi3single crystals with Tc= 4.06\nK. The magnetoresistance results indicate the absence of sc attering usually associated with ferro-\nmagnetic metals, suggesting the absence of bulk long range m agnetic order below 300 K. However,\ntheelectron spin resonance results demonstrate that ferro magnetism fluctuationsexist on thesurface\nof the crystal below 150K.\nPACS numbers: 75.70.-i, 74.70.Ad, 76.30.-v\nI. INTRODUCTION\nFerromagnetism (FM) and superconductivity (SC) are\ntwo fundamental condensed matter phenomena. The for-\nmer one favors all spins parallel, while in the latter one\ncarriers condense in Cooper pairs where spins are an-\ntiparallel in classical BCS scenario. The superconduct-\ning properties of ferromagnetic superconductors are cur-\nrently under debate. The p-wave superconductivity was\nproposed to coexist with itinerant ferromagnetism.1So\nfar, the coexistence of FM and SC has been discovered\nin UGe 2,2URhGe,3and UCoGe.4A binary intermetal-\nlic compound, ZrZn 2was reported to exhibit coexistence\nof ferromagnetism and superconductivity.5Interestingly,\nsuperconductivity originates from the surface alloy rich\nin Zn.6In addition, the spin-singlet pairing supercon-\nductivity and magnetic order parameters entwine each\nother in a spatially modulated pattern, which allows for\ntheir mutual coexistence in for example borocarbides,7\nCeCoIn 5in high magnetic fields,8,9HoMo 6S8,10P doped\nEuFe2As2,11and ErRh 4B4.12\nNi-Bi based compounds show complex physics phe-\nnomena including antiferromagnetism and superconduc-\ntivity. LaNiBiO is isostructural to LnFeAsO ( Lnrep-\nresents the rare earth elements) iron based high tem-\nperature superconductors. CeNiBi 2is an antiferromag-\nnetic metal, whose structure is similar to that of iron\nbased superconductors.13Interestingly, when Ni defi-\ncient, CeNi 1−xBi2shows superconductivity.13Recently,\nNiBi3, an intermetallic superconductor with T c= 4.06\nK, attracted some attention, due to possible coexistence\nof ferromagnetism (Curie temperature <750 K) and\nsuperconductivity in polycrystals.14Coexistence of fer-\nromagnetism and superconductivity is also observed in\nnano structured crystals below 30 K.15The polycrystal\nNiBi3is reported to have large field dependent thermal\ntransport properties and magneto-resistance (MR).16In\naddition, the magnetic ions doping of Co results in Tc\nenhancement and onset of resistivity drop around 10 K\nin NiBi 3.17These results suggest that magnetic order is\nrelated to superconductivity in NiBi 3. However, other\ntwo groups reported that no magnetism was observed inNiBi3bulkcrystals.15,18Itshouldbenotedthatmagnetic\nimpurities cannot be avoidedduring synthesis process for\nnominal stoichiometric 1:3 composition according to the\nNi-Bi phase diagram.19Thus, it is necessary to investi-\ngate the magnetism of NiBi 3in impurity-free single crys-\ntals. Here, we report the comprehensive study of NiBi 3.\nWe show evidence of the ferromagnetism fluctuations in-\nduced at the surface below 160 K. No large MR or field\ndependantheatcapacityisobserved,whichindicatesthat\nno bulk magnetism exists in NiBi 3.\nII. EXPERIMENTAL\nSingle crystals of NiBi 3were grown from self-flux\nmethod with Ni:Bi = 1:10 mol ratio. This method\navoids the NiBi phase and other magnetic impurity in\nthe solid state reaction with of stoichiometric Ni:Bi (1:3)\naccording to the Ni-Bi phase diagram.20High purity Ni\nshot and Bi pieces were mixed and sealed in an evacu-\nated quartz tube. The tube was heated to and soaked\nat 1150◦C for 2 hours, then cooled down to 400◦C\nwith 5◦C/h. Finally, the tube was spun in a cen-\ntrifuge to separate the Bi flux. As is shown in Fig. 1\n(b), needle like, silver colored single crystals with size\nof∼2 mm×0.2 mm ×0.2 mm (with baxis of the\nlongest dimension) were obtained. The single X-ray data\nwere collected using the BrukerAPEX2 softwarepackage\nApex2 on a Bruker SMART APEX II single crystal X-\nray diffractometer with graphite-monochromatedMo K α\nradiation ( λ= 0.71073 ˚A) at room temperature. The\nobtained lattice parameters are a= 0.8955(0.0019) nm,\nb= 0.4153(0.0012) nm and c= 1.158(0.002) nm, which\nis consistent with the previous results.18The directions\nof axis were also determined.\nElectrical resistance measurements were performed us-\ning a four-probe configuration, with the applied current\nalong the b- axis. Thin Pt wires were attached to electri-\ncalcontactsmade ofepoxy. Electricaltransportandheat\ncapacity measurements were carried out in Quantum De-\nsign PPMS-9. The rotating sample holder was used to\nadjust the direction of the magnetic field. During the2\nFIG. 1. (a) Temperature dependence of ρb(open circle) and\nits fitted curves (dashed line) for NiBi 3. (b) The photograph\nof single crystal of NiBi 3. (c) Temperature dependence of ρb\nfor NiBi 3vicinity the superconducting transition. The ellipse\nmarks the initial resistivity drop.\nMR measurements, His perpendicular to the b- axis.\nElectron spin resonance (ESR) measurement was carried\nout in Bruker EMX-plus model spectrometer ( ν∼9.4\nGHz).\nIII. RESULTS AND DISCUSSIONS\nFigure 1(a) shows the temperature ( T) dependence of\nresistivity along the b-axis (ρb) for NiBi 3single crystal.\nIts superconducting transition can be seen in Fig. 1(c).\nThe obtained Tc0is about 4.06 K, which is consistent\nwith former results. The superconducting transition is\nsharp with a transition width ∆ Tc∼0.04 K. Interest-\ningly, the ρb−Tcurve has a initial drop around T=\n4.3 K (Fig. 1(c)). Therefore, it rules out the possibility\nof the NiBi (S.G. P63/mmc, NiAs type) impurity with\nTc= 4.25 K as the origin of the blunt specific heat ( C)\ntransition in the region from 4.1 K to 4.2 K in the poly-\ncrystalline sample.21The residualresistivity ratio(RRR)\nis 19.7, higher than that of the polycrystalline sample.21\nThe initial drop is reminiscent of filamentary supercon-\nductivitythatprecedesbulkphasecoherence,asobserved\nin for example quasi-1D metal Nb 2Se3.22Theρbrises\nwith positive curvature with increasing temperature be-\nlow 22 K, and increases with a linear ρb−Trelation up\nto∼60 K. Then, ρbincreases with a negative curvature\nabove 60 K, and shows a saturation tendency towards\nhigh temperatures. The overall ρb(T) follows the empir-\nical model initially applied to A15 superconductors such\nas Nb3Sn.23The model gives\nρ=ρ0+ρ1·T+ρ2·exp(−T0\nT) (1)0 5 10 15 20 25 30 35 40050100150200250300\n0 25 50 75 100 125 1500204060801001202 3 4 5-10-50510152025\n012345678\n H = 0 kOe\n H = 20 kOe\n fiting \n  C/T(mJ K-2mol-1)\nT2(b)\n  C (J mol-1 K-1)\nT (K)3NR(a)\n(c) \n ∆Ces/T (mJ K-1mol-1)\nT (K)∆Ces ρb (µΩ cm)\nFIG. 2. (a) T2dependence of C/Tof NiBi 3forH= 0 kOe\n(solid circle) and H= 20 kOe (open circle); the linear fitting\nbetween 3 K and 6 K for H= 20 kOe. (b)The Tdependence\nof ∆Ces=C(H= 0kOe)−C(H= 20 kOe))(Left panel) and\ntheTdependence of ρb(right panel) for NiBi 3. (c)The T\ndependence of Cfor NiBi 3. The dashed line represents the\n3NR, where N and R is the number of atoms in molecule and\nuniversal gas constant, respectively.\nwhere,ρ0istheresidualresistivity; ρ1,ρ2, andT0arema-\nterial dependent parameters. The third term arises from\nthe phonon-assisted scattering between the two Fermi-\nsurface sheets. The fitted curve is shown in Fig. 1(a)\nas the dashed line. ρ0= 7.453±0.016µΩ·cm,ρ1=\n0.244±0.004µΩ·cmK−1,ρ2= 96.2±1.5µΩ·cm, and\nT0= 103.8±0.7K are determined from the fitted results.\nThese values are comparable with those of Nb 3Sn.23\nFigure 2(a) shows the T2dependence of specific heat\n(C) divided by ( T) of NiBi 3single crystal for H= 0 kOe\nandH= 20 kOe. Obviously, no magnetic specific heat\ncontributioncanbeobserved. Around4K,aspecificheat\njump due to superconducting transition can be observed.\nThe superconducting transition can be suppressed by H\n=20kOe. Interestingly,ifweusetheSommerfeld-Debye\nexpression C=Ce+Cl=γT+βT3(γTis the electron\ncontribution, and βT3is the lattice contribution) to fit\nthe curve, the linear C/T−T2relation between 3 K and\n6 K gives unphysical negative γ. This result is consistent\nwith the previous reports.18,21Yet,β= 8.255±0.02 mJ\nmol−1K−4is obtained from the linear fitting between 3\nK and 6 K. Θ D= 98.0±0.1K can be estimated from the\nΘD=3/radicalbig\n12π4NR/5β, where N is the number of atoms\nper molecule. In addition, the C/T−T2deviates from\nlinearity below 3 K. Adding the anharmonic contribution\n∼T5does not improve the fit. This is most likely due to\nthe low Debye temperature (Θ D∼100 K).21Most likely\nmeasurements of heat capacity below 2 K are necessary\nin order to reliably estimate γ. At high temperature,\nCfor NiBi 3approaches the ideal value of 3NR as the\nDulong-Petit law predicts (Fig. 2(c)).3\n10-11001011021E-30.010.1110\n0 200 400 600 800 10000.00.51.01.52.02.53.03.54.0\n H/ρxx(0T) (kOe Ω-1cm-1) T = 5K\n T = 10K\n T = 20K\n T = 40K\n T = 60K∆ρxx/ρxx(0T)×10-2~ (H/ ρxx(0T))2 T = 80K\n T = 100K\n T = 150K\n T = 200K\n T = 250K\n T = 300K ∆ρxx/ρxx(0T)×10-3\nH/ρxx(0T) (Oe Ω-1cm-1)\nFIG. 3. (a) MR the Kohler’s log-log plot in single crystals\nof for NiBi 3atT= 5, 10, 20, 40, 60 K. (b) MR at different\ntemperatures and the Kohler’s plot in single crystals of for\nNiBi3atT= 80, 100, 150, 200, 250, 300 K.\nIn order to investigate the specific heat jump of super-\nconducting transition, C(H=0 kOe)- C(H= 20kOe) is\nintroduced to estimate the electronic specific heat in the\nsuperconductingstate. Figure2(b)showsthe∆ Ces(C(H\n= 0 kOe) - C(H= 20 kOe)) and ρb−Tcurves. The\n∆Ces/T∼20 mJ mol−1K−2. From the McMillan for-\nmula\nλe−p=µ∗ln(1.45Tc\nΘD)−1.04\n1.04+ln(1.45Tc\nΘD)(1−0.62µ∗),(2)\nwe estimate the electron-phonon coupling constant λe−p\n∼0.91 by assuming µ∗= 0.13, which is a typical value\nfor the Coulomb pseudo-potential.26This indicates that\nNiBi3is a strongly electron-phonon coupled supercon-\nductor.\nSemiclassical transport theory predicts that Kohler’s\nrule will be valid if there is a single species of charge\ncarrier and the scattering time τis the same at all points\non the Fermi surface. MR at different temperatures can\nbe scaled by the expression eq.(3) with the assumption\nthat scatteringrate 1 /τis proportionalto ρ(T), andωcis\nthe frequency at which the Hcauses the charge carriers\nto sweep across the Fermi surface:\n∆ρxx(H,T)\nρxx(0,T)=f(ωcτ) =f(H\nρ(0,T)),(3)\nand the corresponding plots are known as Kohler’s plots.\nFigure 3(a) depicts the Koler’s plot for NiBi 3from 5\nK to 60 K. Even at H⊥b= 9 T, the MR is only about\n4.5% atT= 5 K. This value is verytypical for a nonmag-\nnetic metal. All curves collapse onto a single line, sug-\ngesting that the Kohler’s rule is valid (∆ ρxx(H)/ρxx(0)\n∝(µ0H/ρxx(0))2). Figure 3(b) depicts the MR data\n∆ρxx(H)/ρxx(0) as a function of µ0H/ρxx(0) above 80\nK. Apparently, Kohler’s rule is violated above 80 K. Due\nto the crystal structure, NiBi 3should be a quasi-one di-\nmensional system, whose Fermi surface topology is not0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 80 1 2 3 4 5 6 7 80 20 40 60 80 100 120 140\n H ⊥ b - axis \n \nH (kOe)T = 300K\nT = 250K\nT = 200K\nT = 160K\nT = 100K\nT = 50K\nT = 10KIntensity (a.u.) (a)\nT = 10KT = 50KT = 100KT = 160KT = 200KT = 250K \n Intensity (a.u.) \nH (kOe)T = 300KH || b - axis\n(c) (b)H || b -axisH ⊥ b -axis\nT  = 1.85 K \n Intensity (a.u.) \nH (kOe)H || b -axisH ⊥ b -axis \n Intensity (a.u.)\nH (Oe)\nFIG. 4. (a)The ESR spectra of NiBi 3measured at 1.85 K\nwith sweeping Hparallel to the b- axis and perpendicular to\nb-axis. Inset shows the amplified plots in the low field region.\nThe ESR spectra of NiBi 3measured at different temperatures\nwith sweeping Hparallel to the b- axis (b) and perpendicular\ntob- axis (c). The curves are shifted for clarity. The dashed\nlines mark the maximum of the ESR spectrums.\nspherical. The possible explanation is that multiband\nconductivity in NiBi 3. Below 60 K, the contribution of\none band (with large ωcτ) dominates whereas the con-\ntribution of other bands is negligible. The system can\nthen be regarded as a single band conductor. Above 80\nK the contributions are comparable, leading to deviation\nof the Kohler’s rule. Similar phenomenon has been ob-\nserved in low dimensional system, such as LiFeP.27What\nis important, the MR is positive below 300 K, indicating\nthat NiBi 3is not ferromagnetic in bulk.\nFigure 4(a) shows the ESR ( dP/dH) spectra of NiBi 3\nmeasured at 1.85 K with sweeping Hparallel to the b-\naxis (H∝bardblb) and perpendicular to b-axis (H⊥b). Ob-\nviously, anisotropy can be observed. Both spectra show\ntypical ESR signals for supercondcuting state. As shown\nin the inset, a stepwise sharp signal appears at low fields\ntypical for the magnetic-shielding feature, indicating the\nappearanceof superconductivity. Then, ESR signalshow\na hump below Hc2with increasing H.\nFigure 4 (b) and (c) show the ESR spectra of NiBi 3\nmeasured at different temperatures (above Tc) with\nsweeping H∝bardblbandH⊥b, respectively. No obvious\nresonance signal around H= 3200 Oe for paramagnetic4\nelectron can be observed. Interestingly, the ESR spectra\nof NiBi 3show abnormal diplike signal at low field below\n160 K, which can be suppressed by increasing tempera-\nture. As marked by the dashed lines in Fig. 4 (b) and\n(c), the magnetic fields corresponding to diplike signal\nare anisotropic. In heavy fermion system CeRuPO, FM\ncorrelationsis define bythe ESRsignal.28At lowtemper-\nature below 10 K (Curie temperature = 15 K), ESR of\nCeRuPOshowssimilardiplikesignalsat lowfields aswell\nas NiBi 3. This diplike signal should be related to ferro-\nmagnetic momentfluctuations. At high temperature, the\ndiplike signal of NiBi 3disappears. The microwave signal\nof ESR can only exist on the surface of a metal (in µm\nlength).29This strongly suggests that ferromagnetic mo-\nments fluctuations exist the surface on NiBi 3below 160\nK, and that NiBi 3is non-magnetic in bulk. The crystal\nstructure of NiBi 3can be regarded as packing of NiBi 3\nrods along the b- axis. In the bulk, the coordination of\nNiBi3rods has perfect translational symmetry; while on\nthe surface, the translational symmetry is broken. The\nFM fluctuations in NiBi 3could be attributed to the sur-\nface effect since surface tension modifies the electronic\nband structure, favoring FM on submicron length scale\nin nanostructured samples.15\nFIG. 5. Temperature dependence of the resistivity ρb(T) of\nNiBi3for (a)H/bardbla, (b)H/bardblband (c)H/bardblca at the various\nmagnetic fields. (d) Temperature dependence of the upper\ncritical field Hc2obtained from the midpoint and 10% on the\nρb−Tcurves for H/bardbla,H/bardblbandH/bardblc.\nFigure 5 (a), (b), and (c) show the ρb−Tcurves\nat the various magnetic fields for H∝bardbla,H∝bardblb\nandH∝bardblc, respectively. The upper critical field\n(Hc2) is obtained from the midpoint and 10% resistiv-\nity on the ρb−Tcurves, which is depicted in Fig. 5\n(d). Obviously, Hc2show linear temperature depen-\ndence in the experimental temperature range. The ob-\ntained [dHmid\nc2/dT]|Tc([dH10%\nc2/dT]|Tc) are 1.48(1.22),\n3.13(2.60), and 1.17(1.14) kOe/K for H∝bardbla,H∝bardblb,TABLE I. Parameters for NiBi 3.\nTc0 4.06 K\nρ0 7.453±0.016µΩ·cm\nρ1 0.244±0.004µΩ·cmK−1\nρ2 96.2±1.5µΩ·cm\nT0 103.8±0.7 K\nρsat 328±2µΩ cm\nAac 0.886±0.001µΩ·cmK−1\nλe−p ∼0.91\nΘD 98.0±0.1K\nHc(0) ∼430 Oe\na b c\ndHc2/dT|Tcmid 1.48 3.13 1.17 kOe/K\n10% 1.22 2.60 1.14 kOe/K\nHc2(0)mid 6.0 12.7 4.75 kOe\n10% 4.95 10.56 4.62 kOe\nHc1(0)mid 65 41 75 Oe\n10% 71 47 74 Oe\nξ(0)mid 18.1 38.3 14.3 nm\n10% 18.3 39.0 17.1 nm\nκ(0)mid 9.9 20.9 7.8\n10% 8.14 17.4 7.6\nλGL(0)mid 180 800 112 nm\n10% 149 678 130 nm\nma:mb:mcmid 4.48 : 1 : 7.15\n10% 4.58 : 1 : 5.02\nandH∝bardblc, respectively. According to the conven-\ntional one-band Werthamer-Helfand-Hohenberg (WHH)\ntheory,30which describes the orbital limited Hc2of dirty\ntype-II superconductors, Hc2= 0.693[dHc2/dT]|TcTc.\nThenHa\nc2(0),Hb\nc2(0),Hc\nc2(0) can be obtained as 6(4.95),\n12.7(10.56), and 4.75(4.62) kOe. The data in the brack-\nets here and the following discussion are within the 10%\nresistivity results.\nIn the anisotropic superconducting system, coherence\nlength (ξi\nGL, whereiis the axis) can be estimated from\nHa\nc2=Φ0\n2πξbξc,Hb\nc2=Φ0\n2πξaξc,Hc\nc2=Φ0\n2πξaξb. The calcu-\nlatedξa\nGL,ξb\nGL,ξc\nGLare 18.1(18.3) nm, 38.3(39.0) nm,\nand 14.3(17.1) nm, respectively. The effective mass ten-\nsor is related to the GL coherence length as ma:mb:mc\n= 1/ξa\nGL2: 1/ξb\nGL2: 1/ξc\nGL2. The calculated effective\nmass ratio ma:mb:mc4.48:1:7.15(4.58:1:5.02). Ob-\nviously, NiBi 3shows a quasi one dimensional behavior\nalong the b- axis, with effective mass ratio m∗\na≃m∗\ncand\nm∗\ni/m∗\nb≃5 (irepresents aandc).\nThe thermodynamic critical magnetic field at T\n= 0 K ( Hc(0)) can be estimated through Hc(0) =\n[4πN(EF)∆2(0)]1/2. According to the band calculation,\nthe density of states at Fermi surface N(EF) is 2.55\nstates/eV f.u.18By assuming the ∆(0) = 1 .76kBTc,Hc\nis estimated to be ∼430 Oe. This value is consistent\nwith experimental results.18GL parameter κ, the lower\ncritical field at zero temperature ( Hc1(0)), and the pen-\netration depth λGLatT= 0 K can be estimated from:\nHi\nc2=√\n2κiHci,Hi\nc1=Hclnκi−0.18√\n2κi,and λi\nGL=κiξi\nGL,\nwhereirepresents the direction of a,b,c. The estimated5\nparameters in the superconducting state and the param-\neters in the normal state mentioned above are listed in\nTab. I.\nIV. SUMMARY\nWe investigated anisotropy in superconducting and\nnormal state properties of NiBi 3. The carrier scattering\nmechanism is dominated by electron-phonon scattering\nwhereas MR suggests the absence of magnetic scatter-\ning commonly observed in bulk FM materials. The heat\ncapacity results indicates it is a strongly e−pcoupling\nsuperconductor. However, FM fluctuations are detected\nat the surface of the crystal. Finally, we give anisotropicsuperconducting parameters.\nV. ACKNOWLEDGEMENT\nWe thank Dr. Wei Tong for the help on ESR measure-\nments. Work at Brookhaven National Laboratory (H. L.\nandC.P.)wassupportedbytheUS DOEunderContract\nNo. DE-AC02-98CH10886. Work at High magnetic field\nlab (Hefei) was supported by the State Key Project of\nFundamental Research, China (2010CB923403) and Na-\ntional Basic Research Program of China (973 Program),\nNo. 2011CBA00111.\n* Email:xdzhu@hmfl.ac.cn\n1D. Fay and J. Appel, Phys. Rev. B 22, 3173 (1980).\n2S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K.\nW. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S.\nR. Julian1, P. Monthoux, G. G. Lonzarich, A. Huxley, I.\nSheikin, D. Braithwaite, and J. Flouquet, Nature 406, 587\n(2000).\n3D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flou-\nquet, J.-P. Brison, E. Lhotel, and C. Paulsen, Nature 413,\n613 (2001).\n4N. T. 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Lett. 100, 066401 (2008).\n29Skin effect depth =/radicalbig\nρ/πfµ, where ρis the resistivity\nof the conductor in m,fis frequency in Hertz, and µis\nthe absolute magnetic permeability of the conductor. The\nabsolute magnetic permeability µ=µ0·µr, whereµ= 4π×\n10−7H/m, and µris the relative magnetic permeability.\nSince NiBi 3is not magnetic in bluk, here we use µr= 1 to\nestimate the Skin effect depth. As for NiBi 3,f= 9.4Ghz,\nρ= 8 - 150 ×10−8ohm m. The skin depth should be 1.5 -\n6.4µm.\n30N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Phys.\nRev.147, 295 (1966)."    },    {        "title": "1005.0452v3.Effect_of_disorder_studied_with_ferromagnetic_resonance_for_arrays_of_tangentially_magnetized_sub_micron_Permalloy_discs_fabricated_by_nanosphere_lithography.pdf",        "content": "arXiv:1005.0452v3  [cond-mat.mes-hall]  2 Jul 2010Effect of disorder studied with ferromagnetic resonance for arrays of\ntangentially magnetized sub-micron Permalloy discs fabri cated by nanosphere\nlithography\nN. Ross,1,a)M. Kostylev,1,b)and R. L. Stamps1\nSchool of Physics, University of Western Australia, Crawle y, WA, Australia\n(Dated: 17 May 2022)\nTangentially magnetized trigonal arrays of sub-micron Permalloy dis cs are characterized with ferromagnetic\nresonance to determine the possible contributions to frequency a nd linewidth from array disorder. Each array\nis fabricated by a water-surface self-assembly lithographic techn ique, and consists of a large trigonal array of\n700 nm diameter magnetic discs. Each arrayis characterizedby a diff erent degree of ordering. Two modes are\npresent in the ferromagneticresonance spectra: a large amplitud e, ‘fundamental’ mode and a lower amplitude\nmode at higher field. Angular dependence of the resonance field in a v ery well ordered array is found to be\nnegligible for both modes. The relationship between resonance freq uency and applied magnetic field is found\nto be uncorrelated with array disorder. Linewidth is found to increa se with increasing array disorder.\nKeywords: ferromagnetic-resonance, linewidth, nanosphere-lit hography, sub-micron-disc\nI. INTRODUCTION\nThere is currently an intense interest in the magnetic\nproperties of nanoscale and sub-micron discs of low as-\npect ratio. Such discs have potential applications in data\nstorage,1–5spintronics,6,7and medicine,8,9and areother-\nwise viewed as simple model systems by which the prop-\nerties of more exotic nanostructures can be understood.\nIn many fundamental studies and in data storage and\nspintronics applications in particular, the dynamic mag-\nnetic propertiesofthe disc structuresareparticularlyim-\nportant. Where arraysofdiscs aredensely packed, dipole\ncoupling between discs can have a significant impact on\ndynamic behaviour. There have been a variety of exper-\nimental and theoretical studies of the dynamic magnetic\nproperties of arrays of dipole-coupled magnetic discs,\nwith recent publications concerning resonance frequency\npositionwithrespecttoappliedmagneticfield,10–13mode\nstructure in the vortex state,14,15and mode structure\nfor in-plane12,16–25and out-of-plane11,26,27applied mag-\nnetic fields. Recognition of the importance of spin-wave\ndamping for many applications has led to a number of\nstudies into ferromagnetic resonance (FMR) linewidth\nbroadening in such systems,28–30and recently in a study\nof microwave-assisted switching of magnetic array nano-\nelements.31To date, however, an experimental study of\nthe effects of array packing on damping of spin wave\nmodes observable with FMR has been lacking.\nOne approach to such a study is to measure FMR\nlinewidth damping in arrays that were patterned by a\n‘top-down’ method like Focussed Ion Beam lithography\nin such a way that they had identical disc geometries but\nvaried pitch (inter-disc spacing). Such fabrication tech-\nniques are time-intensive and therefore often only suit-\na)Electronic mail: rossn2282@gmail.com\nb)Electronic mail: kostylev@cyllene.uwa.edu.auable for the production of small arrays ( <1 mm2). The\nFMRresponsesofsmallarraysarecorrespondinglysmall.\nSuchsmallsignalsareoften beyondthe sensitivityofcon-\nventional FMR techniques, so that the previous experi-\nmental studies of damping in arrays of nano-discs uti-\nlized time-resolved Kerr microscopy28,30and ‘meander-\nline’ FMR29for the detection of spin wave modes . In\nthis study, nanosphere lithography has been used to fa-\ncilitate the production of large-areaarrays, thus allowing\nmeasuring using conventional Vector Network Analyzer\nFMR (VNA-FMR).\n‘Nanosphere lithography’ refers broadly to techniques\nin which the tendency of colloidal particles of sub-micron\nor nanoscale size to self-assemble on hydrophilic surfaces\nis exploited to produce a lithographic mask; the rele-\nvant literature is replete with various examples of such\ntechniques.32–43The most pronounced limitation of such\ntechniques is the difficulty of achieving long-range order-\ning in the colloidal mask.\nIn this study, this tendency towards partial disorder\nwas used to fabricate a series of array samples with var-\nied degrees of array ordering. A nanosphere lithographic\ntechnique involving assembly of colloidal particles into a\nmonolayer on a water surface41was used to produce ar-\nrays of sub-micron magnetic discs with deliberately var-\nied degrees of array order but the same local disc-to-disc\nspacing and disc geometry. The FMR mode structures\nand linewidths of this series of array samples were mea-\nsured by VNA-FMR.\nII. EXPERIMENT\nFour sub-micron magnetic disc array samples with dif-\nferent degrees of ordering were fabricated by a technique\nderived from that of Weekes et al.41Three continuous\nNi81Fe19(Permalloy) films, thickness L= 27±3 nm,\nwereradio-frequencymagnetronsputteredonsiliconsub-\nstrates with 40 nm Ta seed layers: these films are de-2\nnoted f1, f2, and f3 in this study. Sputtered thickness\nwas verified by the measurement of step edges of inde-\npendently sputtered films using white-light optical pro-\nfilometry. Each of these films was placed on a raised\nplatform in a narrow bath filled with 18 MΩ·cm wa-\nter, so that the Permalloy face of the film was under\nthe water surface but as close to it as possible. Sodium\nDodecyl Sulphate (SDS) was added to this bath to a con-\ncentration of 150 µg L−1: it has been demonstrated in\nthe literature that SDS can aid the assembly of reagent\nmicrospheres on a water surface.42,43780 nm diame-\nter carboxylate-terminated polystyrene reagent micro-\nspheres obtained from Duke Scientific were mixed with\nreagent-grade ethanol into a 1:2 microsphere solution :\nethanol mixture by volume. This mixture was intro-\nduced to the water surface via a hydrophilic glass slide\npreviously cleaned in SDS. Teflon sliders placed laterally\nacross the water bath were used to gently agitate the\nwater surface, aiding in the self-assembly of well ordered\ntrigonal monolayers of microspheres. These sliders were\nused to position the monolayers above the film, and the\nwater drained quickly from the bath in order to transfer\nthe monolayer to the film surface. Varying the degrees\nof agitation and of monolayer compression during the\ntransfer stage allowed control over the degree of order-\ning of the resulting lithographic mask. The assembly of\nthe lithographic mask on the water surface is depicted\ndiagrammatically in Figure 1.\nFIG. 1. Schematic cross-section representation of the depo -\nsition of the lithographic mask, followed by agitation of th e\nwater surface with laterally placed teflon sliders to aid ass em-\nbly of trigonal monolayers.\nOnce masked with the trigonal array of microspheres,\nseveral 5 ×8 mm2sections were cut from each of the par-\nent films f1, f2, and f3: one such section from each parent\nfilmwasleftmaskedbutwasnotpatterned; theothersec-\ntionswerepatterned. Inthisstudy, un-patternedsectionsoftheparentfilmaredenotedf1c, f2c, andf3c. Fourfilms\nwith different degrees of mask ordering were patterned:\nin this study they are denoted f1a, f2a, f3a, and f3b to\nmake clear the identity of the film from which each was\npatterned. Each film was patterned by placing it in the\ntarget position of a radio-frequency sputtering chamber\nwhichwasevacuatedtoabasepressureof1 .0×10−6Torr.\nSamples were reactively ion etched in an oxygen atmo-\nsphere of 100 mTorr at a power density of 0 .5 W cm−2in\norder to reduce the diameter of the mask units. Follow-\ning etching, each sample was argon milled at a pressure\nof25mTorrandpowerdensityof4 .25W cm−2toremove\nPermalloy not masked by microsphere material.\nFIG.2. AFMimage representativeofthemicroscale characte r\nof the disc arrays; the high contrast regions in the centre of\nthe discs indicate that some of the polystyrene caps remain.\nThis particular image is a 5 ×5µm image of sample f3b.\nCharacterization of disc geometry was achieved with\ntapping-mode Atomic Force Microscopy (AFM): an ex-\nample AFM image is shown in Figure 2. With the scan\nparametersused, theAFMcouldbeusedtoresolvestruc-\ntures down to 20 nm. The high structures in the cen-\ntre of the discs in the AFM image are the remains of\npolystyrene caps. The discs in all of the arrays have very\nsmooth edges and the local order is very good. Values for\naveragediameterswereextractedbytakingcross-sections\nof rows of dots from several such images from each sam-\nple.\nStatic magnetic properties of the array samples were\nmeasured using a SQuID magnetometer. Hysteresis\nloops were recorded at T = 295 K. Each sample dis-\nplayed the double ‘closed loop’ characteristic of vortex\nnucleation, movement, and annihilation in sub-micron\nmagnetic discs. Shown in Figure 3 is the loop corre-3\n/Minus2000/Minus1000 010002000/Minus0.0004/Minus0.00020.00000.00020.0004\nH/LBracket1Oe/RBracket1Μ/LBracket1emu/RBracket1\nFIG. 3. SQuID-measured hysteresis loop of f3b, typical of\nthe loops recorded for all the patterned films; the fit lines in\nthis figure are linear interpolations that serve as a guide fo r\nthe eye. The hysteresis loop shows the two closed loops char-\nacteristic of vortex nucleation, movement, and annihilati on.\nThe narrowness of the ‘pinching off’ of the two loops provides\nconfirmation for the whole sample of smooth dot boundaries\nseen in Figure 2.\nsponding to sample f3b, the same sample shown in the\nAFM image in Figure 2. Hysteresis loops of the other\nsamples displayed the same characteristics.\nArray regularity of each patterned sample was mea-\nsured using Scanning Electron Microscopy (SEM). Four\nevenly spaced 3600 ×magnification images were taken\nfrom each 1 mm section of the centre line of the long\naxis of the film. Each of these sets of four images were\nstitched together to make a composite image, and the 2-\ndimensional Fourier transform of each composite calcu-\nlated. An example of such a single SEM image with the\nFourier transform of the composite image from a differ-\nent section inset is shown in Figure 4. The length of each\nsample was L= 8 mm. The Fourier transform from each\n1 mm section ofthe sample wasusedto measure∆ φi, the\nvariation of lattice angle over that one millimeter. The\neight separate ∆ φivalues were averaged to yield φ′, the\naverage variation in lattice angle per unit millimeter of\nthe film: φ′=/summationtext8\ni=1∆φi/L.φ′was used to characterize\nthe array ordering of each patterned sample.\nEachofthe continuousandpatternedfilmswascharac-\nterizedusingmicrostrip-waveguideVNA-FMR. Thesam-\nple was placed face-down on an 8 milli-inch copper mi-\ncrostrip waveguide in an in-plane saturating magnetic\nfield. A network analyzer provided microwave excita-\ntion to the waveguide and measurement of transmission\nparameter S21. Negligible reflections allowed S11to be\nignored.44In contrastto how VNA-FMR is typically per-\nformed, the excitation frequency fwas fixed and the ap-\npliedmagneticfield Hvaried, inanalogytoacavityFMR\nmeasurement. The measurements were performed at 1\nGHz intervals in the domain 7-20 GHz. A typical spec-\ntrum is shown in Figure 5. Two modes can be resolved:\na large amplitude mode, and a smaller amplitude mode\nat higher field (lower frequency). Resonance fields Hres\nFIG. 4. An example SEM image with inset 2-D Fourier trans-\nform of a montage of 4 evenly spaced images taken over a one\nmillimeter section of the sample: these particular images a re\nalso of sample f3b. Note that the SEM image shown was not\namong the images used to produce the composite FFT. Im-\nages similar to the inset from each 1 mm section of the sample\nwere used to calculate the average variation in lattice angl e\nper unit length, φ′, for each sample.\nand linewidths ∆ Hwere extracted from these measure-\nments by least-squares fitting Lorentzian curves to the\ntroughs in S21, using at least twenty-five data points on\neach side of the minimum. fvsHresdata for continuous\nsamples f1-3c was fitted with the Kittel equation45with\ndemagnetizing factors Nx=Nz= 0,Ny= 4π:\nω2/γ2=(H+HK+(Ny−Nz)MS)×\n(H+HK+(Nx−Nz)MS).(1)\nand saturation magnetizations 4 πMSextracted. Cav-\nity FMR measurements were performed to verify the ob-\nserved mode structure.\nAfter characterization of all samples, sample f3b was\ncut to obtain a smaller sample (5 ×5 mm2) with a much\nhigher degree of ordering. This sample, denoted f3b*,\nwas then characterized in the same way as the other\nfour samples. Additionally, FMR spectra were recorded\nfor this sample at an excitation frequency of 10 GHz\nfor various relative angles φbetween the static applied\nmagnetic field and the lattice vector defining the line\nbetween nearest neighbours. In addition to providing\na more ordered sample, comparison of FMR results be-\ntween f3b and f3b* allowed the possibility of parent film-\nor patterning-caused differences between array samples\nto be more reliably discounted.4\n210024002700300033000.9900.9920.9940.9960.9981.000\nH/LBracket1Oe/RBracket1S21/LBracket1Normalized/RBracket1\nFIG. 5. Plot of normalized intensity of transmission ampli-\ntudeS21vs applied magnetic field Hfor sample f2a at an\nexcitation frequency of 16 GHz. In addition to the large-\namplitude fundamental mode, there is a small mode at higher\napplied field.\nIII. RESULTS AND DISCUSSION\nA. Film structure\nThe disc diameters as measured by AFM, array order-\ning values as measured by SEM, and parent film magne-\ntization values as measured by VNA-FMR are presented\nin Table I. The variation in disc diameter is on the or-\nder of five percent, which is the manufacturer’s quoted\nuncertainty on the size of the reagent microspheres used\nand therefore the best uniformity achievable by this fab-\nrication technique: the disc sizes of all four samples were\nidentical within the limits of experimental uncertainty.\nThe narrowness of the ‘waist’ of the SQuID hysteresis\nloops of the samples (Figure 3) showed that there were\nvery few uncompensated magnetic moments at rema-\nnence, likely indicating very little roughness to the disc\nboundaries.\nEach of the five patterned samples was distinguished\nby a unique ordering parameter φ′, although there was\nsome overlap between f3a and f2a at the limits of exper-\nimental uncertainty. Each of the three parent films’ sat-\nuration magnetizations MSwere slightly different. The\nlinewidths of these films at a given frequency, ∆ Hwere\nalso different. For this reason, VNA-FMR data for the\nparent continuous films f3, f3, and f1 are included in the\ntables in this section. In Figure 8 the linewidths for the\nparent continuous films are presented alongside those of\nthe corresponding patterned films. In the comparison of\ninhomogeneous broadenings between films shown in Fig-\nure 9 these parent film linewidths have been accounted\nfor.Filmd [nm] φ′[◦mm−1] 4πMS[kOe]\nf3c - - 8.49\nf3b*695±28 2.6±0.4 -\nf3b695±28 6.0±0.8 -\nf3a703±37 9.4±1.1 -\nf2c - - 8.69\nf2a697±31 11.3±1.7 -\nf1c - - 8.85\nf1a699±28 19.9±2.1 -\nTABLE I. Table showing average disc diameter, d, array vari-\nation per unit length φ′, and saturation magnetization 4 πMS\nfor the samples used in this study.\nB. FMR mode structure\nA similar mode to the high-field mode in Figure\n5 has been calculated to exist for isolated discs of\nlarger aspect ratio L/dthat those fabricated for this\nstudy, using OOMMF30and linearized micromagnetic\napproaches.23,24In these publications the mode is calcu-\nlated to be located in the ‘end’ of the disc as defined by\nthe direction external magnetic field. These two studies\nalso report experimental measurements of these modes\nin closely packed disc systems. Such modes have also\nbeen measured elsewhere in systems of comparable disc\ndimensions to those fabicated for this study.12,18VNA-\nFMRlacksthekindofspatialresolutionoftechniqueslike\nmagnetic resonance force microscopy, and in this study\nthepossibleconfinementofthehigh-fieldmodetospecific\nregions of the discs could not be investigated directly.\nSeveral possibilities for the origin of the mode were\nruled out experimentally. The possibility that the mode\nwas caused by a non-uniform excitation field result-\ning from microwave screening by eddy currents in the\nsample47was ruled out by measurement of the mode\nstructure by cavity FMR. In an FMR cavity, the excita-\ntionfield isveryuniform. Figure6showsthecavityFMR\nspectrum. The higher field mode is still present, indicat-\ning that it does not result from non-uniform microwave\nexcitation. The amplitude of this mode as measured by\nVNA-FMR did not vary significantly between samples of\ndifferent degrees of ordering. This ruled out the possibil-\nity of the high-field mode being the result of a collective\nexcitation occurring at discontinuities or around vacan-\ncies in the array, which are more numerousin moredisor-\ndered samples. Finally, the out-of-plane saturated FMR\nspectrum–notincludedhereforbrevity–showedupto five\nwell resolved modes in the structure expected for cylin-\ndrical symmetry,26suggesting that the high-field mode\nwasnot the product ofsome repeated non-cylindricalele-\nment. Remaining possible explanations for the high-field\nmode are that it is an ‘end’ or ‘edge’ mode as in Refer-\nences 18, 24, and 30, or that it arisesout of or is modified\nby some collective effect of the array itself.5\n1000 1200 1400 1600\nH/LBracket1Oe/RBracket1Re/LBracket1dS21/Slash1dH/RBracket1\nFIG. 6. Plot of the real part of dS21/dHvsHfor sample\nf1a at an excitation frequency of 9.55 GHz, as measured by\ncavityFMR. The high-field mode seen in Figure 5is preserved\nintheuniformexcitation fieldoftheFMRcavity. Nounitsare\nincluded on the vertical axes because the scale is dependent\non the particular tuning of the cavity.\nThe relationship between the resonance field Hresof\nboth modes and the relative lattice-applied field angle φ\nwas measured for sample f3b*. Neither mode showedany\nangularvariationbeyondthe the 15 and 20Oe scatterfor\nthe fundamental and high-field modes, respectively. This\nscatter was caused by a combination of small movements\nof the sample in the applied magnetic field when varying\nφand uncertainty in the baseline of the Lorentzian fit\nused to extract the values of Hres. This measurement\nrestricts the angular variation of Hresto be below these\nvalues of uncertainty. Additionally, the amplitudes of\nboth modes were essentially independent of φ.\nIn previous studies by other authors on square arrays\nof sub-micron discs of similar diameter-to-pitch ratio,\nstrong dependencies on the relative array-field angle φ\nof the resonance field have Hreshave been measured for\nboth fundamental and high-field modes.10,18,24Given the\nhigher symmetry of the trigonal system, and the large\ndiameter of the discs with respect to the pitch, it is not\nentirely surprising that such strong dependence was not\nobserved in the fundamental mode of the arrays fabri-\ncated for this study.\nThefvsHresdata for film f1a is shown in Figure 7;\nthis data was representative of equivalent measurements\nof the other four samples. Resonances below 7 GHz were\nnot recorded, so as to ensure that the resonance field\nwas always high enough for the discs to be tangentially\nmagnetically saturated. The data for each of the two\nmodes for each sample was fit with Equation 1, under\nthe assumption Nz=Nx, usingNy−Nzas a fit param-\neter. The results of this process are tabulated in Table\nII. There was no clear variation with φ′inNy−Nzfor\neither mode across the samples.\nSignificant differences between the resonance frequen-\ncies of close-packed and well-isolated discs have been re-01000 2000 3000 40005101520\nHres/LBracket1Oe/RBracket1f/LBracket1GHz/RBracket1\nFIG. 7. Plot of resonance frequency fagainst resonance field\nHresfor sample f1a (squares; circles - high-field mode) and\nf1c (diamonds), a sample of the continuous film from which\nf1a was patterned, as measured byVNA-FMR. No resonances\nwere observed below the saturating field of the dots.\nFilmNy−Nzfundamental Ny−Nzhigh-field\nf3b* 0.958±0.001 0 .740±0.002\nf3b 0.934±0.001 0 .716±0.001\nf3a 0.939±0.001 0 .720±0.002\nf2a 0.947±0.001 0 .724±0.001\nf1a 0.937±0.001 0 .709±0.002\nTABLE II. Table showing fit parameters Nyfor the funda-\nmental and high-field modes, as extracted by fitting Equation\n1.\nported for square arrays in one other paper.16This effect\nwas not observed in this study for the case where ar-\nray ordering–rather than array pitch–was varied: there\nappears to be no correlation for this sample series be-\ntween array order φ′and fitted demagnetizing parameter\nNy−Nz.\nC. FMR linewidth\nThe full-width half-maximum linewidth values ∆ Hex-\ntractedfromLorentzianfitstotheFMRmodesareshown\nin Figure 8. Each plot shows the ∆ Hvsfdata for the\nunpatterned parent film, and both the fundamental and\nhigh-field modes for the patterned films. The uncertain-\nties in theselinewidth valuesarenot shownonthe graphs\nfor clarity: for the fundamental mode these uncertainties\nwere on the order of ±2%, and for the high-field mode\n±20%. In both cases the primary source was uncertainty\nin the baseline of the Lorentzian fit; the uncertainty was\nmore severe for the high-field mode because its smaller\namplitude and proximity to the fundamental mode made\nthe value of the linewidth highly dependent on the num-6\nber of points on either side of the peak used in the fit. It\nis clear from examination that the linewidths of both the\nfundamental and high-field modes increase with increas-\ning array disorder.\nFilmφ′[◦mm−1]α×103∆H0[Oe] ∆H0, α[Oe]\nf3c - 9.28±0.16 0.25±1.20 -\nf3b* 2.6±0.48.61±0.16 8.24±1.26 2.31±3.10\nf3b 6.0±0.810.70±0.18 1.02±1.38 13.68±3.33\nf3a 9.4±1.110.84±0.19 3.54±1.48 17.89±3.41\nf2c - 7.79±0.14 2.48±1.06 -\nf2a 11.3±1.711.00±0.19 3.27±1.50 32.21±3.25\nf1c - 7.77±0.14 2.85±1.06 -\nf1a 19.9±2.19.55±0.21 19.90±1.64 36.08±3.33\nTABLE III. Table showing fit data for each sample to Equa-\ntion 2 for the fundamental mode data, for the case where αis\nallowed to vary for patterned films and for the case when it is\nheld to the value αcntsof the corresponding continuous film.\nFilmφ′[◦mm−1]α×103∆H0[Oe]\nf3b* 2.6±0.45.00±1.14 14.74±9.23\nf3b 6.0±0.86.21±1.06 0.95±8.21\nf3a 9.4±1.16.40±1.11−0.92±8.36\nf2a 11.3±1.76.10±1.12 2.80±8.70\nf1a 19.9±2.18.45±1.61 6.68±12.37\nTABLE IV. Table showing fit data for each sample to Equa-\ntion 2 for the high-field mode data. The large uncertainties o f\nthe fit parameters are the direct result of an estimated 20%\nerror on high-field mode linewidth data.\nIn the context of studies of spin-wave mode broaden-\ning, Shaw et al.have pointed out the importance of sep-\narating the effects of intrinsic damping from inhomoge-\nneous damping30,48in the total linewidth ∆ H, as defined\nby the equation:\n∆H= ∆H0+4πα/(γµ0f) (SI). (2)\nHerefis the frequency of precession and γthe gy-\nromagnetic ratio. αis the intrinsic damping parame-\nter in the Landau-Lifshitz-Gilbert equation,49,50and can\nbe thought of as a ‘viscous’ damping of energy to the\nlattice.51\n∆H0is a term representing inhomogeneous broaden-\ning: an example of such a broadening might be the\ndecay of uniform motion into spin waves having non-\nzero wavevectors, which might themselves decay to the\nlattice.52The ∆Hvsfdata represented in Figure 8 (a)-\n(d)werefitwithequation2. Theextractedfitparameters\nαand ∆H0are given in Table III for the fundamental\nmodes and Table IV for the high-field modes.\nPrevious studies by other authors of FMR linewidth\nbroadening in disc-geometry nanostructures28,30,53haveproduced some conflicting results as to whether pattern-\ning affects the value of the intrinsic damping parameter,\nα. In the study presented here, αvalues did appear to\nchange outside the bounds of experimental uncertainty.\nFor the fundamental mode as measured in this study, α\nwas larger for the patterned samples than for their cor-\nresponding parent continuous films; the exception was\nthe most ordered sample, for which αwas slightly lower.\nConversely, αfor the high-field mode was lower than or\nwithin the range of experimental error of the correspond-\ning value for the fundamental mode. Although the values\nofαfor patterned films appeared to be different than for\nthe corresponding parent films, there was no clear vari-\nation of αfor either mode with the parameter φ′which\ndistinguished the films.\nIt is difficult to interpret these αvalues because of\nthe lack of a complete theoretical description in the lit-\nerature of spin wave modes in closely packed and po-\ntentially dipole coupled tangentially magnetized disc ar-\nrays. This lack is a consequence of the difficulties caused\nby the absence of cylindrical symmetry in the plane of\nthe discs,17,54and complicated for this experimental sys-\ntem by the lack of perfect ordering. For these reasons,\nit is difficult to know that Equation 2 holds in this sys-\ntem. In particular, one can speculate that the inhomoge-\nnous broadening ∆ H0may have a frequency- or field-\ndependence which varies with array ordering, in anal-\nogy to or even contributed to by the field dependence\nof extrinsic damping related to two-magnon scattering\nprocesses.55Such dependency might go some way to ex-\nplaining the apparent differences in αbetween patterned\nand unpatterned films. If ∆ H0is a function of field (or\nfrequency), then the steeper gradientsin ∆ Hseen in Fig-\nure 8 might not require the ‘real’ intrinsic damping pa-\nrameter to differ between patterned and parent films.\nSince the origin of these differing gradients was not\nclear, a second set of fits to Equation 2 was performed,\nwith the slopes of the patterned film data set to be equal\nto those of the parent continuous films: α=αc. The\ndifference ∆ H0, α−∆H0,cbetween this ‘fixed intrinsic\ndamping inhomogeneous broadening’ value ∆ H0, αand\nthe continuous film inhomogeneous broadening, ∆ H0,c,\nis equivalent to the average of the difference between\nlinewidths ∆ Hof the patterned and unpatterned films\nover the range of frequencies measured. It is a conve-\nnient experimental index of the inhomogeneous broad-\nening that takes into account possible changes to αin\nthe frequency domain of interest, rather than a quanti-\ntative measure of the ‘true’ inhomogeneous broadening\nas defined by Equation 2. A plot of this difference of in-\nhomogeneous broadening, ∆ H0, α−∆H0,c, vs ordering\nparameter φ′is shown in Figure 9. The increase of in-\nhomogeneous broadening with increasing array disorder\ncan be clearly seen.\nThe difficulties encountered in fitting the ∆ H0vsf\ndata were even more pronounced for the high-field mode.\nSince this mode is unique to the patterned geometry,\nthere was no analogue from the parent continuous films7\nFIG. 8. Plot of full-width-half maximum linewidth ∆ Hagainst resonance frequency fas measured by VNA-FMR for samples\n(a) f3b*; (b) f3b (open triangles: fundamental mode; open pe ntagons: high-field mode); (c) f2a; and (d)f1a. The filled dia monds\nare the data points for the parent continuous films. For all sa mples except f3b, the open squares denote the fundamental mo de,\nand the open circles the high-field mode of the patterned films listed; the key for f3b has already been listed in parenthese s.\nwith which to compare data. Additionally, there was\na high degree of experimental uncertainty in high-field\nmode ∆Hvalues, which led to high uncertainties in α.\nThis in turn produced ∆ H0uncertainties that were com-\nparable to or greater than the ∆ H0values themselves.\nDespite these uncertainties, the data in Figure 8 suggest\nthat the linewidth for this mode also increases with in-\ncreasing φ′in the frequency range measured.\nDisc uniformity has been maintained as far as exper-\nimentally practicable, as shown by consistency of hys-\nteretic measurements and AFM images. Differences in\nlinewidth and saturation magnetization of parent films\nhave been accounted for in the comparison between pat-\nterned films. The variation in broadening with changing\narray ordering suggests that the discs are dipole coupled\ntogether, andthatthedegreeofcouplingaffectstheFMR\nlinewidth ofboth modes. That sample f3b*showsamore\nnarrow linewidth than the patterned film f3b from which\nit was cut supports this conjecture. It might have been\nspeculated that differently arrangedarraydomains in the\nmore disordered samples, interacting strongly internally\nbut weakly with other domains, were producing differ-\nent resonance fields due to angular dependence of Hres.However, the lack of angulardependence of Hresin either\nmode in sample f3b* suggests that the broadening is not\ndue to the sum of differing resonance fields of different\narray domains, but rather to some less local effect.\nIn the only previous study of the FMR characteristics\nof trigonal arrays fabricated by a self-assembly process,\nit was not clear whether the anomalously large linewidth\nrecorded was due to the fabrication process, a distribu-\ntion of disc parameters, or the quality of the parent con-\ntinuous film.11The results of the present study indicate\nthat there is no dramatic impact on linewidth due to pat-\nterning in such a process: even for the most disordered\narray, f1a, there was less than a factor-of-two broaden-\ning; for the case of near-perfect order represented by\nsample f3b*, there was essentially no broadening. The\nbroad linewidth seen in Reference 11 is therefore likely\nto have been caused by a broad linewidth in the parent\nfilm rather than the patterning process itself. The nar-\nrowness of the ‘waist’ of the vortex hysteresis loops and\nhigh magnetization of the patterned films studied here\nare consistent with this conclusion. The utility of the\nwater-surface self-assembly lithographic process for pat-\nterning disc arrays has thus been partially vindicated:8\n5 10 15 200102030\nΦ'/LBracket1°mm/Minus1/RBracket1/CΑpDeltΑH0,Α/Minus/CΑpDeltΑH0,c/LBracket1Oe/RBracket1\nFIG. 9. Plot of ∆ H0, α−∆H0,cvsφ′for the fundamental\nmode data of patterned films f3b*, f3b, f3a, f2a, and f1a.\nThe data plotted here corresponds to the appropriate data\npresented in tabular format in Table III.\ndespite the reasonable criticisms that the self-assembly\nprocess does not preserve long range order, the impact of\nvariations in that order on the FMR linewidth have been\nshown to be relatively minor.\nIV. CONCLUSIONS\nA series of arrays of sub-micron Permalloy discs was\nfabricated by a water-surface self-assembly lithographic\nprocess. The geometry, static magnetic behavior, and\nordering of the disc arrays were measured. The arrays\nwere characterized by vector network analyzer ferromag-\nnetic resonance. Two modes were present in the ferro-\nmagnetic resonance spectra: a fundamental mode and a\nhigher field, lower frequency mode. The variation of the\nresonance field with relative lattice-to-static field angle\nwas shown to be negligible for both modes. The rela-\ntionshipbetweenresonancefrequencyandresonancefield\nwasfoundtobeuncorrelatedtoincreasingarraydisorder.\nThe linewidth vs frequency data for these arraysrevealed\nthat the self-assembly fabrication process did not dra-\nmaticallyincreasetheferromagneticresonancelinewidth,\nclarifying questions from a previous study. The intrinsic\ndamping parameter for the fundamental mode appeared\nto be higher for the patterned films than in the parent\ncontinuous films, but it was not clear whether this was\na result of field- or frequency-dependencies of the inho-\nmogeneous damping not accounted for in the fit. The\nlinewidth of both the fundamental and high-field modes\nwas found to increase with increasing array disorder.\nACKNOWLEDGEMENTS\nThis workwas supported in part by the Australian Re-\nsearch Council under Discovery Grant “Magnetic nanos-tructures for emerging technologies”. N. Ross is sup-\nported by a University of Western Australia Hackett\nPostgraduate Scholarship. The authors acknowledge the\nfacilities, scientific and technical assistance of the Aus-\ntralian Microscopy & Microanalysis Research Facility at\nthe Centre for Microscopy, Characterisation & Analysis,\nThe University of Western Australia, a facility funded\nby The University, State and Commonwealth Govern-\nments. 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Korenivski2\n1Institute of Magnetism, National Academy of Sciences of Ukr aine, 36 b Vernadsky Blvd., 03142 Kyiv, Ukraine\n2Nanostructure Physics, Royal Institute of Technology, 106 91 Stockholm, Sweden\n(Dated: September 14, 2021)\nFerromagnetic resonance properties of F 1/f/F2/AF multilayers, where weakly ferromagnetic\nspacer f is sandwiched between strongly ferromagnetic laye rs F1and F 2, with F 1being magnet-\nically soft and F 2– magnetically hard due to exchange pinning to antiferromag netic layer AF, are\ninvestigated. Spacer-mediated exchange coupling is shown to strongly affect the resonance fields\nof both F 1and F 2layers. Our theoretical calculations as well as measuremen ts show that the key\nmagnetic parameters of the spacer, which govern the ferroma gnetic resonance in F 1/f/F2/AF, are\nthe magnetic exchange length (Λ), effective saturation magn etization at T= 0 (m0), and effective\nCurie temperature ( Teff\nC). The values of these key parameters are deduced from the exp erimental\ndata for multilayers with f = Ni xCu100−x, for the key ranges in Ni–concentration ( x= 54÷70 at. %)\nand spacer thickness ( d= 3÷6 nm). The results obtained provide a deeper insight into the rmally-\ncontrolled spin precession and switching in magnetic nanos tructures, with potential applications in\nspin-based oscillators and memory devices.\nPACS numbers: 76.50.+g, 75.30.Et, 75.70.-i, 85.70.Kh\nI. INTRODUCTION\nSpin valves, whose central functional part contains two\nferromagnetic layers (F 1, F2) separated by a nonmag-\nnetic spacer, have been the foundation for a wide range\nof applications in nanoelectronics and spintronics.1–3Re-\ncent studies have demonstrated that incorporation of di-\nluted ferromagnetic layer (f) instead of the nonmagnetic\nspacer layer may expand the functionality of the spin\nvalves, yielding nanostructureswith thermally-controlled\nmagnetic properties, of F 1/f/F2generic type.4–7In such\nstructures, the exchangecouplingbetween stronglyferro-\nmagnetic outer layers F 1and F2depends on whether the\ntemperature is higher or lower than the effective Curie\ntemperature of the spacer ( Teff\nC). At low temperatures,\nT < Teff\nC, the direct exchange interaction through the\nspacer in its ferromagnetic state favors the parallel ori-\nentation of the magnetic moments M1andM2of the\nouter layers, F 1and F2. At high temperatures, T > Teff\nC,\nM1andM2are exchange decoupled and their orienta-\ntions can be changed independently by applying a suit-\nableexternalmagneticfield, H. Thus, avariationintem-\nperature and/or field can produce switching between the\nparallel (P) and antiparallel (AP) mutual orientations of\nM1andM2in the system.8,9\nThe key element in the F 1/f/F2sandwich described\nabove (the so-called Curie-switch orCurie-valve ) is the\nweakly ferromagnetic spacer, f, which should have a nar-\nrow ferromagnetic-to-paramagnetic transition and have\ntheTeff\nCvalue tunable in fabrication. Diluted ferromag-\nnetic alloys, such as Ni-Cu, is the natural choice for the\nspacer material, since the Curie temperature of bulk10as\nwell as film11–14samples of Ni xCu100−xalloys depends\nalmost linearly on Ni concentration.\nThe experiments described in Refs. 8 and 9 confirmed\nthe concept of temperature-controlled P to AP switch-\ning in nanostructures F 1/f/F2, in particular containing aNixCu100−x(x= 35÷72 at. %) spacer enclosed by an\nexchange-pinned Co 90Fe10layer and a free Ni 80Fe20(Py)\nlayer: Py/Ni xCu100−x/Co90Fe10/Mn80Ir20(hereinafter –\nF1/f/F2/AF, AF denoting antiferromagnetic Mn 80Ir20).\nSince the earlier work primarily aimed at understanding\nthe switching effect itself, little attention was paid to the\neffect of the spacer-mediated exchange on the ferromag-\nnetic resonance in the structure.\nThis work investigates the magnetic resonance prop-\nerties of the Curie-switch, experimentally and theoreti-\ncally, aiming at understanding the mechanisms involved\nandobtainingtheintrinsicphysicalparametersgoverning\nthe spin dynamics in the system.\nII. THEORY\nConsider an F 1/f/F2/AF multilayer outlined above,\nwhere a weakly ferromagnetic spacer (f) is sandwiched\nbetweenmagneticallysoft(F 1)andhard(F 2)layers,with\nF2exchange pinned by an antiferromagnetic layer (AF).\nThe thicknesses of F 1, F2and f are l1,l2, andd, respec-\ntively.\nOur calculations of the magnetic resonance fields will\nassume that F 1and F 2are single domain. For small\nlayer thicknesses and strong exchange interaction, in the\nweak excitation limit typical of ferromagnetic resonance\n(FMR) experiments, this assumption is well justified.15\nSpacer f with magnetization mprovides a relativelyweak\ncouplingbetweentheouterferromagnets,F 1andF2. The\nanalysis aims to determine the effect of this interlayer\nexchange coupling, variable in strength as a function of\ntemperature, on the FMR.\nWe use the classical Landau-Lifshitz approach16,17to\ndescribetheF 1/f/F2/AFmultilayerandfocusonthecase\nwherethe quasistaticexternalfield Handthealternating\nfieldhareinthe film plane. Axes OxandOyaredirected2\nFIG. 1. (Color online) Schematic of F 1/f/F2/AF multilayer\nand chosen coordinate system in FMR measurements.\nalongHandh, respectively (Fig. 1). With Ozperpen-\ndicular to the film plane, the magnetization vectors of F 1\nand F2can be expressed as\nMi=Mi(sinθicosϕi,sinθisinϕi,cosθi),(1)\nwherei= 1,2;Miis the saturation magnetization of i–\nth layer; θiandϕiare the polar and azimuthal angles,\nrespectively.\nThe exchange bias between F 2and AF can be mod-\nelled using an effective biasing field Hbacting only on\nmagnetization M2.18It will be shown below that for fully\ndescribing the FMR-effects of interest in this work, it is\nsufficient to consider only two cases, for which the exter-\nnal fieldHis either parallel or antiparallel to Hb. Corre-\nspondingly, the value of Hbcan be either positive ( φ= 0\nin Fig. 1) or negative ( φ=π) along the biasing axis.\nThe magnetic energy of the i-th ferromagnetic layer in\nthe above geometrical notations is\nWi=Sliwi, (2a)\nwi= 2πM2\nicos2θi−MiHicosϕisinθi\n−Mihsinϕisinθi,(2b)\nwhereSis the areaof the film surface, wiis thei-th layer\nenergy density, H1=H, andH2=H+Hb. The first\nterms in Eq. (2b) originates from demagnetizing energy,\nwhile the second and third terms describe the energies\nof the interaction of the layers’ magnetizations with the\nquasi-static Hand alternating hexternalmagneticfields,\nrespectively.\nTo simplify the equations, let us recall that in the case\nofathin film, its highout-of-planedemagnetizationfieldspreventthe magnetization vectorfrom stronglydeviating\nfrom the xOyplane. In this case, θican be represented\nasθi=π/2+εi, where|εi| ≪1.\nIn the small-signal approximation relevant for FMR,\nthe magnetizationvectorsofF 1andF2arenearlyaligned\nwith the Oxaxis (the easy axis, also the direction of ex-\nternal field H) and perform only weak oscillations near\nthe ground state under the microwave excitation h. This\nmeans that |ϕi| ≪1. The limits of validity of this ap-\nproximationwill be discussed in the experimental section\nbelow.\nIn the above small-signal thin-film approximation, the\nmagnetic energy density of the system becomes\nwi=−MiHi+(4πM2\ni+MiHi)ε2\ni/2\n+MiHiϕ2\ni/2−Mihϕi.(3)\nF1and F2are exchange coupled through a weakly fer-\nromagnetic spacer f. The case where the spacer is highly\nmagnetically diluted and nominally (in the bulk) is para-\nmagnetic was consideredin Ref. 9, with the interlayerex-\nchange mediated via induced proximity ferromagnetism.\nHere we consider the case where the spacer is diluted\nsuch that it is nominally ferromagnetic and can mediate\ndirect exchange between the outer ferromagnetic layers,\nwith the exchange coupling strength being a steep func-\ntion of temperature near the Curie point of the spacer.\nWedenotethetemperaturedependent saturationmag-\nnetization of the spacer as m. Assuming again that the\nmagnetization is uniform in the xOyplane, the spacer\nenergy density can be written as\nw=αm2\n2/bracketleftigg/parenleftbigg∂θ\n∂z/parenrightbigg2\n+sin2θ/parenleftbigg∂ϕ\n∂z/parenrightbigg2/bracketrightigg\n−mHm\n2−Hmcosϕsinθ−hmsinθsinϕ ,(4)\nwhereαis the constant of exchange interaction, θand\nϕare polar and azimuthal angles, respectively, of the\nspacer magnetic moment m,Hmis the magnetostatic\nfield in the system.\nThe value of the magnetostatic field can be easily\nderived from Maxwell’s equation: div B= div(Hm+\n4πm) = 0. Since both the magnetization and therefore\nmagnetostatic field depend only on one special variable,\nz, the magnetostatic field becomes: Hm=−4πmzez=\n−4πmcosθez, where ezis the unit vector along the z\naxis.\nTaking into account Eq. (4), the Landau-Lifshitz equa-\ntions in the angular form become\n∂2θ\n∂ξ2=−d2\nΛ2/bracketleftbigg\nsinθcosθ+sinθ∂ϕ\n∂τ+H\n4πmcosθcosϕ+h\n4πmcosθsinϕ/bracketrightbigg\n, (5a)3\n∂\n∂ξ/parenleftbigg\nsin2θ∂ϕ\n∂ξ/parenrightbigg\n=d2\nΛ2/bracketleftbigg\nsinθ∂ϕ\n∂τ+H\n4πmsinθsinϕ−h\n4πmsinθcosϕ/bracketrightbigg\n. (5b)\nThe new dimensionless variables, normalized to the char-\nacteristic length and time in the problem, introduced in\nEqs. (5a) and (5b) are ξ=z/dandτ= 4πtγm, wheret\nis the time and γis the gyromagnetic ratio. Λ =/radicalbig\nα/4π\nis the magnetic exchange length.19\nIf the spacer thickness dis much smaller than the\nmagnetic exchange length Λ ( d≪Λ), the right side in\nEqs. (5a) and (5b) becomes a small correction, which in\nthe first approximation can be neglected.\nAs a result, only the exchange terms survive:\n∂2θ\n∂ξ2= 0, (6a)\n∂\n∂ξ/parenleftbigg\nsin2θ∂ϕ\n∂ξ/parenrightbigg\n= 0. (6b)\nThe solution, which satisfies the requirement of con-\ntinuity of the polar and azimuthal components at the\ninterfaces between the layers, has the form:\nϕ(z) =ϕ2+(ϕ1−ϕ2)z/d,\nε(z) =ε2+(ε1−ε2)z/d, (7)\n0≤z≤d.The resulting magnetic energy of the spacer is\nW=SJ/bracketleftbig\n(ϕ1−ϕ2)2+(ε1−ε2)2/bracketrightbig\n/2,(8)\nwhereJ=αm2/d= 4πΛ2m2/d.\nTo determine the resonance conditions for the layered\nsystem under consideration, we express the Lagrange\nfunction in terms of the angle:\nL=2/summationdisplay\ni=1/parenleftbigg\n−SliMi\nγεi∂ϕi\n∂t−Wi/parenrightbigg\n+J/bracketleftbig\n(ϕ1−ϕ2)2+(ε1−ε2)2/bracketrightbig\n/2.(9)\nThe variational equations following from Eg. (9) are\nequivalent to the Landau-Lifshitz equations:\n\niHωH+h1 0 −h1\n4πM1+H+h1−iHω −h1 0\n0 −h2 iHω H+Hb+h2\n−h2 0 4πM2+H+Hb+h2−iHω\n×\nε1\nϕ1\nε2\nϕ2\n=\nh\n0\nh\n0\n (10)\nwherehi=J/Mili=αm2/dMili= 4πΛ2m2/dMili.\nHere,Hω=ω/γ,ω= 2πf,γis the gyromagnetic ra-\ntio,hiis the characteristic field of exchange interaction\nbetween the layers.\nThe characteristic fields of the resonance modes ofthe collective spin dynamics in the system are found by\nequating the determinant of matrix (10) to zero. This\nresults in two branches in the functional form of Hω(H).\nThe first resonance branch, corresponding to the reso-\nnance field of F 1, has the form:\nH2\nω= (Hr1+h1)(4πM1+Hr1+h1)−h1h2/bracketleftbigg\n1−(2πM1+Hr1)(2πM2+Hr1)\n2πHr1(M2−M1)/bracketrightbigg\n−h1h2Hb\nHr14π[π(M2+M1)+Hr1](M2−M1)−(2πM1+Hr1)(2πM2+Hr1)\n[2π(M2−M1)]2,(11)\nwhereHr1is the external field producing FMR in F 1[see\nFig.2(a)]. Onlytermsofordernothigherthanquadratic\ninkiwere kept in Eq. (11).Thevalueof Hr1dependsonwhethertheexternalmag-\nnetic field is parallel ( ↑↑) or antiparallel ( ↑↓) to the ex-\nchangebiasfield Hb. Itiseasytoshowthatthedifference\nin the resonance fields, ∆ Hr1=H↑↓\nr1−H↑↑\nr1, has the form:4\n∆Hr1=H↑↓\nr1−H↑↑\nr1=h1h2Hb\nH0\nr14π/bracketleftbig\nπ(M2+M1)+H0\nr1/bracketrightbig\n(M2−M1)−(2πM1+H0\nr1)(2πM2+H0\nr1)\n(2πM1+H0\nr1)[2π(M2−M1)]2,(12)\nwhereH0\nr1= (H↑↓\nr1+H↑↑\nr1)/2 =/radicalbig\n(2πM1)2+H2ω−\n2πM1−h1.\nIt follows from Eq. (12) that ∆ Hr1is proportional to a\nproduct of h1h2. This means that ∆ Hr1sharply changes\nin the vicinity of the Curie point of the spacer as a result\nof the sharp increase in mat the para-to-ferromagnetic\ntransition (see Eq. (10). Expectedly, ∆ Hr1goes to zero\nasTincreasesabovetheCuriepointofthe spacer. Inthis\nhigh-Tlimit, thereisnocouplingbetweenF 1andF2, and\nEq. (11) describes the resonance field of the decoupled\nsoft outer ferromagnet F 1.\nTo find the resonance fields for F 2, we keep only terms\nof the order not higher than linear in hi. The results for\nH↑↓\nr2andH↑↑\nr2are\nH↑↑\nr2=/radicalbig\n(2πM2)2+H2ω−2πM2−h2−Hb,\nH↑↓\nr2=/radicalbig\n(2πM2)2+H2ω−2πM2−h2+Hb.(13)\nAgain, sharp changes in H↑↑\nr2andH↑↓\nr2are expected in\nthe vicinity of the Curie point of the spacer. At high\ntemperatures where h2→0, the difference between H↑↓\nr2\nandH↑↑\nr2naturally becomes 2 Hb.\nIt follows from Eq. (13) that for sufficiently high values\nofM2andh2, theH↑↑\nr2branchcanfallintonegativefields,\nwhere it cannot be observed experimentally.\nIII. EXPERIMENT\nA. Samples and measurements\nThe experiments were carried out on two sets of\nmultilayered samples, in which either the spacer thick-\nness,d, or its composition, x, were varied. The\nfirst set was Py(10 nm)/Ni 54Cu46(d)/Co90Fe10(5 nm)/\nMn80Ir20(12 nm) (hereinafter – F 1/Ni54Cu46(d)/F2/AF)\nwith the spacer thicknesses d= 3, 4.5, and 6 nm. The\nsecond set was Py(10 nm)/Ni xCu100−x(6 nm)/Co 90Fe10\n(5nm)/Mn 80Ir20(12nm)(hereinafter��F 1/NixCu100−x(6\nnm)/F 2/AF), with x= 54, 62 and 70 at.%. The multi-\nlayers were deposited at room temperature on thermally\noxidized silicon substrates using magnetron sputtering in\nan AJA Orion 8-target system. The exchange pinning\nbetween the ferromagnetic Co 90Fe10and antiferromag-\nnetic Mn 80Ir20was set in during deposition of the mul-\ntilayers using an in-plane magnetic field Hdep≈1 kOe.\nOther fabrication details are similar to those described\nin Refs. 8 and 9.\nIn addition to the multilayers, single-layer Py (10 nm)\nand Co 90Fe10(5 nm) films were prepared under identi-\ncal technological conditions. FMR measurements on the\nsingle-layer films were carried out to extract the mag-\nnetizations of Py and Co 90Fe10layers and use them forsubsequent multilayer-FMR modelling and characteriza-\ntion [e.g., using Eqs. (12) and (13)].\nThe FMR measurements were performed using an X-\nband ELEXSYS E500 spectrometer equipped with an\nautomatic goniometer. The operating frequency was\nf= 9.44 GHz. FMR spectra for various in-plane dc-\nfield angles were studied in the temperature range of 120\nto 400 K.\nB. Results and discussion\n1. Measured FMR spectra\nFig. 2 (a) shows two typical FMR spectra for a\nF1/f/F2/AF multilayer, for which the external magnetic\nfield is parallel (solid line) or antiparallel (red dashed\nline) to the exchange bias field Hb(T= 300 K). The res-\nonancesignalsfrombothF 1andF2layersareclearlyvisi-\nble and are separated in field. As expected [see Eqs. (12)\nand (13)], the resonance conditions for both layers de-\npend on the mutual orientation of HandHb[Fig. 2(b)].\nConsistent with the predicted behavior of Eq. (13), the\nH↑↑\nr2branch for F 2extrapolates into negative fields [see\nFig.2(b)]. Intheremainderofthepaperwethereforedis-\ncuss and in-depth analyzeonly the H↑↓\nr2resonancebranch\nas regards the dynamics of the pinned F 2layer.\nTo understand the details of how the interlayer ex-\nchange coupling affects the spin dynamics of the free\nlayer (F 1), the angular dependence of Hr1was studied\nat various temperatures for various spacer thicknesses.\nThe typical data shown in Fig. 3 indicate that the po-\nsition of the resonance peak is angle-dependent and this\nangular asymmetry becomes strongeras the temperature\nis lowered. At the same time, the Hr1vsφdependence\nbecomes more pronounced as the spacer thickness de-\ncreases. For all the cases shown in Figs. 2 (b) and 3,\neachHr1(φ) data set is well fitted using a model charac-\nteristicofathin filmwith unidirectionalanisotropy(solid\nlines in Fig. 2 (b) and Fig. 3).\nA more detailed analysis shows that there is an addi-\ntional small contribution from uniaxial anisotropy. The\nextracted uniaxial anisotropy field ( ∼4 Oe) is weakly de-\npendent on temperature and spacer composition. This\ncontribution potentially originates from some ordered\nconfiguration at NiCu/Py interface formed during the\nmultilayer deposition in a magnetic field Hdep(used for\nexchange pinning F 2).5\n/s32 /s32/s32/s61/s32/s48 /s40\n/s98/s41\n/s32 /s32/s32/s61/s32/s49/s56/s48 /s40\n/s98/s41\n/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48/s70\n/s49/s47/s78/s105\n/s53/s52/s67/s117\n/s52/s54/s40/s52/s46/s53/s32/s110/s109 /s41/s47 /s70\n/s50/s47/s65/s70\n/s72\n/s114/s50/s72\n/s114/s49\n/s40/s98/s41/s40/s97/s41/s84 /s61/s51/s48/s48/s32/s75\n/s32/s32/s100/s80/s47/s100/s72 /s32/s32/s40/s97/s46/s117/s46/s41\n/s72 /s32/s40/s79/s101/s41/s49/s50/s48/s48 /s49/s50/s53/s48 /s49/s51/s48/s48 /s49/s51/s53/s48/s72\n/s114/s49/s32/s70\n/s49/s32/s32/s40/s80/s121/s41\n/s32/s32\n/s72 /s32/s40/s79/s101/s41\n/s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48\n/s84 /s61/s51/s48/s48/s32/s75\n/s32/s72\n/s114/s50/s32/s40/s70\n/s50/s32/s61/s32/s67/s111\n/s57/s48/s70/s101\n/s49/s48/s32/s41\n/s32/s72\n/s114/s49/s32/s40/s70\n/s49/s32/s61/s32/s80/s121/s41/s72\n/s114/s32/s32/s40/s79/s101/s41\n/s32/s40/s100/s101/s103/s46/s41\nFIG. 2. (Color online) (a) FMR spectra for F 1/Ni54Cu46(4.5\nnm)f/F 2/AF for parallel (solid line) and antiparallel (red\ndashed line) orientations of the external magnetic field H\nwith respect to the exchange biasing field Hb. The upper\ninset shows an enlarged view of the signal from F 1. (b) The\ndependence of the resonance fields of F 1and F 2on the angle\nφbetween HandHbextracted from data sets such as those\nillustrated in (a).\n2. FMR modelling procedure\nExpressions(12) and (13) were used for calculating the\ntemperature dependence of the resonance field asymme-\ntry ∆Hr1=H↑↓\nr1−H↑↑\nr1for F1(Py), and of the resonance\nfieldH↑↓\nr2for F2(Co90Fe10). These expressions contain\nthe values of the saturation magnetization M1andM2\nfor the Py and Co 90Fe10layers, respectively, which are\ntemperature dependent. The M1(T) andM2(T) depen-\ndences, used in the data analysis to follow, are shown in\nFig. 4 and were obtained from the FMR data taken on\nsingle-layer Py (10 nm) and Co 90Fe10(5 nm) films pre-\npared under the same conditions as the multilayers. The\nKittel’s formulas for isotropic thin films20,21were used to\ncalculate the M1(T) andM2(T) shown.\nThe key quantity determining the behavior of ∆ Hr1\nandH↑↓\nr1, is the spacer magnetization m, averaged over\nthe layerthickness [seeEq. (10)]. The spaceris ferromag-\nnetic below the Curie point and nominally paramagnetic/s56/s48/s48/s57/s48/s48/s49/s48/s48/s48/s49/s49/s48/s48/s49/s50/s48/s48\n/s32/s40/s97/s41/s100/s32/s61/s32/s51/s32/s110/s109/s72\n/s114/s49/s32/s40/s79/s101/s41\n/s32/s49/s50/s48/s32/s75/s32\n/s32/s50/s57/s48/s32/s75/s70\n/s49/s47/s78/s105\n/s53/s52/s67/s117\n/s52/s54/s40/s100 /s41/s47/s70\n/s50/s47/s65/s70\n/s45/s49/s56/s48 /s45/s57/s48 /s48 /s57/s48 /s49/s56/s48/s49/s49/s48/s48/s49/s50/s48/s48/s49/s51/s48/s48/s49/s52/s48/s48\n/s32/s49/s50/s48/s32/s75/s32\n/s32/s50/s57/s48/s32/s75/s100/s32/s61/s32/s54/s32/s110/s109\n/s40/s98/s41/s72\n/s114/s49/s32/s40/s79/s101/s41\n/s32/s32/s40/s100/s101/s103/s46/s41\nFIG. 3. (Color online) Angular dependences of the resonance\nfields of the soft layer (F 1) in F 1/Ni54Cu46(d)/F2/AF with\nthe spacer thickness d= 3 nm (a) and 6 nm (b).\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s52/s48/s48/s54/s48/s48/s49/s52/s48/s48/s49/s54/s48/s48\n/s32/s32 /s80/s121 \n/s32/s32 /s67/s111\n/s57/s48/s70/s101\n/s49/s48/s77 /s32/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41\n/s84 /s32/s32/s40/s75/s41\nFIG. 4. (Color online) Temperature dependences of the mag-\nnetizations of Py and Co 90Fe10layers obtained from the FMR\ndata on the respective single-layered films.\naboveit. It has previouslybeen shown, however,that the\nproximity effect at the interface with a strong ferromag-\nnet induces noticeable magnetization in a paramagnetic\nor weakly ferromagnetic metal and may give rise to an\nincrease in its Curie point.9,22,23The proximity length\nis an order of magnitude greater than the atomic spac-\ning and for the case where the spacer thickness dis of6\nthe order of a few nanometers, the induced magnetiza-\ntion penetrates through the spacer thickness.8,9For this\nreason, to account for the proximity effect in our calcula-\ntions, it was assumed that (i) there is an additional field\nHproxwhich acts on the spacer magnetization mand (ii)\nfor the spacer sandwiched between strong ferromagnets\nF1and F 2, the Curie point Teff\nCdiffers from that in the\nbulk.\nThem(T,Hprox) dependence due to the prox-\nimity effect was modelled using the mean-field\napproximation,15,24wherethe magnetizationis described\nby the Brillouin function:\nm\nm0=Bj(m) =ξ1cosh/bracketleftbigg\nξ1/parenleftbigg\nζ1m\nT+ζ2Hprox\nT/parenrightbigg/bracketrightbigg\n−ξ2cosh/bracketleftbigg\nξ2/parenleftbigg\nζ1m\nT+ζ2Hprox\nT/parenrightbigg/bracketrightbigg\n;\nξ1=2j+1\n2j;ξ2=1\n2j;ζ1=3j\nj+1Teff\nC\nm0;ζ2=νm0; (14)\nwherejis the total angular momentum per Ni xCu100−x\nformula unit, m0is the saturation magnetization at\nT= 0 K,Hproxis the effective field reflecting the prox-\nimity effect at the interfaces with the strong ferromag-\nnets F 1and F 2, andTeff\nCis the effective Curie tempera-\nture. Coefficient νequalsµ/(ρNA), where µandρare\nthe molar mass and density of Ni xCu100−x, respectively,\nNAis the Avogadro constant, and kBis the Boltzmann\nconstant.15,24\nThe coefficient νfor our Ni xCu100−xalloy was esti-\nmated to be near 8 .2×10−8cm3K erg−1forxin the\nvicinity of 60 at.%. The initial values of jandm0were\nchosen based on the data calculated in Ref. 8 for bulk\nNixCu100−x, and the value of jwas kept fixed through-\nout the analysis. Since the magnetizationand Curie tem-\nperature of the Ni xCu100−xspacer are expected to differ\nfrom those in the bulk, specifically due to the proximity\neffect,m0was chosen as one of the variable parameters\nin fitting the experimental data.\nThe proximity effect is expected to be most pro-\nnounced in the vicinity of Teff\nC. Theab-initio calculations\nof this effect for F 1/NixCu100−x(x,d)/F2/AF atT∼Teff\nC\nwere detailed in Ref. 9. Based on a comparison of the\nvalues for the average magnetic moment /angbracketleftm/angbracketrightobtained in\nRef. 9 and the mcalc(T) obtained using Eq. 14, it was\nfound that mcalcatT∼Teff\nCis approximately equal to\n/angbracketleftm/angbracketrightforHprox≈100 kOe. This value of Hproxwas kept\nfixed in all subsequent calculations.\nAnother important quantity affecting the spin dy-\nnamics in the system is the exchange bias field\nHb. Based on the magnetometry measurements on\nF1/NixCu100−x(x,d)/F2/AF reported in Ref. 8, Hbwas\nobtained for a range of xanddvalues (for 300 K). These\nand the additional data reported in Ref. 9 make it pos-\nsible to conclude that for our samples with x >52,Hb\nis only weakly temperature dependent. The calculation\ntherefore assumed Hb(T) = const. The specific fixed jandHbvalues used in the calculations, among other pa-\nrameters and variables, are presented in Table I.\nSummarizing, the variable parameters used to fit the\ntheoretical ∆ Hr1(T) andH↑↓\nr2(T) to the experimental\ndata were the effective Curie temperature ( Teff\nC) and sat-\nuration magnetization at T= 0 (m0) of the Ni xCu100−x\nspacer, as well as the characteristic magnetic exchange\nlength (Λ). It will be shown below that for the case un-\nder study, the resulting values of Λ are about two times\ngreater than the spacer thickness d. This is within the\nlimits of the approximation d≪Λ used in the analysis.\n3. FMR data analysis\nFigs. 5 (a) and (b) show the temperature dependences\nofH↑↓\nr1−H↑↑\nr1andH↑↓\nr2for F1/Ni54Cu46(d)/F2/AF sam-\nples obtained from the measured FMR spectra as well as\nthe respective theoretical fits using the above analysis.\nA good agreement between the experiment and theory is\nobtained for realistic values of the fitting parameters.\nThe temperature dependence of the spacer magneti-\nzationm/m0obtained from fitting the resonance fields\nis shown in Fig. 5 (c) for different values of the spacer\nthickness. The proximity of the strongly ferromagnetic\nlayers F 1and F 2has essentially no effect on the low-\ntemperature magnetization of the spacer but is the dom-\ninant factor determining its effective Curie point Teff\nC.\nThe changes in Teff\nCstrongly depend on the spacer thick-\nness: the smaller the dand, therefore, the stronger the\nproximity effect of the interfaces, the higher the Teff\nC.\nFig. 6 (a, b) shows the measured resonance fields\nH↑↓\nr1−H↑↑\nr1andH↑↓\nr2as a function of temperature\nfor F1/NixCu100−x(6 nm)/F 2/AF with different Ni-\nconcentration in the spacer, fitted to theory using\nEqs. (12) and (13). The agreement is good, includ-\ning the case of the highest Ni-concentration with non-\nmonotonous temperature dependence of the resonance\nfield asymmetry (70 at.% Ni in Fig. 6 (a)).\nThe dependence of the spacer magnetization on tem-\nperature extracted from fitting the data in Figs. 6 (a, b)\nis shown in Fig. 6 (c) for different Ni-concentration of the\nspacer. It is clear in this case that the proximity of the\nstrongly ferromagnetic outer layers affects both the low-\ntemperature magnetization m0and the effective Curie\npointTeff\nCof the spacer – the greater the x, the higher\nthem0andTeff\nC.\nHaving performed the data analysis, it is now informa-\ntive to discuss the accuracyof the theoretical assumption\nmade in Section 2 in describing the spin dynamics in an\nF1/f/F2/AF spin-thermionic system. One key assump-\ntion was that the magnetization vectors of the F 1and\nF2layers are parallel to the external field Hand perform\nonly weak oscillation under the influence of the alternat-\ningfieldcomponent h. Thisassumptionisstrictlycorrect\nfor the case where His parallel to Hb, but the opposite\nantiparallelcase( H↑↓Hb)mustbeconsideredwithcare.7\nTABLE I. Magnetic parameters of F 1/NixCu100−x(x,d)/F2/AF multilayers.\nx(at.% Ni) d(nm) jaHb(Oe)bTeff\nC(K) m0(emu/cm3) Λ (nm)\n0.54 3.0 0.19 180 450 120 11 ±2\n0.54 4.5 0.19 240 320 120 11 ±2\n0.54 6.0 0.19 300 250 120 11 ±2\n0.62 6.0 0.21 280 350 140 12 ±2\n0.70 6.0 0.23 210 550 150 13 ±2\naValues calculated from data of Ref. 9 under assumption that L andeg-factor equals 2.\nbValues obtained from magnetic hysretesys loops at room temp erature of Ref. 8.\n/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s32/s100/s32 /s61/s32/s51/s32/s110/s109\n/s32/s100 /s32/s61/s32/s52/s46/s53/s32/s110/s109\n/s32/s100 /s32/s61/s32/s54/s32/s110/s109\n/s32/s32/s72\n/s114/s49/s32 /s32/s72\n/s114/s49/s32/s32/s40/s79/s101/s41\n/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s32\n/s40/s99/s41/s40/s97/s41/s72\n/s114/s50/s32/s32/s40/s79/s101/s41\n/s32/s32/s70\n/s49/s47/s78/s105\n/s53/s52/s67/s117\n/s52/s54/s40/s100 /s41/s47/s70\n/s50/s47/s65/s70\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s100 /s32/s61/s32/s51/s32/s110/s109\n/s52/s46/s53/s32/s110/s109\n/s54/s32/s110/s109/s40/s98/s41/s109 /s47/s109\n/s48/s32/s32/s40/s97/s46/s117/s46/s41\n/s32/s32\n/s84 /s32/s32/s40/s75/s41\nFIG.5. (Coloronline)Temperaturedependencesof H↑↓\nr1−H↑↑\nr1\n(a) andH↑↓\nr2(b) for F 1/Ni54Cu46(d)/F2/AF samples for three\nspacerthicknesses(symbols). Boldsolid linesshowtheore tical\nH↑↓\nr1−H↑↑\nr1andH↑↓\nr2vsTdependences fitted to the measured\ndata using Eqs. (12) and (13), respectively. (c) Temperatur e\ndependence of the normalized spacer magnetization m/m0\nobtained using the above fitting of the resonance fields.\nThevaluesof HbforF1/NixCu100−x(x,d)/F2/AFmul-\ntilayers are listed in Table I. As follows from the M−H\ndata presented in Refs. 8 and 9, the magnetization of\nthe F1/f/F2/AF multilayer fully saturates when the ap-\nplied reversing field H↑↓exceeds (1.2 ÷1.5) times Hb.\nThus, the above assumption should be valid when both\nresonance fields, H↑↓\nr1andH↑↓\nr2, are higher than (1.2 ÷/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s120 /s32/s61/s32/s53/s52/s32/s97/s116/s46/s37/s120 /s32/s61/s32/s54/s50/s32/s97/s116/s46/s37/s120/s32 /s61/s32/s55/s48/s32/s97/s116/s46/s37/s32/s120 /s61/s55/s48/s32/s97/s116/s46/s37\n/s32/s120 /s61/s54/s50/s32/s97/s116/s46/s37\n/s32/s120 /s61/s53/s52/s32/s97/s116/s46/s37/s72\n/s114/s49/s32 /s32/s72\n/s114/s49/s32/s32/s40/s79/s101/s41\n/s32/s32\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s40/s98/s41\n/s32/s32/s109 /s47/s109\n/s48/s32/s40/s97/s46/s117/s46/s41\n/s84 /s32/s40/s75/s41/s70\n/s49/s47/s78/s105\n/s120/s67/s117\n/s49/s48/s48/s45 /s120/s40/s54/s32/s110/s109/s41/s47/s70\n/s50/s47/s65/s70\n/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s32\n/s40/s99/s41/s40/s97/s41/s72\n/s114/s50/s32/s32/s40/s79/s101/s41\n/s32/s32\nFIG.6. (Coloronline)Temperaturedependencesof H↑↓\nr1−H↑↑\nr1\n(a) and H↑↓\nr2(b) for F 1/NixCu100−x(6 nm)/F 2/AF samples\nfor three Ni concentrations (symbols). Bold solid lines sho w\ntheoretical H↑↓\nr1−H↑↑\nr1andH↑↓\nr2vsTdependences fitted to\nthe measured data using Eqs. (12) and (13), respectively. (c )\nTemperature dependences of the normalized spacer magneti-\nzationm/m0obtained usingtheabovefittingoftheresonance\nfields.\n1.5)Hb. For the samples under study, H↑↓\nr1is greater\nthan 900 Oe, so the first condition, H↑↓\nr1>(1.2÷1.5)Hb,\nis well satisfied. The data in Figs. 5 and 6 indicate that\nthe second condition, H↑↓\nr2>(1.2÷1.5)Hb, is also well\nsatisfied for temperatures above ∼180 K.\nThe developed approach, combining theory and ex-8\n/s53/s52 /s54/s50 /s55/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48\n/s32/s32 /s70\n/s49/s47/s78/s105\n/s120/s67/s117\n/s49/s48/s48/s45 /s120/s40/s54/s32/s110/s109/s41 /s47/s70\n/s50/s47/s65/s70/s84/s101/s102/s102\n/s67/s32/s32/s40/s75/s41\n/s120 /s32/s32/s40/s97/s116/s46/s37/s32/s78/s105/s41/s51/s46/s48 /s52/s46/s53 /s54/s46/s48\n/s32/s32/s32 /s70\n/s49/s47/s78/s105\n/s53/s52/s67/s117\n/s52/s54/s40/s100 /s41/s47/s70\n/s50/s47/s65/s70\n/s32/s100 /s32/s40/s110/s109/s41/s32\nFIG. 7. (Color online) DependenceoftheeffectiveCurie poin t\nTeff\nCof the spacer Ni xCu100−xon its thickness d(at fixed com-\nposition x= 54 at.% Ni) and Ni content x(at fixed thickness\nd= 6 nm).\nperiment, makes it possible to extract and analyze the\nthickness and composition dependence of the character-\nistic parameters of the spacer, which are summarized in\nTable I.\nFig. 7 presents the model-extracted dependence of the\neffective Curie point Teff\nCof the spacer on its thickness\ndand Ni content x. We conclude that small variations\nin bothdandxresult in significant variations in Teff\nC.\nWe also note that the values of the Curie temperature\nof NixCu100−xsandwiched between strong ferromagnets\nare much greater than the corresponding values in the\nbulk form of this diluted magnetic alloy – ( Tbulk\nC≈120\nK and 300 K for x= 54 and 70 at.% Ni, respectively)10.IV. CONCLUSION\nFerromagnetic resonance properties of F 1/f/F2/AF\nmultilayers, where spacer f has a low Curie point com-\npared to strongly ferromagnetic F 1and F 2, have been\nanalyzed theoretically and investigated experimentally.\nThe spacer-mediated exchange coupling is shown to\nstrongly affect the resonance fields of both F 1and F 2\nlayers. It is found that the key magnetic parameters of\nthe spacer which govern the magnetic resonance in the\nsystem are the magnetic exchange length (Λ), the effec-\ntive saturation magnetization at T= 0 (m0), and the\neffective Curie temperature ( Teff\nC) of the spacer.\nBy theoretically fitting the measured FMR data, the\nvalues of Λ, m0, andTeff\nCare obtained for the technolog-\nically significant ranges in Ni-concentration ( x= 54÷70\nat.% Ni) and thickness ( d= 3÷6 nm) of the spacer.\nThe values thus obtained are entirely different from the\ncorresponding quantities in the bulk. The developed ap-\nproach to spin dynamics in the system enables such de-\ntailed quantitative characterization that is otherwise is\ndifficult or impossible obtain in terms of direct measure-\nments due to the built-in strong proximity effect.\nTheinferredmagnetisminthekeyelementofthestruc-\nture – the spacer, acting as an interlayer exchange-spring\n– shows a great sensitivity and thereby high tunability\nof its properties versus the degree of magnetic dilution,\ngeometry, and temperature. These results should be use-\nful for designing high-speed nanodevices based on spin-\nthermionic control.\nACKNOWLEDGMENTS\nSupport from the Swedish Stiftelse Olle Engkvist\nBuggm¨ astare,SwedishResearchCouncil(VRgrant2014-\n4548), the Science and Technology Center in Ukraine\n(project P646), and the National Academy of Sciences\nof Ukraine (projects 0115U003536 and 0115U00974) are\ngratefully acknowledged.\n∗anatolii@kth.se; Correspondence author\n1B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D.\nR. Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297 (1991).\n2Nanomagnetism and Spintronics edited by by T. Shinjo\n(Elsevier, Amsterdam, 2009).\n3A. Hirohata and K. Takanashi, J. Phys. D: Appl. Phys.\n47, 193001 (2014).\n4S. Andersson and V. Korenivski, IEEE Trans. Magn. 46,\n2140 (2010).\n5S. Andersson and V. Korenivski, J. Appl. Phys. 107,\n09D711 (2010).\n6A. M. Kadigrobov, S. Andersson, D. Radi´ c, R. I. Shekhter,\nM. Jonson, and V. Korenivski, J. Appl. Phys. 107, 123706\n(2010).7A.M. Kadigrobov, S.Andersson, HeeChulPark, D.Radi´ c,\nR. I. Shekhter, M. Jonson, and V. Korenivski, J. Appl.\nPhys.111, 044315 (2012).\n8A. F. Kravets, A. N. Timoshevskii, B. Z. Yanchitsky, M.\nA. Bergmann, J. Buhler, S. Andersson, and V. Korenivski,\nPhys. Rev. B 86, 214413 (2012).\n9A. F. Kravets, Yu. I. Dzhezherya, A. I. Tovstolytkin, I. M.\nKozak, A. Gryshchuk, Yu. O. Savina, V. A. Pashchenko,\nS. L. Gnatchenko, B. Koop, and V. Korenivski, Phys. Rev.\nB90, 104427 (2014).\n10D. J. Chakrabarti, D. E. Laughlin, S. W. Chen, and Y.\nA. Chang, in Phase Diagrams of Binary Copper Alloys\nedited by P. Subramanian, D. Chakrabarti, and D. Laugh-\nlin (ASM International, Materials Park, OH, 1994), p. 276.9\n11A. P. Thakoor and K. L. Chopra, J. Appl. Phys. 48, 3850\n(1977).\n12I.Bakonyi, E.T´ oth-K´ ad´ ar, J.T´ oth, T. Becsei, T.Tarn´ o czi,\nand P. Kamasa, J. Phys.: Condens. Matter 11, 963 (1999).\n13A. Rusanov, R. Boogaard, M. Hesselberth, H. Sellier, and\nJ. Aarts, Physica C 369, 300 (2002).\n14A. F. Kravets, A. N. Timoshevskii, B. Z. Yanchitsky, O.\nYu. Salyuk, S. O. Yablonovskii, S. Andersson, and V. Ko-\nrenivski, J. Magn. Magn. Mater. 324, 2131 (2012).\n15A. Aharoni, Introduction to the Theory of Ferromagnetism\n(Oxford University Press, Oxford, 1996).\n16L. D. Landau and E. M. Lifshitz, in The Collection of\nWorksedited by L. D. Landau (Nauka, Moscow, 1969)\n[in Russian], Vol. 1, p. 128.\n17C. Serpico, I. D. Mayergoyz, and G. Bertotti, J. Appl.\nPhys.93, 6909 (2003).18J. Nogues, J. Sort, V. Langlais, V. Skumryev, S. Surinach,\nJ. S. Munoz, and M. D. Baro, Phys. Rep. 422, 65 (2005).\n19Gavin S. Abo, Yang-Ki Hong, Jihoon Park, Jaejin Lee,\nWoncheol Lee, andByoung-ChulChoi, IEEE Trans. Magn.\n49, 4937 (2013).\n20Ch. Kittel, Phys. Rev. 73, 155 (1948).\n21A. G. Gurevich and G. A. Melkov, Magnetization Oscilla-\ntions and Waves (CRC Press, Boca Raton, FL, 1996).\n22N. Garcia and A. Hernando, J. Magn. Magn. Mater. 99,\nL12 (1991).\n23I. Navarro, M. Ortuno, and A. Hernando, Phys. Rev. B\n53, 11656 (1996).\n24S. Chikazumi, Physics of Ferromagnetism (Oxford Univer-\nsity Press, New York, 2005)."    },    {        "title": "2308.11996v1.Parallel_simulation_of_the_magnetic_moment_reversal_within_the__φ_0__Josephson_junction_model.pdf",        "content": " 1 PARALLEL SIMULATION OF THE MAGNETIC MOMENT \nREVERSAL  WITHIN THE 0-JOSEPHSON JUNCTION MODEL  \nM. Bashashina,b,*, E. Zemlyanayaa,b, I. Rahmonova,b,c \naJoint Institute for Nuclear Research , Joliot -Curie str. 6, 141980  Dubna, Russia  \nbDubna  State University , University str. 19, 141980  Dubna, Russia  \ne-mail: bashashinmv@jinr.ru  \ncMoscow Institute of Physics and Technology , 14170 0, Dolgoprudny , Russia  \nReceived  \nAbstract  — Periodic structure of the magnetization reversal domains is studied within the superconductor –\nferromagnetic –superconductor  φ0-junction  model. The model is described by the Cauchy problem for the system \nof nonlinear ordinary equations which is numerically s olved by means of the 2 -step Gauss –Legendre method. Two \nversions of parallel implementation on the basis of MPI and OpenMP techniques have been developed. Efficiency \nof both versions is confirmed by test calculations. An effect of frequency of ferromagneti c resonance  on the \nconfiguration MR domains  has been investigated. The calculations have been performed at the HybriLIT Platform \nof the  JINR Multifunctional information and computing complex . \nINTRODUCTION  \nIn the superconductor –ferromagnetic –superconductor structures, the spin -orbit coupling \nin ferromagnetic layer without inversion symmetry provides a mechanism for a direct (linear) \ncoupling between the magnetic moment and the superconducting cu rrent [1]. Such Josephson \njunctions are called φ0-junction. A perspective to control  magnetic properties of φ0-junction s \nby means of the superconducting current as well as effect of magnetic dynamics on the \nsuperconducting current is of great interest from  the viewpoint of practical applications in \nnanoelectronic  devises  [1-3]. One of the important directions in this field is a study  of \nphenomenon of magnetization reversal  (MR) in φ0-junction  [4]. Different scenarios f or the MR \nrealization  and the periodic structure of  MR intervals  are demonstrated  in [5] . This contribution \naims to study  an influence of frequency of ferromagnetic resonance  on the configuration  MR \ndomains. The basic formulae of the mathematical model  are given . Numerical approach \nincluding the parallel implementation is  briefly described . Results of numerical simulations of \nthe MR phenomenon depending on the parameters of model are presented . Also,  the results of \ntest calculations demonstrating  effect of parallel implementation  on the basis of MPI and \nOpenMP techniques are given.   2 THEORETICAL MODEL  \nWe use the theoretical framework as in [ 4]. The model of the  superconductor –\nferromagnetic –superconductor φ0-Josephson  junction  is based on the Landau –Lifshitz –Gilbert \nequation which describes t he dynamics of the magnetization  vector 𝑚⃗⃗  in ferromagnetic layer in \nthe φ 0-junction : \n𝑑𝑚⃗⃗ \n𝑑𝑡=−𝜔𝐹\n1+𝑚⃗⃗ 2𝛼2{[𝑚⃗⃗ ×𝐻⃗⃗ ]+𝛼[𝑚⃗⃗ (𝑚⃗⃗ 𝐻⃗⃗ )−𝐻⃗⃗ 𝑚⃗⃗ 2]}, (1) \nwhere α is damping parameter,  ωF is normalized frequency of ferromagnetic resonance , 𝐻⃗⃗  is an \neffective magnetic field vector with the components  \n{𝐻𝑥=0                                          \n𝐻𝑦=𝐺𝑟sin(𝜑(𝑡)−𝑟𝑚𝑦(𝑡))\n𝐻𝑧=𝑚𝑧(𝑡)                                 , (2) \nG – relation of Josephson energy to energy of magnetic anisotropy, r – the spin -orbit coupling \nparameter , mx,y,z  corresponds the x,y,z-component of mag netization vector  𝑚⃗⃗ . The Josephson \nphase difference φ can be found as follows:   \n𝑑𝜑\n𝑑𝑡=𝐼𝑝𝑢𝑙𝑠𝑒(𝑡)−sin(𝜑−𝑟𝑚𝑦), (3) \nwhere  the pulse current Ipulse is given by  \n𝐼𝑝𝑢𝑙𝑠𝑒={𝐴𝑆,     𝑡∈[𝑡0−12∆𝑡,𝑡0+12∆𝑡]⁄ ⁄                 \n0,        otherwise. (4) \nHere As is the amplitude of the pulse current, and Δt is the time interval, in which the pulse \ncurrent is applied.  The system of equations (1) with effective fiel d (2),  (3) and with the pulse \ncurrent (4) describes the dynamics of the φ0-junction.  The initial conditions are following : \n𝑚𝑥 (0)=0,𝑚𝑦 (0)=0,𝑚𝑧 (0)=1, 𝜑0=0. (5) \nProblem (1 -5) is a Cauchy problem for a system of ordinary differential equations . Numerical \nsolution is based on  the implicit two -step Gauss –Legendre method  which was shown in [6] to \nprovide more accurate calculations in comparison with t he explicit Runge –Kutta scheme .  \nRESULTS OF NUMERICAL SIMULATIONS  \nMagnetic reversal is an effect when the component mz(t) change s its initial value mz(0)=+1  and \nstabilizes on the value  mz(Tmax)=–1 for sufficiently large T max. Numerical study is red uced to \nmassive numerical simulations the Cauchy problem  (1-5) with varied parameters of model. In \n[5], the results of such simulations at the planes of parameters ( G, α) and ( G, r) are presented \nwhile all other parameters have the following values: at As = 1.5; r = 0.1; t0 =25; Δt = 6. \nSimulations were carried out with the time stepsize ht=0.01 while t≤Tmax=1000 . MR was  3 indicated using the condition |mz(Tmax)+1|<ε, where ε=0.0001. It was shown in [ 5] that the MR \nrealization is periodic  when  the value of  G changes, i.e. there are intervals on G where MR \noccurs  and intervals where MR  is absent . The purpose of this work is to demonstrate a \ndependence of configuration of the MR realization intervals on the parameter  ωF (normalized \nfrequency of ferromagnetic resonance).  To this end, massive numerical simulations were \ncarried out on the  (G, α) and ( G, r) planes  in order to reveal  the domains where the MR takes \nplace . Three values of  ωF were used in calculations : ωF=0.1, 1, and 10. The results are presented \nin Figures 1 and 2.  \nFigure 1 demonstrate s the MR domains at the (G, α)-plane. Colored areas  correspond the \npairs of parameters G and α where the MR occur s. One sees that at small ωF=0.1 we obtained \na solid domain of G between 0 and 60  where the  MR takes place. In case ωF=1, the  MR domains \nare in the form of wide bands alternating with areas where  the MR does not occur . As for the \ncase of large  ωF=10 – the MR domains structure looks rather chaotic due to  the very narrow \nalternating intervals  of presence and absence of the magneti zation  reversal.  \n \nFig. 1.  Magnetization reversal domains at the (G, α)-plane  depending on value of ωF. The \ncalculations  were carried out  with the G-stepsize ΔG=0.1, α-stepsize Δα=0.001 .  \n \nFig. 2.  Magnetization  reversal domains at (G,r)-plane  depending on value of ωF. The \ncalculations were carried out with the G-stepsize Δ G = 0.1 , r-stepsize Δ r = 0.005 .  \nFigure 2 shows the MR domains at the (G, r)-plane. As in Figure 1, one observes that the \nmagnetization reversal intervals (colored) alternate with the areas where the MR does not occur. \nIntervals of presence and absence of MR become narrow as ωF grows from 0.1 to 10.  \n 4 EFFECT OF PARALLEL IMPLEMENTATI ON \nSince  numerical investigation of the MR effect requires massive calculation s for numerical \nsolution of the C auchy problem (1 -5) at a large number of p airs of parameters at (G,α) and ( G,r) \nplanes , the parallel implementation have been developed to reduce  the execution time  and to \naccelerate the numerical study . In addition to the  MPI parallel implementation  presented in [7], \nwe developed the OpenMP version to expand the possibilities of numerical research on different \ncomputing systems . In both MPI and OpenMP ve rsions, t he paralleli sm is based on a \ndistribution of the points of (G,α) or (G,r) plane between parallel processes . The calculations \nwere made on two platforms  of the Multifunctional Information Computing Centre [8] – the \nHybriLIT cluster and the  Govorun supercomputer. The number P of parallel MPI processes and \nOpenMP threads was varied from 1 to 32.  Both MPI and OpenMP implementation s are \nconfirmed to be quite effective and provide almost the same execution time in calculations at \nboth HybriLIT c luster and Govorun supercomputer.   \nFigure 3 demonstrates that the  execution time in test calculations at the (G,r) plane  \ndecreases hyperbolically with increasing a number P of parallel MPI processes and OpenMP \nthreads.  Weak oscillations of MPI -curves are explained by the strengthening of the interaction \nof parallel processes with increasing P (for details, see the recent work [9]).   \n \nFig. 3.  The execution time of MPI and OpenMP calculations in depend ence on P. \nSUMMARY  \nMassive  numerical simulations i n the wide range of parameters within the 0-junction \nmodel allowed to  obtained domains where the magnetic moment is reversed.  An influence of  \nthe normalized frequency of the magnetic resonance ωF on the  periodic structure of MR \ndomains  has been studied. Test calculations confirm that both MPI and OpenMP parallel \nversions of the C++ computer code can provide the high-performance investigation of  the MR \nphenomenon within the 0-junction model.  \n 5 The work was supported by the Russian Science Foundation grant № 18-71-10095 \n(https://rscf.ru/project/21 -71-03009/ ). \nREFERENCES  \n1. Buzdin A.  I. Direct Coupling Between Magnetism and Superconducting Current in the \nJosephson φ0 Junction  // Physical Review Letters . 2008 . V.101 ,107005 . \n2. Buzdin A. I . Proximity effects in superconductor -ferromagnet heterostructures  // Rev. Mod. \nPhys. 2005 . V. 77 , 935. \n3. Konschelle F., Buzdin A. Magnetic Moment Manipulation by a Josephson Current // Phys. \nRev. Lett. 2009 , 102, 017001 . \n4. Shukrinov  Yu. M., Rahmonov  I. R., Sengupta  K., Buzdin A.  Magnetization reversal by \nsuperconducting current in Josephson junctions Appl. Phys.  Lett. 2017 . 110, 182407 . \n5. Atanasova, P.K., Panayotova, S.A., Rahmonov, I.R. Shukrinov, Yu. M., Zemlyanaya  E. V., \nBashashin, M. V., Periodicity in the Appearance of Intervals of the Reversal of the Magnetic \nMoment of a 0 Josephson Junction  // JETP Letters . 2019 , 110(11):722 -726. \n6. Atanasova P. K., Panayotova S. A., Shukrinov Yu. M. and Rahmonov I. R. Numerical \nSimu lation of the Stiff System of Equations within the Spintronic Model  // Lecture Notes \nin Computer Sciences . 2019 . 11189 , 301–308. \n7. Stefani Panayotova, Maxim Bashashin, Elena Zemlyanaya, Pavlina Atanasova, Yury \nShukrinov, Ilhom Rahmonov. Parallel Numerical Si mulation of the Magnetic Moment \nReversal within the ф0-Josephson Junction Spintronic Model. // European Physics Journal, \nWeb of Conferences . 2020 .Vol. 226, 02018  \n8. Adam Gh., et al. IT -ecosystem of the HybriLIT heterogeneous platform for high \nperformance comp uting and training of IT -specialists  // CEUR Workshop Proceedings. \n2018 . 2267 , 638-644. \n9. Bashashin M.V., Zemlyanaya E.V. Comparative  performance analysis of MPI - and \nOpenMP -programs on the example of parallel calculations in the framework of the nucleus -\nnucleus potential model and the φ0-spintronic model // Modern Information Technologies \nand IT -Education. 2022. Vol. 18, №3 , 545-557. \n "    },    {        "title": "1906.00249v1.Magnetic_fluctuations_in_the_itinerant_ferromagnet_LaCrGe3_studied_by_139La_NMR.pdf",        "content": "arXiv:1906.00249v1  [cond-mat.str-el]  1 Jun 2019Magnetic fluctuations in the itinerant ferromagnet LaCrGe 3studied by139La NMR\nK. Rana,1H. Kotegawa,2R. R. Ullah,3J. S. Harvey,3S. L.\nBud’ko,1P. C. Canfield,1H. Tou,2V. Taufour,3and Y. Furukawa1\n1Ames Laboratory, U.S. DOE, and Department of Physics and Ast ronomy, Iowa State University, Ames, Iowa 50011, USA\n2Graduate School of Science, Kobe University, Kobe 657-8501 , Japan\n3Department of Physics, University of California, Davis, CA 95616, USA\n(Dated: June 4, 2019)\nLaCrGe 3is an itinerant ferromagnet with a Curie temperature of Tc= 85 K and exhibits an\navoided ferromagnetic quantum critical point under pressu re through a modulated antiferromag-\nnetic phase as well as tri-critical wing structure in its tem perature-pressure-magnetic field ( T-p-H)\nphase diagram. In order to understand the static and dynamic al magnetic properties of LaCrGe 3,\nwe carried out139La nuclear magnetic resonance (NMR) measurements. Based on the analysis\nof NMR data, using the self-consistent-renomalization (SC R) theory, the spin fluctuations in the\nparamagnetic state are revealed to be isotropic ferromagne tic and three dimensional (3D) in nature.\nMoreover, the system is found to follow the generalized Rhod es-Wohfarth relation which is expected\nin 3D itinerant ferromagnetic systems. As compared to other similar itinerant ferromagnets, the\nCr 3delectrons and their spin fluctuations are characterized to h ave a relatively high degree of\nlocalization in real space.\nPACS numbers:\nI. INTRODUCTION\nRecently muchattention has been paid toitinerant fer-\nromagnetic(FM) compounds because ofthe observations\nofunconventionalsuperconductivity(SC) aswellaschar-\nacteristic magnetic properties related to FM quantum\ncriticality under application of pressure ( p) and magnetic\nfield (H)[1–7]. Interestingly, in itinerant FM compounds,\nthe FM quantum critical point (QCP) under pis always\navoided, which has been of great interest in experimen-\ntal and theoretical studies. Usually, when the second\norder paramagnetic (PM)-FM phase transition temper-\nature (Tc) is suppressed by the application of p, the or-\nder of the phase transition changes to the first order at\nthe trictrtical point (TCP) before Tcreaches 0 K at the\nquantum phase transition (QPT). This is known as the\navoided QCP [8–10]. Here the QCP is a second order\nquantum phase transition at T= 0 K. [11] When the\nPM-FM transition becomes of the first order at the TCP\nin thep-Tplane, the application of magnetic field ( H)\nleads to a tricritical wing (TCW) structure in the T-p-H\nthree dimensional phasediagram[see, Fig. 1(a)] asfound\nin UGe 2[8, 9] and ZrZn 2[10]. A PM-FM QCP can also\nbe avoided by the appearance of an antiferromagnetic\n(AFM) ordered state under pnear the putative QCP, as\nactually observed in CeRuPO[12, 13] and MnP[14, 15].\nIn this case, no wing structure has been reported and the\nAFM state is suppressed by the application of moderate\nH, as schematically shown in Fig. 1(b).\nSuch interesting phase diagrams have been theoreti-\ncally well studied, and PM-FM QCP in clean itinerant\nFM systems is suggested to be always avoided following\nthe above two scenarios. It has also been suggested that\nthe coupling of quantum fluctuations of particle-hole ex-\ncitations with the order parameter in metallic systems\ncan generically render the prior second order phase tran-\nFIG. 1: Generic temperature-pressure-magnetic field ( T-p-\nH) phase diagrams for clean itinerant ferromagnets. (a)\nSchematic T-p-Hphase diagram with a tricritical wing\n(TCW) structure. The second order phase transition (solid\nline) becomes first order (dashed line) at a tricritical poin t\n(TCP). Under magnetic field, the TCW structure shown in\nred emerge from the TCP. (b) Schematic T-p-Hphase dia-\ngram with an antiferromagnetic (AFM) phase in blue. The\nAFM phase emerges from the Lifshitz point (LP) that is sup-\npressed under H.\nsitions first order [16–20] or drive the system towards\nincommensurate ordering states [17, 18]. In addition,\nmagnetic fluctuations may also play an important role\nto avoid FM QCP as some systems exhibit phases such\nas incommensurate magnetic ordered states [12–15] and\nunconventional superconductivity [3–7]. Therefore, the\nexperimental characterization of the nature of quantum\nfluctuations, including magnetic fluctuations, in these2\nclasses of materials is important to illuminate the un-\nderlying mechanism that connects magnetism, quantum\ncritically and superconductivity.\nRecently, the itinerant ferromagnet LaCrGe 3has been\ndiscovered to be a new class of itinerant ferromagnets\nexhibiting the remarkable T-p-Hphase diagram where\nboth the TCW structure and AFM phase are observed\n[21–23]. LaCrGe 3crystallizes in the hexagonal BaNiO 3-\ntype structure [space group P63/mmc(194)][24]. At am-\nbientp, LaCrGe 3is FM below the Curie temperature Tc\n= 85 K with an ordered magnetic moment at low tem-\nperatures of 1.25 µB/Cr aligned along the caxis [21].\nThis small value of the magnetic moment compared with\nthe effective moment above Tc(µeff= 2.4µB/Cr) [25] in\nthe PM state suggests some degree of delocalization of\nthe Cr 3dspins.Tccan be suppressed with pleading to\na weakly modulated AFM state around 1.5 GPa and T\nclose to 50 K [21]. Furthermore, it was also reported to\nexhibit a TCP where the second order FM transition be-\ncomes of the first order at Tof around 40 K and pclose\nto 1.8 GPa [22] yielding a TCW structure under H.\nMotivated by the novel magnetic properties in\nLaCrGe 3, we carried out nuclear magnetic resonance\n(NMR) measurement which is a powerful technique to\ninvestigate the magnetic and electronic properties of ma-\nterials from a microscopic point of view. It is known that\nthe temperature dependence of the nuclear spin-lattice\nrelaxation rate (1/ T1) reflects the wave vector q-summed\ndynamical susceptibility. On the other hand, NMR spec-\ntrum measurements, in particular the Knight shift K,\ngive us information on local static magnetic susceptibil-\nityχ. Thus from the temperature dependence of 1/ T1T\nandK, onecan obtain valuableinsights intospin fluctua-\ntions in materials. In this paper, we report the results of\n139La NMR measurements performed to investigate the\nspin fluctuations in LaCrGe 3. Our analysis, based on the\nself-consistent renomalization (SCR) theory, reveals that\nFM spin fluctuations due to Cr 3 dspins are of3D nature.\nIn addition, LaCrGe 3is well characterizedby a relatively\nhigh degree of localization of 3 dspins although the sys-\ntem is itinerant. Furthermore, the spin fluctuations are\nalsorevealedtohaveamorelocalizednaturein realspace\ncompared to other similar itinerant FM materials.\nII. EXPERIMENTAL DETAILS\nNeedle-like shaped single crystals of LaCrGe 3were\ngrown out of high temperature solutions, the details of\nwhich are reported in Ref. [25]. Plural single crystals\nused for NMR measurements were placed in parallel on\na glass plate (5 ×5×0.2 mm3) to align their directions.\nThe crystalline caxis and the abplane are parallel and\nperpendicular to the needle direction of the crystal, re-\nspectively. An NMR coil was tightly wound around the\ncrystals including the glass plate to reduce a loss of the\nfilling factor that is estimated to be about 0.6. NMR\nmeasurements of139La (I=7\n2,γN\n2π= 6.0146 MHz/T,Q= 0.21 barns) nuclei were conducted using a lab-built\nphase-coherent spin-echo pulse spectrometer. The139La\nNMR spectra were obtained by sweeping Hat fixed fre-\nquencies or by sweeping frequency under constant H.H\nwas applied parallel to either the crystalline caxis or\ntheabplane. The zero-shift position corresponding to\nthe Larmor field for each resonance frequency was deter-\nmined by31P NMR in H 3PO4solution or63Cu NMR in\nCu metal.\nThe139La nuclear spin-lattice relaxation rate (1/ T1)\nwas measured with a saturation recovery method. 1 /T1\nat each temperature ( T) was determined by fitting the\nnuclear magnetization Mversus time tusing the ex-\nponential function 1 −M(t)/M(∞) = 0.012e−t/T1+\n0.068e−6t/T1+0.206e−15t/T1+0.714e−28t/T1, whereM(t)\nandM(∞) are the nuclear magnetization at time tafter\nthesaturationandtheequilibrium nuclearmagnetization\natt→ ∞, respectively, for the case of magnetic relax-\nation [26]. The observed recovery data in the paramag-\nnetic state were well fitted by the function, indicating\nthat the nuclear relaxation is mainly induced by fluctu-\nations of the hyperfine field at the139La site. For the\nanalysis of NMR data, we measured the magnetic sus-\nceptibility χ(T) of the single crystal at H= 7 T applied\nparallel to the caxis and to the abplane in a commercial\nQuantum Design superconducting quantum interference\ndevice magnetometer.\nIII. RESULTS AND DISCUSSION\nA.139La NMR spectrum\nFigure 2(a) shows the field-swept139La-NMR spectra\nof LaCrGe 3atT= 230 K for Hparallel to the caxis\n(H||c) and to the abplane (H||ab). The typical NMR\nspectrum for a nucleus with spin I= 7/2 with Zeeman\nand quadrupolar interactions can be described by a nu-\nclear spin Hamiltonian H=−γ¯hI·Heff+hνQ\n6[3I2\nz−I2+\n1\n2η(I2\n++I2\n−)], where Heffis the effective field at the nu-\nclear site, his Planck’s constant, and ηis the asymmetry\nparameter of electric field gradient (EFG) at the nuclear\nsite. The nuclear quadrupole frequency for I= 7/2 nu-\nclei is given by νQ=e2QVZZ/14h, whereQis the nu-\nclear quadrupole moment and VZZis the EFG at the La\nsite. When the Zeeman interaction is much greater than\nthe Quadrupole interaction, this Hamiltonian produces a\nspectrum with a central transition line flanked by three\nsatellite peaks on either side. The observed spectra are\nwell reproduced by simulated spectra (red lines) from the\nsimple Hamiltonian with νQ= 0.66 MHz and η∼0. The\ntiny extra peaks around 6.80 T and 7.66 T for H||cin\nFig. 1(a) could be due to mis-orientation of some of the\ncrystals while being attached on the glass plate and also\nmay be due to slightly different qualities of the crystals.\nThe values of νQandηestimated from the main seven\npeaks are found to be independent of temperature in the\nPM state above TC= 85 K. From the spectrum analy-3\nFIG. 2: (a) Typical field-swept139La-NMR spectra of\nLaCrGe 3in the PM state at f= 42.5 MHz and T= 230\nK forH||candH||ab. (b) Temperature dependence of139La-\nNMR spectrum for H||c. (c) Field-swept139La-NMR spectra\nin the FM state ( T= 1.65 K) for H||candH||ab. Blue and\ngreen curves represent spectra for H||candH||ab, respec-\ntively, and the red curves are the calculated spectra with νQ\n= 0.66 MHz for the PM state and νQ∼0.49 MHz for the FM\nstate. The vertical dashed black lines in (a) and (b) represe nt\nthe zero-shift position ( K= 0).\nsis where θs are found to be 0 and π/2 forH/bardblcand\nH/bardblab, respectively, it is clear that the principal axis\nof the EFG at the La site is along the caxis. Since the\nLa site in LaGeCr 3does not have a local axial symmetry\n(¯6m2), one may expect finite value of η. However, ηis\nfound to be very close to zero within our experimental\nuncertainty.\nAs shown in Fig. 2(b), with decreasing temperatures,\neach line becomes broader due to inhomogeneous mag-\nnetic broadening and the spectra show less clear features\nof the quadrupolar split lines below ∼155 K. At the\nsame time, nuclear spin-spin relaxation time T2becomes\nshort at a wide range of temperatures close to TC. Those\nmakeNMRspectrummeasurementsdifficultbelow ∼110\nK. However, when the temperature is decreased down to\n1.65K,wellbelow TC=85K,wewereabletoobservethe\n139La NMR spectrum in the FM state as shown at the\nbottom of Fig. 2(b), where the spectrum largely shifted\nto higher magnetic field by ∼4 T. The shift is due to\nthe internal magnetic induction ( Bint) at the La site pro-\nduced by the Cr spontaneous magnetic moments in the\nFM state. The νQ∼0.49 MHz estimated from the spec-\ntra under two different magnetic field directions ( H||c\nandH||ab) shown in Fig. 2(c) is slightly smaller than\n0.66 MHz observed in the PM state. From the spectrum,\ntheBintis estimated to be −4 T and−4.2 T for H||cand\nH||ab, respectively. Here we took the zero-shift position\n(K= 0) as the origin of Bint. Unfortunately, the spec-\ntrum is measurable only around 1.6 K since its intensity/s45/s51/s46/s48 /s45/s51/s46/s53 /s45/s52/s46/s48 /s45/s52/s46/s53/s72 /s124/s124 /s99\n/s32/s61/s32/s48/s111\n/s102/s32/s61/s32/s49/s56/s46/s56/s49/s32/s77/s72/s122\n/s102/s32/s61/s32/s50/s50/s46/s55/s32/s77/s72/s122\n/s102/s32 /s61/s32/s51/s49/s32/s77/s72/s122\n/s32/s32/s83/s112/s105/s110/s32/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32\n/s102/s47\n/s78/s45/s72 /s32/s40/s84/s41/s102/s32/s61/s32/s52/s50/s46/s53/s32/s77/s72/s122/s54 /s56 /s49/s48 /s49/s50/s45/s51/s46/s48/s45/s51/s46/s53/s45/s52/s46/s48/s45/s52/s46/s53/s45/s53/s46/s48\n/s32/s72 /s124/s124 /s99\n/s32/s72 /s124/s124 /s97/s98\n/s32/s32/s66\n/s105/s110/s116/s32/s40/s84/s41\n/s72\n/s114/s101/s115/s32/s32/s40/s84/s41\nFIG. 3: (a)139LaNMR spectra at various frequencies for H||c\natT= 1.65 K. Inset: Resonance magnetic field ( Hres) depen-\ndence of BintforH||ab(green circles) and H||c(blue squares).\nThe arrows show the positions of the central transition line\nof the spectra measured at different frequencies\ndecreases rapidly by raising the temperature.\nIn order to check whether the Bintis induced by the\nspontaneous Cr magnetic moments in the FM state and\nnot due to NMR shift produced by the application of\nmagnetic field, wedetermined the externalmagnetic field\ndependence of Bintby measuring the spectra with differ-\nent frequencies as shown in Fig. 3. Although the signal\nintensity decreases with decreasing resonance frequency,\nwe observed the spectrum down to f= 18.81 MHz and\nfound that Bint∼ −4 T and −4.2 T for H||candH||ab\nare nearly independent of resonance frequency (i.e. ex-\nternal magnetic field), confirming that these Bintvalues\ncan be attributed to the hyperfine field at the La sites\nproduced by the magnetic field independent spontaneous\nmagnetic moment of the Cr ions in LaCrGe 3in the FM\nstate.\nFigure 4 shows the Tdependence of the139La-NMR\nshift in the PM state for H/bardblabplane (Kab) andH/bardblc\naxis (Kc) determined from the simulated spectra, where\nbothKabandKcare nearly the same and decrease on\nlowering temperature. The NMR shift consists of tem-\nperature dependent spin shift Ks(T) andTindependent\norbital shift K0:K(T) =Ks(T) +K0whereKs(T) is4\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s51/s48/s45/s50/s53/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48\n/s84 /s32/s32/s40/s75/s41/s32\n/s32/s75\n/s99/s44/s32 /s75\n/s97/s98/s32/s40/s37/s41/s32/s75\n/s97/s98\n/s32/s75\n/s99\n/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51 /s48/s46/s48/s52/s45/s50/s53/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48\n/s32/s32/s75\n/s99/s44/s32 /s75\n/s97/s98/s32/s40/s37/s41\n/s99/s44/s32\n/s97/s98/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\nFIG. 4: Temperature dependence of139La Knight shift for\nbothH||ab(green) and H||c(blue) directions measured at H\n∼7 T. Inset: Kabvs.χabandKcvs.χcplots. The solid\nlines are linear fits as described in the text.\nproportional to the spin part of magnetic susceptibil-\nityχs(T) via hyperfine coupling constant A,Ks(T) =\nAχs(T)\nNA. HereNAis Avogadro’s number. The hyperfine\ncoupling constants are estimated to be Aab=−32±1\nkOe/µBandAc=−27±1 kOe/µBforH||abandH||c,\nrespectively, from the slopes in the so-called K-χplots\nshown in the inset of Figure 4. The intercepts for the fits\nare almost zero for both the directions. This indicates\nthat the observed Knight shifts are mainly attributed to\nKs(T).Bintis proportional to Ahf<µ>whereAhfis the\nhyperfine coupling constant and <µ>is the ordered Cr\nmagnetic moment. Using Bint=−4.0 T and −4.2 T for\nH||candH||ab, respectively, and Aab=−32kOe/µBand\nAc=−27 kOe/µB,<µ>are estimated to be 1.30 µBfor\nH||aband 1.50 µBforH||cwhich are slightly higher but\nin good agreement with 1.22 µB(along the caxis) re-\nported by the neutron diffraction measurements [27] and\n1.25µB(along the caxis) from magnetization measure-\nments [23, 25].\nB.139La spin lattice relaxation time\nIn order to investigate the magnetic fluctuations in\nLaCrGe 3, we measured the139La spin-lattice relaxation\nrate (1/T1) at the peak position of the spectra for both\nthe magnetic field directions. Figure 5 showsthe temper-\nature dependence of 1/ T1Twhere 1/ T1Tincreases with\ndecreasing temperature from room temperature to 125 K\nwith no anisotropy in T1.\nBased on the T1and the spin part of the Knight shift\n(Ks) data, we discuss the magnetic fluctuations in the\nPM state of LaCrGe 3. First we tentatively employ the\nmodified Korringaratioanalysis. In Fermi liquid picture,\n1/T1TandKsare determined by the density of states at/s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s49/s48/s49/s48/s48/s49/s48/s48/s48\n/s49/s48/s49/s48/s48/s49/s48/s48/s48\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s49/s47/s40 /s84\n/s49/s84/s75 /s41/s32/s32/s32/s32 /s105/s110/s45/s112/s108/s97/s110/s101/s44/s32 /s32/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101\n/s49/s47/s40 /s84\n/s49/s84/s75/s51/s47/s50\n/s41/s32 /s32/s105/s110/s45/s112/s108/s97/s110/s101/s44/s32 /s32/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32\n/s32/s49/s47/s40 /s84\n/s49/s84/s75/s51/s47/s50\n/s41/s32/s32/s32/s40/s115/s75/s41/s45/s49/s49/s47/s40 /s84\n/s49/s84/s75 /s41/s32/s32/s32/s32/s40/s115/s75/s41/s45/s49\n/s84 /s32/s32/s40/s75/s41/s32/s49/s47 /s84\n/s49/s84 /s32/s32/s32/s32/s40/s115/s75/s41/s45/s49\n/s32/s72 /s124/s124 /s97/s98\n/s32/s72 /s124/s124 /s99\n/s84 /s32/s32/s40/s75/s41\nFIG. 5: Temperature dependence of139La 1/T1TforH||ab\n(green circles) and H||c(blue squares). The inset shows the\nsemi- log plots of 1/( T1TKs) and 1/( T1TK3/2\ns) for two differ-\nent magnetic fluctuation directions (in-plane and out-of-p lane\ndirections).\nthe Fermi energy D(EF). TheT1has relation with Ks\nthat can be described as T1TK2\ns= (¯h/4πkB)(γe/γn)2≡\nS. Hereγeis the electronic gyromagnetic ratio. The\nKorringa ratio α(≡S/T1TK2\ns) between an experimen-\ntal value of T1TK2\nsand the non-interacting electron sys-\ntemScan reveal information about electron correlations\nin materials [28, 29]. α∼1 represents the situation\nof uncorrelated electrons. However, enhancement of χs\n(q/negationslash= 0) increases 1/ T1Tbut has little or no effect on\nKswhich probes only the uniform χs(q= 0) where q\nrepresents wave vector. Thus α >1 indicates antifer-\nromagnetic (AFM) spin correlations. In contrast, FM\nspin correlations produce α <1. Since 1/ T1Tprobes\nmagneticfluctuationsperpendiculartothe magneticfield\n[28], in general, one should consider the Korringa ratio\n1/(T1,⊥TK2\ns,ab), where 1/( T1,⊥T) = 1/(T1T)H||c, when\nexamining the character of magnetic fluctuations in the\nabplane (in-plane direction). One also needs to con-\nsider the Korringa ratio 1/( T1,||TK2\ns,c) for magnetic fluc-\ntuations along the caxis (out-of-plane direction). Here,\n1/(T1,/bardblT) is estimated from 2/( T1T)H||ab−1/(T1T)H||c.\n[30]. However, since both 1/ T1and139Kare nearly\nisotropic, 1/( T1,⊥TK2\ns,ab) and 1/( T1,||TK2\ns,c) are almost\nthe same, clearly indicating isotropic magnetic fluctua-\ntions in LaCrGe 3.α/bardblandα⊥decrease from ∼0.07 at\nroom temperature to less than ∼0.008 around 120 K, in-\ndicatingdominantFM spincorrelationsbetweenCrspins\nin the compound.\nItshouldbenotedthat, however,theKorringaanalysis\nusually applies for PM materials where electron-electron\ninteraction is weak. Since LaCrGe 3exhibits a FM or-\nder, we also analyze NMR data based on self-consistent\nrenormalization(SCR)theory. Asshownabove,themag-\nnetic fluctuations are governed by FM spin correlations.5\nIn this case, according to SCR theory for weak itiner-\nant ferromagnets, 1/( T1TKs) and 1/( T1TK3/2\ns) are ex-\npected to be independent of Tfor three dimensional (3D)\nor two-dimensional (2D) FM spin fluctuations, respec-\ntively [31, 32]. The inset of Fig. 5 shows the Tde-\npendence of 1/( T1TKs) and 1/( T1TK3/2\ns) for the two\ndirections. Both the 1/( T1,/bardblTKs,c) and 1/( T1,⊥TKs,ab)\nare nearly constant, while both the 1/( T1,/bardblTK3/2\ns,c) and\n1/(T1,⊥TK3/2\ns,ab) increase with increasing temperature.\nThis indicates that the FM spin fluctuations are char-\nacterized as 3D in nature.\nC. Spin fluctuations\nThe Curie Weiss behavior of 1/ T1Tin itinerant ferro-\nmagents is well described in the SCR theory in terms of\n3DFM spin fluctuations [33–37], as described above. Ac-\ncording to SCR theory, there are two important param-\neters to characterize spin fluctuation for itinerant ferro-\nmagnets: T0andTAcorresponding to the widths of the\nspin fluctuations in frequency ( ω) space at wave vector\n(q) = 0 and the width of the distribution of static suscep-\ntibility in qspace at ω= 0, respectively [38]. The former\ncan be obtained from the following equation of 1/ T1Tas\nderived by Corti et al.[39],\n1\nT1T=3¯hγ2\nNHhf\n16πµBK\nT0(1)\nHereHhfis the hyperfine field experienced at the La site\nper spin [39]. From the experimentally determined val-\nues,T0is estimated to be 89 K and 75 K for the in-plane\nand out-of-plane directions of spin fluctuations, respec-\ntively. On the other hand, TAcan be estimated using the\nfollowing equation given by Takahashi [38]:\nTA= 20C4/3/bracketleftBigTc\np3/2\nsT1/4\n0/bracketrightBig4/3\n(2)\nwhereTc= 85 K, psis the saturated moment and\nC4/3=1.006089 is a constant. Utilizing these values, TA\nis estimated to be 998 K and 793 K for the in-plane and\nout-of-plane directions of spin fluctuations, respectively.\nThe estimated values of TAseem to be comparable or\nslightly greater than those in uranium based compounds\nsuch as URhGe ( TA= 568 K) [40] and UGe 2(TA= 442\nK) [40], but are smaller than those in 3 delectron itin-\nerant ferromagnets such as ZrZn 2(TA= 7400 K) [38],\nNiAl3(TA= 3670 K) [38] and MnSi ( TA= 1690 K) [39].\nIn order to compare LaCrGe 3with other clean itiner-\nant ferromagnets that avoid the FM QCP with the TCW\nstructure, we plotted their spin fluctuation parametersin\nFig. 6 along with those in LaCrGe 3. The plot is known\nas the Generalized Rhodes-Wohlfarth plot. The xaxis\nof this plot is the ratio of TCandT0and theyaxis the\nratio of the effective paramagnetic moment ( peff) andps./s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s76/s97/s67/s114/s71/s101\n/s51/s32/s79/s80/s85/s82/s104/s71/s101\n/s32/s32/s77/s110/s83/s105\n/s32/s90/s114/s90/s110\n/s50/s32/s32\n/s85/s82/s104/s65/s108/s32 /s32\n/s32/s112\n/s101/s102/s102/s32/s47/s32/s112\n/s115\n/s84\n/s99/s32/s47/s32 /s84\n/s48/s85/s71/s101\n/s50/s32\n/s76/s97/s67/s114/s71/s101\n/s51/s32/s73/s80\nFIG. 6: Generalized Rhodes-Wohlfarth plot for the in-plane\n(blue) and out-of-plane (green) directions for LaCrGe 3along\nwith other itinerant ferromagnets that avoid FM-PM QCP\nthrough tricritical wings under pressure. The values of pa-\nrameters are estimated from Ref. [38] for ZrZn 2and MnSi\nand Ref. [40] for UGe 2, URhGe and URhAl. The red line\nrepresents the generalized Rhodes-Wohlfarth relation in t he\n3D system: peff/ps= 1.4 (TC/T0)−2/3[38].\nAccording to SCR theory, in the case of itinerant ferro-\nmagnets, magnetic fluctuations contribute to peffin PM\nstate. Therefore, one can expect that the ratio of peff/ps\nbecomes large, when spin fluctuations are important in\nthese systems. As for T0, based on SCR theory, TC/T0is\nclose to unity in the case of localized ferromagnets, while\nthe ratio becomes less than unity if itinerant character\nbecomes significant. These situations are predicted the-\noretically through the generalized Rhodes-Wolfarth rela-\ntion ofpeff/ps= 1.4 (TC/T0)−2/3(red line), and the rela-\ntion has shown great empirical agreement with materials\nhaving 3D FM fluctuations [38]. As shown in the Fig. 6,\nLaCrGe 3and other typical itinerant ferromagnets seem\nto follow the relation. However, the values of LaCrGe 3\nfor both the in-plane and out-of-plane directions of mag-\nnetic fluctuations are located close to unity, indicating a\nrelatively high degree of localization in Cr 3 delectrons\neven though the system is itinerant. In addition, the re-\nsults also indicate that the spin fluctuations in LaCrGe 3\nshow a more localized nature in real space than the other\nsimilar itinerant FM materials compared.\nIV. SUMMARY\nIn summary, we carried out139La NMR in the itiner-\nant ferromagnet LaCrGe 3to characterize the magnetic\nproperties from a microscopic point of view. The prin-\ncipal axis of the electric field gradient at the La site has\nbeen shown to be alongthe caxis with νQof 0.66 MHz in\nthe PMstate andabout 0.49MHz in the FM state.139La6\nNMR spectra measurements in the FM state confirmed\nthe FM ordered state below TC= 85 K with a magnetic\nmomentof ∼1.4µB/Cr,consistentwithpreviousreports.\nBased on the Korringaratio analysisand the SCR theory\nusing the resultsof 1/ T1TandK, spin fluctuations in the\nPM state are revealed to be isotropic FM and three di-\nmensional in nature. In addition, the system is found to\nfollow the generalized Rhodes-Wohfarth relation which\nis expected in 3D itinerant FM systems. TC/T0is found\nto be close to unity, indicating that there is a relatively\nhigh degree of localization in the 3 dCr electrons even\nthough the system is itinerant. It would be interesting\nif these uniqueness in magnetic fluctuations in LaCrGe 3\nis related to the appearance of both tricritcal wings and\nAFM state under pressure and magnetic fields. Analysis\nof spin fluctuations in those materials where FM QCP is\navoided by the appearance of AFM phase, and further\nstudies on LaCrGe 3under pressure are important to in-\nvestigate this connection, which are now in progress.V. ACKNOWLEDGMENTS\nThe authors would like to thank Q.-P. Ding, R.\nTakeuchi and Y. Kuwata, for help in conducting exper-\niments and T. Matsui for fruitful discussions. The re-\nsearch was supported by the U.S. Department of Energy,\nOffice of Basic Energy Sciences, Division of Materials\nSciences and Engineering. 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B 96, 035125\n(2017)."    },    {        "title": "0808.0892v2.Electron_spin_resonance_in_Kondo_systems.pdf",        "content": "arXiv:0808.0892v2  [cond-mat.str-el]  15 Oct 2008Electron spin resonance in Kondo systems\nElihu Abrahams1and Peter W¨ olfle2\n1Center for Materials Theory, Serin Physics Laboratory, Rut gers University, Piscataway, NJ 08854-8019\n2Institut f¨ ur Theorie der Kondensierten Materie,\nUniversit¨ at Karlsruhe, D-76128 Karlsruhe, Germany\n(Dated: October 31, 2018)\nWe calculate the dynamical spin response of Kondo impurity a nd Kondo lattice systems within\na semiphenomenological Fermi liquid description, at low te mperatures T < T K, the Kondo tem-\nperature, and low magnetic fields B≪kBTK/gµB. Fermi liquid parameters are determined by\ncomparison (i) with microscopic theory (numerical renorma lization group) for the impurity model\nand (ii) with experiment for the lattice model. We find in the i mpurity case that the true impurity\nspin resonance has a width of the order of TKand disappears altogether if the g-factors of impurity\nspin and conduction electron spin are equal. However, there is an impurity-induced resonance con-\ntribution at the conduction electron resonance. The latter is broadened by spin lattice relaxation\nand is usually unobservable. In contrast, for the Anderson l attice in the Kondo regime we find\nasharpESR resonance line only slightly shifted from the local reso nance and broadened by spin\nlattice relaxation, the latter significantly reduced by bot h the effects of heavy fermion physics and\nferromagnetic fluctuations. We conjecture that our findings explain the sharp ESR-lines recently\nobserved in several heavy fermion compounds.\nPACS numbers: 71.27.+a, 75.20Hr, 76.30.-v\nI. INTRODUCTION\nThe Kondo effect is arguably the best-studied many-\nbody effect in condensed matter physics1. In its initial\nform2,3, it involves a local “impurity” spin in a d- orf-\norbital, antiferromagnetically coupled to the spins of a\nconduction band in a dilute magnetic alloy. At temper-\naturesTbelow the dynamically generated energy scale\nTK, the Kondo temperature, this interaction causes a lo-\ncal spin 1 /2 to be fully screened. This behavior should\nbe noticeable in the T-dependence of the spin dynam-\nics of the system, as probed by electron spin resonance\n(ESR). In fact, the local spin resonance in dilute Kondo\ncompounds at T≫TKhad been observed even before\nthe Kondo effect was understood. Afterward there were\na number of systematic experimental investigations and\nperturbative calculations for the ESR at T≫TKin di-\nlute Kondo systems4.\nAt low temperatures T≪TK, on the other hand, neu-\ntron scattering studies revealed the existence of a broad\nspin excitation peak of width TK, interpreted as the\nKondo bound state5. Within the isotropic s-dexchange\n(Kondo) model the total spin is conserved. Therefore, in\nthe limit of equal g-factors of local moments and conduc-\ntion electron spins one expects a single spin resonance\nline at all temperatures, only broadened by spin lattice\nrelaxation. As we shall show below, in this limit the\nweight of the broad local spin resonance tends to zero.\nIn several recent experiments6,7low-temperature ESR\nhas been observed in some heavy-fermion metals, in par-\nticular YRh 2Si2(YRS)8. The phase diagram of YRS has\na magnetic-field induced quantum critical point and is a\nmodel system for the study of quantum criticality in the\nKondolattice. Consequently, the observationofa narrow\nESR resonance in this compound aroused great interest,especially since it was commonly believed that heavy-\nfermion ESR would be unobservable due to an enormous\nintrinsic linewidth ∆ Bof order kBTK/gµB6. HereTK\nis the lattice coherence (“Kondo”) temperature for the\nonset of heavy-fermion behavior and gµBis the gyro-\nmagnetic ratio for the resonance. These were the first\nobservationsofESR in Kondolattice systems at T < T K.\nInYRS,theobservednarrowdysonian9ESRlineshape\nwas originally interpreted6as indicating that the reso-\nnance was due to local spins at the Yb sites. Therefore,\ninitially the authors speculated that the appearence of a\nnarrow ESR line might indicate the suppression of the\nKondo effect near the quantum critical point, since, as\nexplained above, carrying over Kondo impurity physics\nto the Kondo lattice one might expect the local spins to\nbe screened by the Kondo effect, giving rise only to a\nbroad spin excitation peak, too wide to be observed in\nESR experiments. However, a closer look10revealed that\nitinerant (heavy) electron ESR could give rise to a simi-\nlar line shape since the carrier diffusion in YRS is quite\nslow. Thus, whether the resonance was that of localized\nor itinerant spins remains an open question.\nNow, a common feature of the compounds in which\nESR has been observed appears to be the existence of\nferromagnetic fluctuations7These findings challenge our\nunderstanding of heavy fermion compounds: How does\na sharp electron spin resonance emerge despite Kondo\nscreeningand spin lattice relaxation, and why is this pro-\ncess influenced by ferromagnetic fluctuations? We shall\naddress these questions in the framework of Fermi liquid\ntheory, taking the relevant parameters from numerical\nstudies and experiment.2\nII. ANDERSON IMPURITY MODEL IN THE\nKONDO SCREENED REGIME\nIn the Kondo regime an impurity spin is screened by\nthe conduction electron spins at (or near) the impurity.\nThe dynamics of the impurity spin is governed by the\nenergy scale of the corresponding many-body resonance,\nthe Kondo temperature TK. Nonetheless the conduc-\ntion electrons in the vicinity of the impurity show the\ninfluence of the Kondo screened state in their dynami-\ncal behavior. In the Anderson model, the local spin is\nthat of a localized felectronWe assume that the Zeeman\nsplittings ωfandωcinduced by a magnetic field acting\non the local and conduction electron spins are small com-\npared to the Kondo temperature TK. Then the Kondo\nscreened state is only weakly perturbed by the magnetic\nfield. At temperatures T≪TK, the spin resonance be-\nhavior of the impurity may then be described by Fermi\nliquid theory11.\nWe start from the bare Anderson model Hamiltonian\nH=Hc+/summationdisplay\nk,σǫkσc+\nkσckσ+/summationdisplay\nσǫfσnfσ+Unf↑nf↓\n+V/summationdisplay\nk,σ(f+\nσckσ+h.c.), (1)\nwhereHcis the conduction electron Hamiltonian and\nc+\nkσ,f+\nσarecreationoperatorsofthe conductionelectrons\nin momentum and spin eigenstates ( kσ), and of electrons\nin the local flevel, respectively. The operator nfσ=\nf+\nσfσcounts the number of electrons on the local level,\nandǫfσ=ǫf−ωfσ/2, σ=±1.\nThe effect ofthe interaction Uis torenormalizethe pa-\nrameters ǫfσ,U,Vto/tildewideǫfσ,/tildewideU,/tildewideVintherenormalizedFermi\nliquid type low energy Hamiltonian, Eq. (1) with the\nrenormalized parameters that may be calculated using\nthenumericalrenormaliztiongroup(NRG)method12. To\nkeep the algebra simple, we assume particle-hole symme-\ntry in the following. Then /tildewideǫfσ=−(/tildewideU+ωfσ)/2. The\nhybridization of the local level with the conduction band\nleads to an f-level broadening /tildewideΓ =π/tildewideV2N0∼TKwith\nN0= 1/Wthe local conduction electron density of states\n(DOS) at the Fermi level (in the model with flat DOS, W\nis the bandwidth). The initially rather large bare level\nwidth is renormalized down to the very narrow width of\nthe Kondo resonance. The NRG calculation shows that\n/tildewideU=π/tildewideΓ and/tildewideV2is O(TK/TF), where TFis the Fermi\ntemperature of the conduction electrons.\nIn the framework of Fermi liquid theory, the interac-\ntion has two major consequences: (i) it gives rise to a\nmolecular field renormalizing the collective response of\nthe system (ii) it leads to a finite lifetime of quasiparti-\ncles. However, the quasiparticle relaxationrate is limited\nby the available phase space and vanishes quadratically\nas the excitation energy goes to zero. Therefore, at tem-\nperatures T≪TKthe Landau quasiparticles are well-\ndefined. The quasiparticle decay contributes to the spin\nrelaxation rate. As we shall show, the local moment re-laxation is governed by rapid spin flips on the frequency\nscaleofTK, occuringaspartofthemanybodyresonance.\nThen at temperatures T≪TKwe may neglect the addi-\ntional relaxation caused by the quasiparticle decay.\nWe now consider the effects of the molecular field\ncaused by the Fermi liquid interaction /tildewideU. We treat the\ninteraction term in the Hamiltonian in mean field ap-\nproximation: /tildewideUnf↑nf↓≈1\n2/tildewideU[∝an}bracketle{tnf∝an}bracketri}htnf−∝an}bracketle{tmf∝an}bracketri}htmf+const],\nwhere we defined the density and spin density operators\nnf=nf↑+nf↓andmf=nf↑−nf↓. In the case\nof particle-hole symmetry, when ∝an}bracketle{tnf∝an}bracketri}ht= 1, the density\nterm is cancelled by the single-particle energy. The spin-\ndensity term gives rise to an effective magnetic field so\nthat/tildewideǫfσ+1\n2/tildewideU[∝an}bracketle{tnf∝an}bracketri}ht−σ∝an}bracketle{tmf∝an}bracketri}ht] =−σωf, which amounts to\na doubling of the Zeeman energy. Here we have used\nthat the spin polarization is given by m=∝an}bracketle{tmf∝an}bracketri}ht=\nχ+−\nff(0)ωf/2 =ωf/π/tildewideΓ, withχ+−\nff(0) = 2/π/tildewideΓ the static\nsusceptibility of the local spin and /tildewideU=π/tildewideΓ, as obtained\nfrom NRG calculations. Then the local electron Green’s\nfunction, including the coupling to the conduction elec-\ntrons, is given by\nGfσ(iωn) = [iωn+σωf+i/tildewideΓsign(ωn)]−1(2)\nThe local spectral function is given by Afσ(ω) =\nImGfσ(ω+i0) =/tildewideΓ/[(ω+σωf)2+/tildewideΓ2], describing the\nKondo resonance. We see that in a magnetic field the\nresonance is shifted from its zero field position ω= 0 to\nthe spin dependent position ω=−σωf, which is double\nthe Zeeman shift.\nWe use the definitions ωf=gfµBB,ωc=gcµBBand\ntakeχ+−\nff, etc. to be response functions of spin 1 /2 oper-\nators. The dynamical conduction electron susceptibility\nχccis characterized by a resonance peak broadened by\nspin-lattice relaxation. We follow Barnes and Zitkova-\nWilcox13and model the spin-lattice relaxation mecha-\nnism by a local random magnetic field hithat fluctuates\nin both direction and magnitude. Then the conduction\nelectron Hamiltonian is\nHc=/summationdisplay\nk,σǫkσc+\nkσckσ\n+/summationdisplay\nk,σ,k′,σ′/summationdisplay\nihi·c+\nkσσσσ′ck′σ′ei(k−k′)·Ri(3)\nThe random field components are assumed to be Gaus-\nsian correlated, with ∝an}bracketle{thi∝an}bracketri}ht= 0 and ∝an}bracketle{thm\nihn\nj∝an}bracketri}ht=δijδmnh2.\nIn Born approximation the average conduction electron\nGreen’s function is then given by\nG0\nc↑(k,iωn) = [iωn−εk+ωc/2+iγsign(ωn)]−1,(4)\nwhereγ=πN0h2.\nThe impurity induced component of the dynamical\ntransverse susceptibility χ+−\nimp(Ω), where Ω is the fre-\nquency of an a.c. electromagnetic field polarized trans-\nverse to the static magnetic field, is a sum of three con-\ntributions, from the conduction electrons (cc), the local3\nelectrons (ff) and the mixed response of conduction elec-\ntrons to a spin polarization of the local electrons or vice\nversa (cf):\nχ+−\nimp(Ω) =µ2\nB{g2\nfχ+−\nff(Ω)+2gcgfχ+−\ncf(Ω)\n+g2\nc[χ+−\ncc(Ω)−χbulk\ncc(Ω)]} (5)\nThe partial susceptibilities may be calculated using stan-\ndardmanybodytechniques, seetheAppendix. Onefinds\nresonances at the two frequencies ωfandωc. They have\nvery different widths: the local electron spin resonance is\nbroadened by /tildewideΓ, whereas the bulk and the impurity in-\nduced conduction electron resonances are broadened by\n4γ. The results are given in the Appendix, Eqs. (A1-A3).\nAssuming that /tildewideΓ>>4γ, it makes sense to consider the\nbehavior at higher frequencies Ω ≫(ωf,c,γ) (regime I)\nand low frequencies (regime II) separately. In regime I\none finds:\nχ+−\nimp(Ω) =2µ2\nB(gf−gc)2\nπ/tildewideΓi/tildewideΓ\nΩ−ωf+i/tildewideΓ.(6)\nNeutron scattering data on diluted magnetic alloys show\na broad resonance in χ+−\nimp(Ω) of width TK5, in accor-\ndance with the above result. Note that this broad peak\nvanishes in the case of equal g-factors, as a consequence\nof the conservationof magnetization in that case (leaving\naside spin lattice relaxation for the moment). The result\nin regime II is:\nχ+−\nimp(Ω) =2µ2\nB\nπ/tildewideΓ{(gf−gc)2+gc(3gf−2gc)−ωc\nΩ−ωc+4iγ\n+g2\nc−ωc(ωf−ωc)\n(Ω−ωc+4iγ)2} (7)\nThe last term carries vanishing spectral weight. The sec-\nond term on the r.h.s. represents an impurity induced\nenhancement (3 gf>2gc) or reduction of the bulk con-\nduction electron spin resonance. This contribution van-\nishes ifgf=2\n3gc. The static susceptibility takes the form\nχ+−\nimp(0) = 2µ2\nBg2\nf/(π/tildewideΓ) .\nTo summarize, the impurity induced component of the\ntotaldynamicalspinsusceptibilityofaKondoionischar-\nacterized by a broad excitation peak of width /tildewideΓ≃TKat\nΩ =ωfand a narrow peak or dip of width 4 γat Ω =ωc,\nwhereγis the conduction electron relaxation rate. The\nrelativeweightsofthetwocomponentsdependsensitively\non the ratio of g-factors. This structure is not easily de-\ntected in an ESR experiment. The narrow resonance line\nhas the same position and width as the bulk ESR line.\nIts weight per atom is, however, enhanced by the large\nfactorTF/TK, which comes from the renormalized sus-\nceptibility scale prefactor ∝1//tildewideΓ. Therefore, the ESR\nresponse of a diluted magnetic alloy with a concentra-\ntion of Kondo ions c > T K/TFwill be dominated by\nthe impurity contribution determined in the above Eqs.\n(6,7).III. ANDERSON LATTICE MODEL IN THE\nKONDO SCREENED REGIME.\nThe Hamiltonian of the simplest Anderson lattice\nmodel, assuming momentum independent hybridization\nand an isotropic conduction band with flat density of\nstates is given by\nH=Hc+/summationdisplay\ni,σǫfσf+\niσfiσ+U/summationdisplay\ninfi↑nfi↓\n+V/summationdisplay\ni,k,σ(eik·Rif+\niσckσ+h.c.) (8)\nHereHcandǫfσ,V, Uhave been introduced in Sec. II\nand the Riare lattice site vectors. The single particle\nGreen’s functions are given by Dyson’s equation\n/parenleftbigg\niωn−ǫfσ−Σfσ(iωn,k) −V\n−V iω n−ǫkσ−Σcσ(iωn,k)/parenrightbigg\nG= 1,\nwhere G=/parenleftbigg\nGff\nkσGcf\nkσ\nGfc\nkσGcc\nkσ/parenrightbigg\n. (9)\nWe assume Fermi liquid theory to hold. Then the self\nenergy Σ fσ(ω,k) has a power series expansion in ωat\nthe Fermi surface, and its imaginary part is small ∝ω2,\nand may be neglected in lowest order. One may use\nω−ǫfσ−Σfσ(ω,kF) =z−1\nσ[ω−/tildewideǫfσ], with the quasipar-\nticle weight factor zσ= [1−(∂Σfσ(ω,kF)/∂ω)0]−1and\nthe renormalized energy /tildewideǫfσ=zσ[ǫdσ+Σfσ(0,kF)]. The\nconduction electron self-energy may be approximated by\nΣcσ(ω+i0,k) =−iγ, whereγis the spin-lattice relax-\nation rate defined earlier. Then for low energies one\nhas a quasiparticle description, with Gff\nkσ(ω) =zσ/tildewideGff\nkσ,\nGcf\nkσ=√zσ/tildewideGcf\nkσand the renormalized hybridization\namplitude /tildewideV2=zσV2. The full matrix of quasiparticle\nGreen’s functions is given by\n/parenleftbigg/tildewideGff\nkσ/tildewideGcf\nkσ/tildewideGfc\nkσGcc\nkσ/parenrightbigg\n=1\ndet/parenleftbigg\nω−ǫkσ+iγ/tildewideV\n/tildewideV ω −/tildewideǫfσ/parenrightbigg\n,(10)\nwhere det = ( ω−/tildewideǫfσ)(ω−ǫkσ+iγ)−/tildewideV2= (ω−ζ+\nkσ)(ω−\nζ−\nkσ). The complex energy eigenvalues are given by\nζ±\nkσ=1\n2(/tildewideǫfσ+ǫkσ−iγ)±/radicalbigg\n1\n4(/tildewideǫfσ−ǫkσ+iγ)2+/tildewideV2\n=ǫ±\nkσ−iγ±\nkσ, (11)\nwhereǫ±\nkσ= Reζ±\nkσandγ±\nkσ=−Imζ±\nkσ. We note γ±\nkσ>\n0. There are two energy bands separatedby an (indirect)\ngap (ǫmin,max\nkσdenote the minimum or maximum of the\nconduction band):\n∆gσ=ǫmin\nkσ−ǫmax\nkσ+/radicalBig\n(/tildewideǫfσ−ǫmax\nkσ)2+4/tildewideV2\n+/radicalBig\n(/tildewideǫfσ−ǫmin\nkσ)2+4/tildewideV2≫ωf,c.(12)\nWe assume that the renormalized f-level/tildewideǫfσis inside the\nconduction band, close to the Fermi level, and consider4\nthe case of almost half-filling, i.e. n/lessorsimilar2 electrons per\nlattice site. Then only the lower band is occupied in the\nground state. We assume an isotropic band structure for\nsimplicity. Then near the Fermi level at k=kF, the\nquasiparticle energy (we drop the spin dependence) has\nthe form\nǫ−\nk−ǫ−\nkF=1\n2(k−kF)vF\n1+(/tildewideǫf−ǫkF)/radicalBig\n(/tildewideǫf−ǫkF)2+4/tildewideV2\n\n≃(k−kF)v∗\nF, (13)\nwhere the renormalizedFermi velocity is defined by v∗\nF=\nvF/tildewideV2/(/tildewideǫf−ǫkF)2=vF(m/m∗)≃vFzV2/(ǫkF)2, and we\nused the fact that ǫkF≫/tildewideǫf. Note that ǫkFis the bare\nconduction band energy at k=kF, which is far above\nthe true Fermi energy. When z≪1, the Fermi velocity\nis renormalized to very low values and one has a “heavy\nfermion liquid” (effective mass m∗≫m). The effective\nFermi temperature of the heavy quasiparticles is given\nbyT∗\nF=1\n2kFv∗\nFσ≪TF.\nTo first order in γthe level widths are given by\nγ±\nkσ=1\n2γ/bracketleftbigg\n1∓/tildewideǫfσ−ǫkσ\nǫ+\nkσ−ǫ−\nkσ/bracketrightbigg\n. (14)\nWe note that the hybridization induced width /tildewideΓ of\nthef-electron energy level in the impurity problem is\nnow absorbed in the electronic band structure: the co-\nherent superposition of contributions from all lattice\nsites to Σ fσ(ω+i0,k) removes the large constant i/tildewideΓ.\nThe remaining imaginary part of the self-energy at fi-\nnite temperatures may be approximated by a constant\nΓFL=cT2/T∗\nF. We shall comment on the effect of quasi-\nparticle scattering on the ESR-linewidth at the end. Us-\ning partial fraction decomposition, we get the retarded\nGreen function\n/tildewideGff\nkσ(ω+i0) =aff,+\nkσ\nω−ζ+\nkσ+aff,−\nkσ\nω−ζ−\nkσ(15)\nand similar expressions for /tildewideGcf\nkσandGcc\nkσ, where, with\nukσ=ζ+\nkσ−ζ−\nkσ,\naff,±\nkσ=±(ζ±\nkσ−/tildewideǫkσ)/ukσ,\nacc,±\nkσ=±(ζ±\nkσ−ǫfσ)/ukσ,\nacf,±\nkσ=±/tildewideV/ukσ.\nFor sufficiently small spin-lattice relaxation, γ≪\n(/tildewideV,/tildewideǫfσ), we may neglect the imaginary parts in the\nweight factors aff,±\nkσ,...and replace ζ±\nkσbyǫ±\nkσ.\nThe quasiparticles interact via the Fermi liquid inter-\naction. For ESR, the relevant component of the Fermi\nliquidinteractionisthe isotropicspin-antisymmetricpart\ndescribed by the Landau parameter Fa\n0. An important\ncontribution to Fa\n0comes from the renormalized value\n/tildewideUof the bare interaction U, Fa\n0=−2N0/tildewideU, similar tothe single impurity case discussed in Sec. II. For the lat-\ntice case, exact numerical results on /tildewideUare not available.\nWe note however, that the onsite repulsion Uis likely\nto be screened down to a positive value of order N−1\n0,\nwhich would lead to a ferromagnetic Landau parameter\n0> Fa\n0/greaterorsimilar−1. Additional contributions to Fa\n0may be\ngenerated by nonlocal interactions like the RKKY inter-\naction, which may be ferro- or antiferromagnetic. We\nemphasize that this Fermi liquid interaction always leads\nto a ferromagnetic contribution to the fluctuation spec-\ntrum, which may be more or less important depending\nupon the other contributions.\nFollowing the way in which the interaction was in-\ncluded in the impurity model, we may express the fully\nscreened f-electron susceptibility in terms of the un-\nscreened one\nχ+−\nff(iΩm) =χ+−\nff,H(iΩm)/[1−/tildewideUχ+−\nff,H(iΩm)],(16)\nwhere\nχ+−\nff,H(iΩm) =−T/summationdisplay\nωn/summationdisplay\nk/tildewideGff\nk↓,H(iωn+iΩm)/tildewideGff\nk↑,H(iωn)\n(17)\nTheone-to-onecorrespondenceofquasiparticlesandbare\nparticles, on which Landau’s Fermi liquid theory rests,\nallows to calculate the spin susceptibility from the quasi-\nparticle Green’s functions defined above, without taking\nthe incoherent parts into account. Here the subscript H\nindicates that the Zeeman energy ωfis replaced every-\nwhere by\n/tildewideωf=ωf[1+/tildewideUχ+−\nff(0)] =ωf[1−/tildewideUχ+−\nff,H(0)]−1(18)\nUsing the representation of /tildewideGff\nk↓,Hin terms of eigen-\nstates, the summation in Eq. (17) on ωnandkmay be\ndone. In the case that only the lower band is occupied,\nthe low frequency response is given by, see the Appendix,\nEq. (A4):\nχ+−(Ω+i0) =χ+−(0)(−ωr+iγr)\nΩ−ωr+iγr,(19)\nwhereχ+−(0) is defined in the Appendix, Eq. (A5). The\nmean field shift largely cancels out of the resonance fre-\nquency\nωr=1\n2ωf\n1−(/tildewideǫf−ǫkF)/radicalBig\n(/tildewideǫf−ǫkF)2+4/tildewideV2\n≃ωf(1−m/m∗),\n(20)\nThe linewidth, however, is reduced by the exchange in-\nteraction, provided the interaction is ferromagnetic.\nγr=γ\n1+/tildewideǫf−ǫkF/radicalBig\n(/tildewideǫf−ǫkF)2+4/tildewideV2\n[1−/tildewideUχ+−\nff,H(0)]\n≃2γm\nm∗[1−/tildewideUχ+−\nff,H(0)]≪γ. (21)5\nIt is seen that the main narrowingmechanism is provided\nby the hybridization throughthe renormalizedamplitude\n/tildewideV, which gives the small factor m/m∗. In simple terms,\nthe quasiparticles at the Fermi surface have mainly f-\ncharacter, with only a small admixture (fraction m/m∗)\nof conduction electron component. Since only the con-\nduction electronsfeel the spin lattice relaxation, the total\nspin relaxation is a fraction m/m∗of the spin lattice re-\nlaxation. Vertex correctionsto the spin-lattice relaxation\nare likely to increase γrsomewhat as they do in the im-\npurity case, Appendix Eq. (A1), where 2 γbecomes 4 γ.\nIn order to discuss the temperature and magnetic field\ndependence of the linewidth it is necessary to incorpo-\nrate quasiparticle scattering effects14and inelastic con-\ntributions to the spin-lattice relaxation. In the case that\ntheg-factors are sufficiently different, the contribution\nto the linewidth from quasiparticle scattering will vary\nwith temperature as T2/T∗\nFand with magnetic field H\nasH2/T∗\nF. In the case of equal g-factors the latter con-\ntribution will be cancelled by vertex corrections. Addi-\ntional temperature dependence may arise from coupling\nto phonons.\nIV. CONCLUSION\nThis paper is motivated by the recent observations\nof electron spin resonance at low temperature in some\nheavy-fermion compounds. We have calculated the dy-\nnamical susceptibility, which describes the resonance, at\nlow temperature in the fully screened Kondo regime for\nboth a single Kondo impurity spin as well as for the\nKondo lattice, described here by the Anderson lattice\nmodel.\nWe have not addressed the behavior of the susceptibil-\nities at temperatures in the neighborhood of the Kondo\ntemperature, where linewidths are expected to be very\nlarge due to rapid spin fluctuations in that temperature\nrange. Rather, we deal with the very low temperature\nregime, where a Kondo impurity is fully screened and\nwhere the heavy-electron Fermi liquid has formed in the\nAnderson lattice.\nFor the realistic case in which the g-factors of felec-\ntrons and conduction electrons are different, we find for\nthe single impurity that structure persists at both the f\nelectron and conduction electron resonance frequencies.\nThe impurity resonance continues to have a large width,\nof orderTK, while for the conduction electron resonance\nthere is an impurity-induced contribution that increases\nor decreases the amplitude depending on the ratio of g-\nfactors.\nThe situation is quite different for the lattice case.\nHere, the hybridization of the fand conduction elec-\ntrons and Fermi-liquid interaction lead to modifications\nof the susceptibility that can lead to substantial line nar-\nrowingand hence the possibility ofexperimentalobserva-\ntion. We find a sharp ESR line near the underlying local\nfelectron resonance. The line is substantially narrowedby a factor of the mass ratio m/m∗and by the effect of\nthe Fermi liquid interaction F0\naprovided it is negative\n(ferromagnetic).\nWenotethattheESRhasbeenonlybeenseeninheavy\nfermion compounds for which there is independent evi-\ndence for ferromagnetic fluctuations7,8. We suggest that\nour analysis accounts for this observation.\nAcknowledgments\nWe thank the Aspen Center for Physics, where this\nwork was begun and completed. We thank Q. Si for\npointing us to this problem and we acknowledge helpful\ndiscussions with K. Baberschke, P. Coleman, I. Martin,\nD. Pines. We are grateful to J. Sichelschmidt for his\ngenerous sharing of information and data. This work has\nbeensupportedinpartbytheDFG-CenterforFunctional\nNanostructurres and the DFG-Forschergruppe ”Quan-\ntum Phase Transitions” (PW).\nAPPENDIX\n1. Anderson impurity model in the Kondo screened\nregime: Green’s function approach to χ+−(Ω)\nAs derived in the main text, Eq. (2), the Green’s func-\ntions of conduction electrons and local electrons, includ-\ning the Fermi-liquid interaction, are given by\nGfσ(iωn) = [iωn+σωf+i/tildewideΓsign(ωn)]−1,\nGcσ(k,iωn) = [iωn−εk+σωc/2+iγsign(ωn)]−1.\nThe dynamical transverse susceptibility χ+−(Ω),\nwhere Ω is the frequency of an a.c. electromagnetic field\npolarized transverse to the static magnetic field, is given\nby\nχ+−(Ω) =µ2\nB[g2\ncχ+−\ncc(Ω)+g2\nfχ+−\nff(Ω)+2gcgfχ+−\ncf(Ω)].\nThe partial susceptibilities are obtained by evaluating\nFeynman bubble diagrams dressed by vertex corrections\noftheladdertypereferringtotheFermiliquidinteraction\n(local electrons) and the spin-orbit interaction (impurity\ncorrelation lines for the conduction electrons).\nThe local susceptibility in the absence of vertex cor-\nrections is obtained as\nχ+−\nff,H(iΩm) =−T/summationdisplay\nωnGf↓(iωn+iΩm)Gf↑(iωn)\n=2\nπ/tildewideΓ−ωf+i/tildewideΓ\niΩm−2ωf+2i/tildewideΓ.\nThe vertex corrections are obtained from the Bethe-\nSalpeter equation\nΛ(iΩm) = 1+/tildewideUχ+−\nff,H(iΩm)Λ(iΩm) =Ω−2ωf+2i/tildewideΓ\nΩ−ωf+i/tildewideΓ,6\nwhere we used /tildewideU=π/tildewideΓ. Then\nχ+−\nff(iΩm) =χ+−\nff,H(iΩm)Λ(iΩm) =2\nπ/tildewideΓ−ωf+i/tildewideΓ\niΩm−ωf+i/tildewideΓ.\nThe conduction electron susceptibility consists of four\ncontributions\nχ+−\ncc(iΩm) =χbulk\ncc(iΩm)+3/summationdisplay\ni=1χ(i)\ncc(iΩm)The bulk contribution has the form χbulk\ncc(iΩm) =\nNχ0\ncc(Ω +i0)Φ(iωn,iΩm), where Nis the num-\nber of atoms in the system and χ0\ncc(Ω +i0) =\n−T/summationtext\nωn,kGc↓(k,iωn+iΩm)Gc↑(k,iωn) =N0(−ωc+\n2iγ)/(Ω−ωc+2iγ), where N0is the conduction electron\ndensity of states at the Fermi level. The vertex function\nΦ(iωn,iΩm) is found as solution to the equation:\nΦ(iωn,iΩm) = 1−h2/summationdisplay\nkG0\nc↓(k,iωn+iΩm)G0\nc↑(k,iωn)Φ(iωn,iΩm)\nas\nΦ(iωn,iΩm) =θ(−ωn)θ(ωn+Ωm)iΩm−ωc+2iγ\niΩm−ωc+4iγ+[1−θ(−ωn)θ(ωn+Ωm)]\nNote that the minus sign in front of h2is generated by\nthe Pauli matrices that appear in Hc, Eq. (3) of the main\ntext (in the case ofpotential scatteringthere would be no\nsign change):/summationtext\ni,α,βσi\nα↓σi\n↑βG0\ncαG0\ncβ=−G0\nc↓G0\nc↑. As a\nconsequence, the vertexcorrectionsdouble the linewidth:\n2γ→4γ. In the case of potential scattering the vertex\ncorrections cancel the self-energy induced linewidth, so\nthat potential scattering does not contribute to the spinrelaxation, as expected. Combining the above results we\nfind\nχbulk\ncc(Ω+i0) =NN0−ωc+4iγ\nΩ−ωc+4iγ.(A.1)\nThe remaining contributions are obtained from\nχ(1)\ncc(iΩm) =−V2T/summationdisplay\nωn/summationdisplay\nk/braceleftBig\n[G0\nc↓(k,iωn+iΩm)]2G0\nc↑(k,iωn)Gf↓(iωn+iΩm)\n+G0\nc↓(k,iωn+iΩm)[G0\nc↑(k,iωn)]2Gf↑(iωn)/bracerightBig\n,\nχ(2)\ncc(iΩm) =−T/summationdisplay\nωn/bracketleftBigg\nV2/summationdisplay\nkG0\nc↓(k,iωn+iΩm)G0\nc↑(k,iωn)Φ(iωn,iΩm)/bracketrightBigg2\nGf↓(iωn+iΩm)Gf↑(iωn),\nχ(3)\ncc(iΩm) =−/bracketleftBigg\nV2T/summationdisplay\nωn/summationdisplay\nkG0\nc↓(k,iωn+iΩm)G0\nc↑(k,iωn)Φ(iωn,iΩm)Gf↓(iωn+iΩm)Gf↑(iωn)/bracketrightBigg2\n×[−/tildewideUΛ(iΩm)].\nUsing\n/summationdisplay\nk[G0\nc↓(k,iωn+iΩm)]2G0\nc↑(k,iωn) =N02πi\n(iΩm−ωc+2iγ)2=−/summationdisplay\nkG0\nc↓(k,iωn+iΩm)[G0\nc↑(k,iωn)]2\nand the identity\nGf↓(iωn+iΩm)−Gf↑(iωn) =−(iΩm−2ωf+2i/tildewideΓ)Gf↓(iωn+iΩm)Gf↑(iωn)\nas well as\nΠ(iΩm) =T/summationdisplay\n−Ωm<ωn<0Gf↓(iωn+iΩm)Gf↑(iωn) =1\nπ/tildewideΓiΩm\niΩm−2ωf+2i/tildewideΓ7\nwe get\nχ(1)\ncc(iΩm) =2\nπ/tildewideΓΩ\nΩ−2ωf+2i/tildewideΓ\nχ(2)\ncc(iΩm) =2\nπ/tildewideΓ2/tildewideΓ2\n(iΩm−ωc+4iγ)2Ω\nΩ−2ωf+2i/tildewideΓ\nχ(3)\ncc(iΩm) =−2\nπ/tildewideΓΩ2\n(iΩm−ωc+4iγ)2/tildewideΓ2\n(Ω−2ωf+2i/tildewideΓ)(Ω−2ωf+2i/tildewideΓ)\nAdding the three contributions we find\n3/summationdisplay\ni=1χ(i)\ncc(iΩm) =2\nπ/tildewideΓΩ(Ω−ωf)\n(iΩm−ωc+4iγ)2i/tildewideΓ\nΩ−ωf+i/tildewideΓ(A.2)\nThe mixed susceptibility may be calculated from the bubble diagram beg inning with a conduction electron particle-\nhole line and ending with a local electron p-h line, dressed by vertex co rrections at both ends:\nχ+−\ncf(Ωm) =−T/summationdisplay\nωn,kG0\nc↓(k,iωn+iΩm)G0\nc↑(k,iωn)Φ(iωn,iΩm)V2Gf↓(iωn+iΩm)Gf↑(iωn)Λ(iΩm)\n=2\nπ/tildewideΓ−i/tildewideΓ\niΩm−ωc+4iγiΩm\niΩm−2ωf+2i/tildewideΓ(A.3)\nAfter analytical continuation to the real frequency axis\nand combining the contributions the total impurity sus-\nceptibility is obtained as given in the main text, Eqs.\n(6,7).\n2. Anderson lattice model in the Kondo screened\nregime: Green’s function approach to χ+−(Ω).\nAs derived in the main text Eq. (10), the matrix of\nquasiparticle Green’s functions is given by\n/parenleftbigg/tildewideGff\nkσ/tildewideGcf\nkσ/tildewideGfc\nkσGcc\nkσ/parenrightbigg\n=1\ndet/parenleftbigg\nω−ǫkσ+iγ/tildewideV\n/tildewideV ω −/tildewideǫfσ/parenrightbigg\n,\nwhere det = ( ω−/tildewideǫfσ)(ω−ǫkσ+iγ)−/tildewideV2= (ω−ζ+\nkσ)(ω−\nζ−\nkσ). The complex energy eigenvalues are given by\nζ±\nkσ=1\n2(/tildewideǫfσ+ǫkσ−iγ)±/radicalbigg\n1\n4(/tildewideǫfσ−ǫkσ+iγ)2+/tildewideV2\n=ǫ±\nkσ−iγ±\nkσ,\nwhere, expanding to leading order in γ, as well as in\nωf,ωc,\nǫ±\nkσ= Reζ±\nkσ≃1\n2(/tildewideǫfσ+ǫkσ)±/radicalbigg\n1\n4(/tildewideǫfσ−ǫkσ)2+/tildewideV2\n≃ǫ±\nk−1\n2ω±\nkσ\nand\nγ±\nkσ=−Imζ±\nkσ≃1\n2γ[1±/tildewideǫfσ−ǫkσ\nǫ+\nkσ−ǫ−\nkσ]≃γ±\nk−1\n2η±\nkσ,with\nǫ±\nk=1\n2(/tildewideǫf+ǫk)±1\n2/radicalBig\n(/tildewideǫf−ǫk)2+4/tildewideV2\nω±\nk=1\n2(/tildewideωf+ωc)±1\n2/tildewideǫf−ǫk/radicalBig\n(/tildewideǫf−ǫk)2+4/tildewideV2(/tildewideωf−ωc)\nγ±\nk=1\n2γ[1∓/tildewideǫf−ǫk/radicalBig\n(/tildewideǫf−ǫk)2+4/tildewideV2]\nη±\nk=∓1\n2γ/tildewideωf−ωc/radicalBig\n(/tildewideǫf−ǫk)2+4/tildewideV24/tildewideV2\n(/tildewideǫf−ǫk)2+4/tildewideV2.\nThe susceptibility is, as in the case of the impurity,\ngiven by the sum of three contributions: ff,ccand\n(cf,fc):\nχ+−(Ω) =µ2\nB[g2\ncχ+−\ncc(Ω)+g2\nfχ+−\nff(Ω)+2gcgfχ+−\ncf(Ω)].\nHere the ff-susceptibility is screened by the Fermi liquid\ninteraction\nχ+−\nff(iΩm) =χ+−\nff,H(iΩm)Λ(iΩm),\nwith\nΛ(iΩm) = 1/[1−/tildewideUχ+−\nff,H(iΩm)]\nwhere\nχ+−\nff,H(iΩm) =−T/summationtext\nωn,k/tildewideGff\nk↓,H(iωn+iΩm)/tildewideGff\nk↑,H(iωn)8\nSimilarly,\nχ+−\ncf(iΩm) =χ+−\ncf,H(iΩm)Λ(iΩm),\nwhere\nchi+−\ncf,H(iΩm) =−T/summationtext\nωn,k/tildewideGcf\nk↓(iωn+iΩm)/tildewideGcf\nk↑(iωn)\nand\nχ+−\ncc(iΩm) =χ+−\ncc,H(iΩm)+/tildewideU[χ+−\ncd,H(iΩm)]2Λ(iΩm),\nwhere\nχ+−\ncc,H(iΩm) =−T/summationtext\nωn,kGcc\nk↓(iωn+iΩm)Gcc\nk↑(iωn)\nUsing the representation of the Green’s functions in\nterms of the eigenstates ν=±, and the fact that low\nenergyexcitationsareonlypossibleclosetothe Fermi en-\nergy, which we assume to lie in the lower band ( ν=−),\nonly the ( −)-components contribute to χ+−\nij,H(iΩm) :\nχ+−\nff,H(Ω+i0) =−/summationtext\nkaff,−\nk↓aff,−\nk↑f(ζ−\nk↑)−f(ζ−\nk↓)\nΩ−ζ−\nk↓+ζ−\nk↑+i0,\nwhere in the arguments of the Fermi function f(ǫ),\nthe complex-valued energy ζ−\nk↓appears. Employingζ−\nk↓−ζ−\nk↑≃ω−\nk−2iγ−\nk, and/summationtext\nk[f(ζ−\nk↑)−f(ζ−\nk↓)]≃\n/summationtext\nk(∂f/∂ǫ−\nk)(−ω−\nkF+2iγ−\nkF) =N0(ω−\nkF−2iγ−\nkF), we get\nχ+−\nff,H(Ω+i0) =N0aff,−\nkF↓aff,−\nkF↑−ω−\nkF+2iγ−\nkF\nΩ−ω−\nkF+2iγ−\nkF\nand hence\nχ+−\nff,H(0) =N0aff,−\nkF↓aff,��\nkF↑.\nEquivalent expressions hold for the ffandcfcompo-\nnents. The vertex function follows as\nΛ(Ω+i0) =Ω−ω−\nkF+2iγ−\nkF\nΩ−(ω−\nkF−2iγ−\nkF)(1−/tildewideUχ+−\nff,H(0))\nand the renormalized ff-susceptibility takes the form\nχ+−\nff(Ω+i0) =χ+−\nff(0)−ωr+iγr\nΩ−ωr+iγr,where\nωr−iγr= (ω−\nkF−2iγ−\nkF)(1−/tildewideUχ+−\nff,H(0)) and\nχ+−\nff(0) =χ+−\nff,H(0)/[1−/tildewideUχ+−\nff,H(0)]\nas discussed in the main text, Eqs. (20,21). The total\nsusceptibility consists of two resonance terms:\nχ+−(Ω+i0) =χ+−\nr(0)−ωr+iγr\nΩ−ωr+iγr+g2\nc−ω−\nkF+2iγ−\nkF\nΩ−ω−\nkF+2iγ−\nkF{χ+−\ncc,H(0)+/tildewideU[χ+−\ncf,H(0)]2−ωr+iγr\nΩ−ωr+iγr},(A.4)\nwhereχ+−\nr(0) =g2\nfχ+−\nff(0)+2gcgfχ+−\ncf(0). In the case that γ−\nkF≫γr, the resonance part simplifies to\nχ+−(Ω+i0) =χ+−(0)−ωr+iγr\nΩ−ωr+iγr,whereχ+−(0) =χ+−\nr(0)+g2\nc/tildewideU[χ+−\ncf,H(0)]2(A.5)\n1A.C. Hewson, The Kondo Problem to Heavy Fermions .\n(Cambridge University Press, 1993)\n2J. Kondo, Prog.Theor. Phys. 32, 37 (1964).\n3K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n4For references to the early experimental and theoretical\npapers, see W. Brenig, W. G¨ otze and P. W¨ olfle, Phys.\nRev. B2, 4533 (1970); J. Sweer, D.C. Langreth and J.W.\nWilkins, Phys. Rev B 13, 192 (1976).\n5U. Walter, E. Holland-Moritz and Z. Fisk, Phys Rev B 43,\n320 (1991)\n6J. Sichelschmidt, V.A. Ivanshin, J. Ferstl, C. Geibel and\nF. Steglich, Phys. Rev. Lett. 91, 156401-1 (2003).\n7C. Krellner, T. Forster, H. Jeevan, C. Geibel and J.\nSichelschmidt, Phys. Rev. Lett. 100, 066401 (2008).8Ref. 6 and U. Schaufuß, V. Kataev, A.A. Zvyagin, B.\nB¨ uchner, J. Sichelschmidt, J. Wykhoff, C. Krellner, C.\nGeibel, and F. Steglich, arXiv0804.4105v1. Also, J.G.S.\nDuque, E.M. Bittar, C. Adriano, C. Giles, L.M. Holanda,\nR. Lora-Serrano, P.G. Pagliuso, C. Rettori, C.A. P´ erez,\nR.W. Hu, C. Petrovic, S. Maquilon, Z. Fisk, D.L. Huber,\nand S.B. Oseroff, preprint.\n9George Feher and A.F. Kip, Phys. Rev. 98, 337 (1955).\n10E. Abrahams, unpublished.\n11P. Nozi` eres, J. Low Temp. Phys. 17, 31 (1974).\n12A.C. Hewson, J. Bauer and W. Koller, Phys. Rev. B 73,\n045117 (2006).\n13S.E. Barnes and J. Zitkova-Wilcox, Phys. Rev. B 7, 2163\n(1973).9\n14A.W. Overhauser, Phys. Rev. 89, 689 (1953)."    },    {        "title": "1702.05389v2.Charge_Transport_and_Entropy_Production_Rate_in_Magnetically_Active_Molecular_Dimer.pdf",        "content": "Charge Transport and Entropy Production Rate in Magnetically Active Molecular Dimer\nJ. D. Vasquez Jaramillo and J. Fransson\u0003\nDepartment of Physics and Astronomy, Box 516, 75120, Uppsala University, Uppsala, Sweden\n(Dated: October 23, 2018)\nWe consider charge and thermal transport properties of magnetically active paramagnetic molecular dimer.\nGeneric properties for both transport quantities are reduced currents in the ferro- and anti-ferromagnetic regimes\ncompared to the paramagnetic and e \u000ecient current blockade in the anti-ferromagnetic regime. In contrast, while\nthe charge current is about an order of magnitude larger in the ferromagnetic regime, compared to the anti-\nferromagnetic, the thermal current is e \u000eciently blockaded there as well. This disparate behavior of the thermal\ncurrent is attributed to current resonances in the ferromagnetic regime which counteract the thermal flow. The\ntemperature di \u000berence strongly reduces the exchange interaction and tends to destroy the magnetic control of the\ntransport properties. The weakened exchange interaction opens up a possibility to tune the system into thermal\nrectification, for both the charge and thermal currents.\nI. INTRODUCTION\nThermal transport properties in molecular junctions can be\nof electronic origin or mediated through lattice vibrations [1].\nThere has been an increasing interest in the study of ther-\nmal properties in molecular junctions [2–5], stimulated by\nexperimental observations [2]. Several realizations of tun-\nneling junctions comprising noble metal electrodes and poly-\nmers that absorb, emit, and transmit thermal have been re-\nported [4, 6]. The interest is, moreover, driven from the\nperspective of information science and technology with re-\nspect to entropy production rate [3, 7] and the meaning of the\nthermodynamics in low dimensional systems [3, 8, 9]. This\nwas motivated by the discovery of conducting polymers and\nsolitonic electronic transport mechanisms discovered by Shi-\nrakawa, see [10] and references therein. By this discovery\npolyacetylene became the test bench, bridging the gap be-\ntween organic and inorganic chemistry regarding electronic\ntransport [11–13]. Since then, charge transport has been ex-\ntensively studied theoretically and experimentally in molec-\nular junctions [15–25]. From the same standpoint, thermal\ntransport studies were conceived and came to be conclusive\nin the upcoming years [2]. Subsequently, several theoretical\nstudies demonstrated the possibility to conduct both thermal\nand charge through tunneling junctions [26–34], both in pres-\nence and absence of lattice vibrations, although there is no\ngeneric framework that successfully is capable of describing\nthe thermodynamic properties of nano junctions [8, 14, 35].\nTheoretical predictions suggest all electrical control for\nboth reading and writing spin states in molecular dimers\n\u0003Electronic address: Jonas.Fransson@physics.uu.se[51, 52]. This prediction is based on the (electronically me-\ndiated) indirect exchange interaction between the localized\nspins which controls the charge transport properties [51, 52].\nSimilar e \u000bects were reported in Ref. [6], where the spin\nground state of a single metal complex is electrically con-\ntrolled, imposing transition between high ( S=5=2) and low\n(S=1=2) spin configurations in a three terminal device.\nHere we build on the previous predictions made in Ref.\n[51, 53] for dimers of magnetic molecules in which the ef-\nfective spin-spin interactions are mediated by the properties\nof the delocalized electrons and extend to thermally induced\nmagnetic and transport properties. In particular we study\nthermal transport and its response to changes of the mag-\nnetic configurations. Our set-up pertains to, for instance,\nM-phthalocyanine (MPc), M-porphyrins, where M denotes a\ntransition metal atom [61–64], e.g., Cr, Mn, Fe, Co, Ni, Cu,\nand also to bis(phthalocyaninato)R (TPc 2) [65, 66], where R\ndenotes a rare earth element, e.g., Tb. Such molecules can be\ninvestigated in, for example, mechanically controlled break-\njunctions [66, 67], in carbon nanotube assemblies [65] and\nscanning tunneling microscope [62, 63].\nWe consider thermal and charge transport as the result of\nthe electrothermal control of the junction. Accordingly, we in-\nvestigate the charge and thermal conductance with respect to\nthe bias voltage and thermal gradient across the junction. With\nthis background we, furthermore, consider a non-equilibrium\nanalogue to the Seebeck coe \u000ecient defined as the ratio be-\ntween the di \u000berential conductances with respect to the ther-\nmal gradient and bias voltage, introduced in [68]. By the\nsame token, we consider the ratio of the energy di \u000berentials\nwith respect to the thermal gradient and the bias voltage. Our\npredictions and results are based on non-equilibrium Green\nfunctions defined on the Keldysh contour.\nThroughout this study we consider spin degenerate con-arXiv:1702.05389v2  [cond-mat.mes-hall]  23 Jan 20182\nditions in that we assume non-magnetic metals in the leads\nas well as the absence of externally applied magnetic fields.\nThe advantage with this set-up, compared to designs based\non ferromagnetic leads is that the dipolar and quadrupolar\nfields considered in Ref. [69] here becomes vanishingly\nsmall. Therefore the e \u000bective isotropic electron mediated\nspin-spin interactions dominates the properties and control of\nthe molecular dimer. In this sense, our system would serve as\na representation of paramagnetic spintronics, or paratronics .\nII. METHODS\nA. Magnetic molecular dimer in a junction\nThe specific set-up we address in this article comprises a\ndimer of paramagnetic molecules which are embedded in se-\nries in the junction between normal metallic leads, see Figure\n1 (a). Each paramagnetic molecule comprises a localized spin\nmoment which is embedded in a ligand structure, where the\nsp-orbitals define the spin-degenerate HOMO or LUMO or-\nbitals. We assume that the d-, or, f-orbitals that constitute the\nmolecular spins, hybridize only weakly with the sp-orbitals,\nallowing to consider the spin moment in the localized moment\npicture. As such, the localized moment interacts with the de-\nlocalized electrons only via local exchange. We also neglect\nspin-orbit coupling in the molecular orbitals as well as consid-\nering them in single electron form, which is typically justified\nfor the sp-electrons. The molecular orbitals couple via tun-\nneling both to one another and to the adjacent lead.\nWe model this set-up using the Hamiltonian\nH=HM+Hint+HL+HR+HT: (1)\nHere, the molecular HOMO or LUMO levels are defined by\nHM=X\n\u001b0BBBBBB@X\nm=L;R\"mdy\nm\u001bdm\u001b+Tc(dy\nL\u001bdR\u001b+H:c:)1CCCCCCA;(2)\nwhere dy\nm\u001b(dm\u001b) creates (annihilates) an electron in the\nleft ( L) or right ( R) molecule at the energy \"m=\"0and\nspin\u001b=\";#, whereasTcdefines the tunneling rate be-\ntween the molecules. Internally in molecule m, the local-\nized spin moment Sminteracts with the electron spin sm=P\n\u001b\u001b0dy\nm\u001b\u001b\u001b\u001b0dm\u001b0=2, where \u001bis the vector of Pauli matri-\nces, via exchange\nHint=X\nm=L;Rvmsm\u0001Sm; (3)\nwhere vmis the exchange integral, and we assume that vm=v.\nWe focus on the case with non-magnetic leads,\nHL=R=X\nk\u001b2L=R\"kcy\nk\u001bck\u001b; (4)\nwhere cy\nk\u001bcreates an electron in the left ( L;k=p) or right ( R;\nk=q) lead at the energy \"kand spin\u001b. Tunneling between\nFIG. 1: (Color online) (a) Molecular dimer of paramagnetic\nmolecules. An electron (at energy \"0) in each molecule interacts\nwith the localized spin moment ( Sm,m=L;R) via exchange ( vm)\nwith the electron in the adjacent molecule (tunneling rate Tc) and\nwith electron sin the left /right electrode (coupling \u0000). The left /right\nnonmagnetic electrode is characterized by its electrochemical poten-\ntial (\u0016L=R). E\u000bective molecular orbutals ( \"0\u0006Tc) emerge from in-\ntermolecular tunneling. (b) E \u000bective exchange interaction between\nthe localized spin moments as function of the voltage bias V. (c)\nOccupation of the states in the spin dimer. The green curve repre-\nsents the occupation of the lowest energy eigenstate of the spin dimer\nwhich changes character between spin singlet and spin triplet states\nas function of the voltage bias. Other colors analogously represent\nthe occupation of the consecutively higher energy eigenstates. In the\nregion indicated by the red arrow three states are degenerate and form\na spin triplet. Calculations are made at TL=4 K and TR=9 K for\n\"0\u0000\u0016=1,Tc=1 meV , vm=5 meV , and \u0000 =1, meV . In panels (b)\nand (c), the ferromagnetic and antiferromagnetic regimes of the spin\ndimer are indicated with red and black arrows, respectively.\nthe leads and molecules is described by\nHT=X\np\u001bTLcy\np\u001bdL\u001b+X\nq\u001bTRcy\nq\u001bdR\u001b+H:c: (5)\nand we define the voltage bias Vacross the junction by eV=\n\u0016L\u0000\u0016R, where\u0016\u001f,\u001f=L;R, denotes the electrochemical po-\ntential of the lead \u001f.\nIn this wayH0=HL+HR+HT+HMprovides a spin-\ndegenerate background electronic structure which mediates\nthe exchange interactions between the localized spin moments\ninHint. The spin-degeneracy implies that these interactions\nare purely isotropic [51, 53], such that we retain the Heisen-\nberg model only for the spins.\n1. Exchange interactions\nIt has been shown that the Hamiltonian in Eq. (1) gives rise\nto the e \u000bective spin model [53]\nHS=X\nmn\u0010\nJmnSm\u0001Sn+Dmn\u0001[Sm\u0002Sn]+Sm\u0001Imn\u0001Sn\u0011\n:(6)\nHere, the parameter Jmndenotes the isotropic Heisenberg\ninteraction, whereas DmnandImnrespectively denotes the\nDzyaloshinskii-Moriya and Ising interactions which both in-\ntroduce anisotropic interactions into the system. In the present3\nFIG. 2: (Color online) The exchange interaction J(units: meV) be-\ntween the molecular spins as function of voltage bias Vand temper-\nature di \u000berence \u0001T=TR\u0000TLfor di \u000berent gating conditions, such\nthat\"0\u0000\u0016=0;1;2;3 meV . Other parameters are as in Figure 1.\nsystem, where the there is no spin-dependence imposed, either\nby external or internal forces, the anisotropic interactions van-\nish,Dmn=0,Imn=0, and we retain the isotropic Heisenberg\ninteraction only.\nIn this paper, we treat the spin dimer as a closed system\nin the sense that we require a conserved number of particles\nfor which the occupations of the states are given by the Gibbs\ndistribution. This is a valid approximation for localized spins\nwhere the hybridization with the surrounding itinerant elec-\ntrons is small such that nature of the localized electrons can be\ndescribed in terms of a Kondo-like model rather than a fluc-\ntuating spin model in the sense of the Hubbard or Anderson\nmodels.\nThe spin-spin interactions are, nevertheless, influenced by\nthe tunneling current that flows through the molecular com-\nplex. In such set-ups, the e \u000bective magnetic interaction pa-\nrameter Jbetween the two spins can be calculated using the\nexpression, see Refs. [51, 53],\nJ=\u0000T2\nc\n8\u0019v2X\n\u001fZ\n\u0000\u001ff\u001f(!)(!\u0000\"0)(!\u0000\"0)2\u0000T2\nc\u0000(\u0000=8)2\nj(!\u0000\"0+i\u0000=8)2\u0000T2cj4d!;\n(7)\nwhere \u0000 =P\n\u001f=L;R\u0000\u001f, with \u0000\u001f=2\u0019P\nk\u001b2\u001fT2\n\u001f\u000e(!\u0000\"k), is the\ncoupling to the leads, and f\u001f(!) is the Fermi function at the\nchemical potential \u0016\u001f. Here, we have assumed that the molec-\nular level\"m=\"0,m=L;R, and that the local exchange cou-\npling vm=v, which is reasonable for equivalent molecules.\nWe remark here that for a more general molecular assembly,\nthat is, non-equivalent molecules and asymmetric couplings\nto the leads, the above expression for the exchange interac-\ntion should be replaced by the general formulas given in Ref.\n[53]. While this article is focused on dimers with equivalent\nmolecules coupled symmetrically to the leads, we briefly dis-\ncuss deviations from this case below.\nNotice that while the exchange interaction is defined\nin terms of the two-electron propagator ( \u0000i)hTs(t)s(t0)i\n[53], we employ the de-coupling approximation\n(\u0000i)sp\u001bG12(t;t0)\u001bG21(t0;t). Here, Gmn(t;t0) is a single-\nelectron Green function for the molecular orbitals projectedonto the sites mandn, whereas sp defines the trace over spin\n1=2 space. This approximation is justified when neglect-\ning local correlations; here the couplings to the localized\nspins. An obvious improvement would be to include these\ncorrelations and solve the two-electron propagator in a\nself-consistent scheme along with the Dyson equation for\nthe single electron Green functions. However, since such\nan approach unavoidably also would include self-consistent\ncalculations of the exchange and the spin configurations, and\nas we are mainly interested in qualitative e \u000bects, this is far\nbeyond the scope of the present article.\n2. Non-equilibrium variations\nThe salient features of the voltage bias dependence of Jand\nthe corresponding occupation of the spin states in the dimer\nare plotted in Figure 1 (b), (c) for the case with \"0\u0000\u0016=1 and\na temperature di \u000berence of 5 K (for other parameters, see the\nfigure caption). For the details about the interdependence be-\ntween the electronic structure in the molecular orbitals and the\nspin-spin exchange we refer to Refs. [51, 53]. The tempera-\ntures of the leads are included in the respective Fermi function\non which the exchange interaction within the spin dimer de-\npends, see Eq. (7). In addition to the dependencies on the\ntemperature of the leads, the exchange also depends on the\nenergy of the molecular levels. These dependencies are il-\nlustrated in Figure 2, which shows the exchange interaction\nenergy as function of the voltage bias and temperature di \u000ber-\nence for di \u000berent energies \"0\u0000\u0016. There is a clear distinction\nbetween the situation in panel (a) compared to panels (b) –\n(d), which originates in the level position relative to \u0016.\nFirst, whenever the localized level \"0\u0000\u0016=0, the molecu-\nlar bonding (at the energy: \"0\u0000T c) and anti-bonding (at the\nenergy:\"0+Tc) orbitals are centered symmetrically around\n\u0016which leads to that the influences from the left and right\nleads is symmetric with respect to voltage bias and indepen-\ndent of whether the source or drain electrode is warmer than\nthe other. The situation is schematically depicted in Figure 3\n(a), (b). The first thing to notice is a weakening of the ex-\nchange interaction caused by the increased thermal spread of\nthe electrons in the hot reservoir, compared to the cold reser-\nvoir. In equilibrium, then, in comparison to the cold reservoir\nthe hot reservoir contains a larger number of electrons above\nand below both molecular orbitals which contribute to an anti-\nferromagnetic interaction and simultaneously a smaller num-\nber of electrons that contribute to the ferromagnetic exchange.\nThis results in a weaker ferromagnetic interaction, which is\nverified in the simulations, see Figure 2 (a). The analo-\ngous argument holds for finite biases in the anti-ferromagnetic\nregime, however, utilized in the opposite way. Due to the\nthermal distribution of the electrons, there are electrons in\nthe hot reservoir contributing to both the ferromagnetic and\nanti-ferromagnetic exchange which results in an e \u000bectively\nweaker anti-ferromagnetic exchange, see Figure 2 (a). More-\nover, since the thermal distribution of the electrons is symmet-\nric around the pertinent chemical potential, the e \u000bect is equal\nregardless of the polarity of the voltage bias across the junc-4\ntion, which basically indicates voltage bias symmetric trans-\nport properties of the junction under these conditions.\nNext, whenever the molecule is gated such that the local-\nized level\"0is o\u000b-set from the chemical potential \u0016, a fi-\nnite temperature di \u000berence do generate an asymmetry with\nrespect to the voltage bias, see Figure 2 (b) – (d). One can\nconvince oneself that the exchange is symmetric with respect\nto the voltage bias at vanishing temperature di \u000berence, how-\never, in general the asymmetry is conspicuous. Under the\ngiven conditions, \"0\u0000\u0016>0 and variable TRwith TL=4 K\nfixed, the ferromagnetic regime at positive voltages remains\nnearly una \u000bected by changes in the temperature di \u000berence.\nThis can be schematically understood from the sketch in Fig-\nure 3 (c), showing \u0016Llying between the molecular orbitals.\nThen, the narrow thermal spread of the electrons in the left\nlead maintains a strong contribution to the ferromagnetic ex-\nchange while the contribution from the right lead is negligible\nmore or less independently of its temperature. By the same\ntoken one can understand that the ferromagnetic regime for\nnegative voltages is extremely sensitive to the temperature of\nthe right lead, see Figure 3 (d). Under these conditions, the\nexchange interaction is dominated by the contribution from\nthe right lead while the contribution from the left lead is neg-\nligible. Hence, at elevated temperatures there are competing\ncontributions to the exchange of both ferromagnetic and anti-\nferromagnetic nature which leads to a very weak ferromag-\nnetic exchange that weakens with increasing temperature dif-\nference.\n3. Deviations from perfectly equivalent and symmetric dimers\nIn realistic situations one can expect that the local parame-\nters vary from molecule to molecule, although they are meant\nto be considered as equivalent. Here, we briefly discuss some\nimplications on the indirect exchange interaction under di \u000ber-\nent local exchange integrals vm, finite level o \u000b-set\u0001 =\"L\u0000\"R,\nbetween the molecular levels \"m, and asymmetric couplings to\nthe leads.\nFor instance, for non-equivalent molecules such that \"1,\n\"2, the exchange interaction Jtends to become strongly asym-\nmetric with respect to the voltage bias, see for instance Ref.\n[51]. It may even lead to situations where Jis ferromag-\nnetic (negative) for one polarity of the voltage and anti-\nferromagnetic (positive) for the opposite. In those situations,\none would expect the resulting transport properties to be sig-\nnificantly di \u000berent for the two voltage polarities, something\nthat was also verified by a strong rectification property [51].\nExcept for the strong dependence of the exchange inter-\naction on the level o \u000b-set between the two molecules, the\ndimer structure is remarkably robust against small variations\nand asymmetries in the couplings \u0000\u001fto the leads as well as to\ninequalities in the local Kondo exchange couplings vm. Dif-\nferences of up to 20 % between the couplings of the two\nmolecules do not leads to any essential variations in the ex-\nchange interaction and are, therefore, not expected to be detri-\nmental to the transport properties either.\neV\n/uni03BCR/uni03BCLeV/uni03BCR\n/uni03BCL(a) (b) V>0 V<0 TL\nTLTRTR\neV\n/uni03BCR/uni03BCLeV/uni03BCR\n/uni03BCL(c) (d) V>0 V<0 TL\nTLTRTRFIG. 3: (Color online) The combined role of the voltage bias and\ntemperature di \u000berence on the tunneling conditions for (a), (c) pos-\nitive and (b), (d) negative bias, and (a), (b) symmetric and (c), (d)\nasymmetric molecular orbitals around \u0016(dashed). The left (right)\nlead is defined at the temperature TL(TR) and chemical potential \u0016L\n(\u0016R); in the figure TL<TRand\u0016L\u0000\u0016R=eV. A low (high) tem-\nperature sustains a sharp (fuzzy) boundary between the occupied and\nunoccupied electron states in the metal.\n4. Spin expectation values\nFor later purpose we introduce the notation hSz\n\u001fifor the\nexpectation values which are projections onto the molecule\n\u001fof the total spin expectation value hSiof the dimer. The\nexpectation value hSiis calculated with respect to the spin\nHamiltonianHS=JSL\u0001SR, with Jobtained from Eq. (7). As\nwe are considering the individual moments to remain in their\nground states under all conditions, the restricted Hilbert space\ncorresponding to this set-up is four dimensional labeled by\nthe states, say,fj\"\"i;j\"#i;j#\"i;j##ig , where the first (second)\nentry corresponds to the left (right) spin. Then, the projected\nexpectation value is calculated by the expression hSz\n\u001fi\u0011\nh\u001b\u001b0jexpf\u0000\favE\u001b\u001b0gSz\n\u001fj\u001b\u001b0i=h\u001b\u001b0jexpf\u0000\favE\u001b\u001b0gj\u001b\u001b0i,\nwhere the operator Sz\n\u001fis expressed in the basis of the dimer.\nHere, also 1 =\fav=kB(TL+TR)=2, represents the average\ntemperature of the two leads. Although this appears to be\na severe simplification, it turns out that the spin occupation\nnumbers vary slowly as function of the temperature in the\ncurrent set-up, which justifies this simplistic treatment.\nWe remark here that any temperature mediated by the tun-\nneling electrons vanish under the present condition in the em-\nployed approximation. To clarify this point, we notice that the\ne\u000bective spin model introduced in Eq. (6) in principle also5\ncontains the contribution\nH\u001c=\u0000g\u0016BB\u001c\u0001X\nmSm; (8)\nwhere the magnetic field B\u001cis proportional to the current\nthrough the junction, see, e.g., Refs. [53–55, 74–76]. This\nfield provides a torque on the loclized spin, however, only\nwhen the spin-degeneracy is broken in the current. In other\nwords, this current induced magnetic field vanishes whenever\nthe tunneling current is spin-degenerate. Hence, since our\nset-up is spin-degenerate by construction, the current induced\nmagnetic field does not generate any renormalization of the\nlocalized spin and, in particular, the temperatures of the leads\nare not transmitted via the charge current to the spins.\nB. Transport formulas\nHere, we will briefly go through and summarize the ap-\nproach we employ to study the transport properties governed\nby the molecular spin dimer. First, the currents under interest\nare the charge and thermal currents, which are defined for the\nflows through interface \u001fas\nI\u001f\nc=\u0000eI\u001f\nN; (9a)\nI\u001f\nQ=I\u001f\nE\u0000\u0016I\u001f\nN; (9b)\nrespectively, where I\u001f\nE(N)denotes the energy (particle) current\nthrough the junction. Second, we notice that the energy and\nparticle currents through the system are defined as the rate of\nchange of the energy and the particle number, respectively.\nConcerning the particle flux, conservation of particles ensures\nthat the rate of particles passing through one of the interfaces\nequals the corresponding rate at the other interface. However,\nthe component in the thermal current that is generated by the\nparticle flux is not a conserved quantity [70]. This problem\ncan be avoided by considering the local entropy production\nrate in the molecular dimer, expressed through IQ=IL\nQ\u0000IR\nQ=\n(IL\nE\u0000\u0016LIL\nN)=2\u0000(IR\nE\u0000\u0016RIR\nN)=2. Then, since the particle contri-\nbution can be written as \u0000(\u0016L+\u0016R)IL\nN=2=\u0000\u0016IL\nN=2 and since\nwe measure all energies relative to the chemical potential \u0016,\nwhich is e \u000bectively zero, the particle contribution to the ther-\nmal current vanishes.\nWe remark here that the terminology entropy production\nrate for the quantity IQis justified in the stationary regime\nsince then the heat current I\u001f\nQ,\u001f=L;R, flowing through the\nleft or right interface equals the corresponding entropy flow\nI\u001f\nSmultiplied by the temperature, T\u001fI\u001f\nS. This can be un-\nderstood by considering that the rate of change of the non-\nequilibrium grand potential vanishes in the stationary regime,\n@t\n = 0, which yields T@tS=@thHi\u0000\u0016@thNi. On the one\nhand, the entropy production rate is given by the di \u000berence\nof the entropies flowing through the left and right interface,\n@tSL\u0000@tSR, with dimensions energy divided by temperature\nand time. On the other hand, the well defined and calculable\nquantity is here defined by the thermal currents I\u001f\nQand, due\nto the strong intimacy between the two quantities @tS\u001fandI\u001f\nQ, we shall use the term energy production rate for IQwith\ndimensions energy divided by time.\nWe express the fluxes at the interface \u001fin the generic forms\nI\u001f\nE=d\ndthH\u001fi=\u00002T\u001fImX\nk\u001b\"k(\u0000i)F<\nk\u001f\u001b(t;t); (10a)\nI\u001f\nN=d\ndthN\u001fi=\u00002T\u001fImX\nk\u001b(\u0000i)F<\nk\u001f\u001b(t;t); (10b)\nwhere we have defined the lesser form F<\nk\u001f\u001b(t;t0) of the trans-\nfer Green function Fk\u001f\u001b(t;t0)=(\u0000i)hTck\u001b(t)dy\n\u001f\u001b(t0)i. Using\nstandard theory [71] and restoring ¯h, the currents can be com-\npactly written on the form\nI\u001f\nE=(\u0000i)\u0000\u001f\n¯hspZ\n!\u0010\nf\u001f(!)G>\n\u001f(!)+f\u001f(\u0000!)G<\n\u001f(!)\u0011d!\n2\u0019;(11a)\nI\u001f\nN=(\u0000i)\u0000\u001f\n¯hspZ\u0010\nf\u001f(!)G>\n\u001f(!)+f\u001f(\u0000!)G<\n\u001f(!)\u0011d!\n2\u0019:(11b)\nHere, sp denotes the trace over spin 1 /2 space whereas G<=>\n\u001f,\n\u001f=L;R, denotes the 2\u00022-matrix Green function of the\nmolecule adjacent to the lead \u001f.\n1. Derivation of the molecular Green function\nWe make further analytical progress by constructing an ex-\nplicit expression for the Green function G\u001f. To this end we\ninclude the broadening e \u000bects from the couplings to the leads\nas well as the inter-molecular tunneling and intra-molecular\nexchange interactions with the localized spin moments. These\npresumptions lead to that we can write the retarded /advanced\nGreen functions G\u001fweighted on molecule \u001fas [51]\nGr=a\nL(R)(!)=1\n2˜TcX\ns=\u0006˜Tc\u001b0\u00002svhSz\nR(L)i\u001bz\n!\u0000Es\u0006i\u0000=8; (12)\nwhere the excitation energies E\u0006=\"0\u0006˜Tc=2 and ˜T2\nc=\nv2hSz\nL\u0000Sz\nRi2+4T2\nc.\nThe Green function Gfor the full dimer system is a 4 \u00024-\nmatrix G=fG\u001f\u001f0g\u001f;\u001f0=L;R, in which each entry is a 2 \u00022-matrix\nG\u001f\u001f0=fG\u001f\u001b\u001f\u001b0g\u001b;\u001b0=\";#. Here, the subscripts \u001f;\u001f0refer to the\nleft (right) molecule if \u001f\u001f0=LL(\u001f\u001f0=RR) and coupling be-\ntween the molecules for \u001f\u001f0=LRor\u001f\u001f0=RL. For brevity,\nwe write G\u001f\u001f=G\u001f. Each entry is defined by\nG\u001f\u001b\u001f0\u001b0(z)=hhd\u001f\u001bjdy\n\u001f0\u001b0ii(z)\n=Z\n(\u0000i)hTd\u001f\u001b(t)dy\n\u001f0\u001b0(t0)ie\u0000iz(t\u0000t0)dt0: (13)\nThe equation of motion for Gcan be summarized in the Dyson\nequation\nG=G0+G0\u0006G; (14)\nwhere G0is the bare Green function for the coupled\nmolecules, however, without couplings to the leads, whereas6\nFIG. 4: (Color online) (a) Molecule ( L,R) and spin projected (indi-\ncated by white arrows) densities of electron states of the left and right\nnon-gated (\u0016=0) molecules, calculated using GLL(RR)\u001bgiven in Eq.\n(16), respectively, as function of the voltage bias Vand energy !.\n(b) Molecule projected eq-fullGreenFunction of the localized spins\nhSz\n\u001fiand the total magnetic moment hSz\nL+Sz\nRi. Parameters are as in\nFigure 1.\u0006defines the self-energy generated by the couplings to\nthe leads. It can be noticed that since molecule 1 (2)\nonly couples to the left (right) lead, the retarded form of\nthis self-energy can be written as the diagonal matrix \u0006=\n(\u0000i)diagf\u0000L\n\";\u0000L\n#;\u0000R\n\";\u0000R\n#g=2. Considering spin-degenerate\nconditions, we can set \u0000\u001f\n\u001b= \u0000\u001f=2. As an e \u000bect of the Dyson\nequation for G, the corresponding lesser /greater forms are\ngiven by G<=>=Gr\u0006<=>Ga, where the lesser /greater forms\nof the self-energy is given by\n\u0006<=>(!)=(\u0006i)1\n4 \nfL(\u0006!)\u0000L\u001b00\n0 fR(\u0006!)\u0000R\u001b0!\n: (15)\nIn order to find the analytical forms of the local Green func-\ntion, we notice that since the spin degrees of freedom are\nuncoupled, we can write Gin block diagonal form. In this\nrepresentation, the blocks are distinguished by the spin index\nwhereas each block can be written on the form\nn\nGr=a\n\u001f\u001f0\u001b(!)o\n\u001f;\u001f0=L;R\n=\" \n!\u0000\"0\u0006i\u0000L=4\u0000Tc\n\u0000Tc!\u0000\"0\u0006i\u0000R=4!\n\u0000v\u001bz\n\u001b\u001b hSz\nLi0\n0hSz\nRi!#\u00001\n= !\u0000\"0\u0000vhSz\nRi\u0006i\u0000R=4Tc\nTc !\u0000\"0\u0000vhSz\nLi\u0006i\u0000L=4!\n(!\u0000Er=a\n+)(!\u0000Er=a\n\u0000);\n(16)\nwhere Er=a\ns=\"0+s˜Tc=2\u0007i(\u0000L+\u0000R)=8=\"0+s˜Tc=2\u0007i\u0000=8 and\nwhere we have put \u0000\u001f= \u0000=2 in the last equality. Accordingly,\nthe Green function weighted on the left molecule with spin \u001b\nis, therefore, given by the entry Gr=a\nLL\u001b(!), which can also be\nwritten on the form\nGr=a\nL\u001b(!)=1\n2˜TcX\ns=\u0006˜Tc\u00002svhSz\nRi\u001bz\n\u001b\u001b\n!\u0000Es\u0006i\u0000=8; (17)\nwith E\u0006=\"0\u0006˜Tc=2.\nWhen putting these results into the combination of the\nlesser and greater Green functions in the transport formulas,\nEq. (11), we retain only\nf\u001f(!)G>\n\u001f(!)\u0000f\u001f(\u0000!)G<\n\u001f(!)\n=(\u0000i)\u0000R\n2\u0010\nf\u001f(!)\u0000f\u001f0(!)\u0011\nGr\n\u001f\u001f0(!)Ga\n\u001f0\u001f(!);(18)\nout of which the only finite terms are the ones that couple the\ntwo molecules to one another and it is, therefore, necessary to\nstudy the forms of the site o \u000b-diagonal Green functions GLR\nandGRL. By a straight forward calculation one finds that theseGreen functions can be written as\nGr=a\n\u001f\u001f0(!)=Tc\n2X\ns=\u00061\u001b0\u0000s\u001bz\n!\u0000\"0\u0000svhSz\n\u001fi\u0006i\u0000=8Gr=a\n\u001f0(!)\n=Tc\n4˜TcX\nss0=\u00061[\u001b0\u0000s\u001bz][˜Tc\u001b0\u00002s0vhSz\n\u001fi\u001bz]\n[!\u0000\"0\u0000svhSz\n\u001fi\u0006i\u0000=8][!\u0000Es0\u0006i\u0000=8]:\n(19)\nIII. RESULTS AND DISCUSSION\nRegarding the charge transport across the junction, most of\nits properties can be understood in terms of the degree of de-\nlocalization of the electronic density in the molecular dimer.\nAs have been discussed in a previous publication [51], the\nmagnetic states and configurations lead to qualitatively dis-\ntinct regimes of the electronic properties of the dimer, some-\nthing that is conveyed over to the inherit transport properties\nof the molecular dimer. Accordingly, the conductance in the\nferromagnetic regime is expected to better, or, higher than in\nthe anti-ferromagnetic regime. This conjecture is based on the\nfact that in the anti-ferromagnetic regime the spin projected\nelectronic density in the bonding state is strongly localized to\noneorthe other molecule, see Figure 4 (a), and in the opposite7\n-6 -3\nvoltage bias (mV)3 6 002 IQ (MeV/s)\n-6 -3\nvoltage bias (mV)3 6 0-2×10-1(c) (d)-303IC (nA)\n-6 -3\nvoltage bias (mV)3 6 0 -6 -3\nvoltage bias (mV)3 6 0(a) (b)\nTR-TL=0 K\nTR-TL=10 K\nFIG. 5: (Color online) (a), (b), The charge current IC(units: nA)\nand (c), (d), entropy production rate IQ(units: MeV /s) as function\nof voltage bias VforTR\u0000TL=0 (black – faint) and TR\u0000TL=10\nK (red – bold) for (a), (c), \"0\u0000\u0016=0 meV and (b), (d), \"0\u0000\u0016=1\nmeV . Other parameters are as in Figure 1. In panels (a) and (b),\nthe ferromagnetic and anti-ferromagnetic regimes are indicated with\ngreen and blue arrows, respectively.\nfashion for the anti-bonding orbital. Therefore, an electron re-\nsiding in one of the molecules has a nearly vanishing probabil-\nity to tunnel over the other molecule which leads to a strongly\nsuppressed conductance. In the ferromagnetic regime, how-\never, the electronic density is more delocalized throughout the\nmolecular dimer which allows for resonant tunneling between\nthe molecules and, in turn, leads to an increased conductance.\nThese features and disparities of the two magnetic regimes\ncan be observed in the current, see Figure 5 (a), (b), showing\nthe charge current Icas function of the voltage bias at zero\ntemperature di \u000berence (black – faint) and TR\u0000TL=10 K (red\n– bold). Especially for vanishing temperature di \u000berence, the\nanti-ferromagnetic and ferromagnetic regimes (indicated by\nblue and green arrows, respectively) are strikingly separated\nby sharp current resonances which originates from a vanish-\ning exchange between the spin. The absence of the exchange\ninteraction leads to uncorrelated spins and a completely delo-\ncalized charge density in the molecular dimer, which allows\nfor a large current flow in the same way as in the high voltage\nregime where J!0.\nAt zero temperature di \u000berence across the junction, the\ncurrent-voltage characteristics is necessarily symmetric when-\never then electronic structure of the molecular dimer are sym-\nmetrically distributed, as is the case we consider here. In-\ntroduction of a finite temperature di \u000berence changes this sce-\nnario, however, a necessary condition for breaking the sym-\nmetric current-voltage characteristics is that the bonding and\nanti-bonding orbitals do not surround \u0016symmetrically. This\ncan be seen in the red – bold traces in Figure 5. In panel (a),\nthe molecular level \"0=\u0016which accordingly leads to a sym-\nmetric current as function of the voltage bias. In panel (b), in\ncontrast, the molecular level satisfies \"0\u0000\u0016=1 meV , which\nFIG. 6: (Color online) (a), (c), The charge current IC(units: nA)\nand (b), (d), entropy conductance IQ(units: MeV /s) as function of\nvoltage bias Vand temperature di \u000berence TR\u0000TLfor (a), (b), \"0\u0000\n\u0016=0 meV and (c), (d), \"0\u0000\u0016=1 meV . Other parameters are as in\nFigure 1\nthen generates a striking asymmetry in the current for finite\ntemperature di \u000berences. This result can be traced back to\nthe dramatically changed, weakened, exchange interaction be-\ntween the spins which causes an increased degree of delocal-\nization of the electronic density in the molecular dimer. In ef-\nfect, therefore, both the ferromagnetic and anti-ferromagnetic\nregimes are essentially destroyed for negative voltages, see\nFigure 5 (b) (red – bold). Hence, under the temperature di \u000ber-\nence between the leads the molecular dimer functions partially\nas a rectifying device where the magnetically active regimes\ncan be employed in the forward direction while the system\nbehaves like a normal conductor in the backward. In Fig-\nure 6 (a), (c), we show contour plots of the charge current\nas function of the voltage bias and temperature di \u000berence for\n(a)\"0\u0000\u0016=0 meV and (c) \"0\u0000\u0016=1 meV . The properties dis-\ncussed in detail above can be seen to be verified in the larger\npicture, varying continuously with the variations of the exter-\nnal conditions. Especially in the latter case ( \"0\u0000\u0016=1 meV),\nthe magnetically active regimes for negative voltage biases are\nseen to be destroyed already for small temperature di \u000berences,\nclosely following the behavior of the exchange interactions,\nc.f., Figure 2.\nNext, in the discussion of the entropy production rate, we\nagain notice that most of the expected features can be ex-\nplained in terms of the properties of the electronic structure\nin the molecular dimer analogous as with the charge current.\nHowever, for vanishing temperature di \u000berence between the\nleads a clear distinction compared to the charge current is that\nthe finiteness of the entropy production rate strongly depends\non the energy of the localized level in the molecular assem-\nbly. This can be traced back to the product of the energy and\nthe distribution functions ( G<=>\n\u001f) of the molecular dimer in,\nfor instance, Eq. (11a). One can make a simple compari-\nson between the qualitative properties of the two currents by\nassuming a Lorentzian model, 1 =[(!+\u0016\u0000\"0)2+(\u0000=2)2], for\nthe device embedded between the leads at low temperatures.\nThen, under the voltage bias \u0016L=R=\u0016\u0006eV=2, the two currents8\nbehave as\nIc\u00181\n\u0000=2\u0012\narctaneV=2\u0000\"0\n\u0000=2+arctaneV=2+\"0\n\u0000=2\u0013\n; (20a)\nIQ\u00181\n2ln(eV=2\u0000\"0)2+(\u0000=2)2\n(eV=2+\"0)2+(\u0000=2)2\n+\"0\u0000\u0016\n\u0000=2\u0012\narctaneV=2\u0000\"0\n\u0000=2+arctaneV=2+\"0\n\u0000=2\u0013\n:(20b)\nHence, while both the charge current and entropy production\nrate have normal on-sets associated with the energy \"0, de-\nscribed by the arctan-component, the entropy production rate\nis also logarithmically resonant at \"0. In addition, for the en-\ntropy production rate the on-set at \"0is weighted by the posi-\ntion of\"0relative to\u0016and, therefore, this contribution to the\nentropy production rate is expected to the small whenever \"0\nlies in the vicinity of \u0016. The logarithmic contribution tends to\nbe small for large voltage biases, since it leads to an increas-\ningly symmetric argument, and is accordingly only significant\nfor voltages such that either eV=2\u0000\"0oreV=2+\"0is small.\nExtrapolating this discussion to our present case with the\nmolecular dimer, we can verify these expected features under\nthe conditions of vanishing temperature di \u000berence. This can\nbe be seen in Figure 5 (c), (d) (black – faint). At \"0\u0000\u0016=0,\npanel (a), the entropy production rate is identically zero while\na finite o \u000b-set from\u0016, panel (b), yields a finite entropy pro-\nduction rate. In the latter case, the entropy production rate is\nsmall in the magnetically active regime and grows large only\nat large voltages where the exchange interaction vanishes. The\nentropy production rate tends to be e \u000eciently transmitted only\nwhen the molecular dimer is in its fully conjugated state, that\nis, when the exchange interaction is small. This is contrast to\nthe charge current where the less localized electronic density,\nin the molecular dimer, in the ferromagnetic regime yields a\nsignificant di \u000berence compared to the anti-ferromagnetic.\nThe reason for this qualitative di \u000berence between the\ncharge and entropy production rate, see Eq. (20), can be\nfound in the properties of the two currents in the ferromag-\nnetic regimes. In particular, the logarithmically resonant con-\ntribution to the entropy production rate, which is absent in the\ncharge current, peaks near the energies of the molecular bond-\ning and anti-bonding orbitals, something which occurs within\nthe ferromagnetic but not in the anti-ferromagnetic regime.\nThereto, these resonant peaks are oppositely directed com-\npared to the contributions varying like arctan x. As for the\nentropy production rate under finite temperature di \u000berence be-\ntween the leads, see Figure 5 (c), (d) (red – bold) and Figure\n6 (b), (d), this behavior is verified. This feature can be traced\nback to an increased degree of delocalization in the ferromag-\nnetic regime due to the increased thermal broadening from the\nhotter reservoir.\nIn the case where the localized level is o \u000b-set from\u0016, there\nis an interesting observation to be made in the temperature and\nvoltage dependence of both the charge and entropy production\nrate. At finite temperature di \u000berence and voltage bias, there\nis a strong asymmetry with respect to zero voltage which is\nclear signature that the magnetically active dimer potentially\ncan be used for current rectification. Previously it was shown\nT (K)048current (nA)\n-4 4 0\nT (K)-4 4 001020entropy prod. (MeV/s)\nT (K)-4 4 0012exchange (meV)(a) (b) (c)V=3 mV\nV=2.5 mV\nV=2 mV\nV=1.5 mV\nV=1 mV\nV=0.5 mVFIG. 7: (Color online) Thermal rectification. (a) Charge current, (b)\nentropy production rate, and (c) exchange interaction, as function of\nthe temperature di \u000berence for di \u000berent voltage biases. Here, \"0\u0000\u0016=\n2 andTc=3 meV , while other parameters are as in Figure 1.\nin Ref. [51] that the charge current can be rectified by intro-\nducing a level o \u000b-set between the localized levels in the two\nmolecules, something which possibly can be obtained by us-\ning di \u000berent types of molecules. The asymmetry induced by\nthe temperature di \u000berence provides a di \u000berent mechanism to\nrectification which is also viable for the entropy production\nrate. Indeed, the plots in Figure 6 (c), (d), illustrate that both\nthe charge and the entropy production rate is rectified at fi-\nnite temperature di \u000berences upon changing the polarity of the\nvoltage bias. For this to be successful, the dimer has to be\nset-up with a finite level o \u000b-set from\u0016and a finite voltage\nbias. Moreover, the parameters of the molecular dimer have\nto be tuned such that the o \u000b-set between the bonding and anti-\nbonding orbitals is larger than the thermal energy fed into the\ndimer from the hotter electrode. This enables a crossover from\nthe anti-ferromagnetic regime to either the ferromagnetic or\nparamagnetic regime, under changes in the temperature dif-\nference, which leads to strong variations in the currents. In\nprinciple, the stronger anti-ferromagnetic exchange one can\nobtain for one polarity of the temperature di \u000berence and the\nweaker the exchange can be in the opposite, regardless of sign,\nthe larger the ratio between the currents for the two polarities.\nIn Figure 7 we plot the (a) charge and (b) entropy produc-\ntion rate as function of the temperature di \u000berence for di \u000berent\nvoltage biases. We apply half the di \u000berence to the tempera-\nture in each lead such that TL=T\u0000\u0001=2 and TR=T+ \u0001=2,\nwhere the base temperature T=4 K. The calculations confirm\nthe argument that variations of the exchange interaction from\nstrongly anti-ferromagnetic to weakly ferromagnetic [ V\u00192\nmV , see Figure 2 (c)] indeed yields the larger ratio between\nthe large and small current regimes.\nFinally, we briefly discuss the di \u000berential conductances,\nboth charge conductance and di \u000berential entropy production\nrate and with respect to both voltage bias and temperature dif-\nference. Hence, we calculate @Ix=@Vand@Ix=@\u0001T, where\nx=c;Q, see Figs. 8 (a) – (d) and 9 (a) – (d). In Figs. 8\nand 9 we plot 1 =(1+expf\u001bFg), whereFis one of@Ix=@V,\n@Ix=@\u0001T, and Sx, in order to lower the values of the high am-\nplitudes. First we notice that both currents have ranges with\nfairly rapid variations with the voltage bias and with the tem-\nperature di \u000berence. While these properties can be traced back\nto the corresponding variations of the exchange interaction,\nwe can predict a few consequences of these features on the ra-9\nFIG. 8: (Color online) (a) – (d) Di \u000berential charge conductance (a),\n(c), and entropy production rate (b), (d), with respect to the voltage\nbias (a), (b), and temperature di \u000berence (c), (d). (e) Scand (f) SQ.\nHere,\"0\u0000\u0016=0, while other parameters are as in Figure 1. All con-\ntours show (1 +expf\u001bFg)\u00001, whereFis one of@Ix=@V,@Ix=@\u0001T,\nandSx, whereas\u001bis a scaling parameter.\ntios of the di \u000berential conductance with respect to the temper-\nature di \u000berence and the di \u000berential conductance with respect\nto the voltage bias, that is,\nSc=\u0000@Ic=d\u0001T\n@Ic=dV; (21)\nwhere the notation Scis used since the ratio coincides with the\nSeebeck coe \u000ecient in the limit V!0,\u0001T!0 [68]. In this\nsense the ratio Scis a non-equilibrium analogue of the See-\nbeck coe \u000ecient, however, we shall refrain from that nomen-\nclature for sake of not causing confusion with the concepts.\nNontheless, as rapid variations in the charge current yield\na large corresponding conductance and, oppositely, for slow\nvariations lead to a small conductance, we would expect that\nSctypically is large in the regions with weak variations of the\ncharge current. One is therefore led to think that Scmight\nbe large in the anti-ferromagnetic regime since the current is\nboth small and slowly varying with the voltage bias in those\nregimes, see Figs. 8 (a) and 9 (a). In addition, within the anti-\nferromagnetic regimes, @Ic=@\u0001Tvaries rapidly, including pos-\nsible sign changes, near zero temperature di \u000berence between\nthe leads. This conjecture is fairly well corroborated in the\nplots of the Scshown in Figs. 8 (e) and 9 (e). In particular, it\ncan be noticed in Figure 8 (e) that Scacquires large values in\nthe anti-ferromagnetic regime (voltages roughly in the ranges\n(\u00004;\u00002) and (2;4) mV) for temperature di \u000berences between\n0 K and 5 K. Except for these small regions of large Sc, it\ntends to be small in the remainder of the magnetically active\nregimes although not vanishing. In the case with a finite level\no\u000b-set, Figure 9 (e), however, Sctends to be large in the re-\ngions where the anti-ferromagnetic coupling is destroyed by\nFIG. 9: (Color online) (a) – (d) Di \u000berential charge conductance (a),\n(c), and entropy production rate (b), (d), with respect to the voltage\nbias (a), (b), and temperature di \u000berence (c), (d). (e), (f), Seebeck\ncoe\u000ecients associated with the (e) charge, Sc, and (f) heat, SQ, cur-\nrents. Here, \"0\u0000\u0016=1, while other parameters are as in Figure 1. All\ncontours show (1 +expf\u001bFg)\u00001, whereFis one of@Ix=@V,@Ix=@\u0001T,\nandSx, whereas\u001bis a scaling parameter.\nthe increased temperature di \u000berence. There are clearly visi-\nble domains at negative voltages which can be correlated with\nthe cross over between the anti-ferromagnetic and paramag-\nnetic regimes. At positive voltages where the dimer remains\nmagnetically active in a larger temperature range, Scis again\nsmall. We conclude that Scin both the ferromagnetic and anti-\nferromagnetic regimes is small and typically becomes finite at\nthe crossovers to the paramagnetic regime. Hence, the spin-\nspin interaction provides a way to not only control the charge\ncurrent but also Scin the system.\nIn analogy with the definition of Scassociated with the\ncharge current, we can define a coe \u000ecient SQfor the entropy\nproduction rate through\nSQ=\u0000@IQ=d\u0001T\n@IQ=dV: (22)\nAlthough we cannot give this coe \u000ecient an as simple interpre-\ntation as with the thermopower, we nevertheless find it useful\nin the analysis of the influence of the magnetic configurations\non the thermal properties. It can be seen in Figs. 8 (b) and\n9 (b) that the di \u000berential entropy conductance, with respect\nto the voltage bias, has a non-trivial dependence on the volt-\nage bias and temperature bias. Moreover, the dependence on\nthe temperature di \u000berence is intriguing. We can, however,\nnotice regarding the generic features of SQthat its more or\nless vanishing in the magnetically active regimes except in\nthe crossover between the di \u000berent regimes, where the en-\ntropy conductance varies slowly with the voltage bias but not\nnecessarily with the temperature di \u000berence. The qualitative\ndi\u000berence of the entropy conductance compared to the charge\ncurrent in the ferromagnetic regime, leads to that SQprovides10\na complimentary piece of information about the system under\ninvestigation in addition to Sc.\nIV . CONCLUSIONS\nIn summary we have theoretically studied the transport\nproperties of a dimer of paramagnetic molecules with local-\nized spins embedded in a junction between metallic leads. In\nparticular, we have addressed the charge and entropy conduc-\ntance flowing through the junction. It is demonstrated that\nthe indirect exchange interaction between the localized spins,\nwhich previously has been shown to depend on the voltage\nbias and temperature di \u000berence across the junction, acquires\na strongly asymmetric voltage bias dependence under finite\ntemperature di \u000berence between the leads. This property was\nsubsequently is predicted to have implications on, for exam-\nple, thermal rectification for both the charge and entropy con-\nductance. Simultaneously, our calculations suggest that the\ntemperature of the source electrode has a stronger influence\non the properties of the indirect exchange than the drain. It\nwas found, for instance, that while a voltage drop from the\nhotter to the colder reservoir tends to e \u000bectively destroy the\ntunable properties of the indirect exchange, these properties\nare stable under the opposite voltage polarity.\nThe transport properties of the dimer are intimately related\nto the indirect exchange, where the charge current is nearly\nblockaded for anti-ferromagnetic exchange, whereas it is fi-\nnite for ferromagnetic exchange, and maximal whenever the\nexchange is negligible. These three regimes can be explained\nby di \u000berent characteristics of the electronic density. In the\nanti-ferromagnetic regimes, the spin projections of the den-\nsity is strongly localized to one molecule such that transport\nbetween the molecules is suppressed. When in the ferro-\nmagnetic regime, the electron density is partially delocalized\nwhich leads to an enhanced conductance, whereas the cur-\nrent can flow freely in the paramagnetic regime due to a com-\npletely delocalized density. The entropy conductance follows\nthe same characteristics, as the charge current, in the anti-\nferromagnetic and paramagnetic regimes. In the ferromag-\nnetic regimes, however, the entropy conductance is strongly\nsuppressed by a contribution which is large, and nearly can-\ncels the regular entropy conductance contribution, near volt-\nage biases corresponding to the energy of the resonant states in\nthe molecular structure. By necessity, this resonant behavior\noccurs in the ferromagnetic regimes which leads to a strongly\nsuppressed entropy conductance there as well.\nWe, further, demonstrated that the non-equilibrium ther-\nmopower, in general is finite at the cross-over between\nregimes of di \u000berent indirect exchange associated with small\ndi\u000berential conductance. Typically, this behavior can be sum-\nmarized in that the thermopower is low within both the ferro-\nand anti-ferromagnetic regimes. Analogously, we introduced\na thermal coe \u000ecient as the ratio between the di \u000berential en-\ntropy conductance with respect to the temperature di \u000berence\nand voltage bias. While some of its properties closely fol-\nlow those ratios, additional features can be extracted, espe-\ncially at the cross over between the ferro- and antiferromag-netic regimes. In this sense, this ratio provides a complemen-\ntary sensitivity to the transport measurement which may show\nuseful in the analysis of the internal properties of the system.\nWe remark that while our results presented in this arti-\ncle are quite qualitative, they are realistic from the follow-\ning point of view. The results for the exchange interaction\nJpresented in the previous section, are obtained using the\nbare single electron Green functions for the molecular lev-\nels. This means that the back-action e \u000bect from the local\nspin moment is not included. In the calculations of the trans-\nport properties, on the other hand, the presence of the local\nspin moments are included, something which is crucial in\norder to investigate possible signatures in the transport data\nthat originates from the spin configurations. As for the trans-\nport calculations we could have chosen to simply demonstrate\nhow the charge and thermal transport depend on the nature\nof the exchange interaction, whether it is ferromagnetic (neg-\native), anti-ferromagnetic (positive), or paramagnetic (zero).\nGiven the approximation in which we calculate the local elec-\ntronic Green function of the molecular dimer, we would obtain\nthe transport characteristics that are presented in the Results.\nHowever, instead of making assumptions about the values of\nthe exchange interaction, we use the values as calculated by\nthe formulas provided in Exchange. In this way we incorpo-\nrate the voltage bias and thermal gradient dependence of the\nexchange also on the transport properties. While this approach\ncertainly has its limitations, we notice that a more thorough\nstudy of the correspondence between the regimes with dif-\nferent spin couplings and the associated transport properties\nrequires self-consistent calculations. Such calculations are,\nhowever, beyond the scope of the present investigation.\nConsidering the limitations of the method, we yet believe\nthat our findings are realistic and relevant to existing molecu-\nlar structures. The values of the local exchange interaction\nbetween localized spin and delocalized electrons can vary\nbetween 0.5 – 20 meV [72, 73], which leaves a large win-\ndow of tuning freedom with respect to couplings to the leads\nand HOMO /LUMO level o \u000b-set of the molecules. Moreover,\nsince our predictions are stable with respect to di \u000berences in\nthe local exchanges as well as the couplings to the leads, this\nalso allows for flexibility in the design of potential experi-\nments.\nThe predictions discussed in this paper opens the possibil-\nity to design nanoscale structures, in particular using mag-\nnetic molecules, that have a strong sensitivity on the local\nspin states of the system, which can be measured through the\ncharge and thermal transport characteristics. In ways, this sug-\ngests an alternative utilization of the spin degrees of freedom,\ncompared to the conventionally implemented spintronics, in\nwhich external magnetic fields are absent. The absence of\nsuch fields, in turn, leads to that the spin-spin interactions are\nisotropic which implies a truly magnetically isotropic (param-\nagnetic) device. Experimental confirmation of our predictions\nare feasible using state-of-the-art technology.11\nV . ACKNOWLEDGEMENT\nWe thank S. 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In this system, magneti-\nzation precession driven by a microwave generates direct-c urrent (dc) and alternate-current (ac)\nspin accumulation in the nonmagnetic layer by the spin pumpi ng. The ac spin accumulation is\ncoupled with the microwave magnetic field through a dynamica l Hanle spin precession, giving rise\nto rectified spin accumulation comparable with the dc spin ac cumulation directly generated by the\nspin pumping.\n∗ando@appi.keio.ac.jp\n1M(t)\nzyxh(t)\nmy(t)δmzδHanle \nFIG. 1: A schematic illustration of the spin pumping and dyna mical Hanle effect. M(t) andh(t)\nare the magnetization and microwave magnetic field, respect ively. The spin pumping creates ac\nspin accumulation δmy(t), which is rectified through the spin precession induced by h(t), giving\nrise to dc spin accumulation δmHanle\nz.\nRectification effects are fundamental in electrical, optical, and mag netic systems. An\nelectrical rectifier is used to convert an alternating current to a d irect current. The\nrectifier essentially strips the high-frequency or alternating part from the incoming cur-\nrent and delivers a low-frequency current, which is based on the tr igonometric relation:\ncosω1tcosω2t= (1/2)[cos(ω1−ω2)t+cos(ω1+ω2)t]; when two waves of frequencies ω1and\nω2are combined, the difference-frequency and sum-frequency ter ms appear. If the frequen-\ncies satisfy ω1=ω2, thetermcos( ω1−ω2)tgives risetoadirect-current (dc) signal. Asimilar\ndynamic response to the product of alternate-current (ac) spin accumulation and a time-\ndependent magnetic field is the origin of a spin rectification effect indu ced by a dynamical\nHanle effect presented in this paper.\nTheHanleeffect refersto thevariationofspin accumulation δminresponse toa magnetic\nfieldHapplied perpendicular to the spin-polarization direction1–3; when a nonmagnet is\nexposed to a magnetic field, spins in the nonmagnet start to preces s around Has∂δm/∂t=\n−γδm×H.\nIn this letter, we show rectification of ac spin accumulation through spin precession in-\nduced by a time-dependent magnetic field: a dynamical Hanle effect. The dynamical Hanle\neffect is discussed in a ferromagnetic/nonmagnetic ( F/N) junction under ferromagnetic res-\nonance, where uniform magnetization precession is driven by a micro wave magnetic field.\nIn this system, the magnetization precession generates dc and ac spin accumulation, or spin\ncurrents, in the Nlayer by the spin pumping.4The dc-component of the spin pumping has\nbeen intensely studied inrecent years.5–11Incontrast, the ac-component ofthe spinpumping\n20200 400 600 800 1000 mzdc \nδ myac \nδ /\n10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 \nτ(s) 10 -3 10 -2 10 -1 0246810 mz    (10 3)dc \nδ myac \nδ /\nατ=10 -13 s\nFIG. 2: The ratio |δmy(t)|/δmdc\nzof the magnitude of the ac spin accumulation |δmy(t)|to that\nof the dc spin accumulation δmdc\nzdirectly induced by the spin pumping as a function of the spin\nrelaxation time τin the nonmagnetic layer for 4 πMs= 0.745 T,h= 0.01 mT,α= 0.01, and\nf= 9.4 GHz. The inset shows Gilbert damping constant αdependence of |δmy(t)|/δmdc\nz.\nhas been directly observed only recently.12Here, we show that the ac-component of the spin\npumping not only generates an ac electric voltage through the inver se spin Hall effect but\nalso generates, through the dynamical Hanle spin precession, larg e dc spin accumulation\ncomparable with the dc spin accumulation directly generated by the d c-component of the\nspin pumping. This finding offers a way to generate a large dc electric v oltage from the\nac-component of the spin pumping using the inverse spin Hall effect.\nThe spin pumping in a F/Njunction generates a spin current js(t) and spin accumulation\nδm(t) in the Nlayer.4–11,13–18The spin current density created by the spin pumping is\nexpressed as13,14\njs(t) =/planckover2pi1\n4πg↑↓\neff1\nM2s/parenleftbigg\nM(t)×dM(t)\ndt/parenrightbigg\n, (1)\nwhereg↑↓\neffis the spin pumping conductance and Msis the saturation value of magnetization\nM. Equation (1) shows that the spin pumping generates two types of spin currents; a dc\nspin current with the spin polarization direction along the zaxis, i.e. the magnetization\nprecession axis, and an ac spin current whose spin polarization direc tion oscillates in the\nx-yplane under the ferromagnetic resonance (FMR) condition [see Fig. 1]. In aF/Nthin\nfilm, where the magnetocrystalline anisotropy can be neglected and a microwave magnetic\nfieldh(t) = (hcosωt,0,0) is applied along the xaxis, the dc and ac spin currents with the\nspin polarization direction along zandyaxis at the resonance condition are obtained from\n3Eq. (1) with the Landau-Lifshitz-Gilbert equation by ignoring the se cond-order contribution\nof the precession amplitude as19\njz\ns=g↑↓\neffh2γ2/planckover2pi1/parenleftBig\n2πMsγ+/radicalbig\n4π2Ms2γ2+ω2/parenrightBig\n16πα2/parenleftbig\n4π2Ms2γ2+ω2/parenrightbig , (2)\njy\ns(t) =jy\ns1cosωt−jy\ns2sinωt, (3)\nwhere\njy\ns1=g↑↓\neffhγ/planckover2pi1/parenleftBig\n2πMsγ+/radicalbig\n4π2Ms2γ2+ω2/parenrightBig\n8πα/radicalbig\n4π2Ms2γ2+ω2, (4)\njy\ns2=g↑↓\neffhγ/planckover2pi1ω\n8π/radicalbig\n4π2Ms2γ2+ω2. (5)\nHere,γandαare the gyromagnetic ratio and the Gilbert damping constant, resp ectively. In\ntheNlayer, the dynamics of spin accumulation δm(x,t) = (δmx(t),δmy(t),δmz(t)) induced\nby the spin pumping is obtained from20\n∂δm(x,t)\n∂t=−γδm(x,t)×Heff(t)−δm(x,t)\nτ+D∇2δm(x,t)+js(t)δ(x),(6)\nwhereHeffis the effective magnetic field, τis the spin relaxation time, and Dis the diffusion\ncoefficient in the Nlayer. For simplicity, in the following discussions, we neglect the exter nal\nmagnetic field and consider the system where the Nlayer is thin enough so that the spin\ndiffusion term can be neglected.\nThe magnitude of the ac and dc spin accumulation directly generated by the spin pump-\ning, i.e., thespinaccumulationintheabsenceofthespinprecession: δm×Heff=0, depends\ncritically on the spin relaxation time τin theNlayer. In the absence of the spin precession,\nthe ac spin accumulation δmy(t) induced by the spin pumping is obtained from Eqs. (3) and\n(6) as\nδmy(t) =τ\n1+(ωτ)2[(jy\ns1+ωτjy\ns2)cosωt+(ωτjy\ns1−jy\ns2)sinωt]. (7)\nUsing Eq. (7), we plot the ratio of the ac to dc spin accumulation, |δmy(t)|/δmdc\nz, in Fig. 2,\nwhere the dc spin accumulation δmdc\nzdirectly generated by the spin pumping is obtained\nfrom Eq. (6): dδmzdc/dt= 0 =−δmzdc/τ+jz\ns. Here, 4 πMs= 0.745 T,h= 0.01 mT,\nγ= 1.86×1011T−1s−1,α= 0.01, andf= 9.4 GHz were used for the calculation, where\nω= 2πfandfis the microwave frequency. Figure 2 shows that |δmy(t)|/δmdc\nzchanges\ncritically around τ≃1/ω= 1.7×10−11s. In a material with τ≫1/ω, where the time scale\n410 -13 10 -12 10 -11 10 -10 10 -9 10 -8 \nτ(s) mzdc \nδ mzHanle \nδ (%) /α\n0.001 \n  0.005 0.01 0.02 0.04 0.06 0.08 = 0.1 \n04\n268\nFIG. 3: The ratio δmHanle\nz/δmdc\nzof the dc spin accumulation induced by the Hanle effect δmHanle\nz\nto that induced by the spin pumping δmdc\nzcalculated for different Gilbert damping constants α\nwith 4πMs= 0.745 T,h= 0.01 mT, and f= 9.4 GHz.\nof the oscillation of the ac spin accumulation 1 /ωis much faster than the spin relaxation\ntimeτ, the ac spin accumulation induced by the spin pumping is almost averag ed out,\nwhereas the dc spin accumulation can be accumulated in the time scale ofτ. In contrast,\nin a material with τ≪1/ω, the spins injected into the Nlayer relaxes before cancelling\nout the ac spin accumulation. Here note that the ac spin current jy\nsis much larger than\nthe dc spin current jz\ns, since the cone angle of the magnetization precession is typically\nless than 1◦, resulting the large ac spin accumulation |δmy(t)|compared with the dc spin\naccumulation δmdc\nzin this system. In fact, as shown in the inset to Fig. 2, |δmy(t)|increases\nwith the Gilbert damping constant α, showing that the ac spin accumulation is significant\nat small amplitudes of magnetization precession, since the precess ion amplitude decreases\nwith increasing α.\nThe acspin accumulation δmy(t) induced by the spin pumping creates rectified spin accu-\nmulation δmHanle\nzwith thespin polarizationdirection along the zaxis throughthe dynamical\nHanleeffect; δmy(t)isrectifiedbytheappliedmicrowave magneticfield h(t) = (hcosωt,0,0)\nbecause of the spin-precession term in Eq. (6): −γδm(x,t)×h(t). By neglecting a contri-\nbution from δmzonδmy, thezcomponent of the spin accumulation due to the precession\nofδmy(t) induced by h(t) is obtained from dδmz(t)/dt=γδmy(t)hcosωt−δmz(t)/τ. The\nspin precession term γδmy(t)hcosωtwith the term δmy(t)∝cosωtgives rise to rectified dc\n5010 20 30 40 50 60 70 mzdc \nδ mzHanle \nδ (%) /\n6 4 21.5 1 0.8 0.6 f = 0.45 GHz \n8 10 \n10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 \nτ(s) 2 4 6 8 10 030 60 \nf (GHz) mzdc \nδ mzHanle \nδ (%) /\nFIG. 4: δmHanle\nz/δmdc\nzcalculated for different microwave frequencies fas a function of the spin\nrelaxation time τ. 4πMs= 0.745 T,h= 0.01 mT, and α= 0.05 were used for the calculation. The\ninset shows fdependence of the maximum value of δmHanle\nz/δmdc\nz, i.e.,δmHanle\nz/δmdc\nzatτ= 1/ω.\nspin accumulation δmHanle\nz= (ω/2π)/integraltext2π/ω\n0δmz(t)dt:\nδmHanle\nz=hγτ2(jy\ns1+ωτjy\ns2)\n2/bracketleftbig\n1+(ωτ)2/bracketrightbig. (8)\nIn Fig. 3, we show the ratio of the dc spin accumulation δmHanle\nzinduced by the dynamical\nHanleeffecttothedcspinaccumulation δmdc\nz=τjz\nsdirectlyinducedbythedcspinpumping:\nδmHanle\nz\nδmdc\nz=ατ/parenleftbig\n4π2Ms2γ2+ω2/parenrightbig/bracketleftBig\n2πMsγ+ατω2+/radicalbig\n4π2Ms2γ2+ω2/bracketrightBig\n/bracketleftbig\n1+(ωτ)2/bracketrightbig/bracketleftBig\nω2+2πMsγ/parenleftBig\n2πMsγ+/radicalbig\n4π2Ms2γ2+ω2/parenrightBig/bracketrightBig (9)\nfor 4πMs= 0.745 T and f= 9.4 GHz. Figure 3 shows that δmHanle\nz/δmdc\nzis maximized\naroundτ≃1/ω= 1.7×10−11s. Here note that τ≃1/ωis the spin relaxation time where\nthe ratio of the ac to dc spin accumulation, |δmy(t)|/δmdc\nz, changes drastically [see Fig. 2],\nsuggesting that it is important to consider separately the two situa tions:τ≪1/ωand\nτ≫1/ω. In a material with τ≪1/ω, spins injected into the Nlayer relax before the\nprecession of δmy(t) due toh(t), resulting the suppression of δmHanle\nzas Fig. 3. In contrast,\nin a material with τ≫1/ω,δmHanle\nzis suppressed by the suppression of |δmy(t)|/δmdc\nz\n[see Fig. 2], resulting the peak structure of δmHanle\nz/δmdc\nzwith respect to τ. Therefore,\nthe dynamical Hanle effect is maximized in a material with the spin relaxa tion time of\nτ≃1/ω, showing that the rectification can be controlled by tuning the micro wave frequency\nf=ω/(2π).\n6The rectification effect is also sensitive to the Gilbert damping consta ntαof theF\nlayer as shown in Fig. 3; the dynamical Hanle spin precession is efficient in a system with\nlargeα. This is due to the fact that the ac spin accumulation δmy(t) induced by the spin\npumping becomes significant compared with the dc spin accumulation δmdc\nzwith increasing\nαas shown in the inset to Fig. 2. At first sight, this result seems to imply that the spin\nrectification depends also on the amplitude of the microwave magnet ic fieldh, since the\ncone angle of the magnetization precession decreases with decrea singh, or|δmy(t)|/δmdc\nz\nincreases with decreasing h. However, this compensates the decrease of the spin-precessio n\nterm; by decreasing h, the spin-precession term δm×Heffalso decreases, resulting that the\nrectification of the spin accumulation is independent of has described in Eq. (9).\nThe magnitude of the rectified spin accumulation δmHanle\nzcan be comparable with that\nof the dc spin accumulation δmdc\nzdirectly induced by the dc spin pumping at low microwave\nfrequencies. Figure 4 shows δmHanle\nz/δmdc\nzat different microwave frequencies fcalculated\nusing Eq. (9) with 4 πMs= 0.745 T and α= 0.05. Figure 4 shows that the spin relaxation\ntimeτmaxat which δmHanle\nzis maximized increases with decreasing fbecause of τmax≃1/ω.\nNotably, the maximum value of δmHanle\nz/δmdc\nzincreases with decreasing f[see also the inset\nto Fig. 4], and δmHanle\nzexceeds 50%of δmdc\nzatf= 0.45 GHz. Since the rectification depends\non the magnetic damping αas shown in Fig. 3, δmHanle\nzcan be further enhanced by selecting\na spin injector ferromagnet with a large magnetic damping α, providing a route for realizing\nδmHanle\nz/δmdc\nz>100%.\nIn summary, we have shown that ac spin accumulation generated by the spin pumping\nis rectified through spin precession induced by a microwave magnetic field: the dynamical\nHanle effect. The magnitude of the rectified spin accumulation can be comparable with\nthat of the dc spin accumulation directly induced by the spin pumping. 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Saitoh, Nature Mater. 10, 655 (2011).\n8"    },    {        "title": "2009.04423v4.Superconductivity_enhanced_spin_pumping__Role_of_Andreev_resonances.pdf",        "content": "Superconductivity-enhanced spin pumping: Role of Andreev resonances\nMostafa Tanhayi Ahari and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe describe a simple hybrid superconductor jferromagnetic-insulator structure manifesting spin-\nresolved Andreev bound states in which dynamic magnetization is employed to probe spin related\nphysics. We show that, at low bias and below Tc, the transfer of spin angular momentum pumped\nby an externally driven ferromagnetic insulator is greatly a\u000bected by the formation of spin-resolved\nAndreev bound states. Our results indicate that these bound states capture the essential physics\nof condensate-facilitated spin \row. For \fnite thicknesses of the superconducting layer, comparable\nto the coherence length, resonant Andreev bound states render highly transmitting subgap spin\ntransport channels. We point out that the resonant enhancement of the subgap transport channels\nestablishes a prototype Fabry-P\u0013 erot resonator for spin pumping.\nIntroduction .|Spatial variations in the superconduct-\ning order in a \fnite region lead to the formation of spin-\ndegenerate Andreev bound states (ABSs) with discrete\nexcitation energies below the superconducting gap [1].\nAn externally applied magnetic \feld or proximity to a\nferromagnetic order, on the other hand, can induce spin\nsplitting in the ABSs that results into spin-resolved ABSs\n[2]. In this paper, we consider a normal metal (N) sand-\nwiched between a superconductor (S) and a ferromag-\nnetic insulator (FI) that serves as a simple platform with\nspin-resolved ABSs, which are localized in the N layer.\nThe nonequilibrium pure spin current engendered from\nthe externally driven FI|spin pumping|is utilized to\nprobe spin transport in an S jNjFI hybrid structure. The\nspin pumping generated from a time-dependent magne-\ntization, on the other hand, is a \row of spin angular\nmomentum into adjacent materials that dissipates en-\nergy of the ferromagnet [3]. We suggest that the mag-\nnetic damping increase in superconducting hybrid multi-\nlayers NbjNi80Fe20and NbNjGdN reported, respectively,\nin Refs. [4] and [5] may be attributed to the resonant\nenhancement of the spin pumping discussed here.\nIn the context of superconducting spintronics, combin-\ning ans-wave superconducting order, favoring electrons\nto form a singlet state, with a ferromagnetic order, fa-\nvoring spin alignment, leads to a powerful enhancement\nor reduction of angular momentum transfer [6, 7]. The\nangular momentum transfer, as a central e\u000bect in spin-\ntronics, is greatly modi\fed on account of two major un-\nderlying causes: the itinerant spin-polarized quasiparti-\ncles (QPs) with long spin-coherence lengths [8, 9] and\nthe creation of spin-triplet Cooper pairs [10{12] induced\nat highly spin-active regions or complex magnetic mul-\ntilayers [13]. Here, the spin pumping in an S jNjFI hy-\nbrid structure, however, is an interplay between spin-\npolarized QPs and spin-triplet Cooper pairs, which are\ndynamically generated by the excited FI [14, 15]. The\nsubgap ABSs accommodate the spin-polarized QPs and\nspin-triplet Cooper pairs that for a su\u000eciently thin S\nlayer can tunnel across and contribute to the spin current.\nTo collect the spin current we have placed a spin reser-\nvoir N r, comprising an N rjSjNjFI structure (see Fig. 1).\nFIG. 1. Schematic sketch for an N rjSjNjFI hybrid structure.\nThe FI region with precessing magnetization m(t) injects spin\nangular momentum, Is, that is carried by the QPs into the\nreservoir N r(spin pumping). For \fnite thicknesses of the S\nand N layers ( dS;dN\u0018coherence length) the resonant ABSs\nprovide highly transmitting transport channels for spin pump-\ning. Here, for example, we show four resonant channels with\ntwo positive-energy ABSs. The energies are measured with\nrespect to the Fermi energy EFdepicted in the middle.\nWhile spin pumping has been considered as a new probe\nof spin dynamics in a superconducting thin \flm [16] and\ntunable pure spin supercurrents [17], here we propose to\nstudy spin-resolved Andreev resonances by means of spin\npumping.\nSpin-resolved ABS .|We will determine the subgap\nspin-resolved ABSs by studying the resonant conditions\nfor the spin transport in an N rjSjNjFI hybrid struc-\nture. In order to establish the pumped spin current in\nthe presence of superconducting and dynamic ferromag-\nnetic orders, we solve a time-dependent scattering prob-\nlem [15, 18], which accounts for the relevant processes\nsuch as Andreev re\rection (AR), ABS, and a dynamic\ntriplet-paring generation. For subgap energies, the full\nscattering matrix develops peak structures marking res-\nonant bound states [19] (ABSs), which result in highly\ntransmitting subgap transport channels for the spin \row.\nWhile we identify the Andreev resonances in the ballistic\nregime, we do not expect their perturbation by a weak\ndisorder to signi\fcantly modify the spectroscopic aspects\nof spin pumping (which should be governed by the un-\nderlying dynamic mixing between the superconductingarXiv:2009.04423v4  [cond-mat.supr-con]  19 Mar 20212\nFIG. 2. Pro\fle of Re g\"#fordN=\u00180= 1. (a) Assuming \fxed\nS layer thickness ( dS=\u00180= 1:5), the spectral overlap of the\nspin-split Andreev levels can be modi\fed by \u0001 ex. (b) At the\n\fnite thicknesses of the S, dS&\u00180, the Andreev resonances\nestablish highly transmitting transport channels. Here, we\nhave set \u0001 ex=EF= 0:4.\nsinglet and triplet components). To capture the essen-\ntial e\u000bect of interfacial scattering on the spin-pumping\nenhancement, we will consider an interface disorder at\nx=dS.\nWe proceed with establishing notations and key fea-\ntures of the scattering. The incident QPs from the reser-\nvoir onto the N rjS interface can either transmit across\nthe S or re\rect as holes back into the reservoir, a process\nknown as AR. The AR amplitude for an incident QP with\nenergy\"is given by\nr1\nA=(\ne\u0000iarccos (\"=\u0001);j\"j<\u0001;\n\"\u0000\"p\n1\u0000\u00012=\"2\n\u0001;j\"j>\u0001;(1)\nwhere \u0001 is the superconducting pair-potential [2]. The\nAR amplitude given in Eq. (1), by focusing on the inter-\nface, assumes a bulk S (a thick S layer) [15]. A nonzero\nprobability of QP transmission for the thin S as well as\nan interfacial disorder can reduce the AR probability,\nwhich we address below. The transmitted QPs with spin\n\u001b2f\";#gpropagate through the normal metal junction\nand acquire a spin-dependent phase ei#\u001bupon re\rection\nfrom the FI interface, where the FI region is considered\nan exchange-splitting insulator for the QPs [21]. This\nrenders the full scattering matrix to be merely a re\rec-\ntion matrix, which is block diagonal in the spin space\ndue to the conservation of the QP spin during each indi-\nvidual scattering event. Consequently, multiple ARs at\nthe NjS along with spin-dependent re\rections at the N jFI\ninterfaces constitute the full scattering matrix.\nThe spin-active interface N jFI, upon re\rection, rotates\na noncollinear QP spin around the FI magnetization axis,\nwhich in turn for a driven magnetization leads to genera-\ntion of a nonequilibrium spin current detected in N r[22].\nThe conductance determining the transport of this spin\ncurrent, known as the mixing conductance g\"#, is givenby\ng\"#=X\nn;m\u0000\n\u000em;n\u0000r\"\nee;mnr#\u0003\nee;mn +r\"\nhe;mnr#\u0003\nhe;mn\u0001\n;(2)\nwherer\u001b\neeandr\u001b\nherepresent the total spin- \u001belectron-to-\nelectron and electron-to-hole re\rection matrices in the\nSjNjFI hybrid structure, respectively [14, 18]. The in-\ndicesnandmrefer to transport channels the number\nof which can be determined by transverse thickness of\nthe normal metal layer [2]. The re\rection matrices in\nEq. (2) are written in the basis where the spin quan-\ntization axis is parallel to the magnetization in the FI\n(the exact expressions for the re\rection amplitudes in\nthe ballistic regime can be found in the Supplemental\nMaterial [15]). In the following, we provide the results\nfor the single-channel scattering and postpone discussing\nthe case of multichannel scattering to the Supplemental\nMaterial [15].\nGenerically, the mixing conductance given in Eq. (2) is\na complex number the real part of which governs the av-\nerage spin pumping and the associated Gilbert damping\n[18]. After a straightforward calculation in the ballistic\nregime, three distinct regimes are recognized for subgap\nenergies:\nRe g\"#\u00198\n><\n>:1\u0000cos# dS\u001c\u00180\n\r2=(\r2+\f2)dS\u0018\u00180\n0 dS\u001d\u00180; (3)\nwhere\r = exp[\u00002p\n1\u0000\"2=\u00012\n0dS=\u00180],\f =\u0002\ncos (4dN\"=\u00180\u00010+ 2\u0012A)\u0000cos#\u0003\n=4 sin#sin2\u0012A, and\n\u0012A\u0011 \u0000 arccos (\"=\u00010). The mixing angle de\fned as\n#\u0011#\"\u0000##is controlled by the FI exchange interaction\n\u0001ex[15]. Here, \u0001 0and\u00180are the superconducting gap\nand coherence length at zero temperature, respectively.\nEvidently, Re g\"#displays a resonant behavior associated\nwith the intermediate S thickness. The resonant energy\nlevels correspond to the ABS energies \"Adetermined by\n(\f= 0):\n\"A= \u0001 0signh\nsin\u00122dN\"A\n\u00180\u00010\u0006#\n2\u0013i\ncos\u00122dN\"A\n\u00180\u00010\u0006#\n2\u0013\n:\n(4)\nWhen the exchange interaction is absent, that is, #= 0,\nEq. (4) yields a pair of solutions ( \u0000\"A;\"A), which is a\nconsequence of the particle-hole symmetry imposed on\nthe scattering formalism [15]. A nonzero mixing angle\n#, on the other hand, by lifting the spin degeneracy of\neach level results in the spin-resolved ABSs with the fol-\nlowing energies (\u0000\"\u0006\nA;\"\u0006\nA). As an outcome, the spectral\noverlap of the spin-split bound states can be controlled\nby the exchange interaction of the FI [e.g., see Fig. 2(a)].\nWe emphasize that, following Eq. (3), only at dS\u0018\u00180\nthe resonant ABSs establish highly transmitting trans-\nport channels for the spin \row, which is greatly enhanced3\nFIG. 3. Geometric resonances of Re g\"#. The Fabry-P\u0013 erot\noscillations happen when the corresponding AR probability\njrAjis nonzero (the inset plot shows the probability of AR\nvs energy). For a \fxed \u0001 ex, the Fabry-P\u0013 erot oscillations are\ndetermined by dNthat for\" > \u00010get modulated with a\nfrequency determined by dS(i.e., energies for which jrAj= 0).\nHere, we have adopted the following parameters: dS=\u00180= 1,\ndN=\u00180= 10, and \u0001 ex=EF= 0:2.\ncompared to either a bulk or no S layer [see Fig. 2(b)].\nThis is one of the main results of this paper.\nIn the limit where dN!0, Eq. (4) can be reduced\nto the well known result [23] for ABSs with magnetically\nactive interfaces, \"A=\u00010=\u0006cos#=2. We highlight the\nfact that Eq. (4) is a condition for a constructive quan-\ntum interference in a Rowell{McMillan process for the\nQPs [12], that is, four times crossing Nwith two An-\ndreev conversions as well as two re\rections from FI, once\nas electron and once as hole. It is clear that a construc-\ntive interference for QPs inside the normal layer remains\nintact as long as the probability of the AR is nonzero.\nThis captures the essential physics of a two-mirror Fabry-\nP\u0013 erot resonator with a resonator length 2 dN, which we\nshall describe now. The N jFI and SjN interfaces operate\nas \\mirrors\" for the QPs that give rise, respectively, to\na spin-dependent specular re\rection (a spin-dependent\nmirror) and a phase-conjugating mirror, which retrore-\n\rects electrons with energy EF+\"Aas holes with energy\nEF\u0000\"A[24]. The resonant enhancement of a Fabry-\nP\u0013 erot device occurs when its mirrors have a near unity\nre\rection probability [25]. Here, the N jFI interface re-\n\rects all the incident QPs with probability 1, while the\nretrore\rection probability of the S jN interface, on the\nother hand, is determined by the AR probability. There-\nfore, the Fabry-P\u0013 erot enhancement is in accordance with\nthe AR amplitude, which for an S with a thickness dS\n[15] is given by\nrA=(1\u0000\r)r1\nA\n1\u0000\r(r1\nA)2: (5)\nFor subgap energies, in the limiting case of dS> \u00180, we\nget\r\u001c1 or equivalentlyjrAj\u00191, for which the resonant\nFIG. 4. The normalized e\u000bective conductance, ~ g\"#\ne\u000b, for sub-\ngap temperatures and intermediate thickness dS\u0018\u00180showing\na signi\fcant enhancement relative to the normal case near Tc.\nAs the S gap shrinks, with the increase of temperature, the\nenhancement peak location creeps up to higher dS, which ul-\ntimately diminishes for higher temperatures near Tc. Here,\nwe have set dN=\u00180= 2:5. The inset shows that the enhance-\nment feature happens for all 0 \u0014dN=\u00180\u00145 whendS=\u00180= 1.\nFor the above plots, we have used \u0001 ex=EF= 0:2.\nenhancement of a Fabry-P\u0013 erot device is expected. For en-\nergies above the gap, on the other hand, the AR re\rection\namplitude decreases. This results in highly transmitive\nchannels near \"&\u00010, which rapidly decline for higher\nenergies (see Fig. 3).\nIn addition to the QP interference in the normal layer\n(Rowell{McMillan resonance), which leads to the mix-\ning conductance oscillation, above the gap a quantum\ninterference inside the S layer can take place. An in-\ncident electronlike QP interferes with a holelike QP re-\n\rected from the other S jN interface, which may result in\njrAj= 0. The process known as Tomasch oscillations [26]\noccurs for QP energies \"n=\u00010=p\n1 + (n\u0019\u00180=2dS)2with\nintegern, which modulates the amplitude for the Fabry-\nP\u0013 erot oscillations. The Rowell{McMillan and Tomasch\nbased geometric resonances of Re g\"#are shown in Fig. 3.\nSpin pumping enhancement .|The spin pumping is\ngenerated by the variations in the magnetization direc-\ntionm(t) [3, 10, 27]. For su\u000eciently slow variations,\nto the \frst order in the pumping parameter frequency\nj@tmj, the spin pumping can be written in terms of the\ninstantaneous mixing conductance in the magnetization\ncoordinate system, that is, Eq. (2). Consequently, the\nspin-pumping current, assuming no voltage bias [18], is\ngiven by\nIs(t) =1\n4\u0019g\"#\ne\u000bm\u0002@tm; (6)\nwhere the e\u000bective mixing conductance is de\fned as fol-4\nlows:\ng\"#\ne\u000b\u0011Z1\n\u00001d\"@\"f(\") Re g\"#: (7)\nHere,f(\") = (1 +e\"=kBT)\u00001is the Fermi-Dirac distri-\nbution and in order to properly take into account the\ntemperature dependence of the S order, we have consid-\nered a temperature{dependent S gap \u0001( T) [21].\nIn accordance with Eq. (6), the spin{pumping current\nin the direction of m\u0002@tmis simply determined by g\"#\ne\u000b,\nwhich can be regarded as the mixing conductance in the\ntemperature domain. In order to focus on the role of ABS\nresonances, we have de\fned normalized e\u000bective conduc-\ntance ~ g\"#\ne\u000b\u0011g\"#\ne\u000b(dS)=g\"#\ne\u000b(dS= 0). Here, g\"#\ne\u000b(0) yields\nthe e\u000bective conductance in the normal state. The re-\nsultant normalized conductance is plotted in Fig. 4. We\n\fnd that, for subgap temperatures, ~ g\"#\ne\u000bshows a signi\f-\ncant enhancement at the \fnite thickness of the S layer,\ni.e.,dS\u0018\u00180. The enhancement is optimal at the midgap\ntemperatures and diminishes down to unity (the normal\ncase) upon approaching Tc.\nBefore closing this paper, we explore the e\u000bect of a bar-\nrier potential at the S jN interface. Physically, this can\noriginate from a thin oxide layer or a localized disorder\non the interface. The essential e\u000bects of the interfacial\nscattering, caused by this layer, can be captured by a\npotential of the form Z~vF\u000e(x\u0000dS), where the dimen-\nsionless parameter Zdetermines strength of the barrier.\nHere,vFis the Fermi velocity and \u000e(x) is the Dirac delta\nfunction.\nAt the SjN interface, a nonzero Zresults in reduced\ntransmission and AR probabilities by introducing ordi-\nnary electron and hole re\rections [3]. This partially re-\n\rective interface can lead to the formation of normal\nbound states localized in the N layer. In contrast, the\ncase of zero barrier ( Z= 0) leads to ABSs only. For sub-\ngap energies, with the increase of Zthe ordinary re\rec-\ntion probability surpasses the AR probability [3], which\nleads to ABSs to be pushed away from zero (towards the\ncontinuum of states above the gap) and replaced with\nordinary bound states [15]. Consequently, in the strong\nbarrier limit Z > 1, the subgap transport channels are\ndue to the normal bound{state resonances, where super-\nconductivity suppresses the spin pumping (see Fig. 5).\nWe point out that the e\u000bective conductance is normal-\nized with respect to g\"#\ne\u000b(dS= 0), which accounts for the\ncontribution of the normal bound{state resonances. The\nweak barrier limit ( Z < 1), on the other hand, can ef-\nfectively be described within the zero{barrier limit by\nan increased dNand \u0001 ex(due to the multiple ordinary\nre\rections) [see Fig. 5 (inset)].\nConclusion and discussion .|We have shown that su-\nperconductivity can greatly a\u000bect spin pumping due to\nthe formation of the resonant ABSs, which result in\nhighly transmitting spin transport channels when dS\u0018\nFIG. 5. The normalized e\u000bective conductance is shown for\nvarious barrier potential strengths Z, which are plotted by\nincreasingZin increments of 0 :5 from 0 to 2 :5. The enhance-\nment feature ceases to exist for strong barriers Z > 1. The\ninset plot shows that the simultaneous increase of dNand\n\u0001exwithZ= 0 can account for the conductance in the weak\nbarrier case Z <1. The red plots are generated by simultane-\nously increasing both the exchange interaction \u0001 ex=EFfrom\n0:1 to 0:6 in increments of 0 :1 anddN=\u00180from 3 to 4 :5 in\nincrements of 0 :3. In the above plots, we have set dN=\u00180= 3,\n\u0001ex=EF= 0:1.\n\u00180. This can be manifested experimentally by an in-\ncrease in Gilbert damping of the FI dynamics. The\nGilbert damping enhancement in an NbN jGdN struc-\nture has been observed to peak for subgab temperatures\n[5], where NbN is an s{wave superconductor with coher-\nence length of\u00185 nm that is adjacent to a ferromag-\nnetic{insulator \flm, GdN. The enhancement takes place\nfordS= 10 nm and it is suppressed for dS= 2 nm. We\nbelieve that this is in good agreement with our results,\ne.g., Fig. 4. On the other hand, unfolding N rjSjNjFI e\u000bec-\ntively maps our setup to an N rjSjFjSjNrstructure, where\nF stands for a normal metal with a ferromagnetic order.\nThe hybrid structure N rjNbjNi80Fe20jNbjNrstudied in\nRef. [4] shows a signi\fcant spin{pumping enhancement\nonly when the Nb thickness is roughly equal to its coher-\nence length ( dS\u0019\u00180= 30 nm). In order to reveal a large\nenhancement, they have utilized a range of spin{sink ma-\nterials N rwith spin{orbit interaction, such as Pt, W, or\nTa. In Ref. [17] the same hybrid structure of Ref. [4] has\nbeen used except for a particularly magnetized spin sink\nPt/Co/Pt. It is shown that the spin{pumping e\u000eciency\nacross Nb is tunable by controlling the magnetization di-\nrection of Co.\nFurthermore, hybrid Josephson junctions realizing\nABSs with near unity transmission probability for charge\ntransport have been proposed to coherently manipu-\nlate quantum{information devices such as Andreev{level\nqubits [29, 30]. From this standpoint, unfolding our setup\nrealizes a magnetically active Josephson junction [8] with\nresonant transport channels, which in turn can provide a\nspintronic paradigm for a coherent manipulation of quan-5\ntum{information devices involving ABSs. Note added :\nRecently, we became aware of an interesting and closely\nrelated work by Silaev [31].\nIt is a pleasure to acknowledge discussions with C. Ci-\nccarelli and W. A. Robinson who drew our attention to\nthis problem. MTA wishes to thank S. Tanhayi Ahari for\nuseful comments on the paper. 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Cambridge: Cambridge University\nPress (2009)\n[21] We consider an energy gap of 1 :1EFfor up-spin and\n1:1EF+ 2\u0001 exfor down-spin electrons. The Fermi en-\nergyEFis assumed to be identical for the di\u000berent lay-\ners,EF= 5 eV. Furthermore, we have assumed Nb as\nthe S with Tc= 9 K,\u00180= 30 nm, and \u0001 0= 3 meV.\nWe have adopted the following empirical parametrization\nfor a temperature-dependent superconductor gap for Nb,\n\u0001(T) = \u0001 0(1\u0000T=Tc )1:94(1 + 2:17T=Tc ), that is taken\nfrom R. C. Dougherty, J. D. Kimel. Superconductivity Re-\nvisited. (CRC Press, Boca Raton, FL, 2012)\n[22] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000)\n[23] F. H ubler, M. J. Wolf, T. Scherer, D. Wang, D. Beck-\nmann, and H. v. L ohneysen, Phys. Rev. Lett. 109, 087004\n(2012)\n[24] H. van Houten, C. W. J. Beenakker, Physica B: Con-\ndensed Matter 175, Issues 1{3, (1991)\n[25] N. Ismail, C. C. Kores, D. Geskus, and M. Pollnau, Op-\ntics Express Vol. 24, Issue 15, (2016)\n[26] W. J. Tomasch, Phys. Rev. Lett. 16, 16 (1966).\n[27] K. Xia, P. J. Kelly, G. E. W. Bauer, and I. Turek, Phys.\nRev. Lett. 89, 166603 (2002)\n[28] G. E. Blonder, M. Tinkham, and T. M. Klapwijk. Phys.\nRev. B 25, 4515 (1982)\n[29] F. Nichele, E. Portoles, A. Fornieri, A. M. Whiticar, A.\nC. C. Drachmann, S. Gronin, T. Wang, G. C. Gardner, C.\nThomas, A. T. Hatke, M. J. Manfra, and C. M. Marcus,\nPhys. Rev. Lett. 124, 226801 (2020)\n[30] J. Michelsen, V. S. Shumeiko, and G. Wendin Phys. Rev.\nB77, 184506 (2008); C. Janvier, L. Tosi, L. Bretheau, C.\nO. Girit, M. Stern, P. Bertet, P. Joyez, D. Vion, D. Esteve,\nM. F. Go\u000bman, H. Pothier, C. Urbina, Science 349, Issue\n6253, (2015)\n[31] M. A. Silaev, Phys. Rev. B. 102, 180502(R) (2020)1\nSupplementary Material for\nSuperconductivity-enhanced spin pumping: Role of Andreev resonances\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nIn this supplementary material, we discuss a QP scattering formalism that takes into account the dynamic mag-\nnetization of the FI and the s-wave order of the S in an N rjSjNjFI hybrid structure, see Fig. S1. We will derive\nanalytical expressions for the scattering amplitudes, ABS modi\fcation under an interfacial disorder, and elaborate on\nthe dynamic generation of the triplet Cooper pairs in our setup.\nSpin-dependnet scattering .|In the context of QP scattering in hybrid structures generic boundary conditions perti-\nnent to interface scattering and disorder can be taken into account by insertion of a perfect spacer N between adjacent\nmaterials [1, 2]. As such, the full scattering process can be broken down into studying N rjSjN, and NjFI structures\nseparately. We bear in mind that propagation through the N layer mixes neither particle-hole nor spin degrees of\nfreedom. For the N rjSjN structure, following BTK [3], we assume a uniform superconducting pair-potential \u0001 inside\nthe S layer that vanishes in the N layer. This way the scattering process takes place only at the interfaces, which\nessentially contains QPs propagating into the S and Andreev retrore\rection. In a more realistic (dirty) N jS interface,\nhowever, there is a possibility of a specular re\rection for electrons and holes. In order to model the partially re\rective\ninterface, we can consider a barrier potential in the form of V=Z1~vF\u000e(x) andZ2~vF\u000e(x\u0000dS). A \fniteZ1, leads to\nquantum interference with the Fermi wavelength and does not a\u000bect ABS. In the following, we set Z1= 0 and focus\non the e\u000bect of a nonzero Z2\u0011Z.\nInitially, we study point contact scattering in a 1D wire. Physically, this can be justi\fed by imposing a constriction,\nfor which the transverse thickness of the N layer is on the order of the Fermi wavelength [1]. Later on, we will relax\nthis condition by considering a 3D geometry for which the incident QPs can have a transverse momentum.\nWe begin with the BdG formalism that accounts for symmetries of the superconducting part for the electron and\nhole wave functions,\nHBdG\t =\"\t;withHBdG=\u0012^H0^\u0001\n^\u0001y\u0000^H\u0003\n0\u0013\n; (S1)\nwhere ^H0=\u0000\np2=2m+V\u0000EF\u0001\nis the single-electron Hamiltonian matrix and ^\u0001 = \u0001i\u001by. Here, we assume a real-\nvalued pairing potential \u0001. The Pauli matrix \u001byis acting on the spin space f\";#gof conduction electrons and holes.\nAbove eigenvalue equation admits the following basic electron and hole QP modes with spin \u001b:\n\t\u0006\nn;e\u001b=\u0012\n\u001f\u001b\nA(\")(\u0000i\u001by)\u001f\u001b\u0013\ne\u0006iqexeikk\u0001rk;\n\t\u0006\nn;h\u001b=\u0012A\u0003(\u0000\")(\u0000i\u001by)\u001f\u001b\n\u001f\u001b\u0013\ne\u0006iqhxeikk\u0001rk; (S2)\nwhere\u001f\"=\u00001\n0\u0001\n,\u001f#= (\u0000i\u001by)\u001f\", are two dimensional spinors, and we have\n~2\n2mq2\ne;h=EF\u0000En+\u0010e;h\n;\n\u0011(\nip\n\u00012\u0000\"2; ifj\"j<\u0001\n\"p\n1\u0000\u00012=\"2;ifj\"j>\u0001; A (\") =(\ne\u0000iarccos (\"=\u0001);j\"j<\u0001\n\"\u0000\"p\n1\u0000\u00012=\"2\n\u0001;j\"j>\u0001; (S3)\nwhereEn\u0011~2k2\nk=2m, and\u0010e;h=\u00061. In the normal layer, the wave number is given by ke;h=qe;h(\u0001!0). The\nconservation of the QP spin leads to a block diagonal full scattering matrix in the spin space. Whence, we can work\nwith the following 2 dimensional eigen vectors,\n\t\u0006\nSne=\u00121\nA(\")\u0013\ne\u0006iqexeikk\u0001rk;\n\t\u0006\nSnh=\u0012A(\")\n1\u0013\ne\u0006iqhxeikk\u0001rk: (S4)\nIt is clear that outside of the S, where \u0001 !0, we getA(\")!0, leading to normal region's basis elements,\n\t\u0006\nNne=\u00121\n0\u0013\ne\u0006ikex\bn(y;z);\n\t\u0006\nNnh=\u00120\n1\u0013\ne\u0006ikhx\bn(y;z); (S5)2\n𝑐\"#\n𝑐\"$\n𝑐%$\n𝑐%#𝑐\"#\n𝑐\"$\n𝑐%$\n𝑐%#\n𝑍' 𝑍(Superconductivity-enhanced spin pumping: The role of Andreev bound-state\nresonances\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe study spin pumping in a hybrid superconductor |ferromagnetic-insulator structure. We show\nthat, at low bias and below Tc, the transfer of spin angular momentum pumped by an externally\ndriven ferromagnetic insulator is a ↵ected by the formation of Andreev bound states. These bound\nstates capture the essential physics of condensate-facilitated spin flow. For finite thicknesses of the\nsuperconducting layer, comparable to the coherence length, Andreev bound states are resonant and\ndominate the subgap spin current, rendering a significant enhancement in spin current over a wide\nrange of parameters. The resonant enhancement of the subgap transport channels establishes a\nprototype Fabry-P´ erot resonator for spin pumping.\nCombining an s-wave superconducting order, favoring\nelectrons to form a singlet state, with a ferromagnetic\norder, favoring spin alignment, leads to a powerful en-\nhancement or reduction of angular momentum transfer\nin spintronic devices [ 1]. This happens through various\nprocesses such as the formation of Andreev bound states\n(ABSs) [ 2–9], the generation of spin-triplet pairing [ 10–\n15], and crossed Andreev reflections (CARs) [ 16]. In this\nLetter, we present an experimentally feasible structure\nmanifesting spin current enhancement due to resonant\nABSs. We argue that the ABS resonances studied here\nmay be responsible for the magnetic damping increase\nin a superconducting hybrid multilayer reported in Ref.\n[17]\nTime-dependent ferromagnetic magnetization, when it\nis driven by an external rf field, generates spin pumping–\na spin angular momentum flow–into adjacent materials\nthat dissipates the energy of the ferromagnet [ 18]. Insert-\ning a thick superconductor (S) in between, however, can\nsuppress spin pumping due to the superconducting gap\nopening for quasiparticles (QPs). Consequently, in spin\ntransport studies with a bulk S, it is common to have\nbias voltages to inject QPs with energies above the gap\n[19]. Generically, for a thick superconducting layer, it is\nexpected that at low temperatures spin current relies on\na spin-polarized (triplet Cooper pairing) supercurrent in-\nduced at highly spin-active regions or complex magnetic\nmultilayers [ 20]. For a thin S layer (thickness ⇠coherence\nlength), on the other hand, subgap currents can be car-\nried via evanescent QPs tunneling through the S layer.\nIn this work, we study a hybrid structure consisting of\na thin S layer in proximity to a driven ferromagnetic in-\nsulator (FI) separated by a normal metal region (N). To\ncollect the spin current we have placed a spin reservoir\nNr, comprising an N r|S|N|FI structure, see Fig. 1. In or-\nder to establish the pumped spin current in the presence\nof superconducting and dynamic ferromagnetic orders,\nwe solve a time-dependent scattering problem [ 21,22],\nwhich accounts for the relevant processes such as AR,\nCAR, ABS, and dynamic triplet-paring generation. For\nsubgap energies, the full scattering matrix develops peakS!(#)\nFI%& xz\ny\nN' NdS \t\t dN \t\t\nFIG. 1. Schematic sketch for an N r|S|N|FI hybrid structure.\nThe FI region with precessing magnetization m(t) injects spin\nangular momentum that is carried by the QPs into the reser-\nvoir N r(spin pumping). To investigate the e ↵ect of supercon-\nductivity on the magnetically pumped spin flow, we utilize\nthe scattering formalism for incident and reflected QPs. For\nfinite thicknesses of the S and N layers ( dS,dN⇠coherence\nlength) the resonant ABSs provide highly transmitting trans-\nport channels for spin pumping. Here, for example, we show\nfour resonant channels by two positive-energy ABSs. The\nenergies are measured with respect to the Fermi energy EF\ndepicted in the middle.\nstructures marking resonant bound states (ABSs), which\nengender highly transmitting subgap transport channels\nfor the spin flow. Due to a resonant enhancement of spin\nflow for certain energies, as we will discuss, our setup can\nbe considered a prototype Fabry-P´ erot resonator for the\nspin pumping.\nThe spin pumping generated by the variations in the\nmagnetization direction m(t), assuming no voltage bias,\nis given by Is(t)=1\n4⇡g\"#\ne↵m⇥@tm[13,21,23], where the\ne↵ective spin-mixing conductance is defined as follows,\ng\"#\ne↵⌘Z1\n\u00001d\"@\"f(\")R e⇥\ng\"#⇤\n. (1)\nThe quantity x, y, z Re⇥\ng\"#⇤\nrefers to the real part of the\nspin-mixing conductance g\"#=T r⇥\n1\u0000r\"\neer⇤#\nee+r#\"\nher⇤\"#\nhe⇤\n,\nwhere r\u0000\neeand r\u0000\u0000\u0000\nhe(\u00002{\",#}) represent the electron-Superconductivity-enhanced spin pumping: The role of Andreev bound-state\nresonances\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe study spin pumping in a hybrid superconductor |ferromagnetic-insulator structure. We show\nthat, at low bias and below Tc, the transfer of spin angular momentum pumped by an externally\ndriven ferromagnetic insulator is a ↵ected by the formation of Andreev bound states. These bound\nstates capture the essential physics of condensate-facilitated spin flow. For finite thicknesses of the\nsuperconducting layer, comparable to the coherence length, Andreev bound states are resonant and\ndominate the subgap spin current, rendering a significant enhancement in spin current over a wide\nrange of parameters. The resonant enhancement of the subgap transport channels establishes a\nprototype Fabry-P´ erot resonator for spin pumping.\nCombining an s-wave superconducting order, favoring\nelectrons to form a singlet state, with a ferromagnetic\norder, favoring spin alignment, leads to a powerful en-\nhancement or reduction of angular momentum transfer\nin spintronic devices [ 1]. This happens through various\nprocesses such as the formation of Andreev bound states\n(ABSs) [ 2–9], the generation of spin-triplet pairing [ 10–\n15], and crossed Andreev reflections (CARs) [ 16]. In this\nLetter, we present an experimentally feasible structure\nmanifesting spin current enhancement due to resonant\nABSs. We argue that the ABS resonances studied here\nmay be responsible for the magnetic damping increase\nin a superconducting hybrid multilayer reported in Ref.\n[17]\nTime-dependent ferromagnetic magnetization, when it\nis driven by an external rf field, generates spin pumping–\na spin angular momentum flow–into adjacent materials\nthat dissipates the energy of the ferromagnet [ 18]. Insert-\ning a thick superconductor (S) in between, however, can\nsuppress spin pumping due to the superconducting gap\nopening for quasiparticles (QPs). Consequently, in spin\ntransport studies with a bulk S, it is common to have\nbias voltages to inject QPs with energies above the gap\n[19]. Generically, for a thick superconducting layer, it is\nexpected that at low temperatures spin current relies on\na spin-polarized (triplet Cooper pairing) supercurrent in-\nduced at highly spin-active regions or complex magnetic\nmultilayers [ 20]. For a thin S layer (thickness ⇠coherence\nlength), on the other hand, subgap currents can be car-\nried via evanescent QPs tunneling through the S layer.\nIn this work, we study a hybrid structure consisting of\na thin S layer in proximity to a driven ferromagnetic in-\nsulator (FI) separated by a normal metal region (N). To\ncollect the spin current we have placed a spin reservoir\nNr, comprising an N r|S|N|FI structure, see Fig. 1. In or-\nder to establish the pumped spin current in the presence\nof superconducting and dynamic ferromagnetic orders,\nwe solve a time-dependent scattering problem [ 21,22],\nwhich accounts for the relevant processes such as AR,\nCAR, ABS, and dynamic triplet-paring generation. For\nsubgap energies, the full scattering matrix develops peakS!(#)\nFI%& xz\ny\nN' NdS \t\t dN \t\t\nFIG. 1. Schematic sketch for an N r|S|N|FI hybrid structure.\nThe FI region with precessing magnetization m(t) injects spin\nangular momentum that is carried by the QPs into the reser-\nvoir N r(spin pumping). To investigate the e ↵ect of supercon-\nductivity on the magnetically pumped spin flow, we utilize\nthe scattering formalism for incident and reflected QPs. For\nfinite thicknesses of the S and N layers ( dS,dN⇠coherence\nlength) the resonant ABSs provide highly transmitting trans-\nport channels for spin pumping. Here, for example, we show\nfour resonant channels by two positive-energy ABSs. The\nenergies are measured with respect to the Fermi energy EF\ndepicted in the middle.\nstructures marking resonant bound states (ABSs), which\nengender highly transmitting subgap transport channels\nfor the spin flow. Due to a resonant enhancement of spin\nflow for certain energies, as we will discuss, our setup can\nbe considered a prototype Fabry-P´ erot resonator for the\nspin pumping.\nThe spin pumping generated by the variations in the\nmagnetization direction m(t), assuming no voltage bias,\nis given by Is(t)=1\n4⇡g\"#\ne↵m⇥@tm[13,21,23], where the\ne↵ective spin-mixing conductance is defined as follows,\ng\"#\ne↵⌘Z1\n\u00001d\"@\"f(\")R e⇥g\"#⇤. (1)\nThe quantity x, y, z Re⇥g\"#⇤refers to the real part of the\nspin-mixing conductance g\"#=T r⇥1\u0000r\"\neer⇤#\nee+r#\"\nher⇤\"#\nhe⇤,\nwhere r\u0000\neeand r\u0000\u0000\u0000\nhe (\u00002{\",#}) represent the electron-Superconductivity-enhanced spin pumping: The role of Andreev bound-state\nresonances\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe study spin pumping in a hybrid superconductor |ferromagnetic-insulator structure. We show\nthat, at low bias and below Tc, the transfer of spin angular momentum pumped by an externally\ndriven ferromagnetic insulator is a ↵ected by the formation of Andreev bound states. These bound\nstates capture the essential physics of condensate-facilitated spin flow. For finite thicknesses of the\nsuperconducting layer, comparable to the coherence length, Andreev bound states are resonant and\ndominate the subgap spin current, rendering a significant enhancement in spin current over a wide\nrange of parameters. The resonant enhancement of the subgap transport channels establishes a\nprototype Fabry-P´ erot resonator for spin pumping.\nCombining an s-wave superconducting order, favoring\nelectrons to form a singlet state, with a ferromagnetic\norder, favoring spin alignment, leads to a powerful en-\nhancement or reduction of angular momentum transfer\nin spintronic devices [ 1]. This happens through various\nprocesses such as the formation of Andreev bound states\n(ABSs) [ 2–9], the generation of spin-triplet pairing [ 10–\n15], and crossed Andreev reflections (CARs) [ 16]. In this\nLetter, we present an experimentally feasible structure\nmanifesting spin current enhancement due to resonant\nABSs. We argue that the ABS resonances studied here\nmay be responsible for the magnetic damping increase\nin a superconducting hybrid multilayer reported in Ref.\n[17]\nTime-dependent ferromagnetic magnetization, when it\nis driven by an external rf field, generates spin pumping–\na spin angular momentum flow–into adjacent materials\nthat dissipates the energy of the ferromagnet [ 18]. Insert-\ning a thick superconductor (S) in between, however, can\nsuppress spin pumping due to the superconducting gap\nopening for quasiparticles (QPs). Consequently, in spin\ntransport studies with a bulk S, it is common to have\nbias voltages to inject QPs with energies above the gap\n[19]. Generically, for a thick superconducting layer, it is\nexpected that at low temperatures spin current relies on\na spin-polarized (triplet Cooper pairing) supercurrent in-\nduced at highly spin-active regions or complex magnetic\nmultilayers [ 20]. For a thin S layer (thickness ⇠coherence\nlength), on the other hand, subgap currents can be car-\nried via evanescent QPs tunneling through the S layer.\nIn this work, we study a hybrid structure consisting of\na thin S layer in proximity to a driven ferromagnetic in-\nsulator (FI) separated by a normal metal region (N). To\ncollect the spin current we have placed a spin reservoir\nNr, comprising an N r|S|N|FI structure, see Fig. 1. In or-\nder to establish the pumped spin current in the presence\nof superconducting and dynamic ferromagnetic orders,\nwe solve a time-dependent scattering problem [ 21,22],\nwhich accounts for the relevant processes such as AR,\nCAR, ABS, and dynamic triplet-paring generation. For\nsubgap energies, the full scattering matrix develops peakS!(#)\nFI%& xz\ny\nN' NdS \t\t dN \t\t\nFIG. 1. Schematic sketch for an N r|S|N|FI hybrid structure.\nThe FI region with precessing magnetization m(t) injects spin\nangular momentum that is carried by the QPs into the reser-\nvoir N r(spin pumping). To investigate the e ↵ect of supercon-\nductivity on the magnetically pumped spin flow, we utilize\nthe scattering formalism for incident and reflected QPs. For\nfinite thicknesses of the S and N layers ( dS,dN⇠coherence\nlength) the resonant ABSs provide highly transmitting trans-\nport channels for spin pumping. Here, for example, we show\nfour resonant channels by two positive-energy ABSs. The\nenergies are measured with respect to the Fermi energy EF\ndepicted in the middle.\nstructures marking resonant bound states (ABSs), which\nengender highly transmitting subgap transport channels\nfor the spin flow. Due to a resonant enhancement of spin\nflow for certain energies, as we will discuss, our setup can\nbe considered a prototype Fabry-P´ erot resonator for the\nspin pumping.\nThe spin pumping generated by the variations in the\nmagnetization direction m(t), assuming no voltage bias,\nis given by Is(t)=1\n4⇡g\"#\ne↵m⇥@tm[13,21,23], where the\ne↵ective spin-mixing conductance is defined as follows,\ng\"#\ne↵⌘Z1\n\u00001d\"@\"f(\")R e⇥\ng\"#⇤\n. (1)\nThe quantity Re⇥\ng\"#⇤\nrefers to the real part of the spin-\nmixing conductance g\"#=T r⇥\n1\u0000r\"\neer⇤#\nee+r#\"\nher⇤\"#\nhe⇤\n,w h e r e\nr\u0000\neeand r\u0000\u0000\u0000\nhe(\u00002{\",#}) represent the electron-to-Superconductivity-enhanced spin pumping: The role of Andreev bound-state\nresonances\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe study spin pumping in a hybrid superconductor |ferromagnetic-insulator structure. We show\nthat, at low bias and below Tc, the transfer of spin angular momentum pumped by an externally\ndriven ferromagnetic insulator is a ↵ected by the formation of Andreev bound states. These bound\nstates capture the essential physics of condensate-facilitated spin flow. For finite thicknesses of the\nsuperconducting layer, comparable to the coherence length, Andreev bound states are resonant and\ndominate the subgap spin current, rendering a significant enhancement in spin current over a wide\nrange of parameters. The resonant enhancement of the subgap transport channels establishes a\nprototype Fabry-P´ erot resonator for spin pumping.\nCombining an s-wave superconducting order, favoring\nelectrons to form a singlet state, with a ferromagnetic\norder, favoring spin alignment, leads to a powerful en-\nhancement or reduction of angular momentum transfer\nin spintronic devices [ 1]. This happens through various\nprocesses such as the formation of Andreev bound states\n(ABSs) [ 2–9], the generation of spin-triplet pairing [ 10–\n15], and crossed Andreev reflections (CARs) [ 16]. In this\nLetter, we present an experimentally feasible structure\nmanifesting spin current enhancement due to resonant\nABSs. We argue that the ABS resonances studied here\nmay be responsible for the magnetic damping increase\nin a superconducting hybrid multilayer reported in Ref.\n[17]\nTime-dependent ferromagnetic magnetization, when it\nis driven by an external rf field, generates spin pumping–\na spin angular momentum flow–into adjacent materials\nthat dissipates the energy of the ferromagnet [ 18]. Insert-\ning a thick superconductor (S) in between, however, can\nsuppress spin pumping due to the superconducting gap\nopening for quasiparticles (QPs). Consequently, in spin\ntransport studies with a bulk S, it is common to have\nbias voltages to inject QPs with energies above the gap\n[19]. Generically, for a thick superconducting layer, it is\nexpected that at low temperatures spin current relies on\na spin-polarized (triplet Cooper pairing) supercurrent in-\nduced at highly spin-active regions or complex magnetic\nmultilayers [ 20]. For a thin S layer (thickness ⇠coherence\nlength), on the other hand, subgap currents can be car-\nried via evanescent QPs tunneling through the S layer.\nIn this work, we study a hybrid structure consisting of\na thin S layer in proximity to a driven ferromagnetic in-\nsulator (FI) separated by a normal metal region (N). To\ncollect the spin current we have placed a spin reservoir\nNr, comprising an N r|S|N|FI structure, see Fig. 1. In or-\nder to establish the pumped spin current in the presence\nof superconducting and dynamic ferromagnetic orders,\nwe solve a time-dependent scattering problem [ 21,22],\nwhich accounts for the relevant processes such as AR,\nCAR, ABS, and dynamic triplet-paring generation. For\nsubgap energies, the full scattering matrix develops peakS!(#)\nFI%& xz\ny\nN' NdS \t\t dN \t\t\nFIG. 1. Schematic sketch for an N r|S|N|FI hybrid structure.\nThe FI region with precessing magnetization m(t) injects spin\nangular momentum that is carried by the QPs into the reser-\nvoir N r(spin pumping). To investigate the e ↵ect of supercon-\nductivity on the magnetically pumped spin flow, we utilize\nthe scattering formalism for incident and reflected QPs. For\nfinite thicknesses of the S and N layers ( dS,dN⇠coherence\nlength) the resonant ABSs provide highly transmitting trans-\nport channels for spin pumping. Here, for example, we show\nfour resonant channels by two positive-energy ABSs. The\nenergies are measured with respect to the Fermi energy EF\ndepicted in the middle.\nstructures marking resonant bound states (ABSs), which\nengender highly transmitting subgap transport channels\nfor the spin flow. Due to a resonant enhancement of spin\nflow for certain energies, as we will discuss, our setup can\nbe considered a prototype Fabry-P´ erot resonator for the\nspin pumping.\nThe spin pumping generated by the variations in the\nmagnetization direction m(t), assuming no voltage bias,\nis given by Is(t)=1\n4⇡g\"#\ne↵m⇥@tm[13,21,23], where the\ne↵ective spin-mixing conductance is defined as follows,\ng\"#\ne↵⌘Z1\n\u00001d\"@\"f(\")R e⇥g\"#⇤. (1)\nThe quantity Re⇥g\"#⇤refers to the real part of the spin-\nmixing conductance g\"#=T r⇥1\u0000r\"\neer⇤#\nee+r#\"\nher⇤\"#\nhe⇤,w h e r e\nr\u0000\neeand r\u0000\u0000\u0000\nhe (\u00002{\",#}) represent the electron-to-Superconductivity-enhanced spin pumping: The role of Andreev bound-state\nresonances\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe study spin pumping in a hybrid superconductor |ferromagnetic-insulator structure. We show\nthat, at low bias and below Tc, the transfer of spin angular momentum pumped by an externally\ndriven ferromagnetic insulator is a ↵ected by the formation of Andreev bound states. These bound\nstates capture the essential physics of condensate-facilitated spin flow. For finite thicknesses of the\nsuperconducting layer, comparable to the coherence length, Andreev bound states are resonant and\ndominate the subgap spin current, rendering a significant enhancement in spin current over a wide\nrange of parameters. The resonant enhancement of the subgap transport channels establishes a\nprototype Fabry-P´ erot resonator for spin pumping.\nCombining an s-wave superconducting order, favoring\nelectrons to form a singlet state, with a ferromagnetic\norder, favoring spin alignment, leads to a powerful en-\nhancement or reduction of angular momentum transfer\nin spintronic devices [ 1]. This happens through various\nprocesses such as the formation of Andreev bound states\n(ABSs) [ 2–9], the generation of spin-triplet pairing [ 10–\n15], and crossed Andreev reflections (CARs) [ 16]. In this\nLetter, we present an experimentally feasible structure\nmanifesting spin current enhancement due to resonant\nABSs. We argue that the ABS resonances studied here\nmay be responsible for the magnetic damping increase\nin a superconducting hybrid multilayer reported in Ref.\n[17]\nTime-dependent ferromagnetic magnetization, when it\nis driven by an external rf field, generates spin pumping–\na spin angular momentum flow–into adjacent materials\nthat dissipates the energy of the ferromagnet [ 18]. Insert-\ning a thick superconductor (S) in between, however, can\nsuppress spin pumping due to the superconducting gap\nopening for quasiparticles (QPs). Consequently, in spin\ntransport studies with a bulk S, it is common to have\nbias voltages to inject QPs with energies above the gap\n[19]. Generically, for a thick superconducting layer, it is\nexpected that at low temperatures spin current relies on\na spin-polarized (triplet Cooper pairing) supercurrent in-\nduced at highly spin-active regions or complex magnetic\nmultilayers [ 20]. For a thin S layer (thickness ⇠coherence\nlength), on the other hand, subgap currents can be car-\nried via evanescent QPs tunneling through the S layer.\nIn this work, we study a hybrid structure consisting of\na thin S layer in proximity to a driven ferromagnetic in-\nsulator (FI) separated by a normal metal region (N). To\ncollect the spin current we have placed a spin reservoir\nNr, comprising an N r|S|N|FI structure, see Fig. 1. In or-\nder to establish the pumped spin current in the presence\nof superconducting and dynamic ferromagnetic orders,\nwe solve a time-dependent scattering problem [ 21,22],\nwhich accounts for the relevant processes such as AR,\nCAR, ABS, and dynamic triplet-paring generation. For\nsubgap energies, the full scattering matrix develops peakS!(#)\nFI%& xzy\nN' NdS \t\t dN \t\t\nFIG. 1. Schematic sketch for an N r|S|N|FI hybrid structure.\nThe FI region with precessing magnetization m(t) injects spin\nangular momentum that is carried by the QPs into the reser-\nvoir N r(spin pumping). To investigate the e ↵ect of supercon-\nductivity on the magnetically pumped spin flow, we utilize\nthe scattering formalism for incident and reflected QPs. For\nfinite thicknesses of the S and N layers ( dS,dN⇠coherence\nlength) the resonant ABSs provide highly transmitting trans-\nport channels for spin pumping. Here, for example, we show\nfour resonant channels by two positive-energy ABSs. The\nenergies are measured with respect to the Fermi energy EF\ndepicted in the middle.\nstructures marking resonant bound states (ABSs), which\nengender highly transmitting subgap transport channels\nfor the spin flow. Due to a resonant enhancement of spin\nflow for certain energies, as we will discuss, our setup can\nbe considered a prototype Fabry-P´ erot resonator for the\nspin pumping.\nThe spin pumping generated by the variations in the\nmagnetization direction m(t), assuming no voltage bias,\nis given by Is(t)=1\n4⇡g\"#\ne↵m⇥@tm[13,21,23], where the\ne↵ective spin-mixing conductance is defined as follows,\ng\"#\ne↵⌘Z1\n\u00001d\"@\"f(\")R e⇥g\"#⇤. (1)\nThe quantity Re⇥g\"#⇤refers to the real part of the spin-\nmixing conductance g\"#=T r⇥1\u0000r\"\neer⇤#\nee+r#\"\nher⇤\"#\nhe⇤,w h e r e\nr\u0000\neeand r\u0000\u0000\u0000\nhe (\u00002{\",#}) represent the electron-to-Superconductivity-enhanced spin pumping: The role of Andreev bound-state\nresonances\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe study spin pumping in a hybrid superconductor |ferromagnetic-insulator structure. We show\nthat, at low bias and below Tc, the transfer of spin angular momentum pumped by an externally\ndriven ferromagnetic insulator is a ↵ected by the formation of Andreev bound states. These bound\nstates capture the essential physics of condensate-facilitated spin flow. For finite thicknesses of the\nsuperconducting layer, comparable to the coherence length, Andreev bound states are resonant and\ndominate the subgap spin current, rendering a significant enhancement in spin current over a wide\nrange of parameters. The resonant enhancement of the subgap transport channels establishes a\nprototype Fabry-P´ erot resonator for spin pumping.\nCombining an s-wave superconducting order, favoring\nelectrons to form a singlet state, with a ferromagnetic\norder, favoring spin alignment, leads to a powerful en-\nhancement or reduction of angular momentum transfer\nin spintronic devices [ 1]. This happens through various\nprocesses such as the formation of Andreev bound states\n(ABSs) [ 2–9], the generation of spin-triplet pairing [ 10–\n15], and crossed Andreev reflections (CARs) [ 16]. In this\nLetter, we present an experimentally feasible structure\nmanifesting spin current enhancement due to resonant\nABSs. We argue that the ABS resonances studied here\nmay be responsible for the magnetic damping increase\nin a superconducting hybrid multilayer reported in Ref.\n[17]\nTime-dependent ferromagnetic magnetization, when it\nis driven by an external rf field, generates spin pumping–\na spin angular momentum flow–into adjacent materials\nthat dissipates the energy of the ferromagnet [ 18]. Insert-\ning a thick superconductor (S) in between, however, can\nsuppress spin pumping due to the superconducting gap\nopening for quasiparticles (QPs). Consequently, in spin\ntransport studies with a bulk S, it is common to have\nbias voltages to inject QPs with energies above the gap\n[19]. Generically, for a thick superconducting layer, it is\nexpected that at low temperatures spin current relies on\na spin-polarized (triplet Cooper pairing) supercurrent in-\nduced at highly spin-active regions or complex magnetic\nmultilayers [ 20]. For a thin S layer (thickness ⇠coherence\nlength), on the other hand, subgap currents can be car-\nried via evanescent QPs tunneling through the S layer.\nIn this work, we study a hybrid structure consisting of\na thin S layer in proximity to a driven ferromagnetic in-\nsulator (FI) separated by a normal metal region (N). To\ncollect the spin current we have placed a spin reservoir\nNr, comprising an N r|S|N|FI structure, see Fig. 1. In or-\nder to establish the pumped spin current in the presence\nof superconducting and dynamic ferromagnetic orders,\nwe solve a time-dependent scattering problem [ 21,22],\nwhich accounts for the relevant processes such as AR,\nCAR, ABS, and dynamic triplet-paring generation. For\nsubgap energies, the full scattering matrix develops peakS!(#)\nFI%& xz\ny\nN' NdS \t\t dN \t\t\nFIG. 1. Schematic sketch for an N r|S|N|FI hybrid structure.\nThe FI region with precessing magnetization m(t) injects spin\nangular momentum that is carried by the QPs into the reser-\nvoir N r(spin pumping). To investigate the e ↵ect of supercon-\nductivity on the magnetically pumped spin flow, we utilize\nthe scattering formalism for incident and reflected QPs. For\nfinite thicknesses of the S and N layers ( dS,dN⇠coherence\nlength) the resonant ABSs provide highly transmitting trans-\nport channels for spin pumping. Here, for example, we show\nfour resonant channels by two positive-energy ABSs. The\nenergies are measured with respect to the Fermi energy EF\ndepicted in the middle.\nstructures marking resonant bound states (ABSs), which\nengender highly transmitting subgap transport channels\nfor the spin flow. Due to a resonant enhancement of spin\nflow for certain energies, as we will discuss, our setup can\nbe considered a prototype Fabry-P´ erot resonator for the\nspin pumping.\nThe spin pumping generated by the variations in the\nmagnetization direction m(t), assuming no voltage bias,\nis given by Is(t)=1\n4⇡g\"#\ne↵m⇥@tm[13,21,23], where the\ne↵ective spin-mixing conductance is defined as follows,\ng\"#\ne↵⌘Z1\n\u00001d\"@\"f(\")R e⇥g\"#⇤. (1)\nThe quantity Re⇥g\"#⇤refers to the real part of the spin-\nmixing conductance g\"#=T r⇥1\u0000r\"\neer⇤#\nee+r#\"\nher⇤\"#\nhe⇤,w h e r e\nr\u0000\neeand r\u0000\u0000\u0000\nhe (\u00002{\",#}) represent the electron-to-Superconductivity-enhanced spin pumping: The role of Andreev bound-state\nresonances\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe study spin pumping in a hybrid superconductor |ferromagnetic-insulator structure. We show\nthat, at low bias and below Tc, the transfer of spin angular momentum pumped by an externally\ndriven ferromagnetic insulator is a ↵ected by the formation of Andreev bound states. These bound\nstates capture the essential physics of condensate-facilitated spin flow. For finite thicknesses of the\nsuperconducting layer, comparable to the coherence length, Andreev bound states are resonant and\ndominate the subgap spin current, rendering a significant enhancement in spin current over a wide\nrange of parameters. The resonant enhancement of the subgap transport channels establishes a\nprototype Fabry-P´ erot resonator for spin pumping.\nCombining an s-wave superconducting order, favoring\nelectrons to form a singlet state, with a ferromagnetic\norder, favoring spin alignment, leads to a powerful en-\nhancement or reduction of angular momentum transfer\nin spintronic devices [ 1]. This happens through various\nprocesses such as the formation of Andreev bound states\n(ABSs) [ 2–9], the generation of spin-triplet pairing [ 10–\n15], and crossed Andreev reflections (CARs) [ 16]. In this\nLetter, we present an experimentally feasible structure\nmanifesting spin current enhancement due to resonant\nABSs. We argue that the ABS resonances studied here\nmay be responsible for the magnetic damping increase\nin a superconducting hybrid multilayer reported in Ref.\n[17]\nTime-dependent ferromagnetic magnetization, when it\nis driven by an external rf field, generates spin pumping–\na spin angular momentum flow–into adjacent materials\nthat dissipates the energy of the ferromagnet [ 18]. Insert-\ning a thick superconductor (S) in between, however, can\nsuppress spin pumping due to the superconducting gap\nopening for quasiparticles (QPs). Consequently, in spin\ntransport studies with a bulk S, it is common to have\nbias voltages to inject QPs with energies above the gap\n[19]. Generically, for a thick superconducting layer, it is\nexpected that at low temperatures spin current relies on\na spin-polarized (triplet Cooper pairing) supercurrent in-\nduced at highly spin-active regions or complex magnetic\nmultilayers [ 20]. For a thin S layer (thickness ⇠coherence\nlength), on the other hand, subgap currents can be car-\nried via evanescent QPs tunneling through the S layer.\nIn this work, we study a hybrid structure consisting of\na thin S layer in proximity to a driven ferromagnetic in-\nsulator (FI) separated by a normal metal region (N). To\ncollect the spin current we have placed a spin reservoir\nNr, comprising an N r|S|N|FI structure, see Fig. 1. In or-\nder to establish the pumped spin current in the presence\nof superconducting and dynamic ferromagnetic orders,\nwe solve a time-dependent scattering problem [ 21,22],\nwhich accounts for the relevant processes such as AR,\nCAR, ABS, and dynamic triplet-paring generation. For\nsubgap energies, the full scattering matrix develops peakS!(#)\nFI%& xz\ny\nN' NdS \t\t dN \t\t\nFIG. 1. Schematic sketch for an N r|S|N|FI hybrid structure.\nThe FI region with precessing magnetization m(t) injects spin\nangular momentum that is carried by the QPs into the reser-\nvoir N r(spin pumping). To investigate the e ↵ect of supercon-\nductivity on the magnetically pumped spin flow, we utilize\nthe scattering formalism for incident and reflected QPs. For\nfinite thicknesses of the S and N layers ( dS,dN⇠coherence\nlength) the resonant ABSs provide highly transmitting trans-\nport channels for spin pumping. Here, for example, we show\nfour resonant channels by two positive-energy ABSs. The\nenergies are measured with respect to the Fermi energy EF\ndepicted in the middle.\nstructures marking resonant bound states (ABSs), which\nengender highly transmitting subgap transport channels\nfor the spin flow. Due to a resonant enhancement of spin\nflow for certain energies, as we will discuss, our setup can\nbe considered a prototype Fabry-P´ erot resonator for the\nspin pumping.\nThe spin pumping generated by the variations in the\nmagnetization direction m(t), assuming no voltage bias,\nis given by Is(t)=1\n4⇡g\"#\ne↵m⇥@tm[13,21,23], where the\ne↵ective spin-mixing conductance is defined as follows,\ng\"#\ne↵⌘Z1\n\u00001d\"@\"f(\")R e⇥\ng\"#⇤\n. (1)\nThe quantity Re⇥\ng\"#⇤\nrefers to the real part of the spin-\nmixing conductance g\"#=T r⇥\n1\u0000r\"\neer⇤#\nee+r#\"\nher⇤\"#\nhe⇤\n,w h e r e\nr\u0000\neeand r\u0000\u0000\u0000\nhe (\u00002{\",#}) represent the electron-to-The Supplemental Material\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nTHEORY OF A SPIN-DEPENDENT\nSCATTERING WITH SUPERCONDUCTOR\nIn the following, we study a time-dependent scatter-\ning in ballistic regime for a Nr|S|N|FIhybrid structure,\nsee Fig. 1. Our scattering process is carried out by spin-\nful QPs, i.e., electrons and holes, whose amplitudes are\ngiven by the complex numbers c±\ne,h.T h e s u p e r s c r i p t ±\nsign indicates the direction of the group velocity (normal\ncomponent) given by ±⇣e,hˆx, for which we have defined\n⇣e= +1, and ⇣h=\u00001. In the absence of magnetic in-\nhomogeneity and spin-orbit interaction, the spin is con-\nserved along magnetization m.\nIt is customary that at the interfaces, generic bound-\nary conditions pertinent to interface scattering and dis-\norder can be taken into account by an insertion of a per-\nfect spacer N between adjacent materials[ 1,2]. This way,\nthe full scattering process is broken down into studying\nNr|S|N, and N|FIstructures separately. We bear in\nmind that propagation through the N layer mixes nei-\nther particle-hole nor spin degrees of freedom. For the\nNr|S|Nstructure, following BTK[ 3], we assume a uni-\nform superconducting pair-potential \u0000inside the S layer\nthat vanishes in the N layers. This way the scattering\nprocess takes place only at the interfaces, which essen-\ntially contains QPs propagating into the S and Andreev\nretro-reflections. In a more realistic (dirty) normal metal-\nS interface, however, there is a possibility of a specular\nreflection for electrons and holes. Hence, to model par-\ntially reflective interfaces, we have incorporated barrier\npotentials in the form of Z1~vF\u0000(x), and Z2~vF\u0000(x\u0000dS)\nwhose strength are parametrized by Z1,Z2,(vFis the\nFermi velocity).\nInitially, we assume no barrier potentials and study\na single-channel scattering in a 1D wire. Physically, this\ncan be justified by imposing a constriction, that is, trans-\nverse thickness of the N layers to be on the order of a\nFermi wavelength[ 1]. Later on, we will relax these condi-\ntions by considering a 3D geometry with partially reflec-\ntive 2D interfaces ( Z1,Z26= 0), for which we integrate\nout all transverse modes.\nAs n a p s h o tc o n fi g u r a t i o ns c a t t e r i n gp r o c e s s\nA comparison between a typical FI precession period\n⇠10\u000010s (FMR period) with the time scales of elec-\ntron and hole dynamics, ( dS+dN)/vF⌧10\u000010s, re-\nveals that the magnetization dynamics can be taken as\nan adiabatic evolution during the scattering process. Ini-!\"#\n!\"$\n!%$\n!%#\nSm(t)\ndS \t\txz\n!\"#\n!\"$\n!%$\n!%#ke,h  k∥\nFIN\nNr \t\tdN\t\t() (*\nFIG. 1. A snapshot configuration for magnetization m(t)i n\nanNr|S|N|FIstructure with incoming and outgoing modes\n(black and red, respectively). The propagation direction for\nelectrons and holes with scattering amplitudes c±\ne,c±\nhis given\nby±ˆxfor electrons, and ⌥ˆxfor holes. The shaded slabs at\nx=0a n d dSrepresent barrier potentials whose strengths are\nparametrized by Z1andZ2.T h e i n c i d e n t Q P s ’ w a v e v e c t o r\nis given by ( ke,h,kk).\ntially, this reduces the problem to a time-independent\nprocess carried out with a snapshot configuration for the\nmagnetization. One can generalize[ ?] to incorporate\nmagnetization precession dynamics with a simple SU(2)\nrotation to the rotating frame of m(t). Including the\nprecessional motion of the magnetization, which is rotat-\ning around z-axism(t)=\u0000\nsin✓cos!t,sin✓sin!t,cos✓\u0000\n,\ncan be accomplished by a spinor rotation[ ?] asS(\",t)=\nU†(t)S(\")U(t). The time-dependent unitary rotation\nU(t) is a transformation from the lab frame to rotating\nframe of m, that is given by\nU= \nei✓\n2\u0000y0\n0 ei✓\n2\u0000y!✓\nei!t\n2\u0000z 0\n0 e\u0000i!t\n2\u0000z◆\n. (1)\nThe precessional angular speed is given by !.\nTo start, let us begin with the scattering process in\nNr|S|Nstructure[ 4]. For a S with a thickness given by\ndS, applying continuity conditions for wave function and\ndiscontinuity of its derivative on the interfaces as well\nas current continuity inside the S, we can relate the in-\ncoming modes⇥\nc+\ne(Nr),c\u0000\ne(N),c\u0000\nh(Nr),c+\nh(N)⇤Tto the\noutgoing modes⇥\nc\u0000\ne(Nr),c+\ne(N),c+\nh(Nr),c\u0000\nh(N)⇤Tin the\nNr|S|Nstructure (for details of this calculation see Ap-\npendix ??).\nFor the normal metal Na width of dN, the electron\nand hole propagate through and reflect back o ↵the FI\ninterface without mixing particle-hole and spin spaces.The Supplemental Material\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nTHEORY OF A SPIN-DEPENDENT\nSCATTERING WITH SUPERCONDUCTOR\nIn the following, we study a time-dependent scatter-\ning in ballistic regime for a Nr|S|N|FIhybrid structure,\nsee Fig. 1. Our scattering process is carried out by spin-\nful QPs, i.e., electrons and holes, whose amplitudes are\ngiven by the complex numbers c±\ne,h.T h e s u p e r s c r i p t ±\nsign indicates the direction of the group velocity (normal\ncomponent) given by ±⇣e,hˆx, for which we have defined\n⇣e= +1, and ⇣h=\u00001. In the absence of magnetic in-\nhomogeneity and spin-orbit interaction, the spin is con-\nserved along magnetization m.\nIt is customary that at the interfaces, generic bound-\nary conditions pertinent to interface scattering and dis-\norder can be taken into account by an insertion of a per-\nfect spacer N between adjacent materials[ 1,2]. This way,\nthe full scattering process is broken down into studying\nNr|S|N, and N|FIstructures separately. We bear in\nmind that propagation through the N layer mixes nei-\nther particle-hole nor spin degrees of freedom. For the\nNr|S|Nstructure, following BTK[ 3], we assume a uni-\nform superconducting pair-potential \u0000inside the S layer\nthat vanishes in the N layers. This way the scattering\nprocess takes place only at the interfaces, which essen-\ntially contains QPs propagating into the S and Andreev\nretro-reflections. In a more realistic (dirty) normal metal-\nS interface, however, there is a possibility of a specular\nreflection for electrons and holes. Hence, to model par-\ntially reflective interfaces, we have incorporated barrier\npotentials in the form of Z1~vF\u0000(x), and Z2~vF\u0000(x\u0000dS)\nwhose strength are parametrized by Z1,Z2,(vFis the\nFermi velocity).\nInitially, we assume no barrier potentials and study\na single-channel scattering in a 1D wire. Physically, this\ncan be justified by imposing a constriction, that is, trans-\nverse thickness of the N layers to be on the order of a\nFermi wavelength[ 1]. Later on, we will relax these condi-\ntions by considering a 3D geometry with partially reflec-\ntive 2D interfaces ( Z1,Z26= 0), for which we integrate\nout all transverse modes.\nAs n a p s h o tc o n fi g u r a t i o ns c a t t e r i n gp r o c e s s\nA comparison between a typical FI precession period\n⇠10\u000010s (FMR period) with the time scales of elec-\ntron and hole dynamics, ( dS+dN)/vF⌧10\u000010s, re-\nveals that the magnetization dynamics can be taken as\nan adiabatic evolution during the scattering process. Ini-!\"#\n!\"$\n!%$\n!%#\nSm(t)\ndS \t\txz\n!\"#\n!\"$\n!%$\n!%#ke,h  k∥\nFIN\nNr \t\tdN\t\t() (*\nFIG. 1. A snapshot configuration for magnetization m(t)i n\nanNr|S|N|FIstructure with incoming and outgoing modes\n(black and red, respectively). The propagation direction for\nelectrons and holes with scattering amplitudes c±\ne,c±\nhis given\nby±ˆxfor electrons, and ⌥ˆxfor holes. The shaded slabs at\nx=0a n d dSrepresent barrier potentials whose strengths are\nparametrized by Z1and Z2.T h e i n c i d e n t Q P s ’ w a v e v e c t o r\nis given by ( ke,h,kk).\ntially, this reduces the problem to a time-independent\nprocess carried out with a snapshot configuration for the\nmagnetization. One can generalize[ ?] to incorporate\nmagnetization precession dynamics with a simple SU(2)\nrotation to the rotating frame of m(t). Including the\nprecessional motion of the magnetization, which is rotat-\ning around z-axism(t)=\u0000sin✓cos!t,sin✓sin!t,cos✓\u0000,\ncan be accomplished by a spinor rotation[ ?] asS(\",t)=\nU†(t)S(\")U(t). The time-dependent unitary rotation\nU(t) is a transformation from the lab frame to rotating\nframe of m, that is given by\nU= \nei✓\n2\u0000y0\n0 ei✓\n2\u0000y!✓ei!t\n2\u0000z 0\n0 e\u0000i!t\n2\u0000z◆\n. (1)\nThe precessional angular speed is given by !.\nTo start, let us begin with the scattering process in\nNr|S|Nstructure[ 4]. For a S with a thickness given by\ndS, applying continuity conditions for wave function and\ndiscontinuity of its derivative on the interfaces as well\nas current continuity inside the S, we can relate the in-\ncoming modes⇥c+\ne(Nr),c\u0000\ne(N),c\u0000\nh(Nr),c+\nh(N)⇤Tto the\noutgoing modes⇥c\u0000\ne(Nr),c+\ne(N),c+\nh(Nr),c\u0000\nh(N)⇤Tin the\nNr|S|Nstructure (for details of this calculation see Ap-\npendix ??).\nFor the normal metal Na width of dN, the electron\nand hole propagate through and reflect back o ↵the FI\ninterface without mixing particle-hole and spin spaces.The Supplemental Material\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nTHEORY OF A SPIN-DEPENDENT\nSCATTERING WITH SUPERCONDUCTOR\nIn the following, we study a time-dependent scatter-\ning in ballistic regime for a Nr|S|N|FIhybrid structure,\nsee Fig. 1. Our scattering process is carried out by spin-\nful QPs, i.e., electrons and holes, whose amplitudes are\ngiven by the complex numbers c±\ne,h.T h e s u p e r s c r i p t ±\nsign indicates the direction of the group velocity (normal\ncomponent) given by ±⇣e,hˆx, for which we have defined\n⇣e= +1, and ⇣h=\u00001. In the absence of magnetic in-\nhomogeneity and spin-orbit interaction, the spin is con-\nserved along magnetization m.\nIt is customary that at the interfaces, generic bound-\nary conditions pertinent to interface scattering and dis-\norder can be taken into account by an insertion of a per-\nfect spacer N between adjacent materials[ 1,2]. This way,\nthe full scattering process is broken down into studying\nNr|S|N, and N|FIstructures separately. We bear in\nmind that propagation through the N layer mixes nei-\nther particle-hole nor spin degrees of freedom. For the\nNr|S|Nstructure, following BTK[ 3], we assume a uni-\nform superconducting pair-potential \u0000inside the S layer\nthat vanishes in the N layers. This way the scattering\nprocess takes place only at the interfaces, which essen-\ntially contains QPs propagating into the S and Andreev\nretro-reflections. In a more realistic (dirty) normal metal-\nS interface, however, there is a possibility of a specular\nreflection for electrons and holes. Hence, to model par-\ntially reflective interfaces, we have incorporated barrier\npotentials in the form of Z1~vF\u0000(x), and Z2~vF\u0000(x\u0000dS)\nwhose strength are parametrized by Z1,Z2,(vFis the\nFermi velocity).\nInitially, we assume no barrier potentials and study\na single-channel scattering in a 1D wire. Physically, this\ncan be justified by imposing a constriction, that is, trans-\nverse thickness of the N layers to be on the order of a\nFermi wavelength[ 1]. Later on, we will relax these condi-\ntions by considering a 3D geometry with partially reflec-\ntive 2D interfaces ( Z1,Z26= 0), for which we integrate\nout all transverse modes.\nAs n a p s h o tc o n fi g u r a t i o ns c a t t e r i n gp r o c e s s\nA comparison between a typical FI precession period\n⇠10\u000010s (FMR period) with the time scales of elec-\ntron and hole dynamics, ( dS+dN)/vF⌧10\u000010s, re-\nveals that the magnetization dynamics can be taken as\nan adiabatic evolution during the scattering process. Ini-!\"#\n!\"$\n!%$\n!%#\nSm(t)\ndS \t\txz\n!\"#\n!\"$\n!%$\n!%#ke,h  k∥\nFIN\nNr \t\tdN\t\t() (*\nFIG. 1. A snapshot configuration for magnetization m(t)i n\nanNr|S|N|FIstructure with incoming and outgoing modes\n(black and red, respectively). The propagation direction for\nelectrons and holes with scattering amplitudes c±\ne,c±\nhis given\nby±ˆxfor electrons, and ⌥ˆxfor holes. The shaded slabs at\nx=0a n d dSrepresent barrier potentials whose strengths are\nparametrized by Z1andZ2.T h e i n c i d e n t Q P s ’ w a v e v e c t o r\nis given by ( ke,h,kk).\ntially, this reduces the problem to a time-independent\nprocess carried out with a snapshot configuration for the\nmagnetization. One can generalize[ ?] to incorporate\nmagnetization precession dynamics with a simple SU(2)\nrotation to the rotating frame of m(t). Including the\nprecessional motion of the magnetization, which is rotat-\ning around z-axism(t)=\u0000\nsin✓cos!t,sin✓sin!t,cos✓\u0000\n,\ncan be accomplished by a spinor rotation[ ?] asS(\",t)=\nU†(t)S(\")U(t). The time-dependent unitary rotation\nU(t) is a transformation from the lab frame to rotating\nframe of m, that is given by\nU= \nei✓\n2\u0000y0\n0ei✓\n2\u0000y!✓\nei!t\n2\u0000z 0\n0 e\u0000i!t\n2\u0000z◆\n. (1)\nThe precessional angular speed is given by !.\nTo start, let us begin with the scattering process in\nNr|S|Nstructure[ 4]. For a S with a thickness given by\ndS, applying continuity conditions for wave function and\ndiscontinuity of its derivative on the interfaces as well\nas current continuity inside the S, we can relate the in-\ncoming modes⇥\nc+\ne(Nr),c\u0000\ne(N),c\u0000\nh(Nr),c+\nh(N)⇤Tto the\noutgoing modes⇥\nc\u0000\ne(Nr),c+\ne(N),c+\nh(Nr),c\u0000\nh(N)⇤Tin the\nNr|S|Nstructure (for details of this calculation see Ap-\npendix ??).\nFor the normal metal Na width of dN, the electron\nand hole propagate through and reflect back o ↵the FI\ninterface without mixing particle-hole and spin spaces.The Supplemental Material\nMostafa Tanhayi Ahari, Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nTHEORY OF A SPIN-DEPENDENT\nSCATTERING WITH SUPERCONDUCTOR\nIn the following, we study a time-dependent scatter-\ning in ballistic regime for a Nr|S|N|FIhybrid structure,\nsee Fig. 1. Our scattering process is carried out by spin-\nful QPs, i.e., electrons and holes, whose amplitudes are\ngiven by the complex numbers c±\ne,h.T h e s u p e r s c r i p t ±\nsign indicates the direction of the group velocity (normal\ncomponent) given by ±⇣e,hˆx, for which we have defined\n⇣e= +1, and ⇣h=\u00001. In the absence of magnetic in-\nhomogeneity and spin-orbit interaction, the spin is con-\nserved along magnetization m.\nIt is customary that at the interfaces, generic bound-\nary conditions pertinent to interface scattering and dis-\norder can be taken into account by an insertion of a per-\nfect spacer N between adjacent materials[ 1,2]. This way,\nthe full scattering process is broken down into studying\nNr|S|N, and N|FIstructures separately. We bear in\nmind that propagation through the N layer mixes nei-\nther particle-hole nor spin degrees of freedom. For the\nNr|S|Nstructure, following BTK[ 3], we assume a uni-\nform superconducting pair-potential \u0000inside the S layer\nthat vanishes in the N layers. This way the scattering\nprocess takes place only at the interfaces, which essen-\ntially contains QPs propagating into the S and Andreev\nretro-reflections. In a more realistic (dirty) normal metal-\nS interface, however, there is a possibility of a specular\nreflection for electrons and holes. Hence, to model par-\ntially reflective interfaces, we have incorporated barrier\npotentials in the form of Z1~vF\u0000(x), and Z2~vF\u0000(x\u0000dS)\nwhose strength are parametrized by Z1,Z2,(vFis the\nFermi velocity).\nInitially, we assume no barrier potentials and study\na single-channel scattering in a 1D wire. Physically, this\ncan be justified by imposing a constriction, that is, trans-\nverse thickness of the N layers to be on the order of a\nFermi wavelength[ 1]. Later on, we will relax these condi-\ntions by considering a 3D geometry with partially reflec-\ntive 2D interfaces ( Z1,Z26= 0), for which we integrate\nout all transverse modes.\nAs n a p s h o tc o n fi g u r a t i o ns c a t t e r i n gp r o c e s s\nA comparison between a typical FI precession period\n⇠10\u000010s (FMR period) with the time scales of elec-\ntron and hole dynamics, ( dS+dN)/vF⌧10\u000010s, re-\nveals that the magnetization dynamics can be taken as\nan adiabatic evolution during the scattering process. Ini-!\"#\n!\"$\n!%$\n!%#\nSm(t)\ndS \t\txz\n!\"#\n!\"$\n!%$\n!%#ke,h  k∥\nFIN\nNr \t\tdN\t\t() (*\nFIG. 1. A snapshot configuration for magnetization m(t)i n\nanNr|S|N|FIstructure with incoming and outgoing modes\n(black and red, respectively). The propagation direction for\nelectrons and holes with scattering amplitudes c±\ne,c±\nhis given\nby±ˆxfor electrons, and ⌥ˆxfor holes. The shaded slabs at\nx=0a n d dSrepresent barrier potentials whose strengths are\nparametrized by Z1andZ2.T h e i n c i d e n t Q P s ’ w a v e v e c t o r\nis given by ( ke,h,kk).\ntially, this reduces the problem to a time-independent\nprocess carried out with a snapshot configuration for the\nmagnetization. One can generalize[ ?] to incorporate\nmagnetization precession dynamics with a simple SU(2)\nrotation to the rotating frame of m(t). Including the\nprecessional motion of the magnetization, which is rotat-\ning around z-axism(t)=\u0000\nsin✓cos!t,sin✓sin!t,cos✓\u0000\n,\ncan be accomplished by a spinor rotation[ ?] asS(\",t)=\nU†(t)S(\")U(t). The time-dependent unitary rotation\nU(t) is a transformation from the lab frame to rotating\nframe of m, that is given by\nU= \nei✓\n2\u0000y0\n0 ei✓\n2\u0000y!✓\nei!t\n2\u0000z 0\n0 e\u0000i!t\n2\u0000z◆\n. (1)\nThe precessional angular speed is given by !.\nTo start, let us begin with the scattering process in\nNr|S|Nstructure[ 4]. For a S with a thickness given by\ndS, applying continuity conditions for wave function and\ndiscontinuity of its derivative on the interfaces as well\nas current continuity inside the S, we can relate the in-\ncoming modes⇥\nc+\ne(Nr),c\u0000\ne(N),c\u0000\nh(Nr),c+\nh(N)⇤Tto the\noutgoing modes⇥\nc\u0000\ne(Nr),c+\ne(N),c+\nh(Nr),c\u0000\nh(N)⇤Tin the\nNr|S|Nstructure (for details of this calculation see Ap-\npendix ??).\nFor the normal metal Na width of dN, the electron\nand hole propagate through and reflect back o ↵the FI\ninterface without mixing particle-hole and spin spaces.\nFIG. S1. A snapshot con\fguration for the magnetization m(t) in an N rjSjNjFI structure with the incoming and outgoing\nmodes into the S (black and red, respectively). The propagation direction for electrons and holes with scattering amplitudes\nc\u0006\ne;c\u0006\nhare given by\u0006^x,\u0007^x, respectively. The shaded slabs at x= 0 anddSrepresent the barrier potentials whose strengths\nare parametrized by Z1andZ2. The incident QP wave vector is given by ( ke;h;kk).\nTherefore, the incident and re\rected modes in N and S layers can be expanded as\n\tN=c+\ne(N)\t+\nNne+c\u0000\ne(N)\t\u0000\nNne+c+\nh(N)\t+\nNnh+c\u0000\nh(N)\t\u0000\nNnh; (S6)\nand\n\tS=c+\ne(S)\t+\nSne+c\u0000\ne(S)\t\u0000\nSne+c+\nh(S)\t+\nSnh+c\u0000\nh(S)\t\u0000\nSnh: (S7)\nThe scattering matrix across the S is obtained by imposing the following conditions: (1) continuity of the wave\nfunctions and their derivative at the interfaces x= 0;dS, (2) continuity of the current inside the S, that is, 0 <x<dS.\nA straight forward algebra, under Andreev approximation \u0001 =EF\u001c1, yields\n0\nBB@c\u0000\ne(Nr)\nc+\ne(N)\nc+\nh(Nr)\nc\u0000\nh(N)1\nCCA=1\n1\u0000A2\r0\nBB@0\u0014\ru\u00002A(1\u0000\r) 0\n\u0014\ru\u000020 0 A(1\u0000\r)\nA(1\u0000\r) 0 0 \r\u0014\u00001u\u00002\n0A(1\u0000\r)\r\u0014\u00001u\u0000201\nCCA0\nBB@c+\ne(Nr)\nc\u0000\ne(N)\nc\u0000\nh(Nr)\nc+\nh(N)1\nCCA; (S8)\nwhere\n\u0014=eikFdS; \r =ei2\ndS=~vF;\nu2=\"\n\n(1\n2+\n2\"); r1\nA\u0011A: (S9)\nThe quantity r1\nAis the AR amplitude at the interface of a normal metal with a bulk S [1]. For a thin S layer, as it\ncan be checked from the QP scattering matrix for the N rjSjN structure, this is modi\fed to\nrA=(1\u0000\r)(r1\nA)\n1\u0000\r(r1\nA)2: (S10)\nThe scattering matrix can be generalized [4] to include a nonzero barrier potential at the interface x=dS. We note\nthat, within the Andreev approximation \u0001 =EF\u001c1, the transmission amplitude for electrons is given by 1 =(1 +iZ).\nWe study the case of nonzero barrier numerically. The electrons and holes propagate through the normal metal and\nre\rect back o\u000b the FI interface,\nc\u0000\ne(N) = r\u001b\nec+\ne(N);\nc+\nh(N) = r\u0000\u001b\nhc\u0000\nh(N); (S11)\nwhere the electron re\rection coe\u000ecient is given by r\u001b\ne(\")\u0019ei(kF+\"=~vF)2dNei#\u001b, and particle-hole symmetry yields\nr\u001b\nh(\") =\u0002\nr\u001b\ne(\u0000\")\u0003\u0003. The spin-dependent phase shift #\u001bis given by\nei#\u001b=p\nE\u0000ipV\u001b\u0000Ep\nE+ipV\u001b\u0000E; E <V \u001b: (S12)3\nThe energy of an incoming electron or hole is given by E=~2k2\ne;h=2m. Combining the scattering matrix of N rjSjN\nwith Eq. (S11), the full scattering matrix with single channel obtains ( Z= 0),\n\u0012c\u0000\ne(Nr)\nc+\nh(Nr)\u0013\n=\u0012r\u001b\neer\u0000\u001b\neh\nr\u001b\nher\u0000\u001b\nhh\u0013\u0012c+\ne(Nr)\nc\u0000\nh(Nr)\u0013\n; (S13)\nwhere\nr\u001b\nee=\u00142\r\n\u0011r\u001b\ne; r\u0000\u001b\nhh=\u0014\u00002\r\n\u0011r\u0000\u001b\nh;\nr\u001b\nhe=r\u0000\u001b\neh=r1\nA(1\u0000\r)\n\u0011h\n1\u0000(r1\nA)2\r\u0000\u0000\n(r1\nA)2\u0000\r\u0001\nr\u001b\ner\u0000\u001b\nhi\n:\nIn the above equations we have de\fned\n\u0011\u0011\u0000\n1\u0000(r1\nA)2\r\u00012\u0000(r1\nA)2(1\u0000\r)2r\u001b\ner\u0000\u001b\nh: (S14)\nIt is instructive to note that a constructive Rowell{McMillan interference takes place when\nr\u001b\ner\u0000\u001b\nh(r1\nA)2=ei2n\u0019; n= 0;\u00061;\u00062;\u0001\u0001\u0001; (S15)\nThe extra terms in \u0011stem from multiple tunneling and AR re\rection possibilities due to \fnite thickness of the S\n(\r6= 0). In short, the scattering matrix with all the nonzero elements can be written as follows:\nS(\") =0\nBBB@r\"\nee(\") 0 0 r#\neh(\")\n0r#\nee(\")r\"\neh(\") 0\n0r#\nhe(\")r\"\nhh(\") 0\nr\"\nhe(\") 0 0 r#\nhh(\")1\nCCCA: (S16)\nTime-dependent scattering .|A comparison between a typical FI precession period \u001810\u000010s (FMR period) with\nthe time scales of electron and hole dynamics, ( dS+dN)=vF\u001c10\u000010s, reveals that the magnetization dynam-\nics can be taken as an adiabatic evolution during the scattering process. Initially, this reduces the problem to a\ntime{independent process carried out with a snapshot con\fguration for the magnetization. Nonetheless, one can\ngeneralize to incorporate magnetization precession dynamics with a simple time-dependent SU(2) rotation to the lab\nframe in which the magnetization direction is varying [5]. Including the precessional motion of the magnetization,\nm(t) =\u0000\nsin\u0012cos!t;sin\u0012sin!t;cos\u0012\u0001\n, can be accomplished by a spinor rotation [5] given by\nU(t) = \nei\u0012\n2\u001by0\n0ei\u0012\n2\u001by! \nei!t\n2\u001bz 0\n0e\u0000i!t\n2\u001bz!\n; (S17)\nwhere!is the precessional angular speed. Consequently, the time dependent scattering matrix in the lab frame\nobtains,S(\";t) =Uy(t)S(\")U(t).\nAndreev resonances .| The condition given in Eq. (S15) corresponds to the resonant behavior in the mixing\nconductance, which determines Andreev bound-state excitation energies \"A. It can be checked that, the low energy\nABSs can only form when dN>0. This indicates that, in the limit of zero barrier, the insertion of a normal metal\nlayer not only provides a level of control but is a necessary element for the enhancement of spin \row. By increasing\nthe barrier potential strength, however, subgap normal bound states begin to form and ABSs are pushed away from\nthe subgap region (see Fig. S2). In order to see this, we have de\fned\nfABS(Z)\u0011Re g\"#(dS;dN;\"res;Z)\u0000Re g\"#(dS;dN;\"res;Z1): (S18)\nHere,\"resrepresent resonance energies, for which the real part of the conductance is maximum. Note that, \"res!\"A\nwhenZ!0. SinceZ1\u001d1, Re g\"#(dS;dN;\"res;Z1) includes only normal bound{state resonances.\n3D geometry .|Due to the translation symmetry on the interfaces, the full 3D scattering problem can be inferred\nfrom our scattering matrix given in Eq. (S16). As we have mentioned earlier, the wave number for the electron and\nhole propagating at energy \"is given by\nkn\ne;h=r\n2m\n~2p\nEF\u0000En+\u0010e;h\"; (S19)4\nFIG. S2. (a) The resonant bound{state energies, \"res, for various Z. WhenZ= 0, we get \"res=\"A. (b) For \fnite but\nsmall barrier strength, Re g\"#has contribution from normal and Andreev bound{state resonances. (c) For the stronger barrier,\nwhen the AR probability is negligible, the resonant energies are due to normal bound states. In (a), (b), and (c), we have set\ndS=\u00180= 1:5 and \u0001 ex= 0:4EF. (d)fABS(Z) is shown for Z1= 2, where the blue, orange, and green curves correspond to\ndS=\u00180= 1, 1:5, and 2, respectively. In this plot, we have \u0001 ex= 0:4EFanddN=\u00180= 1.\nFIG. S3. The normalized conductance averaged over N(\") = 20 channels in ballistic regime for various barrier strengths Z.\nHere, we have set \u0001 ex=EF= 0:1,T=Tc= 0:4, anddN=\u00180= 2:5.\nwhereEnis the transverse energy of the incident electron or hole. Here, the channel number index nis considered\nto distinguish QPs propagating through di\u000berent channels at energy \". The multichannel e\u000bective conductance is\nobtained by summing over all the channels present at energy \". We consider N(\") to denote the total number of\nsuch channels. In Fig. S3, we have shown normalized e\u000bective conductance for multichannel case in the presence of\ninterface barrier potential.\nDynamic generation of triplet Cooper pairs |We utilize a heuristic argument to describe the triplet pairing gen-\neration in our setup. For a thin S layer, dS\u0018\u00180, QPs with the resonant energies can escape the Nlayer either by5\ntunnelling through the S layer or forming Cooper pairs inside the S layer. For the latter case, the Cooper pairs contain\nthe spin{dependent phases as follows [6]:\nei\u000bj\"#i\u0000e\u0000i\u000bj#\"i; (S20)\nwhere\u000b\u0018#. These Cooper pairs contain a spin{triplet component, j\"#i+j#\"i, with an amplitude determined by sin \u000b.\nThe precessing magnetization mcan dynamically induce spin{polarized triplet Cooper pairs j##iandj\"\"i, which is\ncaused by the variations in the direction of the quantization axis for spins [6]. In other words, the spin{polarized\ntriplet components arise when the spin{triplet component j\"#i+j#\"iis written in a new rotated basis [6].\n[1] C. W. J. Beenakker. Rev. Mod. Phys. 69, 731 (1997)\n[2] Y. Nazarov, Y. Blanter. Quantum Transport: Introduction to Nanoscience. Cambridge: Cambridge University Press (2009)\n[3] G. E. Blonder, M. Tinkham, and T. M. Klapwijk. Phys. Rev. B 25, 4515 (1982)\n[4] O. Entin-Wohlman, Y. Imry, and A. Aharony. Phys. Rev. B 78, 224510 (2008)\n[5] H. J. Skadsem, A. Brataas, J. Martinek, and Y. Tserkovnyak, Phys. Rev. B 84, 104420 (2011)\n[6] M. Eschrig, T. L ofwander. Nature Physics, 4, (2008)\nJ. Linder, J. W. A. Robinson, Nature Physics, 11, (2015)"    },    {        "title": "2403.17976v1.Efecto_Hall_de_espin_inverso_en_peliculas_de_Nb_Mo_y_Bi_por_bombeo_de_espin.pdf",        "content": " EFECTO HALL DE ESPÍN INVERSO EN PELÍCULAS DE Nb, Mo y Bi \nPOR BOMBEO DE ESPÍN  \n \n \n \nD. Ley Domínguez1, J. A. Matutes -Aquino2 \n \n1Universidad Tecnológica de Ciudad Juárez, Av. Universidad Tecnológica # 3051.Col. Lote Bravo II,  Cd. Juárez \nChihuahua, México , 32695 . \n2Centro de Investigación en Materiales Avanzados, S.C., Miguel de Cervantes 120, Complejo Industrial Chihuahua, \nChihuahua, México , 31109 . \nAutor Corresponsal : david_ley@utcj.edu.mx  \n \n \nResumen:  El efecto Hall de espín inverso utilizado para la detección de corrientes de espín fue observado mediante \nmedidas de voltaje en bicapas  de metal normal (MN)/metal ferromagnético (MF), utilizando Nb, Mo y  Bi como \nmetal normal y Permalloy (Py, Ni 81Fe19) como metal ferromagnético. La corriente de espín fue g enerada por el \nefecto de bombeo de espín con resonancia ferromagnética. Las muestras fueron depositadas  por pulverización \ncatódica (sputtering) con magnetrón de corriente continua a temperatura ambiente  sobre sustratos de Si (001). Las \ntres bicapas de Nb/P y, Mo/Py y  Bi/Py  tuvieron un acoplamiento espín -órbita lo suficientemente grande para poder \nobservar la generación de voltaje por efecto Hall de espín inverso . \n \nPalabras clave: Corriente de espín pura, efecto Hall de espín , efecto de bombeo de espín , resonancia \nferromagnética . \n \nAbstract : The inverse spin Hall effect  used for detection of spin currents was observed by voltage measurements in \nbilayers of normal metal (NM)/ferromagne tic metal (FM), using Nb, Mo  and Bi as normal metal and Permalloy (Py, \nNi81Fe19) as ferromagnetic metal. The  spin current was generated by the spin pumping effect with ferromagnetic \nresonance.  The samples were deposited by dc magnetron sputtering at room temperature on Si (001) substrates.  The \nthree bilayers of Nb/Py, Mo/Py  and Bi/Py  had a spin -orbit coupling large enough to observe the voltage generation \nby spin Hall effect.  \n \nKeywords:  Pure spin current,  spin Hall effect , spin pumping effect , ferromagnetic resonance . \n \n1. Introducción  \nLa espintrónica  es una disciplina que estudia una propiedad intrínseca de los electrones que es el \nespín, para mejorar la eficiencia y velocidad de los dispositivos electrónicos. Este nuevo tipo de \nelectrónica trabaja no solo con la carga de los electrones sino también c on su espín. Los \ndispositivos espintrónicos combinan la microelectrónica convencional con los efectos derivados \nde la interacción entre el espín del electrón y las propiedades magnéticas del material. El enfoque \npara utilizar el espín está basado en su ali neación (ya sea “arriba” o “abajo”) respecto a una \nreferencia (un campo magnético aplicado o la orientación de la magnetización d e una película \nferromagnética). Los dispositivos operan con cierta cantidad de corriente eléctrica que depende \nde una manera pr edecible de la dirección de alineación del espín. La implementación de  la \nalineación de los espines a la electrónica convencional aña de a los dispositivos una mayor \ncapacidad y velocidad con bajo consumo de energía en los dispositivos electrónicos.  Actualm ente \nlos discos duros de computadora y las memorias MRAM funcionan con dispositivos de \nespintrónica [ 1-3].  \n \nLos aspectos clave  en la espintrónica  son la generación y detección de corrientes de espín, en la \nelectrónica convencional la orientación del espín  del electrón de la corriente eléctrica o corriente  de carga está completamente al azar, como se esquematiza en la figura 1 a), en otras palabras, el \nespín del electrón no tiene ningún rol en los dispositivos electrónicos, por lo que se ignora por \ncompleto  el espín del electrón. En contraste con la corriente eléctrica, la corriente de espines \npolarizados  toma en consideración la orientación del espín pero también puede estar asociada con \nla corriente de carga,  como se ilustra en la figura 1 b), o puede ser solo un flujo de espín sin flujo \nneto de carga, llamada corriente de espín  pura , ilustrada en la figura 1 c). La corriente de espín \npura se puede entender como un flujo de electrones polarizados solo con espín hacia arriba (spin -\nup) sumánd ole a éste un flujo de electrones igual pero polarizados con espín hacia abajo (spin -\ndown) y fluyendo en dirección opuesta, por esta razón no hay un flujo neto de carga, la corriente \nde espín  pura se puede ver como la diferencia entre el flujo de espín arr iba y espín abajo, en \ncontraste con la corriente de carga (corriente eléctrica) donde es la suma de los flujo de espín \narriba y espín abajo.  \n \n \nFigura 1 .- Esquema del flujo de electrones: a) corriente eléctrica o corriente de carga, existe un flujo \nneto de carga y el espín de los electrones está orientado al azar, b) corriente de carga con espines \npolarizados, tiene un flujo neto de carga y la mayoría de los espines están orientados en la misma \ndirección, c) No tiene un flujo neto de carga \n0IIIc  debido a que \n I I y tiene un flujo neto \nde espín \n0 2I IIIs . \n \nLas corrientes de espín pueden ser creadas por fenómeno s recientemente descubiertos como: ( i) \nEl efecto Hall de espín (SHE, siglas del inglés Spin Hall Effect ) como se esquematiza en la figura \n2, consiste en una acumulación de espín en los bordes del material en donde existe un flujo de \ncarga eléctrica donde los espines hacia arriba son desviados a un lado y los espines hacia abajo \nson desviados al otro lado del conductor, si la polaridad de la corr iente eléctrica cambia, la \norientación de los espines también cambia.  \n \n  \nFigura 2. - Esquema del efecto Hall de espín (SHE) y efecto Hall de espín inverso (ISHE), a) una \ncorriente de carga con espines no polarizados fluye a través de un material no magnéti co con \nacoplamiento espín -órbita generando una corriente de espín pura transversal a ésta, b) una corriente de \nespín pura fluyendo en un material no magnético con acoplamiento espín -órbita genera una corriente \neléctrica transversal.  \n \nEl efecto Hall de esp ín en metales no magnéticos  (MN)  se origina en el acoplamiento espín -\nórbita, acoplamiento entre el espín del electrón y su movimiento orbital, generalmente encontrado \nen átomos grandes con un número atómico grande. De manera contraria , si una corriente de espín \npura fluye en un material no magnético que tenga fuerte acoplamiento espín -órbita generará un \nflujo de corriente de carga transversal a la corriente de espín, este efecto es llamado efecto Hall \nde espín inverso (ISHE,  siglas del inglés  Inverse Spin H all Effect ) [4-8]. (ii) El efecto de bombeo \nde espín (SPE, siglas del inglés Spin Pumping  Effect), en donde una corriente de espín pura \npuede ser inyectada en un material no magnético debido a la precesión de un ma terial \nferromagnético adyacente. L a precesión de la dirección de la magnetización en el material \nferromagnético debido al torque ejercido por un campo de RF (radio frecuencia) externo bombea \nespines dentro del metal normar adyacente induciendo una corriente de espín  pura [9-12]. Los \nefectos SHE e ISHE  son requeridos para la creación de dispositivos espintrónicos  siendo el \nPlatino  uno de los elementos más utilizado  en el estudio de estos efectos [13-14]. Debido al alto \ncosto del platino es importante buscar otras alternativas. En este artículo  se reportan medidas  de \nvoltaje obtenido del efecto Hall de espín inverso por la corriente de espín generada mediante el \nefecto de bombeo de espín para los metales de Nb, Mo y Bi.  \n \n2. Experimentación  \nLas películas fueron depositadas por la técnica de pulverización catódica (sputtering) con \nmagnetrón de corriente continua a temperatura ambiente, se utilizaron substratos de silicio \nmonocristalino con orientación (001) con un espesor de 500 µm  previamente  lavados con \nultrasonido , se cortaron en rectángulos  de 1 mm  x 3 mm . Se utilizaran mascaras para depositar \nlos metales normales en el centro de la muestra con una dimensión de 1 mm x 1 mm . La cámara \nde pulverización  fue evacuada a una presión base de 2.2 x  10-7 Torr, después fue presurizada con \nun flujo de argón a 2.4 x 10-3 Torr durante el depósito. Se utilizaron blancos de 2 pulgadas de \ndiámetro de Permaloy (Ni 81Fe19), Niobio , Molibdeno y Bismuto, éstos con una pureza de \n99.95% . Todas las bicapas M N/MF  de las muestras fueron fabricadas con un espesor de Py de 12 \nnm y un tiempo de depósito de 1 minuto a 50 mA para las películas  de MN. Las muestras fueron \npuestas en resonancia ferromagnética con un espectrómetro de banda -X operado a una frecuencia \nde 9.4G Hz, la muestra fue posicionada en el centro de una cavidad resonante. Las muestras \nfueron barridas con un campo magnético externo generado por un electroimán de 9 pulgadas, \naplicado en dirección del plano de las películas . El voltaje generado como resultad o del efecto \nHall de espín inverso se midió  con un nanovoltímetro conectado mediante una interface a una \ncomputadora. En la figura 3 se observa un esquema de la muestra.  \n  \n \nFigura 3. - Representación de la muestra de interface metal -normal /permalloy . \n \n3. Resultados y discusión  \nEn la figura 4 (a -c) se observa  el voltaje detectado  por efecto Hall de espín inverso para l as \nmuestras de  Nb/Py, Mo/Py y Bi/Py . En las tres bicapas se observa un pico de voltaje en \naproximadamente 1.1kOe, que es el campo magnético de resonancia del Py para una frecuencia \nde 9.4GHz.  \nEl pico  de voltaje observado  para las tres muestras se encuentra en el rango de los µV , valores \nsimil ares han sido reportados para interfaces de Py/Py [ 13-14]. El voltaje se genera cuando la s \nbicapa s Nb/Py, Mo/Py y Bi/Py  están  en resonancia ferromagnética, una corriente de espín pura \nes inyectada de la película fer romagnética a la película de  MN por medio  del efecto de bombeo \nde espín. Una vez que existe una corriente de espín pura dentro de la película de MN se genera \nuna acumulación de cargas en las extremidades de la muestra  por el desvío de electrones debido \nal acoplamiento espín -órbita , obteniendo un voltaje  de efecto Hall de espín inverso . \nLa forma del pico de voltaje Hall de espín inverso es la misma que el pico de absorción de \nresonancia ferromagnética debido a que se crea  cuando la película ferromagnética Py se \nencuentra en un campo magnético y fre cuencia adecuada  cumpliendo  con las condiciones de \nresonancia, estando el Py en resonancia ferromagnética  la inyección de corriente de espín hacia \nlas películas de Nb, Mo y Bi  es máxima, obteniendo el máximo voltaje. Una vez saliendo de las \ncondiciones de resonancia el bombeo de espín desaparece, debido a que  fuera de las condiciones \nde resonancia magnética no existe una corriente de espín pura bombeada a la película de MN , el \nefecto Hall de espín inverso no se acciona y no genera una diferencia de potencial.     \n \n  \nFigura 4 .- Voltaje  detectado por  efecto Hall de espín inverso (ISHE) generado por la corriente de espín \ncreada mediante el bombeo de espín de resonancia ferromagnética, para las bicapas (a) Nb/Py, (b) \nMo/Py y (c) Bi/Py . \n \n4. conclusión  \n Se obtuvieron medidas de voltaje por efecto Hall de espín inverso en bicapas de Nb/Py, Mo/Py y \nBi/Py . Por medio de las medidas de voltaje observadas en los resultados  se pudo comprobar que \nlos metales Nb, Mo y Bi tienen un acoplamiento espín -órbita lo sufi cientemente grande para \nobservar el efecto Hall de espín inverso, por lo que pueden ser utilizados como generadores y \ndetectores de corrientes de espín . Debido al alto costo del Platino, s aber que el Nb, Mo y Bi  \ngeneran voltaje por  efecto Hall de espín inverso mediante  bombeo de espín  es una gran \nalternativa para minimizar costos en el área de la espintrónica . Estos elementos pueden ser \nestudiados para desarrollar nuevos dispositivos espintrónicos.  \n  \nReferencias  \n1. Pulizzi  F, Sinova J,  Žutić  I. (2012). Spintronics. Nature materials insight , 11, 367 -371. ISSN #1476 -1122.  \n2. Žutić  I, Jaroslav  Fabian, Das Sarma  S. (2004). Spintronics: Fundamentals and applications.  Rev. Mod. Phys. 76, \n323.ISSN #1539 -0756.  \n3. Wolf S.A , Awschalom  D.D,  Buhrman  R.A, Daughton  J.M, Molnár  S. Von , Roukes  M.L,  Chtchelkanova  A.Y, \nTreger  D.M.  (2001). Spintronics: A Spin -Based Electronics Vision for the Future.  Science , 294, 1488 -1495.  ISSN \n#1095 -9203.  \n4. Dyakonov M. I,  Perel  V.I. (1971). Current -induced spin orientation of electrons in semiconductors. Physics \nletters A, 35, 459 -460. ISSN #0375 -9601.  \n5.  Hirsch J.E. (1999). Spin Hall Effect. Physical Review Letters , 83, 1834 . ISSN #1079 -7114.  \n6. Kato Y.K, Myers  R.C, Gossard  A.C,  Awschalom  D.D. (2004). Observation of the spin Hall effect in \nsemiconductors. Science , 306, 1910. ISSN #1095 -9203.  \n7. Takahashi Saburo, Maekawa Sadamichi. (2008), Spin current, spin accumulation and spin Hall effect. Sci. \nTechnol. Adv. Mater . 9, 014105. ISSN #1878 -5514.  \n8. Vignale G. (2009). Ten Years of Spin Hall Effect. Journal of Superconductivity and Novel Magnetism , 23, ISSN \n#1557 -1947.  \n9. Tserkovnyak Yaroslav, Brataas Arne, Bauer G. E. W. (2002). Spin pumping and magnetization dynamics in \nmetallic multilayers. Physi cal Review B , 66, 224403. ISSN #2469 -9969.  \n10. Tserkovnyak Yaroslav, Brataas Arne, Bauer G. E. W.  (2002). Enhanced Gilbert Damping in Thin Ferromagnetic \nFilms.  Physical review Letters , 88, 117601. ISSN #1079 -7114.  \n11. Azevedo  A, Vilela -Leao  L. H , Rodríguez -Suárez  R. L , Lacerda Santos  A. F , Rezende  S. M.  (2011). Spin \npumping and anisotropic magnetoresistance voltages in magnetic bilayers: Theory and experiment. Physical \nReview B , 83, 144402. ISSN #2469 -9969.  \n12. Azevedo A, Vilela -Leao L. H, Rodríguez -Suárez R. L, Oliveira A. B, Rezende S. M. (2005). dc effect in \nferromagnetic resonance: Evidence of the spin -pumping effect?. Journal of Applied Physics , 97, 10C715. ISSN \n#1089 -7550.  \n13. Nakayama H, Ando K, Harii K, Kajiwara Y, Yoshino T, Uchida K, Ota T, Saitoh E. (2010). Detection or inverse \nspin-Hall effect induced in Pt x-1Mx(M = Cu, Au) thin films. Journal of Physics: Conference Series , 200, 062014. \nISSN #1742 -6596.  \n14. Nakayama H, Ando K, Harii K, Fujikawa Y, Kaj iwara Y, Yoshino T, Saitoh E. (2011). Inverse spin -Hall effect \ninduced by spin pumping in different size Ni 81Fe19/Pt films. Journal of Physics: Conference Series , 266, 012100. \nISSN #1742 -6596.  \n "    },    {        "title": "1803.10925v1.Giant_resonant_nonlinear_damping_in_nanoscale_ferromagnets.pdf",        "content": "Giant resonant nonlinear damping in nanoscale ferromagnets\nI. Barsukov,1,\u0003H. K. Lee,1A. A. Jara,1Y.-J. Chen,1A. M. Gon\u0018 calves,1\nC. Sha,1J. A. Katine,2R. E. Arias,3B. A. Ivanov,4, 5and I. N. Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Western Digital, 5600 Great Oaks Parkway, San Jose, CA 95119, USA\n3Departamento de F\u0013 \u0010sica, CEDENNA, FCFM, Universidad de Chile, Santiago, Chile\n4Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadsky av. 36 B, Kyiv, 03142, Ukraine\n5National University of Science and Technology MISiS, Moscow, 119049, Russian Federation\nMagnetic damping is a key metric for emerging technologies based on magnetic nanoparticles,\nsuch as spin torque memory and high-resolution biomagnetic imaging. Despite its importance,\nunderstanding of magnetic dissipation in nanoscale ferromagnets remains elusive, and the damping\nis often treated as a phenomenological constant. Here we report the discovery of a giant frequency-\ndependent nonlinear damping that strongly alters the response of a nanoscale ferromagnet to spin\ntorque and microwave magnetic \feld. This novel damping mechanism originates from three-magnon\nscattering that is strongly enhanced by geometric con\fnement of magnons in the nanomagnet. We\nshow that the giant nonlinear damping can invert the e\u000bect of spin torque on a nanomagnet leading\nto a surprising current-induced enhancement of damping by an antidamping torque. Our work\nadvances understanding of magnetic dynamics in nanoscale ferromagnets and spin torque devices.\nI. INTRODUCTION\nNanoscale magnetic particles are the core components\nof several emerging technologies such as nonvolatile spin\ntorque memory [1], spin torque oscillators [2{7], targeted\ndrug delivery, and high-resolution biomagnetic imaging\n[8{11]. Control of magnetic damping holds the key to\nimproving the performance of many nanomagnet-based\npractical applications. In biomagnetic characterization\ntechniques such as magnetic resonance imaging [12], re-\nlaxometry [13], and magnetic particle imaging [14, 15],\nmagnetic damping a\u000bects nanoparticles relaxation times\nand image resolution. In spin torque memory and oscil-\nlators, magnetic damping determines the electrical cur-\nrent necessary for magnetic switching [1] and generation\nof auto-oscillations [16] and thereby determines energy-\ne\u000eciency of these technologies. The performance of\nnanomagnet-based microwave detectors is also directly\na\u000bected by the damping [17{19]. Despite its impor-\ntance across multiple disciplines, magnetic damping in\nnanoparticles is poorly understood and is usually mod-\neled as a phenomenological constant [6, 16].\nIn this article, we experimentally demonstrate that a\nferromagnetic nanoparticle can exhibit dynamics quali-\ntatively di\u000berent from those predicted by the constant\ndamping model. We show that nonlinear contributions\nto the damping can be unusually strong and the damp-\ning parameter itself can exhibit resonant frequency de-\npendence. Our work demonstrates that nonlinear damp-\ning in nanomagnets is qualitatively di\u000berent from that in\nbulk ferromagnets and requires a new theoretical frame-\nwork for its description. We show both experimentally\nand theoretically that such resonant nonlinear damping\noriginates from multi-magnon scattering in a magnetic\n\u0003igorb@ucr.edusystem with a discrete spectrum of magnons induced by\ngeometric con\fnement.\nWe also discover that the resonant nonlinear damping\ndramatically alters the response of a nanomagnet to spin\ntorque. Spin torque arising from injection of spin cur-\nrents polarized opposite to the direction of magnetization\nacts as negative damping [2]. We \fnd, however, that the\ne\u000bect of such antidamping spin torque is reversed, lead-\ning to an enhanced dissipation due to the nonlinear res-\nonant scattering. This counterintuitive behavior should\nhave signi\fcant impact on the operation of spin torque\nbased memory [1], oscillators [2{7] and microwave detec-\ntors [17{19].\nII. RESULTS\nA. Spin wave spectroscopy\nWe study nonlinear spin wave dynamics in nanoscale\nelliptical magnetic tunnel junctions (MTJs) that consist\nof a CoFeB free layer (FL), an MgO tunnel barrier, and a\nsynthetic antiferromagnet (SAF) pinned layer [20]. Spec-\ntral properties of the FL spin wave modes are studied in a\nvariety of MTJs with both in-plane and perpendicular-to-\nplane equilibrium orientations of the FL and SAF magne-\ntization. We observe strong resonant nonlinear damping\nin both the in-plane and the perpendicular MTJs, which\npoints to the universality of the e\u000bect.\nWe employ spin torque ferromagnetic resonance (ST-\nFMR) to measure magnetic damping of the FL spin wave\nmodes. In this technique, a microwave drive current\nIacsin(2\u0019ft) applied to the MTJ excites oscillations of\nmagnetization at the drive frequency f. The resulting\nmagnetoresistance oscillations Racsin(2\u0019ft+\u001e) generate\na direct voltage Vmix. Peaks in ST-FMR spectra Vmix(f)\narise from resonant excitation of spin wave eigenmodes\nof the MTJ [21{28]. To improve signal-to-noise ratio,arXiv:1803.10925v1  [cond-mat.mes-hall]  29 Mar 20182\n0 0.5 1 1.5 2H1H2\n36912\nField (kOe)0 0.5 1 1.5 2H2H1\n00.20.4\nField (kOe)Linewidth (GHz)Experiment\nSimulationFrequency (GHz)\n+1\n1a b\n<Vmix>~\nFIG. 1. Spin wave spectra in a nanoscale MTJ. (a) Normalized ST-FMR spectra h~Vmix(f)iof spin wave eigenmodes in a\nperpendicular MTJ device (Sample 1) measured as a function of out-of-plane magnetic \feld. Resonance peaks arising from\nthree low frequency modes of the MTJ free layer j0i,j1i, and j2iare observed. (b) Spectral linewidth of the quasi-uniform\nj0ispin wave mode as a function of out-of-plane magnetic \feld. Strong linewidth enhancement is observed in the resonant\nthree-magnon regime at H1andH2.\nthe magnitude of external magnetic \feld Happlied par-\nallel to the free layer magnetization is modulated, and\na \feld-derivative signal ~Vmix(f) = dVmix(f)=dHis mea-\nsured via lock-in detection technique [20]. Vmix(f) can\nthen be obtained via numerical integration (Supplemen-\ntal Material).\nFigure 1(a) shows ST-FMR spectra ~Vmix(f) measured\nas a function of out-of-plane magnetic \feld Hfor an el-\nliptical 52 nm\u000262 nm perpendicular MTJ device (Sam-\nple 1). Three spin wave eigenmodes with nearly linear\nfrequency-\feld relation fn(H) are clearly visible in the\nspectra. Micromagnetic simulations (Supplemental Ma-\nterial) reveal that these modes are three lowest frequency\nspin wave eigenmodes of the FL (Supplemental Material).\nThe lowest frequency (quasi-uniform) mode j0iis node-\nless and has spatially uniform phase. Each of the two\nhigher-order modes jni(n= 1;2) has a single node at\nthe FL center that is either perpendicular ( n= 1) or\nparallel (n= 2) to the ellipse long axis.\nThe spectral linewidth of the resonances in Fig. 1(a)\ncan be used for evaluation of the mode damping. The\nquasi-uniform mode j0iresonance visibly broadens at\ntwo magnetic \feld values: H1= 0:74 kOe (4 GHz) and\nH2= 1:34 kOe (6 GHz). Near H1, the modej1iresonance\nalso broadens and exhibits splitting, same behavior is ob-\nserved for the mode j2iatH2. At these \felds, the higher-\norder mode frequency is twice that of the quasi-uniform\nmodefn= 2f0. This shows that three-magnon con\ru-\nence [29{33] is the mechanism of the quasi-uniform mode\ndamping increase: two magnons of the quasi-uniform\nmodej0imerge into a single magnon of the higher-ordermodejni.\nThe most striking feature of the quasi-uniform mode\nresonance near H1is its split-peak shape with a local min-\nimum at the resonance frequency. Such a lineshape can-\nnot be \ft by the standard Lorentzian curve with symmet-\nric and antisymmetric components [20]. We therefore use\na double-peak \ftting function (Supplemental Material)\nto quantify the e\u000bective linewidth \u0001 f0of the resonance\npro\fle. For applied \felds su\u000eciently far from H1, the\nST-FMR curve recovers its single-peak shape and \u0001 f0\nis determined as half width of the standard Lorentzian\n\ftting function [20]. Figure 1(b) shows \u0001 f0as a function\nofHand demonstrates a large increase of the linewidth\nnear the \felds of the resonant three-magnon regime H1\nandH2. The stepwise increase of \u0001 f0nearH1is a result\nof the ST-FMR curve transition between the split-peak\nand single-peak shapes. For \felds near H2, the resonance\npro\fle broadens but does not develop a visible split-peak\nlineshape. As a result, \u0001 f0(H) is a smooth function in\nthe vicinity of H2.\nB. E\u000bect of spin torque\nIn MTJs, direct bias current Idcapplied across the\njunction exerts spin torque on the FL magnetization, act-\ning as antidamping for Idc>0 and as positive damping\nforIdc<0 [22, 34]. The antidamping spin torque in-\ncreases the amplitude of the FL spin wave modes [22, 35]\nand decreases their spectral linewidth [36]. We can em-\nploy spin torque from Idcto control the amplitude of spin3\nFIG. 2. E\u000bect of spin torque on spin wave resonance lineshape. (a)-(b) Spin wave resonance lineshapes in the nonresonant\nregime at H > H 1for di\u000berent values of direct bias current Idc. (c)-(d) Spin wave resonance lineshapes in the resonant three-\nmagnon regime at H=H1. (a), (c) Measured ST-FMR spectra (Sample 2). (b), (d) Solutions of Eqs. (3) and (4). Identical\nbias current values Idc(displayed in (a) are used in (a)-(d).\nwave eigenmodes excited in ST-FMR measurements, and\nthereby study the crossover between linear and nonlinear\nregimes of spin wave resonance.\nFigure 2 shows the dependence of ST-FMR resonance\ncurve of thej0imodeVmix(f) onIdcfor a 50 nm\u0002110 nm\nelliptical in-plane MTJ (Sample 2). For in-plane mag-\nnetic \feld values far from the three-magnon resonance\n\feldsHn, the amplitude of ST-FMR resonance curve\nVmix(f) shown in Fig. 2(a) monotonically increases with\nincreasing antidamping spin torque, as expected. At\nH=H1, the antidamping spin torque has a radically\ndi\u000berent and rather surprising e\u000bect on the resonance\ncurve. As illustrated in Fig. 2(c), increasing antidamp-\ning spin torque \frst broadens the resonance at H=H1\nand then transforms a single-peak resonance lineshape\ninto a split-peak lineshape with a local minimum at the\nresonance frequency f0. The data in Fig. 2 demonstrate\nthat the unusual split-peak lineshape of the resonance is\nonly observed when (i) the three-magnon scattering of\nthe quasi-uniform mode is allowed by the conservation of\nenergy and (ii) the amplitude of the mode is su\u000eciently\nhigh, con\frming that the observed e\u000bect is resonant and\nnonlinear in nature.\nFig. 2(c) reveals that antidamping spin torque can in-\ncrease the spectral linewidth and the e\u000bective damping\nof the quasi-uniform spin mode if the mode undergoes\nresonant three-magnon scattering. Figure 3 further illus-\ntrates this counterintuitive e\u000bect. It shows the linewidth\nof the quasi-uniform mode of a 50 nm \u0002110 nm elliptical\nin-plane MTJ (Sample 3) measured as a function of bias\ncurrent. In Fig. 3, blue symbols show the linewidth mea-\nsured at an in-plane magnetic \feld su\u000eciently far fromthe three-magnon resonance \felds Hn. At this \feld, the\nexpected quasi-linear dependence of the linewidth on Idc\nis observed for currents well below the critical current\nfor the excitation of auto-oscillatory magnetic dynamics.\nNear the critical current, the linewidth increases due to\na combination of the fold-over e\u000bect [37{39] and ther-\nmally activated switching between the large- and small-\namplitude oscillatory states of the fold-over regime [22].\nThe red symbols in Fig. 3 show the linewidth measured\nin the resonant three-magnon regime at H=H1. In con-\ntrast to the nonresonant regime, the linewidth increases\nwith increasingjIdcjfor both current polarities. Fur-\nthermore, the maximum linewidth is measured for the\nantidamping current polarity.\nIII. THEORETICAL MODEL\nNonlinear interactions among spin wave eigenmodes\nof a ferromagnet give rise to a number of spectacu-\nlar magneto-dynamic phenomena such as Suhl instabil-\nity of the uniform precession of magnetization [40, 41],\nspin wave self-focusing [42] and magnetic soliton forma-\ntion [43{45]. In bulk ferromagnets, nonlinear interac-\ntions generally couple each spin wave eigenmode to a\ncontinuum of other modes via energy- and momentum-\nconserving multi-magnon scattering [40]. This kinemat-\nically allowed scattering limits the achievable amplitude\nof spin wave modes and leads to broadening of the spin\nwave resonance. These processes lead to a resonance\nbroadening [40, 46{48] and cannot explain the observed\nsplit-peak lineshape of the resonance. In nanoscale ferro-4\nmagnets, geometric con\fnement discretizes the spin wave\nspectrum and thereby generally eliminates the kinemati-\ncally allowed multi-magnon scattering. This suppression\nof nonlinear scattering enables persistent excitation of\nspin waves with very large amplitudes [49] as observed in\nnanomagnet-based spin torque oscillators [2, 50]. Tun-\nability of the spin wave spectrum by external magnetic\n\feld, however, can lead to a resonant restoration of the\nenergy-conserving scattering [31]. The description of\nnonlinear spin wave resonance in the nanoscale ferromag-\nnet geometry therefore requires a new theoretical frame-\nwork. To derive the theory of resonant nonlinear damp-\ning in a nanomagnet, we start with a model Hamilto-\nnian that explicitly takes into account resonant nonlinear\nscattering between the quasi-uniform mode and a higher-\norder spin wave mode (in reduced units with ~\u00111):\nH=!0aya+!nbyb+\t0\n2ayayaa+\tn\n2bybybb (1)\n+( naaby+ \u0003\nnayayb)\n+\u0010\b\nexp(\u0000i!t)ay+ exp(i!t)a\t\nwhereay,aandby,bare the magnon creation and an-\nnihilation operators for the quasi-uniform mode j0iwith\nfrequency!0and for the higher-order spin wave mode\njnimode with frequency !n, respectively. The non-\nlinear mode coupling term proportional to the coupling\nstrength parameter  ndescribes the annihilation of two\nj0imagnons and creation of one jnimagnon, as well as\nthe inverse process. The Hamiltonian is written in the\nresonant approximation, where small nonresonant terms\nsuch asaab,aaayare neglected. The terms proportional\nto \t 0and \t ndescribe the intrinsic nonlinear frequency\nshifts [51] of the modes j0iandjni. The last term de-\nscribes the excitation of the quasi-uniform mode by an\nexternal ac drive with the amplitude \u0010and frequency !.\nWe further de\fne classically a dissipation function Q,\nwhere\u000b0and\u000bnare the intrinsic linear damping param-\neters of the modes j0iandjni[52{54]:\nQ=day\ndtda\ndt(\u000b0+\u00110aya) +dby\ndtdb\ndt(\u000bn+\u0011nbyb) (2)\nFor generality, Eq. (2) includes intrinsic nonlinear\ndamping [16] of the modes j0iandjnidescribed by the\nnonlinearity parameters \u00110and\u0011n. However, our analy-\nsis below shows that the split-peak resonance lineshape\nis predicted by our theory even if \u00110and\u0011nare set equal\nto zero.\nEquations describing the nonlinear dynamics of the\ntwo coupled spin wave modes of the system follow from\nEq. (1) and Eq. (2):\nida\ndt=@H\n@ay+@Q\n@(day=dt)(3)\nidb\ndt=@H\n@by+@Q\n@(dby=dt)(4)\nIt can be shown (Supplemental Material) that these\nequations have a periodic solution a= \u0016aexp (\u0000i!t) and\n100 50 0 50 10000.10.20.30.40.5\nI (A)f0 (GHz)Resonant regime\nNonresonant regimeFIG. 3. E\u000bect of spin torque on linewidth. Linewidth of the\nquasi-uniform spin wave mode as a function of the applied\ndirect bias current (Sample 3): blue symbols { in the non-\nresonant regime H6=H1and red symbols { in the resonant\nthree-magnon regime H=H1. Lines are numerical \fts using\nEqs. (3) and (4).\nb=\u0016bexp (\u0000i2!t), where \u0016a,\u0016bare the complex spin wave\nmode amplitudes. For such periodic solution, Eqs. (3)\nand (4) are reduced to a set of two nonlinear algebraic\nequations for absolute values of the spin wave mode am-\nplitudesj\u0016ajandj\u0016bj, which can be solved numerically.\nSince the ST-FMR signal is proportional to j\u0016aj2(Supple-\nmental Material), the calculated j\u0016aj2(!) function can be\ndirectly compared to the measured ST-FMR resonance\nlineshape.\nWe employ the solution of Eqs. (3) and (4) to \ft the\n\feld dependence of the quasi-uniform mode linewidth in\nFig. 1(b). In this \ftting procedure, the resonance line-\nshapej\u0016aj2(!) is calculated, and its spectral linewidth\n\u0001!0is found numerically. The resonance frequencies !0\nand!nare directly determined from the ST-FMR data\nin Fig. 1(a). The intrinsic damping parameters \u000b0and\n\u000bnnearH1andH2are found from linear interpolations\nof the ST-FMR linewidths \u0001 f0and \u0001fnmeasured at\n\felds far from H1andH2. We \fnd that \u0001 !0weakly\ndepends on the nonlinearity parameters \t and \u0011, and\nthus these parameters are set to zero (Supplemental Ma-\nterial). We also \fnd that the calculated linewidth \u0001 !0\ndepends on the product of the drive amplitude \u0010and\nmode coupling strength  n, but is nearly insensitive to\nthe individual values of \u0010and nas long as\u0010\u0001 n= const\n(Supplemental Material). Therefore, we use \u0010\u0001 nas a\nsingle \ftting parameter in this \ftting procedure. Solid\nline in Fig. 1(b) shows the calculated \feld dependence\nof the quasi-uniform mode linewidth on magnetic \feld.\nThe agreement of this single-parameter \ft with the ex-\nperiment is excellent.\nFigures 2(b) and 2(d) illustrate that Eqs. (3) and\n(4) not only describe the \feld dependence of ST-FMR\nlinewidth but also qualitatively reproduce the spectral5\nlineshapes of the measured ST-FMR resonances as well\nas the e\u000bect of the antidamping spin torque on the line-\nshapes. Fig. 2(b) shows the dependence of the calculated\nlineshapej\u0016aj2(!) on antidamping spin torque for a mag-\nnetic \feldHfar from the three-magnon resonance \felds\nHn. At this nonresonant \feld, increasing antidamping\nspin torque induces the fold-over of the resonance curve\n[37] without resonance peak splitting. The dependence of\nj\u0016aj2(!) on antidamping spin torque for H=H1is shown\nin Fig. 2(d). At this \feld, the resonance peak in j\u0016aj2(!)\n\frst broadens with increasing antidamping spin torque\nand then splits, in qualitative agreement with the ex-\nperimental ST-FMR data in Fig. 2(c). Our calculations\n(Supplemental Material) reveal that while the nonlinear-\nity parameters \t 0,\u00110, \tnand\u0011nhave little e\u000bect on\nthe linewidth \u0001 !0, they modify the lineshape of the res-\nonance. Given that the nonlinearity parameter values\nare not well known for the systems studied here, we do\nnot attempt to quantitatively \ft the measured ST-FMR\nlineshapes.\nEquations (3) and (4) also quantitatively explain\nthe observed dependence of the quasi-uniform mode\nlinewidth \u0001 !0on direct bias current Idc. Assuming an-\ntidamping spin torque linear in bias current [36, 55, 56]:\n\u000b0!\u000b0(1\u0000Idc=Ij0i\nc),\u000bn!\u000bn(1\u0000Idc=Ijni\nc), where\nIjni\nc>Ij0i\ncare the critical currents, we \ft the measured\nbias dependence of ST-FMR linewidth in Fig. 3 by solv-\ning Eqs. (3) and (4). The solid lines in Fig. 3 are the best\nnumerical \fts, where \u0010\u0001 nandIcare used as indepen-\ndent \ftting parameters. The rest of the parameters in\nEqs. (3) and (4) are directly determined from the experi-\nment following the procedure used for \ftting the data in\nFig. 1(b). Theoretical curves in Fig. 3 capture the main\nfeature of the data at the three-magnon resonance \feld\nH1{ increase of the linewidth with increasing antidamp-\ning spin torque.\nIV. DISCUSSION\nFurther insight into the mechanisms of the nonlinear\nspin wave resonance peak splitting and broadening by an-\ntidamping spin torque can be gained by neglecting the in-\ntrinsic nonlinearities \t nand\u0011nof the higher-order mode\njni. Setting \t n= 0 and\u0011n= 0 in Eqs. (3) and (4) allows\nus to reduce the equation of motion for the quasi-uniform\nmode amplitudej\u0016ajto the standard equation for a single-\nmode damped driven oscillator (Supplemental Material)\nwhere a constant damping parameter \u000b0is replaced by\nan e\u000bective frequency-dependent nonlinear damping pa-\nrameter\u000be\u000b\n0:\n\u000be\u000b\n0=\u000b0+\u0014\n\u00110+4\u000bn 2\nn\n(2!\u0000!n)2+ 4\u000b2n!2\u0015\nj\u0016aj2(5)and the resonance frequency is replaced by an e\u000bective\nresonance frequency:\n!e\u000b\n0=!0+\u0014\n\t0+2j nj2(2!\u0000!n)\n(2!\u0000!n)2+ 4\u000b2n!2\u0015\nj\u0016aj2(6)\nEquation (5) clearly shows that the damping parame-\nter of the quasi-uniform mode itself becomes a resonant\nfunction of the drive frequency with a maximum at half\nthe frequency of the higher order mode ( !=1\n2!n). The\namplitude and the width of this resonance in \u000be\u000b\n0(!) are\ndetermined by the intrinsic damping parameter \u000bnof\nthe higher-order mode jni. If\u000bnis su\u000eciently small,\nthe quasi-uniform mode damping is strongly enhanced\nat!=1\n2!n, which leads to a decrease of the quasi-\nuniform mode amplitude at this drive frequency. If the\ndrive frequency is shifted away from1\n2!nto either higher\nor lower values, the damping decreases, which can re-\nsult in an increase of the quasi-uniform mode amplitude\nj\u0016aj. Therefore, the amplitude of the quasi-uniform mode\nj\u0016aj(!) can exhibit a local minimum at !=1\n2!n. Due to\nits nonlinear origin, the tendency to form a local min-\nimum inj\u0016aj(!) at1\n2!nis enhanced with increasing j\u0016aj.\nSincej\u0016ajis large near the resonance frequency !0, tun-\ning!0to be equal to1\n2!ngreatly ampli\fes the e\u000bect of\nlocal minimum formation in j\u0016aj(!). This qualitative ar-\ngument based on Equation (5) explains the data in Fig. 2\n{ the split-peak nonlinear resonance of the quasi-uniform\nmode is only observed when external magnetic \feld tunes\nthe spin wave eigenmode frequencies to the three-magnon\nresonance condition !0=1\n2!n.\nEquation (6) reveals that the nonlinear frequency shift\nof the quasi-uniform mode is also a resonant function of\nthe drive frequency. In contrast to the nonlinear damping\nresonance described by Equation (5), the frequency shift\nresonance is an antisymmetric function of !\u00001\n2!n. The\nnonlinear shift is negative for ! <1\n2!nand thus causes\na fold-over towards lower frequencies while it is positive\nfor!>1\n2!ncausing fold-over towards higher frequencies.\nAt the center of the resonance pro\fle, the three-magnon\nprocess induces no frequency shift. This double-sided\nfold-over also contributes to the formation of the split-\npeak lineshape of the resonance shown in Figs. 2(c) and\n2(d) and to the linewidth broadening. As with the non-\nlinear damping resonance, the antisymmetric nonlinear\nfrequency shift and the double-sided fold-over become\ngreatly ampli\fed when the spin wave mode frequencies\nare tuned near the three-magnon resonance !0=1\n2!n.\nEquations (5) and (6) also shed light on the origin\nof the quasi-uniform mode line broadening by the an-\ntidamping spin torque. The antidamping spin torque in-\ncreases the quasi-uniform mode amplitude j\u0016ajvia transfer\nof angular momentum from spin current to the mode [57].\nSince the nonlinear damping and the nonlinear frequency\nshift are both proportional to j\u0016aj2and both contribute to\nthe line broadening, the antidamping spin torque can in-\ndeed give rise to the line broadening. Equation (5) reveals\ntwo competing e\u000bects of the antidamping spin torque on\nthe quasi-uniform mode damping parameter \u000be\u000b\n0: spin6\ntorque from Idcdecreases the linear component of the\ndamping parameter \u000b0!\u000b0(1\u0000Idc=Ij0i\nc) and increases\nthe nonlinear component via increased j\u0016aj2. Whether the\nantidamping spin torque decreases or increases the spec-\ntral linewidth of the mode depends on the system param-\neters. Our numerical solution of Eqs. (3) and (4) shown\nin Fig. 3 clearly demonstrates that the antidamping spin\ntorque can strongly increase the linewidth of the quasi-\nuniform mode when the three-magnon resonance condi-\ntion!0=1\n2!nis satis\fed. Furthermore, we \fnd that\nthe three-magnon process exhibits no threshold behav-\nior upon increasing amplitude (Supplemental Material)\nor decreasing intrinsic damping.\nThe key requirement for observation of the resonant\nnonlinear damping is the discreteness of the magnon\nspectrum imposed by geometric con\fnement in the\nnanoscale ferromagnet. The split-peak nonlinear reso-\nnance discovered in this work cannot be realized in bulk\nferromagnets because the three-magnon resonance con-\ndition in bulk is not only valid at the uniform mode\nfrequency!0=1\n2!nbut instead in a broad frequency\nrange. Owing to the magnon spectrum continuity in\nbulk, shifting the excitation frequency away from !0does\nnot suppress the three-magnon scattering of the uniform\nmode { it simply shifts it from one group of magnons to\nanother [29, 40]. Therefore, the amplitude of the uni-\nform mode does not increase when the drive frequency is\nshifted away from !0and the split-peak resonance is not\nrealized.\nWe expect that the resonant nonlinear damping dis-\ncovered in this work will have strong impact on the\nperformance of spin torque devices such as spin torque\nmagnetic memory, spin torque nanooscillators and spin\ntorque microwave detectors. Since all these devices rely\non large-amplitude oscillations of magnetization driven\nby spin torque, the amplitude limiting resulting from the\nresonant nonlinear damping is expected to have detri-\nmental e\u000bect on the device performance.V. CONCLUSIONS\nIn conclusion, our measurements demonstrate that\nmagnetic damping of spin wave modes in a nanoscale\nferromagnet has a strong nonlinear component of reso-\nnant character that appears at a discrete set of magnetic\n\felds corresponding to resonant three-magnon scattering.\nThis strong resonant nonlinearity can give rise to unusual\nspin wave resonance pro\fle with a local minimum at the\nresonance frequency in sharp contrast to the properties\nof the linear and nonlinear spin wave resonances in bulk\nferromagnets. The resonant nonlinearity has a profound\ne\u000bect on the response of the nanomagnet to spin torque.\nAntidamping spin torque, that reduces the quasi-uniform\nspin wave mode damping at magnetic \felds far from the\nresonant three-magnon regime, can strongly enhance the\ndamping in the resonant regime. This inversion of the\ne\u000bect of spin torque on magnetization dynamics by the\nresonant nonlinearity is expected to have signi\fcant im-\npact on the performance of nanoscale spin torque devices\nsuch as magnetic memory and spin torque oscillators.\nACKNOWLEDGMENTS\nThis work was supported by the National Science\nFoundation through Grants No. DMR-1610146, No.\nEFMA-1641989 and No. ECCS-1708885. We also ac-\nknowledge support by the Army Research O\u000ece through\nGrant No. W911NF-16-1-0472 and Defense Threat Re-\nduction Agency through Grant No. HDTRA1-16-1-0025.\nA. M. G. thanks CAPES Foundation, Ministry of Educa-\ntion of Brazil for \fnancial support. R.E.A acknowledges\nFinanciamiento Basal para Centros Cienti\fcos y Tec-\nnologicos de Excelencia under project FB 0807 (Chile),\nand Grant ICM P10-061-F by Fondo de Innovacion para\nla Competitividad-MINECON. 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Page, Erick C. Blomberg, Denis V.\nPelekhov, and P. C. Hammel, \\Engineering the spec-\ntrum of dipole \feld-localized spin-wave modes to enable\nspin-torque antidamping,\" Phys. Rev. Appl. 7, 054019\n(2017).\n[57] S. M. Rezende, F. M. de Aguiar, and A. Azevedo,\n\\Magnon excitation by spin-polarized direct currents\nin magnetic nanostructures,\" Phys. Rev. B 73, 094402\n(2006).Supplemental Material:\nGiant resonant nonlinear damping in nanoscale ferromagnets\nI. Barsukov,1H. K. Lee,1A. A. Jara,1Y.-J. Chen,1A. M. Gon¸ calves,1C.\nSha,1J. A. Katine,2R. E. Arias,3B. A. Ivanov,4, 5and I. N. Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Western Digital, 5600 Great Oaks Parkway, San Jose, CA 95119, USA\n3Departamento de F´ ısica, CEDENNA, FCFM, Universidad de Chile, Santiago, Chile\n4Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadsky av. 36 B, Kyiv, 03142, Ukraine\n5National University of Science and Technology MISiS, Moscow, 119049, Russian Federation\nI. METHODS\nA. Linewidth evaluation\nAll measurements presented were carried out with magnetic field applied along the easy axis of the MTJ devices so\nthat the magnetic moments of the free and pinned layers are collinear to each other. In this geometry, the ST-FMR sig-\nnals are dominated by photo-resistance contribution and are proportional to the square of the transverse component of\nthe dynamic magnetization magnetization [1], which allows us to directly compare calculated |a|2(ω) resonance curves\nto measured ST-FMR resonance curves ˜Vmix(f) and toVmix(f) approximated by numerical integration/integraltext˜Vmix(f)df.\nWhenVmix(f) and|a|2(ω) are single-peak curves, they are fit to a sum of symmetric and antisymmetric Lorentzian\ncurves with identical central frequencies and linewidth parameters as described in Ref. [2], and the spectral linewidth\nis determined as half-width at the half-maximum of the symmetric Lorentzian curve.\nIn order to quantify the linewidth of the split-peak resonance profile, we introduce a fitting function that is a sum\nof two Lorentzian curves with different central frequencies separated by δf. The half width of the resonance profile\n∆f0is then defined as the average of the half widths of the two Lorentzians plus δf/2.\nSupplemental Figure 1. Spatial profiles of spin wave eigenmodes. Normalized amplitude and phase of the three lowest frequency\nspin wave eigenmodes of the MTJ free layer, given by micromagnetic simulations.\nB. Micromagnetic simulations\nMicromagnetic simulations were performed using OOMMF software [3, 4]. To account for all magnetic interactions\nin the MTJ, a three dimensional model was employed with three ferromagnetic layers: free, SAF top and SAF bottom.\nWe use material parameters obtained from the measurements and/or their accepted literature values (see Ref. [2] for2\nthe MTJ structure and fabrication details). Magnetization dynamics is excited by a combined pulse of spin torque\nand Oersted field, resulting from a sinc-shaped spatially uniform current pulse. The spatial profile of the Oersted\nfield corresponds to that of a long wire with elliptical cross section. The direction of the spin torque vector acting\non the free layer is determined by the magnetization orientation of the SAF top layer. The spectrum of spin wave\neigenmodes is obtained via fast Fourier transform (FFT) of the time dependent components of the layers’ magnetic\nmoment. Spatial mapping of the resulting Fourier amplitude and phase at a given frequency provides the mode\nprofiles (Supplmental Fig. 1). The observed excitations are confirmed to be spin wave modes localized to the free\nlayer. SAF modes are found at much higher frequencies than the free layer modes, and their frequencies are found to\nbe incommensurable to the free layer quasi-uniform mode frequency [5].\nII. SOLUTION OF THE EQUATIONS OF MOTION\nThe Hamiltonian equations of motion describing the coupled dissipative dynamics of the quasi-uniform ( a) and the\nhigher-order ( b) spin wave modes are:\nida\ndt=∂H\n∂a†+∂Q\n∂(da†/dt)(1)\nidb\ndt=∂H\n∂b†+∂Q\n∂(db†/dt)(2)\nwhereHis the Hamiltonian of the system and Qis the dissipation function, given by:\nH=ω0a†a+ωnb†b+1\n2Ψ0a†a†aa+1\n2Ψnb†b†bb+ (ψ∗\nnaab†+ψna†a†b) +ζ{exp(−iωt)a†+ exp(iωt)a} (3)\nQ=da†\ndtda\ndt(α0+η0a†a) +db†\ndtdb\ndt(αn+ηnb†b) (4)\nBy using Eq. (3) and Eq. (4) in Eq. (1) and Eq. (2), the Hamiltonian equations can be written as:\nida\ndt−(α0+η0a†a)da\ndt=ω0a+ 2ψna†b+ Ψ 0a†aa+ζexp(−iωt) (5)\nidb\ndt−(αn+ηnb†b)db\ndt=ωnb+ψ∗\nnaa+ Ψnb†bb (6)\nUsing a periodic ansatz a= ¯aexp(−iωt) andb=¯bexp(−2iωt) in Eq. (5) and Eq. (6), where ¯ aand¯bare complex\namplitudes, reduces the Hamiltonian equations to a set of two algebraic equation for the complex amplitudes:\n/parenleftbig\nω−ω0−Ψ0|¯a|2+i(α0+η0|¯a|2)ω/parenrightbig\n¯a−2ψn¯a∗¯b=ζ (7)\n/parenleftbig\n2ω−ωn−Ψn|¯b|2+ 2i(αn+ηn|¯b|2)ω/parenrightbig¯b=ψ∗\nn¯a2(8)\nWe solve Eq. (8) for ¯band multiply the numerator and denominator of this expression by the complex conjugate of\nthe denominator:\n¯b=ψ∗\nn¯a2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n−i2(αn+ηn|¯b|2)ω\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(9)\nthen we multiply Eq. (9) by2ψn¯a∗\n¯aand evaluate the real and imaginary parts.\n/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=|ψn|2|¯a|22/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(10)\n/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=|ψn|2|¯a|2−4(αn+ηn|¯b|2)ω\n/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(11)\nBy taking the modulus of Eq. (8), we obtain:\n|¯a|2=|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2 (12)3\nUsing Eq. (12) in Eqs. (10-11), we derive:\n/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(13)\n/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg\n=−4(αn+ηn|¯b|2)ω|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2(14)\nTaking the modulus squared of Eq. (7):\n/braceleftBigg/parenleftbigg\nω−ω0−Ψ0|¯a|2−/Rfractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg/parenrightbigg2\n+/parenleftbigg\n(α0+η0|¯a|2)ω−/Ifractur/bracketleftbigg2ψn¯a∗¯b\n¯a/bracketrightbigg/parenrightbigg2/bracerightBigg\n|¯a|2=ζ2(15)\nand using Equations (12)–(14) in Eq. (15) gives us an algebraic equation for the absolute value of the higher order\nmode amplitude|¯b|:\n\n\n\nω−ω0−Ψ0|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2−2/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig\n|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2\n2\n+\n\n/parenleftbigg\nα0+η0|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2/parenrightbigg\nω−−4(αn+ηn|¯b|2)ω|ψn||¯b|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2\n2\n\n×\n|¯b|\n|ψn|/radicalBig/parenleftbig\n2ω−ωn−Ψn|¯b|2/parenrightbig2+ 4(αn+ηn|¯b|2)2ω2=ζ2\n(16)\nAfter numerically solving Eq. (16) for |¯b|, and using it in Eq. (12), we can calculate the amplitude of the quasi-uniform\nmode|¯a|.\nIII. EFFECTS OF THE DRIVE AMPLITUDE AND INTRINSIC NONLINEARITITES\nTo understand the impact of the intrinsic nonlinearity parameters (Ψ 0, Ψn,η0,ηn) on the quasi-uniform spin wave\nmode resonance, we plot the numerical solution of Eq. (16) in Supplemental Figure 2. Each panel of this figure shows\na reference lineshape of the resonance calculated with all intrinsic nonlinearity parameters set to zero (red curve) and\na lineshape calculated with one of the intrinsic nonlinearity parameter different from zero (blue curve). This figure\nreveals that increasing η0decreases the mode amplitude and slightly increases the linewidth. Increasing ηndecreases\nthe degree of the double-peak lineshape splitting. Increasing Ψ nincreases the lineshape asymmetry. Increasing Ψ 0\nincreases lineshape asymmetry and induces fold-over.\nSupplemental Figure 3 shows the linewidth as a function of the drive amplitude for three scenarios, where the\nintrinsic nonlinearities Ψ 0, Ψn,η0,ηnare set to zero for simplicity. If the coupling parameter is zero, ψn= 0, the\nlinewidth does not depend on the drive amplitude, as expected for a single-mode linear oscillator. The second case\ndemonstrates that the linewidth remains constant when the product ψn·ζis constant. For a constant non-zero\ncoupling parameter, the linewidth shows an increase with the drive amplitude. This observation allows us to employ a\nsingle fitting parameter ( ψn·ζ) to fit the data in Fig. 1b. This conjecture can be confirmed analytically by introducing\na normalized spin wave amplitude ˆ a=ψn¯a, which allows us to rewrite Eq. (16) omitting all intrinsic nonlinearities\ninto the following form:\nω/bracketleftbigg\n1 +iα0+i4αn|ˆa|2\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\nˆa−ω0ˆa−2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2|ˆa|2ˆa=ψnζ (17)\nThis equation describes an effective single-mode nonlinear oscillator with renormalized excitation amplitude ψnζ.4\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nη0\n2 2.5 3 3.500.010.02\nFrequency (GHz)|a|20\nηn\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nΨ0\n2 2.5 3 3.500.010.02\nFrequency (GHz)0\nΨn|a|2|a|2|a|2a b\nc d\nSupplemental Figure 2. Effect of intrinsic nonlinearities on the quasi-uniform spin wave resonance lineshape. Spectral lineshape\nof the quasi-uniform spin wave mode resonance |¯a|2(ω) at the three-magnon resonance condition 2 ω0=ωncalculated by\nnumerically solving Eq. (16). The red curve is a reference lineshape calculated with all intrinsic nonlinearity parameters\n(η0,ηn,Ψ0,Ψn) set to zero. The blue lineshape in each panel is calculated with one of the intrinsic nonlinearity parameters set\nto a non-zero value: (a) η0= 1.325·10−24J, (b)ηn= 3.313·10−24J, (c) Ψ 0= 1.325·10−24J, (d) Ψ n= 1.325·10−23J. Other\nparameters employed in the calculation are: ω0= 2π·2.63 GHz,ωn= 2π·5.26 GHz;α0= 0.02662,αn= 0.03042 atIdc= 0;\nψn·ζ=h2·0.006 GHz2, wherehis the Planck constant.\n0 0.05 0.1 0.15 0.20.150.20.25\nζ h-1 (GHz)∆f0 (GHz)\n(i)(ii)(iii)\nSupplemental Figure 3. Effect of the drive amplitude on linewidth in the resonant three-magnon regime. Calculated linewidth\nof the quasi-uniform spin wave mode as a function of the drive amplitude ζfor different values of the mode coupling parameter\nψn. (i) Green: ψn= 0, (ii) red: variable ψnwith a constraint ψn·ζ=h2·0.006 GHz2, and (iii) blue: ψn=h·0.1 GHz. All\nintrinsic nonlinearity parameters: Ψ 0, Ψn,η0andηnare set to zero. his the Planck constant. Other parameters employed in\nthe calculation are: ω0= 2π·2.63 GHz,ωn= 2π·5.26 GHz;α0= 0.02662 andαn= 0.03042 atIdc= 0.5\nIV. EFFECTIVE SINGLE-MODE NONLINEAR OSCILLATOR APPROXIMATION\nIf we neglect intrinsic nonlinearities Ψ nandηnof the higher order spin wave mode, Eq. (16) can be reduced to a\ncubic equation for ¯ aand solved analytically. This approximation allows us to obtain several important qualitative\ninsights into the properties of the resonant nonlinear damping of the quasi-uniform mode. By setting Ψ n= 0 and\nηn= 0 in Eq. (8), we obtain an exact solution for ¯b:\n¯b=ψ∗\nn¯a2\n2ω(1 +iαn)−ωn(18)\nUsing this result, we reduce Eq. (16) to a cubic algebraic equation for ¯ a:\nω/bracketleftbigg\n1 +i(α0+η0|¯a|2) +i4|ψn|2αn|¯a|2\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n¯a−ω0¯a−/bracketleftbigg\nΨ0+2|ψn|2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2¯a=ζ (19)\nThis equation describes the amplitude ¯ aof an effective single-mode nonlinear oscillator.\nIt is evident from Eq. (19) that the frequency of the quasi-uniform mode experiences a nonlinear shift:\nωeff\n0=ω0+/bracketleftbigg\nΨ0+2|ψn|2(2ω−ωn)\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2(20)\nThe nonlinear frequency shift has a well-pronounced antisymmetric resonant character near the resonance frequency\nωn/2, that arises from the resonant three-magnon scattering.\nFurther, it is clear from Eq. (19) that the effective damping of the quasi-uniform mode also acquires a term arising\nfrom the three-magnon interaction:\nαeff\n0=α0+/bracketleftbigg\nη0+4|ψn|2αn\n(2ω−ωn)2+ 4α2nω2/bracketrightbigg\n|¯a|2(21)\nThe last term describes a resonant enhancement of the nonlinear damping by three-magnon scattering near the\nresonance frequency ωn/2. Strikingly, the magnitude of the resonant damping enhancement at ωn/2 increases when\nthe intrinsic damping of the higher order mode αndecreases. In the limit αn→0, the effective damping becomes\nαeff\n0→α0+/bracketleftbigg\nη0+2π|ψn|2\nωδ(2ω−ωn)/bracketrightbigg\n|¯a|2(22)\nwhereδis Dirac’s delta function. Equation (21) suggests that the effective damping of the quasi-uniform mode αeff\n0\ncan increase with increasing antidamping spin torque applied to the nanomagnet. Indeed, the antidamping spin torque\ntends to increase the amplitude [6] of the quasi-uniform mode |¯a|and decrease the intrinsic damping parameter of the\nhigher order mode αn→αn(1−Idc/I|n/angbracketright\nc), both enhancing the nonlinear damping term in Eq. (19). For a sufficiently\nlarge mode coupling parameter ψn, the enhancement of the nonlinear damping term by the antidamping spin torque\ncan exceed the reduction of the linear damping parameter α0→α0(1−Idc/I|0/angbracketright\nc) by the torque, leading to an increase\nofαeff\n0byIdc>0 and broadening of the quasi-uniform mode resonance by the antidamping spin torque. This scenario\nis indeed realized in the MTJ devices studied here as demonstrated by the data and calculations in Fig. 3.\nV. MODE COUPLING PARAMETER\nIn this Supplementary Note, we discuss how the coupling parameter between the spin wave modes, ψnin Eq. (3),\ncan be calculated. We consider a very thin, magnetically soft ferromagnetic disk with elliptical cross section, that\nis magnetized in-plane. Within a classical micromagnetic model, we include Zeeman, dipolar and exchange terms in\nthe free energy. An applied field Halong thexdirection (long axis of the ellipse) magnetizes the sample to a nearly\nuniform state. Through a classical Holstein-Primakoff transformation [7] we introduce variables c(/vector x,t) andc∗(/vector x,t) to\ndescribe the magnetization such that the magnetization magnitude is conserved:\nmx= 1−cc∗, m +=c√2−cc∗, m−=c∗√2−cc∗, (23)\nwhere/vector m=/vectorM/Ms, andm±≡mz±imy. Approximating the exchange energy to the fourth order in candc∗, the\nnormalized free energy of the disk, U≡E/4πM2\ns, is given by\nU/similarequal−hx/integraldisplay\n(1−cc∗) dV+ (lex)2/integraldisplay/bracketleftbigg\n/vector∇c·/vector∇c∗+1\n4c2(/vector∇c∗)2+1\n4c∗2(/vector∇c)2/bracketrightbigg\ndV−1\n2/integraldisplay\ndV/vectorhD(/vector m)·/vector m, (24)6\nwithhx≡H/4πMs,lex≡/radicalbig\nA/2πM2sis the exchange length, and /vectorhD(/vector m) =/vectorHD(/vector m)/4πMsis the normalized\ndemagnetizing field. The Landau-Lifshitz equations of motion in the new variables are: i˙c=δU/δc∗,i˙c∗=−δU/δc\nwitht/prime= 4πMs|γ|t.\nAssuming the normal modes involved in three magnon scattering dominate the magnetization dynamics, the free\nenergy in Eq. (24) can be written in terms of amplitudes of these modes, by expressing cin terms of aandb:\nc(/vector x,t)/similarequala(t)f(/vector x) +a∗(t)g(/vector x) +b(t)p(/vector x) +b∗(t)q(/vector x) (25)\nThe functions f,g,p,q can be determined from calculating the linear modes of oscillation of the sample. The terms of\nthe free energy proportional to aab∗anda∗a∗bdescribe the three-magnon process and the magnitude of these terms\ngives the coupling parameter ψn.\nIf the magnetization state is approximated as exactly uniform, the dipolar energy for a very thin film may be\napproximated as UD=m2\nz/2 = (c+c∗)2(1−cc∗/2), and in this case all three-magnon terms are zero. However, when\nthe effects due to the sample edges (such as spatial inhomogeneity of the demagnetization field and edge roughness)\nare taken into account, the equilibrium magnetization configuration is generally nonuniform. In this case, there are\nnon-zero three-magnon terms in the free energy expression. An explicit calculation of the corresponding overlap\nintegrals is necessary for a quantitative prediction of ψn. Refs. [8, 9] show such extensive calculations for circular disks\nand include explicit expressions for the exchange and dipolar energies.\n[1] Michael Harder, Yongsheng Gui, and Can-Ming Hu, “Electrical detection of magnetization dynamics via spin rectification\neffects,” Phys. Rep. 661, 1–59 (2016).\n[2] A. M. Gon¸ calves, I. Barsukov, Y.-J. Chen, L. Yang, J. A. Katine, and I. N. Krivorotov, “Spin torque ferromagnetic\nresonance with magnetic field modulation,” Appl. Phys. Lett. 103, 172406 (2013).\n[3] M. J. Donahue and D. G. Porter, OOMMF User’s Guide (National Institute of Standards and Technology, Gaithersburg,\nMD, 1999).\n[4] Robert D. McMichael and Mark D. Stiles, “Magnetic normal modes of nanoelements,” J. Appl. Phys. 97, 10J901 (2005).\n[5] P. S. Keatley, V. V. Kruglyak, A. Neudert, R. J. Hicken, V. D. Poimanov, J. R. Childress, and J. A. Katine, “Resonant\nenhancement of damping within the free layer of a microscale magnetic tunnel valve,” J. Appl. Phys. 117, 17B301 (2015).\n[6] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, “Spin-transfer-driven\nferromagnetic resonance of individual nanomagnets,” Phys. Rev. Lett. 96, 227601 (2006).\n[7] T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58,\n1098–1113 (1940).\n[8] D. Mancilla-Almonacid and R. E. Arias, “Instabilities of spin torque driven auto-oscillations of a ferromagnetic disk mag-\nnetized in plane,” Phys. Rev. B 93, 224416 (2016).\n[9] D. Mancilla-Almonacid and R. E. Arias, “Spin-wave modes in ferromagnetic nanodisks, their excitation via alternating\ncurrents and fields, and auto-oscillations,” Phys. Rev. B 95, 214424 (2017)."    },    {        "title": "1507.03308v1.Effective_90_degree_magnetization_rotation_in_Co2FeAl_thin_film_Piezoelectric_system_probed_by_microstripline_ferromagnetic.pdf",        "content": "Effective 90-degree magnetization rotation in Co 2FeAl thin film/Piezoelectric system\nprobed by microstripline ferromagnetic resonance\nM. Gueye1, F. Zighem1,\u0003M. Belmeguenai1, M. Gabor2, C. Tiusan2, and D. Faurie1\n1LSPM-CNRS, Université Paris XIII, Sorbonne Paris Cité, 93430 Villetaneuse, France and\n2Center for Superconductivity, Spintronics and Surface Science, Technical University of Cluj-Napoca,\nStr. Memorandumului No. 28 RO-400114, Cluj-Napoca, Romania\n(Dated: July 7th2015)\nMicrostripline ferromagnetic resonance technique has been used to study the indirect magneto-\nelectric coupling occurring in an artificial magnetoelectric heterostructure consisting of a magne-\ntostrictive thin film cemented onto a piezoelectric actuator. Two different modes (sweep-field and\nsweep-frequency modes) of this technique have been employed to quantitatively probe the indirect\nmagnetoelectric coupling and to observe a voltage induced magnetization rotation (of 90 degree).\nThis latter has been validated by the experimental frequency variation of the uniform mode and by\nthe amplitude of the sweep-frequency spectra.\nIn recent decades, the development and the sophisti-\ncation of growth and characterization on the atomic and\nnanometric scale, has led to the elaboration of smart ma-\nterials which reveal captivating phenomena. In the midst\nof these materials, multiferroics magnetoelectric materi-\nals, currently at the cutting edge of spintronics, have at-\ntracted the attention of many groups [1–5]. A flurry of\nresearch activities has been launched in order to explore\nthe physics behind these materials. From the applica-\ntions standpoint, an alternative approach to overcome\nthe scientific impediments of weak magnetoelectric cou-\npling in single-phase multiferroics are two-phase systems\nofferromagneticandferroelectricconstituents[6]. Oneof\nthe major advantages of ferromagnetic/ferroelectric ex-\ntrinsic multiferroics as reported in the literature is the\nflexibility in the choice of the materials [6]. One of the\nmain idea behind the fabrication of artificial magneto-\nelectric multiferroics is the electric or voltage control of\nthe magnetization using electric fields for low power and\nultra fast next generation electronics [7, 8]. The straight-\nforward heterostructure allowing such a control appears\nto be the one presenting a piezoelectric/magnetostrictive\ninterface wherein the interaction vector is the voltage in-\nduced in-plane strains[6, 9–14].\nThe present study concerns the experimental observa-\ntion of the voltage induced magnetization rotation in an\nartificial magnetoelectric heterostructure by probing the\nmagnetic resonance of the uniform precession mode. A\nmethod based only on microstripline ferromagnetic res-\nonance (MS-FMR) experiments is presented [15]. The\ntwo different modes of this technique are presented and\nused to perform a quantitative characterization of the\neffective magnetoelectric coupling, as well as of the volt-\nage inducedmagnetization-rotation. Figures1-a) and-b)\nshow the artificial heterostructure composed of a 25 nm\nCo2FeAl(CFA)thinfilmgrownontoa125 µmthickpoly-\nimide (Kapton ®) flexible substrate and then cemented\non a piezoelectric actuator. The CFA film was deposited\n\u0003Electronic address: zighem@univ-paris13.fronto the Kapton ®substrate by rf sputtering. The de-\nposition residual pressure was of around 4\u000210\u00009mbar,\nwhile the working Ar pressure was of 1:3\u000210\u00003mbar. A\n5 nm thick Ta cap layer was deposited on the top of the\nCFA film in order to protect it from oxidation. CFA was\nchosen because of its non-negligible magnetostriction co-\nefficient at saturation even in a polycrystalline film with\nnopreferredorientation( \u0015\u001914\u000210\u00006)[16,17]whichwill\nproducestrongindirectmagnetoelectricfieldandalsobe-\ncause it is a serious candidate for spintronic applications\n[18–20]. Moreover, we have shown that Kapton substrate\ndoes not affect the whole system magnetoelectric behav-\nior because of its high compliance [9, 11].\nMS-FMR experiments can be carried out in two dif-\nferent modes: i) by fixing the microwave driving fre-\nquency (f0)and sweeping the applied magnetic field (H)\n[11, 15, 18] or ii) by fixing the applied magnetic field\n(H0)and sweeping the microwave frequency (f) [21].\nA resonance field ( Hresat fixedf0) and a frequency\nresonance ( fresat fixedH0) are extracted from these\ntwo modes each having their own advantage. Indeed,\nthe sweep-field mode is more sensitive than the sweep-\nfrequency one (both modes are of course less sensitive\nthan standard FMR technique where resonant microwave\ncavities are used). However, contrary to the sweep-field\nmode, the sweep-frequency mode presents the advantage\nof performing spectra without disturbing the magneti-\nzation distribution of the studied systems [22], provided\nthat the rf field ~hrfamplitude is weak enough (which is\nthe case here). These two modes have been used in the\npresent study to characterize the indirect magnetoelec-\ntric field and to probe the voltage induced magnetization\nrotation.\nA beforehand characterization of the magnetic prop-\nerties of the heterostructure has been performed in the\nbenchmark state (i. e. absence of applied voltage) in\norder to measure the effective magnetization ( Meff) and\nthe gyromagnetic factor ( \r) as well as the unavoidable\nuniaxial anisotropy ( Hu) due to in-plane non-equibiaxial\nresidual stresses. This initial anisotropy is attributed to\nimperfect flatness of the Kapton substrate during deposi-arXiv:1507.03308v1  [cond-mat.mtrl-sci]  13 Jul 20152\nxM\nHM\nH V Piezoelectric actuator(1cm)Kapton® substrate (125µm)Co2FeAl(25nm)Ta(5nm)\na) b)\n400 800 1200 1600 20004812\n0 90 180 270 3601.141.161.18\nFrequency (GHz)\nH=90\nH=0\nAppliedfield(Oe)Hres(kOe)H(degree)y\nc) d)\nFigure 1: a) Depiction of the cross section view of the in-\nvestigated system illustrating the CFA/Kapton ®system de-\nposited onto a piezoelectric actuator. b) Top view sketch of\nthe system showing the different angles used in the text. c)\nAngular dependence of the resonance field measured at 10\nGHz. d) Experimental and calculated frequency dependence\nof the uniform mode measured along the easy and hard axis\n('H= 90and'H= 0).\n01002003004000 50 100 150 200\n1000 1200 1400 1600‐100‐50050100Appliedvoltage(V)\nExperiments\nLinearfitHR(Oe)\nAppliedfield(Oe)Intensity(µV)\n200V 100V 0Vx\nHH V\ny°=90H °=90H\nFigure 2: Up-graph: Resonance field shift ( \u000eHres =\nHres(V)\u0000Hres(0)) variation as function of the applied volt-\nage. Circles refer to experimental data while solid line repre-\nsents to the best fit. Down-graph: typical MS-FMR spectra\nrecorded at different applied voltage (0 V, 100 V and 200 V).\ntion and/or cementation on the actuator [9]. The quan-\ntitative evaluation of this initial uniaxial anisotropy is\nperformed through the angular dependence of the uni-\nform mode resonance field (see Figure 1-c)) measured at\nf0= 10GHz [15]. Furthermore, one can note that this\nanisotropy is aligned along the y-axis since the resonance\nfield minimum reaches a minimum at 'H= 90°. In ad-\ndition, the frequency dependencies of the uniform mode,\nmeasured along the easy ( 'H= 90°) and hard ( 'H= 0°)\nxyrfh\nMoutP\ninPH\nxyrfh\nM\nMoutPoutP\ninPinPHH\nx\ny\ninPoutP\nMrfh\nH x\ny\ninPinPoutPoutP\nMMrfh\nHHa) b)\nc)\n6.4 6.6 6.8 7.0 7.2‐10‐50510\nHighsensitivity\nLowsensitivity\nLorentzian derivative fits\n6.4 6.6 6.8 7.0 7.2‐0.50.00.5\nFrequency (GHz)Differentiial FMRabsorption (V)Figure 3: a) Sweep-frequency FMR spectra obtained accord-\ning to the two configurations depicted in b) and c) where Pin\nandPoutaretheinjectedandtransmittedradiofrequencycur-\nrent. The only difference comes from the pumping field ( ~hrf)\nwhich either parallel (red squares) or perpendicular (blue cir-\ncles) to~Mdirection giving rise to the high and low sensitivi-\nties geometries. The inset shows a zoom of the marked area.\nNote that a saturating field of 500 Oe was applied to ensure a\nuniform magnetization distribution. The solid lines show fits\nto the first derivative of the standard Lorentzian curve.\naxes are shown in Figure 1-d). The full lines in Figure\n1-c) and -d) are the best fits to the experimental data\nand allowed the determination of the following parame-\nters:Meff= 720emu.cm\u00003,\r= 1:835\u0002107Hz.Oe\u00001\nandHu\u001830Oe, which are close to the values previously\nobtained for similar films [16].\nSweep-field MS-FMR spectra have been recorded as a\nfunction of the applied voltage (for f0= 10GHz). This\nlatter was varied from 0 V to 200 V by 10 V steps. Figure\n2 presents typical spectra measured at different applied\nvoltages (0V, 100 V and 200 V) with an applied mag-\nnetic field along yaxis ('H= 90°, i. e. along the ini-\ntial easy axis). A shift of the resonance field, defined as\n\u000eHres=Hres(V)\u0000Hres(0), isclearlyobserved. Thevolt-\nage dependence of \u000eHresis shown in Figure 2, where the\nsolid line corresponds to a linear fit showing that \u000eHresis\npositive and almost proportional to V. This means that\ntheydirection gradually becomes a hard magnetization\naxis, or a less easy one for the magnetization direction\n[11, 15, 17]. This is due to the positive magnetostriction\ncoefficient of Co 2FeAl[16, 17] and to the nearly uniaxial\ntensile stress (i. e. \u001bxx(V)>0) along the xdirection\napplied by the piezoelectric actuator to the film (for this\nrange of applied voltage: 0 to 200V[23]). Note that a non\nlinear and hysteretic behavior is observed if the voltage\nis applied backward from 200 V to 0 V (not shown here)\ndue to the intrinsic properties of the piezoelectric mate-\nrial [9–11, 15, 23]. It should be noted that the \u000eHres(V)\ncurve presented in Figure 2 has been measured several\ntimes and all the obtained curves were superimposed.\nFurthermore, the resonance field shift can also be viewed\nas an indirect magnetoelectric field Hme(V) =\u000eHres(V)\naligned along xdirection, i. e. perpendicular to the ini-3\ntial easy axis induced by the deposition conditions. The\nevaluation of the voltage dependence of Hmewill serve\nto quantitatively study the magnetization rotation using\nthe sweep-frequency mode of the MS-FMR setup.\nThe sweep-frequency-mode has been used to probe the\nmagnetization direction as function of the applied volt-\nage by measuring both the resonance frequency and the\nspectra amplitudes. For this purpose, the selective rule\nillustrated in Figure 3 has been be used. In this Fig-\nure, two spectra performed with a similar applied mag-\nnetic field (of 500 Oe) along yare presented; the only\ndifference comes from the direction of the weak rf field\n~hrf(also called pumping field) emanating from the mi-\ncrostripline, which is either parallel ( ~hrfk~M) or perpen-\ndicular (~hrf?~M) to the static magnetization. Indeed,\nthe efficiency of the pumping field is maximum when\n~hrf?~M(Figure 3-b) and minimum when ~hrfk~M(Fig-\nure 3-b). In the present experimental conditions, a factor\n20 is found between these two geometries. Furthermore,\nCounil et al.[24, 25] demonstrated both theoretically\nand experimentally that for a pure Lorentzian profile of\nthe signal, the amplitude of this signal is directly propor-\ntional to sin2', where'is the angle between the pump-\ning field and the static magnetization direction. There-\nfore, the voltage-induced rotation of the magnetization\nhas been probed by measuring the evolution of the rel-\native amplitude of the signal as function of the applied\nvoltage.\nFor this purpose, the pumping field was applied paral-\nlel toxdirection (~hrfk~ x), i. e. perpendicular to the\ninitial easy axis (see Figure 1). In order to ensure an ini-\ntial uniform magnetization along the yaxis (initial easy\naxis), a small bias field of 40 Oe was applied. In this\ncondition and in absence of applied voltage, ~Mis sup-\nposed to be uniform and aligned along yaxis. Then, the\nvoltage applied to the actuator, in steps of 10 V, leads\nto the development of an indirect magnetoelectric filed\nparallel to the xaxis, which tends to align ~Malong this\naxis. The normalized component of the magnetization\nalong the indirect magnetoelectric field mx(Hme)is rep-\nresented in Figure using the following formula deduced\nfrom the model presented by Counil et al.[24]:\nmx(Hme) =cos\u0010\narcsin(p\nAN(V)\u0011\ncos\u0010\narcsin(p\nAN(150V)\u0011(1)\nwhereAN(150V)andAN(V)are the normalized (by\nthe0Vamplitudespectrum)amplitudeofthe150Vspec-\ntrum and of the applied voltage spectra, respectively.\nThe red solid line in Figure corresponds to the theo-\nretical value of the normalized x-component of the mag-\nnetization as function of V. One can note that the rota-\ntion is complete at around 40 V ( Hme= 70Oe) where\nthe magnetoelectric field totally compensates the small\nbias field (Hb= 40Oe) and the initial anisotropy field\n(30 Oe). The confrontation between the model and the\n0 25 50 75 100 125 15001234\n0.000.250.500.751.000 40 80 120 160 200 240Frequency (GHz)\nAppliedvoltage(V)\nM(Hme)/MSHme(Oe)\ny\nx\ninPoutP\nM\n0V\nrfhy\nxy\nx\ninPinPoutPoutP\nM\n0VMM\n0V\nrfh\nrfh\ninPoutP\nM\n20V\nrfh\ninPinPoutPoutP\nMM\n20V\nrfh\nrfh\ninPoutP\nM\nrfh 40V\ninPinPoutPoutP\nM\nM\nrfh\nrfh 40VFigure 4: Voltage dependencies of the uniform mode fre-\nquency (blue symbols) and of the normalized x-component\nof the magnetization mx(Hme)with a pumping field applied\nalongx. Themx(Hme)values have been deduced from equa-\ntion1. Thesolidlinesarecalculatedusingequation2withthe\nfollowing parameters: Meff= 720emu.cm\u00003,\r= 1:835\u0002107\ns\u00001,Hu= 30Oe,Hb= 40Oe. Note that the Hmeis deduced\nfrom Figure 2. The sketches show how the magnetization\ndirection rotates as function of the applied voltage.\nexperimental data proves that a voltage induced mag-\nnetization rotation has been performed. Another proof\nof the effective rotation of the magnetization comes from\nthe voltage dependence of the uniform mode frequency as\nfunctionoftheappliedvoltage. ThebluesolidlineinFig-\nure refers to the simulation of this frequency being given\ntheHme(V)dependence previously determined (see Fig-\nure 2). This dependence highlights the well-known soft-\nening of the frequency when the magnetization rotates\nfrom a hard axis to an easy one [26]; the frequency mini-\nmum directly gives the voltage at which the system is in-\nplane isotropic (i. e. where Hmecompensates the initial\nanisotropy field and the bias field). Both the experimen-\ntal frequency and the amplitude dependencies are well\nfitted by the solid lines and they clearly show a 90 degree\nof the magnetization direction. The theoretical values\nhave been calculated by using a magnetic energy den-\nsity of the film where the indirect magnetoelectric field is\nmodeledbyavoltagedependentuniaxialanisotropyfield.\nIn these conditions, the following analytical expression of\nthe frequency can be derived from the Smit-Beljers equa-\ntion [15]:\n2\u0019f\n\r=p\n(Hme(V)\u0000Hu) cos 2'+Hbsin'\u0002\nq\n4\u0019Ms+Hbsin'+Husin2'+Hme(V) cos2'\n(2)4\nwhere'is evaluated at each increment of the applied\nvoltage. Finally, it should be noted that the spectra pro-\nfiles were not always close to a Lorentzian one, especially\nat high applied voltages, which make this modeling really\nrobust [24, 25].\nIn summary, the indirect magnetoelectric coupling in\nartificial heterostructure was used to rotate the mag-\nnetization direction. MS-FMR was used in two differ-ent modes to: i) quantitatively measure the voltage-\ndependence of the indirect magnetoelectric field (which\nis strain-induced by the piezoelectric actuator) and to ii)\nobserve the rotation of the magnetization. The voltage\ndependence of the Hmewas found to be almost linear (in\nthe range 0V-200V) and was used during the frequency\nand amplitude variations of the sweep-frequency spectra\nmodeling.\n[1] N. A. Spaldin and M. Fiebig, Science, 309, 391 (2005)\n[2] S. -W. Cheong and M. Mostovoy, Nature Materials 6, 13\n(2007)\n[3] R. Ramesh and N. A. Spaldin, Nature Materials 6, 21\n(2007)\n[4] M. Bibes and A. Barthélémy, Nature Materials 7, 425\n(2008)\n[5] W. Eerenstein, N. Mathur, and J. Scott, nature 442, 759\n(2006)\n[6] Ce-Wen Nan, M. I. Bichurin, S. Dong, D. Viehland, and\nG. Srinivasan, Journal of Apllied Physics 103, 031101\n(2008)\n[7] N. X. Sun and G. Srinivasan, Spin, 2, 1240004 (2012)\n[8] J. Ma, J. Hu, Z. Li, Ce-Wen Nan, Advanced Materials\n23, 1062 (2011)\n[9] M. Gueye, F. Zighem, D. Faurie, M. Belmeguenai and S.\nMercone, Applied Physics Letters 105, 052411(2014)\n[10] M. Weiler, A. , S. Geprägs, M. Althammer, M. Opel, C.\nBihler, H. Huebl, M. Brandt, R. Gross, and S. Goennen-\nwein, New Journal of Physics 11, 013021 (2009)\n[11] F. Zighem, D. Faurie, S. Mercone, M. Belmeguenai, H.\nHaddadi, Journal of Applied Physics 114, 073902 (2013)\n[12] J. H. Park, J.-H. Lee, M. G. Kim, Y. K. Jeong, M.-A.\nOak, H. M. Jang, H. J. Choi, and J. F. Scott, Physical\nReview B 81, 134401 (2010)\n[13] C. Thiele, K. Dörr, O. Bilani, J. Rödel, and L. Schultz,\nPhysical Review B 75, 054408 (2007)\n[14] C. Bihler, M. Althammer, A. Brandlmaier, S. Geprägs,\nM.Weiler, M. Opel, W. Schoch, W. Limmer, R. Gross,\nM. S. Brandt, and S. T. B. Goennenwein, Phys. Rev. B\n78, 045203 (2008).\n[15] F. Zighem, A. El Bahoui, J. Moulin, D. Faurie, M.\nBelmeguenai, S. Mercone and H. Haddadi, Journal ofApplied Physics 116, 123903 (2014)\n[16] M. Gueye, B. M. Wague, F. Zighem, M. Belmeguenai,\nM. S. Gabor, T. Petrisor, C. Tiusan, S. Mercone, and D.\nFaurie, Applied Physics Letters 105, 062409 (2014).\n[17] S. Li, J. Xu, Q. Xue, H. Du, Q. Li, C. Chen, R. Yang, S.\nXie, M. Liu, T. Nan, N. X. Sun, and W. Shao, Journal\nof Applied Physics 117, 17B722 (2015)\n[18] M. Belmeguenai, H. Tuzcuoglu, M. Gabor, T. Petrisor\nJr, C. Tiusan, D. Berling, F. Zighem, T. Chauveau, S.\nChérif, and P. Moch, Physical Review B 87, 184431\n(2013)\n[19] W. Wang, H. Sukegawa, R. Shan, S. Mitani, and K. In-\nomata, Applied Physics Letters 95, 182502 (2009)\n[20] M. S. Gabor, M. Belmeguenai, F. Zighem, S. M. Chérif,\nT. Petrisor Jr, T. Petrisor, C Tiusan and M Hehn, Spin,\n4, 1440022 (2014)\n[21] F.Zighem,Y.Roussigné,S.M.ChérifandP.Moch,Jour-\nnal of Physics: Condensed Matter 20, 125201 (2008)\n[22] J. Ben Youssef, N. Vukadinovic, D. Billet and M.\nLabrune, Physical Review B 69, 174402 (2004)\n[23] F. Zighem, M. Belmeguenai, D. Faurie, H. Haddadi and\nJ. Moulin, Review of Scientific Instruments, 85, 103905\n(2014)\n[24] G. Counil, Joo-Von Kim, T. Devolder, C. Chappert, K.\nShigeto and Y. Otani, Journal of Applied Physics 95,\n5646 (2004)\n[25] G. Counil, J.-V. Kim, T. Devolder, P. Crozat, C. Chap-\npert and A. Cebollada, Journal of Applied Physics 98,\n023901 (2005)\n[26] F.Zighem,Y.Roussigné,S.M.ChérifandP.Moch,Jour-\nnal of Physics: Condensed Matter 19, 176220 (2007)"    },    {        "title": "1509.01807v1.Study_of_spin_dynamics_and_damping_on_the_magnetic_nanowire_arrays_with_various_nanowire_widths.pdf",        "content": "1 \n Study of spin dynamics  and damping  on the magnetic nanowire  arrays   \nwith various nanowire widths  \n \nJaehun Cho a, Yuya  Fujii b, Katsunori  Konioshi b, Jungbum Yoon c, Nam -Hui Kim a, Jinyong Jung a, \nShinji Miwa b, Myung -Hwa Jung d, Yoshishige Suzuki b, and Chun -Yeol You  a,* \n \na Department of Physics, Inha University , Inch eon, 402-751, South Korea  \nb Graduate School of Engineering Science,  \nOsaka University,  Toyonaka, Osaka 560 -8531,  Japan  \nc Department of Electrical and Computer  Engineering ,  \nNational University of Singapore , Singapore  117576  \nd Department of Physics, Sogang University, Seoul, 121 -742, South Korea  \n \nAbstract  \nWe investigate the spin dynamics including Gilbert damping in the ferromagnetic  nanowire \narray s. We have measured  the ferromagnetic resonance of ferromagnetic nanowire arrays \nusing vector -network analyzer ferromagnetic resonance  (VNA -FMR)  and analyzed the results \nwith the micromagnetic si mulations . We find excellent agree ment  between the experimental \nVNA -FMR spectra and micromagnetic simulations result  for various applied magnetic fields . \nWe find that  the demagnetization factor for longitudinal conditions, Nz (Ny) increases  \n(decreas es) as decreasing the nanowire width  in the micromagnetic simulation s. For the \ntransverse magnetic field , Nz (Ny) increas es (decreas es) as increasing the nanowire width . We \nalso find that t he Gilbert damping constant increases  from 0.018 to 0.051 as the incr easing \nnanowire width  for the transverse case , while it is almost constant as 0.021  for the \nlongitudinal case . \n 2 \n * Corresponding author.  FAX: +82 32 872 7562.  \nE-mail address:  cyyou@inha.ac.kr  \nKeywords : Nanowires , Ferromagnetic Resonance , Micromagnetic  Simulations , Gilbert \ndamping  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 3 \n  \nFerromagnetic nanostructures have recently attracted much interest for the wide potential \napplications in high density spintronic information storage , logic devices and various spin \norbit torque phenomena .1,2,3,4,5 It is well known that the detail spin dynamics of nanostructure \nis far from the one of the bulk’s because of many reasons, different boundary conditions, \nchanges of the magnetic properties including the saturat ion magnetization, anisotropy energy, \nand exchange stiffness constant, etc. Since the magnetic properties are usually sensitive \nfunctions of the sample fabrication conditions, it has been widely accepted that the detail \nsample fabrications are also importa nt in the study of spin dynamics. However, the relatively \nless caution has been made for the boundary conditions of the spin dynamics in the \nnanostructure.  \nIn the spin transfer torque magnetic random access memory  (STT -MRAM), the magnetic \ndamping constant is important because the switching current density  is proportional to the \ndamping constant .6 In the nanowire, damping constant also plays crucial role in the spin \ndynamics including domain wall motion with magnetic field7 and spin transfer torque .8 \nFurthermore, it is the most important material parameter in spin wave (SW) dynamics .9 \nDespite of the importance of the damping constant, many studies about spin dynamics in \nferromagnetic nanowires  have not taken into account the damping constant  properly .10,11,12 \nOnly a few studies paid attention to the magnetic  damping in the nanowires  spin 4 \n dynamics .13,14 \nIn this study, arrays of CoFeB nanowire s are prepared by e -beam lithography , and they are  \ncovered coplanar  wave -guide  for the ferromagnetic resonance  (FMR) measurement  as shown \nin Fig. 1 . We measured FMR signal with longitudinal (wire direction) and transverse \nmagnetic field s in order to investigate the spin dynamics with different boundary conditions. \nAlso w e extract Gilbert damping constant  using micromagnetic  simulations  with the different \napplied magnetic field directions in various nanowire arrays . We find the damping constant \ndecreas es with increasing the nanowire width for the transverse magnetic field  with constant \ninput damping consta nt in micromagnetic simulations, while we obtain almost constant \ndamping constant for the longitudinal field.  \nThe film s were  prepared using DC magnetron sputtering. The stack s consist of Ta (5  \nnm)/Co 16Fe64B20 (30 nm)/Ta (5  nm) on single crystal MgO (001) substrate s. The film s are \npatterned as 100 -nm-width wire array s with 200 -nm-space each wires using e -beam \nlithography and an Ar ion milling technique  as shown in Fig. 1. The width is determined with \na scanning electron microscope (SEM).  These nanowire arrays are covered by coplanar wave \nguide in order to characterized with the Vector Network Analyzer ( VNA )-FMR technique \ndescribed elsewhere .15 We prepare nanowire arrays  as shown in Fig. 1 , and external DC \nmagnetic field direction  for FMR measurement  is also depicted.  \nWe use  VNA -FMR spectra to measure  imaginary parts of the susceptibility of the samples.16 5 \n The measured imaginary parts of the susceptibility raw data are calibrated with the careful \ncalibration procedures .16 The calibrated  imaginary parts of the susceptibility are shown in Fig \n2(a) and (b)  for an applied magnetic field at 0.194  T for the nanowire arrays . The un -\npatterned thin film is also examined for the reference. We find two resonance frequencies, \n17.2 and 26 .4 GHz, as shown in Fig. 2(a) for the nanowire array, while  there is only one peak \nat 16.8 GHz  for the  un-patterned  thin film  as shown in Fig 2(b). We believe that the smaller \npeak  (17.2 GHz) in Fig. 2(a) is originated  from the un-patterned part of the nanowire array, \nbecause the frequency is closed to the un -patterned thin film’s peak (16.8 GHz). Probably, the \nun-patterned part of the nanowire is formed due to poor e -beam lithography processes. On the \nother hand, t he resonance f requency  near 26.0 GHz is calculated from micromagnetic \nsimulation  at an applied  magnetic  field at 0.200 T , as shown in Fig. 2 (c).  We clarif y the \nsource of the main  peak (26.4 GHz) is nanowire  arrays  by using micromagnetic simulation.  \nThese two peaks name d as the uniform FMR  mode (smaller peak position) and nanowire \nmode (higher peak position).  \nIn order to determine the saturation magnetization, the resonance frequencies are measured \nas a function of the applied magnetic field, and the results are fitted with the Kittel ’s \nequation .17 This equation employs the corresponding demagnetization factors of Nx = 0, Ny = \n0 and Nz = 1 for un -patterned film, when applied magnetic field H is x- direction  with \nfollowing equations , 6 \n  \n    2y x s z x s f H N N M H N N M\n    \n.   (1) \n \n Here,  is the gyromagnetic ratio, H is the applied magnetic field, Ms is saturated \nmagnetization, Nx, Ny, and Nz are the demagnetization factor s applying the cyclic permutation \nfor the  applied magnetic field direction .  \nThe micromagnetic simulations are performed by using the Objective -Oriented -\nMicroMagnetic Framework (OOMMF)18 with 2-dimensional  periodic boundary  condition \n(PBC ).19 We select a square slat of 100 nm × 100 nm × 30 nm  nanowire separated 200 nm in \ny- direction with a cell size of 5 nm × 5 nm ×  30 nm. The material parameters of CoFeB used \nin our simulation are summarized as follows: Ms = 15.79 × 105 A/m, the exchange stiffness \n1.5 × 1011 J/m, the gyromagnetic ratio 2.32  × 1011 m/(A ∙s) and we ignore the magneto -\ncrystalline anisotropy. In this simulation, the Gilbert damping constant  of 0.0 27 is fixed. The \nsaturation magnetization and Gilbert damping constant are determined by using VNA -FMR  \nmeasurement for un -patterned thin film . For t he exchange  stiffness constant, experimentally \ndetermined values are range of 0.98 to 2.84 × 1011 J/m which value  has dependence on the \nfabrication processes20 and composition of ferromagnetic materials ,21 while we have picked \n1.5 × 1011 J/m as the exchange  stiffness  constant . The determination  method of Gilbert \ndamping constant will be described later.  7 \n In order to mimic FMR experiments in the micromagnetic simulations , a “sinc”  function\n0 0 0 ( ) sin 2 / 2y H HH t H f t t f t t    \n, with H0 = 10 mT, and field frequency fH = 45 \nGHz, is applied the whole nanowire  area.22 We obtain the FMR spectra in the corresponding \nfrequency range from 0 to 45 GHz . The FMR spectra due to the RF -magnetic field are \nobtained by the fast Fourier  transform  (FFT) of stored My(x) (x, y, t ) in longitudinal  (transverse)  \nH0 field. More details can be described elsewhere .23 \nThe closed blue circles  in Fig. 3 is the calculated values with the fitting parameter using Eq. \n(1) which are fitted with the experimental data of un -patterned thin film. The obt ained Ms is \n15.79 × 105 A/m while gyromagnetic ratio is fixed as 2.32 × 1011 m/(A∙s) . The obtained Ms \nvalue is similar  with vibrating sample magnetometer  method24 which CoFeB structure has Ta \nbuffer layer. The resonance frequencies of uniform FMR mode in nanowire arrays are plotted \nas open red circles in Fig. 3. The resonance frequencies of uniform FMR mode is measured \nby VNA -FMR are agreed well with resonance freq uency of un-patterned thin film measured \nby VNA -FMR. In Fig. 3, the applied field dependences of  the resonance frequencies \nMeasured by VNA -FAM for the nanowire are plotted as open black  rectangular , along with \nthe result of micromagnetics calculated with E q. (1)  as depicted  closed black rectangular . It is \nalso well agreed with the experimental result in nanowire mode and micromagnetic \nsimulation result  in the nanowire arrays.  \nIn order to reveal the effect s of spin dynamics properties with various  nanowire width s, we 8 \n perform micromagnetic simulat ions. The nanowire width s are varied from 50 to 150 nm in \n25-nm step for fixed 200 -nm-space with PBC , it causes changes of the demagnetization \nfactor  of the nanowire . In Fig. 4 (a) shows  the nanowire width  dependences of the resonance \nfrequencies for the longitudinal magnetic field (open symbols) along with the resonance \nfrequencies calculated with Eq. (1)  (solid lines) . The demagnetization factors can be \ndetermined by fitting Eq. (1) while Nx is fixed as 0  to represent infinitely long wire . The \nagreements between the results of micromagnetic simulations  (open circles)  and Eq. (1)  \n(solid lines)  are excellent.  \nFor the transverse  magnetic  field, the direction of applied magnetic field is y - axis, Eq. (1) \ncan be rewritten as follows:  \n \n    2x y s z y s f H N N M H N N M\n    \n.    (2) \n \nIn this equation, we use the relation of demagnetization factors , \n1x y zN N N   , in \norder to remove uncertainty in the fitting procedure . In the transverse field, the \ndemagnetization factors are determined by Eq. (2). The resonance frequencies  for transverse \nmagnetic field  which are obtained by micromagnetic simulation  (open circles)  and \ncalculated  by Eq. (2)  (solid lines)  as a function of the appl ied magnetic field with various \nnanowire width  are displayed in Fig. 4(b). The longitudinal case, when the field direction is 9 \n easy axis, they are saturated with small field. However, the transverse case, when the field \ndirection is hard axis, certain amoun t of field is necessary to saturate along the transverse \ndirections. The narrower nanowire, the larger field is required as shown in Fig. 4 (b).  \nFig. 5(a) and (b) show the changes of demagnetization factors in longitudinal and transverse \nmagnetic  fields as a functi on of  the nanowire width , respectively.  The demagnetization \nfactors play important role in the domain wall dynamics, for example the Walker breakdown \nis determined by the demagnetization factors .25 Furthermore, they are essential physical \nquantities to analyze the details of the spin dynamics. It is clear ly shown  that the Nz (Ny) \nincrease s (decrease s) with increasing the nanowire width  in longitudinal magnetic  field. For \nthe transverse magnetic  field, Nz (Ny) increase s (decrease s) with increasing the nanowire \nwidth , during Nx is almost zero value.  The demagnetization factors both longitudinal and \ntransverse have similar tendency with the effective demagnetization factors of dynamic \norigin26 and the static demagnetization factors for the prism geometry.27  \nNow, let us discuss about the Gilbert damping constant . The relation of the full width and \nhalf maxim a (f) of a resonance peak s as a function of applied field are shown in Fig. 6 for \nlongitudinal (a) and transverse (b). The f is given by15: \n \n,\n,2\n22xy\ns z ex yxN\nf H M N N f\n         \n. (3)  \n 10 \n where, fex is the extrinsic line width  contributions , when  the applied magnetic field  is x-(y-\n)axis for longitudinal (transverse) case . The symbol s are the result s of the micromagnetic \nsimulations and the solid lines are the fitting result of Eq. (3) . We use pre -determined \ndemagnetization factors (Fig. 5) during fitting procedures, and the agree ments are excellent.  \nWe have plotted the Gilbert damping constant as a function of the wavevector in nanowire \nwidth  (\n/ qa , a is the nanowire width ) in Fig. 7. The black open rectangles are data \nextracted from the transverse field and the red open circles are longitudinal field data. We \nfind that the Gilbert damping constant varied from 0.051 to 0.018 by changing  the \nwavevector  in nanowire width  in transverse field. On the other hand, lon gitudinal field case \nthe damping constant is almost constant as 0.021. Let us discuss about the un -expected \nbehavior of the damping constant of transverse case. T he wire width acts as a kind of cut -off \nwavelength of the SW excitations in the confined geome try. SWs whose wavelength are \nlarger than 2 a are not allowed in the nanowire. Therefore, only limited SW can be excited for \nthe narrower wire, while more SW can be existed in the wider wire. For example, we show \ntransverse standing SW as profiled  in the inset of Fig. 6 for 150 -nm width nanowire in our \nmicromagnetic simulations. More possible SW excitations imply more energy dissipation \npaths, it causes larger damping constant. For narrower nanowire (50 -nm), only limited SWs \ncan be excited, so that the damping constant is smaller. However, for the limit case of infinite \na case, it is the same with un -patterned thin films, there is no boundar y so that only uniform 11 \n mode can be excited, the obtained damping constant must be the input  value.  \nIn summary, the VNA -FMR experiments is employed  to investigate the magnetic properties \nof CoFeB nanowire  arrays  and the micromagnetic simulations is proposed to understand the \nmagnetic properties including Gilbert damping constant of various CoFeB nanowire  arrays  \nwidth. We f ind that the demagnetization factors are similar with the dynamic origin and static \nfor the prism geometry. The wire width or SW wavevector dependent damping constants can \nbe explained with number of SW excitation modes.  \n \nACKNOWLEDGMENTS  \nThis work  was supported by the National Research Foundation of Korea  (NRF) Grants  (Nos. \n616-2011 -C00017  and 2013R1A12011936 ). \nReferences  \n                                            \n1 S. S. P. Parkin,  M. Hayashi, and L. Thomas, Science 320, 190 (2008) . \n2 D. A. Allwood,  G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn,  \nScience 309, 1688 (2005) . \n3 I. M. Miron,  G. Gaudin, S. Aufftet, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel and P. \nGambardella, Nature Materials 9, 230 (2010) . \n4 H-R Lee , K. Lee, J. Cho, Y . -H. Choi, C. -Y . You, M. -H. Jung, F. Bonell, Y. Shiota, S. Miwa \nand Y . Suzuki, Sci. Rep.  4, 6548 (2014).  \n5 J. Cho, et al.  Nat. Commun. 6, 7635 (2015).  \n6 S. Ikeda , K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. \nHayakawa, F. Matsukura and H. Ohno , Nat. Mater. 9, 721 (2010).  \n7 J.-S. Kim et al. , Nat. Commun . 5, 3429 (2014).  \n8 L. Thomas, R. Moriya, C. Rettner and S. S. P. Parkin , Science  330, 1810 (2010).  12 \n                                                                                                                                         \n9 J.-S. Kim, M. Stark, M. Klaui, J. Yoon, C. -Y . You, L. Lopez -Diaz and E. Martinez , Phys. \nRev. B 85, 174428 (2012).  \n10 B. Kuanr, R. E. Camleym, and Z. Celinski, Appl. Phys. Let t. 87, 012502 (2005) . \n11 J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. Le tt. \n89, 277201 (2002).  \n12 L. Kraus, G. Infante, Z. Frait and M. Vazquez, Phys. Rev. B 83, 174438 (2011) . \n13 C. T. Boone , J. A. Katine, J. R. Childress, V . Tiberkevich, A. Slavin, J. Zhu, X. Cheng and \nI. N. Krivorotov , Phys. Rev. Let t. 103, 167 601 (2009) . \n14 M. Sharma, B. K. Kuanr, M. Sharma and Ananjan Nasu, IEEE Trans. Magn 50, 4300110 \n(2014).  \n15 D.-H. Kim, H .-H. Kim, and Chun -Yeol You, Appl. Phys. Lett. 99, 072502 (2011).  \n16 D.-H. Kim, H. -H. Kim, Chun -Yeol You and Hyungsuk Kim, J. Magn etics 16, 206 (2011).  \n17 C. Kittel, Introduction to Solid State Physics, 7th ed., pp. 504, (1996) . \n18 M. J. Donahue and D. G. Porter: OOMMF User’ s Guide : Ver. 1.0, NISTIR 6376  (National \nInstitute of Standards and Technology, Gaithersburg, Maryland, United States, 1999 ). \n19 W. Wang, C. Mu, B. Zhang, Q. Liu, J. Wang, D. Xue, Comput. Mater. Sci. 49, 84 (2010) . \nSee: http://oommf -2dpbc.sourceforge.net.  \n20 J. Cho, J . Jung, K .-E. Kim, S .-I. Kim, S .-Y. Park, M .-H. Jung, C .-Y. You, J. Magn. Magn. \nMater. 339, 36 (2013).  \n21 C. Bilzer, T. Devolder, J -V . Kim, G. Counil, C. Chappert, S. Cardoso and P. P. Feitas , J. \nAppl. Phys. 100, 053903 (2006).  \n22 K.-S. Lee, D. -S. Han , S.-K. Kim, Phys. Rev. Let t. 102, 127202 (2009).  \n23 J. Yoon, C. -Y . You, Y . Jo, S. -Y. Park, M. H. Jung , J. Korean Phys. Soc. 57, 1594 (2010) .  \n24 Y . Shiota, F. Bonell, S. Miwa, N. Muzuochi, T. Shinjo and Y . Suzuki , Appl. Phys. Le tt. 103. \n082410 (2013),  \n25 I. Purnama, I. S. Kerk, G. J. Lim  and W. S. Lew , Sci. Rep. 5, 8754 (2015).  \n26 J. Ding, M. Kostylev, and A. O. Adeyeye, Phys. Rev. B. 84, 054425 (2011) . \n27 A. Aharoni, J. Appl. Phys. 83, 3432 (2011) . 13 \n Figure  Captions  \n \nFig. 1. Measurement geometry with SEM image s of the 100 -nm-width nanowires with a gap \nof 200 nm between nanowires . The longitudinal nanowire arrays are shown. After the \nnanowire patterns have been  defined by e -beam lithography, they are covered by co -planar \nwave guides.  \n \nFig. 2. (a) The measured FMR  spectrum of the CoFeB nanowire with H =0.194 T. The red  (lower \npeak)  and blue  (higher peak)  arrows indicate t he resonance frequencies of the  uniform FMR  mode and \nthe nanowire mode, respectively. (b) The measured FMR spectrum of the CoFeB thin  film with H \n=0.194 T. (c) Simulated FMR spectrum of the CoFeB nanowire with H= 0.200 T.  \n \nFig. 3. Measured and calculated FMR frequencies with the applied magnetic field for 100 -\nnm-width nanowire. The open black rectangles  are nanowire mode and open red circles are \nthe uniform FMR mode for CoFeB thin film. The closed black rectangles  are calculated by \nOOMMF and the closed blue circles  are theoretical ly calculated by Eq. (1)  using fitted \nparameters form un -patterned film . \n \nFig. 4. Variation of resonanc e frequencies with the applied magnetic field for the different \nPBC wire width for (a) longitudinal field and (b) transverse field.  Inset: The geometry of 2 -\ndimensional PBC micromagnetic simulation with nanowire width a and a gap of 200 nm \nbetween nanowire s. The black open rectangles, red open circles, green open upper triangles, \nblue open down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, \n100 nm, 125nm, and 150 nm, respectively.  \n \nFig. 5. Demagnetization factor with PBC wire widt h for (a) longitudinal and (b) transverse \nfield.  The black open circles, red open rectangles, blue open upper triangles represent as \ndemagnetization factors, Ny, Nz, and Nx, respectively.  \n \nFig. 6. Full width and half maxim a with the applied magnetic field for (a) longitudinal and (b) \ntransverse field.  The black open rectangles, red open circles, green open upper triangles, blue \nopen down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, 100 \nnm, 125nm, and 150 nm, respectively.  \n \nFig. 7. Damping constants with wavevector for transverse ( the black open rectangles ) and \nlongitudinal ( the red open circles) field with errors. The black line is the input value which is \ndetermined from un -patterned film. Inset presents the profile of the trans verse spin density as SWs.  \n Fig. 1 \n \n \n    \n \n      Fig. 2. \n \n \n \nFig. 3. \n \n \n \n  \nFig. 4 \n \n` \n \n \n \nFig. 5 \n \n \n \nFig. 6 \n \n \n \nFig. 7 \n \n \n"    },    {        "title": "2001.02895v1.Hybridization_between_the_ligand__p__band_and_Fe_3_d__orbitals_in_the_p_type_ferromagnetic_semiconductor__Ga_Fe_Sb.pdf",        "content": " 1 Hybridization between the ligand p band and Fe-3d orbitals in the p-type ferromagnetic semiconductor (Ga,Fe)Sb Takahito Takeda1, Masahiro Suzuki2, Le Duc Anh1,3, Nguyen Thanh Tu1,4, Thorsten Schmitt5,  Satoshi Yoshida6, Masato Sakano6, Kyoko Ishizaka6, Yukiharu Takeda7, Shin-ichi Fijimori7,  Munetoshi Seki1,8, Hitoshi Tabata1,8, Atsushi Fujimori2,9, Vladimir N. Strocov5,  Masaaki Tanaka1,8, and Masaki Kobayashi1,8 1Department of Electrical Engineering and Information Systems, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 3Institute of Engineering Innovation. University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 4Department of Physics, Ho Chi Minh City University of Pedagogy, 280, Au Duong Vuong Street, District 5, Ho Chi Minh City 748242, Vietnam 5Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 6Quantum-Phase Electronics Center and Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan 7Synchrotron Radiation Research Unit, Japan Atomic Energy Agency, Sayo-gun, Hyogo 679-5148, Japan 8Center for Spintronics Research Network, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 9Department of Applied Physics, Waseda University, Okubo, Shinjuku, Tokyo 169-8555, Japan (Date: January 9, 2019)  ABSTRACT (Ga,Fe)Sb is a promising ferromagnetic semiconductor for practical spintronic device applications because its Curie temperature (TC) is above room temperature. However, the origin of ferromagnetism with high TC remains to be elucidated. Here, we use soft x-ray angle-resolved photoemission spectroscopy (SX-ARPES) to investigate the valence-band (VB) structure of (Ga0.95,Fe0.05)Sb including the Fe-3d impurity band (IB), to unveil the mechanism of ferromagnetism in (Ga,Fe)Sb. We find that the VB dispersion in (Ga0.95,Fe0.05)Sb observed by SX-ARPES is similar to that of GaSb, indicating that the doped Fe atoms hardly affect the band dispersion. The Fe-3d resonant ARPES spectra  2 demonstrate that the Fe-3d IB crosses the Fermi level (EF) and hybridizes with the VB of GaSb. These observations indicate that the VB structure of (Ga0.95,Fe0.05)Sb is consistent with that of the IB model which is based on double-exchange interaction between the localized 3d electrons of the magnetic impurities. The results indicate that the ferromagnetism in (Ga,Fe)Sb is formed by the hybridization of the Fe-3d IB with the ligand p band of GaSb.  I. INTRODUCTION Ferromagnetic semiconductors (FMSs) are alloy semiconductors in which cation sites are partially replaced by a sizable amount of magnetic impurities, leading to ferromagnetic properties. The ferromagnetism of FMSs is considered to originate from the magnetic interaction between the doped magnetic impurities mediated by the spin of the carriers. This nature is called carrier-induced ferromagnetism1. FMSs have attracted much attention as promising materials to apply for semiconductor spintronic devices, because one can control their magnetic properties by changing the carrier concentration. The p-type Mn-doped III-V FMSs, such as (In,Mn)As2,3,4,5 and (Ga,Mn)As6,7,8, have been intensively studied so far as prototypical FMSs showing carrier-induced ferromagnetism.  3 Spintronics devices based on these FMSs have been fabricated9,10. However, these materials are still seriously problematic for practical applications. Firstly, Mn-doped FMSs are only p-type because the doped Mn atoms act as acceptors in III-V semiconductors. Secondly, the reported maximum values of the Curie temperature (TC) of molecular-beam epitaxy (MBE)-grown Mn-doped FMSs are much lower than room temperature. The highest TC of (Ga,Mn)As reported so far is ~ 200 K11 and that of (In,Mn)As is ~ 90 K12. Recently, Fe-doped III-V FMSs such as n-type (In,Fe)As13, n-type (In,Fe)Sb14, and p-type (Ga,Fe)Sb15 have been successfully grown. Since the doped Fe ions are expected to isovalently substitute for the cation (In3+ or Ga3+) sites of III-V semiconductors as Fe3+, one can independently control the concentrations of Fe ions and, by doping other atoms, carriers in Fe-doped FMSs. Furthermore, the previously reported highest TC values n-type (In0.84,Fe0.16)Sb (335 K)16, (In0.65,Fe0.35)Sb (385 K)17, and p-type (Ga0.8,Fe0.2)Sb (> 400 K)18 are well above room temperature. Considering these advantages, Fe-based FMSs are more promising materials for applications to semiconductor spintronic devices operating at room temperature. Understanding the mechanism of the carrier-induced ferromagnetism in FMSs is  4 important for designing functional FMSs materials and for practical applications. Several theoretical models, such as the Zener’s p-d exchange model and impurity band (IB) model have been proposed so far19. The former, based on a mean-field theory, has been proposed as the itinerant limit where the holes are considered to be nearly free carriers20,21. This model indicates that the Fermi level (EF) is located in the valence band (VB) and that the origin of the ferromagnetism is the p-d exchange interactions between the VB holes and localized 3d electrons of the magnetic impurity. In contrast, the latter model has been proposed as the other limit where the hole carriers are localized around the magnetic impurities22,23. This model indicates that EF is located in the IB and that the ferromagnetism arises from the double-exchange interaction between the localized 3d electrons of the magnetic impurities. As described above, the origin of ferromagnetism in FMSs is considered to be related to its band structure near EF. Experimentally, the electronic states of (Ga,Mn)As in the vicinity of EF have been studied by angle-resolved photoemission spectroscopy (ARPES) to unveil the mechanism of carrier-induced ferromagnetism from the viewpoint of electronic band structure24,25,26,27. Soft x-ray (SX) ARPES measurements have been instrumental to  5 directly access the three-dimensional (3D) band structure and Mn-3d IB of (Ga,Mn)As27. Here, we investigate the electronic structure of the Fe-based FMS (Ga,Fe)Sb to reveal the origin of its high TC, and particularly examine the Fe-3d IB in the vicinity of EF, by SX-ARPES measurements. The SX-ARPES results provide an understanding of the carrier-induced ferromagnetism of (Ga,Fe)Sb.  II. EXPERIMENTAL (Ga0.95,Fe0.05)Sb and GaSb thin films a thickness of 30 nm were grown on semi-insulating (SI) GaAs(001) substrates by MBE. To avoid surface contamination, the surfaces of the films were covered with amorphous Sb capping layers. Figure 1(a) shows the schematic sample structure. During the MBE growth, the excellent crystallinity of the samples was confirmed by reflection high-energy electron diffraction. The lattice constant (a) of (Ga0.95,Fe0.05)Sb is 0.608 nm 28. The value of TC of (Ga0.95,Fe0.05)Sb, which was grown by the same condition as the measured one, is about 25 K15. The SX-ARPES experiments were performed at the SX-ARPES end station29 of the ADRESS beamline at the Swiss Light Source. Before the SX-ARPES measurements, the samples were annealed  6 at around 300 oC in the preparation chamber to remove the amorphous Sb capping layer and to expose the clean surface of the samples. The measurements were conducted under an ultrahigh vacuum below 10-10 mbar at a temperature of 12 K, with varying the photon energy (hν) from 500 eV to 1000 eV. The total energy resolution including the thermal broadening was between 50 meV and 200 meV depending on hν. The incident beam with linear-vertical and linear-horizontal polarizations, which respectively correspond to p-polarization and s-polarization configurations30, were used for the measurements. The Fe L2,3 x-ray absorption spectroscopy (XAS) spectra were measured in the total-electron-yield mode.  III. RESULTS AND DISCUSSION A. Constant energy mappings and band dispersion around the Γ point Figures 2(a) - 2(c) show out-of-plane (kz-k//) constant-energy mappings for (Ga0.95,Fe0.05)Sb at different binding energy (EB), where the red solid lines represent the Brillouin zone (BZ) as shown Fig. 1(b). Here, kz and k// (= k[-110]) are out-of-plane (Γ-X-Γ) and in-plane (Γ-K-X) momenta, respectively. The observed band dispersion clearly  7 depends on kz and reflects the symmetry of the BZ. Note here that there is no band that is non-dispersive along the kz direction in the mappings, evidencing the absence of surface (two-dimensional) states in the present SX-ARPES data. These observations demonstrate that the SX-ARPES spectra reflect the 3D band dispersion of the (Ga0.95,Fe0.05)Sb thin film. Figures 3(a) and 3(b) show the out-of-plane and in-plane Fermi surface mappings (FSMs) of the (Ga0.95,Fe0.05)Sb thin film, respectively. It should be noted here that the cut taken at hν = 885 eV contains the Γ point, as shown in Fig. 3(a). Then, the Γ-K-X symmetry line is precisely determined from the in-plane k[-110]-k[-1-10] constant-energy mapping taken at hν = 885 eV as shown in Fig. 3(b). Figures 3(c) and 3(d) show EB vs. momentum (k) plots along the Γ-K-X symmetry line with p and s polarizations, respectively. The light-hole (LH) and split-off (SO) bands show up with p polarization in Fig. 3(c), while the heavy-hole (HH) band is clearly visible in the SX-ARPES image taken with s polarization [Fig. 3(d)]. This linear polarization dependence comes from the wave-function symmetry of these bands31. Following the LH-band peaks of energy distribution curves (EDCs) in Fig. 3(e), the top of the LH band  8 is located at 150 meV below EF. Because the band gap (Eg) of (Ga0.95,Fe0.05)Sb is close to that of GaSb (Eg = 812 meV32)33, the observation evidences that (Ga0.95,Fe0.05)Sb is p-type, which is in agreement with the transport measurements15,28,34.  To examine the Fe-doping effects on the band dispersion, ARPES measurements on the GaSb thin film has also been conducted as a reference. Figure 4(a) shows EB vs k plots for GaSb along the Γ-K-X symmetry line with p polarizations at the hν of 880 eV. GaSb is p-type because the top of the LH band is close to EF. Comparing the band dispersions between (Ga0.95,Fe0.05)Sb in Fig. 3(c) and GaSb in Fig. 4(a), the band dispersion of (Ga0.95,Fe0.05)Sb is nearly identical to that of GaSb. Figure 4(b) shows the EDCs of GaSb (blue line) and (Ga0.95,Fe0.05)Sb (red line) along the Γ-K-X symmetry line with p polarization. The ARPES spectra of (Ga0.95,Fe0.05)Sb are broader than those of GaSb because of the structural disorder due to the Fe doping and/or because of the Zeeman splitting. However, the peak positions of EDCs of (Ga0.95,Fe0.05)Sb are almost the same as those of GaSb. This result indicates that the band dispersion itself originating from the sp-orbitals of GaSb is hardly affected by doping of Fe atoms. Since the position of EF in  9 (Ga0.95,Fe0.05)Sb is approximately the same as that in GaSb, the Fe atoms would isovalently substitute for the Ga site. B. Fe-3d impurity band Note that the Fe-3d IB is hardly seen in the ARPES spectra of (Ga0.95,Fe0.05)Sb near the Γ point in Figs. 3(c) and 3(d) due to the small amount of Fe atoms, although the band dispersion of the GaSb host has been clearly observed. The energy position of the Fe-3d states is a key to understand the ferromagnetism of (Ga0.95,Fe0.05)Sb because its position in the IB model is different from that in the p-d Zener model. To determine the position of the Fe-3d IB in the VB experimentally, we have conducted resonant angle-resolved photoemission spectroscopy (r-ARPES) at the Fe L3 absorption edge. By using r-ARPES, the states derived from the orbitals which are relevant to the absorption process are resonantly enhanced, in our case the Fe d-states. Figure 5(a) shows the XAS spectrum at the Fe L3 edge of the present (Ga0.95,Fe0.05)Sb film decapped by annealing at around 300 oC. Its shape is similar to that of the reported XAS spectra of the (Ga,Fe)Sb thin films capped by an amorphous As layer35. Since the XAS spectra of the capped (Ga,Fe)Sb thin  10 films have been measured without annealing, this result indicates that the annealing hardly changes the 3d states of Fe in the (Ga0.95,Fe0.05)Sb film. A previous x-ray magnetic circular dichroism study of the capped (Ga,Fe)Sb thin films has demonstrated that the XAS peak of 708 eV originates from the ferromagnetic component35. The resonance enhancement of ARPES measured at this hν should therefore reflect the energy position of the ferromagnetic component of Fe-3d states in (Ga,Fe)Sb. Figures 5(b) and 5(c) show the r-ARPES images of the (Ga0.95,Fe0.05)Sb thin films taken at hν = 708 eV (on-resonance) and 704 eV (off-resonance) with p polarizations. A flat band appears in the on-resonance spectrum in the vicinity of EF in Fig. 5(b), while this band apparently disappears in the off-resonance spectrum in Fig. 5(c). The Fe-3d state in the vicinity of EF is consistent with the previous resonant photoemission spectroscopy (RPES) at the Fe L2,3 edge of (Ga,Fe)Sb35. The flat band observed with hν = 708 eV is therefore the Fe-3d IB, which is related to the ferromagnetism. Figure 5(d) shows the EDCs of the on- and off-resonance spectra from k// = 1.5 (k1) to k// = 2.1 (k2) in the reciprocal lattice unit of 2√2π/a. The difference of the on-resonance EDCs from the off-resonance ones [the blue areas between the pink and green dashed  11 curves in Fig. 5(d)] reveals the Fe-3d component of the ARPES spectra. Here, it should be noted here that the on-resonant signal of the IB is much enhanced compared with its real contribution to the density of states (DOS). Generally, the states induced by impurities doped in a single crystal are independent of the wave number owing to the random atomic distribution. In our case, however, the Fe-3d component areas strongly change with k|| along the VB dispersion as shown by the red dots in Figs. 5(b) and 5(d). This indicates that the intensity of this LH band is also enhanced at the Fe 2p-3d resonance through hybridization between the Fe-3d orbital and the ligand p band27. Therefore, the present observation provides experimental evidence that the Fe-3d IB hybridizes with the ligand p band. C. Discussion   Finally, based on the experimental findings, we will discuss the band structure and the origin of the ferromagnetism of (Ga,Fe)Sb. From the observations described above, we have found that the Fe-3d IB located just above the VB maximum (VBM) crosses EF and hybridizes with the p band of GaSb, and that the LH, HH and SO bands of  12 (Ga0.95,Fe0.05)Sb similar to those of the host GaSb are located below EF, as shown in Fig. 6(a). Figure 6(b) shows a schematic diagram of the electronic structure of (Ga0.95,Fe0.05)Sb.  The observation that the Fe-3d IB shows the Fermi cutoff indicates that the Fe-3d IB is partially occupied by electrons. Considering the result that the VBM is located below EF, it is probable that the concentration of hole carriers due to defects, if they exist, is much lower than that of the Fe ions. Additionally, if the Fe ions are ionic Fe3+, the majority(up)-spin states are fully occupied due to the d5 electronic configuration. It follows from these arguments that the number of d electrons in (Ga0.95,Fe0.05)Sb is expected to deviate from the half-filled d5 configuration through the p-d hybridization. The strength of the p-d hybridization is closely related with the symmetry of d electrons in (Ga,Fe)Sb. The five-fold degenerate state of Fe 3d splits into the two-fold degenerate states (e) and the three-fold degenerate states (t2) because the Fe ions of (Ga,Fe)Sb are in the tetrahedral crystal field. Due to the symmetries of the e and t2 states, while the e states do not hybridize with the ligand p bands, the t2 states hybridize with the p bands36. The p-d(t2) hybridization leads to the antibonding (t2a) and bonding (t2b) states. The p-d(t2) hybridization leads to the antibonding (t2a) and bonding (t2b) states, which have both the  13 Fe t2 and the ligand Sb p characters. Figure 6 shows the schematic energy diagram of (Ga,Fe)Sb based on the experimental findings. Here, ↑ and ↓ means majority(up) spin and minority(down) spin, respectively. For the majority-spin states, since the Fe-3𝑑↑ levels are located well below the VBM, the bonding 𝑡&'↑ and antibonding 𝑡&(↑ states have predominantly both the Fe t2 and Sb p characters, respectively. In contrast, since the Fe-3𝑑↓ levels are located above the VBM, the minority-spin bonding 𝑡&'↓ state is mainly composed of the ligand p orbitals, and the antibonding 𝑡&(↓ state primarily consists of the Fe t2 orbitals. Band widths of the bonding t2b and antibonding t2a states with the p characters will be comparable with or narrower than that of VB through the strong p-d(t2) hybridization due to the high covalency of the narrow-gap semiconductor GaSb. Considering that the VBM is located just below EF, it is probable that the majority-spin antibonding 𝑡&(↑ state crosses EF and is partially filled (see the left side of Fig. 6(c)) because the 𝑡&(↑ is located above the VBM. To keep the charge balance of Fe3+, the holes of the majority-spin 𝑡&(↑ state will be compensated by electron occupation of the minority-spin 𝑒↓ and/or 𝑡&(↓ states (see the right side of Fig. 6(c)). This is consistent with the first-principle calculations for (Ga,Fe)Sb that the majority-spin 𝑡&(↑ and the  14 minority-spin 𝑒↓ and/or 𝑡&(↓ states cross EF37,38. It follows from these arguments that the observed narrow Fe-3d IB crossing EF probably originates from the partially occupied minority-spin 𝑒↓ and/or 𝑡&(↓ states, and the observed p-d hybridized states in the vicinity of EF would be the 𝑡&(↑ and 𝑡&'↓ states with the primarily p characters. The present SX-ARPES study indicates that the electronic structure of (Ga0.95,Fe0.05)Sb is consistent with the IB model, where the Fe-3d IB crosses EF. This means that the carriers of (Ga,Fe)Sb are mainly derived from the 3d electrons even if the p-d hybridization is finite. It should be mentioned here that the transport properties of (Ga,Fe)Sb stay semiconducting when the Fe concentration is less than 20%28, although the observed Fe-3d IB of the (Ga0.95,Fe0.05)Sb thin film shows the Fermi cutoff. It is possible that either gap opening, or depletion of DOS near EF which is smaller than the experimental energy resolution occurs in the Fe-3d IB of (Ga,Fe)Sb. Additionally, the actual contribution of the Fe-3d IB to the transport properties is expected to be negligible because the Fermi cutoff of the Fe-3d IB (or the spectral weight at EF) is fairly small in the off-resonance spectra as shown in Fig. 5(b). The DOS of the Fe-3d IB near EF will increase with increasing the Fe concentration, resulting in the increase of the conductivity  15 and TC of (Ga,Fe)Sb with higher Fe concentrations. It follows from these arguments that the double-exchange interaction in the Fe-3d IB crossing EF is the origin of the ferromagnetism in (Ga,Fe)Sb.  It has been reported that the local Fe concentration in (Ga,Fe)Sb fluctuates on the several-nm scale, and zinc-blende Fe-rich clusters are formed in the (Ga,Fe)Sb matrix [10, 22]. Since our (Ga,Fe)Sb thin film with x = 0.05 is expected to have a less fluctuation of the local Fe concentration, further studies including a systemic SX-ARPES measurements of the VB structure of (Ga,Fe)Sb with varying x is necessary to clarify the origin of the magnetic interaction. In addition, not only the conventional valence of Fe but also the intermediate valence of Fe should be taken into account because of the p-d hybridization. To identify the local electronic structure of Fe ions including the electronic structure parameters such as the charge-transfer energy and strength of Coulomb interaction, it is indispensable to conduct other measurements sensitive to the local electronic structure, e.g., resonant inelastic x-ray scattering (RIXS) combined with cluster-model calculation. The effectiveness of RIXS for FMS has been confirmed in (Ga,Mn)As39.   16  IV . SUMMARY We have performed SX-ARPES measurements on a (Ga0.95,Fe0.05)Sb thin film to obtain the information about its VB structure, the location of the Fe-3d IB, and its hybridization with the p band. By capping samples with an amorphous Sb layer and removing the cap by annealing in vacuo just before performing measurements, we have succeeded in observing the 3D bulk VB structure in the samples with clean surfaces. Experimentally, the band dispersion of (Ga0.95,Fe0.05)Sb is similar to that of GaSb. This indicates that the Fe ions hardly affect the band dispersion of the host GaSb. In addition, the non-dispersive Fe-3d IB hybridized with the ligand p bands is located around EF, indicating that the carriers of (Ga,Fe)Sb have the d-like character. Based on our results, the electronic structure of (Ga,Fe)Sb is consistent with the IB model. It is probable that the partially filled Fe-3d IB is composed of the minority-spin 𝑒↓ and/or 𝑡&(↓ states through the p-d hybridization. Thus, double-exchange interaction between Fe ions would be the origin of the ferromagnetic interaction of (Ga,Fe)Sb. To thoroughly study the ferromagnetic mechanism of (Ga,Fe)Sb with higher Fe concentrations, systematic studies  17 of the VB structure and the local electronic state of Fe-3d electrons on the (Ga,Fe)Sb thin films with various Fe concentrations are needed. ACKNOWLEWDGMENTS This work was supported by a Grant-in-Aid for Scientific Research (Grant Nos. 15H02109, 16H02095, 17H04922, 18H05345, and 23000010), Core-to-Core Program A. Advanced Research Networks from JSPS, and CREST of JST (No. JPMJCR1777), Japan. This work was partially supported the Spintronics Research Network of Japan (Spin-RNJ). 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Constant-energy mappings in the kz-k|| plane: (a)-(c) mappings for EB = 1.6 eV, 3.7 eV, and 7.0 eV, respectively and red solid lines represent the BZ in Fig. 1(b).     \nkz  (2π/a)-10k|| (2√2π/a)-10-10141315(a) EB = 1.6 eV(b) EB = 3.7 eV(c) EB = 7.0 eVKXΓ\nLowHigh 21  \n  FIG. 3: Band dispersion around the Γ point of a (Ga0.95,Fe0.05)Sb thin film: (a) FSM in the kz-k|| plane where the yellow curve represents the cut at hν = 885 eV, and the red solid lines represent the BZ; (b) FSM in the k[-110] - k[-1-10] plane taken at hν = 885 eV where purple solid lines represent the BZ; (c), (d) ARPES images along the Γ-K-X line taken with p and s polarizations, respectively. The spectra are measured at hν = 885 eV, and the LH, HH, and SO denote the light-hole, heavy-hole and split-off bands, respectively; (e) EDCs along the Γ-K-X line corresponding to the ARPES spectrum shown in panel (c), and the blue vertical bars are a guide to the eyes tracing the band dispersion of the LH band.     \n86420Binding Energy (eV)\n86420Binding Energy (eV)10k|| (2√2π/a)-101kz  (2π/a)-10k|| (2√2π/a)\n-101k[-1-10] (2√2π/a)10k[-110] (2√2π/a)ΓKX\n p-pol.885 eV\n s-pol.885 eVHHLHSOΓKX\nΓKXEB = 0 eV\nEB = 0 eV885 eV  708 eV  704 eV10151314ΓKX(a)\n(b)(c)\n(d)\nLowHighIntensity (a.u.)420Binding Energy (eV) EDC, p-pol. 885 eV\nΓKX(e) 22  \n  FIG. 4. Band dispersion near the Γ point of GaSb; (a) ARPES image of GaSb along the Γ-K-X line measured with p polarization at hν = 880 eV, (b) EDCs of the (Ga0.95,Fe0.05)Sb (red curves, from Fig. 3(c)) and GaSb (blue curves, from Fig. 4(a)) thin films.      \n86420Binding Energy (eV)10.50k|| (2√2π/a)86420Binding Energy (eV)Intensity (a.u.)ΓKX\n p-pol. 880 eVLHSOGaSbGa0.95Fe0.05SbGaSb\nΓKX(a)(b)\nLowHigh 23  \n  FIG. 5. Resonant ARPES spectra of (Ga0.95,Fe0.05)Sb: (a) Fe L3 XAS spectrum for hv = 708 eV and 704 eV are for the on-resonance and off-resonance, respectively; (b), (c) on- and off-resonant ARPES images measured with p polarization, respectively; (d) EDCs of the on-resonant ARPES (pink solid curves, from Fig. 5(b)) and off-resonant ARPES (green dashed curves, from Fig. 5(c)) between k|| =1.5 (k1) and k|| = 2.1 (k2). Shaded (blue) area denotes the Fe-3d component of ARPES spectra. Red dots on pink solid curves in Figs. 5(b) and 5(d) represent the points where the on-resonant intensity is remarkably stronger than the off-resonant intensity.   \n715710705700Photon Energy (eV)\n43210Binding Energy (eV)210k|| (2√2π/a)210k|| (2√2π/a)Intensity (a.u.)\nIntensity (a.u.)10Binding Energy (eV)(b)p-pol. 708 eV (on)IB(c)p-pol. 704 eV (off)Ga0.95Fe0.05Sb  Fe L3 XASonoff on off\nk2k1(a)(d)\nLowHigh 24  \n  FIG. 6: Electronic structure of (Ga0.95,Fe0.05)Sb: (a) ARPES image from Fig.3(c) with added Fe-3d IB in the vicinity of EF of Fig. 5(b). (b) Schematic diagram for the density of states (DOS) of (Ga0.95,Fe0.05)Sb. (c) Schematic energy diagram of (Ga0.95,Fe0.05)Sb where the p-d hybridization splits t2 states into antibonding (t2a) and bonding (t2b) states. CB means the conduction band.  \n"    },    {        "title": "2004.03784v1.Charge_to_spin_conversion_efficiency_in_ferromagnetic_nanowires_by_spin_torque_ferromagnetic_resonance__Reconciling_lineshape_and_linewidth_analysis_methods.pdf",        "content": "arXiv:2004.03784v1  [cond-mat.mes-hall]  8 Apr 2020Charge-to-spin conversion efficiency in ferromagnetic nano wires by spin torque\nferromagnetic resonance: Reconciling lineshape and linew idth analysis methods\nJun-Wen Xu and Andrew D. Kent∗\nCenter for Quantum Phenomena, Department of Physics, New Yo rk University, New York 10003, USA\n(Dated: April 9, 2020)\nSpinorbit torquesare ofgreat interest for switching thema gnetization direction innanostructures,\nmoving skyrmions and exciting spin waves. The standard meth od of determining their efficiency\nis by spin torque ferromagnetic resonance (ST-FMR), a techn ique that involves analyzing the res-\nonance linewidth or lineshape. On microstuctures these two analysis methods are quite consistent.\nHerewe presentST-FMR results onpermalloy (Ni 80Fe20)nanowires —with widths varyingfrom 150\nto 800nm — that show that the standard model used to analyze th e resonance linewidth and line-\nshape give different results; the efficiency appears greatly e nhanced in nanowires when the lineshape\nmethod is used. A ST-FMR model that properly accounts for the sample shape is presented and\nshows much better consistency between the two methods. Micr omagnetic simulations are used to\nverify the model. These results and the more accurate nanowi re model presented are of importance\nfor characterizing and optimizing charge-to-spin convers ion efficiencies in nanostructures.\nI. INTRODUCTION\nSpin orbit torques (SOTs) are being actively consid-\nered for using in the next generation memory devices\nfor magnetization switching [1, 2], spin oscillators [3, 4]\nand racetrack memories, including those using magnetic\nskyrmions [5–8]. SOTs are fundamentally based on\ncharge-to-spin conversion by the spin-Hall effect [9] or\nRashba effect [10]. These effects result in a spin cur-\nrent or spin accumulation that is transverse to the di-\nrection of charge current flow. An advantage of SOTs\nis that a charge current does not need to flow through\nthe magnetic layer to switch its magnetization direction.\nThe torques can thus be used to switch magnetic insula-\ntors[11] orthe freelayerofamagnetictunnel junction [1]\nwithout current flowing through the insulating layer.\nSOTsareprincipallyofinterestinnanostructuredsam-\nples, samples with minimum dimension less than a mi-\ncron. In such samples the switching current density can\nbe relatively large while the current is still small and the\ntorques on the magnetization associated with chargecur-\nrent induced Oersted fields can be much smaller than the\nSOT. However, the magnitude and form of the torques\nare most often characterized in micron scale samples.\nThe most widely used SOT characterization method is\nspin torque ferromagnetic resonance (ST-FMR) [12, 13].\nIn this technique a radio frequency (rf) charge current\nleads to a rf spin torque that excites magnetization dy-\nnamics, with the largest response occurring at the fer-\nromagnetic resonance (FMR) frequency. The magni-\ntude of the torque can be determined by either ana-\nlyzing the linewidth of the response or the lineshape\nof the resonance [14, 15]. This allows extracting the\neffective charge-to-spin conversion efficiency, which is\nproportional to the spin-Hall angle θSH=js/jc[16].\nThe linewidth and lineshape analysis methods have been\n∗andy.kent@nyu.edushown to give consistent results when applied to mi-\ncrostructures [15, 17–19].\nIn this paper we show that when ST-FMR results on\nnanostructures are analyzed with the standard ST-FMR\nmodel the results are generally not consistent; with the\nlineshape analysis method the efficiency appears greatly\nenhanced in nanowires. We present ST-FMR results on\npermalloy nanowires with linewidths varying from 150\nto 800nm and a model that properly accounts for the\nsample geometry. Our nanowire model provides much\nbetter consistency between the lineshape and linewidth\nanalysis methods.\nII. EXPERIMENT\nExperiments were conducted on Permalloy (Py,\nNi80Fe20) nanowires fabricated from thin films. The\nfilms are grownby dc magnetron sputtering on thermally\noxidized silicon substrates with layer stack SiO 2/Ta\n(3)/Py(5)/Pt(6), where the numbers are the layer thick-\nnesses in nm. Ta is a seed layer [20] and the Pt layer is\nthe main source of the spin current. It also protects the\nPy from oxidation. Electron beam lithography followed\nby argon ion milling is used to define the sample geome-\ntry. We deposit Ti(5)/Au(50) by evaporation and liftoff\nfor the contact pads. An image of the nanowire and con-\ntact pads is shown in the inset of Fig. 1. The width of\nnanowires varies from 150 to 800nm with a fixed aspect\nratio of 20.\nFigure1shows the ST-FMR setup. The nanowire is\noriented at 45 degrees to the applied in-plane field in or-\nderto havea largeST-FMR signal. Arfchargecurrentis\napplied to the sample using a ground-signal (GS) probe\nconnected to contact pads to the nanowire. The rf cur-\nrent is input to the high frequency port of a bias tee,\nwhile the bias tee’s low frequency port is connected to a\ndc current source and a lock-in amplifier.\nThe measurement principal is as follows. The rf cur-\nrent produces an rf torque on the ferromagnetic layer,2\nFigure1. ST-FMRsetup. Thenanowire isoriented45degrees\nto the external field B. An rf and dc current are applied using\na bias tee, while a small field coil is used to modulate the\napplied field at a low frequency.\ncausingthis layer’smagnetizationto precess. The largest\nprecession amplitude occurs at the FMR frequency. The\noscillation in the magnetization leads to oscillations in\nthe nanowire resistance due to the anisotropic magne-\ntoresistance (AMR) of Py. A dc voltage Vmixappears\nacross the nanowire due to the mixing of the rf charge\ncurrent and resistance oscillations at the same frequency.\nIn order to increase the signal-to-noise ratio, a small\namplitude magnetic field of 0 .2mT (an amplitude much\nless than the FMR linewidth) is modulated at low fre-\nquency 727Hz, a frequency far less than the FMR fre-\nquency. The lock-in amplifier is set to detect the signal\natthis lowfrequency. Themodulated voltagesignalmea-\nsured isV(B)∝dVmix/dB[21]. A single measurement\ncorresponds to sweeping the external field from high to\nlow values at fixed rf current amplitude and frequency.\nThe sweep is from high to low field to ensure each mea-\nsurements starts from a saturated magnetic state. For\neach rf frequency, the field is swept multiple times and\nthe signal is averaged in order to further increase the\nsignal-to-noise ratio. The rf frequency is varied from 9\nto 15GHz and magnetic fields up to 0 .3T are applied.\nThe power of our rf source is fixed at 10dBm and we\nhave verified that the resonance response is in the linear\nregime.\nThe lineshape analysis method consists of analyzing\nthepeakshape. Thepeakinthelock-involtageasafunc-\ntion of magnetic field, V(B), can be decomposed into the\nsum of the derivative of Lorentzian and anti-Lorentzian\nfunctions (see the supplementary materials for further\ndetails):\nV(B) =−S(B−B0)∆+A/bracketleftbig\n(B−B0)2−(∆/2)2/bracketrightbig\n[(B−B0)2+(∆/2)2]2,(1)\nwhereB0is the resonance field, Bis the external field\nin vacuum (i.e. B=µ0H, whereµ0is the permeability\nof free space) and ∆ is the resonance full width at half\nmaximum (FWHM). SandAare Lorentzian and anti-\nLorentzian amplitudes, respectively. In contrast, for the\nlinewidth analysis method, a dc current is applied to thesample through the low frequency port of the bias tee.\nThe variation of the resonance peak linewidth with dc\ncurrent is used to determine the efficiency.\nIn the following we denote the standard ST-FMR\nmodel the thin film model, as this model assumes the fer-\nromagnethasaneasy-planemagneticanisotropy,withno\npreferred magnetization axis in the plane. The nanowire\nmodel we present considers the in-plane magnetic shape\nanisotropy that we discuss further below.\nA. Results and analysis with thin film model\nFigure2(a)-(c) show spectra of 150, 400 and 800nm\nlinewidth nanowires at a fixed rf frequency of 12GHz,\nwhere each spectra is normalized to its maximum value.\nIt is clear that the signal-to-noise ratio is more than ade-\nquate for detailed analysis. We fit the spectra to Eq. 1to\ndetermine the Lorentzian Sand anti-Lorentzian Aam-\nplitudes, the resonance field B0and the linewidth ∆.\nIn the thin film model the ratio of the Lorentzian\nand anti-Lorentzian amplitudes is used to compute the\ncharge-to-spin conversion efficiency, ξ, given by [15]:\nξ=S\nA/radicalbigg\n1+µ0Meff\nB0eµ0Mstd\n/planckover2pi1, (2)\nwheretis the thickness of ferromagnetic layer and dis\nthe thickness of heavy metal layer. Meffis the effective\nmagnetization that characterizes the ferromagnet’s easy\nplane anisotropy and Msis its saturation magnetization,\neis the electron charge, and /planckover2pi1is the reduced Planck’s\nconstant. We note that this model assumes a negligible\nfield like SOT; it assumes that the field-induced torque is\nassociatedwiththe Oerstedfield fromthe chargecurrent.\nThis is a reasonableassumption for our samples, as it has\nbeen reported that the field-like SOT decays with mag-\nnetic layer thickness, with a characteristic decay length\nof≃2nm [22]. Thus for our 5nm thick ferromagnetic\nlayer this torque will be far less than the anti-damping\ntorque and the Oersted field induced torque.\nUsing the fitting parameters derived from the data in\nFig.2andtakingMeff=Ms, Eq.2isusedtodetermine ξ.\nThe result is shown in Fig. 2(d). The efficiency increases\ndramatically (i.e. by more than a factor of 2) as the wire\nlinewidth is reduced. For larger nanowire widths ( /greaterorsimilar400\nnm),ξ= 0.65 which is consistent with that reported\nvalue for Pt/Py interfaces [15, 17, 18]. It is reasonable\nto associate the main component of SOT with this in-\nterface because Ta is much more resistive [23] than Pt\nand thinner in our samples, so most of the current passes\nthrough the Pt layer.\nWe also studied the efficiency using the linewidth\nanalysis method, expecting to see a similar trend with\nnanowire width. As noted above, in this method a dc\ncurrent is applied to the nanowire and the ST-FMR\nlinewidth is determined as a function of the current. For\none polarity ofthe current the SOT opposesthe damping\nand leads to a reduced ST-FMR linewidth, while for the3\n00.51.0Data\nLorentzian\nanti-Lorentzian\nSum\n200 400 600 80000.050.100.15\nWidth (nm)ξ00.51.0\n0.1 0.2 0.300.51.0\nB (T)Normalized Voltage\n(d)(a)  150 nm\n(b)  400 nm\n(c)  800 nm\nFigure 2. (a)-(c) Normalized spectra of 150, 400 and 800 nm\nwidth nanowires at 12 GHz. The normalization voltages are\n2.43µV, 1.71µV and 1.48µV respectively. The red and green\ncurves show the anti-Lorentzian and Lorentzian components\nof the resonance peak respectively. (d) The efficiency versus\nnanowire width determined from the lineshape using the thin\nfilm model.\noppositecurrent,SOTincreasesthedampingandtheST-\nFMR linewidth increases [24]. Figure 3shows selected\nspectra at fixed rf frequency for several dc bias currents.\nFitting these spectra to Eq. 1we determine the linewidth\nas function of the dc bias. The results are shown in\nFig.3(b). For positive field, the linewidth decreases with\nincreasing bias current, and vice versa. The slope of the\nlinewidthversusdcbiascurvesis −2.2×10−3mT/µAfor\npositive field and 2 .3×10−3mT/µA for negative field.\nIn the thin film model the efficiency is related to the\nslope of the ST-FMR resonance linewidth versus dc bias\ncurrent [14, 15]:\nξ=γe(B0+µ0Meff/2)Mst\n2π/planckover2pi1fsinφd∆\ndjc, (3)\nwherejcisthe currentdensity in the Pt layerand φis the\nanglebetweenchargecurrentandexternalmagneticfield.\nFigure3(c) shows the resulting ξversus nanowire width;\nξvaries between 0.062 - 0.077. Comparing Fig. 2(d) and\nFig.3(c) it is clear there is a significant discrepancy in\nthe SOT efficiencies deduced from these models. It is\nalso clear that some basic physics in the modeling of the\nefficiency in nanowires is not captured by the standard\nthin film model.0.15 0.16 0.17−1012\nB (T)V (µV)400\n160\n0\n-160\n-400\n−400 0 400192021\nCurrent (µA)∆ (mT)Positive ®eld\nNegative ®eld\n200 400 600 80000.030.060.09\nWidth (nm)ξ(b)\n(c)(a) I (µA)\nFigure 3. (a) ST-FMR spectra of a 300nm width nanowire\nat 12GHz at several dc bias currents. (b) Linewidth as a\nfunction of dc current for both positive and negative field\npolarities. (c) The efficiency as a function of the nanowire\nwidth determined using the linewidth analysis method.\nB. Nanowire model\nWe begin by analyzing the dependence of resonance\nfieldon rffrequencytotest abasicassumptionofthe thin\nfilm model. The model assumes an easy-plane magnetic\nanisotropy with a resonance frequency-field relation:\nf2\nB0=/parenleftBigγ\n2π/parenrightBig2\n(B0+µ0Meff). (4)\nThat isf2/B0should be a straight line when plotted\nversus field B0, with a slope proportional to the gy-\nromagnetic ratio squared and intercept proportional to\nMeff. The experimental data are shown in Fig. 4. It is\nclearthat the 800nm data follows the expectations of the\nmodel. However, the data for the 150nm width wire de-\nviatesstronglyfromthestraightlinetrend(orangecurve)\nfor resonance fields less than about 0 .15T.\nAssumingthatappliedfieldsgreaterthan0 .15Tarere-\nquired to fully saturate the magnetization of the 150nm\nwidth wire, one may omit the lower resonance field data\nfrom the fits to Eq. 4(see the orange line in Fig. 4).\nFollowing this approach for all samples we find the fit-\nting parameters Meffandγ/(2π) given in Table I. For\nwide nanowires the parameters are those expected for4\nWidth (nm) 150 200 300 400 600 800\nµ0Meff(T) 1.35(5) 1.19(1) 1.19(2) 1.11(2) 1.04(1) 0.96(1)\nγ/2π(GHz/T)24.7(4) 26.2(1) 26.3(2) 27.2(2) 28.0(1) 28.9(1)\nTable I. Effective magnetization and gyromagnetic ratio fro m\nfits to Eq. 4 for different width nanowires.\nPy,µ0Meff≈µ0Ms= 1T and γ/(2π) = 28GHz /T.\nHowever, for the narrower samples, µ0Meffis larger and\nγ/(2π) is smaller. This does not make physical sense\nasµ0Meffis associated with the demagnetization field\nperpendicular to the film plane, which decreases as the\nwire width decreases. Without an additional easy plane\nmagnetic anisotropy (e.g. associated with spin-orbit in-\nteractions at the Py interfaces), Meff≤Ms. For exam-\nple, for the 150nm width nanowire, the demagnetization\nfield in the width direction ≃0.05T [25], which is not\nnegligible compared to the applied magnetic field in our\nexperiments (0.1 to 0 .3T).\nIt is thus clear that the in-plane magnetic anisotropy\nassociated with the sample shape is not negligible and\nneeds to be considered in the analysis of the resonance\nfield. The shape anisotropy can be described by a in-\nplane uniaxial anisotropy field BAparallel to the wire\naxis. The resulting resonance field condition becomes:\nf=γ\n2π/radicalbig\n(B0+BA)(B0+BA+µ0Meff).(5)\nThe additional field BAis associated with the average\ndemagnetization field in the width direction, i.e. the in-\nplane magnetic shape anisotropy. The fit to Eq. 5is\nshown as the green curve in Fig. 4. The fit accurately\ncaptures both the low and high field resonance data with\nthe fit parameters for all the nanowires shown in Ta-\nbleII. We now see, as expected, that Meffdecreases as\nthe wire width decreases. Further, the gyromagnetic ra-\ntio is nearly independent of wire width.\n0.10 0.15 0.20 0.258809109409701000\nB0 (T)f2/B0 (GHz2/T)\n150 nm data\n800 nm data\n150 nm Eq. 4\n150 nm Eq. 5\nFigure 4. Resonance field versus rf frequency for 150 and\n800nm nanowires, fit with Eq. 4 and Eq. 5Width (nm) 150 200 300 400 600 800\nµ0Meff(T) 0.80(8) 0.83(7) 0.94(2) 0.94(2) 0.96(4) 0.92(5)\nBA(mT) 14(3) 10(2) 7(1) 3(1) 1(1) 1(1)\nγ/2π(GHz/T)29.6(10) 29 .6(8) 28.7(2) 28.8(2) 28.9(5) 29.4(6)\nTable II. Effective magnetization, in-plane anisotropy fiel d\nand gyromagnetic ratio from fits to Eq. 5 for different width\nnanowires.\nAs a consequence of the shape anisotropy the demag-\nnetization field is not collinear with the applied field in\nour experiment. Thus the precession axis of the magne-\ntization in resonance is also no longer the applied field\ndirection. As a result the angle between the precession\naxis and the current is less than 45 degrees, φ<π/4.φ\ncan be calculated numerically by solving\nBsin(π/4) =Nxµ0Mssinφ+Btanφcos(π/4),(6)\nwhereNxis the demagnetization coefficient in the width\ndirection (see supplementary part 2).\nFurther, the charge current induced Oersted field for\nnanowires is different from the film case as well. The in-\nplane Oersted field decreases at the edge of the wire, as\nits direction becomes more out of the film plane. The\nST-FMR model assumes a uniform magnetization re-\nsponse, so the average in-plane Oersted field is consid-\nered. The average in-plane Oersted field for a very wide\nstrip (BOe=µ0Jd/2) is reduced, multiplied by a factor\n¯Θ(ǫ)<1 (see supplementary part 3):\n¯Θ(ǫ) =1\nπ/bracketleftbigg\n2arctanǫ−1\nǫln/parenleftbig\n1+ǫ2/parenrightbig/bracketrightbigg\n,(7)\nwhereǫ≡w/dis the ratio of width to the thickness of\nthe Pt layer. In the film case, ǫ→ ∞,¯Θ = 1. For a\n150nm nanowire, ǫ= 25 and ¯Θ≈0.89.\nIn the nanowire model lineshape analysis method:\nξ=S\nA/radicalbigg\n1+µ0Meff\nB0+BAeµ0Mstd\n/planckover2pi1¯Θ. (8)\nWhile in the linewidth analysis method, the expression\nis the same as that in Eq. 3. However, φnow needs to\nbe evaluated numerically using Eq. 6.\nFigure5shows the ST-FMR data analysis using our\nnanowire model. The efficiency ξusing both lineshape\nanalysis and linewidth analysis methods for all width of\nthe nanowires is between 0.6-0.7. Also, there is no longer\na significant enhancement in ξat small nanowire widths\nwith the lineshape analysis method.\nC. Nanowire lineshape versus linewidth analysis\nmethods\nWe find that the nanowire lineshape and linewidth\nanalysis methods are in reasonable accord with small\ndifferences, about 10 % differences in ξ, comparable to5\n00.050.100.15\nFilm model\nNanowire model\n200 400 600 80000.030.060.09\nWidth (nm)ξ\nFilm model\nNanowire model(a)\n(b)Lineshape analysis \nLinewidth analysis \nFigure 5. Comparison of the thin film and nanowire models\nusing (a) the lineshape analysis method and (b) the linewidt h\nanalysis method.\nthe measurement error. There are reasons that the line-\nshape method may be more accurate. First, this is be-\ncause the lineshape analysis method is self-calibrated;\nit is a measure of the ratio of the SOT to the Oersted\nfield torque which are both proportional to the current\ndensity in the heavy metal layer. Hence, the ratio of\nLorentzian and anti-Lorentzian peak amplitude is inde-\npendent of the charge current density. In contrast, in\nthe linewidth analysis method, it is important to ac-\ncurately estimate the current density passing through\nthe heavy metal layer. Second, in the linewidth anal-\nysis method, the angle between the magnetization pre-\ncession axis and charge current is estimated assuming a\nuniform demagnetization field for a cuboid shaped sam-\nple [25], which is an approximation (see the last part of\nSec.III). Third, during a field swept spectrum, this angle\nchanges with the external field. Here we take the angle,\nat the resonance field, as an approximation, which leads\nto an additional error. Finally, the most important set\nof data for the linewidth analysis method is the variation\nof the linewidth with the dc bias current. In our study,\nthis change is only 1mT when the current density varies\nby±2×1011A/m2, which is comparable to the 0 .2mT\nmodulation field. This introduces a systematic error in\ndeterminingthelinewidthvariationwithdccurrent. Fur-\nthermore, the external field control is of order 0 .01mT,\nlimiting the signal-to-noise ratio.\nIII. MICROMAGNETIC SIMULATIONS\nIn order to verify our nanowire model, we carried out\nmicromagnetic modeling using MuMax (see supplemen-\ntary part 4 for our code) [26, 27]. We simulate nanowires\nwith a width of 160, 320, 640 and 1280nm and an aspect\nratio of 16. As in the experiment, the magnetization isdriven by an oscillating Oersted field torque and SOT\nassociated with a charge current. After 4ns of simula-\ntion time, the spatial averaged magnetization amplitude\ntakes the form:\nmx(t) =mxcos(2πft−ψ), (9)\nwheremx(t) is the instantaneous xcomponent of mag-\nnetization, mxis the amplitude of oscillation and ψis\nthe phase delay between the magnetization response and\nrf current. We sweep the external field Bat a fixed rf\ncurrent amplitude and frequency. The ST-FMR signal\nresultsfrommixingoftherfcurrentandanisotropicmag-\nnetoresistance:\nVmix=1\nT/integraldisplay\nI(t)R(t)dt\n=−(IδRsinφcosφ)mxcosψ,(10)\nwhere the integral is over one period of the rf signal\nT= 1/fandmxcosψis a sum of a Lorentzian and anti-\nLorentzian function (see supplementary part 1). There-\nfore, we fit the simulation result mxcosψversusBusing\nthe expression\nmxcosψ=S(∆/2)+A(B−B0)\n(B−B0)2+(∆/2)2,(11)\nin order to extract the resonance field ( B0) the linewidth\n(∆), the Lorentzian amplitude ( S) and the anti-\nLorentzian amplitude ( A), as we do for the experimental\ndata.\nFitting the resulting resonance frequency versus mag-\nnetic field with Eq. 4(the thin film model) the trends of\nMeffandγ/(2π) versus wire width are similar to those\nfound for the experimental data (c.f. Table Iand sup-\nplementary part 5 Table 1 for the simulation results);\nMeffincreases with decreasing wire width. Since in the\nsimulations µ0Ms= 1T andγ/(2π) = 28.0GHz/T, this\ndiscrepancy is unphysical. However, using the nanowire\nexpression, Eq. 5, gives reasonable values of Meffand\nγ/(2π) (see supplementary part 5 Table 2 for the simu-\nlation results).\nThe efficiency extracted from the models is plotted in\nFig.6. Note that this should be compared to the ef-\nficiency input into the micromagnetic simulations, ξ=\n0.05. In the wide wire limit, both the thin film model\nandnanowiremodelareingoodagreementusingtheline-\nshapeanalysisandlinewidth analysismethods. However,\nwhenthewirewidthbecomesnarrower,thethinfilmline-\nshape analysis gives a ξthat increases (Fig. 6), as seen\nin experiment (Fig. 2(d)). Using the nanowire model ξ\nis closer to 0.05. The origin of the enhancement of ξ\nis now clear. It is associated with the thin film model\noverestimating Meffand the Oersted field, ¯Θ.\nIn the linewidth analysis method, the thin film model\nshows that ξdecreases in narrower wires. But in the\nnanowire model, there is no longer such a decrease. We\nhave not seen such a decrease in our experimental re-\nsults, Fig. 3(c). This is because in the thin film model6\n0 500 1000 15000.040.050.060.07\nWidth (nm)ξFilm model Film model\nNanowire model Nanowire model\nInput valueLineshape analysis Linewidth analysis\nFigure 6. Simulation results of efficiency found using line-\nshape and linewidth analysis methods with both the thin film\nand nanowire models with for different wire widths. The\ndashed curve is the value of the efficiency used in the sim-\nulation, ξ= 0.05.\n(Eq.3), we overestimate Meffwhile we underestimate\n1/sinφ, which compensate each another.\nFinally, as discussed near Eq. 6, the equilibrium mag-\nnetization angle φwas calculated with demagnetiza-\ntion coefficients associated with a uniformly magnetized\ncuboid. To estimate the error associated with this as-\nsumption, we compared the average magnetization an-\ngle determined in micromagnetics simulations from that\nfound assuming a uniformly magnetized cuboid. For the\nlatter, the demagnetization field is calculated following\nRef. [25] and input into a micromagnetics simulation as\na fixed uniform field (turning off the internal demagneti-\nzation field in the simulation, see supplementary part 6\nfor details). The relation of Band sinφboth using mi-cromagnetics and a uniform demagnetization field with\ndifferent width of nanowires is plotted in supplementary\npart6. Wefindthatassumingauniformdemagnetization\nand following our procedure using Eq. 6produces a neg-\nligible error in the nanowire linewidth analysis method.\nIV. SUMMARY\nIn summary we have introduced an analytic ST-FMR\nmodel for nanowires that accounts for their shape in\na straightforward way. The model gives reliable effi-\nciency results either by analyzing the ST-FMR lineshape\nor the ST-FMR linewidth as a function of current. As\nthe primary interest in spin orbit torques is in exciting\nmagnetization dynamics and switching the magnetiza-\ntion of nanostructures, our model can be of importance\nin reliablecharacterizationandoptimizing charge-to-spin\nconversion efficiencies in such structures. Further, the\nmodel can be extended to other sample shapes and to\ninclude additional physics associated, for example, with\nthe quantization of spin wave modes in nanostructures\nand spin-pumping.\nACKNOWLEDGMENTS\nWe thank Dr. Christopher Safranski for discussions of\nST-FMR analysis methods and Dr. Nahuel Statuto for\nthe guidance in the micromagnetic modeling. This re-\nsearchwas supported by DARPA Grant No. D18AP0000\nand the National Science Foundation under Grant No.\nDMR-1610416. 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Adams\nDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA\nG. Catelani\nDepartment of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA\n(Dated: March 21, 2022)\nWe show that the pairing resonance in the Pauli-limited normal state of ultra-thin superconducting\nAl \flms provides a spin-resolved probe of conduction electron polarization in thin magnetic \flms. A\nsuperconductor-insulator-ferromagnet tunneling junction is used to measure the density of states in\nsupercritical parallel magnetic \felds that are well beyond the Clogston-Chandresekhar limit, thus\ngreatly extending the \feld range of tunneling density of states technique. The applicability and\nlimitations of using the pairing resonance as a spin probe are discussed.\nPACS numbers: 74.50.+r, 75.70.Ak, 85.75.-d,74.78.Db\nThe possibility of incorporating spin degrees of free-\ndom into electronic technologies has led to an explosion\nin research into mechanisms of spin polarization of con-\nduction currents in semiconducting and metallic systems\n[1, 2]. In addition to being technologically important,\nspin polarization in conducting systems remains a fun-\ndamentally interesting many-body problem. Clearly, an\naccurate determination of the conduction electron polar-\nizationP, particularly in thin-\flm magnetic structures,\nis crucial both from the point of view of \\spintronic\"\ndevice development and basic research. Unfortunately,\nhowever, there are few direct probes of P. The three\nmost successful techniques, listed in historical order, have\nbeen Zeeman-split superconducting tunneling density of\nstates spectroscopy (SCTDoS) [3, 4], spin-resolved pho-\ntoemission spectroscopy (SRPES) [5], and point contact\nAndreev re\rection (PCAR) [6, 7]. Each of these tech-\nniques has its own unique advantages and limitations.\nSCTDoS has outstanding resolution \u001810\u0016V, and is\ncompatible with thin-\flm geometries, but is limited to\na narrow range of magnetic \felds and is not easily im-\nplemented with bulk samples [4]. SRPES has a relatively\nlow resolution of\u001810 mV, and does not discriminate well\nbetween itinerant and localized bands. PCAR can be\nused on bulk samples and thick \flms, but it requires very\nlow impedance point contacts, and is also incompatible\nwith magnetic \felds [7, 8]. In this Letter we introduce\nan extension of SCTDoS technique that exploits the spin\nstructure of the Pauli-limited normal state (PLNS) pair-\ning resonance [9, 10] to measure electron polarization in\nmagnetic \felds well above the superconducting critical\n\feld. We show that the technique not only greatly ex-\ntends the range of \felds over which the polarization can\nbe measured, but it can be much less sensitive to \feld\nmisalignment than SCTDoS.\nTedrow and Meservey [4] pioneered the use of super-\nconducting spin-resolved tunneling to directly measure\nthe electron polarization in magnetic \flms. The tech-\nnique utilizes a planar junction geometry consisting ofa superconductor-oxide-ferromagnet (SC-Ox-FM) sand-\nwich in which the superconductor counter-electrode is\npurposely made very thin. The tunnel junction is then\ncooled below the superconducting transition temperature\nand a carefully aligned magnetic \feld is applied parallel\nto the junction plane. If the SC \flm thickness tis much\nless than the superconducting coherence length \u0018, then\nthe Meissner response to the applied \feld is suppressed,\nand the critical \feld of the SC counter-electrode is en-\ntirely Zeeman mediated [11]. If the spin-orbit scattering\nrate of the SC is small, then spin is a good quantum num-\nber, and the spin rotation symmetry of the SC can be ex-\nploited to provide a spin-resolved probe. In the Tedrow\nand Meservey technique the polarization is measured in\nthe superconducting phase by applying a subcritical par-\nallel magnetic \feld to the tunnel junction at low tem-\nperature,T\u001cTc. This induces a Zeeman splitting of\nthe SC quasiparticle DoS spectrum, and the BCS coher-\nence peaks get split into spin-up and spin-down bands\nby the parallel \feld [12]. As will be discussed below, the\nrelative heights of these peaks give a direct measure of\nthe electron polarization in the FM. Here we show that\npolarization can, in fact, be measured in \felds that are\nseveral times larger than the parallel critical of the SC\nby using the PLNS pairing resonance in appropriately\ndesigned tunnel junctions.\nThe SC-Ox-FM tunnel junctions were formed by \frst\ndepositing a thin Al counter-electrode via e-beam depo-\nsition of 99.999% Al stock onto \fre polished glass micro-\nscope slides held at 84 K. The depositions were made at\na rate of\u00180:1 nm/s in a typical vacuum with pressure\n<3\u000210\u00007Torr. After deposition, the counter-electrode\nwas exposed to the atmosphere for 10-20 minutes in or-\nder to allow a thin native oxide layer to form. Then\na 45 \u0017A thick FM \flm was deposited onto the counter-\nelectrode with the oxide serving as the tunneling barrier.\nIn this study the FM was either CNi 3or CCo 3, where\nthe e-beam depositions were made from arc-melted but-\ntons. The counter-electrode thicknesses were chosen soarXiv:0907.1800v1  [cond-mat.supr-con]  10 Jul 20092\nthat their in-plane sheet resistance was \u00181-2 k\n/sq. This\ncorresponded to thicknesses that were typically 22-24 \u0017A.\nThe low temperature parallel critical \felds of the counter-\nelectrodes were\u00186.5 T. The junction area was about\n1 mm\u00021 mm, while the junction resistance ranged from\n15-100 k\n depending on exposure time and other factors.\nOnly junctions with resistances much greater than that\nof the \flms were used. Measurements of resistance and\ntunneling were carried out on an Oxford dilution refriger-\nator using a standard ac four-probe technique. Magnetic\n\felds of up to 9 T were applied using a superconducting\nsolenoid. A mechanical rotator was employed to orient\nthe sample in situ with a precision of \u00180:1\u000e.\nIn the upper panel of Fig. 1 we plot the 70 mK tunnel-\ning conductance of a Al-AlO x-CNi 3tunnel junction in a\nsub-critical parallel magnetic \feld of 4 T. The Zeeman\nsplitting of the BCS DoS spectrum is clearly evident,\nwhere we have labeled the spin moment associated with\neach peak. Normally the respective spin peaks would be\nidentical on either side of the Fermi energy. The asym-\nmetry arises from the unequal spin populations in the\nferromagnetic CNi 3. This is expected since the magneti-\nzation properties of the CNi 3and CCo 3are very similar\nto their elemental counterparts [13]. If one assumes that\nspin is conserved in the tunneling processes, then tun-\nneling currents from the spin-up (down) bands in the Al\nwill only tunnel into corresponding spin-up (down) bands\nin the ferromagnet. Tedrow and Meservey exploited this\nconservation property to extract the polarization of a va-\nriety of transition metal FM \flms [4],\nP=\f\f\f\f\u000e1\u0000\u000e2\n\u000e1+\u000e2\f\f\f\f(1)\nwhere the peak height di\u000berences \u000e1;2are de\fned in\nFig. 1A and the corresponding polarization of the CNi 3\nis\u001818%.\nThe zero-\feld gap energy for the Al counter-electrode\nused in Fig. 1 was determined to be \u0001 o\u00180:49 meV by\n\fts to the superconducting DoS spectrum. Because the\nthickness of the Al counter-electrode is much less than\nthe superconducting coherence length \u0018\u0018150\u0017A, it un-\ndergoes a \frst-order parallel critical \feld transition at the\nClogston-Chandrasekhar critical \feld [14, 15]\nHCC\nc=\u0001op\n1 +G0\np\n2\u0016B(2)\nwhere\u0016Bis the Bohr magneton and G0is the anti-\nsymmetric Landau parameter [16]. In panel (B) of Fig. 1\nwe show the DoS spectrum at the critical \feld. This\nspectrum is in the coexistence region between the su-\nperconducting phase and the Pauli-limited normal state.\nThe new feature that appears at the transition is a mani-\nfestation of the pairing resonance (PR). Finally, in panel\n(C) we show a normal-state spectrum where the BCS\ncoherence peaks have been extinguished. The remain-\ning structure consists of a broad, symmetric, background\n0.80.850.90.9511.05\n-4-3-2-101234V (mV)H|| = 8.5 T(C)0.70.750.80.850.90.9511.05G(V)/G(4mV)H|| = 6.8 T(B)PR00.511.52\nH|| = 4.0 Tδ1δ2(A)FIG. 1: Evolution of the tunneling conductance of a Al-AlO x-\nCNi3 tunnel junction as the parallel critical \feld transition\nis crossed at 70 mK. A: superconducting phase showing an\nasymmetric Zeeman-split DoS spectrum. The arrows denote\nthe spin assignment of the coherence peaks. B: DoS spectrum\nat the parallel critical \feld transition in which the normal-\nstate pairing resonance (PR) coexists with superconducting\ncoherence peaks. C: Pauli-limited normal state in which only\nthe pairing resonance and the zero bias anomaly remain. Note\nthat the positive and negative resonances have di\u000berent mag-\nnitudes.3\nV*PLNSFM\nVH||EF-Ez/2VReverse BiasForward Bias+Ez/2PLNSPM\n(A)(B)\nFIG. 2: A: Schematic of the spin structure of the pairing\nresonance. B: DoS pro\fles of a tunnel junction comprised of\nan Al \flm in the Pauli-limited normal state (PLNS) on one\nside and a ferromagnetic \flm on the other. Note the depletion\nof states in the PLNS due to the pairing resonance and the\nzero bias anomaly.\nwith two small satellite resonances on either side of V =\n0. The background feature is often referred to as the zero\nbias anomaly and is a well documented electron-electron\ninteraction e\u000bect [17, 18]. In contrast, the satellite fea-\ntures represent incoherent Cooper pairing. The fact that\nthe resonance dips have unequal magnitude in Fig. 1 in-\ndicates that they are spin speci\fc. As we show below,\nthey can be used to extract FM polarization in \felds\nwell beyond the Clogston-Chandrasekahr critical \feld of\nEq. (2).\nFor the purposes of measuring polarization the most\nimportant property of the PR is its spin structure. In\nFig. 2 we present a graphic representation of the res-\nonance as it is observed in tunneling into a paramag-\nnetic (PM) metal \flm. Since the PM has no preferred\nspin direction, the resonance is symmetric about V = 0.\nAs is depicted in Fig. 2(A), when a su\u000ecient forward-\nbias voltage is reached, spin-down electrons in the PM\ncan tunnel across to form doubly occupied levels close to\nthe top of the spin-up band in the PLNS. These spin-\nsinglet states can then mix with the unoccupied states\nin the near vicinity to form an evanescent Cooper pair\n[9]. This e\u000bectively produces a small depletion of spin-\ndown quasiparticle states due to the fact that they have\nbeen consumed by the resonance. By particle-hole sym-\nmetry there is a similar depletion of spin-up states at\nthe reverse-bias voltage needed for spin-up electrons ly-\n-4 -3 -2 -1 0 1 2 3 4-0.10-0.08-0.06-0.04-0.020.00∆G/GV (mV)δ+δ‐\nFIG. 3: The red symbols are tunneling spectra taken at 70 mK\nin a parallel \feld of 6.7 T, where the zero bias anomaly back-\nground has been subtracted o\u000b. The orange arrows denote the\nspin assignment of the occupied and unoccupied resonances.\nThe solid black line represents a best \ft to the resonance\ncurve. The dashed blue line is the predicted resonance pro\fle\nat 18 T, extrapolating from the \ftting parameters obtained\nat 6.7 T.\ning just above the PLNS doubly occupied sites to tunnel\nover to the top of the spin-up band in the PM. The pre-\ncise energy of these resonances is \feld dependent,\neV\u0003=1\n2\u0012\nEz+q\nE2z\u0000\u00012\n0\u0013\n; (3)\nwhereEz= (2\u0016BH)=(1 +G0) is the Zeeman energy\nrenormalized by G0.\nThe zero bias anomaly background in Fig. 1(C) is\nindependent of magnetic \feld and varies as ln Vfor\nV&kBT=e. It can easily be subtracted from the data in\norder to isolate the resonances as shown in Fig. 3. The\nred dots are data taken on a Al-AlO x-CCo 3tunnel junc-\ntion at 70 mK in a 6.7 T parallel magnetic \feld. The ar-\nrows depicted the spin assignments of the resonances and\n\u000e\u0006refer to their respective amplitudes. The solid black\nline is a best \ft to the resonance pro\fle using a procedure\nand formalism described elsewhere [19]. As was the case\nfor the superconducting phase, we only need the relative\nmagnitudes of the positive and negative bias resonances\nin order to determine the polarization. If we take the\nPLNS density of states to be Ns, then well away from the\nPR spin rotation symmetry requires Ns\n\"=Ns\n#=Ns=2.\nOn resonance, however, there will be a depletion of one\nspin component at positive bias and the other at negative\nbias,\nNs\n+;\u0000=Ns\n#;\"+Ns\n\";#(1\u0000\u000f) (4)\nwhere 0\u0014\u000f\u00141 represents the strength of the resonance.\nIn general\u000fdepends upon a number of factors, including\nmagnetic \feld, temperature, spin-orbit scattering rate,4\n0102030405060\n0246810CCo3CNi3P (%) \nH (T)\nFIG. 4: Electron polarization of 45 \u0017A CNi 3and CCo 3\flms as\na function of magnetic \feld. The dashed line represents the\napproximate parallel critical \feld of the Al counter-electrodes\nused in these measurements.\nand the dimensionless normal-state transport conduc-\ntance,g[9, 19]. Spin conservation forbids intra-spin-band\ntunneling, so if we assume that the ferromagnet has a ma-\njority spin density Nf\n\"and a minority spin density Nf\n#,\nthen the tunneling conductance is simply proportional to\nthe product of respective spin-speci\fc DoS on either side\nof the tunnel junction as is depicted in Fig. 2(B). The\nmagnitudes of the positive- and negative-bias resonance\nfeatures are\n\u000e+;\u0000= [Ns\n#;\"Nf\n#;\"+Ns\n\";#Nf\n\";#(1\u0000\u000f)]\n\u0000[Ns\n#;\"Nf\n#;\"+Ns\n\";#Nf\n\";#]\n=\u0000\u000fNsNf\n\";#\n2:(5)\nFrom this the polarization follows,\nP=\f\f\f\f\fNf\n\"\u0000Nf\n#\nNf\n\"+Nf\n#\f\f\f\f\f=\f\f\f\f\u000e+\u0000\u000e\u0000\n\u000e++\u000e\u0000\f\f\f\f: (6)\nThe data in Fig. 3 give a polarization of 39.5% which\nis comparable to values reported for pure Co by the early\nwork of Tedrow and Meservey [4] and later measurements\nusing PCAR [7]. We note that Eq. (6) is independent of\nboth the PR's strength \u000fand its width. Thus within the\nlimits of signal-to-noise constraints, one should be able\nto obtain polarization measurements at quite high \felds.\nThe dashed line in Fig. 3 represents a prediction for the\nresonance curve at 18 T obtained by extrapolating thenecessary parameters from \ftting the data at 6.7 T. Al-\nthough the PR is signi\fcantly attenuated and broadened\nby the high \feld, it is still well within the noise level\nof the data. Since the strength of the resonance grows\nas the conductance gis reduced, one can use slightly\nthinner counter-electrodes to increase the visibility of the\nresonance. Previous studies have shown that the PR in\ncounter-electrodes with g\u00146 remains well de\fned in\nperpendicular \felds of a few Tesla, thus eliminating the\nneed for precise parallel alignment [20].\nIn Fig. 4 we present electron polarization as a function\nof magnetic \feld for CNi 3and CCo 3\flms. The verti-\ncal dashed line represents the approximate critical \feld\nof the counter-electrodes used in this study. The data\nshow good agreement between polarization values mea-\nsured just below Hcjjand normal-state values measured\njust above Hcjj. The polarization of CNi 3\flm is inde-\npendent of \feld, as would be expected considering the\nfact that the Zeeman energies associated with \felds of\na few Tesla are less than 0.1 meV, which is much less\nthan exchange energies associated with the magnetiza-\ntion. Interestingly, though, the polarization of the CCo 3\n\flm exhibits a signi\fcant decrease in \felds above 3-4 T.\nThis trend can be seen in both the superconducting phase\nmeasurements and the PR measurements. The origin of\nthis \feld dependence is uncertain.\nIn summary, we have shown that conduction spin po-\nlarization can be determined from the relative amplitudes\nof the occupied and unoccupied pairing resonance fea-\ntures. The technique can be used in \felds that are sev-\neral times higher than the Clogston-Chandrasehkar crit-\nical \feld, thus allowing polarization measurements to be\nmade over a very wide range of magnetic \felds. Prelim-\ninary polarization measurements in CCo 3\flms show a\nstrong suppression of the polarization in \felds above a\nfew Tesla.\nWe gratefully acknowledge enlightening discussions\nwith Ilya Vekhter and Dan Sheehy. This work was\nsupported by the DOE under Grant No. DE-FG02-\n07ER46420.\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molnr, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n[2] G. Prinz, Phys. Today 48, 58 (1995).\n[3] P.M. Tedrow and R. Meservey, Phys. Rev. Lett. 26, 192\n(1971).\n[4] P.M. Tedrow and R. Meservey, Phys. Rep. 238, 173\n(1994).\n[5] P.D. Johnson, Rep. Prog. Phys. 60, 1217 (1997).\n[6] S.K. Upadhyay, A. Palanisami, R.N. Louie, and R.A.\nBuhrman, Phys. Rev. Lett. 81, 3247 (1998).\n[7] J. Soulen, Jr., J. M. Byers, M. S. Osofsky, B. Nadgorny,\nT. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka,\nJ. Nowak, J. S. Moodera, A. Barry, and J. M. D. Coey,5\nScience 282, 85 (1998).\n[8] P. Chalsani, S.K. Upadhyay, O. Ozatay, and R.A.\nBuhrman, Phys. Rev. B 75, 094417 (2007).\n[9] I. L. Aleiner and B. L. Altshuler, Phys. Rev. Lett. 79,\n4242 (1997).\n[10] V. Y. Butko, P. W. Adams, and I. L. Aleiner, Phys. Rev.\nLett. 82, 4284 (1999); H. Y. Kee, I. L. Aleiner, and B. L.\nAltshuler, Phys. Rev. B 58, 5757 (1998).\n[11] P. Fulde, Adv. Phys. 22, 667 (1973).\n[12] R. Meservey, P.M. Tedrow, and P. Fulde, Phys. Rev. Lett.\n25, 1270 (1970).\n[13] D.P. Young, A.B. Karki, P.W. Adams, J.N. Ngunjiri, J.C.\nGarno, H. Zhu, B. Wei, D. Moldovan, J. App. Phys. 103,\n053503 (2008).[14] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).\n[15] B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).\n[16] G. Catelani, X. S. Wu, and P. W. Adams, Phys. Rev. B\n78, 104515 (2008).\n[17] B.L. Altshuler, A.G. Aronov, M.E. Gershenson, and\nYu.V. Sharvin, Sov. Sci. Rev. A. Phys. Vol. 9, 223 (1987).\n[18] Y. Imry and Z. Ovadyahu, Phys. Rev. Lett. 49, 841\n(1982).\n[19] G. Catelani, Y. M. Xiong, X. S. Wu, P. W. Adams, e-\nprint arXiv:0905.2414.\n[20] X.S. Wu, P.W. Adams, and G. Catelani, Phys. Rev. Lett.\n95, 167001 (2005)."    },    {        "title": "1912.02337v2.Spin_pumping_into_a_spin_glass_material.pdf",        "content": "Spin pumping into a spin glass material\nYusei Fujimoto,1Masanori Ichioka,2, 1and Hiroto Adachi2, 1\n1Department of Physics, Okayama University, Okayama 700-8530, Japan\n2Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan\n(Dated: March 24, 2022)\nSpin pumping is a recently established means for generating a pure spin current, whereby spins\nare pumped from a magnet into the adjacent target material under the ferromagnetic resonance\ncondition. We theoretically investigate the spin pumping from an insulating ferromagnet into spin\nglass materials. Combining a dynamic theory of spin glasses with the linear-response formulation\nof the spin pumping, we calculate temperature dependence of the spin pumping near the spin glass\ntransition. The analysis predicts that a characteristic peak appears in the spin pumping signal,\nre\recting that the spin \ructuations slow down upon the onset of spin freezing.\nI. INTRODUCTION\nSpin current is a \row of spin angular momentum [1].\nOver the last two decades, great progress has been made\nin generating, manipulating, and detecting the spin cur-\nrent [2, 3]. With regard to the spin current generation,\nas nicely reviewed in Ref. [4] the spin pumping is now\nestablished as a charge-free and versatile means [5{8]. In\nthis method a pure spin current, which is unaccompa-\nnied by a charge current, is pumped from a ferromagnet\ninto the adjacent spin sink material by a stimulus of mi-\ncrowaves satisfying the ferromagnetic resonance (FMR)\ncondition. Thanks to the advent of the spin pumping\ntechnique, the spin current physics has so far been inves-\ntigated in a variety of spin sink materials, ranging from\nnonmagnetic metals [9{13], semiconductors [14{16], mag-\nnetic metals [17{19], insulators [20], to more exotic sys-\ntems such as graphene [21{23], transition metal dichalco-\ngenides [24], organic materials [25, 26], and strongly spin-\norbit coupled materials [27, 28].\nRecently, the playground of the spin current physics\nhas been extended to disordered magnets or the so-called\nspin glass (SG) materials [29, 30]. The SGs are charac-\nterized by a freezing of random spins [31], and its nature\nhas long been studied both experimentally and theoreti-\ncally [32]. However, despite its long history of research,\nthe interplay of spin current and the SG ordering has\nnot yet been well examined. Thus, it is quite natural to\nask what happens if we inject a pure spin current into a\nSG material by the spin pumping. Experimentally, the\nspin pumping into a SG material was reported in 2011\nusing a Ag 90Mn10/Ni81Fe19bilayer [33]. To the best of\nour knowledge, however, no theoretical work on the spin\npumping into SG materials can be found in the literature.\nTherefore, developing a theory of spin pumping into the\nSG material is highly desirable.\nIn this paper, we theoretically investigate the spin\npumping into SG materials. Although a metallic magnet\nNi81Fe19was used as the spin injecting magnet in the\nprevious experiment [33], we consider here an insulat-\ning ferromagnet such as yttrium iron garnet for the spin\ninjector, since it makes the spin pumping signal more vis-\nible. Our strategy to calculate the spin pumping into SGmaterials is as follows. First, we use a linear-response\napproach to the spin pumping [34, 35]. The notion de-\nrived from the linear response approach, that the spin\npumping is intimately related to the dynamic spin sus-\nceptibility of the spin sink layer, has successfully been\napplied to the spin pumping into a ferromagnet [36], an-\ntiferromagnets [37, 38], and recently it was also applied\nto the spin pumping into superconductors [39{41]. Thus,\nwe relate the spin pumping signal to the dynamic spin\nsusceptibility of the SG layer. Next, we calculate the sus-\nceptibility of the SG layer by employing a dynamic theory\nof SGs [42, 43]. Not only that this theory is known to\nbe an alternative formulation of the static replica the-\nory [44{47], but also that the dynamic theory is more\nsuitable to discuss the dynamic quantity such as the dy-\nnamic spin susceptibility [48].\nIn the literature, the dynamic spin susceptibility near\nthe SG transition was calculated [48], but the result\nwas limited to the Ising case and to an extremely low-\nfrequency regime less than 10 KHz, which is out of the\nFMR condition. In the present paper, we extend the sus-\nceptibility calculation of Ref. [48] to the Heisenberg case\nand GHz frequency regime that is relevant to spin pump-\ning experiments, and calculate temperature dependence\nof the spin pumping into SG materials. With this, we\nshow that a characteristic peak structure appears near\nthe SG transition, which is a consequence of the slowing\ndown of spin \ructuations that is concomitant with the\nspin freezing of the system.\nThe plan of this paper is as follows. In the next section,\nwe introduce our microscopic model, and relate the spin\npumping with the dynamic spin susceptibility of impu-\nrity spins. In Sec. III, on the basis of the dynamic theory\nof SGs, we explain how to calculate the dynamic spin sus-\nceptibility of impurity spins. In Sec. IV, the spin pump-\ning signal into a SG material is calculated as a function\nof temperature. Finally, in Sec. V we discuss and sum-\nmarize our results.arXiv:1912.02337v2  [cond-mat.mes-hall]  13 May 20202\nFIG. 1. Schematics of the system considered in this paper,\nwhere the bilayer is composed of a SG material and a fer-\nromagnetic insulator (FI). Here, \u001bandSare, respectively,\nthe conduction-electron spin and the impurity spin in the SG\nlayer, and \nis the localized spin in the FI layer. A spin cur-\nrent with a helicity opposite to \n\rows from the FI layer to\nthe SG layer.\nII. MODEL\nWe consider a bilayer composed of a ferromagnetic in-\nsulator (FI) and a SG material, as shown in Fig. 1. More\nconcretely, we may think of yttrium iron garnet (YIG)\nfor the FI layer and Mn-doped Cu (Cu:Mn) for the SG\nlayer. We assume that a static magnetic \feld H0=H0^ z\nis applied to the FI/SG bilayer in the lateral direction,\nand that the anisotropy \feld is much smaller than H0\nsuch that it can be discarded.\nWe start from the following Hamiltonian:\nH=HSG+HFI\u0000SG; (1)\nwhere the \frst term,\nHSG=X\np\u0018pcy\npcp+JeSX\nra\u001b(ra)\u0001S(ra); (2)\ndescribes the SG layer [49, 50]. Here, the \frst term on the\nright-hand side describes the conduction electron kinetic\nenergy, and the second term the coupling between the\nconduction electron spin and magnetic impurity at an\nimpurity position ra, whereJeSis thesd-type exchange\ncoupling. Here, cy\np= (cy\np;\";cy\np;#) is the electron creation\noperator for spin projection \"and#,Sis an impurity\nspin,\u001b(r) =cy(r)^\u001bc(r) is the spin density operator with\n^\u001bbeing the Pauli matrices, and c(r) =N\u00001=2\nSGP\npcpeip\u0001r\nwithNSGbeing the number of lattice sites at the SG\nlayer.\nThe second term of Eq. (1),\nHFI\u0000SG=JintX\nrint\u001b(rint)\u0001\n(rint); (3)\ndescribes the interaction between the FI and SG layers.\nHere,Jintis the interfacial sdcoupling between the con-\nduction electron spins in the SG and the localized spins\nin the FI, where rintis a position at the FI/SG interface.\nSG\nFIFIG. 2. Diagrammatic representation of the magnon self-\nenergy giving the spin pumping signal. Here, \u001f(\u001b)is\nthe dynamic spin susceptibility of conduction-electron spins,\nwhereas\u001f(S)is that of impurity spins.\nIn order to investigate the spin pumping in the present\nsystem, we use the linear-response formulation of the spin\npumping [34, 35]. We consider the situation where an\nexternal microwave with the angular frequency !acis ap-\nplied to the FI/SG bilayer that drives the FMR of the\nFI side. The linear-response formulation uses the follow-\ning magnon language. In the absence of the adjacent\nSG layer, the uniform-mode (Kittel mode) magnon has\nan intrinsic damping rate \u000b0!ac, where\u000b0is the intrin-\nsic Gilbert damping constant. In the presence of the SG\nlayer, since the spin-relaxation rate due to the SG layer is\nadditive and hence an additional spin dissipation channel\nopens, there arises an additional magnon damping rate.\nTherefore, the total Gilbert damping constant \u000bfor the\nbilayer is given by\n\u000b=\u000b0+\u000e\u000b; (4)\nwhere\u000e\u000bis the additional Gilbert damping constant.\nThe relationship between this additional Gilbert damp-\ning constant and the spin current Ispumped into the SG\nlayer withz-axis polarization is given by [34]\nIs=\u000e\u000b\r~\nMsV!ac(\rhac)2\n(\rH0\u0000!ac)2+ (\u000b0!ac)2; (5)\nwhere\ris the gyromagnetic ratio, Msthe saturation\nmagnetization, Vthe volume of the magnet, and hacis\nthe amplitude of the external microwave.\nAccording to the linear-response formulation [34, 35],\nthe additional magnon damping rate can be calculated\nfrom the corresponding magnon self-energy \u0006( !) whose\nprocess involves the spin transfer across the interface\n(Fig. 2). In the present situation, up to the lowest or-\nder with respect to Jint, the self-energy is given by\n\u0006(!) =\u0000J2\nintNint\n~2NSGNFIX\nq\u001f(\u001b)\nq(!)JeS\u001f(S)\nq(!)JeS\u001f(\u001b)\nq(!);\n(6)\nwhereNintis the number of the localized spins \nat the\nFI/SG interface, and NFIis the number of lattice sites3\nin the FI layer. In the above equation, \u001f(\u001b)\nq(!) is the\nFourier transform of the retarded susceptibility of the\nconduction-electron spin \u001b, i.e.,\u001f(\u001b)\nq(t\u0000t0) = i\u0002(t\u0000\nt0)h[\u001b\u0000\nq(t);\u001b+\n\u0000q(t0)]i, where \u0002(t) is the step function and\nwe de\fned O\u0006=Ox\u0006iOyfor a vector operator O. By\ncontrast,\u001f(S)\nq(!) is the Fourier transform of the retarded\nsusceptibility of the impurity spin S, i.e.,\u001f(S)\nq(t\u0000t0) =\ni\u0002(t\u0000t0)h[S\u0000\nq(t);S+\n\u0000q(t0)]i,\nUsing the relation \u000e\u000b=\u0000!\u00001\nacIm\u0006(!ac) [34], the addi-\ntional Gilbert damping constant \u000e\u000bis expressed as\n\u000e\u000b\u0019J2\nintNint\n~2NFI\u0010\n\u001f(\u001b)\n0JeS\u001121\n!acIm\u001f(S)\nloc(!ac); (7)\nwhere we introduced a shorthand notation \u001f(\u001b)\n0=\n\u001f(\u001b)\nq=0(!= 0), and\u001f(S)\nloc(!) =N\u00001\nSGP\nq\u001f(S)\nq(!) is the local\nsusceptibility of impurity spins. In obtaining the above\nresult, we made use of the fact that \u001f(S)\nloc(!) is de\fned in\na smallqregionq.2\u0019=bwherebis the average distance\nof two magnetic impurities. In this small qregion, the\nconduction-electron spin susceptibility is approximated\nby the uniform and static component \u001f(\u001b)\n0, where this\nquantity is pure real.\nEquation (7) means that the additional Gilbert damp-\ning constant \u000e\u000bdue to the spin pumping is proportional\nto the imaginary part of the dynamic spin susceptibil-\nity\u001f(S)\nloc(!). In this expression, the strongest temper-\nature dependence upon the SG transition results from\nthe imaginary part of the dynamic spin susceptibility,\nIm[\u001f(S)\nloc(!)]. This means that, as long as the temperature\ndependence is concerned, the part other than Im[ \u001f(S)\nloc(!)]\ncan be regarded as being temperature independent, and\nthe temperature dependence is dominated by that of\nIm[\u001f(S)\nloc(!)]. We adopt this approximation in the nu-\nmerical calculation in Sec. IV.\nThe quantity \u001f(S)\nloc(!) is a correlation function between\ntwo impurity spins, and hence it can be evaluated using\nour knowledge on SGs. In the next section, we evaluate\n\u001f(S)\nloc(!) using a dynamic theory of SGs [42, 43].\nIII. DYNAMIC SPIN SUSCEPTIBILITY OF\nIMPURITY SPINS\nIn this section, by employing a dynamic theory of\nSGs [42, 43], we sketch our procedure for calculating\nthe dynamic spin susceptibility \u001f(S)\nloc(!) appearing in the\nspin pumping signal [Eq. (7)]. We emphasize that we\nuse the dynamic model of the Heisenberg spin glasses\ndeveloped by Sompolinsky and Zippelius [43], which is\nconstructed on top of a similar dynamic theory of Ising\nspin glasses [42].\nWe \frst integrate out the conduction-electron degrees\nof freedom in the Hamiltonian for the SG layer HSG.Then,HSGis transformed into the following form [49, 50]:\nHSG=1\n2X\ni6=jJijS(ri)\u0001S(rj); (8)\nwhereJijis nominally the RKKY interaction of the form\nJij=JeScos(2kFrij)=r3\nijwithrij=jri\u0000rjj. However,\nfollowing the standard approach to the SG problem [32],\nwe regards Jijas Gaussian random variables with zero\nmean and variance [ J2\nij]av=J2=NS, where [\u0001\u0001\u0001]avmeans\nthe random average over the distribution of Jij, andNS\nis the number of impurity spins.\nHamiltonianHSGin Eq. (8) is the same as the vec-\ntor spin version [51] of the Sherrington-Kirkpatrick (SK)\nmodel [45], so that we employ the established dynamical\napproach. Following Sompolinsky and Zippelius [43], we\n\frst replaceHSGwith its soft-spin version:\n\feHSG=1\n2X\n\u000bX\nij(r0\u000eij\u0000\fJij)S\u000b\niS\u000b\nj\u0000\fX\nih\u000b\ni\u0001S\u000b\ni;(9)\nwhere\f= 1=kBTis the inverse temperature, and each\ncomponent of the soft spin varies \u00001<S\u000b\ni<1, where\nthe Greek superscript \u000b=x;y;z speci\fes the direction\nin spin space. Here, we consider a paramagnet/SG tran-\nsition by ignoring any tendency to ferromagnetic order,\nsuch thatr0in Eq. (9) is chosen to be a positive constant.\nNext, we introduce the Langevin dynamics\n\u0000\u00001\n0@tS\u000b\ni=\u0000@(\feHSG)\n@S\u000b\ni+\u0018\u000b\ni(t); (10)\nwhere \u0000 0is the bare relaxation rate. In the above equa-\ntion,\u0018\u000b\ni(t) is a thermal noise represented by a Gaussian\nrandom variable with zero mean and variance\nh\u0018\u000b\ni(t)\u0018\u000b0\nj(t0)i\u0018= 2\u0000\u00001\n0\u000eij\u000e\u000b;\u000b0\u000e(t\u0000t0); (11)\nwhereh\u0001\u0001\u0001i\u0018means the average over \u0018\u000b\ni. In the following\ncalculation, the response function\nG\u000b\u000b0\nij(t\u0000t0) =\"\n@hS\u000b\ni(t)i\u0018\n@h\u000b0\nj(t0)#\nav; (12)\nplays an important role. This is because the spin suscep-\ntibility can be calculated by the relation\n\u001f\u000b\u000b0\nij(!) =\fG\u000b\u000b0\nij(!); (13)\nwhereG\u000b\u000b0\nij(!) is the Fourier transform of G\u000b\u000b0\nij(t\u0000t0).\nBesides, we need to de\fne the correlation function\nC\u000b\u000b0\nij(t\u0000t0) =h\nhS\u000b\ni(t)S\u000b0\nj(t0)i\u0018i\nav: (14)\nTo proceed further, we precisely follow the procedure\nof Refs. [42, 43, 52], which involves a lots of technical\nalgebra. Since reviewing the details of Refs. [42, 43, 52]\nis beyond our scope, we leave it to the original paper4\nand a famous textbook [53], and we brie\ry sketch the\nderivation of the self-consistent dynamical equation in\nthe mean-\feld limit. First, we rewrite Eq. (10) in terms of\na generating functional, and introduce an auxiliary \feld\nbS\u000b\nias was done by Martin, Siggia, and Rose [54]. Next,\nthis generating functional is averaged over Jijwithout\nusing replicas [55], which generates temporally nonlocal\nquartic interactions among S\u000b\niandbS\u000b0\ni. Then, we in-\ntroduce new auxiliary \felds Q\u000b\u000b0\n1;Q\u000b\u000b0\n2;Q\u000b\u000b0\n3andQ\u000b\u000b0\n4\nto decouple the quartic terms, and we evaluate the func-\ntional integral by using the saddle-point approximation.\nFollowing the above procedure, we obtain the new\nequation of motion for S\u000b\nicontaining the local self-\ninteraction, which in the frequency space is written as\n\u0000i!\n\u00000S\u000b(!) =X\n\u000b0h\u0000\n\u0000r0+\fh\u000b(!)\u0001\n\u000e\u000b\u000b0\n+(\fJ)2G\u000b\u000b0(!)i\nS\u000b0(!) +\u001e\u000b(!);(15)\nwhereG\u000b\u000b0(!) is the local response function G\u000b\u000b0\nii(!).\nHere and hereafter, the site index iis discarded. In the\nabove equation, \u001e\u000bis a new noise \feld satisfying\nh\u001e\u000b(!)\u001e\u000b0(!0)i\u001e= 2\u0019\u000e(!+!0)\u00142\u000e\u000b\u000b0\n\u00000+ (\fJ)2C\u000b\u000b0(!)\u0015\n;\n(16)\nwhereC\u000b\u000b0(!) is the Fourier transform of the local cor-\nrelation function C\u000b\u000b0\nii(t\u0000t0), andh\u0001\u0001\u0001i\u001eis the average\nover\u001e\u000b(!).\nIn the approach by Sompolinsky and Zippelius [43] the\nweak external-\feld limit was considered, such that after\nthe random average , physical quantities are diagonal in\nspin space. Therefore, the local correlation function is\nseparated as C(t) =q+\u0001C(t), whereC\u000b\u000b0(t) =C(t)\u000e\u000b\u000b0\nand q=C(t)jt!1. In parallel with this separation, the\nnoise \feld\u001e\u000bis divided into two parts,\n\u001e\u000b(!) =f\u000b(!) +u\u000b(!); (17)\nwhere the \frst term, f\u000b(!), satis\fes\nhf\u000b(!)f\u000b0(!0)i= 2\u0019\u000e(!+!0)\u000e\u000b\u000b0\u00142\n\u00000+ (\fJ)2\u0001C(!)\u0015\n(18)\nwithh\u0001\u0001\u0001i being the usual thermal average over f\u000b(!),\nwhereas the second term, u\u000b(!), satis\fes\n[u\u000b(!)u\u000b0(!0)]u= (2\u0019)2\u000e(!+!0)\u000e(!)\u000e\u000b\u000b0(\fJ)2q;(19)\nwhere [\u0001\u0001\u0001]uis the average over u\u000b(!).\nFrom Eqs. (18) and (19), we \fnd that f\u000b(!) is a usual\nthermal \ructuation, whereas u\u000b(!) represents a frozen\nrandom \feld that breaks the ergodicity. Since u\u000bacts as\na static random \feld, Sompolinsky and Zippelius intro-\nduced the \\unaveraged\" response function over u:\ng\u000b\u000b0(!;u) =@hS\u000b(!)i\n@h\u000b0(!); (20)whereg\u000b\u000b0(!;u) is related to G\u000b\u000b0(!) through\nG\u000b\u000b0(!) = [g\u000b\u000b0(!;u)]u: (21)\nNote that the average over uis separated as\n[\u0001\u0001\u0001]u= [[\u0001\u0001\u0001]^u]u; (22)\nwhere [\u0001\u0001\u0001]^umeans the angular average over a unit vector\n^u=u=u, and from Eq. (19) the average of a quantity\nQ(u) overuis given by\n[Q(u)]u=Z1\n04\u0019u2du\n[2\u0019(\fJ)2q]3=2e\u0000u2=[2(\fJ)2q]Q(u):(23)\nNow, we discuss the Dyson equation for g\u000b\u000b0(!;u). In\ndoing so, we introduce the matrix notation \u0014 g(!;u), where\nthe matrix element of \u0014 g(!;u) equalsg\u000b\u000b0(!;u), i.e.,\n\u0014g(!;u) =\u0010\ng\u000b\u000b0(!;u)\u0011\n: (24)\nWith this matrix notation the Dyson equation for\ng(!;u), which results from Eq. (15) with static random\n\feldu, is given by\n\u0014g(!;u)\u00001=\u0014G(0)(!)\u00001\u0000\u0014\u0006(!;u); (25)\nwhere the bare propagator is given by \u0014G(0)(!)\u00001= [r0\u0000\ni!=\u00000\u0000(\fJ)2G(!)]\u00141, and \u0014\u0006(!;u) is the self-energy com-\ning from the frozen u-\feld. Since we are interested in the\ndynamic behavior of \u0014G(!), we solve the Dyson equation\nby perturbation with respect to \u0001 \u0014G(!) =\u0014G(!)\u0000\u0014G(0).\nWe de\fne \u0014\u0011(!;u) = i!=\u00000+ (\fJ)2\u0001\u0014G(!)\u0000\u0001\u0014\u0006(!;u),\nwhere \u0001 \u0014\u0006(!;u) = \u0014\u0006(!;u)\u0000\u0014\u0006(0;u), and rewrite the\nDyson equation as \u0014 g(!;u)\u00001= \u0014g(0;u)\u00001\u0000\u0014\u0011(!;u). To\nproceed further, we disregard \u0001 \u0014\u0006(!;u) as it brings only\na small change [48]. This allows us to regard \u0014 \u0011(!;u)\nasu-independent, and the matrix \u0014 \u0011becomes diagonal in\nspin space, i.e., \u0014 \u0011(!) =\u0011(!)\u00141. Then, after expanding the\nDyson equation up to \u0014 \u00112and averaging the result over u,\nwe obtain a quadratic equation for \u0001 \u0014G(!) = \u0001G(!)\u00141:\n\u0001G(!)\u0000\n1\u0000(\fJ)2[\u0014g2\n0]u\u0001\n=\u0012\ni!\n\u00000+ (\fJ)2\u0001G(!)\u00132\n[\u0014g3\n0]u\n+i!\n\u00000[\u0014g2\n0]u; (26)\nwhere we introduced the shorthand notation [\u0014 gn\n0]u=\n[\u0014g(0;u)n]u. Note that [\u0014 gn\n0]uis diagnoal in spin space\nafter the average over u, such that it appears in Eq. (26)\nas a c-number. Equation (26) can be solved for \u0001 G(!).\nAfter using the relation in Eq. (13), we obtain\n\u001f(S)\nloc(!) =\u001f(S)\nloc(0)+eT=3\n2[\u0014g3\n0]u\u0012\nD\u0000r\nD2\u0000i4!\n\u00000[\u0014g2\n0]u[\u0014g3\n0]u\u0013\n;\n(27)\nwhereD= (eT=3)2\u0000[\u0014g2\n0]u\u00002i(!=\u00000)[\u0014g3\n0]u,eT=T=Tg.\nNote that, in contrast to the Ising spin glass [45], the5\n0.0 0.5 1.0 1.5 2.0\nT/Tg0.00.10.20.30.40.5(S)\nloc(0)\nFIG. 3. Static spin susceptibility \u001f(S)\nloc(0) as a function of\ntemperature calculated using Eq. (31).\nprenset Heisenberg spin glass has a transition tempera-\ntureTg=J=3kB[51].\nIn the following calculation, we take the mean-\feld\napproximation of the classical Heisenberg spins un-\nder a frozen u\feld. We use hS\u000bi=hSi^u\u000band\nhS\u000bS\u000b0i=\u000e\u000b\u000b0=3, wherehSi= coth(u)\u00001=uis the\nLangevin function. Then, recalling that [ hS\u000bS\u000b0i]u=\n[hSi2]u[^u\u000b^u\u000b0]^u= [hSi2]u\u000e\u000b\u000b0=3, the spin glass order pa-\nrameter qcan be calculated from\nq=1\n3[hSi2]u; (28)\nwhere the quantity [ hSi2s]uis given by\n[hSi2s]u=Z1\n04\u0019u2du\n[2\u0019(\fJ)2q]3=2e\u0000u2=[2(\fJ)2q]\n\u0002\u0012\ncoth(u)\u00001\nu\u00132s\n: (29)\nIn a similar manner, we use [\u0014 g0]u= (1\u00003q)=3, [\u0014g2\n0]u=\n(1\u00006q+ 3[hSi4]u)=9, and [\u0014g3\n0]u= (1\u00009q+ 9[hSi4]u\u0000\n3[hSi6]u)=27. At the SG transition, the Almeida-\nThouless condition [56] holds, which in the present nota-\ntion takes the form\n[\u0014g2\n0]u=\u0010\neT=3\u00112\n: (30)\nThen, following Sompolinsky and Zippelius [42] we as-\nsume that Eq. (30) holds not only at Tgbut also below\nTg, meaning that the SG phase is characterized by the\nmarginal instability condition.\nIV. RESULTS FOR SPIN PUMPING INTO SPIN\nGLASS MATERIALS\nIn this section, using the formalism developed in the\nprevious two sections, we calculate temperature depen-\ndence of the spin pumping signal. The key equation in\n0.0 0.5 1.0 1.5 2.0\nT/Tg024681012141618(T)/(T=2Tg)\n=0.01\n=0.02\n=0.05\n=0.10\n=1.00\nFIG. 4. Temperature dependence of the additional Gilbert\ndamping constant [Eq. (7)] in a SG/FI bilayer (Fig. 1), cal-\nculated for several values of e!ac=!ac=\u00000.\nthe present argument is Eq. (7), which relates the spin\npumping with the dynamic spin susceptibility of SG ma-\nterials.\nBefore presenting our results for the spin pumping that\nis intimately related to the dynamic spin susceptibility\n\u001f(S)\nloc(!), it is instructive to examine the static spin sus-\nceptibility\u001f(S)\nloc(0). This quantity can be calculated us-\ning [57]:\n\u001f(S)\nloc(0) =1\n3eT(1\u00003q); (31)\nwhere the additional factor 3 follows from the relation\n[g\u000b\u000b0]u=\u000e\u000b\u000b0(1\u00003q)=3. Figure 3 shows the static spin\nsusceptibility \u001f(S)\nloc(0) as a function of temperature, cal-\nculated from Eq. (31). The result reproduces the well-\nknown cusp structure at Tg[45, 51].\nNow we discuss the spin pumping into a SG material.\nIn discussing the spin pumping signal in the present case,\nthe important parameter is the ratio of the microwave\nangular frequency !acto the relaxation rate of localized\nspins \u0000 0, i.e.,e!ac\u0011!ac=\u00000. Since the magnitude of\n!acunder the FMR condition is of the order of 60 GHz,\nthe parameter e!acis determined by a material parameter\n\u00000. The spin relaxation time in a prototypical SG mate-\nrial Cu:Mn is reported to be of the order of picosecond\n(corresponding to \u0000 0\u0018103GHz) [58], and hence the\nparametere!acunder the FMR condition is estimated to\nbee!ac\u00180:1. Note that the previous calculation of the\ndynamic spin susceptibility [48] was done for very low\nfrequenciese!ac.10\u00004, which is far out of the FMR\ncondition.\nFigure 4 shows temperature dependence of the spin\npumping signal, calculated from Eqs. (7) and (27). First,\nwe see that a clear peak structure appears at Tg. Second,\nupon the increase of the parameter e!ac, the height of\nthe peak is reduced. This is because the peak structure\noriginates from the critical slowing down of spins that6\ndevelops on the verge of the spin freezing [53], so that\nthe e\u000bects of the slowing down are more prominent when\nthe paramagnetic state has a more rapid dynamics (i.e.,\nlarger \u0000 0) in comparison to the spin frozen state.\nAs shown in Fig. 4, the peak structure is visible for\na parameter region e!ac.0:1. Since the parameter e!ac\nfor a prototypical SG material Cu:Mn is estimated about\ne!ac\u00180:1 [58] as mentioned above, we expect that we can\nobserve a peak structure in the spin pumping signal near\nTg. Therefore, we propose a spin pumping experiment\nfor a Cu:Mn/YIG system in order to test our theoreti-\ncal prediction. Frequency ( !ac) dependence of the peak\nmentioned above, namely, the lower the frequency !ac,\nthe higher the peak, is a key to identify the predicted\nsignal.\nV. DISCUSSION AND CONCLUSION\nThe main result of this paper is the theoretical pre-\ndiction that the spin pumping into a SG material, whose\nsignal is proportional to the additional Gilbert damping\nconstant\u000e\u000b, can be enhanced near the SG transition.\nThe physics behind this enhancement is explained in the\nfollowing way. First, a SG material exhibits the critical\nslowing down upon the spin freezing [53], meaning that\nthe spin relaxation rate of the SG material is reduced.\nThen, recalling that the spin pumping represents an ad-\nditional damping of magnons in the spin injecting mag-\nnet, this type of enhancement in the spin pumping signal\ncan be interpreted as a kind of the inverse of the motional\nnarrowing [59, 60] as discussed in Ref. [61] (see Sec. VI\ntherein). It means that a reduction of the spin relaxation\nrate in the spin sink material results in a broadening of\nthe magnon damping in the adjacent magnet, leading to\nthe enhancement of the spin pumping.\nThe spin pumping has an advantage that it can mea-\nsure the dynamic spin susceptibility of a thin\flm sample.\nSo far, this fact has been applied to the spin pumping into\nferromagnets [34, 36], antiferromagnets [37, 38], and su-\nperconductors [39{41]. Extending the same idea to SGs\nwithin the linear-response approach, we formulated the\nspin pumping into a SG material, and shown that the\nsignal is expressed by using the local spin susceptibility\nof the SG material [Eq. (7)]. Moreover, the spin pumping\ninto a SG material is predicted to exhibit a characteristicpeak around Tg(Fig. 4).\nThe height of the predicted peak in the spin pumping\nsignal is controlled by the ratio of the microwave angu-\nlar frequency to the relaxation rate of impurity spins,\ni.e.,e!ac=!ac=\u00000. That is, a smaller e!acis better for\nan experimental detection of the peak. Conversely, it\nmeans that if the parameter e!acis too large, the peak is\nnot visible. To test our theoretical prediction, we hope\na future spin pumping experiment using an insulating\nmagnet with, e.g., Cu:Mn/YIG structure, since use of an\ninsulating magnet instead of a metallic magnet makes the\nspin pumping signal more visible owing to the smallness\nof the intrinsic Gilbert damping term \u000b0.\nBefore conclusion, we brie\ry comment on the previous\nexperiment of the spin pumping into a SG material us-\ning Ag 90Mn10/Ni81Fe19[33]. In that experiment, the SG\ntransition temperature is estimated to be Tg= 25 K from\nthe cusp in the susceptibility data of a thick Ag90Mn10\n\flm. Note that this thick \flm is di\u000berent from the thin\n\flm used for the spin pumping experiment. In the spin\npumping experiment, a weak temperature dependence of\nthe signal was measured around Tg, but no pronounced\npeak expected from the theoretical calculation (Fig. 4)\ncan be seen. The reason could be either i) the important\nparametere!ac=!ac=\u00000is too large in Ag 90Mn10for the\nenhanced spin pumping to be detected (see Fig. 4), or ii)\nthe SG transition temperature of the thin\flm sample is\nmuch lower than 25 K, since the thin\flm may have lower\nTgthan the thick \flm used to determine Tg= 25 K.\nTo conclude, we have theoretically examined the spin\npumping into a SG material. We have shown that the\ntemperature dependence exhibits a characteristic peak\nnear the SG transition, whose height is controlled by the\ndimensionless angular frequency e!ac=!ac=\u00000. This is a\nconsequence of the critical slowing down of spin \ructua-\ntions upon the spin freezing. Since the spin pumping has\nan advantage of being able to measure the spin dynamics\nof a thin\flm sample [39], we hope that the present the-\nory stimulates further experiments of the spin pumping\ninto SG materials.\nACKNOWLEDGMENTS\nThis work was \fnancially supported by JSPS KAK-\nENHI Grant Number 19K05253.\n[1] S. Maekawa, H. Adachi, K. Uchida, J. Ieda, and E.\nSaitoh, Spin Current: Experimental and Theoretical As-\npects, J. Phys. Soc. Jpn 82, 102002 (2013).\n[2] I. \u0015Zuti\u0013 c, J. Fabian, and S. D. Sarma, Spintronics: Fun-\ndamentals and applications Rev. Mod. Phys. 76, 323\n(2004).\n[3]Spin Current , edited by S. Maekawa, S. O. Valenzuela, E.\nSaitoh, and T. Kimura (Oxford University Press, Oxford,\n2012).[4] I. \u0015Zuti\u0013 c and H. Dery, Taming spin currents, Nat. Mater.\n10, 647 (2011).\n[5] R. Urban, G. Woltersdorf, and B. Heinrich, Gilbert\nDamping in Single and Multilayer Ultrathin Films: Role\nof Interfaces in Nonlocal Spin Dynamics, Phys. Rev. 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We investigate an\nall-optical switching process for ferromagnets, which is close to the one proposed by Qaiumzadeh\net al. [Phys. Rev. B 88, 064416] for the inverse Faraday e\u000bect. By computing the optically driven\ncoherent dynamics together with incoherent scattering mechanisms we go beyond a perturbation\nexpansion in powers of the optical \feld. We \fnd an important contribution of a dynamic Stark\ne\u000bect coupling of the Raman type between the magnetic bands, which leads to a polarization-\ndependent e\u000bect on the magnetization that may support or oppose switching, but also contributes\nto demagnetization via an increase in electronic energy.\nPACS numbers: 75.78.-n, 72.25.Rb, 76.20.+q\nIntroduction. The optical excitation of magnetic sys-\ntems has many facets that have been explored over\nthe last two decades. After demagnetization of 3d-\nferromagnets by ultrashort pulses was discovered,1it was\nrealized that in alloys with anti-ferromagnetically cou-\npled sublattices the excitation by ultrashort pulses can\nlead to a transient ferromagnetic-like state2and even a\ncomplete reversal of the magnetization. These magne-\ntization dynamics can be understood in terms of tran-\nsient heating e\u000bects.3{6More recently, there has also been\nevidence for magnetization switching induced in ferro-\nmagnets7,8where a purely heat-induced e\u000bect should not\nwork. In this case, there should be a microscopic mech-\nanism with which the polarized optical \felds act on the\nmagnetization. Currently most popular explanation is\nthe inverse Faraday e\u000bect (IFE), which has been stud-\nied in the last decade for ultrafast magnetism.9{12How-\never, the name IFE is applied to di\u000berent mechanisms\nand models of light-matter interaction, some of which\nare more or less classical in nature.13{21The IFE was\nintroduced for paramagnetic materials, but there are im-\nportant di\u000berences for this e\u000bect between paramagnetic\nand ferromagnetic materials.22,23\nRecent quantum mechanical calculations of the time-\ndependent IFE for ferromagnets have been performed\nin di\u000berent ways. A perturbation theory in orders of\nthe external \feld for band ferromagnets suggests that\nthe magnetization is most e\u000eciently in\ruenced directly\nby a coherent optical \feld if this \feld is resonant with\ndipole transitions in the system.24,25Calculations for few-\nlevel systems have analyzed a stimulated Raman e\u000bect,\nwhere the \feld is resonant with intermediate levels26and\nwhere angular momentum is transferred from the \feld.\nQaiumzadeh et al.27proposed the spin-dependent opti-\ncal Stark e\u000bect as a mechanism for the inverse Faraday\ne\u000bect that works for o\u000b-resonant optical \felds. In this\nscenario the spin-dependent optical Stark e\u000bect changes\nthe splitting between the spin-up and spin-down bands\nand the actual magnetization change occurs by redistri-bution of electrons between the split bands so that the\nlattice acts as source and sink of angular momentum.\nThe present paper analyzes a mechanism for all-optical\nswitching by o\u000b-resonant \felds that is based on the ideas\nof Qaiumzadeh et al.27but modi\fes and extends their\nmodel to also include angular momentum exchange with\nthe optical \feld and spin-orbit coupling in the magnetic\nbands. It combines the action of a coherent optical\n\feld, incoherent scattering dynamics and k-dependent\nspin-orbit contributions in a ferromagnetic model band\nstructure of itinerant electrons. The main purpose of\nthis paper is to elucidate general properties of the mag-\nnetic switching dynamics due to the interaction with o\u000b-\nresonant optical \felds. In the following, we will generally\nrefer to the optically induced magnetization dynamics as\nall-optical switching and avoid the term \\inverse Fara-\nday e\u000bect,\" because of the di\u000berent meanings attached\nto it.13{19,24{26\nModel. We intend to capture important aspects of\nthe mechanisms underlying all-optical switching by tak-\ning into consideration as essential ingredients spin-orbit-\ncoupling (SOC), electronic scattering processes and a\nferromagnetic exchange splitting that can change in re-\nsponse to the electronic dynamics and thus captures\ntime-dependent magnetic properties. The model band\nstructure is shown in Fig. 1. We consider two partially\n\flled s-like bands j\u0006;ki, which exhibit a magnetic split-\nting of about 20 meV due to a Stoner mean-\feld contribu-\ntion but also include spin-orbit coupling. For the purpose\nof nonlinear optics the magnetic bands are the \\essen-\ntial bands\" and are dipole coupled to two \\non-essential\nbands\"j\u00063\n2;ki, which are modeled as \flled p-like bands\n(with total angular momentum j= 1 andmj=\u00063=2\nand e\u000bective mass 0 :4me)28and are separated from the\nessential magnetic bands by an energy that is large com-\npared to the equilibrium magnetic splitting. In order\nto keep the numerical calculations manageable, we use\na Rashba-type spin-orbit coupling with an e\u000bectively 2-\ndimensional kspace and include magnetism at the levelarXiv:1803.01924v1  [cond-mat.mtrl-sci]  5 Mar 20182\nFIG. 1. Schematic band structure and transitions. The bands\nj+iandj\u0000iare s-like bands with a magnetic splitting due to\na Stoner mean \feld and are coupled by electric dipole transi-\ntions (dotted lines) to two p-like bands. The di\u000berent dipole\ncoupling matrix elements are indicated by the thickness of the\narrows and the angular momentum transferred to the carri-\ners by a left (right) circularly polarized photon is indicated\nby\u0000~(+~). Bothj+iandj\u0000istates are connected to each\nof the p-like state by each circular optical \feld polarization.\nThe arrows for the di\u000berent transitions are o\u000bset to show\nthe Raman-like nature of the scheme, there is no momentum\ntransfer.\nof a Stoner splitting\n^H= \n~2k2\n2m\u0003+U\n2(n\u0000m)\u0000\u000b(ky+ikx)\n\u000b(\u0000ky+ikx)~2k2\n2m\u0003+U\n2(n+m)!\n(1)\nwherendenotes the particle density and mrepresents\nthe spin polarization per site, which is determined from\nthe reduced density matrix, see below. The Stoner ex-\nchange splitting is \u0001 = Um, and the magnetic and spin-\norbit splitting are controlled, respectively, by the Stoner\nand Rashba parameters Uand\u000b. For our model sys-\ntem, the carrier densities and splitting are smaller than\nin metallic ferromagnets, but they are consistent with\neach other, and the model includes the complete carrier\nand magnetic dynamics of the switching process. In par-\nticular, if the carrier distributions in the electronic bands\nchange due to carrier scattering and/or interaction with\noptical \felds, this a\u000bects the magnetization and electron\ndensity, so that the instantaneous energy dispersions de-\nscribed by (1) also change with time.\nThe magnetization dynamics are induced by the cou-\npling to the optical \feld via dipole matrix elements con-\nnecting the magnetic bands and the non-essential bands,\nas shown in Fig. 1. In di\u000berence to Qaiumzadeh et al.,27\nthe Rashba spin mixing in the magnetic states means that\neach circular polarization state of the optical \feld cou-\nples both magnetic bands to a p-like state, which makes\nRaman-like transitions between the magnetic states pos-\nsible, as indicated by arrows in Fig. 1. While the direc-\ntions of the transitions depend on the system parameters\nand excitation conditions, there is an asymmetry betweenopposite circular \feld polarizations due to the di\u000berent\nstrengths of the dipole matrix elements between the mag-\nnetic states and p-like states, as signi\fed by the thickness\nof the arrows in Fig. 1. We describe the coherent dy-\nnamics of the 4 band system sketched in Fig. 1 using the\nreduced single-particle density matrix \u001a\u0016\u0017\nk=h^cy\nk;\u0017^ck;\u0016i,\nwhere ^cy\n\u0016;kcreates a particle in a Bloch state labeled by\nband index \u0016, which runs over the 4 bands shown in\nFig. 1, and crystal momentum k:\n@\n@t\u001a\u0017\u0016\nk=i\n~(\u000f\u0016(k)\u0000\u000f\u0017(k))\u001a\u0016\u0017\nk\n+X\n\u00160\u0002\n\u001a\u0017\u00160\nk\n\u00160;\u0016\nk(t)\u0000\n\u0017;\u00160\nk(t)\u001a\u00160\u0016\nk\u0003\n+@\n@t\u001a\u0016\u0017\nk\f\f\f\nscat;(2)\nHere, \n\u0016\u0017\nk(t) =d\u0016;\u0017(k)\u0001E(t) are the matrix elements of\nthe Rabi energy, which contains the real electric \feld vec-\ntorE(t) and the matrix elements of the dipole operator\nd=\u0000er.\nWe assume that these vanish between the magnetic\nbands and use the Rabi energy as a measure of the \feld\namplitude. Importantly, the band energies \u000f\u0016(k) and ba-\nsis statesj\u0016;kiused for the matrix elements in (2) are\nthe instantaneous eigenenergies of the mean-\feld hamil-\ntonian (1). In each timestep, the basis of single particle\nstatesj\u0016;kicorresponding to the instantaneous values of\nnandmis used. This procedure includes the in\ruence\nof the coherent optical \feld and the change of the in-\nstantaneous quasiparticle band structure via incoherent\nredistribution and relaxation processes.\nWe compute the incoherent redistribution of carriers,\ni.e., scattering, only for the magnetic bands, as we as-\nsume that only this is relevant for the magnetization\ndynamics. To this end we introduce scattering contri-\nbutions to the elements \u001a\u0016\u0017with\u0016;\u0017=\u0006of Eq. (2),\nin the form of a relaxation-time ansatz designed for\nthe treatment of systems with spin-orbit coupling and\nmean-\feld magnetism. We use a generalized relaxation\ntime ansatz d\u001a\u0016\u0017\nk=dtjscat=\u0000\u001a\u0016\u0017\nk\u0000~\u001a\u0016\u0017\nk\n\u001cin (2), where ~ \u001a\u0016\u0017\nk\nindicates the elements of a suitably determined quasi-\nequilibrium spin density matrix.29In order to model\nelectron-electron scattering, we determine ~ \u001asuch that\nthe relaxation-time ansatz conserves spin polarization,\nenergy density and charge density of the carriers in\nthe essential bands. The quantity \u001cplays the role of\nan e\u000bective scattering time for the incoherent electron-\nelectron-scattering. The spin polarization of the mag-\nnetic bands, which we take as the magnetization, is cal-\nculated via m=P\nk;\u0017;\u00170h\u0017;kj~ \u001bj\u00170;ki\u001a\u0017\u00170\nkand includes\nthe time-dependent spin expectation values of the eigen-\nstatesj\u0017;kiof (1).\nResults. We determine a self-consistent magnetic\nground-state for the model (1) with initial magnetization\ndirection +z. For the calculations presented in the follow-\ning, we assume a Stoner parameter of 50 meV, a Rashba\nparameter of 20 meV nm, m\u0003=me, and a equilibrium\ntemperature for the mean-\feld ground state calculation3\n−0.200.20.4\n2.152.22.242.292.33spin polarizationenergy density [meV nm−2]\ntime [ps]spin polarization\n0 0.25 0.5 0.75 10.330.340.350.360.37\nFIG. 2. (a) Time dependent magnetization (spin polariza-\ntion) of the ferromagnetic model system excited by a optical\n\feld with right-circular (dashed line) and left-circular (solid\nline). Magnetization switching occurs only for left-circularly-\npolarized excitation. (b) corresponding energy density. (c)\nSame calculation as (a) without electronic redistribution.\nofT= 70 K. Starting from this ground state, we com-\npute the dynamical distribution functions and states by\nsolving (2) together with (1). The excitation is charac-\nterized by coupling to the non-essential bands, which are\nseparated by 1 :5 eV. For the e\u000bective electron-electron\nscattering time we take \u001c= 50 fs.\nWe consider the switch-on of a CW \feld which is\nramped up with a rise time of 40 fs at t= 0 and to an am-\nplitude corresponding to a Rabi energy of ~\n = 15 meV,\nfor a transition between a non-essential band state and\na magnetic band as shown in Fig. 1. The frequency\nof the optical \feld is detuned by 200 meV with respect\nto the transition to bottom of the lower band. We\nalways assume a vanishing dephasing contribution, i.e,\nd\u001a\u0016\u0017=dtjscat\u0011\r!0 for the o\u000b-diagonal elements of the\ndensity matrix corresponding to the optical transitions,\nwhich is well justi\fed for a large detuning.28Fig. 2(a)\nshows the time-dependent magnetization including scat-\ntering contributions for o\u000b-resonant left polarized and\nright polarized optical \felds. For the left-polarized opti-\ncal \feld the magnetization is \frst reduced to about 1/4\nof its initial value, and then the magnetization directionis reversed rapidly in about 50 fs. After magnetization\nswitching is completed at about 250 fs, there is a slower\ndecay of the magnetization. For right polarized light we\n\fnd in Fig. 2(a) essentially only demagnetization. Com-\nparison of the two light polarization states shows that\nthis setup realizes o\u000b-resonant all-optical switching in a\nmodel ferromagnet.\nTo explain the behavior of the magnetization for the\ndi\u000berent polarization states of the optical \feld, we take\na closer look at how the optical \felds in\ruence the elec-\ntronic dynamics in the spin-orbit coupled essential bands.\nIn addition to the magnetization, we characterize the\nelectronic dynamics by the density and total energy den-\nsitye=P\nk\u0016\u000fk\u0016\u001a\u0016\u0016\nk+Un\"n#. The latter quantity is\nshown in Fig. 2(b). When the optical \feld is switched\non, the energy increases rapidly and subsequently shows\na slow increase. The comparatively large initial increase\nby 5% is due to our choice of model system whose density\nof states is smaller than in a metallic ferromagnet. The\nincrease in energy is accompanied by an 0.5% increase of\nthe electronic density in the magnetic bands (not shown).\nThe initial change in the energy and electronic density\nis due to the 40 fs switch-on of the optical \felds, which\nleads to Fourier components of the \feld that are reso-\nnant with transitions between the \flled bands and the\nmagnetic bands, even though we use a vanishing dephas-\ning\r!0 of the optical polarization. One could call\nthis a stimulated Raman process. We stress that this\nstimulated Raman process involves photons of the same\npolarization, so that it is fundamentally di\u000berent from\nthe process studied by Popova et al.30\nIn addition to the initial rise of electronic energy in\nthe magnetic bands with the switching-on of the \feld,\nthere is a subsequent slower increase in electronic energy\nvisible in Fig. 2(b) for both right and left-polarized op-\ntical \felds. After the switch-on, the \feld is essentially\ncontinuous-wave, so that there is no stimulated Raman\ntransition and the in\ruence of the optical \feld is better\ndescribed as dynamic Stark e\u000bect coupling of the Ra-\nman type.31These Raman-like transitions are indicated\nby arrows in Fig. 1. This is similar to the mechanism\nsuggested by Qaiumzadeh et al., but in their model the\nelectronic bands are not spin mixed and the Raman tran-\nsition is between the non-essential bands instead of the\nmagnetic electron bands in our case.\nTo assess the in\ruence of the incoherent contributions\nto the dynamics, we compare Fig. 2(a) with Fig. 2(c),\nwhich shows the results if we repeat the calculation of\nFig. 2(a) with identical parameters but without scat-\ntering/carrier redistribution contributions. In this case\nthere is a small change of spin polarization, i.e., the mag-\nnetization, in opposite directions for the di\u000berent light\npolarizations (the di\u000berence will be discussed below),\nwhich proves that the direct e\u000bect on the magnetization\ninduced by the polarized coherent optical \felds is indeed\nsmall and incoherent scattering is needed for all-optical\nmagnetization switching to occur. Without scattering,\nthe charge and energy dynamics, which are not shown4\nhere, are very similar from the case with scattering shown\nin Fig. 2(b). The carrier redistribution thus converts the\nenergy increase due to the dynamic Stark e\u000bect coupling\ninto a demagnetization, as suggested by Qaiumzadeh et\nal.27In our microscopic calculation it does this via the\ninterplay of k-dependent spin-orbit coupling and a spin\nconserving scattering processes.32{34Together, they act\nas an Elliott-Yafet-type process, which leads to a demag-\nnetization whenever the energy in the electronic system is\nincreased. Only when this demagnetization process is in-\ncluded in the calculation we \fnd a pronounced di\u000berence\nbetween the magnetization dynamics for the di\u000berent po-\nlarization states of the optical \feld.\nWe calculate the whole magnetization dynamics, which\nleads to a reversal of the magnetization direction for the\nleft-polarized optical \feld and which is due to a combina-\ntion of the coherent \feld, the change in the band struc-\nture, and the incoherent redistribution dynamics. Thus\none cannot simply determine an e\u000bective magnetic \feld\nthat is responsible for the switching dynamics, but we can\ncompare to the calculation without carrier redistribution\n(scattering) processes, which is shown in Fig. 2(c). In this\ncase, the left-polarized light leads to a steady state with\nreduced magnetization, whereas the right-polarized op-\ntical \feld leads to a steady state with slightly increased\nmagnetization. We therefore \fnd that for the left po-\nlarized optical \feld, the dynamic Stark-e\u000bect coupling\nworks in the direction of magnetization reversal, but is\nnot su\u000ecient to switch the magnetization by itself. Only\nwhen the magnetic splitting is reduced so that the band\nstructure is closer to that of a pure Rashba system, the\noptical \feld can switch the direction of the magnetization\nby reversing the small zcomponent of the single-particle\nexpectation value h\u0016kj\u001bzj\u0016kiof the\u0016= + and\u0016=\u0000\nbands, respectively.29For the right-polarized optical \feld\nthe dynamic Stark-e\u000bect coupling alone would give rise\nto an increase in magnetization, but the concomitant\nincrease in energy e\u000bectively leads to demagnetization.\nWhile we cannot compare this result directly to the mea-\nsurement of El Hadri et al.35, it shows qualitatively they\ncorrect behavior as the experiment also \fnds an e\u000bec-\ntive demagnetization for the light polarization opposite\nto the one that leads to switching. The last observation in\nconnection with Fig. 2 concerns the behavior for longer\ntimes (>1 ps). There the magnetization dynamics for\nthe left and right polarized pulses become symmetric be-\ncause for the right polarized \feld, the helicity of the light\nand the magnetization direction are aligned for the whole\ndynamics and for the left polarized case the helicity an\nmagnetization are aligned in the same \u0000zdirection.\nFinally, we would like to investigate the connection to\nperturbation theory with respect to the external \feld,\nwhich is often regarded as the sole contribution of the\ncoherent optical \feld. We can achieve this by computing\nthe magnetization dynamics for a \fxed band structure.\nIn this case, we do not update the eigenenergies and\neigenstates of hamiltonian (1), i.e., we keep the Stoner\nmean-\feld spliting \fxed, so that the change in magneti-\ntime [ps]spin polarization\n0 0.1 0.2 0.3 0.4 0.50.330.340.350.360.37FIG. 3. Magnetization (spin polarization) dynamics com-\nputed without carrier redistribution and for a \fxed band\nstructure with a left-polarized (solid line) and a right-\npolarized (dashed line) optical \feld. The CW optical \feld\nis closer to resonance with a detuning of 5 meV. This setup\nis designed to approximate the change of the density matrix\ndue to the coherent optical \feld and leads to a reduction in\nmagnetization for both circular polarizations of the optical\n\feld.\nzation comes from the dynamics of the reduced density\nmatrix under the in\ruence of the optical \feld. Such a\ncalculation for right and left-circularly polarized optical\n\feld is shown in Fig. 3. For a detuning of 200 meV the\ne\u000bect on the magnetization is vanishingly small, so that,\nin addition to keeping the band structure \fxed, we now\nuse a \feld closer to the resonance between magnetic and\nnon-essential bands, namely a detuning of 5 meV. The\nother parameters are kept the same as in Fig. 2. In this\nscenario, the magnetization approaches di\u000berent steady\nstates with a reduced magnetization (compared to equi-\nlibrium) for both right and left-polarized optical \felds.\nAs the density matrix includes the in\ruence of the co-\nherent \feld to all orders of the \feld, this result should\ncorrespond to a mechanism for the inverse Faraday e\u000bect\nanalyzed by Oppeneer and coworkers in the framework\nof 2nd-order perturbation theory for close-to-resonance\n\felds. Indeed, Berritta et al.36have also found that the\nin\ruence of optical \felds with opposite circular polariza-\ntion may lead to a reduced steady-state magnetization\nwith respect to the equilibrium magnetization. This is\nsomewhat contrary to what one expects from older the-\nories of the inverse Faraday e\u000bect which yielded exactly\nopposite e\u000bective magnetic \felds for opposite circular po-\nlarizations of the optical \feld. This \\antisymmetry\" be-\ntween the magnetic \feld and the circular optical polar-\nization is not present because of the \fnite equilibrium\nmagnetization, which breaks the symmetry between the\ndynamics induced by right and left-circularly polarized\n\felds.37In our case, we \fnd that the asymmetry is pro-\nnounced for the close-to-resonance case in Fig. 3 because\nthere the optical \feld \\sees\" the ferromagnetic splitting.\nIn the o\u000b-resonant case in Fig. 2(c), the ferromagnetic\nsplitting should play a smaller role, and indeed the mag-\nnetization dyamics are closer to realizing antisymmetry,\nas the magnetization is changed in opposite directions for5\nopposite polarizations.\nConclusion. We introduced a microscopic dynamical\nmodel to study the all-optical magnetization switching\nprocess in a simple ferromagnetic band structure, includ-\ning spin-orbit coupling and incoherent carrier redistribu-\ntion/scattering processes. For o\u000b-resonant excitation we\nfound that the switching process is a combination of de-\nmagnetization and the in\ruence of the o\u000b-resonant \feld\nin the form of a dynamical Stark e\u000bect. The main an-\ngular momentum change is supplied by the lattice via\nan Elliott-Yafet like magnetization change, which results\nfrom the combination of spin-orbit coupling and scatter-\ning processes. Even for a continuous-wave excitation, we\nfound that the \feld acts directly on the magnetization\nvia an o\u000b-resonant Raman-like processes, which is closelyrelated to the dynamical Stark e\u000bect. This Raman-like\nprocess leads to a decrease/increase of the electronic spin\npolarization, i.e., the magnetization, as one would expect\nfrom the angular momentum supplied by the left/right\ncircularly optical \feld. 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I.\nRukman, Zh. Eksp. Teor. Fiz 86, 1376 (1986)."    },    {        "title": "1803.05038v1.Ultrafast_light_switching_of_ferromagnetism_in_EuSe.pdf",        "content": "arXiv:1803.05038v1  [cond-mat.mtrl-sci]  13 Mar 2018DRAFTUltrafast light switching of ferromagnetism in EuSe\nA. B. Henriques,1X. Gratens,1P. A. Usachev,1∗V. A. Chitta,1and G. Springholz2\n1Instituto de F´ ısica, Universidade de S˜ ao Paulo, 05508-09 0 S˜ ao Paulo, Brazil\n2Institut f¨ ur Halbleiter und Festk¨ orperphysik, Johannes Kepler Universit¨ at Linz, 4040 Linz, Austria\nWe demonstrate that light resonant with the bandgap forces t he antiferromagnetic semiconducor\nEuSe to enter ferromagnetic alignment in the picosecond tim e scale. A photon generates an electron-\nhole pair, whose electron forms a supergiant spin polaron of magnetic moment of nearly 6,000 Bohr\nmagnetons. By increasing the light intensity, the whole of t he sample can be fully magnetized.\nThe key to the novel large photoinduced magnetization mecha nism is the huge enhancement of the\nmagnetic susceptibility when both antiferromagnetic and f erromagnetic interactions are present in\nthe material, and are of nearly equal magnitude, as is the cas e in EuSe.\nPACS numbers:\nThe ultrafast control of the magnetic state of matter\nis a topic of vast current interest for the development of\napplications and for advancing the knowledge in the field\nof light-matter interaction [1–3]. Here we report on the\ndiscovery of an ultrafast mechanism of light switching\nof antiferromagnetic EuSe into the ferromagnetic phase,\nwhichisefficientintheproximityofitsN´ eeltemperature.\nThe EuSe samples were grown by molecular beam epi-\ntaxy (MBE) onto (111) BaF 2substrates. Because of the\nalmost perfect lattice constant matching ( a= 6.191˚A\nanda= 6.196˚AforEuSeandBaF 2, respectively), nearly\nunstrained bulklike EuSe reference layers with µm thick-\nness were obtained directly by growth on BaF 2. The\ntime-resolved photoinduced Faraday was measured using\natwo-colorpump-probetechnique using 2ps light pulses,\nasillustrated in Fig. 1. The Faradayrotationangleofthe\nprobe light pulse, ∆ θF, induced by the pump pulse, was\nmeasured using lock-in techniques with a resolution bet-\nter than 10−7radians. The image of the excitation spot\non the sample had a diameter of 150 µm, about twice the\ndiameter of the probe spot. For continuous wave (CW)\nmeasurements the same setup was used, except that the\npump source was a doubled Nd:Yag laser or a Xe lamp\ncoupled to a monochromator, and the probe source was a\nsemiconductor laser of energy below the EuSe bandgap.\nAll measurements were performed using an optical cryo-\nstat containing a superconducting coil for magnetic fields\nup to 8 Teslas, applied in the Faraday geometry. Fig-\nure 2 shows the photoinduced Faradayrotation (PFR) as\na function of applied magnetic field, for CW excitation\nwith intensity p= 400 mW/cm−2, atT= 12 K. The\nPFR signal has all the characteristics expected for pho-\ntoinduced magnetic polarons: no signal at zero magnetic\nfield, because polarons are radomly oriented and produce\nzero magnetization, and a very rapid increase when a\nmagnetic field is applied, due to the complete alignment\nofthepolaronensemblealongthefieldduetotheZeeman\n∗Permanent address: Ioffe Institute, 194021 St. Petersburg, Russia\nFIG. 1: Scheme of the setup for measuring the time-resolved\nphotoinduced Faraday rotation angle. The linearly polariz ed\nprobe pulse arrives at the sample when a time ∆ thas elapsed\nafter the arrival of the pump pulse. We measured ∆ θF, the\nFaraday rotation of the probe, induced by the pump illumi-\nnation.\ntorque acting on particles with a largemagnetic moment.\nFig. 2 shows that when the applied magnetic field is in-\ncreased beyond 1T, the PFR effect gradually decreases\ntowards zero; this is because large magnetic fields pro-\ngressively force the EuSe lattice spins into ferromagnetic\norder, and ultimately the formation of spin polarons is\nquenched. Fig. 3 shows that the PFR signal vanishes if\nthe excitation photon energy is less than the bandgap,\nand increases abruptly at the bandgap energy, demon-\nstrating that the PFR effect is associated with photogen-\nerated conduction band electrons.\nBecause the photoexcited electrons are bound to very\nheavyphotoexcitedholes[4], thespin polaronsareimmo-\nbile and are confined to a layer below the surface of the\ncrystal whose thickness equals the penetration depth of\nthe excitation light, 1 /α, whereα∼=15µm−1is the EuSe\nabsorption coefficient for the excitation photon energy\n[4]. From elementary kinetics, the steady-state density\nof photoinduced polarons is given by[5]\nnPol=χpατPol\nhν≤pατPol\nhν, (1)\nwhereχ≤1 is the quantum efficiency for spin polaron\nphotogeneration, and τPolis the spin polaron lifetime.DRAFT2\nFIG. 2: Photoinduced Faraday rotation in EuSe. The max-\nimum PFR is indicated by ∆ θSAT\nF.\nFIG. 3: PFR excitation spectrum.\nThe spin polaron lifetime was deduced from the depen-\ndenceofthePFRamplitudeonthemodulationfrequency\nof the excitation light, shown in Fig. 4, following the pro-\ncedure described in 6, giving τPol=1.6µsec. The mean\ndistance between polarons, d, in units of the EuSe lattice\nconstant, is therefore\nd\na≥2\na/parenleftbigg3\n4πhν\npατPol/parenrightbigg1/3\n(2)\nSubstituting all parameters into (2), we obtain dPol≥\n36a. Thus,dis at least one order of magnitude greater\nthan the typical radius of a spin polaron in europium\nchalcogenide [8], RPol∼3. Being so far apart, the po-\nlarons are non-interacting. Moreover, above the N´ eel\ntemperature the spin polarons orientation floats freely,\nFIG. 4: PFR as a function of the pump modulation fre-\nquency.\nFIG. 5: PFR as a function of internal magnetic field. The\nratioχ⊥/χ||, used to compute the internal magnetic field, is\nshown in the inset, as a function of temperature.\ntherefore they form a superparamagnetic gas, whose\nmagnetization dependence on magnetic field and tem-\nperature obeys a Langevin function [9]. Because PFR is\nproportionalto the magnetization [10], the photoinduced\nFaraday rotation angle will be given by\n∆θF= ∆θSAT\nFL/parenleftbiggµPolB\nkBT/parenrightbigg\n(3)\nThe magnitude of the magnetic moment of the spin\npolaron at a given temperature can be determined ac-\ncurately by fitting the experimental curve with (3), be-\ncause it is the sole adjustable parameter determining the\nsharpness of the step. However, we must first convert the\napplied magnetic field into internal one, which is smallerDRAFT3\nFIG. 6: Magnetic moment of a photoinduced polaron as a\nfunction of temperature. Full circles, empty circles and in -\nverted triangles correspond to probe wavelength of 665, 730\nand 760 nm, respectively. The dashed line shows the para-\nmagnetic approximation, with JXf= 0.075 eV obtained from\nthe linear fit shown in the inset.\nthan the former due to the demagnetizing field [11]. For\ntheFaradaygeometryusedhere,wherethemagneticfield\nis normal to a very thin epitaxial layer, the ratio of the\ninternal magnetic field to the applied one is equal to the\nratio of the magnetic susceptibility measured when the\napplied field is perpendicular to the layer, ( χ⊥), to the\nsusceptibility measured when the applied field is parallel\nto it (χ||). The susceptibilities χ⊥andχ||were mea-\nsured using a SQUID magnetometer, which had a mag-\nnetic moment resolution better than 10−11Am2. Figure\n5 shows the measured PFR as a function of internal mag-\nnetic field. The inset shows the measured ratio between\nsusceptibilities χ⊥andχ/bardbl, used to convert the applied\nmagnetic field into internal one, above the N´ eel temper-\nature. The solid line in Fig. 5 shows the fit of the theory,\nusing (3), which yields the magnetic moment of the spin\npolaron,µPol= 5,590 Bohr magnetons ( µB). This mag-\nnetic moment is 10 times greater than the giant photoin-\nduced spin polarons known so far [5, 6], which allows us\nto categorize the spin polarons discovered as supergiant.\nThe magnetic moment of the spin polaron as a function\nof temperature is shown in Fig. 6. Increasing the tem-\nperature quenches spin polaron formation, due to the de-\ncreasing magnetic susceptibility of the lattice spins. At\ntemperatures much higher than the N´ eel temperature,\nandµEuB/kBT≪1, the magnetization of lattice spins\ncanbewelldescribedbytheparamagneticapproximation\nM=NµEuS+1\n3SµEuB\nkBT. (4)\nwhereN= 4/a3is the density of Eu sites in the crystal\nandµEu=gµBSis the magnetic moment of an Eu atom,whereg= 2,S= 7/2. The exchange interactionbetween\nthe photoexcited electron, whose wavefunction is ψ(r),\nandthelatticespins,isdescribedbyaneffectivemagnetic\nfield,BXf, acting on the lattice spins [12]\nBXf=JXfS\nNµEu|ψ(r)|2, (5)\nwhereJXfistheexchangeinteractionintegralbetweenthe\nelectronformingthespinpolaronandthelatticespins[8].\nSubstituting (5) in (4), and integrating in the whole vol-\nume, the high temperature magnetic moment of a spin\npolaronin the paramagneticcrystal approximationis ob-\ntained\nµPol(T) =gµBS(S+1)JXf\n3kBT, (6)\nAfitofthehightemperaturetail, T >40K,ofµPolversus\nTwith (6), whereby JXfis the only adjustable parameter,\nis shown in the inset of Fig. 6, and gives JXf= 75 meV.\nThisJXfvalue is about the same as measured for EuTe,\nmeaning that the mean exchange field generated by the\nphotoexcited electron in EuSe is about the same as for\nEuTe, i.e. about 1 Tesla [8]. Fig. 6 shows that when the\ntemperature is decreased below 20K and approaches the\nN´ eel temperature, µPolincreases much faster than the\nparamagnetic approximation, and can reach almost an\norder of magnitude greater thanµPolthat would be ob-\nserved if the exchange interaction was switched off (the\nparamagnetic approximation). This is in sharp contrast\nto EuTe, where µPolis always smaller than the param-\nagnetic limit [6]. The reason for the very large increase\nin EuSe is the near equal absolute values of the first and\nsecond neighbor exchange interaction constants, J1and\nJ2. Whereas J1>0 favors ferromagnetism, J2<0 fa-\nvors antiferromagnetism, which predominates only be-\ncause|J2|is marginally larger than |J1|. This also im-\nplies than around the N´ eel temperature only a very small\ninternal magnetic field, of about 0.1 T is sufficient to\npromote a phase transition into the ferromagnetic state,\nas demonstrated experimentally in Ref. 13. Bearing in\nmind that within a spin polaron the effective magnetic\nfield acting on the spins is about 1 T, this implies that\nthe within the polaron the crystal lattice attains ferro-\nmagnetic alignment, which explains why the magnetic\nmoment of the polaron is so large in EuSe. Such reason-\ning is supported quantitatively by the Weiss mean field\napproximation (MFA) with first and second neighbor in-\nteraction at T= 0K, according to which in the AFM-II\nphase the magnetic field to induce ferromagnetic align-\nment is given by [8, 12, 14]\nBSAT=24|J1+J2|S\ngµB, (7)\nand in the range B < B SATthe magnetization is given\nbyM=NµEuB/B SAT. Using (5) and (7), and integrat-\ningMin the volume, the magnetic moment of the spinDRAFT4\nFIG. 7: PFR as a function of pump excitation power.\npolaron in the mean field approximation is obtained\nµMFA\nPol(T= 0K) =JXf\n24|J1+J2|gµB. (8)\nTable I displays the result produced by formula (8) for\nEuTe and EuSe. The calculated values agree by order of\nmagnitude with the measured values, and demonstrate\nthat the very large µPolin EuSe is due to the near can-\ncellation of the ferro and antiferromagnetic lattice inter-\nactions.\nTABLE I: EuSe and EuTe parameters, µPolmeasured at T=\n5 K and estimated using the MFA for T= 0 K.\nµPol(µB)\nJ1(K)J2(K)JXf(eV)Experiment MFA\nEuTe 0.043 -0.15 0.083 600 750\nRef. 15 12 5\nEuSe 0.29 -0.30 0.075 5,590 7,300\nRef. 16 this work\nThe MFA does not provide any information about the\ninternal structure of the spin polaron. A more accurate\ndescription of the spin polaron, and a description of its\ninternal structure, is provided by a self-consistent calcu-\nlation for an AFM-II type system at T= 0 K [12]. In\nperforming such a calculation for EuSe, the input param-\neters were a conduction band effective mass m∗= 0.3m0\n[17], a dielectric constant ε= 9.4 [7], as well as the pa-\nrameters given in Table I. The self-consistent calculation\nproduces a spin polaron containing a large ferromagnetic\ncore of radius 2.8 lattice parameters, containing 370 fer-\nromagnetically aligned Eu atoms, with a calculated spin\npolaron of magnetic moment of µPol= 4,920µB.\nFigure 7 shows ∆ θSAT\nF, and the corresponding polaron\npopulation, deduced using the Verdet constant as de-\nFIG. 8: PFR as a function of the delay, ∆ t, between pump\nand probe pulses.\nscribed in Ref.10, as a function of excitation pump power\ndensity. The solid line is a fit using (1) with χas the\nonly fitting parameter, which yields χ= 0.053. In dra-\nmatic contrast to EuTe, which shows a saturation of the\npolaron population, in EuSe the population grows lin-\nearly with excitation even at concentrations far above\nexpected residual deffect concentrations, indicating that\nthe photoinduced polarons are intrinsic, and therefore\nis should be possible to fully magnetize the layer pene-\ntrated by light simply by using enough excitation power.\nThe maximum pump intensity shown in Fig.7 generated\nabout 2×1017cm−3spin polarons, and taking into ac-\ncount their ferromagnetic core radius of about 3 a, this\ncorresponds to a magnetization of 5 ×10−3of the satura-\ntion value. We could not exploit greater pump intensities\nwithout heating of the sample, because we used an opti-\ncal chopper with a high (50%) duty cycle.\nFigure 8 shows the PFR signal as a function of the\ndelay, ∆t, between the pump and probe pulses (Fig.1).\nBefore the arrival of the pump pulse (negative delay), no\nPFR signal is detected, however, after the arrival of the\npump pulse the PFR signal grows exponentially with a\ncharacteristic rise time of 62 ps, which classifies the pho-\ntomagnetization process as ultrafast. After achieving the\nmaximum, the PFR remained approximately constant\nwithin the maximum delay available of our experiment,\nof a few nanoseconds, which is explained by the long life-\ntime of the polarons, τPol= 1.6µs, reported above.\nIn conclusion, we have demonstrated that light can be\nusedtoconvertazeromagnetizationstateofaEuSecrys-\ntal into a completely polarized ferromagnetic state in the\nultrafast time scale, through the photogeneration of su-\npergiantintrinsicspinpolarons. Thismagetizationmech-\nanism is made possible because the exchange interaction\nbetween lattice spins in EuSe contains both ferromag-DRAFT5\nnetic and antiferromagnetic components, which nearly\ncancel each other out.\nThis work was supported by the Brazilian agencies\nCNPq (Project 304685/2010-0) and FAPESP (Project\n2016/24125-5).\n[1] J. Xie, H. Qin, Y. Hao, B.Cheng, W. Liu, L. Liu, S. Ren,\nG. Zhou, Z. Ji, and J. Hu, Sci. Rep. 7, 45642 (2017).\n[2] T. Satoh, R. Iida, T. Higuchi, Y. Fujii, A. Koreeda,\nH.Ueda, T. Shimura, K. Kuroda, V. I. Butrim, and B. A.\nIvanov, Nature Communications 8, 638 (2017).\n[3] I. Radu et al, Nature 472, 205 (2011).\n[4] A. B. Henriques, A. Wierts, M. A. Manfrini,\nG. Springholz, P. H. O. Rappl, E. Abramof, and A. Y.\nUeta, Phys. Rev. B 72, 155337 (2005).\n[5] A. B. Henriques, A. R. Naupa, P. A. Usachev, V. V.\nPavlov, P. H. O. Rappl, and E. Abramof, Phys. Rev. B\n95, 045205 (2017).\n[6] A. B. Henriques, G. D. Galgano, P. H. O. Rappl, and\nE. Abramof, Phys. Rev. B 93, 201201 (2016).[7] A. Mauger and C. Godart, Phys. Rep. 141, 51 (1986).\n[8] A. B. Henriques, G. D. Galgano, E. Abramof, B. Diaz,\nand P. H. O. Rappl, Appl. Phys. Lett. 99, 091906 (2011).\n[9] C. P. Bean and J. D. Livingston, J. Appl. Phys. 30, 120S\n(1959).\n[10] A. B. Henriques and P. A. Usachev, Phys. Rev. B 96,\n195210 (2017).\n[11] S. Blundell, Magnetism in Condensed Matter (Oxford\nUniversity Press, 2001), Appendix D.\n[12] A. B. Henriques, F. C. D. Moraes, G. D. Galgano, A. J.\nMeaney, P. C. M. Christianen, J. C. Maan, E. Abramof,\nand P. H. O. Rappl, Phys. Rev. B 90, 165202 (2014).\n[13] R. T. Lechner, G. Springholz, T. U. Sch¨ ulli, J. Stangl,\nT. Schwarzl, and G. Bauer, Phys. Rev. Lett. 94, 157201\n(2005).\n[14] W. S¨ ollinger, W. Heiss, R. T. Lechner, K. Rumpf,\nP. Granitzer, H. Krenn, and G. Springholz, Phys. Rev.\nB81, 155213 (2010).\n[15] P. Wachter, C R C Critical Reviews in Solid State Sci-\nences3, 189 (1972).\n[16] X. Gratens et al, unpublished (2018).\n[17] S. J. Cho, Phys. Rev. B 1, 4589 (1970)."    },    {        "title": "1011.2788v1.Spin_Torque_Ferromagnetic_Resonance_Induced_by_the_Spin_Hall_Effect.pdf",        "content": " 1Spin Torque Ferromagnetic Resonance Induced by the Spin Hall Effect \nLuqiao Liu, Takahiro Moriyama, D. C. Ralph, and R. A. Buhrman \nCornell University, Ithaca, New York, 14853 \n \nABSTRACT \nWe demonstrate that the spin Hall effect in a thin film with str ong spin-orbit scattering \ncan excite magnetic precession in an adjacent ferromagnetic film. Th e flow of alternating current \nthrough a Pt/NiFe bilayer generates an oscillatin g transverse spin current in the Pt, and the \nresultant transfer of spin angular momentum to  the NiFe induces ferromagnetic resonance (FMR) \ndynamics. The Oersted field from the current also  generates an FMR signa l but with a different \nsymmetry. The ratio of these two signals allows a quantitative determinati on of the spin current \nand the spin Hall angle.   \n  2The spin Hall effect (SHE), the conversion of a longitudinal charge current density JC \ninto a transverse spin current density / 2SJe= , originates from spin-orbit scattering [1-4], \nwhereby conduction electrons with  opposite spin orientations in  a nonmagnetic metal [5] or \nsemiconductor [6] are deflected in opposite di rections. The SHE has attracted widespread \ninterest because it can generate pure spin currents from  a nonmagnetic source, a phenomenon \nthat could find important applications in future  spintronic devices. Seve ral techniques [5, 7, 8] \nhave been developed to determine the magnitude of the SHE, which is generally characterized by \nthe spin Hall angle, θSH = JS/JC. For thin-film Pt, estimates of θSH obtained using different \napproaches differ by more than an order of magnit ude [8-10], but already there have been efforts \nto utilize the spin current that arises from the SHE, first to tune the damping coefficient in a \nferromagnetic metal [8], and, most recently, to indu ce a spin wave oscillation in a ferrimagnetic \ninsulator having small damping [11]. Here we show  that the SHE can also be used to excite \ndynamics in an ordinary metallic ferromagnet.  Our experiment also allows a quantitative \ndetermination of the SHE strength that is self-calibrated, as explained below, enabling \nmeasurements of the spin currents generated by th e SHE with small experimental uncertainties.  \nWe study Pt/Permalloy bilayer films with a microwave-frequency (RF) charge current \napplied in the film plane (Permalloy = Py = Ni 81Fe19). An oscillating transv erse spin current is \ngenerated in the Pt by the SHE and injected into the adjacent Py (Fig. 1(a)), thereby exerting an \noscillating spin torque (ST) on the Py th at induces magnetization precession. When the \nfrequency and field bias satisfy  the FMR condition for the Py, st rong resonant precession results \nin a significant oscillation of  the bilayer resistance due to the anisotropic magnetoresistance \n(AMR) of the Py. This generates a DC voltage signal across the sample from the mixing of the \nRF current and the oscillating re sistance, similar to the signal th at arises from ST induced FMR  3in spin valves and magnetic tunne l junctions [12-15]. The resonan ce properties enable a direct \nquantitative measure of the spin  current absorbed by the Py. \nOur measurement setup is shown in Fig. 1(c). Pt/Py bilayers were grown by DC \nmagnetron sputter deposition. The in dividual layer thicknesses we re 4-15 nm, with specific \nvalues stated below. The star ting material for the Pt was  99.95% pure. Highly resistive Ta (1 nm) \nwas employed as the capping layer to preven t oxidation of the Py. The bilayers were \nsubsequently patterned into microstrips usi ng photolithography and ion milling. The samples’ \nwidths ranged from 1 to 20 μm and the lengths from 3 to 250 μm. By using a bias tee, we were \nable to apply a microwave current to our sample and at the same time me asure the DC voltage. A \nsweeping magnetic field Hext was applied in the film plane, with the angle θ between Hext and \nmicrostrip kept at 45° unless otherwise indi cated. The output power of  the microwave signal \ngenerator was varied from 0 to 20 dBm and th e measured DC voltage was proportional to the \napplied power, indicating that the induced precession was in the small angle regime. All the \nmeasurements we present were performed at room temperature with a power of 10 dBm.  \nWe model the motion of the Py magnetic moment ˆm by the Landau-Lifshitz-Gilbert \nequation containing the ST term [16]:  \n,\n0ˆˆˆˆ ˆ ˆ ˆ ˆ ()2eff S RF RF\nSdm dmmH m J m m mHdt dt e M tγα γ σ γμ=− × + × + × × − ×GG =.  (1) \nHere γ is the gyromagnetic ratio, α is the Gilbert damping coefficient, μ0 is the permeability in \nvacuum, Ms is the saturation magnetization of Py,  t is the thickness of the Py layer, ,/2SR FJe=  \nrepresents the oscillating spin curr ent density injected into Py,  HRF is the Oersted field \ngenerated by the RF current, Heff is the sum of Hext and the out-of-plane demagnetization field \n  4πMeff, and ˆσ is the direction of the injected spin moment. The third and fourth terms on the  4right hand side of Eq. (1) are the result of in-plane spin torque and the out-of-plane torque due \nto the Oersted field, respectiv ely (Fig. 1(a)).  The magnetic-re sonance mixing signal in response \nto a combination of in-plane a nd out-of-plane torques has been calculated in the context of ST-\ndriven FMR [14, 15], which we can  translate to our notation as: \nVmix=−1\n4dR\ndθγIRFcosθ\nΔ2πdf/dH()Hext=H0SFS(Hext)+AFA(Hext) [] ,    (2) \nwhere FSHext() =Δ2/Δ2+Hext−H0 ()2⎡⎣⎤⎦ is a symmetric Lorentzian  function centered at the \nfield H0 with linewidth Δ, FAHext() =FSHext() Hext−H0 () /Δ is an antisymmetric Lorentzian, \n(),0/2SR F s SJ e M t μ == , A=HRF1+4πMeff/Hext ()⎡⎣⎤⎦1/2, R is the resistance of the stripline, IRF \nis the microwave current through the stripline, and f is the resonance frequency. We therefore \nexpect the resonance signal to consist of two pa rts, a symmetric Lorentzian peak proportional to \nthe spin current density and an an tisymmetric peak proportional to HRF.   \nThe Oersted field HRF can be calculated from the geometry of the sample. Since the \nmicrowave skin depth is much greater than the Py thickness the current de nsity in the Py should \nbe spatially uniform, and in this case the Oersted field from the charge current density in the Py \nshould produce no net torque on the Py [see Fig.  1(b)]. The Oersted field can therefore be \ncalculated entirely from the current density JC,RF in the Pt layer. The microstrip width is much \nlarger than the Pt thickness, so the sample can  be approximated as an infinitely wide conducting \nplate and the Oersted field determined by Ampère's law, HRF=JC,RFd/2, where d is the Pt \nthickness. We checked HRF by numerical integration and the di fference is less than 0.1% from \nthe infinite plate approximation. Using this result , the ratio of the spin current density entering \nthe Py to the charge current density in the Pt can  then be determined quantitatively in a simple  5way from the ratio of the symmetric and anti symmetric components of the resonance curve \n()1/2, 0\neff ext\n,14 /SR F S\nCR FJ eM t dSMHJAμπ =+ ⎡ ⎤⎣ ⎦=.     ( 3 )  \nAll of the parameters entering Eq . (3) are either fundamental cons tants or quantities that can be \nmeasured directly, so this expression allows a measurement of JS,RF/JC,RF with small \nexperimental uncertainties. The measurement is se lf-calibrated in the sense that the strength of \nthe torque from the spin current is m easured relative to the torque from HRF, which can be \ncalculated easily from the geometry of the sample. \nAn additional contribution to the DC voltage can arise from spin pumping by the \nprecessing moment in combination w ith the inverse SHE in the Pt la yer, as observed in Ref. 10.  \nHowever, this effect is second order in θSH in our geometry and we calculate that it should \ncontribute a negligible voltage, about two orders of magnitude smaller than the signals we \nmeasure. \nFigure 2(a) shows the ST-FMR signals m easured on a Pt(6)/Py(4) (thicknesses in \nnanometers) sample for f = 5-10 GHz. As expected from Eq. (2), the resonance peak shapes can \nbe very well fit by the sum of symmetric and antisymmetric Lorentzian curves with the same linewidth for a given f (fits are shown as lines in Fig. 2(a)). The fact that the symmetric peak \nchanges its sign when H\next  is reversed (inset of Fig. 2(a)) ag rees with the form of spin torque \nˆˆˆST mm τσ∝× ×G given in Eq. (1), and excludes the possi bility that the signal is due to an \nunbalanced perpendicular Oerste d field torque, in direction ˆˆRF mH⊥× , which would yield \nsymmetric peaks with the same sign for opposite Hext. The resonant peak positions are \nsummarized in Fig. 2(b), and agree well with the Kittel formula \n() ( )1/2\n00 e f f /2 4 fH H Mγπ π=+ ⎡⎤⎣⎦. From a one-parameter fit to the resonance frequencies we  6determine that the demagnetization field 4πMeff= 0.805 ± 0.005 T for the Pt(6)/Py(4) bilayers.  \nWe have also measured th e saturation magnetization MS = 6.4 × 105 A/m in test samples [17].   \nTo verify the SHE origin of field-symmetr ic components of the FMR signals, we have \nstudied several different types of  control samples. In Fig. 2(c) we compare the FMR signals \nmeasured at 8 GHz for a Pt(15)/Py(15) and a Pt (6)/Py(4) sample. The signal for the Pt(6)/Py(4) \nsample contains a sizable field-symmetric component, with S/A = 0.63. Due to the increased \nthicknesses of the two layers, we expect from Eq. (3) that S/A for the Pt (15)/Py (15) should be \ngreatly reduced, approximately ∝1/td if the spin Hall currents in the two samples are similar. \nS/A for the Pt(15)/Py(15) is very small, S/A = 0.08 ± 0.05, near the noise floor for the fits of the \nsymmetric component (the uncertainty reflects  the standard deviation over five samples \nmeasured). The difference between the change in the S/A ratio expected from Eq. (3) (a factor of \n11.2, taking into account a small change in 4πMeff) and the measured reduction by a factor 8.0 \nmay be associated with a change in the magnit ude of the spin Hall current generated by the \ndifferent thicknesses of the Pt films when this thickness is comparable to the spin diffusion length (see below). \nWe also studied control samples with the laye rs Cu(6)/Py(4) and 4 nm of Py alone, with \nresults as shown in Fig. 2(d). The Cu/Py bila yer sample gives a purely antisymmetric signal, \nindicating that only the Oersted-fi eld contribution is present, as expected because of the very \nsmall SHE in Cu in comparison to that in Pt . For the Py(4)  sample, we would expect no \nresonance signal at all, since ther e is no SHE and as noted above if  the current density in the Py \nis uniform there should also be no net effect of  the Oersted field on the Py dynamics. However, \nwe do observe a very small, purely antisymmetric signal in the 4 nm Py sa mple. We suspect that \nthis may arise from an Oersted field due to non-unif orm current flow at the ends of the Py due to  7the electrode contacts. The lack of field-symmet ric components in the re sonance curves for the \ncontrol samples provides strong su pport that the symmetric component  we observe in Pt(6)/Py(4)  \ndoes indeed arise from the SHE in the Pt.   \nWith   4πMeff and MS determined , we can use Eq. (3) and the measured values of S/A to \ncalculate JS,RF/JC,RF. The results are shown in Fig. 2(e)  for the resonance curves spanning 5-\n10 GHz shown in Fig. 2(a).  We find JS,RF/JC,RF = 0.056 ± 0.005 for Pt(6)/Py(4). We \nmeasured more than ten Pt(6)/Py(4) samples w ith different lateral dimensions and the total \nvariation of JS,RF/JC,RF was < 15%. The dominant experimental uncertainty [and the small \nvariation with Hext visible in Fig. 2(e)] may be associated  with Oersted fields from non-uniform \ncurrents at the sample ends, as noted above for the single-layer Py sample.  Note that according to Eq. (2) S/A should not depend upon the angle of the a pplied DC field, as confirmed by the \nresults shown in Fig. 2(f).  \nAs an independent check we also employed an alternative method for determining the \nspin current density absorbed by the Py  layer, by measuring the FMR linewidth \nΔ as a function \nof DC current, similar to the technique introduced in Ref. 8. According to the theory of ST, a \nDC spin current IS,DC will increase or decrease the effective magnetic damping and hence Δ, \ndepending upon its relative or ientation with respect to the magnetic moment: [18] \neff 02s i n\n(2 ) 2S\next SJ f\nHM M t eπθαγπ μ⎛⎞Δ= +⎜⎟+ ⎝⎠=         (4) \nOur results obtained with a Pt(6)/Py(4) sample  ~1 μm wide are shown in Fig. 3. The measured \ndamping coefficient at zero current ( α ≈ 0.028) is significantly higher than that measured in a \nspin valve nanopillar sample ha ving a 4 nm Py free layer ( α≈ 0.01 ) [19]. This can be \nexplained by the spin pumping e ffect previously observed in the Py/Pt system [20, 21]. For a  8negative applied field ( Hext applied -135° from the current di rection in the microstrip), the \nlinewidth is broadened when I DC ramps from -0.7 mA to 0.7 mA; while for a positive field \n(Hext applied 45° from the current direction), th e trend is the opposite. By fitting the data \nshown in Fig. 3, and calculating the charge cu rrent density in the Pt using the measured \nresistivities 20Pt cm ρμ=Ω and 45Py cm ρμ=Ω , we find 10/ (0.9 0.012) 10cJα−Δ= ± ×  (A/cm2)-1. \nWith Eq. (4), this yields / 0.048 0.007scJJ =±  for Pt(6)/Py(4), which agrees well with the \nvalue 0.056 ± 0.005 determined from the FMR lineshape.  \nOur experiments yield values for JS/JC, the ratio of the spin current density (in units of \ncharge) absorbed by the Py to the charge current  density in the Pt film. For many applications, \nthis is the figure of merit of direct interest. Ho wever, for comparing to other experiments, it is \nalso of interest to determine the spin Hall angle θSH, the ratio of the spin current density inside \nbulk Pt to the charge current density. For a perf ectly transparent Pt/Py interface and for a Pt \nlayer much thicker than the spin diffusion length λsf, the quantities JS/JC and θSH should be \nequal. However, because our Pt/Py interface is li kely not perfectly transparent, and because our \nPt layers likely do not have thicknesses \u0015λsf, our results may underestimate the transverse \nspin current density appropriate  to bulk Pt. Therefore, our measurements imply a lower bound, \nθSH >  0.056 ± 0.005 for our Pt material. In the lim it of a transparent Pt/P y interface, for which \nthere should be no spin accumulation transverse to  the Py moment at the interface, we calculate \nusing drift-diffusion theory [22] that the spin Hall current density in a Pt film of thickness d \nshould be reduced from the bulk value by JSd()/JS∞()=1−sech d/λsf(). Using this \nexpression, our best estimate, based on compar ison between the Pt(15)/Py(15) and Pt(6)/Py(4) \nsamples is that 3sf nm λ≈ , and we can set an upper bound of 6sf nm λ< , lower than the low  9temperature value measured previously [23]. This gives a best estimate of 0.076SHθ= , and \nbounds 0.056 0.005 0.16SHθ ±< < , again in the limit of a tr ansparent Pt/Py interface.  \nWe mentioned above that pr evious measurements of θSH in Pt have differed by over an \norder of magnitude. Kimura et al. [9], using a Pt/Cu/Py lateral nonlocal geometry reported \nθSH= 0.0037. However, their 4-nm-thick Pt wires are in contact to 80-nm -thick Cu wires. We \nbelieve that the Cu likely shunted the charge current flowing in the Pt, resulting in a large \nunderestimation of θSH. Ando et al.  [8] by measuring magnetic damp ing in Pt/Py versus current, \nreported JS/JC = 0.03 and estimated 0.08SHθ= . We have shown that a technique closely \nrelated to the method of Ref. 8 gives result s that agree with our FMR method, although we \ndiffer with Ref. 8 regarding the form of our Eq. (4) and the drift-di ffusion analysis. Mosendz et \nal. [10], using a technique based on spin pump ing together with the inverse SHE, reported θSH= \n0.0067, later refined to θSH = 0.013 [24]. This result relied on an assumption that λsf= 10 nm \nfor Pt. Their value for θSHwould be 3 times larger, and in much better accord with our value, \nusing our estimate that λsf= 3 nm.  \nIn summary, we demonstrate th at spin current generated by the SHE in a Pt film can be \nused to excite spin-torque FMR in an adjacen t metallic ferromagnet (Py) thin film. This \ntechnique allows a straightforward determination of the efficiency of spin current generation, \nJs/Jc (the spin current density absorbed by the Py di vided by the charge curre nt density in the Pt), \nthat is self-calibrated, in that the torque due to  the spin current can be measured relative to the \ntorque from the Oersted field generated by the sa me charge current density in the Pt layer. We \nfind Js/Jc = 0.056 ± 0.005 for Pt(6)/Py(4), implying θSH >  0.056 for bulk Pt . This simple \ntechnique is an excellent soluti on for the quantitative measuremen t of the SHE efficiency in any  10metallic film that can be produced as part of  a ferromagnetic/non-magnetic metal bilayer. The \nrelatively large efficiency of spin current generation that we observe for Py/Pt is promising for applications which might utilize the SH E to manipulate ferromagnet dynamics. \nThis research was supported in part by th e Army Research Office, and by the NSF-NSEC \nprogram through the Cornell Center for Nanoscale Sy stems.  This work was performed in part at \nthe Cornell NanoScale Facility, which is  supported by the NSF through the National \nNanofabrication Infrastructure Network and bene fitted from the use of the facilities of the \nCornell Center for Materials Research , supported by the NSF-MRSEC program. 11REFERENCES \n1. J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). \n2. S. F. Zhang, Phys. Rev. Lett. 85, 393 (2000). \n3. J. Sinova  et al. , Phys. Rev. Lett. 92, 126603 (2004). \n4. S. Murakami, N. Nagaosa and S. C. Zhang, Science 301, 1348 (2003). \n5. S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). \n6. Y. K. Kato  et al. , Science 306, 1910 (2004). \n7. E. 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Lett. 58, 2271 (1987). \n23. H. Kurt  et al. , Appl. Phys. Lett. 81, 4787 (2002). \n24. A. Hoffman, private communication. \n \n  12FIGURE CAPTIONS \nFig. 1 (color online): (a) Schematic of a Pt/Py bi layer thin film illustrating the spin transfer \ntorque τSTT, the torque τH induced by the Oersted field HRF , and the direction of the damping \ntorque τα. θ denotes the angle betw een the magnetization M and the microstrip. Hext is the applied \nexternal field. (b) Left side view of the Pt/P y system, with the solid line showing the Oersted \nfield generated by the current flowing just in the Py layer, which should produce no net effect on \nthe Py AMR. (c) Schematic circuit for the ST-FMR measurement.   Fig. 2 (color online): (a) Spectra of ST-F MR on a Pt(6)/Py(4) sample measured under \nfrequencies of 5-10 GHz. The sample dimension is 20 μm wide × 110 μm long. Inset: ST-FMR \nspectrum of 8 GHz for both positive and negative H\next. (b) Resonance frequency f as a function \nof the resonant field H0. The solid curve represents a fit to  the Kittel formula. (c) FMR spectra \nmeasured for two Pt/Py bilayer samples, with fits  to Eq. (2). The data were taken at 8 GHz. (d) \nFMR spectra ( f = 8 GHz) on the Pt(6)/Py(4) sample (blue triangles) as well as control samples \nconsisting of Cu(6)/Py(4) (red circles) and Py(4) (black squares). (e) JS,RF/JC,RF values \ndetermined from the FMR analysis [Eq. (3)] at different f. (f) FMR signals measured for different \nexternal field angles θ (f = 8 GHz). The mixing voltages Vmix are normalized and offset to enable \ncomparison of the lineshapes.   Fig.3 (color online): The change of the FMR linewidth (left y axis) and Gilbert damping coefficient (right y axis) as a function of I\nDC for two orientations of the Py magnetization relative \nto the current direction. The data are taken at f = 8 GHz.  13\n \n  14 \n0 1000 2000-2002040\n  Vmix (μV)\nHext (Oe)5 GHz\n6 GHz\n7 GHz\n8 GHz\n9 GHz\n10 GHz\n  \n0 500 1000 15000246810\n  f (GHz)\nH0 (Oe)\n-600 -300 0 300 6000.00.81.62.43.2 \n \n75o60o45oNormalized V mix\nHext - H0 (Oe)θ=15o\n30o\n400 800 12000.000.020.040.06\n  Js / Jc\nH0 (Oe)0 1000 2000-20-1001020\n  Vmix (μV)\nHext (Oe) Pt (6) / Py (4)\n Cu (6) /Py (4)\n Py (4)\n0 1000 2000-40-2002040\n  Vmix (μV)\nHext (Oe)Pt (15) / Py (15)\nPt (6)/Py (4)(b)\n(f)(c) (d)\n(e)(a) 15    \n-0.8 -0.4 0.0 0.4 0.87578818487\n2.72.82.93.0 \n positive field\n negative fieldlinewidth (Oe)\nIDC (mA)X 10-2\n α   \n "    },    {        "title": "2311.05868v1.Observation_of_the_out_of_plane_orbital_antidamping_like_torque.pdf",        "content": "Observation of the out-of-plane orbital antidamping -like torque  \nZeyang Gong1†, Fu Liu1†, Xinhong  Guo1, Changjun Jiang1* \n1 Key Laboratory for Magnetism and Magnetic Materials, Ministry of Education, Lanzhou \nUniversity, Lanzhou 730000, China  \n \n†These authors contributed equally to this work.  \nCorresponding author. E -mail address: *jiangchj@lzu.edu.cn  \n \n \nAbstract  \nThe out -of-plane antidamping -like orbital torque fosters great hope for high -\nefficiency spintronic devices. Here w e report experimentally the observation of out -of-\nplane antidamping -like torque that could be generated by z-polarized orbital current  in \nferromagnetic -metal/oxidized Cu  bilayers , which is presented unambiguously by t he \nmagnetic field angle dependence of spin-torque ferromagnetic resonance signal . The \noxidized Cu thickness  dependence of orbital torque ratios highlights the interfacial \neffect would be responsible for the generation of orbital current . Besides that, the \noxidized Cu thickness  dependence of damping parameter further proves the observation \nof antidamping -like torque. This result contributes to enriching the orbital -related \ntheory of the generation mechanism of the orbital to rque. \n \n  1 Introduction  \nOrbital current, a flow of orbital angular momentum  (OAM) , have attract ed wide \ninterest  in the recent past .1-4 Orbital current is defined as  the directional flow of carriers \ncarrying OAM , which carriers angular momenta polarized  in different directions \naccording to the different direction of their own OAM .5 Generally, there are two \nmechanisms giving rise to the generation of orbital current, orbital Hall effect  (OHE)6-\n7 and orbital Rashba effect (ORE) .8-9 OHE refers to a longitudinal electric field induc ing \nan excitation of orbital current in a transverse direction. While the ORE allows the \ngeneration of chiral OAM texture in k space via OAM dependence of energy splitting  \nanalogous  to spin Rashba  effect (SRE). Notablely, unlike SRE, the ORE can still exist \nwithout spin -orbit coupling (SOC), with the broken inversion symmetry alone adequate \nto its emergence .10 Generally, the resulting orbital current is polarized along y direction \nwhen the charge current along x direction, while the orbital current flows toward z \ndirection  similar to spin current .11 This orbital current  (to distinguish it from spin \ncurrent, denot ing as oy) can generate an in -plane antidamping -like torque (orbital torque , \nOT) 𝜏DL  that contributes to highly efficient magne tization switching  of thin films  \nsimilar to spin torque .12 Besides, the out -of-plane antidamping -like torque possess \npotential applications, in which the generation mechanism of this torque is worthing \nstudying  for the  development of  orbital engineering . \nRecently, the out -of-plane antidamping -like torque is widely reported  in spin-\nrelated systems that generated by σz-polarized spin current , which prevails in specific \nsystem with lower crystalline or magnetic symmetry, such as anti-ferromagnetic  (AFM)  \nmaterials  Mn 3GaN13, two-dimensional semimetals WTe 214 as well as \nferromagnet/ferroelectric multiferroic  (FM/ FE) heterostructures interface .15 Meanwhile, \nthe confirmation of orbital  current  becomes more popular issue, where more and more \nsystems are confirmed that exist orbital torque, such as Ta/Ni1, Ru/Ni16, Nb/Co17, etc. \nIn th ese systems,  the orbital current induc ed by OHE and the OT is proved to exist in \nthe form of in -plane damping -like torque, which is generated by oy. Besides, the orbital \ncurrent  has been  theoretically and experimentally verified in naturally oxidized Cu (CuOx) film induced by  ORE18-19, which the form of orbital current  is still oy. However, \nthe out -of-plane polarized orbital current ( oz) is rarely reported despite its rich orbital \nphysics in orbitronics.  \nHerein, the oz-polarized orbital current and out -of-plane antidamping -like torque \nare observed in FM/CuO x bilayers  via spin -torque ferromagnetic resonance (ST -FMR). \nThe orbital current  generated by ORE in the FM/CuO x interface as shown in Fig. 1a, \nthe induced orbital current are injected into FM and next converted to spin through SOC \nof FM. From the magnetic field dependence of ST -FMR spectra, excepting for oy-\npolarized orbital current, we further observe out -of-plane antidamping -like torque \ngenerated by oz-polarized orbital current. After fitting the  ST-FMR antisymmetric \nresonance component Va vs in-plane magnetic -field angle φ and symmetric component \nVs vs φ, respectively, we conclude the magnitude of various OT and the equivalent OT \nefficiency. Besides that, we also characterize the FM species dependence of OT \nefficiency, which result coincides with orbital -related theory.  \n2. Experimental  \nThe FM/Cu thin films were grown on thermally oxidized Si substrates via \nmagnetron sputtering at room temperature. For clarif ying the physical mechanism of \nthe OT, two different types of bilayers were prepared: Co(8 nm)/Cu( tCu) and Fe 20Ni80(8 \nnm)/Cu(tCu) (Fe 20Ni80 = Permalloy = Py) . The devices were prepare d by exploiting \nshadow mask prior to the deposition and the dimension is 100 × 600 μm2 rectangular \nstrips. The FM/CuO x was obtained via the FM/Cu bilayers exposed to the laboratory \nambient.  \n \nFig. 1 (a) Schematic illustration for the mechanism of orbital torque in FM/CuO x. (b) Schematic of \nthe device geometry and the circuit for ST -FMR measurements.  \nTo measure the charge -to-orbital  conversion, we use the ST -FMR technique  as \nx y\nz(a) (b)shown in Fig. 1b . In the ST -FMR measurement, a radio frequency (RF) microwave \ncurrent was applied in the FM/CuO x interface produc ing alternating torques into the \nFM layer, which excites the FM magnetization to precess and drives the resistance \noscillations owing to the anisotropic magnetoresistance (AMR) of the FM layer  when \nthe microwave frequency and the external magnetic field satisfy the ferromagnetic \nresonance (FMR) condition20,  \n                      𝑓=𝛾\n2𝜋√𝐻(𝐻+4𝜋𝑀𝑒𝑓𝑓).                       (1) \nHere, 𝑀𝑒𝑓𝑓 is the effective saturation magnetization and 𝛾 is the gyromagnetic ratio.  \nWe measure a direct current ( DC) voltage Vdc across the stripe that couples the \noscillating resistance and the RF current. In the ST -FMR measurements, this DC signal \nVdc can be fitted with the equation21: \n                    𝑉𝑑𝑐=𝑉𝑎∆𝐻(𝐻−𝐻𝑟)\n(𝐻−𝐻𝑟)2+∆𝐻2+𝑉𝑠∆𝐻2\n(𝐻−𝐻𝑟)2+∆𝐻2,               (2) \nwhere ∆H is the resonance  linewidth, and Hr is the resonance field. Vs and Va are the \nmagnitude of the symmetric and antisymmetric component, respectively. The out -of-\nplane ( 𝜏⊥) and in -plane ( 𝜏ǁ) torques defined in Fig . 1b could be obtained individually, \nas the  antisymmetric ST -FMR resonance component Va and symmetric component Vs \nare proportional to the amplitude of the out -of-plane and in -plane torques. These \ncomponents relate to the two torques as follows13: \n𝑉𝑠=−𝐼𝑅𝐹\n2(𝑑𝑅\n𝑑𝜑)1\n𝛼𝛾(2𝐻𝑟+𝜇0𝑀𝑒𝑓𝑓)𝜏ǁ,                 (3)  \n            𝑉𝑎=−𝐼𝑅𝐹\n2(𝑑𝑅\n𝑑𝜑)√1+𝜇0𝑀𝑒𝑓𝑓/𝐻𝑟\n𝛼𝛾(2𝐻𝑟+𝜇0𝑀𝑒𝑓𝑓)𝜏⊥.                 (4)  \nwhere φ is the in -plane magnetic field angle defined in Fig . 1b. dR/dφ is due to the \nAMR in the FM. IRF is the radio microwave current. 𝜇0 is the permeability in vacuum, \nα is the Gilbert damping coefficient.  \n2 Results and discussion  \nSimilar to  spin-related systems, considering only orbital  Hall effect (or the orbital \nRashba effect and Oersted field), the out-of-plane and in -plane torques could be \ngenerally given 𝒎×𝒐𝒚 and 𝒎×(𝒐𝒚×𝒎), respectively .22 Under the circumstances , \nanalogous to  systems  with 2 -fold rotational symmetry, if m is reversal by applying a negative magnetic field equivalent to rotating the in -plane magnetic field angle φ (in \nregard to x direction) by  180º, Vdc must satisfy same amplitude but opposite sign, that \nis Vdc(H) = -Vdc(-H). Any difference in resonance line shape between Vdc(H) and -Vdc(-\nH) represents the existence of an additional torque. Generally, besides oy-induced OT,  \nthe oz-induced OT , with features of  more comparable Vs signals (both the magnitude \nand sign are taken into account) and distinct  Va signals , which may be considered.  \n \nFig. 2 ST-FMR resonance signal for (a) Co/CuO x(3 nm), (b) Co/CuO x(10 nm), (c) Py/CuO x(3 nm) \nand (d) Py/CuO x(10 nm). \nFig. 2a shows the representative ST-FMR resonance signal Vdc of the Co/CuO x(3 \nnm) at frequenc y of 10 GHz with the current flow for the magnetic field angle 𝜑= 45° \n(sweeping from the positive and negative magnetic field) at room temperature. \nAccording to previous analysis, comparing the ST -FMR resonance signals under \npositive and negative magnetic field, Va signals with opposite signs reflects an existence  \nof oz-induced OT . Moreover,  as presented in Fig. 2b for Co/CuO x(10 nm), Vs signals \nwith same signs  but smaller magnitude relative to Va means  weak er contribution from \noz-induced OT  when the thickness of CuO x reaching to 10 nm. One representative \ncharacteristic of the magnitude of OT is strongly dependent on material variation of a \nFM.2 Thus, we perform similar ST -FMR measurements for Py/CuO x(3 nm)  bilayers as \nshown in Fig. 2c, distinct Vs and Va signals in magnitude provides obvious evidence for \nthe existence of oz-induced OT. While as for Py/CuO x(10 nm) bilayers, there is no \nobvious  difference between Vdc(H) and -Vdc(-H), which indicates the weak oz-induced \nOT.  \n  \nFig. 3 In-plane magnetic field angle φ dependence of Vs and Va for (a) Co/CuO x(3 nm), (b) \nCo/CuO x(10 nm), (c) Py/CuO x(3 nm) and (d) Py/CuO x(10 nm).  \nTo analyse the torque components and magnitude in FM/ CuO x systems, we \nconsider the full angular dependence of the ST -FMR signal as an external magnetic \nfield. The ST -FMR measurement is conducted for values of the in -plane magnetic -field \nangle φ from 0º to 360º at an excitation frequency of 1 0 GHz . To obtain Vs and Va, the \nresonance spectr a are fitted via Eq. (2). The angular dependence of Vs and Va are \nexhibited  in Fig. 3a  for Co/CuO x(3 nm) bilayers , which can be fitted by the variation of \nEq. (3) and (4)  that is  𝑉𝑠 ∝sin(2𝜑)𝜏ǁ  and 𝑉𝑎 ∝sin(2𝜑)𝜏⊥ . Traditionally, \nanalogous  to spin-related systems, the current -induced orbital torques follow a cos ( φ) \nbehavior due to the orbital  Hall effect and/or Oersted field20,23, which leads to a total \nangular dependence of the form sin (2 φ) cos ( φ) for both Vs and Va. However, it is \nnotable that the angular dependence both Va and Vs for the  Co/CuO x(3 nm) bilayers \nneed to be well fitted through adding extra term, that is:  \n𝑉s=𝐴ssin(2𝜑)cos(𝜑)+𝐵ssin(2𝜑) (5) \n𝑉a=𝐴asin(2𝜑)cos(𝜑)+𝐵asin(2𝜑) (6) \n0 90 180 270 360-0.4-0.20.00.20.4\nCo-CuOx \ntCuOx = 3 nmVdc (V)\n () Vs     Assin2cos+ Bssin2\n  \n Va     Aasin2cos+ Basin2\n0 90 180 270 360-6-3036\nPy-CuOx \ntCuOx = 3 nmVdc (V)\n () Vs     Assin2cos+ Bssin2\n  \n Va     Aasin2cos+ Basin2\n0 90 180 270 360-46-2302346\nPy-CuOx \ntCuOx = 10 nmVdc (V)\n () Vs     Assin2cos+ Bssin2\n  \n Va     Aasin2cos+ Basin2\n0 90 180 270 360-4-2024\nCo-CuOx \ntCuOx = 10 nmVdc (V)\n () Vs     Assin2cos+ Bssin2\n  \n Va     Aasin2cos+ Basin2(a) (b)\n(c) (d)where As(a) and Bs(a) represent the magnitude of OT τy,AD(FL)  and τz,FL(AD)  generated by \norbital current that are polarized along y direction ( oy) and z direction ( oz), respectively. \nIt is essential to point that the out -of-plane field -like torque τy,FL is deemed to the \ncontribution from current -induced Oersted field .13 In other words, the magnitude of \ncorresponding OT could be obtained by fitting Va(s) as functions of φ. As shown in Fig. \n3a, better fit through two terms further indicates the existence of oz-induced OT.  \nHowever, as for Co/CuO x(10 nm) bilayers  in Fig. 3b , small er Vs and a trend of close to \nsin (2 φ) cos ( φ) for Va means weaker oz-induced OT. Moreover, in Py/ CuO x bilayers, \nsimilar behavior has been  observed , that is, strong oz-induced OT appears at a CuO x \nthickness  (tCuOx) of 3 nm  in Fig. 3c , while weaken at 10 nm  in Fig. 3d.  \n \nFig. 4 (a) 𝜏𝑦,𝐴𝐷\n𝜏𝑦,𝐹𝐿 and (b) 𝜏𝑧,𝐴𝐷\n𝜏𝑦,𝐹𝐿 corresponding to in -plane antidamping -like torque and out -of-plane \nantidamping -like torque generated by orbital polarization along y and z direction  for Co/CuO x and \nPy/CuO x bilayers , respectively.  \nFurthermore, to obtain the proportion of antidamping -like torque in  the generated \ntorques , we employ  the OT ratios  𝜏𝑦,𝐴𝐷\n𝜏𝑦,𝐹𝐿  and 𝜏𝑧,𝐴𝐷\n𝜏𝑦,𝐹𝐿  corresponding to in-plane \nantidamping -like torque and out -of-plane antidamping -like torque generated by orbital \npolarization along y and z direction , respectively. We extract the  magnitude of OT from \nVa(s) vs φ under different tCuOx for Co/CuO x and Py/CuO x bilayers  and conclude the OT \nratios , showing  tCuOx dependence of 𝜏𝑦,𝐴𝐷\n𝜏𝑦,𝐹𝐿 in Fig. 4a.  when the tCuOx from 2 nm to 3 nm, \nthe oy-induced OT ratios 𝜏𝑦,𝐴𝐷\n𝜏𝑦,𝐹𝐿 increase s dramatically and peak s when tCuOx reaches 3 \nnm, highlighting that the orbital current  stems from FM/CuO x interface.  For thicker \nsamples ( tCuOx > 4 nm), the system becomes FM/Cu/CuO x, it exhibits a monotonic \n3 6 9 120.000.050.100.150.20\ntCuOx (nm)z,AD/y,FL Co-CuOx\n Py-CuOx\n3 6 9 120.00.61.21.82.4\ntCuOx (nm)y,AD/y,FL Co-CuOx\n Py-CuOx(a) (b)decrease of OT ratios as tCuOx increases, which can be attributed to  either the current \nshunting of low resistivity Cu o r the blocking  of the orbital current  in non -oxidized Cu \nwith filled  3d-shell .18 Moreover, the oz-induced  OT ratios  (𝜏𝑧,𝐴𝐷\n𝜏𝑦,𝐹𝐿) also shows similar \ntrend as shown in Fig. 4b . It is obvious that there exists out-of-plane antidamping -like \ntorque when tCuOx is 3 nm. We contribute the generation of orbital current  to ORE19 and \nspin-vorticity coupling (SVC) .24 Note that the oxygen gradient has a vital effect on OT \nand the modulation of orbital current as the electronic configuration of CuO x alters to \nd8 or d9 not d10. According to above result, we conclude that thinner Cu layer ( tCu < 4 \nnm) is all oxidized to CuO x and adjacent  to FM layer, where the OAM is absorbed by \nthe magnetization of FM layer. However, the thicker Cu layer, where the  system \nexhibits the structure of FM/Cu -CuO x due to shorter length scale of the oxygen gradient \n(~ 3-5 nm), hence  the orbital current  cannot efficiently diffuse  through the non -oxidized \nCu due to intrinsic quenching of the OAM , which could be ascribed to the negligible d \ncharacter of non-oxidized Cu near the Fermi level .19 \n \nFig. 5 (a) Frequency dependence of linewidth ΔH for different CuO x thicknesses. (b)  CuO x thickness \ndependence of damping parameter α. \nTo further identify the existence the antidamping -like torque in the system, we \ncarry out microwave frequency  f dependence of ST-FMR measurement for \nPy/CuO x(tCuOx) bilayers to obtain the variety of damping .25 Following, we extract \nresonance linewidth  ∆𝐻  under different frequency for each sample, showing the f \ndependence of ΔH in Fig. 5a  and fitted by following equation26: ∆𝐻=∆𝐻0+4𝜋𝛼𝑓 /𝛾, \nwhere ∆𝐻0  and α can be  quantitatively obtained by linear fitting. Almost -zero \ninhomogeneous  linewidth broadening ∆𝐻0 implies small roughness of each sample.  \n0 5 10 1511223344\nPy-CuOx(t)  2 nm\n 3 nm\n 5 nm\n 6 nm\n 8 nm\n 10 nm\n 12 nmH\nFrequency (GHz)\n0 3 6 9 124.24.54.85.1\nPy-CuOx (x10-3)\ntCuOx (nm)(a) (b)Moreover, w hen the thickness of CuO x gradually increase, the slope of ΔH vs f decrease , \nwhich means a similar  trend of α and further characterize s in Fig. 5b.  With the \nincreasing CuO x thickness , we find a monotonic decrease of damping α, which proves \nthe existence of anti -damping25 (decrease of α) and attributes to interfacial OT by ORE \nat Py/CuO x system. We note that similar trend w as observed in Py/Ag/Bi26, Py/Cu/Ta27 \nand Ta/Py25 interfacial Rashba systems , where the decrease of α indicates the generation \nof antidamping -like torque.  \n3 Conclusion s \nIn conclusion , the generation of out -of-plane anti damping -like torque  is \ninvestigated based on orbital Rashba effect in FM/CuO x bilayers by employing spin -\ntorque ferromagnetic resonance. The magnetic field dependence of resonance signal \nproves the existence of both y- and z-polarized orbital current. The oy- and oz- induced \norbital torque  ratios 𝜏𝑦,𝐴𝐷\n𝜏𝑦,𝐹𝐿  and 𝜏𝑧,𝐴𝐷\n𝜏𝑦,𝐹𝐿  all increase dramatically and peak when the \nthickness of CuO x reaches 3 nm  and then monotonically decrease for thicker CuO x, \nhighlighting that the orbital current  originates in  FM/CuO x interface  and strong \nantidamping -like torque present in 3 -nm CuO x systems, which further be proved by \nCuO x thickness dependence of damping parameter α. This result provides a physic \nunderstanding of the generation out-of-plane antidamping -like torque by orbital current  \nand opens a route to generating efficient OT by orbital engineering.  \n \nConflict of Interest  \nThe authors declare no conflict of interest.  \n \nAcknowledgements  \nThis work was supported by the National Natural Science Foundation of China (Grant \nNo. 52271179 ), the Natural Science Foundation of Gansu Province (Grant No. \n21JR7RA472).  \n  Notes and r eferences  \n1 D. Lee, D. Go, H. J Park, W. Jeong, H. W. Ko, D. Yun, D. 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Han, and Ch. Kim, Phys. Rev. Lett. , 2011, 107, \n156803.  \n10 J. Kim, D. Go, H. Tsai, D. Jo, K. Kondou, H. W. Lee, and Y . Otani, Phys. Rev. B , \n2021, 103, L020407.  \n11 J. Sinova , S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. \nPhys. , 2015, 87, 1213.  \n12 A. Manchon, J. Železný, I.M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. \nGarello, and P. Gambardella , Rev. Mod. Phys. , 2019, 91, 035004.  \n13 T. Nan, C. X. Quintela, J. Irwin, G. Gurung, D. F. Shao, J. Gibbons, N. Campbell, \nK. Song, S. -Y . Choi, L. Guo, R. D. Johnson, P. Manuel, R. V . Chopdekar, I. Hallsteinsen, \nT. Tybell, P. J. Ryan, J. W. Kim, Y . Choi, P. G. Radaelli, D. C. Ralph, E. Y . Tsymba l, M. \nS. Rzchowski, and C. B. Eom, Nat. Commun. , 2020, 11, 4671.  \n14 D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park, and D. C. Ralph, Nat. Phys. , 2017, 13, 300 -305. \n15 F. Liu, Y . Jin, Y . Zhao, Ch. Jia, and Ch. Jiang , Appl. Phys. Lett., 2022, 121, 022404 . \n16 A. Bose, F. Kammerbauer, R. Gupta, D. Go, Y . Mokrousov, G. Jakob, and M. Kläui , \nPhys. Rev. B , 2023, 107, 134423 . \n17 F. Liu, B. Liang, J. Xu, Ch. Jia, and Ch. Jiang , Phys. Rev. B , 2023, 107, 054404.  \n18 Sh. Ding, Zh . Liang, D. Go, Ch. Yun, M. Xue, Zh. Liu, S. Becker, W. Yang, H. Du, \nCh. Wang, Y . Yang, G. Jakob, M. Kläui, Y . Mokrousov, and J. Yang,  Phys. Rev. Lett. , \n2022, 128, 067201.  \n19 D. Go, D. Jo, T. Gao, K. Ando, S. Blügel, H. W. Lee, and Y . Mokrousov , Phys. Rev. \nB, 2021, 103, L121113.  \n20 L. Q. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. , 2011, \n106, 03660.  \n21 W. L. Yang, J. W. Wei, C. H. Wan, Y . W. Xing, Z. R. Yan, X. Wang, C. Fang, C. Y . \nGuo, G. Q. Yu, and X. F. Han , Phys. Rev. B , 2020, 101, 064412.  \n22 Y . Wang, R. Ramaswamy, and H. Yang, J. Phys. D: Appl. Phys. , 2018, 51, 273002.  \n23 A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. \nVaezi , A. Manchon, E. A. Kim, N. Samarth, D. C. Ralph, Nature , 2014, 511, 449 -451. \n24 Genki Okano, Mamoru Matsuo, Yuichi Ohnuma, Sadamichi Maekawa, and Yukio \nNozaki , Phys. Rev. Lett. , 2019, 122, 217701.  \n25 N. Behera, S. Chaudhary, and D. Pandya, Sci. Rep. , 2016, 6, 19488.  \n26 M. B. Jungfleisch, W. Zhang, J. Sklenar, W. Jiang, J. E. Pearson, J. B. Ketterson, and \nA. Hoffmann, Phys. Rev. B , 2016, 93, 224419.  \n27 F. Liu, C. Zhou, R. Tang, G. Chai, and C. Jiang, J. Magn. Magn.  Mater. , 2021, 540, \n168462 . "    },    {        "title": "0710.2826v2.Ferromagnetic_resonance_study_of_polycrystalline_Fe__1_x_V_x_alloy_thin_films.pdf",        "content": "arXiv:0710.2826v2  [cond-mat.mes-hall]  17 Oct 2007Ferromagnetic resonance study of polycrystalline Fe 1−xVxalloy thin films\nJ-M. L. Beaujour, A. D. Kent\nDepartment of Physics, New York University, 4 Washington Pl ace, New York, NY 10003, USA\nJ. Z. Sun\nIBM T. J. Watson Research Center, Yorktown Heights, NY 10598 , USA\n(Dated: November 18, 2018)\nFerromagnetic resonance has been used to study the magnetic properties and magnetization dy-\nnamics of polycrystalline Fe 1−xVxalloy films with 0 ≤x <0.7. Films were produced by co-\nsputtering from separate Fe and V targets, leading to a compo sition gradient across a Si substrate.\nFMR studies were conducted at room temperature with a broadb and coplanar waveguide at fre-\nquencies up to 50 GHz using the flip-chip method. The effective demagnetization field 4 πMeffand\nthe Gilbert damping parameter αhave been determined as a function of V concentration. The\nresults are compared to those of epitaxial FeV films.\nI. INTRODUCTION\nA decade ago, it was predicted that a spin polar-\nized current from a relatively thick ferromagnet (FM)\ncould be used to switch the magnetization of a thin FM\n[1]. Since then, this effect, known as spin-transfer, has\nbeen demonstrated in spin-valves [2] and magnetic tun-\nnel junctions [3]. In a macrospin model with collinear\nlayer magnetizations, there is a threshold current den-\nsityJcfor an instability necessary for current-induced\nmagnetization switching of the thin FM layer [1, 4]:\nJc=2eαMstf(Hk+2πMs)\n/planckover2pi1η, (1)\nwhereαisthedampingconstant. tfandMsarethethick-\nness and the magnetization density of the free layer, re-\nspectively. Hkis the in-plane uniaxial anisotropy field. η\nis the currentspin-polarization. In orderfor spin-transfer\nto be used in high density memory devices Jcmust be re-\nduced. From Eq. 1 it is seen that this can be achieved by\nemploying materials with low Msandαin spin-transfer\ndevicesor, equivalently materialswith lowGilbert damp-\ning coefficients, G = αMs(gµB//planckover2pi1).\nVery recently, an experimental study of epitaxial FeV\nalloy thin films demonstrated a record low Gilbert damp-\ning coefficient [5]. This material is therefore of interest\nfor spin transfer devices. However, such devices are gen-\nerally composed of polycrystalline layers. Therefore it is\nof interest to examine polycrystalline FeV films to assess\ntheir characteristics and device potential.\nIn this paper, we present a FMR study of thin poly-\ncrystalline Fe 1−xVxalloy films with 0 ≤x <0.7 grown\nby co-sputtering. The FeV layers were embedded be-\ntween two Ta |Cu layers, resulting in the layer structure\n||5 Ta|10 Cu|FeV|5 Cu|10 Ta||, where the numbers are\nlayer thickness in nm. FeV polycrystalline films were\nprepared by dc magnetron sputtering at room tempera-\nture from two separate sources, oriented at a 45oangle\n(Fig. 1a). The substrate, cut from a Silicon wafer with\n100 nm thermal oxide, was 64 mm long and about 5 mm\nwide. The Fe and V deposition rates were found to vary/s49 /s50 /s51 /s52/s45/s50/s45/s49/s48/s40/s98/s41\n/s32/s86/s32/s116/s97/s114/s103/s101/s116\n/s52/s53/s111\n/s32\n/s32/s115/s117/s98/s115/s116/s114/s97/s116/s101/s70/s101/s32/s116/s97/s114/s103/s101/s116/s40/s97/s41\n/s72\n/s114/s101/s115/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s108/s105/s110/s101/s32/s32/s40/s97/s117/s41\n/s72\n/s97/s112/s112/s32/s32/s40/s107/s79/s101/s41/s49/s52/s32/s71/s72/s122\n/s32/s32/s32\n/s32/s120 /s61/s48/s46/s51/s55/s120 /s61/s48/s46/s53/s50\n/s120 /s61/s48/s46/s49/s57/s72\n/s115/s117/s98/s115/s116/s114/s97/s116/s101/s32/s104/s111/s108/s100/s101/s114\n/s97/s120/s105/s115\nFIG. 1: a) The co-sputtering setup. b) Typical absorp-\ntion curves at 14 GHz for a selection of ||Ta|Cu|7.5 nm\nFe1−xVx|Cu|Ta||films, with x=0.19, 0.37 and 0.52. The res-\nonance field Hresand the linewidth ∆ Hare indicated.\nlinearly across the wafer. The Fe and V rates were then\nadjusted to produce a film in which xvaries from 0.37\nto 0.66 across the long axis of the wafer. The base pres-\nsure in the UHV chamber was 5 ×10−8Torr and the\nAr pressure was set to 3.5 mTorr. The FeV was 7.5 nm\nin thickness, varying by less than 0.3 % across the sub-\nstrate. An Fe 1−xVxfilm 3 nm thick was also fabricated.\nTo produce films with x <0.30 the rate of the V source\nwas decreased. Finally, pure Fe films with a thickness\ngradient ranging from 7 nm to 13.3 nm were deposited.\nTheFMRmeasurementswerecarriedoutatroomtem-\nperature using a coplanar wave-guide (CPW) and the\nflip-chip method. Details of the experimental setup and\nofthe CPWstructuralcharacteristicscan be found in [6].\nAdc magnetic field, up to 10 kOe, wasapplied in the film\nplane, perpendiculartotheacmagneticfield. Absorption\nlines at frequencies from 2 to 50 GHz were measured by\nmonitoring the relative change in the transmitted signal\nas a function of the applied magnetic field.2\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s48/s53/s49/s48/s49/s53/s50/s48\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s50/s46/s48/s53/s50/s46/s49/s48/s50/s46/s49/s53/s49/s50/s51\n/s120\n/s32/s52 /s77\n/s101/s102/s102/s32/s32/s40/s107/s71/s41/s55/s46/s53/s32/s110/s109/s32/s70/s101\n/s49/s45 /s120/s86\n/s120\n/s51/s32/s110/s109/s32/s70/s101\n/s48/s46/s54/s51/s86\n/s48/s46/s51/s55\n/s49/s50/s46/s57/s32/s110/s109/s32/s70/s101\n/s120/s32\n/s32/s103/s45/s102/s97/s99/s116/s111/s114/s40/s98/s41/s40/s99/s41/s32\n/s32\n/s32/s72\n/s114/s101/s115/s32/s32/s40/s107/s79/s101/s41\n/s49/s49/s32/s71/s72/s122/s40/s97/s41\nFIG. 2: a) The resonance field at 11 GHz versus xand b) the\neffective demagnetization field versus x. The solid line is a\nguide to the eye. c) The Land´ e gfactor as a function of x.\nThe dotted line shows the g-factor value of bulk Fe.\nII. RESULTS\nTypical absorption lines at 14 GHz of selected FeV al-\nloyfilmsareshowninFig. 1b. Thelinesarelorentzianfor\nmost frequencies. At a fixed frequency, the FMR absorp-\ntion decreaseswith increasingV content. The FMR peak\nof a film 7.5 nm thick with x= 0.66 is about 100 times\nsmallerthanthatofapureFeofthesamethickness. This\nis accompaniedby a shift of Hrestowardshigher field val-\nues (Fig. 2a). The effective demagnetization field 4 πMeff\nand the Land´ eg-factorgwere determined by fitting the\nfrequency dependence of the resonance field Hresto the\nKittel formula [7]:\nf2=/parenleftBiggµB\nh/parenrightBig2\nHres(Hres+4πMeff),(2)\nwhere the effective demagnetization field is:\n4πMeff= 4πMs−H⊥. (3)\nNote that in the absence of a perpendicular anisotropy\nfieldH⊥, the effective field would be directly related to\nMs. The dependence of 4 πMeffon V concentration is\nshown in Fig. 2c. As xincreases the effective demagneti-\nzation field decreases dramatically, going from about 16\nkG forx= 0 to 1.1 kG for x= 0.66. Note that the effec-\ntive demagnetization field of the 7.5 nm Fe film is about\n25 % lower than that of bulk Fe (21.5 kG). The 12.9\nnm Fe film exhibits a larger 4 πMeff, which is, however,\nstill lower than 4 πMsof the bulk material. Similarly,\nthe 4πMeffof an Fe 0.63V0.37film is thickness dependent:\ndecreasing with decreasing layer thickness.\nThe Land´ eg-factor increases monotonically with in-\ncreasing V concentration (Fig. 2b). The minimum g-\nfactor is measured for the Fe film: g= 2.11±0.01,\nwhich is slightly larger than the value of bulk material\n(g= 2.09). Note that gof a Fe film 12.9 nm thick is\nequal to that of Fe bulk. However, the g-factor of the/s48 /s50/s48 /s52/s48/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\n/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s40/s99/s41/s120 /s32/s61/s32/s48/s46/s53/s50/s120/s32 /s61/s32/s48/s46/s52/s51/s120/s32 /s61/s32/s48/s40/s98/s41/s32/s40/s49/s48/s45/s50\n/s41/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s32/s40/s107/s79/s101/s41/s32\n/s32/s72 /s32/s32/s40/s107/s79/s101/s41\n/s72\n/s48/s72\n/s48/s72\n/s49/s52/s32/s32\n/s32/s40/s97/s41\n/s120/s32 /s32/s102/s32/s32/s40/s71/s72/s122/s41\nFIG. 3: a)Frequency dependence of the linewidth for 7.5 nm\nFe1−xVxalloy film with x=0, 0.43 and 0.52. The solid lines\nare the best linear fit of the experimental data. b) ∆ H14, the\nlinewidth at 14 GHz , and ∆ H0are shown as a function of x.\nc) The magnetic damping parameter versus V concentration.\nFe0.63V0.37, does not appear to be thickness dependent:\nthe 3 nm Fe 0.63V0.37layer has about the same gvalue\nthan the 7.5 nm Fe 0.63V0.37layer.\nThe half-power linewidth, ∆ H, was studied as a func-\ntion of the frequency and of the V concentration. Fig.\n3b shows the dependence of the FMR linewidth on xat\n14 GHz. The general trend is that ∆ Hincreases with\nx. However, there are two regimes. For x >0.4, the\nlinewidth depends strongly on x, increasing by a factor\n5 whenxis increased from 0.4 to 0.66. The dependence\nof the linewidth on xis more moderate for the films with\nx <0.4: it increases by about 30 %. For all samples,\nthe linewidth scales linearly with the frequency. A least\nsquare fit of ∆ H(f) gives ∆ H0, the intercept at zero\nfrequency, and the Gilbert damping parameter αwhich\nis proportional to the slope: d∆H/df= (2h/gµB)α[8].\n∆H0is typically associated with an extrinsic contribu-\ntion to the linewidth and related to magnetic and struc-\ntural inhomogeneities in the layer. For two samples with\nthe highest Vanadium concentration, x= 0.60, 0.66, the\nlinewith is dominated by inhomogeneous broadening and\nit wasnot possible to extract α. Asxincreases, ∆ H0and\nαincreases. The damping parameter and ∆ H0remain\npractically unchanged for x≤0.4 and when x >0.4,\nboth the intercept and the slope of ∆ Hversusfincrease\nrapidly.\nIII. DISCUSSION\nSeveral factors can contribute to the dependence of\n4πMeffon the V concentration. First, the decrease of\nthe effective demagnetizationfield canbe associatedwith\nthe reduction of the alloy magnetization density Mssince\nthe Fe content is reduced. In addition, a neutron scat-\ntering study showed that V acquires a magnetic moment3\nantiparallel to the Fe, and that the Fe atom moment de-\ncreases with increasing V concentration [9]. The Curie\ntemperature of Fe 1−xVxdepends on x. In fact, Tcfor\nx=0.65 is near room temperature [10]. It is important to\nmention that a factor that can further decrease 4 πMeff\nis an out-of-plane uniaxial anisotropy field H⊥(Eq. 3).\nIn thin films, the perpendicular anisotropy field is com-\nmonly expressed as H⊥= 2K⊥/(Mst), where K⊥>0\nis the anisotropy constant and tthe ferromagnetic film\nthickness [11]. In this simple picture, it is assumed that\nK⊥is nearly constant over the thickness range of our\nfilms. This anisotropy can be associated with strain\ndue to the lattice mismatch between the FeV alloy and\nthe adjacent Cu layers and/or with an interface contri-\nbution to the magnetic anisotropy. For Fe films with\nt= 7.5 and 12.9 nm, a linear fit of 4 πMeffversus 1/t\ngives 4πMs= 20.2 kG and K⊥= 2.5 erg/cm2. The\nvalue extracted for 4 πMsis in the range of the value of\nFebulk. AsimilaranalysisconductedonFe 0.63V0.37films\nof thickness t=3 and 7.5 nm gives 4 πMs= 12.2 kG and\nK⊥= 0.1 erg/cm2. The result suggests that the surface\nanisotropy constant decreases with increasing x.\nIV. SUMMARY\nThe effective demagnetization field of the polycrys-\ntalline Fe 1−xVxalloy films decreases with increasing xandalmostvanishesfor x≈0.7. AFMRstudyonepitax-\nial films haveshown a similar xdependence of 4 πMeff[5].\nUsing the value of Mscalculated in the analysis above,\nwe estimate the Gilbert damping constant of a 7.5 nm Fe\nlayer and 7.5 nm Fe 0.63V0.37alloy film to be G Fe= 239\nMHz and G FeV= 145 MHz respectively. The decrease of\nthe effective demagnetization field of Fe 1−xVxwith in-\ncreasing xis accompanied by a decrease of the Gilbert\ndamping constant. A similar xdependence of G was\nobserved in epitaxial films [5]. The authors explained\nthe decrease of G by the reduced influence of spin-orbit\ncoupling in lighter ferromagnets. Note that the Gilbert\ndampingofourfilmsislargerthanwhatwasfoundforthe\nepitaxial films (G=57 MHz for epitaxial Fe 8 nm thick).\nWe note that the Fe 0.63V0.37alloy film, which has\n4πMsapproximatly the same as that of Permalloy, has a\nmagnetic damping constant of the same order than that\nof Py layer in a similar layer structure [12]. Hence, with\ntheir low Msandα, polycrystalline FeV alloy films are\npromising materials to be integrated in spin-tranfer de-\nvices.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159(1-2), L1\n(1996) ; L. Berger, Phys. Rev. B 54(12), 9353 (1996).\n[2] see, for example, J. A. Katine et al., Phys. Rev. Lett. 84,\n3149 (2000) ; B. Oezyilmaz et al., Phys. Rev. Lett. 91,\n067203 (2003).\n[3] see, for example, G. D. Fuchs et al., J. Appl. Phys. 85\n(7), 1205 (2004).\n[4] J. Z. Sun, Phys. Rev. B 62(1), 570 (2000).\n[5] C. Scheck et al., Phys. Rev. Lett. 98, 117601 (2007).\n[6] J-M. L. Beaujour et al., Europhys. J. B, DOI:\n10.1140/epjb/e2007-00071-1 (2007).\n[7] C. Kittel in Introduction to Solid State Physics, Ed. 7,\np.505.[8] see, for example, D. L. Mills and S. M. Rezende in Spin\nDynamics in Confined Magnetic Structures II (Eds. B.\nHillebrands andK.Ounadjela), pp.27-58, (Springer, Hei-\ndelberg 2002).\n[9] I. Mirebeau, G. Parette, and J. W. Cable. J. Phys. F:\nMet. Phys. 17, 191 (1987).\n[10] Y. Kakehashi, Phys. Rev. B 32(5), 3035 (1985).\n[11] Y. K. Kim and T. J. Silva, Appl. Phys. Lett. 68, 2885\n(1996).\n[12] S. Mizukami et al., J. Magn. Magn. Mater. 239, 42\n(2002)."    },    {        "title": "0712.1403v2.Ferromagnetic_Quantum_Critical_Fluctuations_and_Anomalous_Coexistence_of_Ferromagnetism_and_Superconductivity_in_UCoGe_Revealed_by_Co_NMR_and_NQR_Studies.pdf",        "content": "arXiv:0712.1403v2  [cond-mat.supr-con]  31 Dec 2007Typeset with jpsj2.cls <ver.1.2> Full Paper\nFerromagnetic Quantum Critical Fluctuations and Anomalou s Coexistence of\nFerromagnetism and Superconductivity in UCoGe Revealed by Co-NMR and NQR\nStudies\nTetsuya Ohta,1∗YusukeNakai,1Yoshihiko Ihara,1KenjiIshida,1†Kazuhiko Deguchi,2Noriaki K. Sato2\nand Isamu Satoh3\n1Department of Physics, Graduate School of Science, Kyoto Un iversity, Kyoto 606-8502, Japan.\n2Department of Physics, Graduate School of Science, Nagoya U niversity, Nagoya 464-8602, Japan.\n3Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan.\nCo nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) studies\nwere carried out for the recently discovered UCoGe, in which the ferromagnetic and supercon-\nducting(SC) transitions are reported tooccur at TCurie∼3 K andTS∼0.8 K (N. T. Huy et al.,\nPhys. Rev. Lett. 99(2007) 067006), in order to investigate the coexistence of f erromagnetism\nand superconductivity as well as the normal-state and SC pro perties from a microscopic point\nof view. From the nuclear spin-lattice relaxation rate 1 /T1and Knight-shift measurements, we\nconfirm that ferromagnetic fluctuations that possess a quant um critical character are present\naboveTCurieand also the occurrence of a ferromagnetic transition at 2.5 K in our polycrys-\ntalline sample. The magnetic fluctuations in the normal stat e show that UCoGe is an itinerant\nferromagnet similar to ZrZn 2and YCo 2. The onset SC transition is identified at TS∼0.7 K,\nbelow which 1 /T1arising from 30% of the volume fraction starts to decrease du e to the opening\nof the SC gap. This component of 1 /T1, which follows a T3dependence in the temperature\nrange 0.3−0.1 K, coexists with the magnetic components of 1 /T1showing a√\nTdependence\nbelowTS. From the NQR measurements in the SC state, we suggest that th e self-induced\nvortex state is realized in UCoGe.\nKEYWORDS: UCoGe, ferromagnetic superconductor, Co-NMR an d NQR, coexistence\nSince the discovery of superconductivity in ferromag-\nnetic compounds under high pressure,1,2the concept of\nthe interplay between magnetism and superconductiv-\nity was changed, because ferromagnetism and supercon-\nductivity are considered to be mutually exclusive. The\npressure studies in UGe 2have shown that the super-\nconducting (SC) transition temperature TSis higher in\nthe pressure region where the ordered moments are en-\nhanced,and that superconductivityis not observedwhen\nthe ferromagnetism disappears in a further higher pres-\nsure region.3These results suggest that ferromagnetism\nseems to enhance the superconductivity in UGe 2. In ad-\ndition, ambient-pressure ferromagnetic superconductiv-\nity was discovered in URhGe, where TCurie= 9.5 K and\nTS∼0.25 K were reported.4The relation between the\nferromagnetismand superconductivity observedin the U\ncompounds is one of the attractive topics for the com-\nmunity studying strongly correlated electron systems.\nQuite recently, new ambient-pressure ferromagnetic\nsuperconductivity with TCurie= 3 K and TS∼0.8 K\nwas found in UCoGe.5One of the advantages of UCoGe\nisthatitincludes“NMR-active”nucleusCo.IntheNMR\nstudies of ferromagnetic superconductors, Ge in UGe 2is\nreplaced by73Ge with a nuclear spin and73Ge nuclear\nquadrupole resonance (NQR) studies have been carried\nout by Kotegawa et al.6and Harada et al.7,8They have\nreported the SC properties and the relation between fer-\nromagnetism and superconductivity from the73Ge-NQR\n∗E-mail address: t-ohta@scphys.kyoto-u.ac.jp\n†E-mail address: kishida@scphys.kyoto-u.ac.jp0.0 0.2 0.4 0.6 0.8 1.0 Tc mid Tc onsetpolycrystalline sample δ χac   ∝  - δ f   (arb. units) UCoGe \n  \nT (K)  9 kG \n 7 kG \n 6 kG \n 5 kG \n 4 kG \n 3 kG \n 2 kG \n 1 kG \n 0 kG \nFig. 1. (Color online) Temperature dependence of δχacunder\nvarious magnetic fields below 9 kOe.\nresults. In this paper, we show the detailed Co-NMR and\nNQR results of UCoGe, which are compared with the\n73Ge-NQR results of high-pressure UGe 2.\nPolycrystalline UCoGe was prepared by the arc melt-\ning method. The details will be reported in a separate\npaper.9Figure 1 shows the temperature dependence of\nχacin various magnetic fields, which was obtained by\nmeasuring the self-inductance of an NMR coil with pow-\n12 J. Phys. Soc. Jpn. Full Paper Author Name\n10 15 20 25 59 K = 0 \n1.5 K 4.2 K 6 K 8 K 10 K 15 K 20 K 40 K 60 K 100 K 150 K 63 Cu 65 Cu \nf = 18.2 MHz 59 Co-NMR Intensity (arb. units) UCoGe \nMagnetic Field  (kOe) \nFig. 2. (Color online) Field-sweep59Co-NMR spectra obtained\nat 18.2 MHz at various temperatures. The sharp signals shown\nby63Cu and65Cu are the NMR signals arising from the NMR\ncoil.59K= 0 shows the field where the Knight shift of59Co is\nzero. The dashed curve traces the NMR-signal peak arising fr om\nthe 1/2↔ −1/2 transition.\ndered UCoGe. A clear Meissner signal was observed in\nzero and magnetic fields, indicating that superconduc-\ntivity persists up to 9 kOe. In zero field, the onset and\nmidpoint of the SC transition are estimated to be 0.7\nand 0.55 K, respectively. The Meissner behavior in zero\nfield in our sample is consistent with that in a previous\nreport.5\nFigure2showsthe59Co-NMRspectraofourpowdered\nsample obtained by sweeping an external field at various\ntemperatures. A symmetric NMR spectrum with several\nsatellite peaks at higher temperatures, indicative of the\npresence of the electric field gradient (EFG) at the Co\nsite, becomes anisotropic with decreasing temperature.\nThe shift of the gravity of the broad NMR spectra orig-\ninates from the isotropic component of the Knight shift.\nThe NMR peak arising from the 1 /2↔ −1/2 transition\nshifts significantlywith respecttotemperature. Theshift\nof this peak is traced by the dashed curve in Fig. 2.\nFigure 3 shows the temperature dependence of the\nKnight shift59Kevaluated from the peak arising from\nthe 1/2↔ −1/2 transition, along with the magnetiza-\ntionMdivided by Hat 15 kOe. The temperature de-\npendence of59Kis well scaled with M/H. With decreas-\ningT,59Kbecomes so large that it increases up to 35%\nat low temperatures. It is confirmed that the Curie be-\nhavior in M/His microscopically intrinsic. It should be\nnoted that the59Kand Co-NMR spectra do not change\nthrough TCurie∼3 K, showing that the ferromagnetic\ntransition is smeared out in the magnetic fields. This is\ncharacteristic of the ferromagnetic transition. From the\nlinear relationship between59KandM/H, the hyper-1 10 100 010 20 30 40 50 60 \nf = 18.2 MHz \n 59 KUCoGe 59 K   (%) \nT (K) 0.00 0.01 0.02 0.03 0.04 \n   M/H  in 15 kOe \n M/H (emu/mol) 0.00 0.01 0.02 010 20 30 40 59 K   (%) \nM/H  (emu/mol) \nFig. 3. Temperature dependence of the59Co Knight shift eval-\nuated from the 1 /2↔ −1/2 transition of the spectra shown in\nFig. 2. The temperature dependence of M/Hmeasured in 15\nkOe is shown in the same figure. The inset is the59K−M/H\nplot.\n4 5 8 9(c) \n95 mK νQ= 2.85 MHz \nη = 0.52 \nFrequency  (MHz) (b) \n1.5 K 59 Co-NQR Intensity  (arb. units) UCoGe \nν\u0001 ν\u0002ν\u0003(a) \n4.2 K \nFig. 4. Co-NQR spectra at 4.2 K (above TCurie: (a)), 1.5 K (be-\nlowTCurie: (b)), and 95 mK (the lowest temperature in the mea-\nsurements: (c)). The NQR spectrum at95 mKcan be understood\nby the superposition of the two NQR spectra arising from the\nregions with and without the internal field. (Refer to the tex t).\nfine coupling constant Hhfat the Co site and the or-\nbital shift Korbare estimated to be 79 ±1 kOe/µBand\n−0.96±0.22%, respectively. From the positive value of\nHhfand the predominance of the isotropic term in the\nKnight shift, it is considered that the core polarization\neffect by Co-3 delectrons is not dominant but the trans-\nferred contribution from U-5 felectrons by way of Co-4 s\norbitsisdominantattheCo-nuclearsite.TheCoKnight-\nshift results indicate that U-5 felectrons play a crucial\nrole for the ferromagnetism in UCoGe.\nThree Co-NQR signals were observed at the peaks of\n4.8, 5.2, and 8.3 MHz at 4.2 K, as shown in Fig. 4. From\nthese signals, we evaluated the NQR parameters νzz,J. Phys. Soc. Jpn. Full Paper Author Name 3\u0000 \u0004 \u0005 \u0006 \u0007 \b \t \n \u000b\n\f \r\n\u000e\n\u000f \u0010\n\u0011\n\u0012 \u0013\n\u0014\n\u0015 \u0016\n\u0017\u0018 \u0019\n\u001a\n\u001b \u001c\n\u001d\n\u001e \u001f\n \n! \"\n#$ % & ' ( ) * + , - .\n/ 0 1\n2\n345 6 7 8\n9 : ; < =>?@\nA\nBC\nDE\nFG H I J\nKL\nM N O P QR S\nT U VWXY\nZ\n[\\\n]^\n_\n` a b c de fg h i j kl m\nn\no p q r\ns t u v\nw x y\nz{ | } ~      \n           \n   \n  \n¡¢ ∞ \n£−\n¤¥¦§¨©ª« ∞ \n¬ ® ¯ ° ± ² ³ ´ µ ¶\n· ¸ ¹º » ¼ ½ ¾ ¿ ÀÁ  ÃÄ Å Æ Ç È É ÊË Ì Í Î Ï Ð Ñ Ò Ó Ô\nFig. 5. (Color online) The main graph is the temperature depe n-\ndence of 1 /T1measured at the Co-NQR signal of 8.3 MHz. The\nvalues of 1 /T1above (below) 1.5 K are shown by open (gray and\nblack) circles with respect to the right (left) axis due to th e dif-\nference in the 1 /T1values by the measurement conditions (refer\nto the text). The scale of the left-hand-sideaxis is 0.58 tha t of the\nright-hand-side axis. The insets are the recoveries of the n uclear\nmagnetization after a saturation pulse. The dotted curves i n the\ninsets are the fitting curves for evaluating 1 /T1. The recoveries\naboveTSare fitted by a single component of 1 /T1, as shown in\nthe insets of (a) and (b). Two components of 1 /T1are observed\nbelowTS, as shown in the inset of (c). The longer (black circles)\nand shorter (gray circles) components of 1 /T1are plotted in the\nmain figure. The dotted curve below TSrepresents the temper-\nature dependence of 1 /T1calculated using the 3-D polar model\nwith 2∆ 0/kBTS= 4.\nwhich is the NQR frequency along the principal axis of\nthe EFG, to be 2.85 MHz and an asymmetric parameter\nηto be 0.52. The large ηvalue shows that the symme-\ntry of the Co site is lower than an axial symmetry. The\nCo-NQR spectra affected by the internal fields at the Co\nsite were observed at low temperatures. The spectrum\nchange by temperature is discussed below.\nThe nuclear spin-lattice relaxation rate 1/ T1of Co in\nzero magnetic field was measured at the NQR signal ob-\nserved at 8.3 MHz denoted as ν3in Fig. 4. Figure 5\nshows the temperature dependence of 1 /T1of Co over\nthe wide temperature range between 95 mK and 150 K.\nIt was found that the values of 1 /T1below 10 K depend\non the NQR pulse conditions. This is considered to be\ndue to the development of the short-range ferromagnetic\ncorrelations. The details of this will be reported else-\nwhere.10Thus, the values of 1 /T1above and below 1.5\nK are shown by different axes because the measurement\nconditions change around 1.5 K. The insets show the re-\ncovery curves m(t) of the nuclear magnetization M(t)1 10 100 050 100 150 200 250 \n0.00 0.05 0.10 0.15 \n0.00 0.05 0.10 0100 200 \nCo-NQR \nf = 8.3 MHz UCoGe \nχ  (emu / mol) TCurie \nT  (K )1/T 1T  \nÕs\nÖ1 K \n×1\nØ\n1/T 1T  \nÙs-1  K -1 \nÚ\nχ  Ûemu/ molÜ\nFig. 6. (Color online) Temperature dependence of 1 /T1Tin zero\nfield and χbulkmeasured in 1 kOe. The dashed curve shows a\nT−4/3dependence, which is expected at the FM critical point.16\nThe inset is the plot of 1 /T1Tagainstχbulk.\nat a time tafter a saturation pulse at three character-\nistic temperatures. Above TS, the recovery curves were\nalmost consistently determined with a single component\nof 1/T1, as observed in the insets (a) and (b).\nFirst of all, we discuss the magnetic properties in the\nnormal state on the basis of the 1 /T1and59Kresults.\nIn general, 1 /T1divided by temperature 1 /T1Tis related\nto the low-energypart ofthe q-dependent dynamical sus-\nceptibility in compounds. The spin-fluctuation character\nin compounds can be known from the comparison be-\ntween 1/T1Tandχor Knight shift. The main plot of\nFig. 6 shows the temperature dependence of 1 /T1Tin\nzero field, together with that of χbulkin 1 kOe. The good\nlinear relation was found between 1 /T1Tandχbulk, as\nshownintheinset.Thisclearlyprovesthatferromagnetic\n(FM) fluctuations are dominant in UCoGe, since the dy-\nnamical susceptibility revealed by 1 /T1Tis determined\nby theq= 0 susceptibility. The relation1 /T1T∝χbulkis\noften observed in weak and nearly FM compounds,11–13\nand it is interpreted by the SCR theory.14,15Further-\nmore, we point out that the temperature dependence\nof 1/T1Tis close to a T−4/3dependence below 10 K,\nwhich is an expected behavior when a 3-D metal is close\nto the FM instability.16The characteristic energy of\nthe spin fluctuation T0is estimated to be T0∼220 K\nfrom the Sommerfeld coefficient γ,5,17and according to\nthe SCR calculation, 1 /T1Tis close to a T−4/3depen-\ndence below T/T0<0.1.16Therefore, the observation of\n1/T1T∼T−4/3below 10 K is reasonable and suggests\nthatUCoGepossessesanFMquantumcriticalcharacter.\nAlthough a clear anomaly was observed in 1 /T1Tat\nTCurie, the NQR spectra of ν1andν2are almost un-\nchanged even at 1.5 K, as observed in Fig. 4, indicative\nof the absence of the static internal field down to 1.5\nK. This might be related to the FM quantum critical\ncharacter because it is anticipated that ferromagnetism\ndevelops gradually near the quantum critical point. By\nfurther lowering the temperature, new NQR signals af-\nfected by the static fields start to develop below 1 K.4 J. Phys. Soc. Jpn. Full Paper Author Name\nThe Co-NQR spectra at 95 mK shown in Fig. 4 (c) can\nbe understood with the superposition of two NQR spec-\ntra; one is the Co-NQR spectra without the internal field\n(“nonmagnetic” NQR signals), and the other is the Co-\nNQR spectra affected by the internal field at the Co site\n(“magnetic”NQR signal). The magnitude of the internal\nfields of the magnetic NQR signal is approximately 400\nOe, and its direction is almost perpendicular to the prin-\ncipal axis of the EFG. In Fig. 4, the calculated resonance\nfrequencies are shown by arrow.18The NQR spectra in\nthe FM state indicate that there exist two Co sites with\nand without the internal magnetic fields. Furthermore,\nthe temperature variation of the NQR spectrum in the\nFM state shows a first-order-like behavior because the\ninternal field of the magnetic NQR signal is independent\nof temperature and only a fraction of the magnetic and\nnonmagnetic NQR signals changes by due to tempera-\nture.\nNext, we discuss the SC properties revealed by the\n1/T1measurement. Below 0.7 K, which is the onset tem-\nperature of the SC transition, the longer component of\nT1ascribed to the opening of the SC gap was observed in\nm(t), as shown in the inset (c) of Fig. 5. The longer and\nshorter components of 1 /T1are shown in the main panel\nof Fig. 5. The longer component of 1 /T1is estimated\nto be 30% of the total relaxation component, which is\ndetermined by the extrapolation of m(t) tot= 0, and\nit follows a T3dependence below 0.3 K down to 0.1 K.\nTheT3dependence of 1 /T1suggests the presence of line\nnodes in the SC gap, and the complete Tdependence of\n1/T1belowTScan be interpreted by the 3-D polar state\nwith ∆(θ) = ∆ 0cosθ. TheTdependence of 1 /T1calcu-\nlated using the model is shown in Fig. 5 with the dotted\ncurve below TS, where the magnitude of the SC gap is\ntaken as 2∆ 0/kBTS= 4. The magnitude of the SC gap is\nquite comparable to that in UGe 2.6,8It should be noted\nthat an appreciable Korringa ( T1T= constant) behav-\nior was not observed down to 95 mK, indicative of the\nsmall effect of the residual density of state. In contrast,\nthe shorter component of 1 /T1shows a√\nTdependence\nbelowTS, suggestive of the persistence of the magnetic\nfluctuationsdownto95mK.Itisquiteunusualthatmag-\nnetic fluctuations continue to develop in the SC state.\nThe underlying problem is the relation between\nthe magnetism and superconductivity in UCoGe, i.e,\nwhether superconductivity occurs only in the region\nwithout the internal field. To clarify this, we measure\n1/T1at 95mKat variouslow-frequencyNQRpeaks.Fig-\nure 7 shows the recovery curves m(t) measured at differ-\nentfrequencies.Here,the peaksat4.83and5.25MHzare\nthe Co-NQR signals mainly arising from the region with-\nout the internal field, and the peaks at 4.49 and 4.9 MHz\nare the NQR signals appearing below TCurie, which are\naffected by the internal field, i.e. Co-NQR signals arising\nfrom the magnetic region. As observed in Fig. 7, m(t)\nmeasured at the 4 different peaks shows the same behav-\nior,indicatingthatthesuperconductingfractions(30%of\nthe total relaxation component) of the regions with and\nwithout the internal fields are not different. The same\nbehavior of m(t) at the 4 peaks excludes the possibility\nthat superconductivity occurs in one of the regions with0 50 100 150 200 0.01 0.1 1\n4.0 4.5 5.0 5.5 6.0 \n    T=95mK \n 4.49 MHz \n 4.83 MHz \n 4.9   MHz \n 5.25 MHz m(t) = \nÝM\nÞ∞\nß \nà M(t) \ná/M \nâ∞\nã\n time (msec) T = 95 mK Intensity \näarb. units \nå\nFrequency (MHz) \nFig. 7. (Coloronline)Co nuclearrecoveries m(t) measured atvar-\niousfrequencies at95 mK.The insetshowsthe NQR-signalpea ks\nat 95 mK, where the recoveries are measured.\nand without the internal field; rather, superconductivity\noccurs in both regions from the NQR spectra. This sit-\nuation is different from the high-pressure ferromagnetic\nphase in UGe 2, where superconductivity is observed only\nin the ferromagnetic region.7We also point out that the\nshort 1/T1component is quite similar in both regions\nwith and without the internal field. This is considered to\nbe because the ordered moment is as small as 0.03 µBat\nT= 0 K.\nNext, we attempt to understand the presence of the\ntwo components of 1 /T1in the entire sample. We sug-\ngest the possibility ofthe self-induced vortex(SIV) state,\nwhich is considered to be realized when superconductiv-\nity occurs in the ferromagnetic state.19The condition for\nthe realization of the SIV state is Hc1<4πM < H c2,\nwhereMis the net magnetization of a ferromagnet.\nWhen this condition is satisfied, the vortices are gen-\nerated spontaneously. In this case, it is considered that\n1/T1intheSCstatehastwocomponentsarisingfromthe\nSIV and SC regions, respectively. Recently, we observed\ntwowell-separatedcomponentsof1 /T1intheregionclose\nto the SC vortices in LaRu 4P12.20\nAnother interesting possibility to be expected in\nUCoGe is the nonunitary spin-triplet state similar to\nthe A1 state in3He superfluidity, where the SC gap\nopens only in the up-spin band and not in the down-spin\nband.21In this case, it is considered that the Korringa\nbehavior with 1/4 of the magnitude of 1 /T1Tin the nor-\nmal state is anticipated to remain at low temperatures,\nwhich arises from the relaxation process occurring in the\ndown-spin intraband. This possibility has been discussed\nin UGe 2from the73Ge-NQR studies.8However, the ab-\nsence of such Korringa behavior in the low-temperature\nregion seems to exclude this possibility.\nIn conclusion, ferromagnetic and SC transitions were\nidentified at TCurie∼2.5 K and TS∼0.7 K in our\npolycrystalline sample. Ferromagnetic fluctuations thatJ. Phys. Soc. Jpn. Full Paper Author Name 5\npossess a quantum critical character are dominant above\nTCuriefrom the 1 /T1and Knight-shiftmeasurements. We\nfoundthat1 /T1equalto30%ofthe totalrelaxationcom-\nponent startstodecreasebelow TS, accompaniedbya T3\ndependence below 0.3 K, and that 1 /T1of the remaining\ncomponent shows a√\nTdependence, indicative of mag-\nnetic fluctuations even below TS. From the present NQR\nmeasurements in the SC state, we suggest that the self-\ninduced vortex state is realized in UCoGe.\nWe thank Y. Maeno and K. Yoshimura for experi-\nmental support and valuable discussions. We also thank\nN.Tateiwa,H.Kotegawa,Y.Itoh, C.Michioka,H. Ikeda,\nand S. Fujimoto for valuable discussions. This work was\npartiallysupportedbythe21stCOEprogramon“Center\nfor Diversity and Universality in Physics” by MEXT of\nJapan,and“Skutterudite”and “SuperClean”byMEXT\n(Nos.1827006,17071007)andGrants-in-AidforScientific\nResearch from the Japan Society for the Promotion of\nScience (JSPS)(No.18340102).\n1) S. S. Saxena et al.: Nature (London) 406(2000) 587.\n2) T. Akazawa et al.: J. Phys. Condens. Matter 16B (2004) L29.\n3) A. Huxley et al.: Phys. Rev. B 63(2001) 144519.4) D. Aoki et al.: Nature 413(2001) 613.\n5) N. T. Huy et al.: Phys. Rev. Lett. 99(2007) 067006.\n6) H. Kotegawa et al.: J. Phys. Soc. Jpn. 74(2005) 705.\n7) A. Harada et al.: J. Phys. Soc. Jpn. 74(2005) 2675.\n8) A. Harada et al.: Phys. Rev. B 75(2007) 140502.\n9) N. K. Sato: in preparation for publication.\n10) T. Ohta: in preparation for publication.\n11) M. Kontani, T. Hioki, and Y. Masuda: Solid.State. Commun .\n18(1976) 1251.\n12) M. Takigawa and H. Yasuoka: J. Phys. Soc. Jpn. 51(1982)\n787.\n13) K. Yoshimura et al.: J. Phys. Soc. Jpn. 53(1984) 503.\n14) T. Moriya and A. Kawabatra: J. Phys. Soc. Jpn. 34(1973)\n639.\n15) T. Moriya and A. Kawabatra: J. Phys. Soc. Jpn. 35(1973)\n669.\n16) A. Ishigaki and T. Moriya: J. Phys. Soc. Jpn. 65(1996) 3402.\n17) T. Moriya and T. Takimoto: J.Phys. Soc. Jpn. 64(1995) 960.\n18) For the calculation of the NQR frequencies affected by the\ninternal magnetic field, the effect of the internal field is\ntreated as perturbation. When the field was assumed to be\nH= (H0sinθcosφ,H0sinθsinφ,H0cosθ), the frequencies\nshown by the arrows are obtained by parameters of ( H0,θ,φ)\n= (400 Oe, 85◦, 45◦).\n19) M. Tachiki et al.: Solid State Commun. 34(1980) 19.\n20) Y. Nakai et al.: submitted to Phys. Rev. Lett.\n21) T. Ohmi and K. Machida: Phys. Rev. Lett. 71(1993) 625."    },    {        "title": "1601.03133v1.Spin_Vortex_Resonance_in_Non_planar_Ferromagnetic_Dots.pdf",        "content": " \n \n Spin Vortex Resonance in Non -planar Ferromagnetic Dots  \nJunjia Ding, Pavel Lapa,† Shikha Jain,‡ Trupti Khaire, Sergi Lendinez,  Wei Zhang, Matthias B. \nJungfleisch, Christian M. Posada, Volodymyr G. Yefremenko,§ John E. Pearson,  \nAxel Hoffmann, and Valentine  Novosad* \nMaterials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA \n  \n \nAbstract  \nIn planar structures, the vortex resonance frequency changes little as a function of an in -plane \nmagnetic field as long as the vortex state persists. Alte ring the topography of the element leads to \na vastly different dynamic response that arises due to the local vortex core confinement effect. In \nthis work, we studied the magnetic excitations in non -planar ferromagnetic dots using a \nbroadband microwave spec troscopy technique . Two distinct resonance frequency ranges were \nobserved depending on th e position of the vortex core  controllable by applying a r elatively small \nmagnetic field. The micromagnetic simulations are i n qualitative agreement with the.  \nexperime ntal results . \n                                                           \n† On leave from Texas A&M University, College Station, TX 77843, USA.  \n‡ Presently at HGST, A Western Digital Company \n§ Presently at High Energy Physics Division, Argonne National Laboratory.  \n* Correspondence and requests for materials should be addressed to V. N. (email: novosad@anl.gov)   \n \n Introduction  \nThe investigation of spin dynamics in geometrically confined ferromagnets is an important \nresearch topic due to both the technological and the basic science relevance. Some prominent \nexamples include the nanoscale spin textures with non -collinear magnetization  arrangement \nsuch as skyrmions ,1 domain walls ,2,3 and spin vortices.4 A vortex -type magnetization is \nenergetically favorable for micron and sub -micron disks with no crystalline anisotropy. The \nlong-range dipolar forces govern the spins dy namics in the vortex -state. Of particular interest is a \nlow-frequency excitation mode associated with the vortex core gyration . Besides the \nmagnetization of saturation, the vortex eigen frequencies depend on the dot thickness -to-diameter \nratio.5 The excitation spectrum depends only weakly on the value of an in -plane applied field \nbecause the external field is compensated by the demagnetizing field of the shifted vortex.6 \nFurthermore, it has been reported that the intri nsic pinning due to grain boundaries,7 surface \nroughness8 or exchange bias9 could also influence the vortex core gyration frequency. While the \nvast majority of the works on the dynamics o f spin vortices deal with the geometrically flat \nelements, here we have studied the circular elements with intentionally altered topography. This \nwas achieved by using a pre -patterned substrate with the characteristic lateral dimensions \nsignificantly small er that the diameter of the desired magnetic element. In our case, the \nnonmagnetic disks were first defined via lift -off process, and then covered by another patterned \nferromagnetic layer with the same geometric center but a larger diameter. As the result,  we \nobtained a “hat -like” structure of the sample, schematically depicted in Figure 1(a). Then the \ndynamic properties of these non -planar elements were systematically investigated using a  \n \n broadband microwave spectroscopy and micromagnetic modeling as a fun ction of their \ndimensions and amplitude of the external field. As we show below, the introduced disk -shaped \nstep (also referred as a vortex barrier) provides a strong geometric confinement and vortex core \npinning effect.   \n \nResults  \nElement geometry.  Two se ts of samples were fabricated. The first one is a reference sample, a \nflat Permalloy (Py, Ni 80Fe20 alloy) disk with diameter of 1 micron, and thickness of 50 nm. The \nsecond one is a non -planar, or an “engineered” dot, which is also circular in shape, with the same \ndimensions but has altered topography. Shown in Fig. 1(a) is a sketch of the cross -section profile \nof the engineered dot in which the magnetic material at the center has been lifted 25 nm by \npreparing a non -magnetic (Titanium) island prior to the deposition of the ferromagnetic material. \nFigures 1(b) and (c) show representative scanning electron microscope images of the non -planar \ndot with 200 nm diameter of the step and the reference dot. The step height Tb and the dot \nthickness TPy have been fixe d at 25 nm and 50 nm, respectively in all samples presented in this \npaper, while the inner diameter Db varied from 150 to 300 nm.  \nDynamic response.  In order to characterize the dynamic response of the samples, a microwave \ntransmission measurement has been  performed in a broad frequency range using a vector \nnetwork analyzer (VNA). Applying an external magnetic field Happ allowed us to probe how the  \n \n dynamic response of the vortex core changes once it has been displaced from the central area of \nthe disk.  \nFigure 1(d) compares the absorption spectra of the engineered dot taken at remanence ( Happ \n= 0), and in applied field ( Happ = -180 Oe) after saturating the sample with a –1.5 kOe external \nfield. Interestingly, for a higher value of applied field we detect the  resonance frequency of the \nvortex translation mode at ~0.4 GHz, while for the Happ = 0 Oe the frequency increases almost \n50%, to ~0.6 GHz. The same microwave measurements performed for a flat Ni 80Fe20 dot (e.g. \nthe reference sample, without vortex barrier ) are shown in Fig. 1(e). The frequency change when \nthe Happ is decreased from -180 Oe to 0 Oe is insignificant for the reference dot. A slightly higher \nfrequency for shifted vortex state is in fact expected. It originates from additional dipolar and \nexcha nge forces due the vortex spin structure deformation. The striking result is that while the \nresonance frequencies for the shifted vortex  states are the same for both samples, their values in \nremanence  differ significantly. This suggests that the dynamic re sponse of the engineered dot can \nbe fine -tuned by controlling the relative position of the vortex core with respect to the vortex \nbarrier using a relatively small magnetic field.  One could speculate that for the engineered dot \nsubjected to Happ = -180 Oe,  the vortex core is located outside the circumference of the barrier \nand thus it encounters the dipolar fields averaged within the outer boundary of the entire dot. As \nthe |Happ| value is reduced to zero, the vortex core shifts to the dot center and its gy ration is now \ngoverned by the dominant contribution from the magnetic charges confined within the barrier \nboundary. As a result, there appears the ~0.2 GHz frequency difference between these two core  \n \n positions. Obviously, this effect is not observed for th e reference dot (Fig. 1(d)) due to the lack of \nthe barrier.  \nTo further clarify the observed effect we fabricated and systematically investigated the dots \nwith variable barrier diameter. Figures 2 (b - e) are the representative absorption spectra taken at \nremanence ( Happ = 0 Oe) for the 50 nm thick, 1 micron diameter Py dots but with Db varied from \n150 to 300 nm. The vortex mode of all modified dots is characterized by the domination of one \npeak. The frequency of this main mode ( V0) decreases as the barrier  diameter is increased. This \ntrend is similar to the results of previous studies10 for planar Ni 80Fe20 disks where the resonance \nfrequency was reported to decrease with the increase in the dot diameter (for a fixed  thickness of \nthe element). The micromagnetic modeling yields similar results as it is discussed below. The 2D \nabsorption spectra (Fig. 2(a’ – e’)) also contain important information about the magnetization \nevolution process. Let us consider the reference dot ( Db = 0 nm, Fig. 2(a’)) and the dot with Db = \n200 nm (Fig. 2(c’)) as two examples. The vortex resonance corresponding to the translational \nmode appears at Happ ~250 Oe in both structures, indicating that the value of the vortex \nnucleation field is not affected by presence of the vortex barrier. Unlike to the almost field -\nindependent spectra of the reference dot, a discontinuous step -like frequency change is observed \nin the engineered dot. For instance, for Happ = -180 Oe and 0 Oe, the frequency differen ce is \n~200 MHz for the dot with Db = 200 nm (Fig. 1(d)), while it is only ~15 MHz for the reference \ndot (Fig. 1(e)). As it was mentioned before, this jump in the frequency is attributed to the \nchanges of the position of the vortex core. In a high magnetic field, when Happ is just below the \nnucleation field, the vortex core is located close to the outer edge. As the magnetic field Happ is  \n \n gradually decreased, the vortex core progressively displaces towards to the center of the dot. The \nintensity of the reson ance line in an engineered dot remains unchanged till Happ = -150 Oe. In the \nfield range from -150 Oe to -60 Oe the signal disappears (or is below our experimental \nsensitivity limit). We speculate that in this fields range the vortex gyration is significan tly \nsuppressed due to the pinning effect of the barrier edges.  (This vortex core pinning field range \nvaries as a function of the barrier diameter as the vertical line indicated in Fig. 2(b’ to e’).)  The \nresonance reappears again when Happ is decreased belo w -60 Oe suggesting that the vortex core \nis now fully inside the barrier circumference. Stronger dynamic dipolar fields of the barrier cause \nthe gyrotropic mode frequency shift to a much higher values. The frequency does not change in \nsmall positive fields  till the gyration stops when the core reaches the barrier edge again. With \nfurther magnetic field increase, the vortex core overcomes the barrier border and the low \nfrequency resonance line re -emerges again. Similarly, to the nucleation fields, the vortex  \nannihilations fields almost do not differ in the modified and reference dots. Thus, while we have \nseen that altering the dot topography has a profound impact on the low -field vortex core \ndynamics, its overall magnetostatic properties (e.g. the hysteresis loop) remain unaffected.  This \nis different from the case of dot -on-dot structure where bi -stable magnetic states were reported .11 \n  \nMicromagnetic modeling:  A systematic micromagnetic study has been performed to further \ninvestigate the static and dynamic response of the modified dots. The simulations confirm that in \nspite of such significant alteration to the topography o f the disk, the vortex state remains the  \n \n ground state for the system. Figures 3 (a, b) show the magnetization distribution in remanence \nfor the dot with Db = 200 nm and in an in -plane magnetic field Happ = -180 Oe.  \nA significant difference in the eigenfre quencies for these two distinct cases (e.g. the core \nlocated outside and inside of the barrier circumference) were also confirmed micromagnetically.. \nShown in Fig. 3(c) are the calculated relative energy profiles of the engineered dot (triangle \nsymbols) an d reference dot (square symbols) plotted as a function of the displacement of the \nvortex core. A clear difference between the two sets of results indicates the strong effect of the \nvortex confinement when the core is at the center of the element. The energ y profile vs the core \ndisplacement can be approximated as a parabolic function12 E(X) = E(0) +1/2X2, where  is the \neffective stiffness coefficient, E(0) is the energy at the equilibrium position and X is the vortex \ncore displacement. It should be emphasized that the dot self -induced magnetostatic energy of \nmoving vortex provides the dominant contrib ution to E(X). Using the simulation data shown in \nFig. 3(c), one can find the remanent values of stiffness coefficients ( ) as 0.9410-20 J/nm2 and \n1.710-20 J/nm2 for the reference and engineered dots, respectively. Since the frequency is \ndirectly proport ional to ,13 the vortex core gyrates faster when trapped inside the barrier. There \nappears no significant difference between the energy profiles for engineered (tri angle symbols) \nand reference (square symbols) dots when the vortex core is located outside the barrier ( Happ = -\n180 Oe), Figure 3(d). The asymmetry in the energy profile due the vortex structure deformation \nin the shifted state can be accounted by adding a  cubic term.6 \nAs the vortex core is very small, the dipolar forces originating from the dynamic magnetic \ncharges outside the vortex core govern its gyration. These cha rges can be calculated using a so - \n \n called “side -surface charges free” analytical model.14 Within this model, the magnetization \ndistribution of precessing vortex obeys boundary conditions such that there is no net \nmagnetization component perpendicular to the dot's lateral surface. Altering the topograph y that \nleads to formation of a step -like barrier will inevitably impose an additional requirement so the \nmagnetic “charges” on the barrier edges surfaces are minimized as well. Figure 3 (e) shows  \nrepresentative images of temporal changes in the divergence  of the magnetization for flat and \nengineered dots (upper and lower images, respectively). It is clear that the volume magnetostatic \ncharges contributing to the vortex dynamics and defining its eigenfrequency are well -distributed \nacross the dot for the ref erence sample, while their counterparts in the engineered dot are \npredominantly located in the central area circumscribed by the barrier edge. The same simulation \nwas also performed for Happ = -180 Oe as shown in Fig. 3(f). To our surprise, in this case th e \nspatial distribution of changes in the volume charges is almost identical for both samples \nresulting in a similar vortex gyration frequencies.   \nFinally, to further understand the effect of the vortex barrier to the translational mode \nfrequency, systemat ic simulations have been performed as a function of the barrier size. Shown \nin Fig. 4(a) is the summary of the experimental (square symbols) and micromagnetic (solid line) \nfrequencies as a function of Db. While there is a noticeable quantitative discrepanc y between the \nexperimental and computational results, they are in a good qualitative agreement. Figure 4(b) \nshows the simulated frequency plotted as a function of barrier thickness Tb. The frequency \ncontinuously increases with Tb, it almost doubles in comp arison to the reference dot for Tb = 40 \nnm. These results demonstrate that the vortex resonance frequency can be effectively controlled  \n \n by adjusting the barrier geometry. Interestingly, the low field experimental and micromagnetic \nfrequencies for engineere d dots shown in Fig. 3(a, b) scale universally when replotted as a \nfunction the barrier geometric aspect ratio Tb/Db. This is similar to how the translational mode \nfrequency in flat disks scales as a function of the disk thickness to diameter ratio.10  \n \nSummary  \nA nonmagnetic nanodot inserted under a mesoscale Ni 80Fe20 dot was shown to provide a \ngeometric confinement effect causing the changes in the vortex translational mode frequency. \nTwo distinct resonance frequen cy ranges were observed depending on the position of the vortex \ncore (inside or outside of the barrier) controllable by applying a relatively small magnetic field. \nBy comparing the experimental data and micromagnetic simulations if was found that the \nfrequ ency of the gyrotropic mode increases as the thickness -diameter ratio of the barrier is \nincreased. Further studies of such non -planar ferromagnetic  elements will be focused on the \ndetails of the pinning mechanism, its possible impact on the energetics of t he vortex core reversal \nprocess and the high frequency spin dynamics.  \n \nMethods  \nSample fabrication.  The engineered dots were fabricated using a multistep electron -beam (EBL) \nlithography process. First, the disk arrays with diameter in a range of 150 nm to 3 00 nm and \nalignment marks were defined on polymethyl methacrylate (PMMA) resist, accompanied by e - \n \n beam evaporation and lift -off process of a 25 -nm-thick titanium film. The second step EBL \npatterning of 1 -micron diameter disks followed by deposition of 50 -nm-thick Ni 80Fe20 and lift -\noff completes the fabrication process. The barriers and the disk are concentric as is confirmed by \nScanning Electron Microscopy imaging.  \n \nSpectral measurements.  In order to characterize the dynamic properties of the samples, a \ncoplanar waveguide (CPW) with a 3 μm -wide -signal line was fabricated on top of each dot array \nusing optical lithography followed by a Ti(5 nm)/Au(150 nm) sputter deposition and a lift -off \nprocess. Microwave transmission measurements have been performed in th e 0.05 ~ 10 GHz \nfrequency range using a broadband microwave vector network analyzer (VNA). The microwave \ntransmission was measured by sweeping the frequency for fixed magnetic field. Applying an in \nplane magnetic field allowed us to probe how the dynamic r esponse of the vortex core changes \nonce it has been displaced from the central area of the disks. Since the focus of this paper is the \nfundamental gyrotropic vortex mode, all results are presented in the field range of -500 ~ 500 Oe \nand frequency range of 0.05 ~ 1.0 GHz . Prior to the magnetic field sweep, the samples were \nmagnetized at the 1.5 kOe field.  \n \nMicromagnetic simulation details.  Systematic micromagnetic modeling was performed using \nmumax3 code .15 Typical parameters for Ni 80Fe20 (the saturation magnetization Ms = 700 kA/m, \nthe exchange constant A = 13×10-12 J/m, the damping factor 0.01, the gyromagnetic ratio γ = 2.8  \n \n GHz/kOe, negligible magnetocrystalline anisotropy) and 5 ×5×5 nm3 cell size  were used in the \nsimulation.   \n \n References:  \n1. Jiang, W. et al.  Blowing magnetic skyrmion bubbles. Science (80 -. ). 349, 283–286 (2015).  \n2. Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic Domain -Wall Racetrack Memory. \nScience (80 -. ). 320, 190–194 (2008).  \n3. Parkin, S. & Yang, S. -H. Memory on the racetrack. Nat. Nanotechnol.  10, 195–198 (2015).  \n4. Guslienko, K. Y., Kakazei, G. N., Ding, J., Liu, X. M. & Adeyeye, A. O. Giant moving \nvortex mass in thick magnetic nanodots. Nat. Publ. Gr.  1–8 (2015). \ndoi:10.1038/srep13881  \n5. Novosad, V. et al.  Magnetic vortex resonance in patterned ferromagnetic dots. Phys. Rev. \nB 72, 24455 (2005).  \n6. Buchanan, K. S. et al.  Magnetic -field tunability of the vortex translational mode in \nmicron -sized permalloy ellipses: Experiment and micromagnetic modeling. Phys. Rev. B  \n74, 064404 (2006).  \n7. Compton, R. L. & Crowell, P. a. Dynamics of a pinned magnetic  vortex. Phys. Rev. Lett.  \n97, 137202 (2006).  \n8. Compton, R. L., Chen, T. Y. & Crowell, P. a. Magnetic vortex dynamics in the presence \nof pinning. Phys. Rev. B  81, 144412 (2010).  \n9. Sort, J. et al.  Imprinting Vortices into Antiferromagnets. Phys. Rev. Lett.  97, 067201 \n(2006).  \n10. Guslienko, K., Novosad, V., Otani, Y., Shima, H. & Fukamichi, K. Magnetization reversal \ndue to vortex nucleation, displacement, and annihilation in submicron ferromagnetic dot \narrays. Phys. Rev. B  65, 024414 (2001).  \n11. Stebliy, M. E. et al.  High -frequency switching of magnetic bistability in an asymmetric \ndouble disk nanostructure. Appl. Phys. Lett.  104, 112405 (2014).  \n12. Guslienko, K. Y. et al.  Eigenfrequencies of vortex state excitations in magnetic \nsubmicron -size disks. J. Appl.  Phys.  91, 8037 (2002).  \n13. Guslienko, K. Y., Novosad, V., Otani, Y., Shima, H. & Fukamichi, K. Field evolution of \nmagnetic vortex state in ferromagnetic disks. Appl. Phys. Lett.  78, 3848 –3850 (2001).  \n14. Metlov, K. L. & Guslienko, K. Y. Stability of magne tic vortex in soft magnetic nano -sized \ncircular cylinder. J. Magn. Magn. Mater.  242-245, 1015 –1017 (2001).  \n15. Vansteenkiste, A. et al.  The design and verification of MuMax3. AIP Adv.  4, 107133 \n(2014).  \n  \n \n Acknowledgements  \nThis work was supported by the U. S . Department of Energy (DOE), Office of Science, Basic \nEnergy Sciences (BES), under Award # DE -AC02 -06CH11357.   \n \n Figure Captions  \nFig. 1 : (a) A sketch of the dot with Db = 200 nm with the simulated remnant magnetization state \nof the engineered dot. The x -component of the magnetization is represented  from red to blue in a \nrange of +1 to -1. Scanning electron micrographs of the dot with (b) 200 nm barrier diameter ( Db) \nand (c) reference dot. (d) and (e) shows the experimental FMR curves with Happ = -180 Oe and  0 \nOe for Db = 200 nm and 0 nm, respectively  \n \nFig. 2 : (a-e) The experimental remanent FMR absorption curves of the dots with Db = 0 nm \n(reference dot), 150 nm, 200 nm, 250 nm and 300 nm. The corresponding 2D FMR absorptions \nspectrums are shown in (a ’ to e’ ). The vertical lines in (b’ to e’) indicate the field range that \nvortex core jump is pinned at the barrier.  \n \nFig. 3 : The simulated magnetization state (central layer) of the dot with Db = 200 nm for Happ = 0 \nOe (a) and -180 Oe (b). (c) and (d) shows the plots of the simulated energy profile of both the \nreference dot and the engineered dot as a function of the core displacement for Happ = 0 Oe and -\n180 Oe. Solid lines are the fitting results of the profile  with the parabolic function. The simulated \nmagnetic charge density ( M) changes when the core is displaced from the equilibrium position  \n(when it is performing gyrotropic movement) for the reference dot and the engineered dot with \nHapp = 0 Oe (e) and -180 Oe (f). The color code is in log scale.   \n \n  \nFig. 4 : (a) and (b) shows the simulated and experimental results of the gyrotropic mode \nfrequency of the modified dots as a function of the barrier diameter ( Db) and the barrier thickness \n(Tb), respectively. All results have been reformatted and plotted as a function of the barrier  \naspect ratio ( Tb/Db) as shown in Fig. 4(c).  \n  \n \n \nFig. 1\n200 nmDb= 200 nmD= 1000 nm (b)\n(a)\nTb= 25 nm TPy= 50 nm\n  \n0.2 0.4 0.6 0.8 1.0\n  \nHapp= 0 OeHapp= -180 Oe(d)Absorpation (arb. unit)\n  \n0.2 0.4 0.6 0.8 1.0\n  \nHapp= 0 OeHapp= -180 Oe(e)\nFrequency ( GHz)Absorpation (arb. unit)Db= 0 nmDb= 200 nm\nV0V1\nV1\nV1\n200 nmD= 1000 nm (c) \n  \n \n \nFig. 2Microwave Absorpation (arb. unit)\n  (b)\n  (a)\n  \n(c)\n  \n(d)\n0.2 0.4 0.6 0.8 1.0\n  \n(e)Db= 150 nm\nDb= 200 nm\nDb= 250 nm\nDb= 300 nmV0V1\nHapp(Oe)\n0.20.40.60.81.0\n  \n  Db= 150 nm(b’)\n0.20.40.60.81.0\n  \n  \nDb= 200 nm(c’)\n0.20.40.60.81.0\n  \n  \nDb= 250 nm(d’)\n-500 -250 0 250 5000.20.40.60.81.0\n  \n  \nDb= 300 nm(e’)Frequency (GHz)\n0.20.40.60.81.0\n  \n  Db= 0 nm(a’) Db= 0 nm\nReference DotReference Dot\nFrequency ( GHz) \n  \n \n \nFig. 3\n(a)Happ= 0 Oe\n(b)Happ= -180 Oe\n-15 -10 -5 0 5 10 15-50510152025\n  \nCore Displacement ( X, nm)(d)ΔEnergy (10-17J)\n-15 -10 -5 0 5 10 15-50510152025\n  (c)ΔEnergy (10-17J)\nCore Displacement ( X, nm)\n(f)\n Ref. Dot Db= 200 nm Ref. Dot Db= 200 nm\nhigh\nlow(e)\nRef. Dot\nDb= 200 nm\nRef. Dot\nDb= 200 nm \n  \n \n \nFig. 4\n100 200 300 4000.30.40.50.60.70.8\n Vortex Barrier Diameter ( Db, nm)Frequency (GHz)Simulation Results\nExperimental Results(a)\nTb= 25 nm\nHapp= 0 Oe\n0 10 20 30 40\n \nVortex Barrier Thickness ( Tb, nm)(b) Simulation Results\nDb= 200 nm\nHapp= 0 Oe\nDbTb\n0.0 0.1 0.2 0.3\n \nBarrier Aspect Ratio ( Tb/Db)(c)\nFixed Db= 200 nm\nFixed Tb= 25 nm\nExperimental Results "    },    {        "title": "2303.05175v2.Antiferromagnetic_resonances_in_superconductor_ferromagnet_multilayers.pdf",        "content": "arXiv:2303.05175v2  [cond-mat.supr-con]  24 Mar 2023Antiferromagnetic resonances in superconductor-ferroma gnet multilayers\nI. A. Golovchanskiy1,2,3,∗, V. V. Ryazanov1,2,4, V. S. Stolyarov1,2,3\n1Moscow Institute of Physics and Technology, State Universi ty,\n9 Institutskiy per., Dolgoprudny, Moscow Region, 141700,\nRussia;2National University of Science and Technology MISIS, 4 Leni nsky prosp.,\nMoscow, 119049, Russia;3Dukhov Research Institute of Automatics (VNIIA),\n127055 Moscow, Russia;4Institute of Solid State Physics (ISSP RAS),\nChernogolovka, 142432, Moscow region, Russia.\nIn this work, we study magnetization dynamics in supercondu ctor-ferromagnet (S-F) thin-film\nmultilayer. Theoretical considerations supported by the b road-band ferromagnetic resonance spec-\ntroscopy reveal development of acoustic and optic resonanc e modes in S-F multilayers at signifi-\ncantly higher frequencies in comparison to the Kittel mode o f individual F-layers. These modes are\nformed due to antiferromagnetic-like interaction between F-layers via shared circulating supercon-\nducting currents in S-layers . The gap between resonance mod es is determined by the thickness\nand superconducting penetration depth in S-layers. Overal l, rich spectrum of S-F multilayers and\nits tunability opens wide prospects for application of thes e multialyers in magnonics as well as in\nvarious superconducting hybrid systems.\nIntroduction. Hybridization of antagonistic super-\nconducting (S) and ferromagnetic (F) orders offers in\nelectronics and spintronics, which have been repeatedly\ndemonstratedinpastdecades[1]. Recentlytheinterestin\nS-F hybridization has been reinforced by demonstrations\nof its prospects in relation with the magnetization dy-\nnamics phenomena. In particular, interactions between\nmagnetization dynamics and the superconducting vortex\nlattice allow to form and guide the tunable magnonic\nband structure [2], as well as to induce exchange spin\nwaves by the DC electric current [3]. Also, interactions\nbetween magnetization dynamics and superconducting\nMeissner currents in hybrid structures modifies the spin-\nwave dispersion [4, 5], which can be used for creation\nof magnonic crystals [6] or for gating magnon currents\n[7]. Remarkably, low speed of electromagnetic propa-\ngation in superconductor-insulator-superconductor thin-\nfilm structures facilitates achievement of the ultra-strong\nphoton-to-magnon coupling in on-chip hybrid devices\n[8, 9] aiming forthe photon-to-magnonentanglement[10].\nA new strong phenomenon in S-F hybrid structures\nwas reported recently in Refs. [11–13] and investigated\nfurther in Refs. [14, 15]. In superconductor-ferromagnet-\nsuperconductor (S-F-S) thin-film structures in the pres-\nence of electronic interaction between superconducting\nand ferromagnetic layers a radical increase in the ferro-\nmagnetic resonance (FMR) frequency occurs. The mech-\nanism behind the phenomenon constitutes a formation\nof one-dimensional superconducting torque via the in-\nterplay between the superconducting imaginary conduc-\ntance and magnetization precession at S-F interfaces,\nwhich result in induction ofalternatingcirculatingsuper-\nconducting currents in the opposite phase to the magne-\ntization precession.\nIn this work, we generalize the problem and consider\nmagnetization dynamics in arbitrary S-F multilayers.\nCouplingbetweenferromagneticlayersviasuperconduct-\ning currents allows to induce antiferromagnetic-like in-\nteraction between F-layers, which result in acoustic andoptic resonances modes. Theoretical considerations sup-\nported by the broadband ferromagnetic resonance spec-\ntroscopy demonstrate that the spectrum is determined\nby geometrical characteristics of a multilayer as well as\nby the superconducting penetration depth in S-layers.\nTheory. Following Refs. [15, 16], electrodynamics and\nmagnetizationdynamicsinS-Fmultilayersobeysconven-\ntional Maxwell equations supplemented by the Ohm law\nwith imaginary conductance in superconducting layers\nand by the Polder susceptibility in ferromagnetic layers.\nBy neglecting edge effects, the y-component of magnetic\nfield as well as x-components of electric field and of the\ncurrent are functions of the transverse coordinate zonly\n(see Fig. 1). Derivation of the magnetic field in S- and\nF-layers from initial Maxwell equations yields following\ngeneral expressions\nHS\ny(z) =Aiexpz\nλS+Biexp−z\nλS,\nHF\ny(z) =Ciexpz\nλF+Diexp−z\nλF,(1)\nwhere the subscript of coefficients ispecifies the su-\nperconducting layer or the ferromagnetic layer in the\nstack,λSis the superconducting penetration depth, and\nλF=δFΩ is the ferromagnetic penetration depth, δF=/radicalbig\ni/µ0ωσFis the conventional electromagnetic penetra-\ntion depth into a metal with conductivity σF(typically,\nin permalloy σF∼106Ohm−1m−1). The characteristic\ndimensionless frequency Ω is given by\nΩ2=γ2(H+Ha)(H+Ha+Meff)−ω2\nγ2(H+Ha+Meff)2−ω2,(2)\nwhereHis the external field (aligned with the x-axis\nin Fig. 1),Hais the effective field of the uniaxial\nanisotropy aligned with the external field, and Meff=\nMs−2Ku/µ0Msis the effective magnetization, which ac-\ncounts the out-of-plane uniaxial anisotropy with the con-\nstantKu. Notice that the conventional Kittel formula2\nfor the ferromagnetic resonance in thin films in these no-\ntations is provided by Ω = 0. At every S/F interface the\nfollowing boundary conditions are fulfilled\nHS\ny=HF\ny,\n1\nσSdHS\ny\ndz=1\nσFdHF\ny\ndz,(3)\nwhereσS=i/µ0ωλ2\nSis the imaginary conductance in\nsuperconducting layers.\nFIG. 1. Schematic illustration of the interplay between ac\nmagnetic field, magnetization precession and superconduct -\ning currents in S-F-S trilayer. Magnetization precession ( /vectorM,\nblack arrow) at S-F interfaces induces macroscopic superco n-\nducting currents alternating in S-layers along the x-direction\n(JS\nx, blue arrows). These currents form the magnetic field HS\ny\nin the F-layer along the y-direction in opposite phase to the\nprecession of /vectorM.\nFor the S-F-S trilayer, depicted schematicallyin Fig. 1,\nsolution of Eq. 3together with natural boundary condi-\ntions at outer surfaces of S-layers, namely, HS\ny= 0, and\nin the limit dF≪λFyields the following expression for\nferromagnetic resonance frequency:\nΩ2=−dF\nλStanhdS1/λStanhdS2/λS\ntanhdS1/λS+tanhdS2/λS.(4)\nAs example, black and red curves in Fig. 2compares the\ndependence of the resonance frequency on the magnetic\nfieldfr(H), respectively, in a single F-layer and in S-F-\nS trilayer with the following thicknesses: dF1= 50 nm,\ndS1= 150 nm, dS2= 100 nm, λS= 100 nm.\nApplication of the same derivation approach for the\nsymmetric S-F-S-F-S multilayer, depicted schematically\nin Fig.3, yields two resonance modes with the in-phase\n(acoustic mode) and the anti-phase (optic mode) preces-\nsion of ferromagnetic layers, respectively,\nΩ2\na=−dF\nλStanhdSi/λScothdSe/2λS\ntanhdSi/λS+cothdSe/2λS,\nΩ2\no=−dF\nλStanhdSi/λStanhdSe/2λS\ntanhdSi/λS+tanhdSe/2λS,(5)\nwhereSedenotes external superconducting layers (S1\nand S3 in Fig. 3),Sicorresponds to the internal super-\nconducting layer (S2 in Fig. 3), and thicknesses of both\nferromagneticlayersis consideredequal dF1=dF2=dF.\nBlue curves in Fig. 2show acoustic and optic resonance\ncurvesfr(H) in the multilayer with the same thicknesses/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54 /s48/s46/s48/s57 /s48/s46/s49/s50 /s48/s46/s49/s53 /s48/s46/s49/s56 /s48/s46/s50/s49/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102\n/s114/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84\nFIG. 2. Theoretical dependencies of the resonance frequenc y\non the magnetic field fr(H) in conventional F-layer (solid\nblack curve, Kittel formula), S-F-S trilayer (solid red cur ve,\nEq. 4) and symmetric S-F-S-F-S multilayer (solid and dashed\nblue curves, Eq. 5). The following parameters are used for\ncalculations: γ/2π= 29.5 GHz/T, dF1=dF2= 50 nm,\ndS1=dS3=dSe= 150 nm, dS2=dSi= 100 nm,\nλS= 100 nm, µ0Meff= 1 T,Ha= 0.\nas in a single F-layer and in S-F-S trilayer : dF1=dF2=\ndF= 50 nm, dS1=dS3=dSe= 150 nm, dSi= 100 nm,\nλS= 100 nm.\nFIG. 3. Schematic illustration of the interplay between ac\nmagnetic field, magnetization precession andsuperconduct ing\ncurrents in symmetric S-F-S-F-S multilayer. Acoustic (a) a nd\noptic (b) modes a formed.\nIt should be noticed that the acoustic mode (Fig. 3a)\ncan be thought as being formed by the global supercon-\nducting current, which circulates in external supercon-\nducting layers S1 and S3, while the internal S-layer S2\nonly screens the induced magnetic field in conventional\nmanner. In fact, in the limit dSe→ ∞the expression\nfor Ωain Eq.5meets Ω in Eq. 4with the substitution\ndF→2dF. The optic mode (Fig. 3b) can be thought as\nthe resonance in S-F-S trilayer with reduced thickness of\ninternal S-layer: the expression for Ω oin Eq.5meets Ω3\n/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54 /s48/s46/s48/s57 /s48/s46/s49/s50 /s48/s46/s49/s53 /s48/s46/s49/s56 /s48/s46/s50/s49/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s45 /s48/s46/s48/s49/s48\n/s45 /s48/s46/s48/s48/s54\n/s45 /s48/s46/s48/s48/s50\n/s48/s46/s48/s48/s50\n/s48/s46/s48/s48/s54\n/s48/s46/s48/s49/s48\n/s40/s97/s41/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54 /s48/s46/s48/s57 /s48/s46/s49/s50 /s48/s46/s49/s53 /s48/s46/s49/s56 /s48/s46/s50/s49/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s40/s98/s41/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s45 /s48/s46/s48/s49/s48\n/s45 /s48/s46/s48/s48/s54\n/s45 /s48/s46/s48/s48/s50\n/s48/s46/s48/s48/s50\n/s48/s46/s48/s48/s54\n/s48/s46/s48/s49/s48\n/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54 /s48/s46/s48/s57 /s48/s46/s49/s50 /s48/s46/s49/s53 /s48/s46/s49/s56 /s48/s46/s50/s49/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s40/s99/s41/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s45 /s48/s46/s48/s53\n/s45 /s48/s46/s48/s51\n/s45 /s48/s46/s48/s49\n/s48/s46/s48/s49\n/s48/s46/s48/s51\n/s48/s46/s48/s53\n/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54 /s48/s46/s48/s57 /s48/s46/s49/s50 /s48/s46/s49/s53 /s48/s46/s49/s56 /s48/s46/s50/s49/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s40/s100/s41/s32/s79/s112/s116/s105/s99/s32/s109/s111/s100/s101/s44/s32/s101/s120/s112\n/s32/s79/s112/s116/s105/s99/s32/s109/s111/s100/s101/s44/s32/s115/s105/s109\n/s32/s65/s99/s111/s117/s115/s116/s105/s99/s32/s109/s111/s100/s101/s44/s32/s101/s120/s112\n/s32/s65/s99/s111/s117/s115/s116/s105/s99/s32/s109/s111/s100/s101/s44/s32/s115/s105/s109/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54 /s48/s46/s48/s57 /s48/s46/s49/s50 /s48/s46/s49/s53 /s48/s46/s49/s56 /s48/s46/s50/s49/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s32/s79/s112/s116/s105/s99/s32/s109/s111/s100/s101/s44/s32/s101/s120/s112\n/s32/s79/s112/s116/s105/s99/s32/s109/s111/s100/s101/s44/s32/s115/s105/s109\n/s32/s65/s99/s111/s117/s115/s116/s105/s99/s32/s109/s111/s100/s101/s32/s49/s44/s32/s101/s120/s112\n/s32/s65/s99/s111/s117/s115/s116/s105/s99/s32/s109/s111/s100/s101/s32/s49/s44/s32/s115/s105/s109\n/s32/s65/s99/s111/s117/s115/s116/s105/s99/s32/s109/s111/s100/s101/s32/s50/s44/s32/s101/s120/s112\n/s32/s65/s99/s111/s117/s115/s116/s105/s99/s32/s109/s111/s100/s101/s32/s49/s44/s32/s115/s105/s109/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84 /s40/s101/s41\nFIG. 4. a-c) Differentiated transmission spectra dS21/dH(f,H) for samples SF1 (a) and SF2 (b) at temperature 2 K and\nfor sample SF1 at 8 K (c). Colour codes are provided in insets. d,f) Experimental (symbols) and theoretical (solid curves )\nresonance lines for samples SF1 (a) and SF2 (b) at temperatur e 2 K.\nin Eq.4with the substitution dS2=dSi/2. Qualitatively\nit can be concluded that the optic mode is unaffected by\ntheinteractionbetweenF-layerslayers,whiletheacoustic\nmodegainsenergyduetothecoupling. Interestingly,this\nqualitative picture is in direct contradiction with magne-\ntizationdynamicsinexchange-coupledferromagneticlay-\ners [17–21], where regardless the details of the exchange\ninteraction the acoustic mode corresponds to magnetiza-\ntion dynamics in non-interacting magnetic layers. Also,\naccording to Fig. 2and Eq.5the optic mode is observed\nat lower frequencies in comparison to the acoustic mode,\nwhich characterises the coupling between ferromagnetic\nlayers via superconducting layers as antiferromagnetic.\nIn a way, such interaction between ferromagnetic lay-\ners via superconducting currents in adjacent layers re-\nminds interaction of fluxons in superconductor-insulator\nJosephson junction stacks [22–24].\nIn the general case, resonance modes of an arbitrary S-\nF multilayer, whichconsistof Nferromagneticlayersand\nN+1 superconducting layers, can be derived numerically\nfrom the set of equations 3in the matrix form, [ M]×\n[An,Bn,Cn,Dn]T= 0, by finding frequencies ωrthat\nobey the expression\ndet[M(ωr)] = 0. (6)\nExperimental details and results. Experimentallymag-\nnetization dynamicsin S-F multilayersisstudied bymea-\nsuring the ferromagnetic resonance absorption spectrum\nusing the VNA-FMR approach[25–27] and the samechip\nlayout and experimental setup as in Refs [13, 14]. A\nseries of niobium-permalloy(Py=Fe 20Ni80)-niobium (Nb-\nPy) multilayered structures are placed directly on topof the central transmission line of superconducting Nb\nwaveguide. Deposition of Nb-Py multilayers is per-\nformed in a single vacuum cycle ensuring the electron\ntransparency at Nb-Py interfaces. Thin Si or AlO x\nspacing layer is deposited between Nb co-planar waveg-\nuide and the multialyers in order to ensure electrical\ninsulation of the studied samples from the waveguide.\nTwo test samples have been studied: a sample with\ntwo ferromagnetic layers, that consist of Nb-101nm/Py-\n11nm/Nb-41nm/Py-11nm/Nb-41nm, referred to as SF2\nand a sample with three ferromagnetic layers, that\nconsist of Nb-101nm/Py-11nm/Nb-40nm/Py-11nm/Nb-\n40nm/Py-12nm/Nb-41nm, referred to as SF3 . The SF2\nsample is made asymmetric on purpose in order to pro-\nvide a finite dynamic susceptibility of the optic mode,\nwhich otherwise is zero and, thus, does not couple to uni-\nform microwave magnetic field of the transmission line.\nMicrowave spectroscopy of samples was performed\nby measuring the transmission characteristics S21(f,H)\nin the closed-cycle cryostat Oxford Instruments Triton\n(base temperature 1.2 K) equipped with the home-made\nsuperconducting solenoid. Spectroscopy was performed\nin the field range from -0.22 T to 0.22 T, in the fre-\nquency range from 0 up to 20 GHz, and in the tempera-\nture range from 2 to 11 K. Magnetic field was applied in-\nplane along the direction of the waveguide(see Ref. [14]).\nFMR spectra at different temperatures were analysed by\nfittingS21(f) characteristics at specified HandTwith\nthe Lorentz curve and, thus, obtaining the dependencies\nof the resonance frequency on magnetic field fr(H).\nFigure4a-c demonstrates the essence of the studied\nphenomenon: at temperatures below the critical tem-4\nperature of Nb, T < T c, the transmission spectrum for\nSF2 sample consist of two spectral lines (Fig. 4a) and\nfor SF3 sample consist of three spectral lines (Fig. 4b).\nAtT > T c(Fig.4c) FMR spectrum for both samples is\nreduced to a single spectral line, which obeys the con-\nventional Kittel formula (Eq. 2, Ω = 0). For both sam-\nples the fit of FMR curves at T > T cyields negligible\nanisotropy field µ0Ha≈2 mT, the effective magnetiza-\ntionµ0Meff≈1.108T, which is close to typical values of\nthe saturation magnetization of permalloy µ0Ms≈1 T,\nand no noticeable dependence of HaandMeffon tem-\nperature. Temperature dependencies of FMR spectra\nfor both samples yield superconducting critical temper-\naturesTc= 7.7 K for SF2 sample and Tc= 7.9 K for\nSF3 sample. The critical temperature of Nb layers is re-\nduced in comparison to the bulk critical temperature of\nNbTc≈9 K owing to the inverse proximity effect [28].\nAtT < T c(Fig.4a,b) FMR spectrum shifts to higher\nfrequencies and splits to spectral lines in accordance to\nthe number of F-layers in the stack. The strongest line,\nthe acoustic mode, is observed at the highest frequencies,\nwhile weaker lines at lower frequencies correspond to op-\ntic modes. Resonance lines were modelled with Eq. 6\nusingλSas the fitting parameter (see Fig. 4d,e). The\noptimum fit is obtained with λS= 115 nm for SF2 sam-\nple andλS= 98 nm for SF3 sample. The obtained λS\nis slightly higher than typical values in bulk Nb (about80 nm) due to the inverse proximity effect [15, 28]. A\nbetter fit could be obtained by considering a variation of\nλSin different S-layers. Thus, the provided theoretical\ndescription of the magnetization dynamics phenomenon\nin arbitrary S-F multilayers is verified.\nConclusion. Summarising,wereportastudyofmagne-\ntization dynamics in S-F multialyers. Theoreticalconsid-\nerations supported by experiments in a wide frequency,\nfield, and temperature ranges show that the coupling\nbetween ferromagnetic layers via superconducting layers\nresults in formation of antiferromagnetic interaction be-\ntween F-layers with the strength that depends of thick-\nness and superconducting properties of S-layers. This in-\nteraction between ferromagnetic layers is formed via su-\nperconducting currents and result in formation of acous-\ntic and optic spectral branches. These results open wide\nprospects for application of S-F multialyers in magnon-\nics and also bridges magnetization dynamics phenomena\nwith various superconducting circuits [29–31], hybrid de-\nvices [8, 9], and metamaterials [32]. Moreover, resonance\nproperties of S-F multilayers by changing the supercon-\nducting state of S-layers optically[15, 33, 34] or via elec-\ntric currents.\nAcknowledgements The authors acknowledge Dr M.\nSilaev for fruitful discussions. The research study was\nfinancially supported by the Russian Science Foundation\n(grant N 22-22-00314).\n[1]J. Linder and J. W. A. Robinson, Superconducting spin-\ntronics, Nat. Phys. 11, 307 (2015).\n[2]O. V. Dobrovolskiy, R. Sachser, T. Bracher, T. Fischer,\nV. V. Kruglyak, R. V. Vovk, V. A. Shklovskij, M. Huth,\nB. Hillebrands, and A. V. Chumak, Magnon-fluxon inter-\naction in a ferromagnet/superconductor heterostructure,\nNat. Phys. 15, 477 (2019).\n[3]O. V. Dobrovolskiy, Q. Wang, D. Y. Vodolazov,\nB. Budinska, R. Sachser, A. V. Chumak, M. Huth, and\nA. I. Buzdin, arXiv , 2103.10156 (2021).\n[4]I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nV. V. Bolginov, V. V. Ryazanov, A. A. Golubov, and\nA. V. Ustinov, Adv. Funct. Mater. 28, 1802375 (2018).\n[5]I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nV.V.Ryazanov, A.A.Golubov, andA.V.Ustinov,Mod-\nified dispersion law for spin waves coupled to a supercon-\nductor, J. Appl. Phys. 124, 233903 (2018).\n[6]I. A. Golovchanskiy, N. N. Abramov, V. S. Stol-\nyarov, P. S. Dzhumaev, O. V. Emelyanova, A. A.\nGolubov, V. V. Ryazanov, and A. V. Ustinov, Ferro-\nmagnet/superconductor hybridmagnonic metamaterials,\nAdv. Sci. 6, 1900435 (2019).\n[7]T. Yu and G. E. W. Bauer, Phys. Rev. Lett. 129, 117201\n(2022).\n[8]I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nM. 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Lett. 114, 142601\n(2019)."    },    {        "title": "0906.4863v1.Conductance_oscillation_due_to_the_geometrical_resonance_in_FNS_double_junctions.pdf",        "content": "arXiv:0906.4863v1  [cond-mat.mes-hall]  26 Jun 2009Conductance oscillation due to the geometrical resonance i nFNSdouble junctions\nHiroyuki Ohtori1,2, Hiroshi Imamura2\n1Institute of Applied Physics, University of Tsukuba, Tsuku ba 305-8573, Japan\n2NRI-AIST, Central 2, 1-1-1 Umezono, Tsukuba 305-8568, Japa n\nWe theoretically analyzed the Andreev reflection in ferroma gnetic metal / nonmagnetic metal\n/ superconductor double junctions with special attention t o the electron interference effect in the\nnonmagnetic metal layer. We showed that the conductance osc illates as a function of the bias\nvoltage due to the geometrical resonance. We found that the e xchange field and therefore the\nspin polarization of the ferromagnetic metal can be determi ned from the period of the conductance\noscillation, which is proportional to the square-root of th e exchange field.\nRecently much attention has been focused on the An-\ndreev reflection (AR) in ferromagnetic metal (FM) / su-\nperconductor (SC) contacts1,2,3,4,5,6,7,8,9,10since the spin\npolarization of conduction electrons is measured through\nthe suppression of the conductance below the supercon-\nducting gap. This method is called point contact An-\ndreev reflection (PCAR) spectroscopy.\nOn the other hand, the quasiparticle (QP) interfer-\nence in nonmagnetic metal(NM) / SC junctions has\nbeen extensively studied in the past11,12,13,14,15,16,17,18.\nAs shown in Refs.12,13,14, the interference of QPs in\nthe SC layer brings about the oscillation of the den-\nsity of states against the QP energy, which is called a\nTomasch oscillation. The interference in the NM layer\nalso brings about the oscillation of the density of states\nin the NM layer11,15, which is known as the deGennes-\nSaint-James bound state or the McMillan-Rowell oscilla-\ntion. NesherandKorenmeasuredthedynamic resistance\nof YBa 2Cu3O6.6/ YBa 2Cu2.55Fe0.45Oy/ YBa 2Cu3O6.6\njunctions and determined the renormalized Fermi veloc-\nity of QPs in the YBa 2Cu2.55Fe0.45Oylayer from the pe-\nriod of the McMillan-Rowell oscillation16.\nIn this paper, we theoretically analyze the Andreev re-\nflection in a FM/NM/SC double junction system with\nspecial attention to the electron interference effect in the\nNM layer. Following the work of Blonder, Tinkham, and\nKlapwijk (BTK)19, we solve the Bogoliubov-de Gennes\n(BdG)20equations and calculate the conductance. We\nshow that the conductance due to the Andreev reflection\noscillates as a function of the bias voltage because of the\ngeometrical resonance predicted by deGennes and Saint-\nJames. We obtain the analytical expression of the prob-\nability of the Andreev reflection under the Andreev ap-\nproximation and find that the period of the conductance\noscillation is proportional to the square-root of the ex-\nchange field. Therefore, we can determine the exchange\nfield and therefore the spin polarization of the FM layer\nfrom the period of the conductance oscillation.\nThe system we consider is comprised of the\nFM/NM/SC double junctions shown in Fig. 1(a). The\ncurrentflows alongthe x-axis, and the interfacesbetween\nFM/NM and NM/SC are located at x= 0 and x=d,\nrespectively. The system is described by the followingBdG equation20:\n/parenleftbigg\nH0−h(x)σ∆(x)\n∆∗(x)−H0−h(x)σ/parenrightbigg/parenleftbigg\nfσ(r)\ngσ(r)/parenrightbigg\n=E/parenleftbigg\nfσ(r)\ngσ(r)/parenrightbigg\n,\n(1)\nwhereH0≡ −(/planckover2pi12/2m)∇2+V(x)−µFis the single parti-\ncle Hamiltonian, Eis the QP energy measured from the\nFermi energy µF,V(x) is the interfacial barrier21, and\nσ= +(−) represents the up-(down-)spin band. The ex-\nchange field function h(x) is given by h(x) =h0[1−Θ(x)]\nwhereh0represents the exchange field in the FM layer\nand Θ(x) is the Heaviside step function. We employed\nthe two-band Stoner model for the FM layer for simplic-\nity. The superconducting gap function is expressed as\n∆(x) = ∆ 0Θ(x−d), where ∆ 0represents the supercon-\nducting gap in the SC layer. We assume that the system\nFIG. 1: (a) Schematic diagram of aFM/NM/SC double junc-\ntion. An NM with a thickness of dis sandwiched by FM and\nSC layers. (b) Schematic diagrams of energy vs. momen-\ntumof the FM/NM/SC double junction for aspin-upincident\nelectron are shown. The open circles denote holes, the filled\ncircles electrons, and the arrows point in the direction of t he\ngroup velocity. The incident electron with up-spin is denot ed\nby 0, along with the resulting scattering processes: Andree v\nreflection (1), normal reflection (2) at the FM/NM interface,\ntransmission to the NM (3, 4) and reflection at the NM/SC\ninterface (5, 6), transmission as a electron-like quasi-pa rticle\nto the SC (7) and that as a hole-like quasi-particle (8). (c)\nSchematic diagrams of energy vs. momentum in the FM layer\nfor a spin-down incident electron are shown.2\nhas translational symmetry in the transverse ( yandz)\ndirection, and therefore the wave vector parallel to the\ninterface k/bardbl≡(ky,kz) is a conserved quantity.\nThe general solutions of the BdG equation (1) in the\nFM (NM) layer are given by\nΨ±k+\nFM(NM) ,σ(r) =/parenleftbigg\n1\n0/parenrightbigg\ne±ik+\nFM(NM) ,σxSk/bardbl(r/bardbl),(2)\nΨ±k−\nFM(NM) ,σ(r) =/parenleftbigg\n0\n1/parenrightbigg\ne±ik−\nFM(NM) ,σxSk/bardbl(r/bardbl),(3)\nwhere S k/bardbl(r/bardbl) represents the eigen function in the trans-\nverse direction in the k/bardblchannel and k+(−)\nFM(NM) ,σis thex\ncomponent of the wave number of an electron (hole) with\nσ-spin defined as k±\nFM,σ=√\n2m\n/planckover2pi1/radicalbigµF±E+σh0−E/bardbl\nandk±\nNM=√\n2m\n/planckover2pi1/radicalbigµF±E−E/bardbl,whereE/bardbl=/planckover2pi12\n2mk2\n/bardbl. In\nthe SC layer, we have\nΨ±k+\nSC(r) =/parenleftbigg\nu0\nv0/parenrightbigg\ne±ik+\nSCxSk/bardbl(r/bardbl), (4)\nΨ±k−\nSC(r) =/parenleftbigg\nv0\nu0/parenrightbigg\ne±ik−\nSCxSk/bardbl(r/bardbl), (5)\nwhereu0andv0are the coherence factors expressed as\nu2\n0= 1−v2\n0=1\n2/bracketleftBig\n1+√\nE2−∆2\nE/bracketrightBig\n,andk+(−)\nSCis thexcom-\nponent of the wave number of an electron-(hole-)like QP\ndefined as k±\nSC=√\n2m\n/planckover2pi1/radicalBig\nµF±√\nE2−∆2−E/bardbl.\nThe wave function of the FM/NM/SC double junction\nis given by the linear combination of the above general\nsolutions. Let us consider the scattering of an electron\nin thek/bardblchannel with σ-spin injected into the NM from\nthe FM, the eight processesshown in Fig.1 (b) areactive.\nTherefore, the wave function in the FM layer ( x <0)\ntakes the form\nΨFM\nσ,k/bardbl(r) =/bracketleftBigg/parenleftbigg\n1\n0/parenrightbigg\neik+\nFM,σx+aσ,k/bardbl/parenleftbigg\n0\n1/parenrightbigg\neik−\nFM,σx\n+bσ,k/bardbl/parenleftbigg\n1\n0/parenrightbigg\ne−ik+\nFM,σx/bracketrightBigg\nSk/bardbl(r/bardbl). (6)In the NM layer (0 ≤x < d), we have\nΨNM\nσ,k/bardbl(r) =/bracketleftBigg\nασ,k/bardbl/parenleftbigg\n1\n0/parenrightbigg\neik+\nNMx+βσ,k/bardbl/parenleftbigg\n0\n1/parenrightbigg\ne−ik−\nNMx\n+ξσ,k/bardbl/parenleftbigg\n1\n0/parenrightbigg\ne−ik+\nNMx+χσ,k/bardbl/parenleftbigg\n0\n1/parenrightbigg\neik−\nNMx/bracketrightBigg\nSk/bardbl(r/bardbl),\n(7)\nand in the SC layer ( x≥d)\nΨSC\nσ,k/bardbl(r) =/bracketleftBigg\ncσ,k/bardbl/parenleftbigg\nu0\nv0/parenrightbigg\neik+\nSCx\n+dσ,k/bardbl/parenleftbigg\nv0\nu0/parenrightbigg\ne−ik−\nSCx/bracketrightBigg\nSk/bardbl(r/bardbl).(8)\nThe coefficients aσ,k/bardbl,bσ,k/bardbl,cσ,k/bardbl,dσ,k/bardbl,ασ,k/bardbl,βσ,k/bardbl,ξσ,k/bardbl,\nandχσ,k/bardblare determined by matching the wave func-\ntion at the boundary of the contact x= 0 and d. Fol-\nlowing the BTK theory19the probabilities of the AR\nand the normal reflection are given by Aσ,k/bardbl(E) =\n(k−\nFM,σ/k+\nFM,σ)a∗\nσ,k/bardblaσ,k/bardblandBσ,k/bardbl(E) =b∗\nσ,k/bardblbσ,k/bardbl, re-\nspectively. Since we assume that the temperature is\nzero, the conductance at bias voltage Vis given by\nG=e\nh/summationtext\nσ,k/bardbl/bracketleftbig\n1+Aσ,k/bardbl(eV)−Bσ,k/bardbl(eV)/bracketrightbig\n,whereweas-\nsume that the voltage drop occurs at the NM/SC inter-\nface for simplicity. Below the superconducting gap, i.e.,\neV <∆0, the probabilities Aσ,k/bardbl(E) andBσ,k/bardbl(E) sat-\nisfy the relation that 1 −Bσ,k/bardbl(E) =Aσ,k/bardbl(E) and then\nwe have\nG= 2e\nh/summationdisplay\nσ,k/bardblAσ,k/bardbl(eV). (9)\nLet us first consider the most idealistic case where the\ninterfacial scattering potential, V(x), is assumed to be\nzero. This simplification enables us to obtain the analyt-\nical expression of Aσ,k/bardbl(E) under the Andreev approxi-\nmation.\nAσ,k/bardbl(E)≃4(1−ζ)/radicalbig\n1−η2\n2/bracketleftBig\n(1−ζ)2+(1−ζ)/radicalbig\n(1−ζ)2−η2)/bracketrightBig\n−η2/bracketleftbig\nǫ2+(1−2ǫ2)cos2[(k+\nNM−k−\nNM)d]−ǫ√\n1−ǫ2sin[2(k+\nNM−k−\nNM)d]/bracketrightbig,\n(10)\nwhere we introduced the normalized parameters η=\nh0/µF,ζ=E/bardbl/µF, andǫ=E/∆0. We note that Eq.\n(10) contains trigonometric functions in the denomina-\ntor. For the FM/SC junction, i.e., d=0, the trigono-\nmetric functions become constant and Eq. (10) repro-\nduces the deJong and Beenakker results of the zero-bias\nconductance1. FortheFM/NM/SCjunctionswith d/negationslash= 0,thetrigonometricfunctionsinEq. (10)giverisetotheos-\ncillationofthe conductanceagainstthe biasvoltage. The\norigin of the conductance oscillation is the interference of\nelectrons in the NM layer11,15. In the FM/NM/SC dou-\nble junctions, the injected electron propagates across the\nNM layer to the interface as an electron (3 in Fig.1) and\nis scattered into a hole (6 in Fig.1) by the superconduct-3\nFIG. 2: (a) Probability of the AR Aσ,0(eV) for the\nFM/NM/SC double junction with d=1µm is plotted against\nthe bias voltage V. The exact numerical results and the value\nof Eq. (10) are plotted by lines and circles, respectively. T he\nvalue of the exchange field is taken to be h= 0.3, 0.6, 0.9 µF\nfrom top to bottom. (b) Same plot for d=10µm.\ning gap. The superconducting gap can pair an excited\nelectron with an electron inside the Fermi sea, leaving\na hole excitation. The hole propagates back across the\nNM layer; however, it cannot interfere with the original\nelectron. In order for interference to occur the hole must\nbe reflected at the FM/NM interface, propagate to the\nNM/SC interface, be scattered into the electron state (5\nin Fig.1) by the superconducting gap, and propagate as\nan electron in the NM layer. It can interfere with the\noriginal electron (3 in Fig.1). This interference produces\nan oscillation of the conductance against the bias volt-\nage, and the period of the oscillation is determined by\nthe thickness of the NM layer.\nIn order to analyze the interference effect on the con-\nductance oscillation, we consider the AR probability\nAσ,k/bardbl(E) of the one-dimensional system; i.e., only the\ntransverse channel with k/bardbl= 0 is considered. In Fig.\n2 (a), 2 (b) we plot the probability Aσ,k/bardbl(eV) of the\nFM/NM/SC double junctions with d=1µm and 10 µm,\nrespectively, as a function of the bias voltage. Since\nEq.(10) is an even function of normalized value of the\nexchange field η,Aσ,k/bardbl(eV) is independent of the spin di-\nrectionσ; i.e.,A↑,k/bardbl(eV) =A↓,k/bardbl(eV). TheFermienergy\nand the superconducting gap are assumed to be µF= 3.8\neV (kF= 1.0˚A−1) and ∆ 0= 1.5meV, respectively. The\nexact numerical results and the approximate values of\nEq. (10) are plotted by lines and circles, respectively.\nThe value of the exchange field is taken to be h0= 0.3,\n0.6, 0.9µFfrom top to bottom.\nAs shown in Figs. 2 (a) and 2 (b), Eq. (10) and\ntherefore the Andreev approximation are valid for all\nvalues of the exchange field. According to Eq. (10),\nthe period of the oscillation is determined by the condi-\ntion that k+−k−=nπ, wherenis an integer. Since\nk±≃kF(1±E/2µF), the period is obtained as\n∆V1D≃/planckover2pi12πkF\n2med, (11)\nwhich is inversely proportional to the thickness of the\nNM layer, d. For the one-dimensional system, the period\n∆V1DisindependentoftheexchangefieldoftheFMlayeras shown in Figs. 2 (a) and 2 (b), and in Eq.(11). How-\never,asweshallshowlater,theperiodoftheconductance\noscillation due to the geometrical resonance depends on\nthe exchange field of the FM layer because the number\nofk/bardblchannels available for the AR is restricted by the\nexchange field.\nTheperiodoftheoscillationof Aσ,k/bardbl(eV)withfinite k/bardbl\nis given by ∆ Vk/bardbl≃/planckover2pi1π√\n2med/radicalbigµF−E/bardbl. From Eq.(9) the\nconductanceisobtainedbysummingup Aσ,k/bardbl(eV) forall\navailable k/bardbl. Since the spin of the Andreev reflected hole\nis opposite to that of the incident electron, the maximum\nvalue of k/bardbland therefore E/bardblis limited by the exchange\nfieldh0as maxE/bardbl=µF−h0. We assume that oscilla-\ntions ofAσ,k/bardbl(eV) with different periods cancel out each\nother and the period of the sum/summationtext\nσ,k/bardblAσ,k/bardbl(eV) is de-\ntermined by the shortest period. Thus, the period of the\nconductance oscillation of the three-dimensional system\nis obtained as\n∆V3D≃min∆Vk/bardbl=/planckover2pi1π√\n2med/radicalbig\nh0.(12)\nIn Figs. 3 (a), 3 (b), and 3 (c) we plot the conductance\nof the FM/NM/SC junction, GFNS, normalized by that of\nthe FM/NM/NM junction, GFNN, against the bias volt-\nage. As shown in Fig. 3 (a), the oscillation due to the ge-\nometrical resonance does not appear in the conductance-\nvoltage curve if the thickness of the NM layer, d, is less\nthan or of the order of nm. The conductance-voltage\ncurve is indistinguishable from that of the FM/SC junc-\ntion. Hence, we can use the conventional PCAR analysis\nfor a FM film, the surface of which is coated by a thin\nFIG. 3: (a) The conductanceoftheFM/NM/SC double junc-\ntion,GFNSwithd=1nm is plotted against the bias voltage.\nThe conductance is normalized by that of the FM/NM/NM\njunction, GFNS. (b) Same plot for d=1µm. (c) Same plot for\nd=10µm. (d) The period of the conductance oscillation of the\nFM/NM/SC double junction with d=10µm is plotted against\nthe exchange field h0.4\nFIG. 4: (a) The normalized conductance, GFNS/GFNNfor\nd=1µm, Z=0.2 is plotted against the bias voltage. (b) Same\nplot ford=10µm, Z=0.2.\n(less than a few nm) NM layer.\nFigures 3 (b) and 3 (c) show the conductance-voltage\ncurves for the FM/NM/SC double junctions with d=\n1µm andd= 10µm, respectively. One can see that the\nperiod does depend on the exchange field, h0, in the FM\nlayer as well as the thickness of the NM layer. The pe-\nriod is a decreasing function of the exchange field. We\ncan easily confirm that the period is proportional to the\nsquare-root dependence of the exchange field by looking\nat Fig. 3 (d). The exact numerical results (filled cir-\ncles) agree well with Eq. (12) (dotted line). The results\nsuggest that we can determine the exchange field and\ntherefore the spin polarization of the FM layer from the\nperiod of the conductance oscillation.\nNext we consider the effect of the interfacial scattering\npotential at the NM/SC interface. We assume that the\ninterfacial potential is represented by the delta-function\nasV(x) = (/planckover2pi12kFZ/m)δ(x−d), where Zis the dimension\nless parameter which characterize the strength of the in-\nterfacial scattering potential. For simplicity, we neglect\nresistances of the FM/NM interface and the FM layer\nwhich do not change the oscillation period of the conduc-tance but reduce the amplitude of it. In Figs.4 (a) and 4\n(b), we show the normalized conductance-voltage curves\nfor junctions with d=1µm andd=10µm. The parameter\nZis assumed to be 0.26. Because the conductance os-\ncillation is due to the geometrical resonance in the NM\nlayer, the period of the oscillation is not affected by the\ninterfacial scattering potential and is given by Eq.(12).\nIn the present analysis we employed the simplest BdG\napproach and consider the clean FM/NM/SC junctions\nwith perfect interfaces. In the real experiments the con-\nductance oscillation we predicted might be smeared out\ndue to the interface roughness and imperfections. How-\never, recent advances in fabrication technology enables\nus to fabricate epitaxial FM/NM/SC trilayers of high\nquality22. We expect that the conductance oscillation\nwe predicted can be observed in such epitaxial trilay-\ners. For further understanding of the transport prop-\nerties of FM/NM/SC trilayers, we have to take into ac-\ncounttheeffectsoffinitemeanfreepath, bandstructures,\nand selfconsistent determination of the electron’s distri-\nbution function and electric-pottential, which is beyond\nthe scope of this Brief Report.\nIn summary, we studied the conductance oscillation\nduetothegeometricalresonanceinaFM/NM/SCdouble\njunction theoretically. We showed that the conductance\ndue to the Andreev reflection oscillates as a function of\nthe bias voltage due to the geometrical resonance. We\nfound that the exchange field and therefore the spin po-\nlarizationoftheFM layercanbedetermined fromthepe-\nriod of the conductance oscillation because the period of\nthe conductance oscillation is proportionalto the square-\nroot of the exchange field.\nThe authors would like to acknowledge the valuable\ndiscussions they had with K. Matsushita and N. Yokoshi.\n1M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. 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Rev. Lett. 16,\n85 (1966).\n15J. M. Rowell and W. L. Macmillan, Phys. Rev. Lett. 16,\n453 (1966).\n16O. Nesher and G. Koren, Phys. Rev. B 60, 9287 (1999).\n17H.-S. Chang, M.-H. Bae, and H.-J. Lee, Physica C 408-\n410, 618 (2004).\n18L. Shkedy, P. Aronov, G. Koren, and E. Polturak, Phys.\nRev. B69, 132507 (2004).\n19G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys.\nRev. B25, 4515 (1982).\n20P. G. de Gennes, Superconductivity of Metals and Alloys\n(W. A. Benjamin, New York, 1966), chap. 5.\n21L. R. Tagirov and N. Garc´ ıa, Superlattices Microstruct.\n41, 152 (2007).\n22H. Yamazaki, N. Shannon, and H. Takagi, Phys. Rev. B\n73, 094507 (2006)."    },    {        "title": "2105.09023v1.Determination_of_the_spin_Hall_angle_by_the_inverse_spin_Hall_effect__device_level_ferromagnetic_resonance__and_spin_torque_ferromagnetic_resonance__a_comparison_of_methods.pdf",        "content": "   1 Determination of the spin Hall angle by the inverse spin Hall effect, device level ferromagnetic resonance, and spin torque ferromagnetic resonance: a comparison of methods   Ranen Ben-Shalom1, Nirel Bernstein1, See-Hun Yang2, Amir Capua1* 1Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel  2IBM Research Division, Almaden Research Center, 650 Harry Rd., San Jose, California 95120, USA *e-mail: amir.capua@mail.huji.ac.il   Abstract: The spin torque ferromagnetic resonance (STFMR) is one of the popular methods for measurement of the spin Hall angle, 𝜽𝑺𝑯. However, in order to accurately determine 𝜽𝑺𝑯 from STFMR measurements, the acquired data must be carefully analyzed: The resonance linewidth should be determined to an accuracy of a fraction of an Oe, while the dynamical interaction leading to the measured response consists of the conventional field-induced ferromagnetic resonance (FMR), spin-torque induced FMR, and of the inverse spin Hall effect (ISHE). Additionally, the signal often deteriorates when DC current is passed through the device. In this work we compare the STFMR method with two other FMR-based methods that are used to extract 𝜽𝑺𝑯. The first is a device-level FMR and the second is based on the ISHE. We identify artefacts that are caused by the noise floor of the instrumentation that make the measurement of 𝜽𝑺𝑯 illusive even when the signal to noise ratio seems to be reasonable. Additionally, we estimate a 𝟏𝟎% error in  𝜽𝑺𝑯 that results from neglecting the magnetic anisotropies as in conventional measurements. Overall, we find the STFMR to be the most robust of the three methods despite the complexity of the interaction taking place therein. The conclusions of our work lead to a more accurate determination of 𝜽𝑺𝑯 and will assist in the search of novel materials for energy efficient spin-based applications.       2 Efficient generation of spin currents is key to spintronics. Over the last decade the majority of the efforts was focused on finding new physics for polarizing the transported spins. Such an example is the intensive study of the spin Hall effect (SHE) [1,2] that relies on the spin orbit coupling (SOC) or on non-trivial topologies of the band structure [3]. In addition to the challenges in generating spin currents, their accurate quantification is not straight forward. For example, in Pt, which is one of the main materials used to investigate the SHE, a large variation in the SHE efficiency was reported [4-7]. The charge to spin current conversion rate is referred to as the spin Hall angle [8], 𝜃(), and is defined by 𝜃()=2𝑒𝐽./ℏ𝐽1\twhere 𝑒 is the electron charge, ℏ is Planck’s constant, 𝐽1 is the charge current density, and 𝐽( is the spin current density. Measurement of 𝜃() can be carried out by various methods such as the cavity ferromagnetic resonance (FMR) [9,10], spin pumping [11-15], optical FMR [16,17], anomalous and planar Hall effect [18-21], spin torque ferromagnetic resonance (STFMR) [5,22-25], and nonlocal spin transport [8,26-28]. Among these methods, the STFMR is most commonly used due to its simplicity and applicability. In the STFMR, an AC charge current is injected into a bilayer of a ferromagnet (FM)/heavy metal (HM) and excites spin precessions. Due to the anisotropic magneto resistance (AMR) [29], a rectified DC voltage builds up. Accordingly, the measured FMR response stems from three different processes: 1) SHE in the HM that generates AC spin currents that are responsible for an antisymmetric lineshape. 2) A symmetric lineshape that stems from the stray AC Oersted field generated by the current passing in the HM. 3) An additional symmetric resonance response that originates in spin pumping [30] by the precessing magnetization 𝑀44⃗ [22] in the FM layer. Therefore, analysis of the measured lineshape is not trivial and requires separating the different contributions, each resolved to a fraction of an Oe. This, in turn, usually requires solving a five-parameter lineshape fitting optimization problem [24]. In this work, we compare the STFMR technique with the inverse SHE (ISHE) [31] and device-level ferromagnetic resonance (DLFMR) methods for measurement of 𝜃(). All techniques are applied on the same device. Each method has relative advantages such as the elimination of the spin transfer torque (STT) [22] or in the simplicity of the analysis of the measured responses. Our study shows that the anisotropies that are usually neglected, e.g. the shape anisotropy, may play a considerable role [32]. However, we show that the conventional optimization algorithm used to analyze the STFMR response is robust. In addition, our measurements indicate a significant electrical background noise that affects all methods. This noise often deteriorates the ISHE and DLFMR signals and in the worst case leads to unnoticeable measurement artefacts that result in erroneous values of 𝜃(). We conclude that despite the relative advantages of the ISHE and DLFMR methods and the challenge of analyzing the STFMR response the STFMR is a more robust and reliable technique.     3  Fig. 1. Experimental setup. (a) Patterned SHE device with a micro-antenna. (b) Schematic of the measurement setup that includes the STFMR, ISHE, and DLFMR.   The experimental system is presented in Fig. 1 and can be converted between the STFMR, ISHE, and DLFMR modes of operation so that all methods are applied on the same device. In the STFMR a microwave signal is generated by the RF signal generator and is driven into the device by connecting the RF signal generator to node ‘A’ indicated in the figure. The injected AC current drives the FMR in the device in the presence of an externally applied DC magnetic field,\t𝐻789. The oscillatory driving torque stems from the Oersted stray field and the AC STT that is generated by the SHE. Due to the AMR and the magnetization precessions, the AC current is rectified and a DC voltage builds up on the device. In addition, the precessions of 𝑀44⃗ pump spin current by the spin pumping mechanism [30] into the HM which is converted into a DC charge current by the ISHE. Once the FMR is excited a DC charge current, 𝐼;, that is converted into a DC spin current is injected and exerts an anti-damping torque that is reflected on the resonance linewidth from which 𝜃() is extracted in the usual manner [22]. The DC voltage building upon the device in the STFMR configuration is given by 𝑉(=>?@=𝐼;𝑅;+𝐼C()D(𝐻789)⋅𝑅;+HI⋅𝑖;⋅(∆𝑅L7(𝐻789)+∆𝑅(==(𝐻789)) where 𝑖;, 𝑅;, 𝐼C()D, ∆𝑅L7, and ∆𝑅(== are the amplitude of AC charge current, the DC resistance, the DC ISHE charge current that is pumped by 𝑀44⃗, the amplitude of the AC resistance that originates in the AC Oersted excitation, and the amplitude of the AC resistance that originates from the STT, respectively. Both ∆𝑅L7 and ∆𝑅(== stem from the AMR. To extract the resonance linewidth, following a data fitting procedure, the measured resonances are decomposed to the symmetric and antisymmetric components that originate in the Oersted and STT excitations. The fitting process requires solving a five-parameter optimization problem of the following expression:   \nAntenna Device \nFigure captions a b \n   4 𝑉(=>?@=𝐴∆)N()OPQR)SOT)NU∆)N+𝐵∆)N()OPQR)SOT)NU∆)N)OPQR)SOT∆)\t+𝐶   (1) In Eq. 1 𝐻X7Y is the resonance field, ∆𝐻 is the resonance width, and A, B, and C are the coefficients of the symmetric, antisymmetric, and DC background level, respectively. The contribution of the ISHE in 𝑉(=>?@ is neglected in our analysis [22]. Our experimental results below indicate that this approximation is valid, although according to Ref. [33] the ISHE contribution was found to be significant. In the other two configurations of the DLFMR and ISHE, the AC current is driven into a micro-antenna that is patterned in proximity to the device (Fig. 1) rather than being injected into it. In this manner we excite the FMR without the contribution of the AC STT. Therefore, the dynamical processes taking place in the ISHE and the DLFMR measurements are greatly simplified as compared to the STFMR. Consequently, the resonance responses of the ISHE and DLFMR are purely symmetric and in the data fitting procedure the optimization algorithm is applied to solve a four-parameter problem.  The ISHE and DLFMR differ in the manner at which the FMR signal is read. In the ISHE spin pumping induces a DC voltage signal by the ISHE in the HM layer that is probed. The measured voltage is given by 𝑉C()D=𝐼;𝑅;+𝐼C()D(𝐻789)⋅𝑅;. In the DLFMR configuration the FMR signal is read at the fundamental RF harmonic. Due to the AMR effect, the resistance of the device is modulated at the RF frequency and in the presence of the applied DC current a voltage signal is generated at the fundamental frequency. A homodyne detection scheme using an RF mixer is then used to down-convert this signal and read it as illustrated in Fig. 1. The measured DLFMR signal is expressed by 𝑉Z[>?@=\\𝐼;+𝐼C()D(𝐻789)]⋅∆𝑅L7(𝐻789). The DLFMR signal contains also an ISHE contribution, 𝐼C()D, however, it is three orders of magnitude weaker than 𝐼;. To further eliminate ISHE component, in the experiment we used a lock-in detection scheme in which 𝐼; was modulated.  In order to model the dynamics with a minimal set of approximations we derived the theoretical model based on the Smit-Beljiers-Suhl formalism [17,34,35]. Accordingly, the Landau-Lifshitz-Gilbert equation in spherical coordinates becomes:       𝜃̇=𝛾𝐻`,\t\t\t\t𝑠𝑖𝑛𝜃𝜙̇=−𝛾𝐻f               (2) Here, 𝛾 is the gyromagnetic ratio while 𝐻f and 𝐻` are the effective fields along the polar and azimuthal unit vectors, respectively. These originate from the conserving energies, 𝐸hijY7Xk., that account for 𝐻789, the demagnetizing field, the crystalline anisotropy, the microwave field, and the field-like spin orbit torque (SOT):\t𝐸hijY7Xk.=−𝑀44⃗⋅\\𝐻44⃗\t789+𝐻44⃗L7+ℎ4⃗L7]−12𝑀44⃗⋅𝐻44⃗Z−𝐾psinI𝜃−𝑀44⃗⋅ℎ4⃗Xt−𝐻>[𝑀44⃗⋅𝑠̂. 𝐻44⃗L7, ℎ4⃗L7,    5 𝐻44⃗Z, ℎ4⃗Xt, 𝐻44⃗>[, 𝑠̂, and 𝐾p are the stray DC Oersted field stemming from the HM, the Oersted field generated by AC current, the demagnetization field, the RF field of amplitude ℎXt, the field-like STT, the spin polarization of the injected spins, and the magnetocrystalline anisotropy constant, respectively. In addition, 𝐻f and 𝐻` also stem from the nonconserving energies, 𝐸jijhijY7Xk. that account for the magnetic Gilbert damping losses and the damping-like SOT:\t𝐸jijhijY7Xk.=(𝛼/2𝛾𝑀Y)𝑀44⃗̇I+𝐻()D𝑀44⃗̇⋅\\𝑀44⃗×𝑠̂] where 𝛼 is the Gilbert damping and 𝑀Y is the saturation magnetization. 𝐻f and 𝐻` are then determined from the effective field, 𝐻44⃗7tt, that is given by 𝐻44⃗7tt=−∇?44⃗𝐸hijY7Xk.−∇?44⃗̇𝐸jijhijY7Xk.. The spin Hall parameter 𝐻()D is given by ℏ𝜃()𝐽1/2𝑒𝑀(𝑡\twhere 𝑡 is the thickness of the ferromagnetic layer. 𝐸hijY7Xk. and 𝐸jijhijY7Xk. have units of energy density and energy density flow. Solving for small deviations, (∆𝜃,∆𝜙) around the equilibrium state, (𝜃;,𝜙;), in phasor-space we find the in-plane component of 𝑀44⃗ at the angular frequency 𝜔 which is given by:  ∆𝜙={(HU|N)}~f\\NRN]U∆\\NRN]NUN∆N\\ℎf+𝛼ℎ`]𝑖𝜔+{?TH}~fℎf𝜁`f+ℎ`𝜁ff      (3) with: 𝜔;I=𝛾I𝑀YI(1+𝛼I)sinI𝜃;𝜁``𝜁ff−𝜁`f𝜁f` ∆𝜔=𝛾𝑀Y(1+𝛼I)sin𝜃;𝛼sin𝜃;𝜁ff+𝛼sin𝜃;𝜁``+𝜁`f−𝜁f` And 𝜁 defined by 𝜁f`=𝐸f`,𝜁`f=𝐸`f+𝑀Y𝐻()D\\𝑠8cos𝜙;−𝑠sin𝜙;] 𝜁ff=𝐸ff+𝑀Y𝐻()Dsin𝜃;\\−𝑠8cos𝜃;sin𝜙;+𝑠cos𝜃;cos𝜙;] 𝜁``=𝐸``+𝑀Y𝐻()Dsin𝜃;\\𝑠8sin𝜃;cos𝜙;+𝑠sin𝜃;sin𝜙;−𝑠cos𝜃;] In Eq. (3) 𝜔; is the resonance frequency, ∆𝜔 is the resonance linewidth, and ℎf and ℎ` are the polar and azimuthal components of the RF excitation. 𝑠8,\t𝑠, and 𝑠\tare the projection of 𝑠̂ on the cartesian axis. 𝐸``,𝐸`f,𝐸f`, and 𝐸ff are the second order derivatives of (𝐸hijY7Xk.+𝐸jijhijY7Xk.), with respect to 𝜃 and 𝜙. The equilibrium state was evaluated without the contribution of the SOT.     6 To compare the methods we used the well-studied Pt based system: 60 Pt/5 Py/60 Co/20 TaN (units in Å). The film was grown by magnetron sputtering and patterned to a 5\t𝑋\t40\t𝜇𝑚 device. The Au micro-antenna of 100\t𝑛𝑚 thickness and 2\t𝜇𝑚 width was deposited at a distance of 1.5\t𝜇𝑚. The AC current density driven into the device in the STFMR experiment was 10⋅10HI\t𝐴/𝑚I and 1.4⋅10HI\t𝐴/𝑚I into micro-antenna in the ISHE and DLFMR experiments. To maximize the signals, in the STFMR and DLFMR experiments the current was injected at 45° with respect to 𝐻789 and at 90° in the ISHE experiment. The frequency dispersion relation and linewidth vs. frequency as obtained from the STFMR method are presented in Fig. 2. From these measurements the magnetic parameters of the sample were extracted using Kittel’s formula, 𝑓;=𝛾/2𝜋⋅𝐻X7Y(𝐻X7Y+4𝜋𝑀Y), with 𝑓; being the resonance frequency, and the frequency dependent linewidth, ∆𝐻=2α/γ⋅𝜔+∆𝐻C), with ∆𝐻C) being the inhomogeneous broadening. Accordingly, the values of 𝛼=0.0048,\t𝑀Y=1384\t𝑒𝑚𝑢/𝑐𝑚£, and ∆𝐻C)=16\t𝑂𝑒 were extracted. \n Fig. 2. (a) STFMR Frequency-field dispersion relation. Open blue circles indicate the measurement. Solid red line indicates Kittel’s formula. Inset illustrates the measured STFMR responses for 5 - 18 GHz. (b) Resonance linewidth as a function of frequency. Solid red line indicates the linear fit.  The measurement of 𝜃() using the STFMR and ISHE experiments are presented in Fig. 3. The experiments were carried out at 8 GHz. The figure presents ∆𝐻 and 𝐻X7Y as function of 𝐽1. Fig. 3(a) and 3(b) present the STFMR results. The shift of 𝐻X7Y with 𝐽1 originates from the stray DC Oersted field stemming from the DC current and agrees well with a calculation of the expected stray field. The linewidth measurements were fitted using Eq. (3) from which a 𝜃() of 23.1% was extracted. This value is within the range of previously measured 𝜃() values [4,17,36,37]. Interestingly, it is readily seen that the noise level increases as |𝐽1| is a b 5GHz 18GHz    7 increased. This behavior is marked by the shaded area in the figure and is typical of an intrinsic Johnson noise and the shot noise limited processes. These processes take place either in the device or the measurement instrumentation, or both. This noise limits the maximal DC current that can be injected. A similar behavior was also seen in Ref. [38]. There, it was shown that this limitation can be overcome by optically probing 𝑀44⃗. \n Fig. 3. (a) 𝑯𝒓𝒆𝒔 as a function of 𝑱𝑪 for the STFMR measurements. (b) ∆𝑯 as a function of 𝑱𝑪 for the STFMR measurements. Shaded blue area guides the eye to the noise level. Black dashed line indicates ∆𝑯 at 𝑱𝑪 = 0. (c) 𝑯𝒓𝒆𝒔 as a function of 𝑱𝑪 for the ISHE measurements. (d) ∆𝑯 as a function of 𝑱𝑪 for the ISHE measurements. Black dashed line marks ∆𝑯 for 𝑱𝑪 = 0. In (a) - (d) the open blue circles indicate the measurement and solid red lines indicate the linear fits. Data presented at 8 GHz.   STFMR \nISHE ISHE STFMR a b \nc d \n   8 The overall broadening that results from the anti-damping torque is only of a few Oe. Therefore, in order to extract 𝜃(), the linewidth should be determined to the accuracy of a fraction of an Oe. However, since the optimization algorithm is applied on a multi-parameter problem, five in our case, it is possible that it would converge into a local minimum. Therefore, we tested the convergence of the algorithm. This was carried out by providing the algorithm with initial conditions that are far off from the actual solution. We found that a shift larger than 100 Oe in the resonance frequency and 50 Oe in the linewidth was the limit after which the optimization algorithm converged to a wrong solution. This illustrates that the fitting algorithm is robust.  The dependence of 𝐻X7Y and ∆𝐻 on the injected DC current in the ISHE measurement are shown in Fig. 3(c) and 3(d). The ISHE measurements are significantly noisier. Consequently, a wrong value of 𝜃()=46.8 that is larger by a factor of two as compared to the STFMR method was measured. Despite the noise, we can readily see that the shift in 𝐻X7Y as function of 𝐽1 is much smaller than in the STFMR measurement. In the STFMR the overall shift is ~ 35 Oe while in the ISHE the shift is only ~ 11 Oe. Since the STFMR measurements are carried out at 45° and the ISHE measurements at 90° we can attribute the difference to the anisotropy fields which are usually neglected. However, our model which includes the anisotropy fields could not fully account for the differences and we believe there is an additional contribution such as the existence of magnetic domains. From our model, we found that neglecting the anisotropy fields results in an error of ~10% in 𝜃(). The DLFMR configuration is the most direct measurement of the FMR. The FMR is excited externally by the micro-antenna and is read directly on the device using a homodyne detection scheme. In contrast to the other configurations, in the DLFMR the signal arises only from the AC resistance that is modulated by the AMR. Additionally, the homodyne detection scheme eliminates the DC noise which is responsible for a significant portion of the noise as shown above. For these reasons we expect the DLFMR experiment to be more accurate. In reality this is not the case and the DLFMR measurement (Fig. 4(a)) was noisy to the extent that we could not extract 𝜃() reliably. We speculate that the noise originates from thermal effects in the RF mixer.  In order to compare the STFMR with the ISHE and DLFMR methods we plot only the symmetric part of the STFMR measurement that originates in the Oersted field. The symmetric part of the STFMR response has the same functional form as the DLFMR and ISHE lineshapes. A comparison of all lineshapes overlaid on top of each other is presented in Fig. 4. Each trace was normalized to its maximum. The ISHE resonance frequency is higher by ~ 50 Oe as compared to the STFMR, and DLFMR. This stems from the difference    9 in the direction at which 𝐼; is applied with respect to 𝐻789. This behavior demonstrates once more that the anisotropies are not as negligible as conceived. The shift in the resonance field resulting from the different angle of applied 𝐻789 is also seen in the STFMR and ISHE measurements of 𝐻X7Y as function of 𝐽1\tin Fig. 3. Fig. 4(a) shows that the DLFMR and ISHE are much noisier as compared to the STFMR. At the origin of the different noise levels of the ISHE and the DLFMR lies the physical mechanism used to read out the signal. In the ISHE, the signal originates from the spin pumping and in the DLFMR and STFMR it originates in the AMR. These differences translate to a STFMR signal that is 26 times stronger than the ISHE signal and, therefore, lead to a higher signal to noise ratio (SNR). For this reason, conclude that the contribution of the ISHE is negligible in the STFMR experiment and can be neglected. In the DLFMR the signal originates in the AMR as in the STFMR. Hence, the magnitude of the signals readout is similar. In the DLFMR, the absolute noise level is higher than in the STFMR because of the mixer and possibly impedance mismatching effects that are not present in the DC measurements. Therefore, the SNR is significantly lower than in the STFMR measurement. To understand how the measurement is affected by the poor SNR we simulated a Lorentzian resonance function that is submerged in a background noise level as shown in Fig. 4(b). We then fitted the resultant signal to an ideal Lorentzian. It is seen that the fitted resonance is narrower than the real resonance. The narrower linewidth of the ISHE and DLFMR is also seen in our measurements when the ISHE lineshape is overlaid on the STFMR and DLFMR lineshapes as illustrated by the black dashed line in Fig. 4(a). Clearly, the DLFMR and ISHE resonances appear narrower than the STFMR resonance indicating the significant contribution of the noise. We evaluate the ISHE noise floor to be 0.8𝜇𝑉 and the DLFMR noise floor to be 140𝜇𝑉. Both are 7% of the resonance peak. We can see the same effect in the linewidth measurements of the STFMR and ISHE in Fig. 3. The absolute width of the ISHE spectrum is smaller than the STFMR width as indicated by the vertical black dashed lines of Fig. 3(b) and 3(d). This illusive artifact illustrates the challenges of reliably extracting 𝜃().    10  Fig. 4. (a) Symmetric part of the STFMR measurement (blue) together with the responses of the DLFMR (red) and ISHE (yellow) measurements and the shifted ISHE response (dashed black). Traces are normalized to the peak value. (b) Noise simulation. Blue solid line indicates an ideal Lorentzian lineshape. Red solid line illustrates the calculated Lorentzian lineshape together with a simulated noise floor. Black solid line is a fit of the simulated noisy Lorentzian to a noise-free response.    To summarize, in this work we have examined the intricate details of the commonly used STFMR technique and compared it to the ISHE and DLFMR methods that sense more directly the FMR response in the presence of spin currents. We found that despite the physical complexity of the dynamical interaction in the STFMR, it is more robust than the ISHE and DLFMR. We showed that it is necessary to consider the anisotropies, e.g. the shape anisotropy in the device to resolve more accurately 𝜃(). Our study shows that the background noise distorts the measured signal even when the SNR is high enough, resulting in erroneous 𝜃() values and may be one of the reasons for the large scattering in the values reported in the literature for common materials. Although the DLFMR is expected to be the most reliable method, there seems to be a fundamental limitation to the measurement. Our work marks another step towards utilizing the SHE in realistic practical applications.     Lorentz. line Lorentz. + noise a b STFMR symmetric part ISHE DLFMR fitShifted ISHE    11 References [1] J. E. Hirsch, \"Spin Hall Effect\", Physical Review Letters 83, 1834 (1999). [2] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, \"Spin Hall effects\", Reviews of Modern Physics 87, 1213 (2015). [3] Y. Zhang, J. Železný, Y. Sun, J. van den Brink, and B. 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"    },    {        "title": "1403.0656v1.Off_Resonant_Manipulation_of_Spins_in_Diamond_via_Precessing_Magnetization_of_a_Proximal_Ferromagnet.pdf",        "content": "O\u000b-Resonant Manipulation of Spins in Diamond via Precessing Magnetization of a\nProximal Ferromagnet\nC. S. Wolfe*,1V. P. Bhallamudi*,1H. L. Wang,1C. H. Du,1S.\nManuilov,1A. J. Berger,1R. Adur,1F. Y. Yang,1and P. C. Hammel1,\u0003\n1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\n(Dated: September 29, 2018)\nWe report the manipulation of nitrogen vacancy (NV) spins in diamond when nearby ferrimagnetic\ninsulator, yttrium iron garnet, is driven into precession. The change in NV spin polarization, as\nmeasured by changes in photoluminescence, is comparable in magnitude to that from conventional\noptically detected magnetic resonance, but relies on a distinct mechanism as it occurs at a microwave\nfrequency far removed from the magnetic resonance frequency of the NV spin. This observation\npresents a new approach to transferring ferromagnetic spin information into a paramagnet and\nthen transducing the response into a robust optical signal. It also opens new avenues for studying\nferromagnetism and spin transport at the nanoscale.\nPACS numbers: 72.25.Mk, 75.76.+j, 75.78.-n, 75.78.-n, 75.78.-n\nKeywords: NV, magnetization dynamics, FMR, spin transport, YIG\nUnderstanding the transport of spin and energy be-\ntween dissimilar materials is a topic of intense current\ninterest re\recting both the scienti\fc richness of the topic\nas well as it technological potential[1{6]. Metal/metal in-\nterfaces have been extensively studied, and to a lesser de-\ngree metal/semiconductor and metal/insulator systems.\nHowever, the transfer of angular momentum between two\ninsulating materials has been more challenging to study\ndue to the lack of suitable detection methods.\nNV centers in wide band-gap insulating diamond pro-\nvide an exceptional platform for performing spin-based\nmeasurements. The paramagnetic NV center is optically\nactive, and its photoluminescence (PL) is dependent on\nthe relative occupation of the lowest lying electronic spin\nstate of the defect center [7, 8]. This enables optical mea-\nsurement of NV-center spin state with excellent sensitiv-\nity, making optically detected magnetic resonance of NV\ncenters an area of intense research activity [9{21]. How-\never, work done thus far relies on manipulation of NV\nspins using magnetic resonance.\nHere we present experimental evidence that the NV\ncenter state can be modi\fed non-resonantly ( i.e.,by ir-\nradiation with microwave magnetic \felds at frequencies\nfar from NV center Larmor frequencies) by coupling to\nthe dynamics of a proximal ferromagnetic insulator. This\nchange can be detected as a change in the NV center\nPL, as is done for conventional microwave-driven reso-\nnant spin manipulations. This new e\u000bect promises valu-\nable insights into interactions between spins in adjacent\ndissimilar materials. This is a particular case in which\nboth materials are insulating and the interaction is ef-\nfective over long distances ( >300 nm). More generally,\n* C. Wolfe and V. Bhallamudi contributed equally to this work.\nFIG. 1. Experimental Schematic : The sample is a 20 nm\nthick single crystal YIG \flm with nanodiamonds dispersed on\ntop with a thickness of about 500 nm. To apply microwave\n\felds to the sample a silver microwire is patterned on the YIG.\nGreen laser light is focused onto nanodiamonds near the wire,\nand the intensity of the resulting photoluminescence from the\nNV centers is measured. Inset is an SEM image of dispersed\nnanodiamonds.\nthis provides a novel method for manipulating NV center\nspins and could enable sensitive spatially resolved imag-\ning of ferromagnetic phenomena by means atomic scale\nNV centers [9, 12, 15].\nYttrium Iron Garnet (YIG), Y 3Fe5O12, was chosen for\nthis experiment as a well known ferrimagnetic insulator\nwith exceptionally low damping [22, 23]. Here an epitax-\nial YIG \flm, 20nm thick, was grown on a gadolinium\ngallium garnet (GGG) (111) substrate by o\u000b-axis sput-\ntering [24{26]. Continuous wave microwave \felds are ap-\nplied to the sample by means of a 300 nm thick and 30 \u0016m\nwide silver microstrip line patterned on top of the YIG.arXiv:1403.0656v1  [cond-mat.mes-hall]  4 Mar 20142\nNanodiamonds, 50-200 nm in size (as shown by SEM im-\nage analysis) and containing up to a few thousand NV\ncenters each, were dispersed on top of a lithographically\nde\fned microstrip line as shown in Fig. 1. AFM mea-\nsurement indicates that the nanodiamonds form a 500\nnm thick \flm. Photoluminescence is excited in the NV\ncenters using a 532 nm laser beam and is collected by a\nphotodiode. A lock-in measurement is performed on the\nphotodiode signal by modulating the amplitude of the\napplied microwave \feld.\nThe lock-in measurement of the resulting modulation\nof the PL intensity is presented in Fig. 2 as a function\nof an applied in-plane magnetic \feld and the applied mi-\ncrowave frequency. Data for a control sample with nan-\nodiamonds on a GGG substrate without YIG is shown\nin Fig. 2 (a). We observe the intrinsic and well-known\nmagnetic resonances of the NV center ground and excited\nstates, starting at 2.87 GHz and 1.43 GHz respectively.\nShown in Fig. 2 (b) is the same data overlaid with the\ntheoretically expected resonance conditions for the NV\ncenters, which are obtained by solving the NV hamilto-\nnian in the presence of magnetic \feld parallel and per-\npendicular to the NV axis (see supplementary informa-\ntion of [9]). We also see several features in the PL below\n1.25 GHz (Fig. 2 (a)) that are not related to the normal\nground- and excited-state resonances of the NV centers.\nThese will be discussed in a forthcoming publication.\nThe data obtained from the nanodiamonds on top of\nthe YIG \flm is shown in Fig. 2 (c). The key di\u000berence be-\ntween Fig. 2 (a) and (c) is the feature in (c) that extends\nup from the lower left-hand corner. This is highlighted in\nFig. 2 (d) where the data from (c) is overlaid with a solid\nblue curve showing the YIG ferromagnetic resonance con-\ndition. The blue dots show the YIG resonance condition\nas measured by re\rected microwave power. These data\nare \ft (blue curve) using the equation for the uniform\nFMR mode in a thin \flm [27], and we obtain a magneti-\nzation,\u00160Ms, of 183 mT (see supplementary information\nfor more information). As can be seen, the intensity of\nthe PL from the NV centers strongly changes precisely\nwhen the YIG FMR is excited.\nWe have considered the e\u000bect of heating, caused by the\nFMR absorption in YIG, on the PL of NV centers. Sev-\neral control experiments and estimates of the possible\ne\u000bect render this potential explanation highly unlikely.\nMore details can be found in the supplementary infor-\nmation.\nThere are two key points to note about the FMR-\ninduced feature in the PL signal. First, it is seen at fre-\nquencies and \felds well separated from the NV center's\nown resonance conditions. This is in clear contrast to\nthe quantum computing and magnetometry techniques\nbeing developed, where the spin-state of NV centers is\ncoherently manipulated by microwave \felds meeting the\nmagnetic resonance conditions [9{20]. Instead, we see a\nchange in the PL that correlates to the excitation of the\n3020100\nMagnetic Field (mT)\n3020100\nMagnetic Field (mT)4.0\n3.5\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5Microwave Frequency (GHz)\n4.0\n3.5\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5Microwave Frequency (GHz)\n(a) (b)\n(c) (d) NV || H \n NV /s94 H  \n YIG data\n  Fit\nOn YIGOn GGG\n1.00.50.0\nChange in NV PL (a.u.)FIG. 2. NV PL Data while YIG undergoes FMR : (a)\nRaw data showing the change in the intensity of the PL from\nthe NV centers in nanodiamonds dispersed on top of GGG,\nwith no YIG, as a function of microwave frequency and mag-\nnetic \feld. (b) The same data overlaid with theoretical reso-\nnance conditions for the NV centers. The black dashed lines\nshow the resonance condition for an NV center with the mag-\nnetic \feld parallel to the NV axis for the ground and excited\nstates. The grey crosses show the resonance condition for an\nNV center with the magnetic \feld perpendicular to the NV\naxis. (c) Similar data from nanodiamonds dispersed on the\nYIG sample with the distinct feature corresponding to the\nYIG FMR condition. (d) The data from (c) with the FMR\npeaks measured using re\rected microwave power (blue dots).\nAlso shown is a \ft (blue line) to the calculated dispersion re-\nlation for YIG \flm FMR with the magnetic \feld in plane (see\nmain text).3\nFIG. 3. NV center PL Data on YIG and on Wire\nAbove YIG : (a) Data taken with the laser spot focused on\ntop of the \u0018300 nm thick patterned microwire. The data\nshows that the signal corresponding to the YIG FMR is still\npresent, though reduced. The coupling must extend at least\n300 nm through the wire. (b) Data from the nanodiamonds\ndirectly on top of the YIG for comparison. (c) Line cuts of\nthe data in (a) and (b) at 16.5 mT. The FMR-induced peak is\nreduced by 3 times relative to the NV excited state peak for\nthe case of nanodiamonds on top of the microstrip, compared\nto the the case directly on YIG.\nYIG magnetization into precession by means of ferro-\nmagnetic resonance. It is remarkable that excitations at\nenergies as much as three to six times smaller than any\nNV center resonance have such a large e\u000bect on the NV\ncenter spin state. Second, the FMR-induced feature is\ncomparable in amplitude to the intrinsic NV resonances.\nThe large amplitude of the signal implies that a signif-\nicant number of NV centers in our laser spot must be\ncontributing to the signal. This suggests that since the\nnanodiamond \flm is 500 nm thick, the coupling must\nbe either long range (extending hundreds of nanometers)\nor that spin transport by means of spin di\u000busion plays a\nrole [28].\nTo probe the spatial extent of the coupling, we re-\npeated the measurement with our laser spot focused on\nthe nanodiamonds on top of the microwire, where the\nnanodiamonds are separated from the YIG by more than300 nm. This data can be seen in Fig. 3 (a) and compared\nto the signal when the diamond is directly on top of the\nYIG in Fig. 3 (b). Linecuts (at 16.5 mT), presented in\nFig. 3 (c), show that the FMR-induced feature is reduced\nbut clearly persists even when the nanodiamonds are not\nin direct contact with the YIG.\nWhile a clear explanation for the e\u000bect is not forthcom-\ning from our experimental results, we can make a few\nobservations. First, the insulating nature of both ma-\nterials rules out long range carrier-mediated transport\nof spins. Second, one of the unique aspects of this ex-\nperiment is that both the YIG magnetization and the\nNV center spins are out of equilibrium when the YIG\nis on FMR: YIG due to its resonance and the NV cen-\nters due to the hyperpolarizing action of the laser ex-\ncitation [29, 30]. This is in contrast to other systems\nwhere angular momentum is transferred between spin\nsub-systems, e.g. dynamic nuclear polarization or spin\npumping [1, 2, 31], where spins in one spin sub-system\nrelax by transferring polarization to another spin sub-\nsystem. The transfer of angular momentum from the\nYIG to the NV centers could result in relaxation towards\nequilibrium of both the spin systems, and thus be highly\ndesirable for the overall system.\nThis phenomenon o\u000bers the opportunity of probing\nspin transport in the absence of conductors and employ-\ning an all-optical readout. These advantages dramat-\nically reduce the potential confounding e\u000bects encoun-\ntered in studies of spin transport in metallic systems.\nThus this system o\u000bers an attractive and powerful ap-\nproach to better understanding microscopic details of re-\nlaxation and spin transport in magnetic heterostructures.\nExperiments are underway on other ferromagnets and\nstructures to gain more insight on this e\u000bect and how\nangular momentum is transferred between spin systems.\nIn summary, we have shown that the NV center spin\nstate can be manipulated by a dynamic coupling to the\nmagnetization of YIG. The availability of a wide selec-\ntion of ferromagnetic materials and structures (e.g., satu-\nration magnetization and anisotropies) potentially o\u000bers\na high degree of control in manipulating the NV spin\nstate. The potential for ultra high resolution imaging\nof ferromagnetic phenomena using individual NV centers\ncan have a signi\fcant impact as well. It should in\ru-\nence the \felds of spintronics and quantum information\nby combining the sensitivity of the NV center and the\ntunability and scope of ferromagnetism.\nThe authors wish to thank Yaroslav Tserkovnyak for\nuseful discussions. Funding for this research was provided\nby the Center for Emergent Materials at the Ohio State\nUniversity, an NSF MRSEC (Award Number DMR-\n0820414, ARO (Award number W911NF-12-1-0587) and\nDOE (Award number DE-FG02-03ER46054).4\n\u0003hammel@physics.osu.edu\n[1] C. P. Slichter, Principles of Magnetic Resonance\n(Springer-Verlag, New York, 1989).\n[2] F. Meier and B. P. Zakharchenya, Optical Orientation\n(North-Holland, Amsterdam, 1984).\n[3] P. C. Hammel, M. L. Roukes, Y. Hu, T. J. Gramila,\nT. Mamiya, and R. C. Richardson, Phys. Rev. Lett. 51,\n2124 (1983).\n[4] F. Pulizzi, ed., Nat. Mater. (Insight Issue:Spintronics) ,\nVol. 11 (Nature Publishing Group, 2012).\n[5] I. \u0014Zuti\u0013 c, J. Fabian, and S. D. Sarma, Rev. Mod. 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Lett. 88, 117601 (2002)."    },    {        "title": "1506.05879v2.Coherent_manipulation_of_a_Majorana_qubit_by_a_mechanical_resonator.pdf",        "content": "Coherent manipulation of a Majorana qubit by a mechanical resonator\nP. Zhang1and Franco Nori1, 2\n1CEMS, RIKEN, Saitama 351-0198, Japan\n2Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA\n(Dated: March 21, 2022)\nWe propose a hybrid system composed of a Majorana qubit and a nanomechanical resonator,\nimplemented by a spin-orbit-coupled superconducting nanowire, using a set of static and oscillating\nferromagnetic gates. The ferromagnetic gates induce Majorana bound states in the nanowire, which\nhybridize and constitute a Majorana qubit. Due to the oscillation of one of these gates, the Majorana\nqubit can be coherently rotated. By tuning the gate voltage to modulate the local spin-orbit\ncoupling, it is possible to reach the resonance of the qubit-oscillator system for relatively strong\ncouplings.\nPACS numbers: 71.10.Pm, 07.10.Cm\nI. INTRODUCTION\nMajorana bound states (MBS)1,2are recently attract-\ning increasing interest both theoretically and experimen-\ntally. These have been predicted to exist in arti\fcial\nstructures, such as nanowires with spin-orbit coupling\n(SOC) in proximity to a superconductor,3,4ferromag-\nnetic atom chains on top of a superconductor,5topo-\nlogical insulator/superconductor hybrid structures,6{11\nand superconducting circuits.12Recently, possible sig-\nnatures of MBS have been reported in nanowires,13{15\natom chains,16and topological insulator/superconductor\nstructures.17The MBS attract considerable atten-\ntion partly due to their hypothetical non-abelian\nanyonic statistics, which might allow the realiza-\ntion of topologically-protected quantum information\nmanipulation.2,18{20In parallel to the ongoing search of\nsome unambiguous con\frmation of MBS,21,22there are\nalso numerous theoretical studies on how to e\u000eciently\nexploit these MBS.\nOne promising application of MBS is to construct Ma-\njorana qubits.18It has been suggested that Majorana\nqubits might be robust against local perturbations and\nare hence promising to store quantum information.18,23,24\n(Note that Majorana qubits are not totally protected\nfrom decoherence, as studied in, e.g., Refs. 25{28.)\nFurthermore, Majorana qubits could be rotated by\ntopologically-protected braiding operations.19,29There-\nfore, among various realizations of qubits,30{36Majo-\nrana qubits are considered to be promising candidates for\nbuilding blocks of quantum information processors. The\nbraiding operations alone are insu\u000ecient to realize a uni-\nversal quantum gate based on a Majorana qubit.18For\nthe implementation of arbitrary qubit rotations, other\nnon-topological operations are required. Several schemes\nof such non-topological operations assisted by, e.g., phase\ngates,37,38quantum dots,39,40\rux qubits,41,42or mi-\ncrowave cavities,43have been proposed in the literature.\nNanomechanical resonators44could also be used to\nstudy non-topological operations of a Majorana qubit.\nFor example, quite recently, Kovalev et al.45have pro-posed to rotate a Majorana qubit by a magnetic can-\ntilever. Indeed, nanomechanical resonators have been\nutilized to couple to a wide range of quantum systems,\nincluding electric circuits,46optomechanical devices,47\natoms,48Cooper-pair boxes,49spin qubits,50or mi-\ncrowave cavities.51With the assistance of nanomechani-\ncal resonators, it is possible to perform important appli-\ncations such as quantum manipulations, quantum mea-\nsurements, as well as e\u000ecient sensing. These applications\nexploit the advantages of nanomechanical resonators,\ne.g., their large quality factors (103-106), high natural\nfrequencies (MHz-GHz), as well as the feasibility of reach-\ning the quantum ground states by cooling methods.52{54\nRecently, nanomechanical resonators have also been ex-\nploited to measure or manipulate the MBS.55{57Never-\ntheless, the study of hybrid systems58coupling nanome-\nchanical resonators to Majorana qubits is quite limited.\nThis work aims to contribute to this \feld. In this paper,\nwe propose another Majorana qubit-nanomechanical res-\nonator hybrid system in the framework of the spin-boson\nmodel,59based on a semiconductor nanowire in proxim-\nity to an s-wave superconductor. We show that a strong\ncoupling between a nanomechanical resonator and a Ma-\njorana qubit can be achieved, allowing an e\u000ecient trans-\nfer of quantum information between these two quantum\nsystems. Further, with braiding operations, it should be\npossible to realize a universal quantum gate based on a\nMajorana qubit.\nThis paper is organized as follows. First, we describe\nthe Majorana qubit and its coupling to a nanomechani-\ncal resonator. Afterwards, we numerically study the cou-\npling strength and the resonance condition of the hybrid\nsystem. Then, we solve the qubit-phonon dynamics and\nachieve a coherent control of the Majorana qubit. Fi-\nnally, we summarize our results.\nII. MODEL AND HAMILTONIAN\nAs illustrated in Fig. 1, we consider a semiconductor\nnanowire with a Rashba SOC of strength \u000bR\n0on the sur-\nface of an s-wave superconductor with a superconductingarXiv:1506.05879v2  [cond-mat.mes-hall]  24 Aug 20152\ngap \u0001. Three ferromagnetic gates, FM1, FM2 and FM3,\nare placed on top of and along the nanowire. Among\nthese gates, FM1 and FM3 are static while FM2 is free\nto harmonically oscillate along the nanowire (with a mass\nMand an oscillation frequency !0). The gates FM1\nand FM3 are su\u000eciently long (of the order of 1-10 \u0016m)\nwhile FM2 in between is relatively short (of the order of\n100 nm). These ferromagnetic gates induce a local Zee-\nman splitting in the nanowire. For simplicity, we take\nthe Zeeman splitting under the three gates to be identi-\ncal, with a magnitude of B0. An electric voltage Vcan\nbe applied on the gates to modulate the Rashba SOC lo-\ncally, e.g., from \u000bR\n0to\u000bR\nV. In our study, we consider the\ncase withB2\n0>(\u00012+\u00162), where\u0016is the chemical poten-\ntial in the nanowire. Therefore in the nanowire, the parts\nsubject to the Zeeman splitting (under the three ferro-\nmagnetic gates) are in the topological ( T) region. The\nremaining parts, without the Zeeman splitting, are in the\nnon-topological ( N) region.4,19As a result, the nanowire\nhas anN-T-N-T-N-T-Ndomain structure where the\nthreeTdomains are under the gates. At the six bound-\naries between the NandTdomains in the nanowire,\nMBS arise. As the two outer MBS are far apart, only the\nfour inner ones, schematically labeled as \r1-\r4in Fig. 1,\nare coupled due to hybridization arising from their small\nseparation60{63and are hence relevant to our considera-\ntion.\nTo lowest order, the hybrid system constructed above\ncan be described by the Hamiltonian\nH=HM+Hosc; (1)\nwhere the mutual coupling Hamiltonian of the MBS60{63\nHM=ign[l12(t)]\r1\r2+igt(l23)\r2\r3+ign[l34(t)]\r3\r4;\n(2)\nand the nanomechanical oscillator Hamiltonian\nHosc=p2\n2M+1\n2M!2\n0x2\n0(t): (3)\nThe coupling strengths gn;tdepend on the domain\nlengthslij. Due to the oscillation of the gate FM2,\nl12(t) =l0\n12+x0(t) andl34(t) =l0\n34\u0000x0(t) are time\ndependent. Here, x0(t) stands for the displacement of\nthe gate FM2 from its balance position, which is much\nsmaller than the static domain lengths l0\n12;34. Therefore,\nto \frst order in x0, one has\nHM=i[gn(l0\n12) +x0(t)g0\nn(l0\n12)]\r1\r2+igt(l23)\r2\r3\n+i[gn(l0\n34)\u0000x0(t)g0\nn(l0\n34)]\r3\r4: (4)\nThe four MBS, satisfying f\ri;\rjg=\u000eij, can be used\nto construct a Majorana qubit as follows.2,18At \frst we\nde\fne two Dirac fermion operators64c\"= (\r1+i\r4)=p\n2\nandc#= (\r2+i\r3)=p\n2. The Hilbert space of HMcan\nthen be spanned by states jn\";n#i, with the fermion oc-\ncupation numbers n\"=cy\n\"c\"andn#=cy\n#c#. Due to the\nxy\nγ1γ2γ3γ4\nFM1\nFM3\nsuperconductorl\n   12\nFM2\nω0\nV(a)\nl\n   23l\n   34\n(b)\nx(µm)\nB/B 0|Ψ|2(arb. units)2\n1\n0\n1 0 -13\n2\n1\n0FIG. 1: (Color online) (a) Schematic diagram of the proposed\nMajorana qubit-nanomechanical resonator hybrid system. A\nsemiconductor nanowire is placed on the surface of an s-wave\nsuperconductor. Three ferromagnetic gates are on top of the\nnanowire, of which FM1 and FM3 are su\u000eciently long (of the\norder of 1-10 \u0016m) and static, while FM2 is relatively short (of\nthe order of 100 nm) and free to oscillate as a harmonic oscil-\nlator. The ferromagnetic gates induce a local Zeeman split-\ntingB0in the underlying nanowire, and can also be used to\nmodulate the local Rashba SOC strength by applying an elec-\ntric voltage V. (b) Wave amplitude j\tj2of the four coupled\nMBS at a static state in an InSb nanowire with the set-up\nshown in (a). The red dashed and red solid curves respec-\ntively correspond to the wave amplitudes of the lowest two\neigenstates (close to the zero energy). These two states con-\nstitute the Majorana qubit. The dotted curve with the scale\non the right-hand side of the frame indicates the pro\fle of the\ninhomogeneous Zeeman splitting along the nanowire. In the\ncalculation l12=l34= 150 nm, l23= 400 nm, B0= 1 meV,\nand \u0001 = 0:5 meV. The gate voltage Vis zero and the Rashba\nSOC is homogeneous along the nanowire, with a strength\n\u000bR\n0= 20 meV nm.\nconservation of fermion parity, the states fj0;1i;j1;0ig\nandfj0;0i;j1;1igform two decoupled (odd and even)\nsectors.2,41,43We assume that there is no high-energy\nexcitation (e.g., no Cooper-pair breaking in the super-\nconducting substrate) and restrict our study to the odd\nsector with n\"+n#= 1. For convenience, we de\fne\npseudo-spinsj\"i=j1;0iandj#i=j0;1i, and use them\nas the two logical states of the Majorana qubit.39,41,43,45\nIn this pseudo-spin space, i\r1\r2=\u0000i\r3\r4=\u0000\u001byand\ni\r2\r3=\u0000\u001bz. The nanomechanical oscillator is quan-3\ntized in the Fock space fjnigwithjni=(ay)n\np\nn!j0i, where\na=q\nM! 0\n2~(x0+i\nM! 0p) is the annihilation operator of\nphonons. Consequently, in the space fj\"#i\njnig, the\nhybrid system can be simply described by the spin-boson\nHamiltonian,59\nHe\u000b=\u0000\"\n2\u001bz\u0000\u000e\u001by+g(ay+a)\u001by+n~!0; (5)\nwith the constant omitted. Here \"= 2gt(l23),\u000e=\ngn(l0\n12)\u0000gn(l0\n34), andg=\u0000~x0[g0\nn(l0\n12) +g0\nn(l0\n34)], where\n~x0= [~=(2M! 0)]1=2is the zero-point motion of the oscil-\nlator.\nIII. HYBRIDIZATION OF MAJORANA BOUND\nSTATES\nIn this section, we study the MBS and their mu-\ntual coupling. In the static state, the inhomoge-\nneous nanowire can be described by a tight-binding\nmodel. Using the Bogoliubov-de Gennes basis \t j=\n(fj\";fj#;fy\nj#;\u0000fy\nj\"), wherefj\u0011stands for the fermion op-\nerator of a spin- \u0011(\u0011=\",#) electron on the j-th lattice\nsite, the particle-hole Hamiltonian reads5\nHBDG =1\n2X\nj[\ty\nj^hj\tj+ (\ty\nj^tj\tj+1+ H:c:)];(6)\nwhere\n^hj= (2t0\u0000\u0016)s0\u001cz+ \u0001s0\u001cx+Bjsy\u001c0; (7)\n^tj=t0s0\u001cz+i\u000bjsz\u001cz: (8)\nIn the above Hamiltonian, the Pauli matrices \u001cx;y;z act\non the particle-hole space and sx;y;z act on the real\nspin space. The spin-diagonal hopping energy is t0=\n~2=(2m\u0003a2), and the spin-o\u000b-diagonal hopping energy is\n\u000bj=\u000bR\nj=(2a). HereBjand\u000bR\njare the on-site Zeeman\nsplitting and Rashba SOC, respectively, m\u0003is the e\u000bec-\ntive electron mass, and ais the lattice spacing in the\ndiscretized tight-binding model. In the T(N) domains\nBj=B0(Bj= 0) and\u000bR\nj=\u000bR\nV(\u000bR\nj=\u000bR\n0). When the\ngate voltage Vis zero,\u000bR\nV=\u000bR\n0.\nHere, to lowest order, we follow Refs. 45 and 11 to\ninvestigate the coupling strength gn(gt) approximately\nin an isolated T-N-T(N-T-N) three-domain structure.\nIn such a simpli\fed model, the inner N(T) domain has\na \fnite length, while the outer two T(N) domains are\nassumed to be in\fnitely long. By numerically diagonal-\nizing this three-domain system, one can obtain the en-\nergy splitting of the two MBS localized at the two T/N\nboundaries. This energy splitting is precisely caused by\nthe coupling of the MBS. With gnandgtknown numeri-\ncally, the Majorana qubit can be well described by \"and\n\u000e, and the qubit-phonon coupling gcan be obtained also\nfromg0\nn[refer to Eq. (5)]. Moreover, by exactly diagonal-\nizing the Hamiltonian of the genuine N-T-N-T-N-T-Ndomain structure as shown in Fig. 1(a), one can obtain\nthe hybrid four MBS under consideration.\nIn this work, we consider an InSb quantum wire13with\nan e\u000bective electron mass m\u0003= 0:015me, a Rashba SOC\n\u000bR\n0= 20 meV nm, and a large Landau factor gL\u001950.\nWe choose the superconducting gap \u0001 = 0 :5 meV, the\nlocal Zeeman splitting B0= 1 meV, the chemical poten-\ntial\u0016= 0, and the lattice constant a= 10 nm. The total\nlattice site number is chosen as 1000 for the numerical\nconvergence. In Fig. 2, we show the dependence of the\nMajorana coupling strength gn(gt) on the length of the\nN(T) domainln(lt), as well as the derivative g0\nnver-\nsusln. Further, as an example, in Fig. 1(b) we present\nthe wave amplitude j\tj2of the four hybrid MBS, when\nl12=l34= 150 nm and l23= 400 nm. In Fig. 1(b), the\nred dashed and red solid curves stand for the wave am-\nplitudes of the lowest two eigenstates (close to the zero\nenergy) in the static inhomogeneous nanowire. The state\ncorresponding to the red solid (dashed) curve is mainly\ncontributed by the \r2and\r3(\r1and\r4) MBS. Here,\nthese two states form the Majorana qubit.\ng′\nngtgn\nln(t)(nm)\ng′\nn(µeV/nm)gn(t)(µeV)0\n-0.5\n-1\n-1.5\n-2\n500 400 300 200 100 0100\n0\n-100\n-200\nFIG. 2: (Color online) Majorana coupling strength gn(gt)\nversusln(lt), the length of the inner N(T) domain between\nthe two outer T(N) domains. The derivative g0\nnversuslnis\nalso shown, with the scale on the right hand side of the frame.\nThe necessary parameters for the calculation are speci\fed in\nthe main text.\nIV. QUBIT-PHONON COUPLING AND\nRESONANCE\nWe now look into the qubit-phonon coupling and the\nresonance condition. We assume that the nanomechan-\nical oscillator FM2 has a mass M= 10\u000015Kg and an\noscillation frequency !0= 5 MHz. With these parame-\nters, the zero-point motion of the oscillator is calculated\nto be ~x0= 0:1 pm. We consider the symmetric case\nwithl0\n12=l0\n34= 150 nm, and hence we have \u000e= 0 and\ng= 0:2 MHz in Eq. (5). The longitudinal length l23of\nthe FM2 gate is chosen as 400 nm, such that the Rabi4\nresonance condition \"\u0019\u0000!0can be easily satis\fed, e.g.,\nby further subtly adjusting the gate voltage Vwhich con-\ntrols the local Rashba SOC strength \u000bR\nV. In Fig. 3, we\npresent the variation of \"as well asgversus\u000bR\nV. It is\nshown that when slightly adjusting V, and hence \u000bR\n0,\nthe resonance point \"\u0019\u0000!0can be reached while the\nqubit-phonon coupling gremains almost invariant. This\nqubit-phonon coupling is relatively strong, in view of the\nlong lifetime of the Majorana qubit and the high qual-\nity factor of the nanomechanical oscillator. In principle,\nthe qubit-phonon coupling can be stronger when the do-\nmain length l12(as well as l34) becomes smaller (refer\nto Fig. 2). However, if the two edge modes \r1and\r2\n(as well as \r3and\r4) are too close and hence their hy-\nbridization becomes quite strong, the model Hamiltonian\n(2) describing four distinguishable MBS might fail.\ng/ω 0ε/ω 0\nresonance\nαR\nV/αR\n0\ng/ω 0ε/ω 00.1\n0\n-0.1\n1.0685 1.068 1.067520\n10\n0\n-10\n-20\nFIG. 3: (Color online) Qubit splitting \"and the qubit-phonon\ncouplingg(the scale is on the right-hand side of the frame)\nversus\u000bR\nV, the Rashba SOC strength in the topological ( T)\ndomains modulated by the gate voltage V.\nV. QUBIT-PHONON DYNAMICS\nHere we study the dynamics of the qubit-phonon hy-\nbrid system. To achieve this, we make use of the Python-\nbased Qutip software package65,66to solve the Lindblad\nmaster equation,\n_\u001a(t) =\u0000i\n~[He\u000b;\u001a(t)] +1\n2X\nkn\n[Lk;\u001a(t)Ly\nk]\n+ [Lk\u001a(t);Ly\nk]o\n: (9)\nIn this equation, \u001ais the density matrix of the qubit-\nphonon system, and Lkare the Lindblad operators ac-\ncounting for the dissipation of the hybrid system due to\nits coupling to the environment. The relaxation of the\nMajorana qubit is taken into account by L1=p\n1=T1\u001b\u0000,\nwhile the dissipation of the nanomechanical resonator isincluded by L2=p\n(\u0016n+ 1)!0=Qa andL3=p\n\u0016n!0=Qay.\nHere \u0016n= [exp( ~!0=kB~T)\u00001]\u00001is the thermal phonon\nnumber in equilibrium with the environmental tempera-\nture ~T,Qis the quality factor of the nanomechanical os-\ncillator, and T1is the usual relaxation time of the qubit.\nBy solving the master equation, one can obtain the time\nevolution of the qubit and phonon occupations.\nIn our model, the temperature ~Tis set as 10 mK and\nhence the thermal phonon number \u0016 nis as large as 258.\nTherefore, an additional cooling of the oscillator52{54is\nrequired, e.g., as also applied in a proposed nanomechan-\nical resonator{nitrogen-vacancy center hybrid system.51\nWe assume that after side-band cooling52{54the phonons\nthermally occupy the lowest several quantum states with\na small phonon number, e.g., n= 0:3. The initial state of\nthe Majorana qubit is set as j\"i, implying that a single\nelectron is splitted into the \r1and\r4Majorana fermions.\nExperimentally, this initial state might be realized when\nonly the FM1 and FM3 gates are in proximity to the\nnanowire before inserting the middle FM2 gate. The re-\nlaxation time of the Majorana qubit depends on the con-\ncrete set-up and environment. Following Refs. 26 and 27,\nwe typically set T1around 100 \u0016s.\nIn Fig. 4, we plot the time evolution of the occupations\nof the qubit and phonons respectively, with di\u000berent val-\nues ofT1andQ. As indicated by the \fgure, quantum\ninformation can be e\u000bectively transferred back and forth\nbetween the Majorana qubit and the nanomechanical res-\nonator. During this process, the single electron in the\nnanowire alternatively occupies (back and forth) the pair\nof MBS:\r1and\r4, or\r2and\r3. Inversely, this quan-\ntum information transfer can also modulate the motion of\nthe oscillator, e.g., the oscillation amplitude. In fact, as\nthe nanomechanical resonator is near its quantum ground\nstate, the oscillation amplitude hx2\n0i, which might be ob-\nservable, is almost linearly related to the phonon number.\nThis is becausehx2\n0i/h (ay+a)2i\u00192hayai+ 1 = 2n+ 1.\nTherefore, the dashed curves in Fig. 4, representing the\ntime evolution of the phonon number, also supply in-\nformation on the change of the oscillation amplitude of\nthe resonator due to its coupling with the qubit. This\nphenomenon signi\fes the presence of a Majorana qubit.\nCertainly, for better performance of this hybrid system\n(e.g., with a higher \fdelity), a higher quality factor of\nthe resonator and a longer relaxation time of the qubit\nare preferred.\nVI. DISCUSSION\nHere we brie\ry compare our model to the one pro-\nposed by Kovalev et al. ,45where a vibrating cantilever is\nutilized to rotate a Majorana qubit. The e\u000bective Hamil-\ntonian in their model [Eq. (7) in Ref. 45] is in fact equiv-\nalent to the one in our manuscript [Eq. (5)]. This is\nunderstandable as both are in the framework of the spin-\nboson model. Note that for both cases there exists a\nstatic o\u000b-diagonal term [for our case, that is the \u000eterm5\n(b)T1= 150 µs\ngt(π)Occupation\n5 4 3 2 1 01\n0.5\n0(a)T1= 50 µs\nphononsQ= 3×105: qubitphononsQ= 106: qubit1.5\n1\n0.5\n0\nFIG. 4: (Color online) Time evolution of the occupations\nof the Majorana qubit (solid curves) and phonons (dashed\ncurves) which are in Rabi resonance. The qubit relaxation\ntime is set as 50 \u0016s in (a) and 150 \u0016s in (b). The calcula-\ntions for both (a) and (b) are performed with two di\u000berent\nresonator quality factors: Q= 106andQ= 3\u0002105.\nin Eq. (5)] coupling the two levels of the qubit in the\nHamiltonian. To neglect this term, in order to simplify\nthe theoretical analysis, some conditions have to be satis-\n\fed. Speci\fcally, in Ref. 5, a certain equilibrium angle ( \u00120\nthere) of the vibrating cantilever has to be established. In\nour opinion, exactly solving this angle and then adjust-\ning the experimental setup correspondingly45are chal-\nlenging. However, in order to neglect the constant o\u000b-\ndiagonal term in our case, the experimental setup must\nbe mirror-symmetric about the middle point of the FM2gate, i.e.,l0\n12=l0\n34. Therefore, we think that our model\nis more easily accessible by experiments and hence more\nadvantageous.\nVII. CONCLUSIONS\nIn conclusion, we have proposed a hybrid system com-\nposed of a Majorana qubit and a mechanical resonator,\nimplemented by a semiconductor nanowire in proxim-\nity to an s-wave superconductor. In this proposal, three\nferromagnetic gates are placed on top of and along the\nnanowire; the two outer gates are static and the inner\none is free to oscillate harmonically as a mechanical res-\nonator. These ferromagnetic gates induce a local Zeeman\nsplitting and give rise to four Majorana bound states,\nconstituting a Majorana qubit in the nanowire. The\ndynamical hybridization of the Majorana bound states,\narising from the motion of the oscillating gate, results in\na coherent coupling between the Majorana qubit and the\nmechanical resonator.\nThis hybrid system can be adjusted to be in resonance,\ne.g., with the assistance of a gate voltage on the ferro-\nmagnetic gates, which controls the Rashba SOC locally in\nthe nanowire. Our study reveals that under resonance, a\nstrong coupling between the qubit and the resonator can\nbe achieved. Consequently, quantum information can be\ne\u000bectively transferred from the Majorana qubit to the\noscillator and then back to the qubit. 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Commun. 184, 1234 (2013)."    },    {        "title": "1809.06836v2.Local_manipulation_of_quantum_magnetism_in_1D_ultracold_Fermi_gases_across_narrow_resonances.pdf",        "content": "arXiv:1809.06836v2  [cond-mat.quant-gas]  19 Sep 2018Local manipulation of quantum magnetism in 1D ultracold Fer mi gases across narrow\nresonances\nLei Pan,1,2Xiaoling Cui,1,∗and Shu Chen1,2,3, †\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China\n3Collaborative Innovation Center of Quantum Matter, Beijin g, China\n(Dated: April 5, 2022)\nEffective range is a quantity to characterize the energy depe ndence in two-body scattering\nstrength, and is widely used in cold atomic systems especial ly across narrow resonances. Here we\nshow that the effective range can significantly modify the mag netic propertyof one-dimensional (1D)\nspin-1/2 fermions in the strongly repulsive regime. In particular, the effective range breaks the large\nspindegeneracy inthehard-corelimit, andinducesaHeisen bergexchangeterminthespinchainthat\nis much more sensitive to the local density than that induced by the bare coupling. With an external\nharmonic trap, this leads to a very rich magnetic pattern whe re the anti-ferromagnetic(AFM) and\nferromagnetic (FM) correlations can coexist and distribut e in highly tunable regions across the trap.\nFinally, we propose to detect the range-induced magnetic or der in the tunneling experiment. Our\nresults can be directly tested in 1D Fermi gases across narro w resonance, and suggest a convenient\nroute towards the local manipulation of quantum magnetism i n cold atoms.\nIntroduction. Energy dependent interaction is com-\nmon in nature, which roots deeply in the renormaliza-\ntion group theory. In cold atomic systems, such energy\ndependence appears naturally in the Feshbach resonance\nwhich essentially relies on the energy difference of (open)\natomic and (closed) molecular channels[ 1]. To describe\nthe low-energy physics, an effective range expansion is\nusually introduced to incorporate the energy dependence\nof coupling strength g(E), which reads\n1\ng(E)=1\ng(0)+r0E, (1)\nwhereEis the energy of two colliding particles in the\ncenter-of-mass frame, and g(0) is coupling strength at\nthreshold energy. Here r0is the effective range, which\ncrucially depends on the width of resonance. In par-\nticular, for narrow resonances, r0is typically large as\nto be comparable with the inter-particle distance. Pre-\nvious studies have revealed interesting effects of finite\nrange in cold atoms systems, including the generation\nof stronger interaction effects[ 2–6], the modification of\nFermi superfluids[ 7,8], the sub-leading high-momentum\ntail[9–14], and the stabilization of repulsive polaron[ 15]\nand p-wave system[ 16].\nIn this work, we reveal the significant effect of effective\nrangetothequantummagnetismofspin-1 /2Fermigases.\nHere we take the 1D Fermi gas in strongly repulsive\nregime, which features an impenetrable nature and thus\nsupports a hidden “lattice” structure. In this regime, the\nsystem is well described by an effective Heisenberg spin-\nchain Hamiltonian which exhibits exotic quantum mag-\nnetic properties and provides additional insights into the\nsimulation of quantum magnetism without lattice[ 17–30]\nand suchspin-chainsystemswererealizedexperimentally\nrecently[31]. Previous studies based on zero-range inter- AFMFM Mixed \nPhase \n0/10<g 0/10>g0\nFIG. 1. (Color online). Schematic diagram of magnetic order\nofaspin-1 /2fermion systemwithafinite-range( r0/negationslash= 0) inter-\naction in a harmonic trap. As we tune the interaction (1 /g0)\nfrom the repulsive to the attractive side, the system adiaba t-\nically goes from a AFM- correlated spin chain to FM-AFM-\nFM mixed and finally to fully FM-correlated chain, which\nis in sharp contrast to the zero-range case with either AFM\nphase (1 /g0>0) or FM phase(1 /g0<0). The red and green\nbonds represent the AFM and FM couplings between spins,\nrespectively.\nactions have shown that the system can host either an\nanti-ferromagnetic (AFM) or a ferromagnetic (FM) spin\ncorrelation, depending on the sign of coupling strength\nor other small perturbations. Moreover, at the infinite\ncoupling (hard-core) limit, spin and charge are fully de-\ncoupled and the system exhibits huge spin degeneracy, at\nwhich point the FM transition is predicted[ 32]. Here, we\nwill show that the inclusion of a finite effective range can\nqualitatively change above conclusions and brings much\nricher magnetic structures to the system.\nOur results can be summarized in Fig. 1. In the pres-\nence of a finite range r0, as tuning the inverse coupling\n1/g0from the repulsive to attractive side, the system2\nadiabatically goes from a AFM-correlated spin chain to\nFM-AFM-FM mixed and finally to fully FM-correlated\nchain. The spatially modulated magnetic correlation is\ndue to the range-modified Heisenberg coupling in an ef-\nfective spin chain model, which is more sensitive to the\nlocal density as compared to that induced by bare cou-\npling (see Eq. 10). Our results demonstrate an adiabatic\nformationofFM domainsfromthe AFM state, whichhas\nnot been achieved up to date. We further propose to ver-\nify these magneticpropertiesinthe tunneling experiment\nof tilted harmonic potential.\nRange-modified effective spin chain. We begin with de-\nriving an effective spin chain model for strongly repulsive\nspin-1/2 fermions( ↑,↓) in the presence of a finite range.\nThe original Hamiltonian is H=H(0)+U, (here/planckover2pi1= 1)\nH(0)=/summationdisplay\ni/parenleftbigg\n−1\n2m∂2\n∂x2\ni+1\n2mω2\nhox2\ni/parenrightbigg\n,(2)\nU=/summationdisplay\ni,jgijδ(xi↑−xj↓). (3)\nHeregijfollows the effective range expansion: 1 /gij=\n1/g0+r0Eij, withg0the bare coupling constant and Eij\nthe relative energy of two colliding particles xi↑andxj↓.\nThepresentstudywillfocusonthenearresonanceregime\nwith large g0and small r0. Here the confinement length\nis defined as aho= (mωho)−1/2.\nTo highlight the range effect, let us first consider the\ncase ofg0=∞. In this case, without the range ( r0= 0)\nthe collision of atoms is forbidden due to hard-core in-\nteraction and the spins can distribute in an arbitrary\norder in coordinate space, giving the large spin degen-\neracy. When turn on the range ( r0/ne}ationslash= 0), however, the\natoms only experience hard-core interaction at zero rel-\native energy( Erel= 0) but not at finite Erel, and the\nfinite-Erelinteraction causes a super-exchange of spins\nat neighboring orders in the coordinate space, giving rise\nto an effective spin chain model. Following the standard\nprocedure, we obtain the effective Heisenberg spin-chain\nsolely induced by r0:\nHr\neff=r0/summationdisplay\nlJr\nl/parenleftbigg\nsl·sl+1−1\n4/parenrightbigg\n, (4)\nherelis the order index of particles in coordinate space,\nand the Heisenberg coupling is\nJr\nl=2N!\nm2/integraldisplay\ndxEij/vextendsingle/vextendsingle/vextendsingle∂D\n∂xij|xij=0/vextendsingle/vextendsingle/vextendsingle2\nθ(···< xi=xj<···),(5)\nwherexij=xi−xj;D({xi}) is the Slater determinant of\nN fermions occupying the lowest N-level of H(0), andEij\nis the relative collision energy of two particles ( xi, xj)\nin theD({xi}); in the θ-function xi(=xj) is with order\nindexl.\nTo verify the range-induced spin-chain model, we have\nexactly solved the two-body ( ↑↓) and three-body ( ↑↑↓)r0\nFIG. 2. (Color online). Energy spectrum for 1D trapped\n↑↓and↑↑↓systems as a function of effective range r0at 1D\nresonance (1 /g0= 0). The linear fit is based on the effective\nspin-chain model ( 4). Here the energy and length units are\nωhoandahorespectively.\nproblems in a trapped system with tunable range. In\nFig.2, we plot the obtained energy spectra of these sys-\ntems as a function of r0at 1/g0= 0, in comparison with\nthe linear fit from the model Hr\neff. We can see that the\neffective model can well predict the real spectra of both\nsystems for small r0(≤0.1aho). This validates the effec-\ntive spin-chain model used for larger systems.\nIn combination with the spin-chain model for small\n1/g0[18–28], we write the final effective model in the\nlimit of large g0and small r0as:\nHeff=/summationdisplay\nl/parenleftbigg1\ng0Jg\nl+r0Jr\nl/parenrightbigg/parenleftbigg\nsl·sl+1−1\n4/parenrightbigg\n,(6)\nwithJg\nl:\nJg\nl=2N!\nm2/integraldisplay\ndx/vextendsingle/vextendsingle/vextendsingle∂D\n∂xij|xij=0/vextendsingle/vextendsingle/vextendsingle2\nθ(···< xi=xj<···).(7)\nPhysically, both the bare coupling ( g0) and effective\nrange (r0) produce the same isotropic Heisenberg term\nis because both of them take effect in the spin-singlet in-\nteraction channel, thus the effective model is determined\nby the same spin-projection operator [ 27].\nHeisenberg couplings. Before studying the quantum\nmagnetism, we first examine the density dependence\nofJg\nl, Jr\nlfor a homogeneous large system, and the\ntrapped case can be deduced from the local density ap-\nproximation(LDA). Previously, Jg\nlwas shown to depend\non the cubic density ( ∼n3)[18,33], by extrapolating\nthe nearest-neighboring exchange coupling in Hubbard\nmodel to continuum[ 34,35]. Here we point out an alter-\nnative way to derive Jg\nlandJr\nlfrom Eqs.( 7,5) through3\n−3 −2 −1 0 1 2 30510 15 20 25 \n−3 −2 −1 0 1 2 30246LDA v.s Exact\n  LDA v.s Exact\nglJrlJ\n2 Nl− 2 Nl−(a) (b)\nFIG. 3. (Color online). Heisenberg couplings Jg\nl(a) and\nJr\nl(b) (in the units of1\nm2a3\nhoand1\nm3a5\nhorespectively) in a\nharmonic trap. The star, circle, diamond, square, and trian -\ngular points are exact solutions of Eqs.( 8,9) for total number\nN= 2,3,4,5,6. The red lines are from analytical expressions\n(10) together with Thomas-Fermi density (see text).\nthe momentum averaging below the Fermi sea:\nJg=2n\nm2/angbracketleftBig/parenleftbiggk1−k2\n2/parenrightbigg2/angbracketrightBig\n, (8)\nJr=2n\nm3/angbracketleftBig/parenleftbiggk1−k2\n2/parenrightbigg4/angbracketrightBig\n, (9)\nhere/an}bracketle{tF(k1,k2)/an}bracketri}ht ≡/integraltext /integraltext\nF(k1,k2)dk1dk2//integraltext /integraltext\ndk1dk2, and\nthe integration is for k1,k2∈[−kF,kF], with kF\nthe Fermi momentum determined by the density n=\nk3\nF/(6π2). The essence of Eqs.( 8,9) is to reformulate the\nmany-fold integration into the combination of the local\ndensity and a pair-averaged function in terms of the rel-\native momentum of two particles within the Fermi sea.\nThe procedure leads to\nJg=2π2n3\n3m2, Jr=4π4n5\n15m3. (10)\nRemarkably, here the range-induced coupling Jrhas a\nmuchmoresensitivedependenceonthelocaldensitythan\nJg, which we will show below to significantly affect the\nquantum magnetism in the trapped system. In Fig. 3,\nwe show Eq. 10can well reproduce the exact solutions of\nJg\nl, Jr\nlfrom Eqs.( 8,9) for trapped systems up to N= 6,\nhere for the local density we have used Thomas-Fermi\napproximation nl→n(¯xl) =1\nπ/radicalBig\n2m(NωT−1\n2mω2\nT¯x2\nl),\nwith ¯xl=/an}bracketle{t(xl+xl+1)/2/an}bracketri}ht.\nTo this end we can rewrite the spin-chain Hamiltonian\n(6) asHeff=/summationtext\nlJeff\nl/parenleftbig\nsl·sl+1−1\n4/parenrightbig\n, where the effective\ncoupling depends on the coupling, range, and local den-\nsitynl:\nJeff\nl=2π2n3\nl\n3m2/parenleftbigg1\ng0+2π2\n5mn2\nlr0/parenrightbigg\n. (11)\nSpatially modulated quantum magnetism. From the ex-\npression of Jeff\nl, we can see that its sign can be effectivelytuned by local density nl, distinct from the zero-range\ncase where the magnetic property is solely determined by\nthe sign of coupling strength. This immediately leads to\ntwo effects for trapped system with inhomogeneous den-\nsity. First, the system is no longer described by a single\ncoupling strength, and there is no exact hard-core limit\nwith largespin degeneracy. Secondly, given r0>0 and in\nthe regime of 1 /g0<0, particles at different regions in-\nside the trap may experience different signs of Jeff, which\nmeans that the AFM and FM magnetic correlation can\ncoexist in the system, i.e., the quantum magnetism can\nbe locally manipulated.\n   0.25    0.2    0.15    0.1   0.05 7.85 7.9 7.95 88.05 \n1/ g0E(a)\n\u0012\n\u0013\n\u0014\n\u0015 USJQMF\u0001EFHFOFSBUF EPVCMF\u0001EFHFOFSBUF \u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\"'. \n\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001'. \u0012\u0013\u0014\u0015\u0016\n\u0016\nOPO\u000eEFHFOFSBUF \n@(b) \nFIG. 4. (Color online). (a) Energy spectrum of four fermions\n(N↑=N↓= 2) as a function of −1/g0(in the unit of maho) in\na trapped system. Here we take r0= 0.05aho. (b) Schematics\nof AFM (red lines) and FM (green lines) magnetic correlation\nin the chain at different coupling strengths as marked in (a).\nThe dashed lines with crossings refers to the zero effective\ncoupling Jeff\nl= 0.\nIn Fig.4(a), we show the energy spectrum of four\nfermions ( N↑=N↓= 2) by solvingthe spin-chainHamil-\ntonian (6) at a given r0= 0.05aho. Totally there are\nsix energy levels, two with total spin S= 0, three with\nS= 1 and one with S= 2, similar to the zero-range\ncase[5]. With a finite range, the six levels no longer cross\neach other at 1 /g0= 0, while the ground state transition\n(fromS= 0 toS= 2) moves to the negative coupling\nside atgc<0.\nNow we analyze the adiabatic change of the ground\nstate (with S= 0) before the transition (in the regime\n−1/g0<−1/gc). In the positive coupling side of reso-\nnance, the ground state holds the AFM correlations in\nthe spin chain given by all positive Jeff\nl(marked as 1○\nin Fig.4(a,b)). As increasing −1/g0across resonance to\nnegative coupling side, two states becomes degenerate at\npoint2○, where the Heisenberg coupling at the edges of\nthe chain touches zero( Jeff\n1=Jeff\n3= 0) and the edge par-\nticlesaredecoupledfromthechain, givingtwodegenerate\nspin states ( S= 0 and S= 1). Further increasing 1 /g0\nbeyond this point, the edge spins have FM correlation\nwith negative coupling, while the center are AFM corre-\nlated with positive coupling (see 3○). When reaching 4○,4\nthe center coupling becomes zero ( Jeff\n2= 0), and the sys-\ntem is divided into two independent magnetic domains\nwith FM correlation. At this point, three spin states are\ndegenerate, correspondingto twoFM(triplet) pairsform-\ning total spin S= 0,1,2. Beyond this point, the ground\nstate of the system changes to S= 2 FM state, and all\nsites are with FM correlations ( 5○).\n0 0.005 0.01 0.015 0.02 −0.1−0.08−0.06−0.04−0.020\nr01/g 0\nFM AFM \nMixed\nFIG. 5. (Color online). Phase diagram of magnetic or-\nder in terms of the bare coupling and effective range. Here\nN↑=N↓= 4. The units of 1 /g0andr0aremahoandaho\nrespectively.\nAbove picture can be generalized to an arbitrary num-\nber of particles. Take the spin-balanced case N↑=N↓=\nN/2 for example, as increasing −1/g0the ground state of\nthe system(with S= 0) is expected to cross a sequence\nof degeneracies with other spin states until the transition\nto FM ( S=N/2) state. The first degeneracy occurs\nat 1/g0=−2π2\n5mn2\n1r0, whenJeff\n1=Jeff\nN−1= 0 and two\nedge spins are separated from the system giving a two-\nfold degeneracy, see the upper line in Fig. 5. Increasing\n−1/g0beyond this point, the Heisenberg couplings tran-\nsit from pure AFM type to FM-AFM-FM mixed type.\nIn this mixed phase, the trap center shows the AFM\ncorrelation (with Jeff>0) because of higher density,\nwhile the trap edge shows FM correlation with Jeff>0\nbecause of lower density. In this regime, a m+ 1-fold\n(1≤m≤N/2) degeneracy occurs at 1 /g0=−2π2\n5mn2\nmr0\nwhenJeff\nm=Jeff\nN−m= 0. Continuously increasing −1/g0,\nthe regions with FM correlations becomes enlarged while\nAFM correlation becomes reduced, giving the increasing\n/an}bracketle{tS2\nLS2\nR/an}bracketri}ht, whereSL/Ris the total spin of left/right part\nof the trapped system. The ground state transition to\nFM state happens exactly at the N/2 + 1 fold degen-\neracy point(see lower phase boundary in Fig. 5), when\n1/g0=−2π2\n5mn(0)2r0withn(0) the density at trap cen-\nter. At this point/radicalbig\n/an}bracketle{tS2\nLS2\nR/an}bracketri}htreaches the maximum value\nN/4(N/4 + 1), suggesting two large and separated FMdomains formed at the left and right regions of the trap.\nTunneling experiment . Now we come to the exper-\nimental detection of the range-induced magnetic or-\nders, using the tunneling techniques as established in\nexperiments[ 31]. By varying the magnetic field gradi-\nent and tilting the potential barrier, one can control the\nnumber of atoms tunneling out of the trap and probe the\nspin structure. Here we propose the measure the possi-\nbility of having all spin- ↓atoms tunneling from the right\nside of the trap, which is given by the weight of the full\nspin separated configuration ( | ↑↑...↓↓.../an}bracketri}ht) in the wave\nfunction, as denoted by P↓in following discussions.\n   0.1    0.05 0 −0.05 −0.100.050.10.150.20.250.30.33\n  \n4 5 6 7 8 9 10 00.05 0.1 0.15 0.2 0.25 0.3 0.33 \n  P\nP\n-1/g 0 NMixed(r0=0.02)\nFM(r0=0)\nAFM(r0=0)(a) (b) Mixed \nFM \nAFM \nFIG. 6. (Color online). (a) Probabilities of all spin- ↓atoms\ntunneling from the right side of the titled trap, denoted by\nP↓. Red solid line is for system with range r0= 0.02aho, by\nadiabatically following the ground state in g0>0 side to g0<\n0 side. The cross point marks the location of FM transition,\nwhenP↓shows a maximum. Blue dashed and green dotted\nlines are for zero-range systems adiabatically following t he\nAFM ground state( S= 0) and the FM state( S=N/2). Here\nN↑=N↓= 2, the unit of 1 /g0ismaho. (b) The maximum\nofP↓as a function of particle number Nfor spin-balanced\nfermions. Red, blue and green lines are the same as in (a).\nIn Fig.6(a), we show P↓as a function of −1/g0for\nN↑=N↓= 2 system, by adiabatically following certain\nstatefromthe g0>0side. Without range,weseethat P↓\nis very small ( ∼4%) if following the AFM ground state,\nandcanbeaslargeas16 .7%iffollowingFMstate, consis-\ntent with the experiment [ 31]. Turning on the range and\nfollowing the ground state in g0>0 side,P↓is no longer\na constant but varies sensitively with −1/g0, suggesting\nthesignificantchangeofmagneticstructures/correlations\nin the trap. In particular, we see that P↓reaches a max-\nimum near the N/2+1-fold degeneracy point, where the\neffective coupling at the trap center touches zero and the\nsystem is composed of two FM domains (each with spin\nS=N/4). The maximum value can be then deduced by\nexpanding the S= 0 state by two spins with S=N/4:\n|S= 0,Sz= 0/an}bracketri}ht=N\n4/summationdisplay\nm=−N\n4C(N)\nm|S1=N\n4,m/an}bracketri}htN\n2|S2=N\n4,−m/an}bracketri}htN\n2,\n(12)5\nthenP↓is exactly given by the Clebsch-Gordor coeffi-\ncients as:\nP↓=/vextendsingle/vextendsingle/vextendsingleC(N)\nN\n4/vextendsingle/vextendsingle/vextendsingle2\n=1\nN/2+1. (13)\nIn Fig.6(b), we have verified this analytic result by nu-\nmerically calculations from the spin-chain model. Re-\nmarkably, Eq. 13produces a much larger P↓in compar-\nison to the FM state, where P↓=N\n2!N\n2!\nN!≃√\nπN\n2\n2Nis\nexponentially small. This reflect the distinct magnetic\nstructure of a pure FM state and a composition of two\nFM domains. This can serve as experimental evidence\nto identify the spatially-modulated quantum magnetism\ndue to the finite range effect.\nFinal remark. Our results reveal the significant effect\nof a finite effective range in the strong coupling regime\nof 1D trapped spin-1 /2 fermions. The sensitive density-\ndependence of Heisenberg coupling induced by the finite\nrange suggests a convenient route towards the local ma-\nnipulation of quantum magnetism. In particular, by en-\ngineering the density distribution of cold atomic gases\nthrough the laser potentials, one may get access to an ar-\nbitrary configuration of local Heisenberg coupling in the\ncoordinate space, and thus an arbitrary type of magnetic\norder may be achieved. This concept can be generalized\nto higher spins and other composition ofatomic mixtures\nin the 1D strong coupling regime.\nAcknowledgment. The work is supported by the Na-\ntional Key Research and Development Programof China\n(2018YFA0307600, 2016YFA0300603), and the National\nNatural Science Foundation of China (No.11622436,\nNo.11425419, No.11421092, No.11534014).\n∗xlcui@iphy.ac.cn\n†schen@iphy.ac.cn\n[1] C. Chin, R. Grimm, P. Julienne, and E. Teisinga, Rev.\nMod. Phys. 82, 1225 (2010).\n[2] T.-L. Ho, X. Cui, W. Li, Phys. Rev. Lett. 108, 250401\n(2012).\n[3] E. L. Hazlett et al, Phys. Rev. Lett. 108, 045304 (2012).\n[4] R. Qi and H. Zhai, Phys. Rev. A 85, 041603(R) (2012).\n[5] X. Cui, Phy. Rev. A 86, 012705 (2012).\n[6] R. Qi and X. Guan, Europhysics Letters 101, 40002\n(2013).[7] V. Gurarie and L. Radzihovsky, Annals of Physics 322,\n2 (2007).\n[8] Y. Nishida, Phys. Rev. Lett. 109, 240401 (2012)\n[9] S. B. Emmons, D. Kang and L. Platter, Phys. Rev. A 94,\n043615 (2016).\n[10] Z. Yu, J. H. Thywissen, S. Zhang, Phys. Rev. Lett. 115,\n135304 (2015); see also Erratum: Phys. Rev. Lett. 117,\n019901 (2016).\n[11] C. 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Massignan, J. Levinsen, and M. M. Parish, Phys. Rev.\nLett.115, 247202 (2015).\n[23] H. Hu, L. Guan, and S. Chen, New J. Phys. 18, 025009\n(2016).\n[24] H. Hu, L. Pan, and S. Chen, Phys. Rev. A 93, 033636\n(2016).\n[25] L. Yang and H. Pu, Phys. Rev. A 94, 033614 (2016).\n[26] F. DeuretzbacherandL.Santos, Phys.Rev.A 96, 013629\n(2016).\n[27] L. Yang and X. Cui, Phys. Rev. A 93, 013617 (2016).\n[28] L. Yang, X.-W. Guan, X. Cui, Phys. Rev. A 93, 051605\n(R) (2016).\n[29] F. Deuretzbacher, D. Becker, J. Bjerlin, S. M. Reimann,\nand L. Santos, Phys. Rev. A 95, 043630 (2017).\n[30] L. Pan, Y. Liu, H. Hu, Y. Zhang, and S. Chen, Phys.\nRev. B96, 075149 (2017).\n[31] S. Murmann, F. Deuretzbacher, G. Z¨ urn, J. Bjerlin, S.\nM. Reimann, L. Santos, T. Lompe, S. Jochim, Phys. Rev.\nLett. 115, 215301 (2015).\n[32] X. Cui and T.-L. Ho, Phys. Rev. A 89, 023611 (2014).\n[33] O. V. Marchukov, E. H. Eriksen, J. M. Midtgaard, A. A.\nS. Kalaee, D. V. Fedorov, A. S. Jensen, and N. T. Zinner,\nEur. Phys. J. 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Xiao1  \n1Department of Physics and Astronomy, Univer sity of Delaware, Newark, DE 19716, USA \n2Department of Electrical and Computer Engineeri ng, University of Delaware, Newark, DE 19716, USA \n3Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA \n \n \n \nAbstract: \nWe report the electrical detection of magnetization dynamics in an Al/AlO x/Ni80Fe20/Cu tunnel \njunction, where a Ni 80Fe20 ferromagnetic layer is brought into precession under the \nferromagnetic resonance (FMR) conditions. The dc  voltage generated across the junction by the \nprecessing ferromagnet is enhanced about an  order of magnitude  compared to the voltage signal \nobserved when the contacts in this type of mu ltilayered structure are ohmic.  We discuss the \nrelation of this phenomenon to magnetic spin pumping and speculate on other possible \nunderlying mechanisms responsible for the enhanced electrical signal. \n \nPACS numbers:  76.50.+g, 72.25.Mk, 72.25.Hg \n  2  In recent years, basic and applied research  in metal-based spintronics has shifted \nincreasingly from the static to the dynamic magnetic properties in hybrid nanostructures \ncomposed of ferromagnetic and normal metal layers  [1-5]. A variety of experimentally observed \nphenomena involving nonlocal magnetization dynamics  in magnetic multilayers are due to two \ncomplementary effects: (i) the transfer of spin angular momentum accompanying charge \ncurrents driven by the applied bias voltage between  ferromagnetic layers results in torques that \n(for sufficiently high current densities) ge nerate spontaneous magnetization precession and \nswitching [1]; and (ii) the precessi ng magnetization of  a ferromagnet ( FM) pumps spins into \nadjacent normal metal layers ( NM) with no applied bias [2, 5, 6]. In particular, the spin pumping \neffect [5] is a promising candidate for realizing a spin battery device [7] as a source of elusive \npure spin currents (not accompanied by any net charge transport) emitted at the FM/NM  \ninterface, where steady magnetization precession of the FM layer is sustained by the absorption \nof external rf radiation under the FMR conditions. Another promising application of \nmagnetization dynamics is microwave-assist ed reduction of the switching field of FM, which \ncould play an important role in  advanced recording media [4]. \nThus far, however, the spin pumping effect has been demonstrated mostly indirectly  as \nan additional contribution to the FMR linewidth in FM/NM  multilayers (where the NM is Pt, Pd, \nCu, etc.) that can be described as the inte rface-induced enhancement of the Gilbert dumping \nconstant [8-11]. The vigorous pursuit of direct el ectrical detection of spin pumping has led to \ntheoretical proposals [6] to use a single precessing FM as both the source and detector  of \npumped spin accumulation in NM layers. This adroit scheme has b een realized in a very recent \nexperiment [2] measuring the diff erence in voltages of the orde r of several hu ndred nanovolts  3  between two FM/NM  interfaces of a NM1/FM/NM2  trilayer.  \nIn this Letter, we report measuremen ts of the dc voltage across Al/AlO x/Ni80Fe20/Cu \ntunnel junctions with precessing magnetization of Ni 80Fe20. The surprisingly large observed \nvoltage is about μV , which seems to qualitatively  agree with the spin pumping theory [5, 6] but \nrequires an unreasonably large  spin-mixing conductance of the FM/tunnel-barrier contact. We \nconclude that a new nonequilibrium phenome non, which dynamically couples the spin and \ncharge degrees of freedom, exists in tunneling stru ctures. The rest of this  Letter presents details \nof our experiment, and we conclude by speculati ng on possible theoretical scenarios responsible \nfor these surprising experimental results. \nThe experimental setup is illustrated in Fi g. 1. A tunnel junction was fabricated on the \nsignal conductor of a coplanar waveguide (C PW) transmission line. The tunnel junction \nstructure of  Cu (100nm)/Al (10nm)/AlO x (2.3nm)/Ni 80Fe20 (20nm)/Cu (70nm)/Au (25nm) \nwas fabricated on a Si substrate with a 1 μm thick thermal oxide layer, by using magnetron \nsputtering deposition and conventional microfab rication processing. The bottom-most 100 nm \nCu layer was patterned into the CPW designed to have 50 Ω characteristic impedance in the \nabsence of the tunnel structure. The aluminum  oxide tunnel barrier was formed by plasma \noxidation. The size of the tunnel junction pillar was 50×50 μm2, and the dc junction resistance \nwas measured to be 67 k Ω. A microwave signal from a vect or network analyzer (Agilent \n8753B) was introduced through the CPW and generated a microwave magnetic field rfHthat \nwas linearly polarized in the plane of the tunne l junction. The external dc magnetic field (-120 \nOe to 120 Oe) was swept along the axis of the CPW (the y-direction), so that the magnetization \nchanged its direction within the x-y plane. The precessing spin mainly rotated around the y-axis.  4  Two electrical probe tips were  used to measure the dc voltage across the junction. The \nmicrowave input signal was varied from 0.7 GHz to 4 GHz, with power up to 18 dBm \namplitude-modulated with a 400 Hz sinusoidal signal to allow for lock-in detection. It should be \nmentioned that there was always a few tens of microvolt background voltage at the detector due \nto microwave noise. We found th at the background voltage was dire ctly proportional to the input \nmicrowave power.  We can thus maintain the sa me power applied to the device at different \nfrequencies by slightly tuning the nominal input power (±1 dB m) to maintain the same \nbackground voltage within 20% error.  This ma kes it possible to compare the data between \ndifferent frequencies without concerning the frequency dependence of the CPW impedance \nwhich changes the power input to the device slightly.   \nFigure 2 shows the dc voltage as a functi on of the external magnetic field in the \nAl/AlO x/Ni80Fe20/Cu tunnel junction. At each microwav e frequency, the voltage peaks of \nmagnitude VΔappear symmetrically at positive and negative fields. The peak field as a \nfunction of the microwave frequency shown in Fi g. 3 agrees well with the values we obtained \nfrom the flip-chip CPW FMR measurements. Th e Kittel formula [12] fits the data with \nreasonable parameters, 4 9 kGsMπ= , 19 OekH= , and gyromagnetic ratio \n-1 -10.0176 s Oeγ= , confirming that the dc voltage peak appears at the uniform FMR mode of \nthe Ni 80Fe20 layer. The peak magnitude reaches about 1 μV at 2GHz which is much larger than \nthe maximum value of about 250 nV at 14.5 GHz reported in Re f. [2] for a Pt/Ni 80Fe20/Al \nstructure. Figure 4 shows microwav e power and frequency dependence of VΔ, which increases \nwith increasing microwave power [Fig. 4 (a)]. We also plot VΔ as a function of precession \ncone angle in Fig. 4(b). Th e precession cone angle of Ni 80Fe20 was determined by the change in  5  the tunnel resistance at the FMR field in IrMn/Fe 70Co30/AlO x/Ni80Fe20 magnetic tunnel junctions \n[13] with 20mV bias voltage so that the dc voltage effect of the microvolt order we are \ndiscussing here can be neglected.  A clear dip in an tiparallel states and a pe ak in parallel states \nare observed corresponding to FMR fields, and the precession angle θ can be determined from \n()1c o s RR θ Δ∝ − .  At 10 dBm power input, the precessi on cone angle was around 17° [13].  \nAs the applied frequency increased, VΔ increases almost linearly,  as shown in Fig. 4(c). \nBefore attempting to interpret our results, we have to examine carefully the \nrectification effects which could induce similar dc voltage response.  Possible rectification effect \nmay arise from both the time-dependent anis otropy magnetoresistance (AMR) effect and the \nanomalous Hall effect (AHE) discussed in Refs.[ 14]. The current due to these two effects is \ngiven by \n()2RMρσρΔ′=− ⋅ + ×jj MM jM,   (1) \nwhere j is the current, σ is the conductivity, ρ is the resistivity, ρΔ is the \nmagnetoresistive anisotropy, and R is the anomalous Hall constant. The magnetization \nprecessing around the y-axis is described by the vector \n()s i ns i n ,c o s ,c o ss i nMt M M tωθθω θ =M . The microwave-induced current across the \njunction along the microwave electric fi eld direction (z-axis) is given by \n()() αω+ = t jcos,0,0j , where αis the phase difference with respect to the phase of spin \nprecession . The z -component of Eq. (1) is of in terest to our experiments, \n() θωαωρρ2 2sin cos cos ' t t j jz + Δ−= . The time average of zj′ is zero, allowing us to \nconclude that there is no dc component in the z -direction in our sample. The result holds even if  6  the precession axis fluctuates in the x-y plane.  Thus, in our sample c onfiguration we expect no \ndc voltage generated due to the rectification effect. We purposely broke  the tunnel barrier to \ninvestigate Al/Ni 80Fe20/Cu contact and also made Cu/AlO x/Al/Ni 80Fe20/Cu junctions. In both \ncases, no VΔ was observed within our measurement se nsitivity of about 100 nV . This implies \nthat the large VΔ in Al/AlO x/Ni80Fe20/Cu was indeed developed due to the AlO x/Ni80Fe20 \ninterface. \nLet us now try to interpret our results w ithin the framework of the standard spin \npumping theory [5-7].  At the FMR, a steady precession of the magneti zation of the FM layer \npumps a spin current into the adjacent NM according to [5, 7] \nRe Im4pump\nsddggdt dt π↑↓ ↑↓⎛⎞=× +⎜⎟⎝⎠mmIm=,  (2) \nwhere m is the unit vector along th e instantaneous direction  of the precessing magnetization \nand Reg↑↓ (Img↑↓) is the real (imaginary) part of the dimensionless interfacial spin-mixing \nconductance (in units of 2eh ) which describes spin tr ansport perpendicular to mat the \nFM/NM  interface [5, 15, 16]. For transparent intermetallic FM/NM contacts Img↑↓ is \ntypically neglected because of  being much smaller than Reg↑↓[5-7], while for low transparent \ncontacts we find  Re 0.5gg↑↓= and Im 0.5gg↑↓\u0011  (using the simple Stoner model for \nFM and random binary alloy w ith a gap in the energy spectru m for the tunnel barrier as NM \nlayer; g is the total charge conductanc e of the junction).  The possi bility for non-negligible  \nImg↑↓ for tunneling interfaces is also highlighted by recent first principles  calculations [17]. \nThe injected spin current builds up a spin accumulation sμ in the NM layer (close to the \nFM/NM  interface) when the spin -flip relaxation rate in NM is smaller than the spin injection rate.  7  This, in turn, drives a backward flowing spin current back\nsIinto the precessing FM [5]. The \nbackward flowing spin current parallel to the magnetization can be absorbed by the FM, in the \npresence of spin-flip processes . Due to spin-dependent bulk and interface conductances, this \nabsorbed spin current is converted  into charge accumulation at the FM/NM  interface [18]. The \nmaximum value of the voltage drop dcVacross the FM/NM  interface, at fixed frequency ω \nand cone angle θof magnetization precession (assuming the frequency is much greater than the \ncharacteristic spin-flip rate in the normal me tal and the ferromagnet is thicker than its bulk \nspin-diffusion length), is obtained for NM layer thickness much sma ller than its spin-diffusion \nlength dN << λdsN  as [6]:  \n()2\ndc 22 2 2sin cos\n2 () s i n ( 1 ) c o sF\nFN FpgVe gp gg p g g gω\nωω ω ω ω ωθθ ω\nηθ θ↑↓=\n−+ + + −=,  (3) \nwhere ()tanhN\nNN N s dgg dω η λ↑↓=  with the thickness dN and the spin diffusionN\nsdλ of the \nNM layer, Fg and Ng parameterize the transport properties of the bulk FM and NM, gω↑↓ \nis the real part of the eff ective spin-mixing conductance, ()( ) p gg ggω ωω ωω↑↓↑ ↓=− +  is the \ninterfacial spin-polarization, gggωωω↑↓=+  is the sum of spin-up and spin-down effective \nconductances of the FM/NM  interface. The “ef fective” interfacial transport quantities are in \ngeneral frequency-dependent since they have to be  evaluated for the interface resistance in series \nwith a NM resistor of length NM LDω ω=  over which the oscillati ng transverse components \nof spin accumulation in NM (with diffusion constantNMD) are averaged to zero, although this is \nnot important in practice for high-impeda nce tunnel barriers.  The voltage drops 1,2\ndcVemerging \nat each of the two FM/NM contacts will differ from each other when conductances gω↑↓ and/or \nspin-flip diffusion lengthsN\nsλon two sides of the multilayer ar e substantially different, as  8  observed by measuring 12\ndc dc VV VΔ= − on lateral Pt/Ni 80Fe20/Al device in a recent experiment \n[2].  \nTo compare the spin pumping theory with our results, we assume that spin-mixing \nconductance is governed by the AlO x/Ni80Fe20 interface, but there is no spin flipping inside the \nbarrier and spin accumulation is induced in the Al layer. Since the interface conductance gω \nfor the low transparency tunnel barri er is much smaller than smallFg, the first term in the \ndenominator of Eq. (3) reduces to ()2sinFNgηθ+ , so that Eq. (3) can be parameterized \nwithggωω↑↓,Nη,pω, and θ. Using 0.3 pω= for the AlO x/Ni80Fe20 interface, Fig. 4(b) shows \nthe best fitting (solid line) of our results by Eq. (3), where we extract 3.4 ggωω↑↓≈ and \n0Nη≈ from the fit. We found that Nη has to be set to zero in order to fit our data, which \nrequires that Ng is roughly comparable or smaller than gω↑↓, while we expect the former to be \nseveral orders of magnitude larger than the latter, using the measured tunneling conductance. \nThis is the first discrepancy between our results and an attempt to explai n them using standard \ninterfacial spin pumping theory originally devel oped [5, 7] and experimentally confirmed [2] for \nintermetallic FM/NM  contacts. On the other hand, linear fitting of VΔvs. frequency in Fig. 4(c) \nyields the slope of 0.85 μV/GHz, and 6.1 ggωω↑↓≈ at precession cone angle of 17° by Eq. (3). \nThese values are larger than the typical value 1 ggωω↑↓≈ for transparent intermetallic contacts, \nwhich is highly unexpected  when compared to standard estimates [16] of ggωω↑↓ for trivial \n(non-magnetic) tunnel contacts. \nV oltage generation based on the spin pumping mechanism [6] is based on spin injection \ninto the normal metal, across the FM/NM  interface, with its subsequent  diffusion, relaxation, and \nbackflow into the ferromagnet, which is ultimately responsible for the build-up of the voltage  9  drop across the contact. A tunnel barrier exponen tially impedes electron flows (and thus spin \ncurrents) across the FM/NM  contact, and one, therefore, would not expect  a significant voltage \ngeneration by the spin-pumping mechanism. This is  the reason why we were  not able to reach a \nquantitative agreement with the theory. The tunne l barrier essentially cuts off the normal metal \nfrom the FM, while a voltage probe may now be thought  of as a nonintrusive probe of dynamic \nprocesses within the FM. If the magnetization dynamics can  generate nonequilibrium spin \naccumulation inside the ferromagnet, in analogy with the pumped spin generation in the normal \nmetal (presumably requiring spin-orbit or other spin-flip processes in the FM), the voltage \nmeasured by the FM may in fact be probing this spin accumulation rather than a nonlocal spin \npumping process. Exploring this possibility re quires further theoretical  analysis and other \nnonlocal probes of the magnetization dynamics. Fi nally, we note that our theoretical discussion \ncompletely disregarded many-body effects due to  electron-electron inte ractions, which may \nmodify substantially the predictions of the standa rd spin pumping theory, especially if we drive \nthe magnetization dynamics beyond the linearized regime. \nIn conclusion, we observed a large dc voltage , of the order of microvolts, across the \nAl/AlO x/Ni80Fe20/Cu tunnel junctions, due to a dynamic spin and charge coupling driven by the \nprecessing magnetization of a single Ni 80Fe20 ferromagnetic layer at ferromagnetic resonance. \nBy short circuiting the tunnel ba rrier, we demonstrated that the observed dc voltage mainly \narises from the Al/AlO x/Ni80Fe20 contact. The phenomenon appears qualitatively similar to the \npredictions of the spin pumping formalism, but  a quantitative analysis shows a number of \ndiscrepancies with the standard theory. This  suggests a new nonequilibrium mechanism for the \nspin and charge coupling, which is responsible fo r the voltage generation much larger than that  10  observed very recently for intermetallic interfaces  [2]. We speculate on the role of intrinsic \ndynamic processes in the ferromagnet and the effect s of the electron-electr on interactions, as \npossible culprits for our observations, but a more t horough theoretical analysis is desirable in the \nfuture. \nWe thank M. D. Stiles and S.-T. Chui fo r illuminating discussions. This work was \nsupported by NSF DMR Grant No. 0405136, and DOE DE-FG02-07ER46374. 11   \nReferences:  \n[1] S. I. Kiselev, et al., Nature 425, 380 (2003). \n[2] M. V . Costache, et al., Phys. Rev. Lett. 97, 216603 (2006). \n[3] J. Grollier, et al., Phys. Rev. B 73, 060409 (2006). \n[4] T. Moriyama, et al., Appl. Phys. Lett. 90, 152503 (2007). \n[5] Y . Tserkovnyak, et al., Rev. Modern Phys. 77, 1375 (2005). \n[6] X. H. Wang, et al., Phys. Rev. Lett. 97, 216602 (2006). \n[7] A. Brataas, et al., Phys. Rev. B 66, 060404 (2002). \n[8] S. Mizukami, et al., J. Magn. Magn. Mater. 226, 1640 (2001). \n[9] B. Heinrich, et al., Phys. Rev. Lett. 90, 187601 (2003). \n[10] B. Heinrich, et al., J. Supercond. Novel Magn. 20, 83 (2007). \n[11] T. Gerrits, et al., J. Appl. Phys. 99, 023901 (2006). \n[12] C. Kittel, Introduction to solid state physics  (Wiley, Hoboken, NJ, 2005), 8th ed., Chap. 13. \n[13] T. Moriyama, et al., unpublished. \n[14] W. G. Egan, et al., J. Appl. Phys. 34, 1477 (1963). \n[15] K. Xia, et al., Phys. Rev. B 65, 220401 (2002). \n[16] A. Brataas, et al., Phys. Rep. 427, 157 (2006). \n[17] I. Turek, et al., J. Phys.: Condens. Matter 19, 365203 (2007). \n[18] S. Takahashi, et al., Phys. Rev. B 67, 052409 (2003). \n \n  12  Figure captions:  \nFIG. 1. (Color online) Schema tic diagram of the sample stru cture (a) and the measurement \ngeometry (b). The arrow in the Ni 80Fe20 layer indicates the magnetiza tion direction. An external \ndc field exH is applied in the y-direc tion and the rf magnetic field rfH is applied along the \nx-direction. A coplanar microwave probe f eeds microwave signals through the coplanar \nwaveguide. DC voltage across the junction (TJ) is measured between top of the junction and \nsignal line of the CPW.  \n \nFIG. 2. (Color online) The dc voltage VΔ generated across the Al/AlO x/Ni80Fe20/Cu tunnel \njunction as a function of the ex ternally applied static magnetic  field. The frequency of the \napplied rf field ranges from 1.8 to 2.8 GH z. The background voltage is subtracted for \ncomparison purpose. \n \nFIG. 3. (Color online) The frequency dependence of  static magnetic field at which the dc voltage \npeak (circles) appears in Fig. 2. The crosses label the frequency depe ndence of the resonance \nfield obtained from FMR measurem ents on flip-chip structures in the CPW line. The curve is a \nfit to the Kittel formula [12]. \n \nFIG. 4. The amplitude of the dc voltage VΔ measured across Al/AlO x/Ni80Fe20/Cu device as \nfunction of: (a) microwave power;  (b) precession cone angle; and (c) microwave frequency at 10 \ndBm. Solid lines in (b) and (c) are the f it to Eq. (3) as described in the text. \n  13   \n \n \nFig.1 T. Moriyama et al. 14  \n \nFig. 2 T. Moriyama et al. 15  \n \nFig.3 T. Moriyama et al.  16  \n \nFig. 4 T. Moriyama et al. \n "    },    {        "title": "0810.4286v3.Magnetic_moment_manipulation_by_a_Josephson_current.pdf",        "content": "arXiv:0810.4286v3  [cond-mat.supr-con]  8 Jan 2009Magnetic moment manipulation by a Josephson current\nF. Konschelle∗and A. B uzdin†\nCondensed Matter Theory Group, CPMOH, Universit´ e de Borde aux and CNRS. F-33405 Talence, France\n(Dated: November 2, 2018, Document published in Phys. Rev. L ett.102, 017001 (2009).)\nWe consider a Josephson junction where the weak-link is form ed by a non-centrosymmetric fer-\nromagnet. In such a junction, the superconducting current a cts as a direct driving force on the\nmagnetic moment. We show that the a.c. Josephson effect gener ates a magnetic precession provid-\ning then a feedback to the current. Magnetic dynamics result in several anomalies of current-phase\nrelations (second harmonic, dissipative current) which ar e strongly enhanced near the ferromagnetic\nresonance frequency.\nMany interesting phenomena have been observed re-\ncently in the field ofspintronics: the spin-dependent elec-\ntric current and inversely the current-dependent magne-\ntization orientation (see for example [1, 2]). Moreover,\nit is well known that spin-orbit interaction may be of\nprimary importance for spintronic, namely for systems\nusing a two-dimensional electron gas [3]. In the super-\nconductor/ferromagnet/superconductor(S/F/S)Joseph-\nson junctions, the spin-orbit interaction in a ferromagnet\nwithout inversion symmetry provides a mechanism for a\ndirect (linear) coupling between the magnetic moment\nand the superconducting current [4]. Similar anomalous\nproperties have been predicted for Josephson junctions\nwith spin-polarized quantum point contact in a two di-\nmensional electron gas [5]. S/F/S junctions are known\nto reveal a transition to π-phase, where the supercon-\nducting phase difference ϕin the ground state is equal\ntoπ[6]. However, the current-phase relation (CPR) in\nsuchaπ-junction hasausualsinusoidalform I=Icsinϕ,\nwhere the critical current Icdepends in a damped oscilla-\ntorymanneron themodulus ofthe ferromagnetexchange\nfield. In a non-centrosymmetric ferromagnetic junction,\ncalled hereafter ϕ0-junction, the time reversal symme-\ntry is broken and the CPR becomes I=Icsin(ϕ−ϕ0),\nwhere the phase shift ϕ0is proportional to the magnetic\nmoment perpendicular to the gradient of the asymmetric\nspin-orbit potential [4]. Therefore, manipulation of the\ninternal magnetic moment can be achieved via the super-\nconducting phase difference ( i.e.by Josephson current).\nIn the present work we study theoretically the spin dy-\nnamics associated with such ϕ0-junctions. Though there\nis a lot of experimental progress in studying the static\nproperties of S/F/S junctions, little is known about the\nspin-dynamics in S/F systems. Note here the pioneering\nwork [7] where a sharpening of the ferromagnetic reso-\nnancewasobservedbelowthe superconductingtransition\nin Nb/Ni 80Fe20system. Theoretically, the singlespin dy-\nnamicsinterplaywithaJosephsoneffecthasbeenstudied\nin [8, 9, 10, 11]. More recently, the dynamically induced\ntriplet proximity effect in S/F/S junctions was studied in\n[12, 13], while the junctions with composite regions (in-\ncluding several F regions with different magnetization)\nwere discussed in [14, 15]. Here we consider a simple\nS/F/Sϕ0-junction in a low frequency regime ℏωJ≪Tc(ωJ= 2eV/ℏbeing the Josephson angular frequency\n[16]), which allows us to use the quasi-static approach\nto treat the superconducting subsystem in contrast with\nthe case analyzed in [12, 13]. We demonstrate that a\nd.c. superconducting current may produce a strong ori-\nentation effect on the F layer magnetic moment. More\ninterestingly, the a.c. Josephson effect, i.e.applying a\nd.c. voltage Vto theϕ0-junction, would produce cur-\nrent oscillations and consequently magnetic precession.\nThis precession may be monitored by the appearance of\nhigher harmonics in CPR as well as a d.c. component of\nthe current. In particular regimes, a total reversal of the\nmagnetization could be observed. In the case of strong\ncoupling between magnetic and superconducting subsys-\ntems, complicated non-linear dynamic regimes emerge.\nTo demonstrate the unusual properties of the ϕ0-\njunction, we consider the case of an easy-axis magnetic\nanisotropy of the F material (see Fig.1). Both the easy\naxis and gradient of the asymmetric spin-orbit potential\nnare along the z-axis. Note that suitable candidates for\nthe F interlayer may be MnSi or FeGe. In these systems,\nthe lack of inversion center comes from the crystalline\nstructure, but the origin of broken-inversion symmetry\nmay also be extrinsic, like in a situation near the surface\nofathin Ffilm. Inthefollowing, wecompletelydisregard\nthe magnetic induction. Indeed the magnetic induction\nin the (xy) plane is negligible for the thin F layer consid-\nered in this paper, whereas the demagnetization factor\ncancels the internal induction along the z-axis (N= 1).\nThe coupling between F and S subsystems due to the or-\nbital effect has been studied in [17] and it appears to be\nvery weak, and quadratic over magnetic moment Mfor\nthe case when the flux of Mthrough the F layer is small\nin comparison with flux quantum Φ 0=h/2e.\nThe superconducting part of the energy of a ϕ0-\njunction is\nEs(ϕ,ϕ0) =EJ[1−cos(ϕ−ϕ0)], (1)\nwhereEJ= Φ0Ic/2πis the Josephson energy, Icis the\ncritical current and ϕ0is proportional to the Mycom-\nponent of the magnetic moment (see Fig.1). Therefore,\nwhen the magnetic moment is oriented along the z-axis,\nwe have the usual Josephson junction with ϕ0= 0. As-\nsuming the ballistic limit we may estimate the charac-2/BY\n/CB /CB\n/DE /BN /D2/DD/DC\n/C5/DD\n/B4 /D8 /B5\n/C5/DE/BDFIG. 1: Geometry of the considered ϕ0-junction. The ferro-\nmagnetic easy-axis is directed along the z-axis, which is also\nthe direction nof the gradient of thespin-orbit potential. The\nmagnetization component Myis coupled with Josephson cur-\nrent through the phase shift term ϕ0∝n.(M∧∇Ψ), where\nΨ is the superconducting order parameter ( ∇Ψ is along the\nx-axis in the system considered here).\nteristic Josephson energy as [6] Φ 0Ic/S∼Tck2\nFsinℓ/ℓ\nwithℓ= 4hL/ℏvF, whereS,Landhare the section, the\nlength and the exchange field of the F layer, respectively.\nThe phase shift is\nϕ0=ℓvso\nvFMy\nM0(2)\nwhere the parameter vso/vFcharacterizes the relative\nstrength of the spin-orbit interaction [4]. Further on we\nassume that vso/vF∼0.1. If the temperature is well be-\nlowthe Curietemperature, M0=/bardblM/bardblcanbe considered\nasaconstantequaltothesaturationmagnetizationofthe\nF layer. The magnetic energy contribution is reduced to\nthe anisotropy energy\nEM=−KV\n2/parenleftbiggMz\nM0/parenrightbigg2\n, (3)\nwhereKis an anisotropy constant and Vis the volume\nof the F layer.\nNaturally, we may expect that the most interesting\nsituation corresponds to the case when the magnetic\nanisotropy energy does not exceed too much the Joseph-\nson energy. From the measurements [18] on permal-\nloy with very weak anisotropy, we may estimate K∼\n4.10−5K.˚A−3. On the other hand, typical value of L\nin S/F/S junction is L∼10nm and sin ℓ/ℓ∼1. Then,\nthe ratio of the Josephson over magnetic energy would\nbeEJ/EM∼100 forTc∼10K. Naturally, in the more\nrealistic case of stronger anisotropy this ratio would be\nsmaller but it is plausible to expect a great variety of\nregimes from EJ/EM≪1 toEJ/EM≫1.\nLet us now consider the case when a constant current\nI < Icis applied to the ϕ0-junction. The total energy is\n(see,e.g.[16]):\nEtot=−Φ0\n2πϕI+Es(ϕ,ϕ0)+EM(ϕ0),(4)and both the superconducting phase shift difference ϕ\nand the rotation of the magnetic moment My=M0sinθ\n(whereθisthe anglebetweenthe z-axisandthe direction\nofM) are determined from the energy minimum condi-\ntions∂ϕEtot=∂ϕ0Etot= 0. It results in\nsinθ=I\nIcΓ with Γ =EJ\nKVℓvso\nvF, (5)\nwhich signifies that a superconducting current provokes\nthe rotation of the magnetic moment Myin the (yz)\nplane. Therefore, for small values of the rotation, θ(I)\ndependence is linear. In principle, the parameter Γ can\nbe larger than one. In that case, when the condition\nI/Ic≥1/Γ is fulfilled, the magnetic moment will be ori-\nented along the y-axis. Therefore, applying a d.c. super-\nconducting current switches the direction of the magneti-\nzation, whereasapplyingana.c. currentona ϕ0-junction\ncould generate the precession of the magnetic moment.\nWe briefly comment on the situation when the direc-\ntion of the gradient of the spin-orbit potential is perpen-\ndicular (along y) to the easy axis z. To consider this\ncase we simply need to take ϕ0=ℓ(vso/vF)cosθ. The\ntotal energy (4) has two minima θ= (0,π), while apply-\ning the current removes the degeneracy between them.\nHowever, the energy barrier exists for the switch from\none minimum into another. This barrier may disappear\nif Γ>1 and the current is large enough I > Ic/Γ. In\nthis regime the superconducting current would provoke\nthe switching of the magnetization between one stable\nconfiguration θ= 0 and another θ=π. This corresponds\nto the transitions of the junction between + ϕ0and−ϕ0\nstates. The read-out of the state of the ϕ0-junction may\nbe easily performed if it is a part of some SQUID-like\ncircuit (the ϕ0-junction induces a shift of the diffraction\npattern by ϕ0).\nIn fact, the voltage-biased Josephson junction, and\nthus the a.c. Josephson effect provides an ideal tool\nto study magnetic dynamics in a ϕ0-junction. In such\na case, the superconducting phase varies with time like\nϕ(t) =ωJt[19]. IfℏωJ≪Tc, onecanusethestaticvalue\nfor the energy of the junction (4) considering ϕ(t) as an\nexternal potential. The magnetization dynamics are de-\nscribed by the Landau-Lifshitz-Gilbert equation (LLG)\n[20]\ndM\ndt=γM×Heff+α\nM0/parenleftbigg\nM×dM\ndt/parenrightbigg\n,(6)\nwhereHeff=−δF/VδMis the effective magnetic field\napplied to the compound, γthe gyromagnetic ratio, and\nαa phenomenological damping constant. The corre-\nsponding free energy F=Es+EMyields\nHeff=K\nM0/bracketleftbigg\nΓsin/parenleftbigg\nωJt−rMy\nM0/parenrightbigg\nˆy+Mz\nM0ˆz/bracketrightbigg\n,(7)\nwherer=ℓvso/vF. Introducing mi=Mi/M0,τ=ωFt3\n/s45/s49 /s48 /s49/s45/s49/s48/s49\n/s61 /s61/s49/s44 /s61/s48/s46/s49/s44 /s114/s61/s49/s109\n/s122\n/s109\n/s120/s99/s41\n/s45/s49 /s48 /s49/s45/s49/s48/s49\n/s61 /s61/s49/s44 /s61/s48/s46/s48/s53/s44 /s114/s61/s53/s109\n/s122\n/s109\n/s121 /s100/s41/s48 /s49 /s50 /s51/s45/s49/s48/s49/s109\n/s122/s40 /s41/s32/s110/s117/s109/s101/s114/s105/s99\n/s32/s97/s110/s97/s108/s121/s116/s105/s99\n/s61/s53 /s44 /s61/s53/s44 /s61 /s114/s61/s48/s46/s49/s97/s41\n/s45/s49 /s48 /s49/s45/s49/s48/s49\n/s61 /s61/s48/s46/s53/s44 /s61/s48/s46/s54/s44 /s114/s61/s49\n/s109\n/s121 /s109\n/s122/s98/s41/BDFIG. 2: Results of numerical analysis of the magnetic moment dynamics of the ϕ0-junction. a) Reversal of mzfrom analytical\nexpression Eq.(15) (dashed curve) and numerical one (norma l curve). The other curves are related to the Mtrajectory: b)\nin strong damping case c) and d) in the strong coupling regime revealing incomplete and complete magnetic moment reversa l,\nrespectively.\n(ωF=γK/M2\n0is the frequency of the ferromagnetic res-\nonance) in LLG equation (6) leads to\n\n\n˙mx=mz(τ)my(τ)−Γmz(τ)sin(ωτ−rmy)\n˙my=−mz(τ)mx(τ)\n˙mz= Γmx(τ)sin(ωτ−rmy),(8)\nwhereω=ωJ/ωF. The generalization of Eq.(8) for\nα/negationslash= 0 is straightforward. One considers first the ”weak\ncoupling” regime Γ ≪1 when the Josephson energy EJ\nis small in comparison with the magnetic energy EM. In\nthis case, the magnetic moment precess around the z-\naxis. If the other components verify ( mx,my)≪1, then\nthe equations (8) may be linearized, and the correspond-\ning solutions are\nmx(t) =ΓωcosωJt\n1−ω2andmy(t) =−ΓsinωJt\n1−ω2.(9)\nNear the resonance ωJ≈ωF, the conditions of lineariza-\ntion are violated and it is necessary to take the damping\nintoaccount.Theprecessingmagneticmomentinfluences\nthe current through the ϕ0-junction like\nI\nIc= sinωJt+Γr\n21\nω2−1sin2ωJt+...,(10)\ni.e., in addition to the first harmonic oscillations, the\ncurrent reveals higher harmonics contributions. The am-\nplitude oftheharmonicsincreasesneartheresonanceandchanges its sign when ωJ=ωF. Thus, monitoring the\nsecond harmonic oscillations of the current would reveal\nthe dynamics of the magnetic system.\nThe damping plays an important role in the dynamics\nof the considered system. It results in a d.c. contribu-\ntion to the Josephson current. Indeed, the corresponding\nexpressionfor my(t) in the presence of damping becomes\nmy(t) =ω+−ω−\nrsinωJt+α−−α+\nrcosωJt,(11)\nwhere\nω±=Γr\n2ω±1\nΩ±andα±=Γr\n2α\nΩ±,(12)\nwith Ω ±= (ω±1)2+α2. It thus exhibits a damped\nresonance as the Josephson frequency is tuned to the fer-\nromagnetic one ω→1. Moreover, the damping leads\nto the appearance of out of phase oscillations of my(t)\n(term proportional to cos ωJtin Eq.(11)). In the result\nthe current\nI(t)≈IcsinωJt+Icω+−ω−\n2sin2ωJt+\n+Icα−−α+\n2cos2ωJt+I0(α) (13)4\nacquires a d.c. component\nI0(α) =αΓr\n4/parenleftbigg1\nΩ−−1\nΩ+/parenrightbigg\n. (14)\nThis d.c. current in the presence of a constant voltage V\napplied to the junction means a dissipative regime which\ncan be easily detected. In some aspect, the peak of d.c.\ncurrent near the resonance is reminiscent of the Shapiro\nsteps effect in Josephson junctions under external r.f.\nfields. Note that the presence of the second harmonic\ninI(t) Eq.(13) should also lead to half-integer Shapiro\nsteps inϕ0-junctions [21].\nThe limit of the ”strong coupling” Γ ≫1 (butr≪1)\ncan also be treated analytically. In this case, my≈0 and\nsolutions of Eq.(8) yields\n\n\nmx(t) = sin/bracketleftbiggΓ\nω(1−cosωJt)/bracketrightbigg\nmz(t) = cos/bracketleftbiggΓ\nω(1−cosωJt)/bracketrightbigg,(15)\nwhich are the equations of the magnetization reversal, a\ncomplete reversal being induced by Γ /ω > π/2. Strictly\nspeaking, these solutions are not exact oscillatory func-\ntions in the sense that mz(t) turns around the sphere\ncenter counterclockwise before reversingits rotation, and\nreturns to the position mz(t= 0) = 1 clockwise, like a\npendulum in a spherical potential (see Fig.2.c).\nFinally, we have performed numerical studies of the\nnon-linear LLG Eq.(6) for some choices of the parame-\nters when the analytical approaches fail. To check the\nconsistency of our numerical and analytical approaches,\nwe present in Fig.2.a) the corresponding mz(t) depen-\ndences for low-damping regimes. They clearly demon-\nstrate the possibility of the magnetization reversal. In\nFigs.2b-d), some trajectories of the magnetization vec-\ntorsarepresentedforgeneralcouplingregimes. Thesere-\nsults demonstrate that the magnetic dynamics of S/F/S\nϕ0-junctionmaybe prettycomplicatedand stronglynon-\nharmonic.\nIf theϕ0-junction is exposed to a microwave radia-\ntion at angular frequency ω1, the physics that emerge\nare very rich. First, in addition to the Shapiro steps\natωJ=nω1, half-integer-steps will appear. Secondly,\nthe microwave magnetic field may also generate an addi-\ntional magnetic precession with ω1frequency. Depend-\ning on the parameters of ϕ0-junction and the amplitude\nof the microwave radiation the main precession mecha-\nnism may be related either to the Josephson current or\nthe microwave radiation. In the last case the magnetic\nspin-orbit coupling may substantially contribute to the\namplitude of the Shapiro steps. Therefore, we could ex-\npect a dramatic increase of this amplitude at frequencies\nnear the ferromagnetic resonance. When the influence\nof the microwave radiation and Josephson current on theprecession is comparable, a very complicated regime may\nbe observed.\nIn the present work we considered the case of the easy-\naxis magnetic anisotropy. If the ferromagnet presents an\neasy-planeanisotropythanqualitativelythemainconclu-\nsions ofthis article remain the same because the coupling\nbetween magnetism and superconductivity depends only\non theMycomponent. However, the detailed dynamics\nwould be strongly affected by a weak in-plane anisotropy.\nTo summarize, we have demonstrated that S/F/S ϕ0-\njunctions provide the possibility to generate magnetic\nmoment precession via Josephson current. In the regime\nof strong coupling between magnetization and current,\nmagnetic reversal may also occur. These effects have\nbeen studied analytically and numerically. We believe\nthat the discussed properties of the ϕ0-junctions could\nopen interesting perspectives for its applications in spin-\ntronics.\nThe authors are grateful to Z. Nussinov, J. Cayssol,\nM. Houzet, D. Gusakova, M. Roche and D. Braithwaite\nfor useful discussions and comments. This work was sup-\nported by the French ANR Grant N◦ANR-07-NANO-\n011: ELEC-EPR.\n∗Electronic address: f.konschelle@cpmoh.u-bordeaux1.fr\n†Also atInstitut Universitaire de France .\n[1] I.ˇZuti´ c, J. Fabian, and S. Das Sarma. Rev. Mod. Phys.\n76, 323 (2004).\n[2] J. Hauptmann, J. Paaske, and P. Lindelof. Nature\nPhysics4, 373 (2008).\n[3] R. Winkler. Spin-orbit coupling effects in two-\ndimensional electron and hole systems (Springer, New\nYork, 2003).\n[4] A. Buzdin. Phys. Rev. Lett. 101, 107005 (2008).\n[5] A.A.Reynoso et al.Phys.Rev.Lett. 101, 107001 (2008).\n[6] A. I. Buzdin. Rev. Mod. Phys. 77, 935 (2005).\n[7] C. Bell et al.Phys. Rev. Lett. 100, 047002 (2008).\n[8] J.-X. Zhu, and A. V. Balatsky. Phys. Rev. B 67, 174505\n(2003).\n[9] L. Bulaevskii et al.Phys. Rev. Lett. 92, 177001 (2004).\n[10] J.-X. Zhu et al.Phys. Rev. Lett. 92, 107001 (2004).\n[11] Z. Nussinov et al.Phys. Rev. B 71, 214520 (2005).\n[12] S. Takahashi et al.Phys. Rev. Lett. 99, 057003 (2007).\n[13] M. Houzet. Phys. Rev. Lett. 101, 057009 (2008).\n[14] X. Waintal, and P. W. Brouwer. Phys. Rev. B 65, 054407\n(2002).\n[15] V. Braude, and Y. M. Blanter. Phys. Rev. Lett. 100,\n207001 (2008).\n[16] K. K. Likharev. Dynamics of Josephson junctions and\ncircuits(Gordon and Beach Science Publishers, 1986).\n[17] S. Hikino et al.J. Phys. Soc. Jap. 77, 053707 (2008).\n[18] A. Y. Rusanov et al.Phys. Rev. Lett. 93, 057002 (2004).\n[19] B. D. Josephson. Superconductivity (in two volumes),\nvol. 1, chapter 9, Weakly coupled superconductors (R.D\nParks, Marcel Dekker, Inc., 1968).5\n[20] E. M. Lifshitz, and L. P. Pitaevskii. Course of theoret-\nical physics, Volume 9 : Theory of the condensed state\n(Butterworth Heinemann, 1991).[21] H. Sellier et al.Phys. Rev. Lett. 92, 257005 (2004)."    },    {        "title": "1905.09241v1.Floquet_Second_Order_Topological_Superconductor_Driven_via_Ferromagnetic_Resonance.pdf",        "content": "Floquet Second-Order Topological Superconductor\nDriven via Ferromagnetic Resonance\nKirill Plekhanov, Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja1\n1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n(Dated: May 23, 2019)\nWe consider a Floquet triple-layer setup composed of a two-dimensional electron gas with spin-\norbit interactions, proximity coupled to an s-wave superconductor and to a ferromagnet driven\nat resonance. The ferromagnetic layer generates a time-oscillating Zeeman \feld which competes\nwith the induced superconducting gap and leads to a topological phase transition. The resulting\nFloquet states support a second-order topological superconducting phase with a pair of localized\nzero-energy Floquet Majorana corner states. Moreover, the phase diagram comprises a Floquet\nhelical topological superconductor, hosting a Kramers pair of Majorana edge modes protected by an\ne\u000bective time-reversal symmetry, as well as a gapless Floquet Weyl phase. The topological phases\nare stable against disorder and parameter variations and are within experimental reach.\nIntroduction. Over the last decade topological states\nof matter [1{5] have attracted a lot of attention. Re-\ncently, particular interest has been raised by higher-\norder topological insulators and superconductors [6{8],\nwhich host topologically protected gapless modes on their\nhigher-order faces ( e.g. corners in d > 1, hinges in\nd > 2, withdbeing the spatial dimension of the sys-\ntem). However, such systems were studied mostly at the\nstatic level [9{19], with the out-of-equilibrium dynam-\nics [20{26] rarely addressed. At the same time, Floquet\nengineering [1{4], based on applying time-periodic per-\nturbations, has been serving as a powerful tool to gener-\nate exotic phases of quantum matter, including topolog-\nical and Chern insulators [31{50], as well as non-Abelian\nstates such as Majorana bound states (MBSs) [51{59].\nSuch Floquet phases can be generated by time-dependent\nelectromagnetic \felds [31{38, 57, 58]. Strong oscillating\nelectric \felds are easy to obtain for this purpose, but\nthe direct coupling to the spin degree of freedom is more\nchallenging, typically achieved only indirectly via strong\nspin orbit interaction [60, 61]. Hence, \fnding ways to\ngenerate strong oscillating magnetic \felds which couple\ndirectly to the spin is important. One possible solution\nconsists in using magnetic proximity e\u000bects, which, for\nthe static case, have already been widely studied for su-\nperconductors [62, 63] and topological insulators [64{66].\nIn this work we consider a driven triple-layer setup\nwhich allows us to engineer a Floquet higher-order topo-\nlogical superconductor (FHOTS). Its key component is\na two-dimensional electron gas (2DEG) with spin-orbit\ninteraction (SOI) sandwiched between an s-wave super-\nconductor (SC) and a ferromagnet (FM), as shown in\nFig. 1(a). The FM layer is resonantly driven by an ex-\nternal \feldH(t), which results in the generation of an\noscillating magnetic \feld B(t), giving rise to strong Zee-\nman coupling in the 2DEG. The out-of-plane component\nofB(t) competes with the proximity-induced supercon-\nducting gap and leads to a topological phase transition\nin the 2DEG. The topological phase is identi\fed as Flo-\nquet helical topological superconductor (FHeTS), char-\nacterized by the presence of a Kramers pair of gapless\nFIG. 1. (a) Schematics of the Floquet triple-layer setup con-\nsisting of a 2DEG (blue layer) with SOI of strengths \u000bx\nand\u000by, proximity coupled to an s-wave SC (red layer) at\nthe bottom and a FM at the top (green layer) which is\nresonantly driven by an external time-dependent magnetic\n\feldH(t) to generate an oscillating magnetic \feld B(t) =\nB?cos(!t)ez+Bksin(!t)ukin the 2DEG. (b) Band structure\nof the 2DEG in the isotropic regime \u000bx=\u000by. The chemical\npotential\u0016is \fxed below the crossing point of the spin-split\nbands (indicated by the blue area). The driving frequency !\nis tuned to achieve resonance at the smallest Fermi momen-\ntum (represented by the two green circles) between the two\nFloquet bands labeled by \u001c=\u00061.\nhelical Majorana edge modes protected by an e\u000bective\ntime-reversal symmetry. A moderate in-plane component\nofB(t) opens a gap in the helical edge modes through\na mass term that changes sign at the corners in a rect-\nangular geometry, resulting in Majorana corner states\n(MCSs) { a distinct feature of the FHOTS. In the regime\nwhere the in-plane component of B(t) is strong, the sys-\ntem moves into a gapless Weyl phase. The topological\nphase diagram is robust against moderate local disorder,\ndetuning from resonance, and static magnetic \felds.\nModel. The triple-layer setup is composed of a 2DEG\nwith strong Rashba and Dresselhaus SOIs, proximity\ncoupled to an s-wave SC and a FM, as shown in Fig. 1(a).\nWe assume that the SOI vector is along zdirection, which\nis perpendicular to the xy2DEG-plane. The SOI cou-\npling strengths \u000bx=\u000bR+\fDand\u000by=\u000bR\u0000\fDare\nexpressed in terms of the Rashba and Dresselhaus SOI\ncoe\u000ecients \u000bRand\fDfor a proper choice of the coordi-\nnate system [67{74]. Introducing a creation operator  y\n\u001bkarXiv:1905.09241v1  [cond-mat.mes-hall]  22 May 20192\nacting on an electron with momentum k= (kx;ky) and\nspin component \u001balong thezaxis, the corresponding\nHamiltonian reads\nH0=X\n\u001b\u001b0Z\ndk y\n\u001bk\u0014~2k2\n2m\u0000\u0016 (1)\n\u0000\u000bxkx\u001by+\u000byky\u001bx\u0015\n\u001b\u001b0 \u001b0k:\nHere,\u001bjare the Pauli matrices acting in spin space.\nThe chemical potential \u0016is calculated from the cross-\ning point of the two spin-split bands at k= 0. In\nthe following, it will be convenient to introduce the\nSOI energy Eso=~2k2\nso=(2m) and the SOI momentum\nkso=m\u000bx=~2. We note that this Hamiltonian e\u000bectively\ndescribes a 2D array of coupled Rashba wires if the mass\nmis also chosen to be anistropic in the xyplane such\nthatmy6=mx[38, 75{86].\nThe proximity e\u000bect between the 2DEG and the SC is\ndescribed by the following Hamiltonian\nHsc=\u0001sc\n2X\n\u001b\u001b0Z\ndk\u0010\n y\n\u001bk[i\u001by]\u001b\u001b0 y\n\u001b0(\u0000k)+ H.c.\u0011\n;(2)\nwhere \u0001 scis the induced SC gap. The resulting 2DEG-\nSC heterostructure is placed in the vicinity of a FM layer,\nand the setup is subjected to an external magnetic \feld\nH(t). Under the FM resonance condition (see discussion\nbelow), the FM generates an oscillating demagnetizing\n\feld which adds up to H(t) to produce a total magnetic\n\feldB(t) =B?cos(!t)ez+Bksin(!t)ukin the 2DEG.\nHere,!= 2\u0019=T denotes the oscillation frequency and\nthe 2D vector uk= (ux;uy) indicates the orientation of\nthe magnetic \feld in the xyplane. The FM proximity\ne\u000bect is described by the following Floquet-Zeeman term\nHZ(t) = 2X\n\u001b\u001b0Z\ndk y\n\u001bk\u0010\nt?\nZcos(!t)[\u001bz]\u001b\u001b0\n+tk\nZsin(!t)[uk\u0001\u001b]\u001b\u001b0\u0011\n \u001b0k;(3)\nwheret\u0017\nZ=\u0016Bg\u0017B\u0017=2 (with\u0017=k;?) are two Floquet-\nZeeman amplitudes. The anisotropy in the g-factors,\nwhich leads to gkandg?, arises from the quantum con-\n\fnement and the intrinsic strain of the 2DEG [87{89].\nThe resulting time-dependent problem can be solved\nusing the Floquet formalism [1{4], by writing the quasi-\nenergy operator H=H0+Hsc+HZ(t)\u0000i~@tin the\nFloquet-Hilbert space generated by T-periodic states\n l\u001bk= exp(\u0000il!t) \u001bk,l2Z. In this basis Hacquires\na simple block-diagonal form, where each block, also re-\nferred to as a Floquet band, is composed of the modes\nwith the same index l. The static term acts within the\nsame block and receives an additional constant energy\nshift~!l, while the oscillating term couples di\u000berent\nblocks. We assume that the chemical potential is re-\nstricted to\u0000Eso< \u0016 < 0, so that the frequency !can\nbe resonantly tuned to the energy di\u000berence between the\n(a) (b)\ntrivialFHeTS Weyl\nFIG. 2. (a) Energy spectrum E(kx) in the FHeTS phase\nwitht?\nZ=\u0001sc= 3 and\u000bx=\u000by. The helical edge modes are\nlocalized at the edges. (b) Phase diagram showing the gap to\nthe \frst excited bulk state (in units of Eso) as a function of\nthe ratios\u000by=\u000bxandt?\nZ=\u0001sc. Blue lines indicate the phase\nboundaries. In the isotropic regime, \u000bx=\u000by, the phase tran-\nsition occurs at the critical point \u0001 sc=t?\nZ. When\u000bx6=\u000by,\nthe critical point is transformed into a gapless Weyl phase,\nand the value t?\nZ=\u0001screquired to reach the FHeTS increases\nup to the point of strong anisotropy beyond which the FHeTS\ncannot be reached. Remaining parameters in both simula-\ntions aretk\nZ= 0, \u0001 sc=Eso= 0:2,ksoLx=ksoLy= 80, and\n\u0016=\u0000Eso=2.\ntwo spin-split bands. This allows us to treat the oscil-\nlating terms at low energies by only taking into account\nthe coupling between the modes at l= 0, to which we\nassociate a Floquet band index \u001c= 1, and the modes at\nl=\u00001 with\u001c=\u00161. As a result, in the Nambu basis \ty\nk\n= ( y\n1\"k, y\n1#k, y\n\u00161\"k, y\n\u00161#k, 1\"\u0000k, 1#\u0000k, \u00161\"\u0000k, \u00161#\u0000k),\nthe total Hamiltonian reads as H=R\ndk\ty\nkHk\tk=2,\nwith the Hamiltonian density\nHk=\u0014~2k2\n2m\u0000\u0016+~!(\u001cz\u0000\u001c0)\n2\u0015\n\u0011z+ \u0001 sc\u0011y\u001by+\u000byky\u001bx\n\u0000\u000bxkx\u0011z\u001by+t?\nZ\u0011z\u001cx\u001bz+tk\nZ(ux\u001cy\u001bx+uy\u0011z\u001cy\u001by) (4)\nwith the Pauli matrices \u001ci(\u0011i) acting in the Floquet\n(particle-hole) space. In the following, we analyze dif-\nferent topological phases of the system as a function of\nthe parameters appearing in Eq. (4).\nFloquet Helical Topological Superconductor. In order\nto determine the phase diagram of our model, we \frst\nconsider the e\u000bect of the out-of-plane component of B(t)\nby imposing tk\nZ= 0 in Eq. (4). In the isotropic regime\nwith\u000bx=\u000by, the Fermi surface is composed of two con-\ncentric circles and the problem depends only on the mag-\nnitude of the momentum jkj, as shown in Fig. 1(b). The\nresonance condition for the frequency !is satis\fed along\nthe entire circle with the smallest Fermi momentum (see\nthe Supplemental Material (SM) [90] for more details).\nThe Hamiltonian is linearized close to the Fermi surface\nand provides the eigenenergies E2\n1= (~vF\u000ek)2+ \u00012\nscand\nE2\n2;\u0006= (~vF\u000ek)2+(t?\nZ\u0006\u0001sc)2, withvF=\u000bx=~the Fermi\nvelocity and \u000ekthe radial distance from the Fermi sur-\nface. The phase diagram consists of two gapped phases\nseparated by the gapless line \u0001 sc=t?\nZ. The topologically3\ntrivial (topological) phase is identi\fed with the regime\n\u0001sc> t?\nZ(\u0001sc< t?\nZ). Fortk\nZ= 0, the system is char-\nacterized by an emerging e\u000bective time-reversal symme-\ntryTe\u000b=\u0000i\u001cz\u001byK, a particle-hole symmetry P=\u0011xK,\nand a chiral symmetry UC=PTe\u000b, withKthe complex-\nconjugation operator. Thus, the system belongs to the\nDIII symmetry class with Z2topological invariant.\nThe topological phase, denoted as FHeTS, hosts gap-\nless boundary modes { a Kramers pair of Floquet Ma-\njorana fermions j\b\u0006i, obeying Pj\b\u0006i=j\b\u0006iand\nTe\u000bj\b\u0006i=\u0006j\b\u0007i. The Kramers partners propagate in\nopposite directions along the same edge, forming a pair\nof helical modes protected by the e\u000bective time-reversal\nand particle-hole symmetries. We have veri\fed the pres-\nence of these modes numerically in the discretized version\nof the model (see the SM [90]) de\fned on a rectangular\nlattice of size Lx\u0002Lywith periodic boundary conditions\nalongx, as shown in Fig. 2(a).\nIf the rotation symmetry is broken ( \u000bx6=\u000by), the res-\nonance condition can be satis\fed only along a particular\ndirection in momentum space, resulting in an o\u000b-set \u000e!\nin the resonance condition almost everywhere except at\na few points on the Fermi surface. While a small \u000e!in\nthe weak anisotropy regime hardly a\u000bects the phase di-\nagram and can be compensated by increasing t?\nZ, strong\nanisotropy e\u000bects are more drastic. In Fig. 2(b), we cal-\nculate numerically the energy of the lowest bulk state as\na function of the ratios \u000by=\u000bxandt?\nZ=\u0001sc. We see that\nthe critical point \u0001 sc=t?\nZtransforms into a gapless Weyl\nphase at a \fnite value of anisotropy ( \u000by\u0000\u000bx)=\u000bx. This\ngapless regime is characterized by a semi-metal energy\nstructure with four Weyl cones. The nodes of the Weyl\ncones appear \frst on the Fermi surface and move further\nin the reciprocal space, when the parameters t?\nZ=\u0001scand\n\u000by=\u000bxare modi\fed. We also note that if we decrease the\nratiot?\nZ=\u0001scup to a point of reaching the phase transi-\ntion to the trivial phase, the low energy physics becomes\ninsensitive to the anisotropy.\nFloquet Majorana corner states. Next, we analyze the\ne\u000bect of an oscillating in-plane magnetic \feld that breaks\nthe e\u000bective time-reversal symmetry Te\u000band, thus, gaps\nout the helical edge modes of the FHeTS. Nevertheless,\nthe system remains topologically non-trivial as it now\nhosts a set of zero-energy MCSs, characteristic for the\nFHOTS phase [91]. The presence of such MCSs is uncov-\nered by focussing on the low-energy degrees of freedom\nexpressed in terms of the Majorana edge modes j\b\u0006i.\nThe in-plane Zeeman \feld Bkcouples the two helical\nmodes, leading to the following low-energy Hamiltonian\ndensity:\nHedge=~ve\nFjkj\u001az+~tk\nZ\u001ay; (5)\nwhere the Pauli matrices \u001aiact in the space of\nj\b\u0006i,ve\nFis the velocity of the helical edge modes,\nand~tk\nZ=tk\nZImh\b\u0000jux\u001cy\u001bx+uy\u0011z\u001cy\u001byj\b+iis the `mass\n(a)\n(d)\nWeyl\ntrivialFHOTSFHeTS\n(c)\n(b)\nFIG. 3. (a,b,d) Probability density of the lowest energy state\nin the FHOTS phase for t?\nZ=\u0001sc= 3,tk\nZ=\u0001sc= 2:5, and\n\u000bx=\u000by. The inset shows the 10 lowest eigenenergies. (a)\nForuk= (1;1)=p\n2, the in-plane Zeeman \feld Bkopens gaps\nin all edge modes such that zero-energy MCSs emerge at two\nopposite corners. (b) The vector uk= (0;1) is parallel to the\nedges along which the system stays gapless. (c) Phase dia-\ngram showing the bulk gap (color coded in units of Eso) as a\nfunction of the ratios t?\nZ=\u0001scandtk\nZ=\u0001scwithuk= (1;1)=p\n2.\nThe critical point \u0001 sc=t?\nZattk\nZ= 0 merges into a gapless\nWeyl phase at \fnite tk\nZ. As a result, higher value of t?\nZ=\u0001sc\nare required to reach the topological phase. The various\nphase boundaries correspond to \u000by=\u000bx= 0:8 (red dotted),\n0:9 (purple dashed), and 1 :0 (blue solid). (d) The MCSs are\nstable against moderate external perturbations and disorder:\n\u000e!=p\n3S\u0016= \u0001 Z= 0:10Eso. Rest of parameters are the\nsame as in Fig. 2.\nterm'. Thus, our system is described by the well-known\nJackiw-Rebbi model [7, 8].\nFrom the symmetry of the modes j\b\u0006i, we deduce that\nthe value of the mass term ~tk\nZonly depends on the com-\nponent of the in-plane \feld Bkwhich is perpendicular to\nthe corresponding edge in a rectangular geometry (see\nthe SM [90]). Generally, the sign of ~tk\nZis opposite on two\nparallel edges at x= 0 (y= 0) andx=Lx(y=Ly).\nHence, ~tk\nZhas to change its sign at two opposite corners\nof the 2DEG, leading to the emergence of domain walls at\nthese corners that host zero-energy MCSs, see Fig. 3(a).\nIn the special case when ukis parallel to one of the edges,\nthe corresponding edge modes stay gapless, see Fig. 3(b).\nThe simple boundary description in terms of the\nJackiw-Rebbi model is expected to work in the regime\nwhere the amplitude of the in-plane magnetic \feld is\nsmall. In order to construct the full phase diagram, we\ncalculated numerically the gap to the \frst excited bulk\nstate as a function of the ratios t?\nZ=\u0001scandtk\nZ=\u0001sc, see\nFig. 3(c). The FHOTS phase emerges from the FHeTS\nphase at non-zero tk\nZ. However, if tk\nZis large, the system\nenters into the gapless Weyl phase. To observe MCSs, the4\nFM\n2DEG\nSC\nFIG. 4. Setup of the FM layer to generate the desired\noscillating magnetic \feld in the 2DEG. The 2DEG-SC het-\nerostructure is placed in the vicinity of the FM. The system is\nsubjected to an external magnetic \feld H(t) =H0+h(t) (yel-\nlow lines) generating FM resonance (shown not to scale). This\ninduces the precession of the FM magnetization M(t) (black\nlines inside the FM) and demagnetizing \feld D(t) (black lines\noutside of the FM). Close to the surface of the FM layer, the\nstatic components of D(t) and H(t) cancel out, so that the\ndynamics of the total \feld B(t) =H(t) +D(t) in the 2DEG\nlayer is close to a full 360\u000erotation.\nFloquet-Zeeman \feld perpendicular to the 2DEG plane\nshould dominate such that the condition tk\nZ< t?\nZis ful-\n\flled. All the results remain qualitatively the same for a\nweak anisotropy ( \u000bx6=\u000by), which shifts the topological\nphase transition line only slightly.\nExperimental feasibility. Next, we discuss the stabil-\nity of our setup. We check numerically that the topo-\nlogical phases are robust against an o\u000b-set \u000e!in the res-\nonance frequency !. Similarly, we check the stability\nwith respect to an on-site disorder by adding a \ructu-\nating chemical potential randomly chosen from a uni-\nform distribution with standard deviation S\u0016and with\nrespect to a static magnetic \feld by adding a Zeeman\nterm with strength \u0001 Zdirected both in-plane and out-of-\nplane. The result of the calculations is shown in Fig. 3(d).\nThe topological phases are stable against the perturba-\ntions of a strength comparable to the gap. The e\u000bect of\nthe out-of-plane component of the static magnetic \feld\nis also less important: the MCSs can be observed even\nfor \u0001 Z\u0018t?\nZand disappear only when the static term\nbecomes stronger.\nIn experiments, the proximity induced SC gap \u0001 scis\nexpected to be of the order of 0 :05 meV, depending on\nthe properties of the SC and the strength of the cou-\npling between the SC and the 2DEG [94{96]. As shown\nin this work, the strength of the Floquet-Zeeman am-\nplitudet?\nZshould exceed \u0001 scto reach the topological\nregimes. Hence, assuming that the 2DEG material has\nan electron g-factorg?= 15, the amplitude of the mag-\nnetic \feldB?should be of the order of 0 :1 T. At the\nsame time, the static component of the magnetic \feld\nshould be smaller than the dynamic one and the oscilla-\ntion frequency !should be in the GHz range. The FMlayer in the setup is proposed to generate the required\nZeeman \felds as follows. Applying an external magnetic\n\feldH(t) =H0+h(t) withjh(t)j\u001cjH0jandh(t)?H0\nunder FM resonance condition induces a precession of the\nFM magnetization M(t) [11, 12]. The precession cone of\nM(t) depends on the angle between the FM easy axis and\nH0, while the resonance frequency is determined by the\nmagnitudejH0j. Outside of the FM, the total \feld B(t)\nis equal to the sum of the external \feld H(t) and an os-\ncillating demagnetizing \feld D(t), see Fig. 4. Hence, by\ncarefully choosing the system geometry, the static com-\nponent ofB(t) could be adjusted close to zero over a\nlarge region of space in the proximity of the FM surface\nincluding the 2DEG. The amplitude of the remaining os-\ncillating component overcomes the threshold of 0 :1 T in-\nside the 2DEG, as we have con\frmed by micromagnetic\nsimulations (see the SM [90]). Promising candidates for\nsuch FMs are e.g. EuS [64], GdN [65], and YiG [66].\nAlternatively, the fast switching or the sustained oscil-\nlation of the FM magnetization has already been achieved\nexperimentally by shining optical light on a FM (via an\nall-optical magnetization reversal) [99{101], by applying\npiezostrain [102, 103] or by injecting a spin-polarized cur-\nrent (via a spin-orbit torque) [104{108]. This domain of\nresearch is currently under an active exploration because\nof its crucial role in the implementation of magnetic mem-\nory and logic devices. We also note that in our setup, the\nmagnetic \feld in the 2DEG can originate from both the\nFM demagnetizing \feld at a moderate range and from\nthe exchange interactions at atomic distances.\nConclusions. We have considered a Floquet triple-\nlayer setup of a 2DEG proximity coupled to a SC and\na FM. Under resonant drive the FM induces an oscillat-\ning Zeeman \feld in the 2DEG. The out-of-plane com-\nponent of the magnetic \feld competes with the proxim-\nity induced SC gap and leads to the emergence of the\nFHeTS hosting an e\u000bective Kramers pair of gapless he-\nlical edge modes. Moreover, the in-plane component of\nthe magnetic \feld enters into the low-energy description\ncorresponding to the e\u000bective Jackiw-Rebbi model as a\nmass term and opens a gap in the edge mode spectrum.\nChange in the sign of the mass term, which inevitably\noccurs at two opposite corners of the system in a rectan-\ngular geometry, leads to the emergence of Floquet MCSs.\nWe argued that the proposed setup is within experimen-\ntal reach combining available magnetic, semiconducting,\nand superconducting materials.\nAcknowledgments. 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Topological criterion in the isotropic limit\nIn the isotropic limit, \u000bx=\u000by, when the in-plane component of the oscillating magnetic \feld is zero, tk\nZ= 0, the\nsystem described by the Hamiltonian given in Eq. (4) has a continuous rotation symmetry in the xyplane. The\ncorresponding symmetry operator reads Urot\nJz(\u0012) = exp (\u0000i\u0012\u0011zJz=~), with\u0012being the rotation angle and Jzthez\ncomponent of the orbital angular momentum operator. Thus, the energy structure depends only on the absolute\nvalue of the momentum jkj=q\nk2x+k2yand can be calculated along a particular direction in the xyplane. For\nsimplicity we perform the calculations along the axis ky= 0 [see Fig. S1]. This results in the following Hamiltonian\ndensity\nHk(ky= 0) =\u0012~2k2\nx\n2m\u0000\u0016\u0013\n\u0011z\u0000\u000bxkx\u0011z\u001by+ \u0001 sc\u0011y\u001by+t?\nZ\u0011z\u001cx\u001bz+~!\u0011z\u0012\u001cz\u0000\u001c0\n2\u0013\n: (S1)\nHere, similar to the main text, we work in the Nambu basis \ty\nk= ( y\n1\"k, y\n1#k, y\n\u00161\"k, y\n\u00161#k, 1\"\u0000k, 1#\u0000k, \u00161\"\u0000k,\n \u00161#\u0000k). We de\fne by \u0011ithe Pauli matrices (the 2 \u00022 identity matrix for i= 0) acting on the particle-hole space, \u001ci{ on\nthe Floquet space, \u001bi{ on the space associated with the two spin components in the zdirection. The total Hamiltonian\nacts in the corresponding tensor-product space and for notational simplicity we suppress the explicit writing of the\ntensor product sign and the identity matrices. The system is characterized by an e\u000bective time-reversal symmetry\nTe\u000b=UTK=\u0000i\u001cz\u001byKwithUy\nTHkUT=H\u0003\n\u0000k, a particle-hole symmetry P=UPK=\u0011xKwithUy\nPHkUP=\u0000H\u0003\n\u0000k,\nand a chiral symmetry UC=PTe\u000b=\u0000i\u0011x\u001cz\u001bywithUy\nCHkUC=\u0000Hk. From this we deduce that the system belongs\nto the DIII symmetry class, characterized by Z2topological invariants, and has two distinct topological phases.\nIn the following, it will be convenient to change to the spin basis with quantization axis along the ydirection, using\nthe unitary rotation in the yzplane described by the operator Urot\n\u001bx= exp (\u0000i\u0019\u0011z\u001bx=4). This transformation satis\fes\n~\tk=Urot\n\u001bx\tkand ~Hk=Urot\n\u001bxHkUrot\n\u001bxy, with\n~Hk(ky= 0) =\u0012~2k2\nx\n2m\u0000\u0016\u0013\n\u0011z\u0000\u000bxkx\u001bz+ \u0001 sc\u0011y\u001by+t?\nZ\u001cx\u001by+~!\u0011z\u0012\u001cz\u0000\u001c0\n2\u0013\n: (S2)\nIn the new spin basis the Rashba SOI term acts on the spin component along the zaxis and we denote by \u001b= 1;\u00161\nits two possible orientations. In order to achieve the resonance condition, we \frst \fx the chemical potential \u0016to be\nsmaller in absolute value than the SOI energy, \u0000Eso<\u0016< 0. For instance, we restrict the discussion to the Floquet\nband\u001c= 1 only. As a result of the Rashba SOI, which lifts the spin degeneracy, there are four Fermi momenta\nkF\n1\u001b\u0006=\u001bkso\u0006k\u0016;withk\u0016=r\n2m(Eso+\u0016)\n~2; kso=m\u000bx\n~2;andEso=~2k2\nso\n2m: (S3)\nThe resonance condition is \fxed at the momentum kres=kF\n11\u0000, where the spin-split band \u001b= 1 crosses the chemical\npotential\u0016, as shown in Fig. S1. The resonance frequency is tuned to the energy di\u000berence between the two spin-split\nbands. Since the energy of the band \u001b= 1 atkresis zero, ~!is simply equal to the energy of the band \u001b=\u00161, which\ncan be written as\n~!=~2(kres+kso)2\n2m\u0000~2(kres\u0000kso)2\n2m= 4Eso \n1\u0000r\nEso+\u0016\nEso!\n: (S4)\nAccording to the Floquet formalism [1{4], the second Floquet band \u001c=\u00161 is shifted in energy with respect to the \frst\nband\u001c= 1 by an energy \u0000~!. Hence, it crosses the chemical potential \u0016at four momenta\nkF\n\u00161\u001b\u0006=\u001bkso\u0006(2kso\u0000k\u0016): (S5)\nWhen!is at resonance, the following identities hold true: kF\n\u00161\u00161+=kF\n11\u0000=kresandkF\n\u001611\u0000=kF\n1\u00161+=\u0000kres.2\nFIG. S1. The band structure of the 2DEG with SOI in the isotropic regime \u000bx=\u000byexpressed as a function of kxatky= 0\nand chemical potential \u0016(dashed line at bottom). The frequency !of a periodic drive is chosen resonantly such that the two\nFloquet bands (corresponding to \u001c= 1;\u00161) have the same smallest Fermi momentum kres(indicated by green circles) for the\ntwo spin-split bands (represented by the blue and red colors). Operators R\u001c\u001bandL\u001c\u001bcorrespond to the slowly varying left\nand right movers. The symbols colored in green indicate the operators involved in the resonant process at kres.\nTo see the e\u000bect of t?\nZand \u0001 scanalytically, we linearize the spectrum around the Fermi momenta kF\n\u001c\u001b\u0006and represent\nthe original operators in terms of slowly varying left and right moving \felds [5, 6] using the following relations\n~ 1\u001b(x) =~R1\u001b(x)e\u0000ikF\n1\u001b+x+~L1\u001b(x)e\u0000ikF\n1\u001b\u0000x;\n~ \u00161\u001b(x) =~R\u00161\u001b(x)e\u0000ikF\n\u00161\u001b+x+~L\u00161\u001b(x)e\u0000ikF\n\u00161\u001b\u0000x: (S6)\nIn the basis associated with the slowly varying \felds, the Hamiltonian density becomes\n~Hk(ky= 0) = ~vF\u000ek\u001az+ \u0001 sc\u001ax\u0011y\u001by+t?\nZ\n2\u001cx(\u001ax\u001by\u0000\u001ay\u001bx): (S7)\nHerevF=\u000bx=~is the Fermi velocity assumed to be equal for both Floquet bands, \u000ek=kx\u0000kF\n\u001c\u001b\u0006is the distance from\nthe Fermi momenta kF\n\u001c\u001b\u0006, and the Pauli matrices \u001aiact on the space of left and right movers. The bulk spectrum of\nthe linearized problem is given by\nE2\n1= (~vF\u000ek)2+ \u00012\nsc; E2\n2;\u0006= (~vF\u000ek)2+ (t?\nZ\u0006\u0001sc)2: (S8)\nBoth the Floquet-Zeeman and the superconducting terms induce an opening of the gap at the Fermi momenta \u0006kres,\nwhich leads to a competition between the two terms. In contrast, the gap at the remaining Fermi momenta is\nopened only by the superconducting term. When the Floquet amplitude t?\nZbecomes of the same strength as the\nsuperconducting pairing amplitude \u0001 sc, we observe the closing of the gap, indicating a topological phase transition.\nWhen \u0001 sc>t?\nZthe system is in the topologically trivial phase, while in the regime \u0001 sc<t?\nZit hosts a Kramers pair\nof helical Majorana edge modes, connected one to another via Te\u000b, as shown in Fig. 2(a) of the main text.\nS2. Boundary description of the FHeTS and FHOTS phases\nIn order to better understand the two topological phases emerging in our model, we study its low-energy behavior\nwith focus on the boundary of the system. For simplicity, we again consider only the isotropic case \u000bx=\u000byhere.\nWe start with the FHeTS phase which is characterized by the presence of helical gapless edge modes, as shown in\nFig. 2(a). We consider a strip geometry with periodic boundary conditions along yand open boundary conditions\nalongx. In such a geometry two pairs of helical gapless modes have to be localized at the two edges at x= 0 and\nx=Lx. As a result of the interplay between the time-reversal symmetry Te\u000band the particle-hole symmetry P, the\nenergy of these states at ky= 0 has to be equal to zero. Hence, in order to \fnd such states, we look for a zero-energy\nsolution of the real-space version of the Hamiltonian (S2). For convenience, we keep working in the spin basis with\nquantization axis along ydirection. We identify the momentum kxwith the spatial derivative \u0000i~@xand linearize\nthe Hamiltonian around the momenta kF\n\u001c\u001b\u0006in the basis of slowly varying \felds of Eq.(S6), which leads to\n~H(x;ky= 0) =\u0000i~vF\u001az@x+ \u0001 sc\u001ax\u0011y\u001by+t?\nZ\n2\u001cx(\u001ax\u001by\u0000\u001ay\u001bx): (S9)3\nWe now solve the equation ~H(x;ky= 0)j~\b\u0006i= 0, assuming that j~\b\u0006iare two states connected to each other\nthrough the e\u000bective time-reversal transformation Te\u000bj~\b\u0006i=\u0006j~\b\u0007i. They can be explicitly written as j~\b\u0006i=P\nx;j~\bj\n\u0006(x)jx;jiin the orthonormal basis of spatially localized states jx;jiwith an orbital (Nambu) index j. We\nare particularly interested in solutions of the form Pj~\b\u0006i=j~\b\u0006iassociated with self-adjoint Majorana operators.\nWe also focus on the edge at x= 0, which requires imposing vanishing boundary conditions ~\bj\n\u0006(x= 0) = 0 for all\nj. We \fnd that such solutions exist only in the topological phase \u0001 sc<t?\nZand are expressed as follows (suppressing\nnormalization constants):\n~\b+(x) =\u0010\nf1;g1;f\u00161;g\u00161;f\u0003\n1;g\u0003\n1;f\u0003\n\u00161;g\u0003\n\u00161\u0011T\n;\n~\b\u0000(x) =\u0010\n\u0000g\u0003\n1;f\u0003\n1;g\u0003\n\u00161;\u0000f\u0003\n\u00161;\u0000g1;f1;g\u00161;\u0000f\u00161\u0011T\n;\nf\u001c(x) =\u0000ig\u0003\n\u001c(x) =e\u0000ixkF\n\u001c\u001b+e\u0000x=\u0018\u0000e\u0000ixkF\n\u001c\u001b\u0000e\u0000x=\u0018\u0000: (S10)\nHere,\u0018=~v0\nF=\u0001scand\u0018\u0000=~v0\nF=\u0000\nt?\nZ\u0000\u0001sc\u0001\nare two correlation lengths. Functions f\u001candg\u001care related to each\nother as a result of the presence of an additional spatial unitary symmetry in the system. This symmetry reads as\nUMF=\u0011y\u001bx, withUMF~Hk(ky= 0)UMFy=\u0000~Hk(ky= 0) andUMFj~\b\u0006i=j~\b\u0006i. Whenkyis non-zero, it also maps\ntheycomponent of the momentum, i.e., ky!\u0000ky, e\u000bectively exchanging the two helical components of the gapless\nedge mode.\nIn the second part of this section we show how the modulated in-plane magnetic \feld with an amplitude t?\nZcouples\nthe two helical Majorana fermion edge modes j~\b\u0006iliving on the edge x= 0 and leads to the emergence of the FHOTS\nphase hosting MCSs. First, we write down the corresponding Hamiltonian density in the y-spin basis as\n~Hk\nZ=tk\nZ(ux\u001cy\u001bx+uy\u001cy\u001bz) =Urot\n\u001bxyHk\nZUrot\n\u001bx=Urot\n\u001bxyh\ntk\nZ(ux\u001cy\u001bx+uy\u0011z\u001cy\u001by)i\nUrot\n\u001bx: (S11)\nUsing the particle-hole symmetry P, we deduce that all the diagonal components of ~Hk\nZin this basis of states j~\b\u0006i\nare exactly zero:\nh~\b\u0006j\u001cy\u001bxj~\b\u0006i=h~\b\u0006jP\u001cy\u001bxPj~\b\u0006i=\u0000h~\b\u0006j\u001cy\u001bxj~\b\u0006i= 0;\nh~\b\u0006j\u001cy\u001bzj~\b\u0006i=h~\b\u0006jP\u001cy\u001bzPj~\b\u0006i=\u0000h~\b\u0006j\u001cy\u001bzj~\b\u0006i= 0: (S12)\nEmploying the e\u000bective time-reversal symmetry Te\u000b, we also \fnd that all the o\u000b-diagonal terms are purely imaginary:\nh~\b\u0000j\u001cy\u001bxj~\b+i=\u0000h~\b+jTe\u000b\u001cy\u001bxTe\u000bj~\b\u0000i=\u0000h~\b+j\u001cy\u001bxj~\b\u0000i=\u0000h~\b\u0000j\u001cy\u001bxj~\b+i\u0003;\nh~\b\u0000j\u001cy\u001bzj~\b+i=\u0000h~\b+jTe\u000b\u001cy\u001bzTe\u000bj~\b\u0000i=\u0000h~\b+j\u001cy\u001bzj~\b\u0000i=\u0000h~\b\u0000j\u001cy\u001bzj~\b+i\u0003: (S13)\nThis allows us to rewrite the term ~Hk\nZto the zeroth order in perturbation theory as ~Hk\nZ=~tk\nZ\u001ay, with the Pauli\nmatrix\u001aiacting in the space of j~\b\u0006iand~tk\nZ=tk\nZImh~\b\u0000jux\u001cy\u001bx+uy\u001cy\u001bzj~\b+i. Moreover, using the spatial unitary\nsymmetryUMF, we \fnd that\nh~\b\u0000j\u001cy\u001bzj~\b+i=\u0000h~\b\u0000j\u001cy\u001bzPTe\u000bj~\b\u0000i=ih~\b\u0000j\u0011x\u001cy\u001cz\u001bz\u001byj~\b\u0000i=ih~\b\u0000j\u0011x\u001cx\u001bxj~\b\u0000i\n=ih~\b\u0000jUy\nMF\u0011x\u001cx\u001bxUMFj~\b\u0000i=\u0000ih~\b\u0000j\u0011x\u001cx\u001bxj~\b\u0000i= 0: (S14)\nHence, the ycomponent of the magnetic \feld does not contribute to the emergence of the mass term ~tk\nZon the edge\nx= 0. As a result, the mass term can be simply expressed using the functions f\u0011andg\u0011as\n~tk\nZ= 2tk\nZuxZ1\n0dxIm(f\u0003\n1f\u0003\n\u00161\u0000f1f\u00161\u0000g1g\u00161+g\u0003\n1g\u0003\n\u00161) =\u00008tk\nZuxZ1\n0dxIm(f1f\u00161): (S15)\nFinally, after including the kinetic term linear in momentum, the total e\u000bective Hamiltonian density describing the\nx= 0 edge of the FHeTS under the applied magnetic \feld takes on the form\nHedge=~ve\nFky\u001az+~tk\nZ\u001ay; (S16)\nwhereve\nFis the velocity of the Majorana fermions j~\b\u0006i. We recover the Hamiltonian density of the Jackiw-Rebbi\nmodel [7, 8].4\nStarting from the above expression of Hedgeatx= 0, it is straightforward to generalize the result to other edges\nusing the spatial symmetries of the problem. In particular, for the opposite edge at x=Lx, one has to apply the\nre\rection symmetry, which simply corresponds to changing the direction of the magnetic \feld: ux!\u0000ux. Under\nthis transformation the mass term ~tk\nZchanges sign. The change in sign of the mass term in the Jackiw-Rebbi model\nimplies the presence of zero-energy domain wall bound states [7, 8], which we identify with MCSs. Hence, an even\nnumber of such corner states has to be present in the system (i.e. MBSs come in pairs). Alternatively, the same result\ncan be obtained using the in-plane rotation symmetries. We recall that in the basis with quantization axis along z\ndirection the rotation operator in the xyplane reads Urot\nJz(\u0012) = exp (\u0000i\u0012\u0011zJz=~), while, the spin rotation operator is\ngiven byUrot\nSz(\u0012) = exp (\u0000i\u0012\u0011zSz=~), withSz=~\u001bz. The eigenstates j~\b\u0006(\u0012)iliving on the edge rotated by an angle\n\u0012with respect to the x= 0 axis, are simply expressed as j~\b\u0006(\u0012)i=~Urot\nJz+Sz(\u0012)j~\b\u0006i=~Urot\nJz(\u0012)~Urot\nSz(\u0012)j~\b\u0006i, where all\nthe rotations are expressed in the y-spin basis In particular, the opposite edge at x=Lxis obtained for \u0012=\u0019. The\ntwo perpendicular edges at y= 0 andy=Lxin a rectangular geometry with open boundary conditions along both x\nandyare calculated using \u0012=\u0006\u0019=2. We note that in all the cases only the magnetic \feld component perpendicular\nto the edge contributes to the mass term ~tk\nZ.\nS3. Anisotropic regime\nIn this section we study more closely the e\u000bect of the anisotropy on the topological phase diagram shown in Fig. 2(b).\nWe also consider a more general scenario when the mass mis anisotropic in the xyplane with mx6=my. Such kind\nof anisotropy is more relevant to the experimental setups of coupled Rashba wires, where the strength of both the\ninter-wire SOI and the hopping term scales with the distance between the neighboring wires. First of all, we notice,\nthat when\u000bx6=\u000byand (or)mx6=my, the Fermi surface of the 2DEG is deformed since the rotation symmetry in\nthexyplane is broken. This can be seen by looking at the energy structure, which is equal to\nE(kx;ky) =~2k2\nx\n2mx+~2k2\ny\n2my\u0000\u0016\u0006q\n\u000b2xk2x+\u000b2yk2y: (S17)\nIn the following it will be convenient to go to the polar coordinate system with k=q\nk2x+k2yand\u0012= arctan(ky=kx).\nThe eigenvalues E(k;\u0012) =E(kx;ky) expressed in terms of the new coordinates read\nE(k;\u0012) =~2k2\n2\u0012cos2\u0012\nmx+sin2\u0012\nmy\u0013\n\u0000\u0016\u0006kq\n\u000b2xcos2\u0012+\u000b2ysin2\u0012=~2k2\n2m(\u0012)\u0000\u0016\u0006\u000b(\u0012)k; (S18)\nwhere we de\fned m(\u0012) =mxmy=(mxsin2\u0012+mycos2\u0012) and\u000b(\u0012) =q\n\u000b2xcos2\u0012+\u000b2ysin2\u0012. The Fermi surface\ncorresponds to the solutions of kat a given\u0012such thatE(k;\u0012) = 0. These solutions can be explicitly written as\nkF\n1\u0006(\u0012) =jkso(\u0012)\u0006k\u0016(\u0012)j;withk\u0016(\u0012) =r\n2m(\u0012) [Eso(\u0012) +\u0016]\n~2; kso(\u0012) =m(\u0012)\u000b(\u0012)\n~2;andEso(\u0012) =~2k2\nso(\u0012)\n2m(\u0012):(S19)\nThe result of the calculations is represented on Fig. S2. In the following we will be interested only in the solutions\nwith the smallest amplitude kF\n1\u0000(\u0012). The resonance condition for the frequency !has to be \fxed with respect to a\nparticular axis in the xyplane, following the construction procedure presented in Section S1. In this work we always\n\fx!with respect to the ky= 0 axis, such that Eq. (S4) is veri\fed for m=mxand\u0012= 0. Once the choice of the\naxis and the frequency !are \fxed, one can calculate the solution of the equation E(k;\u0012) =!of the form kF\n\u00161\u0006(\u0012) to\ndeduce the Fermi surface of the second Floquet band \u001c=\u00161. By analogy to Section S1, we denote the solution with\nthe smallest amplitude as kF\n\u00161+(\u0012), as shown in Fig. S2(a). The resonance condition corresponds to kF\n1\u0000(\u0012) =kF\n\u00161+(\u0012).\nAs a result of the anisotropy, the Fermi surfaces of the two Floquet bands have a mismatch, \u000ek(\u0012) =kF\n1\u0000(\u0012)\u0000kF\n\u00161+(\u0012).\nThis leads to the emergence of a frequency o\u000b-set \u000e!\u0019vF(\u0012)\u000ek(\u0012), wherevF(\u0012) =@kE(k;\u0012)=~is the Fermi velocity\nevaluated at the Fermi surface. The e\u000bect of such a frequency o\u000b-set is weaker than the one studied in Fig. 3(d) of\nthe main text, since it is not uniform along the entire Fermi surface and \u000e!vanishes (by construction) for some \u0012.\nIn order to study this e\u000bect more precisely, we use the numerical simulations [see Fig. S2(b)-(d)]. We calculate the\nphase diagram (the gap to the \frst excited bulk state) as a function of the ratios t?\nZ=\u0001sc,\u000by=\u000bx, andmx=my. For\nsimplicity, we assume that the two last terms scale as a power-law: mx=my=r,\u000by=\u000bx=rq. We \fnd that for small\nvalues of the ratio rthe gapped regime adiabatically connected to the 1D topological phase transforms into a gapless\nregime with several Weyl points. We also notice that for big values of rthe transition to the FHeTS phase is observed,5\nWeyl\n1D topological\n(b)\ntrivial(a)\n(c)\n(d)\nFHeTS\nFIG. S2. (a) Schematic presentation of the resonance o\u000b-set in the anisotropic regime \u000bx6=\u000byshowing the SOI momentum\nkso(\u0012) (black solid line), the two Fermi surfaces kF\n1\u0006(\u0012) (red solid and green dotted lines) of the Floquet band \u001c= 1 and the\nFermi surface kF\n\u00161\u0000(\u0012) (blue dashed line) of the Floquet band \u001c=\u00161. The coupling between the two Floquet bands occurs in the\nneighborhood of the Fermi surfaces kF\n1\u0000(\u0012) andkF\n\u00161+(\u0012) but, due to the SOI anisotropy, a mismatch \u000ek=kF\n\u00161+(\u0012)\u0000kF\n1\u0000(\u0012) (orange\n\flled surface) emerges. (b) Phase diagram showing the gap to the \frst excited bulk state (in units of Eso) as a function of the\nratiosr=mx=my= (\u000by=\u000bx)1=qandt?\nZ=\u0001scforq= 1. Di\u000berent colors and line types of the phase boundaries correspond to\ndi\u000berent power laws with q= 1 (blue solid), q= 2 (purple dashed), q= 3 (red dotted). (c)-(d) Probability density of the lowest\nenergy state and the band structure E(kx) (in the inset) in di\u000berent regions of the phase diagram. The choice of parameters\nt?\nF=\u0001scandris indicated by the red square and the green star in (b). Remaining parameters in all the simulations are tk\nZ= 0,\n\u0001sc=Eso= 0:2,ksoa= 0:2,ksoLx=ksoLy= 80, and\u0016=\u0000Eso=2.\nsimilarly to Fig. 2(b). In order to better visualize some phases emerging in the phase diagram, in Fig. S2(c)-(d) we\ncalculate the probability density of the lowest energy state and the bulk spectrum E(kx;ky). We see that the bulk\nof the FHeTS is gapped and the boundary hosts gapless helical edge modes, while at moderate rthe gap closes and\nfour Weyl points emerge in the Weyl phase.\nS4. Discretized model\nIn the discretized version of our model, the creation (annihilation) operators  y\n\u001c\u001bmn ( \u001c\u001bmn ) of an electron with spin\ncomponent \u001balong thezaxis, in a Floquet band \u001c, are de\fned at discrete coordinate sites nandm. For simplicity,\nwe assume that the lattice constant ais the same in xandydirections. The Hamiltonian describing the anisotropic\n2DEG, or equivalently the array of coupled Rashba wires, corresponds to\nH0=X\nmnX\n\u001cnh\n\u0000tx\u0010\n y\n\u001c\"m(n+1) \u001c\"mn+ y\n\u001c#m(n+1) \u001c#mn\u0011\n\u0000ty\u0010\n y\n\u001c\"(m+1)n \u001c\"mn+ y\n\u001c#(m+1)n \u001c#mn\u0011\n\u0000~\u000bx\u0010\n y\n\u001c\"(m+1)n \u001c#mn\u0000 y\n\u001c\"mn \u001c#(m+1)n\u0011\n+ ~\u000byi\u0010\n y\n\u001c\"(m+1)n \u001c#mn\u0000 y\n\u001c\"mn \u001c#(m+1)n\u0011\n+ H.c.i\n+X\n\u001b(2tx+ 2ty\u0000\u0016) y\n\u001c\u001bmn \u001c\u001bmno\n: (S20)\nHere,tx=~2=(2mxa2) andty=~2=(2mya2). The spin-\rip hopping amplitudes ~ \u000bxand ~\u000byare related to the\ncorresponding SOI strengths of the continuum model via \u000by=~\u000by=\u000bx=~\u000bx= 2a. The proximity induced s-wave6\nsuperconducting term is expressed as\nHsc=\u0000\u0001sc\n2X\nmnX\n\u001c\u0010\n y\n\u001c\"mn y\n\u001c#mn\u0000 y\n\u001c#mn y\n\u001c\"mn+ H.c.\u0011\n: (S21)\nThe out-of-plane component of the Floquet-Zeeman coupling is\nH?\nZ=t?\nZX\nmn\u0010\n y\n1\"mn \u00161\"mn\u0000 y\n1#mn \u00161#mn+ H.c.\u0011\n\u0000~!X\nmnX\n\u001b y\n\u00161\u001bmn \u00161\u001bmn; (S22)\nwhere the second term describes the constant energy shift of the second Floquet band with respect to the \frst one.\nIt incorporates the i~@t-term which is present in the expression of the quasi-energy operator written in the basis of\nT-periodic states. We \fnally express the in-plane component of the Floquet-Zeeman term as\nHk\nZ=tk\nZX\nmnX\n\u001c\u001b\u001b0 y\n\u001c\u001bmn\u0002\nuk\u0001\u001b\u0003\n\u001b\u001b0 \u0016\u001c\u001b0mn: (S23)\nIn the presence of translation symmetry, the total Hamiltonian H=H0+Hsc+H?\nZ+Hk\nZcan be diagonalized in\nmomentum space as H=P\nk\ty\nkHk\tk=2, leading to\nHk=\u0010\n2tx[1\u0000cos(kxa)] + 2ty[1\u0000cos(kya)]\u0000\u0016\u0011\n\u0011z\u00002~\u000bxsin(kxa)\u0011z\u001by+ 2~\u000bysin(kya)\u001bx\n+\u0001sc\u0011y\u001by+t?\nZ\u0011z\u001cx\u001bz+tk\nZ(ux\u001cy\u001bx+uy\u0011z\u001cy\u001by) +~!\u0011z\u0012\u001cz\u0000\u001c0\n2\u0013\n: (S24)\nThe explicit representation of the Hamiltonian as 8 \u00028 matrix reads\nHk=0\nBBBBBBBBBBBBB@\u0015k\u000bky+i\u000bkxt?\nZtk\nZ(\u0000uy\u0000iux) 0 \u0000\u0001sc 0 0\n\u000bky\u0000i\u000bkx\u0015ktk\nZ(uy\u0000iux)\u0000t?\nZ\u0001sc 0 0 0\nt?\nZtk\nZ(uy+iux)\u0015k\u0000~! \u000bky+i\u000bkx0 0 0 \u0000\u0001sc\ntk\nZ(\u0000uy+iux)\u0000t?\nZ\u000bky\u0000i\u000bkx\u0015k\u0000~! 0 0 \u0001 sc 0\n0 \u0001 sc 0 0 \u0000\u0015k\u000bky\u0000i\u000bkx\u0000tZ?tZk(uy\u0000iux)\n\u0000\u0001sc 0 0 0 \u000bky+i\u000bkx\u0000\u0015ktk\nZ(\u0000uy\u0000iux)t?\nZ\n0 0 0 \u0001 sc\u0000t?\nZtk\nZ(\u0000uy+iux)\u0000\u0015k+~! \u000bky\u0000i\u000bkx\n0 0 \u0000\u0001sc 0tk\nZ(uy+iux)t?\nZ\u000bky+i\u000bkx\u0000\u0015k+~!1\nCCCCCCCCCCCCCA;\n(S25)\nwhere we introduced the notations \u0015k=tx[1\u0000cos(kxa)] +ty[1\u0000cos(kya)]\u0000\u0016=2,\u000bkx= ~\u000bxsin(kxa), and\n\u000bky= ~\u000bysin(kya).\nS5. Micromagnetic simulations\nAs shown in the main text, to reach the topological phase transition one needs to generate a magnetic \feld B(t)\nwith a magnitude of the order of 0 :1 T oscillating in the GHz frequency range. Moreover, the oscillating component\nofB(t) should be greater than the static one. One possible solution to generate such a magnetic \feld consists in\nplacing the 2DEG layer in proximity to a FM slab. In this section we provide details of this construction.\nAt equilibrium, the magnetization Mof the FM is aligned with the easy axis of the FM. For simplicity, we assume\nthat the easy axis lies in the xyplane, parallel to the bottom (and top) surface of the FM slab. We consider a\ntime-dependent protocol, where the system at an initial time is subjected to a static external magnetic \feld H0,\nwhich makes an angle with the magnetization Mbut also lies in the xyplane. Moreover, an additional small\ntime-dependent magnetic \feld h(t) withjh(t)j\u001cjH0jis applied perpendicularly to H0such that the total applied\nexternal magnetic \feld is H(t) =H0+h(t). As a result, the magnetization M(t) will start to precess, described by\nthe Landau-Lifshitz-Gilbert (LLG) equation [9, 10]\ndM(t)\ndt=\u0000\rM(t)\u0002B(t)\u0000\u000b\r\nMsM(t)\u0002[M(t)\u0002B(t)]: (S26)\nHere,\ris the gyromagnetic ratio, \u000bthe dimensionless damping factor, and Msthe saturation magnetization. The\ntotal magnetic \feld B(t) =H(t)+D(t) comprises the applied static \feld H0, the small time-dependent \feld h(t), and7\nFIG. S3. OOMMF simulations of the total e\u000bective \feld B(t) in a 50 nm height pocket above the FM at four di\u000berent\nmoments of time in the interval [0 ;T]. Arrows represent the xandzcomponents of B(t), while the intensity plot represents\nitsycomponent. In the region inside the green box the static component of B(t) is much smaller than the oscillating part,\nand the e\u000bective \feld makes a full rotation in the yzplane, corresponding to the optimal arrangement for our Floquet setup.\nIn the region outside the green box the left (right) boundary of the FM is too close and no sizable precession of B(t) can\ndevelop there. The amplitude of the magnetic \feld is shown on the scale [0 ;0:124] T. Parameters of the simulation are\nLx\u0002Ly\u0002Lz= 200\u0002200\u0002100 nm andjH0j= 0:124 T.\nthe demagnetizing \feld D(t) generated by the FM. We note that the demagnetizing \feld has in general a complicated\nstructure that depends on the shape of the FM as well as the crystalline anisotropy. The LLG equation can be solved\nunder simpli\fed conditions, by neglecting the small \feld h(t) and assuming that the demagnetizing \feld is constant:\nD(t) =D. Such a solution describes a damped precession of the magnetization with the frequency determined by\nthe Kittel formula [11, 12]\n!=\rp\njH0jjH0+Dj: (S27)\nFinally, if the small \feld h(t) is resonant at the frequency !, the FM absorbs energy, establishing a steady-state with\na \fxed precession angle of the magnetization.\nThe protocol described above allows one to generate a substantial large oscillating magnetic \feld through the\ndemagnetizing \feld of the FM. However, generically the static component of such a magnetic \feld will dominate over\nthe dynamic one, which is not desired for our purpose. This problem can be removed by adjusting the geometry of\nthe setup (the size of the FM) and the strength of the applied static \feld H0, so that the static component of the\ndemagnetizing \feld D(t) cancelsH0exactly. In Fig. S3 we show the result of a calculation, based on the numerical\nsolution of the LLG equation using the \fnite di\u000berence micromagnetic solver OOMMF [13]. We \fnd that for a FM\nslab of the size Lx\u0002Ly\u0002Lz= 200\u0002200\u0002100 nm there is a wide region of the space distanced from the left and\nright boundaries of the FM, where the total e\u000bective \feld B(t) oscillates at the frequency != 2\u0019=T = 63 GHz and\nmakes a full 360\u000erotation in the yzplane (which is optimal for our purpose). The initial parameters of the simulation\nare chosen such that the static \feld H0withjH0j= 0:124 T is directed along the xaxis. At the initial time the\nmagnetization Mmakes an angle of 11 :3\u000ewith the vector H0. The amplitude of the resulting oscillating \feld B(t)\nis of the order of 0 :1 T.\n1J. H. Shirley, Phys. Rev. 138, 979 (1965).\n2N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027 (2014).\n3M. Bukov, L. D'Alessio, and A. Polkovnikov, Advances in Physics 64, 2, 139 (2015).\n4A. Eckardt and E. Anisimovas, New. J. Phys. 17, 093039 (2015).\n5J. Klinovaja and D. Loss, Phys. Rev. B 86, 085408 (2012).\n6J. Klinovaja and D. Loss, Eur. Phys. J. B 88, 62 (2015).\n7R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976).\n8R. Jackiw and J. Schrie\u000ber, Nucl. Phys. B 190, 253 (1981).\n9L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion. 8, 153 (1935).\n10T. L. Gilbert, Phys. Rev. 100, 1235 (1955).\n11C. Kittel, Phys. Rev. 73, 155 (1948).\n12C. Kittel, J. Phys. Radium 12, 3 (1951).\n13M. J. Donahue and D. G. Porter, \\OOMMF User's Guide, Version 1.0\", NISTIR 6376 (1999)."    },    {        "title": "1005.2087v1.Proposal_for_a_ferromagnetic_multiwell_spin_oscillator.pdf",        "content": "arXiv:1005.2087v1  [cond-mat.mes-hall]  12 May 2010Proposal for a ferromagnetic multiwell spin oscillator\nChristian Ertler∗,1,2Walter P¨ otz,1and Jaroslav Fabian2\n1Institute of Theoretical Physics, Karl-Franzens Universi ty Graz, Universit¨ atsplatz 5, 8010 Graz, Austria\n2Institute for Theoretical Physics, University of Regensbu rg,\nUniversit¨ atsstrasse 31, D-93040 Regensburg, Germany\nThe highly nonlinear couplingof transport andmagnetic pro perties in a multiwell heterostructure,\nwhich comprises ferromagnetic quantum wells made of dilute d magnetic semiconductors, is theo-\nretically investigated. The interplay of resonant tunneli ng and carrier-mediated ferromagnetism in\nthe magnetic wells induces very robust, self-sustained cur rent and magnetization oscillations. Over\na large window of steady bias voltages the spin polarization of the collector current is oscillating\nbetween positive and negative values, realizing a spin osci llator device.\nMagnetic resonant tunneling structures are prominent\nspintronic devices [1], which are proposed for spin valves,\nspin filtering [2–6], or for realizing digital magnetoresis-\ntance [7]. In nonmagnetic multiwell heterostructures in-\nteresting dynamiceffects such asthe formation ofelectric\nfield domains and the motion of charge dipoles through\nthe structure can be observed [8, 9]. Recently, it has\nbeenpredictedthatinstructureswithparamagneticwells\nthese phenomena can be controlled by an external mag-\nneticfield[10–12]. Theparamagneticsystemscanbehave\nas a spin oscillator device, in which the moving charge\ndipoles generate spin-polarized current oscillations in the\nMHzrange[12–14]. Inferromagneticmultiwellstructures\nmade of dilute magnetic semiconductors (DMS) in addi-\ntion also the magnetic order will become dynamic [15].\nThis additional degree of freedom enriches the dynamic\ncomplexity, which up to now is still largely unexplored.\nDMSs [16–18] are made magnetic by doping them with\ntransition metal elements, e.g., by incorporating Mn in\na GaAs crystal host. In several experiments ferromag-\nnetism has been generated in the bulk by tailoring the\nactual particle density electrically or optically [19, 20].\nIn 2d-confined systems made of DMSs the magnetic or-\nder depends strongly on the local spin density, which is\ninfluenced by the tunneling current [21–23]. This inter-\nplay leads to a novel mechanism of generating robust\nself-sustained current oscillations.\nIn this article a detailed study of the dynamics in a\nferromagnetic four-well system, as a prototype of the ex-\npected multiwell functionality, is provided. The carrier\ndynamics is described by a self-consistent sequential tun-\nneling model which includes momentum and spin relax-\nation inside the wells. The feedback effects of both the\ncarriers Coulomb interaction and the magnetic exchange\ncoupling with the magnetic ions are described within a\nmean-field picture. We give an qualitative explanation\nfor the occurrence of spin-polarized current oscillations,\nwhichrealizesthe functionalityofaspinoscillatordevice.\nThe band profile of a four-well heterostructure with\ntwo ferromagneticquantum wells made of a DMS, e.g., of\n∗email:christian.ertler@uni-graz.atGaMnAs GaAs \nGaMnAs \nGaAs emitter \ncollectorAlGaAs \nFIG. 1: (Color online) Schematic scheme of the band profile\nof the ferromagnetic four-well structure. The exchange int er-\naction of the magnetic ions in the second and third well is\nmediated by the carriers tunneling in and out of the wells.\nGaMnAs, issketchedinFig.1. Intheferromagneticwells\nthe ferromagnetic order of the magnetic ions is mediated\nbytheitinerantcarriers. Describingthe mutualexchange\ninteractionwithinamean-fieldpictureallowstoderivean\nanalytic expression for the exchange splitting ∆0\niof the\nsubbands in the ferromagnetic wells [1, 21, 22, 24]\n∆0\ni=Jpd/integraldisplay\ndznimp(z)|ψ0(z)|2\n×SBS/bracketleftBigg\nSJpds(n↑\ni−n↓\ni)|ψ0(z)|2\nkBT/bracketrightBigg\n,(1)\nwhereJpdis the coupling strength between the impurity\nspin and the carrier spin density (in case of GaMnAs p-\nlike holes couple to the d-like impurity electrons), zis the\nlongitudinal (growth) direction of the structure, nimp(z)\nis the impurity densityprofile ofmagneticallyactive ions,\nψ0(z) labels the well ground wave function, and s= 1/2\nis the particles spin. The Brillouin function of order S\nis denoted by BS, whereSis the impurity spin, which\nfor Mn equals 5/2. We assume a homogenous impurity\ndistribution making the spin polarization ξi=s(n↑\ni−n↓\ni)\nthe determining factor for the exchange splitting ∆0\ni.\nThe well spin polarization can be rapidly changed by\nin- and out-tunneling particles. However, the magnetic2\nimpurities need some time to respond until the corre-\nsponding mean field value ∆0is established. In the case\nof GaMnAs, experimental studies of the magnetization\ndynamics revealed typical response times τ∆of about\n100 ps [25]. We model the magnetization evolution in\nthe ferromagnetic wells i= (2,3) within the relaxation\ntime approximation, d∆ i/dt=−(∆i−∆0\ni)/τ∆.\nThe transportthroughweaklycoupled multiwell struc-\ntures is well described by a sequential tunneling model,\nwhichispresentedindetailinRefs.[9,12]. Thetransport\nmodel includes the following assumptions: (i) The wells\nare weakly coupled and tunneling can be described by\nan transfer Hamiltonian formalism. (ii) The particles in\neach well always stay in a quasiequilibrium state due to\nelastic and inelastic scattering, which is assumed to con-\nstitute the shortest time scale in the problem ( τscatt≈1\nps). (iii) Particles in the first excited subband relax\nrapidly to the ground level by inelastic scattering pro-\ncesses, e.g., due to emission of LO-phonons. This hap-\npens within a few ps, which is much smaller than the\nother relevant time scales occurring in the problem. (iv)\nThe spectral function of the subbands is a Lorentzian\nwith the broadening given by γ=/planckover2pi1/2τscatt. (v) Dur-\ning tunneling processes the parallel momentum and spin\nare conserved, i.e., effects of interface roughness are not\ntaken into account. (vi) Spin flipping processes inside\nthe well can be described by a single spin relaxation time\nτs. Based on these approximations rate equations for the\nspin-dependent particle densities nσ\ni,(σ=↑,↓=±1/2) in\ntheith well can be found\ndnσ\ni\ndt=Jσ\ni−1→i−Jσ\ni→i+1−nσ\ni−n0,σ\ni\nτs.(2)\nwhere the tunneling currents Ji→i+1are derived mi-\ncroscopically within the Kubo-formalism [9, 26]. The\nquasiequilibrium particle spin densities n0,σ\niare calcu-\nlated from the two conditions n↑\ni+n↓\ni=n0,↑\ni+n0,↓\niand\nn0,↑\ni−n0,↓\ni=ξ0\ni/s, withξ0\niresulting from inverting Eq. 1\nfor the actual ∆ i.\nFinally, the electron-electron interaction is taken into\naccount within a discretized Poisson equation, yielding\nFi−Fi−1=e\nε/parenleftBigg/summationdisplay\nσnσ\ni−Nback/parenrightBigg\n(3)\nwithFidenoting the electric field at the ith barrier,eis\nthe elementary charge, εdenotes the dielectric constant,\nandNbacklabels the background charge.\nThetunneling currentsarestronglysensitivetotherel-\native alignmentof adjacentsubbands and since the quan-\ntum well levels Eσ\ni,j=Ej+φi−σ∆iexplicitly depend\non the magnetic exchange splitting and the electrostatic\npotentialφiall equations are coupled highly nonlinearly.\nFor the numerical simulations we used generic parame-\nters representing GaAs and GaMnAs quantum wells, re-\nspectively, m= 0.5m0,εr= 12.9,Vbar= 300 meV,\nd= 20˚A,w= 20˚A,E0= 80 meV, E1= 188 meV,00.10.20.30.401000200030004000\nV (V)j (A/cm2)\nFIG. 2: Current-Voltage (IV)-characteristics of the four w ell\nstructure. Self-sustained current oscillations appear in the\nvoltage range of V= 0.22 V to V= 0.42 V, in which both\nthe maximum and minimum values of the oscillations are in-\ndicated by the solid and dashed lines, respectively.\nµe= 100 meV, nimp= 0.5×1020cm−3,Jpd= 0.15 eV\nnm3[24],γ= 2 meV, and τs= 0.1 ns, where mis the\neffective mass with m0denoting the free electron mass,\nandεris the relative permittivity, Vbaris the barrier\nheight,dandware the barrierand quantum well widths,\nE0andE1are the energies of the two lowest subbands,\nandµeis the emitter Fermi energy. The background\nchargeNback= 0.1nimpw= 1012cm−2is considered to\nbe only of about 10% of the nominal Mn doping den-\nsity [27] and the lattice temperature is T= 4.2 K. The\nboundary currents J0→1andJN→N+1withNdenoting\nthe total number of quantum wells are determined by\nthe tunneling from or into a three-dimensional Fermi sea\nat the emitter and collector side, respectively. For these\ncurrents analytic expressions can be derived [8, 26].\nFigure 2 shows the current-voltage( I−V) characteris-\ntics of the investigated structure. We obtain the typical\nsawtooth-like structure, as can be expected for a multi-\nwell heterostructure. However, in a broad voltage range\nfromV= 0.22 V toV= 0.42 V the current does not\nreach a steady state value; instead self-sustained current\noscillations occur. In this region also the spin polariza-\ntion of the collector current JN→N+1is oscillating be-\ntween positive and negative values realizing in this way a\nspin oscillator device. The effect could serve to generate\nac spin currents for investigating ac spin injection phe-\nnomena. The maximum amplitude is found at V= 0.32\nV, as illustrated in Fig. 3. Along with the current, also\nthe well magnetization is oscillating, as shown in Fig. 4.\nThe phase shift between the magnetizations (one well\npoints into up-direction while the other is down) is al-\nmost independent of the applied bias.\nBy analyzing the local spin densities and the flow-\ning tunneling currents in-between adjacent wells a qual-\nitative understanding for the underlying mechanism of\nthese oscillations can be found: (i) In the bias region, in3\n0.20.250.30.350.4−50050\nV (V)Pj (%)\nFIG. 3: Spin polarization of the collector current versus ap -\nplied bias Vin the voltage range in which oscillations are\noccurring. The maximum and minimum values of the oscilla-\ntions are indicated by the solid and dashed lines, respectiv ely.\n0 510 15 20−20020∆ (meV)\n0 510 15 20−50050\nt (ns)Pj (%)(a)\n(b)\nFIG. 4: (a) Exchange splitting ∆ iin the second (solid line)\nand third (dashed line) well versus time tatV= 0.32 V.\n(b) Spin polarization of the collector current versus time tat\nV= 0.32 V. The oscillations frequency is 0.25 GHz.which self-sustained oscillations are occurring, the cur-\nrentsJ1→2andJ2→3are governed by tunneling from the\nground state to the first excited subband of the neigh-\nboring wells, as shown schematically in Fig. 1. (ii) Ac-\ncording to an initial small positive energy splitting, spin\nup particles accumulate in the second well and, hence,\n∆2increases. (iii) During this accumulation, particles\nstart to tunnel to the third well. At some point the spin\nextraction to the third well overwhelms the spin injec-\ntion from the first well. Therefore, with time the spin up\nparticles in the second well are completely extracted and\nspin down particles become the majority species therein.\nThis spin inversion comes along with an inversion of the\nexchange splitting ∆ 2∝(n↑−n↓) from positive to neg-\native values. (iv) Now the whole process starts from the\nbeginningwiththeroleofspinupandspindownparticles\nbeing interchanged.\nThe frequency of the oscillations is of the order of\nabout 0.25 GHz and depends strongly, as can be ex-\npected, on the particle tunneling time τt, the spin relax-\nation time τs, and on the response time of the Mn-ions\nτ∆. Sinceτ∆is usually the largesttime-scale in the prob-\nlem, it determines the final frequency of the oscillations.\nIn summary, we haveshownthat in ferromagneticmul-\ntiwell structures an interesting dynamic interplay of res-\nonant tunneling and carrier mediated ferromagnetism\nemerges, which results in stable self-sustained current\noscillations. The underlying mechanism based on spin\ninversion is completely different from the one in para-\nmagnetic multiwell structures, in which moving charge\ndipoles are formed. The ferromagnetic heterostructure\nmay be useful as a spin oscillator device.\nThis work has been supported by the DFG SFB 689\nand the FWF project P21289-N16.\n[1] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and\nˇZuti´ c, Acta Physica Slovaca 57, 565 (2007).\n[2] A. Slobodskyy, C. Gould, T. Slobodskyy, C. R. Becker,\nG. Schmidt, and L. W. Molenkamp, Phys. Rev. Lett. 90,\n246601 (2003).\n[3] A. G. Petukhov, A. N. Chantis, and D. O. Demchenko,\nPhys. Rev. Lett. 89, 107205 (2002).\n[4] C. Ertler and J. Fabian, Appl. Phys. Lett. 89, 242101\n(2006).\n[5] E. Likovich, K. Russell, W. Yi, V. Narayanamurti, K.-\nC. Ku, M. Zhu, and N. Samarth, Phys. Rev. B 80,\n201307(R) (2009).\n[6] S. Ohya, I. Muneta, P.-N. Hai, and M. Tanaka, Phys.\nRev. Lett. 104, 167204 (2010).\n[7] C. Ertler and J. Fabian, Phys. Rev. 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Phys. 78, 809 (2006).\n[19] H.Ohno,D.Chiba, F.Matsukura, T.O.E.Abe, T.Dietl,\nY. Ohno, and K. Ohtani, Nature408, 944 (2000).\n[20] H.Boukari, P. Kossacki, M. Bertolini, D. Ferrand, J. Ci b-\nert, S. Tatarenko, A. Wasiela, J. A. Gaj, and T. Dietl,\nPhys. Rev. Lett. 88, 207204 (2002).\n[21] T. Dietl, A. Haury, and Y. M. d’Aubign´ e, Phys. Rev. B\n55, R3347 (1997).\n[22] T. Jungwirth, W. A. Atkinson, B. H. Lee, and A. H.\nMacDonald, Phys. Rev. B 59, 9818 (1999).[23] B. Lee, T. Jungwirth, and A. H. MacDonald, Semi. Sci.\nTechn.17, 393 (2002).\n[24] B. Lee, T. Jungwirth, and A. H. MacDonald, Phys. Rev.\nB61, 15606 (2000).\n[25] J. Wang, I. Cotoros, K. M. Dani, X. Liu, J. K. Furdyna,\nand D. S. Chemla, Phys. Rev. Lett. 98, 217401 (2007).\n[26] H. Xu and S. W. Teitsworth, Phys. Rev. B 76, 235302\n(2007).\n[27] S. Das Sarma, E. H. Hwang, and A. Kaminski, Phys.\nRev. B67, 155201 (2003)."    },    {        "title": "1711.02263v1.Dzyaloshinskii_Moriya_interaction_induced_extrinsic_linewidth_broadening_of_ferromagnetic_resonance.pdf",        "content": "arXiv:1711.02263v1  [cond-mat.mtrl-sci]  7 Nov 2017Dzyaloshinskii-Moriya interaction induced extrinsic lin ewidth broadening of\nferromagnetic resonance\nJung Hyun Oh,1Ik-Sun Hong,2and Kyung-Jin Lee1,2,∗\n1Department of Materials Science and Engineering, Korea Uni versity, Seoul 02841, Korea\n2KU-KIST Graduate School of Conversing Science and Technolo gy, Korea University, Seoul 02841, Korea\n(Dated: September 28, 2018)\nFor a thin ferromagnetic film with the Dzyaloshinskii-Moriy a interaction (DMI), we derive an\nexpression of the extrinsic ferromagnetic resonance (FMR) linewidth in a quantum mechanical way,\ntaking into account scatterings from structural inhomogen eity. In the presence of the DMI, the\nmagnon dispersion exhibits rich resonant states, especial ly in small external magnetic fields and\nstrong DMI strength. It is found that the FMR linewidth shows several characteristic features such\nas a finite linewidth at zero frequency and peaks in the low fre quency range.\nI. INTRODUCTION\nMagnetic damping1–7, which is parameterized by the\nGilbert damping constant α, describes the energy dissi-\npation rate of magnetization dynamics and determines\nthe performance of magnetic devices including magnetic\nrandom access memories and magnetic sensors8–11. The\nferromagnetic resonance (FMR) provides a way to mea-\nsure the damping αfrom the FMR linewidth12–16. A\nwidely adopted method for the estimation of αis to mea-\nsure the frequency-dependent change in the linewidth,\nassuming that the only intrinsic damping contribution\nis proportional to the frequency. However, this method\nis not always straightforward as the measured linewidth\nincludes not only intrinsic damping contribution but\nalsoextrinsiccontributionsoriginatingfrominhomogene-\nity of the sample. Especially, the contribution of two-\nmagnon scattering17–25to the linewidth also depends on\nthefrequency, demandingadetailedunderstandingofthe\nfrequency-dependent linewidth of two-magnonscattering\norigin21,25.\nRecently, therehasbeenmuchinterestinthespin-orbit\ncoupling effect for a thin ferromagnetic layer in contact\nwith a heavy metal layer because it permits the con-\ntrol and manipulation of electronic spin degree of free-\ndom26–39. In the presence of spin-orbit interaction com-\nbined with inversion symmetry breaking, the antisym-\nmetric exchange interaction, known as Dzyaloshinskii-\nMoriya interaction (DMI), emerges40,41in addition to\nthe symmetric exchange interaction responsible for fer-\nromagnetism. The DMI affects equilibrium spin textures\nby stabilizing chiral orders such as spin spirals, chiral\ndomain walls, and magnetic skyrmions42–60because the\nDMI energetically favors an orthogonal configuration of\ntwo neighboring spins with a fixed rotating sense. The\nDMI also affects spin wave properties by differentiating\nthe right- and left-moving spin waves because of their\ndifferent chirality61–72. In this respect, it is important\nto investigate the contribution of DMI inhomogeneity to\nthe extrinsic broadening of FMR linewidth through two-\nmagnon scattering process.\nIn this work, we theoretically investigate the FMR\nlinewidth of a thin ferromagnetic layer with DMI in aquantum mechanical way. A two-magnon model is used\nto describe an extrinsic source for the FMR linewidth,\ntaking into account scatterings caused by structural in-\nhomogeneities. The inhomogeneity may arise from a\nwidevarietyofmicrostructuralorigins. Here, weconsider\natomic-sizedstructuralimperfectionsandassociatedfluc-\ntuation of DMI strength in space.\nII. CALCULATIONAL MODEL\nWe consider a thin ferromagnetic film with the thick-\nnessdas illustrated in Fig. 1. We chose the film surface\nparallel to the zxplane and the y-axis normal to the sur-\nface. An external magnetic field Hextis applied in the yz\nplane with an angle θHfrom the z-axis. Then, the equi-\nlibrium magnetization is also in that plane with a tilt\nangleθM, determined by the competition of magnetic\nenergies.\nWe assume that the magnetization spatially varies in\nthezx-plane while it is uniform in the y-direction due to\nthe small thickness. Thus, we describe magnon modes\nwith a wave vector, k=k(cosφkˆ ez+ sinφkˆ ex) with a\nwave vector modulus, k. In this case, a demagnetization\nfactor of the system is given by Nd= (1−e−kd)/kd[73].\nWe also make use of the ˆ x1,2,3coordinate system. As\nshown in Fig. 1, the ˆ x3axis is coincided with the equilib-\nriummagnetizationdirectionwhilethe ˆ x2axisisdirected\nto the negative x-axis. Thus, H1(k) = cosθMHy\next(k)\nandH2(k) =−Hx\next(k).\nTo be specific, the Hamiltonian H0consists of the de-\nmagnetization, exchange, anisotropy, and Zeeman ener-\ngies;H0=Hdemag+Hex+HA+HZ. In a second-\nquantized form, H0is summarized as20,22,75\nH0=/summationdisplay\nk\n/summationdisplay\nκκ′=1,2mκ(−k)ǫκκ′(k)mκ′(k)\n−/radicalbig\n2/planckover2pi1γMS/summationdisplay\nκ=1,2mκ(k)µ0Hκ(−k)/bracketrightBigg\n,(1)2\nwhere\nǫ11=ǫ0\n11+ǫM(sin2θMcos2φk−cos2θM)+ǫexk2d2,\nǫ22=ǫ0\n22+ǫM(sin2φk+sin2θM)+ǫexk2d2,\nǫ12=ǫMsinφkcosφksinθM+iǫDMkdsinφkcosθM,\nǫ21=ǫ∗\n12. (2)\nHere, we define ǫM=/planckover2pi1γµ0MS(1−Nd) with the saturation\nmagnetization MSand the gyromagnetic ratio γ,ǫex=\n2/planckover2pi1γA/M Sd2with the exchange stiffness constant A, and\nǫDM= 4/planckover2pi1γD/M Sdwith the DMI strength D.ǫ0\n11andǫ0\n22\narek-independent termsandoriginatefrombothZeeman\nand surface anisotropy energies with an anisotropy field\nHS;\nǫ0\n11=/planckover2pi1γµ0[H3+(MS+HS)cos2θM],(3)\nǫ0\n22=/planckover2pi1γµ0/bracketleftbig\nH3−(MS+HS)sin2θM/bracketrightbig\n.(4)\nwhereH3=Hextcos(θH−θM) is the ˆ x3-component of\nmagnetic field Hext. By requiring the Mx= 0 in equi-\nlibrium, the magnetization angle θMis determined by\n2Hextsin(θH−θM) = (HS+MS)sin2θMfor a given mag-\nnetic field.\nWe express transverse components of the magnetiza-\ntion,M1,2=√2/planckover2pi1γMSm1,2with bosonic annihilation\nand creation operators approximated from the Holstein-\nPrimakoff setup74;\nm1(k) =1\n2/parenleftBig\nck+c†\n−k/parenrightBig\n, (5)\nm2(k) =1\n2i/parenleftBig\nck−c†\n−k/parenrightBig\n, (6)\nsatisfying usual bosonic commutation relations of\n[ck,ck] = 0, [c†\nk,c†\nk] = 0, and [ ck,c†\nk] =δk,k′.\nEigenenergy of magnon modes can be obtained by the\nBogoliubov transformation of the Hamiltonian H0. In\nterms of ǫκκ′, the eigenenergy at a kpoint is given by,\nǫ(k) = Im[ǫ12(k)]+/radicalbig\nǫ11(k)ǫ22(k)−Re[ǫ12(k)]2.(7)\n-65 55\n -30.0 M=x3Hextyx1\nx2\nx\nzk kθM\nθHφk\n-65 55\n -30.0 M=x3Hextyx1\nx2\nx\nzk kθM\nθHφk\nFIG. 1. (color online) Schematic representation of aferrom ag-\nnetic thin film. We choose a coordinate system for applied\nmagnetic field and equilibrium magnetization to be in the yz-\nplane. A local coordinate system ˆ x1,2,3is shown, where the\nˆ x3axis is coincide to the equilibrium magnetization directio n.0.0 0.2 0.4 0.6\nkd3035404550f (GHz)D=1.5 mJ/m2\nφk=0.3π\n0.1π\n 0.0\n-0.2π\n-0.5π\n0.0 0.2 0.4 0.6\nkd3035404550f (GHz)D=1.5 mJ/m2\nφk=0.3π\n0.1π\n 0.0\n-0.2π\n-0.5π\n-0.4 -0.2 0.0 0.2 0.4\nkzd-0.6-0.4-0.20.0kxd0.0\n0.3\n0.5\n1.0\nD=1.5 mJ/m2\n-0.4 -0.2 0.0 0.2 0.4\nkzd-0.6-0.4-0.20.0kxd0.0\n0.3\n0.5\n1.0\nD=1.5 mJ/m2\nFIG. 2. (color online) Magnon dispersion of a ferromagnetic\nthin film in the presence of DMI ( D= 1.5mJ/m2) is plotted\nfor various kdirections (upper panel). Equi-energy lines at a\nfixed energy equal to ǫ(k=0) are plotted in the ( kz,kx) plane\nfor several DMI strengths (lower panel). Used parameters ar e\nd= 5nm, A= 1.3×10−11J/m,µ0MS= 1.0 T,µ0HS=\n−0.5T, and µ0Hext= 1.0 T with θH= 0.\nFor two-magnon scattering of FMR experiment, we fo-\ncus on the energy ǫ(k= 0) =/radicalbig\nǫ0\n11ǫ0\n22and its resonant\nstates, namely a set of states at k/ne}ationslash= 0 with the same\nenergy. We plot the dispersion of the system at various\nDMI strengths in the upper panel of Fig. 2. In the fig-\nure, we display those resonant states with solid circles,\nsatisfying the relation of ǫ(k) =/radicalbig\nǫ0\n11ǫ0\n22. We find that\nthe calculated dispersion is different from the DMI-free\ncasemainly in two points; the resonantstate with the the\nlargest|k|occurs around the angle φk=−π/2 (in the\ncase ofD <0,φk=π/2) and moreover, its value of kdis\nnot small, but can be comparable to one. In the absence\nof DMI, resonant states are determined from the compe-\ntition between exchange and demagnetization energies.\nIn this case, the resonant state with the largest koccurs\natφk= 0. In the presence of DMI, on the other hand, an\nadditionaldecreasingenergyis incorporatedin theregion\nofksinφk<0through ǫ12(k) ofEq.(2). Consequently, as\nkincreases, this energy competes with an increasing part\n(i.e., the exchange energy) and eventually, the largest- k\nresonant state is formed around φk=−π/2.\nIn order to show the tendency clearly, we plot the evo-\nlution of resonant states in the lower panel of Fig. 2 for\nseveral DMI strengths. In a DMI-free case ( D= 0), one\nfinds two lobes having the largest distances at φk= 0\nandπ, respectively. As we increase DMI strength, the\ntwo lobes become one at a certain DMI energy (say ǫ0\nDM)3\nby creating a resonant state at φk=−π/2. Eventually,\nfor a sufficiently strong DMI, the resonant state with the\nlargestkappears at φk=−π/2.\nAccording to the two-magnon theory, inhomogeneities\nallow a magnon mode at k= 0 to scatter into modes\nwith finite wave vectors and couple these modes with\ndifferent wave vectors. To describe this coupling, we add\na perturbation term into the Hamiltonian as, H=H0+\nHpertub;\nHpertub=/summationdisplay\nk,k′/summationdisplay\nκ,κ′=1,2mκ(−k)Uk,k′\nκκ′mκ′(k′).(8)\nIn general, the potential Uk,k′\nκκ′is not diagonal in the k-\nspace and thus causes a coupling between different k-\nstates. The perturbation potential from inhomogeneities\nis zeroin average,but it providesappropriatemomentum\nand energy for magnon energy levels to be shifted and\nbroaden.\nIn this work, we focus on DMI-induced perturbations.\nInferring from the imaginary part of ǫ12(k) in Eq. (2) rel-\nevanttotheDMI,weconsidertwofeasiblemechanismsas\nperturbation sources. One is a random spatial variation\nof the parameter Dwhile the other is an abrupt change\nofthe magnetizationvectors, forexample, atedgesofter-\nraceonsamplesurfaces, nearmagneticandnon-magnetic\nimpurities, etc. If structural defects whose characteristic\nsizeis largerorequaltothe film thickness d, the magneti-\nzation will be slowly varying and the former is dominant\nfor the fluctuation, whereas, in a scale much less than\nd, the latter is dominant. In both cases, the fluctuation\npotential is modeled as,\nUk,k′\n11=Uk,k′\n22= 0,\nUk,k′\n12=−Uk′,k\n21=/summationdisplay\nje−i(k−k′)·RjUa(k,k′),(9)\nwhereRjis a random position vector of impurity (local-\nized inhomogeneity). By defining λcas a characteristic\nlength scale of the impurity, we model its localized po-\ntential with Fourier components of (see Appendix A for\ndetails),\nUa(k,k′)=ǫDMfR(k−k′)cosθMW(k,k′),(10)\nwhereW(k,k′) =i(k′\nx+kx)d/2 forλc/greaterorsimilard, whereas W=\nd/λcforλc≪d. Here,fR(k) is aformfactorofthe local-\nized potential; for example, fR(k) =e−k2λ2\nimpπλ2\nimp/L2\nif we assume Gaussian shapes with an impurity size λimp\ndistributed in the ( L×L×d)-sized ferromagnetic film.\nIII. SUSCEPTIBILITY\nWe shall calculate FMR linewidth by examining the\nsusceptibility of the system. According to equation of\nmotion for m1,2(k,t), the susceptibility for an externalfield with wave-number k′and time-varying eiωtis given\nby,\n/parenleftbigg\nχ12χ11\nχ22χ21/parenrightbigg\n(k,k′;ω) = [G−1δk,k′−V]−1,(11)\nwhere\nG(k;ω) =/planckover2pi1γMS/parenleftbigg\n−i/planckover2pi1ω+ǫ21(k)ǫ22(k)\nǫ11(k)i/planckover2pi1ω+ǫ12(k)/parenrightbigg−1\n,\nV(k,k′) =−1\n/planckover2pi1γMS/parenleftbigg\nV21(k,k′)V22(k,k′)\nV11(k,k′)V12(k,k′)/parenrightbigg\n(12)\nwithVκκ′(k,k′) = (Uk,k′\nκκ′+U−k′,−k\nκ′κ)/2.\nA. Average over impurities\nIn order to average the susceptibility over an ensemble\nof impurity, we expand the susceptibility with perturba-\ntion potential V;\n/parenleftbigg\nχ12χ11\nχ22χ21/parenrightbigg\n=G+G/an}bracketle{tV/an}bracketri}htimpG+G/an}bracketle{tVGV/an}bracketri}htimpG+···\n= [G−1δk,k′−C]−1. (13)\nwhere/an}bracketle{t.../an}bracketri}htimpdenotesanaverageoverimpurityandusu-\nally,/an}bracketle{tV/an}bracketri}htimp= 0 due to the randomness of V. We ap-\nproximate a self-energy as C=/an}bracketle{tVGV/an}bracketri}htimpby neglecting\ncrossing terms. Because the fluctuating potential is con-\ntributed from randomly located sites at {Rj}and thus\nVκκ′(k,k′) =/summationtext\njVa\nκκ′(k,k′)e−i(k−k′)·Rj, we arrive at the\nself-energy of\nC(k,k′;ω) =δk,k′1\n/planckover2pi1γMS/parenleftbigg\nΣ21Σ22\nΣ11Σ12/parenrightbigg\n(k;ω) (14)\nwhere\n/parenleftbigg\nΣ21Σ22\nΣ11Σ12/parenrightbigg\n(k;ω) =Nimp\n/planckover2pi1γMs/summationdisplay\nk1/parenleftbigg\nVa\n21Va\n22\nVa\n11Va\n12/parenrightbigg\n(k,k1)\nG(k1;ω)/parenleftbigg\nVa\n21Va\n22\nVa\n11Va\n12/parenrightbigg\n(k1,k) (15)\nwith the number of impurities Nimp. We note that the\nself-energy of C(k,k′;ω) is now diagonal over the k-\nspace, meaning that a translation symmetry is recovered\nthrough the impurity average. Finally, the susceptibility\nis given as,\n/parenleftbigg\nχ12χ11\nχ22χ21/parenrightbigg\n(k,k′;ω) =/planckover2pi1γMS\n/parenleftbigg\n−i/planckover2pi1ω+˜ǫ21˜ǫ22\n˜ǫ11i/planckover2pi1ω+˜ǫ12/parenrightbigg−1\nδk,k′(16)\nwith renormalized energy components of ˜ ǫκκ′(k,ω) =\nǫκκ′(k)−Σκκ′(k;ω).4\nB. Linewidths\nNow we focus on the longitudinal susceptibility along\nthex-direction, χxx=χ22, from Eq. (16);\nχxx(k,k;ω) =/planckover2pi1γMS˜ǫ11\n[i/planckover2pi1ω−˜ǫ21][i/planckover2pi1ω+˜ǫ12]+˜ǫ11˜ǫ22.(17)\nIn the absence of DMI, this result is consistent with that\ninRef. [25]. Atthisstage,weinserttheintrinsicdamping\ncontribution by adding i/planckover2pi1ωαtoǫ11(k) andǫ22(k). In\nFMR experiments, sincethe frequency ωis fixedto ωFMR\nand the dc field Hextis swept, a linewidth of the FMR\nresonance is equal to Lorentzian broadening of Eq. (17)\natk= 0. A straightforward calculation leads to the\nlinewidth of,\n/planckover2pi1γµ0cos(θM−θH)∆Hext=α/planckover2pi1ωFMR\n+Im[ǫ0\n11Σ22(k,ω)+ǫ0\n22Σ11(k,ω)]k=0\nω=ωFMR\n(ǫ0\n11+ǫ0\n22).(18)\nUsing the perturbation potential of Eq. (10), this result\ncan be summarized as\n/planckover2pi1γµ0∆Hext=α\ncos(θM−θH)/planckover2pi1ωFMR+/planckover2pi1γµ0∆H(2),(19)\nwhere ∆ H(2)stands for the extrinsic contribution from\nthe two-magnon scattering. Detailed forms of those\nquantities depend on the model of perturbation poten-\ntial. For instance, in the case of Eq. (10), the extrinsic\ncontribution is given by,\n/planckover2pi1γµ0∆H(2)=/parenleftBigg\nNimp\nL2(πλ2\nimp)2\nλ2cǫ2\nDM\nǫexcos2θM/parenrightBigg\nΓ(ω).\n(20)\nHere, the amplitude (the part enclosed by the parenthe-\nsis) depends on the model of scattering potential and a\ndimensionless function Γ( ω) is determined purely by the\nmagnon dispersion and a cut-off function through,\nΓ(ω) =1\ncos(θM−θH)ǫex\n(ǫ0\n22+ǫ0\n11)d2\nL2Im/summationdisplay\nkfc(k)\nǫ0\n11ǫ11(k)+ǫ0\n22ǫ22(k)\n[i/planckover2pi1ω−ǫ21(k)][i/planckover2pi1ω+ǫ12(k)]+ǫ11(k)ǫ22(k),(21)\nwherefc(k) =e−k2λ2\nimp(kλc)2for a large-sized impu-\nrity model ( λc/greaterorsimilard) andfc(k) =e−k2λ2\nimpfor a small-\nsized case ( λc≪d). Other scattering models, such as\nanisotropy-inducedscatteringpotentialbysurfacerough-\nness, can be reproduced by simply replacing fc(k) = 1,\nbut with the modified amplitude that includes detailed\nform-factors of impurity21. In calculating the function\nΓ(ω), the summation over kis mainly contributed by\nthosek-pointswherethedenominator, [ i/planckover2pi1ω−ǫ21(k)][i/planckover2pi1ω+\nǫ12(k)] +ǫ11(k)ǫ22(k), approaches zero. These k-points\ncorrespond to the resonant states discussed in the previ-\nous section. Hence, the total number of resonant state is\nimportant in determining Γ( ω) and eventually the FMR\nlinewidth.IV. RESULTS AND DISCUSSION\nNow we proceed to study behavior of the linewidth\nby varying the DMI strength D. In order to present\ncalculated results in a simple way, the perturbation po-\ntential model by a small sized impurities of λc≪dis\nmainly discussed for a fixed film thickness of d= 5 nm.\nWe note that Γ( ω) is much smaller for a large size im-\npurity of λc≫d. As an example, we choose param-\neters of impurity potential such as impurity density of\nNimp/L2= 0.03/(πλ2\nimp),λc= 1nm, and λimp= 0.5nm.\nFor the exchange stiffness constant A= 1.3×10−11J/m\nand DMI strength D= 1.5 mJ/m2, the chosen parame-\nters give 1 .186cos2θMµeV for the amplitude of Eq. (20),\nwhich corresponds to 102 .5cos2θMG with/planckover2pi1γ= 0.1158\nmeV/T.\nIn Fig. 3(a), we show the extrinsic linewidth broaden-\ning ∆H(2)as a function of resonant frequency for various\nDMI strengths. Because of ǫ2\nDMdependence in Eq. (20),\noverall amplitudes of the linewidth become larger with\nan increased DMI strength, whereas, for a given DMI\nstrength, the DMI-limited scattering potential exhibits\ncharacteristic behavior of the linewidth as a function of\nfrequency. Namely, with increasing the DMI strength\nfrom zero, the linewidth changes slightly in a high fre-\nquency range while bumps or peaks are developed in a\nlow frequency range. Furthermore, in a strong DMI case,\nfor example, D= 2.3 mJ/m2(orǫex/ǫDM∼1.0) in the\nfigure, one can seethat the linewidth isfinite evenat zero\nfrequency.\nWe first analyze the results in the range of high fre-\nquency in Fig. 3(a), where the linewidth shows ǫ2\nDMbe-\nhavior as a function of frequency. In fact, as well as the\nǫ2\nDMdependence, it is found that the linewidth has struc-\ntured behavior resulted from the function Γ( ω) as shown\nin Fig. 3(b). For a given frequency, one can see that\nΓ(ω) increaseswith an increased DMI and then decreases\nabove a certain DMI strength. Through detailed numeri-\ncal examinations, we find that, at the turning point (call\nthe DMI energy ǫ0\nDM), the two lobes in equi-energy lines\nillustrated in the lower panel of Fig. 2 are about to be\none by separating magnon modes at φk=−π/2 into two\nresonant states. We derive ǫ0\nDMin terms of various ener-\ngies in Eq. (B3). In the case of θH= 0, the DMI energy\nǫ0\nDMis approximately given by,\nǫ0\nDM=/planckover2pi1γµ0(MS+Hs)ǫMs\n2/planckover2pi1ωFMR(22)\nwithǫMs=/planckover2pi1γµ0MS/2.\nIn the case of ǫDM/lessorsimilarǫ0\nDM, we find that lobes of the\nequi-energylineconsistofsmall kpoints, namely kd≪1.\nThus, in this weak DMI regime, we find that a calculated\nlinewidth function Γ( ω) is consistent with the free-DMI\nresults by Ref. [21]. A detailed expressionof Γ( ω) for the\nweak DMI is written in Eq. (C2). According to the equa-\ntion, the increasing behavior in the weak DMI regime\n(ǫDM/lessorsimilarǫ0\nDM) of Fig. 3(b) originates from an enlarged5\nsize of the equi-energy lobes in the k-space by increasing\nthe DMI strength. In other words, the enlarged lobes are\ndirectly related to more available resonant states to give\nrise to the increased linewidth.\nIn the strong DMI case of ǫDM/greaterorsimilarǫ0\nDM, on the other\nhand, Γ( ω) shows a decreasing behavior as a function\nof DMI strength even though the size of equi-energy\nlobes becomes larger as shown in Fig. 2(b). In this\nregime, there are many resonant states with large k-\npoints, namely kd/greaterorsimilar1. At these kpoints, the magnon\ndispersion is dominantly governed by the exchange en-\nergy,ǫexk2d2. This is a rapidly increasing function of kd\nand thus, results in small magnon density of states or\nless-available resonant states. This explanation is also\nconsistent with the expression of Γ( ω) forkd/greaterorsimilar1 in\nEq. (C4).\nWe next discuss the origin of bumps (appeared for the\ncaseD <1.2 mJ/m2) and peaks (appeared for the case\nD≥1.2 mJ/m2) in Fig. 3(a). We find that the bumps\nappear when the DMI energy equal to ǫ0\nDM. Thus, the\nappearanceof resonantstate at φk=−π/2 is responsible\nfor the bumps. On the other hand, the peaks in Fig. 3(a)\nhave somewhat different origins, more complicated be-\nhavior of the energy dispersion. At the points where the\npeaksaredeveloped, we find that therearedual solutions\nof the cubic equation derived from ǫ(k)=/radicalbig\nǫ0\n11ǫ0\n22. For\nexample, in the case of D= 1.5 mJ/m2, we plot equi-\n0 10 20 30\nf (GHz)020406080∆H(2) (G)(a)\nD=2.3 (mJ/m2)\n1.5\n1.2\n1.0\n0.7\n0 10 20 30\nf (GHz)020406080∆H(2) (G)(a)\nD=2.3 (mJ/m2)\n1.5\n1.2\n1.0\n0.7\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nD (mJ/m2)0.120.150.180.21Γ(2πf)(b)\nf (GHz)\n15\n20\n30\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nD (mJ/m2)0.120.150.180.21Γ(2πf)(b)\nf (GHz)\n15\n20\n30\nFIG. 3. (color online) In (a), the calculated linewidths for\nvarious DMI strengths Dare plotted as a function of FMR\nfrequency. In (b), we show the variation of Γ( ω) with increas-\ning DMI strength by fixing FMR frequency. The exchange\nstiffness constant, A= 1.3×10−11J/m is used and the ex-\nternal magnetic field is in-plane, θH= 0. Other parameters\nare equal to those in Fig. 2.0.00 0.04 0.08 0.12\nkzd-0.20-0.15-0.10-0.050.00kxdµ0Hext=456.0 G\nµ0Hext=470.0 G\n0.00 0.04 0.08 0.12\nkzd-0.20-0.15-0.10-0.050.00kxdµ0Hext=456.0 G\nµ0Hext=470.0 G\nFIG. 4. (color online) In order to resolve the origin of the\npeak in Fig. 3(a), equi-energy lines are plotted in the case o f\nD= 1.5 mJ/m2around the external magnetic field (464 .4 G)\nexhibiting the peak. We increase the magnetic field by 2 .0 G\nand use the same parameters as those in Fig. 3.\nenergy lines (the right lobe in Fig. 2) in Fig. 4 around\nthe external magnetic field exhibiting the peak. As the\nexternal magnetic field increases, the equi-energy lines\nare evolved to have a distorted shape, and eventually\nto touch the other (the left lobe). Then, the left and\nright lobes are separated into outer and inner ones. This\nmodification of the lobes occurs at a lower magnetic field\nthan that giving the appearance of a resonant state at\nφk=−π/2. At the magnetic field where the two lobes\ninteract each other, the peak is developed in the func-\ntion Γ(ω). This is because such formation of the lobes\ngives rise to a very small group velocity of ∂ǫ(k)/∂k. In\nother words, this means that magnon density of states at\nthe point is very abundant and provides much possibil-\nity into which two magnons are scattered to increase the\nFMR linewidth.\nAs for a finite linewidth at ωFMR= 0, we find\nthat a DMI strength is very large, roughly ǫDM>/radicalbig\n/planckover2pi1γµ0(MS+Hs)ǫex. Under this condition, there are\nlower energy states than a uniformly magnetized state\neven atHext= 0, making resonant states still possible.\nThis can be understood by reminding the fact that spi-\nral states are a ground state rather than a ferromagnetic\nstate for systems accompanying a large DMI.\nWe now turn to the discussion about the linewidth\nin the out-of-plane geometry of applied magnetic fields.\nFigure 5 is the linewidth for different tilted angles of the\nexternal magnetic field. Because we consider a strong\nDMI case ( ǫDM≃ǫex), the peaks appear in the linewidth\ncurves. We find that the peak positions are nearly un-\nchanged with the tilting angle of magnetic field. In the\nrange of high frequency apart from the peak structures,\nthereisacriticalfrequencyabovewhichthelinewidth be-\ncomes zero, a similar behavior found in DMI-free cases25.\nIn Fig. 5(b), we plot the linewidth as a function of\nthe tilting angle when the FMR frequency is fixed. For\na small DMI case, the critical angle above which the\nlinewidth is zero is determined by Eq. (B2) and is simi-6\n0 10 20 30\nf (GHz)04080120∆H(2) (G)(a)\nθH (deg.)\n 0 \n60 \n70 \n80 \n0 10 20 30\nf (GHz)04080120∆H(2) (G)(a)\nθH (deg.)\n 0 \n60 \n70 \n80 \n0 20 40 60 80\nθH (deg.)0102030∆H(2) (G)(b)\nf (GHz)\n7\n15\n20\n30\n0 20 40 60 80\nθH (deg.)0102030∆H(2) (G)(b)\nf (GHz)\n7\n15\n20\n30\nFIG. 5. (color online) For a given DMI strength D= 1.5\nmJ/m2, we compare ∆ H(2)for different angles of applied\nmagnetic field. Other parameters are the same as those in\nFig. 2.\nlar to that in the work of Ref. [25]. On the other hand,\nthe critical angle in a strong DMI case is determined by\nEq. (B3).\nV. CONCLUSION\nIn this work, we have developed the theory of\ntwo-magnon scattering for a thin ferromagnetic film\nwith DMI as well as dipole-dipole, exchange, surface\nanisotropy, and Zeeman energies. In a quantum mechan-\nical way, we derive a expression of the FMR linewidth,\ntaking into account scattering from structural inhomo-\ngeneity, for example, DMI fluctuation induced by mi-\ncroscopic structural imperfection. We present the ex-\ntrinsic FMR linewidth in terms of various energy con-\ntributions, Eq. (21). In the presence of the DMI term,\nthe magnon dispersion exhibits rich resonant states, es-\npecially in small external magnetic fields. Furthermore,\ndue to the competition between the exchange and DMI\nenergies, the resonant states are formed at large kpoints\n(namely kd≃1). Different from the DMI-free case\n(kd≪1), it makes difficult to derive analytical expres-\nsions from the FMR linewidth.\nWe find that the characteristic linewidth broadening\nwith the DMI are twofold. One is the appearance of\npeak in the low frequency range when the DMI is strong.\nThis usually occurs when twolobes of the resonantstates\ntouch each other in magnon dispersion curves. The other\nis a finite value of the FMR linewidth at zero magneticfield. This may be not possible if a uniform ferromag-\nnetic system such as DMI-free case is the ground state.\nHowever, in the case of strong DMI, a non-colinear spin\nstate like spiral configuration has a lower energy than a\nferromagnetic one. Consequently, in the presence of the\nstrong DMI, there are still resonant states even at zero\nmagnetic field and thus a uniform spin state excited by\nmicrowaves in FMR experiments is still scattered into\nresonant states by inhomogeneities scattering.\nVI. ACKNOWLEDGEMENTS\nWe acknowledge R. D. McMichael for a fruitful discus-\nsion. This work was supported by the National Research\nFoundation of Korea (NRF-2015M3D1A1070465, NRF-\n2017R1A2B2006119) and KU-KIST Graduate School of\nConverging Science and Technology Program.\nAppendix A: DMI perturbation potential\nLet’s consider fluctuation from a Rashba-type spin-\norbit interaction51;\n∆ED=4/planckover2pi1γ\nMS/integraldisplay\ndrDd(r)/bracketleftbigg\nˆz·m×∂m\n∂x−ˆx·m×∂m\n∂z/bracketrightbigg\nwhereDd(r) is local DMI strength for each defect.\nOneoffeasiblemodelsforthe fluctuationofthisenergy\nisarandomspatialvariationofthe parameter Dd(r) with\nkeeping the magnon propagating vector m. This case\nmay occur structural defects whose characteristic size is\nlarger than or equal to the film thickness dwith weak\nDMI. Then, the fluctuation energy becomes, in Fourier\nspace,\n∆ED=/summationdisplay\nk,k′Vd(k−k′)ik′\nxd[m(−k)×m(k′)]·ˆz\n−/summationdisplay\nk,k′Vd(k−k′)ik′\nzd[m(−k)×m(k′)]·ˆx,(A1)\nwhere the defect potential energy is defined by,\nVd(r) =4/planckover2pi1γ\nMSdDd(r) =/summationdisplay\nkVd(k)eik·r.\nAs another model, we can take account of an abrupt\nchange of the magnetization vectors, for example, at\nedges of terrace on sample surfaces, near magnetic and\nnon-magnetic impurities, etc. Moreover, it is also known\nthat the tilt magnetization state (spiral or skyrmion\nphases) rather than a ferromagnetic state is more sta-\nble for a strong DMI strength.76For those cases, we ex-\npect that magnon modes are additionally modulated as\nˆm(r)ei(k+q)·rnear the defect ( qis a pitch vector, for ex-\nample, in a spiral state). Then, the fluctuation energy\nbecomes similar to Eq. (A1), but replacing kx→kx+qx\nandkz→kz+qz. By considering small defects with a7\ncharacteristic length λc(order of an impurity size ≪d),\nwe further set kx→1/λcandkz→1/λcin Eq. (A1).\nWe assume that the defect potential energy Vd(r) is\ncontributed from atom-like and Gaussian-shaped func-\ntions located at random sites;\nVd(r) =/summationdisplay\njva(r−Rj), va(r) =4/planckover2pi1γD\nMSde−r2/λ2\nimp(A2)\nwith a defect size, λimp. Then, by inserting Eq. (A2)\ninto Eq. (A1), the fluctuation energy becomes, in the\nlocal coordinates,\n∆ED=/summationdisplay\nk,k′Vd(k−k′)i(kx+k′\nx+2/λc)d\n2cosθM\nˆx3·[m(−k)×m(k′)].\n=/summationdisplay\nk,k′/summationdisplay\nκ,κ′=1,2mκ(−k)Uk,k′\nκκ′mκ′(k′) (A3)\nwhere\nUk,k′\n11=Uk,k′\n22= 0,\nUk,k′\n12=−Uk′,k\n21=/summationdisplay\nje−i(k−k′)·RjUa(k,k′).\nHere,Ua(k,k′) is given by,\nUa(k,k′) =2/planckover2pi1γD\nMSdfR(k−k′)W(k,k′)cosθM(A4)\nwhereW(k,k′) =i(k′\nx+kx)d/2 forλc/greaterorsimilard, whereas\nW=d/λcforλc≪d. Here,fR(k) =e−k2λ2\nimpπλ2\nimp/L2\nis a form factor of the atomic potential ( Lis a side length\nof the ferromagnetic film).\nAppendix B: Critical angle of external magnetic\nfields\nAbove a certain angle of external magnetic field, the\nlinewidth of FMR becomes zero. The critical angle is\ndetermined by demanding zero resonant states, or tiny\nsize of lobes, in the FMR condition of ǫ(k) =/planckover2pi1ωFMR. By\nrequiring a small value of k, it is straightforward to find\nkR(φk)d=/parenleftBig\nC1\n2C2/parenrightBig2\n+C0\nC2−/parenleftBig\nC1\n2C2+sinφk/parenrightBig2\nD2/C2(B1)\nwhere\nC0=ǫMs/bracketleftbig\nǫ0\n22cos2θM−(ǫ0\n11+ǫ0\n22)sin2θM/bracketrightbig\nC1= 2/radicalBig\nǫ0\n11ǫ0\n22ǫDMcosθM,\nC2=ǫMs/bracketleftbig\nǫ0\n11−ǫ0\n22sin2θM/bracketrightbig\n,\nD2= (ǫ0\n11+ǫ0\n22)ǫex+ǫ2\nMssin2θM(sin2θM−cos2θM)\n−1\n2sin2φkcos2θM[2ǫ2\nDM+ǫ2\nMs(1+cos2 θM)].Thus, for a small DMI ofC1\n2C2≤1, the critical angle is\ndetermined by solving the equation,\n/parenleftbiggC1\n2C2/parenrightbigg2\n+C0\nC2= 0 (B2)\nwhile,C1\n2C2>1,\n/parenleftbiggC1\n2C2/parenrightbigg2\n+C0\nC2−/parenleftbiggC1\n2C2−1/parenrightbigg2\n= 0.(B3)\nAppendix C: Evaluation of the linewidth function\nΓ(ω)\nThe linewidth function Γ( ωFMR) can be rewritten as,\nfrom Eq. (21),\nΓ(ω) =1\ncos(θM−θH)ǫex\n(ǫ0\n22+ǫ0\n11)πd2\nL2/summationdisplay\nkfc(k)\nǫ0\n11ǫ11(k)+ǫ0\n22ǫ22(k)/radicalbig\n4ǫ11(k)ǫ22(k)−[ǫ12(k)+ǫ21(k)]2δ(/planckover2pi1ω−ǫ(k)).(C1)\nFor a weak DMI ofC1\n2C2≤1 and due to kd≪1, the\ndelta function in the equation can be approximated by,\nδ(/planckover2pi1ω−ǫ(k))≃2/radicalbig\nǫ0\n11ǫ0\n22\n|C0|dδ(k−kR(φk))\nfrom Eq. (B1) and then, Γ( ω) is further simplified to\nΓ(ω)≃ǫex([ǫ0\n22]2+[ǫ0\n11]2)\n2πcos(θM−θH)(ǫ0\n22+ǫ0\n11)|C0|/integraldisplayφb\nφadφkR(φ)d\n(C2)\nwhere\nsinφa=−C1\n2C2−/radicalBigg/parenleftbiggC1\n2C2/parenrightbigg2\n+C0\nC2\nsinφb=−C1\n2C2+/radicalBigg/parenleftbiggC1\n2C2/parenrightbigg2\n+C0\nC2.\nNow we consider a strong DMI ofC1\n2C2≫1 where the\ndipole energy ǫMsis relatively unimportant. In this case,\neigenenergy of magnon can be approximated by, from\nEq. 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Commun.\n4, 1463 (2009)."    },    {        "title": "2301.03256v1.X_ray_detected_ferromagnetic_resonance_techniques_for_the_study_of_magnetization_dynamics.pdf",        "content": "X-ray detected ferromagnetic resonance techniques\nfor the study of magnetization dynamics\nGerrit van der Laan1and Thorsten Hesjedal2\n1Diamond Light Source, Harwell Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom\n2Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU, United Kingdom\n(Dated: January 10, 2023)\nElement-speci\fc spectroscopies using synchrotron-radiation can provide unique insights into\nmaterials properties. The recently developed technique of X-ray detected ferromagnetic resonance\n(XFMR) allows studying the magnetization dynamics of magnetic spin structures. Magnetic\nsensitivity in XFMR is obtained from the X-ray magnetic circular dichroism (XMCD) e\u000bect, where\nthe phase of the magnetization precession of each magnetic layer with respect to the exciting\nradio frequency is obtained using stroboscopic probing of the spin precession. Measurement of\nboth amplitude and phase response in the magnetic layers as a function of bias \feld can give a\nclear signature of spin-transfer torque (STT) coupling between ferromagnetic layers due to spin\npumping. Over the last few years, there have been new developments utilizing X-ray scattering\ntechniques to reveal the precessional magnetization dynamics of ordered spin structures in the GHz\nfrequency range. The techniques of di\u000braction and re\rectometry ferromagnetic resonance (DFMR\nand RFMR) provide novel ways for the probing of the dynamics of chiral and multilayered magnetic\nmaterials, thereby opening up new pathways for the development of high-density and low-energy\nconsumption data processing solutions.\nKeywords: FMR, XMCD, X-ray scattering, X-ray re\rectivity, spin structures\nI. INTRODUCTION\nMagnetization dynamics is at the heart of high fre-\nquency magnetic nanoscale devices based on spin waves,\nspin pumping, and spin-torque oscillators in the GHz\nfrequency range. Traditionally, ferromagnetic resonance\n(FMR) has been a work horse technique to determine the\nfundamental parameters for magnetic resonance and re-\nlaxation in thin \flms. The recent growing complexity of\nmany modern magnetic materials and devices requires\nthe development of advanced measurement techniques\nthat more directly reveal the microscopic origin of the\ndynamical magnetic interactions that are at play.\nThe novel techniques of X-ray detected FMR (XFMR)\nenables studying the magnetization dynamics of indi-\nvidual layers, where element-speci\fc magnetic contrast\nis obtained using the X-ray magnetic circular dichroism\n(XMCD) e\u000bect [1]. Not only can the FMR signal be mon-\nitored in X-ray absorption, it can also be done in X-ray\ndi\u000braction and re\rectivity, using techniques termed as\nDFMR and RFMR, respectively [2]. In these X-ray mea-\nsurements, time-resolved FMR gives both the amplitude\nand phase of the spin precession for the di\u000berent chemical\nelements, and hence di\u000berent layers, in the sample. The\nchallenge of such measurements lays in the fact that the\nprecession cone angle is small ( <1\u000e) and that the preces-\nsion frequency is on the order of GHz. The solution is\nto use lock-in techniques and to detect the phase of the\nprecession stroboscopically by using the time structure\nof the X-ray pulses from the synchrotron ( \u0018500 MHz,\ni.e., corresponding to a period between the pulses of 2\nns). The radio frequency (RF) \feld applied to drive the\nspin precession is synchronized with the X-ray pulses us-\ning the clock of the synchrotron. Therefore, each X-raypulse measures the magnetization cone at precisely the\nsame point in the precession cycle. Hence, XFMR com-\nbines the techniques of FMR and XMCD. Thus, the spin\nprecession along the bias \feld is pumped by the RF \feld\nto generate the magnetic resonance (i.e., FMR), whose\namplitude and phase is probed by the synchronized X-\nray pulses using the XMCD e\u000bect. The time dependence\nis recorded using a delay line to vary the phase of the RF\nsignal with respect to the X-ray pulses.\nDuring the last few years, many XFMR studies either\nin time-averaged or time-resolved mode have been re-\nported [3{52]. The \frst element-speci\fc and time-depen-\ndent measurement of the magnetization dynamics using\npump-probe XMCD was reported by Bailey et al. [3] on a\npermalloy (Py = Ni 80Fe20) thin \flm, where the moments\non the Ni and Fe sites were found to precess together at\nall frequencies, and by Arena et al. [4] on a Py/Cu/CoZr\ntrilayer, where at resonance, a weak ferromagnetic cou-\npling was found in the phase and amplitude response of\nindividual layers across resonance.\nThe amplitude and phase response of the magnetic\nprobe layer measured by XFMR provides a signature for\neither static exchange interaction in strongly exchange-\ncoupled bilayers or spin-transfer torque (STT) coupling\ndue to spin pumping. Marcham et al. [25] \frst evidenced\nSTT in a CoFe/Cu/Py spin valve using XFMR where the\n\feld dependence of the \fxed layer phase showed a clear\nsignature of STT due to spin pumping. Using XFMR,\nBaker et al. [28] reported a strong anisotropy of the spin\npumping, providing new opportunities for device appli-\ncations.\nPreviously, time-resolved XFMR has been reviewed in\ngreat detail in Ref. [1]. Here, we present a timely up-\ndate, especially emphasizing the newly developed time-arXiv:2301.03256v1  [cond-mat.mtrl-sci]  9 Jan 20232\nresolved FMR techniques in X-ray re\rectivity and di\u000brac-\ntion.\nThe outline of the remainder of this paper is as follows.\nSec. II gives a brief theoretical background of magnetiza-\ntion dynamics and STT. Sec. III describes the experimen-\ntal setup, conditions, and considerations for the various\nXFMR techniques. Sec. IV highlights several recent ex-\namples of XFMR, DFMR, and RFMR experiments and\nmentions their scienti\fc impact. Finally, conclusions are\ndrawn in Sec. V.\nII. BACKGROUND ON FMR AND STT\nA. Ferromagnetic resonance (FMR)\nBefore presenting the experimental details and show-\ncasing several recent examples, we will brie\ry introduce\nsome relevant background material.\nFMR arises when the energy levels of a quantized sys-\ntem of electronic moments are Zeeman split by a uniform\nmagnetic \feld and the system absorbs energy from an os-\ncillating magnetic \feld [53]. A resonance occurs when the\ntransverse AC \feld is applied at the Larmor frequency\ncorresponding to the energy di\u000berence between the mag-\nnetic levels, i.e., ~!= \u0001E. The spin precession in a\nsingle-domain magnetic material can be described with\nthe equation of motion, the so-called Landau-Lifshitz-\nGilbert (LLG) equation,\n_ m=\u0000\rm\u0002He\u000b+\u000b(m\u0002_ m); (1)\nwhere the e\u000bective \feld He\u000b=\u0000@F(M)=@Mis ob-\ntained by minimization of the free energy Fwith re-\nspect to the magnetization M. The free energy contains\nterms such as the exchange, Dzyaloshinskii-Moriya, de-\nmagnetization, magnetocrystalline anisotropy, magneto-\nstatic, external Zeeman \feld, and elastic energy. Further,\n_ m=\u000em=\u000et; the reduced magnetization is m=M=Ms,\nwhereMs=jmjis the saturation magnetization; and\n\r=g\u0016B=~is the gyromagnetic ratio, where \u0016Bis the\nBohr magneton and gis the Land\u0013 e (spectroscopic split-\nting) g-factor. The dimensionless damping parameter\n\u000b\u001c1 (typically 10\u00003{10\u00002for 3dmetals) determines\nthe width of the resonance absorption peak.\nThe \frst right-hand term in Eq. (1) corresponds to\nthe torque due to the e\u000bective \feld He\u000b. In a classical\npicture,\u001c=dS=dtequates to the time change in an-\ngular momentum S, which leads to the spin precession.\nThe second right-hand term corresponds to the damp-\ning, which can also be written in the form of the Gilbert\ndamping term\u0000\u000b\r(m\u0002m\u0002He\u000b). Both torque and\ndamping are vectorially sketched in Fig. 1(a). Without\nexternal RF excitation, the magnetization would relax to\nthe steady state given by Brown's equation, m\u0002He\u000b= 0.\nLinearization of the LLG equation gives the relation\nbetween the frequency \u00170(or circular frequency !0) and\n\feld, which in the form of the Kittel equation is writtenas [1]\n2\u0019\u00170\u0011!0=\rp\nHe\u000bBe\u000b=\rp\nHe\u000b(Ms+He\u000b):(2)\nB. Spin-transfer torque (STT)\nThe layer selectivity of XFMR makes this technique\na unique probe to investigate STT and related spin cur-\nrents in multi-layered spin valves [1]. STT is the e\u000bect\nin which the spin direction in a magnetic layer can be\nmodi\fed using a spin-polarized current [54, 55].\nSpin pumping occurs when the precessing magnetiza-\ntion vector generated by FMR in a ferromagnetic (FM)\nlayer emits a pure spin current into an adjacent normal\nmetal (NM) layer [56]. Traditionally, spin currents have\nbeen probed using indirect measurements. For instance,\nin the metals through which they \row they can create an\nelectrical voltage drop perpendicular to the spin current\ndirection, or a torque that bends the magnetization di-\nrection. However, such indirect measurements are often\nambiguous because they are also in\ruenced by other fac-\ntors, such as magnetic proximity e\u000bects at the interface.\nSTT gives an extra term in the LLG equation, which\nis (anti)-parallel to the (anti)-damping (see Fig. 1(a)).\nAccording to Slonczewski [54], the adiabatic torque is\n\u001cs=\u000bsm\u0002_ m, where\u000bsis the STT damping. The\nspin current pumped across a FM/NM interface due to\nprecession is [56]\nIs=~\n4\u0019g\"#\ne\u000bm\u0002_ m; (3)\nwhereg\"#\ne\u000bis the e\u000bective spin-mixing conductance. The\nspin pumping depends critically on the FM/NM inter-\nface (the material-dependent g\"#\ne\u000b) and the spin di\u000busion\nlength in the NM.\nFor two FM layers iandjwith di\u000berent resonance fre-\nquencies and coupled by both spin pumping (dynamic ex-\nchange coupling) and static exchange coupling, the cou-\npled LLG equations are\n_ mi=\u0000\rmi\u0002He\u000b;i+\u000b0\nimi\u0002_ mi\n+\u000bs\nimi\u0002(_ mi\u0000_ mj) +Aexmj\u0001mi;(4)\nand equivalently when exchanging i$j, where miis\nthe magnetization direction, He\u000b;ithe e\u000bective \feld, \u000b0\ni\nthe Gilbert damping, and \u000bs\nithe STT damping in layer\ni. The spin pumping induced coupling is determined by\n\u000bs\niand the static exchange coupling by Aex.\nIII. EXPERIMENTAL\nXFMR provides an element-speci\fc and time-resolved\nmeasurement of the precessional dynamics of each FM\nlayer on a ps time scale, where the spin precession in-\nduced by a driving RF signal is detected using the XMCD3\nHeffh(t) m(t)Sample\nCPW(b)\nm×Heffz\nyHeff\nm(t)m ×m ×Heff\nSTT\nx(a)\nFIG. 1. (a) Precession, damping, and spin transfer torque\n(STT) in FMR. The precession m\u0002He\u000baround the e\u000bective\n\feldHe\u000bis damped by the Gilbert term m\u0002m\u0002He\u000b. The\nspin-transfer torque is parallel (antiparallel) to the Gilbert\ndamping term, and can enhance (oppose) the latter depend-\ning on the direction of the spin current. (b) Schematics of\nthe sample geometry for XFMR. The sample (red disk) is\nmounted on the signal line (the central strip) of a coplanar\nwaveguide (CPW). The magnetization m(t) precesses about\nHe\u000b, driven by the in-plane continuous RF \feld h(t) in the\nCPW. The cone angle of precession is exaggerated for clar-\nity; its typical magnitude is \u00181\u000e. Circularly polarized X-ray\npulses from the synchrotron impinge at an grazing incidence\nangle on the sample in transverse geometry in order to enable\nstroboscopic detection of the oscillatory component of m(t)\nat variable phase delays.\ne\u000bect [57]. However, before performing the XFMR ex-\nperiment, the static magnetization of the samples has\nto be precharacterized with standard techniques, such as\nsuperconducting quantum interference device (SQUID)\nmagnetometry to measure the hysteresis loops along the\neasy and hard direction, followed by standard FMR mea-\nsurements.\nA. VNA-FMR\nVector network analyzers (VNA)-FMR measurements\nare used to characterize the magnetic resonances in order\nto judge whether these are suitable and intense enough\nfor the XFMR measurements at the synchrotron. VNA-\nFMR is a broadband FMR technique, where the sample is\nmounted onto a coplanar waveguide (CPW) and driven\nby an external RF \feld, while under a static magnetic\nbias \feld. Measurement of the S-parameters of the sam-\nple results in a frequency-\feld map, where the resonances\nappear in the form of Kittel curves (Eq. (2)). The angu-\nlar dependence of the resonances gives information about\nthe magnetic anisotropy [53]. It allows us to chose the\nbest azimuthal angle of the applied \feld with respect to\nthe crystallographic axes to separate the magnetic res-\nonances at the optimal distance for detecting STT [28].\nHence, at a given RF frequency, this gives us the cor-\nresponding \feld values of the resonances in the XFMR\nexperiments. Conventional FMR will normally probe the\nwhole thickness of a thin \flm since the skin depth of, e.g.,metallic iron at 10 GHz, is on the order a micron.\nB. XMCD\nAt the synchrotron, \frst the static XMCD is measured\nby sweeping the photon energy across the absorption\nedge of the magnetic elements. This allows us to se-\nlect the \fxed photon energies suitable for XFMR. The\nstatic XMCD is obtained from the di\u000berence between\ntwo X-ray absorption spectra recorded with the helic-\nity vector of the circularly polarized X-rays parallel and\nantiparallel, respectively, to the applied magnetic \feld\n[58]. The XMCD signal is proportional to the projection\nof the helicity vector, which is along the incident beam\ndirection ^k, onto the sample magnetization M, hence\nIXMCD/^k\u0001M.\nThe XMCD at the soft X-ray absorption edges, such\nas the Fe, Co, and Ni L2;3, is very strong [59], which\nhelps to compensate for the small changes in magneti-\nzation direction due to the limited cone angle ( <1\u000e) of\nthe precession in XFMR. The X-ray penetration length,\nwhich limits the sampling depth, is in the nm range, e.g.,\nfor pure Fe it is\u001820 nm at the Fe L3maximum at\u0018707\neV [57]. By tuning the photon energy away from the\nabsorption maximum the penetration length can be in-\ncreased (\u0018600 nm below the edge at 700 eV). Note that\nthe length scale of the probing depth is well matched to\nthe thickness of the magnetic layers in spin valves. The\ntypical lateral spot size of the X-ray beam on the sample\nis 200\u000220\u0016m2, again well suited for small devices.\nC. Time-resolved measurements\nThe measurement of the projected magnetic moment\nin XFMR does not require to take the di\u000berence between\nopposite circular polarizations as done in XMCD. In-\nstead, a change in the projection of the magnetization\nprecession is measured using a \fxed circular polarization.\nXFMR can be measured in two distinctly di\u000berent ge-\nometries, namely ( i) time-averaged in longitudinal ge-\nometry [12, 20] or ( ii) time-resolved in transverse geom-\netry [8, 25, 28]. In longitudinal geometry, a shortening\nof the magnetization vector along the z-axis (parallel to\nthe X-ray beam direction) leads to a di\u000berence \u0001 Mz=\nMs(1\u0000cos\u0012)\u00191\n2Ms\u00122, where\u0012is the small cone angle of\nthe magnetization precession. The time-averaged XFMR\nrequires no synchronization with the synchrotron clock,\ntherefore it can be done at an arbitrary frequency, but\nit needs a larger RF power which can lead to nonlinear\ne\u000bects.\nOnly measurements in transverse geometry give access\nto the precessional phase. This geometry is depicted in\nFig. 1(b). The transverse component of the magnetiza-\ntion precession will give a sinusoidal variation on top of\nthe static X-ray absorption signal. With the incident X-\nray beam perpendicular to the bias \feld, the oscillating4\ncomponent of the magnetization precession is measured\nwith a magnitude jMyj=Mssin\u0012\u0019Ms\u0012. Thus, for a\ntypical cone angle of \u0012\u00191\u000e, the transverse geometry\ngives a signal that is larger by a factor of \u0018200 com-\npared to the longitudinal geometry. Due to the shape\nanisotropy of the \flm, the precession is strongly ellipti-\ncal, often with a larger in-plane amplitude. This favors\na measurement geometry with the X-rays at grazing in-\ncidence. A good compromise is an X-ray incidence angle\nof\u001835\u000ewith respect to the plane of the sample, which\nensures that the signal is sensitive to the larger in-plane\ncomponent of the magnetization precession.\nUsing a vector magnet system, such as the portable oc-\ntupole magnet system (POMS) at Diamond [57], where\nthe \feld can be applied in any direction, permits a simple\nchange of the \feld from ( i) parallel to the photon direc-\ntion, as needed for static XMCD scans, to ( ii) orthogonal\nto both X-ray beam and RF \feld direction, as required\nfor time-resolved XFMR.\nThe detection of the X-ray absorption can be done by\neither X-ray transmission [27, 31], \ruorescence yield [23],\nor X-ray scattering or re\rectivity [2, 35, 36, 60, 61]. How-\never, RF plays havoc with total-electron yield. In the\ncase of transmission, the incident X-rays impinge on the\nsample through a hole in the signal line of the CPW.\nAfter passing through the sample, the transmitted X-\nrays are detected with X-ray excited optical luminescence\n(XEOL) emerging from the MgO or sapphire (Al 2O3)\nsubstrate using a photodiode placed behind the sample.\nNote that not all substrates, such as non-transparent ones\nlike Si, are suitable for XEOL detection [62].\nTime resolution is established by using the periodic\nX-ray pulses from the synchrotron (normally operating\nin multibunch or hybrid mode). To enable stroboscopic\nprobing, the RF driving \feld is taken as a harmonic of\nthe X-ray pulse frequency, hence the resonance is driven\nat multiples of the master oscillator clock of the stor-\nage ring. These harmonics are generated using an RF\ncomb generator (Atlantic Microwave) driven by the mas-\nter oscillator clock, which has a frequency of 499.65 MHz\n(at the DLS, ALS, and BESSY synchrotron). This cor-\nresponds to\u00182 ns intervals between consecutive X-ray\npulses, which have a pulse width of \u001835 ps (at DLS and\nBESSY) or\u001870 ps (at ALS). The desired frequency is\nselected using \flters and ampli\fers to drive a narrow\nband, high power (25{30 dBm) RF \feld to the CPW.\nA programmable delay line (Colby Instruments) enables\nphase shifting of the RF oscillation with respect to the\nX-ray pulses with a step resolution of \u00180.5 ps. De-\npending on the speci\fc technique either the transmitted,\ndi\u000bracted, or re\rected X-rays are measured using a pho-\ntodiode. Fig. 2 show a schematic representation of the\nsetup for DFMR; for XFMR and RFMR the electronics\nis very similar. The signal is obtained using a lock-in\nampli\fer (LIA) by switching the signal at a given audio\nfrequency. There are two usual modulation modes. In\namplitude modulation the LIA measures the di\u000berence\nbetween signals obtained with the RF signal on and o\u000b.\nFIG. 2. Schematic of the setup for DFMR measurements in\nthe RASOR di\u000bractometer at the Diamond Light Source. The\nsample is placed on the CPW, which is mounted on the cold\n\fnger inside the di\u000bractometer. Incident circularly or linearly\npolarized X-rays are scattered o\u000b the sample and detected via\na photodiode in a #-2#geometry. A variable magnetic \feld\nis applied in the scattering plane via a pair of permanent\nmagnets whose distance can be controlled externally. An RF\nsignal is fed to the CPW to drive the ferromagnetic resonance\nin the magnetic sample. As the synchrotron gives X-ray pulses\nat a frequency of \u0018500 MHz, a comb generator is used to\nproduce higher harmonics, which are selected and fed to the\nCPW. To probe the time dependence of the scattered X-ray\nintensity, a tunable delay line is used, which shifts the phase\nbetween the pump (the RF signal) and the probe (the pulsed\nX-rays). (Adapted from Ref. [63]).\nIn 180\u000e-phase modulation, the LIA measures the di\u000ber-\nence between signals obtained with the RF of opposite\nphase.\nD. XFMR\nIn order to record the time-resolved XAS signal with\ncircular polarization at \fxed photon energy, the RF fre-\nquency is locked to a multiple of the synchrotron clock.\nThen at \fxed angles and for given temperature, this\nleaves two free scanning parameters, namely the mag-\nnetic bias \feld strength and the delay time between X-ray\npulses and RF \feld.\nMagnetic \feld scans record the signal by sweeping the\nbias \feld at a constant delay time. The signal contains\nboth real and imaginary parts of the magnetic suscepti-\nbility, whose relative contributions strongly change across\nresonance. By measuring two \feld scans, which di\u000ber by5\n90\u000ein phase (obtained using the corresponding time de-\nlays), and \ftting these scans simultaneously using the\nKramers-Kronig relation, gives a good apprehension of\nthe \feld dependence of the resonances [40].\nDelay scans record the signal for each of the magnetic\nlayers at constant bias \feld by sweeping the delay time.\nAs an example, Fig. 3(a) shows a series delay scans over\ntwo periods of the phase taken at di\u000berent bias \felds (40{\n200 mT) across the Co resonance in a magnetic tunnel\njunction (MTJ), in more detail discussed in Sec. IV A.\nThe solid lines represent sinusoidal \fts to the experimen-\ntal data (dots), from which the amplitude and relative\nphase of the magnetization precession can be extracted.\nA sinusoidal function of the form\nS(t) =Xsin(2\u0019\u0017t) +Ycos(2\u0019\u0017t); (5)\nis \ftted to the delay scan, where tis the time delay and\n\u0017the frequency of the RF. This procedure is repeated\nfor various \feld strengths and directions. By extracting\nthe coe\u000ecients XandYin Eq. (5) from the delay scans,\nthe amplitude Aand phase of the oscillations can be\ndetermined using the relationships\nA=p\nX2+Y2;  = 2 arctan\u0012Y\nA+X\u0013\n:(6)\nXFMR precessional plots are assembled by combining\nthe amplitudes and phases extracted from the delay scans\nmeasured over a range of bias \felds. This gives the \feld\ndependence of the amplitude and phase for each element\n(e.g., for Co and Ni in Fig. 3(b)), from which the type\nof coupling between layers can be assessed. By normaliz-\ning the XFMR signal to the static XMCD, the amplitude\nof the signal can be obtained per atom for each chemi-\ncal element in the sample. This enables a quantitative\ndecomposition of the resonance features [31].\nThe static coupling (i.e., exchange interaction) and dy-\nnamic coupling (i.e., spin pumping) give a very di\u000ber-\nent XFMR response, as can be understood from Eq. (4).\nConsider a pump layer FM1 that is free to rotate, and a\nprobe layer FM2 that is pinned. Using XFMR at a \fxed\nfrequency, we scan the \feld across the entire resonance.\nAt resonance, FM1 will show a symmetric peak for the\namplitude, while the phase is 90\u000edelayed with respect\nto the RF driving \feld. Across the entire resonance, the\nphase will change by 180\u000e. To investigate the type of\ncoupling between both layers we measure the XFMR re-\nsponse of FM2 at the resonance condition of FM1.\nFor static exchange coupling, E=\u0000Aexm1\u0001m2, so\nthatHe\u000b;2/m1. This means that the e\u000bective \feld in\nthe second layer is aligned along the magnetization of the\n\frst layer. Then the \feld dependent precession of FM2\nwill show a dispersive (bipolar) peak in the amplitude\nand a symmetric (unipolar) peak in the phase.\nOn the other hand, for dynamic exchange coupling\nHe\u000b;2/_ m2=\u0000i!m1. The magnetic \feld is imagi-\nnary, resulting in a 90\u000ephase change. In this case, the\n\feld dependent precession of FM2 will show a unipolar\npeak in the amplitude and bipolar peak in the phase.This behavior means that XFMR can distinguish be-\ntween static and dynamic coupling by their amplitude\nand phase signature in the precessional plot, and thus\ndetermine the relative contribution of these couplings.\nThis has previously been utilized for, e.g., exchange cou-\npled layers [4, 31], spin values [25], MgO magnetic tunnel\njunctions [29, 45], topological insulators [27, 30, 41], spin\nvalve with\u000e-layer [26], Heusler alloys [40], NiO antiferro-\nmagnetic interlayer [42], exchange springs [44], and \u000b-Sn\nthin \flms [47].\nE. DFMR and RFMR\nDFMR and RFMR measurements have been per-\nformed in the RASOR soft X-ray di\u000bractometer on beam-\nline I10 at the Diamond Light Source [57] (see setup in\nFig. 2). Incident X-rays with wavevector kiilluminate\nthe sample, while the scattered beam ( ks) is detected us-\ning a photodiode. The scattering geometry is con\fgured\nto probe the sample at certain di\u000braction or specular re-\n\rectivity conditions. The sample in the di\u000bractometer is\nmounted on a CPW that is connected to a liquid He cryo-\nstat arm which can reach temperatures down to 12 K. A\nbias magnetic \feld is applied by two permanent magnets,\nwhich can be positioned to vary both the \feld strength\nup to 200 mT and the orientation within the scatter-\ning plane. Perpendicular to the bias \feld, a transverse\nRF \feld around the central conductor of the CPW is\ngenerated, which excites the magnetization dynamics in\nthe system. In contrast to conventional XFMR measure-\nments, where the sample is mounted \rip-chip onto the\nCPW, in the scattering geometry the sample is mounted\nface up to allow for the X-ray beam to probe its sur-\nface. To ensure good coupling between the CPW and the\nprobed top surface, the sample must either be thinned,\nor in the case of multilayers, grown on a thin substrate\nof the order of 100 \u0016m.\nIn DFMR, where the detector is aligned to a Bragg\npeak or magnetic scattering peak, the stroboscopic sig-\nnal is used to measure delay scans for di\u000berent linear or\ncircular polarization of the incident X-rays, to give infor-\nmation about the periodic spin structure.\nIn RFMR where the photo diode detector accepts the\nre\rected beam the stroboscopic signal is used to measure\ndelay scans for di\u000berent values of the scattering vector\nQz, to obtain depth information. An advantage of RFMR\nover DFMR is that it can be done on thin \flms and\nmultilayers, and no single crystals are needed.\nIV. X-RAY BASED FMR EXAMPLES\nA. XFMR of spin-current mediated exchange\ncoupling in MgO-based MTJs\nMagnetic tunnel junctions composed of ferromagnetic\nlayers which are mutually interacting through a nonmag-6\nnetic spacer layer are at the core of magnetic sensor and\nmemory devices. G ladczuk et al. [45] used layer-resolved\nXFMR to investigate the coupling between the magnetic\nlayers of a Co/MgO/Py MTJ. Two magnetic resonance\npeaks were observed for both magnetic layers, as probed\nat the Co and Ni L3X-ray absorption edges.\nFigure 3 shows XFMR delay scans for the Co layer in\nthe Co/MgO/Py MTJ at 80 K continuously driven at 4\nGHz. The curves in Fig. 3(a) show a strong increase in\namplitude as well as a large phase shift across the reso-\nnance at\u001890 mT. The amplitude and phase of the pre-\ncession, which are extracted using Eq. (6), are shown in\nFig. 3(b) and (c), respectively, for both Co (orange) and\nNi (blue) as a function of the bias \feld. The amplitude\ncurves show that the Ni resonance originating from the\nPy layer around \u0018120 mT is strongly coupled with the\nCo layer. On the other hand, the Co resonance around\n\u001890 mT is only weakly present in the Py layer. Instead\nof plotting amplitude Aand phase , one can also plot\nthe FMR signal in the ( X,Y)-plane as a function of \feld\n[45]. Since the sine and cosine functions in Eq. (5) are\northogonal, the estimators of XandYare given by pro-\njections to orthogonal subspaces.\nA theoretical model based on the Landau-Lifshitz-Gil-\nbert-Slonczewski equation (Eq. (4)) was developed, in-\ncluding exchange coupling and spin pumping between the\nmagnetic layers. Fits to the experimental data were car-\nried out, both with and without a spin pumping term,\nand the goodness of the \ft was compared using a likeli-\nhood ratio test. This rigorous statistical approach pro-\nvided an unambiguous proof of the existence of interlayer\ncoupling mediated by spin pumping through MgO [45].\nIt was also found that spin pumping is more e\u000bective\nat lower temperatures, which agrees with the theoretical\nunderstanding.\nB. XFMR of coherent spin currents in\nantiferromagnetic NiO\nAntiferromagnets have recently gained large interest in\nthe \feld of spintronics, as they allow for faster and more\nrobust memory operation than present technologies and\nas they can carry spin current over long distances. How-\never, many fundamental physics questions about these\nmaterials regarding their spin transport properties still\nremain unanswered [64]. A spin current generated by\nspin pumping should have a single wave mode, carrying\nthe coherent magnetization excitation. In contrast, spin\ncurrents generated by thermal gradients produce incoher-\nent currents with a continuum of spin excitation modes.\nThe magnetic excitations in antiferromagnets typically\nhave THz frequencies, while the resonant excitation of\nthe ferromagnetic injector is in the GHz range. Con-\nventional spin pumping experiments measure only the\ntime-averaged DC component of the spin current, i.e.,\nthey cannot distinguish between GHz and THz frequen-\ncies, which is needed to determine how the spin current\nFIG. 3. Time resolved precession. (a) Series of XFMR de-\nlay scans for the Co layer in a Co/MgO/Py MTJ at 80 K\ncontinuously driven at 4 GHz. As expected, the period of\nthe precession is 250 ps, and the delay scan covers two peri-\nods. For clarity, the data points (circles) obtained at di\u000berent\nmagnetic \feld values (between 40 and 200 mT) are shifted\nby a constant o\u000bset and have been di\u000berently colored. The\ndrawn lines represent the \ftted sinusoidal functions. Their\namplitude and phase as a function of magnetic \feld strength\nis plotted in panels (b) and (c), respectively, for both the Co\n(orange) and Py (blue) layers. (Adapted from G ladczuk et\nal.[45]).\npropagates. Alternative techniques such as XFMR are\nneeded to measure the time-varying AC spin current.\nDabrowski et al. [42] used XFMR to study the coher-\nent spin current propagation in a device with three layers,\nwhere the top (injector) and bottom (sink) layers were\nferromagnetic NiFe and FeCo, respectively, and the mid-\ndle layer was epitaxial NiO (001). The phase and am-7\nFIG. 4. DFMR delay scans of the structural and magnetic\npeaks as a function of linear polarization angle. Measure-\nments of (a,b) the anisotropic mode B at 6 GHz and (c,d)\nthe isotropic mode A at 2 GHz. The results for the magnetic\npeak and the structural (0,0,3) peak are shown in the left\nand the right column, respectively. The magnetic resonance\nmodes are probed with linearly polarized light for the range\nof incident polarization angles \u0011between 0{180\u000e. (Adapted\nfrom Ref. [60]).\nplitude of the magnetization precession within adjoining\nsource and sink FM layers were detected, from which\nthe injection and transmission of pure AC spin current\nthrough NiO can be inferred. It was found that magne-\ntization modes in the FM layers oscillate in phase. Fur-\nthermore, the e\u000eciency of the spin transfer varied with\nthe thickness of the antiferromagnet, with a maximal ef-\n\fciency for a 2-nm-thick layer. These results indicate\nthat a spin current propagates coherently through the\nantiferromagnetic NiO layer. The AC spin current is en-\nhanced for NiO thicknesses of less than 6 nm, both with\nand without a nonmagnetic spacer layer inserted into the\nstack, in a manner consistent with previously reported\nexperimental measurements of DC spin current and the-\noretical studies [65]. The XFMR results show that the\npropagation of spin current through NiO layers is medi-\nated by evanescent antiferromagnetic spin wave modes at\nGHz frequencies, rather than THz frequency magnons.\nC. DFMR for mode-resolved detection of\nmagnetization dynamics\nRecent scienti\fc interest has shifted towards more\ncomplex magnetically ordered materials, which are\npromising for high-density and low-energy consumption\ndevices. These systems contain chiral magnetic phases\nsuch as helical, conical, or skyrmion spin structures,\noriginating from the Dzyaloshinskii-Moriya interaction(DMI) found in noncentrosymmetric bulk materials, as\nwell as in systems where symmetry breaking occurs at\na ferromagnetic/heavy metal interface. Such spin struc-\ntures are much more complex than simple ferromagnetic\nstructures, especially their dynamic behavior is so far ill-\nunderstood.\nThe periodic structure of magnetically ordered sys-\ntems can be probed by resonant elastic X-ray scattering\n(REXS), making use of interference e\u000bects from the reg-\nularly repeating magnetization density variations. This\nleads to pure magnetic X-ray scattering peaks which give\ninformation about the static magnetic structure. Analy-\nsis of these magnetic peaks in REXS measurements using\nsynchrotron radiation has led to signi\fcant progress in\nthe understanding of chiral magnetic systems [66, 67].\nIn a pioneering DFMR experiment, Burn et al. [60]\ninvestigated the complex dynamic behavior of the chi-\nral spin structure in Y-type hexaferrite Ba 2Mg2Fe12O22.\nVNA-FMR measurements of this material showed a \feld-\nfrequency map containing two ferromagnetic resonance\nmodes. While mode A is isotropic, i.e., its \feld value is\nindependent of the direction of the applied \feld, mode B\nis anisotropic, showing greater absorption at increasingly\nhigher \felds as the \feld direction rotates out-of-plane.\nREXS at the Fe L2;3absorption edge was used to char-\nacterize the static magnetic structure of the hexaferrite\nand to determine its \feld dependence. Static REXS mea-\nsurements along (0,0, `) in zero \feld show a (0,0,3) struc-\ntural peak decorated with two incommensurate magnetic\nsatellites.\nThe DFMR signal was measured by pointing the pho-\ntodiode at the scattered beam, selecting either the struc-\ntural or the magnetic satellite peak (Fig. 2). Delay scans\nwere measured as a function of applied \feld using linearly\npolarized X-rays. Sinusoidal \fts to the measured data en-\nables the extraction of amplitude and phase. Fig. 4 shows\nthe delay scans of the structural and magnetic peaks of\nthe Y-type hexaferrite for variable incident linear polar-\nization angles \u0011. The panels (a,b) in the top row refer to\nthe anisotropic mode B at 6 GHz, and the panels (c,d)\nin the bottom to the isotropic mode A at 2 GHz. The\nleft and right column refer to the results for the mag-\nnetic peak and the structural (0,0,3) peak, respectively.\nThe results were compared to computer simulations of\nthe Y-type hexaferrite to obtain insight in the periodic\nspin structure of this material.\nA second example of the use of DFMR for mode re-\nsolved detection concerns the dynamic behavior of topo-\nlogical spin textures and chiral magnets, which is an area\nof signi\fcant interest and key to the development of fast\nand e\u000ecient spintronics devices. DFMR measurements\nby Burn et al. [63] revealed how the time-dependence\nof the magnetization dynamics relate to the complex\nspin texture in the well-known chiral magnetic system\nCu2OSeO 3. Using polarized soft X-rays, the dynamic\nexcitations in all three dimensions were probed, which\nrevealed phase shifts that were previously undetectable\nand indistinguishable using conventional FMR.8\nFIG. 5. (a) Pseudo-3D plot of the RFMR signal and its pro-\njection showing the dynamic contribution to the re\rectivity\nfor a [CoFeB/MgO/Ta] 4multilayer as a function of pump-\nprobe time delay. The measurements were carried out with\nleft-circularly polarized X-rays at the Fe L3resonance (707.7\neV) and in an out-of-plane \feld of 29 mT using RF excitation\nat 2 GHz. The various delay curves are shown for di\u000berent Qz,\nranging between 0 and 0.6 nm\u00001. The color scale represents\nthe normalized intensity for each delay scan, highlighting the\nsinusoidal dependence and the shift in phase as Qzis varied\nwhen the intensity is small. (b) Static and dynamic re\rectiv-\nity, and (c) phase of the dynamic re\rectivity as a function of\nQz. The phase point size is scaled by the strength of the dy-\nnamic signal amplitude, and the blue-shaded regions indicate\nwhere the 180\u000ephase shifts have been subtracted to reveal the\notherwise smooth phase variation. (Adapted from Ref. [61]).\nD. RFMR on a [CoFeB/MgO/Ta] 4multilayer\nX-ray re\rectivity with the photon energy tuned to the\nabsorption edge has become a valuable tool for character-\nizing the depth-dependent structure of layered materials.\nThe X-ray re\rectivity is measured as a function of the\nscattering vector Qz=ks\u0000ki= (4\u0019=\u0015) sin#, where ki\n(ks) is the ingoing (outgoing) wavevector of the X-rays\nwith incident angle #and wavelength \u0015. The scattering\nlength density, which gives the scattering strength of the\nchemical and magnetic species within the depth pro\fle of\nthe \flm, is obtained through \ftting the re\rectivity data.\nBurn et al. [61] revealed the depth dependence of the\nmagnetization dynamics in a [CoFeB/MgO/Ta] 4multi-layer system. The structural depth pro\fle was charac-\nterized through static X-ray re\rectometry. The dynamic\nre\rectivity was probed with stroboscopic DFMR using\nan out-of-plane saturating \feld of HBias= 29 mT and\nan RF \feld generated by the CPW beneath the sample.\nThe RF \feld was phase-locked to the fourth harmonic of\nthe\u0018500 MHz synchrotron master clock at 2 GHz. The\ntime dependence of the re\rectivity during precession was\nmapped out as a function of the time delay between the\nRF pump and X-ray probe. Fig. 5(a) shows a color map\nof the sinusoidal variation in the re\rected signal with a\n500 ps period, corresponding to the 2 GHz excitation.\nThe amplitude and phase of the dynamic signal are ex-\ntracted by \ftting the sinusoidal delay scans. The ampli-\ntude is plotted in Fig. 5(b) alongside the static re\rectivity\nfor the di\u000berent values of Qz, ranging between 0 and 0.6\nnm\u00001, and the phase in Fig. 5(c).\nBoth the static intensity and the amplitude of the\ndynamic signal in Fig. 5(b) show re\rectivity fringes re-\nsulting from interference e\u000bects arising from the layered\nchemical and magnetic structure. Additional minima are\nobserved in the dynamic case. The phase of the dynamic\nsignal in Fig. 5(c) shows variations with two contribu-\ntions. Firstly, abrupt 180\u000ephase jumps occur, coincid-\ning with minima in the amplitude of the dynamic signal.\nThese 180\u000ejumps correspond to inversion of the sign of\nthe XMCD signal measured at di\u000berent scattering con-\nditions. In addition, there are smoother variations in\nthe phase, which can be attributed to variations in the\nmagnetization dynamics occurring between the magnetic\nlayers in the multilayered structure.\nTo reveal the depth dependent magnetization dynam-\nics, the experimental results were compared with model-\ning of the dynamic behavior In all layers, the magnetiza-\ntion precesses about a nominal static state when excited\nby an RF \feld. It was shown that inclusion of a small,\nbut signi\fcant phase lag of 5\u000ebetween the four layers\nis necessary to explain the observed change in phase of\nthe dynamic signal. In contrast, a single slab of mag-\nnetic thin \flm material shows a coherent precession of\nthe magnetization as a function of depth.\nWith RFMR, the dynamics from di\u000berent layers con-\ntaining the same element can be explored, and this tech-\nnique has the potential to study the dynamics of inter-\nfacial layers and proximity e\u000bects in complex thin \flm\nand multilayer materials for future magnetic memory and\nprocessing device applications.\nV. CONCLUSIONS\nAlthough conventional FMR is a powerful technique\nto study magnetic resonances in thin \flms and multi-\nlayers, the measured response corresponds to an aver-\nage over the entire magnetic structure of the sample. In\ncontrast, X-ray based FMR techniques allow for time-\nresolved measurements of the magnetization dynamics,\nand, in addition, o\u000ber the bene\fts of XMCD, such as9\nelement-, site-, and shell-speci\fcity [57]. The time reso-\nlution is achieved by stroboscopic probing using higher\nharmonics (1-10 GHz) of the synchrotron master clock.\nXFMR can be used to study spin-transfer torque, dipo-\nlar \feld strength, magneto-crystalline anisotropy, inter-\nlayer exchange coupling, gyromagnetic ratio and damp-\ning constants. It can be applied to study the behav-\nior of spintronics systems, e.g., spin pumping in mag-\nnetic multilayers, heterostructures, spin valves, MTJ, etc.\nThe amplitude and phase of the magnetic resonances\nextracted from the \feld-dependence of the precessional\nplots enable us to distinguish between static and dy-\nnamic exchange coupling and to quantify their relative\ncontributions. Apart from measuring the signal in ab-\nsorption, XFMR can also be detected in di\u000braction and\nre\rectivity; each of these techniques bringing unique ad-vantages. DFMR reveals the dynamical spin modes at\nthe probed magnetic wavevectors, and RFMR gives the\ndepth-resolved dynamics in magnetic multilayers. Future\nXFMR studies can be envisaged to investigate vortex dy-\nnamics, spatial resolution imaging, and X-ray hologra-\nphy.\nVI. ACKNOWLEDGMENTS\nThe XFMR experiments were carried out on beamline\nI10 at the Diamond Light Source (Oxfordshire, United\nKingdom). We like to acknowledge valuable collabo-\nrations with Alex A. Baker, David M. Burn, Maciej\nDabrowski, Adriana I. Figueroa, Lukasz Gladczuk, and\nRobert J. Hicken.\n||||||||{\n[1] G. van der Laan, J. Electron Spectrosc. Relat. Phenom.\n220 (2017) 137{146.\n[2] D. M. Burn, S. L. Zhang, G. van der Laan, T. Hesjedal,\nAIP Advances 11 (2021) 015327.\n[3] W. E. Bailey, L. Cheng, D. J. Keavney, C.-C. Kao,\nE. Vescovo, D. A. Arena, Phys. Rev. B 70 (2004) 172403.\n[4] D. A. Arena, E. Vescovo, C. C. Kao, Y. Guan, W. E.\nBailey, Phys. Rev. B 74 (2006) 064409.\n[5] Y. Guan, W. E. Bailey, C.-C. 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Schreck,2and G.V. Shlyapnikov1, 2, 4, 5\n1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,\nWuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China\n2Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam,\nScience Park 904, 1098 XH Amsterdam, The Netherlands\n3Department of Theoretical Physics, Research School of Physics and Engineering,\nAustralian National University, Canberra ACT 0200, Australia\n4LPTMS, CNRS, Univ. Paris-Sud, Universit\u0013 e Paris-Saclay, 91405 Orsay, France\n5Russian Quantum Center, Novaya Street 100, Skolkovo, Moscow Region 143025, Russia\nWe study a one-dimensional two-component atomic Fermi gas with an in\fnite intercomponent\ncontact repulsion. It is found that adding an attractive resonant odd-wave interaction breaking\nthe rotational symmetry one can make the ground state ferromagnetic. A promising system for\nthe observation of this itinerant ferromagnetic state is a 1D gas of40K atoms, where 3D s-wave\nandp-wave Feshbach resonances are very close to each other and the 1D con\fnement signi\fcantly\nreduces the inelastic decay.\nItinerant ferromagnetism of degenerate spin-1/2\nfermions is an intriguing problem promoting our under-\nstanding of strongly correlated systems [1]. The origin\nof such ferromagnetic states is deeply rooted in quantum\nmechanics. In contrast to ultracold bosons, degenerate\nfermions try to avoid the ferromagnetic state because it\nrequires them to have a signi\fcantly higher kinetic en-\nergy than in non-ferromagnetic states. Ultracold gases\nof atomic fermions which are in two internal states can\nbe mapped onto spin-1/2 fermions treating the internal\nenergy levels as pseudo-spin states. The ferromagnetic\nphase is the one where all atoms are in the same super-\nposition of the two internal states, and one has a system\nof identical fermions. The kinetic energy is then higher\nthan, for example, in the paramagnetic phase represent-\ning a statistical mixture of the two spin components.\nItinerant ferromagnetism for fermions is studied since\nthe 1930-ths, when the Stoner criterion for ferromag-\nnetism in a free electron gas was introduced [2, 3]. In\nthree dimensions the ground state can be ferromagnetic\nif there is a strong intercomponent repulsion in the para-\nmagnetic state, which compensates the large di\u000berence in\nthe kinetic energies of the ferro- and paramagnetic states.\nFor atomic Fermi gases the Stoner mechanism was dis-\ncussed in a number of papers [4] and was then tested by\nMonte Carlo calculations [5, 6] which found an instability\non approach to the strongly interacting regime. The ef-\nforts to stabilize the ferromagnetic state experimentally\ndid not succeed [7, 8]. The reason is [7{9] that in three\ndimensions a large intercomponent repulsion corresponds\nto a very large and positive s-wave scattering length, and\nthere is a weakly bound dimer of two fermions belonging\nto di\u000berent internal states. A composition of such dimers\nhas a lower energy than the ferromagnetic state even if\nthe dimers overlap with each other, and at common gas\ndensities the formation of the dimerized phase out of the\ncloud of atoms is very fast.\nIn one and two dimensions the di\u000berence in the ki-netic energies of the ferro- and non-ferromagnetic states\nis even larger than in 3D. For example, in the 1D two-\ncomponent Fermi gas even at an in\fnite intercomponent\ncontact repulsion the energies of these states are equal,\nand at any \fnite repulsion the antiferro- or paramag-\nnetic states have a lower energy [10]. Therefore, it looks\nsuch that in low dimensions making the ground state\nferromagnetic is harder than in 3D. However, we show\nthat in 1D the odd-wave interaction (analog of p-wave in\nhigher dimensions) can drastically change the situation\nand make the ground state ferromagnetic. This interac-\ntion is momentum dependent and does not fall into the\nclass of interactions satisfying the conditions of the Lieb-\nMattis theorem [11] which states that in the 1D (two-\ncomponent) Fermi system the ground state can not be\nferromagnetic.\nTo make the odd-wave interaction signi\fcant one needs\na Feshbach and/or con\fnement-induced odd-wave reso-\nnance. At the same time, another resonance is needed in\norder to make the even-wave contact interaction strongly\nrepulsive. Here we have a \\present from nature\". In\nthe case of40K atoms the s-wave resonance for the in-\nteraction between 9 =2;\u00007=2 and 9=2;\u00009=2 states occurs\nat the magnetic \feld of 202.1 Gauss, and is very close\nto thep-wave resonance for the interaction between two\n9=2;\u00007=2 atoms at 198.8 Gauss [12, 13]. In the \felds\nbetween 198.8 Gauss and 202.1 Gauss the s-wave inter-\naction is repulsive and the p-wave interaction is attrac-\ntive. Moreover, the reduction of dimensionality to 1D\ndecreases the inelastic decay even not far from the res-\nonances, which is very promising for achieving itinerant\nferromagnetism in the 1D gas of40K.\nThe even-wave scattering of identical fermions requires\nthat the spatial wavefunction of colliding atoms is sym-\nmetric, and the spinor part is antisymmetric, i.e. it oc-\ncurs through the singlet channel. On the contrary, in\nthe odd-wave scattering the spatial part of the fermionic\nwavefunction is antisymmetric, and the spinor part isarXiv:1512.01392v1  [cond-mat.quant-gas]  4 Dec 20152\nsymmetric, which corresponds to the triplet state. If the\nodd-wave interaction is the same in all triplet states, and\nif it is relatively weak (such that in the expression for the\ncorresponding interaction amplitude we can keep only the\nlowest term, which is proportional to k2, withkbeing\nthe relative momentum of interacting atoms), while the\neven-wave repulsion is in\fnitely strong, then the prob-\nlem is exactly solvable and can be mapped onto two-\ncomponent bosons with SU(2) spin rotation symmetry.\nFor the latter case the ground state is known to be ferro-\nmagnetic [14{16]. However, the spin rotation symmetry\nbreaks if the odd-wave interaction is resonant, since it\nthen depends on the spin projections of colliding par-\nticles. Therefore, in this regime the exact solution is\nno longer available, and in order to make conclusions\nabout the character of the ground state we have to em-\nploy many-body perturbation theory.\nWe consider a 1D two-component Fermi gas in free\nspace and assume that the intercomponent contact (even-\nwave) interaction is in\fnitely repulsive. Since it takes\nplace between two particles with zero total (pseudo)spin,\nit is only present in the non-ferromagnetic phases. There-\nfore, omitting the odd-wave interaction, the ferromag-\nnetic phase represents an ideal single-component Fermi\ngas, with the Fermi momentum kF=\u0019n, and the total\nenergy is equal to the kinetic energy:\nEf=Ekin=\u00192~2n2\n6mN=1\n3EFN; (1)\nwherenis the 1D density, Nis the total number of par-\nticles,mis the mass of a particle, and EF=~2k2\nF=2mis\nthe Fermi energy. The non-ferromagnetic phases in this\ncase are described by the exactly solvable Yang-Gaudin\nmodel [17, 18], and for any \fnite contact repulsion they\nhave a lower energy than the ferromagnetic phase. If the\nrepulsion is in\fnite, then the wavefunction vanishes when\ntwo particles approach each other at zero distance, and\nall spin con\fgurations are degenerate with the energy\nequal toEkin[17{19].\nThe odd-wave interaction can be both inter- and in-\ntracomponent. However, we are interested in the regime\nwhere this interaction is resonant. Since for the most\nimportant case of40K atoms the resonance in the odd-\nwave channel is present only between two atoms in the\n9=2;\u00007=2 states we con\fne ourselves to the odd-wave in-\nteraction between these states. Below the state 9 =2;\u00007=2\nis denoted as spin- \", and the state 9 =2;\u00009=2 as spin-#.\nMoreover, we assume that although the odd-wave inter-\naction is resonant, it is not too strong (a more precise\ncondition will be given later), and still can be treated as\nperturbation. The fact that one can use a perturbative\napproach in a 1D odd-wave interacting system (in con-\ntrast to the even-wave interaction) \fnds its origin in the\nabsence of a weakly bound state in a su\u000eciently shallow\nattractive potential. For the ferromagnetic many-body\nsystem our perturbative results perfectly agree with theexisting Bethe Ansatz solution [20].\nThus, to zero order the kinetic energy Ekinis the same\nin any spin con\fguration (as a consequence of the in\f-\nnite even repulsion) and gives the main contribution to\nthe total energy Eof the system, while the odd-wave in-\nteraction provides a small correction, which we derive up\nto the second order in perturbation theory. For the non-\nferromagnetic phases we employ the single-component\nmomentum distribution functions N\"(k) andN#(k) that\nwe obtain by solving numerically the Bethe Ansatz equa-\ntions for the Yang-Gaudin model at an in\fnite intercom-\nponent repulsion (see Supplemental Material). Consid-\nering equally populated \"and#internal states, N\"(k) =\nN#(k) =N(k), we have\n\u0002+1\n\u00001dk\n2\u0019N(k) =n\n2; (2)\n\u0002+1\n\u00001Ldk\n2\u0019~2k2\n2mN(k) =Ekin=2 =\u00192~2n2\n12mN; (3)\nwithLbeing the size of the system. For the ferromag-\nnetic phase we use the Fermi step momentum distribu-\ntionN(k) =\u0012(kF\u0000jkj)=2. The kinetic energy obtained\nby using the calculated non-ferromagnetic distributions\ndirectly in Eq. (3) di\u000bers from Ekinby less than 0 :3% in\nthe antiferromagnetic phase and by approximately 0 :5%\nin the paramagnetic phase.\nIn order to develop many-body perturbation theory we\nfollow the method used in Refs. [21, 22]. We de\fne the\no\u000b-shell scattering amplitude\nf(k0;k) =\u00021\n\u00001dxe\u0000ik0xV(x) k(x); (4)\nwhereV(x) is the interaction potential,  k(x) is the true\nwavefunction of the relative motion with momentum k=\n(k1\u0000k2)=2, withk1,k2andk0\n1,k0\n2being the particle mo-\nmenta in the incoming and outgoing scattering channels.\nForjk0j=jk0\n1\u0000k0\n2j=2 =jkjwe have the on-shell ampli-\ntude. Then, the total energy is E=Ekin+~E(1)+~E(2),\nwhere the \frst- and second-order corrections are given by\n(see Supplemental Material):\n~E(1)=1\nLX\nk1;k2~fodd(k)N(k1)N(k2); (5)\n~E(2)=\u00001\nL2X\nk1;k2;k0\n14m\n~2~fodd(k0;k)~fodd(k;k0)\nk2\n1+k2\n2\u0000k02\n1\u0000k02\n2\n\u0002N(k1)N(k2)N(k0\n1);(6)\nwithk1+k2=k0\n1+k0\n2. The amplitude ~foddis di\u000ber-\nent from the odd-wave part of (4) by the absence of the3\nimaginary term in the denominator (see Supplemental\nMaterial). The terms ~E(1)and ~E(2)are the two-body\n(mean-\feld) and the many-body, or beyond mean-\feld,\ncontributions to the interaction energy. As the 1D regime\nis obtained by tightly con\fning the motion of particles in\ntwo directions to zero point oscillations, the odd-wave o\u000b-\nshell scattering amplitude in the vicinity of the resonance\nis given by (see [23] and Supplemental Material):\n~fodd(k0;k) =2~2\nmk0klp\n1 +\u0018plpk2; (7)\nwhere the parameters lpand\u0018pof the 1D odd-wave scat-\ntering can be expressed through the parameters of the\n3Dp-wave scattering as\nlp= 3a?\u0014a3\n?\nw1+A\u0015\u00001\n; \u0018p=\u000b1a2\n?\n3: (8)\nHerew1and\u000b1are the 3D scattering volume and e\u000bec-\ntive range, respectively, a?=p\n~=(m!?) is the exten-\nsion of the wavefunction in the directions tightly (har-\nmonically) con\fned with frequency !?, and the numeri-\ncal constant isA=\u00003p\n2\u0010(\u00001=2)\u00190:88, with\u0010(\u00001=2)\nbeing the Riemann zeta function. For40K atoms near\nthep-wave Feshbach resonance the magnetic \feld depen-\ndence ofw1and\u000b1has been measured in the JILA ex-\nperiments [24]. Near the resonance in 3D the scattering\nvolumew1changes from in\fnitely negative to in\fnitely\npositive, whereas the e\u000bective range \u000b1remains practi-\ncally constant and equal to 4 \u0002106cm\u00001. On the positive\nside of the resonance ( w1>0) in 3D one has the forma-\ntion of rapidly decaying p-wave molecules [25{29], and\na similar phenomenon is expected in 1D. The issue of\ninelastic losses is discussed in more detail below, but in\nwhat follows we consider only the case of attractive odd-\nwave interaction, i.e. lp<0. The energy corrections (5)\nand (6) can be rewritten as (see Supplemental Material)\n~E(1)=\u0000Ekin\u001a1\n2\u0019\u0011+3\n16\u0019\u0014\u00112I(Q)\u001b\n; (9)\n~E(2)=3\n4\u00192\u00112J(Q)Ekin; (10)\nwhere\u0011=kFjlpj,\u0014=kF\u0018p,Q=\u0011\u0014, and we took into\naccount that in any spin con\fguration the momentum\ndistribution is a universal function of k=kF. The inte-\ngralsI(Q) andJ(Q) are given by\nI(Q) = +1\n\u00001dx1dx2N(x1)N(x2)(x1\u0000x2)4\n1\u0000Q\n4(x1\u0000x2)2;(11)\nJ(Q) = +1\n\u00001dx1dx2dx3N(x1)N(x2)N(x3)\nx1\u0000x3\n\u0002(x1\u0000x2)2\n\u0010\n1\u0000Q\n4(x1\u0000x2)2\u0011(x1+x2\u00002x3)2\n\u0010\n1\u0000Q\n4(x1+x2\u00002x3)2\u0011;(12)withxi=ki=kFbeing a dimensionless momentum, and\nthe symbol\u001f\ndenoting the principal value of the inte-\ngral. The choice of a particular spin con\fguration is\nencoded in the momentum distribution N(k=kF), and\nfrom Eqs. (9)-(12) it is evident that the odd-wave in-\nteraction splits the energies of di\u000berent phases only if\n\u00146= 0. The unperturbed momentum distributions for the\nferro-, antiferro-, and paramagnetic states are displayed\nin Fig.1. At k\u001dkFthe momentum distributions in the\nnon-ferromagnetic states behave as N(k)!C=k4, where\nCis Tan's contact [30, 31]. In the aniferromagnetic phase\nits value isC=k4\nF= 2 ln(2)=3\u00192\u00190:047 [31], and in the\nparamagnetic phase we obtain C=k4\nF\u00190:016.\nFIG. 1. Momentum distributions in the ferro-, antiferro-, and\nparamagnetic phases. The latter two are for the Yang-Gaudin\nmodel at an in\fnite repulsion.\nIn the limit of Q!0 we haveI= 2(1 + 3D)=3\nandJ= 1=2, whereD=\u00011\n\u00001dx\u0002\nN(x)x4\u0000C=k4\nF\u0003\n. For\na \fniteQwe calculate the integrals (11) and (12) numer-\nically.\nRealization of the 1D regime requires the Fermi energy\nto be much smaller than the tight con\fnement frequency:\nEF=~2k2\nF\n2m\u001c~!?: (13)\nFor realistic con\fnement frequencies !?in the range\nfrom 50 to 150 kHz, the condition (13) requires the Fermi\nmomentum kF.105cm\u00001(which corresponds to densi-\ntiesn.3\u0002104cm\u00001andEF.1\u0016K). The con\fnement\nlengtha?for such frequencies is from 400 to 700 \u0017A. Then,\ntaking the potassium value 4 \u0002106cm\u00001for the e\u000bective\nrange\u000b1and using relations (8) we see that the param-\neter\u0014ranges from 1 to 5. In the perturbative regime we\nrequire\u0011=\u0019=njlpj\u001c1, i.e. one should not be too close\nto the resonance, and in order to stay within the limits\nof perturbation theory we put \u0011<0:8.\nWe then calculate the total energy of the gas up\nto the second order in perturbation theory for the\nferro-, antiferro-, and paramagnetic phases. The results\nare presented in Fig.2, which shows that the di\u000berence4\nbetween the energies of the ferro- and non-ferromagnetic\nstates is the largest at \u0011\u00181=\u0014and decreases signi\fcantly\nas\u0014grows.\nFor a gas of40K atoms with a density n\u00193\u0002\n104cm\u00001(EF\u0019540 nK) under the transverse con-\n\fnement with frequency !?\u0019100 kHz we have\n\u0014=\u0019n\u000b 1~=(3m!?)\u00193:1, and at \felds slightly lower\nthan 199 G the parameter \u0011is approximately 0 :36. Then\nthe ferromagnetic state has the lowest energy, and the\nenergies of antiferro- and paramagnetic states are close\nto each other. The energy di\u000berence ( Ep\u0000Ef)=N\nis about 0:03EFor 16 nK. Increasing the con\fnement\nstrength to !?\u0019120 kHz, we obtain \u0014\u00192:6, and\nwith\u0011\u00190:43 at the magnetic \feld B\u0019199 G the energy\ndi\u000berence ( Ep\u0000Ef)=Nbecomes approximately 20 nK.\nThus, the ferromagnetic state can be observed at tem-\nperatures below 20 nK.\nThe regimes described above ensure that the even re-\npulsion is in\fnitely strong, although at \frst sight the\ncorresponding magnetic \felds are not close enough to\nthes-wave resonance. However, in the 1D geometry ob-\ntained by tightly con\fning particles in two directions, the\ncoupling constant for the even contact interaction is [32]\ng1D=2~2a\nma?(a?\u00001:03a): (14)\nFor the con\fning frequencies of 100 and 150 kHz the\nharmonic length a?is about 500 and 400 \u0017A, respec-\ntively. In a \feld close to 199 Gauss the scattering length\nisa\u0019400\u0017A. Thus, due to the con\fnement-induced\nresonance, one can achieve an in\fnite contact repulsion\ng1D!1 .\nIt is well known that in three dimensions p-wave Fesh-\nbach resonances are su\u000bering of rapid inelastic losses [25{\n29]. There are two types of inelastic collisional processes.\nThe \frst one is three-body recombination, which is espe-\ncially pronounced if there are weakly bound dimer states.\nHowever, weakly bound dimer p-wave states are expected\nonly on the positive side of the resonance ( lp>0), and\non the negative side ( lp<0 and attractive interactions)\nthe three-body recombination should not be very danger-\nous, at least slightly away from the resonance. Another\ndecay process is two-body relaxation. The internal state\n9=2;\u00007=2 has a higher energy than 9 =2;\u00009=2. Therefore,\nthe state 9=2;\u00007=2 can undergo collisional relaxation to\nthe 9=2;\u00009=2 state. In 3D the rate of this process is\nespecially high very near the resonance [25, 33].\nIn \felds slightly higher than 199 G the mea-\nsured rate constant of three-body recombination in 3D\nis\u000b3D\nrec\u001810\u000025cm6/s and the rate constant of two-body\nrelaxation is \u000b3D\nrel\u001810\u000014cm3/s [25]. The measure-\nments were done in a pure gas of 9 =2;\u00007=2 atoms, and\nin a mixture of 9 =2;\u00007=2 and 9=2;\u00009=2 states it can be\nsomewhat higher. The temperature in the experiment\nwas from 1 \u0016K to 3\u0016K, so that one expects about the\nFIG. 2. The total energy (in units of Ekin=EFN=3) versus\u0011\nin the ferro-, antiferro- and paramagnetic state for \u0014= 3,\n\u0014= 4, and\u0014= 5.\nsame rate constants at EF\u00181\u0016K and much lower tem-\nperatures. In order to transform these results to 1D one\nshould recall that the inelastic processes occur at atomic\ninterparticle distances. We thus may integrate out the\nmotion in the tightly con\fned directions [34]. This leads\nto\u000b1D\nrel\u0019\u000b3D\nrel=2\u0019a2\n?[35]. With the above speci\fed \u000b3D\nrel\nand densities n\u0018104or 3\u0002104cm\u00001we obtain a relax-\nation time of about a second.\nRegarding the three-body recombination the situation\nis more peculiar. For the collision of three particles in one\nand the same internal state in 1D one has an extra sup-\npression by a factor of EF=E\u0003compared to 3D [36]. The\nquantityE\u0003is a typical energy in the molecular problem,\nand one has E\u0003\u0018~2=mR2\ne, whereRe\u001850\u0017A is the ra-5\ndius of interaction between particles. So, E\u0003\u00181 mK or\neven larger and there is an extra suppression by 3 orders\nof magnitude for EF\u00181\u0016K. Integrating out the par-\nticle motion in the tightly con\fned direction we obtain\n\u000b1D\nrec\u0018\u000b3D\nrec(EF=E\u0003)=3\u00192a4\n?[35]. Again, for the above\nmentioned parameters we obtain the decay time about a\nsecond at 1D densities in between 104and 3\u0002104cm\u00001.\nThe ferromagnetic state can be viewed as a composition\nof identical fermions, and hence this estimate remains\nvalid in this phase.\nWe thus see that in 1D the issue of inelastic decay\nprocesses is not as crucial as in 3D. In this respect the\nsituation is somewhat similar to the one in recent exper-\niments with strongly interacting bosons [37, 38].\nIn conclusion, we showed that there is a realistic pos-\nsibility to \fnd itinerant ferromagnetic states in 1D two-\ncomponent Fermi gases, and a promising system is the\ngas of40K atoms. This will require \fne tuning of the in-\nteraction between particles by varying the magnetic \feld\nand the strength of the tight con\fnement. The required\ntemperatures are about 10 to 20 nanokelvin, which is\nachievable with present facilities.\nWe would like to thank V. Gritsev, D. Petrov, and\nM. 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Lett. 107, 230404 (2011).\nSupplemental Material: Itinerant ferromagnetism in 1D two-component Fermi gases\nIn the Supplemental Material we \frst discuss the scattering of identical fermions in 1D, obtain the expression\nfor the odd-wave o\u000b-shell scattering amplitude, and derive two-body and many-body contributions to the odd-wave\ninteraction energy of a 1D Fermi gas. We then describe our calculation of the momentum distribution in the antiferro-\nand paramagnetic states.\nScattering amplitude in 1D. The o\u000b-shell scattering amplitude, de\fned by Eq. (4) of the main text, is the sum of\nthe even-wave and odd-wave partial amplitudes:\nf(k0;k) =feven(k0;k) +fodd(k0;k); (S.1)\nwhere\nfeven(k0;k) =\u00021\n\u00001dx0cosk0x0V(x0) even(k;x0); fodd(k0;k) =\u0000i\u00021\n\u00001dx0sink0x0V(x0) odd(k;x0); (S.2)\nand even, oddare the partial wavefunctions of the relative motion in the even-wave and the odd-wave channels,\nrespectively. For jk0j=jkjone has the on-shell amplitudes, which follow from from Eq. (S.2) putting k0=k.\nThese amplitudes enter the asymptotic expression for the wavefunction of the relative motion at interparticle separa-\ntionsjxj!1 [S1]:\n k(x) =eikx\u0000im\n2~2keikjxj\u0000\nfeven(k) + sign(x)fodd(k)\u0001\n: (S.3)\nIn the quasi1D regime obtained by tightly con\fning the motion of particles in two directions to zero point oscillations,\nthe on-shell scattering amplitude in the odd-wave channel was calculated in Ref. [S2], and it reads:\nfodd(k) =2~2\nmk2\n1=lp+ik+\u0018pk2; (S.4)\nwhere parameters lpand\u0018pare given by Eq. (8) of the main text.\nWe now proceed with the derivation of the odd-wave o\u000b-shell scattering amplitude. The asymptotic form of the\nwavefunction  odd(k;x) atjxj!1 can be written as\n odd(k;x)/sin\u0000\nkjxj\u0000\u000eodd(k)\u0001\n; (S.5)\nwhere\u000eodd(k) is the scattering phase shift. The relation between the odd-wave phase shift and the corresponding\nscattering amplitude follows from Eq. (S.5) and the odd-wave part of Eq. (S.3):\nfodd(k) =2~2k\nmtan\u000eodd(k)\n1 +itan\u000eodd(k); (S.6)\nwith tan\u000eodd(k) =lpk=\u0000\n1 +lp\u0018pk2\u0001\n. Let us rewrite the wavefunction of the odd-wave channel  odd(k;x) in the\nfollowing form:\n odd(k;x) =i~ odd(k;x)\n1 +itan\u000eodd(k); (S.7)\nwhere ~ odd(k;x) = sinkx\u0000sign(x) tan\u000eodd(k) coskxis real. Then, the odd-wave o\u000b-shell scattering amplitude can\nbe represented as\nfodd(k0;k) =~fodd(k0;k)\n1 +itan\u000eodd(k); (S.8)2\nwhere ~fodd(k0;k) is also real:\n~fodd(k0;k) =\u00021\n\u00001dx0sink0x0V(x0)~ odd(k;x0): (S.9)\nAssuming that the phase shift is small, one obtains the following relation:\nfodd(k0;k) =~fodd(k0;k)\u0000im\n2~2k~f2\nodd(k0;k): (S.10)\nSubstituting k0=kinto Eqs. (S.8)-(S.10) we obtain similar relations for the on-shell amplitudes fodd(k) and ~fodd(k).\nIn the case of low-energy scattering we may put sin k0x0\u0019k0x0and sinkx0\u0019kx0in the expressions for ~fodd(k0;k) and\n~fodd(k), which shows that in the odd-wave channel one obtains the o\u000b-shell amplitude from the on-shell amplitude\nby simply replacing kin (S.6) with k0. Then, using Eqs. (S.6) and (S.8) we arrive at Eq. (7) of the main text for the\nodd-wave o\u000b-shell scattering amplitude ~fodd(k0;k).\nTwo-body and many-body contributions to the interaction energy. An in\fnitely strong even-wave contact repulsion\ncan be transferred to the boundary condition for the wavefunction. The interaction part of the Hamiltonian then\ncontains only the odd-wave interaction, which in our case is present solely between \"-state particles:\n^Hint=1\n2LX\nk1;k2;qV(q) ^ay\nk1+q^ay\nk2\u0000q^ak2^ak1; (S.11)\nwhere ^ay\nk;^akare the creation and annihilation operators of \"-fermions, and V(q) is the Fourier transform of the\ninteraction potential in the odd-wave channel:\nV(q) =\u00021\n\u00001dxV(x)e\u0000iqx=\u00021\n\u00001dxV(x) cosqx; (S.12)\nwhere we took into account that V(x) is even. Then, the \frst-order correction is given by the diagonal matrix element\nof^Hint:\nE(1)=1\n2LX\nk1;k2[V(0)\u0000V(k2\u0000k1)]N(k1)N(k2): (S.13)\nThe second-order correction to the energy of a state jjiis given by\nE(2)\nj=X\nm6=jjVjmj2\nEj\u0000Em; (S.14)\nwhere the summation is over the eigenstates of the non-interacting system, and the non-diagonal matrix element\nVjm=hmj^Hintjjiis related to the scattering of two particles from the initial state k1;k2to an intermediate state k0\n1;k0\n2.\nIn our case, the symbol jcorresponds to the ground state, and the symbol mto excited states. Then, taking into\naccount the momentum conservation law k1+k2=k0\n1+k0\n2, we obtain the following expression for the quantity jVjmj2:\njVjmj2=1\n(2L)2X\nk1;k2jV(k0\n1\u0000k1)\u0000V(k0\n2\u0000k1)j2N(k1)N(k2)\u0000\n1\u0000N(k0\n1)\u0001\u0000\n1\u0000N(k0\n2)\u0001\n; (S.15)\nand the second-order correction becomes:\nE(2)=1\n(2L)2X\nk1;k2;k0\n1jV(k0\n1\u0000k1)\u0000V(k0\n2\u0000k1)j2\n~2(k2\n1+k2\n2\u0000k02\n1\u0000k02\n2)=2mN(k1)N(k2)\u0000\n1\u0000N(k0\n1)\u0001\u0000\n1\u0000N(k0\n2)\u0001\n: (S.16)\nIt is evident that the second-order correction diverges at large k0\n1because of the term proportional to N(k1)N(k2).\nThis arti\fcial divergence can be eliminated if one expresses E(1)andE(2)in terms of a real physical quantity |\nthe scattering amplitude. The relation between the Fourier component of the interaction potential and the o\u000b-shell\nscattering amplitude is given by [S1]\nf(k0;k) =V(k0\u0000k) +1\nLX\nk00V(k0\u0000k00)f(k00;k)\nEk\u0000Ek00+i0; (S.17)3\nwhereEkandEk00are relative collision energies, and we have Ek\u0000Ek00=~2(k2\n1+k2\n2\u0000k002\n1\u0000k002\n2)=2m, withk1;k2(k00\n1;k00\n2)\nbeing the momenta of colliding particles in the initial (intermediate) state. Equation (S.17) allows us to rewrite the\n\frst-order correction as\nE(1)=1\n2LX\nk1;k2[f(k;k)\u0000f(\u0000k;k)]N(k1)N(k2)\u00001\n2L2X\nk1;k2;k0[V(k\u0000k0)\u0000V(\u0000k\u0000k0)]f(k0;k)\n~2(k2\u0000k02+i0)=mN(k1)N(k2):(S.18)\nIn the second term of Eq. (S.18) we represent f(k0;k) according to Eq. (S.1) and keep only the odd-wave\npartfodd(k0;k), because the quantity [ V(k\u0000k0)\u0000V(\u0000k\u0000k0)] is odd in k0, and the terms containing feven(k0;k)\nvanish after the integration over dk0. Then, we replace the Fourier components by the exact scattering amplitudes,\nwhich gives [ V(k\u0000k0)\u0000V(\u0000k\u0000k0)] = [f(k;k0)\u0000f(\u0000k;k0)] = 2fodd(k;k0), and the \frst-order correction becomes:\nE(1)=1\nLX\nk1;k2fodd(k)N(k1)N(k2)\u00001\nLX\nk1;k2N(k1)N(k2)\u00021\n\u00001dk0\n2\u0019m\n~2fodd(k0;k)fodd(k;k0)\nk2\u0000k02+i0: (S.19)\nThe contribution of the pole at k0=kto the integral in Eq. (S.19) gives \u0000imf2\nodd(k)=2~2kfor each term in the sum\noverk1,k2, and we can use here ~fodd(k) instead of fodd(k). At the same time, the second term of the right hand side\nof Eq. (S.10), being substituted into the \frst term of Eq. (S.19), cancels the contribution of the pole in the second\nterm of Eq. (S.19). Therefore, we use the amplitude ~fodd(k) in the \frst term of Eq. (S.19) and take the principal\nvalue of the integral in the second term. This leads to the following expression for the \frst-order correction:\nE(1)=1\nLX\nk1;k2~fodd(k)N(k1)N(k2)\u00001\nL2X\nk1;k2;k0\n12m\n~2~fodd(k0;k)~fodd(k;k0)\nk2\n1+k2\n2\u0000k02\n1\u0000k02\n2N(k1)N(k2): (S.20)\nThe second-order correction can also be expressed in terms of the scattering amplitude. Taking into ac-\ncount that V(k0\n1\u0000k1) =V(k0\u0000k) andV(k0\n2\u0000k1) =V(\u0000k\u0000k0), we writejV(k0\n1\u0000k1)\u0000V(k0\n2\u0000k1)j2=\n[V(k\u0000k0)\u0000V(k+k0)] [V(k0\u0000k)\u0000V(\u0000k\u0000k0)], where we used the fact that Fourier components V(q) are real\nand even functions. Then for the second-order correction we obtain:\nE(2)=1\nL2X\nk1;k2;k0\n12m\n~2fodd(k0;k)fodd(k;k0)\nk2\n1+k2\n2\u0000k02\n1\u0000k02\n2N(k1)N(k2)\u0000\n1\u0000N(k0\n1)\u0001\u0000\n1\u0000N(k0\n2)\u0001\n; (S.21)\nwhere we may use the amplitudes ~fodd(k0;k) and ~fodd(k;k0) because the contribution of tan \u000eodd(k) in the denominator\nof Eq. (S.8) is negligible. Then, the divergent term proportional to N(k1)N(k2) in Eq. (S.21) and the (divergent)\nsecond term of Eq. (S.20) exactly cancel each other. Note that in Eq. (S.21) the term proportional to the product of\nfour occupation numbers vanishes, since its numerator is symmetrical and the denominator is antisymmetrical with\nrespect to an interchange of k1;k2andk0\n1;k0\n2. Two terms containing the product of three occupation numbers are\nequal to each other, because the expression (S.21) for E(2)is symmetrical with respect to an interchange of k0\n1and\nk0\n2. Therefore, the sum of the \frst- and second-order corrections can be written as E(1)+E(2)=~E(1)+~E(2), where\n~E(1)and ~E(2)are given by\n~E(1)=1\nLX\nk1;k2~fodd(k)N(k1)N(k2) =2~2\nmLX\nk1;k2lpk2\n1 +lp\u0018pk2N(k1)N(k2); (S.22)\n~E(2)=\u00001\nL2X\nk1;k2;k0\n14m\n~2~fodd(k0;k)~fodd(k;k0)\nk2\n1+k2\n2\u0000k02\n1\u0000k02\n2N(k1)N(k2)N(k0\n1)\n=\u000016~2\nmL2X\nk1;k2;k0\n1l2\npk02k2\nk2\n1+k2\n2\u0000k02\n1\u0000k02\n2N(k1)N(k2)N(k0\n1)\n(1 +lp\u0018pk02)(1 +lp\u0018pk2):(S.23)\nWe then reduce Eqs. (S22) and (S23) to equations (9)-(12) of the main text.\nMomentum distributions for the antiferro- and paramagnetic states. The momentum distribution functions N\"(k)\nandN#(k) for the antiferro- and paramagnetic states can be calculated from the Bethe Ansatz wave functions of the4\nYang-Gaudin model with an in\fnite repulsion. To this end, we will use the method proposed by Ogata and Shiba [S3].\nThe one-dimensional two-component Fermi gas with contact interactions is described by the Hamiltonian [S4, S5]\n^H=\u0000PN\ni=1@2=@x2\ni+2cP\ni<j\u000e(xi\u0000xj), wherec=mg1D=~2is the interaction strength for the even-wave scattering.\nThe Bethe Ansatz wavefunction for this model is given by [S5]\n\t\u001b(x) =X\nQP\u0002(Q)A(Q;P)eiP\njkPjxQj; (S.24)\nwhereQ= (Q1;Q2;:::;QN ) andP= (P1;P2;:::;PN ) are two permutations of integers f1;2;\u0001\u0001\u0001;Ngand \u0002(Q)\ndenotes the step function, i.e. \u0002( Q) =\u0012(xQ2\u0000xQ1)\u0012(xQ3\u0000xQ2)\u0001\u0001\u0001\u0012(xQN\u0000xQN\u00001) with\u0012(x) = 1 forx > 0\nwhereas\u0012(x) = 0 forx < 0. In the above equations, we denoted x=fx1;x2;\u0001\u0001\u0001;xNg,\u001b=f\u001b1;\u001b2;\u0001\u0001\u0001;\u001bNgand\nA=A\u001b1\u0001\u0001\u0001\u001bN(Q;P) are superposition coe\u000ecients. Here \u001bj=\u00061=2 stand for the spin projection of the j-th particle.\nFor periodic boundary conditions the wave numbers fkjgwithj= 1;2:::;N are subject to the following Bethe\nAnsatz equations (BAE):\neikjL=MY\ni=1kj\u0000\u0016i+ ic=2\nkj\u0000\u0016i\u0000ic=2; (S.25)\nNY\nj=1\u0016i\u0000kj\u0000ic=2\n\u0016i\u0000kj+ ic=2=\u0000MY\ni0=1\u0016i\u0000\u0016i0\u0000ic\n\u0016i\u0000\u0016i0+ ic:\nHereMis the number of down-spin fermions, and \u0016iwithi= 1;:::;M are spin rapidities.\nWe observe that the BAE (S.25) decouple into two parts in terms of charge and spin degrees of freedom as c!1 .\nThe second set of equations in the BAE (S.25) reduces to the BAE for the spin-1 =2 Heisenberg XXX model with the\nscaling\u0016\u000b!\u0016\u000bc. In this limit, the wave function (S.24) can be simpli\fed as\n\t\u001b(x) =X\nQP(\u00001)Q+P\b(y1;:::;yM) eiP\njkPjxQj: (S.26)\nHere\b(y1;:::;yM) is the eigenstate of the spin XXX model with Mdown-spins in the N-site lattice [S3, S5].\nUsing the wave function (S.26), we \frst calculate the momentum distribution in an analytical fashion. Then the \fnal\nmomentum distributions can be obtained by further numerical calculation. The momentum distribution is de\fned as\nN\u001b(k) =h^ y\n\u001b(k)^ \u001b(k)i, which is the Fourier transform of the density matrix N\u001b(k) =\u0003\ndydy0N\u001b(y0;y)eik(y\u0000y0). The\ndensity matrix is given by\nN\u001b(y0;y) =h y\n\u001b(y0) \u001b(y)i=N!\u0002x3\n0dx2\u0002x4\n0dx3\u0001\u0001\u0001\u0002xN\n0dxN\u00001\u0002L\n0dxNX\n\u001b2\u0001\u0001\u0001\u001bN\t\u0003\n\u001b(x(l))\t\u001b(x(r));(S.27)\nwhere we used the following notations:\nx(l)=fy0;x2;x3;\u0001\u0001\u0001;xNg;x(r)=fy;x2;x3;\u0001\u0001\u0001;xNg;\u001b=f\u001b;\u001b;\u001b 3;\u0001\u0001\u0001;\u001bNg: (S.28)\nIn the limit c!1 , the superposition coe\u000ecients satisfy the relations [S3]:\n~A(Q(ab);P(ab)) =^Pab~A(Q(ba);P(ba));~A(Q(ab);P(ab)) =\u0000~A(Q(ab);P(ba));\nwhere\nQ(ab)=fQ1;Q2;\u0001\u0001\u0001;Qa;Qb;\u0001\u0001\u0001;QNg;Q(ba)=fQ1;Q2;\u0001\u0001\u0001;Qb;Qa;\u0001\u0001\u0001;QNg:\nSubstituting the wave function (S.26) into (S.27), the density matrix is thus rewritten as\nN\u001b(y0;y) =N!\u0002x3\n0dx2\u0002x4\n0dx3\u0001\u0001\u0001\u0002L\n0dxNX\nP;P0(\u00001)P0+P+Q+Q0e\u0000iPN\nj=1kP0\njx(l)\nQ0eiPN\nj=1kPjx(l)\nQ\n\u0002X\n\u001b2;\u0001\u0001\u0001;\u001bN\u0002\nAy\fy(Q0)\u0003\n\u001b;\u001b2;\u0001\u0001\u0001;\u001bN\u0002\n\f(Q)A\u0003\n\u001b;\u001b2;\u0001\u0001\u0001;\u001bN: (S.29)\nHere the positions yandy0in the domain x2<\u0001\u0001\u0001<xNsatisfy two inequalities:\nx2<x3<\u0001\u0001\u0001<x\u0011<y0<x\u0011+1<\u0001\u0001\u0001<xN;\nx2<x3<\u0001\u0001\u0001<x\u0018<y<x\u0018+1<\u0001\u0001\u0001<xN:5\nAccordingly, the permutations QandQ0are given by\nQ0=f2;3;\u0001\u0001\u0001;\u0011;1;\u0011+ 1;\u0001\u0001\u0001;Ng;Q=f2;3;\u0001\u0001\u0001;\u0018;1;\u0018+ 1;\u0001\u0001\u0001Ng: (S.30)\nThe values of \u0011and\u0018are \fxed by the values of yandy0in the set offxjgwithj= 2;3;\u0001\u0001\u0001;N. In equation (S.29)\nwe de\fne the operator \fas\n\f(Q0) =^P\u00111^P\u0011\u00001;1\u0001\u0001\u0001^P3;1^P2;1; (S.31)\n\f(Q) =^P\u00181^P\u0018\u00001;1\u0001\u0001\u0001^P3;1^P2;1: (S.32)\nWhere ^Pi;jis the permutation operator.\nDe\fning an identity order 1=f1;2;\u0001\u0001\u0001;Ngfor the permutation operator, we \fnd that \f(Q0)1=Q0and\f(Q)1=\nQ. In Eq. (S.29) the superposition coe\u000ecient A\u0011A(1;1) is an eigenstate of the XXX Heisenberg spin chain, i.e.\nHxxxA=ExxxA. It hasCM\nNcomponents of spin wavefunction for each Q. Each of them is characterized by the\ncoordinates of Mdown-spins. In the following, we use the eigenstate of the Heisenberg spin chain to evaluate the\nintegral in Eq. (S.29). For convenience, we denote that\nw\u001b\n\u0011;\u0018=Ay^W\u001bA=X\n\u001b2;\u0001\u0001\u0001;\u001bN\u0002\nAy\fy(Q0)\u0003\n\u001b;\u001b2;\u0001\u0001\u0001;\u001bN\u0002\n\f(Q)A\u0003\n\u001b;\u001b2;\u0001\u0001\u0001;\u001bN;\nwith\n^W\u001b= ^n\u001b\n\u0018^P\u0018;\u0018+1^P\u0018+1;\u0018+2\u0001\u0001\u0001^P\u0011\u00001;\u0011: (S.33)\nThusw\u001b\n\u0011;\u0018can be regarded as the expectation value of the operator ^W\u001bof the XXX spin chain. In the above equation\nwe also denoted the operators ^ n\"= ^\u001b+^\u001b\u0000, ^n#= ^\u001b\u0000^\u001b+. By using the above notations, the density matrix is rewritten\nas\nN\u001b(y0;y) =N!\u0002x3\n0dx2\u0002x4\n0dx3\u0001\u0001\u0001\u0002L\n0dxNw\u001b\n\u0011;\u0018X\nP;P0(\u00001)P0+P+Q+Q0e\u0000iPN\nj=1kP0\njx(l)\nQ0eiPN\nj=1kPjx(l)\nQ:(S.34)\nDue to the translational symmetry of the system, it is convenient to introduce x=y\u0000y0; \u001c=j\u0011\u0000\u0018jandw\u001b\n\u0011;\u0018=w\u001b\n\u001c.\nThen the density matrix is simpli\fed as\nN\u001b(x) =N! eik0xNX\ni;j(\u00001)i+jei\u0001kIjxX\nQ(\u00001)Qfm(x)nY\ni=11\u0000e\u0000idi\u0001kx\n\u0000idi\u0001k; (S.35)\nThe quantities fm; di, and \u0001kused in (S.35) are explained below.\nIn the ground state the quasi-momenta take the following values [S3]\nkj=Ij\u0001k+k0;\u0001k=2\u0019\nL;\nIj=\u0000N\u00001\n2;\u0000N\u00003\n2;\u0001\u0001\u0001;N\u00001\n2: (S.36)\nIt is worth noting that the quantum numbers Ijare di\u000berent form the quantum numbers de\fned in the usual Bethe\nAnsatz equations [S3]. This notation is convenient for our calculation. The index i(j) in Eq. (S.35) indicates the\nquasi-momentum of the i-th (j-th) particle which is separated from other k's, i.e.\nfk1;k2;\u0001\u0001\u0001;ki\u00001;ki+1;\u0001\u0001\u0001;kNg;fk1;k2;\u0001\u0001\u0001;kj\u00001;kj+1;\u0001\u0001\u0001;kNg: (S.37)\nThe corresponding quantum numbers of these two sets of k's read\nI=fI1;I2;\u0001\u0001\u0001;Ii\u00001;Ii+1;\u0001\u0001\u0001;INg;J=fJ1;J2;\u0001\u0001\u0001;Jj\u00001;Jj+1;\u0001\u0001\u0001;JNg: (S.38)\nFor \fxing the \frst set of quantum numbers I, there are ( N\u00001)! permutations Qfor the second set J. The summation\nofQin Eq. (S.35) is carried out for ( N\u00001)! permutations. Each of these permutations gives a set of d's\nfd0\n1;\u0001\u0001\u0001;d0\nN\u00001g=I\u0000QJ: (S.39)6\nAssuming that there are mzero elements in the set fd0\n1;\u0001\u0001\u0001g we haven=N\u00001\u0000mnonzerod's. We denote the\nnonzero ones asfd1;d2;\u0001\u0001\u0001;dng. The function fm(x) introduced in (S.35) is derived explicitly:\nfm(x) =N\u00001X\n\u001c=0w\u001b\n\u001c(\u00001)\u001c\u001cX\nt=0Cj\nmC\u001c\u0000t\nn(L\u0000x)t(\u0000x)m\u0000t; (S.40)\nwhereCa\nbstands for the combinatory. This is the key simpli\fcation of the momentum distribution (S.34).\nIn the main text, we presented the interaction energy of the ferro- and non-ferromagnetic states, calculated through\nthe corresponding momentum distribution functions. For the antiferromagnetic state, the momentum distribution\nfunction (S.35) can be calculated by using the corresponding ground state wave function of the XXX spin chain model,\ndenoted asAanti. We performed our calculation from the Bethe Ansatz roots of the XXX spin chain with a \fnite size\nand \fnite number of down-spins.\nThe paramagnetic state is more peculiar. It is a mixed state which consists of all possible spin states classi\fed by\nthe total spin of the system [S6, S7]. We denote the ferromagnetic state of the XXX spin chain as A(m)\nferro/(^S\u0000)mj0i\nwithm= 0;1;\u0001\u0001\u0001N. Here the state j0i=j\";\";\u0001\u0001\u0001;\"iis the highest state and the total spin operator is given by\n^S\u0000=^S\u0000\n1+^S\u0000\n2+\u0001\u0001\u0001+^S\u0000\nN. Each state A(m)\nferrogives rise to the density matrix n(m)\nferro(x). Here we chose the state\nwith zero spin projection, i.e., Sz= 0, as the ferromagnetic state. In the main text, the momentum distribution\nof the ferromagnetic state refers to n(N=2)\nferro(x). For the paramagnetic state, we have to consider a mixed state that\ncontains all possible spin con\fgurations for the spin chain, i.e. using all the eigenstates Amwithm= 1;2;\u0001\u0001\u00012N.\nThe corresponding density matrix is denoted as nm(x). Then the density matrix for the paramagnetic state is equally\nweighted as npara(x) =P\nmnm(x)=2N.\n[S1] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory (Butterworth-Heinemann, Oxford, 1999).\n[S2] L. Pricoupenko, Phys. Rev. Lett. 100, 170404 (2008).\n[S3] M. Ogata, H. Shiba, Phys. Rev. B 41, 2326 (1990).\n[S4] M. Gaudin, Phys. Lett. A 24, 55 (1967).\n[S5] C.-N.Yang, Phys. Rev. Lett. 19, 1312 (1967).\n[S6] E. Eisenberg and E. H. Lieb, Phys. Rev. Lett. 89, 220403 (2002).\n[S7] X.-W. Guan, M.T. Batchelor, and M. Takahashi, Phys. Rev. A 76, 043617 (2007)."    },    {        "title": "1808.03966v2.Resonant_Driving_induced_Ferromagnetism_in_the_Fermi_Hubbard_Model.pdf",        "content": "Resonant Driving induced Ferromagnetism in the Fermi Hubbard Model\nNing Sun,1Pengfei Zhang,1and Hui Zhai1, 2\n1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China\n2Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China\n(Dated: September 18, 2018)\nIn this letter we consider quantum phases and the phase diagram of a Fermi Hubbard model under periodic\ndriving that has been realized in recent cold atom experiments, in particular, when the driving frequency is\nresonant with the interaction energy. Due to the resonant driving, the e ffective Hamiltonian contains a correlated\nhopping term where the density occupation strongly modifies the hopping strength. Focusing on half filling, in\naddition to the charge and spin density wave phases, large regions of ferromagnetic phase and phase separation\nare discovered in the weakly interacting regime. The mechanism of this ferromagnetism is attributed to the\ncorrelated hopping because the hopping strength within a ferromagnetic domain is normalized to a larger value\nthan the hopping strength across the domain. Thus, the kinetic energy favors a large ferromagnetic domain\nand consequently drives the system into a ferromagnetic phase. We note that this is a di fferent mechanism\nin contrast to the well-known Stoner mechanism for ferromagnetism where the ferromagnetism is driven by\ninteraction energy.\nRecently, with the help of quantum gas microscope for\nfermions [1–6], tremendous experimental progresses have\nbeen made on quantum simulation of the Fermi Hubbard\nmodel. These progresses include the observation of equilib-\nrium properties such as short-range antiferromagnetic corre-\nlations [7–9], hidden antiferromagnetic correlations [10], in-\ncommensurate spin correlations [11], canted antiferromag-\nnetic correlations [12] and pairing correlations [13] in sev-\neral di fferent circumstances. In particular, the antiferromag-\nnetic quasi-long-range order has been successfully observed\nthrough entropy engineering [14]. These progresses also in-\nclude the study of non-equilibrium transport behaviors such as\nthe measurement of optical conductivity [15], and the spin and\ncharge transport behavior in the strongly interacting regime\n[16, 17].\nStudying Fermi Hubbard model with cold atoms also al-\nlows us to open up new avenue beyond the traditional con-\ndensed matter paradigm. One of such examples is the period-\nically driven Fermi Hubbard model [18, 19]. Since the typ-\nical parameters of a Hubbard model is the hopping strength\nJand the on-site interaction U, both of which are of the or-\nder of electron volt in strongly correlated solid-state materi-\nals, it is therefore hard to drive a solid-state material with fre-\nquency resonant with any of these two energy scales. How-\never, in cold-atom optical lattice realization of the Fermi Hub-\nbard model, the typical energy scales for these two parameters\nare both of the order of thousand Hertz, and it is quite easy\nto drive the optical lattices with such a frequency. When the\ndriving frequency is resonant with the interaction parameter\nU, the driving can strongly modify the Fermi Hubbard model.\nAs observed in a recent experiment from the ETH group, the\nshort-range antiferromagnetic correlation can be reduced, or\nenhanced, or even switch sign to become ferromagnetic cor-\nrelation [20]. Similar experiment has also been performed by\ndriving the Hubbard model with two-photon Raman transition\n[21]. Hence, by combining such a resonant driving with the\nquantum gas microscope, it is very promising to study novel\nphysics induced by periodic driving that cannot be accessed\nJ\nUShaking:A,!\n<latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit>Shaking:A,!\n<latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit>FIG. 1: A schematic of the Fermi Hubbard model on a two-\ndimensional square lattice. Jdenotes the hopping strength. Uis the\non-site interaction. The system is time-periodically modulated by\nshaking the lattice in a sinusoidal way with frequency ωand ampli-\ntude A. Arrows indicate the direction of shaking. Balls with di fferent\ncolors and arrows indicate fermions with di fferent spins.\nin a static system. The goal of this letter is therefore to pre-\ndict quantum phases and phase diagram of the resonant driven\nFermi Hubbard model that is newly realized in cold atom ex-\nperiments.\nModel. We consider a two-dimensional square lattice un-\nder similar driving as realized in experiments [20, 28]. The\nlattice is periodically modulated along the ˆ xand ˆydirections\nwith a frequency ωand an amplitude A, whose single particle\nHamiltonian can be written as\nˆH0(t)=p2\n2m+ˆV(x+Acos(ωt),y+Acos(ωt)) (1)\nwhere mis the mass of an atom. We can now preform a unitary\ntransformation [22]\nˆU(t)=exp(ip·r0(t)), (2)arXiv:1808.03966v2  [cond-mat.quant-gas]  17 Sep 20182\nwhere r0(t)=−Acos(ωt)(1,1) are two-dimensional vectors.\nThis unitary transformation transfers position r0(t) into the\ncomoving frame where the lattice becomes static but an extra\ntime-dependent gauge field is introduce. The resulting Hamil-\ntonian is written as\nˆH0(t)=(p−A(t))2\n2m+ˆV(r), (3)\nwith A(t)=mr0(t). In principle, the Hamiltonian Eq. 3 is\nequivalent to the one Eq. 1 but it is more convenient for later\npurpose.\nNow we consider a single-band tight-binding model with\nthe nearest neighboring tunneling coe fficient Jand on-site\nHubbard interaction strength U. The Hamiltonian in a sec-\nond quantized form can be written by the Peierls substitution\nas\nˆH(t)=−J/summationdisplay\n/angbracketlefti,j/angbracketright\nσ=↑,↓eidij·A(t)ˆc†\niˆcj+U/summationdisplay\ni/parenleftBigg\nˆni↑−1\n2/parenrightBigg/parenleftBigg\nˆni↓−1\n2/parenrightBigg\n.(4)\nwhere ˆ ciσ(ˆc†\niσ) is the fermionic annihilation (creation) opera-\ntor on site iwith spinσ, ˆniσis the density operator on site i\nwith spinσ,/angbracketleft.../angbracketrightdenotes the nearest neighboring sites, and\ndij=di−djwithdibeing the position of the ith lattice site.\nThroughout this work we focus on the half-filling case and the\nchemical potential is set to zero.\nIf the modulation frequency ωis the largest energy scale\nof the problem, one can make a high-frequency expansion\nto obtain an e ffective time-independent Hamiltonian [23, 24].\nThe e ffective Hamiltonian takes the same form as the normal\nHubbard model and the only modification is that the tunnel-\ning coe fficient is renormalized by the oscillating gauge field\nas˜J=JB0(A), where we useBlto denote the lth Bessel\nfunction andA=mAωdis the normalized shaking amplitude\nhereinafter. dis the distance of two Wannier wave packets in\nthe nearest neighboring lattice sites.\nHowever, this expansion falls down when the modulation\nfrequencyω, or lth multiple of it, is comparable to one of\nthe energy scale of the problem, say, the Hubbard interaction\nstrength U. That is to say, l/planckover2pi1ω≈U, and we call it the lth\nresonance. Note that in this case, because U−l/planckover2pi1ωis a small\nenergy scale, we should apply another unitary transformation\nˆR(t)=exp(i/summationdisplay\njlωtˆnj↑ˆnj↓), (5)\nwhich alters the interaction strength to an e ffective one ˜U=\nU−l/planckover2pi1ω. Moreover, since ˆR(t) does not commute with the\nhopping term, it introduces an additional density dependence\nto the hopping term, and e ffectively it changes the gauge field\nto a spin and density dependent one as\n˜Ai j,σ(t)=A(t)−lωt\nd2dij((1−ˆni¯σ)ˆnj¯σ−(1−ˆnj¯σ)ˆni¯σ)).(6)\nNow the high frequency expansion can be safely applied, and\nto the lowest order it again results in a time-independent ef-fective Hamiltonian written as\nˆHeff=/summationdisplay\n/angbracketlefti,j/angbracketright,σ−ˆJ/angbracketlefti j/angbracketright\neff,σˆc†\niσˆcjσ+˜U/summationdisplay\ni/parenleftBigg\nˆni↑−1\n2/parenrightBigg/parenleftBigg\nˆni↓−1\n2/parenrightBigg\n.(7)\nHere ˆJ/angbracketlefti j/angbracketright\neff,σis defined as\nˆJ/angbracketlefti j/angbracketright\neff,σ=J0ˆai j¯σ+J1ˆbi j¯σ, (8)\nwhere ¯σdenotes the complement of σ,J0=JB0(A),J1=\nJBl(ηi jA) (ηi j=±1 for ( ix,iy)=(jx±1,jy) or ( ix,iy=\njx,jy±1)), and\nˆai jσ=(1−ˆniσ)(1−ˆnjσ)+ˆniσˆnjσ, (9)\nˆbi jσ=(−1)l(1−ˆniσ)ˆnjσ+ˆniσ(1−ˆnjσ). (10)\nIn above, the site dependence of J1is made implicitly. Note,\nhowever, that for even lthe Bessel function Blis an even func-\ntion, in which case ηi jcan be simply dropped and J1becomes\na constant. Compared to the o ff-resonance case, now the hop-\nping strength depends on site occupation of fermions. As we\nwill show below, this correlated hopping plays a key role in\nthe emergent new mechanism for ferromagnetism phase.\nSymmetry. Before we discuss how to solve this e ffective\nHamiltonian, let us first comment on the symmetry of this\nproblem. Note that the original Hubbard model possesses a\nSO(4) symmetry [25], which is composed of a spin SU(2),\ngenerated by ˆSz=(1/2)/summationtext\niˆc†\ni↑ˆci↑−ˆci↓ˆci↓,ˆS+=/summationtext\niˆc†\ni↑ˆci↓\nand ˆS−=ˆS†\n+, and a charge SU(2), generated by ˆLz=\n−(1/2)/summationtext\niˆc†\ni↑ˆci↑+ˆc†\ni↓ˆci↓+Ns/2,ˆL+=/summationtext\ni(−1)iˆci↑ˆci↓and ˆL−=\nˆL†\n+.Nsis the total number of sites. The spin SU(2) ensures\nthat the direction of spin-density-wave (SDW) order parame-\nter can be taken along any direction, while the charge SU(2)\nensures the degeneracy of a charge-density-wave (CDW) or-\nder and the fermion pairing order (P).\nIn the presence of periodic modulation, considering the\ntime-dependent Hamiltonian Eq. 4, it is straightforward to\nshow that the spin SU(2) symmetry stays, yet the charge\nSU(2) symmetry no longer holds because ˆLzdoes not com-\nmute with the/summationtext\ni,σfi(t)ˆniσterm. However, considering the\ntime-independent e ffective Hamiltonian Eq. 7, one can show\nthat the charge SU(2) symmetry is recovered for even lcase\nthough not for odd lcase [26]. Hereafter we focus only on the\neven lcase which possesses the same SO(4) symmetry as the\noriginal Hubbard model. In addition, the e ffective Hamilto-\nnian also possesses particle-hole symmetry at half-filling.\nPhase Diagram. We present our results on the phase dia-\ngram following from a standard mean-field treatment based on\nthis e ffective Hamiltonian Eq. 7, which is known to be quali-\ntatively reliable for a normal Hubbard model [26, 27]. Thanks\nto the SO(4) symmetry, we can choose SDW along ˆ zdirection\n(i.e. si=/angbracketleftˆni↑−ˆni↓/angbracketright) and CDW (i.e. ci=/angbracketleftˆni↑+ˆni↓/angbracketright−1) as the\norder parameters in our mean-field theory. Note that when we\nobtain the CDW order, it means that the system can have either\nCDW order or fermion pairing order, or an arbitrary combina-\ntion of them, as the order parameter of the degenerate ground3\n-� -� -� � � � �-�-���\n�/����/��\n���/� ������/�\n+����� FM1\n2\n3\nFIG. 2: Phase diagram for the e ffective Hamiltonian Eq. 7 of a\nresonantly driven Fermi Hubbard model with even lat half filling.\nThe phase diagram is controlled by two dimensionless parameters,\nJ1/J0and ˜U/J0. “CDW” and “SDW” denote charge and spin den-\nsity wave order. “P” denote fermion pairing order. “FM” denotes\nferromagnetism, “PS” denotes phase separation into high and low\ndensity regimes. Red lines denote the first order transition and the\nblue dashes lines denote the second order transition. Dashed arrows\nmark the lines along which the order parameters are plotted in FIG.\n3.\nstates. Higher order e ffect will break the degeneracy between\nCDW and fermion pairing order, but it is beyond the scope of\ncurrent work.\nThe phase diagram is shown in Fig. 2. Setting J0as the\nenergy unit, the phase diagram is controlled by two param-\neters of J1/J0and ˜U/J0, both of which can be easily tuned\nfrom positive to negative via control of ωandA. As breach-\nmark of our calculation, first of all, note that when J1/J0=1,\nbecause ˆ ai jσ+ˆbi jσ=ˆI, the kinetic energy term becomes\n−J0/summationtext\n/angbracketlefti j/angbracketright,σˆc†\niσˆcjσand the Hamiltonian recovers the usual Hub-\nbard model. In this case (labeled by 2 in Fig. 2), the result\nfor the normal Hubbard model is retrieved where we obtain a\nCDW order of ( π,π) with attractive interaction ( ˜U<0) and a\nSDW order of ( π,π) with repulsive interaction ( ˜U>0). Ex-\nplicitly, the order parameters are chosen as si=(−1)ix+iysand\nci=(−1)ix+iyc, and s(c) gradually vanishes as ˜Uapproaches\nzero from the positive (negative) side. As a result, a second or-\nder phase transition occur at ˜U=0 (gray dot in FIG. 2). This\ncan be seen from the order parameters plotted as a function of\n˜U, shown as green curves in Fig. 3.\nSince both CDW and SDW are ordered phases, the more\ngeneric situation should be either a first order transition or a\nphase co-existence regime in between. As marked by the red\nsolid lines in Fig. 2, the phase boundary of CDW and SDW\n��/��=�����/��=�����/��=-���\n-�-�-� �������������������\n�/������� ���������\nSolid: SDW\nDahsed: CDWFIG. 3: The CDW order parameter c(dashed lines) and the SDW\norder parameter s(solid lines) as a function of ˜Ufor three represen-\ntative cases labeled as 1-3 in Fig. 2, with J1/J0=1.5, 1.0 and−2.0,\nrespectively.\nat large|J1/J0|is a first order transition. The order parame-\nters are shown with orange lines in Fig. 3 for a representative\ncase (labeled by 1 in Fig. 2), where the CDW or SDW order\nparameter jumps from a finite value to zero at ˜U=0. At cer-\ntain regime of J1/J0, a CDW and SDW co-existence regime\nshows up in between as displayed by the yellow regime in Fig.\n2. The order parameters are shown with blue lines in Fig. 3\nfor a representative case (labeled by 3 in Fig. 2).\nThe most notable feature in Fig. 2 is the green region. In\nthis region a mean-field ansatz of CDW or SDW orders with\nordering vector at ( π,π) may not yield any ordered solution.\nHowever, when we consider the case of enlarged 2 ×2, 3×3,\nup to L×Ldomains, and within each domain the CDW and\nSDW order parameters are uniformly chosen as candswhile\nin its neighboring domain they are taken as −cand−s, the\nmean-field ansatz does yield ordered solutions. It can be seen\nfrom Fig. 4 which shows that the mean-field ground state en-\nergy decreases monotonically as Lincreases. It indicates that\nthe ground state will form large domains with opposite order\nparameter values. Moreover, minimizing ground state energy\nyields c=0 and s/nequal0 at positive ˜Uandc/nequal0 and s=0\nat negative ˜U. Hence, the system at positive ˜Upossesses\nspin order, and the increasing of the domain size means the\ndecreasing of the spin ordering wave vector. Eventually, the\nwave vector decreases toward zero, and the ground state be-\ncomes a ferromagnetic state. In another word, as the domain\nsize becomes larger and larger, the system is essentially made\nof ferromagnetic domains. For negative ˜Uthe system tends\nto phase separation with high density in one domain and low\ndensity in its neighboring domain. The transition between the\nferromagnetic phase to the SDW phase, as well as the transi-\ntion from phase separation to CDW, is a first order transition.\nIt should be emphasized that both the ferromagnetism and\nthe phase separation regime occur at small ˜U. In fact, it is\npurely due to the correlated hopping e ffect in the e ffective4\nFIG. 4: The mean-field ground state energy as a function of the do-\nmain size L. The insets show the configuration for 2 ×2, 3×3 and\n4×4 blocks where the CDW and SDW order parameters are uni-\nformly distributed within a domain and take opposite values between\nneighboring domains. For comparison, the solid line is the mean-\nfield energy when order parameters are all zero.\nHamiltonian Eq. 7, which originates essentially from the res-\nonant driving. Considering the mean-field configurations as\nshown in the insets of Fig. 4, let us look at the mean-field\nvalue of the e ffective hopping strength /angbracketleftˆJ/angbracketlefti j/angbracketright\neff,σ/angbracketrightwhich quanti-\nfies how the particle occupations a ffect the hopping strength\nand thus a ffect the bandwidth. We define Jintra\neff,σas/angbracketleftˆJ/angbracketlefti j/angbracketright\neff,σ/angbracketrightwith\nboth iand jin the same domain, and Jinter\neff,σas/angbracketleftˆJ/angbracketlefti j/angbracketright\neff,σ/angbracketrightwith i\nand jacross two neighboring domains. It is straightforward\nto write down both Jintra\neff,σandJinter\neff,σas\nJintra\neff,σ=1\n2/parenleftBig\nJ0[1+(c∓s)2]+J1[1−(c∓s)2]/parenrightBig\n, (11)\nJinter\neff,σ=1\n2/parenleftBig\nJ0[1−(c∓s)2]+J1[1+(c∓s)2]/parenrightBig\n, (12)\nwhere∓corresponds to di fferent spin component. One can\nshow that when|J1|<|J0|,|Jinter\neff,σ|is always smaller than |Jintra\neff,σ|.\nHence, the size of the domain tends to increase such that\nthere are more intra-domain links than inter-domain links, and\ntherefore the e ffective bandwidth on average becomes larger.\nFor a given filling, a larger bandwidth leads to more kinetic\nenergy gain.|J1|<|J0|is precisely the regime where we find\nferromagnetism or phase separation in the phase diagram of\nFig. 3 with arbitary weak interaction. This regime can be\neasily accessed when Ais small.\nAn alternative way to understand the emergence of this fer-\nromagnetism is to consider a uniform system with ˜U=0,\nwhere the Hamiltonian contains only the correlated tunneling\nterm. Substitute ˆJ/angbracketlefti j/angbracketright\neff,σby its mean-field value, it is straight-\nforward to compute the kinetic energy of this uniform sys-\ntem that depends on ni↑andni↓, where n↑=(n+sz)/2 and\nn↓=(n−sz)/2. We plot the kinetic energy as a function of\nnforsz=0 in Fig. 5(a) and as a function of szforn=1\nin Fig. 5(b) for two representative cases with J1/J0=−0.8\nand−1.8. One can see from Fig. 5(a) that for J1/J0=−0.8,\nthere are two local minima with one at n>1 and the other at\nn<1, who locate symmetrically on two sides of n=1, while\n��/��=-�����/��=-���\n��� ��� ��� ��� ���-���-���-���-���-����-���-������\n�������� �������(� ↑+�↓)��������� ������ℰ(�)/� �\n��� ��� ��� ��� ���-���-���-���-���-���-���-������\n�������� �������(� ↑+�↓)��������� ������ℰ(�)/� �FIG. 5: The mean-field energy with ˜U=0 (a) as a function of total\ndensity nwith sz=0 fixed, and (b) as a function of szwith n=1\nfixed. Solid line is for J1/J0=−0.8 and the dashed line is for J1/J0=\n−1.8.\nforJ1/J0=−1.8 there is only one minimum located at n=1.\nSimilarly, in Fig. 5(b) for J1/J0=−0.8, there are two lo-\ncal minima with one at positive szand the other at negative sz,\nsymmetrically distributed around sz=0, and for J1/J0=−1.8\nthere is only one minimum at sz=0. Thus, when the system\nis constrained with the average n=1 and sz=0, for the\ncase with J1/J0=−0.8, it will actually phase separate into\ndomains with either di fferent nor different sz, corresponding\nto phase separation and ferromagnetism, respectively. The\nchoice is made by the sign of ˜Uwhen a small but finite ˜U\nperturbation is turned on.\nConclusion and Outlook. The most significant finding of\nthis work is to provide an alternative mechanism for the on-\nset of ferromagnetism in the model with correlated hopping,\nwhich roots in the cooperation between the spin order and the\ncorrelated hopping. It is driven by the kinetic energy term, and\noccurs in the weakly interacting regime. This is in contrast to\nthe well-known Stoner ferromagnetism mechanism which is\ndriven by large interaction energy and occurs when the inter-\naction strength is beyond certain critical value. This is also\ndifferent from the ferromagnetism due to the super-exchange\nprocesses discussed in the experiment of Ref. [20] which re-\nquires ˜Uto be negative.\nFinally we shall comment on the experimental observation\nof this ferromagnetism. First of all, the system itself has been\nrealized with cold atoms, and moreover, a very recent experi-\nment shows that the heating is insignificant in the presence of\ndriving and the life time of the system can be about one sec-\nond [28]. Second, because this ferromagnetism is driven by\nthe kinetic energy, and as one can see from Fig. 5, the energy\ngain is of the order of bandwidth, thus one expects this fer-\nromagnetism to be observed when temperature is of the order\nof bandwidth, which can be accessed now by cold atom ex-\nperiments. Thirdly, the quantum gas microscope techniques\nmentioned at the beginning can be used to detect real space\nferromagnetic domains. Hence, it is quite promising to verify\nthis theory experimentally in very near future.\nAcknowledgment. This work is supported MOST un-\nder Grant No. 2016YFA0301600 and NSFC Grant No.5\n11734010.\n[1] E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G.\nD. Bruce and S. Kuhr, Single-atom imaging of fermions in a\nquantum-gas microscope , Nature Physics 11, 738 (2015).\n[2] L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V . V . Ra-\nmasesh, W. S. Bakr, T. Lompe, and M. W. Zwierlein, Quantum-\nGas Microscope for Fermionic Atoms , Phys. Rev. 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Gross, Direct observation of incommen-\nsurate magnetism in Hubbard chains , arXiv: 1803.08892.\n[12] P. T. Brown, D. Mitra, E. Guardado-Sanchez, P. Schaub, S. S.\nKondov, E. Khatami, T. Paiva, N. Trivedi, D. A. Huse and W. S.\nBakr, Spin-imbalance in a 2D Fermi-Hubbard system , Science\n357, 1385-1388 (2017).[13] D. Mitra, P. T. Brown, E. Guardado-Sanchez, S. S. Kondov,\nT. Devakul, D. A. Huse, P. Schaub and W. S. Bakr, Quantum\ngas microscopy of an attractive Fermi-Hubbard system , Nature\nPhysics 14, 173 (2018).\n[14] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanasz-\nNagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif and M.\nGreiner, A cold-atom Fermi-Hubbard antiferromagnet , Nature\n545, 462-466 (2017).\n[15] R. Anderson, F. Wang, P. Xu, V . Venu, S. Trotzky, F. Chevy,\nand J. H. Thywissen, Optical conductivity of a quantum gas ,\narXiv:1712.09965.\n[16] M. A. Nichols, L. W. Cheuk, M. Okan, T. R. Hartke, E.\nMendez, T. Senthil, E. Khatami, H. Zhang, and M. W. 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Esslinger, Floquet dynamics in driven Fermi-\nHubbard systems , arXiv:1808.00506.Supplementary Material: Resonant Driving induced Ferromagnetism in the Fermi Hubbard Model\nNing Sun,1Pengfei Zhang,1and Hui Zhai1, 2\n1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China\n2Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China\n(Dated: September 18, 2018)\nIn this supplementary material, we first discuss the symmetry of the resonantly driven Fermi Hubbard model.\nThe SO(4) symmetry [1] of the e ffective Hamiltonian for even lis proved. Then we establish the mean-field\ntheory adopted in this work for even l, which is consistent with the text-book results [2] for J0=J1.\nSYMMETRY\nIn this section we dicuss the symmetry of the resonantly driven Fermi Hubbard model. The symmetry is of vital significance\nsince it greatly simplifies the formulation of the mean-field description.\nSO(4) symmetry.— A usual bipartite Fermi-Hubbard model, written as ˆH=−J/summationtext\n/angbracketlefti,j/angbracketright,σˆc†\niσˆcjσ+˜U/summationtext\ni(ˆni↑−1/2)(ˆni↓−1/2),\npossesses SO(4) symmetry [1]. The SO(4) symmetry is resolved into two SU(2) symmetries, the spin SU(2) and the charge\nSU(2). The generators of spin SU(2) are given by:\nˆSz=1\n2/summationdisplay\niˆc†\ni↑ˆci↑−ˆci↓ˆci↓,ˆS+=/summationdisplay\niˆc†\ni↑ˆci↓, (1)\nand\nˆS−=ˆS†\n+,ˆSx=ˆS++ˆS−\n2,ˆSy=ˆS+−ˆS−\n2i, (2)\nwho satisfy the commutation relation of the SU(2) algebra. Due to the contraction between spin indices, the very beginning\ntime-dependent Hamiltonian ˆH(t) (Eq.(4) in the main text) is invariant under this SU(2) symmetry operations, is also the unitary\ntransformations ˆR(t) (Eq.(5) and (6) in the main text). Since the time average does not alter this attribute, one arrives at the\nconclusion that the e ffective static Hamiltonian ˆHeffin Eq.(7) of the main text also possess this spin SU(2) symmetry no matter l\neven or odd. This can also be verified directly by checking/bracketleftBigˆHeff,ˆSα/bracketrightBig\n=0 for anyα=x,y,z. The spin rotational symmetry hence\nallows us to automatically get a spin-balanced system without any additional magnetic field.\nHowever, regarding the charge SU(2) symmetry, the time-dependent Hamiltonian lacks it for the appearance of the time-\ndependent onsite energy term (the last term in Eq.(4) of the main text). As a result we do not expect the SO(4) symmetry in\ngeneral. Nevertheless, it emerges in the e ffective Hamiltonian with leven. We introduce the charge SU(2) as follows [1].\nˆLz=−1\n2/summationdisplay\niˆc†\ni↑ˆci↑+ˆc†\ni↓ˆci↓+1\n2N, (3)\nˆL+=/summationdisplay\ni(−1)iˆci↑ˆci↓=/summationdisplay\niexp(iQ·xi)ˆci↑ˆci↓ (4)\nwhere Q=(π,π),Nis the total number of lattice sites hereinafter, and\nˆL−=ˆL†\n+,ˆLx=ˆL++ˆL−\n2,ˆLy=ˆL−−ˆL−\n2i. (5)\n{ˆLz,ˆLx,ˆLy}forms an SU(2) algebra. It can be verified straightforwardly that the e ffective Hamiltonian ˆHeffwith leven commutes\nwith all these generators. This fact can also be seen from the following transformation\nˆP: ˆci↑→(−1)iˆc†\ni↑,\nwho maps ˆHeffwith leven of interaction +˜Uto the same class but of e ffective interaction−˜U, exchanging the role played by {Li}\nand{Si}. As a result, the spin SU(2) invariance for a model with −˜Uindicates the charge SU(2) symmetry for +˜U. And vice\nversa. In addition, this transformation also implies the phase diagram should be symmetric under +˜U↔− ˜Utogether with the\ninterchange of charge and spin order. However, in the lodd cases, this mapping falls down because of the additional minus sign\ninˆbi jσ.arXiv:1808.03966v2  [cond-mat.quant-gas]  17 Sep 20182\nParticle-hole symmetry.— The half-filled e ffective Hamiltonian ˆHeffholds particle-hole symmetry no matter leven or odd.\n(i) When lis even, define particle-hole transformation ˆC: ˆciσ→(−1)iˆc†\niσ. The Hamiltonian is invariant under ˆC. [ˆC,ˆHeff]=0.\n(ii) When lis odd, we further define the bipartite transformation ˆS, which switches the A/Bsublattices. Then the Hamiltonian\nis invariant under the combination of ˆCand ˆS: [ˆCˆS,ˆHeff]=0. This symmetry allow us to fix chemical potential µ=0 during\nthe mean-field calculation at half-filling situation.\nMEAN-FIELD TREATMENT\nIn this section we establish the mean-field theory adopted in this work. As explained in the main text, assuming no canted\norder, we can consider only the charge density wave (CDW) order and spin density wave (SDW) order in the zdirection because\nof the SO(4) symmetry. We focus on the half-filled spin-balanced system, in which case the total charge density and total\nmagnetic momentum is given by\n/angbracketleftˆn/angbracketright=1,/angbracketleftˆS/angbracketright=0. (6)\nPath integral approach.— Here we provide a derivation of the mean-field Hamiltonian Eq. (19) via a path integral approach.\nIn the path integral language, the real-time partition function of the system is given by:\nZ=/integraldisplay\nDψD¯ψexp(i/integraldisplay\ndtL) (7)\nL=/summationdisplay\ni,σi¯ψiσ∂tψiσ+/summationdisplay\n/angbracketlefti,j/angbracketright,σ/parenleftBig\nJ0ai j¯σ(¯ψψ)+J1bl\ni j¯σ(¯ψψ)/parenrightBig¯ψiσψjσ\n−˜U/summationdisplay\ni¯ψi↑ψi↑¯ψi↓ψi↓. (8)\nHereψcorresponds to the fermionic field operators, and ai j¯σandbl\ni j¯σare given by replacing the operators in ˆ ai j¯σandˆbl\ni j¯σwith\nfields. Since the e ffective Hamiltonian (Eq. (7) in the main text) contains six-fermion terms, the traditional decoupling based on\nHubbard-Stratonovich transformation does not applied directly. However, an alternative way of decoupling based on the similar\nspirit of Hubbard-Stratonovich transformation should be adopted here. Specifically, by inserting a delta function, we directly\nintroduce the auxiliary bosonic field:\nZ=/integraldisplay\nDψD¯ψDn/productdisplay\niσδ(ni,σ−¯ψi,σψi,σ) exp( i/integraldisplay\ndtL) (9)\nL=/summationdisplay\ni,σi¯ψiσ∂tψiσ+/summationdisplay\n/angbracketlefti,j/angbracketright,σ/parenleftBig\nJ0ai j¯σ(n)+J1bl\ni j¯σ(n)/parenrightBig¯ψiσψjσ\n−˜U/summationdisplay\nini↑ni↓. (10)\nThanks to the delta function inserted, one could replace all ψ†ψbynin the action. As an equivalence check, if we integrate out\nnfields first, we get our original action back. Now we introduce another field ηto absorb the delta function into an integral:\nZ=/integraldisplay\nDψD¯ψDnDηexp(i/integraldisplay\ndtL) (11)\nL=/summationdisplay\ni,σi¯ψiσ∂tψiσ+/summationdisplay\n/angbracketlefti,j/angbracketright,σ/parenleftBig\nJ0ai j¯σ(n)+J1bl\ni j¯σ(n)/parenrightBig¯ψiσψjσ\n−˜U/summationdisplay\nini↑ni↓−/summationdisplay\niσηiσ(ni,σ−¯ψi,σψi,σ) (12)\nAs a result, the fermion becomes quadratic. In general, one can integrate out all the fermions to get an e ffective action of wholly\nbosonic degrees of freedom. Whereas, the mean-field approximation is to say that all the bosonic fields will be replaced by their\nsaddle point solutions.\nMean-field Hamiltonian.— By doing a Legendre transformation and replacing the fermion bilinears with its expectation\nvalues, we obtain the mean-field Hamiltonian\nˆH=/summationdisplay\n/angbracketlefti,j/angbracketright,σ−/parenleftBig\nJ0ai j¯σ(n)+J1bl\ni j¯σ(n)/parenrightBigˆψ†\niσˆψjσ+˜U/summationdisplay\nini↑ni↓+/summationdisplay\niσηiσ(ni,σ−ˆψ†\ni,σˆψi,σ) (13)3\nFIG. 1: Schematic of the two cases. Left panel: case 1. Right panel: typical examples of case 2 when L=2 and L=3, respectively, as labeled.\nLattice sites of the same color have the same local particle density ciand local spin density si.\nwhere the following replacement is carried out:\nˆai jσ=(1−ˆniσ)(1−ˆnjσ)+ˆniσˆnjσ,→(1−niσ)(1−njσ)+niσnjσ (14)\nˆbi jσ=(−1)l(1−ˆniσ)ˆnjσ+ˆniσ(1−ˆnjσ)→(−1)l(1−niσ)njσ+niσ(1−njσ) (15)\nFor the half-filled spin-balanced situation, we focus on the following two cases here. Notice that, in a half-filled spin-balanced\nsystem, the local particle density can be expressed in terms of local charge density and local spin density as\nni↑=1\n2(1+ci+si),ni↓=1\n2(1+ci−si) (16)\nwhere/summationtext\nici=0 and/summationtext\nisi=0 . The two cases are then:\n1.ci=(−1)ix+iyc,si=(−1)ix+iys\n2.ci=(−1)⌈ix/L⌉+⌈iy/L⌉c,si=(−1)⌈ix/L⌉+⌈iy/L⌉s\nwhere candsare CDW and SDW order parameters introduced, and ⌈x⌉denotes the ceiling function of x. Substituting these two\nexpressions into the original mean-field Hamiltonian Eq.(13) will yield the mean-field Hamiltonian in each case.\nCase 1. The first case is actually the ( π,π) order, where the lattice is naturally divided into A/Bsublattices and the unit cell\nof the lattice consists of 2 sites, with one from Aand the other from B. The schematic is shown in FIG. 1 (Left).\nTo be specific, we explicit show here the workflow in the first case. By substitution,\nnA≡/angbracketleftˆni∈A/angbracketright=1+c,sA≡/angbracketleftˆSz\ni∈A/angbracketright=s/2, (17)\nnB≡/angbracketleftˆni∈B/angbracketright=1−c,sB≡/angbracketleftˆSz\ni∈B/angbracketright=−s/2, (18)\nand the mean-field Hamiltonian is explicitly written as:\nˆHmf=/summationdisplay\n/angbracketlefti,j/angbracketright,σ{−J0[(1−nA¯σ)(1−nB¯σ)+nA¯σnB¯σ]−J1[(1−nA¯σ)nB¯σ+nA¯σ(1−nB¯σ)]}ˆcA†\niσˆcB\njσ+H. C. +˜UN\n2(nA↑nA↓+nB↑nB↓)\n+ηc/bracketleftBigg\nc−(ˆnA↑+ˆnA↓)−(ˆnB↑+ˆnB↓)\n2/bracketrightBigg\n+ηs/bracketleftBigg\ns−(ˆnA↑−ˆnA↓)−(ˆnB↑−ˆnB↓)\n2/bracketrightBigg\n(19)\nwhere ˆ cA\niσ(ˆcB\niσ) are the annihilation operators of isite and spin σonA(B) sublattice, ˆ nµσ=1\nN/2/summationtext\ni∈µˆniσ,µ=AorB, and\nnµσ=/angbracketleftˆnµσ/angbracketright. Hereηcandηsare Lagrangian multipliers of the CDW and SDW orders, respectively, who are also variational\nparameters to optimize the ground-state energy of ˆHmfto get a mean-field solution.\nAfter some substitution and simplification, the mean-filed Hamiltonian in momentum space is written as\nˆHmf=/summationdisplay\nkσ−Pσ(c,s)Q(k)ˆcA†\nkσˆcB\nkσ+H.c.+˜UN\n2(1+c2−s2)\n+ηc/bracketleftBigg\nc−(ˆnA↑+ˆnA↓)−(ˆnB↑+ˆnB↓)\n2/bracketrightBigg\n+ηs/bracketleftBigg\ns−(ˆnA↑−ˆnA↓)−(ˆnB↑−ˆnB↓)\n2/bracketrightBigg\n(20)4\nwhere ˆ cA\nkσ(ˆcB\nkσ) are the annihilation operators on A(B) sublattice of quasi-momentum kand spinσ,Q(k)=/summationtext\niexp(ik·di),{di}\nare the lattice vectors, and\nP↑(c,s)=J0\n2/bracketleftBig\n1−(c−s)2/bracketrightBig\n+J1\n2/bracketleftBig\n1+(c−s)2/bracketrightBig\n(21)\nP↓(c,s)=J0\n2/bracketleftBig\n1−(c+s)2/bracketrightBig\n+J1\n2/bracketleftBig\n1+(c+s)2/bracketrightBig\n(22)\nThe summation of kin Eq.(20) is over the first Brillioun zone. Minimization the energy for all parameters {c,s,ηc,ηs}yields a\nset of mean-field equations that is solved by numerical iteration in this work.\nCase 2. The second case actually includes a series of circumstances of Lbeing 2,3,4,...,∞. Typical examples of L=2\nandL=3 are shown in the right panel of FIG. 1. In this case, for each independent L, doing the similar substitution as above and\nsolving optimization problem in a numerical way returns us a set of self-consistent mean-field solutions in each L. The ground\nstate should be the one with the lowest ground state energy.\nFIG. 2, 3, 4, 5 in the main text are based on the mean-field numerics of above two cases.\nCheck the formulation for J 0=J1.— Finally, we explain why our mean-field theory reduces to a traditional one appearing\nin common text books, e.g. [2]. In fact, the Pσin this case is just a constant with no candsdependence. Thus the variation of c\nandsyields:\nηc=−˜UNcηs=−˜UNs. (23)\nUsing this relation, we see the mean-field Hamiltonian reduces to a familiar one that appears in textbooks [2].\n[1] C. N. Yang and S. C. Zhang, SO4symmetry in a Hubbard model , Mod. Phys. Lett. B 4, 759 (1990).\n[2] A. Altland, and B. D. Simons. Condensed matter field theory, second edition, Cambridge University Press, 2010."    },    {        "title": "2104.11878v2.Theory_of_magnetic_inertial_dynamics_in_two_sublattice_ferromagnets.pdf",        "content": "Signatures of magnetic inertial dynamics in\ntwo-sublattice ferromagnets\nRitwik Mondal1;2\n1Department of Spintronics and Nanoelectronics, Institute of Physics ASCR, v.v.i.,\nCukrovarnická 10, Prague 6, 162 53, Czech Republic\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120\nUppsala, Sweden\nE-mail: mondal@fzu.cz\nAbstract. The magnetic inertial dynamics have been investigated for one sublattice\nferromagnets. Here, we develop the magnetization dynamics in two-sublattice\nferromagnets including the intra- and inter-sublattice inertial dynamics. First, we\nderive the magnetic susceptibility of such a ferromagnet. Next, by finding the poles of\nthe susceptibility, we calculate the precession and nutation resonance frequencies. Our\nresults suggest that while the resonance frequencies show decreasing behavior with\nthe increasing intra-sublattice relaxation time, the effect of inter-sublattice inertial\ndynamics is contrasting.\n1. Introduction\nUltrafast manipulation of electrons’ spin remains at the heart of future generation spin-\nbased memory technology [1–3]. It has been observed that a fs laser pulse is capable of\ndemagnetizing a ferromagnetic material [4–6]. On the other hand, using these ultrashort\npulses, magnetic switching has been reported in ferrimagnetic [7–9] and ferromagnetic\nmaterials [10, 11]. These observations have been explained through the spin dynamics\nwithin Landau-Lifshitz-Gilbert (LLG) equation of motion [12–15].\nThe phenomenological LLG spin dynamics consists of spin precession and a\ntransverse damping [16–18]. Such an equation of motion has been derived from a\nrelativistic Dirac theory, where the transverse damping is found to originate from spin-\norbit coupling [19–22]. However, at ultrashort timescales, the traditional LLG equation\nneeds to be supplemented by several other spin torque terms [23]. Especially, at the\nultrafast timescales, the magnetic inertia becomes particularly relevant [24]. The effect\nof magnetic inertia has been incorporated within extended LLG dynamics as a torque\ndue to the second-order time derivative of the magnetization M(r;t). The inertial LLG\n(ILLG) equation of motion reads [25–27]\n@M\n@t=M\u0002\u0014\n\u0000\rH+\u000b\nM0@M\n@t+\u0011\nM0@2M\n@t2\u0015\n; (1)arXiv:2104.11878v2  [cond-mat.other]  1 Sep 2021Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 2\nwhereM0andHdefine the ground state magnetization and an effective field,\nrespectively. The first and second terms in Eq. (1) represent the traditional LLG\nequation [18]. The inertial spin dynamics in the last term of Eq. (1) gives rise to\nthe spin nutation [28, 29]. The ILLG equation signifies the fact that the dynamics of\na magnetic moment shows precession with nutation at ultrafast timescales, followed by\ntransverse damping [24]. The ILLG equation has schematically been depicted in Fig. 1.\nA simple dimension analysis shows that the transverse damping is characterized by\na dimensionless parameter \u000b, and the inertial dynamics are strengthened by inertial\nrelaxation time \u0011. The ILLG dynamics have been derived within the relativistic Dirac\nM(r, t)H\nPrecession\nNutationDamping\nFigure 1: Schematic depiction of ILLG equation of motion.\nframework as well, where it shows that the Gilbert damping \u000band inertial relaxation\ntime\u0011are tensors [30]. In particular, the relativistic theory derives that the Gilbert\ndamping dynamics is associated with the imaginary part of the susceptibility, while the\ninertial dynamics is given by the real part [31]. Such findings are found to be consistent\nwith a linear response theory of ferromagnet [32]. The inertial dynamics have also been\nderived within classical mechanics of a current loop [33]. Eq. (1) has been applied to\na single sublattice ferromagnet beyond ferromagnetic resonance (FMR), observing an\nadditional peak due to nutation resonance [34–36]. While the FMR peak appears at the\nGHz regime, the nutation resonance peak appears at the THz regime [37]. The ILLG\nequation has also been applied to antiferromagnets and ferrimagnets, and it has been\npredicted that the spin nutation should be better detected in antiferromagnets as it is\nexchange enhanced [38].\nRecently, the spin nutation resonance has been observed for ferromagnets in the\nexperiment [39]. Indeed, the nutation resonance peak has been seen at around 0.5\nTHz. Note that the experiment was performed in two-sublattice ferromagnets namely\nCoFeB and NiFe. For two-sublattice ferromagnet, the inter-sublattice exchange energies\nbecome important. Here, we describe the inertial effects in a two-sublattice ferromagnet\ncoupled by the Heisenberg exchange interaction. We follow the similar procedureSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 3\nof Ref. [38] and derive the magnetic susceptibility. We not only consider the intra-\nsublattice inertial dynamics, but alsothe inter-sublattice dynamics. Our results suggest\nthat there are two precession resonance peaks: one at GHz regime and another at THz\nregime. Similarly, two nutation peaks can also be observed, both are at the THz regime.\nBy calculating the precession and nutation resonance frequencies, we observe that the\nresonance frequencies decrease with increasing intra-sublattice relaxation time, however,\nthe scenario is different for inter-sublattice inertial dynamics.\n2. Theory of intra- and inter-sublattice inertial dynamics in two-sublattice\nferromagnets\nThe inertial dynamics for antiferromagnets have been introduced in Ref. [38]. For\ntwo-sublattice magnetic systems having magnetization MAandMB, forAandB\nrepresenting the two-sublattice, the ILLG equations of motion can be recast as\n@MA\n@t=\u0000\rA(MA\u0002HA) +\u000bAA\nMA0\u0012\nMA\u0002@MA\n@t\u0013\n+\u000bAB\nMB0\u0012\nMA\u0002@MB\n@t\u0013\n+\u0011AA\nMA0\u0012\nMA\u0002@2MA\n@t2\u0013\n+\u0011AB\nMB0\u0012\nMA\u0002@2MB\n@t2\u0013\n(2)\n@MB\n@t=\u0000\rB(MB\u0002HB) +\u000bBB\nMB0\u0012\nMB\u0002@MB\n@t\u0013\n+\u000bBA\nMA0\u0012\nMB\u0002@MA\n@t\u0013\n+\u0011BB\nMB0\u0012\nMB\u0002@2MB\n@t2\u0013\n+\u0011BA\nMA0\u0012\nMB\u0002@2MA\n@t2\u0013\n(3)\nIn each ILLG dynamics, the first term represents the spin precession around an effective\nfieldHA=B. The intra- and inter-sublattice Gilbert damping dynamics have been\ndenoted by the second and third terms, respectively. Similarly, the last two terms define\ninertial dynamics. While the intra-sublattice Gilbert and inertial dynamics have been\nweighed by \u000bAA=BBand\u0011AA=BB, the same for inter-sublattice dynamics are denoted by\n\u000bAB=BAand\u0011AB=BA. From a simple dimension analysis, it is clear to show that the\nGilbert damping parameters \u000bare dimensionless, in contrast, the inertial relaxation\ntimes\u0011have a dimension of time [26, 30]. It is worth mentioning that the Gilbert\ndamping\u000bhas been calculated for several materials within ab initio frameworks [32, 40–\n52], while there are also proposals to calculate the inertial relaxation time within\nextended breathing Fermi surface model [53–55]. These ILLG equations have been\ncontemplatedtoforecastthesignaturesofinertialdynamicsincollinearantiferromagnets\nand ferrimagnets [38].\nWe consider that the two-sublattice ferromagnet is aligned collinear at the ground\nstate such that MA=MA0^zandMB=MB0^z. The ferromagnetic system is under\nthe application of an external Zeeman field H0=H0^z. Then, the free energy of the\nconsidered two-sublattice system can be considered as the sum of Zeeman, anisotropy,Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 4\nand exchange energies as\nF(MA;MB) =\u0000H0(MAz+MBz)\u0000KA\nM2\nA0M2\nAz\u0000KB\nM2\nB0M2\nBz\u0000J\nMA0MB0MA\u0001MB;\n(4)\nwhereKAandKBare anisotropy energies and Jis the isotropic Heisenberg exchange\nwithJ > 0for ferromagnetic coupling. To calculate the linear response properties of\nthe system, we consider that the small deviations of magnetization mA(t)andmB(t)\nwith respect to the ground state are induced by the transverse external field hA(t)and\nhB(t). We calculate the effective field in the ILLG equation as the derivative of free\nenergy in Eq. (4) to the corresponding magnetization\nHA=\u0000@F(MA;MB)\n@MA=\u0012\nH0+2KA\nM2\nA0MAz\u0013\n^z+J\nMA0MB0MB\n=1\nMA0(H0MA0+ 2KA+J)^z+J\nMA0MB0mB;(5)\nHB=\u0000@F(MA;MB)\n@MB=\u0012\nH0+2KB\nM2\nB0MBz\u0013\n^z+J\nMA0MB0MA\n=1\nMB0(H0MB0+ 2KB+J)^z+J\nMA0MB0mA:(6)\nWe then expand the magnetization around the ground state in small deviations,\nMA=MA0^z+mA(t)andMB=MB0^z+mB(t). Essentially, with the effective fields\nin Eqs. (5) and (6) along with the magnetization, the linear response for sublattice A\nprovides\n@mA\n@t=\u0000\rA\nMA0(H0MA0+ 2KA+J) [mAy^x\u0000mAx^y]\u0000\rAJ\nMB0[mBx^y\u0000mBy^x]\n\u0000\rAMA0[hAx^y\u0000hAy^x] +\u000bAA\u0014@mAx\n@t^y\u0000@mAy\n@t^x\u0015\n+\u000bABMA0\nMB0\u0014@mBx\n@t^y\u0000@mBy\n@t^x\u0015\n+\u0011AA\u0014@2mAx\n@t2^y\u0000@2mAy\n@t2^x\u0015\n+\u0011ABMA0\nMB0\u0014@2mBx\n@t2^y\u0000@2mBy\n@t2^x\u0015\n; (7)\nobtaining the dynamics for two components xandyas\n\rAMA0hAx=\rA\nMA0(H0MA0+ 2KA+J)mAx\u0000\rAJ\nMB0mBx+\u000bAA@mAx\n@t+\u000bABMA0\nMB0@mBx\n@t\n\u0000@mAy\n@t+\u0011AA@2mAx\n@t2+\u0011ABMA0\nMB0@2mBx\n@t2; (8)\n\rAMA0hAy=\rA\nMA0(H0MA0+ 2KA+J)mAy\u0000\rAJ\nMB0mBy+\u000bAA@mAy\n@t+\u000bABMA0\nMB0@mBy\n@t\n+@mAx\n@t+\u0011AA@2mAy\n@t2+\u0011ABMA0\nMB0@2mBy\n@t2: (9)\nIn the circular basis defined by mA\u0006=mAx\u0006imAyandhA\u0006=hAx\u0006ihAy, the equationsSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 5\ncan be put together\n\rAMA0hA\u0006=\rA\nMA0(H0MA0+ 2KA+J)mA\u0006\u0000\rAJ\nMB0mB\u0006+\u000bAA@mA\u0006\n@t+\u000bABMA0\nMB0@mB\u0006\n@t\n\u0006i@mA\u0007\n@t+\u0011AA@2mA\u0006\n@t2+\u0011ABMA0\nMB0@2mB\u0006\n@t2: (10)\nSimilarly, one can calculate the linear response of the sublattice B in the circular basis\ndefined bymB\u0006=mBx\u0006imByandhB\u0006=hBx\u0006ihByas\n\rBMB0hB\u0006=\rB\nMB0(H0MB0+ 2KB+J)mB\u0006\u0000\rBJ\nMA0mA\u0006+\u000bBB@mB\u0006\n@t+\u000bBAMB0\nMA0@mA\u0006\n@t\n\u0006i@mB\u0007\n@t+\u0011BB@2mB\u0006\n@t2+\u0011BAMB0\nMA0@2mA\u0006\n@t2: (11)\nWe define the response functions mA\u0006;mB\u0006;hA\u0006;hB\u0006/e\u0006i!tand \nA=\n\rA\nMA0(H0MA0+ 2KA+J)and \nB=\rB\nMB0(H0MB0+ 2KB+J). To simplify the\nexpressions, we introduce the following: \u0000AA=\rAMA0,\u0000BB=\rBMB0,\u0000AB=\rAMB0\nand\u0000BA=\rBMA0such that \u0000AA\u0000BB= \u0000AB\u0000BA. The linear response Eqs. (10) and\n(11) can be written in a matrix formalism\n\u0010hA\u0006hB\u0006\u0011\n=0\nBB@1\n\u0000AA(\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!)\u00001\n\u0000AB\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u0013\n\u00001\n\u0000BA\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u00131\n\u0000BB(\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!)1\nCCA\u0010mA\u0006mB\u0006\u0011\n:\n(12)\nFor finding the susceptibility, we recall m\u0006=\u001f\u0006\u0001h\u0006such that the susceptibility matrix\nderives as\n\u001fAB\n\u0006=1\nD\u00060\nBB@1\n\u0000BB(\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!)1\n\u0000BA\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u0013\n1\n\u0000AB\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u00131\n\u0000AA(\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!)1\nCCA;\n(13)\nwhere the determinant is expressed as\nD\u0006=1\n\u0000AA\u0000BB\u0000\n\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!\u0001\u0000\n\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!\u0001\n\u00001\n\u0000AB\u0000BA\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u0013\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u0013\n:(14)\nNote that the intra-sublattice dynamical parameters enter in the diagonal elements of\nthe susceptibility matrix, however, the inter-sublattice dynamics are reflected in the off-\ndiagonal elements. Such a susceptibility matrix has been obtained with intra- and inter-\nsublattice Gilbert damping dynamics for antiferromagnets [56]. To find the resonanceSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 6\nfrequencies, one has to solve the equation setting D\u0006= 0. Therefore, a fourth-order\nequation in frequency is obtained\nA\u0006!4+B\u0006!3+C\u0006!2+D\u0006!+E\u0006= 0; (15)\nwith the following coefficients\nA\u0006=\u0011AA\u0011BB\u0000\u0011AB\u0011BA; (16)\nB\u0006= (\u0011AA+\u0011BB)\u0007i (\u000bAA\u0011BB+\u000bBB\u0011AA)\u0006i (\u000bAB\u0011BA+\u000bBA\u0011AB); (17)\nC\u0006= 1\u0007i (\u000bAA+\u000bBB)\u0000(\nA\u0011BB+ \nB\u0011AA)\u0000\u000bAA\u000bBB\n\u0000\u0012\rA\nMA0\u0011BA+\rB\nMB0\u0011AB\u0013\nJ\u0000\u000bAB\u000bBA; (18)\nD\u0006=\u0000(\nA+ \nB)\u0006i (\nA\u000bBB+ \nB\u000bAA)\u0006i\u0012\rA\nMA0\u000bBA+\rB\nMB0\u000bAB\u0013\nJ;(19)\nE\u0006= \nA\nB\u0000\rA\rB\nMA0MB0J2: (20)\nTheanalyticalsolutionoftheabove-mentionedequationisverycumbersome. Therefore,\nwe adopt the numerical techniques for solving Eq. (15). The solution of the above\nequation results in four frequencies, two of them correspond to the precession resonance\n(!p)ofeachsublatticeandtheothertwobelongtothenutationresonance( !n). Thereal\nand imaginary parts of the resonance frequency are denoted by ReandIm, respectively.\nFor example, the precession resonance frequencies are !p=Re(!p) + i Im(!p), while\nthe nutation resonance frequencies are !n=Re(!n) + iIm(!n). Comparing Eq. (15), a\nsimilar equation has been obtained for antiferromagnets and ferrimagnets [38], however,\nwithout the inter-sublattice inertial dynamics. We mention that the inter-sublattice\nGilbert damping dynamics have extensively been discussed [56, 57]. Therefore, we will\nnot consider in the following discussions. In particular, we allow \u000bAB=\u000bBA= 0, and\ncalculate the inertial effects on precession and nutation resonances.\n3. Numerical results\nTo calculate the resonace frequencies, we numerically solve the Eq. (15) for two-\nsublattice ferromagnets having same magnetic moments in each sublattice i.e., MA0=\nMB0. We use the following parameters: \rA=\rB= 1:76\u00021011T\u00001-s\u00001,J= 10\u000021J,\nKA=KB= 10\u000023J,\u000bAA=\u000bBB=\u000b= 0:05,\u000bAB=\u000bBA= 0. Theconsideredexchange\nand anisotropy energies have similar order of magnitude as typical ferromagnets e.g., Fe\n[58]. The chosen Gilbert damping \u000b= 0:05is within the ab initio reported values [51].\nFor inertial relaxation times, even though, the ab initio calculation suggests about fs\ntimescales for transition metals [55], the recent experiment predicts it to be a higher\nvalue up to several hundreds of fs [39]. Therefore, in what follows, we have considered\nthe inertial relaxation times ranging from fs to ps.Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 7\n10−1100101102103\nη(fs)10−1100101102103ωFM\n±/2π(THz)\n1/η(a)\nRe/parenleftbig\nωu\np+/parenrightbig\nRe/parenleftbig\nωl\np+/parenrightbig\nRe/parenleftbig\nωu\nn−/parenrightbig\nRe/parenleftbig\nωl\nn−/parenrightbig\n10−1100101102103\nη(fs)0.000.010.020.030.040.05Im(ω±)/Re(ω±)|FM\n(b)\nIm/parenleftbig\nωu\np+/parenrightbig\n/Re/parenleftbig\nωu\np+/parenrightbig\nIm/parenleftbig\nωl\np+/parenrightbig\n/Re/parenleftbig\nωl\np+/parenrightbig\nIm/parenleftbig\nωu\nn−/parenrightbig\n/Re/parenleftbig\nωu\nn−/parenrightbig\nIm/parenleftbig\nωl\nn−/parenrightbig\n/Re/parenleftbig\nωl\nn−/parenrightbig\nFigure 2: The calculated resonance frequencies as a function of intra-sublattice inertial\nrelaxation time for two-sublattice ferromagnets using MA0=MB0= 2\u0016B. (a) The\nprecession and nutation resonance frequencies and (b) the effective Gilbert damping\nhave been plotted.\n3.1. Intra-sublattice inertial dynamics\nTo focus on the intra-sublattice inertial dynamics, we set the inter-sublattice relaxation\ntime to zero i.e., \u0011AB=\u0011BA= 0, keeping the same inertial relaxation time in two-\nsublattice\u0011AA=\u0011BB=\u0011. Withthissetofspecifications, thecalculatedfrequencieshave\nbeen shown in Fig. 2. One can see that there exist two precession resonance frequencies\n(positive) and the corresponding two nutation resonance frequencies (negative). We\ndenote these two positive precession frequencies as !u\np+and!l\np+, while the two\nnegative nutation frequencies are !u\nn\u0000and!l\nn\u0000. The superscripts “u” and “l” denote\nthe upper and lower frequencies, respectively. These results are in contrast with the\nobservation in antiferromagnets or ferrimagnets, where one positive and one negative\nprecession (and nutation) frequencies are expected [38]. Nevertheless, the quantitative\ncomparison of the calculated frequencies agrees with those of the ferrimagnets, where\nthe upper (THz), and lower (GHz) frequency precession resonances are called an\nexchange and ferromagnetic modes, respectively [38, 59]. Similar to antiferromagnets\nand ferrimagnets [38], the resonance frequencies decrease with the intra-sublattice\ninertial relaxation time in the case of two-sublattice ferromagnets. Especially, the lower\nnutation resonance frequency scales with 1=\u0011, while the upper one shows deviation\nfrom 1=\u0011at higher relaxation times. This deviation from 1=\u0011has been noticed in\ntwo nutation modes for antiferromagnets and ferrimagnets [38]. An interesting feature\nis that the precession and nutation frequencies cross each other at certain inertial\nrelaxation times in ferromagnets. Such crossing was not observed in antiferromagnets\nand ferrimagnets [38]. The crossing happens especially with the upper precession mode\nwith lower nutation mode as seen in Fig. 2(a). However, we note that crossing of theseSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 8\n12345MA0/MB00.51.01.52.02.53.0Re°!up+¢/2º(THz)(c)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\n12345MA0/MB00.010.020.030.040.05Im°!up+¢/Re°!up+¢(d)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps12345MA0/MB00.030.040.050.06Re°!lp+¢/2º(THz)(a)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\n12345MA0/MB00.030.040.05Im°!lp+¢/Re°!lp+¢(b)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\nFigure 3: The calculated precession resonance frequencies as a function of MA0=MB0\nfor two-sublattice ferromagnets, at several intra-sublattice inertial relaxation times.\n(a) The real part of the lower precession resonance frequencies, (b) the effective\ndamping of lower resonance mode, (c) the real part of the upper precession resonance\nfrequencies, (d) the effective damping of upper resonance mode, has been plotted.\ntwo modes have positive and negative frequencies, meaning that the upper precession\nmode (!u\np+) has a positive rotational sense, however, the lower nutation mode ( !l\nn\u0000) has\nthe opposite rotational sense in circular basis.\nThe inertial dynamics affect the effective Gilbert damping in a system. This has\nbeen demonstrated in Fig. 2(b) for two-sublattice ferromagnet by the ratio of imaginary\nand real parts of the calculated frequencies. We have used the same Gilbert damping\nfor both the sublattices \u000b\u00180:05and therefore, the effective damping remains the same\nat smaller inertial relaxation times. However, the effective damping decreases with\nincreased relaxation times, a fact that is consistent with the results of antiferromagnets\n[38]. It is observed that the decrease in effective damping is exactly the same for\nprecession and corresponding nutation modes. Moreover, the upper precession mode\nis influenced strongly, which has already been observed for ferrimagnets [38].Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 9\nNext, we calculate the influence of different sublattice magnetic moment ( MA06=\nMB0) on inertial dynamics. In particular, we compute the precession resonance\nfrequencies as a function of the ratio of magnetic moments ( MA0=MB0), at several\ninertial relaxation times in Fig. 3. We observe that the resonance frequencies decrease\nwith increasing difference in the magnetic moments. Such reduction is less visible in\ncase of lower precession frequencies e.g., Fig. 3(a), however, more prominent in upper\nprecession frequencies in Fig. 3(c). However, the difference of frequencies calculated\nat several relaxation times are similar for MA0=MB0andMA06=MB0. The latter\nsuggests that the inertial dynamics do not get quantitatively influenced by the same or\ndifferent sublattice magnetic moments. A similar conclusion can also be made from the\ncomputation of effective damping in Figs. 3(b) and 3(d). The effective damping for the\nupper and lower precession modes remains almost constant (with a very small positive\nslope) for a higher ratio of MA0=MB0.\n3.2. Inter-sublattice inertial dynamics\n10−1100101102\nη/prime(fs)10−1100101102103ωFM\n±/2π(THz)\n(a)\nRe/parenleftbig\nωu\np+/parenrightbig\nRe/parenleftbig\nωl\np+/parenrightbig\nRe/parenleftbig\nωu\nn−/parenrightbig\nRe/parenleftbig\nωl\nn−/parenrightbig\n10−1100101102\nη/prime(fs)0.000.010.020.030.040.05Im(ω±)/Re(ω±)|FM\n(b)\nIm/parenleftbig\nωu\np+/parenrightbig\n/Re/parenleftbig\nωu\np+/parenrightbig\nIm/parenleftbig\nωl\np+/parenrightbig\n/Re/parenleftbig\nωl\np+/parenrightbig\nIm/parenleftbig\nωu\nn−/parenrightbig\n/Re/parenleftbig\nωu\nn−/parenrightbig\nIm/parenleftbig\nωl\nn−/parenrightbig\n/Re/parenleftbig\nωl\nn−/parenrightbig\nFigure 4: The calculated resonance frequencies as a function of inter-sublattice inertial\nrelaxation time for two-sublattice ferromagnets using MA0=MB0= 2\u0016B. The\nintra-sublattice inertial relaxation time was kept constant \u0011= 100fs. (a) The\nprecession and nutation resonance frequencies and (b) the effective Gilbert damping\nhave been plotted.\nTo investigate the inter-sublattice inertial dynamics, we set the intra-sublattice\nrelaxation time as \u0011AA=\u0011BB=\u0011= 100fs. Such a relaxation time is lower than the\nexperimental findings in two-sublattice ferromagnets [39]. In fact, the direct comparison\nof Eq. (2) with the Eq. (2) of Ref. [39] provides \u0011\u0018\u000b\u001c. With the experimental\nfindings for CoFeB, \u000b= 0:0044and\u001c= 72ps (see Table 1 in Ref. [39]), we calculate\n\u0011= 316fs. We compute the effect of inter-sublattice inertial dynamics as a function of\n\u0011AB=\u0011BA=\u00110in Fig. 4 considering \u00110<\u0011. As we mentioned earlier, the overlapping ofSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 10\nprecession ( !u\np+) and nutation ( !l\nn\u0000) frequencies at the intra-sublattice relaxation time\n\u0011= 100fs can be seen. We observe that the upper precession resonance frequency ( !u\np+)\nincreases, while the lower one ( !l\np+) decreases very small with inter-sublattice relaxation\ntimes. A similar conclusion can be made for nutation frequencies. This is in contrast to\nthe observation of intra-sublattice inertial dynamics as discussed above. A divergence\nin the upper nutation frequency can be noticed at the limit \u00110!\u0011. Such divergence can\nbe explained through the coefficient Ain Eq. (15). At the limit \u00110!\u0011, the coefficient of\nfourth power in frequency becomes A=\u0011AA\u0011BB\u0000\u0011AB\u0011BA=\u00112\u0000\u001102!0, which brings\nthe fourth-order equation into an effective third-order equation in frequency.\nA similar observation can also be concluded from the calculation of effective\ndamping in Fig. 4(b). Similar to the intra-sublattice inertial dynamics, the effective\ndamping of the precession and corresponding nutation mode behaves exactly the same\nfor the inter-sublattice inertial dynamics. We observe that the damping of upper\nprecession and nutation modes increases with inter-sublattice inertial relaxation time,\nhowever, it is the opposite for lower precession and nutation modes. Therefore, we\nconclude that the effect of intra- and inter-sublattice inertial dynamics are contrasting.\n4. Conclusions\nTo conclude, we have incorporated the intra- and inter-sublattice inertial dynamics\nwithintheLLGequationofmotionandcalculatedtheFMRresonancefortwo-sublattice\nferromagnets. To this end, we first derive the magnetic susceptibility that is a tensor.\nTo calculate the resonance frequencies, we find the poles of the susceptibility. Without\nthe inertial dynamics, there exist two precession modes in a typical two-sublattice\nferromagnet. The introduction of inertial dynamics shows two nutation resonance\nfrequencies corresponding to the precession modes. We note that these precession\nand nutation resonances can be excited by right and left circularly polarised pulses,\nrespectively, and vice-versa within a circular basis. The precession and nutation\nfrequenciesdecreasewiththeintra-sublatticerelaxationtimeasalsohasbeenseeninthe\ncase of antiferromagnets in previous work [38]. However, at certain relaxation times, the\nprecessionandnutationfrequenciesoverlapwitheachother. Notethattheseoverlapping\nprecessionandnutationfrequencieshaveoppositerotationalsenseincircularbasis, thus,\nthey can be neatly realised in the experiments. The inter-sublattice inertial dynamics\nincrease the resonance frequencies and effective damping for upper precession mode,\nhowever, have opposite effect on lower precession mode in two-sublattice ferromagnets.\n5. 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B 64(17)\n174402 URL https://link.aps.org/doi/10.1103/PhysRevB.64.174402\n[59] Schlickeiser F, Atxitia U, Wienholdt S, Hinzke D, Chubykalo-Fesenko O and Nowak\nU 2012 Phys. Rev. B 86(21) 214416 URL https://link.aps.org/doi/10.1103/\nPhysRevB.86.214416"    },    {        "title": "1002.4889v1.Ferromagnetic_Resonance_of_Co_Gd_and_Co_Tb_Multilayers.pdf",        "content": "1Ferromagnetic Resonance of Co/Gd and Co/Tb Multilayers\nS. Demirtas1, I. Harward1, R. E. Camley1and Z. Celinski1\n1Department of Physics, University of Colorado at Colorado Springs,\nColorado Springs, CO 80918 USA\nM. R. Hossu2and A. R. Koymen2\n2Department of Physics, University of Texas at Arlington, Arlington, TX 76019 USA\nC. Yu3and M. J. Pechan3\n3Department of Physics, Miami University, Oxford, OH 45056 USA\nAbstract\nThe in-plane dynamics of ferrimagnetic Co/Gd multilayers are investigated by \nmeans of ferromagnetic resonance, magneto-optical Kerr effect and SQUID \nmagnetometry. The power absorbed from these multilayers is strongly temperature \ndependent. For example, the resonant peak for a (Co 40 Å /Gd 40 Å) 8multilayer vanishes \napproximately 50 K below room temperature. We have further investigated Gd/Co/Gd\nand Tb/Co/Tb trilayers with different thicknesses of Gd (5-7 Å), Tb (1-7 Å) and Co (30-\n40 Å). At room temperature, these Co-based trilayers show a shift of approximately 600 \nOe at 24 GHz in the uniform ferromagnetic resonance field, compared to pure Co film, \nindicating the exchange coupling between the Co and Gd. The shift in the field for the \nresonance increases as the temperature is decreased. Furthermore the resonance linewidth \nincreases as the temperature is decreased. The experimental results are in good agreement \nwith our theoretical calculations.2Introduction\nMagnetic multilayers have been an area of active research for a considerable \nperiod, partly because their properties, such as giant magnetoresistance, have lead to \napplications in the recording industry.  In addition to the common multilayers involving \ntransition metal ferromagnets with nonmagnetic spacers, there has also been a substantial \nstudy of transition metal / rare earth multilayers where both materials are ferromagnetic.  \nThese systems exhibit multiple magnetic configurations and phase transitions between \nthese states at different external magnetic fields and temperatures.  \nThe Co/Gd and Fe/Gd multilayer systems [1-21] have a number of known \nmagnetic states. There is a low temperature Gd-aligned state where the Gd magnetization \nis aligned with the external field (and the Co is opposite) and there is a high temperature \nCo-aligned state where the Co magnetization is aligned with the external field (and Gd is \nopposite). Finally there is also a \"twisted\" or canted state where the moments in each \natomic layer have a different angle with respect to the applied field. Therefore, Co/Gd \nand Fe/Gd structures are artificial ferrimagnets where the effective contribution of the \ntwo magnetic components is controllable by changing the layering pattern. At a particular \ntemperature, which is called compensation temperature (T comp), the magnetic moment of \nthe Co and Gd layers are equal in magnitude and total moment goes to zero due to the \nantiparallel alignment.\nAlthough there have been many investigations into the static configurations found \nin Fe/Gd and Co/Gd multilayers, there have been relatively few studies on the dynamical \nmodes [22-31] found in these artificial ferrimagnets. In particular, the ferromagnetic \nresonance (FMR) of alternating ferromagnetic films that are antiferromagnetically \ncoupled at the interfaces may be interesting since the magnetizations of the two materials \nrespond very differently to temperature changes. For example, in the Co/Gd multilayer \nthe average Gd magnetic moment changes from about 7 B to zero when the temperature \nis varied from 0 to around 300 K, while the moment of the Co changes only slightly over \nthe same temperature range.\nMost of the earlier work on the dynamic modes [22-30] was done on FeGd and \nCoGd alloys and concentrated on determining the g-value, the exchange stiffness \nconstant, the magnetic damping parameter and the anisotropy constants. An interesting 3property found for these amorphous alloys was the observation of multi-resonance peaks. \nA reasonable explanation [23] was that the non-uniformity of these amorphous alloys \ncreates the multi peak FMR spectra since the structurally different portions of the \nsamples may absorb the microwave energy at different fields due to the different \nanisotropy constants. High temperature annealing can change the multi-peak spectra of \nthese alloys. An alternative explanation [26] was made under the assumption that the \nsurface of these alloys can act differently than the bulk of the thin (<5000 A) film. \nTherefore surface and the bulk of the film can resonate at two different fields. \nOur attention here focuses on the strong antiferromagnetic coupling at the \ninterface of the Co/Gd multilayers and its effects on the FMR absorption. Furthermore we \ninvestigate the temperature dependence of the FMR signal because there are significant \nchanges in magnetic structure with varying temperature. Indeed, our FMR results are \nhighly sensitive to temperature changes.  In fact we will see that the FMR signal for some \nmultilayer structures simply vanishes 50 K below the room temperature, while the FMR \nsignal for other structures can be observed down to very low temperatures.\nIn this paper we measure the FMR of Co/Gd multilayers, Gd/Co/Gd and \nTb/Co/Tb trilayers. Our main results demonstrate substantial shifts in the resonance field \ndue to the antiferromagnetic exchange interaction between the Co and Gd. We show that \nthe shift in the resonant field increases as the temperature is reduced and the linewidth of \nthe absorption peak increases as the temperature is decreased. We have also performed \ntheoretical calculations which explain this behavior and which are in reasonable \nquantitative agreement with the experimental data. \nPart of the motivation for this study is to produce device materials which operate \nat higher resonance frequencies. Such materials are of interest, for example, in high \nfrequency signal processing [32-34] where a large operational frequency is desired at a \nlow external magnetic field. The shift in the resonance field discussed above corresponds \napproximately to a 10-15 GHz boost in resonance frequency at low fields, and represents \na substantial increase of the FMR frequency at low fields.  4Experimental Details\nA variety of Gd/Co multilayers, Gd/Co/Gd and Tb/Co/Tb trilayers were prepared \nin a dc magnetron sputtering system at room temperature. The unbaked base pressure of \nthe UHV deposition chamber was 910Torr. Ultra high purity Argon gas pressure was 3\nmTorr during the deposition. The samples were deposited on Corning glass substrates \nand 50 Å Ag layers were used as buffer and protective cap layers in all samples. The \ndeposition thicknesses were monitored in situ by a quartz thickness gage which was\ncalibrated by a stylus profilometer. Magnetization measurements were taken using a \nSQUID magnetometer and a Magneto-Optical Kerr Effect (MOKE) system starting from \n300 K under a constant in-plane external magnetic field. The dynamic response of the \nsamples was measured by a 24 GHz Ferromagnetic Resonance (FMR) spectrometer \nbetween 30 and 300 K. Measurements were made using a cylindrical resonant cavity \nwhich operates in the TE 012mode. External magnetic field is parallel to the sample \nsurface. The peak to peak linewidth was measured from the differential FMR signal.\nTo begin the magnetic characterization of the samples, we measured the \nmagnetization as a function of temperature.  A typical curve of the ferrimagnetic (Co 40 \nÅ /Gd 40 Å) 8multilayer is shown in Fig. 1.  The results display significant thermal \nhysteresis, and the compensation temperature, T comp, is around 175 K as seen from the \nmidpoint of the \"bow-tie\" in Fig. 1. Details of the tunable thermal hysteresis can be found \nelsewhere [35-38].\nWe confirmed the thermal hysteresis results of Fig. 1 by measuring the MOKE \nsignal as a function of temperature as shown in Fig. 2.  Unlike the complicated thermal \nhysteresis results in Fig. 1, this measurement gives a nearly square hysteresis loop which \nmight have applications in magnetic recording.  The reason for the differences in Fig. 1 \nand Fig 2 is that the MOKE rotation angle is more sensitive to the 3d Co spins than the \nlocalized 4f Gd spins. This occurs because the MOKE signal originates from spin-orbit \ncoupling. Since there is LS coupling in Co while in Gd there no spin-orbit contribution \n(L=0), the standard MOKE experiment essentially provides element specific results (Co\nin this case).5Experimental Results\nTo understand the FMR spectra [39-40] of the Co/Gd multilayers, it is helpful to \ncompare the results for a pure Co layer with those of the multilayers. In Fig. (3a) we \npresent the FMR spectrum for a single 400 Å Co film at room temperature. There is a \nnarrow peak centered at 3.5 kOe indicating an effective magnetization of 1.25 kOe for the \npure Co film. This magnetization is found using the formula \n)M4 H(H fS res res   . (1)\nHere the gyromagnetic ratio is = 2.92 GHz/kOe, H resis the field at which the FMR \nshows maximum absorption and f = 23.93 GHz is the operation frequency.  \nThe FMR spectra for a (Co 40 Å /Gd 40 Å) 8multilayer are shown in Fig. (3b-d)\nfor different temperatures. Compared to the pure Co film, the position of the uniform \nFMR peak is lower by about 1 kOe at room temperature. The reason for this shift is the \nexchange coupling between the Co and Gd as we will discuss later on. As the temperature \nis lowered from the room temperature (Fig 3c and 3d), there are some small changes in \nthe resonance field and the linewidth increases substantially. Around 250 K the FMR \nsignal essentially vanishes.\nIt is not immediately clear what changes should occur as the temperature is \ndecreased. At lower temperatures, the effective Gd moment increases in its thermal \naveraged magnitude and this would reduce the magnetization of the entire structure as \nseen in Fig. 1. Using Eq (1), the reduced magnetization would shift the resonance field to \nhigher values. However, this can not be the only effect. As we have seen, the exchange \nfield produced by the Gd acts on the Co and provides an effective field which \ndramatically lowers the resonance field of the Co/Gd multilayer compared to the pure Co \nfilm. At lower temperatures, one would expect that this exchange field would be larger \nand that the resonance position for the multilayer would be further lowered. For some \nCo/Gd multilayer samples we observed multi-peak spectra where the multiple peaks \noccurred above the uniform resonance field however this was not the case for every \nmultilayer combination. These multi-peak spectra may have different sources as indicated 6in the introduction section. Indeed for example the curve in Fig. 3c can be better fitted by \ntwo Lorentzians numerically instead of one. \nThe in-plane angular dependence of the FMR field for the (Co 40 Å /Gd 40 Å) 8\nmultilayer at 24 GHz is shown in Fig. 4. The (Co 40 Å /Gd 40 Å) 8multilayer can be \nconsidered as polycrystalline since it does not show any easy or hard axis anisotropies. \nThis is consistent with the previous X-ray measurement where the (Co 40 Å /Gd 40 Å) 8\nmultilayer does not show any preferred orientations [36]. In most of our samples we did \nnot observe any significant in-plane anisotropy.\nPart of the motivation to proceed further with covering the Co with atomically \nthin Gd is to better focus on the coupling effects on the resonance properties. This is due \nto the fact that increasing Gd moment with decreasing temperature has the effect of \neliminating the FMR signal as shown in Fig. 3b-d in a small (~50 K) temperature interval \nand this 50 K temperature interval below room temperature is not sufficiently large to \nprocess the data. Our theoretical calculations lead us to use different thickness \ncombinations of the Co/Gd layers. Our goal was to reach large, temperature independent,\nshifts of the FMR field. As mentioned earlier, the motivation for the large FMR field \nshifts is to use this structure in microwave device applications in which the external \nmagnetic field requirement would be substantially lower [38-40]. We therefore created \ntrilayers of Co and Gd where the Co film (30-40 Å) is sandwiched between a few\nmonolayers (ML) of Gd (5-7 Å) and Tb (1-7 Å). To increase the signal we repeated the \ntrilayer structure 10 times putting a 50 Å thick nonmagnetic Ag spacer between the \ntrilayers. The experimental results are shown in Figs. 5-7.\nAs shown in Fig. 5, the ferromagnetic resonance field for a 400 Å thick single \nfilm of Co is around 3.5 kOe at room temperature where resonance frequency f = 23.93 \nGHz. When we sandwiched the 40 Å Co film with 5 Å (~1.5 monolayers) of Gd on both \nsides of the Co the FMR peak shifts about 0.6 kOe lower to 2.92 kOe. The FMR peak \nfield for (Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å) 10is slightly lower still around 2.86 kOe.\nThis is in agreement with theoretical calculations which show that adding additional Gd \nlayers enhances the downward shift. The (Gd 5 Å/Co 30 Å/Gd 5 Å/Ag 50 Å) 10\nmultilayers have a FMR peak position near 2.9 kOe. As will be discussed later, the reason \nfor these field shifts is the strong antiferromagnetic coupling of the Co and Gd layers. 7Antiferromagnetic exchange coupling acts like an extra effective field that goes into the \nEq. (1) and, at a fixed frequency; this reduces the amount of the applied field which is \nrequired for the resonance.\nWhen we repeated the same experiment with atomically thin Tb films on the \noutside of the 40 Å thick Co film, we could only observe an FMR signal when the Tb \nlayer thickness was 1 Å (~0.3 monolayer). Thicker Tb films above 1 Å damped the FMR \nsignal so that no response was visible. The resonance field shift for the (Tb 1 Å/Co 40\nÅ/Tb 1 Å/Ag 50 Å) 10multilayer is on the order of 0.25 kOe from the Co peak.\nThe temperature dependence of the FMR field for the trilayer structures are\nshown in Fig. 6. The FMR field for Co is nearly constant over the entire temperature \nrange. In contrast, the multilayers show a slight decrease in the FMR fields as the \ntemperature is reduced. This may be due to an increased effective exchange field \nproduced by the Gd and Tb as the temperature is reduced. Again this feature is in \nreasonable agreement with the theoretical results.\nThe FMR linewidth as a function of temperature is shown in Fig. 7 for all the \nsamples. The linewidths generally increase monotonically as the temperatures decreases. \nHowever there is an abrupt discontinuity for the sample in Fig. 7 (top panel) below 150 K \nwhere the Gd thickness is 5 Å. This may be due to the non-uniformity of that particular \nsample. Although the resonance field values shown in Fig. 6 for the trilayer with 30 Å\nand 40 Å of Co when the Gd is 5 Å thick are similar at room temperature the gap widens \nat low temperatures and smaller fields for the resonance are observed when the Co is 30 \nÅ thick. This is meaningful owing to the forthcoming theoretical discussion since the \nproportional amount of the Gd is slightly larger when the Co layer is thinner. However a \nreverse effect is shown on the linewidth pattern shown in Fig. 7 as the temperature is \ndecreased. This time trilayer with the thicker Co deflected more as the temperature is \ndecreased. Furthermore temperature dependent linewidth for the (Gd 5 Å/Co 30 Å/Gd 5 \nÅ/Ag 50 Å) 10multilayer has smaller values than that of the bare 400 Å Co film which we \nuse as the reference film. Therefore correct comparison for the linewidth for the trilayer \ncontaining 30 Å thin Co film can be made most probably with a thinner reference Co \nfilm. Nevertheless the general features of trilayer containing 30 Å of Co with 5 Å of Gd \non each side are in good agreement with other films and also with the theoretical 8discussion such as the downward shift of the resonance field compared to bare Co film \nand the monotonic decrease (increase) of the resonance field (linewidth) on the large \ntemperature range. The FMR linewidth depends on a number of factors including the \ndamping mechanisms and the uniformity of the structure. It is well known that Tb can \nsignificantly increase the damping [41], and indeed we find that even 1 Å of Tb on both\nsurfaces of the Co film causes the linewidth to increase by a factor of two at room \ntemperature and a factor of three at low temperatures. However, Gd doping in transition \nmetals [42] does not produce such a dramatic increase. Nonetheless we find that as the \ntemperature is lowered there is a distinct increase in the linewidth even for the Gd/Co/Gd \nsamples. This is in agreement with the theoretical calculations and will be discussed in \nthe next section. \nTheoretical Calculations\nThere are a number of factors which need to be considered in order to understand \nhow the resonance field measured in the Co film is influenced by the thin Gd layers.  \nFirst, if the Gd spins were rigidly coupled to the Co spins by exchange coupling, one \nmight expect either no shift in the resonance field or an upward shift in the resonance \nfield. The reason for this is that, in this case, the Gd moments would be strictly \nantiparallel to the Co moments and the exchange field of the Gd would not contribute to \nthe torque on the Co. Then the net effect of the Gd would be to reduce the net \nmagnetization of the structure. Then using the standard formula for resonance in a thin \nfilm, Eq (1), the required field for resonance would have to be increased. If the \ntemperature would be reduced, the net magnetization would be further reduced and the \nresonance field would have to be further increased. This is not what is seen \nexperimentally.\nExperimentally, one finds a downward shift in the resonance field and the shift is \nincreased as the temperature is reduced, therefore the Co and Gd moments can not be \nstrictly antiparallel. There can be several causes for this.  First, the magnetization of the \nGd film is very different from that in a Co film. As is well known, the precession in a thin \nferromagnetic film is elliptical due to the demagnetizing fields. For example, in an 9isolated film the ratio of the x and y amplitudes of the magnetization when an external \nfield H is parallel to the surfaces of the film is given by\nHM41MM\nyoxo  .              (2)\nSince M is not the same in the Gd and Co, one would not get the same ellipticity in two \nmaterials and the moments in the two materials will not always be antiparallel. This \neventually produces an effective field from the Gd onto Co, thus reducing the value of the \nexternal field H necessary to create the ferromagnetic resonance condition. These \nconsiderations agree well with the experimental results.\nNow we want to consider how temperature influences the measured resonance \nfield. It is well known that the effective exchange coupling between the Co and Gd has \nenergy proportional to J I SGdSCo. The interface exchange constant J I is much larger than \nthe exchange constant within Gd. As a result, the Co magnetization can stabilize the Gd \nmoments right at the interface above the usual Curie temperature of the Gd. The \ncalculations show that when there is only one atomic layer of Gd the ratio of the thermal \naveraged Gd moment at 300 K to that at 0 K is about 0.53. In a bulk Gd sample, of \ncourse, this ratio would be zero since the Curie temperature is 293 K. This explains how \nthere can be a substantial shift in H reseven at room temperature. Furthermore, as the \ntemperature is reduced, the thermal averaged Gd moment increases, thereby increasing \nthe effective field and reducing the resonance field. Again, this behavior is in good \nagreement with the experimental results. \nWe can also qualitatively understand what happens as the number of Gd layers is \nincreased. Adding additional Gd layers also helps stabilize the Gd layer right next to the \nCo. For example, when there are two atomic layers of Gd the ratio of the thermal \naveraged Gd moment at 300 K to that at 0 K is about 0.6, compared to the 0.53 seen \nwhen there is only one atomic Gd layer. Thus one expects to see a larger shift in the \nresonance field. This will be evident in the theoretical results, but the experimental results \nare less clear on this point.  10Finally we would like to understand the behavior of the FMR linewidth as a \nfunction of temperature. The linewidth measures many different factors including \nrelaxation mechanisms and sample uniformity. In the calculations we have chosen the \nlinewidth of the two materials to be the same, so we are investigating the question of how \nthe linewidth is influenced by the coupling of the two materials. Without the Gd, all the \nCo moments can precess in a uniform motion; the effective field acting on each Co is the \nsame. When the Gd is added the outer Co spins see a different field than the interior ones \nand this nonuniformity leads to an enhancement of the linewidth. As the temperature is \nreduced the effective exchange field becomes larger, leading to a larger nonuniformity \nand a larger linewidth. \nBecause the interfacial coupling has an energy proportional to J I <SGd><S Co>, one \nexpects the strength of the effective field acting on the Co at the interface is proportional \nto J I<SGd>. As the temperature is reduced <S Gd> increases and so does the effective \nexchange field. This increase in effective field then reduces the resonance field at low \ntemperatures, in agreement with the experimental results.  \nThe calculations are done using the Landau-Lifshitz-Gilbert (LLG) equations.  \nThis technique can give information on both the static and dynamic properties of \nmagnetic multilayers. We write a set of LLG equations for each atomic layer in the \nmultilayer\n                 \n\n\ndtSdSS)HS (dtSdi\ni\nii ii \n(3) \nHere H iis the effective field acting on the magnetization, S i, in layer i. is the \ngyromagnetic ratio and is a dimensionless damping parameter. The total effective field \nis a sum of the external field, exchange fields, the oscillating field, and dipolar fields and \nis given by\n       dti\n1i1i , i 1i1i , i o i h xˆ he S J S JzˆH H   \n \n  (4)11Here H ois the static magnetic field in the z direction, J i,i+1is the exchange coupling \nconstant between layer i and layer i+1. <S i> is the thermal averaged magnitude of the \nspin in layer i. The oscillating field h has an angular frequency and is applied in the x\ndirection. We assume that the main contribution to the dipole field h dis the \ndemagnetizing field of a thin layer acting on itself; thus, yˆM4 hy d , where y is the \ndirection perpendicular to the film surface. This approximation works well in the thin \nfilm or long wavelength limit and is appropriate for FMR calculations. \nOne picks an initial spin configuration and integrates the coupled set of equations \nforward in time using a differential equation solver such as the 4thorder Runge-Kutta \nmethod. Thus one obtains S x(t), S y(t) and S z(t). In order to get sensible results appropriate \nto FMR one has to wait for the system to come into dynamic equilibrium with the driving \nfield. The power absorbed in a period T is then given by\nT\n0y\nT1dtdt) t ( dM\n) t ( h P (5)\nand the integral is done numerically.  \nThe temperature dependence is included by renormalizing the magnitude of the \nmagnetization using the Brillouin function. For example the magnetic moment for the Co \nis given by <m Co> = gBSCoBS(gBSCoHeff/kT), and a similar equation holds for the \nmagnetic moment in the Gd. Here <m Co> is the thermal averaged magnitude of the \nmagnetic moment in Co, and B s(x) is the Brillouin function given by\n.S2xcothS21\nS2x ) 1 S2 (cothS2) 1S2 () x ( Bs \n\n\n\n\n\n\n\n (6)\nThe effective field, H eff, is the same field that acts in Eq (4), but here the exchange field \nthat has the dominant contribution. In fact, the exchange constants within the Gd and Co \nare set by requiring that the bulk Curie temperatures for the two materials are given \ncorrectly. Initially each step of the time iteration also involves a renormalization of the 12magnetic moments for the Co and Gd, but after about 100,000 time steps the magnitudes \nno longer change substantially. \nThe results of the calculation are present in Figs. 8-9. The key parameters for the \ncalculation are given by M Gd(T=0) = 2.06 kG, M Co = 1.46 kG, J Co= 9,680 kG,                 \nJGd= 198 kG, and J I= -1100 kG. We assume that g = 2 in both materials and S Gd= 3.5 \nand S Co= 0.86. The Fig. 8 shows the calculated FMR absorption curves for a 40 Å Co \nfilm and for the same film with 1 or 2 ML of Gd on each of the outside surfaces of the \nCo. The results are quite close to those found experimentally. The films with the Gd on \nthe outside exhibit a reduced resonance field and are smaller and broader than the FMR \ncurve for the Co alone. The shift between the pure Co resonance field and the system \nwith 1 ML of Gd on the outside is 0.59 kOe, in good agreement with the experiment. This \nshift has been used to obtain the value for the interface coupling constant J I. It is \nreassuring that this value is, in fact, quite close to values based on magnetization results.  \nThe theoretical results for how the linewidth and resonance field depend on \ntemperature are shown in Fig. 9. We find that as T is reduced the resonance field \ndecreases slightly, about 0.5 kOe over the entire temperature range. Again, this is in \nreasonable agreement with the experimental results, and indicates the strengthening of the \nexchange field as the temperature is reduced. The linewidth also changes with \ntemperature, increasing substantially as the temperature is reduced. Again the values are \nin reasonable agreement with the experimental results and with the general discussion \nabove.  \nSummary and Conclusions\nWe have performed static and dynamic measurements and also self consistent \ntheoretical calculations for Co/Gd and Co/Tb films. Resonance field for the Co/Gd \nmultilayers are substantially reduced compared to the pure Co film. This shift is due to \nthe antiferromagnetic coupling between the Co and Gd. As the temperature is decreased \nfrom room temperature the effective Gd magnetic moment increases and this reduces the \ntotal magnetic moment of the Co/Gd multilayer system. We have observed the \nbroadening of the FMR linewidth for Co/Gd trilayer and multilayers as the temperature \nreduces from room temperature and the Gd becomes magnetic. FMR signal vanishes 13approximately 50 K below room temperature for Co/Gd multilayers when the Gd layer is \nthicker than a few atomic layers. Therefore Gd/Co/Gd trilayers where the thickness of the \nGd is 1-2 monolayers are better choice to study the temperature dependence of the \nantiferromagnetic coupling dynamically. These Gd doped trilayers also showed a \nsubstantial downward shift of the FMR field on the order of 600 Oe at 24 GHz compared \nto that of pure Co film at room temperature. Likewise multilayers of Co/Gd with thicker \nGd layer shows even greater downward shifts on the order of 1 kOe compared to pure Co \npeak at room temperature. Our theoretical calculations relate this substantial downward \nshift of the FMR field to the effective field from the Gd onto the Co due to the \nantiferromagnetic exchange coupling between them. As the temperature decreases \nresonance peak position at room temperature slightly decreases and the associated \nlinewidth increases gradually at the same time. When the same experiments are repeated \nfor the Tb/Co/Tb trilayers the FMR signal is only observable if the Tb layer thickness is \non the order of 1 Å or less. Higher thicknesses over 1 Å of Tb damp the FMR signal. The \nresults of Tb/Co/Tb trilayers are supportive and comparable with the arguments used for \nthe case of Gd/Co/Gd trilayers. The experimental results are in good agreement with the \ntheoretical calculations.\nAcknowledgements\nThe work at UCCS was supported by DOD Grant # W911NF-04-1-0247. The \nwork at UTA is supported by a grant (No. Y-1215) from The Welch Foundation. 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Thermal hysteresis with increasing external magnetic fields for the (Co 40 Å/Gd \n40 Å) 8multilayer.\nFig. 2. An example of MOKE thermal hysteresis for (Co 40 Å /Gd 40 Å) 8multilayer. \nCooling and heating curves are measured under a constant 150 Oe inplane magnetic field \nis applied. Drifts on either side of the hysteresis are artifacts from the electronics of the \nsetup. \nFig. 3. FMR spectrum of a) 400 Å Co film b-d) (Co 40 Å/Gd 40 Å) 8multilayer as a \nfunction of temperature.\nFig. 4. In plane angular dependence of FMR field for the (Co 40 Å /Gd 40 Å) 8multilayer \nat room temperature.\nFig. 5 . Room temperature FMR spectra for Co 400 Å (full squares), (Gd 5 Å/Co 40\nÅ/Gd 5 Å/Ag 50 Å) 10(full circles), (Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å) 10(empty circles),\n(Gd 5 Å/Co 30 Å/Gd 5 Å/Ag 50 Å) 10(full triangles) and (Tb 1 Å/Co 40 Å/Tb 1 Å/Ag 50\nÅ)10(empty squares) films.\nFig. 6. FMR field as a function of temperature for various films. For the multilayer films \nthere is a gradual decrease in the FMR field as the temperature is reduced. \nFig. 7. Linewidth of the FMR field as a function of temperature for various films. The \nlinewidth of the multilayer films generally increases as the temperature is reduced. \nFig. 8. Theoretical results for FMR absorption for a 40 Å Co film sandwiched between 0, \n1 or 2 monolayers of Gd on each side.  \nFig. 9 . Theoretical results showing how the resonance field and the linewidth changes as \na function of temperature for a 40 Å Co film and for a 40 Å Co film with one or two ML \nof Gd on each side.  17-0.00020.00000.00020.00040.0006\n0.00000.00020.00040.0006\n0 50 100 150 200 250 3000.00000.00020.00040.0006Magnetic Moment [emu]100 Oe(Co 40 Å/Gd 40 Å)8\n200 Oe\nTemperature [K]400 Oe\nFig. 1. Thermal hysteresis with increasing external magnetic fields for the (Co 40 Å/Gd \n40 Å) 8multilayer.1850 100 150 200 250 300 3500.000.020.040.060.080.100.120.14 KerrAngle []\nTemperature [K]150 Oe(Co 40 Å/Gd 40 Å)8\nFig. 2. An example of MOKE thermal hysteresis for (Co 40 Å /Gd 40 Å) 8multilayer. \nCooling and heating curves are measured under a constant 150 Oe inplane magnetic field \nis applied. Drifts on either side of the hysteresis are artifacts from the electronics of the \nsetup.190 1 2 3 4 5-0.00050.00000.00050.00100.00150.00200.00250.0030\nMagnetic Field [kOe]Diode Voltage [V]a) Co 400 Å \n295K\n0.00000.00060.0012\n0.00000.00060.0012\n0 1 2 3 4 5-0.00060.00000.00060.0012295K b)Diode Voltage [V]\n270Kc)\nMagnetic Field [kOe]255K(Co 40 Å/Gd 40 Å)8\nd)\nFig. 3. FMR spectrum of a) 400 Å Co film b-d) (Co 40 Å/Gd 40 Å) 8multilayer as a \nfunction of temperature.20Fig. 4. In plane angular dependence of FMR field for the (Co 40 Å /Gd 40 Å) 8multilayer \nat room temperature.-90 -60 -30 0 30 60 902200230024002500260027002800\n(Co 40 Å/Gd 40 Å)8In Plane Angular DependenceFMR Field [Oe]\nAngle [degree]211 2 3 4 5-0.00050.00000.00050.00100.00150.00200.00250.0030\n(Gd 5Å/Co 40Å/Gd 5Å/Ag 50Å)10(Gd 7 Å/Co40 Å/Gd 7 Å/Ag 50 Å)10Diode Voltage [V]\nMagnetic Field [kOe]Co 400 Å\n1 2 3 4 5-0.00020.00000.00020.00040.00060.00080.0010\n(Gd 5Å/Co 30Å/Gd 5Å/Ag 50Å)10(Tb 1Å/Co 40Å/Tb 1Å/Ag 50Å)10\nMagnetic Field [kOe]Diode Voltage [V]\nFig. 5 .  Room temperature FMR spectra for Co 400 Å (full squares), (Gd 5 Å/Co 40\nÅ/Gd 5 Å/Ag 50 Å) 10(full circles), (Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å) 10(empty circles), \n(Gd 5 Å/Co 30 Å/Gd 5 Å/Ag 50 Å) 10(full triangles) and (Tb 1 Å/Co 40 Å/Tb 1 Å/Ag 50\nÅ)10(empty squares) films.220 50 100 150 200 250 30028003000320034003600\nTemperature [K]FMR Field [Oe]\n(Gd 5Å/Co 30Å/Gd 5Å/Ag 50Å)10(Tb 1Å/Co 40Å/Tb 1Å/Ag 50Å)100 50 100 150 200 250 30028003000320034003600FMR Field [Oe]\nTemperature [K](Gd 5Å/Co 40Å/Gd 5Å/Ag 50Å)10\n(Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å)10Co 400Å\nFig. 6.   FMR field as a function of temperature for various films. For the multilayer films \nthere is a gradual decrease in the FMR field as the temperature is reduced. 230 50 100 150 200 250 300200400600800100012001400\nTemperature [K]Linewidth [Oe] (Gd 5Å/Co 40Å/Gd 5Å/Ag 50Å)10\n(Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å)10\nCo 400Å\n0 50 100 150 200 250 300200400600800100012001400\nTemperature [K]Linewidth [Oe]\n(Gd 5Å/Co 30Å/Gd 5Å/Ag 50Å)10(Tb 1Å/Co 40Å/Tb 1Å/Ag 50Å)10\nFig. 7.   Linewidth of the FMR field as a function of temperature for various films.  The \nlinewidth of the multilayer films generally increases as the temperature is reduced. 240 1 2 3 4012345678\nGd / Co / Gd\n2 ML Gd1 ML GdNo GdAbsorption [arb units]\nMagnetic Field [kOe]\nFig. 8. Theoretical results for FMR absorption for a 40 Å Co film sandwiched between 0, \n1 or 2 monolayers of Gd on each side.  2550 100 150 200 250 3000.00.40.81.22 ML Gd\n2 ML GdH  [kOe]\nTemperature [K]1 ML Gd1.52.02.53.03.5\nNo Gd\n1 ML GdHres [kOe]Gd/Co/Gd\nFig. 9 . Theoretical results showing how the resonance field and the linewidth changes as \na function of temperature for a 40 Å Co film and for a 40 Å Co film with one or two ML \nof Gd on each side.  "    },    {        "title": "1105.5626v2.Resonances_in_a_dilute_gas_of_magnons_and_metamagnetism_of_isotropic_frustrated_ferromagnetic_spin_chains.pdf",        "content": "arXiv:1105.5626v2  [cond-mat.str-el]  23 Nov 2011Resonances in a dilute gas of magnons and metamagnetism of is otropic frustrated\nferromagnetic spin chains\nM. Arlego,1F. Heidrich-Meisner,2A. Honecker,3G. Rossini,1and T. Vekua4\n1Departamento de F´ ısica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina\n2Physics Department and Arnold Sommerfeld Center for Theore tical Physics,\nLudwig-Maximilians-Universit¨ at M¨ unchen, 80333 M¨ unch en, Germany\n3Institut f¨ ur Theoretische Physik, Georg-August-Univers it¨ at G¨ ottingen, 37077 G¨ ottingen, Germany\n4Institut f¨ ur Theoretische Physik, Leibniz Universit¨ at H annover, 30167 Hannover, Germany\n(Dated: May 27, 2011; revised November 22, 2011)\nWe show that spin- Schains with SU(2)-symmetric, ferromagnetic nearest-neig hbor and frustrating\nantiferromagnetic next-nearest-neighbor exchange inter actions exhibit metamagnetic behavior under\nthe influence of an external magnetic field for small S, in the form of a first-order transition to the\nfully polarized state. The corresponding magnetization ju mp increases gradually starting from\nanS-dependent critical value of exchange couplings and takes a maximum in the vicinity of a\nferromagnetic Lifshitz point. The metamagnetism results f rom resonances in the dilute magnon gas\ncaused by an interplay between quantum fluctuations and frus tration.\nI. INTRODUCTION\nQuantum spin systems at low temperatures share\nmany of the macroscopic quantum behaviors with sys-\ntems of bosons such as Bose-Einstein condensation, su-\nperfluidity (see Ref. 1 and references therein), or macro-\nscopic quantum tunneling,2and may indeed be viewed\nas quantum simulators of interacting bosons.1Spin sys-\ntems with frustration, in particular, realize exotic phases\nof strongly correlated bosons, such as spin liquids3or\nsupersolids.4Experimentally, such systems can be stud-\nied both in low-dimensional quantum magnets (see, e.g.,\nRef. 3) and in ultra-cold atomic gases.5,6In the latter\ncase, interest in the many-body physics has been excited\nby the extraordinary control over interactions between\nbosons via Feshbach resonances.7These can, in particu-\nlar, be used to tune interactions from repulsive to attrac-\ntive. In the attractive regime, the collapse of an ultra-\ncold gas of bosons was observed in experiments.8In this\nwork, we discuss a mechanism by which the same can be\nachieved in spin systems.\nIn the large- Slimit, spins map onto bosons with a\nfinite but large Hilbert space, justifying a description in\nterms ofsoft-corebosons. Yet, in many cases, similarities\nbetween spins and soft-core bosons may even exist for\nS= 1/2. In three dimensions (3D), systems of spins\nand bosons resemble each other the more the smaller the\ndensity of bosons is, whereas 1D spin- Santiferromagnets\nclose to saturation behave as spinless fermions, or hard-\ncore bosons. In spin systems, the external magnetic field\nhtunes the density of magnons. The limit of a dilute gas\nof magnons is then realized as the fully polarized state\n(the vacuum of magnons) is approached from below.\nIn this work, we argue that resonances can play a cru-\ncialroleinfrustratedquantumspinsystemslargelydeter-\nmining their low-energybehaviorin an externalmagnetic\nfield. We consider the frustrated ferromagnetic (FM)\nspin-Schain and show that upon changing system pa-\nrameters such as the coupling constants or h, one cantune the effective interaction between magnons from re-\npulsive to attractive by exploiting the existence of res-\nonances. As a main result, we demonstrate that, close\nto resonances, where the scattering length is much larger\nthan the lattice constant, and in the case of attractive\neffective interactions, the intrinsic hard-coreness of spins\ndoes not play a significant role in the limit of a dilute gas\nofmagnons. Abehaviorresemblingthecollapseofattrac-\ntively interacting bosons9,10therefore exists in such spin\nsystems close to their fully polarized state. The thermo-\ndynamic instability of collapsed states causes jumps in\nthe magnetization curve just below saturation. This has\nto be contrasted with the scattering length being of the\norder of a few lattice sites. In that case, which is real-\nized for spin 1/2, the formation of mutually repulsive,\nmulti-magnon bound states can be observed (see, e.g.,\nRefs. 11–15).\nOur work demonstrates that the mapping of a purely\n1Dspin system close to saturation to an effective theory\nof a dilute Bose gas properly accounts for the physics of\nthe model. This is accomplished by connecting the scat-\ntering vertices of the microscopic lattice model with the\ncoupling constants of the effective theory. Furthermore,\ndespite the purely 1D nature of our problem, a 1 /Sex-\npansion is a valuable tool and yields the correct physics.\nConcretely, we study the following system:\nHS=L/summationdisplay\ni=1/bracketleftBig\nJ/vectorSi·/vectorSi+1+J′/vectorSi·/vectorSi+2−hSz\ni/bracketrightBig\n.(1)\n/vectorSi= (Sx\ni,Sy\ni,Sz\ni) is a spin- Soperator acting on site i\nandLis the number of sites. J <0 is the FM, nearest-\nneighbor exchange interaction and J′= 1 is the antifer-\nromagnetic (AFM), next-nearest-neighbor exchange in-\nteraction setting the energy scale. In the absence of an\nexternal field h,HShas a FM ground state for J≤ −4\nfor allS(see Ref. 16). J=−4 is a ferromagnetic Lifshitz\npoint where the quadratic term in the dispersion of the\nmagnons vanishes.2\nSystemswithcompetingFMandAFMinteractionsare\nof timely interest,17in particular, the spin-1 /2 version of\nEq. (1),11–14,18–20motivated by the experimental real-\nizations in, e.g., LiCuVO 4(Refs. 21,22) and Li 2ZrCuO 4\n(Ref. 23). The 1D case of J <0 andS >1/2 is largely\nunexplored (for J >0 andS >1/2, see Refs. 19 and 24).\nWe are mainly interested in the region −4< J <0\nand magnetization M=Sz/(SL) (Sz=/summationtext\ni/an}b∇acketle{tSz\ni/an}b∇acket∇i}ht) close\nto saturation M= 1, for general spin S. We will proceed\nin three steps: First, in Sec. II, we discuss the solution\nof the two-magnon problem and introduce the scatter-\ning length. Second, in Sec. III, we map the low-energy\nlimit of Eq. (1), close to saturation, to a dilute 1D gas of\ntwo species of bosons interacting via an effective short-\nrange interaction.25,26We then calculate the interaction\nverticesinthiseffectivetheoryusinga1 /Sexpansion. Fi-\nnally, in Sec. IV, we compare the analytical results with\nexact numerical ones using the density matrix renormal-\nization group (DMRG) method27,28and exact diagonal-\nization (ED). We put a particular focus on the case of\nS= 1. A summary of our results is presented in Sec-\ntion V, while technical details of the mapping to a di-\nlute gas and of the 1 /Sexpansion are given in Appendix\nA. A comparison of DMRG results for open boundary\nconditions vs. results for periodic boundary conditions is\nshown in Appendix B.\nII. TWO-MAGNON PROBLEM\nA. Solution of the two-magnon problem\nWe now solve the interacting two-magnon problem,\nstarting with the thermodynamic limit (for S= 1/2, see\nRef. 29). On a chain of finite length Lwith periodic\nboundary conditions the total momentum Kis a good\nquantum number due to the translational invariance of\nthe Hamiltonian HSin Eq. (1). Thus, it is convenient to\nuse a basis separating momentum subspaces\n|K,r/an}b∇acket∇i}ht=L/summationdisplay\nl=1eiK(l+r/2)S−\nlS−\nl+r|F/an}b∇acket∇i}ht, (2)\nwhere|F/an}b∇acket∇i}htis the fully polarized state, K= 2qπ/L\n(q= 0,1,···,L−1) andris the relative distance of\ntwo magnons. The allowed values of rdepend onSand\nthe parityof Landq. Forinstance, inthe caseof S >1/2\nandLeven,r= 0,1,..,L/2−1,(L/2) forqodd (even).\nWe expand a general two-magnon state with momen-\ntumKinto the (unnormalized) basis of Eq. (2) as\n|Ψ2M/an}b∇acket∇i}ht=/summationdisplay\nrCr|K,r/an}b∇acket∇i}ht (3)\nanddetermine Cranalyticallybysolvingthetwo-magnon\nSchr¨ odinger equation\nHS|Ψ2M/an}b∇acket∇i}ht=E2M|Ψ2M/an}b∇acket∇i}ht. (4)This leads to the recurrence relations\nΩ0C0=S/radicalbig\nS(2S−1)(ζ1C1+ζ2C2)\n(Ω0−J)C1=(2S−1)3/2\nS3/2ζ1C0+ζ1C2\n+ζ2(C1+C3)\n(Ω0−1)C2=(2S−1)3/2\nS3/2ζ2C0+ζ2C4\n+ζ1(C1+C3)\nΩ0Cr=ζ1(Cr+1+Cr−1)\n+ζ2(Cr+2+Cr−2),forr≥3,(5)\nwhereζ1= 2SJcos(K/2),ζ2= 2Scos(K). When\n|Ψ2M/an}b∇acket∇i}htis a bound state, Ω 0=Eb−4S(1+J2/8) whereEb\nis the (negative) binding energy (defined as the bound-\nstate energy minus the energy of the minimum of the\ntwo-magnon scattering states).\nThe (unnormalized) two-magnon bound states for a\ngivenKare constructed with the ansatz\nCr=e−κ−r+ve−κ+r(r≥1), (6)\nwhich, inserted in Eq. (5), leads to a characteristic quar-\ntic equation for r≥3\nΩ0z2−ζ1(z3+z)−ζ2(z4+1) = 0,(7)\nzbeing any of e−κ±with Re[κ±]>0. The remaining\nunknown quantities C0,vandEbare determined from\nthe remaining relations listed in Eq. (5).\nB. Scattering length in the lattice problem\nForS >1/2, bound states with energies below the\nminimum of the two-magnon scattering continuum ex-\nist only for K≃ ±2kcland only in a finite window of\ncouplings\n−4<J <J cr(S), (8)\nwithS-dependent critical values Jcr(S), as illustrated in\nTable I. The critical value Jcr(S), which is Jcr≈ −2.11\nforS= 1, quickly approaches Jcr(S)≃ −4 with increas-\ningS(see Tab. I). In fact, the 1 /Sanalysis to be pre-\nsented in Sec. III suggests the existence of an Scrbeyond\nwhich this window disappears completely.\nWe define the scattering length of bound states, in the\nthermodynamic limit L→ ∞, from their spatial extent\nS 1 3/2 2 5/2\n−Jcr(S)2.11 (2.95) 3.31 (3.42) 3.68 (3.66) 3.84 (3.80)\nTABLE I: Critical exchange couplings Jcr(S) for the existence\nof metamagnetism in Eq. (1) derived from solving the two-\nmagnon problem (values in parenthesis: Results from the 1 /S\nexpansion of Sec. III).3\n-4 -3.5 -3 -2.5 -2\nJ-3000300600scattering length aS/aS=1/2\nS=1\nS=3/2\nJcr(S=1) Jcr(S=3/2)\nFIG. 1: (Color online) 1D scattering length aS/a(a: lattice\nspacing) for S= 1/2,1,3/2 (solid, dashed, dot-dashed line).\n(in analogy to the continuum problem of particles inter-\nacting via a short-range, attractive potential):\naS=1\nmin{Re[κ±]}. (9)\nThebindingenergytakesitslowestvalue( i.e., the largest\nabsolute value) for K=K∗≃ ±2kcl=±2arccos( −J/4)\nand this quantity, with extremely high accuracy, is re-\nlated to the scattering length by\nEb(K∗)≃ −1\nma2\nS, (10)\nwheremis the one-magnon mass,\nm=2\nS(4−J)(4+J). (11)\nThe relation Eq. (10) between the binding energy and\nthe scattering length that holds for our microscopic lat-\ntice model is typical for a 1D Bose gas in the continuum\ninteracting via an attractive contact potential, the Lieb-\nLiniger model.30\nThe scattering length aScan also be determined from\nthe scattering problem of two magnons for general Sin\nthe thermodynamic limit. We have solved this problem,\nwith the momenta of the two magnons (which partici-\npate in scattering) taken in the vicinity of the same dis-\npersion minimum, k1=kcl+kandk2=kcl−k. From\nthe asymptotic form of the two magnon scattering state\nwavefunction we extract the scattering phase shift δS(k)\nfor anyS,\nlim\nr→∞Cr∼cos(rk+δS(k)). (12)\nToextractthescatteringlengthfromthescatteringphase\nshiftweusethesamerelationasina1Dcontinuummodel\nof particles interacting via a short-range potential\naS= lim\nk→0cot(δS(k))\nk. (13)\nThis allowsus to calculatethe scatteringlength in the re-\npulsive regime aS<0 as well (when two-magnon boundstates are not formed below the minimum of the scat-\ntering continuum). In the attractive regime aS>0, the\nscattering lengths obtained from both approaches [ i.e.,\nEq. (9) and Eq. (13)] are in excellent agreement with\neach other. As a side note on terminology, we call at-\ntractive (repulsive) regime the one in which the effective\ninteraction between magnons is attractive (repulsive).\nWe can now generalize the procedure31of mapping the\nantiferromagnetic (unfrustrated) spin- Schain close to\nsaturation onto the low-density limit of the Lieb-Liniger\nmodel with a coupling constant\ng0=−2\nmaS. (14)\nHowever, in our model the single-magnon dispersion\nhas two minima. The effective theory thus will be a two-\ncomponent (two species) Lieb-Liniger model. There are\ntwo types of low-energy scattering processes, first when\nmomenta of two magnons are in the vicinity of the same\ndispersion minimum that we have presented above (in-\ntraspecies scattering), and second, when the momenta\nk1andk2of two magnons are in the vicinity of different\nminima of dispersion, i.e.,k1=kcl+kandk2=−kcl−k\n(interspecies scattering). For the latter case we can re-\npeat all steps presented above and extract another cou-\npling constant ˜ g0from the interspecies scattering length,\n˜aSin analogy with Eq. (14). We obtain that ˜ g0>0 (im-\nplying that bound states with total momentum K= 0\nare never formed below the scattering continuum), and\n˜g0> g0for anySin the region −4< J <0. Since\nthe relation ˜ g0> g0always holds, the relevant scatter-\ning length at low energies is the intraspecies scattering\nlengthaS.\nThe scattering length, shown in Fig. 1, can be well\ndescribed, for small S >1/2, by a sum of two terms\n(resonances): aS≃λ−\nS/(4+J)+λ+\nS/(Jcr(S)−J) [where\nλ±\nSare numerical prefactors]. We emphasize that, for\nS≥1, the scattering length is in general much larger\nthan the lattice spacing, as is evident from Fig. 1: For\nS= 1,aStakes a minimum at J≃ −3.3 withaS≃80a.\nAdditionally, the emergence of bound states manifests\nitselfby a divergingscatteringlength at Jcr(S), whereaS\nchanges its sign jumping from −∞to +∞. Thus, bound\nstates are typically shallow, with a binding energy given\nby Eq. (10). In addition, the minima in their dispersion\noccur at incommensurate momenta K∗.\nForS= 1/2, anyJ <0 induces a two-magnon bound\nstate with total momentum K=π,18,20i.e.,aS>0\nand there is no resonance at −4< J <0 forS= 1/2.\nHence,S= 1/2 is very different from the S >1/2 case\nwhere bound states with K=πare never below the two-\nmagnon scattering continuum, and, as discussed above,\na resonance exists for −4<J <0 and 1≤S <S cr.\nIn order to analyze the finite-size effects with respect\nto results in the thermodynamic limit, we have numer-\nically diagonalized HSin the basis given in Eq. (2)\n(see Ref. 12 for details of the procedure). We obtained\nthe full spectrum, i.e., the scattering continuum and4\nL 1000 2000 4000∞\n−Jcr(S= 1) 2.252.162.132.11\nTABLE II: Finite-size dependence of critical values Jcr(S)\nfor the emergence of bound states below the minimum of the\ntwo-magnon continuum of scattering states, for S= 1.\nbound/antibound states if present, for selected values\nofJ∈[−4,0] in systems with up to L= 4000 sites\nand several values of S. Table II shows the numerical\ndetermination of Jcr(S) forS= 1 and different sys-\ntem sizes. Although finite-size effects are apparent, a\nquadratic fit in 1 /Lto numerical data for Jcr(S) extrap-\nolates toJcr≃ −2.11 in agreement with the result de-\ntermined directly in the thermodynamic limit (see the\npreceding discussion and Table I).\nIII. MAPPING OF THE SPIN HAMILTONIAN\nTO A DILUTE GAS OF BOSONS\nIn this section, we describe our effective theory in the\nthermodynamic limit, for the case of a finite (though\nvanishingly small) density of magnons. The mapping\nto a dilute gas of bosons is motivated by the following\nobservation: For S >1/2, we have shown that aSis\nlarge. Hence in the dilute limit, we can safely neglect\nthe hard-core constraint and take the continuum limit.\nForS= 1/2, on the contrary, the scattering length aSis\ntypically of the order of a few lattice constants and only\nfor−4< J <−3.9 doesaSbecome comparable to the\nsmallest value of the scattering length for S= 1.\nForS≥1, close to saturation, and in the low-energy\nlimit, we therefore map our system onto a dilute two-\ncomponent gas of bosons interacting with effective short-\nrange interactions. Many-body effects will be incorpo-\nrated by properly shifting the two-body T matrix off-\nshell as explained in Ref. 32. We show that, while the\ninterspecies interaction is always repulsive and stronger\nthan the intraspecies interaction, the latter undergoes a\nsign change. When the intraspecies interaction becomes\nnegative, the bosons are unstable against a collapse. We\nshowthata1 /Sexpansioncapturesthisphysicscorrectly\nand, similar to the case of the (unfrustrated) Heisenberg\nchain,33is applicable to the present problem, albeit its\none-dimensional nature.\nA. Effective Hamiltonian\nUsing the Dyson-Maleev transformation34(the Dyson-\nMaleev representation is used here for convenience. We\nhave checked that the explicitly hermitean Holstein-\nPrimakoff representation,35to leading order 1 /S, pro-vides equivalent results)\nSz\ni=S−a†\niai, S+\ni=√\n2Sai,\nS−\ni=√\n2Sa†\ni(1−a†\niai/2S), (15)\nwe map Eq. (1) onto a bosonic problem:\nH=/summationdisplay\nk(2Sǫk−µ)a†\nkak+/summationdisplay\nk,k′,qΓ0(q;k,k′)\n2La†\nk+qa†\nk′−qakak′,\n(16)\nwhere\nǫk=Jcosk+cos2k−(Jcoskcl+cos2kcl)≥0\nis the single-magnon dispersion and kcl= arccos( −J/4).\nNote that in our normalization, the minima of the single-\nparticle dispersion are at zero energy: ǫ±kcl= 0. The\nbare interaction vertex Γ 0is given by Γ 0(q;k,k′) =\nVq−1\n2(Vk+Vk′) withVk= 2Jcosk+ 2cos2k. The\nchemical potential is µ=hcl\ns−h, wherehcl\nsis the clas-\nsical saturation field value hcl\ns=S(J+ 4)2/4. We are\ninterested in the dilute regime µ→0.\nConcentratingonthe low-energybehaviorwearriveat,\nvia a Bogoliubov procedure,25,26a two-component Bose\ngas interacting via a δ-potential with Hamiltonian den-\nsity\nHeff=/summationdisplay\nα−|∇ψα|2\n2m+g0(S)\n2(n2\n1+n2\n2)+˜g0(S)n1n2.(17)\nHereψα,α= 1,2 describe bosonic modes with mo-\nmentacloseto ±kcl(i.e., theFouriertransformsof ψαare\nψ1(k→0)≈akcl+k,ψ2(k→0)≈a−kcl+k) andnα=ψ†\nαψα\nare the corresponding densities. The bare coupling con-\nstants of the effective 1D model of the two-component\nBose gas,g0(S) and ˜g0(S), are, in the dilute limit of\nbosons, related to the renormalized vertices of the micro-\nscopic model Eq. (16) through\nΓ(0;kcl,kcl) =g0(S)\n1+g0(S)√\n2m/(π√µ), (18)\nΓ(0;kcl,−kcl)+Γ(−2kcl;kcl,−kcl)=˜g0(S)\n1+˜g0(S)√\n2m\nπ√µ.\nThe relations Eq. (18) follow from a generalization of the\ncorresponding equation for the case of a one-component\nBose gas32,36to the two-component case using an RG\nanalysis37(see Appendix Afor detailsofthe calculation).\nB. 1/Sexpansion\nNext, we apply a 1 /Sexpansion to calculate the inter-\naction vertices and extract the coupling constants g0(S)\nand ˜g0(S). Using a standard ladder approximation the\nBethe-Salpeter equation for the vertices Γ reads:\nΓ(q;k,k′) = Γ 0(q;k,k′) (19)\n−1\n2SL/summationdisplay\npΓ0(q−p;k+p,k′−p)\nǫk+p+ǫk′−pΓ(p;k,k′).5\nSetting the transferred momentum q= 0 in Γ and the\nincoming momenta to k=k′=kcl, we get (see Appendix\nA for details):\nΓ(0;kcl,kcl)/bracketleftBigg\n1+V0−Vkcl\n2SL/summationdisplay\np1\nǫkcl+p+ǫkcl−p/bracketrightBigg\n=\nV0−Vkcl+1\n2SL/summationdisplay\np/bracketleftbigg\n1−Vp−V0\nǫkcl+p+ǫkcl−p/bracketrightbigg\nΓ(p;kcl,kcl)\n−1\n2SL/summationdisplay\np(V0−Vkcl)[Γ(p;kcl,kcl)−Γ(0;kcl,kcl)]\nǫkcl+p+ǫkcl−p.\n(20)\nNow, in the spirit of the 1 /Sexpansion, we replace the\nrenormalized vertex with the bare vertex on the right\nhand side of Eq. (20), Γ( p;kcl,kcl)→2ǫp, which is possi-\nble since there are no infrared divergences. Regularizing\nthe left hand side of Eq. (20) as in Refs. 32,36, and using\nEq.(18) weextractthe couplingconstantsofthe effective\nmodel. The analytical expression for g0(S) is\ng0(S) =F\n1−F(J2−8)\n|J|S(16−J2)3/2(21)\n(the derivation of Eq. (21) can be found in Appendix A\nand the constant Fis given in Eq. (A12)). To leading\norder inJ+4,\nlim\nJ→−4+g0(S)≃S−Scr\n4S(J+4)2+O/parenleftBig\n(J+4)5/2/parenrightBig\n.(22)\nTo first order in 1 /S, we obtain Scr= 6, which is not\nthat large a number, hence corrections beyond 1 /Smay\naffectScr. In the same way we calculate ˜ g0(S) as\n˜g0(S) =˜F\n1+J2−8\n16S(23)\n(the derivation is presented in detail in Appendix A; see\nEq. (A25) for the expression for ˜F).\nFrom Eqs. (21) and (23), we notice that g0(S)<˜g0(S)\nfor−4< J < 0. Thus the state below saturation is\na single-component one. Provided the interactions are\nrepulsive, the ground state is a translationally invariant\nchiral state,19where bosons prefer to ‘condense’ at the\nsameminimumofthesingle-particledispersionsincethey\nexperienceaminimalrepulsionthere.38Wealsonotethat\n|g0|m≪1 forJ <0, hence interactions between bosons\nare generically weak. In particular, even though m→\n∞when approaching the ferromagnetic Lifshitz point,\n|g0|m→0.\nThe effective bare intraspecies interaction is depicted\nin Fig. 2 and behaves as g0(S)∼[J−Jcr(S)] forJ→\nJcr(S). The scattering length is related to the effective\ncoupling constant by Eq. (14), signaling a resonance at\nJcr(S). Thus, we see that for S < S crthere is a finite\nregion near J≃ −4 whereg0(S)<0 and bosons attract\neach other, producing a collapsed state.-4 -3.95 -3.9\nJ-2 10-3-10-3010-32 10-3g0(S)\nS=3\nS=10\nS=∞-4-3.5-3-2.5-2\nJ-0.08-0.040g0(S)S=3/2\nS=1\nFIG. 2: (Color online) Effective bare intraspecies interact ion\ng0(S), forS= 3 (solid line), representative of the generic\nbehavior for S < S cr,S= 10 (dashed curve), representative\nofS > S cr[dot-dashed line: g0(S=∞)]. Inset: g0forS= 1\nand 3/2.\nTo corroborate this, using ED for Eq. (1) and S= 1\nwith periodic boundary conditions, we have calculated\nthe ground-state momentum of the states with a small,\nbut finite number of magnons, which is incommensurate,\nsupporting the picture of a uniform chiral state in the\nrepulsive case J > J cr, and a collapsed state in the at-\ntractive case J < J crat one of the two minima of the\nsingle-particle dispersion.\nIn the attractive case, ∂2E0/∂n2<0, whereE0and\nnare the bosons’ ground-state energy and density, re-\nspectively. In the language of spins, the inverse magnetic\nsusceptibility at saturation becomes negative, and hence,\nfollowing standard arguments,39we conclude that there\nis a first-order transition at M= 1,i.e., a jump in the\nmagnetization curve just below saturation.\nAs pointed out above, the case of S= 1/2 is special\nsince the scattering length is typically of the order of the\nlatticeconstanthere. Themappingofthe S= 1/2caseto\na two-component Bose gas (by the procedure presented\nabove forS >1/2) can be trusted only for J→ −4,\nwhere the scattering length becomes much larger than\nthe lattice constant. In that case we can easily incorpo-\nrate the exact hard-core constraint into our formalism26\nand again expect that S= 1/2 also shows metamag-\nnetic behavior. This conclusion is in agreement with\nDMRG results for S= 1/2.14Note that a metamagnetic\njump can also be stabilized for spin 1 /2 with suitable\nanisotropicexchangeinteractions.40,41However,with our\nprocedure we cannot account for the formation of stable\ntwo-, three-, and four-magnon bound states that is char-\nacteristic for most of the region J >−4 in the spin-1 /2\nfrustrated ferromagnetic Heisenberg chain.12–14,20\nGoing back to S >1/2, at lower M, corresponding to\nhigherdensitiesofmagnons,thehard-corenatureofspins\neventually prevails as well, resulting in a uniform ground\nstate at a nonzero momentum. However, as alreadymen-\ntioned, from the finite-size analysis of the two-magnon\nproblem,weobservethatboundstatesdisappearwithde-\ncreasingL, suggestingthat the attractive effective poten-\ntial (in the limit of a small magnon density) can become6\n0 0.1 0.2 0.3 0.4 0.50.6\nh00.20.40.60.81M(h)\nJ=-3.5\nJ=-3\nJ=-2.5(a) S=1, L=128\n∆Mjump\n0.07 0.08 0.09 0.1\nh0.50.60.70.80.91\nM(h)\n00.511.522.5\nh00.20.40.60.81M(h)(c) S=3/2, J=-3.5 (b) S=1, J=-1\nhc\nFIG. 3: (Color online) (a) Magnetization curves M(h) for\nS= 1 atJ=−2.5,−3,−3.5. (b)M(h) forS= 1,J=−1\n(c)M(h) forS= 3/2 atJ=−3.5 (all for L= 128).\nrepulsive upon increasing the magnon densities. Thus,\nthe state below the jump ( i.e., 0< M < 1−∆Mjump,\nwhere ∆Mjumpis the height of the jump) will be similar\nto the one encountered in the case of J > J cr,i.e., it is\na translationally uniform chiral state.\nIV. DMRG RESULTS\nNext, we turn tonumericalresultsforthe caseof S= 1\n(unless stated otherwise), solving for the ground state of\nEq. (1) in a finite magnetic field h, using ED where pos-\nsible or DMRG.42We present data from DMRG simu-\nlations using up to 1200 states, for L≤128 sites, and\nfor open boundary conditions (OBC), unless stated oth-\nerwise.\nA. Magnetization curves and magnetization\nprofiles\nThe main result of this work, namely, the metamag-\nnetic transition from a gapless finite-field phase to full\nsaturation, is clearly seen in the magnetization curves\nshown in Fig. 3(a). For S= 1, we observe the appear-\nance of this jump for −4< J/lessorsimilar−2, while for S= 3/2\n[an example is shown in Fig. 3(c)] the jump exists in a\nmuch narrower window J/lessorsimilar−3. This is consistent with\nour analytical results for Jcr, listed in Tab. I. Moreover,\nfor−2/lessorsimilarJ <0, we resolve a plateau in M(h) atM= 0,\nwhich is due to the Haldane gap43in thisS= 1 system\n[see Fig. 3(b)]. This gap defines the critical field hcthat0 50 100\nsite i00.10.20.30.40.51-<Sz\ni>J=-3 (jump at Sz=97)\nJ=-1 (no jump)S=1, L=128\nFIG. 4: (Color online) Magnetization profiles for S= 1,L=\n128 atJ=−1 (no jump, dashed lines) and J=−3 (jump at\nSz= 97) for Sz= 80,90,110,120 (top to bottom).\nseparates the gapped Haldane phase from the finite- M\nphase with a smooth M(h)-behavior for hc<h<h sat.\nThe collapse of magnons manifests itself in the magne-\ntization profiles (1 −/an}b∇acketle{tSz\ni/an}b∇acket∇i}htvs. sitei) using OBC, displayed\nin Fig. 4 for J=−3 andJ=−1. In the former case,\nthere is a jump, but in the latter, there is none. Clearly,\nin the states that get skipped over ( Sz>97 forJ=−3),\nmagnons collapse into the center of the system, whereas\nfor actual ground states below the metamagnetic transi-\ntion, themagnetizationprofilesbecomeflat. Bycontrast,\nin the case of J=−1 where the transition to the fully\npolarized state is smooth and continuous, allprofiles are,\napart from boundary effects, flat.\nB. Central charge\nOur effective theory developed in Section III suggests\nthat the gapless phase in the region hc< h < h satis a\none-component phase (where hsatis the saturation field).\nTo substantiate this result, one can make use of entan-\nglement measures such as the von-Neumann entropy to\nextract the central charge, which directly yields the num-\nber of components of the gapless state.\nThe von-Neumann entropy is defined as\nSvN(l) =−tr(ρllnρl), (24)\nwhereρlis the reduced density matrix of a subsystem of\nlengthlof our one-dimensional chain of length L. In a\ngapless state that is conformally invariant, the landL\ndependence of the von-Neumann entropy is given by44,45\nSvN(l) =c\n3ln/parenleftbiggL\nπsin(π\nLl)/parenrightbigg\n+g, (25)\nwhich is valid for systems with periodic boundary con-\nditions (PBC). PBC are preferable for the calculation of\nthe central charge from Eq. (25) since for OBC, there\nmay be additional oscillatory terms. gis a non-universal\nconstantthatdependson M. AsDMRGdirectlyaccesses\nthe eigenvalues of these reduced density matrices,28it is7\n01020 30 40 5060\nl01234SvN(l)\n1020 30 40 5060\nl1020 30 40 5060\nlDMRG\nFit with c~1Spin S=1, PBC\n(a) J=-1, M=1/2(b) J=-2, M=1/2 (c) J=-3, M=5/16\nFIG. 5: (Color online) DMRG results for the von-Neumann\nentropySvN(l) in the gapless phase hc< h < h satof theS= 1\nsystem: (a) J=−1,M= 1/2, (b)J=−2,M= 1/2, (c)\nJ=−3,M= 5/16 (symbols). The lines are fits to Eq. (25),\nresulting in c= 1.0±0.1 in all cases (we exclude SvN(l) for\nl <10 andl >54 from the fit). In this figure, we display\nresults for periodic boundary conditions and L= 64 sites.\nstraightforward to measure SvN(l) with this numerical\nmethod.\nSome typical DMRG results (squares) for systems of\nL= 64 and periodic boundary conditions are presented\nin Fig. 5. We have fitted the expression Eq. (25) to our\nnumericaldata(shownassolidlinesin thefigure)andob-\ntainc= 1.0±0.1 in all examples. Therefore, we expect\nthe gapless phase to be a (chiral) one-component liquid.\nNote, though, that at both small Mand|J|, where the\nconvergence of DMRG is notoriously difficult, we can-\nnot completely rule out the presence of a c= 2 region,\nwhich, however, is irrelevant for the main conclusions of\nour work.\nC. Phase diagram for S= 1\nOur results for the S= 1 chain are summarized in the\nhvs.Jphase diagram Fig. 6. We identify three phases:\n(i) a gapped M= 0 phase at h < h c(similar to the\nDouble-Haldane phase known for J >0, see Ref. 46), (ii)\na gapless (chiral) finite-field phase for hc<h<h sat, and\n(iii) the fully polarized state at hsat< h(withhsat= 0\nforJ <−4).\n∆Mjumpis plotted in the inset of Fig. 6: the jump sets\nin atJ/lessorsimilar−2 (close to where the zero-field gap becomes\nsmall renderingit difficult to resolveit numerically), con-\nsistent with our theory.\nSince in the limit of J→0, one has two spin-1\nchains with antiferromagnetic interactions which both\nseparately have a Haldane gap at zero field,43upon\ncoupling the chains, one obtains the so-called Double-\nHaldane phase (in contrast to the regular Haldane phase\nthat is inherited from a single spin-1 chain with antifer-\nromagnetic interactions). Both phases, Double-Haldane\nand Haldane phase, are realized in the frustrated, anti-\nferromagnetic spin-1 chain,46yet in our case, only the\nDouble-Haldane phase exists. The determination of the-4 -3 -2 -1 0\nJ01234 hhsat\nhc-4 -3 -2\nJ00.20.40.6\n∆MjumpL=64\nL=128Spin S=1\n(Double-)HaldaneFully polarizedgapless (chiral) \n[t!]\nFIG. 6: (Color online) Phase diagram of Eq. (1) for S= 1\n(circles: saturation field hsat; squares: spin gap hc). Inset:\nheight ∆ Mjump of the metamagnetic jump vs. J. The com-\nparison of L= 64 (open symbols) and L= 128 (solid symbols)\nas well as finite-size scaling (not shown here) supports that\nboth ∆Mjump andhcare finite in extended regions of J.\ncorresponding spin gap hcis a bit subtle. Namely, a Hal-\ndane chain with open boundaries gives rise to spin-1/2\nexcitations at the open ends.47Since we have two chains\n(for small |J|), we have a total of four spin-1/2 end spins.\nHence, the spin gap in Fig. 6 is determined from\nhc=E(Sz= 3)−E(Sz= 2), (26)\nwhereE(Sz) is the ground-state energy in a sector with\na given total Sz.\nIt is worth emphasizing several differences with the\nphase diagram of the spin-1/2 version of Eq. (1). First,\nforS= 1, there are no multipolar phases, which oc-\ncupy a large portion of the corresponding spin-1/2 phase\ndiagram.11–14,20Second, the spin-1/2 system features an\ninstability towards nematic order,11,12,18which can be\nexcluded on general grounds for integer spin- Schains,19\neven for 0<M≪1.\nV. SUMMARY\nIn conclusion, we showed that resonances can play a\ncrucial role in determining the low-energy behavior of\nfrustrated quantum spin systems subject to a magnetic\nfield. The proximityofresonancescaused by an interplay\nbetween frustration ( J >−4) and quantum fluctuations\n(1/2<S <S cr) results in extremely large values of the\n1D scattering length that allows to develop an effective\ntheory of a weakly interacting two-component Bose gas.\nThe quasi-collapse of the dilute gas of magnons provides\nthe physical origin of the emergent metamagnetism.\nThe preditictions of our analytic theory were verified\nby numerical data. We focused on the case S= 1 since\nthere the jump in the magnetization curve is the most\npronounced. As a by-product, we obtained the phase\ndiagram for the S= 1J-J′chain with antiferromagnetic\nJ′and ferromagnetic Jin a magnetic field. This phase\ndiagram is remarkablysimple. In particular, there are no8\nmultipolar phases in the case S= 1, in marked contrast\nto the spin-1/2 case.11–14,20\nAcknowledgments\nWe thank H. Frahm, A. Kolezhuk, R. Noack, and D.\nPetrov for fruitful discussions. A.H. acknowledges finan-\ncial support from the DFG via a Heisenberg fellowship\n(HO 2325/4-2). T.V. is supported by the Center of Ex-\ncellence QUEST. G.R. and M.A. are partially supported\nby CONICET (PIP 1691) and ANPCyT (PICT 1426).\nAppendix A: Details on the mapping to a dilute\nBose gas\nIn Sec. III, we mapped the spin Hamiltonian Eq. (1)\nclose to saturation to an effective, bosonic field theory\nEq. (17). Here we provide the details of generalizing the\nprocedure of obtaining the coupling constants from the\nmany-body T-matrix for an effective single-component\nBose gas model, which is described in Ref. 32, to the\ncase relevant to us here, namely an effective theory of a\ntwo-component Bose gas.\nWe introduce the ‘mean-field interaction coefficients’,\ng(S) = Γ(0;kcl,kcl) (A1)\n˜g(S) = Γ(0;kcl,−kcl)+Γ(−2kcl;kcl,−kcl),(A2)\nwhere Γ(q;k,k′) is the full interaction vertex of the\nHamiltonian Eq. (16). They satisfy mean-field like re-\nlations for ˜g0>g0,\nµ=g(S)n (A3)\nand for ˜g0<g0\nµ= (g(S)+ ˜g(S))n/2, (A4)\nwheren=/an}b∇acketle{t/summationtext\nαnα/an}b∇acket∇i}ht →0 is the total density of bosons.\nTo get the connection between g(S) and ˜g(S) on the one\nhand and the bare coupling constants of the effective 1D\nmodel of a two-component Bose gas, g0(S) and ˜g0(S) on\nthe other hand, we generalizethe correspondingequation\nfor the case of a one-component Bose gas32to the two-\ncomponent case, consistent with the SU(2) symmetry of\ntheRGfixedpointofadilute, two-componentBosegas,37\nlimµ→0g(S) = lim µ→0˜g(S) =π√µ/√\n2m,\ng(S) =g0(S)\n1+g0(S)√\n2m/(π√µ),(A5)\n˜g(S)=˜g0(S)\n1+ ˜g0(S)√\n2m/(π√µ).(A6)\nTo zeroth order in 1 /S, we have\ng(S=∞) =V0−Vkcl=(J+4)2\n4(A7)and\n˜g(S=∞) =V2kcl+V0−2Vkcl>g(S=∞).(A8)\nThus, classically ( S→ ∞,m→0), we have ˜ g0> g0for\n−4< J <0. We will show below that incorporating\nquantum fluctuations does not modify this relation.\nIn the following we will calculate the effective in-\ntraspecies and interspecies interaction constants, to see\nhow 1/Scorrections modify g0(S) and ˜g0(S). In order to\nobtaing(S) wehaveto set in Γ( q,k,k′) givenbyEq.(19),\nq= 0 andk=k′=kcl,\ng(S) =V0−Vkcl+1\n2SL/summationdisplay\npΓp\n−1\n2SL/summationdisplay\npVp−V0+V0−Vkcl\nǫkcl+p+ǫkcl−pΓp,(A9)\nwherewehavedenoted Γ( p;kcl,kcl) = Γp. After straight-\nforward manipulations we obtain,\ng(S)/bracketleftBigg\n1+V0−Vkcl\n2SL/summationdisplay\np1\nǫkcl+p+ǫkcl−p/bracketrightBigg\n=\nV0−Vkcl+1\n2SL/summationdisplay\npΓp/bracketleftbigg\n1−Vp−V0\nǫkcl+p+ǫkcl−p/bracketrightbigg\n−1\n2SL/summationdisplay\np(V0−Vkcl)(Γp−Γ0)\nǫkcl+p+ǫkcl−p. (A10)\nNow, on the right hand side of Eq. (A10), we plug\nin the zeroth order vertex (in the 1 /Sexpansion),\nΓ(p;kcl,kcl)→2ǫp(which is possible because there are\nno infrared divergencesanymore), and use that Vp−V0=\n2(ǫp−ǫ0). Equation (A10), after passing to infinite sys-\ntem size, becomes\ng(S)\n1+F\n4πSπ/integraldisplay\n−π1\nǫkcl+p+ǫkcl−p\n=F,(A11)\nwhere\nF=V0−(1+1\n2S)Vkcl−1\nπSπ/integraldisplay\n−πdpǫp(ǫp−ǫ0)\nǫkcl+p+ǫkcl−p\n−V0−Vkcl\n2πSπ/integraldisplay\n−πdpǫp−ǫ0\nǫkcl+p+ǫkcl−p(A12)\nandwehaveusedthat V0−Vkcl=F+O(1/S). According\ntothe schemethat wefollowthe two-bodyTmatrixmust\nbecalculatedoff-shell, thusthedenominatorinEq.(A11)\nmust be understood as ǫp→ǫp+Cµ/4S, where the exact\nvalue of the numerical constant is C=π2/8.32This leads\nto (compare Eqs. (18) and (A1))\ng(S) =g0(S)\n1+g0(S)√\n2m/(π√µ),(A13)9\nwhere we have introduced the intraspecies Lieb-Liniger\ncoupling constant g0(S) as in Eq. (21). Note that all\nintegrals presented in this section are evaluated analyti-\ncally, which is a nice feature of the 1 /Streatment of our\nproblem.\nNow we outline the calculation of ˜ g(S). We denote\nΓ(p;kcl,−kcl) =˜Γpand obtain the Bethe-Salpeter equa-\ntion for\n˜Γ0=V0−Vkcl+1\n4πSπ/integraldisplay\n−πdp˜Γp−1\n4πSπ/integraldisplay\n−πdpV−p−Vkcl\n2ǫkcl+p˜Γp\nand\n˜Γ−2kcl=V−2kcl−Vkcl+1\n4πSπ/integraldisplay\n−πdp˜Γp\n−1\n4πSπ/integraldisplay\n−πdpV−2kcl−p−Vkcl\n2ǫkcl+p˜Γp.(A14)\nAdding these two equations gives ˜ g(S) =˜Γ0+˜Γ−2kcl,\n˜g(S) =V0+V−2kcl−2Vkcl+2\n4πSπ/integraldisplay\n−πdp˜Γp\n−1\n4πSπ/integraldisplay\n−πdpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p˜Γp.(A15)\nWe divide the last term in Eq. (A15) into two pieces,\n−1\n4πSπ/integraldisplay\n−πdpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p˜Γp=I1+I2,(A16)\nwhere\nI1=−1\n4πS−kcl/integraldisplay\n−π−kcldpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p\n×(˜Γp−˜Γ−2kcl+˜Γ−2kcl) (A17)\nand\nI2=−1\n4πSπ−kcl/integraldisplay\n−kcldpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p\n×(˜Γp−˜Γ0+˜Γ0). (A18)\nNote that, for convenience, we have shifted the first Bril-\nlouin zone, ( −π,π)→(−π−kcl,π−kcl). Shifting the\nT-matrix off-shell, we have (the integral with a dash de-\nnotes the principal value)\nI1=−1\n4πS−kcl\n−/integraldisplay\n−π−kcldpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p(˜Γp−˜Γ−2kcl)\n−˜Γ−2kcl\n4πS−kcl/integraldisplay\n−π−kcldpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p+Cµ/2S(A19)and\nI2=−1\n4πSπ−kcl/integraldisplay\n−kcldpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p(˜Γp−˜Γ0)\n−˜Γ0\n4πSπ−kcl/integraldisplay\n−kcldpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p+Cµ/2S.(A20)\nThe last terms in I1can be written as\n−˜Γ−2kcl\n4πS−kcl/integraldisplay\n−π−kcldpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p+Cµ/2S=\n−˜Γ−2kcl\n4πS−kcl\n−/integraldisplay\n−π−kcldpVp+V−2kcl−p−(V0+V−2kcl)\n2ǫkcl+p\n−˜Γ−2kcl\n4πS−kcl/integraldisplay\n−π−kcldpV0+V−2kcl−2Vkcl\n2ǫkcl+p+Cµ/2S.(A21)\nSimilarly, for the last terms in I2we get,\n−˜Γ0\n4πSπ−kcl/integraldisplay\n−kcldpVp+V−2kcl−p−2Vkcl\n2ǫkcl+p+Cµ/2S=\n−˜Γ0\n4πSπ−kcl\n−/integraldisplay\n−kcldpVp+V−2kcl−p−(V0+V−2kcl)\n2ǫkcl+p\n−˜Γ0\n4πSπ−kcl/integraldisplay\n−kcldpV0+V−2kcl−2Vkcl\n2ǫkcl+p+Cµ/2S. (A22)\nNoting that\n−1\n4πSπ−kcl\n−/integraldisplay\n−kcldpVp+V−2kcl−p−(V0+V−2kcl)\n2ǫkcl+p\n=−1\n4πS−kcl\n−/integraldisplay\n−π−kcldpVp+V−2kcl−p−(V0+V−2kcl)\n2ǫkcl+p\n=J2−8\n16S, (A23)10\nand gatheringall contributions, Eq. (A15) takes the form\n˜g(S)\n1+J2−8\n16S+V0+V−2kcl−2Vkcl\n16πSπ/integraldisplay\n−πdp\nǫp+Cµ/4S\n\n=V0+V2kcl−2Vkcl+8+J2\n4S\n−1\n4πS0\n−/integraldisplay\n−πdpVp−kcl+Vp+kcl−2Vkcl\n2ǫp(˜Γp−kcl−˜Γ−2kcl)\n−1\n4πSπ/integraldisplay\n0dpVp−kcl+Vp+kcl−2Vkcl\n2ǫp(˜Γp−kcl−˜Γ0).(A24)\nNow we can plug the zeroth-order vertices into the\nright hand side of Eq. (A24), in the spirit of the 1 /S\nexpansion: ˜Γp→2ǫp. Noting that V0+V−2kcl−2Vkcl=\n˜F+O(1/S), where\n˜F=V0+V2kcl−2Vkcl+8+J2\n4S(A25)\n−1\n2πS0\n−/integraldisplay\n−πdpVp−kcl+Vp+kcl−2Vkcl\n2ǫp(ǫp−kcl−ǫ−2kcl)\n−1\n2πSπ/integraldisplay\n0dpVp−kcl+Vp+kcl−2Vkcl\n2ǫp(ǫp−kcl−ǫ0),\nin the same way as for g(S) we obtain\n˜g(S) =˜g0(S)\n1+ ˜g0(S)√\n2m/(π√µ)(A26)\nwith ˜g0(S) given in Eq. (23).\nWe see that in order 1 /S, quantum fluctuations do not\nmodify the relation ˜ g0(S)>g0(S) forJ <0. Therefore,\nthe interspecies interaction is always positive (repulsive)\nfor anyJandS >1/2, and behaves as ˜ g0(S)∼(J+4)2\nforJ→ −4 for allS. A hard-core constraint, stemming\nfrom the mapping of spins to bosons, is not included in\nour theory. While it is well-known how to treat the hard-\ncore constraint in an exact way for S= 1/2,26this is not\nthe case for S >1/2.\nAppendix B: DMRG results for periodic boundary\nconditions\nThe DMRG results presented in the main text were\ncomputed for open boundary conditions (OBC), except\nfor those in Fig. 5 where we used periodic boundary con-\nditions (PBC). Alternatively, one may compute all the\ndata,e.g., the magnetization curves M(h) using PBC.\nThis approach is generally expected to suffer from (i)\nslower convergence with respect to the number of states\nkept in the DMRG runs28and (ii) large finite-size ef-\nfects due to the incommensurability in the problem. We00.05 0.1\nh00.20.40.60.81M(h)\n00.511.522.53\nh00.51\nM(h)S=1, L=48, PBC\nhc(a) J=-3.5\n∆Mjump(b) J=-1\nFIG. 7: (Color online) Magnetization curves M(h) forS= 1\n(a)J=−3.5 and (b) J=−1 (L= 48), calculated with\nDMRG on systems with periodic boundary conditions. In\npanel (a), we clearly see the metamagnetic jump of height\n∆Mjump and in panel (b), the Haldane gap that defines hc\nshows up as a zero-field magnetization plateau.\nhere demonstrate that all main features can be seen with\nboth OBC and PBC, namely the existence of the meta-\nmagnetic jump and the Haldane gap. Moreover, the\nquantitative results are comparable, except for the ex-\npected finite-size effects due to the incommensurability.\n1. Magnetization curves from periodic boundary\nconditions for S= 1\nFirst, we discuss some examples of magnetization\ncurves obtained for PBC. We kept up to m= 1600 states\nfor the PBC DMRG computations presented here. In\naddition, we used exact diagonalization for those sectors\nwhich are sufficiently small. Therefore, in particular the\ndata close to the saturation field are free of truncation\nerrors.\nFigure 7 shows the magnetization curves for J=−3.5\n[panel (a)] and J=−1 [panel (b)] for L= 48 and PBC.\nAs in the results for OBC, we observe the presence of the\nmetamagnetic jump (here in the case of J=−3.5) and\nthe Haldane gap (see the J=−1 curve), which manifests\nitself as a plateau in the magnetization curve at M= 0.\nThe latter defines the critical field hcthat separates the\ngapped Double-Haldane phase from the gapless finite- M\nphase (compare Fig. 5).\nNote that at intermediate M, the PBC data show spu-\nrious small steps with ∆ Sz>1. We have checked that\nthese features disappear as one goes to larger system\nsizes.\n2. Comparison of OBC vs PBC for S= 1\nFigure 8 contains the results for the Haldane gap that\ndefines the critical field hcseparating the gapped zero-\nfield phase from the gapless phase at finite magnetiza-\ntions, the saturation field hsat, and the jump height (in-\nset), comparing data for OBC (open symbols) with data11\n-4 -3 -2 -1 0\nJ01234 hhc (PBC)\nhsat (PBC)\nhsat\nhc-4 -3 -2\nJ00.20.40.6\n∆MjumpPBC\nOBCSpin S=1, L=64\n(Double-)HaldaneFully polarizedgapless (chiral) \nFIG. 8: (Color online) Phase diagram of the frustrated ferro -\nmagnetic S= 1 chain, comparing DMRG results from systems\nwith OBC (open symbols) to results from systems with PBC\n(solid symbols). L= 64 in all cases. Lines are guides to the\neye.\nfromPBC(solidsymbols). HerewechooseachainlengthofL= 64 for both PBC and OBC.\nFor PBC we can determine the spin gap from\nhPBC\nc=E(Sz= 1)−E(Sz= 0),(B1)\nwhere, asin Eq.(26), E(Sz) is the ground-stateenergyin\na sector with a given total Sz. The good agreement be-\ntween the OBC and PBC results for hcin Fig. 8 confirms\nthat it is indeed appropriate to use Eq. (26) for OBC.\nThe PBC data for ∆ Mjumpin Fig. 8 suffer from the\npresence of several peaks, resulting in a non-monotonic\ndependence on J. This is due to the incommensurabil-\nity in the finite-magnetization region, which is incom-\npatible with the lattice vectors of a system with peri-\nodic boundary conditions that is translationally invari-\nant. Therefore, in the main text we focus on the discus-\nsion of DMRG data from systems with OBC. The results\nfrom OBC and PBC data for the saturation field hsat,\nhowever, agree very well with each other.\n1T. Giamarchi, C. R¨ uegg, and O. Tchernyshyov, Nature\nPhys.4, 198 (2008).\n2E. M. Chudnovsky and L. Gunther, Phys. Rev. Lett. 60,\n661 (1988).\n3L. Balents, Nature 464, 199 (2010).\n4K. Liu and M. Fisher, J. Low Temp. Phys. 10, 655 (1973).\n5J. Struck, C. ¨Olschl¨ ager, R. Le Targat, P. Soltan-Panahi,\nA. Eckardt, M. 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We find a substantial decrease in th e nucleation field when microwave fields are \napplied, in comparison to pulse fields. A clear resonance peak is observed in the frequency dependence of the nucleation field, which coincides with the uniform mode ferromagnetic \nresonance frequency. Owing to the well-define d nucleation process, the switching field \ndistribution is small in contrast to previous reports. Our results show that localized microwave field provides an efficient tool for injecting domain walls into magnetic nanowires. \n \n  \n \n  \n \n  \n \n  \n \n*E-mail address: hayashi.masamitsu@nims.go.jp2 \n   Recent progress in the understanding of spin transfer torque and its related phenomena \nenabled development of domain wall based memory1 and logic devices2.  In particular, a \nmagnetic shift register mimicking the so called “Racetrack memory”1 has been demonstrated \nusing localized magnetic fields to inject doma in walls and spin polarized current to move \nthem3-5.  One challenge of exploring materials for domain wall based device applications is \nthe difficulty in injecting domain walls into the nanowire since it typically requires large \nlocalized magnetic field3,6.  Solid understanding of the nucleation process is thus required \nfor further developmen t of such devices. \n  Microwave assisted magnetization reversal (MAMR) provides a unique tool to lower the \nmagnetization switching field7,8.  The concept has been demonstrated in a number of \nsystems7,9-18 but the understanding of the underlying physics is still developing.  In most of \nthe systems, it is likely that the magnetization reversal process involves nucleation and \npropagation of domain walls (exception is the coherent rotation of magnetic moments12).  \nBoth domain wall (reversed domain) nucleation and propagation can be enhanced by the \napplication of microwave field.  However, its response may differ depending on the applied \nmicrowave frequency and thus may cause broadening of the switching field9,10,17,18.  Thus to \ngain further insight into MAMR, it is essential to distinguish the effect of microwave field \napplication on domain wall nucleation19 and domain wall propagation20,21.   \nHere we design a system to study the effect of microwave field application on a \nwell-defined domain wall nucleation process in permalloy naonwires.  Substantial drop in the nucleation field is observed for a microwave field frequency of ~3 GHz.  We find a clear \nresonance peak in the frequency dependence of  the nucleation field, in which the peak \nposition corresponds to the ferromagnetic resonance uniform excitation mode.  The domain 3 \n wall nucleation field is reduced by ~50% at the resonance condition compared to the pulse \nfield excitation, which shows the high efficiency of microwave wave assisted domain wall \nnucleation process. \n  P e r m a l l o y  ( N i 81Fe19) nanowires are made from film stack of 10 Ni 81Fe19|5 Ru (units in nm) \ndeposited on single crystal MgO substrate using magnetron sputtering.  Figures 1(a) shows \nan optical microscopy image of the device an d schematic illustration of the measurement \nsetup.  Electron beam (e-beam) lithography and Ar ion milling are used to define the \nmagnetic nanowire.  Typical width and length of the nanowire are 600 nm and 32 m, \nrespectively.  Contacts to the nanowire are made from 5 Ta|85 Au using e-beam lithography \nand a liftoff process.  An isolation pad, 30 nm thick SiO 2 layer, is inserted between the \npermalloy nanowire and the field generation lines (lines A and B in Fig. 1(a)).  The width of \nline A parallel to the permalloy nanowire is set to ~2 m, large enough to cover the 600 nm \nwide nanowire.  A signal generator or a pulse generator is connected to field generation lines \nA and/or B to apply localized magnetic field.  Note that the Oersted field generated at line A \n(B), when current is passed along the line, is pointing transverse (parallel) to the wire’s long \naxis.  The rise time of the pulse generator is typically ~500 ps.  We use a 7 GHz bandwidth \ndigitizing oscilloscope, connected to line C, D in Fig. 1(a), to monitor the change in the \nvoltage drop across this section (C-D) of the wire.  DC current (~500 A) is supplied to the \nnanowire via a bias tee in order to read out the wire’s resistance through the scope voltage.   \nPropagation of domain walls are studied using time resolved voltage measurements.  We \ncycle the external magnetic field to first reset the magnetic state of the wire by applying a \nlarge magnetic field -H SAT, and then reduce the field to H in order to nucleate and drive a \ndomain wall along the wire, and finally apply -H SAT to avoid the electromagnet to get 4 \n demagnetized.  Positive field points along the +x direction in Fig. 1(a).  When the field is at \nH, we apply a voltage pulse (180 ns long) from the pulse generator to apply a localized \nOersted field around the field generation line. The direction of the Oersted field, determined \nby the polarity of the voltage pulse, is set to point along H (for pulses applied to line B) to \nassist the magnetization reversal process.  We monitor the scope voltage when the voltage \npulse is applied.  The field cycling is repeated for 256-512 times in order to increase the \nsignal to noise ratio. \nWe first examine the domain wall nucleation fiel d when the local Oersted field is directed \nparallel to the wire’s long axis.  Here, a pulse generator is connected to line B.  Figures 1(b) \nand 1(c) show representative scope traces captured when the amplitude of the voltage pulse \n(VP) applied to the field generation line is varied ( pulse length is fixed to 180 ns).  Due to the \nanisotropic magnetoresistance (AMR) of permal loy, the scope signal drops from zero to a \nnegative value when a domain wall enters section C-D22.  For V P=-0.8 V (Fig. 1(b)), we \nobserve signatures of domain wall nucleation and injection into section C-D for all H shown.  \nHowever, for V P=-0.5 V (Fig. 1(c)), no domain wall injection is observed when H is 35 Oe.  \nNote that the propagation field of this wire, i.e. the minimum field needed to move a domain \nwall along the wire, is ~ 5 Oe.  These results indicate that when H=35 Oe and V P=-0.5 V , \ncombination of H and the Oersted field is not enough to nucleate a domain wall, thereby \nresulting in no domain wall passing by section C-D. \nWe define the “assisting field” as the minimum H needed to assist the domain wall \nnucleation process.  The assisting field is plotted in Fig. 1(d) as a function of the current \npassed along the line B.  For smaller voltage pulse, larger assisting field (H) is needed to \nnucleate a domain wall.  As a reference, the dot ted line in Fig. 1(d) represent the field (H) 5 \n needed to reverse the wire magnetization with quasi-static field.  This field (defined as \nHS~77 Oe hereafter) represents the minimum field needed to nucleate a reversed domain (i.e. \na domain wall) likely from the edges of the wire.  Once a domain wall is created, the wall \ncan propagate along the wire as long as the field is larger than ~5 Oe, which results in \nmagnetization reversal.  Note that the total fi eld required to locally nucleate a domain wall, \ni.e. the nucleation field, is larger than H S in all cases.  For example, the nucleation field for \nVP=-0.8 V is ~125 Oe, which includes the assisting field of ~15 Oe and the current induced \nOersted field at line B of ~110 Oe (the Oersted field is calculated using the Biot-Savart law23).   \nWe next turn to the effect of the transverse field (perpendicular to the wire’s long axis and \nH) on the domain wall nucleation process. Here  we use line A to generate the transverse \nOersted field (pulse generator is  connected to line A).  Similar to the method described in \nFig. 1(d), we extract the assisting field H need ed to nucleate a domain wall.  The assisting \nfield is represented by the red circles in Fig. 2( e), plotted as a function of current passed along \nline A.  As observed with the parallel field, the assisting field decreases with the magnitude \nof current.   \nThese assisting fields are compared with those obtained using the microwave Oersted field. \nA signal generator (microwave source) is connected to line A and a pulse generator is \nconnected to line B.  Since the rise time of the microwave source is slow (more than ~1 s), \nthe source is turned on throughout the field cy cling.  The microwave Oersted field is small \nenough such that it does not affect the saturation process when H SAT is applied.  A voltage \npulse (-2.0 V , 180 ns) is applied from line B to nucleate a domain wall, as in Fig. 1(d).  \nExemplary scope traces are shown in Fig. 2(a-c) for three different microwave frequencies f RF.  \nThe microwave power P RF is fixed to 0 dBm (Vrms~0.22 V).  For microwave frequencies of 6 \n 0.1 GHz (Fig. 2(a)) and 6.0 GHz (Fig. 2(c)), signature of domain wall nucleation and \npropagation is observed for all H shown.  In contrast, when f RF=3.0 GHz (Fig. 2(b)), drop in \nthe scope signal associated with the presence of domain wall in section C-D is absent when H \nis above 35 Oe.  This indicates that the microwave field creates a domain wall and reverses \nthe magnetization direction of the wire via domain wall propagation prior  to the injection of \nthe voltage pulse to line B when the field is at H (nucleation of the wall by the microwave \nfield is confirmed by monitoring the microwav e field induced nucleation process in real-time \n(results not shown)). Note that the propagating sp in waves generated at line A via parametric \npumping can influence domain wall propagation24-27  However, using a wire with different \nconfiguration of the field generation lines (i.e. line A is placed to the right of line C-D) we \nobserve no change in the nucleation field and th us consider that the effect of spin waves on \nthe nucleation process is negligible.   \n  The dependence of the amplitude of the scope signal, as defined in Fig. 2(a), is plotted as a function of H for various f\nRF in Fig. 2(d).  Since each scope signal is an average of a few \nhundred independent measurements, the amplitude represents the probability of a domain wall \nmoving into section C-D.  In addition, the amplitude also reflects the mean size of the \ndomain wall as it propagates.  The gradual increase in the amplitude with the field reflects \nthe variation of the wall width28-30: the wall structure typically gets elongated at higher \nfields31.  The Walker breakdown field here is close to the domain wall propagation field (~5 \nOe).  For a given f RF, we look for H above which the amplitude drops to zero.  This field is \ndefined again as the “assisting field” and is plotted in Fig. 2(e) for f RF=0.1 GHz and 3.0 GHz \nas a function of the current applied to line A.  The current shown here is the amplitude of the \nsinusoidal current (not the rms value).  Interestingly, when f RF=3.0 GHz, the assisting field 7 \n needed is, in most cases, smaller than that when a pulse Oersted field is used. This trend is \npronounced when the current applied to the field generation line is smaller. For f RF=0.1 GHz, \nthe required assisting field is always larger than that of the pulse excitation (the difference \nbetween the two is rather large and to understand the reason behind this requires further \ninvestigation). \nSince the width of the field generation lines are different between lines A and B, the \nOersted field generated from each line at a given current is different.  We thus calculate the \nOersted field in each geometry and attempt to directly compare the efficiency of domain wall nucleation between the two.  These results are summarized in Fig. 3(a).  Here the \n“nucleation field”, defined by the sum of the assi sting field and the calculated Oersted field, is \nplotted as a function of the Oersted field.  Nucleation fields for the pulse excitation using \nlines A (Oersted field transverse to the easy axis) and B (Oersted field is parallel to the easy \naxis) are shown by the red circles and black squares, respectively.  The triangles in Fig. 3(a) represent the nucleation fields for the microw ave excitation from line A (transverse Oersted \nfield).  The nucleation field is always smaller when the Oersted field is pointing transverse \nto the easy axis as compared to the parallel Oersted field.  The nucleation field is the \nsmallest among all the geometries used when microwave current of f\nRF=3.0 GHz is applied to \nline A (transverse Oersted field).  Note that the nucleation field tends to increase when the \nOersted field amplitude is increased, particular ly for the parallel pulse excitation (squares).  \nThis feature may depend on the details of the nucleation process and further investigation is \nrequired to clarify its origin. \nTo further illustrate the effect of the microwave field on the nucleation field, we show the \nexcitation frequency dependence of the nucleatio n field in Fig. 3(b).  The microwave power 8 \n is set to 0 dBm (Oersted field of ~27 Oe), as shown by the vertical dashed line in Fig. 3(a).  \nWe clearly observe a drop in the nucleation field when f RF~3.0 GHz, indicating a resonance \neffect on the nucleation process.  This is in contrast to previous results where the nucleation \nfield shows little dependence with the microwave field application19.  Note that the change in \nthe nucleation field with respect to f RF is asymmetric: such effect has been predicted by \nmicromagnetic simulations but not been observed experimentally19. \nTo understand the origin of the “resonance frequency” at which the nucleation field takes \nits minimum, we measure the ferromagnetic re sonance (FMR) in a similar nanowire (same \nwire width).  A coplanar waveguide (signal line width: 2 m, ground line width: 4 m, \ndistance between the two lines: 1.4 m) made of 5 Ta|85 Au (units in nm) is formed on top of \na permalloy Hall bar (wire width: 600 nm, wire length: 60 m, Hall bar width: 500 nm).  See \nFig. 3(c) for the experimental setup.  A 30 nm thick SiO 2 layer is inserted between the \npermalloy Hall bar and the waveguide to avoid shorting the Hall bar with the waveguide.  Direct current (100 uA) is applied to the Hall bar and we use the AMR effect to probe the \ndynamics of the magnetic moments\n32.  Sinusoidal current is applied to the waveguide to \ngenerate in-plane oscillating field to the perma lloy layer.  The inset of Fig. 3(d) shows an \nexemplary in-plane field dependence of th e permalloy Hall bar resistance.  Here, a \nmicrowave current (9.5 GHz, 6 dBm) is applied to the waveguide.  Large negative peaks \nappearing at ~±750 Oe is the uniform excitation mode and the smaller peaks observed at \nsmaller fields are likely to do with the quantized spin wave modes33. The peak fields of the \nuniform mode (absolute mean of the positive and negative peak fields) are plotted as a \nfunction of the sinusoidal current frequency in Fig.  3(d).  At small magnetic fields (~27 Oe \nin Fig. 3(a)), the frequency of the uniform mode is ~3 GHz, similar to what we find for the 9 \n “resonance frequency” shown in Fig. 3(b).   \n  A number of studies have reported that microwave assisted magnetization switching is \nrather stochastic9 and the switching field shows broad distribution9,10,17,18.  This is likely to \ndo with the presence of multiple nucleation sites with different internal fields (and thus \ndifferent resonant frequencies) that can contribute to the switching process. In addition, the \ninfluence of microwave field application on domain wall propagation may also add to the \nswitching field distribution20,21.  As evident from the plots in Fig. 2(d), the amplitude (which \nis proportional to one minus the nucleation probability) shows an abrupt drop above a certain H for each f\nRF which indicates that the switching field distribution is rather small.  Since we \nuse a localized magnetic field, we consider that domain wall nucleation always occurs from a \nfixed position, likely at the edge of the field generation lines, which contributes to the small \ndistribution in the nucleation field.  In addition, the effect of microwave field application on \ndomain wall propagation is excluded in our syst em, which eliminates another factor that adds \nto the broadening of the switching field. \nIn summary, we have studied domain wall nucleation in permalloy nanowires under pulse \nand microwave Oersted field excitations.  Cont rary to previous reports where microwave \nfield likely acted on both domain wall nucleation and propagation, here we design a system to \nstudy only the former process.  We find that the nucleation field is reduced by a factor of ~2 \nwhen a resonant transverse microwave Oersted field, whose amplitude is roughly a quarter of \nthe off-resonant nucleation field, is applied.  The resonance frequency is ~3 GHz, which is \nclose to the ferromagnetic resonance frequency of the uniform excitation mode.  Owing to \nthe well-defined nucleation process, the nucleation field upon microwave field application \nshows small distribution.  Our results show that localized microwave fields are effective in 10 \n injecting domain walls in magnetic nanowires.   \n \nAcknowledgments \nThis work is partly supported by a Grant-in -Aid for Scientific Research (No. 22760015) from \nMEXT, Japan.11 \n References  \n1 S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). \n2 D. A. 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W e g e l i n ,  S .  A .  N e p i j k o ,  H .  J .  E l m e r s ,  G .  S c h o n h e n s e ,  M .  \nBolte, and C. M. Schneider, Phys. Rev. Lett. 95, 207201 (2005). \n22 M. Hayashi, L. Thomas, Y. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, and \nS. S. P. Parkin, Phys. Rev. Lett. 96, 197207 (2006). \n23 T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, J. Appl. Phys. 85, \n7849 (1999). \n24 S .  M .  S e o ,  K .  J .  L e e ,  H .  Y a n g ,  a n d  T .  O n o ,  P h y s .  R e v .  L e t t .  102, 147202 \n(2009). \n25 D.-S. Han, S.-K. Kim, J.-Y. Lee, S. J. Hermsdoerfer, H. Schultheiss, B. \nLeven, and B. Hillebrands, Appl. Phys. Lett. 94, 112502 (2009). \n26 M. Jamali, K. J. Lee, and H. Yang, New J. Phys. 14, 033010 (2012). \n27 J. S. Kim, M. Stark, M. Klaui, J. Yo on, C. Y. You, L. Lopez-Diaz, and E. \nMartinez, Phys. Rev. B 85, 174428 (2012). \n28 G .  S .  D .  B e a c h ,  C .  N i s t o r ,  C .  K n u t s o n ,  M .  T s o i ,  a n d  J .  L .  E r s k i n e ,  N a t .  \nMater. 4, 741 (2005). \n29 M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and S. S. P. Parkin, Nat. \nPhys. 3, 21 (2007). \n30 J.-Y. Lee, K.-S. Lee, S. Choi, K. Y. Guslienko, and S.-K. Kim, Phys. Rev. B \n76, 184408 (2007). \n31 M. Hayashi, S. Kasai, and S. Mitani, Appl. Phys. Express 3, 113004 (2010). \n32 N. Mecking, Y. S. Gui, and C. M. Hu, Phys. Rev. B 76, 224430 (2007). \n33 J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y. \nGuslienko, A. N. Slavin, D. V. Berkov, and N. L. Gorn, Phys. Rev. Lett. 88, \n047204 (2002). \n 13 \n Figure captions \nFig. 1.  (a) Optical microscopy image of  the nanowire device along with schematic \nillustration of the experimental setup.  The area trimmed by the dotted line indicates where \nthe 30 nm thick SiO 2 isolation pad is inserted. (b,c) Typical waveforms captured by the \noscilloscope for various fields H and pulse amplit udes: (b) -0.8 V and (c) -0.5 V .  (d) Field H \nneeded to assist domain wall nucleation (assisting field) plotted against the current that flows \nalong line B during the pulse application.   \nFig. 2. (a-c) Representative scope waveforms for different fields H and microwave \nfrequencies.  The microwave power is set to 0 dBm.  (d) Scope signal amplitude plotted as a function of field H for various f\nRF.  The arrow illustrates the “assisting field” for f RF=3.0 \nGHz.  (e) Assisting field versus the current amplitude that flows through line A when the \nmicrowave current is applied. The arrow shows the point that corresponds to the arrow shown in (d).  \n \nFig. 3. (a) The nucleation field, i.e. the sum of the Oersted field and the assisting field (shown in Figs. 1(d) and 2(e)), plotted as a function of  the Oersted field at the field generation line.  \nSquares: parallel pulse field, circles: transverse  pulse field, up triangles: transverse microwave \nfield (0.1 GHz), down triangles: transverse microwave field (3.0 GHz). (b) Nucleation field versus the microwave frequency when the microwave Oersted field at line A is set to ~27 Oe \n(microwave power of 0 dBm), as shown by the vertical dotted line in (a).  (c) Optical \nmicrograph of the device used to study the ferromagnetic resonance frequency.  The blue dotted lines represent the permalloy Hall bar.  (d) Microwave frequency applied to the \nwaveguide versus the absolute mean of the field where the largest peaks are observed in the \nFMR measurements. Line is guide to the eye. Inset: wire resistance versus magnetic field when microwave current (frequency: 9.5 GH z) is applied to the waveguide.  \n 0 5 10 15 20020406080100\n-0.8 V-0.5 V-0.2 V\n  Assisting field (Oe)\nCurrent (mA)01 0 0-2002040\n01 0 0\n  Voltage (uV)\nTime (ns)66 Oe39 Oe35 Oe-0.5 V -0.8 V\n35 Oe\n39 Oe\n  \nTime (ns)66 Oe\nPulse \ngeneratorSignal generatorScopeDC source\nABCD Isolation\npad\nXY\n(d)(b) (c)(a)\nFig. 1Field0 5 10 15 20020406080100\n  Assisting field (Oe)\nCurrent (mA) pulse\n 0.1 GHz\n 3.0 GHz01 0 0-2002040\n01 0 0\n  Voltage (uV)\nTime (ns)70 Oe35 Oe32 Oe3.0 GHz 0.1 GHz\n32 Oe\n35 Oe\n  \nTime (ns)70 Oe\n0 1006.0 GHz\n  \nTime (ns)70 Oe35 Oe32 Oe(a) (b) (c)\n(e)\nFig. 2Amp.\n02 0 4 0 6 00102030\n Amplitude ( /g80V)\nField (Oe)Freq. (GHz)\n      4.0\n      3.4\n      3.2\n      3.0\n      2.8\n      2.0\n      1.0(d)0 400 800 1200036912\n Frequency (GHz)\nField (Oe)0 50 100 150050100150\n  Nucleation field (Oe)\nOersted field (Oe) para. pulse\n trans. pulse\n trans. 0.1 GHz\n trans. 3.0 GHz\n0246050100150\n  \nFrequency (GHz)(a) (b)\nFig. 3(c)\n-1000 0 1000-2-10 \n R (arb. unit)\nField (Oe)9.5 GHz\nV100 /g80A(d)\nField"    },    {        "title": "1508.02093v1.Dynamic_detection_of_electron_spin_accumulation_in_ferromagnet_semiconductor_devices_by_ferromagnetic_resonance.pdf",        "content": "arXiv:1508.02093v1  [cond-mat.mes-hall]  9 Aug 2015Dynamic detectionof electronspin accumulationin\nferromagnet-semiconductordevices by ferromagneticreso nance\nChangjiang Liu,1Sahil J. Patel,2Timothy A. Peterson,1Chad C. Geppert,1\nKevin D. Christie,1Gordon Stecklein,1Chris J. Palmstrøm,2,3and Paul A. Crowell1,∗\n1SchoolofPhysicsandAstronomy,UniversityofMinnesota,M inneapolis,Minnesota55455,USA\n2Department of Materials, University of California,\nSanta Barbara, California 93106, USA\n3Department of Electrical and Computer Engineering,\nUniversity of California, Santa Barbara, California 93106 , USA\nAbstract\nAdistinguishing feature ofspin accumulation inferromagn et-semiconductor devices isprecession ofthe\nnon-equilibrium spin population of the semiconductor in a m agnetic field. This is the basis for detection\ntechniques such as the Hanle effect, but these approaches be come less effective as the spin lifetime in\nthe semiconductor decreases. For this reason, no electrica l Hanle measurement has been demonstrated in\nGaAs at room temperature. We show here that by forcing the mag netization in the ferromagnet (the spin\ninjector and detector) to precess at the ferromagnetic reso nance frequency, an electrically generated spin\naccumulation canbe detected from 30to300K.Atlow temperat ures, the distinct Larmorprecession of the\nspinaccumulation inthesemiconductor canbedetected byfe rromagnetic resonance inanoblique field. We\nverifytheeffectivenessofthisnewspindetectiontechniq uebycomparingtheinjectionbiasandtemperature\ndependence of the measured spin signal to the results obtain ed using traditional methods. We further show\nthat this new approach enables a measurement of short spin li fetimes (< 100 psec), a regime that is not\naccessible in semiconductors using traditional Hanle tech niques.\n∗crowell@umn.edu\n1An effective means to electrically detect electron spin acc umulation in a semiconductor is es-\nsential to the development of semiconductor spintronics1,2. A common detection method, known\nas the Hanleeffect3, relies on theprecession ofthespin accumulationin thesem iconductorby an\nexternal magnetic field. This approach has been employed in b oth the non-local4,5and the local\n3-terminal (3T)6measurement geometries. As the spin lifetime decreases at h igher temperatures,\na significantly higher magnetic field is required in order for the Hanle effect to dephase the spin\naccumulation7. The ordinary magnetoresistance in semiconductors in such large fields produces\nlarge quadratic backgrounds6that mask any truly spin-dependent effects. As a result, no e lectri-\ncal Hanle measurement has been demonstrated in n-GaAs at room temperature. Although the 3T\ntechnique has recently been widely used to study spin transp ort in new material systems8–10at\nhigher temperatures, it has been shownthat the3T Hanle meas urement is sensitiveto a variety of\nmagneticfield assistedphenomena11–13, whichoften cannot beseparated fromtheHanleeffect.\nWe introduce here a new detection technique, which utilizes the precession of the magneti-\nzation under ferromagnetic resonance (FMR) to dynamically detect the spin accumulation in a\nsemiconductor. The approach is similar in some aspects to a H anle measurement, in that the ob-\nserved signal corresponds to a suppression of the component of the spin accumulation parallel to\nthe magnetization. In this case, however, the suppression i s due to the onset of FMR rather than\nprecession of the spin accumulation in an orthogonal magnet ic field. The narrow FMR linewidth\nandthesensitivitytoboththeprecessionofthemagnetizat ionintheferromagnetaswellasthedy-\nnamicsofthespinaccumulationinthesemiconductorprovid eanimmunitytothefield-dependent\nbackgrounds that affect Hanle measurements, particularly in the 3T case. At low temperatures,\ntheLarmor precession ofthespin accumulationin thesemico nductorcan be detected by an FMR\nmeasurement in an oblique magnetic field. We verify the effec tiveness of this approach by com-\nparingthebiasandtemperaturedependenceofthemeasureds pinsignaltoresultsobtainedonthe\nsame heterostructures using traditional spin detection me thods. Finally, we show that this FMR-\nbased spin detection technique enables one to determine the spin relaxation rate in GaAs at room\ntemperature, atwhich an ordinaryHanlecurvecannotbemeas ured.\nThe devices used in this electrical spin injection and detec tion experiment consist of ferro-\nmagnet (FM) /semiconductor (SC) heterostructures. A spin current acros s the FM/SC interface\nis generated by a current source14,15, leading to a spin accumulation, which has previously been\ndetected potentiometrically5,16or by using a spin filter to detect the polarized current17. In the\npotentiometric approach, the voltage detected by a FM conta ct due to spin accumulation can be\n2expressedas5\nV=ηPFM1\ne∂µ\n∂ns·ˆm, (1)\nwhereηis the spin detection efficiency, PFMis the spin polarization at the Fermi level in the FM,\neistheelectroncharge, and µandnarethechemicalpotentialandnumberdensityofelectronsi n\ntheSC. Thespinaccumulation sintheSChas amagnitudedefined as |s|=n↑−n↓,wheren↑and\nn↓refer to the numberdensityof spin-up and spin-downelectro ns, andˆmis the unit vectorof the\nmagnetization mintheFM.Thedotproductinequation(1)accountsforthepro jectionofthespin\naccumulation, which must be averaged over the area of the con tact, onto the magnetization. In a\nHanle measurement, one applies a magnetic field perpendicul ar toˆmto precess and dephase the\ninjected spins, thus reducing the magnitude and changing th e orientation of s. In our experiment,\nwe force the magnetization of the FM to precess instead of the spin accumulation in the SC. We\ninjectspinsfromtheFMandsimultaneouslydrivethemagnet izationintheFMtoprecessbyFMR.\nDuring this process, s·ˆmalso changes, which allows for detection of the spin accumul ation. As\nwillbeexplainedindetailinthemodelingsection,theimpo rtanttimescalesinthisexperimentare\nthe FMR precession period and the electron spin lifetime τsin the SC. Together these determine\nthesteadystatevalueof s·ˆm, whichchanges at resonance.\nResults\nExperimental setupandroomtemperature measurement. Figure1aisaschematicdiagramof\nthe FMR-spin detection experiment. The devices used in our e xperiment are fabricated from epi-\ntaxial FM /n-GaAs (100) heterostructures. Recently, the utilization of the ferromagnetic Heusler\nalloys19has greatly improved the spin injection efficiency into GaAs20,21. In our experiment, we\nuse the Heusler alloys Co 2MnSi and Co 2FeSi as spin injectors. The FM film is 5 nm thick with\nlateraldimensionsof5 ×50µm. Then-GaAsisdopedwithaconcentrationof n=3×1016cm−3.\nThejunctionregionconsistsofahighlydoped n+(5×1018cm−3)layertothintheSchottkytunnel\nbarrier in order to achieve higher spin injection efficiency22,23. A 120 nm thick gold wire, which\nforms a microwavestripline, is deposited directly on top of the FM contact. The microwaveOer-\nstedfieldfromthestriplinedrivestheresonance. Themicro wavefrequencyusedintheexperiment\nranges from 5 Ghz to 20 Ghz. The applied dc magnetic field is in t he sample plane as indicated\nbyHin Fig. 1a. A forward bias current across the FM /SC interface injects a spin current into\nthen-GaAs24. The microwave field is modulated at low frequency (100 Hz) an d a lock-in ampli-\nfiersynchronizedwiththemodulationfrequencymeasuresth edifferenceininterfacevoltagewith\n3900 1200 1500 1800 2100-21.6-20.4-19.2-18.0\nCMS\nT = 300 K\nf = 10.5 GHzV (μV)\nH (Oe)T = 300 K\nj = 1600 A/cm2\nFigure 1.Schematic diagram of the experiment and a representative pl ot of the spin accumulation\ndata measured by FMR at room temperature. (a) A schematic diagram of the FMR-spin detection ex-\nperiment. The red contact in the middle of the device is ferro magnetic. On top of the ferromagnet is the\nmicrowave stripline (shown in yellow). The two remote conta cts (shown in blue) are ohmic and are fab-\nricated from non-magnetic Cu-Ge18.Handm(t)represent the applied magnetic field and the precessing\nmagnetization of the FM, respectively. Iis the applied interface bias current, which generates a spi n accu-\nmulation in n-GaAs. Spins (shown in purple) injected by the precessing ma gnetization at different times\nhave different orientations, and they diffuse in the n-GaAs channel. ( b) Spin signal measured by FMR as\na decrease in voltage at room temperature. The solid line is a Lorentzian fit to the data. CMS refers to the\nsample with Co 2MnSi asthe FM .\nmicrowaveson and off.\nWhen the bias current generates a spin accumulation in the n-GaAs and the magnetization is\ndriven on resonance, a dip in the three-terminal voltage is o bserved. The raw data in Figure 1b\nshow the resonance observed at room temperature. The resona nce field measured as a function\nof the microwavefrequency agrees with the FMR spectrum as ca lculated from Kittel’s formula25\nusing the anisotropy and magnetization of each sample (see S upplementary Fig. 1). The non-\nzero background voltage in the plot is due to the nonlinear IV characteristic of the FM /n-GaAs\nSchottky contact26, which rectifies the microwave current generated in the devi ce. We see that\nthe interface voltage decreases on resonance, which, as wil l be explained in detail below, is the\nresultofthedecreaseoftheprojectionof sontoˆm. In previouswork,wefoundthatthetunneling\nanisotropic magnetoresistance (TAMR) effect at the FM /SC interface also produces a change in\nthe interface voltage at FMR27. The size of the TAMR effect depends on the bias voltage acros s\n4the FM/SC interface27,28. When a high forward bias current is applied to the sample gen erating\na spin accumulation in the SC, the FMR signal due to the TAMR ef fect is much smaller than that\nproduced by the spin accumulation27. In the data analysis below, the small contribution to the\nFMRsignalfrom theTAMReffect has been removed.\nDetecting the Larmor precession. A distinguishing feature of spin is the Larmor precession\naboutan appliedmagneticfield. Theprecession frequencyis ωL=gµBB/¯h, whereµBistheBohr\nmagneton,the gfactoris−0.44forelectronsinGaAs29and¯hisPlanck’sconstant. Weexpectthat\nthe Larmor precession in the SC will influence the spin accumu lation measured by FMR when\nωLis larger than the spin relaxation rate. In these samples, th is condition can be achieved at low\ntemperatures (30 K), at which the spin lifetime for this dopi ng is longer than 10 ns30, while the\nthe Larmor precession period is about 1 ns for a magnetic field of 2000 Oe. We tilt the magnetic\nfield out of the sample plane such that the perpendicular comp onent of the magnetic field is still\nmuch smaller than the magnetic anisotropy field of the FM film. This allows the magnetization\nto remain nearly in the sample plane, while the spins in the se miconductor precess about the\noblique magnetic field. In steady state, the spin accumulati on can be viewed as an ensemble\nof spins injected at different times3. WhenωL>τ−1\ns,as is the case at 30 K, the orientations\nof electron spins are therefore distributed uniformly on th e precession cone, which is depicted in\npurpleinFig.2b. Inotherwords,the transverse componentofthespinaccumulationiscompletely\ndephased, and there is no component of the steady-state spin accumulation perpendicular to the\nappliedfield.\nFigure 2a shows that the FMR spin accumulation signal decrea ses significantly as the out-of-\nplane angle θof the magnetic field increases. To verify that this change is not simply due to the\nmagnetizationdynamicsoftheFM,weconfirmthattheamplitu deoftheprecessionconeanglein\nthe FM does not change significantly with increasing angle by applying a reversebias across the\nFM/n-GaAsinterface. Underreversebias,thereisnospinaccumu lationgeneratedinthe n-GaAs,\nas we have verified in traditional non-local spin valve measu rements, and the interface voltage at\nresonanceisdueonlytotheTAMR.TheTAMRcaninturnberelat edverysimplytothegeometry\noftheprecessingmagnetizationintheFM(seeSupplementar yNote2). Figure2cshowsalmostno\nchangein thesizeofthevoltagepeak when thefield istiltedo utofplane, and hence weconclude\nthattheamplitudeoftheprecessionconeangleintheFM isne arly constant.\nThespinaccumulationdetectedbytheFMisproportionalto /angbracketlefts·ˆm/angbracketright,wherethebracketsindicate\nthe time average in steady state. Because the transverse spi n is completely dephased under the\n50 20 40 60 80-24-18-12-60\n FitDataVFMR (μV)\nθ (deg)1200 1800 2400 3000-50-40-30-20-10\n60\n50\n40(μV)\nCMS\nf = 8 GHz\nj = 400 A/cm2V\nH (Oe)T = 30 Kθ =0 deg\n1200 1800 2400 3000-15-10-505\nj = - 1.6 A/cm2f = 8 GHzT = 30 KV (μV)\nH (Oe)θ =0 deg40 50 60\nFigure 2.Experimental results in the oblique configuration at 30 K and illustration of the Larmor\nprecession ofthespinaccumulation. (a) TheFMRsignal measured atdifferent out-of-plane angleso fthe\nappliedfieldwhenaspinaccumulation existsinthe n-GaAs. Notethedecreaseinthesignal amplitude with\nincreasing angle. ( b) The applied magnetic field H, represented by the grey arrow, is tilted out-of-plane\nby an angle θ. The injected spins precess about Hand their directions are uniformly distributed on a cone\nwhenτs≫2π\nωL,asisthecaseat30K.( c)FMRmeasured under reversebias, atwhichonlytheTAMReffe ct\nis present. The out-of-plane angles are the same as in Fig. 2a . In this case the amplitude is independent of\nangle. (d)Angledependence ofthemagnitudeofthespinaccumulation signal measuredbyFMR(Fig.2a),\nand a fit based on the Larmor precession model. Because of the l arge demagnetizing field, the precession\naxisofthemagnetization remainsnearly inthesampleplane . Therawdatain( a,c)areplotted asafunction\nof thetotal magnetic field, andsotheresonance fieldpositio ns andthewidthsof thepeaks scale as1 /cosθ\nrelative tothe in-plane ( θ=0) case.\nconditions being considered, the magnitude of the spin accumulation is reduced by a factor of\ncosθrelative to its magnitude s0when the field is in the sample plane. As a result, s=s0cosθˆh,\n6whereˆhisaunitvectorparalleltothemagneticfield. Becausethede tectedvoltageisproportional\nto/angbracketlefts·ˆm/angbracketrightandnoneoftheprefactorsinequation(1)dependonangle,th espinaccumulationsignal\natresonanceshouldthereforevaryascos2θ. Figure2dshowstheamplitudeoftheresonantsignal\nVFMR(themagnitudesofthenegativepeaks in Fig. 2a)as a functio nofθ.Thesolidlineis a fit to\ncos2θ, andso thebehaviorexpectedfor aspinaccumulationsignal is observed.\nComparison to other measurement techniques. To obtain further evidence that the observed\nFMR signal is due to the spin accumulation in the n-GaAs layer, we measure the dependence\nofVFMRon the bias current and temperature. The resulting data can t hen be compared to those\nobtained using othertechniques. We first use the3T Hanle mea surement, carried out in a perpen-\ndicular field, to measure the spin accumulation at different injection bias currents6,31. Figure 3a\nshows the measured 3T signal V3Tat 30 K for different forward bias currents, and Fig. 3b shows\nV3Tat a fixed forward bias current for different temperatures. T he asymmetric lineshape and the\nnarrowing of the peak at low magnetic field in Fig. 3a are cause d by the hyperfine interaction be-\ntweentheinjectedspinsandthepolarizednuclei32. Atlowtemperature, V3Tisnearlyproportional\ntothespinaccumulation6,andsoweexpectthebiasdependencesof V3TandVFMRtobethesame.\nIn Fig. 3c, V3Tand the FMR signal at 30 K are plotted as a function of the bias v oltage. The\ntwo measurement techniques agree up to a single bias-indepe ndent scale factor. Unfortunately,\nthis comparison cannot be made at high temperatures, at whic h a spin-dependent contribution to\nV3Tcannot be measured, as can be seen from the 300 K data in Fig. 3b . We have also performed\nnon-local Hanle measurements5on this sample (see Supplementary Fig. 2) . Although consist ent\nwithV3Tup to120K, thenon-localHanlesignalalsocannot bemeasure d at highertemperatures.\nThesteady-statespin accumulation in the semiconductorde pends on thespin lifetimeand dif-\nfusion constant7, which are strong functions of temperature. The Hanle techn iques fail at high\ntemperaturesbecausethemagneticfieldscalecorrespondin gtoωLτs∼1growsdramaticallyas τs\ndecreases. Asaresult,theHanlecurvesbroadenandcannotb edistinguishedfromfield-dependent\nbackgrounds. This limitation does not impact the FMR signal , as we can see by comparison\nwith non-local spin valve measurements5. Spin valves with source-detector separations between\n250 nm and 2 µm were fabricated for each heterostructure by electron beam lithography. In this\ncase, the field is swept in the plane, and the spin accumulatio n is inferred from the magnitude of\nthe jump VNLSVin the voltage when the source and detector magnetizations s witch from parallel\nto antiparallel. Data for Co 2FeSi/n-GaAs at a source-detector separation of 250 nm are shown in\nFig. 3d. This spacing is smaller than the spin diffusion leng th of approximately 800 nm at room\n70.2 0.3 0.4 0.5 0.6 0.7-18-12-60\n(μV)CMS\nVFMR\nV3TVFMR (μV) \nVbias (V)T = 30 K\nV3T\n-1350-900-4500\n-2 -1 0 1 2040080012001600 VCMS\n80 160j = 400 A/cm2T = 30 K\nH (kOe)(μV)\n-4 -2 0 2 40200400600\niT=300iT=150 j = 800 A/cm2\n T = 60 KCMS\nH (kOe)\n-1000 -500 0 500 1000-400-2000200400CFS\nT = 300 K\n150 \n50 \nH (Oe)X 2V(μV)\n0 60 120 180 240 300-3.0-2.4-1.8-1.2-0.60.0\nVFMR (CMS)\nVNLSV (CFS)VNLSV (CMS,  x 0.67)VFMR (CFS,  x 0.64)\nT (K)VFMR (arb. u.)\n-600-450-300-1500\nVNLSV (μV)\nFigure 3.Injection bias and temperature dependence of the spin accum ulation measured by tra-\nditional methods and their comparison to the results obtain ed by the FMR technique. (a) 3T data\nmeasured as a function of perpendicular field at 30 K for diffe rent injection bias currents. ( b) 3T signal\nmeasured at different temperatures for a fixed bias current. The spin signal cannot be resolved using the\nHanle measurement at 300 K. ( c) Magnitude of the FMR peak, VFMR(black squares, left axis), and V3T\n(red circles, right axis) plotted as a function of injection bias voltage at 30 K. Note that the two sets of\nmeasurements differ byonlyasinglebias-independent scal efactor. ( d)Non-local spinvalvemeasurements\natdifferent temperatures, obtained atasource-detector s eparation of250nm. The300Kdataaremultiplied\nby a factor of 2 for clarity. The arrows indicate the relative orientations of the magnetization in the source\nanddetector FM.In a,banddasecond orderbackground hasbeensubtracted fromtherawda ta. Curvesin\naanddare offset for clarity. ( e)VFMR(black solid symbols, left axis) and VNLSV(red open symbols, right\naxis) measured as a function of temperature. CMS and CFS refe r to samples with Co 2MnSi and Co 2FeSi\nasthe FM,respectively.\ntemperature (see Supplementary Fig. 4), and so it is possibl e to measure VNLSVover the entire\n8temperature range of this experiment. Figure 3e compares VFMRwithVNLSVfor both Co 2FeSi/n-\nGaAs and Co 2MnSi/n-GaAs from 30 K to 300 K. The temperature dependences are simi lar for\nboth sets of measurements on each heterostructure. Note tha t the choice of FM should not matter\ninthiscase, asthetemperaturedependenceisgovernedprim arilybypropertiesoftheSC.\nModeling and measuring the frequency dependence to study sp in relaxation. The data of\nFig. 3 demonstrate that VFMRis proportional to the spin accumulation. We now consider th e\nfrequency dependence of VFMRand its relationship to τs. Remarkably, we find that the FMR\ntechnique can providea measurement of the spin lifetime in t he regime ωFMRτs∼1, which turns\noutto beaccessibleat hightemperatures. In thislimit,a spininjectedfrom theFM rela xes before\nthe magnetization precesses through a complete cycle. This physical scenario is illustrated in\nFig.4a,inwhichthespinaccumulationisrepresentedbyase riesofspins(inpurple)thathavebeen\ninjectedintothesemiconductoratthetimesindicated,whe ret0correspondstothepresent,and −t3\nthrough−t1represent previoustimesas labeledon thetrajectory oftheprecessingmagneti zation,\nwhich is shown in red. The spins shown are oriented approxima tely parallel to the magnetization\natthetimeatwhichthosespinswereinjected. Precessionin thesemiconductorisignoredbecause\nωL<ωFMR/10 for theentirefrequency and magneticfield range covered b y theexperiment. The\nsmaller number of spins present from the earlier times is due to the relaxation of spins overtime.\nThe average spin accumulation, shown by the large purple arr ow in Fig. 4a, is therefore a vector\nwhichlags behindtheprecessingmagnetization.\nThe spin accumulation signal under these conditions can be c alculated by an appropriately\nweightedtimeaverageofequation(1), incorporatingspinr elaxationand diffusion6:\nV(t)=ηPFMjs\ne∂µ\n∂ntˆ\n−∞ˆm(t)·ˆs(t′)/radicalbig\n2πD(t−t′)exp[−t−t′\nτs]dt′, (2)\nwherejsis the injected spin current density, while Dandτsare the spin diffusion constant and\nthe spin lifetime in n-GaAs. Here ˆm(t)andˆs(t′)represent the time-dependent orientations of the\nmagnetization and the injected spins, respectively. The in tegral over t′evaluates the projection of\nthespinsinjectedatallprevioustimes,reduced bytheeffe ctsofdiffusionandrelaxation,ontothe\npresentmagnetizationdirection ˆm(t). Thisisthe instantaneous voltagedetectedbytheFM.Inthe\nFM thin film, the precessing magnetization traces out an elli pse, with an in-plane cone angle ϕin\nand an out-of-plane angle ϕout. From geometric analysis (ignoring the Larmor precession i n the\nSC as notedinthepreviousparagraph), ˆm(t)·ˆs(t′)takestheform\n90 5 10 15 20 25 30CMS\n300 KVFMR  (arb. u.)\nf (GHz)220 K\n100 K\n-2.0-1.5-1.0-0.50.0\nx 1x 2.2x 4.730 Kx 9.1\nFigure 4.Illustration of the spin accumulation in the short spin life time regime and the frequency\ndependence of VFMRat different temperatures. (a) Injected spins (small purple arrows in n-GaAs) are\nparallel to the instantaneous orientation of the magnetiza tion (red arrow on top of the FM) at the time they\nare injected. Earlier injected spins, e.g., at t=−t1,−t2or−t3have progressively smaller magnitudes\ndue to spin relaxation and diffusion. The orientation of the average spin accumulation, indicated by the\nlarge purple vector S, lags behind the magnetization m, which decreases s·ˆmrelative to the static case\n(ω=0). (b)VFMRmeasured as a function of frequency at different temperatur es. For clarity, signal sizes\nare multiplied by the factors shown in the plot. The solid lin es are fits based on equation (4) derived in the\nmain text, from which the electron spin lifetime in the SC is o btained. For the 30 K data, the solid line is\ncalculated from the expression of VFMR(ω)by using theknown spin lifetime τs=10 ns.\nˆm(t)·ˆs(t′)=1−ϕ2\nin\n2(cosωt−cosωt′)2\n−ϕ2out\n2(sinωt−sinωt′)2, (3)\n(seeSupplementaryNote3),inwhich ωisprecessionfrequencyintheFM.Thedcvoltage /angbracketleftV(t)/angbracketright\nin the presence of a microwave field is the average value of V(t)over a precession period. The\nmagnitude of VFMR(ω)is the difference of /angbracketleftV(t)/angbracketrightbetween resonance (ω=ωFMR)and off reso-\nnance, which can bedeterminedbysetting ω=0 inequation(2). We thenobtain\nVFMR(ω)=1\n2ηPFMjs\neN(Ef)(ϕ2\nin+ϕ2\nout)/radicalBig\nτs\n2D\n×/parenleftBigg/radicalbigg\n1\n2√\n1+ω2τ2s+1\n2(1+ω2τ2s)−1/parenrightBigg\n. (4)\nFrom equation (4) we obtain lim\nω→0VFMR(ω) =0. This corresponds to the dc limit, in which the\norientation of the spin accumulation is always able to follo w the magnetization. As ωincreases,\n10the angle by which the spin accumulation “lags” the magnetiz ation increases, and VFMR(ω)be-\ncomesmorenegative. Theexpressionfor VFMR(ω)suggeststhatintheregime ωτs∼1,VFMR(ω)\nwill have considerable frequency dependence. This regime c orresponds to short spin lifetimes,\nie.,τs∼16 ps for a typical FMR frequency of f=ω/2π=10 GHz. As ωincreases further,\nVFMR(ω)→ −1\n2ηPFMjs\neN(Ef)(ϕ2\nin+ϕ2\nout)/radicalBig\nτs\n2Dand saturates (no frequency dependence). This sit-\nuation is equivalent to the long τslimit which happens at low temperature. We note that the\nmagnitudeof VFMR(ω)in this case is proportional to the steady-state spin accumu lation, with the\nproportionalitydeterminedbythemagnitudeoftheprecess ionconeangles. Thisexplainsthedata\npresented inFigs 2and 3.\nTo test this model, we measure the frequency dependence of VFMRat different temperatures.\nDuringthemeasurement,theinjectionbiascurrentisfixed, andtheamplitudeofthemagnetization\nprecession, which is measured from the TAMR at reverse bias27, is fixed at a constant value by\nadjusting the microwave power at each frequency. Figure 4b p resents the experimental results.\nTheFMRsignalshowsapronouncedfrequencydependenceathi ghtemperatures. Themagnitude\nof the signal increases as the FMR frequency increases, and t he sign of the effect is negative as\nexpected from the decrease in /angbracketlefts·ˆm/angbracketright. The solid lines in Fig. 4b are fits to VFMR(ω)given by\nequation (4), except for the 30 K data, for which τsis fixed at the value (10 nsec) inferred from\nnon-localspinvalvemeasurements. Besidesanoverallscal efactor,thespinlifetime τsistheonly\nfitting parameter ( Dis absorbed into the prefactor in equation (4)). We obtain a s pin lifetime of\nτs=35±10 ps inn-GaAs at room temperature. This value is comparable to previ ous theoretical\ncalculations based on D’yakonov and Perel’ (DP) spin relaxa tion in bulk GaAs33as well as the\nvalue ofτS(50±10 ps) extracted from non-local spin valve measurements (se e Supplementary\nFig. 4). The frequency dependence in Fig. 4b disappears as th e temperature is lowered to 30 K.\nIn this limit, the magnetization precesses through multipl e cycles before the spin accumulation\nrelaxes,and VFMRshouldthereforebeinsensitiveto frequency,as isfoundex perimentally.\nDiscussion\nInsummary,wehavedemonstratedanewtechniquebasedonFMR tomeasurespinaccumulation\nin ferromagnet-semiconductordevices. Theprecession oft hemagnetizationat resonanceleads to\nameasurablephaselagbetweenthespinaccumulationintheS CandthemagnetizationintheFM.\nBecause thevoltagedetected bytheFM isproportionalto the projectionofthespinaccumulation\nonto the magnetization, the spin accumulation voltage decr eases at resonance. The typical FMR\n11frequency is larger than the spin relaxation rate, and so the spin accumulation can be measured\nup to room temperature, making this approach more effective than the traditional Hanle effect,\nwhich has previouslylimitedspin lifetimemeasurementsin GaAs-based spin transportdevices to\ntemperatureslessthanapproximately150K.FMRoccurswith inanarrowmagneticfieldwindow,\nandthespinaccumulationsignaldetectedbytheFMissensit ivetotheprecessionofthemagneti-\nzation. Togetherthesemakethetechniqueessentiallyimmu netothefield-dependentbackgrounds\nthat plague Hanle measurements, particularly at high tempe ratures. By measuring the frequency\ndependence of the spin accumulation signal, spin relaxatio n in the SC can be probed directly.\nWith this method, we obtain an electron spin lifetime of a few tens of picoseconds in n-GaAs\n(n=3×1016cm−3)at room temperature. The FMR-based spin detection techniqu e developed\nhere can be used above room temperature and applied to other m aterial systems, such as metals,\ninwhich thespinlifetimesare shortand traditionalHanlem easurementsareimpractical.\nMethods\nSamplegrowth. TheFM/n-GaAsheterostructuresinvestigatedinthisexperimentwe regrownby\nmolecular beam epitaxy on GaAs (001) substrates. The growth started with a 500 nm undoped\nGaAs buffer layer, followed by 2500 nm of Si-doped n-GaAs (n= 3×1016cm��3). The junction\nregion consists of a 15 nm n→n+-GaAs transition layer followed by 18 nm n+(5×1018cm−3)\nGaAs5. The 5 nm thick FM film was then deposited epitaxially, follow ed by 10 nm thick Al\nand Au capping layers. The deposition temperatures for Co 2MnSi and Co 2FeSi were 220◦C and\n270◦C, respectively.\nDevice fabrication. Ion milling was used to define the 5 µm×50µm FM contacts. The remote\n50µm×50µm non-magneticcontacts were fabricated by depositing40 nm Cu and then 40 nm\nGe using electron beam evaporation. The CuGe contacts were p repared in pairs and annealed by\npassing a current through them in order to make them ohmic. Th is step avoids thermal annealing\nof the FM /n-GaAs Schottky contact. A n-GaAs channel was defined by wet etching down to\nthe substrate. 80 nm SiN was deposited over the semiconducto r using PECVD at a substrate\ntemperature of 100◦C and then lifted off in order to expose the CuGe and FM contact s. Finally,\nTi/Au electrodes and bonding pads were fabricated by depositin g 25 nm Ti and 120 nm Au by\nelectron beamevaporation.\nMeasurement. The FMR-spin detection measurement was carried out using a m icrowave gen-\nerator, a lock-in amplifier and a current source, as shown in F ig. 1(a). The microwave excitation\n12signal was coupled to the FM injector through a coaxial cable and an ordinary wire bond. A\n120 nm thick Au wire patterned over the FM contact functioned as a stripline, which was termi-\nnated at ground. The dc current generating the spin accumula tion was fed to the sample using a\nconventional twisted pair, with one lead connected to one of the ohmic contacts. The differential\nvoltage between a second ohmic reference electrode and the F M contact (connected through a\nbias tee) was measured using a lock-in amplifier. The microwa vesignal was modulated using the\n100 Hz reference signal of the lock-in amplifier. The circuit therefore measured the difference in\nthespinaccumulationwithand withoutmicrowaveexcitatio n.\n1. Chappert, C.,Fert, A.&VanDau, F.N. Theemergence ofspin electronics indatastorage. Nat.Mater.\n6, 813–823 (2007).\n2. Dery, H., Cywi ´nski, L. & Sham, L. J. Spin transference and magnetoresistan ce amplification in a\ntransistor. Phys. Rev. B 73, 161307 (2006).\n3. F.Meier, B.P.Z.(ed.) Optical Orientation (North-Holland, Amsterdam, 1984).\n4. Jedema, F. 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Rev.B 64, 161301 (2001).\nAcknowledgments\nThis work was supported by the National Science Foundation ( NSF) under DMR-1104951, C-\nSPIN, one of the six centers of STARnet, a SRC program sponsor ed by MARCO and DARPA,\nthe Materials Research Science and Engineering Centers (MR SEC) program of NSF under DMR\n08-19885,andtheNSF NNIN program.\nAuthor contributions\nC.L. fabricated the FMR devices, performed the FMR-spin det ection experiment, developed the\ntheoretical model and wrote the paper with P.A.C.. S.J.P. gr ew all the FM /n-GaAs heterostruc-\ntures used in this work. T.A.P. fabricated the non-local spi n valve devices and took the non-local\nspin valve measurement. C.C.G. and K.D.C. took the non-loca l Hanle measurement. G.S. took\nsome of the TAMR measurements. C.C.G. developed software fo r performing the experiment\nand analyzing the data. P.A.C. provided the initial concept , and P.A.C. and C.J.P. supervised the\nproject.\nAdditional information\nCompetingfinancial interests: Theauthorsdeclare nocompe tingfinancial interests.\n15"    },    {        "title": "2105.10875v1.Multifunctional_operation_of_the_double_layer_ferromagnetic_structure_coupled_by_a_rectangular_nanoresonator.pdf",        "content": "Multifunctional operation of the double-layer ferromagnetic structure coupled by a rectangular nanoresonator\nMultifunctional operation of the double-layer ferromagnetic structure\ncoupled by a rectangular nanoresonator\nPierre Roberjot,1Krzysztof Szulc,1,a)Jarosław W. Kłos,1and Maciej Krawczyk1,b)\nInstitute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, Pozna´ n,\nUniwersytetu Pozna´ nskiego 2, 61-614 Pozna´ n, Poland\n(Dated: May 25, 2021)\nThe use of spin waves as a signal carrier requires developing the functional elements allowing for multiplexing and\ndemultiplexing information coded at different wavelengths. For this purpose, we propose a system of thin ferromagnetic\nlayers dynamically coupled by a rectangular ferromagnetic resonator. We show that a single and double, clockwise\nand counter-clockwise, circulating modes of the resonator offer a wide possibility of control of propagating waves.\nParticularly, at frequency related to the double-clockwise circulating spin-wave mode of the resonator, the spin wave\nexcited in one layer is transferred to the second one where it propagates in the backward direction. Interestingly, the\nwave excited in the second layer propagates in the forward direction only in that layer. This demonstrates add-drop\nfiltering, as well as circulator functionality. Thus, the proposed system can become an important part of future magnonic\ntechnology for signal routing.\nA modern society is experiencing a rapidly-growing de-\nmand for interconnected wireless facilities, known as Internet\nof Things (IoTs),1,2which requires the development of faster\nways to communicate and lower energy consumption to sat-\nisfy the environmental sustainability.3IoTs are devices receiv-\ning and sending back information using microwaves, thus the\nspin waves (SWs) are perfect candidates to be used in these\ndevelopments.4It is due to the same range of frequencies as\nmicrowaves and a few orders shorter wavelengths, enabling\nminiaturization down to nanoscale. The SWs allow process-\ning both the analog and digital signals at low energy cost, be-\ning inductively coupled to microwaves and compatible with\nCMOS technology.5–7\nA possibility of frequency-dependent SW routing in a mul-\ntiport device is essential to build any complex magnonic\nsystem.8Therefore, implementation of the magnonic counter-\npart of the channel add-drop filter, which was demonstrated\nin photonics,9,10is highly desirable. The crucial element of\nthe add-drop filter is the resonator, e.g., a dot, a stripe or a\nring, where the resonant modes mediate the coupling between\nthe waves in unconstrained conduits. The resonance related\nto the fundamental mode of the stripe is widely explored to\ncontrol the SW propagation and the interesting effects related\nto the chirality of the magnetostatic-stray-field coupling have\nbeen already demonstrated.11–13From the other side, the ring-\nshaped resonators, extensively investigated in photonics, have\nbeen tested as a coupler between two SW waveguides, only\nrecently.14However, a single solid element with the circulat-\ning modes for SW routing remains unexplored.\nIn our paper, we investigate theoretically a system com-\nposed of two Co ferromagnetic films coupled through a\nmultimode resonant element—a stripe of the rectangular\ncross-section made from Py (Ni 80Fe20). We observe var-\nious kinds of SW eigenmodes in the stripe. In particu-\na)Author to whom correspondence should be addressed: Krzysztof Szulc,\nkrzysztof.szulc@amu.edu.pl\nb)Author to whom correspondence should be addressed: Maciej Krawczyk,\nkrawczyk@amu.edu.pllar, we found resonance modes in the form of single and\ndouble, clockwise (CW) and counter-clockwise (CCW), cir-\nculating waves. Our study goes along the exploitation of\nthese modes for various functionalities, including add-drop\nfiltering and a circulator. Our study is in line with re-\ncent activities in magnonics aiming in prototyping functional\nmagnonic devices, like diodes,15circulators,16couplers,17,18\nmultiplexers,19and logic gates.20–24\nThe paper is organized as follows. First, we present the\ngeometry of the structure. Then, we describe the dispersion\nrelations of the subsystems and analyze the eigenmode spec-\ntra. Finally, we demonstrate the operation of the system at\ndifferent frequencies and present a summary.\nWe consider a system composed of two parallel, infinitely\nextended in the xy-plane, 5-nm-thick Co films and a resonant\nelement in the form of a Py stripe of the thickness of 50 nm\nand the width of 100 nm placed between the films and oriented\nalong the z-axis, as presented in Fig. 1(a). The stripe is sepa-\nrated from the films by the 10-nm-thick nonmagnetic spacers.\nThe system is magnetized by a uniform in-plane bias magnetic\nfield of magnitude m0H0=0:05 T directed along the stripe and\nperpendicularly to the direction of SW propagation. We as-\nsume the magnetization saturation and the exchange constant\nof Co to be MS=1100 kA/m and A=20 pJ/m, respectively,\nand for Py MS=850 kA/m and A=13 pJ/m. The gyromag-\nnetic ratio for the whole structure is g=175:95 rad/(sT). We\nconsider the Damon-Eshbach geometry, where the SW group\nvelocity and the magnetostatic coupling are significant.5\nWe consider the system shown in Fig. 1(a) as a four-\nterminal magnonic device, from which one is selected as an\ninput (I1 or I3) and the others as outputs (O1, O2, O3, O4),\nas shown in Fig. 1(b). We want to guide the signal between\npair of selected terminals through the stripe element by proper\ntuning the frequency of the excited SW.\nTo describe the propagation in the system, we solve numer-\nically the Landau-Lifshitz-Gilbert equation\n¶M\n¶t=\u0000gm0M\u0002Heff+a\nMSM\u0002¶M\n¶t; (1)\nwhere M= (mx;my;mz)is the magnetization vector, m0isarXiv:2105.10875v1  [cond-mat.mes-hall]  23 May 2021Multifunctional operation of the double-layer ferromagnetic structure coupled by a rectangular nanoresonator 2\n𝐻0Py NM\n𝐻0𝑦\n𝑥\n𝑧(a)\n(b)spin-wave propagation\nresonatorO1\nO3 O4O2\nI𝑦\n𝑥Co 𝑀\nI\nFigure 1: (a) The geometry of the bilayer structure under\ninvestigation. The system consists of two Co layers and a Py\nstripe separated by a nonmagnetic material. The separation\nbetween the layers and the stripe is s=10 nm, the thickness\nand the width of the stripe is 50 nm and 100 nm, respectively,\nand the thickness of the layers is 5 nm. (b) Schematic\nrepresentation of the four-terminal device. The possible\npositions for the inputs, i.e., the sources of SWs, are marked\nin green, while the four outputs in gray.\nthe magnetic permeability of vacuum, ais the dimensionless\ndamping parameter, and Heffis the effective magnetic field:\nHeff=H0+Hm+Hex; (2)\nwhich is a sum of the external magnetic field H0=H0ˆz,\nthe magnetostatic field Hm=\u0000Ñj, and the exchange field\nHex=2A\nm0M2\nSÑ2M.jis the scalar magnetic potential that ful-\nfill Maxwell equations in the magnetostatic approximation:\nÑ2j=Ñ\u0001M: (3)\nWe linearized Eq. (1), assuming the harmonic time depen-\ndence exp (\u0000iwt)andH0saturating the sample. We split the\nmagnetization and the magnetostatic field into the static com-\nponents parallel to the z-axis, mz=MS,Hm;z=0, and the\ndynamic components laying in the xy-plane, m= [mx;my;0],\nhm= [\u0000¶xj;\u0000¶yj;0]. With these approximations, we have\nperformed the finite-element method (FEM) simulations of\nthe magnetization dynamics using COMSOL Multiphysics.\nFirst, we solve eigenproblem based on linearized Eq. (1)\nusing the frequency-domain solver in order to obtain the dis-\npersion relation for SWs. In these simulations, the Floquet-\nBloch boundary conditions are applied along the x-direction\nat the edges of the unit cell, which reproduce the effect of in-\nfinite layers. We neglect damping in these studies.\nIn the second approach, we solve linearized Eq. (1) in the\ntime domain and for the finite structure. We used this ap-\nproach to demonstrate functionality of the proposed devices.\nWe excite the SWs in the input port with an antenna positioned\n20 nm from the stripe edge. The antenna generates a dynamic\nmagnetic sinusoidal signal of small amplitude\nSant(t) =10\u00007gm0M2\nSsin(2pf0t) (4)\n/uni00000019/uni00000013\n /uni00000017/uni00000013\n /uni00000015/uni00000013\n /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013\n/uni0000003a/uni00000044/uni00000059/uni00000048/uni00000059/uni00000048/uni00000046/uni00000057/uni00000052/uni00000055/uni00000003/uni0000000b/uni00000055/uni00000044/uni00000047/uni00000012 /uni00000050/uni0000000c\n/uni0000001b/uni00000014/uni00000013/uni00000014/uni00000015/uni00000014/uni00000017/uni00000014/uni00000019/uni00000014/uni0000001b/uni00000015/uni00000013/uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni0000000cFigure 2: Dispersion relation of SWs in the subsystem\ncomposed of two Co layers separated by 70-nm-thick\nnonmagnetic material (red and blue solid lines) and the\neigenfrequencies of the the Py stripe of 100-nm width and\n50-nm thickness (horizontal dashed black lines).\nwhere f0is the frequency of excitation and the antenna’s width\nwant=p=kz0, with kz0being a wavenumber at the excitation\nfrequency determined numerically from the dispersion rela-\ntion obtained in frequency-domain simulations. In order to\navoid reflection of the SWs from the system edges, we as-\nsume a linearly increasing damping at the edges of the Co\nlayers. The excitation last 3 ns, which is sufficient to observe\nthe SW propagation and a resonant behavior, if present.\nIn order to understand the impact of the stripe on the prop-\nagation of the SWs in bilayered structure, we calculate the\ndispersion relations and the eigenmode spectra for two addi-\ntional systems. The first one is made of two 5-nm-thick Co\nlayers, separated by 70-nm-thick nonmagnetic material. The\nsecond one is a single Py stripe of width 100 nm and thickness\n50 nm, i.e., the system in Fig. 1(a) without the Co layers.\nDispersion relation of the Co bilayer without a resonator is\nshown in Fig. 2. There are two bands with the frequency dif-\nference proportional to the dynamical coupling between the\nSWs in the layers.16,25,26The low- (blue) and high-frequency\n(red) modes can be assigned to antisymmetric and symmetric\noscillations in the top and the bottom Co layer, respectively.\nWe notice that these branches are getting closer with the in-\ncrease of the frequency (and the wavevector), which means\nthe coupling between the two layers is getting weaker with\ndecreasing wavelength of the propagating waves. This mag-\nnetostatic coupling between layers means that the SW excited\nin one layer will transfer between the layers periodically dur-\ning the propagation. The distance on which the SW migrates\nbetween the layers is defined by the coupling strength. This\neffect was exploited in the directional couplers.17,18,26–28\nThe spectrum of the eigenmodes of the second subsystem\nis presented in Fig. 2 with dashed black lines and indicates\nSW resonant modes of the infinitely-long Py stripe with the\nmagnetization saturated along the z-axis. In the investigated\nfrequency range, the six modes are present with two pairs of\nthe modes very close in frequency, at about 15 and 19 GHz. ToMultifunctional operation of the double-layer ferromagnetic structure coupled by a rectangular nanoresonator 3\n/uni00000013/uni00000014\n/uni00000010\n/uni00000013\n/uni00000026/uni00000026/uni0000003a\n/uni00000015/uni0000005b/uni00000026/uni00000026/uni0000003a\n/uni00000026/uni0000003a\n/uni00000015/uni0000005b/uni00000026/uni0000003a\n/uni00000016/uni0000005b/uni00000026/uni00000026/uni0000003a\n/uni0000005b/uni0000005c\n/uni0000005d\n/uni00000024/uni00000050/uni00000053/uni0000004f/uni0000004c/uni00000057/uni00000058/uni00000047/uni00000048/uni00000003/uni0000000b/uni00000044/uni00000055/uni00000045/uni00000011/uni00000003/uni00000058/uni00000051/uni0000004c/uni00000057/uni00000056/uni0000000c\n/uni00000033/uni0000004b/uni00000044/uni00000056/uni00000048/uni00000003/uni0000000b/uni00000055/uni00000044/uni00000047/uni0000000c\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 /uni0000000b/uni00000045/uni0000000c /uni00000024/uni00000045/uni00000056/uni0000000bmy/uni0000000c /uni00000024/uni00000055/uni0000004a/uni0000000bmy/uni0000000c\nFigure 3: Amplitude (a) and phase (b) of the mycomponent\nof the magnetization in the isolated Py stripe for the modes\nshown by dashed black lines in Fig. 2. The CW and CCW\nmodes are indicated in (b), the continuous phase change and\ntwo zeros of the amplitude at 15.01 and 19.43 GHz indicate a\ndouble CCW and CW circulating modes, respectively.\nanalyze the types of resonant excitations, we plot in Fig. 3 the\namplitude and phase of the y-component of the magnetization,\nAbs( my) and Arg( my), respectively. We can see a fundamental\nmode with the in-phase oscillations in the stripe cross-section\nat 15.28 GHz. Interestingly, all other modes are CW or CCW\ncirculating SWs, as indicated by continuous change of the SW\nphase in Fig. 3(b), and marked with the white arrows. We\nfound also the modes composed of the two CCW (at 15.01\nGHz) or two CW (19.43 GHz) circulated oscillations, we will\ncall them double-CCW and double-CW circulating modes, re-\nspectively. All considered CCW modes have lower frequency,\nthan the corresponding CW modes, which points at the pres-\nence of a nonreciprocity in this system. As shown in Fig. 3(b),\nthe double circulating modes present circulations possessing\nrelative phase shift by 180\u000eat the top and the bottom edges of\nthe stripe. These two properties shall cause different coupling\nof the circulating resonant modes to the modes propagating in\nthe same direction in the upper and lower ferromagnetic lay-\ners or alternatively, a different coupling between the modes\npropagating in the same layer, but in the opposite directions.\nIndeed, this feature is used to control waves in the waveg-\nuides coupled with the ring resonators and whispering gallery\nmodes in photonics29–31and magnonics14,32. We will use this\nproperty for demonstration of the SW circulator.\nThe gradient of the phase on the sides of the resonator de-\npends on the number of the circulation areas, it is one for sin-\ngle CW/CCW and two for double-CW/double-CCW modes\n[see, Fig. 3(b)]. We expect that if the gradient of the phase\nat the top and bottom edge of the resonator match to the\nwavenumber and the phase change of the SW propagating inthe ferromagnetic layer, there will be an enhanced transmis-\nsion of SW in one and suppressed in the opposite direction of\npropagation.\nBefore studying the transfer of SWs between Co layers\nat frequencies close to the circulating resonances of the Py\nstripe, we analyzed also the dispersion relation of the base\nsystem, i.e., bilayered structure with the Py stripe in-between,\nwith periodic boundary conditions along the x-axis. The spec-\ntra (shown in the Supplementary Material) has many band\ngaps between more or less dispersive bands and it is signif-\nicantly different from the spectra of the two Co layers shown\nin Fig. 2. It indicates a strong and complex coupling between\nthe propagating modes in the bilayer and the resonant modes\nof the Py stripe.\nTo demonstrate how different resonances in the Py stripe in-\nfluence the coupling and transfer of SWs between the Co lay-\ners, we perform time-domain simulations for the four cases.\nCase 1 at 15 GHz, it is around the resonance of the fundamen-\ntal and the double-CCW mode. For the case 2, we selected the\nfrequency 19 GHz, which is close to the double-CW circular\nmode of the stripe (mode at 19.43 GHz in Fig. 3). Case 3, 13\nGHz, was chosen to be close to the single-CCW circulating\nmode of the stripe (12.87 GHz in Fig. 3), and the case 4 at 17\nGHz, the frequency close to the single-CW circulating mode\n(mode 17.42 GHz).\nCase 1 at 15 GHz is shown in Fig. 4(a) and (b) for the SW\nsource placed in the top and the bottom layer, respectively.\nThe SW is transferred to the Py resonator but then transfers\nback to the Co layer, propagating in the same direction. We\ncan observe two phenomena. First, the SW changes its phase\nupon the movement through the resonator. Second, the trans-\nfer between Co layers is delayed. During propagation, the\nSW transfers between the layers coupled by the stray dynamic\nmagnetic field, similarly to the directional couplers.17,26,27,33\nSuch behavior is present also at low frequencies, i.e., below\nthe fundamental mode of the resonator, where the dynami-\ncal magnetostatic coupling between Co layers is strongest and\nthus the period of the transfer between the layers is shortest.\nCase 2, add-drop filtering and circulator. In this case, we\nfound a functionality analogous to the circulator, as demon-\nstrated in Fig. 4(c) and (d). In this device, the wave excited\nin one terminal has to be transferred only to one of remain-\ning terminals according to the assumed clockwise or counter-\nclockwise circulation rule.16We found the 19-GHz frequency\noptimal for such a functionality, close to the double-CW circu-\nlating mode resonance of the Py stripe (Fig. 3, mode at 19.43\nGHz). At this frequency, the wave propagating from the an-\ntenna can excite the resonant mode of the stripe which will\nact as an energy harvester due to the magnetization circulating\nmode. Indeed, when the wave is excited on the left side of the\ntop layer (the input port I1) [Fig. 4(c)], the Py resonator redi-\nrects it to the bottom layer as a wave propagating to the left,\nto the output port O3. Importantly, an isolation of the ports\nO2 and O4, exceeding 5 times of the signal directed to O3,\nexists over 600 MHz range. When the input port is placed in\nthe bottom layer on the left (I3) from the resonator [Fig. 4(d)],\nthe wave propagates directly to the port O4 in the same layer.\nFor the wave excited in I4, the resonator redirect to O2, whileMultifunctional operation of the double-layer ferromagnetic structure coupled by a rectangular nanoresonator 4\nFigure 4: Spin-wave amplitude plots showing propagation in the investigated structure at selected excitation frequencies to\ndemonstrate different functionalities. The wave excited in the top (a) or bottom (b) layer at 15 GHz weakly couples with the\nresonator. The observed transfer of the wave from the top to the bottom layer is an effect of the direct coupling between the Co\nlayers. This is case 1. (c) The excitation at 19 GHz in the top-left part of the layer is transferred via resonator to the bottom\nlayer and propagates to the left. (d) The excitation is located in the bottom layer on the left, the wave propagates straight to the\nright without coupling. This is operation of the magnonic add-drop filter and circulator, case 2. (e) The wave excited at 13 GHz\nis transferred through the resonator to the bottom layer and propagates in both direction, case 3. (f) The wave excited in the top\nlayer at 17 GHz is reflected back from the resonator, case 4. The simulations were performed in the time domain, the plots\nshow the SW amplitude, i.e., mycomponent of the magnetization, after 2 ns from excitation.\nexcited in I2 will propagate to O1. We observe similar func-\ntionality also at 13 GHz but with larger leakage of the energy\nto the other ports. The presented operation can be used for a\ndesign of magnonic circulator or add-drop filtering.\nCase 3 at 13 GHz is shown in Fig. 4(e). The wave intro-\nduced in the port I1 couples with the single-CCW circulating\nmode in the Py stripe which allows to make transfer to the\nbottom layer and form waves propagating to the both ports lo-\ncated there, O3 and O4. Although the intensity of the waves\nreaching the ports O3 and O4 differs, the clear splitting of the\nwave propagating in the top layer has been demonstrated.\nCase 4. In this case, we introduce the wave at 17 GHz in\nthe port I1. As can be seen from the Fig. 4(f), the excited wave\nis transferred to the Py stripe and the circulating mode passes\nit to different ports but mostly go back to the top-left output\nO1. The Py resonance involved in this operation is related to a\nsingle-CW circulating mode, thus the functionality can be ex-\npected to be similar to the coupling with the ring resonators14\nor of the circulator shown in the case 2. However, the mis-\nmatch between the wavelength of the propagating SW and the\nresonance length results in a weak coupling and unclear oper-\nation.\nInterestingly, the coupling of SWs excited in the left part of\nthe bottom Co layer (I3 port) with the Py stripe is suppressedat all considered frequencies. We attribute this to the unidirec-\ntional magnetization precession and the chirality of the mag-\nnetostatic stray field coupling11,12.\nThe investigations presented above were made without\ndamping in order to have clear visualization of the coupling\nprocesses. We performed also the same simulations but with a\ndamping, we assume in Co layers aCo=0:01 and in Py stripe\naPy=0:005. We observed the presence of the same phenom-\nena only with an attenuation. The results are presented in the\nSupplementary Material.\nUsing a stripe of 100-nm width and 50-nm thickness, we\nhave demonstrated some of the possibilities to control the\npropagation of the SWs in thin ferromagnetic films at differ-\nent frequencies. The energy plots shown in the Supplementary\nMaterial additionally support these demonstrations. However,\nthe demonstrations were done for selected materials, the one\nrelative magnetization orientation and at the fixed geometrical\nparameters. The functionality of the resonance element de-\npends on the wavelength of the propagating wave and its rela-\ntion to the size of the stripe, and chirality of the magnetization\nprecession, thus further investigations and model developing\nis necessary to optimize the operation of the proposed solu-\ntions.\nIn conclusion, we explored numerically the SW behavior inMultifunctional operation of the double-layer ferromagnetic structure coupled by a rectangular nanoresonator 5\na structure composed of two 5-nm-thick Co layers separated\nby a rectangular Py stripe, all homogeneously magnetized par-\nallel to the stripe axis. We show that its dispersion relation is\nsignificantly different as compared to a bilayer structure due\nto dynamical coupling of propagating SWs in Co layers with\nthe resonant modes of the Py stripe. Interestingly, the low-\nfrequency part of the isolated-stripe spectra consists of the\nfundamental mode and a number of single and double, non-\ndegenerated CW and CCW circulating modes. The interaction\nbetween propagating SWs in layers and modes circulating in\nthe resonator allows for design add-drop filters and circulators\nfor SWs.\nIn particular, we show that depending on the direction of\nthe SW propagation and on the excitation frequency, the SWs\ncan transmit in four possible routs: direct propagation be-\ntween the two Co layers, circulation (SWs are propagating\nfrom one layer to the other keeping the direction of propaga-\ntion), reflection (forth and back to the excitation point), and\nuncoupled propagation (SWs are propagating in the one layer\nonly). Especially effective is the use of double-CW circulating\nmode of the stripe allowing us to demonstrate the functional-\nity of the magnonic circulator. This shows that the circulating\nmodes in the ferromagnetic stripe are promising for design of\nsignal-processing magnonic devices—especially multiplexers\nand demultiplexers.\nACKNOWLEDGMENTS\nThe research leading to these results has received funding\nfrom the Polish National Science Centre, project no. UMO-\n2018/30/Q/ST3/00416.\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\nREFERENCES\n1F. Mattern and C. Floerkemeier, “From the internet of computers to the in-\nternet of things,” in From Active Data Management to Event-Based Systems\nand More: Papers in Honor of Alejandro Buchmann on the Occasion of\nHis 60th Birthday , edited by K. Sachs, I. Petrov, and P. Guerrero (Springer\nBerlin Heidelberg, Berlin, Heidelberg, 2010) pp. 242–259.\n2J. Lin, W. Yu, N. Zhang, X. Yang, H. Zhang, and W. 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In both measurements, \nsamples were enclosed tightly inside  a copper strip and impedance  or reflection coefficient of this \ncopper ‘strip coil’ was measured as a function of applied magnetic field for different frequencies \nof current ( f = 0.1 to 2.5 GHz). The direction of the applied magnetic field was perpendi cular to \nthe a lternating  magnetic field produced by the coil . In the ferromagnetic sample ( x = 0.1), \nmagnetoresistance  shows a peak around zero field for lower frequencies but a peak appears at H > \n0 at higher frequencies. The position of the peak shifts towards higher fields with increasing \nfrequency. A similar trend is al so found for the paramagnetic sample (x = 0.06) but the peak occurs \nat a higher field compared to the  ferromagnetic sample  for the same frequency . Microwave power \nabsorption also show s features similar to magnetoresistance.  Line shape analysis of the data w as \nperformed by fitting the data to a Lorentzian function.  It is concluded  that the observed features \nare imprints of ferromagnetic resonance in x = 0.1 and paramagnetic resonance in x = 0.06 samples.  \n \n1 Author for correspondence(phyrm@nus.edu.sg)  2 \n 1. Introduction  \nThe most  remarkable propert y of manganese perovskite  oxides  (R1-xAxMnO 3 where R = \nLa, Pr , Nd etc. , and A = Sr, Ba, Ca) is the occurrence of colossal negative magnetoresistance  \naround ferromagnetic Curie temperature and below. Magnetoresistance is usually measured by a \nfour-probe method  with a direct current (dc) or a low frequency ( f < 500 Hz ) current  using a lock -\nin amplifier . On the other hand, m agnetoresistance in GHz range is  estimated  from the change s in \nmicrowave reflectivity of a cavity resonator loaded with a manganite sample while  an externally \napplied dc magnetic field  is swept. The change in microwave reflectivity is related to  an alteration \nin surface resistance of the sample. Dominguez et al.1 and Srinivas et al.2 reported  large microwave \nmagnetoresistance at low fields  (~80% for H = 600 Oe) compared to magnetoresistance measured \nwith a dc current  in ferromagnetic samples of Nd0.7Sr0.3MnO 3 and La 0.7A0.3MnO 3 (A= Sr, Ba) , \nrespectively.  However, t hose measurements were restricted to a single microwave frequency ( f = \n9.8 GHz) , and multiple cavities had to be used for different  frequencies.  Very recently, we have \nreport ed that the magnitude of magnetoresistance in samples of  La0.7Ca0.3-xSrxMnO 3 series is not \nonly enhanced if  a radio frequency ( RF)/ microwave ( MW) current (f = 0.01 to 3 GHz) is directly \npassed through the sample, but the field -dependence of high frequency magnetoimpedance  (f = 0.1 \nto 3 GHz) also shows features of ferromagnetic and paramagnetic resonances  (FMR and EPR) .3 \nThe possibility of FMR detection via magnetoimpedance was proven for Co -based amorphous \nalloy s but not for EPR4,5. Electrical detection of EPR and FMR  using a simple magnetoimpedance \nmethod that does not make use of microwave cavit ies or coplanar waveguides is advantag eous for \napplications because of the simplicity of the technique.  However, it  is important to confirm these  \nresults by alternative methods . For this purpose, in this work, we have used two different 3 \n techniques making use of two  different instruments  (An RF impedance analyzer and a vector \nnetwork analyzer) . \n2. Experimental details  \nPolycrystalline samples of La 0.7Ca0.3-xSrxMnO 3 (x = 0 to 0.3) were prepared by solid state \nreaction  method  and characterized earlier by X -ray diffraction, magnetization and \nmagnetoimpedance. Previously, we  reported that the ferromagnetic Curie temperature ( TC) \nincrease d with Sr content3. The samples with x > 0.06 are ferromagnetic and x < 0.06 are \nparamagnetic at room temperature. We choose a sample of La0.7Ca0.24Sr0.06MnO 3 which is \nparamagnetic at room temperature (TC = 292  K) and a sample of  La0.7Ca0.2Sr0.1MnO 3 which is \nferromagnetic  (TC =309 K) at room temperature  (T=298 K ) for the present work . These samples \nwere cut into a  rectangular shape ( 5 mm length x 3 mm width x 2 mm thick ). A copper strip of \nsimilar dimensions made from 0.2 mm thick copper sheet was folded  into a  cuboidal coil. The \nsample was firmly held inside  the strip coil . The inner surface of the strip coil was covered with a \nKapton tape layer to electrically isolate  the sample from the copper strip. The two ends of the \ncopper strips were connected to a  radio -frequency impedance analyzer (Agilent model E4991A) \nor to a vector  network analyzer ( Agilent model N5230A ) using high frequency cables from \nHUBER+SUHNER . These instruments were calibrated by performing  the standard open -short -\nload procedure.  The strip coil  with the sample was placed at the center of poles of an \nelectromagnet. Resistance of the strip  coil was measured by the impedance analyzer for each dc \nmagnetic fields for different frequencies of alternating  current  (ac). Magnetoresistance is \ncalculated from the standard definition: MR ac = [R(H, f)-R(0,f)]/R(0,f) where R(H, f) is the \nresistance in a field H for a frequency ( f) of the current.  Data were taken without and with the \nsample inside the strip  coil for same set of frequencies and magnetic fields.  Later, t he empty coil 4 \n data was subtracted from the MR ac data taken  with the sample.  The magneti c field dependence of \nthe microwave power absorption  (MWPA ) was measured  using a vector network analyser (VNA) \nover a broad frequency range (10 0 MHz - 2.5 GHz).  Here also, the empty coil data were subtracted.  \nPreviously, Korenivski et al.6 had employed a similar copper strip to measure impedance up to 1 \nGHz using impedance analyzer HP4191A. For MWPA measurement , reflection coefficient of \nelectromagnetic  waves  in the copper strip is measured via the S11 scatterin g parameter . S11 \nrepresents the power that is reflected from the device  connected to the VNA . To estimate  the power \nabsorbed by the sample  in the strip coil  due to the application of a magnetic field, we record  ∆P(H) \n= S11(H, f)-S11(Hmax, f) where , Hmax is the maximum value of the applied dc magnetic field.  \n3. Results  \nFigs. 1(a) and (b) display the magnetic field dependence of the alternating current ( ac) \nmagnetoresistance ( MR ac) for the samples x = 0.06 and 0.1, respectively at room temperature in \nthe frequency range: f = 0.1 to 2.5 GHz. The MR ac at f = 0.1 GHz  and 0.5 GHz  for the paramagnetic \nsample x = 0.06 displays a single peak at the origin Hdc = 0. Surprisingly , MR ac for f > 0.5 GHz  \nabruptly increases and exhibit a peak at Hdc = ±Hc on either sides of the zero field . Both the peaks \nat ±Hc shift towards higher value of Hdc with increas e in the frequency of the ac current.  On the \nother hand, MR ac at f = 0.1 GHz for the ferromagnetic sample x = 0.1 also shows a single peak at \nHdc = 0. However, as the frequency of the current  increases a bove f = 1 GHz , the single peak at \nHdc= 0 splits into two symmetrical peaks at Hdc = ±Hp on either sides of Hdc = 0. In addition,  both \nthese peaks  also move  towards higher value of Hdc as the frequency of the current  increases further, \nsimilar to the response observed in the  paramagnetic sample. However, the peaks in MR ac occurs \nat higher values of Hdc for the paramagnetic sample (0. 784 kOe at 2 GHz ) in comparison to the \nferromagnetic sample ( Hp = 0.402 kOe at 2 GHz ). 5 \n Figs. 2 (a) and (b) compare  the magnetic field dependence of the normalized power \nabsorption ( P) for x = 0.06 and 0.1 samples, respectively at room temperature for selected  \nfrequencies of the microwave  elect romagnetic field between f = 0.1 GHz and 2.5 GHz . P at f = \n0.1 GHz for x = 0.06 sample exhibits a single peak at Hdc = 0 which splits into two symmetrical \nsharp peaks at Hdc = ±Hc for f ≥0.5 GHz. Both t he peak s in P shift towards higher  Hdc rapidly as \nthe frequency of the microwave electromagnetic field increases  similar to the behavior of MR ac. \nConversely, P at f = 0.1 GHz  for the ferromagnetic sample x = 0.1 shows a single peak at Hdc = \n0 but splits into double peaks at Hdc = ±Hp for f ≥ 1.5 GHz. Similar to the behavior of MR ac, both \nthe peaks in P also move apart from each other towards higher values of Hdc as the frequency of \nthe microwave e lectromagnetic field increases. The peaks in P are positioned at higher values of \nHdc for x = 0.06  in contrast to x = 0.1 . Thus, both ac MR  and P exhibit distinct features for the \nferromagnetic and paramagnetic samples.  \n4. Discussion  \nThe flow of RF current in the copper strip coil generates axi al RF magnetic field at the \ncenter of the coil and hence the sample’s magnetization oscillates with the frequency of the RF \nmagnetic field , which in turns affects the complex impedance  (Z) of the strip coil, The real (R) and \nimaginary ( X) comp onents of the complex e lectrical impedance ( Z) of the copper strip  coil depend \non the relative permeability ( r) of the sample through the relations 𝑅=𝐾√(𝜔𝜇0𝜇𝑟′′) and 𝑋=\n𝐾√(𝜔𝜇0𝜇𝑟′),  where,  is the angular frequency, o is the free space permeability, rꞌ and rꞌꞌ are \nthe real and imaginary components of relative permeability of the sample and K is a constant \nrelated to the geometry of the strip coil. Thus, changes in permeability of the sample in response \nto variations of magnetic field and frequency will give rise to change s in impedance .  The RF 6 \n magnetic field and the applied dc magnetic field  (Hdc) are perpendicular to each other  in our \nexperiment as like the  field configuration in  a conventional microwave cavity based EPR/FMR \nspectrometer. Both core spins ( t2g3: S = 3/2) and eg elect ron’s spin  (S = 1/2) of Mn ions contribute \nto the total magnetization of the sample. The magnetization re-orient s itself  to align along the \ndirection of the dc magnetic field but the RF magnetic field , which is transverse to the dc magnetic \nfield induces precession of the magnetization about the dc magnetic field.  When  resonance \ncondition is met, the sample  absorbs a maximum power from the RF electromagnetic field. T he \npower absorbed  (P) from the RF magnetic field by a sample of volume V is 𝑃= 1\n2𝑉𝜇 𝑟\"𝜔ℎ𝑟𝑓2 where, \nhrf is the amplitude of the RF magnetic field.  Magnetic field dependence of r″ goes through  a \npeak  value at the resonance field  (Hres), hence, power absorption  P(Hdc) will also show  a peak  at \nHres. As R depend s on r″, the anomalous features in MR ac in the ferromagnetic ( x = 0.1) and \nparamagnetic ( x = 0.06) samples  can be considered as  manifestations of  maximum power \nabsorption by the sample due to ferromagnetic resonance (FMR) and electron paramagnetic \nresonance (EPR) , respectively . \nSince both absorption and dispersive signals are mixed in power absorption for conducting \nsamples during resonance, we choose to fit  the MR ac line shapes for the paramagnetic sample x = \n0.06 using the following equation,  \n𝑀𝑅 𝑎𝑐(𝐻𝑑𝑐)= 𝐴𝑆𝑦𝑚(∆𝐻\n2)2\n(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2+𝐴𝐴𝑠𝑦𝑚∆𝐻\n2(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)\n(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2+𝐴0            (1) \nwhere, ASym and AAsym are the coefficients of symmetric and antisymmetric Lorentzian \nfunctions , Hres and H are the resonance field and linewidth  (Full width at the half maximum) , \nrespectively and A0 is a constant.  The symmetric Lorentzian function accounts for the absorption \ncomponent and the antisymmetric Lorentzian function is rela ted to the dispersive component7. The 7 \n fitting of the MR ac line shape  at f = 2 GHz for the x = 0.06 sample is shown on the left y-scale of \nFig. 3(a). We have also analyzed the  normalized microwave power absorption ( P (Hdc)) line \nshape  for the paramagnetic x = 0.06 sample  using  a linear combination of the symmetric and the \nantisymmetric Lorentzian functions which can be expressed as , \n∆𝑃(𝐻𝑑𝑐)= 𝑃𝑆𝑦𝑚(∆𝐻\n2)2\n(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2+𝑃𝐴𝑠𝑦𝑚∆𝐻\n2(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)\n(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2+𝑃0                        (3)  \nwhere, PSym and PAsym are the coefficients of symmetric and antisymmetric Lorentzian \nfunctions and P0 is a constant. T he fitting of the P line shape  at f = 2 GHz is shown on the right \ny-scale of Fig. 3(a).  Hres and H were obtained using these fits and analyzed. H for the \nferromagnetic x=0.1 sample was too broad ( H > Hres) and could not be obtained using Eq. 2 or \nEq. 3. H for the paramagnetic x=0.06 sample was obtained and presented in Fig. 3(b). H \nobtained from MR ac measurements ( open  circles) and P measurements ( closed  spheres) both \nincreased with increase in frequency  of the current in the strip coil . Solid lines represent the linear \nrelationship of f vs H which agrees  with the Landau -Lifshitz -Gilbert equation  for magnetization \nprecession given by Eq. 4.   \n                         𝑑𝑴\n𝑑𝑡=𝛾𝜇0(𝑴×𝑯)+𝛼\n𝑀(𝑴×𝑑𝑴\n𝑑𝑡)                                   (4) \n  It can be observed , for the paramagnetic sample , that Hres increases linearly with f and hence \nfollows the resonance condition for EPR, i.e., fres = (/2) Hdc, where   is the gyromagnetic ratio . \nA linear fit to the fres vs. Hdc curve yields /2 values to be around 2.68±0.05 MHz/Oe using the \nimpedance analyzer method and 2.75 ±0.02 MHz/Oe  using the VNA. These values  slightly deviate  \nfrom the free electron value of 2.8 MHz/Oe . This difference could be due to limitation of the \nmaximum frequency (2.5 GHz) in our experiment compared to commercial EPR spectro meters 8 \n which usually operate at 9.8 GHz. Depending on measurement parameters, /2 values can be \ndifferent and a more accurate determination of /2 requires measurements carried out several tens \nof GHz8. Using these   values, 𝛼 was estimated to be 0.0 402 and 0. 0550  from the MR ac and P \nmeasurements  respectively . \nFig. 3  (c) illustrates the  fres vs. Hdc curve s for the ferromagnetic sample x = 0.1 . We fitted \nthe fres vs. Hdc curve s with the Kittel’s equation , 𝑓𝑟𝑒𝑠= 𝛾\n2𝜋√[(𝐻𝑑𝑐+𝐻𝑘)(𝐻𝑑𝑐+𝐻𝑘+𝑀𝑆)] where, Hk is \nthe anisotropy  field, MS is the saturation magnetization and  is the gyromagnetic ratio . Fitting the  \nfres vs Hdc curve s with the Kittel’s equation yield s /2 = 2.52±0.02 MHz/Oe  for the MR ac line \nshape and 2. 59±0.02 MHz/Oe for the P line shape . These distinct results  for paramagnetic and \nferromagnetic samples  presented above using the strip coil method where the chosen sample was \nsubjected to RF magnetic field induced by the alternating  current in the strip coil indeed confirms \nour earlier results obtained w hen RF  current s were  directly  passed  through the samples3. An added \nadvantage of the strip coil technique is that it can also be used for insulating magnetic samples.  \n5. Summary  \nIn summary, we have investigated the change in electrical impedance in a paramagnetic ( x \n= 0.06) and ferromagnetic ( x = 0.1) samples in La 0.7Ca0.3-xSrxMnO 3 series as a function of  dc \nmagnetic field and frequency  using a radio -frequency impedance analyzer  by measuring the \nimpedance of  a strip coil enclosing  the sample inside it  at room temperature. Our \nmagnetoimpedance results  show features  of electron paramagnetic resonance and ferromagnetic \nresonance in x = 0.06 and 0.1, respectively which w ere validated  by microwave absorption \nmeasurements using a vector network analyzer . Hence, this low -cost magnetoimpedance technique 9 \n can potentially investigate spin dynamics in various nanostructured magnetic materials  or \ninsulators . \nAcknowledgements:  R. M. thanks the Ministry of Education for supp orting this work \n(Grant number:  R144 -000-381-112).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 10 \n Figure Captions  \n \nFIG. 1 Magnetic field dependence of the  ac magnetoresistance ( MR ac) for different frequencies \nbetween f = 0.1 GHz and 2.5 GHz for the (a) paramagnetic ( x = 0.06) and (b) ferromagnetic ( x = \n0.1) samples in La 0.7Ca0.3-xSrxMnO 3 series  room temperature.  \n \nFIG. 2 Magnetic field dependence of the normalized power absorption ( P) for (a) x = 0.06 and \n(b) x = 0.1 at room temperature for different frequencies between f = 0.1 GHz and 2.5 GHz.      \n \nFIG. 3 (a) Lorentzian fits to the line shape of MR ac (left y -scale) and P (right y -scale) for the x = \n0.06 sample at f = 2 GHz, (b) Line width  () as a function of frequency ( f), obtained from the \nline shape analysis of MR ac (open  circles)  and P (closed  spheres)  measurements. The solid lines \nare the linear fits to the data. (c)Plot of fres vs Hdc obtained from the MR ac (open symbol ) and P \n(closed symbol ) line shape analysis for x = 0.06 (circles) and 0.1 (squares) samples.   The solid \nlines depict the Kittel equation fitting for all the samples.  \n \n \n \n \n \n 11 \n References  \n1 M. Dominguez, S. M. Bhagat, S. E. Lofland, J. S. Ramachandran, G. C. Xiong, H. L. Ju, T. Venkatesan, \nand R. L. Greene, Euro. Phys. Lett.  32, 349 (1995).  \n2 V. V. Srinivasu, S. E. Lofland, S. M. Bhagat, K. Ghosh, and S. Tyagi, J. Appl. Phys. 86, 1067 (1999) . \n3 U. Chaudhuri, R Mahendiran, J. Mag n. Mag n. Mat er. 488 , 165377 (2019).  \n4 M. R. Britel, D. Ménard, L. G. Melo, P. Ciureanu, A. Yelon, R. W. Cochrane, M. Rouabhi, and B. Cornut, \nAppl. Phys. Lett. 77, 2737 (2000).  \n5 J. M. Barandiarán, A. García -Arribas, and D. de Cos, J. Appl. Phys. 99, 103904 (2006).  \n6 V. Korenivski , V., R. B. Van Dover, P. M. Mankiewich, Z -X. Ma, A. J. Becker, P. A. Polakos, and V. J. \nFratello, IEEE Trans. Magn. 32, 4905, (1996).  \n7 M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C. -M. Hu, Phys. Rev. B 84, 054423 (2011).  \n8 J. M. Shaw, H. T. Nembach, T. J. Silva, and C. T. Boone, J. Appl. Phys. 114, 243906 (2013) . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  12 \n  \n \n \nFIG. 1 Magnetic field dependence of the  ac magnetoresistance ( MR ac) for different frequencies \nbetween f = 0.1 GHz and 2.5 GHz for the (a) paramagnetic ( x = 0.06) and (b) ferromagnetic ( x = \n0.1) samples in La 0.7Ca0.3-xSrxMnO 3 series  room temperature.  \n \n \n \n \n \n \n13 \n  \n \n \nFIG. 2 Magnetic field dependence of the normalized power absorption ( P) for (a) x = 0.06 and \n(b) x = 0.1 at room temperature for different frequencies between f = 0.1 GHz and 2.5 GHz.      \n \n \n \n \n \n \n \n \n \n \n14 \n  \n \n \n \n \n \n \nFIG. 3 (a) Lorentzian fits to the line shape of MR ac (left y -scale) and P (right y -scale) for the x = \n0.06 sample at f = 2 GHz, (b) Line width  () as a function of frequency ( f), obtained from the \nline shape analysis of MR ac (open  circles)  and P (closed  spheres)  measurements. The solid lines \nare the linear fits to the data. (c)Plot of fres vs Hdc obtained from the MR ac (open symbol ) and P \n(closed symbol ) line shape analysis for x = 0.06 (circles) and 0.1 (squares) samples.   The solid \nlines depict the Kittel equation fitting for all the samples.  \n \n"    },    {        "title": "1811.07467v1.Electronic_structure_of_the_novel_high__T___rm_C___ferromagnetic_semiconductor__Ga_Fe_Sb__x_ray_magnetic_circular_dichroism_and_resonance_photoemission_spectroscopy_studies.pdf",        "content": "arXiv:1811.07467v1  [cond-mat.str-el]  19 Nov 2018Electronic structure of the novel high- TCferromagnetic semiconductor\n(Ga,Fe)Sb: x-ray magnetic circular dichroism and resonanc e\nphotoemission spectroscopy studies\nS. Sakamoto,1N. T. Tu,2Y. Takeda,3S.-i. Fujimori,3P. N. Hai,4L. D. Anh,2Y. K. Wakabayashi,2\nG. Shibata,1M. Horio,1K. Ikeda,1Y. Saitoh,3H. Yamagami,3,5M. Tanaka,2,6and A. Fujimori1\n1Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan\n2Department of Electrical Engineering and Information Syst ems,\nThe University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan\n3Materials Sciences Research Center, Japan Atomic\nEnergy Agency (JAEA), Sayo-gun, Hyogo 679-5148, Japan\n4Department of Physical Electronics, Tokyo Institute\nof Technology, Meguro-ku, Tokyo 152-0033, Japan\n5Department of Physics, Kyoto Sangyo University, Kyoto 603- 8555, Japan\n6Center for Spintronics Research Network,\nThe University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan\n(Dated: November 20, 2018)\nAbstract\nTheelectronic structureandthe magnetism of thenovel ferr omagnetic semiconductor (Ga,Fe)Sb, whose\nCurie temperature TCcan exceed room temperature, were investigated by means of x -ray absorption spec-\ntroscopy (XAS), x-ray magnetic circular dichroism (XMCD), and resonance photoemission spectroscopy\n(RPES). The line-shape analyses of the XAS and XMCD spectra s uggest that the ferromagnetism is of\nintrinsic origin. The orbital magnetic moments deduced usi ng XMCD sum rules were found to be large,\nindicating that there is a considerable amount of 3 d6contribution to the ground state of Fe. From RPES,\nwe observed a strong dispersive Auger peak and non-dispersi ve resonantly enhanced peaks in the valence-\nband spectra. The latter is a fingerprint of the correlated na ture of Fe 3 delectrons, whereas the former\nindicates their itinerant nature. It was also found that the Fe 3dstates have finite contribution to the\nDOS at the Fermi energy. These states presumably consisting of majority-spin p-dhybridized states or\nminority-spin estates would be responsible for the ferromagnetic order in t his material.\n1I. INTRODUCTION\nTo create functional devices exploiting the spin degree of freedom in semiconductors has been\none of the major challenges in the field of electronics1–3. Under such circumstances, magneti-\ncally doped semiconductors, or diluted magnetic semiconductors (D MSs), have attracted much\nattention since they possess both magnetic and semiconducting pr operties4–9. Mn-doped III-V\nsemiconductors such as (In,Mn)As10,11and (Ga,Mn)As12–14have been extensively studied because\nthey exhibit carrier-induced ferromagnetism, where the ferroma gnetic interaction between the Mn\nmagnetic moments is mediated by hole carriers, and it is possible to con trol the ferromagnetism\nthrough changing the carrier concentration by gate voltage15,16or light irradiation17. Despite those\nattractivefeatures, theyhavealsoshortcomingsforpractical applications: theirCurietemperatures\n(TC) are much lower than room temperature, 90 K for (In,Mn)As18and 200 K for (Ga,Mn)As19;\nonlyp-type conductivity is realized since Mn always acts as an acceptor at the substitutional In3+\nor Ga3+sites.\nRecently Fe-doped ferromagnetic III-V semiconductors (In,Fe) As:Be20–22, (Ga,Fe)Sb23–25,\n(Al,Fe)Sb26, and (In,Fe)Sb27–29were synthesized, and exhibit some advantages over the Mn-dope d\nones. If Fe substitutes for the In3+or Ga3+site and takes the stable valence of 3+ with the\n3d5(4sp)3configuration, no charge carrier will be provided and hence both n- andp-type conduc-\ntion will be possible via additional carrier doping. In fact, (Al,Fe)Sb is in sulating, (In,Fe)As:Be\nn-type, where doped interstitial Be atoms act as double donors, an d (Ga,Fe)Sb p-type, where\nnative charged defects such as Ga anti-sites are thought to act a s acceptors and provide holes.\nTheTC’s of these materials are relatively high, 70 K for (In,Fe)As:Be22, 40 K for (Al,Fe)Sb26, 335\nK for (In,Fe)Sb27, and 340 K for (Ga,Fe)Sb25. It has been found that the distribution of Fe atoms\nis non-uniform in the zinc-blende crystal structure of these mate rials, which seems to play an\nimportant role in stabilizing the ferromagnetic order24,26,30. However, the microscopic origin of the\nferromagnetism in terms of their electronic structures has not be en clarified yet and remains to\nbe investigated. For this purpose, we have performed soft x-ray absorption spectroscopy (XAS),\nx-ray magnetic circular dichroism (XMCD), and resonance photoem ission spectroscopy (RPES)\nstudies of (Ga,Fe)Sb.\nXAS and XMCD at the L2,3absorption edges of the 3 dtransition metals are very powerful\nmethods for the purpose of clarifying the electronic structures r elated to the ferromagnetism.\nSince x-ray absorption takes place at a specific constituent atom, one can obtain element-specific\ninformation about the electronic structure and its relation to the m agnetism, excluding extrinsic\n2effects such as diamagnetic contribution from the substrate. XMC D sum rules make it possible to\nobtain the spin and orbital magnetic moments of the constituent at oms separately31,32. In addition\nto XAS and XMCD, resonance photoemission spectroscopy (RPES) has been frequently employed\nas a direct probe to examine the electronic structure of materials. RPES provides the information\nabout the partial density of states (PDOS) of 3 dtransition-metal element, and has been used to\nstudy the electronic structures of ferromagnetic semiconducto rs (FMSs) such as (Ga,Mn)As33,34,\nGe:Fe35, and (Ba,K)(Zn,Mn) 2As236,37. Moreover, the combination of RPES and XMCD yields\nthe PDOS of only ferromagnetically active components, and is suitab le for studying FMSs, where\ndoped magnetic atoms are often oxidized at the surface.\nII. EXPERIMENT\nGa1−xFexSb films with two different Fe contents x= 6.0 and 13.7%(referred to as sample A and\nB, respectively) were grown on GaAs(001) substrates using the lo w-temperature molecular beam\nepitaxy (LT-MBE) method. To relax the lattice mismatch between (G a,Fe)Sb and GaAs, three\nbuffer layers were inserted; initially GaAs(50nm) andAlAs(10nm) laye rs were successively grown\nat the substrate temperature TSof 550◦C, and then an AlSb (100 nm) layer at TS= 470◦C. After\ngrowing the buffer layers, the (Ga,Fe)Sb layer of 50 nm thickness wa s grown. Here, the TSwas set\nto 200◦C for sample A and 250◦C for sample B. Lastly sub-nanometer-thick amorphous As cap\nlayer was deposited to prevent surface oxidation. Note that samp le A was paramagnetic down to 5\nK, and sample B was ferromagnetic with TC= 170 K. In order to remove the oxidized surface, we\netched the sample by hydrochloric acid (HCl) (2.4 mol/L) for 5 second s and subsequently rinsed\nit by water just before loading the sample in the vacuum chamber of t he spectrometer35,38.\nAll the measurements were performed at beam line BL23SU of SPring -8. Circularly polarized x\nrays of 690 - 780 eV were used for both absorption and photoemiss ion measurements. For XMCD\nmeasurements, a magnetic field was applied parallel to the incident x r ays and perpendicular to\nthe sample surface. Absorption signals were taken in the total elec tron yield (TEY) mode, and\ndichroic signals were measured by reversing the helicity of x rays with 1 Hz frequency at each\nphoton energy under a fixed magnetic field. In order to eliminate spu rious XMCD signals, the\nscans were repeated with opposite magnetic field directions. Each X MCD spectrum was obtained\nas the average, namely, (( σ+,h−σ−,h) +(σ−,−h−σ+,−h)), where σdenotes the absorption cross-\nsections, the first subscript the helicity of x rays, and the second subscript the sign of the magnetic\nfield. XASwasobtainedasthesummationofall thefourterms. Note that two-stepinverse tangent\n3function representing the Fe L2,3-edge jumps were subtracted from each term39.\nFor RPES measurements, the sample temperature was set to 100 K , and the energy resolution\nwas about 150 meV. The samples were placed so that the [-110] direc tion became parallel to the\nanalyzer slit and perpendicular to the beam. Photoelectrons were c ollected in the normal emission\ngeometry with 45-degree light incidence.\nIII. RESULTS AND DISCUSSION\nA. XAS and XMCD\nFigures 1(a) and 1(b) show XAS and XMCD spectra of the present ( Ga,Fe)Sb films after the\nHCl etching compared with those of (In,Fe)As:Be30, Fe metal39, FeCr 2S440, andγ-Fe2O341. Note\nthat the Fe 3 delectrons in FeCr 2S4andγ-Fe2O3are localized, and that the valence of Fe is 2+ in\nFeCr2S4and 3+ in γ-Fe2O3.\nThe line shapes of the spectra of the two (Ga,Fe)Sb films resemble th ose of bcc Fe metal rather\nthan the sharper spectra of FeCr 2S4, manifesting the itinerant nature of the Fe 3 delectrons in\n(Ga,Fe)Sb. Although the spectral line shapes of the two (Ga,Fe)Sb films look almost identical,\nrepresenting nearly the same electronic structure, the intensity of the XMCD signals was signifi-\ncantly suppressed for the 6% Fe-doped sample compared with the 1 3.7% Fe-doped sample simply\nbecause only the 13.7% Fe-doped sample exhibits ferromagnetism wh ile the 6% Fe-doped one is\nparamagnetic at 5 K. Such an insensitivity of the line shape to transit ion-metal content was also\nreported for (Ga,Mn)As42, (In,Fe)As:Be30, etc.\nIn addition to the main peak at ∼708 eV, there is a shoulder at ∼710 eV, which is more evident\nin XAS than in XMCD. This can be attributed to Fe3+oxides formed at the surface because\nthe feature at ∼710 eV was prominent before etching and disappeared almost comple tely after\netching, as shown in Fig. 2. Here, the spectra of both paramagnet ic and ferromagnetic samples\nafter etching resemble those of Ge:Fe and (In,Fe)As:Be. Using thes e two spectra containing the\ndifferent degrees of contribution from surface oxides, it was poss ible to deduce the intrinsic spectra\nas [XAS] int∝[XAS] a−p[XAS] b. Here, [XAS] aand [XAS] bdenote the XAS spectrum after and\nbefore etching, respectively, and pwas chosen so that the shoulder at ∼710 eV vanished, or the\nsecond derivative of [XAS] intdid not show a peak at ∼710 eV as shown in the inset of Fig. 2. The\nextrinsic contribution from the surface oxides to the XAS spectra was also extractable in a similar\nmanner as [XAS] ext∝[XAS] a−q[XAS] b, whereqwas chosen so that [XAS] extbecame identical\n4/s32/s33/s34/s32/s35/s34/s32/s36/s34/s37/s36/s38/s34/s37/s34/s38/s39/s37/s34/s38/s40/s37/s34/s38/s41/s37/s34/s38/s35/s34/s38/s34/s34/s38/s35/s42/s43/s44/s45/s46/s47/s48/s49/s50/s38/s46/s51/s52/s53/s54/s55\n/s37/s56/s37/s41/s37/s33/s37/s35/s37/s36/s34/s36/s42/s57/s58/s46/s47/s48/s49/s50/s38/s46/s51/s52/s53/s54/s55\n/s32/s33/s34/s32/s35/s34/s32/s36/s34\n/s59/s60/s61/s54/s61/s52/s46/s62/s52/s63/s49/s64/s65/s46/s47/s63/s66/s55/s67/s48/s68/s46/s69/s63/s46/s32/s35/s70/s33/s46/s42/s57/s58\n/s46/s71/s37/s69/s63/s35/s72/s33/s69/s63/s46/s73/s63/s54/s48/s74/s67/s50/s68/s46/s69/s63/s46/s32/s35/s70/s33/s46/s42/s43/s44/s45\n/s46/s33/s34/s34/s75/s56/s76/s70/s46/s35/s75/s35/s34/s77\n/s78/s46/s34/s38/s56\n/s78/s46/s34/s38/s56/s67/s79/s48/s70/s69/s63/s68/s58/s50/s46\n/s46/s46/s46/s46/s46/s46/s46/s69/s63/s46/s40/s80\n/s67/s79/s48/s70/s69/s63/s68/s58/s50/s46\n/s46/s46/s69/s63/s46/s36/s33/s38/s32/s80/s33/s34/s34/s75/s36/s76/s70/s46/s35/s75/s56/s77\n/s67/s81/s52/s70/s69/s63/s68/s57/s82/s83/s84/s63\n/s69/s63/s44/s49/s35/s58/s41/s69/s63/s46/s56/s80\n/s78/s46/s34/s38/s56/s69/s63/s35/s85\n/s69/s63/s33/s85\nFIG. 1. XAS(a) andXMCD (b) spectraof (Ga,Fe)Sb comparedwit h thoseof (In,Fe)As:Be30, bccFe (Ref.\n37)39, FeCr 2O440, andγ-Fe2O341. The XMCD spectra of bcc Fe, FeCr 2O4, andγ-Fe2O3are multiplied\nby 0.5 for ease of comparison.\nto the spectra of α-Fe2O3shown by the green dashed curve. Thus obtained intrinsic and extr insic\ncomponents are separately shown in Fig. 2 by red and orange dashe d curves, respectively, for both\nspectra before and after etching. From this procedure, it was fo und that the extrinsic contribution\nto the XAS was almost ∼60 % before etching, and was significantly reduced to ∼4% after etching,\nwhich guarantees the efficiency of HCl etching.\nOne may suspect that, because the spectra look similar to those of bcc Fe, the magnetism\nmay originate from nanoscale metallic Fe precipitates in the samples. I n order to rule out this\npossibility, details of the “intrinsic XAS” and XMCD spectra at the Fe L3edge are shown in the\ninset of Fig. 2 together with the second derivative XAS spectrum to highlight weak shoulders\non the XAS spectrum labeled a−d. Here, the signs of XMCD and the second derivative XAS\nspectra are reversed for ease of comparison. The XAS spectrum of (Ga,Fe)Sb mainly consists\nof two features candd, which do not coincide with the feature in XMCD (labeled b). This is\nnot the case for bcc Fe, where the XAS spectrum consists of a bro ad single peak, and the peak\npositions of XAS and XMCD indeed coincide30. Such a peak-position difference was also found in\n(In,Fe)As:Be30and Ge:Fe43, and might be a universal spectral feature of Fe-doped semicond uctors.\nNote that there is also a shoulder at ∼710 eV in the XMCD spectra originating from Fe3+oxides,\n5/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s88/s65/s83/s32/s91/s97/s114/s98/s46/s32/s117/s110/s105/s116/s93\n/s55/s51/s48 /s55/s50/s53 /s55/s50/s48 /s55/s49/s53 /s55/s49/s48 /s55/s48/s53\n/s80/s104/s111/s116/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s91/s101/s86/s93/s48\n/s55/s49/s49 /s55/s49/s48 /s55/s48/s57 /s55/s48/s56 /s55/s48/s55 /s55/s48/s54\n/s80/s104/s111/s116/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s91/s101/s86/s93\n/s98/s101/s102/s111/s114/s101/s32/s101/s116/s99/s104/s105/s110/s103\n/s97/s102/s116/s101/s114/s32/s101/s116/s99/s104/s105/s110/s103/s32/s97/s45/s70/s101/s50/s79/s51\n/s32/s105/s110/s116/s114/s105/s110/s115/s105/s99/s32/s88/s65/s83\n/s32/s101/s120/s116/s114/s105/s110/s115/s105/s99/s32/s88/s65/s83/s76/s51/s32/s101/s100/s103/s101/s105/s110/s116/s114/s105/s110/s115/s105/s99\n/s32/s32/s32/s32/s32/s88/s65/s83\n/s88/s77/s67/s68/s40/s71/s97/s44/s70/s101/s41/s83/s98\n/s32/s70/s101/s32/s49/s51/s46/s55/s32/s37\n/s88/s65/s83/s32/s50/s110/s100/s32/s68/s101/s114/s105/s118/s46/s97\n/s98\n/s99/s100\nFIG. 2. XAS and XMCD spectra of the 13.7% Fe-doped (Ga,Fe)Sb b efore and after HCl etching. Intrinsic\nandextrinsiccomponentsarealsoseparatelyshownbyredan dorangedashedcurves, respectively, together\nwith the spectrum of α-Fe2O3. Inset shows the intrinsic XAS and XMCD spectra expanded at t he FeL3\nedge. Second derivative of the XAS spectrum is also shown to e mphasize weak features, labeled as a-d.\nHere, the signs of the XMCD and the second derivative spectru m have been reversed.\nTABLEI.SpinandorbitalmagneticmomentsofFein(Ga,Fe)Sb incomparisonwiththoseof(In,Fe)As:Be,\nGe:Fe, and bcc Fe metal. All the values have been estimated us ing the XMCD sum rules [Eq.(1) and\nEq.(2)] except for the ones in the third and the forth rows, wh ich were computed using the GGA and\nGGA+Umethods.\nmorb/mspinmorbmspin\nGa0.94Fe0.06Sb∗0.13±0.01 0.05†0.37†\nGa0.863Fe0.137Sb∗0.13±0.01 0.14†1.07†\nIn0.9Fe0.1As:Be∗300.10±0.02 0.17‡1.75‡\nGe0.935Fe0.065∗430.11±0.03 0.14†1.29†\nFe bcc390.043±0.001 0.085 1.98\n†Values at µ0H= 1 T and T= 5 K;‡Values at 5 T and 20 K\n∗Fe2+configurations was assumed;\nbut the contribution is much smaller.\nBy applying the XMCD sum rules, we have estimated the spin and orbita l magnetic moments\n6of Fe31,32as follows.\nmorb=−4/integraltext\nL2,3XMCD d ω\n3/integraltext\nL2,3XAS dωnh, (1)\nmspin=−6/integraltext\nL3XMCD d ω−4/integraltext\nL2,3XMCD d ω\n/integraltext\nL2,3XAS dωnh, (2)\nwheremorbandmspinaretheorbitalandspinmagneticmomentsinunitsof µB, respectively, and nh\nthe number of 3 dholes. Here, we have ignored the magnetic dipole term, which is negligib ly small\nfor an atomic site with high symmetry such as TdorOh44, andnhwas set to 4 assuming the valence\nof Fe is 2+ with six 3 delectrons as implied by the density functional theory calculation45. The\ncorrection factor of 0.875 for Fe2+ion46,47was used to estimate the spin magnetic moment. Note\nthat if we assume the Fe3+state with five 3 delectrons and the correction factor to be 0.685, the\nspin and orbital magnetic moment would be changed by a factor of 1.6 (= (0.875/0.685)×(5/4)).\nThe raw XAS and XMCD spectra after etching are used for the sake of simplicity because extrinsic\ncontribution was only a few percent.\nTable I summarizes the spin and orbital magnetic moments of various Fe compounds including\nother Fe-doped semiconductors and bcc Fe. It was found that th emorb/mspinratio of (Ga,Fe)Sb\nis substantially larger than that of bcc Fe. This may be due to the str onger localization of Fe\n3delectrons48in bulk (Ga,Fe)Sb, or at the interface between the Fe-rich and Fe-p oor regions,\nwhere the translational symmetry is broken49. The large value of morb/mspinsuggests that a con-\nsiderable fraction of Fe atoms have the valence of 2+ with six 3 delectrons because the orbital\nmagnetic moments would be quenched if all the Fe atoms took the high -spin Fe3+(3d5) config-\nuration. Nevertheless, there probably exist a significant amount o f Fe3+ions as well considering\nthe inhomogeneous nature of the material. Such a large orbital mag netic moment was also ob-\nserved for (In,Fe)As:Be30and Ge:Fe43. This fact and the similar spectral line shapes among the\nabove-mentioned Fe-doped FMSs imply that the local electronic str ucture of Fe is similar and the\nferromagnetism has a common origin. Note that the morb/mspinratio is more unambiguous than\nthe absolute values of morbandmspinsince it does contain neither the uncertainty in estimating\nnhnor extrinsic contributions to the XAS area.\nThe deduced spin magnetic moments at H= 1T and T= 5 K for both the paramagnetic and\nferromagnetic samples are small (0.37 µB/Fe and 1.07 µB/Fe, respectively) compared with the\nionic value of 4 µB50and the experimental saturated magnetic moment of 2.4-2.9 µBmeasured at\nthe same temperature of 5 K24. The cause of the small deduced magnetic moment is unclear at\nthis stage, but it can be attributed to the possible existence of Fe2+oxides component in the XAS\n7/s49/s52/s48\n/s49/s50/s48\n/s49/s48/s48\n/s56/s48\n/s54/s48\n/s52/s48\n/s50/s48\n/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s91/s97/s114/s98/s32/s46/s117/s110/s105/s116/s93/s55/s50/s48\n/s55/s49/s56\n/s55/s49/s54\n/s55/s49/s52\n/s55/s49/s50\n/s55/s49/s48\n/s55/s48/s56\n/s55/s48/s54\n/s55/s48/s52/s80/s104/s111/s116/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s91/s101/s86/s93\n/s52\n/s50\n/s48\n/s45/s49/s50 /s45/s56 /s45/s52 /s48\n/s69/s32/s45/s32/s69/s70/s32/s91/s101/s86/s93/s120/s32/s49/s53\n/s79/s102/s102/s32/s114/s101/s115/s111/s110/s97/s110/s99/s101/s70/s101/s32/s49/s51/s46/s55/s37\n/s70/s101/s32/s76/s51\n/s55/s48/s52/s32/s101/s86/s40/s101/s41/s79/s102/s102/s32/s114/s101/s115/s111/s97/s110/s110/s99/s101\n/s115/s117/s98/s116/s114/s97/s99/s116/s101/s100\n/s32/s97/s32/s98/s65/s117/s103/s101/s114\n/s84/s32/s61/s32/s49/s48/s48/s32/s75/s88/s65/s83 /s40/s100/s41 /s40/s102/s41/s51/s53\n/s51/s48\n/s50/s53\n/s50/s48\n/s49/s53\n/s49/s48\n/s53\n/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s91/s97/s114/s98/s32/s46/s117/s110/s105/s116/s93/s55/s50/s48\n/s55/s49/s56\n/s55/s49/s54\n/s55/s49/s52\n/s55/s49/s50\n/s55/s49/s48\n/s55/s48/s56\n/s55/s48/s54\n/s55/s48/s52/s80/s104/s111/s116/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s91/s101/s86/s93\n/s52\n/s50\n/s48\n/s45/s49/s50 /s45/s56 /s45/s52 /s48\n/s69/s32/s45/s32/s69/s70/s32/s91/s101/s86/s93/s70/s101/s32/s54/s37 /s88/s65/s83\n/s84/s32/s61/s32/s49/s48/s48/s32/s75\n/s79/s102/s102/s32/s82/s101/s115/s111/s110/s97/s110/s99/s101/s55/s48/s52/s32/s101/s86/s70/s101/s32/s76/s51/s32/s98\n/s32/s97/s65/s117/s103/s101/s114\n/s120/s32/s50/s79/s102/s102/s32/s114/s101/s115/s111/s97/s110/s110/s99/s101\n/s115/s117/s98/s116/s114/s97/s99/s116/s101/s100/s40/s97/s41\n/s40/s98/s41/s40/s99/s41\n /s32/s33/s34\n/s32/s33/s35\n/s32/s33/s36\n/s32/s36/s37\n/s32/s36/s38\n/s39/s33/s35/s39/s37/s39/s34/s36/s32\n/s33/s40/s41/s42/s43/s44 /s45/s42/s46\n/s47/s43/s48\n/s38/s49/s36/s50\n/s32/s33/s34\n/s32/s33/s35\n/s32/s33/s36\n/s32/s36/s37\n/s32/s36/s38\n/s39/s33/s35/s39/s37/s39/s34/s36\n/s32/s33/s34/s33/s32/s40/s41/s42/s43/s44/s45/s40/s43/s41\n/s33/s46/s47/s32/s48\n/s32\n/s33/s49/s50/s51/s43/s52 /s53/s54/s55\nFIG. 3. (a), (d) Resonance photoemission spectroscopy (RPE S) spectra of (Ga,Fe)Sb with 6% Fe and\n13.7% Fe taken with photon energies across the Fe L3absorption edge. The color and the base line\npositions of the spectra represent the photon energies as de picted by triangles on the XAS spectra in\npanels (c) and (f). Here, the off-resonance spectra shown in pa nels (b) and (e) have been subtracted from\nall the spectra. (g), (h) False color plots of the second deri vatives of the RPES spectra in panels (a) and\n(d). The dispersive normal Auger peak and the non-dispersiv e resonance features are indicated by white\ndashed lines and red solid lines, respectively.\nspectra because the TEY-mode detection of XAS and XMCD signals is surface sensitive with the\nprobing depth of ∼3-5 nm.\nB. Resonance Photoemission\nFigures 3(a) and 3(d) show RPES spectra of the 6% and 13.7% Fe-do ped samples, respectively,\ntaken across the Fe L3absorption edge from 705 eV to 716 eV. The color and the baseline po sitions\nrepresent the photon energies indicated by triangles on the XAS sp ectra in Figs. 3(c) and 3(f).\nHere, off-resonance spectra taken with the photon energy of 70 4 eV shown in Figs. 3(b) and 3(e)\nhave been subtracted to emphasize the resonance behavior. Not e that the units of the vertical\naxes of Figs. 3(a), 3(b), 3(d), and 3(e) are the same. For both s amples with different Fe contents,\na strong normal Auger peak dispersing with incident photon energy was observed. This suggests\nthat the Fe 3 delectrons in (Ga,Fe)Sb have itinerant character, as already sugge sted by the XAS\nmeasurements, since Auger decay is a consequence of faster scr eening of the core hole than the\n8recombination of the excited electron with the core hole and is norma lly observed in metallic\nsystems51,52. It is worth mentioning that such a strong Auger peak was also obse rved in the case\nof Fe-doped Ge35. In addition to the Auger peak, there are resonantly enhanced fe atures around\n-1.7 eV and -10.3 eV, denoted by αandβ, respectively. Unlike the Auger peak, these peaks do\nnot disperse with incident photon energy, representing a local exc itation of Fe 3 dstates including\ncharge transfer from ligand orbitals accompanying the photoemiss ion process. Such resonance\nfeatures are widely observed in transition-metal oxides, where th e 3delectrons are well localized\nand strongly correlated53–56. Those local excitation peaks were almost absent in the case of Fe-\ndoped Ge (Ref.35), suggesting that Fe 3 delectrons in (Ga,Fe)Sb are more correlated or more\nlocalized than those in Ge:Fe. Considering that the Fe 3 ddensity of states (DOS) should not\nbe located as deep as 10 eV below EF, at least feature βcan be attributed to a charge-transfer\nsatellite. In order to highlight subtle features, the second derivat ive image of the RPES spectra\nare shown in Figs. 3(g) and 3(h), where dispersive normal Auger pe ak and resonantly enhanced\npeaksαandβcan be seen as described above. These features are commonly see n in both samples,\nhowever, there exists a difference as indicated by red dashed lines in Figs. 3(g) and 3(h). For the\n6% Fe sample, two features are observed around 6.8 eV and 3.1 eV in t he entire photon energy\nrange, while for the 13.7% Fe sample, there is only one feature aroun d 4 eV. This may imply the\ndifference in the electronic structure between the two samples. Pr obably the 3 dstates are more\nlocalized or more correlated in the 6% Fe-doped sample because the d istance between adjacent Fe\natoms are longer and the Fe 3 dband width would be narrower than in the 13.7% Fe-doped sample.\nFor the purpose of examining the nature of resonance enhanceme nt, it is useful to plot the\nintensity at a fixed binding energy as a function of incident photon en ergy, namely, a constant-\ninitial-state (CIS) spectrum. In order to eliminate the effect of the overlapping Auger peak from\nthe CIS spectra and to extract the resonance behavior of featu resαandβ, we have employed\ncurve fitting as shown in Fig. 4(a). In the figure, resonance peaks αandβare fitted by Gaussian\nfunctions and the Auger peak with its tail is fitted by an asymmetric G aussian function introduced\nin Ref.57. CIS spectra for features αandβwere obtained from the peak areas of the Gaussian\nfunctions. Thisisan adhocprocedure, butstillitprovidesareasonabledescriptionoftheres onance\nenhancement.\nFigures 4(b) and 4(c) show the RPES spectra on an expanded scale nearEF. The data of the\n6% Fe-doped sample have been smoothed and multiplied by 3.5 for ease of comparison with the\ndata of the 13.7% Fe-doped sample. As can be seen from the figures , the spectral intensities at EF\n9/s32/s33/s34/s35/s32/s35/s34/s36/s35/s35/s34/s36/s33/s34/s35/s37/s38/s39/s40/s38/s41/s42/s39/s43/s44/s45/s46/s47/s48/s34/s44/s49/s38/s42/s39/s50\n/s51/s33/s52/s51/s33/s53/s51/s33/s35/s51/s35/s54/s51/s35/s55\n/s56/s57/s58/s39/s58/s38/s44/s59/s38/s40/s47/s60/s43/s44/s45/s40/s61/s50/s32/s33/s53 /s32/s54 /s32/s52 /s35\n/s32/s33/s34/s33/s32/s62/s44/s45/s40/s61/s50/s53\n/s33\n/s35/s37/s38/s39/s40/s38/s41/s42/s39/s43/s44/s45/s46/s47/s48/s34/s44/s49/s38/s42/s39/s50\n/s32/s35/s34/s36 /s35\n/s32/s33/s34/s33/s32/s62/s44/s45/s40/s61/s50/s32/s35/s34/s36 /s35/s44/s35/s36/s62/s40/s44/s33/s63/s34/s51/s64\n/s37/s38/s44/s65/s44/s51/s35/s51/s44/s40/s61 /s44/s66/s67/s68\n/s44/s32/s62\n/s44/s35\n/s33/s36\n/s33/s67/s49/s60/s40/s47/s62/s40/s44/s55/s34/s35/s64/s44/s69/s46/s70\n/s69/s48/s70/s71/s56/s59/s68\n/s62/s40/s44/s33/s63/s34/s51/s64/s44/s69/s72/s70/s69/s73/s70\n/s68/s74/s58/s58/s39/s57/s40/s73\n/s75/s44/s63/s34/s36/s62/s40/s44/s55/s34/s35/s64 /s62/s40/s44/s33/s63/s34/s51/s64\n/s51/s35/s36/s44/s40/s61/s51/s33/s36/s44/s40/s61/s37/s38/s44/s65/s44/s51/s35/s54/s44/s40/s61\n/s51/s33/s35/s44/s40/s61/s67/s49/s60/s40/s47\nFIG. 4. (a) RPES spectrum of the 13.7% Fe-doped sample taken w ithhν= 707 eV (grey thick curve)\nand fitting results (black curve). Fitting components of two Gaussian functions for features αandβand\none asymmetric Gaussian function for Auger peak are also sho wn separately. (b), (c) RPES spectra of\nthe 6% and 13.7% Fe-doped samples near EF. Here, the color is the same as those used in Figs. 3(a) and\n3(d). Note that the RPES spectra of the 6% Fe-doped sample hav e been smoothed and multiplied by 3.5\nfor easy comparison. (d) Constant-initial-state (CIS) spe ctra for features αandβ, and the intensity at\nEFand the Auger peak intensity as functions of photon energy. F or comparison, XAS spectra are also\nplotted.\nfor both samples are enhanced on resonance, suggesting that Fe 3dstates have finite contribution\nto the DOS at EF. Here, the CIS spectrum at EFis defined as the area of RPES spectra between\nE−EF=−0.6 eV and 0.2 eV.\nThus obtained CIS spectra for features α,β, and the intensity at EFare plotted in Fig. 4(d)\ntogether with the Auger peak height as a function of photon energ y and the XAS spectra. The\nCIS spectrum for feature αis peaked at hν= 707.5 eV, but that for feature βis peaked at a\nhigher photon energy of hν= 708 eV for both samples. The difference in the CIS peak positions\nimplies that the broad XAS spectra actually consist of different kinds of excitations, which may\ncorrespond to peaks canddin the inset of Fig. 2, probably involving different types of 3 dorbitals,\ni.e.,t2andeorbitals, rather than excitation into a single kind of broad metallic ban ds as in the\ncase of bcc Fe metal. We note that the CIS spectra of features αandβvanish around the photon\nenergy of 710 eV, at which the extrinsic Fe3+shoulder exists on each XAS spectrum. Therefore,\n10Fe in IIIV semiconductor\nConduction Band\nMinority-spin Majority-spin Valence Bandt2↓\ne↓ ta↑\ne↑\ntb↑t2↑p-d hybridized\nd levelp-d hybridizedd level ta↓\ntb↓Fe 3+ \nFe 2+ \nFIG. 5. Schematic energy diagram of Fe-doped III-V semicond uctors. The dlevels of Fe are shown on\nthe sides of the figure, and the valence band and the conductio n band of the host GaSb are shown by\ngreen boxes in the middle. Due to p-d(t2) hybridization, the t2orbitals are split into the bonding ( tb) and\nanti-bonding ( ta) states having both pandt2characters. States with predominant pandt2characters\nare indicated by green and gray boxes, respectively. The ele ctron occupancy of Fe3+state is illustrated\nby red and black arrows while that of Fe2+by blue and black arrows.\nthe observed resonance features are not from surface oxides b ut are most likely intrinsic.\nIV. DISCUSSION\nFigure 5 illustrates the basic electronic structure of Fe in GaSb. The valence and conduction\nbands of GaSb are shown in the middle by large green boxes, and the F e 3dlevels are shown on\nboth right-hand and left-hand sides. Each arrow represents an e lectron with a specific spin. At\nthe substitutional sites, the Fe 3 dlevel is split into the doubly-degenerate lower elevel and the\ntriply-degenerate higher t2level due to the crystal field of Tdsymmetry. Due to the strong p-d(t2)\nhybridization, the Fe t2orbitals and the ligand Sb porbitals form bonding ( tb) and anti-bonding\n(ta) states with mixed t2andpcharacters. Those levels with predominant t2andpcharacters\nare indicated by gray and green boxes, respectively. As for the ma jority-spin states, where the t2↑\nlevel is located well below the valence-band maximum (VBM) because o f the relatively high Sb\n5p-derived valence-band position58, the bonding states have predominant Fe t2character, and the\nanti-bonding states predominant Sb pcharacter. On the other hand, the minority-spin bonding\nstates (t2↓) consist primarily of Sb porbitals, and the anti-bonding states of Fe t2orbitals.\nIf Fe takes the valence of 3+ with five 3 delectrons, the ta↑level is fully occupied and there\n11exists no minority-spin 3 delectrons (except for the small contribution in the tb↓states). On the\nother hand, if the valence of Fe is 2+, the sixth 3 delectron occupies the e↓level and one hole\nresides in the ta↑level. The e↓level would be at the Fermi level since it is doubly degenerate and\nhalf occupied. In Fig. 5, the Fe3+electronic structure is represented by red and black arrows,\nwhile Fe2+by blue and black arrows.\nFrom the XMCD measurements, it was suggested that there is a con siderable 3 d6contribution\nto the ground state of Fe as in (In,Fe)As:Be30,34and Ge:Fe43. This suggests that there may exist\nthelong-range p-dexchange interactionmediated byholesin ta↑states asinthecaseof(Ga,Mn)As.\nConsidering the coexistence of Fe2+and Fe3+states, the short-range double exchange interaction\nwithin the e↓orbitals would also be present. Note that the finite Fe PDOS at the Fe rmi level\nfound by the RPES measurements can be attributed to either ta,↑ore↓states in this scenario. If\nthis is the case, the double exchange interaction would be more impor tant than the p-dexchange\ninteraction considering that the ferromagnetism with similar TCis also observed in the n-type\n(In,Fe)As and (In,Fe)Sb, where the s-dexchange interaction is significantly weaker than the p-d\nexchange interaction. In real (Ga,Fe)Sb samples, it was reported that there is Fe concentration\nfluctuation especially when Fe is heavily doped24,25. The double exchange interaction would be\nlocally strong in Fe-rich regions and stabilize the local ferromagnetic order. This may explain the\nobserved convex line shape of M-Tcurves24indicating the existence of superparamagnetism even\naboveTC.\nThe Fe2+scenario described above, however, may not explain the fact that the carrier concen-\ntration of (Ga,Fe)Sb obtained by Hall measurements is not more tha n∼1019cm−3(Ref.24), two\norders of magnitude smaller than doped Featoms. One possible expla nation is that the carriers are\nstrongly trapped inside the Fe-rich regions and macroscopic carrie r transport occurs via hopping\nbetween those Fe-rich regions. Such a model was introduced by Ka minski and Das Sarma59and\napplied to Ge:Mn60, Ge:Fe61, and (Zn,Cr)Te62to describe their insulating/semiconducting natures\nandlowcarrierconcentrations ( ∼1018cm−3forGe:Mn60andGe:Fe63,∼1015cm−3for(Zn,Cr)Te64).\nNote that, although the low carrier concentration of (Ga,Fe)Sb wo uld be explained by Fe3+sce-\nnario instead, where only the p-dexchange interaction is present, it seems difficult to explain why\nferromagnetism is universally observed in the other Fe-doped FMSs regardless of the carrier type.\nIt is worth mentioning that a recent theoretical calculation65has pointed out the important role\nof superexchange interaction. In order to resolve the puzzle and to fully understand the peculiar\nnature of the ferromagnetism in the Fe-doped FMSs, further the oretical and experimental studies\n12are necessary.\nV. SUMMARY\nIn the present work, we have studied the electronic structure an d the magnetism of the novel\np-type ferromagnetic semiconductor (Ga,Fe)Sb, whose Curie temp erature exceeds room tempera-\nture, using XAS, XMCD, and RPES. The line shapes of XAS and XMCD sp ectra suggested that\nthe ferromagnetism is of intrinsic origin. XMCD sum rules yielded an unq uenched large orbital\nmoment, implying the3 d6configurationofFe. Thevalence-bandRPESspectrashowed adisp ersive\nAuger peak and non-dispersive resonantly enhanced peaks, which are fingerprints of the itinerant\nand correlated nature of the Fe 3 delectrons, respectively. It was also found that there is a finite Fe\nPDOSat the Fermi level. This hasbeen attributed to majority-spin a ntibonding p-d(t2) hybridized\nstate and/or minority spin estate, both of which can play a role in stabilizing the ferromagnetic\norder through p-dexchange interaction and double exchange interaction, respectiv ely.\nACKNOWLEDGMENTS\nThis work was supported by Grants-in-Aid for Scientific Research f rom the JSPS (Grants No.\n15H02109, No. 15K17696, and No.16H02095), and CREST of JST (G rant No. JPMJCR1777).\nThe experiment was done under the Shared Use Program of JAEA Fa cilities (Proposal No. 2015B-\nE24 and No. 2016A-E27) with the approval of the Nanotechnology Platform Project supported by\nMEXT. The synchrotron radiation experiments were performed at the JAEA beamline BL23SU\nin SPring-8 (Proposal No. 2015B3881 and No. 2016A3831). S.S., L.D .A., Y.K.W., and M.H.\nacknowledge support from the Program for Leading Graduate Sch ools; L.D.A., Y.K.W., and M.H.\nacknowledge support from the JSPS Fellowship for Young Scientists . 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Hitachi Global Storage Technologies, San Jose, Cali fornia 95135  \n3.  Department of Physics, Oakland University, Rocheste r, Michigan 48309  \n \nAbstract \n  \nWe use spin torque ferromagnetic resonance to measu re the spectral properties of dipole-\nexchange spin waves in permalloy nanowires. Our mea surements reveal that geometric \nconfinement has a profound effect on the damping of  spin waves in the nanowire geometry. \nThe damping parameter of the lowest-energy quantize d spin wave mode depends on \napplied magnetic field in a resonant way and exhibi ts a maximum at a field that increases \nwith decreasing nanowire width. This enhancement of  damping originates from a \nnonlinear resonant three-magnon confluence process allowed at a particular bias field \nvalue determined by quantization of the spin wave s pectrum in the nanowire geometry.Boone et al.  Page 2 of 10 October 15, 2009 Spin transfer torque (STT) [1-4] is emerging as a n ew tool for studies of magnetization \ndynamics in nanostructures [5-13]. In this Letter, we use STT to measure properties of dipole-\nexchange spin waves in permalloy (Py = Ni 86Fe 14) nanowires. Our measurements reveal that \ngeometric confinement of spin waves in nanowires ha s a profound effect on the spin wave \ndamping [14].  We find a resonant dependence of the  damping parameter of the lowest-energy \nspin wave mode on external magnetic field − a maximum of damping is observed at a field that \nincreases with decreasing nanowire width. We explai n this resonant enhancement of damping by \na nonlinear three-magnon confluence process in whic h two quanta of the first ( n=1) spin wave \nmode merge into a single quantum of the third ( n =3) mode. \nBrillouin light scattering [15,16], ferromagnetic r esonance (FMR) [17-20] and magneto-\noptical techniques [21,22] have been applied to stu dies of spin waves in ferromagnetic wires. \nHere we use spin torque FMR (st-FMR) [5,6], to make  measurements of the spectral properties \nof dipole-exchange spin waves in Py nanowires. In t his technique, a ferromagnetic nanocontact \nis used to apply an ac spin-polarized current, Iac , to a metallic nanowire on a nonmagnetic \nmetallic substrate as illustrated in Fig. 1. When t he frequency of the current flowing \nperpendicular to the nanowire into the substrate is  equal to the eigen-frequency of a spin wave \nmode, this mode is resonantly excited by the ac STT  from the current.  The excited mode induces \nresistance oscillations at the nanocontact due to t he giant magnetoresistance effect and generates \na rectified voltage, Vdc , proportional to the mode amplitude [5,6].  As the  frequency of the ac \ncurrent, f, is swept through a spin wave eigen-frequency, a r esonance peak or a trough appears in \nthe plot of Vdc  versus f. The peaks and troughs in the st-FRM spectrum, Vdc (f), give information \nabout spin wave frequencies, amplitudes and damping  parameters. Boone et al.  Page 3 of 10 October 15, 2009  \n \nFig. 1. (color online): (a) Scanning electron micro scopy (SEM) image \nof a Py nanowire device with five current-polarizin g SAF nanocontacts \nand Au/Ta top leads. (b) Schematic side view of the  sample. We make st-FMR measurements for a set of eight 6-nm  thick and 100-250 nm wide Py \nnanowires lithographically defined on a Cu/Ta under layer that serves as the bottom electrical \nlead. Spin-polarized current is injected into the n anowire through a Co 50 Fe 50 / Ru/ Co 50 Fe 50 / \nIr 20 Cr 3Mn 77  synthetic antiferromagnet (SAF) nanopillar pattern ed on top of the Py nanowire and \nseparated from the nanowire \nby an 8-nm thick Cu spacer, as \nshown in Fig. 1. The top Au/Ta \nlead is connected to the SAF \nfixed layer and current is \napplied between the top and \nthe bottom leads. Since the Cu spacer layer above t he Py layer is only partially milled away to \nprotect the Py nanowire surface, some lateral curre nt spreading in the spacer takes place. Five \nspin current injector nanocontacts are attached to the Py nanowire. In this Letter, we report \nsingle-contact st-FMR measurements. The device in F ig. 1(a) is made in a multi-step \nnanofabrication process starting from a Ta(5)/ Cu(3 0)/ Ta(3)/ Cu(30)/ Ta(5)/ Cu(3)/ Py(6)/ \nCu(8)/ Co 50 Fe 50 (3.5)/ Ru(0.8)/ Co 50 Fe 50 (3.5)/ Ir 20 Cr 3Mn 77  (7)/ Cu(10)/ Ru(5)/ Ta(2.5) multilayer \n(thicknesses are in nanometers). The values of satu ration magnetization of the Py ( M = 580 \nemu/cm 3) and Co 50 Fe 50  (1480 emu/cm 3) films are measured by vibrating sample magnetomet ry. \nThe nanowire axis is perpendicular to the SAF excha nge bias direction in order to maximize the \nmagnitude of STT [6].  For st-FMR measurements, we apply modulated microwave current to \nthe nanocontact and measure the resulting rectified  voltage, Vdc , using lock-in detection [6]. The \nfrequency of the current is swept in the 1–15 GHz r ange and, after subtraction of a background \nvoltage due to Ohmic heating [13], the resulting Vdc (f) gives the st-FMR spectrum. Fig. 2(a) Boone et al.  Page 4 of 10 October 15, 2009 \n \nFig. 2. (color online): (a) Evolution of st-FMR spe ctra with increasing \nmagnetic field applied parallel to a 240 nm wide na nowire. The curves \nare vertically offset for clarity. (b) st-FMR spect rum showing three \nspin wave resonances. (c), (d): Mode frequency vers us field applied \nparallel to the nanowire for a 140 nm wide wire and  a 240 nm wide \nwire. Symbols are data and lines are Eq. (1) fits w ith the nanowire \nwidth w and the pinning parameter δ as fitting parameters. Inset in (c) \nis the dependence of the damping parameters of thre e spin wave \nmodes in the 240 nm wide nanowire on magnetic field .  Inset in (d) \nshows the dispersion relation calculated from Eq. ( 1) for a 140 nm \nwire at 400 Oe.  The dashed arrows indicate the two -magnon \nscattering channels and the solid arrow shows the kx=0 three-magnon \nconfluence process active only at the field for whi ch the frequency of \nthe n=1 mode is half the frequency of the n=3 mode.  shows a series of st-FMR spectra measured at severa l fields applied along the nanowire axis of a \n240 nm wide Py nanowire at T = 4.2 K. These spectra  exhibit three resonances (peaks and \ntroughs) corresponding to the lowest three spin wav e modes in the nanowire, as shown in Fig. \n2(b). All three modes shift to higher frequencies, and the frequency spacing among the modes \ndecreases with increasing magnetic field [15]. The observed resonances are due to spin waves in \nthe Py nanowire since the lowest-frequency SAF acou stic mode at H = 0 is expected at f >13 \nGHz − well above the \nfrequencies of the resonances in \nFig. 2(b) [23].  The line shape of \neach mode can be fit to a sum of \na symmetric and an \nantisymmetric Lorentzians, \nindicating that not only STT but \nalso magnetic field from the top \nlead contribute to excitation of \nspin waves [13]. From the \namplitude Vdc (f1)  of the lowest-\nenergy mode we estimate the \nprecession cone angle of this \nmode to be 4.6 °/mA. In our \nsample geometry, spin waves \nwith wave vectors ranging from 0 to max \nxkcan be excited, where kx is the wave vector along the \nnanowire axis, and the value of  max \nxk is determined by the extent of current spreading i n the Cu Boone et al.  Page 5 of 10 October 15, 2009 spacer. Therefore, the line shape of each mode in t he st-FMR spectra is a convolution of line \nshapes of spin waves with kx < max \nxk. For our samples this inhomogeneous line broadenin g does \nnot exceed line broadening due to intrinsic spin wa ve damping as discussed later in this Letter. \nThis allows us to identify positions of minima and maxima of the resonances in st-FMR spectra \nwith the frequencies of spin wave modes at  k x = 0.  \nTo quantify the effect of confinement on spin waves  in nanowires, we compare the \nmeasured spin wave frequencies to a theoretical exp ression for the frequency of quantized \ndipole-exchange spin wave width modes in thin-film strips of rectangular cross section [24,25]: \n( ) ( )\n\n − −+ +\n\n\n\n\n\n\n\n − −−+ + =dd\nMA\nMH k\ndd\nMA\nMHM f\nnn\nn\nnyn \nnn\nn nκκκ\nπ π κ κκκ\nπ πππγ exp 1\n2 4exp 11\n2 4422\n2 22\n2\n2    (1) \nIn this equation, fn is the frequency of n-th mode, γ /2 π  = 2.95 MHz/Oe is the gyromagnetic \nratio, d is the nanowire thickness, H is the magnetic field along the strip axis, A = 10 -6 erg/cm is \nthe exchange constant, and 2\nnκ= 2\nxk + 2\nyn k where kyn is the quantized spin-wave wave vector \nalong the width of the strip. In our measurements, both even and odd modes are excited due to \ninhomogeneous current injection over the nanowire w idth arising from the current injector being \nmisaligned with the nanowire center as shown in Fig . 1(a). The quantized wave vector, kyn, \ngenerally can be expressed as kyn  =( )\nwnnπδ−, where  w  is the nanowire width and 0 ≤ δn ≤ 1 is a \npinning parameter determined by the boundary condit ions for dynamic magnetization at the \nnanowire edges ( δn = 0 corresponds to complete pinning, whereas δn = 1 describes free spins). \nWe fit Eq. (1) to the fn(H)  data using δn and w as fitting parameters, assuming that δn is \nindependent of n and is identical for all studied nanowires, δn = δ. We use w as a fitting \nparameter because we find ~ 20% spread in the spin w ave frequencies for nanowires of Boone et al.  Page 6 of 10 October 15, 2009 nominally identical width. This fitting procedure g ives high-quality fits shown in Fig. 2(c) and \n2(d) for fn(H)  with δ ≈ 0.5 and the nanowire width w similar to the nanowire width measured \nwith SEM (135 nm and 225 nm with ~ 10% standard devi ation for two groups of the nanowires \nstudied). A recent theory of boundary conditions fo r dipole-exchange spin waves in thin-film \nelements [25] predicts values of δ smaller ( δ ≈ 0.2, corresponding to stronger edge pinning) than \nδ = 0.5 found in our experiment. We expect some edge  pinning reduction for nanowires with \nthicknesses comparable to or smaller than the excha nge length of Py (~ 5 nm) (see Fig. 2 in \n[25]). The edge pinning in our system can be also r educed because the magnetic properties of Py \nnear the nanowire edges are modified by ion milling  [26, 27], and, therefore, the assumptions of \nspatially homogeneous magnetization and exchange us ed in [25] are not satisfied for our system.  \nFitting a mode spectral line \nshape to a linear combination of \nsymmetric and antisymmetric \nLorentzians [13] gives the mode \nhalf width at half maximum, ∆fn, \nwhich carries information about \nthe mode damping parameter, αn.  \nDue to inhomogeneous line \nbroadening caused by excitation of \nspin waves with kx in the (0 − kxmax ) \nrange, ∆fn sets an upper bound on \nαn: αn  ≤( ) M Hfn\nπ γπ\n22\n+∆≡nα~[28]. \n \nFig. 3 (color online): (a), (b) Damping parameter  of the first mode \n1~α versus field for 140 nm and 125 nm nanowires, show ing a \nmaximum of damping at a field Hmax  (425 Oe and 550 Oe, \nrespectively).  (c) Damping versus ac drive current  for the 125 nm \nwire at H = Hmax  (550 Oe) and H < Hmax  (375 Oe). (d)  H max  versus \nfrequency of the first mode at H = Hmax . Squares are data, line shows \nthe field at which frequency of the third mode is e qual to twice the \nfrequency of the first mode as calculated from Eq. (1), data labels \nshow the nanowire width obtained from Eq. (1) fit o f the fn(H)  data.  Boone et al.  Page 7 of 10 October 15, 2009 The inset in Fig. 2(c) shows nα~ for the first three modes of a nanowire with w = 240 nm as a \nfunction of field. The high-field ( H > 250 Oe) value of 1~α=0.013 ± 0.002 does not significantly \nexceed the value of Gilbert damping parameter α ≈ 0.01 for thin Py films [28]. Using the \ndispersion relation of Eq. (1) and assuming intrins ic damping α =0.008 [12], inhomogeneous line \nbroadening gives 1~α=0.013 for kxmax  = 2 ×10 5 cm -1. The width of the region where spin torque is \napplied to the nanowire, π/kxmax =160 nm, is approximately twice as large as the phy sical width of \nthe current injector, which can be explained by cur rent spreading in the Cu spacer. We observe a \nlarge increase of nα~for H < 250 Oe for all modes. A low-field increase of da mping of similar \nmagnitude was previously observed in thin Py films [28,29]. This damping enhancement can be \nattributed to enhanced inhomogeneous broadening [30 ] and to incomplete saturation of \nmagnetization at low fields [27].  \nThe inset in Fig. 2(c) also demonstrates that the d amping parameters of the n ≥ 2 modes \nare enhanced compared to damping of the n=1 mode. This increase of damping can be explained \nby two-magnon scattering of spin waves [31,32]. Hig her order ( n ≥ 2) width modes can scatter \ninto lower order modes with the same frequency and kx ≠0 via a two-magnon process leading to \nthe linewidth broadening. A likely mechanism for th e observed large two-magnon scattering is a \nrandom magnetic potential induced by the nanowire e dge roughness or by magnetization pinning \ncenters. In contrast, two-magnon scattering of the n=1 mode to other modes is prohibited by \nconservation of energy, and the high-field value of  the damping parameter for this mode is close \nto the Py thin-film value. \nFor nanowires with w < 220 nm, we observe a non-monotonic dependence of  damping of \nthe n=1 mode on magnetic field as shown in Fig. 3(a) and  3(b). The damping as a function of Boone et al.  Page 8 of 10 October 15, 2009 field exhibits a maximum at a field Hmax  that increases with decreasing w. This damping \nmaximum is a nonlinear effect as demonstrated in Fi g. 3(c) that shows 1~α versus Iac  measured at \nHmax =550 Oe and at H = 375 Oe < Hmax  for a 125-nm wide nanowire. The damping increases \nlinearly with Iac  (and thus with the mode amplitude) for all field v alues, but the rate of increase is \nsignificantly larger at H = Hmax , indicating that an additional nonlinear damping-en hancing \nprocess is active at H = Hmax .  Fig. 3(d) shows the dependence of Hmax  on frequency of the first \nmode at this field, max \n1f, for nanowires of different widths. We also note t hat at Hmax , the \nfrequency of the lowest-energy mode, f1, is equal to half the frequency of the third mode,  2 f1 = f3. \nThe solid line in Fig. 3(d) shows the field at whic h 2 f1 = f3 as a function of f1 calculated from Eq. \n(1). This theoretical curve is in excellent agreeme nt with the observed dependence of Hmax  \non max \n1f. Therefore, we conclude that the observed enhancem ent of damping at  Hmax  is due to a \nthree-magnon confluence process in which two spin w aves of the n= 1 mode merge into a single \nspin wave of the n= 3 mode [33,34]. Due to quantization of the spin wav e spectrum in nanowires, \nthis energy- and momentum-conserving scattering cha nnel has a resonant character and exists \nonly at a particular magnetic field, Hmax . \nIn summary, we use st-FMR to study spin waves in Py  nanowires and find that two-\nmagnon scattering enhances the damping of higher or der quantized spin wave width modes \ncompared to the damping of the lowest-energy mode. We also observe that the damping of the \nlowest-energy width mode resonantly depends on the bias magnetic field and exhibits a \nmaximum at a particular field value. We attribute t his maximum to a nonlinear three-magnon \nconfluence process in which two magnons of the firs t width mode merge into a single magnon of \nthe third width mode. This resonant dependence of t he spin wave damping parameter on \nmagnetic field originates from quantization of the spin wave spectrum in the nanowire geometry. Boone et al.  Page 9 of 10 October 15, 2009 Our work demonstrates that the nonlinear three-magn on confluence process creates an additional \nresonant damping channel for the lowest-energy spin  wave mode of a ferromagnetic nanowire \nand gives rise to an unusual dependence of spin wav e damping on external magnetic field. This \nwork was supported by the NSF, and by the NRI throu gh the Western Institute of \nNanoelectronics. Boone et al.  Page 10 of 10 October 15, 2009 References \n                                                 \n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159 , L1 (1996) \n[2] L. Berger, Phys. Rev. B 54 , 9353 (1996) \n[3] D. C. Ralph, M. D, Stiles, J. Magn. Magn. Mater ., 320 , 1190 (2008) \n[4] J. A. Katine et al.,  Phys. Rev. Lett. 84 , 3149 (2000) \n[5] A. A. Tulapurkar et al.,  Nature 438 , 339 (2005) \n[6] J. C. Sankey et al.,  Phys. Rev. Lett. 96 , 227601 (2006) \n[7] S. I. Kiselev et al.,  Nature 425 , 380 (2003) \n[8] W. H. Rippard et al.,  Phys. Rev. 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Kalinikos and A. N. Slavin, J. Phys. C 19 , 7013 (1986) \n[25] K. Y. Guslienko, A. N. Slavin, Phys. Rev. B 72 , 014463 (2005)  \n[26] M. P. Kostylev et al.,  Phys. Rev. B 76 , 054422 (2007) \n[27] R. P. Cowburn et al.,  J. Appl. Phys. 87 , 7067 (2000)  \n[28] J. P. Nibarger, R. Lopusnik, T. J. Silva, Appl . Phys. Lett. 82 , 2112 (2003) \n[29] T. J. Silva et al.,  J. Appl. Phys. 85 , 7849 (1999) \n[30] G. Counil et al. , J. Appl. Phys. 95 , 5646 (2004)  \n[31] R. Arias, D. L. Mills, Phys. Rev. B 60 , 7395 (1999) \n[32] R. D. McMichael, D. J. Twisselmann, A. Kunz, P hys. Rev. Lett. 90 , 227601 (2003) \n[33] M. Sparks, “Ferromagnetic Relaxation Theory”, McGraw-Hill, p. 95 (1964)  \n[34] R. N. Costa Filho, M. G. Cottam, G. A. Farias,  Phys. Rev. B 62 , 6545 (2000) "    },    {        "title": "1406.2491v2.Influence_of_Ta_insertions_on_the_magnetic_properties_of_MgO_CoFeB_MgO_films_probed_by_ferromagnetic_resonance.pdf",        "content": "arXiv:1406.2491v2  [cond-mat.mtrl-sci]  13 Aug 2014Influence ofTa insertions onthe magneticproperties ofMgO /CoFeB/MgOfilms probed by\nferromagnetic resonance\nMariaPatriciaRouelliSabino,Sze TerLim,andMichaelTran\nDataStorage Institute,Agency for Science, Technology and Research, 5Engineering Drive 1,117608 Singapore\n(Dated: April 3,2018)\nAbstract We show by vector network analyzer ferromagnetic resonance measurements that low Gilbert damping α,\ndownto0.006,canbeachievedinperpendicularlymagnetize dMgO/CoFeB/MgOthinfilmswithultrathininsertionsofTa\nintheCoFeBlayer. AlthoughincreasingthenumberofTainse rtionsallowsthickerCoFeBlayerstoremainperpendicular ,\nthe effective areal magnetic anisotropy does not improve withmore insertions, whichcome withan increase in α.\nPerpendicularmagnetic anisotropy (PMA) is the key to furth er downscaling of spin transfer torque magnetoresistive\nrandommemorydevices,asitallowstwokeyrequirementstob esatisfied: lowcriticalcurrent Ic0andhighthermalstability\n∆, the latter of which is proportional to the energy barrier Ebbetween the two stable magnetic states. The spin torque\nswitching efficiency, defined as Eb/Ic0, is commonly used as a metric to account for both requirement s. For a Stoner-\nWohlfarthmodel,itisgivenby[1]( /planckover2pi1/4e)·(η/α),whereαistheGilbertdampingparameter,and ηisthespinpolarization\nfactor, which is related to the tunnel magnetoresistance ra tio (TMR) byη=[TMR(TMR+2)]1/2/[2(TMR+1)]. It thus\nbecomes evident that for high switching e fficiency, one has to decrease αwhile keeping TMR high. Magnetic tunnel\njunctions(MTJs)basedonCoFeB /MgOsystemsarewell knownto providehighTMR[2] andhaverec entlybeenshown\nto possess PMA, which is attributed to the CoFeB /MgO interface.[3] A Ta layer is usually placed adjacent to th e CoFeB\nto induce the proper crystallization necessary for PMA and h igh TMR [4]. In Ta /CoFeB/MgO systems, however, spin\npumping to the Ta increases α. [5] Moreover, the CoFeB layer also needs to be ultrathin (ty pically less than 1.5nm) in\nordertoexhibitPMA.[3]Toimprovethethermalstabilityas devicesarescaleddowntosmallerdiameters,increasingth e\neffectivearealanisotropyenergydensity Kef ftisdesired.\nOne approach to address these issues is the use of double-MgO structures, i.e., those in which both the barrier layer\nand cappinglayer straddlingthe free layerare made of MgO. I mproved Ic0and/or∆have beenreportedin devicesusing\ndoubleMgOfreelayers. [6,7,8,9]Theimprovementintherma lstabilityisattributedtotheadditionalCoFeB /MgOinter-\nface,whereaslower Ic0isassociatedwithlow α. Indeed,αdownto0.005hasbeenmeasuredinin-planeMgO /FeB/MgO\nfilms,[10] which agrees with device measurements.[11] The s tacks investigated in these damping studies, however, did\nnot have the Ta layer used in practical free layers with perpe ndicular anisotropy.[9, 12] In addition, although the inte r-\nfacial anisotropy in the out-of-plane devices measured by T sunegi et al.[11] can be as high as 3.3 mJ /m2, the effective\nperpendicularanisotropywasratherlow ( Kef ft≈0.04mJ/m2) relativetothat ofa Ta /CoFeB/MgOstack[3]. In thiswork,\nweexploretheinfluenceofTainsertionswithintheCoFeBlay erofMgO/CoFeB/MgOfilmsbymagnetometryandvector\nnetwork analyzer ferromagnetic resonance (VNA-FMR) measu rements. The insertion of extremely thin Ta layers (0.3\nnm) inside the CoFeB layer aids crystallization, allowing a larger total CoFeB thickness to remain perpendicular, [13]\nwith aneffectivearealanistropycomparableto thatofTa /CoFeB/MgO.\nTwo sample series were deposited by magnetron sputtering on SiO2substrates with seed layers of Ta 5 /TaN 20/Ta 5\nin an ultrahigh vacuum environment (all thicknesses in nm). The stack configurations of the two sample series are: (1)\nMgO 3/CoFeB 1.0/Ta 0.3/CoFeB 0.5 - 1.5/MgO 3 (“single-insertion”) and (2) MgO 3 /CoFeB 1.0/Ta 0.3/CoFeB 0.5 -\n1.5/Ta0.3/CoFeB1.0/MgO3(“double-insertion”),wheretheCoFeBcompositionis Co40Fe40B20(at%). TheTainsertion\nlayer thickness is in the regime allowing strong ferromagne tic coupling between the CoFeB layers. [16] Two other\nsample serieswere grownasreferences: (a)MgO3 /CoFeB1.0-2.5/MgO3(“zero-insertion”),and(b)seed /CoFeB1.0-\n2.5/MgO3(“single-MgO”).Forallthedouble-MgOsamples,anult rathinCoFeBlayerbelowthebottomMgOlayerwas\nalsodepositedforgoodMgOgrowth. Weconfirmedfromseparat emeasurementsthatthislayerdoesnotcontributetothe\nmagneticsignal. All sampleswerecappedwith15nmofTa forp rotectionandwereannealedpost-growthat 300◦Cfor1\nh in vacuum. Although3 nm MgO is too thick for practical use in MTJs, it was chosen to ensure continuityof the MgO\nlayers and lessen the influence of the layersbeyondit.[10, 1 4, 15] (Measurementsof similar sampleswith 1 nm of MgO\nonbothsidesofthe magneticlayeryieldedthesametrends.)\nMagnetization measurements were performed using an altern ating gradient magnetometer (AGM). The PMA im-\nproveswith doublingof the CoFeB /MgO interface and with increasingnumber of Ta insertions, n, as shown in Fig. 1(a)\nfor samples with a similar total nominal CoFeB thickness tnom≈2.5 nm. We also confirmed that we cannot obtain a\n1perpendicular easy axis in double-MgO structures without T a insertions.[17] The double-insertion sample, on the othe r\nhand,exhibitslargeout-of-planeremanenceas shownin the inset of Fig. 1(a). A coercivefield less than0.01T (inset) is\ntypicalofCoFeBfilmswithPMA[18,16].\n/s49 /s50 /s51 /s52/s49/s50/s51/s52/s53\n/s45/s48/s46/s48/s49 /s48/s46/s48/s49\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s32/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s77/s47/s65/s32/s40/s65 /s109/s50\n/s47/s109/s50\n/s41/s32/s120/s32/s49/s48/s45/s54\n/s116\n/s110/s111/s109/s32/s40/s110/s109/s41/s48/s40/s98/s41\n/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s97/s46/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s40/s84/s41/s116\n/s110/s111 /s109/s32 /s32/s50/s46/s53/s110/s109/s40/s97/s41\nFigure 1: (Color online) (a) Out-of-planeAGM loops for samp les with single-MgO (red squares), zero-insertion(purple\ninvertedtriangles),single-insertion(bluecircles),an ddouble-insertion(greentriangles),withtotalnominalC oFeBthick-\nnesstnom≈2.5nm. Insetshowsalow-fieldout-of-planeloopforthesame doubleinsertionsample. (b)Magneticmoment\nperunitarea( M/A)asafunctionofthetotalnominalCoFeBthicknessforallsa mpleseries. Linearfitsareshownassolid\nlines.MSandtMDLcanbeextractedfromtheslopeand xintercept,respectively,andaresummarizedinTable 1.\nIt is known that Ta can create a magnetically dead layer (MDL) when it is near a magnetic layer.[19] We plot the\nmagneticmomentperarea against tnom[Fig.1(b)]to obtainthethicknessoftheMDL foreachseries fromthexintercept\nofalinearfit. TheresultsaresummarizedinTable1,alongwi ththeMSvaluesobtainedfromtheslope. WefindanMDL\nthickness tMDLof 0.24±0.09 nm for n=1 and 0.7±0.1 nm for n=2. These thicknesses are similar to the total Ta\ninsertionthicknessintherespectiveseries,andareconsi stentwiththepictureofCoFeBintermixingwithTatoproduc ea\nmagneticallydeadvolume. The tMDLvalueforthesingle-MgOsamples(0.26 ±0.08nm)agreeswithvaluesfoundinthe\nliterature.[14, 20, 21] On the other hand, no dead layer was f ound for the zero-insertionsamples, which is similar to the\nresultsinRef. [19].\nTable1: SummaryofMagneticProperties\nSeries tMDL(nm) MS(MA/m) Ki(mJ/m2) Kv(MJ/m3)\nsingle MgO 0 .26±0.08 1.51±0.08 1.61±0.07−0.29±0.08\ndouble MgO, n=0 0.04±0.06 1.29±0.05 0.91±0.09 0.37±0.05\ndouble MgO, n=1 0.24±0.09 1.12±0.05 2.18±0.08−0.34±0.04\ndouble MgO, n=2 0.7±0.1 1.10±0.04 2.4±0.1−0.25±0.03\nVNA-FMR was used to measure the e ffective anisotropy field and damping parameter of the samples . In the VNA-\nFMR setup, the samples were placed face down on a coplanar wav eguide and situated in a dc magnetic field of up to\n1.2 T applied perpendicular to the film plane. The transmission scattering parameter S21was measured at a specific\nfrequency while the dc field was swept. For each sweep, the rea l and imaginary parts of the resonance response were\nfitted simultaneouslyusingthecomplexsusceptibilityequ ation\nχ(H)=Mef f(H−Mef f+i∆H\n2)\n(H−Mef f)2−/parenleftBig2π/planckover2pi1f\ngµB/parenrightBig2+i∆H(H−Mef f)(1)\nwherefis the frequencyofthe ac field, Mef f=MS−H⊥\nK,∆His the fullwidth at half-maximum, H⊥\nKis the anisotropy\nfield perpendicular to the plane, gis the spectroscopic splitting factor, µBis the Bohr magneton, and /planckover2pi1is the reduced\nPlanck’s constant. Nonmagnetic contributions to the S21parameter and a linear time-dependentdrift of the instrume nts\nwere taken into account during the fit. We note that only one re sonance peak is observed within the range studied. A\nrepresentativefitofthesusceptibilitydataisshowninFig .2(a)foradouble-insertionsamplewith tnom=2.5nm. Inusing\nEq.1,a valueof g=2isfirst assumedtoobtainvaluesfor Mef fand∆H, whichdoesnotaffectthe finalresult.\nForeachfrequency,a resonancefield\nµ0Hres(f)=2π/planckover2pi1\ngµBf+µ0Mef f (2)\naccording to Kittel’s equation is calculated and plotted ag ainst the frequency, as shown in Fig. 2(c). A linear fit, now\nwithgandMef fas fitting parameters, is then performed. The e ffective anisotropyenergydensity Kef fcan be calculated\n2from the effective anisotropy field HKef f(=−Mef f) asKef f=HKef fMS/2, noting that a positive anisotropy constant\ncorrespondsto aperpendiculareasyaxis.\nToobtainα,we performa linearfit ofthe measuredFMRlinewidthasafunc tionofthefrequencyto\nµ0∆H(f)=4π/planckover2pi1α\ngµBf+µ0∆H0 (3)\nwhere∆H0is the inhomogeneous linewidth broadening, and the value of gused is the fitted value from Eq. 2. We\nnote that two-magnon scattering contributions to the linew idth are eliminated owing to the perpendicular measurement\nconfiguration[22]. Such a fit is shown in Fig. 2(c). Only data p oints taken well beyond the saturation field for each\nsamplewereusedinthefit, andasymptoticanalysisasdescri bedinRef. [23]fortheaccessiblefrequencyrangewasalso\nperformed.\nWe define an effective thickness tef f=tnom−tMDLand show the calculated Kef ftef f(to which Ebis proportional)\nfor both sample series in Fig. 3(a). The x-axis error bars originate from the fitting error in obtainin gtMDL. We find that\nfortef f>2 nm,double-insertionsampleshavehigher Kef ftef fthan single-insertionsamplesforthe same tef f. However,\nthe maximum Kef ftef fachieved for both the single- and double-insertion series d oes not significantly exceed Kef ftef f\nmeasuredin ourthinnestsingle-MgOsample( tef f=1.0nm),similar tothatobservedin MTJmeasurements.[7]\nTounderstandthisfurther,we considerthedi fferentcontributionsto Kef ftef f,whichisgivenby\nKef ftef f=Ki+(Kv−µ0M2\nS\n2)tef f (4)\nwhereKiis the total interfacial anisotropy constant, including al l CoFeB/MgO interfaces; Kvis the volume anisotropy\nconstant; and the demagnetizing energy is given by the M2\nSterm. We assume that any interfacial anisotropy from the\nTa/CoFeB interface is negligible.[24] Kiis commonly derived from the yintercept of a linear Kef ftef fversustef ffit,\nwhereasKvcan be calculated from the slope if MSis known. Because it is possible that for CoFeB thicknesses b elow\n1.0 nm,Kiis degraded because of Ta reaching the CoFeB /MgO interface,[25] we consider only the linear region of the\ncurve during the fit. The calculated values, given in Table 1, demonstrate that the absence of a Ta insertion leads to the\nlowest value of Ki(0.91±0.09mJ/m2), explaining why n=0 samples did not exhibit a perpendiculareasy axis. On the\nother hand, Kiforn=1 (2.18±0.08mJ/m2) andn=2 (2.4±0.1mJ/m2) are both larger than the single-MgO series\n(1.61±0.07mJ/m2), as would be expected from the additional PMA from the secon d CoFeB/MgO interface. However,\nthe anisotropy per interface did not double with the additional CoFeB /MgO, which may be attributed to the di fferent\ndegrees of crystallization for single- and double-MgO samp les. An indication of better crystallization into CoFe in th e\nsingle-MgO series is its higher MS.Kvis negative and does not vary appreciably in samples where Ta is present, in\ncontrast to the positive value found for zero-insertion sam ples. The role of Ta with regard to Kvis not yet understood,\nas previouslypointed out by Sinha et al. [20], and a detailed study of the amount, proximity,and profile of Ta would be\nnecessarytoclarifythesee ffects.\nTurningourattentionto α,weidentifyasingle-MgOsample( tef f≈0.8nm)andasingle-insertionsample( tef f≈1.3\nnm)withacomparable Kef ftef f≈0.2mJ/m2. Weimmediatelynoticethat αforthesingle-insertionsampleisaroundtwo\ntimeslowerthanthatforthe single-MgOsample.\nThis dramaticdecrease in αmay be attributedto the suppressionof spin pumpingby the Mg O layers straddlingboth\nsides of the precessing magnet.[10, 26] Indeed, measuremen ts of zero-insertion samples show no thickness dependence\n[purple dashed line in Fig. 3(b)] and a low mean value of α=0.0035±0.0002 comparable to the bulk damping of\nCo40Fe40B20[5,27]. However,adecreasein αwithincreasing tef fcanstillbeseeninboththesingle-anddouble-insertion\nseries. One possible reason is the alloying of CoFeB and Ta, a s Ta is known to readily intermix with CoFeB [21], and\nhigher damping may be expected from CoFeBTa alloys [28]. The relative percentage of CoFeBTa alloy decreases with\nincreasing CoFeB thickness, coinciding with the αdecrease. This picture is also consistent with the jump in αfrom\nsingle-to double-insertionsamples, i.e.,thereismoreCo FeBTa alloybecausetherearemoreTa insertions. [11, 10]\nItmayalsobepossiblethatspinpumpingtotheTainsertionl ayeroccurs,asinthecaseofthePdinterlayerinCoFe /Pd\nmultilayers[29]. The complexityof oursystem, however,pr eventsusfromusinga simple multilayermodel. One reason\nisthatthemiddleCoFeBlayer(inthedouble-insertioncase )mayhavedifferentpropertiesfromtheCoFeBlayersadjacent\nto MgO, because CoFeB crystallizes from the MgO interface, [ 30] with which the middle CoFeB has no contact. The\ndegree of Ta intermixing also depends on the deposition orde r and will vary across the structure.[19] At this point, we\ncannot discriminate the mechanism behind the damping behav ior. It may be worthwhile to study the use of CoFeBTa\nalloysasinterlayerstopossiblyhavemorecontroloverthe amountanddistributionofTa inthestack.[31]\nInconclusion,wehavedemonstratedPMAandlowdampingindo uble-MgOstructures. AthinTainsertionlayerwas\nfound to significantly increase the PMA - no perpendicular ea sy axis was realized in our MgO /Co40Fe40B20/MgO films\nwithout Ta - and adding more insertionsallowed thicker CoFe B layers to remain perpendicular. However, the maximum\nKef ftef findouble-MgOsamplesiscomparableonlywiththatofthesin gle-MgOsampleforthisCoFeBcomposition.[9]\nOn the other hand, αfor double MgO films increases with the number of insertions b ut is still lower than that of single\n3MgO films for the entire range. Considering both trends with n, we find that the optimal stack in the range of samples\nwe studiedis a double-MgO, n=1 sample with tnom=1.75nm,which exhibits Kef ftef f=0.27mJ/m2at a low damping\nvalueof0.006.\nAcknowledgement\nWe expressgratitudeforsupportfromtheA*STARGraduateAc ademySINGAProgram.\nReferences\n[1] J. Z. Sun, S. L. Brown, W. Chen, E. A. Delenia, M. C. Gaidis, J. Harms, G. Hu, Xin Jiang, R. Kilaru, W. Kula,\nG.Lauer,L.Q.Liu,S.Murthy,J.Nowak,E.J.Oâ ˘A´ZSullivan,S.S.P.Parkin,R.P.Robertazzi,P.M.Rice,G.Sa ndhu,\nT.Topuria,andD.C. Worledge: Phys.Rev.B88,104426(2013) .\n[2] S. Yuasa: J. Phys.Soc.Jpn.77,031001(2008).\n[3] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and\nH. Ohno: Nat.Mater. 9,721(2010).\n[4] D. C. Worledge, G. Hu, David W. Abraham, J. Z. Sun, P. L. Tro uilloud, J. Nowak, S. Brown, M. C. Gaidis, E. J.\nO’Sullivan,andR. P.Robertazzi: Appl.Phys.Lett.98,0225 01(2011).\n[5] X. Liu,W.Zhang,M.J. Carter,andG. 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Noshiro,C. Yoshida, Y. Yamazaki, A. Taka hashi, Y. Iba, A. Hatada, M. Nakabayashi,T. Takenaga,\nM. Aoki,andT.Sugii.IEDM,29.1.1(2012).\n5/s50/s48/s48 /s50/s50/s48 /s50/s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s40/s99/s41/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s70/s105/s101/s108/s100/s32/s40/s109 /s84/s41/s32\n/s32\n/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s109 /s84/s41/s32/s68/s97/s116/s97 /s32\n/s32/s70/s105/s116\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s71/s72/s122/s41/s32/s68/s97/s116/s97\n/s32/s70/s105/s116/s83\n/s50/s49/s32/s82/s101/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s40/s97/s41\n/s40/s98/s41/s83\n/s50/s49/s32/s73/s109/s97/s103/s46/s32/s40/s97/s46/s117/s46/s41\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41\nFigure 2: (a) Real and (b) imaginary parts of the S21parameter obtained from VNA-FMR measurements for a double-\ninsertion sample with tnom=2.5 nm at 12 GHz while a perpendiculardc magnetic field is swept. The lines are fits to an\nexpressionusingEq.1, takingnonmagneticcontributionst oS21anda lineardriftintoaccount. (c)Field-sweptlinewidth\nandresonancefieldsforthesamesampleasafunctionoffrequ ency. Thelinearfitsdescribedinthetextareusedtoextract\nHKef f(=−Mef f)andα.\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49 /s50 /s51/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s75\n/s101/s102/s102/s116\n/s101/s102/s102/s32/s40/s109/s74/s47/s109/s50\n/s41\n/s40/s97/s41/s40/s49/s48/s45/s51\n/s41\n/s116\n/s101/s102/s102/s32/s40/s110/s109/s41/s40/s98/s41\nFigure3: (a) Kef ftef fand(b)αversuseffectiveCoFeBthickness tef fobtainedfromfield-sweptVNA-FMRmeasurements\nfor all sample series. Solidlines in (a)are linear fits. Purp ledashedline in (b) correspondsto the mean αvalueaveraged\noverall zero-insertionsamples,whichwasfoundtobeconst antwithinerroracrosstheentirethicknessrangestudied.\n6"    },    {        "title": "1604.06332v1.Ferromagnetic_resonance_of_exchange_coupled_perpendicularly_magnetized_bilayers.pdf",        "content": "arXiv:1604.06332v1  [cond-mat.mtrl-sci]  21 Apr 2016Ferromagneticresonance ofexchange-coupled perpendicul arly magnetized bilayers\nThibaut Devolder∗\nInstitut d’Electronique Fondamentale, CNRS, Univ. Paris- Sud, Universit´ e Paris-Saclay, 91405 Orsay, France\n(Dated: June 13, 2021)\nStrong ferromagnetic interlayer exchange couplings Jin perpendicularly magnetized systems are becoming\nincreasingly desirable for applications. We study whether ferromagnetic interlayer exchange couplings can\nbe measured by a combination of broadband ferromagnetic res onance methods and magnetometry hysteresis\nloops. For this we model the switching and the eigenexcitati ons in bilayer systems comprising a soft layer\ncoupled to a thicker harder layer that possesses higher perp endicular magnetic anisotropy. For large J >0the\nswitching fields are essentially independent of Jbut the frequency of the optical eigenmode of the bilayer and\nthe linewidth of the acoustical and optical eigenmode are di rectly sensitive tothe coupling. We derive a corpus\nofcompact analyticalexpressions toanalyze thesefrequen cies, theirlinewidthanddiscuss themeaningthereof.\nWe illustrate this corpus on a system mimicking the fixed laye rs of a magnetic tunnel junction meant for spin\ntorque applications.\nI. INTRODUCTION\nThe next generations of spin-torque operated magnetic\nmemory cells will rely on perpendicularmagnetic anisotrop y\n(PMA) magnetic tunnel junctions (MTJ)1, because of the su-\nperior scalability of this configuration2. The correct sens-\ning of the stored information requires reference layers wit h\naresilientmagnetizationorientationthatshouldbeinsen sitive\nto thermal fluctuations, magnetic fields and spin-torques. I n\npractice thistranslates into referencelayerrequirement sfora\nsufficient anisotropy, a sufficient damping and no stray field .\nIn addition, the layout of the reference system must ensure\na high tunnel magneto-resistance (TMR). As a result, opti-\nmized MTJs3–13rely on composite reference systems, where\neachofthepreviouslymentionedfeatureisoptimizedbysom e\nsub-system. The TMR is generally optimized by using an\nFeCoB14spinpolarizinglayer(layer1 ofthickness t1), while\nthehighanisotropyisusuallyprovidedbyanfcc(111)cobal t-\nbased multilayer (layer 2 of thickness t2). Because these\ntwo layers have different crystal structures a spacer layer is\nneeded.\nHowever the role of the spacer layer is also to promote\na high interlayer exchange coupling Jbetween its adjacent\nlayers 1 and 2. Indeed, the sole criterion of maximal TMR\nwould argue11,15for a large t1, leading to the loss of perpen-\ndicular magnetization at remanence when the demagnetizing\nenergyµ0M2\nS/2overcomes the interface anisotropy energy\nKS/t1. This can be relaxed and one can use thicker spin po-\nlarizing layer if one complements its interface anisotropy by\na ferromagnetic interlayer exchange coupling J/t1with the\nhigh anisotropy layer. As a result, the measurement of large\nand ferromagnetic interlayer exchange coupling in soft lay er\n/ spacer / hard layer composites is of importance. Unfortu-\nnately,conventionalmagnetometrymethodsareineffectiv eto\nmeasure large positive J: strong couplings force the layers 1\nand 2 to switch in synchronyin a rigid manner, such that the\nloopsare not informativewith respect to the amplitudeof th e\ncoupling. Besides, the coercivities of real systems are oft en\nextrinsic and largely influenced by the defects, in addition to\nbeingaffectedby thermalactivation;as a result the switch ing\nfields in hysteresis loopscan only be used to providequalita -\ntiveinformationontheanisotropiesandthe coupling.In this paper, we study whether the ferromagnetic reso-\nnancemodesofthecompositecanbeusedtoquantifythecou-\npling by either looking at the eigenmode frequencies or thei r\nlinewidthinadditiontothehysteresisloops. Wefirstsolve the\nsystemexactlyformaterialpropertiesthataretypicalino rder\ntoclassify the possibleclassof behaviors. We thenderivea p-\nproximate analytical formulas meant to ease the analysis of\nfutureexperimentalstudiesoftheeigenmodefrequenciesa nd\ntheirlinewidth.\nWe model samples consisting of a soft layer / spacer /\nhard layer sequence with perpendicular anisotropies. We de -\nscribe the system as a set of two coupled macrospin labelled\ni= 1,2, with thicknesses ti, magnetizations MSi, damp-\ning parameters αi, magnetocrystalline anisotropy fields Hki,\nnormalizedmagnetizationcomponents {mxi, myi, mzi}and\ncoupled through a bilinear interlayer exchange energy J.\nJis supposed to be much smaller than the intralayer ex-\nchange stiffness so that the macrospin approximations re-\nmain valid within each layer16. For simplicity we will of-\nten gather the anisotropy and the demagnetizing energies\ntogether, by writing Heff\nki=Hki−MSifor each layer.\nThe layer i= 1is chosen as the softest and the thinnest\nlayer. The areal energy of the system (in units of µ0)\nis the sum of the anisotropy and demagnetizing energies\n:(1/2)/parenleftbig\n−Heff\nk1MS1mz12t1−Heff\nk2MS2mz22t2/parenrightbig\nand of the\nZeemanenergy −Hz(Ms1mz1t1+Ms2mz2t2)andtheinter-\nlayer exchangecoupling energy −J0(mx1mx2+my1my2+\nmz1mz2)where we have written J0=J/µ0, withJin J/m2\nand all other terms also expressed in SI units. The configu-\nration of the system is found by minimizing the total energy,\nwhile its eigenexcitations are found by linearizing the mag -\nnetization dynamics equation about that energy minimum.\nTheanalytical expressionsforthe frequenciesand the crit ical\nfieldsare derivedforvanishingdampingparameters.\nII. CLASSIFICATIONOFTHE POSSIBLEBEHAVIORS\nLet us first look at some typical behaviors by solving the\nsystemnumericallyonagivensetofmaterialparameters. Th e\ncalculations are done with parameters mimicking an FeCoB\nlayer as soft layer (i.e. layer 1) and [Co/Pt]Nmultilayer as2\n/s45/s50/s45/s49/s48/s49/s50/s45/s50 /s45/s49 /s48 /s49 /s50 /s45/s49 /s48 /s49 /s50\n/s48/s50/s53/s53/s48/s55/s53/s49/s48/s48\n/s45/s50/s45/s49/s48/s49/s50\n/s48/s50/s53/s53/s48/s55/s53\n/s48/s50/s53/s53/s48/s55/s53\n/s45/s50/s45/s49/s48/s49/s50\n/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50\n/s48/s50/s53/s53/s48/s55/s53/s109\n/s90/s49/s109\n/s90/s50\n/s74/s61/s32/s49/s32/s109/s74/s47/s109/s50/s74/s61/s32/s49/s46/s53/s32/s109/s74/s47/s109/s50/s32/s32\n/s32/s77\n/s90\n/s74/s61/s32/s50/s32/s109/s74/s47/s109/s50/s109\n/s90/s49/s43/s109\n/s90/s50/s32/s80/s101/s114/s112/s46/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s32/s80/s101/s114/s112/s46/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s32\n/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s70/s69/s82/s82/s79/s77/s65 /s71/s78/s69/s84/s73/s67/s32/s67/s79/s85/s80/s76/s73/s78/s71\n/s115/s119/s101/s101/s112\n/s32\n/s32\n/s32/s77\n/s90\n/s32\n/s32\n/s78/s67/s83/s80/s32/s62/s32/s48/s32\n/s32\n/s32/s80/s32/s60/s32/s48/s32/s32/s32/s65/s80/s32\n/s32\n/s32/s77\n/s90\n/s32\n/s32/s77\n/s90\n/s80/s101/s114/s112/s46/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s72/s80/s32/s98/s114/s97/s110/s99/s104\n/s118\n/s72/s65/s80\n/s67/s32 /s80/s60/s48\n/s72/s80\n/s67/s62/s48/s74/s61/s32/s48/s46/s53/s32/s109/s74/s47/s109/s50\n/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41\nFIG. 1. Hysteresis loops (left panels, back and forth field sw eeps)\nand eigenexcitations (right panels, decreasing field only) of a\nsoft/hard PMA composite bilayer for ferromagnetic interla yer ex-\nchange coupling. P,APandNCSstand for parallel, antiparallel\nand non collinear states. The black loops are the sum of the no rmal-\nizedmagnetizations of the twolayers.\nhard layer (i.e. layer 2). In the numerical calculations, th e\nchosen parameters are MS1= 106A/m,MS2= 8×105\nA/m,Hk2= 2×106A/m,t1= 2nm,t2= 5nm. The\ninterlayer exchange coupling was varied from strongly anti -\nferromagnetic ( J=−1.5mJ/m2) to strongly ferromagnetic\n(J= 2mJ/m2). The field is swept from positive (favoring\na so-called Parallel ”P >0” state) to negative (favoring a\nso-called reversed Parallel ”P <0” state). In all figures ex-\ncept Fig. 4, the magneto-cristalline anisotropy of layer 1 is\nHk1= 106A/m, leading to a zero effective anisotropy. In\nFig.4, we shall use Hk1= 6×105A/m (implying negative\neffective anisotropy) in order to describe also the systems in\nwhich the perpendicularremanenceof the layer 1 is obtained\nbythecouplingandnotbyitssoleanisotropy.A. Case of ferromagnetic interlayer coupling\nThe case of ferromagnetic interlayer exchange coupling is\ndisplayed in Fig. 1. The common features for J >0are that\nthecoercivitieshavealwaysthenormalsign(positiveconv en-\ntion)andthat the switchingof a givenlayer is alwaysaccom-\npaniedwithanincreaseofthefrequencyofitseigenexcitat ion.\nLetuslookat thedetails.\nIn case of very strong coupling (Fig. 1, top panels), the soft\nandhardlayersswitchsimultaneouslyandalwayskeepamag-\nnetization fully perpendicular to the sample plane: there i s a\ndirect transition from the P >0to the P>0state. The low-\nestfrequencyeigenmode(theacousticalexcitation)softe nsto\nzeroat thisunique(positive)switchingfield HC,1=HC,2.\nWhenthecouplingisreducedto1.5mJ/m2,thetwolayersdo\nnot switch in synchrony any longer: the soft layer first tilts\nits magnetization creating a Non Collinear State (NCS), and\nit saturatesonlywhen hardlayer switchesalso; thereis a ti ny\nfield interval during which the magnetizations are no longer\ncollineartoeachother,andtheacousticalmodestayssoft( i.e.\nω= 0)duringthatinterval.\nWhen the coupling is further reduced to 1 mJ/m2, the two\nlayers start to switch at distinct fields. NCS magnetization s\nstates tilted from the perpendicular axis are possible for t he\ntwo layers. An AntiParallel (AP) configurationoccursin par t\nof the field interval between the two coercivities. The high-\nest frequency eigenmode (the optical excitation) evolves n on\nlinearly with the applied field in that interval. One of the AP\neigenmodesoftens at the correspondingAP to P <0switch-\ningfield.\nFinally when the coupling is reduced to a weak value of 0.5\nmJ/m2, the two layers switch independently. Only collinear\nstatesarepossible,butalargeintervalallowsforanantip aral-\nlel(AP)situation. TheP >0toAPswitchingofthesoftlayer\ninducesa step-likereductionofthe eigenexcitationfrequ ency\nof the optical mode, whose amplitude is mostly localized in\nthehardlayer.\nB. Case of antiferromagnetic interlayer coupling\nThe cases of zero and antiferromagnetic interlayer ex-\nchangecouplingsare displayedin Fig. 2. Thesoft layercoer-\ncivities is decreasedand its switching is always accompani ed\nwithanincreaseofthefrequencyoftheeigenexcitationoft he\nhardlayer. Letuslookat thedetails.\nWhenthelayersarenotcoupled(toppanels),theyswitchsep -\naratelyat theirrespectiveeffectiveanisotropyfields, an dtheir\ncorresponding ferromagnetic resonance modes are indepen-\ndent. Parallel and antiparallel collinear configurations b oth\noccur during a loop, but the magnetizations always stay per-\npendiculartothe plane.\nWhen the antiferromagnetic coupling is turned on to a weak\nvalue of -0.5 mJ/m2, the soft layer coercivity decreases and\ncanchangesign(switchinghappensata positivebeforereach-\ningzerofield). ThesoftlayerP >0toAPswitchingincreases\nthefrequencyofthehardlayereigenexcitation. Theamplit ude\nofthisfrequencyjumpincreaseswith thestrengthof |J|.3\n/s45/s50/s45/s49/s48/s49/s50/s45/s50 /s45/s49 /s48 /s49 /s50 /s45/s49 /s48 /s49 /s50\n/s48/s50/s53/s53/s48/s55/s53/s49/s48/s48\n/s45/s50/s45/s49/s48/s49/s50\n/s48/s50/s53/s53/s48/s55/s53\n/s48/s50/s53/s53/s48/s55/s53\n/s45/s50/s45/s49/s48/s49/s50\n/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50\n/s48/s50/s53/s53/s48/s55/s53/s109\n/s90/s50/s109\n/s90/s49/s43/s109\n/s90/s50\n/s74/s61/s32/s45/s49/s32/s109/s74/s47/s109/s50/s74/s61/s32/s45/s48/s46/s53/s32/s109/s74/s47/s109/s50/s32/s32\n/s32/s74/s61/s32/s48/s32/s40/s117/s110/s99/s111/s117/s112/s108/s101/s100/s41\n/s115/s119/s101/s101/s112/s32/s80/s101/s114/s112/s46/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s32/s80/s101/s114/s112/s46/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s32/s32/s78/s79/s32/s111/s114/s32/s65 /s78/s84/s73/s70/s69/s82/s82/s79/s77/s65 /s71/s78/s69/s84/s73/s67/s32/s67/s79/s85/s80/s76/s73/s78/s71\n/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32\n/s32\n/s32/s77\n/s90/s77\n/s90/s77\n/s90/s77\n/s90/s32\n/s32/s72/s65/s80\n/s67/s80\n/s32\n/s32/s109\n/s90/s49\n/s32\n/s32\n/s32\n/s32\n/s80/s101/s114/s112/s46/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s72/s65/s80/s32/s98/s114/s97/s110/s99/s104\n/s115/s111/s102/s116/s72/s80/s32/s98/s114/s97/s110/s99/s104\n/s118\n/s72/s80\n/s67/s32 /s65/s80/s74/s61/s32/s45/s49/s46/s53/s32/s109/s74/s47/s109/s50\nFIG. 2. Hysteresis loops (left panels, back and forth field sw eeps)\nand eigenexcitations (right panels, decreasing field only) of a\nsoft/hard PMA composite bilayer for uncoupled films and anti fer-\nromagnetically coupled films.\nWhen the antiferromagnetic coupling is further increased t o\n-1 and -1.5 mJ/m2, non collinear states (NCS) become possi-\nble again. There is a gradual increase of the P >0to NCS\ncoercivity of the soft layer, and of the frequency impact of a\nlayer’sswitchingonthe other.\nFinally for very strong antiferromagnetic coupling (botto m\npanel),NCSoccurattheonsetofthesoftlayerswitching;th is\ncomeswitharoundingofthecorrespondingfrequencyjumps.\nTheAP state occursina largefield interval.\nIn summary, the specific example calculated above illus-\ntrates most of the possible behaviors. Starting from Hz>0,\nthereare4possiblestatesP >0,P<0,APandnoncollinear\nconfigurations NCS. The corresponding state diagram is dis-\nplayedinFig. 3. Theboundariesofthisstatediagramdepends\non the material properties. Noticeably, the NCS states occu r\nmore frequently when the layers’ easy axes are different, as\nillustratedinFig. 4in whichthelayer1hasbeenchosenwith\nan easy plane (i.e. Heff\nk1<0). Let us derivethe boundariesof\nthestate diagramsin ananalyticalmanner.III. ANALYTICALMODELINTHE PARALLELSTATES\nFrom the previous section, we have seen that states with\nparallelmagnetizationsare presentin largefield interval s,es-\npecially in the case of ferromagnetic coupling. With the ob-\njective of easing the analysis of experimental data, we spen d\nthe next sections to derive a corpus of analytical expressio ns\nthat describe the switching fields and the eigenexcitationf re-\nquenciesintheP >0state(thissection)andAP(nextsection)\ncase. The first step is to linearize the magnetization dynam-\nics equation about that configuration when it is the energy\nminimum, and then take the Hessian matrix of the total en-\nergy of the system to find how the effective fields depend on\nthe dynamic magnetization. In the parallel state, the eigen -\nfrequencies ℜ(ω), the half linewidth −ℑ(ω)and the eigen-\nmodes of the soft/hard composite are the complex eigenval-\nuesω=ℜ(ω)+iℑ(ω)andtheeigenvectorsoftheso-defined\ndynamicalmatrix:\n\nα1˜H1˜H10α1J0\nMS1t1−J0\nMS1t10\n−˜H1α1˜H10J0\nMS1t1α1J0\nMS1t10\n0 0 0 0 0 0\nα2J0\nMS2t2−J0\nMS2t20α2˜H2˜H20\nJ0\nMS2t2α2J0\nMS2t20−˜H2α2˜H20\n0 0 0 0 0 0\n\nwhere we have defined ˜Hi=Heff\nki+Hz+J0\nMSiti. Note\nthatasourphysicalproblemshasonly4independentvariabl es\nwhicharetheorientationanglesofthetwomagnetizations, the\nabove6×6matrix is only of rank 4: two of its eigenvalues\narenullandtheeigenvaluesaretwo-by-twodegenerate. Wri t-\ningthe problemin circularcoordinatesisunpracticalbeca use\nthe groundstates are close to the θi= 0andθ=πpositions\nwhere circular coordinates are singular. Our notation is su ch\nthatαi=−ℑ(ωi)\nRe(ωi)for perpendicularly magnetized states in\ntheabsenceofcoupling(i.e. J= 0). Fromtheantisymmetric\ncharacterofthefour 3×3sub-quartersofthedynamicalma-\ntrix, it results that the eigenmodes will have no ellipticit y in\nthisP >0state, and that consequently the frequency versus\nfieldcurveswillhaveaslopebeingthegyromagneticratio γ0.\nTheknowledgeofthecriticalfields Hcritatwhichthesemodes\nsoften is thus enough to describe the eigenmode frequencies\nwhichwill thenfollow ω=γ0(Hz−Hcrit).\nIn the specific case of no coupling, the dynamical matrix\nyields the usual ferromagnetic resonance frequencies for t he\ntwo layers : ωFMR, J=0=γ0(Heff\nki+Hz)which gives (neg-\native) switching fields of HCi=−Heff\nkiwhen these modes\nsoften.\nThesituationismorecomplexwhenthereisafinitecoupling.\nLet us focus on the case of the P >0ground state, and\nfirst find the softening field at which this state looses stabil -\nity. Theyarethefieldsatwhichatleastoneeigenexcitation of\nthesystemsoftensto zerofrequency.4\nA. P>0parallel state softeningfields\n1. Identical layers\ns\nIn the case of identical layers (i.e. MS1=MS2=MS,\nt1=t2andHk1=Hk2=Hk),therearetwosofteningfields\natwhichthehypotheticalP >0stateloosesstability. Thefirst\noneissimply:\nHP>0\nC,1=2=−Heff\nk\nThe stability analysis indicates that this is a real switchi ng\nfieldforferromagneticcoupling,andit inducesatransitio nto\nP<0, AP or NCS. Note that this switching field is indepen-\ndent ofJ, which confirmsthat conventionalmagnetometryis\ninoperativeto quantify J; Qualitatively, the switching field is\nindependentof Jbecause if we virtually ”cut” a layer in two\nhalves, this does not change the physics as long as the two\npartsareferromagneticallycoupled.\nThesecondsofteningfieldis\nHPbranch\nv,1=2=HP>0\nC,1=2−2J0\nMSt(1)\nThe subscript vstands for virtual; indeed generally HP branch\nv,1=2\nis not a switching field since the P >0state has lost stability\n(hence has disappeared) before this field is actually reache d\nduring a field sweep. It is however an interesting quantity as\nit is the zero frequencyextrapolationof the optical excita tion\nof the symmetric bilayer in the P >0branch (i.e. this is the\nanalogousof the PerpendicularStanding Spin Wave in single\nuniformfilms).\n2. Nonidentical layers\nIn practice, we deal with soft and hardlayersthat can have\nvery different properties. To describe this non-symmetric\ncase,it isusefultodefinethenotationsgatheredinTable I.\nTABLEI.Notations used whendescribing non identical layer s.\nPhysicalproperty Formula\nMiscompensation ofmoments ∆Mt=MS1t1−MS2t2\nDifference inanisotropies ∆Heff\nk=Heff\nk1−Heff\nk2\nTotal moment ΣMt=MS1t1+MS2t2\nTotal anisotropy ΣHeff\nk=Heff\nk1+Heff\nk2\n”average” square moment MS2t2=MS1MS2t1t2\nTheexpressionsoftheeigenmodefrequenciesare γ0(Hz−\nHP>0\nsoft,1/negationslash=2), and they soften at critical fields that can be ex-/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50\n/s80/s62/s48/s32/s115/s116/s97/s116/s101\n/s80/s60/s48/s32/s115/s116/s97/s116/s101/s65 /s80/s32/s115/s116/s97/s116/s101\n/s80/s60/s48/s32/s115/s116/s97/s116/s101/s80/s62/s48/s32/s115/s116/s97/s116/s101\n/s65 /s80/s32/s115/s116/s97/s116/s101/s32\n/s32\n/s78/s67/s83/s115/s119/s101/s101/s112\n/s72/s80/s32/s98/s114/s97/s110/s99/s104/s32\n/s118/s105/s114/s116/s117/s97/s108/s70/s105/s101/s108/s100/s32/s40/s84/s41\n/s73/s110/s116/s101/s114/s108/s97/s121/s101/s114/s32/s101/s120/s99/s104/s97/s110/s103/s101/s32/s40/s109/s74/s47/s109/s50\n/s41/s78/s67/s83\nFIG.3. Statediagram derived from the numerical simulation s (sym-\nbols) and from the analytical models (lines). The out-of-pl ane field\nis swept down after a start at Hz>0. The notations P >0, P<0\nAPand NCSstandfor parallel withpositive full remanence, p arallel\nwith negative full remanence, antiparallel and non colline ar states.\nThecriticalfields areEq.2(greenand redlines)andEq.7(bl ue and\nmagenta lines). These linesmerge inthe point definedby Eq.8 .\npressedas:\nHP>0\nsoft,1/negationslash=2 =−ΣHeff\nk\n2−J0ΣMt\n2Ms2t2\n±√\nJ2\n0ΣMt2−2∆Hk∆MtJ0Ms2t2+∆Hk2Ms4t4\n2Ms2t2 (2)\nNote that once again only one of these two fields is a real\nswitching field: it is the first one that is reached during a\nfield sweep. The other softening field is anyway an interest-\ning quantity. Indeed it is the zero frequency extrapolation of\nthe optical excitation of the bilayer. These two fields are di s-\nplayedinredandgreenintheFigs. 3and4. Twolimitsofthe\npreviousequationsareworthmentioning.\na. Non identical layers, limit of large coupling. In the\nlimit of large ferromagnetic coupling (i.e. J2\n0ΣMt2much\ngreaterthantheothertermsunderthesquarerootofeq. 2and\nJ >0),thefirstsofteningfieldisaswitchingfieldawayfrom\nP>0toeitherA P,NCS orP <0),whichreducesto:\nHC,1/negationslash=2J>>1≈ −ΣHeff\nk\n2−∆Hk∆Mt\n2ΣMt+O(1\nJ0)(3)\nTheexpressionof HC,1/negationslash=2J>>1isonlyanasymptoticlimit\n[see Fig. 6(a)] which requires J >>2mJ/m2, i. e. di-\nrectlayer-to-layerexchangecoupling,orpinholemediate dex-\nchange coupling. In that case, the first term of the right hand\nside of the above expression means that the switching field\nis essentially the average of these of the two layers when un-\ncoupled, and the second term is a correction that scale with\nthe asymmetry of their properties. This recalls the results for\nin-plane magnetized systems17in which a so-called bilayer\nscalingparametercanbeusedto describehowthe bilayerco-\ncercivity can be calculated from an average of the propertie s\nofthetwolayers.5\nIn this asymptotic limit of strongferromagneticcoupling, the\nsecondsofteningfieldreducestoa virtualfield thatis:\nHP branch\nv,1/negationslash=2J>>1≈HC,1/negationslash=2J>>1−J0ΣMt\nMs2t2.(4)\nThis field corresponds to the zero frequency intercept of the\noptical(PSSW-like)branchofthebilayer.\nb. Non identical layers, limit of weak coupling. In the\nlimit of weakly exchanged very asymmetric systems (i.e.\nJ2\n0ΣMt2muchsmallerthantheothertermsinthesquareroot\nEq.2, in practice for J <<2mJ/m2), the softening fields\nreduceto:\nHP>0\nC,1/negationslash=2J<<1≈ −Heff\nk1−J0\nMS1t1(5)\nand\nHP branch\nv,1/negationslash=2J<<1≈ −Heff\nk2−J0\nMS2t2(6)\nTheseexpressionsindicatequalitativelythatthetwoeige n-\nmodesaresufficientlydistantforlittlemodehybridationt ooc-\ncur,suchthat a layer iactsonthe layer jasa static exchange\nbiasing field ofJ0\nMSjtjthat needs to be compensated for the\nstability loss of the P >0state. When in the P >0state, a\nferromagnetic interlayer exchange coupling is such that ea ch\nlayer stabilizes the other. The coupling simply increases t he\ncoercivityofthesoftestlayer(seeEq. 5).\nIt is worth noticing the factor of 2 difference in the role\nof the exchangecoupling Jin the case of softening fields for\nstronglyasymmetric(Eq. 6)andsymmetriccases(Eq. 1). The\nfactoroftworecallsthatinthesymmetriccase,theoptical ex-\ncitationinvolvethemotionofthe twolayerswithequalampli-\ntude, while in the strongly asymmetric case, each eigenmode\nis essentially a one-layer oscillation under the bias provided\nbythealmoststaticotherlayer. Thisnumericalfactorthan can\nvary between 1 and 2 depending on the layer-to-layer asym-\nmetrycanleadtoconfusion: indeeditmeansthatthemeasure -\nmentofthesolecriticalfields(Eq. 5and6)cannotinformon\nthevalueof Junlessallothermagneticpropertiesareknown.\nIV. ANALYTICALMODELINTHE CASEOF\nANTIPARALLELMAGNETIZATIONS\nSimilarly, we can derivethe dynamicalmatrix (not shown)\ndescribingtheeigenexcitationsabouttheAPposition {mz1=\n−1,mz2= 1}where we have assumed that the soft layer\nswitchesfirst. Theformalisminthissectionisonlyvalidwh en\nthis AP state is visited during a field sweep; note that this is\nnot the case for strong ferromagnetic coupling when only P\nstatesoccur(Fig. 3).A. Identicallayers\nIn the case of identical layers, two softening fields are ob-\ntained:\nHAP\nC,1=2=±/radicalbigg\nHeff\nk×(Heff\nk−2J0\nMSt)\nThesefieldsarevirtualforstrongferromagneticcoupling( i.e.\n2J0\nMst> Heff\nk) and this AP state is not visited during a field\nsweep. For small or negative J, these two critical fields are\nreal and opposite as intuitively expected for a symmetric bi -\nlayer.\nB. Non identicallayers\nInthecaseofnonsymmetriclayers,thetwosofteningfields\nare:\nHAP\nsoft,1/negationslash=2 =∆Heff\nk\n2+J0∆Mt\n2Ms2t2 (7)\n±√\nJ2\n0∆Mt2−2ΣHeff\nkΣMtJ0Ms2t2+ΣHk2Ms4t4\n2Ms2t2\nwhere±distinguishestherealandthevirtualswitchingfields\nwhose role are interchanged when the sign of Jis changed.\nThetwobranchesofEq. 7areplottedin Fig. 3.\nIt is worth comparing the signs of the first terms of Eq. 7\nand Eq.2. Indeed the former decreases with Heff\nk1while the\nsecond increases, with the consequence that the NCS pock-\nets in the state phase diagrams (Fig. 3) essentially shrink and\nseparate when the layer 1 anisotropy is strengthened, or the\ntwo NCS pockets merge in a single one that enlarges when\nthelayer1anisotropyis decreased(Fig. 4).\nFromthemergingofthetwobranchesofEq. 7itisalsointer-\nestingtosee thatthe conditionforthenonexistenceofanAP\nstate duringthehysteresisloopissimply J0≥JnoAPwith\nJno AP=MS2t2/bracketleftig√MS1t1−√MS2t2\nMS1t1−MS2t2/bracketrightig2\nΣHeff\nk(8)\nIf both layers have positive effective anisotropies, this c on-\ndition ensures that the only possible states are the P states :\nin practice this is a condition of proper pinning of the softe st\nlayer. It is thus the condition that has to fulfilled in applic a-\ntions where the softest layer ( i= 1) is required to stay mag-\nnetized parallel to the hard layer. In our numerical example s,\nthisconditionis µ0JP only= 1.03mJ/m2.\nTwolimitsofEq. 7areworthlookingat.\nc. Non identical layers, limit of large coupling. In the\nlimit of large coupling and/or strong asymmetry (i.e. when\nJ2\n0∆Mt2is much greater than the other terms under the\nsquarerootinEq. 7),the softeningfieldsreduceto\nHAPbranch\nsoft,1/negationslash=2J>>1≈δJ0∆Mt\nMs2t2+∆Heff\nk\n2+ΣHeff\nkΣMt\n2∆Mt+O(1\nJ0)6\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s50/s45/s49/s48/s49/s50/s51/s45/s49 /s48 /s49 /s50\n/s48/s50/s53/s53/s48/s55/s53\n/s45/s50/s48/s50/s45/s50 /s45/s49 /s48 /s49 /s50\n/s45/s50/s48/s50\n/s48/s50/s53/s53/s48/s55/s53\n/s80/s60/s48/s32/s115/s116/s97/s116/s101/s65/s80/s32/s115/s116/s97/s116/s101/s80/s62/s48/s32/s115/s116/s97/s116/s101/s32\n/s32/s115/s119/s101/s101/s112/s70/s105/s101/s108/s100/s32/s40/s84/s41\n/s73/s110/s116/s101/s114/s108/s97/s121/s101/s114/s32/s101/s120/s99/s104/s97/s110/s103/s101/s32/s99/s111/s117/s112/s108/s105/s110/s103/s32 /s40/s109/s74/s47/s109/s50\n/s41/s78/s67/s83/s65/s80/s70/s105/s101/s108/s100/s32/s32/s40/s84/s41\n/s78/s67/s83/s80/s32/s62/s32/s48/s32\n/s32\n/s32/s80/s60/s48/s32\n/s74/s61/s32/s45/s49/s32/s109/s74/s47/s109/s50/s70/s105/s101/s108/s100/s32/s40/s84/s41\n/s32/s77\n/s122/s76/s65/s89/s69/s82/s32/s49/s32/s119/s105/s116/s104/s32/s73/s77/s65\n/s32\n/s32/s77\n/s122\n/s74/s61/s32/s51/s32/s109/s74/s47/s109/s50\n/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41\nFIG. 4. Loops and eigeneiexcitation frequencies when the so ft layer\nis decreased and yields a negative effective anisotropy ( Hk1= 600\nkA/mand MS1=1MA/m)fortwovaluesoftheinterlayerexchange\ncoupling. Bottom panel: state diagram when the out-of-plan e field\nis swept down after a start at Hz>0. The notations P >0, P<0\nAPand NCSstand for parallelwithpositive fullremanence, p arallel\nwith negative full remanence, antiparallel and non colline ar states.\nThe criticalfieldsare Eq.2(green andredlines)and Eq.7(bl ue and\nmagenta lines). These linesmerge inthepoint definedbyEq.8 . The\nblack lineis a numerical resolution.\nwhereδiseither0or1fortherealandvirtualswitchingfields.\nThe convergenceof the expressionat large Jis slow and one\nshould better use Eq. 7for the asymmetries and the coupling\nvaluesencounteredin practice.\nd. Non identical layers, limit of weak coupling. In the\nlimit of weakly exchanged, or nearly symmetric systems, the\nsofteningfieldsreduceto:\nHAP↔NCS\nC,1/negationslash=2J<<1≈Heff\nk1−J0\nMs1t1(9)\nwhichisthefieldatwhichtheAPstatewith mz1=−1looses\nstabilityuponincreasingfield,and\nHAP→P<0\nv,1/negationslash=2J0<<1≈ −Heff\nk2+J0\nMs2t2(10)\nwhich is the field leading to a switching of the hard layer\n(i= 2) towards the P <0state. These two above expres-\nsions recall the correspond ones in Eq. 5for the instability\nof the P>0case. They illustrate that for weak Jor nearly\nsymmetricsystemsa minorloopof the soft layer( i= 1)per-\nformed between the P >0and the AP states has an opening\nof2Heff\nk1and an offset of −J0\nMs1t1. As it was the case in the\nP states, such ways of measuring Jrequires the prior knowl-\nedge the layers’ magneticpropertiesto be certain to be in th e\nweaklyexchangednearlysymmetricregime./s45/s50 /s45/s49 /s48 /s49 /s50/s48/s52/s48/s56/s48/s49/s69/s45/s52/s48/s46/s48/s50/s48/s46/s48/s52\n/s78/s67/s83/s65/s80\n/s80/s62/s48/s70/s77/s82/s32/s40/s71/s72/s122/s41\n/s70/s105/s101/s108/s100/s32/s40/s84/s41/s80/s60/s48/s108/s97/s121/s101/s114/s32/s50/s32/s109/s111/s100/s101\n/s108/s97/s121/s101/s114/s32/s50/s32/s109/s111/s100/s101\n/s40/s98/s41/s49/s61/s32/s48/s46/s48/s49/s44/s32\n/s50/s61/s32/s48/s46/s48/s53/s45/s73/s109/s40 /s41/s47/s82/s101/s40 /s41\n/s49/s61/s32/s48/s46/s48/s49/s44/s32\n/s50/s32/s61/s32/s48/s80/s60/s48/s65/s80/s80/s62/s48\n/s40/s97/s41/s108/s97/s121/s101/s114/s32/s49/s32/s109/s111/s100/s101/s115\nFIG. 5. Eigenexcitation relative linewidth (a) and frequen cies (b)\nversus field curves for samples whose properties are: MS1= 106\nA/m,MS2= 8×105A/m,Hk1= 106A/m,Hk2= 2×106\nA/m,t1= 2nm,t2= 5nm,α1= 0.01andα2= 0(red curves)\nand0.05(greencuves) foranantiferromagnetic couplingof J=−1\nmJ/m2. When in the NCS state, the eigenmodes have a finite ellip-\nticity, which correlates with a substantial increase to the acoustical\nmode linewidth.\nV. EFFECTOFEXCHANGECOUPLING ON THE\nEIGENMODELINEWIDTHS\nIn this last section, we study how the damping of a given\nlayeraffectsthelinewidthofthetwoeigenmodesthankstot he\ninterlayerexchangecoupling J. Inthenumericalevaluations,\nwe useα1= 0.01(mimickingFeCoB18) anda variable α2=\n0−0.1to mimic the effect of the harder layer. As in the\nisolatedlayercase, introducingreasonablevaluesof damp ing\ndoesnotchangetheeigenmodefrequenciesandconsequently\nthecriticalfieldsat whichthe modessoften(Fig. 5).\nHowever, as soon as the layers are coupled, each eigen-\nmodeinvolvesmagnetizationmotionin bothlayers,such that\nthe damping of bothlayers matters to set the linewidth of a\ngiven eigenmode19even at low coupling when the mode can\nbeconsideredtobelongpreferentiallytooneofthelayers. We\nhave illustrated in Fig. 6(a) how the interlayer exchange cou-\npling affects the linewidths of the two modes in a situation\nwith ferromagnetic coupling. Before setting the coupling o f\ncoupling,therelativelinewidths−ℑ(ωi)\nRe(ωi)ofthe twomodesare\nequaltothe αiasexpected. Thechangeofeigenmoderelative\nlinewidth is not proportionalto the coupling. Tiny couplin gs\n(J <<0.5mJ/m2) do not affectmuch the relativelinewidths\nthat appearsto be quadraticwith low J[Fig.6(a)]. This con-\ntrasts with the eigenmode frequencies [Fig. 6(b)] that evolve\nlinearly with Jat low coupling (Eq. 5and Eq.6). At large\ncouplingthe linewidthofthe broadestlinewidthmodeis sub -\nstantially reduced while the linewidth of the narrowest mod e\nis substantially enlarged [Fig. 6(a)]; this continued linewidth\nbroadeningoftheacousticalmodegoesondespitethefactth at\nthe frequencyofthismodedoesonlyevolvein an asymptotic\nway(seeFig. 6(b),greencurve).7\n/s48 /s50 /s52 /s54/s48/s52/s48/s56/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s76/s97/s121/s101/s114/s32/s50\n/s76/s97/s121/s101/s114/s32/s50/s32\n/s32/s70/s77/s82/s32/s102/s114/s101/s113/s115/s46/s32/s40/s71/s72/s122/s41\n/s73/s110/s116/s101/s114/s108/s97/s121/s101/s114/s32/s101/s120/s99/s104/s97/s110/s103/s101/s32/s99/s111/s117/s112/s108/s105/s110/s103/s32/s40/s109/s74/s47/s109/s50\n/s41/s76/s97/s121/s101/s114/s32/s50\n/s104/s97/s114/s100\n/s76/s97/s121/s101/s114/s32/s49/s40/s97/s41\n/s49/s61/s48/s46/s48/s49\n/s40/s98/s41\n/s32/s32/s45/s73/s109/s40 /s41/s32/s47/s32/s82/s101/s32/s40 /s41\n/s67/s111/s117/s112/s108/s101/s114/s50/s61/s48/s46/s49\nFIG. 6. Eigenexcitation relative linewidths (a) and freque ncies (b)\nversus ferromagnetic interlayer exchange coupling at zero applied\nfield. In panel (b) the weak coupling limit is described by Eq. 5 and\nEq.6,while the highcoupling limitcorresponds toEq.3and E q.4.\n/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50/s48/s46/s48/s49/s52/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s40/s98/s41/s45/s49/s32/s109/s74/s47/s109/s50\n/s44/s32/s65/s80\n/s45/s49/s32/s109/s74/s47/s109/s50\n/s44/s32/s80\n/s49/s32/s109/s74/s47/s109/s50\n/s44/s32/s80\n/s68/s97/s109/s112/s105/s110/s103/s32/s111/s102/s32/s108/s97/s121/s101/s114/s32/s50/s97/s99/s111/s117/s115/s116/s105/s99/s97/s108/s32/s109/s111/s100/s101\n/s40/s48/s46/s48/s49/s44/s32/s48/s46/s48/s49/s41/s49/s46/s48/s50/s55/s32\n/s50\n/s111/s112/s116/s105/s99/s97/s108/s32/s109/s111/s100/s101/s48/s46/s57/s51/s57/s32\n/s50/s115/s108/s111/s112/s101/s115/s58\n/s48/s46/s57/s54/s53/s32\n/s50/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s32/s32/s91/s45/s32/s73/s109/s32/s40 /s41/s32/s47/s32/s82/s101/s40 /s41/s32/s93\n/s48/s46/s48/s50/s55/s32\n/s50\n/s48/s46/s48/s54/s54/s32\n/s50/s115/s108/s111/s112/s101/s115/s58\n/s48/s46/s48/s51/s53/s32\n/s50/s40/s97/s41\nFIG.7. RelativeLinewidthoftheoptical(a)andacoustical (b)eigen-\nmode versus the damping parameter of the second layer. The in ter-\nlayerexchange isvariedfromferromagnetic(greencurve)t oantifer-\nromagnetic (blue curve inthe APremanent state) and black cu rve in\nthePstateatsaturation( Hz= 2.5MA/m. Theredarrowsrecallthat\ninthePstatetherelativelinewidthareequaltothedamping provided\nα1=α2.\nLetusseetowhatextentwecanmanipulatethelinewidthof\ntheopticalandacousticaleigenmodeswhenplayingwithonl y\none damping parameter, for instance α2(Fig.7). It appears\nthat the relative linewidths are quasi affine functions of th e\ndamping parameters when in the collinear P and AP states.\nIn the P state situations, an equality of the damping in the\ntwo layers leads a strict equality of the linewidths with the\ndamping for the two modes, as stressed by the red arrows in\nFig.7. Comparatively, the AP states have a larger linewidth,\nlargerthatthedampingofcommontothetwo layers.Indeed the impact of the damping factors onto the\nlinewidths depends significantly on the magnetization ar-\nrangement (P, AP or NCS) [Fig. 5(a)]. This is because the\namplitudes of dynamical magnetization in each layer depend\nontheirarrangements. When α2>> α1thelargestinfluence\nofα2on the acoustical mode linewidth is obtained for ferro-\nmagnetic coupling in the P state [green curve in Fig. 7(b)].\nGenerally the linewidths of the eigenmodes in the AP states\naredifferentfromtheonesoftheP state.\nSuch a configuration-dependent linewidth has been found\nin the past in various in-plane magnetized systems includ-\ning Fe/Au/YIG bilayers20and all-metallic spin-valves21,22; in\nthese weakly exchange-coupled systems, the configuration\ndependenceofthelinewidthwasattributedtoaconfiguratio n-\ndependentcontributionofspinpumpingcoupling23,24,butour\nresultssuggest that it couldalso be partly attributedto th e in-\nterlayerexchangecoupling,inlinewiththeconclusionsdr awn\nfor in-plane magnetized layers in ref. 19. A way to discrimi-\nnatebetweenspin-pumping-inducedandinterlayer-exchan ge-\ninducedconfiguration-dependenceof the linewidth is that t he\nfirst one is substantially enhanced when near a crossing of\nthe acoustical and optical eigenexcitation frequencies, w hile\nthe second one has a broadband impact i.e. it affects the\nlinewidths irrespective of the frequencydistance between the\ntwoeigenmodes.\nOur findings have consequences for the metrology of the\ninterlayer exchange coupling in the ferromagnetic case. In\nthelowcouplinglimit,theeigenmodefrequenciesdependli n-\nearlyonthecoupling,suchthatusingtheeigenmodefrequen -\nciesisbyfarthebestwaytodeducethecouplingstrength. Th e\nsituation is opposite at large couplings where one would lik e\ntousethehighestfrequencymode. Unfortunatelyinmanyex-\nperimentssofarthelimitedsensitivityrendersthedetect ionof\ntheopticalmodechallengingwhiletheacousticalmodecanb e\ncharacterized in greater detail25. The asymptotic dependence\n(see Fig.6(b), greencurve)of the frequencyof the acoustical\nmode at large coupling makes it ineffective to deduce more\nthan a lower boundfor J. An alternativemethod is to benefit\nfromthepronouncedchangeoftheeigenmodelinewidthwith\nthe couplingat large J: the linewidth of the acoustical mode\ncan be used to quantify the coupling provided the damping\nparametersareknown.\nBesides, our findings have also consequences for the ma-\nterial engineering in STT-MRAM technologies, where stable\nfixedlayersareneededforproperdeviceoperation. Here”st a-\nble” means both stable against thermal fluctuations – imply-\ning largeeigenmodefrequencyat remanence(or equivalentl y\nlarge coercivity) – and stable against spin-torque, which r e-\nquires in addition the largest possible linewidth to preven t\nauto-oscillationor switching. The typical situation3–11is that\nlayer 1 is an FeCoB-based spin polarizinglayer ( i= 1) of an\nMTJ.Thisnominallyfixedlayerneedstobefurtherstabilize d,\nwhich is usually3–5,7?–11done by coupling it ferromagneti-\ncallythroughTa26withharderreferencelayers( i= 2)thatare\nusually made of thicker and heavily damped high anisotropy\nmaterials, i.e. α2>> α 1. From Fig. 6(b) we see that the\nthermal stability of the layer 1 can be increased through the\ncoupling,but the zero field frequencyof its acoustical exci ta-8\ntion is subject to saturation. Fortunately it is possible to ob-\ntain a further substantial increase of its stability agains t spin-\ntorque (i.e. further increase of the effective damping) by i n-\ncreasing further the coupling (see Fig. 6(a), green curve). As\na result,alargerstabilizingeffectcanbeobtainedif thei nter-\nlayer exchange coupling is further increased despite the fa ct\nthat the frequency of the acoustical mode converges asymp-\ntotically(Eq. 3)to ahigherlimit.\nVI. SUMMARYAND CONCLUSION\nInthecase ofstrongferromagneticcouplingbetweenmag-\nnetic layers with perpendicular anisotropy, the measureme nt\nof the strength of the coupling can not rely on the measure-\nmentofthecoercivities,astheyareusuallyextrinsicando nly\nweakly depend on the strength of the coupling at large cou-\npling. Instead, we propose to use the frequency of the fer-\nromagnetic resonance modes along the hysteresis loops, be-\ncause their frequencies are always indicative of the intrin sicproperties. One can use the fact that the optical mode has a\nfrequencythat is stronglysensitive to the strength of the c ou-\npling or the fact that the acoustical mode linewidth can also\nbe largelydependenton the coupling. We have givenanalyti-\ncalexpressionsderivedinlimitcasestoanalyzetheferrom ag-\nneticresonancefrequencies,theirlinewidthandtheirmea ning\nfor a soft/hard perpendicular magnetized composite. Our re -\nsults can be used to derive the minimum interlayer exchange\ncoupling needed for the proper pinning of the softest of the\ntwolayers,andtooptimizetheirstabilityagainstspintra nsfer\ntorques.\nACKNOWLEDGMENT\nI acknowledge support from the Samsung Global MRAM\nInnovationprogramespeciallyV. Nikitin, whoprovidedsam -\nplesonwhichwecouldtestandvalidatetheproposedmethod.\nDiscussionswithJoo-VonKimcontributedtoclarifysomeas -\npectsrelatedtoeigenmodelinewidth.\n∗thibaut.devolder@u-psud.fr\n1S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan,\nM. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno,\nNature Materials 9, 721 (2010) .\n2A. V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R. S .\nBeach, A. Ong, X. Tang, A. Driskill-Smith, W. H. Butler, P. B.\nVisscher, D. Lottis, E. 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E -mail address: †cljia@lzu.edu.cn, *jiangchj@lzu.edu.cn  \n \nABSTRACT  \nWe report unambiguous ly experimental evidence of a strong orbital current in Nb films \nwith weak spin -orbit coupling via the spin-torque ferromagnetic resonance (ST -FMR)  \nspectrum for Fe/Nb and Co/Nb bilayers.  The sign change of the damping -like torque in \nCo/Nb demonstrates a large spin -orbit correlation  and thus great efficiency of orbital \ntorque in Co/Nb.  By studying the efficiency as a function of the thickness of Nb \nsublayer, we reveal a long orbital diffusion length (~ 3.1 nm)  of Nb.  Further planar Hall \nresistance (PHE) measurements at positive and negative apply ing current  confirm the \nnonlocal orbital transport in ferromagnetic -metal/Nb heterostructures.  \n \n \n \n \n \n \n \n \n  Spin-related  torque s induc ed by spin current s have drawn considerable interest  in \nspintronic  over the decade  [1-3]. Especially, spin Hall effect (SHE)  prevail s in heavy \nmetal (HM) with strong spin -orbit coupling , where the resulting spin -orbit torque (SOT) \nhas served as an efficient manipulation of magnetization in HM/ferromagnet (HM/FM)  \nheter ostructures  [1, 4-7]. Recently, orbital current based on the flowing  orbital angular \nmomentum  (OAM)  [8] via orbital Hall effect (OHE) receive s quite attention  [9-12]. \nLike SHE, orbital current is generated along the perpendicular direction to the charge \ncurrent. However, there is distinctive  features  of OHE compared  to SHE. Theoretically, \nthe OHE roots in orbital textures  in momentum -space , which means  that the OHE \nallows for the absence of  spin-orbit coupling (SOC)  due to the direct action of charge \ncurrent on orbital degree of freedom  [13], where charge current directly acts on spin  \ndegree of freedom  through strong SOC that results in  SHE.  Moreover , orbital Hall \nconductivity  (OHC)  is much larger than  spin Hall conductivity  (SHC)  in many \ntransition metals , which could be estimated that the orbital torque (OT) efficiency can \nbe comparable to or larger than spin torque efficiency  [10]. Thus, the OT promote s the \nbirth of orbitronics  which holds great  potential for future highly efficient magnetic  \ndevices and a  complement to spintronics .  \nMany alternative materials can be regarded as the orbital source  to induce OT , and \ntheoretically  found  in multiorbital centrosymmetric systems  such as transition metals  \n[10], graphene  [14], semiconductors  [11] as well as two-dimensional  transition metal \ndichalcogenides  [15-17]. Experimentally, the OT was investigated in various systems, \nespecially in thulium iron garnet TmIG/Pt/CuO x [18], FM/Cu/Al 2O3 [19], Py/CuO x [20-\n21], etc. which all originates from the orbital Rashba effect (ORE) and modulates OT \nvia layer design  [22]. In these systems, Pt or oxidized Cu was utilized as a n orbital \nsource , it undoubtedly complicate s the discussion of OT.  Moreover, the intrinsic OHE \nwas observed in Ta/Ni by evaluating the sign of damping -like torque similar to the sign \nof OHC in Ta  [13]. However, the relatively larger SHC for Ta leads to slightly smaller \nOT efficiency in Ta/Ni system. Thus, materials selection without strong SOC for large \nOT efficiency is extremely necessary.  The transition metal  niobium (Nb) is \ncharacterized by weak SOC as well as large enough OHC  (𝜎𝑂𝐻𝑁𝑏) relative to SHC  (𝜎𝑆𝐻𝑁𝑏) and opposite sign of them  [10,], which implies highly possible to observe obvious OHE  \nvia sign relation  and generate expected  OT acting on magnetization of FM  [23]. \nMoreover, the large enough OHC is expected to be accompanied by a large OT \nefficiency .  \nHerein , we present experimental evidence confirm ing the existence of OHE in \nFM/Nb bilayers  by the sign relation of damping -like torque according with OHC  (or \nSHC)  in Nb via spin -torque ferromagnetic resonance (ST -FMR) . After fitting the FM \ndependence of ST -FMR spectra, the variety between the signs of damping -like torque  \nfor Fe/Nb and Co/Nb  bilayers unambiguous ly proves the emergence of orbital torque. \nFurthermore, we support our conclusion that the sign of damping -like torque is \nexamined by utilizing the method of planar Hall resistance  (PHE)  at positive and \nnegative measuring current, which result coincides wi th ST -FMR and the theoretical \ncalculations.  \nGenerally, it is difficult to disentangle  OT and ST due to identical properties. Thus, \nthe total torque usually is ascribed to the combined effect  of ST and OT,  which includes \nfollowing three cases: The first  case corresponds to the  same sign of 𝜎𝑆𝐻 and 𝜎𝑂𝐻 as \nshown in Figure 1(a). The total torque is enhanced by a synergistic effect of  the OT and \nST. In the second  case, the sign of 𝜎𝑆𝐻 varies from 𝜎𝑂𝐻 as shown in the top of Figure \n1(b), while the magnitude of spin Hall contribution ( ASH) is larger than orbital Hall \ncontribution ( AOH), the total torque is dominated by ST. In the final case, the sign of \n𝜎𝑆𝐻 varies from 𝜎𝑂𝐻 in the bottom of Figure 1(b), and the ASH is smaller than AOH that \nprovides a great chance to disentangl e the OT from the ST . Given all this, the 4d \ntransition metal Nb is  an excellent candidate  to observe OT  where 𝜎𝑂𝐻 is opposite sign \nand much larger than 𝜎𝑆𝐻 [10]. Specifically speaking , as shown in Figure 1(c), the \norbital current and spin current are generated by OHE and SHE and injected into  FM, \nrespectively. Next the injected orbital current  is converted to spin via SOC of the FM.  \nA crucial  conversion ratio 𝜂𝐹𝑀 (spin -orbit correlation) [ 24] is used to describe  how \nmuch spin in FM is induced by the  orbital current injected from the Nb.  For most FMs, \nthe sign of 𝜂𝐹𝑀 is positive, such as Fe, Co, Ni, etc. However, the small 𝜂𝐹𝑀 tends to \ninduce ST , while the large 𝜂𝐹𝑀 leads to OT , as demonstrated in Figure 1(c). Based on  the previous theoretical[13], we summary the total torque of FM/Nb systems in Figure \n1(d). Obviously, the Co/Nb bilayer  is a good candidate to clarify the OT, given that  the \nsign of the effective Hall conductivity ( 𝜎𝑆𝐻𝑁𝑏+𝜂𝐹𝑀𝜎𝑂𝐻𝑁𝑏) obtained from  torque coincides \nwith the sign of  𝜎𝑂𝐻𝑁𝑏 . The FM/Nb  thin films were deposited  on MgO substrate by \nmagnetron sputtering at room temperature.  In order to clarify the physical mechanism \nof OT in experiments, f our different types of bilayers were prepared: Fe(tFe)/Nb(6 nm), \nCo(tCo)/Nb(6 nm).  Fe(8 nm)/Nb( tNb) and Co(8 nm)/Nb( tNb). For comparison , the \nsamples Pt(6 nm)/Co( tCo) and Pt(6 nm)/Fe( tFe) were grown on MgO substrate  as well . \nMoreover, Fe(2 nm)/Nb(8 nm) and Co(2 nm)/Nb(8 nm) are prepared to conduct the \nPHE measurements.  \n To measure the OT efficiency, we exploit the spin -torque ferromagnetic resonance  \n(ST-FMR)  measurement  as shown in Figure 2(a), the ST -FMR technique is employed \nbecause it is a well-established  method to determine the charge -spin conversion \nefficiency  as well as  the type, number and efficiency of torques  [1, 25-28]. In this \nmeasurement,  a radio frequency (RF) microwave current is applied along the x -\ndirection , this RF current will induce the  resulting  torque and cause the FM \nmagnetization process, which contributes to the ST -FMR resonance signal. The typical \nST-FMR signal for Fe(8 nm)/Nb(6 nm) is shown in Figure 2(b), which could be fitted \nby symmetric  (Vs) and antisymmetric component  (Va). Generally, the Vs and Va \ncorresponds to different torques, where the Vs arises solely from the damping -like \ntorque ( 𝝉𝐃𝐋∝𝒎×(𝒚×𝒎)) and Va originates jointly from the  field-like torque ( 𝝉𝐅𝐋∝\n𝒎×𝒚   and current -induced Oersted field , respectively  [13]. Furthermore, the \nquantitative  conversion efficiency ξ can be determined by the ratio of Vs and Va based \non the following equation  [1, 13, 29-30]: \n                    𝜉=𝑉𝑠\n𝑉𝑎𝑒𝜇0𝑀𝑠𝑡𝐹𝑀𝑑𝑁𝑏\nℏ[1+(𝑀𝑠𝐻𝑟⁄ )]12⁄,              (1) \nwhere tFM and dNb are the thickness of FM and Nb sublayer s, respectively.  Hr is the \nresonance field . Ms is the effective saturation magnetization  that can be obtained by \nKittel equation:  (2𝜋𝑓)/𝛾=√[𝐻𝑟(𝐻𝑟+𝑀𝑠)] with γ being  the gyromagnetic ratio.   \nAnalogous to torque components of SOT, the 𝝉𝐃𝐋 and 𝝉𝐅𝐋 components of OT can be generated in adjacent magnetic layer  and separately determined  via ST -FMR. In the \npresent study, we are more interested in  the sign of 𝝉𝐃𝐋, which could reflect the sign of \nSHC and /or OHC. As shown in Figure 2(b), a negative  Vs is observed for Fe(8 nm)/Nb(6 \nnm), which mean s the existence of 𝝉𝐃𝐋. Obviously, the sign of Vs is consistent with the \nexpected sign of 𝜎𝑆𝐻𝑁𝑏 and opposite t o the 𝜎𝑂𝐻𝑁𝑏. The sign of 𝝉𝐃𝐋 for Fe/Nb could be \nfurther determined  by the FM layer thickness dependence of the ST-FMR resonance \nsignal Vdc [24] as shown in Figure 2( c). The experimentally measured conversion \nefficiency ξ as a function of tFM provides a route to separately evaluate the damping -\nlike ( 𝜉𝐷𝐿) and field-like (𝜉𝐹𝐿) torque efficiencies as  [13, 20 ] \n                          1\n𝜉=1\n𝜉𝐷𝐿(1+ℏ\n𝑒𝜉𝐹𝐿\n𝜇0𝑀𝑠𝑡𝐹𝑀𝑑𝑁𝑏)                   (2) \nwhere the intercept implies the sign of 𝜉𝐷𝐿. As shown in Figure 2(c), the sign of the \nintercept is in accord with the symmetric component  Vs in Figure 2(b). Hence , it is \napparent that the sign of 𝜉𝐷𝐿  of Fe/Nb bilayer is consistent with the sign of 𝜎𝑆𝐻𝑁𝑏 . \nHowever, we have a sign reversal of the symmetric component  Vs in ST-FMR signal \nfor Co(8 nm)/Nb(6 nm) , i.e., a positive Vs as shown in Figure 2( d). Consequently , the \nFM layer thickness dependence of efficiency for Co/Nb bilayers  [24] gives a positive \n𝜉𝐷𝐿  as plotted in Figure 2(e) , which coincides with  the expected  sign of 𝜎𝑂𝐻𝑁𝑏  but \nopposite to the sign of 𝜎𝑆𝐻𝑁𝑏. On the other hand , for the Pt/Fe and Pt/Co control sample \nas shown in Figures 3, the heavy metal Pt, is acknowledged as a strong spin source \nmaterial [1]. From the experimental result, the positive sign of Vs is consistent with the \nsign of 𝜎𝑆𝐻𝑃𝑡 , and further accords w ith the sign of 𝜉𝐷𝐿 . According to schematic \nillustration for the mechanism of the orbital torque  in Figure 1( c) and  theoretical \ncalculations  [24], the to tal torque in Co/Nb includes conventional SOT arising from \nspin Hall contribution ( 𝜎𝑆𝐻𝑁𝑏 ) and OT attributing to orbital Hall contribution ( 𝜎𝑂𝐻𝑁𝑏 ). \nWhen the spin Hall contribution is much less than orbital Hall contribution, the sign of \nthe total torque is same  as the sign of 𝜎𝑂𝐻𝑁𝑏. In this case, as for Co/Nb bilayer,  the sign \nof OT depends on the 𝜎𝑂𝐻𝑁𝑏  and spin -orbit correlation ˂L·S˃Co [13]. The fact from \ntheoretical calculation is that the orbital Hall effect contribution is larger than the spin \nHall one in magnitude  of Co/Nb , the sign of total torque is in accord with the expected sign of 𝜎𝑂𝐻𝑁𝑏, i.e., the orbital torque .  \nFurthermore, to verify the source of OT, we perform the ST -FMR measurement \nfor Co(8 nm)/Cu(t)/Nb(6 nm) , Figure S4 shows the typical ST -FMR spectra. The \npositive Vs means the existence of OT even if inserting the Cu layer. Cu layer thickness \ndependence of ef ficiency ξ is plotted in Figure 4(a). Enough efficiency exclude s the \ninterface effect at Co/Nb interface and bulk OHE is responsible for the OT . Moreover, \nto obtain the orbital  diffusion length of Nb, a series of ST-FMR samples with different \nNb thickness es of Co (and Fe, t = 8 nm) /Nb(d = 6-15 nm) was fabricated, and the \nconversion efficiency  ξ was characterized by Eq. (1) and summarized in Figure 3(a), \nthe Nb thickness dependence of ξ match with 𝜉=𝜉∞[1−sech  (𝑑/𝜆𝑠)] [31], where  ξ \nis the measured conversion efficiency at different Nb thickness and ξꝏ is the conversion \nefficiency at infinite Nb thickness. From the fitting, λs of Nb is quantitatively \ndetermined to be 3.1 nm. This point sets it apart from conventional heavy metal, such \nas Pt, where spin Hall contribution dominates and shows a slightly small spin diffusion \nlength (~ 1.5 nm) [31].  \nFurthermore, to verify the sign relation of damping -like torque in FM/Nb bila yers, \nwe perform the method of PHE at positive and negative measuring current, which has \nbeen regarded as a reliable and effective method to measure charge -to-spin conversion \n[32-34]. In this measurement, the damping -like torque induces the out -of-plane \neffective field Hp, while the field -like torque induces the in -plane effective field HT as \nshown in Figure 4(a). Here we are just concerned with the sign of Hp. Figure 4(b) and \n(c) exhibits the typical magnetic angle dependence of planar Hall resistance RH as well \nas the corresponding RDH curve at I = ± 0.9 mA  for Fe(2 nm)/N b (8 nm) , and I = ± 1.0 \nmA for Co (2 nm)/Nb(8 nm), respectively  (see Supplemental Material for other currents  \n[24]). Note  that the effective fields Hp and HT can be extracted from the resistance \ndifference RDH and charactered by following equation [ 32-34]: \n   𝑅𝐷𝐻(𝐼,𝜑)=2𝑅𝐻(𝐻𝑇+𝐻𝑂𝑒)\n𝐻𝑒𝑥𝑡(𝑐𝑜𝑠𝜑 +𝑐𝑜𝑠3𝜑)+2𝑑𝑅𝐴𝐻𝐸\n𝑑𝐻𝑝𝑒𝑟𝑝𝐻𝑃𝑐𝑜𝑠𝜑 +𝐶   (3) \nwhere C is the resistance offset, HOe is the Oersted effective field, φ is defined in Figure \n4(a), Hperp is the applied out -of-plane magnetic field, 𝑑𝑅𝐴𝐻𝐸\n𝑑𝐻𝑝𝑒𝑟𝑝 is the slope of  RAHE vs Hperp [24]. After determining relevant parameters in Eq. (3), the out -of-plane effective \nfield Hp is obtained by fitting the φ dependence of RDH. Figure 4(d) shows the linear \nfitting Hp relative to the current  for Fe/Nb and Co/Nb , respectively . The sign of effective \nconversion efficiency can be quantitatively determined by [ 32]:𝜉~𝐻𝑝\n𝐽𝑁𝑀, where JNM is \nthe density of current. Obviously, the sign of 𝜉  depends on the slope of Hp versus \ncurrent I. Any difference in sign of slope for Hp vs I between Fe/Nb and Co/Nb would \nsuggest an existence of orbital torque, which is consistent with ST -FMR result and the \ntheoretical calculations.  \n In conclusion, the generation of orbital current and orbital torque is experimentally \nconfirmed based on orbital Hall effect present in FM/Nb systems measured by spin -\ntorque ferromagnetic resonance. After fitting the FM thicknesses dependence of the ST -\nFMR reso nance sign al Vdc, we characterize the sign of 𝜉𝐷𝐿 for Fe/Nb and Co/Nb is \npositive, which is  consistent with the spin Hall conductivity and  orbital Hall \nconductivity  of Nb from theoretical calculations , respectively . This result can be further \nverif ied by the method of planar Hall resistance . This strategy of generating orbital \ncurrent and orbital torque by experimental method provide a route of enabling \napplications in spintronic and orbitronic  devices  by orbital engineering.  \n  REFERENCES  \n[1] L. Q. 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The resulting torque exerted by t his spin exerts and acts on the magnetization of FM, which \nis referred to as orbital torque (OT). (d) Theoretical calculation for  𝜎𝑆𝐻𝑁𝑏+𝜂𝐹𝑀𝜎𝑂𝐻𝑁𝑏 of FM/Nb \nbilayers ( FM = Fe and Co ). Where 𝜎𝑆𝐻𝑁𝑏, 𝜎𝑂𝐻𝑁𝑏 and 𝜂𝐹𝑀 are spin Hall conductivity, orbital Hall \nconductivity of Nb and orbital -to-spin conversion efficiency of FM, respectively.  \n  \n-80-60-40-20020\n+5\nCo/NbNb\nSH + FMNb\nOH(ћ/e -1cm-1)\nFe/Nb-68(c) (d)(a) (b) \nFigure 2. (a) Schematic of the circuit for ST -FMR measurements.  ST-FMR voltage signal Vdc for \n(b) MgO/Fe (8 nm)/Nb(6 nm)  at 8 GHz , (d) MgO/Co (8 nm)/Nb(6 nm)  at 7 GHz . The inverse of the \nconversion efficiency 1/ ξ as a function of 1/ tFM for (c) Fe/Nb bilayer  and (e) Co/Nb bilayer. The \nsolid line s are the linear fit to the experimental data (grids) . \n0.00 0.05 0.10 0.15 0.20-6061218\nDL = - 0.22 Fe/Nb\n Linear1/\n1/t Fe (nm-1)\n0 200 400 600 800-24-120122436\n Data   Fitting     Va    Vs\n  Vdc (V)\nH (Oe)7 GHzCo(8nm)/Nb(6nm)\n0 300 600 900-2-1012\n Data   Fitting     Va    Vs\n  Vdc (V)\nH (Oe)8 GHzFe(8nm)/Nb(6nm)(b) (c)\n(d) (e)(a)\n0.00 0.07 0.14 0.21 0.28-12-606\nDL = + 0.16\n Co/Nb\n Linear1/\n1/t Co (nm-1) \nFigure 3. ST-FMR voltage signal Vdc for (a) MgO/Pt (6 nm)/Fe(8 nm)  at 7 GHz , (c) MgO/Pt (6 \nnm)/Co(8 nm)  at 7 GHz . The inverse of the conversion efficiency 1/ ξ as a function of 1/ tFM for (b) \nPt/Fe bilayer and (d) Pt/Co bilayer. The girds are the experimental data,  and the solid line is the \nlinear fit to the data.  \n0.00 0.07 0.14 0.21 0.28061218\nDL = + 0.06\n Pt/Co\n Linear1/\n1/t Co (nm-1)\n0.00 0.07 0.14 0.21 0.28071421DL = + 0.05\n Pt/Fe\n Linear1/\n1/t Fe (nm-1)\n0 300 600 900-12-6061218\n Data   Fitting     Va    Vs\n  Vdc (V)\nH (Oe)7 GHzPt(6nm)/Fe(6nm)\n0 300 600 900 1200-10-5051015  Data   Fitting     Va    Vs\n  Vdc (V)\nH (Oe)7 GHzPt(6nm)/Co(6nm)(a) (b)\n(c) (d) \nFigure 4. (a) Cu-layer thickness tCu dependence of efficiency for Co(8 nm)/Cu( tCu)/Nb(6 nm). (b) \nThe conversion efficiency ξ (orange squares) as a function of Nb thickness for Co (8 nm)/Nb(t) as \nwell as Fe  (8 nm)/Nb(t)  systems and a fit (solid orange line).  \n  \n0 1 2 3 4 50.000.060.120.180.240.30\n  \ntCu (nm)Co(8nm)/Cu (t\nCu)/Nb(6nm)\n0 4 8 12 16-0.26-0.130.000.130.26\n Co/Nb\n Fitting Fe/Nb\n Fitting\ntNb(nm)s= 3.1nm(b) (a) \nFigure 5. (a) Current -induced effective fields and schematic illustration of the planar Hall resistance  \nmeasurement srtup.  The external magnetic field angle φ dependence of planar Hall resistance RH of \n(b) Fe(2 nm)/Nb(8 nm) and ( c) Co(2 nm)/Nb(8 nm). The right axis  of (b) and ( c) represents the \ndifference of the Hall resistances RDH at positive and negative currents.  (d) The variety of Hp with \nthe applied current for Fe(2 nm)/Nb(8 nm) and Co(2 nm)/Nb(8 nm) . \n(a)\n0 90 180 270 360-0.2-0.10.00.10.2\nRDH ()\n (°)  +1 mA     -1 mA    DifferenceRH ()\n-0.18-0.090.000.090.18\nCo/Nb\n0 90 180 270 360-0.2-0.10.00.10.2 RH ()\n (°) Fe/Nb\n-0.14-0.070.000.070.14\n RDH () +0.9 mA      -0.9 mA  Difference\n0.0 0.3 0.6 0.9-4004080\n Co/Nb\n Fe/NbHP (Oe)\nCurrent (mA) (b)\n(c) (d)\nHP\nτDLτFL\nHTz\nyx\ny\nx zφIHext\nV"    },    {        "title": "2001.09581v1.Acoustic_ferromagnetic_resonance_and_spin_pumping_induced_by_surface_acoustic_waves.pdf",        "content": "Acoustic ferromagnetic resonance and spin pumping induced by surface acoustic waves\nJorge Puebla,1,\u0003Mingran Xu,1, 2Bivas Rana,1Kei Yamamoto,1, 3Sadamichi Maekawa,1, 3, 4and Yoshichika Otani1, 2,y\n1CEMS, RIKEN, Saitama, 351-0198, Japan\n2Institute for Solid State Physics, University of Tokyo,\n5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8581, Japan\n3Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n4Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100049, Peoples Republic of China\n(Dated: January 28, 2020)\nV oltage induced magnetization dynamics of magnetic thin films is a valuable tool to study anisotropic fields,\nexchange couplings, magnetization damping and spin pumping mechanism. A particularly well established\ntechnique is the ferromagnetic resonance (FMR) generated by the coupling of microwave photons and magneti-\nzation eigenmodes in the GHz range. Here we review the basic concepts of the so-called acoustic ferromagnetic\nresonance technique (a-FMR) induced by the coupling of surface acoustic waves (SAW) and magnetization of\nthin films. Interestingly, additional to the benefits of the microwave excited FMR technique, the coupling be-\ntween SAW and magnetization also offers fertile ground to study magnon-phonon and spin rotation couplings.\nWe describe the in-plane magnetic field angle dependence of the a-FMR by measuring the absorption / trans-\nmission of SAW and the attenuation of SAW in the presence of rotational motion of the lattice, and show the\nconsequent generation of spin current by acoustic spin pumping.\nPACS numbers:\nI. INTRODUCTION\nArguably, one of the most enlightening works by the late\nphysicist Charles Kittel1is the theoretical description of the\nferromagnetic resonance absorption, published first in 1947\n[1] and extended to shape anistropies in 1948 [2]. At these\nearly works, Kittel described the magnetization dynamics ex-\nerted in a ferromagnetic specimen subjected to a strong dc\nfieldHzand a weak perpendicular microwave field Hx, such\nthat the magnetization dynamics can be well described by\n@M=@t=\r[M\u0002H]; where\ris the gyromagnetic ratio, M\nthe magnetization and His the external field with components\n(Hx;Hy;Hz). If the dc field is in-plane and strong enough to\nfully align the magnetization M, then the magnetization pre-\ncess as a single magnetic domain, a phenomenon we know\nnowadays as ferromagnetic resonance (FMR). The corre-\nsponding frequency at the resonance condition f0is described\nby the so-called Kittel formula f0=\r\n2\u0019p\nHz(Hz+\u00160jMj).\nNowadays, the FMR is a versatile tool that allows study-\ning magnetization dynamics in thin films [3], spin waves [4],\nmagnetization switching [5] and spin pumping [6]. Remark-\nably, 10 years after his description of FMR, it was the same\nCharles Kittel who first formulated the coupling of spin waves\n(magnons) and lattice vibrations (phonons) at resonance con-\nditions, giving origin to the acoustic excited FMR [7]. One\ninitial conclusion was that microwave phonons were neces-\nsary for reaching the resonance condition. Conveniently, mi-\ncrowave phonons can be generated by interdigital transduc-\ners fabricated on top of piezoelectric substrates. Additional\nto the previous description of applications of standard FMR\n\u0003Electronic address: jorgeluis.pueblanunez@riken.jp\nyElectronic address: yotani@issp.u-tokyo.ac.jp\n1Charles Kittel passed away the last 15th of May 2019 at the age of 102(microwave field excited), Kittel suggested that magnetic bulk\ncrystals may show nonreciprocal acoustic properties, and in-\nduce strong phonon attenuation.\nHere, we first overview the main characteristics of acoustic\nferromagnetic resonance (a-FMR) excited by GHz frequency\nsurface acoustic waves (SAW). Then, we review the descrip-\ntion of the magnetization coupling with elastic rotation and its\ndependence when varying in-plane magnetic field [8, 9]. Fur-\nthermore, we recalled an example of the generation of spin\ncurrent by a-FMR, the so-called acoustic spin pumping. We\nshow that the order of magnitude of the spin current density is\nof the same order of the more standard spin pumping mech-\nanism excited by microwave photons. Additional to the most\nrecent works of a-FMR [8, 9], we describe the less explored\ncoupling of magnetization with lattice rotation, first published\nmore than 40 years ago by one of the authors of the present re-\nview [10]. Finally, beyond the a-FMR study we conclude with\nan outlook for the coupling of SAW with magnetic and non-\nmagnetic materials for spintronic research and applications.\nII. ACOUSTIC FERROMAGNETIC RESONANCE\nA. Surface acoustic waves\nAs its name suggests, surface acoustic waves (SAW) are\nelastic waves that travel parallel to the surface of an elastic\nmaterial, with a decay perpendicular to the surface into the\nbulk with an approximated decay length equal to the acous-\ntic wavelength \u0015SAW . Figure 1(a) shows the schematics of\na two port interdigital transducer (IDT) device on a piezo-\nelectric substrate (LiNbO 3, Lithium Niobate) for generation\nof SAW, the wavelength of acoustic waves, \u0015SAW , depends\non the periodicity of IDTs as also shown in the schematics.\nThe generation of SAW is done by the inverse piezoelectric\neffect, where elastic deformation is produced by injection ofarXiv:2001.09581v1  [cond-mat.mes-hall]  27 Jan 20202\nrf-voltage. The reading of SAW is done by the direct piezo-\nelectric effect, converting elastic deformation back to voltage\nsignal. The SAW frequency is defined by fSAW =vs\n\u0015SAW,\nwherevsis the sound velocity of the piezoelectric material.\nFigure 1(b) shows a scanning electron microscope (SEM) im-\nage of IDTs with 400 nm width and \u0015SAW = 1:6\u0016m. The\nIDTs are patterned by electron beam lithography and made of\nTi(5nm)/Au(30nm) by the lift-off method. Since the sound ve-\nlocityvs= 4000 m/s in LiNbO 3, we expect the generation of\nSAW with an approximate frequency fSAW = 2:5GHz. Fig-\nure 1(c) shows the characterization of the scattering parame-\nter for transmission S 12by a vector network analyzer (VNA),\ngiving a SAW resonance frequency fSAW = 2:38GHz. The\nvoltage generation of SAW also induces generation of spuri-\nous electromagnetic waves (EMW); since the SAW and EMW\nvelocities are different, it is possible to filter out the EMW\nby a technique called time-gating that employs Fourier trans-\nform operations. IDTs act not only as the generator and re-\nceiver for acoustic phonons, but also as an antenna, receiving\nthe microwave signals through the air. And acoustic phonon\nis propagating in a speed of vs, which is in thousand meters\nper second level, while the velocity of EMW is approximately\n3\u0002108m/s. Hence, SAW signal arrive to the second IDT port\nafter the EMW. As an additional function of our VNA, we are\nable to analyze the signal in time-domain. In order to rule out\nthe noise brought by the EMW, we set a gate in time-domain,\nto take the signal which arrives in the interval expected ac-\ncording the the SAW velocity vsand the lenght of our device.\nAnd by using Fourier transform, we obtain frequency-domain\nsignal. The comparision of the signal with and without time\ngating is presented in figure 1(c).\nB. Magnetization coupling with elastic strain\nWhen SAWs propagate on a ferromagnetic layer it produces\na time variant strain tensor field ^\"(t)in the lattice, which cou-\nples to the local magnetic environment via the magnetoelas-\ntic effect; as a consequence a time varying magnetic field is\nexerted. Here, we describe how this time varying magnon-\nphonon coupling induces a-FMR.\nLet us start by describing first the dynamics exerted on the\nnormalized magnetization vector m=M=jMjwhen an ef-\nfective driving magnetic field \u00160He\u000bis present. Such dy-\nnamics are described by the Landau-Lifshitz-Gilbert (LLG)\nequation\n@m=@t=\rm\u0002\u00160He\u000b+\u000bm\u0002@m=@t (1)\nwhere\u000bis the Gilbert damping constant and the gyromagnetic\nratio\ris taken as negative. In standard FMR the \u00160He\u000b\nis given by the magnetic field generated by microwave pho-\ntons, commonly achieved by coplanar waveguides. In a SAW\nexcited a-FMR, the magnetoelastic coupling links the time\ndependent lattice strain tensor ^\"(t)to the magnetic environ-\nment. To calculate \u00160He\u000bas a function of the strain ten-\nsor components \"ijand magnetization components miwhere\ni;j=x;y;z ; we phenomenologically postulate the ther-\nmodynamic relation of Gibbs \u00160He\u000b=\u0000rmGtot, which\nFIG. 1: (a) Illustration of experiment setup. The surface acoustic\nwaves (SAW) are generated by applying RF voltage on an interdigital\ntransducer port (IDT) via the inverse piezoelectric effect. A second\nIDT port measures the transmittance of SAW via the direct piezoelec-\ntric effect. (b) Scanning electron microscope (SEM) image of IDT\nfingers with a nominal width of 400 nm. (c) Transmittance measure-\nment of SAW by vector network analyzer, the measurement shows a\nSAW resonance frequency, fSAW = 2:38GHz. We show the mea-\nsurement before (red open circles) and after (blue open squares) time\ngating filtering removal of electromagnetic waves.\nshould be valid for linear response around equilibrium: Foll-\nwing Dreher et al. [9], we have introduced Gtot=G+Gme\nwhereGis the free energy density normalized by jMjin a\nferromagnet, which may for instance be taken to be\nG=\u0000\u00160H\u0001m+Bdm2\nz+Bu(m\u0001u)2: (2)\nHereBd=\u00160jMj=2is the shape anisotropy and Buis the\nin-plane uniaxial anisotropy along the unit vector u. Al-\nthough one could further include other terms such as dipole-\ndipole and exchange interactions, the detailed form of Gis\nlargely irrelevant in the following discussions and we omit\nthem here. Gmeis the magnetoelasitc contribution to the\n(jMj-normalized) free energy density, which for a cubic crys-\ntal reads\nGme=b1(\"xxm2\nx+\"yym2\ny+\"zzm2\nz)\n+ 2b2(\"xymxmy+\"yzmymz+\"zxmzmx);(3)\nwhereb1;2are the magnetoelastic coupling constants. We\ndefine a new coordinate system ( x1;x2;x3) so that the mag-\nnetization components are accordingly ( m1;m2;m3); where3\nthe equilibrium direction of the magnetization m0lies inx3\n(m3), whilem1andm2are the small dynamical compo-\nnents perpendicular to each other and perpendicular to m3\n(see schematic in fig. 2(a)). Now we can expand the effective\nfield\u00160He\u000bunder the influence of magnetoelastic coupling\nto first order in m1;2\n\u00160He\u000b=\u00000\n@G11m1+G12m2\nG12m1+G22m2\nG31\nA+0\n@\u00160h1\n\u00160h2\n\u00160h31\nA:(4)\nHereG3=@m3Gjm=m0andGab=@ma@mbGjm=m0,\na;b= 1;2are constants whose details are not needed. The\ntransverse components of the strain induced field h1;2are\ngiven shortly while the longitudinal component h3is irrele-\nvant in our present setup. Since the components of the driving\nfield\u00160He\u000bthat induce a-FMR are those transverse to the\nequilibrium of magnetization m0(m3), we rewrite the LLG\nequation in its matrix form as\n \nG11\u0000G3+i!\u000b\n\rG12\u0000i!\n\r\nG12+i!\n\rG22\u0000G3+i!\u000b\n\r!\u0012\nm1\nm2\u0013\n=\u00160\u0012\nh1\nh2\u0013\n(5)\nSolving for the transverse components ( m1,m2) to the mag-\nnetization in equilibrium m0, we have\njMj\u0012\nm1\nm2\u0013\n=\u001f\u0012\nh1\nh2\u0013\n(6)\nwhere\u001fis the Polder susceptibility tensor which describes the\ndependence on the material parameters and static magnetic\nfield following from the free energy density Gof Eq. (2)\n\u001f=\u00160jMj \nG11\u0000G3+i!\u000b\n\rG12\u0000i!\n\r\nG12+i!\n\rG22\u0000G3+i!\u000b\n\r!\u00001\n:\n(7)\nh1andh2are the transverse components of the driving field\ninduced by elastic strain, such that\n\u00160h1=\u00002b1sin\u0012cos\u0012[\"xxcos2\u001e+\"yysin2\u001e\u0000\"zz]\n\u00002b2[cos 2\u0012(\"xzcos\u001e+\"yzsin\u001e) + 2\"xy\nsin\u0012cos\u0012sin\u001ecos\u001e]; (8)\n\u00160h2=2b1sin\u0012sin\u001ecos\u001e[\"xx\u0000\"yy]\u00002b2[cos\u0012\n(\"yxcos\u001e\u0000\"xzsin\u001e) +\"xysin\u0012cos 2\u001e]: (9)\nWe observe that different from conventional FMR, the a-FMR\nhas a strong dependence in the angles \u0012(out of plane angle)\nand\u001e(in plane angle) between the magnetization m0and the\nstrain components \"ij. Rayleigh waves in SAW contain the\nstrain components \"xx,\"xz, where the dominant component\nis the longitudinal strain ( \"xx). For a pure longitudinal strain\n\"xxwhen the m0is in-plane configuration \u0012= 0\u000e, the max-\nimum value of the driving field \u00160h1(2)is at\u001e= 45\u000ewhere\nthe magnetoelastic torque is larger, and vanishes at \u001e= 0\u000e\nand\u001e= 90\u000e, as schematically shown in fig. 2(b). This an-\ngle dependence gives origin to the now characteristic four-\nfold butterfly shape of the a-FMR excited by SAW [8, 9, 11].\nFIG. 2: (a) Schematic representation of the coordinates of magnetiza-\ntion components m1,m2andm3(m0). (b) Schematic representation\nof stress fields of SAW from a top view. The maximum magnetoe-\nlastic torque is induced at \u001e= 45\u000e, and vanishes at \u001e= 0\u000eand\n\u001e= 90\u000e\nWhile the SAW power absorption \u0001PSAW(\u001e)is in general\nproportional to the imaginary part of (h\u0003\n1;h\u0003\n2)\u001f(h1;h2)T[9],\nits angular dependence in the in-plane configuration is well-\ncaptured by the approximate formula [8]\n\u0001PSAW(\u001e)/[b1\"xxsin\u001ecos\u001e\u00072b2\"xzcos\u001e]2;(10)\nwhere\"xx;xz are the amplitudes of \"xx;xz respectively and\n\u0007corresponds to the SAW propagation along \u0006^xdirections.\nFigure 3(a) shows the spectrum at resonance condition for\nFMR driving of a Ni layer of 10nm within a Ni/Cu/Bi 2O3\nheterostructure (inset), with an external magnetic field angle\n\u001e= 240\u000e. The full in-plane magnetic field angle dependence\nof the SAW power absorption \u0001PSAW shows the four-fold\nbutterfly shape of a-FMR in figure 3(b) which is well-fitted by\nEq. (10).\nC. Possible role of magnetization coupling with lattice rotation\nThe four-fold butterfly shape is expected to be symmetric\nfor pure longitudinal strain \"xx, however, an asymmetric dis-\ntribution may arise due to the contribution of the shear strain\n\"xz. Such asymmetric SAW absorption of a-FMR has recently\nreceived increasing attention [8, 12]. The origin of the asym-\nmetric SAW absorption on these works was explained as inter-\nference of the longitudinal ( \"xx) and shear ( \"xz) strain com-\nponents. Under this scenario one may expect that the shear\nstrain component should be dominant or at least comparable\nto the longitudinal strain [12]. However, in the thin film limit,\nkd<< 1, wherekis the wavevector and dthe film thickness,\nthe shear strain is strongly suppressed and longitudinal strain\nis dominant.\nEven though the shear strain \"xzis strongly suppressed in\nthe thin film limit, Maekawa and Tachiki theoretically demon-\nstrated that a rotational deformation of the lattice !xz=\n1=2(@ux=@z\u0000@uz=@x)survives and can couple to the mag-\nnetic anisotropy (out of plane, i.e. along z-direction) [10],\nwhere the displacement vector components uxanduzare4\nFIG. 3: (a) Power absorption PSAW of SAW at resonance condi-\ntion for FMR driving of a Ni layer of 10nm within a Ni/Cu/Bi 2O3\nheterostructure (inset). (b) Normalized in-plane magnetic field angle\ndependence of \u0001PSAW . Figure (b) is adapted from ref.[8].\ngiven by\nux=Akt\u0014\ne\u0000ktz\u00002q2\nq2+k2\nte\u0000klz\u0015\ncos(qx\u0000!t); (11)\nuz=\u0000Aq\u0014\ne\u0000ktz\u00002ktkl\nq2+k2\nte\u0000klz\u0015\nsin(qx\u0000!t);(12)\nwhere A is the amplitude of the SAW, kt=p\nq2\u0000(!=vt)2,\nkl=p\nq2\u0000(!=vl)2, withvtandvlbeing the transverse and\nlongitudinal sound velocities. One can compute !xzas\n!xz=1\n2Aq2\u00182(e\u0000ktz) cos(qx\u0000!t): (13)\n\u0018is given by the ratio of the velocities vtandvl. Follow-\ning [10], different from the standard magnetoelastic coupling,\nhere the rotational deformation !xzcouples to the uniaxial\ncrystal anisotropy (out of plane, z)Dasociated with a spin Si\nin thei-th site, such that\n\u0000D[S2\niz+!xz(SizSix+SixSiz)] (14)\nIf we insert Eq. (13) in Eq. (14) we obtain the Hamiltonian\ndescribing the interaction between SAW and spins via the ro-\ntational deformation !xz\nH=1\n2Aq2\u00182DX\ni(e\u0000ktliz)(SizSix+SixSiz) cos(qlix\u0000!t)\n(15)\nFIG. 4: Asymmetric power absorption PSAW of SAW in (a)\nNi/Cu/Bi 2O3and (b) Ni/Ag/Bi 2O3. Figures are adapted from ref.[8].\nwherelxiis thex-component of the position vector for the i-th\nsite. The interaction described by Eq. (15) implies that SAW\nmay excite surface magnons via rotational motion, and induce\nSAW attenuation even in the absence of shear strain.\nFigure 4 shows a comparison of the in-plane magnetic\nfield dependence of SAW attenuation for Ni/Cu/Bi 2O3and\nNi/Ag/Bi 2O3. Although, nominally both heterostructures\ncontain similar Ni and Bi 2O3, and Cu and Ag posses simi-\nlar acoustic attenuation, it is possible to observe a significant\ndifference in the asymmetric behavior of SAW absorption be-\ntween the two heterostructures. In a previous report, we at-\ntributed such difference in asymmetric SAW absorption to in-\nterference of longitudinal and shear waves [8], as was also\nsuggested for other systems [12]. However, the authors have\nrecently become aware of a report suggesting enhancement\nof magnetic anisotropy energy in the Ag/Ni interface [14].\nSuch enhancement of magnetic anisotropy most likely is the\nresult of spin reorientation due to intefacial spin orbit cou-\npling; however, further studies are necessary to clarify it. To-\ngether with recent independent experiments [13], this asym-\nmetric absorption in figure 4, may be due to the lattice rotation\ncoupling with magnetization as described by Eq. (15).\nIII. ACOUSTIC SPIN PUMPING\nAs described by Tserkovnyak et al [15, 16], magnetization\ndynamics following the LLG equation can pump spin current\nfrom a ferromagnetic layer into a nonmagnetic metal. Such\nspin pumping is the result of loss of torque acting in the mag-\nnetization vector, and can be directly related to enhancements\nin the Gilbert damping \u000b. The conservation of angular mo-\nmentum indicates that the damping of magnetization preces-\nsion can pump angular momentum or spin current into an ad-\njacent layer. The spin current generated by acoustic ferromag-\nnetic resonance can be converted to charge current by either\ninverse spin Hall effect (ISHE) [17, 18] or inverse Edelstein5\nFIG. 5: (a) Inverse Edelstein effect (IEE) signal rectified at the\nCu/Bi 2O3(red circles) and Ag/Bi 2O3(blue triangles) interfaces. (b)\nAngle dependence of the spin current density J sfor Ni/Cu/Bi 2O3.\nFigures adapted from ref.[8].\neffect (IEE) [8]. Figure 5(a) shows the IEE signal rectified at\nthe Cu/Bi 2O3(red circles) and Ag/Bi 2O3(blue triangles) in-\nterfaces. The opposite signs of rectified signal reflect the op-\nposite Rashba spin splitting at these two interfaces [19]. We\ncan use the IEE signal to estimate the generated spin current\ndensityJswith the following formula taken from [8]\nJs=V(\u001e)\n\u0015IEEwRsin\u0012(16)\nwhereV(\u001e)is the voltage signal detected while the mag-\nnetic field is applied at angle \u001e,\u0015IEE is the inverse Edelstein\nlength,wis the sample width and Rthe electric sample re-\nsistance. Figure 5(b) shows the angle dependence of the spin\ncurrent density for Ni/Cu/Bi 2O3, with the following parame-\nters:\u0015IEE= 0:17nm,w= 10\u0016m,R= 42:87\n. The angle\ndependence of spin current density shows similar behavior of\nthat of the power absorption of SAW presented in figure 3(b).\nThe spin current density is in the order of 108A/m2, same\norder of magnitude to spin current density generated by stan-\ndard microwave photon FMR [6, 20]. For completeness, we\nshow in figure 6 the power dependence of IEE signal rectified\nat the Cu/Bi 2O3interface. As the input power increases the\nIEE voltage increases monotonically. At low input powers the\nsignal can be approximated to a linear increase, however, at\nhigh input power the signal increases nonlinearly.\nFIG. 6: Power dependence of rectified Inverse Edelstein effect (IEE)\nsignal at the Cu/Bi 2O3interface\nIV . CONCLUSION AND OUTLOOK\nWe provided an overview of the basic characteristics of\nSAW and description of the magnon-phonon coupling that\ntriggers acoustic ferromagnetic resonance. We extended our\ndiscussion in a relatively yet unexplored coupling mecha-\nnism of lattice rotation and magnetic anisotropies [10]. Such\ncoupling mechanism offers a novel direction for promoting\nSAW asymmetric attenuation. Enhancement and modulation\nof asymmetric electrical charge conductivity in electronic cir-\ncuits allowed the development of the electronic diode technol-\nogy. The coupling mechanism between magnetic anisotropy\nand lattice rotation paves the way to explore significant en-\nhancements of asymmetric SAW attenuation in the GHz fre-\nquency range, with potential for the development of the\nmagneto-acoustic analog to an electronic diode [13]. We de-\nscribed and showed the spin current density generated by a-\nFMR, the so-called acoustic spin pumping. The order of mag-\nnitude of the spin current density J sis comparable to that\nproduced by standard FMR technique. Here, acoustic wave\nreflectors [21] may represent an opportunity for further en-\nhancement of the generated spin current densities in acoustic\nspin pumping. Beyond the specific topic presented here, SAW\ncan also couple to nonmagnetic layers via the so-called spin\nrotation coupling and generate spin currents in the absence\nof magnetic materials or external magnetic fields [22]. Initial\nexperimental evidence of spin rotation coupling has been re-\ncently reported via spin transfer torque mechanism in Cu/NiFe\nbilayer [23]. However, it would be interesting to demon-\nstrate the generation of SAW induced spin current in an all-\nnonmagnetic structure. Coupling involving SAW has multi-\nple applications that range from wireless technology, sensing,\nbiology to control of elemental charges in condensed matter\nand coupling to quantum states of matter [24], which offers\nopportunities for interdisciplinary research and device devel-\nopments.\nThis work was supported by Grants-in-Aid for Scientific6\nResearch on Innovative Areas (No. 26103001, No. 26103002)\nand JSPS KAKENHI (No. 19H05629). MX was supported\nby JSPS KAKENHI (No. JP19J21720). KY would like to ac-knowledge support of JSPS KAKENHI (JP 19K21040). SM\nwas financially supported by ERATO, JST, and KAKENHI\n(No 17H02927 and No. 26103006) from MEXT, Japan.\n[1] C. 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Rev.\nB 66, 224403 (2002)\n[17] K. Uchida, T. An, Y . Kajiwara, M. Toda, and E. Saitoh, Surface-\nacoustic-wave-driven spin pumping in Y 3Fe5O12/Pt hybrid\nstructure, Appl. Phys. Lett. 99, 212501 (2011)\n[18] M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka, R. Gross,\nand S. T. B. Goennenwein, Spin Pumping with Coherent Elastic\nWaves, Phys. Rev. Lett. 108, 176601 (2012)\n[19] J. Puebla, F. Auvray, M. Xu, B. Rana, A. Albouy, H. Tsai, K.\nKondou, G. Tatara, and Y . Otani, Direct optical observation of\nspin accumulation at nonmagnetic metal/oxide interface, Appl.\nPhys. Lett. 111, 092402 (2017)\n[20] W. Zhang, M. B. Jungfleisch, W. Jiang, J. E. Pearson, and A.\nHoffmann, Spin pumping and inverse Rashba-Edelstein effect\nin NiFe/Ag/Bi and NiFe/Ag/Sb, J. Appl. Phys. 117, 17C727\n(2015)\n[21] T. R. Joseph, K. M. Lakin, and D. Penunuri, Surface acoustic\nwave planar resonator using grating reflectors, Appl. Phys. Lett.\n26, 29 (1975)\n[22] M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. 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Phys. 52 353001 (2019)"    },    {        "title": "1006.5763v1.Micromagnetic_analysis_of_magnetic_noise_in_ferromagnetic_nanowires.pdf",        "content": " 1Micromagnetic analysis of magn etic noise in ferromagnetic \nnanowires \nJungbum Yoon and Chun-Yeol You \nDepartment of Physics, Inha University, Incheon 402-751, Korea \nYounghun Jo and Se ung-Young Park \nDivision of Materials Science, Korea Basi c Science Institute, Daejeon 305-333, Korea \nMyung-Hwa Jung \nDepartment of Physics, Sogang University, Seoul 121-742, Korea \nWe investigate the magnetic thermal noise in magnetic nanowires with and without a domain \nwall by employing micromagnetic simulations. The magnetic thermal noise due to random \nthermal fluctuation fields gives important physical quantities related with the magnetic \nsusceptibility. We find that the re sonance frequency of a domain wa ll is distinguishable from one \nof a magnetic domain itself. For the single domain without a domain wall, the resonance \nfrequency is well described by the Kittel’s form ula considering a ferromagnetic specimen as a \nsimple ellipsoid with demagnetizing factors for various wire widths and thicknesses. However, \nadditional resonance frequencies from the magnetic domain wall show the different dependences \nof the wire width and thickness. It implies that the spins inside the domain wall have different \neffective fields and the spin dynamics. \nPACS number: 05.45.-a, 75.40. Gb., 75.70. kw. \nKeywords: magnetic noise, thermal fluctuation, resonance frequency \nEmail: cyyou@inha.ac.kr , Fax: +82-32-872-7562  2I. INTRODUCTION \nSince the spin-transfer torque (STT) effect [1, 2] al lows the control of magnetization direction by the \nspin-polarized current in nanostructur es, it is raising to the possibility of novel device applications with \nthe STT effect, such as the current induced magnetization switching [3, 4], magnetic domain wall (DW) motion [5-12], magnetization precession [13, 14], and spin wave nucleation [15]. Among the \napplications, the race-track memory (RM) related to the current induced DW motion is attractive as a \nnext generation nonvolatile memory  device which can substitute for the current hard disk drive. The \nRM has advantages of the high density compared with the hard disk drive and is suitable for the mobile \nmemory due to the nonvolatility. Furthermore, it has no mechanical part being a weakness of the \nconventional hard disk drive as a mobile storage [9]. The RM is operated by reading and writing of a \nbit data by moving the DW with the STT in ferroma gnetic nanowires. There are many reports about the \nsuccessful DW motion by the STT [6-10], however, the velocity and reproducibility is far from \nsatisfactory [7, 8], and the high critical current density causes unwanted Joule heating problems [11, \n12]. The underlying physics of the current induced DW motion has not yet been properly investigated. \nMost of the studies related with the current indu ced DW motion are focused on the observation of the \nDW motion itself, by the magnetic domain imaging [6] or the transport measurements [7, 8]. In order to \nelucidate the interaction between localized non-coll inear spins in the DW and the conducting polarized \nelectrons by STT, the basic information of the spin s in the DW is essential and important for the \nefficient control of the DW.  \nThe magnetic thermal noise due to the thermal ma gnetization fluctuations is related with the \nmagnetic susceptibility by the fl uctuation-dissipation theorem [16]. From the magnetic susceptibility, \nwe can extract the important physical information such as the magnetization, anisotropy energy, and damping parameters [17]. In addition, the noise measurement is also connected with the STT effects [18-21]. However, to the best of our knowledge, there is no study about the noise spectra from the DW  3[22]. \nIn this study, we present the magnetic noise due to thermal fluctuations by employing micromagntic \nsimulations in ferromagnetic nanowires. We find that the noise spectra of the DW have a resonance frequency distinguished from that of the single domain. We confirm that the resonance frequencies of \nthe single domain are well agreed with the Kittel’s formula for the nanowires with various widths and \nthicknesses. However, the resonance frequency of the DW cannot be explained by a simple model, and \nit implies the effective field of th e spins in the DW is qu ite different from that of the single domain. \nII. MICROMAGNETICS \nFor studying on magnetic noise in ferromagnetic nanowires, we performed the micromagnetic \nsimulation with object oriented micromagnetic framework (OOMMF) [23]. The spin dynamics is \ndescribed by the Landau-Lifshitz-Gilbert (LLG) equation,  \n)/ (/ ) ( / dtMdMM HM dtMdseff × −×−= α γ .                                                                                    (1) \nWhere γ, α, sM, and effH are the gyromagnetic ratio, Gilb ert damping constant, saturation \nmagnetization, and the effective magnetic field, respectively. The effective magnetic field is defined as \nME Heff ∂∂−=−/1μ . The energy density E is a function of M specified by Brown’s equations [24], \nincluding magnetic anisotropy, exchange, magnetostatic, and Zeeman energies due to the applied \nmagnetic field. Here, the archetypal LLG equation doesn’t consider the motion of spin due to the \nrandom thermal fluctuation field, which always exists in the finite temperature and it is the source of \nthe magnetic thermal noise. In orde r to include the thermal noise effect, we employed the stochastic \nLLG equation with the thermal fluctuation term, \n)/ (/ ))( ( / dtMdMM th H M dtMdsfl eff × −+×−= α γ .                                                                        (2) \nHere )(thfl is a highly irregular field term due to the Gaussian stochastic process. )(thfl has the \nstatistical properties, 0)(,=〉〈 thifl  and ) ( 2 )( )(, , st D s hthij jfl ifl − =〉 〈 δδ , where i and j are Cartesian  4indices, the constant D measures the strength of the thermal fluctuations. Dirac δ represents that above \nfinite temperature the autocorrelation time of )(thfl is much shorter than th e rotational-response time \nof the system, while the Kronecker δ represents that the different components of )(thfl are assumed to \nbe uncorrelated [25].  \nMicromagnetic simulations are calculated for a magnetic structure divided by small cells. If a cell \nsize is small enough, the simulation is accurate but it  takes longer simulation time. So the cell size must \nbe optimized by considering the accuracy and simulation time. Recently, it has been reported that the \nproper renormalization process is required for the physical quantities such as the saturation \nmagnetization [26]. Due to the finite cell size which is much larger than the atomic size, the \ncontribution of smaller wavelength magnon than the cell size cannot be properly considered in \nmicromagnetic simulations. To avoid this problem in the finite temperature, the renormalization \nprocess is necessary. In this work, however, we perform the simulation without the renormalization \nprocess at 300 K. We pay our attention to a qualitative behavior of the magnetic noise spectra and the resonance frequency, not a quantitative analysis. Th e quantitative results are not changed without the \nrenormalization process in our study. \nFor micromagnetic simulations, we select a model system of a 10-nm thick, 80-nm wide, and 1000-\nnm long NiFe (permalloy, Py) nanowire. A tail-to-tail transverse DW is placed at center position with a small notch (5×10 nm\n2), as shown in Fig. 1 (a). The notch trap s the domain wall as a potential well, but \nthe effect on the resonance frequency is negligible when the size of notch is not large. The material \nparameter of Py is as follows : the saturation magnetization 5106.8×=sM  A/m, the exchange stiffness \n121013−×=exA  J/m, and the Gilber t damping constant 01.0=α . We perform our simulations with a \nzero applied magnetic field and take a cell size of 555××  nm3. Also, the thickness and width of the Py \nnanowire is varied from 10 to 30 nm with 5 nm steps for a fixed 80 nm width and from 50 to 120 nm \nwith 10 nm steps for a fixed 10 nm thickness, respectively. The DW structure of narrow magnetic  5nanowires can be one of the transverse, vortex, and asymmetric transverse types depending on the \nthickness, width, and saturation magnetization due to the minimization of the demagnetization energy \n[27]. We limited the DW type as a symmetry transverse type by choosing the proper ranges of the thickness and width in our study. Because of a spatial symmetry of the vortex type DW, the sum of \nmagnetic noise from the whole vortex DW structure cancels each other and gives very small noise \nsignal, while the transverse type DW has better results. We obtained the spin configurations in the Py \nnanowires at each temporal moment for sufficiently long time (10\n-7 sec). The random thermal \nfluctuation field term is updated with the interval of 10-14 sec. The magnetization configurations are \nstored 10-11 sec steps. The resonance frequency due to the magnetic noise can be extracted by the fast \nFourier transform (FFT) from the time varying the transverse magnetization component ( My). \nIII. RESULTS AND DISCUSSION \nThe resonance frequency due to magnetic fluctu ations in the Py nanowires is calculated by \nemploying the OOMMF with the stochastic LLG equati on [28]. First, we consider the 80-nm wide, 10-\nnm thick, and 1000-nm long Py nanowire with a transverse type DW configuration as shown in Fig. 1 \n(a). The temporal variation of the total My and the results of the FFT power of the total My is shown in \nFig. 1 (b) and (c), respectively. The random therma l motion of the magnetization due to the finite \ntemperature is clearly shown in Fig. 1 (b). We find two resonance frequencies at 7.1 and 9.4 GHz, as \nshown in Fig. 1 (c). It must be noted that there is only one peak (9.7 GHz) obtained for the single \ndomain case, as shown in Fig. 1 (d). The resonance frequency is well agreed with the Kittel’s \nequation’s result, 9.3 GHz. This equation employ s the corresponding demagnetization factors of \n00248.0=xN , 10932.0=yN , and 8882.0=zN , with the relation of \n) ) ( () ) ( (2s x y eff s x z eff M N N H M N N H f ×−+××−+×=πγ.                                                          (3) \nHere Nx, Ny, and Nz are demagnetization factors for an corresponding ellipsoid [29]. To clarify the  6source of the additional peak (7.1 GHz), we employed the concept of the local spectra [22]. The local \nspectra are calculated from the limited region around th e DW, as shown the black rectangle in Fig. 1 (a). \nIn order to obtain the local spectra, we consider the sums of only the local My instead of the sums of the \nMy for the whole region. With the local spectra anal ysis, we can confirm that  the additional peaks are \ngenerated due to the DW. The additional peaks from the DW have distinguished resonance frequencies \nfrom that of the single domain. It implies the effective field acting on spins inside of the DW is \ndifferent from the inside of the single domain. Furthermore, the resonance frequency from the DW cannot be described with Kittel’s formula, because the demagnetization factor cannot be determined for \nthe DW. \nIn order to reveal the behavior of the resonance frequency of the DW, we perform the same \nmicromagnetic simulations with various widths and thicknesses. In Fig. 2, we plotted the local spectra \nfrom (a) the transverse-type DW and (b) the single do main part for various wire widths (50 to 120 nm \nwith 10 nm steps). The decrement of the resonance frequencies with the wire-width variation is clearly \nobserved for the domain and DW, respectively. At Fig. 3 (a), the width dependence of the resonance \nfrequencies for the domain and DW is plotted with the calculated resonance frequencies with Eq. (3). \nThe agreements between the results of micromagnetic simulations and Eq. (3) are excellent. The \ncorresponding demagnetization factors are shown in Fig. 3 (b). For all width, \nxN~ 0, and \n1=++z y x N N N  are hold, Eq. (3) can be rewritten as follows: \n) ) 1( (22 2\ns y y eff s eff MN N HM H f −++×=πγ.                                                                                   (4) \nIn this equation, it is clear that the resonance frequency only depends on ) 1(y y N N− term, and the \ncorresponding ) 1(y y N N−  is depicted in the Fig. 3 (b). The relations between the resonance frequency \nand demagnetization factors are clear. However, it must be pointed out that even though the resonance \nfrequency of the DW shows the same trend, decreased with broadening the wire width, the dependence  7of the resonance frequency of the DW is more complicated, which we will discuss later.  \nNext, we perform the same micromagnetic simulations with various thicknesses (from 10 to 30 nm \nwith 5 nm steps) with a fixed width of 80 nm as shown in Fig. 4 (a) and (b).  In this case, the thickness dependence of the resonance frequencies of the single domain and DW shows different behaviors. \nWhile the resonance frequencies of the single domain increase with increasing the wire thickness, those \nof the DW decrease. The agreements between micromagnetic simulations and Eq. (3) are still excellent \nas shown Fig. 5 (a). However, the resonance frequencies of the DW show the opposite behavior. In Fig. \n5 (b), corresponding demagnetization factors with \n) 1(y y N N−  are also depicted. Only the resonance \nfrequencies of the single domain can be explained with Eq. (4) and ) 1(y y N N−  term, but that of the \nDW shows the different behavior. \nSince the resonance frequency of the ferromagnetic body is a function of the effective field and the \nmagnetization, the distinguished resonance frequency implies that the effective field inside of the DW \nis quite different to that of the single domain. Furthermore, the dependence of the width and thickness \nis different. Inside of the single domain, where the magnetizations are collinear, 0=⋅∇M  is satisfied. \nHowever, this condition is no longer valid inside of the DW, where the magnetization directions are \nabruptly changed. Furthermore, there are many free poles at the both side of the nanowire edges, it also \ngives additional stray field. Therefore, different effective fields for the spins inside of the DW are \nunderstandable. However, there is no simple analytic model for the realistic transverse type DW so that \nwe cannot estimate the effective field inside of the DW. In order to obtain better understanding of the \nresonance frequency of the DW, experimental radio-frequency measurements must be carried out. \nIV. CONCLUSION \nThe magnetic noise spectra due to thermal fluctuatio ns in the Py nanowires are investigated by the \nOOMMF with the stochastic LLG equation. The Kittel’s formula well elucidates the resonance \nfrequencies of the magnetic noise spectra with the variation of the wire width and thickness for the  8single domain. We find that the resonance frequencies of the DW are distinguished from that of the \nsingle domain and confirm by the local spectra concept. The distinguishable resonance frequency of the DW implies that the spins inside of the DW have difference effective fields from that of the single domain. We believe that the magnetic noise measurem ent and the analysis of the DW will give a better \nunderstanding of the spin dynamics for the spins inside of the DW. \nACKNOWLEDGEMENT  \nThis research was supported by Nano R&D program (2008-02553, 2009-0082441) through the \nNational Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and \nTechnology  and the IT R&D program of MKE/KEIT (2009-F-004-01). \n \n \n \n \n \n \n \n \n \n \n      9REREFENCES \n[1] J. C. Slonczewski, J. Magn. Magn. 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Korea Magnetic Society 20, 1 (2010). \n[23] See http://math.nist.gov/oommf  \n[24] W. F. Brown, Jr., Micromagnetics  (Krieger, New York, 1978). \n[25] J. L. García-Palacios and F. J. Lázaro, Phys. Rev. B 58, 14937 (1998). \n[26] G. Grinstein and R. H. Koch, Phys. Rev. Lett. 90, 207201-1 (2003). \n[27] Y. Nakatani, A. Thiaville, and J. Miltat, J. Magn. Magn. Mater 290-291 , 750-753 (2005). \n[28] See http://www.nanoscience.de/group_r/stm-spst m/projects/temperature/download.shtml  \n[29] J. A. Osborn, Phys. Rev. 67, 351 (1945). \n \n \n \n \n \n   \n \n \n   11Figure Captions \n \nFig. 1. (a) Magnetization configuration of the Py nanowire with transverse type DW at the center. It is generated by the OOMMF simulation. The in-plane ma gnetization is represented by small arrows and \nthe blue and red direction arrows. The black rectangle represents a local region for magnetic thermal \nnoise spectra of the domain wall. (b) Random thermal motion of the magnetization due to the thermal \nfluctuations at 300 K. Magnetic noi se spectra due to thermal fluctuatio ns of the Py nanowire (c) with \nand (d) without the DW without the external magnetic field. \n \nFig. 2. Local magnetic noise spectra due to thermal fluctuations in (a) the transverse type DW and (b) \nsingle domain. The widths are varied from 50 to 120 nm with 10 nm steps. \n \nFig. 3. (a) Comparison of the resonance frequency from the magnetic noise spectra of the single \ndomain, DW, and from Eq. (3) as a function of the width of the Py nanowires. (b) Corresponding \ndemagnetization factors and N\ny(1-Ny). \n \nFig. 4. Local magnetic noise spectra due to thermal fluctuations in (a) the transverse type DW and (b) \nsingle domain. The thicknesses are varied from 10 to 30 nm with 5 nm steps. \n Fig. 5. (a) Comparison of the resonance frequency from the magnetic noise spectra of the single domain, DW, and from Eq. (3) as a function of th e thickness of the Py nanowires. (b) Corresponding \ndemagnetization factors and N\ny(1-Ny).  \n \n   12Fig. 1. \n0 200 400 600 8004000050000600007000080000\n  My (A/m)\nTime (10-11 sec)(b)\n0 2 4 6 8 1 01 21 40500100015002000\n7.1 GHz\n  FFT power of My (arb. unit)\nFrequency (GHz)9.4 GHz(c)\n0 2 4 6 8 1 01 21 405001000150020002500\n  FFT power of My (arb. unit)\nFrequency (GHz)(d)\n9.7 GHz\n  13Fig. 2. \n0 2 4 6 8 10 12 140200040006000800010000 50 nm\n 60 nm\n 70 nm\n 80 nm\n 90 nm\n 100 nm\n 110 nm\n 120 nm\n  FFT power of My (arb. unit)\nFrequency (GHz)(a)\n \n0 2 4 6 8 10 12 140100020003000400050006000\n 50 nm\n 60 nm\n 70 nm\n 80 nm\n 90 nm\n 100 nm\n 110 nm\n 120 nm\n  FFT power of My (arb. unit)\nFrequency (GHz)(b)\n \n \n \n \n \n \n \n \n  14Fig. 3. \n40 50 60 70 80 90 100 110 120 130681012\n Kittel's formula\n Single domain\n Domain wall(a)\n  Resonance frequency (GHz)\nWidth (nm) \n40 50 60 70 80 90 100 110 120 1300.00.20.40.60.81.0\n Nx\n Ny\n Nz(b) Demagnetizing factor\nWidth (nm)0.040.080.120.160.20\n  Ny(1-Ny)\n \n \n \n \n \n \n \n \n  15Fig. 4. \n0 2 4 6 8 10 12 1402000400060008000\n 10 nm\n 15 nm\n 20 nm\n 25 nm\n 30 nm(a)\n  FFT power of My (arb. unit)\nFrequency (GHz) \n0 2 4 6 8 10 12 14010002000300040005000 10 nm\n 15 nm\n 20 nm\n 25 nm\n 30 nm(b)\n  FFT power of My (arb. unit)\nFrequency (GHz) \n \n \n \n \n \n \n \n  16Fig. 5. \n10 15 20 25 300481216\n Kittel's formula\n Single domain\n Domain wall(a)\n  Resonance frequency (GHz)\nThickness (nm) \n10 15 20 25 30-0.20.00.20.40.60.81.0\n Nx\n Ny\n Nz(b) Demagnetizing factor\nThickness (nm)0.080.120.160.200.24\n Ny(1-Ny)\n \n "    },    {        "title": "1409.6641v1.Combining_ferromagnetic_resonator_and_digital_image_correlation_to_study_the_strain_induced_resonance_tunability_in_magnetoelectric_heterostructures.pdf",        "content": "Combining ferromagnetic resonator and digital image correlation to study the strain\ninduced resonance tunability in magnetoelectric heterostructures\nFatih Zighem1,\u0003Mohamed Belmeguenai1, Damien Faurie1,yHalim Haddadi2, and Johan Moulin3\n1Laboratoire des Sciences des Procédés et des Matériaux,\nCNRS-Université Paris XIII, Sorbonne Paris Cité, Villetaneuse, France\n2Laboratoire MSMP—Carnot Arts, ENSAM ParisTech,\nrue Saint-Dominique, 51006 Châlons-en-Champagne, France and\n3Institut d’Electronique Fondamentale, UMR 8622, Université Paris Sud-CNRS, Orsay, France\n(Dated: July 28 2014)\nThis paper reports the development of a methodology combining microstrip ferromagnetic res-\nonance (MS-FMR) and digital image correlation (DIC) in order to silmuteanously measure the\nvoltage-induced strains and the magnetic resonance in artificial magnetoelectric heterostructures\n(“magnetic films/piezoelectric substrate” or “magnetic films/flexible substrate/piezoelectric actua-\ntor”). The overall principle of the technique and the related analytical modelling are described. It\nis powerful to estimate the magnetostriction coefficient of ferromagnetic thin films and can be used\nto determine the effective magnetoelectric coefficient of the whole heterostructures in addition to\nthe piezoelectric coefficient related to the in-plane voltage-induced strains. This methodology can\nbe applied to system for which the strains are well transmitted at the different interfaces.\nPACS numbers:\nKeywords: indirect magnetoelectric effect, digital image correlation, ferromagnetic resonance\nI. INTRODUCTION\nDuring the last decade, many groups have con-\ncentrated their effort on the study of ferromag-\nnetic resonance (FMR) tunability through electric field\nstrain induced MagnetoElectric (ME) coupling [1, 2].\nMany ME systems based on ferromagnetic/ferroelectric\nheterostructures have been developed, encompassing\nvoltage-tunable microwave signal processing devices [3–\n5], magnetoelectric random access memory devices[6–9]\nand strain-control GMR devices [10–12]. For all these\nkinds of systems, the properties are controlled viathe\nelastic voltage-induced strains transmitted from the fer-\nroelectric medium to the ferromagnetic one. Hence, the\nknowledge of the elastic strains must be known in or-\nder to predict the changes of magnetic state induced by\napplying voltages. Concerning magnetic thin films de-\nposited on substrates, there are two ways to control the\nmagnetization by applying voltage-induced strains : i)\ndeposition of the film on a piezoelectric substrate [13–\n15] and ii) deposition of the film on a “non-piezoelectric”\nsubstrate and subsequent cementation on a piezoelec-\ntric actuator [16–18]. In the first one, the in-plane\nstrain transmission is generally good, the main disad-\nvantage being to transfer existing process to piezoelec-\ntric substrates (PZT, PZN-PT, PMN-PT, ...) ; in the\nsecond one, the main problem is the possibly bad in-\nplane strain transmission when the substrate is rigid (Si,\nGaAs, MgO, ...), the total transmission being attained\nonly with compliant substrates (polymers) [19]. In the\n\u0003Electronic address: zighem@univ-paris13.fr\nyElectronic address: faurie@univ-paris13.frtwo cases of full transmission (“film/piezoelectric sub-\nstrate”, “film/polymer substrate/piezoelectric actuator”),\nthe knowledge of the in-plane strains in the piezoelec-\ntric substrate or actuator is sufficient to know the in-\nplane strains in the whole system. In order to mea-\nsure simultaneously the in-plane strains and ferromag-\nnetic resonance in such systems, we have developed a\nmethodology combining in situmicrostrip ferromagnetic\nresonance (MS-FMR) with the Digital Image Correla-\ntion (DIC) technique. In this paper, we will show the\ngeneral principle of the technique including the analyti-\ncal formalism, before describing the experimental details.\nThen we will show an example of study on a “Finemet ®\nfilm/Kapton ®substrate/piezoelectric actuator” system\nhighlighting the potentialities of the method. Especially,\nwe will show that this technique is very efficient to de-\ntermine the magnetostriction coefficient at saturation of\nthe film, in complement to recently developed techniques\nfor thin films, such as magnetoelastic measurement setup\nwith a MOKE magnetometer [20], substrate deflection\nmethod under magnetic field [21], laser Doppler vibrom-\netry [22], applied whole wafer curvature [23], strain mod-\nulated ferromagnetic resonance [24], or vibrating sample\nmagnetometer coupled with bending[25].\nII. PRINCIPLE\nA. Methodology\nThevoltagestrain-inducedmagnetoelectriccouplingin\nartificial multiferroic systems, consisting on a magnetic\nfilm deposited onto a piezoelectric substrate, is based\non a two step process: i) the application of an electric\nfield inside the piezoelectric medium induces mechani-arXiv:1409.6641v1  [cond-mat.mtrl-sci]  23 Sep 20142\nxyz\nHHMM\nMagnetostrictive film\nPiezoelectric substrateHHMMMM\nFigure 1: Schematic illustration showing angles, fields and\ncoordinate systems used in the text.\ncal deformations to the magnetic film viapiezoelectric\neffect and ii) magnetostriction of the magnetic film in-\nduces a magnetoelastic anisotropy which can be strain(or\nvoltage)-tunnedinamplitudeanddirection. Inthesecon-\nditions, a quantitative evaluation of this indirect magne-\ntoelectric coupling requires both the determination of the\nstrains induced by the piezoelectric medium to the ferro-\nmagneticfilmandtheinducedmagnetoelasticanisotropy.\nThese parameters can be evaluated with our experimen-\ntal setup. Indeed, with our setup, when applying a volt-\nage to the piezoelectric medium, it is possible to simul-\ntaneously measure the induced mechanical deformation\nand the magnetoelastic anisotropy thanks to a combi-\nnation of digital image correlation and broadband fer-\nromagnetic resonance techniques. Indeed, the uniform\nprecession mode resonance field is directly linked to the\nmagnetoelastic anisotropy when the film is submitted to\nexternal stresses. In these conditions, it is not only possi-\nbletomeasureaneffectivemagnetoeletriccouplingwhich\ndo not necessary require the strains evaluation inside the\nmagnetic film but also the magnetostriction coefficients.\nThe following paragraph presents a basic analytical mod-\nelling of the resonance field variation as function of ex-\nternal applied stresses.\nB. Theoretical background\nIn this paragraph, the resonance field of the uniform\nprecession mode of a magnetostrictive film submitted to\nin-plane external stresses is derived in the macrospin ap-\nproximation ( i. e.a uniform magnetization is consid-\nered). In our system, these in-plane stresses are ap-\nplied by the piezoelectric medium to the ferromagnetic\nfilm (see figure 1). For simplicity, the magnetostric-\ntive and elastic properties of this thin film are consid-\nered as isotropic (it will be the case in the studied sys-\ntem). With these assumptions (isotropic behavior and\nmacrospin approximation), the magnetostriction coeffi-cient, the Young’s modulus and the Poisson’s ratio of the\nthin film are scalars parameters. The magnetic energy of\nthe thin film, using the coordinates system presented in\nFigure 1 can be written as:\nF=Fzee+Fdip+Fme (1)\nWhere the two first terms stand for the Zeeman and\nthe dipolar contributions, respectively. The last term\ncorresponds to the magnetoelastic anisotropy term and\ncan be written as:\nFme=\u00003\n2\u0015\u0010\u0000\ncos2'Msin2\u0012M\u00001\n3\u0001\n\u001bxx\n+\u0000\nsin2'Msin2\u0012M\u00001\n3\u0001\n\u001byy\u0011\n(2)\n\u001bxxand\u001byybeing the in-plane principal stress tensor\ncomponents while \u0012Mand'Mstand for the polar and\nthe azimuthal angles of the magnetization ~M. Finally,\u0015\nis the isotropic magnetostriction coefficient at saturation\nof the thin film. The relation between the principal stress\ncomponents ( \u001bxx,\u001byy) and strains ( \"xx,\"yy) tensors is\ngiven by the isotropic Hook’s law where Eis the Young’s\nmodulus and \u0017is the Poisson’s ratio:\n\u001bxx=\u0012E\n1 +\u0017\u0013\u00121\n1\u0000\u0017\"xx+\u0017\n1\u0000\u0017\"yy\u0013\n(3)\n\u001byy=\u0012E\n1 +\u0017\u0013\u00121\n1\u0000\u0017\"yy+\u0017\n1\u0000\u0017\"xx\u0013\n(4)\nThe resonance field of the uniform precession mode eval-\nuated at the equilibrium is obtained from the following\nexpression:\n\u00122\u0019f\n\r\u00132\n=\u00121\nMSsin\u0012M\u00132 \n@2F\n@\u00122\nM@2F\n@'2\nM\u0000\u0012@2F\n@\u0012M'M\u00132!\n(5)\nWherefis the microwave driving frequency and \ris\nthe gyromagnetic factor ( \r=g\u00028:794\u0002106s\u00001.Oe\u00001).\nThe different energy derivatives are evaluated for the\nequilibrium direction of the magnetization. For an in-\nplane applied magnetic field, the equilibrium polar angle\nis'M=\u0019\n2because of the large effective demagnetizing\nfield associated with the planar film geometry and an\nexplicit expression is obtained:\n\u00122\u0019f\n\r\u00132\n=h\n4\u0019Ms+Hrescos('M\u0000'H)+\n3\u0015\nMs\u0010\n\u001bxxcos2'M+\u001byysin2'M\u0011i\n\u0002\nh\nHrescos('M\u0000'H)+3\u0015\nMs\u0010\n\u001bxxcos2'M+\u001byysin2'Mi\n(6)3\n52\n1\n4\n63\nPin\nPout7\nSpeckledpattern\n(spray‐painted)Microstripline\nV\na)b)\nc)\nFigure 2: a) Setup image of the combined MS-FMR/DIC experiment. The circled numbers correspond to: 1○: Keithley Model\n2400;2○: CCD camera (AVT-Pike-f421b); 3○: objective lens for the CCD camera; 4○: white light source; 5○: electromagnet; 6○:\nSchottky detector and 7○: modulation coils. PinandPoutare the injected and transmitted radio frequency current. b) Zoom in\nshowing the sample mounted onto the microstripline. A typical calculated strain field map is present at the top of the sample.\nc) Zoom in of the speckled pattern (spray-painted) at the top of the sample necessary for the strain fields calculations.\nHere'Histheanglebetweenthein-planeappliedmag-\nnetic field and xdirection (see figure 1). The analysis can\nbe simplified if the resonance field is larger than the mag-\nnetoelastic anisotropy field ( ~Hme=\u0000~r~MFme). In this\ncondition, the magnetization direction is almost aligned\nalong the applied magnetic field ( 'M\u0018'H). The reso-\nnance field is thus given by:\nHres=\ns\n\u0000\n2\u0019Ms+3\u0015\n2Ms(\u001bxxsin2'H+\u001byycos2'H)\u00012+\u00102\u0019f\n\r\u00112\n\u00002\u0019Ms\u00003\u0015\n4Ms\u0010\n\u001bxx(1+3 cos 2'H)+\u001byy(1\u00003 cos 2'H)\u0011\n(7)\nThe two first terms represents an almost constant shift\nin the resonance field baseline because 2\u0019Msand2\u0019f\n\rare\nfound to be larger than the magnetoelastic and the mag-\nnetoelastic anisotropy field. The last term correspond to\nthe angular variation of the resonance field due to the\nvoltage induced magnetoelastic anisotropy field.III. EXPERIMENTAL SETUP\nThe experimental setup presented in figure 2 has been\ndeveloped in order to in situstudy the indirect magneto-\nelectric effect occurring in artificial magnetoelectric mul-\ntiferroics heterostructures such as the ones presented in\nintroduction. Our setup combines microstrip ferromag-\nnetic resonance (MS-FMR) and digital image correlation\n(DIC) techniques. The MS-FMR characterization is per-\nformed with the help of a field modulated FMR setup us-\ning HP 83752B generator, operating in the 0.01-20 GHz\nfrequency range, to generate a radio frequency field ( ~hrf)\nof variable power (-100 dBm to +20 dBm). The sam-\nple (figure 2b)) is mounted on a 0.5 mm microstrip line\n(the film side is in direct contact with the microstripline)\nconnected to the HP generator viaa semi-flexible SMA\ncable and to a lock-in amplifier (Stanford research sys-\ntem SR830) to derive the field modulated measurements\nviaa Schottky detector. The microstrip line (MS), com-\nposed of 0.5 mm Cu-strip grown on Cu-back side metal-\nized Al 2O3substrate, is designed to have 50 Ohm char-\nacteristic impedance and broadband. The MS and the\nsample are inserted inside the gap of an electromagnet\nconnected to a DC power supplier to generate in-plane4\nVPiezoelectric actuatorKapton® (125 µm)Finemet® (500 nm)xy\nFigure 3: Sketch of the studied heterostructure showing the\n500 nm thick Finemet ®film deposited onto a 125 µm thick\nKapton ®substrateandgluedontothepiezoelectricactuator.\nmagnetic fields up to 5 kOe with a resolution of 0.1 Oe.\nThis external magnetic field is modulated at a frequency\nof 170 Hz with an amplitude varying from 1 to 8 Oe al-\nlowing lock-in detection to be used in order to increase\nthe signal-to-noise ratio. This broadband MS-FMR of-\nfers a high sensitivity allowing to detect a net magnetic\nmoment down to 10\u00005emu.\nThe piezoelectric media is connected to a Keithley\npower supplier (Model 2400) allowing to apply DC volt-\nages in the range [-200 V;+200 V] with 0.001 V resolu-\ntion. For each applied voltage, a MS-FMR spectrum and\nan image of the top surface of the sample are recorded\n(note that the spectrum recording takes from 1 minutes\nto several hours depending on the magnetic field step and\non the total width of the spectrum). The images record-\ning at each step will serve to determine the in-plane field\nstrains. Furthermore, being given the initial “homoge-\nneous” surface of the sample, a speckle pattern has been\nspray painted in order to generate a contrast which will\nserve to calculate the strain fields (an image of a typical\nspeckled pattern (spray-painted) is presented in Figure\n2c)). In our setup, the images are recorded thanks to a\n2048\u00022048pixels CCD AVT-Pike f421b camera verti-\ncally positioned in the top of the surface sample. The ob-\njective lens has been chosen in order to be sufficiently far\nfrom the electromagnet (20 cm) with a field view around\n1\u00021cm2as shown on Figure2b). The strain fields calcu-\nlations(fromthedifferentimages)havebeenperfumedby\ndigital image correlation. Several commercial and open\nsource softwares based on digital image correlation tech-\nnique are available (Ncorr [26], Moire [27], Aramis [28],\nMatchid [29], Correli [30], ...) for the determination of\nstrains fields. In the following illustrative results, the\nDIC calculations have been performed by using Aramis.\nFinally, it is worth to mention that this setup is piloted\nviaa Labview program providing flexibility of a real time\ncontrol of the magnetic field and the DC voltage sweeps,\nstep and rate, real time data acquisition and visualiza-\ntion.IV. ILLUSTRATIVE RESULTS\nThe methodology is illustrated with a “magnetic\nfilm/polymer substrate/piezoelectric actuator” system.\nFigure 3 presents a cross-section sketch of the fabricated\nheterostructure. An amorphous 530 nm thick Finemet ®\nfilm has been deposited onto a 125 \u0016m thick polyimide\nflexiblesubstrate(Kapton ®)byradiofrequencysputter-\ning. Thedepositionconditionswerearesidualpressureof\naround 10\u00007mbar, a working Ar pressure of 40 mbar and\na RF power of 250 W. Prior to the Finemet ®deposition,\na 10 nm thick Ti buffer layer , was deposited on the sub-\nstrate to ensure a proper adhesion of the Finemet ®film.\nFinally, another 10 nm thick Ti cap layer was deposited\non the top of the Finemet ®film in order to protect it\nfrom oxidation. The composition of the film has been\nmeasured by EDS (Energy Dispersive Spectroscopy) and\nisclosetothatofthetarget(Fe 73:5Cu1Nb3Si15:5B7)while\nthe thickness of the film (530 nm) has been measured\nby Scanning Electron Microscopy. After deposition, the\nfilm/substrate system has been glued onto a piezoelectric\nactuator.\nThefigure4showsthemapofthein-planestrains( \"xx,\n\"yy,\"xy) for a few voltages applied to the piezoelectric\nactuator (from 0 V to +150V). For the three in-plane\nstrain components, the whole distribution is estimated to\nbe about 10\u00004so that the strain heterogeneities in the\npiezoelectric actuator are relatively weak. Obviously, to\ninterpret the FMR results, we use the mean values of the\nelasticstrains. Whenweapplyavoltage, themeanvalues\nof\"xxand\"yyvary with\"yy\n\"xx\u0018\u00000:65, while the in-plane\nshear strain \"xyremains unchanged. This behavior is\nillustrated in figures 5-a and 5-b where the mean strains\n\"xxand\"yyare plotted as function of the voltage.\nIn 5-a, the curve corresponds to a simple electric\nloading-unloading (0V - 150V - 0V) ; the non-linearity\nof the curve is due to the specific piezoelectric behav-\nior of the actuator, which is reversible. In figure 5-b, we\nshow an unloading from 150V to -150V and a subsequent\nloading from -150V to 150V. We observe the so-called\n“butterfly” behavior that is due to polarization switching\nat about -60V during unloading and 60V during load-\ning. In addition, we have checked that the analysis of\n\"xxand\"yyon the thin film gives same mean values[19].\nThus, the transmission of the in-plane strains between\nthe actuator and the surface of the film is close to 100%.\nThe voltage induced anisotropy has been studied in a\nspecific configuration (magnetic field\u0000 !Happlied along y\naxis). The influence of the applied voltage on the mag-\nnetic properties of the thin film has been probed by MS-\nFMR technique. Figures 5-c and 5-d show MS-FMR re-\nsults obtained from experimental spectra recorded at 8\nGHz with an applied magnetic field along the yaxis ( i.\ne.'H=\u0019\n2) and for different applied voltages; typical ex-\nperimental spectra are shown on figure 6-a. In these con-\nditions, the deduced resonance fields are in a magnetic\nsaturating regime so that we can deduce here the mag-\nnetostriction coefficient at saturation. It clearly appears5\n10V\n30V\n50V\n90V\n110V\n130V\n150V\n70Vxxyy\nâ10‐3‐2.0‐1.6 ‐1.2 ‐0.8 ‐0.4 0.4 0.8 1.2 1.6 2.0 0.0xy\n\n10V\n30V\n50V\n\n70V\n90V\n110V\n\n130V\nAppliedmagnetic field(Oe)FMRspectra\n400 600 800 1000\n\n150V\nFigure 4: Calculated in-plane strains fields ( \"xx,\"yyand\"xy) and corresponding FMR spectra for specific applied voltages\n(from 0 V to +150V). Histograms of the in-plane strains fields are represented for each applied voltage. Note the almost zero\nvalue of the shear strain ( \"xy).\n0 50 100 150‐1.0‐0.50.00.51.01.5\n\nxx\nyyStrain(x10‐3)\nAppliedvoltage(V)‐150‐100‐50 0 50 100 150‐1.0‐0.50.00.51.01.5xx\nyyStrain(x10‐3)\nAppliedvoltage(V)updownup\ndown1\n23\n1\n32a) b)\n‐150‐100‐50 0 50 100 150‐100‐50050100150Experiments\nTheoretical model\nHres(Oe)\nAppliedvoltage(V)0 50 100 150050100150\nExperiments\nTheoretical modelHres(Oe)\nAppliedvoltage(V)updown1\n23c) d)\nFigure 5: Mean in-plane strains ( \"xxand\"yy) and\u000eHres\nvariations as function of the applied voltage (a,c: from 0 to\n+150V and back to 0 V; b,d: from +150V to \u0000150V and\nback to +150V). The continuous lines in c) and d) are cal-\nculated by using equation 7 with the following parameters:\n\r= 1:885\u0002107s\u00001.Oe\u00001,MS= 965emu.cm\u00003,E= 145\nGPa and\u0017= 0:27and\u0015= 17:5\u000210\u00006.\nthat the resonance field increases with the applied volt-\nage, which indicates that the yaxis is harder and the x\naxis easier for the magnetization direction when a posi-\ntive voltage is applied. This is consistent with a positive\n‐150‐100‐50 0 50 100 150160170180190200\nFMRlinewidth (Oe)\nAppliedvoltage(V)‐150‐100‐50 0 50 100 150‐1012\n+150V\n+100V\n+50V\n0V\n‐50V\n‐56VIntensity(a.u.)\nH‐Hres(Oe)b) a)\n1\n23Figure 6: a) Typical FMR spectra obtained at 8 GHz show-\ning the linewidth variation as function of the applied voltage.\nFor clarity, they are centered around 0 Oe (shifted by their\nrespective resonance field). b) Corresponding FMR linewidth\n(\u0001Hpp) variation as function of the applied voltage (from\n+150V to \u0000150V and back to +150V).\nmagnetostriction coefficient, as expected for amorphous\nFinemet®alloys [31]. The shift of the resonance fields\nas function of the applied voltage are presented in fig-\nures 5-c and 5-d and corresponds to the tests shown re-\nspectively in figures 5-a and 5-b for the strain analysis.\nObviously, the observed non-linear and hysteretic vari-\nations of the resonance field is due to the variations of\n\"xxand\"yyas function of the applied voltage (see figure\n3). In order to quantitatively bind the resonance field\nvariations to the in-plane strains induced by the applied\nvoltage, we can calculate the uniform precession mode\nfrequency as function of the applied voltage (strain) by6\nadding a magnetoelastic energy term Fmeto the total\nmagnetic energy density Fof the film, as described pre-\nviously. The solid lines in Figures 5-c and 5-d are fits to\nthe experimental data calculated by using the parame-\nters previously determined by MS-FMR ( \r= 1:885\u0002107\ns\u00001.Oe\u00001,MS= 965emu.cm\u00003) and Brillouin light scat-\ntering (E= 145GPa and\u0017= 0:27) [32]. Actually, the\nmagnetostriction coefficient of the thin film is the sole\nparameter to be determined using this experiment and\nhas been estimated to be \u0015= 17:5\u000210\u00006, slightly lower\nthat the bulk counterpart ( \u0015= 23\u000210\u00006) [31].\nIn addition, in figure 6-b, the FMR peak to peak\nlinewidth ( \u0001Hpp), defined as the field difference between\nthe extrema of the sweep-field measured FMR spectra,\nis plotted as function of the applied voltage using 8 GHz\ndriving frequency. This figure shows that \u0001Hppis signif-\nicantly enhanced by the applied voltage and interestingly\npresents similar behavior as the resonance field shift and\nthe strain (5-b and 5-d) suggesting its correlation with\nthe voltage induced strain. The modelling of the strain\nbehavior of the FMR linewidth is out of this paper scope\nand will be addressed in forthcoming papers. However,\nthis strain tuning of the FMR linewidth remains a simple\nand promising manner to artificially choose the desired\ndamping depending on the aimed application.\nV. CONCLUSION AND PERSPECTIVES\nWe have shown a methodology that combines mi-\ncrostrip ferromagnetic resonance (MS-FMR) and digitalimage correlation (DIC) in order to study the voltage-\ninduced strains effect on magnetic anisotropy in thin\nfilms. The elastic strains are measured on the actua-\ntor (or substrate) side while the magnetic resonance field\nis measured in the thin films. This technique allows de-\ntermining the magnetostriction coefficient of the film and\ncan also be used to determine the effective magnetoelec-\ntric coefficient of the whole system and the piezoelectric\ncoefficientrelatedtothein-planevoltage-inducedstrains.\nThis methodology can be applied to system for which the\nstrains are well transmitted at the different interfaces\n(“film/piezoelectric substrate” system or “film/polymer\nsubstrate/piezoelectric actuator” system). Moreover, the\nstrain tuning of the FMR linewidth is promising for spin-\ntronics applications. Indeed, the magnetic damping con-\ntrols how fast the magnetization reverses and therefore\nis interesting technological parameter.\nAcknowledgments\nThe authors gratefully acknowledge the CNRS for\nhis financial support through the “PEPS INSIS” pro-\ngram (FERROFLEX project) and the Université Paris\n13 through a “Bonus Qualité Recherche”.\n[1] Y. K. Fetisov and G. Srinivasan, Appl. Phys. Lett. Appl.\nPhys. 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Phys. 115,\n17E309 (2014)[26] http://www.ncorr.com/\n[27] http://www.opticist.org/\n[28] http://www.gom.com/\n[29] http://www.matchid.org/\n[30] G. Besnard, F. Hild, S. Roux, Exp Mech 46, 789 (2006).\n[31] J. Moulin, I. Shahosseini, F. Alves, F. Mazaleyrat, Jour-\nnal of Micromechanics and Microengineering 21, 074010\n(2011)\n[32] A. Fillon, C. Jaouen, A. Michel, G. Abadias, C. Tro-\nmas, L. Belliard, B. Perrin, P. Djemia, Phys. Rev. B 88,\n174104 (2014)"    },    {        "title": "0903.2285v1.Observation_of_localized_ferromagnetic_resonance_in_a_continuous_ferromagnetic_film_via_magnetic_resonance_force_microscopy.pdf",        "content": "arXiv:0903.2285v1  [cond-mat.mtrl-sci]  12 Mar 2009Observation of localized ferromagnetic resonance in a cont inuous ferromagnetic film\nvia magnetic resonance force microscopy\nE. Nazaretski1, D.V. Pelekhov2, I. Martin1, M. Zalalutdinov3,\nD. Ponarin4, A. Smirnov4, P. C. Hammel2, and R. Movshovich1\n1Los Alamos National Laboratory, Los Alamos, NM 87545 USA,\n2Department of Physics, Ohio State University, Columbus OH 4 3210,\n3SFA Inc., Crofton, MD 21114,4Department of Chemistry,\nNorth Carolina State University, Raleigh, NC 27695\n(Dated: September 25, 2018)\nWe present Magnetic Resonance Force Microscopy (MRFM) meas urements of Ferromagnetic Res-\nonance (FMR) in a 50 nm thick permalloy film, tilted with respe ct to the direction of the external\nmagnetic field. At small probe-sample distances the MRFM spe ctrum breaks up into multiple\nmodes, which we identify as local ferromagnetic resonances confined by the magnetic field of the\nMRFM tip. Micromagnetic simulations support this identific ation of the modes and show they are\nstabilized in the region where the dipolar tip field has a comp onent anti-parallel to the applied field.\nPACS numbers: 07.79.Pk, 07.55.-w, 76.50.+g, 75.70.-i\nMagnetic resonanceforce microscopy(MRFM) [1] pro-\nvides a route to detection of magnetic resonance with\nexcellent spin sensitivity and spatial resolution and has\nreceived considerable attention. Detection of electron\nspin resonance [2], nuclear magnetic resonance [3] and\nferromagnetic resonance (FMR) [4, 5, 6, 7] with this ap-\nproachhasbeen reported. The highsensitivityofMRFM\nwhen combinedwith the imagingmechanismofMagnetic\nResonance Imaging (MRI) allows excellent spatial res-\nolution in non-interacting spin systems [8]. Studies of\npatterned films show FMR can be detected in individ-\nual microstructures with high sensitivity [4, 5, 6, 9]. As\na consequence of the strong interactions between spins\nin a ferromagnet the excitations are collective (magne-\ntostatic) modes, so the assumptions on which MRI is\nbasedareinvalidated, howeverrecentworkshowsMRFM\nis also a promising technique for spatially resolved FMR:\nObukhov et al.,at OSUdemonstrated that the largefield\ngradient of the micromagnetic probe enables ferromag-\nnetic resonance imaging of the magnetostatic modes of a\n2µm diameter permalloy dot with ∼250nm resolution\n[10].\nHere we argue that the micromagnetic probe locally\nstabilizes magnetostatic modes detected in FMR thus in-\ndicating the potential for scanned probe imaging of mag-\nnetic properties of extended ferromagnetic films. Due to\nthe strong interaction between the spins the resonance\nmodesarecollectiveexcitationsofspinsintheentiresam-\nple, and in the case of the microfabricated samples, the\nFMR modes are defined by the geometry of the sample.\nWe study the dependence of the mode resonance field\nontheorientationofthefilmrelativetotheexternalmag-\nnetic field, and on the probe-sample separation. We find\nthat localized FMR modes are spatially confined imme-\ndiately below the MRFM probe magnet. This occurs be-\ncause off the axis of the micromagnetic probe its dipolar\nfield is negative and, so reduces the magnitude of the to-tal magnetic field stabilizing a confined mode. To under-\nstand the nature of these modes and their behavior, we\nperformed micromagnetic simulations based on a finite\nelement solution of the Landau-Lifshitz (LL) equation\nwith damping. The simulations reveal that the degree\nof localization can be controlled not only by the sample-\nprobe separation, but that it is also very sensitive to the\ntilt anglebetweenthefilm andtheappliedmagneticfield.\nThese results demonstrate confinement of the magneto-\nstatic mode by the localized inhomogeneous field of the\nprobe, a mechanism distinct from geometrical confine-\nment such as was studied in Ref. 10.\nmHproben Hext\nMs /c89/c81\nFIG. 1: (Color online) Schematic of the experiment. The\nsample is tilted and placed in the uniform external magnetic\nfieldHext. Here ˆnis the normal to the sample surface, Θ\nis the sample tilt angle with respect to Hextand Ψ is the\ntilt angle of the equilibrium magnetization Ms. The MRFM\nprobe field is aligned with the direction of Hextand its dipolar\nfield reduces the magnitude of Hextin the region off to the\nside of the magnetic tip m.\nThe MRFM experiments were performed with the\nlow temperature dual MFM/MRFM apparatus at Los\nAlamos National Laboratory. We used a commercially\navailable silicon nitride cantilever (resonance frequency\nfc≈8.0 kHz and spring constant k∼10 mN/m). The\nmicromagnetic probe tip is a 2.4 µm diameter spherical\nNd2Fe14B particle. Its silicon nitride tip was removed by\nfocused ion milling, and the magnetic sphere was manu-\nally glued to the cantilever in the presence of an align-2\ning magnetic field of a few kOe. The saturation mag-\nnetization of Nd 2Fe14B powder was determined to be\n4πMs≈13.2 kG at 10 K using SQUID magnetometery.\nThe spatial field profile produced by the probe magnet\nhas been carefully characterized [11]. For a detailed de-\nscription of the MRFM apparatus we refer the reader to\nRef. 12.\nThe 50 nm thick permalloy film was deposited on a\n100µm thick silicon wafer on top of a 20 nm thick Ti\nadhesion layer, and capped with a protective 20 nm Ti\nlayer. Anapproximately1.9 ×1.9mm2samplewasglued\nto the strip line microwave resonator. The temperature\nof the sample was stabilized at T = 11 K (temperature\nstability was better than 5 mK) and the frequency of the\nmicrowavefield was set to fRF= 9.75 GHz. Fig. 1 shows\na schematic diagram of the experiment.\nThe in-phase component of the lock-in detected\nMRFM signal is shown in Fig. 2 as a function of the\nprobe-sample separation for three values of the tilt angle\nΘ between the normal to the film plane and the direc-\ntion ofHext(as shown in Fig. 1). The sample tilt angle\nwas determined by measuring the deflection of a laser\nbeam reflected off the film surface at room temperature\nand found to be 1 .0◦, 3.4◦and 5.4◦. While the abso-\nlute uncertainty of the sample tilt at low temperatures is\n≈ ±1.0◦, mainly due to thermal contraction of the glue\nused to hold the sample in place, the relative uncertainty\nof the tilt angles at low temperature is better than 0.1\ndegree. These tilt angleswereconfirmedbythe measured\nvalues of the resonance field Hextat large probe-sample\nseparations (discussed further below).\nAll spectra shown in Fig. 2 exhibit very similar evo-\nlution as a function of probe-sample separation. A sin-\ngle fundamental resonance mode is observed at distances\ngreater than a few micrometers. The resonance field of\nthis mode changes as the probe approaches the sample\nsurface (Fig. 2a). At larger values of Θ (Fig. 2b, c) split-\nting of the fundamental mode is observed at small sep-\narations. The magnitude of the splitting increases with\nincreasing Θ (Fig. 2c).\nIn order to better understand the evolution of the\nMRFM signal, we performed FMR studies as a function\nof Θ in a conventional FMR spectrometer. In Fig. 3 we\nshow the angular dependence of the resonance field for\nthe permalloy sample. Conventional FMR experiments\nwere performed at NCSU. The data were obtained in\na uniform applied field Hextand are equivalent to the\nMRFM resonance spectra acquired at large values of the\nprobe-sample separation. In fact, the MRFM and con-\nventional FMR sets of data exhibit excellent agreement\nas shown in Fig. 3. The value of the ferromagnetic res-\nonance field in a continuous film for a given Θ is deter-\nmined by the dispersion relation [13]:\n/parenleftBig\nω\nγ/parenrightBig2\n= [Hextcos(Θ−Ψ)−4πMscos(2Ψ)] ×Probe□-□film□separation□[ m]\n/c109\n012345\n10.5 11 11.5 1201234\n13.2 13.4 13.6 13.80246\n14.1 14.3 14.5\nField□[kOe](c)(b)(a)\n04812\n13.1 13.3 13.5 13.7MRFM□signal\nField□[kOe]In-phase□MRFM□signal□amplitude□[arb.□units]\nFIG. 2: (Color online) The in-phase component of the MRFM\nsignal. Each spectrum is plotted with an offset proportional\nto the probe-sample separation. This separation can be read\noff of the right-hand vertical axis far from resonance. Spect ra\nwere recorded at three different angles Θ: a) 1 .0◦, b) 3.4◦\nand c) 5 .4◦at T = 11 K and fRF= 9.75 GHz. Note the\ndifferent horizontal axis scales for each panel. The red dott ed\nline traces the evolution of the fundamental resonance mode .\nSpectra were scaled for clarity of presentation. The inset i n\npanel b) shows the shift of the entire resonance structure wi th\nthe change of the microwave frequency from 9.75 to 9.65 GHz\nat a probe-film separation of ∼760 nm.\n[Hextcos(Θ−Ψ)−4πMscos2(Ψ)]; (1)\nHextsin(Θ−Ψ)+2πMssin(2Θ) = 0 .(2)\nHereω= 2πfRFis the RF frequency, γis the gyromag-\nnetic ratio, 4 πMsis the saturation magnetization and\nΨ is the tilt angle of the equilibrium magnetization as\nshown in Fig. 1. We neglect the anisotropy contribution\nbecause it is typically small ( ∼few Gauss) in uniform\npermalloy films [14, 15]. Equations 1 and 2 cannot be\nsolved analytically, so we solved them numerically; the\nresults are shown in Fig. 3 by the solid line. We used\nthe conventional FMR data set to verify the values of\n4πMs= 11.3 kG, and γ= 2.89±0.05 GHz/kOe. With\nconventional FMR we never observed any multiple reso-\nnance structuresreminiscent ofthose appearingin Fig. 2.\nThe origin of the isolated resonance modes seen in\nthe MRFM studies can be understood in the following\nway. Conventional FMR theory assumes a uniform mag-\nnetic field throughout the sample [16]. The LL analysis\nshows that the presence of a position dependent inho-\nmogeneous magnetic field of the probe may result in a3\nMRFM□data\n0246810121416\n0 20 40 60 80Resonance□Field□[kOe]\nAngle [degrees] /c81Conventional□FMR\nMicromagnetic□modelSolution□to□Eq.□1\nFIG. 3: (Color online) Angular dependence of the resonance\nfield for the uniform Hext. Four data sets: conventional FMR\ndata (△), solution to Eg. 1 (solid line), MRFM data at large\nprobe-sample separation ( •), and the results of the micro-\nmagnetic modeling ( /square) are in good agreement. The numeri-\ncal solution and the micromagnetic modeling were done using\nthe same simulation parameters: 4 πMs=11.3 kG, fRF= 9.5\nGHz,γ= 2.89±0.05 GHz/kOe. MRFM data were acquired\natfRF= 9.75 GHz.\nformation of localized FMR modes confined to the re-\ngion where the field inhomogeneity exceeds the value of\n∆Hmax≃2πMs(L/a) (where Lis the thickness of the\nfilm and ais the radius of the localization) [10]. The\nMRFM probe produces an approximately dipolar field in\nthe plane of the sample. In the simplest geometry Hext\nand the tip’s magnetic moment mare parallel to each\nother. In this case the total magnetic field directly un-\nder the tip (uniform external field plus the dipolar field\ndue to the tip) is higher than the external field, but off\nto the side it is lower than Hext(see Fig. 1). In these\ntwo loci the field at which resonance occurs is shifted to\nhigher and lower fields, respectively relative to resonance\nin the externalfield alone. While the modes shifted down\nin field may overlap with the bulk magnetostatic modes,\nthe modes shifted up in field arewell isolated, so they can\nreadily be used to perform local MRFM spectroscopy.\nIn order to quantify this picture we have conducted\nmicromagnetic simulations based on the Landau-Lifshitz\n(LL) equation with damping:\n∂M\n∂t=γM×Htot−λ\nM2sM×M×Htot,(3)\nwhereλis the damping coefficient. The total magnetic\nfieldHtotexperienced by the sample magnetization M\nconsists of the external, probe, dipolar, exchange and\nanisotropy fields. The numerical solution of Eq. 3 for\nthis field configuration provides the information about\nthe spatial profile of the FMR modes excited in the sam-\nple and their corresponding resonance fields. The details\nofthe numericalapproachand its applicationto avariety\nof systems will be presented elsewhere [17].\nAs numerical simulation of an infinite ferromagnetic\n/c109m/c109m\n/c109m/c109m\n/c109m/c109m(a)\n(b)\n(c)\nFIG. 4: (Color online) Numerically calculated spatial profi le\nof the excited transverse magnetization of the fundamental\nFMR mode in a 250 ×250µm2permalloy film. The thickness\nof the film is 50 nm and it is tilted at an angle Θ with respect\nto the direction of the applied magnetic field. The probe\nmagnetis a2.4 µm diametersphereplacedat adistance dover\nthe center of the film, (0,0,d) a) Θ = 0◦, no probe magnet b)\nΘ = 5.99◦,d= 1.0µm, localized FMR mode at the location\nof the probe can be seen. c) Zoom-in onto the localized mode.\nDue to the sample tilt, the peak mode is localized to the side\nof the probe. All modes were calculated for the value of Hext\ncorresponding to fRF= 9.75 GHz.\nfilm with an inhomogeneous external field profile is dif-\nficult we simulated a finite ferromagnetic thin film that\nwas large compared to the phenomena of interest (250 ×\n250µm2squarewith thickness50nm). Themagneticpa-\nrameters chosen for simulations were obtained from the\nconventionalFMR experiment: 4 πMs=11.3kGand γ=\n2.89 GHz/kOe. For the damping we used λ= 0.005s−1,\na typical value for permalloy. Eq. 3 was linearized on a\nvariabledensitypixelgridwith higherpixeldensityinthe\nvicinity of the probe magnet. The rare earth probe mag-\nnet was modeled as a 2.4 µm diameter uniformly mag-\nnetized sphere (4 πMs= 13.2 kG) placed at a height d\nmeasured from the surface of the sphere. The probe was\npositioned over the center of the film. Probe magnet pa-\nrameters were experimentally determined in Ref. [7, 11].\nThe orientation of the probe magnetic moment mis as-\nsumed to be aligned with the direction of Hext. The va-\nlidityofthemodelingapproachwasverifiedbycomparing\nthemodelingresultsforatiltedfilm(withoutMRFMtip)\nwith the conventional FMR data (see Fig. 3). In the ab-\nsenceofthelocalizedprobefield, thefundamental(lowest\nfrequency) resonance mode has a well known bell-shaped4\n0123\n11 11.5 12 12.5 13 13.5 14 14.5Probe-film□distance□[ m] /c109\nField□[kOe]4\n/c81=5.990/c81=3.940/c81=2.060\nFIG. 5: (Color online) Comparison of the numerically cal-\nculated evolution of the fundamental FMR mode (solid line)\nwith data points taken from Fig. 2 (filled circles). The micro -\nmagnetic modeling was performed for the tilt angles obtaine d\nfrom the resonance fields of the fundamental mode at large\nprobe-sample separations. To obtain the agreement shown in\nFig. 5 we had to increase the measured probe-sample separa-\ntion by≈1µm(depending on the tilt angle). The origin of\nthe offset is the short range probe-sample interaction which\ncauses the cantilever to snap to the film at probe-sample dis-\ntances less than a micron. The nature of the interaction has\nbeen discussed in Ref. [19, 20] and the magnitude of the offset\nhas been quantified in Ref. [11].\nspatial profile, as shown in Fig. 4a, and spans the entire\nsample. This resultis in goodagreementwith earlierthe-\noretical work [18]. In the presence of the probe magnet,\nhowever, the mode changes its shape dramatically and is\nspatially confined to the region immediately beneath the\nprobe magnet (see Fig. 4b). A closer look at the mode\nlocalization (see Fig. 4c) reveals confinement on the or-\nder of a few µm2. This is the region of the sample were\nthe dipolar field of the probe magnet opposes the exter-\nnally applied field Hext. The asymmetry in the shape\nof the localized FMR mode with respect to the position\nof the probe magnet is due to the tilt of the sample, as\nillustrated in Fig. 1.\nIn Fig. 5 we compare the micromagnetic simulations of\nthe evolution of the fundamental mode with the experi-\nmental points taken from the dotted line in Fig. 2. The\nvaluesofthe sampletilt angleΘused in simulationwhere\nchosen to match the values of the resonance field Hextat\nlarge probe-sample separations. It can be seen that the\nvalue of Hextshifts towards higher values as the probe\napproaches the sample. This shift also increases with in-\ncreasing tilt angle of the sample. This is consistent with\nthe increase of the probe’s negative field, which would re-\nquire higher values of Hextto satisfy the spin resonance\ncondition at fRF= 9.75 GHz.\nIn conclusion, we have observedFMR modes stabilized\nin a confined region defined by the localized field of mi-\ncromagnetic probe field. These are observed when the\nsample is tilted with respect to the orientation of the\nprobe axis thus subjecting the sample to the negative\ndipolar field produced off the axis of the probe magnet.\nOur micromagnetic simulations accurately describe the\nexperiments and find the modes to be confined to a re-gion of order a micron across. Our experiments point\nto the possibility of spatially resolved FMR (e.g. imag-\ning and characterization of defects and magnetic inho-\nmogeneities) in continuous ferromagnetic films.\nWe thank Dr. T. Mewes for stimulating discussions\nand Dr. B. Houston and Dr. J. W. Baldwin for sample\nfabrication. The workperformedat LosAlamosNational\nLaboratory was supported by the US Department of En-\nergy, Center for Integrated Nanotechnologies. The work\nat OSU was supported by the US Department of Energy\nthrough grant DE-FG02-03ER46054. This work was also\nsupported by the Office of Naval Research through the\nInstitute for Nanoscience at Naval Research Laboratory.\nThe work at NCSU was supported by NSF grant ECS-\n0420775.\n[1] J. A. Sidles, Appl. Phys. Lett. 58, 2854 (1991).\n[2] D. Rugar, C. S. Yannoni, and J. A. Sidles, Nature 360,\n563 (1992).\n[3] D. Rugar and O. Z¨ uger and S. Hoen and C. S. Yannoni\nand H. M. Vieth and R. D. Kendrick, Science 264, 1560\n(1994).\n[4] Z. Zhang, P. C. Hammel, and P. E. Wigen, Appl. Phys.\nLett.68, 2005 (1996).\n[5] G. de Loubens, V.V. Naletov, O. Klein, J. B. Youssef, F.\nBoust, and N. Vukadinovic, Phys. Rev. Lett. 98, 127601\n(2007).\n[6] K. Wago, D. Botkin, C. S. Yannoni, and D. Rugar, Appl.\nPhys. Lett. 72, 2757 (1998).\n[7] E. Nazaretski, I. Martin, R. Movshovich, D. V. Pelekhov,\nP. C. Hammel, M. Zalalutdinov, J. W. Baldwin, B. Hous-\nton, and T. Mewes, Appl. Phys. Lett. 90, 234105 (2007).\n[8] H. J. Mamin, M. Poggio, C. L. Degen, and D. 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C. Hammel, preprint\n(2007).\n[18] G. N. Kakazei, P. E. Wigen, K. Yu. Guslenko, V.\nNovosad, A. N. Slavin, V. O. Golub, N. A. Lesnik, and5\nY. Otani, Appl. Phys. Lett. 85, 443 (2004).\n[19] M. Saint Jean, S. Hudlet, C. Guthmann, and J. Berger,\nJ. Appl. Phys. 86, 5245 (1999).[20] S. Hudlet, M. Saint Jean, C. Guthmann, and J. Berger,\nEuropean Physical Journal B 2, 5 (1998)."    },    {        "title": "1208.0575v2.Anomalous_Fermi_level_behavior_in_GaMnAs_at_the_onset_of_ferromagnetism.pdf",        "content": "Anomalous Fermi level behavior in GaMnAs at the onset of\nferromagnetism\nIriya Muneta,a)Hiroshi Terada, Shinobu Ohya, and Masaaki Tanakab)\nDepartment of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,\nTokyo 113-8656, Japan\n(Dated:)\nWe present the systematic study of the resonant tunneling spectroscopy on a series of ferromagnetic-semiconductor\nGa1\u0000xMnxAs with the Mn content xfrom\u00180.01 to 3.2%. The Fermi level of Ga 1\u0000xMnxAs exists in the band gap\nin the whole xregion. The Fermi level is closest to the valence band (VB) at x=1.0% corresponding to the onset of\nferromagnetism near the metal-insulator transition (MIT), but it moves away from the VB as xincreases or decreases\nfrom 1.0%. This anomalous behavior of the Fermi level indicates that the ferromagnetism and MIT emerge in the\nMn-derived impurity band.\nThe origin of the ferromagnetism and the metal-insulator\ntransition (MIT) has been a long-debated issue in the pro-\ntotype ferromagnetic semiconductor GaMnAs.1–4Previously,\nthe valence band (VB) conduction picture has been widely ac-\ncepted in this material,5where the MIT of GaMnAs was un-\nderstood by the Fermi level crossing over the VB2,6similarly\nto p-type GaAs doped with non-magnetic acceptors such as\nBe or Zn. The ferromagnetism in GaMnAs has been thought\nto be induced by the VB holes interacting with the localized\ndelectrons of the Mn atoms.5,7However, recently, many ex-\nperiments have shown the strong evidence that the Fermi level\nexists in the impurity band (IB) in the band gap,4,8–21which\nrequires reconsideration on the above scenario. Therefore, to\nclarify the origin of the ferromagnetism and MIT in GaMnAs,\nit is essential to precisely investigate the Fermi level position\nand the VB structure of GaMnAs in the low Mn content region\nincluding the onset of ferromagnetism and MIT.\nResonant tunneling spectroscopy is a powerful method to\ninvestigate the VB structure and the Fermi level position in\nGaMnAs with a precision of several meV . The advantage of\nthis method is that we can detect energy bands only with the\nsame wave-function symmetry as that of the p-wave func-\ntions of the VB holes. Also, it is sensitive to the effective\nmass of the energy bands. Furthermore, by carefully ana-\nlyzing the quantum-well (QW) thickness dependence of the\nresonant levels, we can avoid the unwanted effects induced\nat the surface, which prevent the precise determination of the\nVB or the Fermi level position.3,22Recently, from the reso-\nnant tunneling experiments in the double-barrier (DB) QW\nheterostructures14,15and the single-barrier structures with an\nultra-thin surface GaMnAs layer,16it was shown that the\nFermi level exists in the band gap in Ga 1\u0000xMnxAs with the\nMn content xhigher than\u00185%. In this Letter, we carefully\nanalyze the VB structure and the Fermi level position in a\nseries of Ga 1\u0000xMnxAs from the unexplored insulating region\n(x=\u00180.01%) to the metallic region ( x=3.2%). We find that the\nVB structure is not largely affected by the Mn doping and that\nthe Fermi level never crosses over the VB near the MIT: The\na)Electronic mail: muneta@cryst.t.u-tokyo.ac.jp\nb)Electronic mail: masaaki@ee.t.u-tokyo.ac.jp\n(a)\n(b) (c)GaAs:Be (100 nm)GaMnAs QW ( d nm)\nAlAs (6 nm)AlAs (6 nm)\np+GaAs (001) Substrated1d2d3200 µm\nd1Fermi level Fermi level\nEVd2d3 d d2d3EnergyEnergyGa0.94Mn0.06As(20 nm)\nheavy hole\nlight hole\nd1Fermi level\nEVAlAs AlAsAlAsAlAsAlAsAlAsFIG. 1. (a) Schematic device structures investigated in this study. d1,\nd2, and d3are the GaMnAs QW thickness. (b) VB lineups of AlAs /\nGaMnAs QW / AlAs when the QW thickness is d1,d2, and d3:The\nblack solid and red dash-dotted lines correspond to the VB top and\nFermi level. The blue and orange lines correspond to the quantum\nlevels in heavy and light hole bands, respectively. (c) ddependence\nof the quantum levels, where the blue and orange curves correspond\nto the heavy and light hole bands, respectively.\nFermi level becomes closest to the VB at x=1.0% at the onset\nof the ferromagnetism, but it moves away from the VB as x\nincreases or decreases from 1.0%. This anomalous behavior\nof the Fermi level is completely different from that of GaAs\ndoped with other shallow acceptors.\nFigure 1(a) shows the schematic device structures investi-\ngated in this study. We grew DB-QW heterostructures com-\nposed of (from the top to the bottom) Ga 0:94Mn 0:06As (20 nm)\n/ AlAs (6 nm) / Ga 1\u0000xMnxAs QW ( dnm) / AlAs (6 nm) /\nGaAs:Be (100 nm, Be concentration: 1 \u000218 cm\u00003) by low-\ntemperature molecular-beam epitaxy (MBE). Table I shows\nthe characteristics of the GaMnAs QW samples examined in\nthis study. The Mn content xof sample D-H, F2, and G2 was\ndetermined by the x-ray diffraction measurements for the 100\nnm-thick reference GaMnAs samples grown in the same con-\ndition as that for the GaMnAs QW in each DB-QW sample.\nWe estimated xof sample A-C and C2 from an Arrhenius plot\nof the Mn flux vs.the K-cell temperature obtained from the\nflux data of sample D-H, F2, and G2. In this study, xwas\nvaried from\u00180.01% to 3.2%. In this Mn content range, as x\nincreases, GaMnAs changes from insulating paramagnetic to\ninsulating ferromagnetic at x=\u00181%. At 2%, it starts to showarXiv:1208.0575v2  [cond-mat.mtrl-sci]  24 Apr 20132\nTABLE I. Parameters and characteristics of the DB-QW samples\ninvestigated in this study. xis the Mn content of the Ga 1\u0000xMnxAs\nQW. TSis the growth temperature of the top AlAs barrier and the\nGaMnAs QW. dandTCare the thickness and Curie temperature of\nthe GaMnAs QW, respectively. EFis the estimated energy distance\nbetween the Fermi level and the VB top at the Gpoint in the GaMnAs\nQW.\nSample. x(%) TS(\u000eC)d(nm) TC(K) EF(meV)\nSample A\u00180.01 400 10 - 16 paramagnetic 60\nSample B\u00180.1 350 10 - 16 paramagnetic 35\nSample C\u00180.3 330 10 - 16 paramagnetic 25\nSample C2\u00180.3 330 4 - 10 paramagnetic 25\nSample D 1.0 265 10 - 16 25 4\nSample E 1.2 260 10 - 16 30 6\nSample F 1.6 250 10 - 16 40 10\nSample F2 1.6 250 4 - 10 25 7\nSample G 2.3 240 10 - 16 45 17\nSample G2 2.3 240 4 - 10 30 12\nSample H 3.2 230 10 - 16 60 25\nthe metallic behavior, in which the resistivity decreases as\ntemperature Tdecreases at T<TC(Curie temperature). The\nTCvalues of the Ga 1\u0000xMnxAs QW in sample D-H, F2, and\nG2 were obtained from the temperature dependence of tun-\nneling magnetoresistance (TMR) observed in these devices.\nAs shown in Table I, the Ga 1\u0000xMnxAs QW with x=1-3% has\nlower TCthan that of the surface Ga 0:94Mn 0:06As (60-70 K)\ndue to the low xin the QWs. To obtain the resonant tunnel-\ning diodes with various GaMnAs QW thickness don a same\nwafer, we linearly moved a shutter in front of the substrate\nwhile growing the Ga 1\u0000xMnxAs QW, in which dwas varied\nfrom 10 nm to 16 nm within the wafer of 15 \u000210 mm2.23\nAlso, we prepared sample C2 ( x=\u00180.3%), F2 ( x=1.6%), and\nG2 (x=2.3%) with dfrom 4 nm to 10 nm, where the growth\nconditions of sample C2, F2, and G2 are the same as those\nof sample C ( x=\u00180.3%), F ( x=1.6%), and G ( x=2.3%) except\nford, respectively. The values of dwere estimated from the\nspeed of the shutter. Although simple structures (GaMnAs /\nAlAs / GaAs:Be single-barrier heterostructures) were used in\nRef. 16, they are not applicable for this study due to the low\nMn contents <\u00182% in the GaMnAs QW that we are using\nhere. The low Mn contents in the surface GaMnAs induce a\nlarge depletion thickness at the surface, making an electrical\ncontact to the GaMnAs layer very difficult. Thus, the DB-QW\nstructures are the best structure for our current purpose. The\ndetails about the preparation of the samples are described in\nthe supplemental material.24\nFigure 1(b) shows the schematic VB profiles of the DB-\nQW devices shown in Fig. 1(a). The black solid and red\ndash-dotted lines indicate the VB top and Fermi level, respec-\ntively. The blue and orange lines are the resonant levels of\nthe heavy hole (HH) and light hole (LH), respectively. Here,\nthe small exchange splitting is neglected for simplicity. Fig-\nure 1(c) shows the schematic ddependence of the resonant\nlevels. The resonant levels converge on the VB top energy EV\nof the bulk GaMnAs as dincreases. Since the Fermi level cor-\nresponds to the zero bias condition, the Fermi level position\nGaAs:Be AlAs AlAsGaMnAs\nQWGaMnAsEnergyEVEFFermi level\n0.55 eV\nd (nm) 6 nm 6 nmBand gap\ndepth (from surface to substrate)~7 meVFIG. 2. VB diagram of the DB-QW devices used in the theoretical\ncalculation. The black solid and red dash-dotted lines are EVand the\nFermi level, respectively. The gray region is the band gap.\ncan be determined by measuring bias voltage corresponding\ntoEVin sufficiently wide d. The bias polarity is defined by\nthe voltage of the top GaMnAs electrode with respect to the\nsubstrate.\nFigure 2 shows the VB diagram of the investigated DB-\nQW devices assumed in the following theoretical calculation\nof the resonant levels. The black solid and red dash-dotted\nlines are EVand the Fermi level, respectively. The gray re-\ngion is the band gap. For the quantum level calculation, we\nused the 4\u00024 Luttinger-Kohn k\u0001pHamiltonian25and transfer\nmatrix method.26The VB offset between AlAs and GaMnAs\nwas assumed to be 0.55 eV . We assumed that the in-plain wave\nvector kkis0during the tunneling, because holes are injected\nfrom the GaAs:Be electrode in the negative bias region and\nholes in the GaAs:Be electrode exist only around the Gpoint.\nFor simplicity, the small band bending was neglected.24\nFigure 3(a)-(h) show the color contour maps of the\nd2I=dV2-Vas a function of din sample A, C2, C, D, F2, F, G,\nand H measured at 3.5 K in the negative bias region, respec-\ntively. Here, the color-coded intensities are extrapolated from\nthe measured data at the dvalues corresponding to the white\ndots shown at the top of these figures. The black square and\nwhite triangle dots are calculated resonant peak bias voltages\nVRof the HH and LH bands, respectively. EFis the energy\ndistance between EVand the Fermi level. Clear oscillations\ndue to the resonant tunneling are observed in all the d2I=dV2-\nVcurves. Moreover, the resonant peak bias voltages become\nsmaller as dincreases, which is well reproduced by the calcu-\nlated ddependence of the resonant peak bias voltages. The EF\nvalue is estimated by the following procedure. As dincreases\nto infinity, these resonant levels converge on a certain voltage\nVVBcorresponding to the valence band edge. The Fermi level\nposition corresponds to the zero bias. Experimentally, the\nmeasured voltage Vis proportional to the energy E(relative to\nthe Fermi level); V=sE. Therefore, VVB=scorresponds to the\nenergy distance EFbetween the Fermi level and EV. Then, the\nddependence of VRis calculated from the equation VR=sERin\nindividual d. Here, ERis quantum level energy with respect to\nthe Fermi level. In the fitting, we slightly shifted the dvalue\ntod0in order to obtain the perfect fit. The value of d0used in\nthe calculation was shown as the upper red ruler of the contour\nmaps. The necessity of this shift of dis due to the difficulty in\naccurately controlling the relative positions between the sub-\nstrate shutter and the sample wafer inside the MBE chamber.\nWe note that, however, the thickness difference between dand3\nEF=25 meV\n12 14 16d’ (nm)\nd (nm)\nd2I/dV2 (A/V2)0.35 -0.5\n(c) Sample C\nx=~0.3%\n12 14 16\nEF=25 meV\n8 9 10 11d’ (nm)\nd (nm)\nd2I/dV2 (A/V2)0.35 -0.5\n5 6 7 8 9 10(b) Sample C2\nx=~0.3%\n0.30\n0.25\n0.20\n0.15\n0.10\n0.000.05-V(V)FE=10 meV\n16 18 14d’ (nm)\nd (nm)\n2 2 2dI/dV  (A/V )0.15 -0.3\n12 14 16(f) Sample F\nx=1.6%\nEF=7 meV\n8 9 11 1210d’ (nm)\nd (nm)\nd2I/dV2 (A/V2)0.2 -0.6\n5 6 7 8 9(e) Sample F2\nx=1.6%\nparamagnetic paramagnetic TC: 40 K TC: 25 K\nLHHHCalc.\nEF=60 meV\n16 12 14d’ (nm)\nd (nm)\nd2I/dV2 (A/V2)0.82 -0.91\n11 13 15(a) Sample A\nx=~0.01%\nparamagnetic\nEF=4 meV\n18 14 16d’ (nm)\nd (nm)\nd2I/dV2 (A/V2)0.1 -0.23\n11 13 15(d) Sample D\nx=1.0%\nTC:25 K\nEF=17 meV\n14 15 16 17d’ (nm)\nd (nm)\nd2I/dV2 (A/V2)0.4 -0.8\n111213 14 15(g) Sample G\nx=2.3%\nTC:45 K\n10 - 16 nm 10 - 16 nm 10 - 16 nm 10 - 16 nm 4 - 10 nm 4 - 10 nm 10 - 16 nm\nd (nm)\nd2I/dV2 (A/V2)0.11 -0.18\n11 1213 14 15EF=25 meV\n11 12 13d’ (nm)(h) Sample H\nx=3.2%\nTC:60 K10 - 16 nm\nFIG. 3. (a)-(h) The comparison between the calculated resonant levels and the experimentally obtained d2I=dV2data of (a) sample A, (b)\nsample C2, (c) sample C, (d) sample D, (e) sample F2, (f) sample F, (g) sample G, and (h) sample H as functions of \u0000Vandd. The d2I=dV2\nintensity is expressed by color. Here, these color intensities are extrapolated from the measured data with dcorresponding to the white dots\nshown at the top of these figures. The connected black and white dots are the calculated resonant peak bias voltages VRof the HH and LH\nbands, respectively. EFandd0are defined in the main text.\nd0is only less than\u00183 nm. The svalue corresponds to the ra-\ntio of the total bias voltage to the voltage applied at the bottom\nAlAs barrier. The svalues are assumed to be 2.0 in this study.\nConsidering that sis 2 in ideal DB tunnel junctions26and that\nthe interstitial Mn and its distribution variation in the QW are\nsufficiently suppressed in the low xwith the relatively high\ngrowth temperature, the s=2.0 assumed in this study is suffi-\nciently reasonable. The color contour maps of the d2I=dV2in\nall the samples are shown in the supplemental material.24\nIn Fig. 4, the blue circles are the EFvalues obtained in this\nstudy. The green squares are the ones reported in Ref. 16.\nThe black pentagons are the thermal-activation energy Eaof\nthe Mn acceptors in GaAs obtained by the magneto-transport\nmeasurements in Ref. 27. The pink curve is the calculated\nEavalues obtained from the equation 0 :11[1\u0000(x=0:8)1=3]\nmentioned in Ref. 27. Here, we selected 0.8% as the in-\ntercept of xto fit the curve to the experimental data. The\ngray dash-dotted, dashed, and solid curves are the calculated\nEFby the valence-band anti-crossing model (VBAC)28and\nthe free-electron approximation. In the VBAC, the impurity\nlevel EImpare assumed to be 0.1, 0.05, and 0.01 eV , respec-\ntively. Also, the anti-crossing coupling constant CMnis as-\nsumed to be 0.18 eV . The free-electron approximation calcu-\nlation for the Fermi level position in the IB is done by roughly\nassuming the hole concentration pto be x/2 and the effec-\ntive mass to be 10 me,11where meis the electron mass. Fig-\nure 4(b)-(d) show the VB and IB diagrams expected from this\nstudy in the (b) insulating paramagnetic ( x<1%), (c) insulat-\ning ferromagnetic ( x=1-2%), and (d) metallic ferromagnetic\n(x>2%) regions, respectively. The black solid curves cor-\nrespond to the VB. The blue dotted lines correspond to the\nupper and bottom edges of the IB. The blue region repre-\nsents the IB region. The red dash-dotted lines correspond\nto the Fermi level. In the paramagnetic GaMnAs ( x<1%),\nEFdecreases as xincreases, which is quantitatively consis-\ntent with the activation-energy-lowering effect observed inthe magneto-transport measurements.27This behavior is the\nsame as the one in the insulating region of the non-magnetic\naccepter-doped p-type GaAs.29The Fermi level behavior in\nthe paramagnetic region is caused by the screening effect due\nto the heavy Mn doping, which makes the IB position close to\nthe VB. On the other hand, EFincreases as xincreases in the\nferromagnetic GaMnAs with x>1%, which means that the\nFermi level moves away from the VB. This behavior is qual-\nitatively explained by the VBAC model [the gray solid curve\nin Fig. 4(a)]. The quantitative discrepancy between the ex-\nperimental and calculated Fermi level is probably because is\nprobably because this calculation does not take into account\nthe screening and many body effects. This Fermi level behav-\nior in the ferromagnetic region means that the IB truly exists\nin the ferromagnetic GaMnAs and the anti-crossing interac-\ntion is induced by the electron-electron interaction.19\nFigure 4(a) shows that the Fermi level exists in the IB in\nthe band gap in whole the xregion, which suggests that the\nferromagnetism is strongly related to the IB. At the MIT bor-\nder (x=\u00182%), the Fermi level is still in the band gap, which\nsuggests that the MIT occurs in the IB. This behavior of the\nFermi level is completely different from that in the case of the\nheavily-doped p-type GaAs with the non-magnetic acceptors\nand contradicts the VB conduction picture, where the IB com-\npletely merges into the VB. The IB still exists in the metallic\nGaMnAs, which means that the impurity states still remain in\nthe band gap. This is probably because the holes are trapped\ninto the acceptor states induced by p-dhybridized orbitals af-\nter the Coulomb potential completely screened.30,31At the on-\nset of the ferromagnetism ( x=\u00181%), the xdependence of the\nFermi level changes, which is thought to be induced by the\nchange of the main effect for determining the IB position from\nthe screening effect to the anti-crossing interaction. This must\ngive a clue to understanding the mechanism of the ferromag-\nnetism. We note that this anomalous behavior of the Fermi\nlevel is not directly related to the MIT because the MIT and4\n(b) insulating para. (c) insulating ferro. (d) metallic ferro.\nVB VB VBIBIB IBEnergy kEFEF\nEF(a)\n0.01 0.1 1 10\nMn content (%)050100 Ea & EF  (meV)Ref. 16.This study.\nRef. 27.\ncalc.EF\ncalc. Ea\nFIG. 4. (a) The blue circles and green squares are the EFvalues\nobtained in this study and in Ref. 16, respectively. The black pen-\ntagons are the Eavalues reported in Ref. 27. The pink curve is the\ncalculated Eaobtained from the equation 0 :11[1\u0000(x=0:8)1=3]. The\ngray dash-dotted, dashed, and solid curves are the calculated EFwith\nrespect to EVby the VBAC model28and the free-electron approxi-\nmation. In the VBAC, EImpare assumed to be 0.1, 0.05, and 0.01\neV , respectively. The red dashed curve connects the EFvalues in\nx>1%. (b)-(d) The VB and IB diagrams of GaMnAs derived from\nthis study in the (b) insulating paramagnetic ( x<1%), (c) insulating\nferromagnetic ( x=1-2%), and (d) metallic ferromagnetic ( x>2%) re-\ngions. The black solid curves are the VB. The blue dotted lines are\nthe upper and bottom edges of the IB. The blue region represents the\nIB region. The red dash-dotted lines are the Fermi level.\nthe turning up of the Fermi level position occur at slightly dif-\nferent value of x(1.5 - 2% and\u00181%, respectively).\nIn summary, we fabricated the DB heterostructures contain-\ning a Ga 1\u0000xMnxAs QW with various x. From the resonant\ntunneling measurements in these structures, we found that the\nFermi level exists in IB in the band gap in the whole range of\nx. When xis less than 1.0%, the Fermi level becomes close\nto the VB top as xincreases, which is induced by the screen-\ning effect. At x=1.0% around the emergence of the ferromag-\nnetism, the Fermi level becomes closest to the VB, where EF\nis 4 meV . When xis larger than 1.0%, the Fermi level goes\nfurther away from the VB top as xincreases, which is qualita-\ntively explained by the anti-crossing interaction. The anoma-\nlous behavior of the Fermi level means that the ferromag-\nnetic GaMnAs is completely different from the non-magnetic\nheavily-doped p-type GaAs, which indicates that the IB in the\nferromagnetic GaMnAs is composed of the p-dhybridized\norbitals. 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MEMSD are produced by bridging the molecular \nchannels across the insulator , in the exposed edge region (s) of a magnetic tunnel junction  \n(MTJ) . The bridged m olecular channels start serving as the dominant exchange coupling \nmedium between the two ferromagnetic electrodes of a MTJ. Present study focus on the effect \nof molecule enhanced exchange coupling on the magnetic properties of the  MTJ. This paper  \nshow s that o rganometallic molecular clusters (OMCs) strongly increased the magnetic coupling \nbetween the two ferromagnetic electrodes . SQUID magnetometer showed that OMCs \ntransformed the typical hysteresis magnetization curve of a Co/NiFe/AlOx/NiFe MTJ into linear \none. Ferromagnetic resonance studies showed that OMC bridges affected the two  fundamental \nresonance peaks of the Co/NiFe/AlOx/NiFe  MTJ. According to magnetic force microscopy, \nOMCs caused the disappearance of magnetic contrast from the Co/Ni Fe/AlOx/NiFe tunnel \njunction area. These three independent and complimentary experiments , suggested the \ndevelopment of extremely strong interl ayer exchange coupling. This  work  delineated a practical \nroute to control the exchange coupling between ferromagnet ic electrodes . Ability to tailor \nmagnetic coupling can lead to the development of molecule based quantum computation device \narchitecture.   \n \nIntroduction:  Molecular spintronics devices (MSDs) are promising to produce new computer \nlogic and advanced spintronics  devices . According to theoretical and experimental studies, \nMSDs are shown to possess several new remarkable attributes which are not observed with \nconventional magneto resistance devices. MSDs’ development will critically depend on the  2 ability to disting uish the role of molecules in bri nging out new device attributes . Conventionally, \nspintronics devices are investigated by performing highly correlated transport and magnetic \nstudies. Magnetic characterizations have played crucial role in the advancement of  magnetic \ntunnel junction (MTJ) based spin valves [1-3]. Similar m agnetic characterizations  will be even \nmore crucial for the development of MSDs . Magnetic characterizations  are required to gauge \nthe effect of molecule(s) on the magnetic coupling between two ferromagnetic (FM) electrodes  \n[4]. It is d ue to the fact that i n a MSD molecules are not merely the spin transport channe ls; \nmolecules  are capable of strengthening  the inter -electrode exchange coupling  [5]. For the study \nof molecul es’ role in a MSD , a test platform that can be studied before and after attaching the \nmolecular channels between two electrodes will be  highly  desirable  [6-8]. Tunnel junction based \nmultilayer edge molecular spintronics devices (MEMSD) are the promising candidate for the \ndevelopmen t of novel MSDs . MEMSD enable s the  insightful magnetic characterizations to study \nthe effect of molecular channels in producing new device attributes  [8]. MEMSDs are produced \nby bridging the molecular channels across the insulator of a MTJ, in the exposed edge region \n(Fig. 1). As a major advantage, a MTJ can be  easily characterized be fore transforming it into a \nMEMSD .  \nEffectiveness of molecule as an exchange coupler can be better understood by the \nevaluation of analogous entities like impurities and nanoclusters. Several research groups \nshowed that MTJs  experie nced significant increase in the exchange coupling due to impurities \n[9] and nanoclusters  [10]. Utilization of molecules can serve the role of nano clusters and \nimpurities to tailor the inter -electrode exchange coupling and yield mass producible devices . \nMolecules are much more reproducible than the impurities and nanoclusters. Precise control \nover molecule’s size, spin [11] and spatial location can lead to an extremely strong inter-\nelectrode exchange coupling  and unprecedented advancement in the field of MSDs [6, 12 -16]. If \nthese coupling become strong beyond a limit [17] then FM electrodes may undergo dramatic \nchanges  [18] and can lead to anomalous phenomenon [5]. Magnetic characterizations  are \nessential t o investigate  the effect of molecule’s in duced exchange coupling  on the MSDs \nproperties. This paper  investigated the effect of molecules on the magnetic properties of a MTJ \nbased  MEMSD  (Fig. 1a -b). In our studies organometallic molecular clusters (OMCs) [8, 19]  were \nutilized to tailor the exchange coupling between the FM electrodes (Fig.1c). Super conducting \nquantum interference device ( SQUID ) magnetometer, ferromagnetic resonance  (FMR) , and \nmagnetic force microscopy (MFM)  were used for the magnetic characterization. Here we \npresent m agnetic studies which showed a strong effect of molecules on the exchange c oupling  3 between two FM electrodes of the  magnetic tunnel junction.  The magnetic studies are utilized in \nexplaining th e dramatic current suppression observed with the MEMSD systems .  \n \nExperimental details:  MEMSDs for the magnetic characterizations were designed so that  to: \n(a) extraneous magnetic electrodes , away from the molecular junction , do not mask the OMCs \neffect  (b) a chip studied by the magnetic characterization s produced unequivocal response \nbefore and after the treatment with OMCs . To achieve the se two objectives MEMSDs were \nproduced in the form of cylindrical dots,  with a typical MTJ configuration and exposed si de \nedges (Fig.1 a-b). Cylindrical dot geometry with a 3 -5 m diameter allow ed OMCs (Fig. 1c) to \nbridge across the insulator of a MTJ along its  perimeter . A typical sample used for the magnetic \ncharacterizations  contained 7000 -21000 MTJs . The inclusion of  a large number of MTJ \ncylindrical  dots ensured the reasonably strong magnetic signal in a variety of magnetic \ncharacterizations . Fabrication of the MEMSD  dots required the creation of cylindrical  cavities in \na photoresist layer  by photolithography. Phot oresi st layer was produced by spin coating Shipley \n1813 photoresist at 3000 rpm speed on a thermally oxidized silicon chip. After spin -coating , \nphotoresist was backed at 90 -100 ºC for 1 -2 minutes. Hardened photoresist was exposed to UV \nlight for ~30 seconds wit h Karl Suss Mask Aligner. Subsequently, photoresist was develop ed in \nShipley MF 150 developer for ~40 -60 seconds. After this step, a typical chip contained a large \nnumber of cavities for the simultaneous deposition of cylin drical MTJs . To fabricate MTJ \ncylinders, sequential ly tantalum (Ta), cobalt (Co), 80% nickel -20% iron alloy (NiFe), alumina \n(AlOx), and NiFe films were sputter deposited in the photoresist cavity molds  (Fig. 1d) . Liftoff \nstep resulted in  an array of MTJ dots with ~4 m inter -dot separation  and \nTa[5nm]/Co[5nm]/NiFe[5nm]/AlOx[2nm]/NiFe[1 2 nm] configuration . To measure the magnetic \nresponse before OMC s attachment MTJs were subjected to numerous magnetic studies.   \nIn order to transform MTJ into MEMSD , OMCs were electroch emically bridged across the \ninsulator of a tunnel junction  (Fig. 1b) . For molecule attachment, MTJ s were submerged in the 2  \nmM OMC solution in dichloromethane. Then two electrical probes were dipped in the drop of \nOMC solution , and ±100 mV alternating volt age with a time interval of 0.02 seconds was \napplied. This step presumably de -protect ed the thiol functional groups present at the end of four \nalkane tethers  (L) of an OMC  (Fig. 1c)  [20]. Those OMCs whose two alkane tethers bridge d \nacross the AlOx insulator of a MTJ (Fig.1b) , become the part of molecular device and yield ed a \nMEMSD cylinder. After the molecule attachment step MEMSDs were  sequentially rinsed in di \ncholoro meth ane, iso -propyl alcohol, and deionized water. Finally, samples were again  4 subjected to magnetic characterizations to measure the effect of OMC s on MTJs’ basic \nmagnetic properties.     \nMagnetic properties of a MEMSD were studied using three measurement tech niques: (a) \nmagnetization study, (b) ferromagnetic resonance (FMR), and (c) magnetic force microscopy \n(MFM). The magnetization studies were performed using the Quantum Design MPMS XL 7T \nSQUID  magnetometer. MEMSD samples, mounted on a glass rod with the hel p of epoxy, were \nintroduced into the helium gas flushed superconducting coil of SQUID magnetometer. Samples \nwere carefully centered in magnetometer sample cavity to get the optimum signal intensity. The \ntemperature during magnetization study was kept at 15 0 K. Very low temperatures in 4 K range \nproduced detrimental thermal stresses which damaged the tunnel barrier . At higher temperature  \nclose to room temperature , typical magnetization data was highly noisy.  \n \nFig. 1: MEMSD cylinders for magnetic characterizations: Schematic of MTJ \n(Ta/Co/NiFe/AlOx/NiFe) (a) before and (b) after bridging of OMC channels across AlOx. (c) \nChemical structure of an  OMC unit, sulfur (S) helped covalent bonding of OMCs to the NiFe \nelectrodes. Fabrication of MEMSD cylinders required (d) the creation of several thousand \ncavities in the photoresist layer on insulating substrate, and (e) the sequential sputter \ndeposition of all the layers through the cavity. The lift -off step produced several thousand \nMTJs. OMC attachments convert each MTJ into (b) MEMSD. (f) 3D AFM image of a \nMEMSD. (g) Optical micrograph of a section of the chip containing ~7000 MEMSDs.       \n  5 For FMR studies Brucker EMX EPR spectrometer eq uipped with Brucker Mirowave \nBridge ER 041MR and Brucker Power Supply ER 081(90/30) was utilized. For all the \nexperiments ~9.7 GHz microwave frequency and room temperature  were  maintained . A \nmagnetic field was applied in the sample plane to study the unifo rm modes of thin films and \nmultilayers. Application of the magnetic field perpendicular to the sample plane did not produce \npromising distinct FMR signal s, for the meaningful study of OMCs  effect  on MTJ . Before every \nFMR study, empty cavity’s spectra was c hecked for the background signal at 5 fold higher gain \nthan that used for the typical MTJ and MEMSD samples.    \nMFM studies were performed using Digital instrument’s multimode AFM. Highly sensitive \nMFM tips (supplied by Nanosensors .Inc) with the following specifications were utilized : Type \nPPP-MFMR, tip side Co coated, force constant  in 0.5-9.5 N/m range , resonance frequency 45 -\n115 kHz.  During MFM scans,  the gap between the probe ’s tip and substrate was 10 -150 nm. \nFor the high resolu tion and minimally interrupted MFM studies  a 100 nm tip sample gap was \nfrequently utilized .    \n \nResults and discussion:  Initially, m agnetization studies were performed to elucidate the effect \nof OMCs on the MTJ and overall MEMSD ’s attributes . During the magnetization studies \nmagnetic field was applied in the sample plane. The first sample, which was subjected to \nmagnetization study, was also studied by FMR and MFM along with other MEMSD samples; \nhence, for easy identification we named it as MEMSD -A. The MTJ (Ta (5 nm)/Co  (5 nm) /NiFe Fig. 2: M agnetization study of MEMSD: (a) magnetization curve of (a) Ta/Co/NiFe/AlOx/NiFe \nMTJ before and after OMCs attachment. Inset show OMCs induced magnetization curve did \nnot saturate up to 30 kOe (~3  T). OMCs did not affect the magnetization of individual (b)  \nbottom (Ta/Co/NiFe) and (c) NiFe top electrode.     \n 6 (5 nm)/AlOx (2 nm)/NiFe (12 nm)) used for MEMSD -A exhibited a saturation magnetization \nvalue of ~1.0  10-4 emu at ~2 kOe. For the comparison of MTJ’s saturation field with that of \nindividual electrode’s saturation field top  and bottom electrodes wer e also subjected to the \nmagnetization study.  The saturation magnetization for the unpatterned top [NiFe (12 nm)] and \nbottom [Ta(5nm)/Co(5 nm)/NiFe (5 nm)] electrodes w ere ~10 Oe and ~60 Oe, respectively (Fig \n2 b-c). A higher coercive field for MTJ dots, as  compared to the unpatterned individual \nelectrodes is supposedly due to the shape anisotropy of a MEMSD cylinder and inter -dot dipolar \ninteractions [21]. \n \nTable 1: Parameters for the calculation of OMCs induce d anti ferromagnetic coupling strength . \nParameters  Values  \nMagnetization of NiFe film(Ma)  1.2x105 A/m \nThickness of  soft film (t)  1.0x10-8 m \nApplied saturation field (H ext) 3T \nMEMSD dot diameter  3.0x 10-6m \nArea  7.1x10-12m2 \nVolume/MEMSD cylinder  2.0x10-19 m3 \nNo. of molecules  1.0x104 \nAntiferromagnetic coupling energy  1.5x10-6erg \nEnergy per unit junction area  21.87 erg/cm2 \nEnergy per molecule (in erg)  2.2x10-3 erg/(cm2 –molecule)  \n \nThe OMCs radically affected  the magnetization of Ta/Co/NiFe/AlOx/NiFe M TJ. OMCs \nbridging reduced the MTJ’s magnetic moment by ~90% and produced a linear magnetization \ncurve  (Fig. 2a) . In essence, OMC s transformed MTJ hysteresis curve into a  linear magnetization \ncurve. A linear magnetization response is the characteristics of paramagnetic materials or two \nantiferromagnetic ally coupled FM electrodes  [22, 23] . The MEMSD configuration, comprising a \nMTJ with two FM electrodes and OMC s, support the hypothesis that OMC induced  linear \nmagnetization is due to the development of strong antiferromagnetic coupling  (AFC)  between \ntwo FM electrodes  [22, 24] . A system comprising two FM electrodes showed  a linear \nmagnetization curve w hen AFC strengths ( J) wa s more  than the product of the anisotropy \nconstant ( K) and FM electrode thickness ( t) [22], i.e.,  J > Kt. A linear magnetization curve can \nbe used for the calculation of AFC strength. In a related study, a MTJ  with iron/silicon/iron  7 configuration  produced the linear magnetization curve for the limited range of magnetic field \n[24], after certain magnetic field the linear magnetization saturated. Magnitude of the saturation \nfield for an AFC induced linear ma gnetization curve , is believed to overcome the inter-electrode \nexchange  strength between the two FM electrodes . Based on the saturation magnetic field \nexchange coupling strength can be expressed  by the following expression:  \next aHMVJ ..\n         (1) \nHere, V= volume of FM electrodes, Ma= saturation magnetic field, and Hext is the \nmagnetic field at which magnetization saturate. In order to utilize this expression to calculate the \nstrength of OMC induced inter -electrode AFC we need to know H ext, exte rnal magnetic field at \nwhich magnetization saturated. However, magnetization of MEMSD -A did not saturate up to 3 T \nexternal magnetic field (inset of Fig. 2a); due to the lack of resources we were unable to raise \nHext>3 T.  Assuming that in our case H is at  least 3 T, we calculated the magnitude of J. Details \nof calculation J are tabulated in Table 1.  \nIt is n oteworthy that the magnitude of  the estimated OMC induced J is consistent with the \nmagnitude of extrapolated J from the analogous systems  [21, 22] . A system  analogous to \nMEMSD and with ferromagne t/nonmagnet/ferromagnet system with 1.54  erg/cm2 AFC strength , \nshowed the saturation of linear magnetization curve at ~0.5 T  [22]. We did not observe  \nsaturation of OMC induced magnetization loop up to 3  T, and hence we assume d that 3T to be  \nat least the lower bound of saturation field for our MEMSD system. Considering that the \nMEMSD ’s linear magnetization loop saturated at 3 T, the linear extrapolation of literature data \n[22] yielded corresponding OMC s induced AFC strength to be 9 erg/cm2. This magnitude of \nextrapolated J (9 erg/cm2) is of the same order of the J directly estimated by  using MEMSD \nspecific parameter s (21.87 erg/cm2). In summary, ~10,000 OMC enhanced the AFC between \ntwo FM electrodes of the Co/NiFe/AlOx/NiFe MTJ to 21.87 erg/cm2. Enhancement of inter -FM \nelectrode coupling per OMC unit is ~2.2 x 10-3 ergs/cm2, which  is close to the inter -electrode \ncoup ling strength of a typical bare MTJ [9]. \nMolecules [25], like quantum dots  [26] and atomic defects  [9], enable a stronger \nexchange coupling between  the two FM electrodes . Strength of molecular coupling  increase s \nwith their population . In the present case , cumulative effect of ~10,000  OMCs becomes  strong \nto the extent that magnetization properties of the Co/NiFe/AlOx/NiFe MTJ changed dramatically. \nAnalogous to MEMSD, Ni breakjunction with C60 magnetic molecule( s) [5], also produced \nunprecedented exchange coupling between two Ni FM electrodes. In both the cases magnetic \nmolecules showed dramatic effect when simultaneously coupled to the two FM electrodes. Can  8 OMCs interaction with the single FM electrode produce the dramatic change in its magnetic \nproperties?  To investigate the answer to this question we studied unpatterned FM electrodes \nbefore and after treating them with OMCs (Fig. 2 b -c). Covalent bonding of OMC s to \nunpatterned Ta/Co/NiFe (bottom electrode material) a nd NiFe (top electrode material) did not \nproduce any noticeable change in magnetic moment (Fig. 2 b -c). This study established  that the \ninteraction between OMCs  and single FM electrode is unable to produce the same dramatic \neffect , which it produce d when s imultaneously interacting with the two FM electrodes.   In order \nto ascertain that observed linear magnetization curve on MEMSD -A was due to development of \nstrong AFC, we performed magnetic moment versus temperature study. If MEMSD’s FM \nelectrodes are anti ferromagnetically coupled then a change in cumulative magnetic moment \nmust occur  when the supplied thermal energy overcome  the OMCs  induced inter-electrode \ncoupling energy [27].  \n \nIn our experiment , magnetic moment first increased up to ~30 K; presumably, increasing \ntemperature enabled the rotation of rather stiff and randomly oriented magnetic domains to \ncause increase in magnetic moment  (Fig. 3a) . The second transition around 330 K ca used \nsharp reduction in the magnetic moment. We argue that heating the sample to 330 K disrupted \nthe OMC s induced AFC between the two FM electrodes of the Ta/CoNiFe/AlOx/NiFe MTJ. To \ndetermine the nature and magnitude of OMC s induced coupling  on MEMSD -A, we plotted the \ninverse of magnetic susceptibility as a function of temperature (Fig. 3b) . The intersection point of \na tangent on the part after the second transition produced the useful information  (Fig. 3b) . This \ntangent intersect ed temperature axis at -330 K [27]; according to Curie -Weiss law this \nintersection on negative temperature axis indicates the  existence of AFC  [27], and represented \nthe Neel temperature . However, we are still unsure if the magnitude of the OMC induced AFC is \n~330 K. Conventional theories are unexpected to reveal the complete understanding  of OMCs Fig. 3: (a) Temperature versus magnetic moment, and (b) inverse of magnetic susceptibility \nversus temperature for the MEMSD -A.  \n 9 effect . OMC s can also produce a second order biquadratic coupling [28] and spin fluctuation \ninduced couplings  [5]. These effects are not encompassed  by the conventional method of \ncalculating AFC strength using moment versus temperature study  [27]. To further understand \nthe OMCs induced strong coupling same MEMSD -A sample was a lso studied by FMR and \nMFM ; results of these studies are discussed elsewhere in this paper .  \nWe found that OMCs effect was prominent when a test bed MTJ had two FM elec trodes. \nCan OMCs also affect the magnetic moment of a tunnel junction with one FM electrode? To \ninvestigate the OMC s effect two MTJ configurations  with Ta/Co/NiFe/AlOx/Pd (palladium) , and \nPd/AlOx/NiFe, were prepared . These configurations involved top or bot tom electrodes of the \nTa/Co/NiFe/AlOx/NiFe MTJ configuration, on which OMCs showed the dramatic effect. OMCs \nproduced highly intriguing effect on MTJ with one FM electrode. OMC increased the magnetic \nmoment of MTJ with Ta/Co/NiFe bottom FM electrode (Fig. 4a). On the other hand , OMCs \ndecreased the magnetic moment of MTJ with NiFe FM electrode (Fig. 4b).  These magnetization \nstudies (Fig. 4) suggested that OMCs interaction is specific to the FM electrode. Present study \nemphasized that a vast variety of MEMSD s can be produced by having a large combination of \nmagnetic molecules and FM electrodes.  By now we have shown three independent samples \nwhich showed OMC induced changes in magnetic moment. Change in magnetic moment can \nimpart dramatic changes in the tran sport characteristics of the MEMSD  [1].     \n \n \nDoes OMCs effect depen d on the nature of preexisting coupling between the two FM \nelectrodes of a MTJ?  This question arise f rom the fact that development of OMCs induced \nexchange coupling was observed on the MTJ with preexisting weak AFC (Fig.1) To further \nprobe the nature of OMC induced coupling we also studied MTJs with the ferromagnetically Fig.4: Effect of OMCs on the magnetization of MTJ with single (a) Ta/Co/N iFe and (b) NiFe \nFM electrode.  \n 10 coupled FM electrodes . As discussed elsewhere in this paper  FMR studies were performed to \ndetermine the nature of coupling prior to OMC attachment. The Ta/Co/NiFe/AlOx/NiFe \nconfiguration possessed weak AFC. Bridging of OMCs dramatically enhanced the AFC strength \n(Fig. 3). Interestingly, just small modification in Ta/Co/NiFe/AlOx/NiFe produced MTJ with \nferromagnetic inter -electrode coupling. The addition of Ta (5 nm) on the top of \nTa/Co/NiF e/AlOx/NiFe MTJ yielded Ta/Co/NiFe/AlOx/NiFe /Ta MTJ configuration with inter -\nelectrode ferromagnetic coupling. Interestingly, OMCs dramatically enhanced the magnetic \nmoment of the Ta/Co/NiFe/AlOx/NiFe/Ta MTJ (Fig. 5). OMC s induced magneti c moment \nincrease was equivalent to the deposition  of additional magnetic material . Our magnetization \nstudies ( Fig. 2 and 5)  exhibited that OMCs  enhanc ed the strength of preexisting inter -electrode \ncoupling. Another utility of the magnetization study reported in Fig. 5 i s to serve as a \ncomplimentary  experiment  to support the magnetization study on MEMSD -A (Fig. 2a). Same \nOMCs which dramatically increased the magnetic moment of MTJ with inter -electrode \nferromagnetic coupling (Fig.5), also dramatically reduc ed the magnetic mom ent of the MTJ with \ninterelectrode antiferromagnetic coupling ( Fig. 2 a). It is apparent that, OMCs are only \nresponsible for the dramatic response from MTJs . It is also noteworthy that four MEMSDs \nutilized in magnetization studies (Fig. 2 -5) were produced u sing identical experimental \nconditions . Additionally, the OMC batch [19] utilized in magnetization studies also produced \ndramatic current suppression [4] and successful multilayer edge molecula r electronics devices \n[8].  \n \nFig. 5: OMCs increased the magnetic moment of MTJ  (Ta/Co/NiFe/AlOx/NiFe/Ta),  showing \nweak ferromagnetic coupling between the two FM electrodes  in bare state .   11  \nFMR characterization was uti lized to investigate the validity of magnetization studies \nperformed on MEMSDs. First FMR study was ca rried out on MEMSD -A, the sample which \nshowed dramatic OMCs effect during magnetization study (Fig. 2 and 3). FMR response from \nMEMSD -A showed a major pea k at ~3.4 kOe and a very small hump around ~1.2 kOe  (Fig. 6a) . \nThe sharp peak at ~ 3.4 kOe was due to the carbon  tape, used for sticking the sample on to the \ncarrier rod. FMR pea k for the carbon tape was used for  monitor ing the variation in experimental \nconditions . The small hump around ~1.2 kOe is due to the MEMSD -A. A typical MTJ show ed \ntwo FMR peaks. However, the observation of only one peak with MEMSD -A is in agreement \nwith the FMR model  developed for analogous system of strongly coupled FM electrodes [17, \n29]. FMR model exhibited that when inter -electrode AFC strength exceed a critical value then \nonly single FMR mode will appear [17]. Moreover, the intensity of MEMSD -A’s single FMR mode \nwas much weaker than the intensity of typical FMR modes of the Co/NiFe/AlOx/NiFe  MTJ (Fig. \n6b). The reduction in intensity can be explained in terms of the following FMR intensity (I) \nexpression  [29]:  \nFig. 6: FMR study of Ta/Co/NiFe/AlOx/NiFe MTJ based MEMSD: FMR spectra of MEMSD -\nA in (a) 0 -6 kOe  and (b) 0 -2 kOe range. (c) FMR spectra of a MTJ before and after \ntransforming it into MEMSD (named as MEMSD -B). (d) FMR spectra zoomed in 0 -2 kOe \nrange.   12 \nKeffKeff\nHHHH MVFI2) ( 2       (2) \nWhere, M is magnetization, V is sample volume, and H is external field. F is the proportionality \nfactor, which  included  strength of FMR experimental factors like RF magnetic field, microwave \nfrequency;  lock in amplifier operational conditions etc. HKeff is effective anisotropy field and given \nby H Keff= (2K/M) -4πM,  where K is the magnetocrystalline anisotropy  [29].  \n \nFig. 7: OMCs induced FMR mode suppression on (a) MTJ with ~2 nm Al Ox and produced in \noptimized experimental conditions, (b) MTJ with ~1 nm thick AlOx, and (c) MTJ treated with \na new batch of molecules and deviated experimental factors.   13 According to the equation (2), the i ntensity of FMR mode depended  on the sample \nmagnetization M. Our magnetization stud ies on MEMSD -A (Fig. 2a) showed that OMCs reduced \nthe magnetization of Ta/Co/NiFe/AlOx/NiFe MTJ by ~ 90% in 0.5 -2 kOe range. Hence, the \nobservation of weak intensity of single FMR mode (Fig. 3a -b) is in agreement with the \nmagnetization study on the same ME MSD -A sample (Fig. 2a). To further strengthen the FMR \nstudy on MEMSD -A, we attempted to compare the FMR response from the same MTJ  before \nand after transforming it into a MEMSD ; the MTJ for MEMSD -A could not be studied prior to \nOMCs treatment because FMR s tudies were planned several mont hs after the magnetization \nstudies  (Fig. 2a). The MTJ utilized in the second FMR study was transformed into a MEMSD \nand named it as MEMSD -B.    \n \nFMR spectra of the  bare Ta/Co/NiFe/AlOx/NiFe MTJ, utilized for MEMSD -B, exhibi ted \nacoustic mode (peak with higher intensity) at ~670 Oe and optical mode (peak with smaller \nintensity) at 1120 Oe. The position of optical mode after the acoustic mode imply an \nantiferromagnetic coupling [30] between the two FM electrodes  via ~2 nm AlOx insulato r. We \nalso compared the two modes of bare MTJ with the FMR modes recorded on the  NiFe and \nTaCoNiFe  electrode s deposited in isolation. MTJ’s acoustic mode (optical mode) was close to \nthe mode position of isolated and separately grown NiFe  (Ta/Co/NiFe) layer ; the difference \nbetween acoustic mode  (optical mode) position and isolated NiFe  (TaCoNiFe) mode position \nwas 40 -70 (30-50) Oe. After the bridging of OMCs, MTJ  transformed in to MEMSD -B and \nexperienced dramatic change in the intensity of its two FMR  peaks ( Fig. 6c -d). Bridging of \nOMCs across Al Ox insulator (inset of Fig. 6c) diminished  the acoustic mode at 0.67 kOe (Fig. Fig. 8: AFM imaging of MEMSD showing FMR modes suppression: (a) 3D image of a \nMEMSD and (b) statistical comparison of MTJ cylinder height before and after transforming it \ninto MEMSD.  \n 14 6d). Only a hump around the optical mode position remained  (Fig. 6d) , which is consistent with \nthe FMR spectra of MEMSD -A (Fig. 6b).  The ap pearance of only a weak FMR peak on \nMEMSD -B represented the development of OMCs induced strong coupling between two FM \nelectrodes  [17]. The FMR result on MEMSD -B agrees with that of MEMSD -A. It is noteworthy \nthat FMR peak intensity for the carbon tape remain ed unchanged during the two experiments, \nsignifying that the parameters used in FMR studies before and after OMC treatment of the MTJ \nwere the same  (Fig. 6d) .   \nFMR studies on  several MEMSDs supported the OM Cs induced dramatic change in \nmagnetic coupling . In a control study a Ta/Co/NiFe/AlOx/NiFe MTJ experienced  the OMC \ninduced FMR mode suppression (Fig. 7a -b). However, OMCs effect varied among control \nMEMSD samples (Fig. 7) . Variation in the experimental con ditions, molecule quality, \nphotolithography conditions produced partial OMC s effect  in several cases (Fig. 7c -f). The AlOx \nsturdiness was the key parameter for invoking full potential of OMC bridges. For instance, the \nMTJ sample  where AlOx thickness was re duced to ~1.0 nm OMCs yielded partial effect  (Fig. 7c -\nd). Presumably, attempt to produce thinner AlOx barrier lead to the production of defective or \nleaky MTJs, which  failed to show complete FMR mode suppression. Two FMR modes of the \nbare MTJ with ~1nm AlO x (Fig. 7b) were  not as clear as observed with thicker AlOx (2 nm) \ntunnel barrier (Fig. 7a) . Another control MTJ  showed partial OMCs induced mode suppression \n(Fig. 7e -f) with the OMCs  batch . This p artial mode suppression commenced with unsound \nOMCs . The sa me batch of OMCs  was also ineffective in yielding conclusive multilayer edge \nmolecular electronics devices [8].  \nTo ensure that severe damage , like  complete etching of MTJs cylindrical dots,  is not the \nreason behind  FMR mode  disappearance, several physical characterizations  were performed. \nFor instance , the control MTJ sample which experienced strong OMCs’ effect (Fig. 7a) was \nsubjected to AFM thickness measurement before and after OMC treatment. More than 140 MTJ \ncylindrical dots were studied from the various part s of the control sample before and after OMCs  \nattachment. During this study, topography and 3D profile of individual MTJ cylinders was \nrecorded (Fig. 8a). Thickness measurement by the AFM showed unnoticeable stati stical \ndifference in the height of cylindrical dots (Fig. 8b)  proving that etching type ph ysical damage \ndid not occur . Similar  physical  thickness before and after OMC  treatment also evidenced  that \noxidation of FM electrodes , which could  have caused volume expansion,  did not take place.  \nAdditionally, h igh magnification optical microscopy with po larized light was extensively \nemployed to monitor the sample integrity. In some cases,  a small population of dots (5 -10%)  15 underwent partial to complete damage during the electrochemical OMC attachment step. FMR \nstudy of such samples did not exhibit more th an 10% change in signal intensity after OMC \ntreatment . A control sample, which was unknowingly processed with deviated photolithography \nparameters, underwent accidental wash off of > 20% MTJ cylindrical dots during OMC \ntreatment. This sample represented  the case of highly damaged MEMSDs , and hence this \nsample was utilized to study the FMR response in the worst -case scenario.  This sample  \nproduced strong FMR modes (Fig. 9a), which were quite different from the stable sample (Fig. \n6). The optical micrographs collected from the different region of this damaged sample showed \nthat a number of MTJs remained in the sound state (Fig. 9b -c).    \n \nTo investigate the potential damage from the electrochemical OMC attachment step two \nadditional control experiments were p erformed. In the first control experiment MTJ cylindrical \ndots were prepared with ~ 4 nm AlOx. Use of ~4 nm thick AlOx disable d the bridging of ~ 3 nm \nOMCs across AlOx insulator . In this case OMCs were not expected to enhance the exchange \ncoupling between the FM electrodes of MTJ. FMR studies before and after OMC treatment \nshowed that OMCs did not produce any significant change in the intensity of FMR modes (Fig. Fig. 9: FMR study of the damaged  MEMSDs showing partial or complete damage to ~30% \nMTJ cylinders during electrochemistry for OMCs attachment. (b) and (c) a re the optical \nmicrographs from two different region s of the damaged MEMSD chip.  \n 16 10a). In the second control experiment, cylindrical dots were made up of only Ta/Co/N iFe \nbottom  electrode materials . The significance of this control experiment is that OMC solution can \naccess the edges of Ta/Co/NiFe cylindrical dots and to etch -sensitive Co therein . FMR studies \nbefore and after OMC treatment showed no change in FMR modes’ intensity  (Fig. 10b) . This \nobservation is consistent with our control experiments in which Co etching rate was <10 \nnm/OMC treatment. A 3  m diameter dot is unexpected to loose significant Co to show \nmeasureable change in FMR mode intensity.    \n \nCan unnoticed oxidation of NiFe produce  antiferromagnetic oxides like  NiO and FeO, \nthus leading to the disappearance of FMR modes ? To check this po ssibility a NiFe electrode \nwas purposefully plasma oxidized to reveal  the nature of changes in the FMR mode . Plasma \noxidation of NiFe did not reduce the intensity of FMR mode recorded before oxidation. \nInterestingly, additional resonance peak emerged at 80 0 Oe  [4]. We attribute d this addition al \npeak to the NiFe surface -oxides.   \nWe also investigated the effect of sample  misalignment during FMR study on the  mode \nintensity. To check this possibility a chip with Ta/Co/NiFe/AlOx/NiFe MTJ cylindrical d ots was \nstudied. This sample was intentionally misaligned by ~ 10º with respect to the direction of \nexternal magnetic field. We were able to precisely place the sample in the plane of applied \nmagnetic field with an accuracy of ±10º. Two major FMR modes rem ained unchanged with small \nvariation  (Fig. 11) . We also s tudied the effect of reinserting the sample in FMR measurement \ncavity . Reinsertion of sample in FMR cavity was essential to do the comparative study before \nand after OMC treatment. No measurable chan ge in FMR mode intensity was observed  when Fig. 10: Monitoring chemical damage due to electrochemical OMC treatment step: (a) FMR \nresponse of MTJs with ~ 4 nm AlOx and (b) FMR study of only bottom FM electrode that \ncontained etch sensitive Co.  \n 17 sample was reinserted  (Fig.11). Additionally, FMR mode for the carbon tape at ~ 3.3 kOe \nremained unchanged with this misalignment and reinserting of the sample. In addition to major \npeaks FMR spectra also showed  the presence of labile walker modes between ~1.3 to 3.3 kOe. \nHowever, these second ary effect s were not utilized to gauge the OMC s effect , and their study is \nbeyond the scope of present work.  \n \nWe observed that OMCs effect was mainly observed on the Ta/CoNiF e/AlOx/NiFe MTJs \nwith inherent weak AFC . Does OMCs effect depend on the initial coupling between FM \nelectrodes of a bare MTJ, i.e. before OMC treatment  step? To answer this q uestion we utilized a \nMTJ with ferromagnetic inter-electrode coupling. Fortunately , a MTJ with ferromagnetic inter -\nelectrode coupling was realized with the minor modification in the thin film configuration showing \nAFC in the bare state. In a serendipitously discovery we found that just the addition of Ta layer \nabove the top NiFe electro de produced Ta/CoNiFe/AlOx/NiFe/Ta MTJ, which possessed the \nferromagnetic coupling  between the two FM electrodes. For Ta/CoNiFe/AlOx/NiFe/Ta MTJ the \nlow intensity FMR mode (optical) occurred at lower magnetic field while high intensity FMR \nmode (acoustic) occurred at higher magnetic field. This particular characteristic of FMR spectra \nrepresents the appearance of ferromagnetic coupling between the two FM electrodes [21]. For \nthe present FMR study we used MEMSD  sample which also showed unambiguous OMCs effect \nin the magnetization study (Fig. 5). FMR spectra of OMCs treated  Ta/CoNiFe/AlOx/NiFe/Ta MTJ  \nproduced only one prominent mode (Fig. 12 a). The reasonably high FMR mode intensity of this \nsingle mode is consis tent with the enhanced magnetization on the same MEMSD (Fig. 5) . In the \nsame manner, very low intensity of single FMR mode on MEMSD -A, utilizing MTJ with Fig. 11: Effect of variation in angle of sample orientation with respect to direction of the \napplied magnetic field.   \n 18 antiferromagnetically coupled FM electrodes, was consistent with the reduced magnetization on \nthis sam ple (Fig. 2a).  According to theoretical modeling the development of single FMR mode is \nindicative of strong ferromagnetic coupling between the two FM electrodes  [17]. Two FM \nelectrodes coupled by the strong ferr omagnetic coupling behave d like a single ferromagnetic \nelectrode  and produced  a single mode  [17]. We were  unable to compare the intensity of FMR \nmodes before and after OMCs attachment  on the same MTJ . FMR studie s were not planned \naround the time of magnetization study , and when all the experimental factors were in the \noptimized state. FMR data for the bare Ta/CoNiFe/AlOx/NiFe/Ta reported here was recorded \nfrom the different sample. In summary,  OMCs produced sing le prominent (very weak) FMR \nmode  after interacting with the bare Ta/CoNiFe/AlOx/NiFe/Ta (Ta/CoNiFe/AlOx/NiFe) MTJ with \nferromagnetic (antiferromagnetic) inter-electrode coupling . The contrasting magnetization (Fig. 2 \nversus  Fig. 5) and FMR (Fig. 12) studi es suggested that OMCs enhanced the pre -existing \ncoupling between two FM electrodes  of a MTJ . Simultaneous evaluation of these results also \nsupports  that the only OMCs , not any artifact , is the reason behind the dramatic changes on \nMTJs.  \n \nReversing the effe ct of molecules can produce an unequivocal support to the observation \nthat only OMCs produced linear magnetization (Fig. 2a) and FMR mode suppression (Fig. 6). \nOMCs effect can be reversed by  the two methods: (a) destroy the OMCs to recover the \nresponse of  a bare MTJ or transform a MEMSD back to MTJ and (b) damage the tunnel barrier \nso that inter -electrode coupling via defects within the barrier supersede the OMCs e ffect, \nFig. 12: Effect of OMCs on FMR modes of (a) Ta/Co/NiFe/AlOx/NiFe/Ta MTJ with weak \nferromagnetic coupling, a nd (b) Ta/Co/NiFe/AlOx/NiFe MTJ with weak antiferromagnetic \ncoupling.   19 however in this case FMR response from damaged MTJ will be much different than that of a \nstable MTJ. We attempted both type of reverse experiments.  \n \nIn the first approach, OMC bridges were burnt in the oxygen plasma  (inset of Fig. 13a -b). \nOxygen plasma was generated by the 50 W RF bias at 60 mTorr pressure of 1:1 Ar:O 2 mixture \nin the sputteri ng machine chamber. For this study the MEMSD -A, on which magnetization and \nFMR study exhibited dramatic changes due to OMCs, was utilized. Prior to the burning of OMCs \nin oxygen plasma, MEMSD -A showed rather featureless FMR spectra (Fig. 13a). FMR study \nafter burning the OMCs  showed the emergence of two unambiguous FMR modes (Fig. 13a). \nThe position and shape of these two modes are in excellent agreement with the acoustic and \noptical modes recorded on a separately grown bare Ta/Co/NiFe/AlOx/NiFe MTJ (Fig. 1 3b). It is \nnoteworthy that w e did not perform the FMR study on the MTJ before treating it with OMCs to Fig. 13: Reversing the OMCs effect: (a) Burning the OMCs on MEMSDS -A in oxygen \nplasma produced two FMR modes (b) Two FMR modes appearing after plasma burning of \nOMCs are consistent with  the typical two FMR modes corresponding to a bare MTJ. (c) \nDamaging AlOx barrier between, without affecting OMCs, produced diffused FMR modes, \n(d) Diffused FMR modes did not match with the modes of bare and stable MTJ.     \n 20 produce MEMSD -A. Because  MEMSD -A was fabricated for the magnetization study and at that \ntime we did not plan for the FMR studies. The FMR spectra of plas ma treated MEMSD -A and a \nbare MTJ are in excellent agreement and confirm following: (a) OMCs only transformed the \nFMR modes, (b) MTJ used for fabricating MEMSD -A possessed the same magnetic \ncharacteristics as observed with the typical bare MTJ.    \nIn the s econd control experiment, we attempted to make OMCs induced coupling \nineffective, but without damaging the OMC channels. Simple strategy we employed was based \non creating stronger inter -electrode coupling  via a planar ~ 2 nm AlOx barrier (inset of Fig. 13c -\nd). To create stronger coupling via defects, AlOx, tunnel barrier was damaged by thermal \nstresses [31]. MEMSD -B was heated at 390 K for 30 min in the flowing helium ambience, to \navoid unintentional oxidation. Subsequently, this sample was cooled to room temperature and \nwas subjected to FMR study. FMR spectra of this sam ple showed a broad FMR mode within 0 -2 \nkOe range (Fig. 13c). We attributed the broadened resonance mode to the emergence of a wide \nrange of exchange coupling strengths between the two FM electrodes via defective AlOx \ninsulator [30]. It is noteworthy that the broad  FMR mode and two distinct FMR modes of the \nbare MTJ utilized by MEMSD -B were in 0 -2 kOe range (Fig. 13d). In essence, we were able to \nrecover magnetic signal from the MEMSD, which suggested that dramatic reduction in FMR \nmode intensity was not due to the permanent structural damage of any kind.  \nMFM characterization was utilized as an approach to investigate the effect of OMCs on \nthe individual MTJ.  The role of MFM is crucial to support the results of magnetization and FMR \nstudies  which were unable to di stinguish the response from the individual MEMSD . The  \nmagnetization and FMR studies exhibited the average response from several thousand MTJs  \nand hence coul d not show the effect of OMCs from  a single MTJ cylindrical dot . MFM images \nwere produced by  plottin g the phase data over the scan ned area. Bare MTJs \n(Ta/Co/NiFe/AlOx/NiFe) produced clear phase contrast and could be unambiguously recognized  \nagainst nonmagnetic background like typical magnetic materials [23]. To understand the OMCs \neffect through MFM study , we mainly focused  on MEMSD -B (Fig. 6) , which showed OMCs \ninduced modes suppression during FMR studies. Several MFM scans from regions of MEMSD -\nB showed  negligible magnetic contrast at the sites of the MEMSDs. H owever, according to \ntopographical scans the negligible MFM contrast was not due to the loss of MEMSDs at those \nsites (Fig. 14a). Topographical scans at the locations of MEMSDs produced unequivocal  and \nuniform height data of ~34 nm , closely matching with the targeted physical thickness of the \nMEMSD cylindrical dot .   21  \nSeveral self -reference d scan s evidenced the accuracy of our MFM experimental \nprocedure investigating  the effect of OMCs . One can argue that source of low MFM contrast \nmay be due to poorly o ptimized experimental factors . However, our MFM experimental data \nassert the accuracy of MFM procedure. Fortunately, in some MFM scans two types of magnetic \nforce images were recorded  (Fig. 14 b). Majori ty of the MEMSDs cylinders showed highly dim \nmagnetic contrast while some of the MEMSDs showed a clear magnetic contrast. High contrast \nMFM images resemble d with that of the typical bare MTJs , and are believed to arise from those \nMTJs which were not affect ed by OMCs . We utilize d high contrast phase images as the \nreference to discuss the majority of low contrast data  (Fig.14b) . The topographical image for the \nMEMSDs in box 1 and 2 are identical (Fig. 14a); the corresponding MFM  images in box -1 and \nbox-2 show ed stark difference (Fig.14b). Such a dramatic change in two magnetic image s was  \nnot possible if dim magnetic contrast was due to improper experimental procedure or any \nartifact . Furthermore, MFM images of three dots in the third row showed that central MEMSD \ndot is of the dim color contrast  (Fig. 14a) , while the two immediate neighbors showed high \nmagnetic contrast (Fig. 14b).  Again, such a dramatic change in magnetic images of the \nneighboring cylindrical dots is not possible if faint magnetic contrast was  due to the improper \nFig. 14: MFM study of MEMSD -B: (a) Topographical image showing height data of \nMEMSDs and (b) corresponding magnetic force images were produced by using phase \ndata.   22 experimental conditions. The simultaneous observation of dim and strong MFM contrast \nsuggested that all the MTJs were not affected by the OMCs (Fig. 14b) . This MFM scan showed \nthat some MTJs remained unaffected by the OMCs and rational ize the FMR studies where \nOMCs showed partial effect (Fig.7c -f). Reduction in FMR signal intensity corresponded to the \npopulation of MTJ affected by OMCs (Fig. 7c -f).    \nOur magnetization study (Fig. 2a) rationalized the observed dim MFM contrast (Fig.14b)  \non MEMSDs. Magnetization study showed that Co/NiFe/AlOx/NiFe MTJs lost ~ 90% magnetic \nmoments  when treated with OMCs for producing MEMSDs, specifically MEMSD -A (Fig. 2a).  In \nthe simple term, the MFM contrast (MFM contrast ) of two antiferromagnetically coup led FM \nelectrodes is presumably due to the difference in magnetic moment s of the top (M Top) and the \nbottom (M Bot) FM electrodes (equation 3) .   \nBot Top Contrast M M MFM \n       (3)   \n \nDramatic change in MFM contrast (Fig. 14) is in agreement  with the equally dr amatic \nchanges in the FMR and magnetization studies. FMR and magnetization studies suggested that \nOMCs produced unprecedented strong AFC between the two FM electrodes of \nCo/NiFe/AlOx/NiFe MTJ. W e hypothesized that OMC s not only produced a strong AFC but al so \nserved as a highly effective spin filter. Potential role of OMC as a spin filter can be justified by its Fig. 15:  (a) 2D image showing OMC channel bridging the AlOx insulator between two FM \nelectrodes. (b) OMC modified energy  band diagram showing additional molecular energy \nlevels in between two FM electrodes and the modified barrier height.  \n 23 chemical structure (Fig.1c). The center of a n OMC is a cluster with a net spin, which is directly \nconnected to FM elec trodes via alkane spin channel s. Alkane spin channels can preserve spin \ncoherence for longer distance and time as compared to AlOx  or inorganic mediums [14]. When \nan OMC become the dominant spin channel then it ’s core  expectedly  start working as a valve .  \nOMC’s core will have the highest occupied molecular orbitals  (HOMO) and the lowest \nunoccupied molecular orbitals (LUMO). It is surmis ed that HOMO and LUMO will  allow the \ntransport of only one type of spin via them. It means , HOMO and LUMO of OMC core wil l work \nas the two spin valves.  This selective passage of one s pin type through HOMO or LUMO siphon \nout one type of spin, e.g. spin up , and pump in opposite  spins , e.g. spin down, at one o f the two \nMEMSD’s FM electrode . This spin filtering will produce two ~100% spin polarized FM \nelectrodes.  As a result in equilibrium  state a MEMSD had two magnetic electrodes with equal \ndensity of states of the opposite spins, located at ~2 nm from each other . In MFM imaging \nmagnetic probe  was ~ 100 nm above the MEMSD . MFM probe interacted with the two \nelectrodes of a MEMSD through the  long range dipolar interaction. For this distant MFM probe \nnet magnetic moment will be the sum of magnetic moment of the two OMCs modified FM \nelectrodes. When two electrodes are antiparallel to each other, the magnetic moments  of two \nelectrodes cancel eac h other to give very faint MFM contrast (Fig. 14). This hypothesis is \nstrongly supported by our magnetization studies, which established that OMCs reduced the \neffective magnetization of MTJ by ~90%. The magnetic moment reduction  in our magnetization \nstudy  (Fig. 2a) supports the reduction of  the magnetic contrast in MFM study (Fig. 15) .  \n \nStudy of the mechanism behind an OMC’s effect is crucial for the advancement of \nMEMSD. Following sections focuses on several perspectives useful in  explain ing the OMCs \ninduced dramatic change s in the inter-electrode coupling . According to our magnetization study \n(Fig. 2a) and c onventional  definition [22] the lower bound of OMCs induced AFC strength was \n~22 erg/cm2 [Table -1]. T his estimate showed that OMCs increased the coupling between two \nFM electrodes of a MTJ by ~5 orders . The typical inter -electrode coupling strength on a MTJ \nwas < 1x10-3 erg/cm2  [9]. However, our  calculation of J did not focus on several potentially \nimportant aspects like: (a) nature of interaction between a OMC and two individual FM \nelectrodes, (b) role of biquadratic coupling, which is expected to emerge when two FM \nelectrodes are coupled by nan oscale objects [28], (c) the role of spin fluctuation s in enhancing \noverall exchange co upling, (d) the critical number of OMCs required to transform a  ~10 m2 \narea MTJ with ~20 nm thick FM metals into a MEMSD.   24 Quantum mechanical calculations on such MEMSD configuration are extremely \nchallenging . Recently, a few  theoretical and experimental studies have been conducted to \nunderstand the magnetic interaction b etween magnetic molecules and a FM electrode [32]. The \ndensity functional theory (DFT) based studies were  performed on a very small system involving  \nsingle molecule and one FM electrode. Such  calculation s on a system of single molecule and \nfew hundred FM atoms necessitated a large number of  approximations [32]. Our MEMSDs \ninvolved ~10,000 OMCs simultaneously coupled with the two microscopic FM electrodes,  and \neach FM electrode involved ~1012 atoms. Though q uantum mechanical calculation s are critical \nto reveal the fundament al level understanding of MEMSD,  but due to complexiti es and \nlargeness of MEMSD such calculations are extremely challenging and  beyond the scope of \npresent work. For MEMSD it is more practical to utilize existing models and experimental \nstudies to discuss the unprecedented coupling observed on MEMSDs.  \nFig. 16: MEMSD introduction: (a) Schematic showing bridging of single OMC between two \nNiFe electrodes. Band diagram of a MTJ (b) before  and (c) after bridging OMC channels \nacross insulator, as shown in the inse t of (a). (d) New band diagram resulting from the \nOMC induced strong exchange coupling. (e) 1D analog of MEMSD and (f) display of \nexchange coupling strength between nearest neighbors of a OMC.   \n 25 To unde rstand the OMCs effect on MEMSD  we focus ed on two aspects: (a) q uantifying \nthe magnitude of OMCs induced inter -electrode exchange coupling and (b) estimating the \nmagnetic moment of OMC affected MTJ. For the quantification of inter - electrode exchange \ncoupl ing OMCs can be viewed in two ways.  In the first role, OMCs can be viewed as the agents \nthat produce intermediate energy levels and modify tunnel barrier height  and thickness (Fig. 15)  \n[8]. OMCs are expected to produce an effective energy band  diagram of the  double barrier \ntunnel junction akin to quantum well s [33]. After the establishment of OMC channels (Fig. 15a) \neffective barrier height changed from ФAlOx to ФOMC and effective barrier thickness reduced from \nphysical thickness of AlOx to the physical length of tether attached to OMCs. To quantify OMCs \ninduced coupling we have utilized Slonczewski’s model derived for MTJs  [34]; we rationalize the \nselection of this model (eq uation 4 ) because our MEMSD ar e the modified MTJs  or customized \nMTJs . \n \ndk Be\nkk kkdkk kkkkkkJ.2\n22 2 22 2 222 2 3\n) () () () )( (\n \n  \n\n     (4) \nHere, d and ФB are the barrier thickness and barrier height, respectively, of the insulating \nmedium betwee n two spin -polarized electrodes.  Whereas, k ↓ and k ↑, are the wave vectors of the \nspin-down and spin -up electrons of the FM electrodes . Equation 4 couple the electrons’ wave \nvectors in the FM electrode with the electrons’ wave vector in the region between FM \nelectrodes.  The wave vector of an electron in the insulating regime is k an d is given by \nk=√(2.ϕ B.m.ћ-2 ). Here, m is the effective mass of the electron and ћ is Planck’s constant. We \nassume that  a OMC , which has net spin state, behaves like a spin-polarized  electrode and \nhence reduces the effective barrier thickness from 2  nm (AlOx thickness) to 1 nm (distance \nbetween center of the OMC  core to the FM electrode edge). However, we are unable to \ndetermine k ↓ and k ↑ for the FM electrodes. k ↓ and k ↑ strongly depends on the interfacial \nproperties and the spacer(s ) between the two FM el ectrodes.  According to our magnetization, \nFMR  (Fig. 6) , and MFM studies,  OMCs produced dramatic changes in the magnetic properties \nof FM electrodes. The calculation of k ↓ and k ↑ for OMCs affected FM electrodes is challenging \nand beyond the scope of presen t work. We  rather choose to work with the following simplified \nform of eq uation 4  [34] given in equation 5.   \n \n2).2 exp(\nddkJ                                 (5)   26 According to the analysis of charge transport data,  barrier thickness (d) chan ged from ~2 \nnm to ~1 nm and barrier height changed from ~1  eV to 0.6 eV [8] after OMCs attachment. \nThese OMCs induc ed modifications in barrier height and barrier thickness increased the \nmagnitude of J by ~5 orders.  \nThis calculation does not explain utility of OMCs;  it is arguable that such  change in barrier \nheight and thickness can also be produced by simply reducing the barrier thickness using a \nthinner insulator or reducing the barrier height by using any appropriate insulating material. \nHowever, to date manipulation of the insulating barrier between FM electrodes have not yielded \ndramatic changes according to the on e reported here. However, modifying barrier properties in \na trilayer thin film configuration with film thickness in few monolayer ranges exhibited intriguing \neffect of inter -electrode exchange coupling. I n a related study, reducing the thickness of non-\nmagnetic metallic spacer  between the two FM electrodes to few monolayers produced dramatic  \nchanges in the Curie temperature of the constituents FM electrodes  [18, 35] . Importantly, this \nphenomenon could not be explained in the conventional theoretical frame work  as utilized for \nthe development of Slonc zewski’s model derived for MTJs [34] (equation 4) ; additional \nconsideration of role of spin fluctuations satisfactorily explained the dramatic change in the basic \nmagnetic properties like Curie temperature of the FM electrodes  [18]. Spin fluctuations are also \nexpected to be crucial for defining the anomalous magnetic response of MEMSD type molecular \nspin devise  as well . In fact theoretical and experimental studies on devices akin to MEMSD  \nhave established the utility of spin fluctuations. In the theoretical study by Martinek et al. [36], \nspin fluctuations assisted coupling produc ed anomalous Kondo level splitting phenomenon \nwhen a magnet ic quantum dot or nanoscale object was inserted between the two FM \nelectrodes . In an experimental study , Pasupathy et al. [5] realized the spin fluctuation assisted \nanomalous Kondo splitting on a molecular spin device  configuration . This molecular spin device \nwas produced by placing a C60 molecule with net spin state in th e nanogap of nickel \nbreakjunction. However, Pasupathy et al. [5] did not report any exclusive magnetic \ncharacterization like magnetization, FMR, and MFM before and after inserting C 60 molecules in \nthe Ni breakjunction. Because of the lack of magnetic studies,  it is uncertain if C 60 induced \nstrong inter -electrode coupling a lso produced dramatic changes in the magnetic properties of Ni \nFM electrodes , as observed on our MEMSDs  when OMCs were bridged between the two FM \nelectrodes  (Fig. 1) .           \nTo quantify  the magnitude of OMCs induced exchange coupling,  we investigated ot her \nanalogous systems. In the f irst comparison , we considered OMCs to be analogous to atomic  27 impurities . Zhuralev et al. [9] have studied the nature and strength of impurities enhanced \nexchange coupling between the two FM electrodes of a MTJ. This theoretical study provide \nsatisfactory explanation to the experimental observation of impurities enha nced exchange \ncoupling and the magnitude of inter -electrode coupling [9, 21] . However, the magnitude of \ntheoretically calculated inter -electrode exchange coupling was ~ 2 orders smaller than he \nmagnitude of  lower bound of OMCs induced inter-electrode exchange coupling on our MEMSDs \n(Table -1).  \nIn the second comparison, we considered OMC s to be analogous to magnetic \nnanoclusters. Wong et al. [10] studied the effect of iron  (Fe) nanoclusters on the inter  electrodes  \nexchange coupling of a MTJ . In this study, Fe nanoclusters were embedded within the \nmagnesium oxide ( MgO ) tunnel barrier, throughout the planar area of a MTJ. The MTJ i n this \nstudy included one hard and one soft FM electrode s, similar to our MEMSD configuration . \nDepending on the Fe nanoclusters position in the MgO tunnel barrier, the inter -electrode \nexchange coupling not only exhibited a significant variations in magnitu de but also  switched \nnature of coupling between antiferromagnetic and ferromagnetic  types [10]. Nanoclusters, \nhowever , only enhanced the int er-electrode coupling  by a factor of < 2. One of t he most \nimportant  finding of this study is the unambiguous demonstration of the effect of using hard and \nsoft FM electrodes for observing the strongest effect of nanoclusters. Nanoclusters induced \ninter-electrode exchange coupling was 160% higher for Fe/MgO/Co (MTJ with hard and soft FM \nelectrodes) than that of Fe/MgO/Fe (MTJ with soft FM electrodes). This study  [10] complement  \nour observation of the st rongest OMC effect when MEMSD employed the MTJs with hard and \nsoft FM electrodes  only. The observation of current s uppression phenomenon discussed \nelsewhere,  and the observation of change in magnetic properties discussed in the present paper  \nrequired Co/ NiFe/AlOx/NiFe MTJ.  In a MEMSD, the Co/NiFe electrode was ~4 times \nmagnetically harder than the NiFe electrode.  In addition to our MEMSD , Pasupathy et al.’s \n[5]experimental observation of anomalous Ko ndo splitting also required a molecular spin device  \nwhere two FM electrodes possessed significantly different magnetic hardn ess.  \nThe interaction of nanoscale objects with the FM electrodes of different magnetic \nhardness governs  the strength of inter-electrode exchange coupling ( J). Wong et al. [10] \nsatisfactorily modeled the effect of nanoclusters by focusing on its interaction with the individual \nFM electrodes . We cu stomized the Wong et al.  [10] model to exp ress the OMC induced \ncumulative J in a MEMSD (equation 6 ):  28 \n2)OMC Hard OMC Soft OMC Hard OMC Soft\nAlOxJ J J J\nJJ     \n    (6) \nHere, JSoft-OMC (JHard -OMC) is the  exchange coupling between OMC and soft (hard) FM electrode, \nand J AlOx is the inter -electrode exchange coupling via ~ 2 nm AlOx insu lator. In the present case \nJAlOx<< JSoft-OMC or J Hard -OMC. Assuming,  J Soft-OMC ≈  J Hard -OMC and using the estimate of OMC s \ninduced J from table 1 we found that J Soft-OMC or  J Hard -OMC  is of the order of ~22 erg/cm2. The \nOMC s induced exchange coupling is 3 -4 orders higher than the exchange coupling produced by \nthe nanoclusters.  \nIn essence,  OMC produce d much stronger effect as compared to impurities and \nnanoclusters.  OMCs ’ strong effect is mainly due to its uniform shape and size that is not \npossible with nanoclusters and impurities. Moreover,  nanoclusters are  separated from  the FM \nelectrodes by MgO  and represent the case of physiosorption interaction; whereas OMC s are \ncovalently attached to FM electrodes via covalent bonding through thio functional group  (Fig. \n16a). The nature of molecule - metal bond ing strongly affect the molecular transport and inter -\nelectrode coupling  [37]. For instance, the cobalt phthalocyanine (CoPc) molecule exhibited ~ \n200 K Kondo temperatures when the nature of molecule -metal interaction c hanged from \nphysiosorption to covalent  bonding  [37]. The n ature of OMC interaction with FM electrodes of \nMEMSD will critically affect  the effective exchange coupling [32]. Accurate estimation of J of \nMEMSD will also necessitate the experimental studies focusing on the determination of OMCs \ninteraction with individual FM electrodes.  \nAfter the discussion on the magnitude of exchange coupling it is imperative to focus on \nthe mechanism by which a strong exchange coupling can affect the magnetic properties of \nMEMSD’s FM electrodes. Our magnetization, FMR and MFM studies showed dramatic changes \nin the inter -electrode coupling and the magnetic moment of  the FM electrodes when a MTJ  was \ntreated with OMCs. To elucidate the effect of OMC bridge s we compared our MEMSD with Ni \nbrekjunction based molecular spin device [5]. In the Ni break junction , one or few C60 molecule s \nconnected the two Ni FM electrodes similar to the case of  MEMSD where ~10,000 OMCs \nconnected the Co/NiFe and NiFe FM electrodes. Pashu paty et al. [5] reported that C 60 \nmolecule(s) invoked spin fluctuati on assisted strong exchange coupling . This strong exchange \ncoupling produce d a local magnetic field , which influence d the correlated Kondo energy level \narising due to the strong overlap between the molecule and metal lic energy level s.  This strong \ncoupling  induced local magnetic field is equivalent to the external magnetic field.  We expect that \nin our case an OMC produce d similar magnetic coupling induced local magnetic field as  29 produced by a C 60 molecule [5]. Pasupathy et al. [5] estimated that only one or few C 60 \nmolecules produced  > 50 T local magnetic field. In our case, a much stronger local magnetic \nfield is expected since ~10,000 OMC s are coupling the FM electrodes of a MEMSD. We \nhypothesized that OMC induced local magnetic field will influence the stoner splitting of FM \nelectrodes . The Stoner splitting determines the energy gap between DOS of spin up and spin \ndown electrons. The loca l magnetic field will influence the stoner splitting in the same manner as \nan external magnetic field will influence the Zeeman splitting.  Molecule induced local magnetic \nfield in turn can produce a difference in chemical potential under the effect of magn etic field to \nyield much different spin DOS and different spin polarization (P).  In addition, OMC’s HOMO -\nLUMO is expected to serve as a spin filter to amass only one type of spin on each of the \nMEMSD’s FM electrode. A plausible mechanism about the OMC depe ndent spin filtering has \nbeen discussed in the context of MFM studies of MEMSD (Fig. 14) In order to determine the \nmagnitude of OMCs induced local magnetic field we have customized the model used by \nPasupathy et al. [5].  In the context of MEMSD with two anti-parallel FM  electrodes the coupling \ninduced magnetic field (B)  can be given by the following model  (equation 7) :  \nBN\nnnBB nTT\ngP P a\nB.) (\n1\n\n\n      (7) \nHere, N is the number of OMCs  bridging the gap between FM electrodes , PT (PB) is the degree \nof spin polarization o f the top (bottom) FM  electrode.  ΓnT (ΓnB) is the coupling energy of the nth \nmolecule with the top (bottom) layer.  g and μ B denote d the gyromagnetic ratio for the molecule \nand Bohr magnetron, respectively.  \nIn order to estimate the magnitude of B, we can assume  N=10,000, aP is 0.15  [26] and \ngμB ~115  μV/T  [38]. Generally molecular coupling energy is ~10 meV  [5]. Without delving into \ncoupling strength of an OMC s with the top and bottom FM electrode, we focus o n the difference \nin coupling energy. Assuming the difference in coupling ene rgy is  0.001 -0.01  mV, we obtain ed B \nin the range of ~100 -1000  T. We assumed, this OMC induced local magnetic field is applied on \nthe FM electrodes to produce Stoner  splitting and can be quantified by the following expression \n[39] (equation 8) . \n∆= ±0.5(Pgµ 0µBH)        (8) \nAssuming P~0.5 for NiFe [1], the magnitude of OMC induced additional Stoner splitting  \nwas found to be in the 30 -300 mV range. This analysis shows that MEMSD’s FM electrodes can  30 indeed undergo reshuffling of DOS  (Fig. 16b -d), which in turn will affect their degree of spin \npolarization. However, this analysis by no mean is the best one , and would need significant \nadjustment s to produce a more realistic estimate. The FM atoms, which are in direct contact of \nan OMC,  are likely to experience the strongest effect. However, our magnetic studies have \nshown that OMCs induced exchange coupling influenced the microscopic junction area , not just \nfew atoms in the vicinity of molecular junctions . It means there must be some mechanism in \nplace to transport the OMC coupling effect from its nearest neighboring FM atoms. We surmise \nthat Heisenberg exchange coupling and dipolar long range coupling among the atoms of a FM \nelectrode transpire the OMC’s coupling effect into the regions , which are even several  m off \nfrom the molecular junction.  In order to validate our hypothesis we set up 1D Monte Carlo \nsimulations on the system representing 1D form of a MEMSD (Fig. 16e). The magnetic ordering \nof FM electrodes (FM1 and FM2) was studied with the va riation of exchange coupling strength \nof OMC with the FM electrodes. During Monte Carlo simulation, we minimized the system \nenergy as given by the following expression  (equation 9) : \n   \nMR RM ML LM iiRN\ni iiLN\ni SSJ SSJ SSJ SSJ E       1 1 1 1 \nIn the above equation: J L, JR, JLM, and J RM are th e Heisenberg exchange coupling strengths for \nthe FM electrode on the right,  FM electrode on  the left, between the left metal and the molecule, \nbetween the right metal and the molecule, respectively (Fig. 16f). S i and S i-1 are the spins of the \nnearest neigh bors. S L and S R are the spins of left and right FM electrode’s atoms, which are the \nnearest neighbor to the molecule.  Initially, we considered only the Heisenberg interaction, and \nemploy ed a periodic boundary condition [40]. Usage of the periodic boundary condition  ensure d \nthat the spin s on the one edge of the Ising lattice are the nearest neig hbor to the corresponding \nspin on the opposite edge [40]. After choosing appropriate values for the Heisenberg exchange \ncoupling coefficient, temperature and random spin states, we set u p a Markov process to \ngenerate new state s. Under the Metropolis algorithm, the spin o f a randomly selected site w as \nflipped to produce a new st ate. New state was  only rejected if difference between the final and \nnew energy (ΔE) satisf ied the following condition  (equation 10) :  \n # )/ exp(,0 rand kTE E \n       (10) \n Following these strategies, we generated new configurations of 1D MEMSD , and \ncalculated the magnitude of observa bles. First, we studied the magnitude and sign of OMC -FM \nelectrode exchange coupling required to observe the long range  OMC’s effect. According to  31 Monte Carlo simulations, the following two conditions about the sign and magn itude  of OMC -FM \nelectrode exchan ge coupling were critical to observe the experimentally observed OMC effect . \nAccording to t he first condition OMC ’s exchange coupling with the two FM electrodes  must be of \nthe opposite sign . According to the  second condition OMC exchange coupling with at l east one \nFM electrode was so that the JLM/JL (or J RM/JR) ratio was in the range of 0.6 to 3.5.  If the J LM/JL \n(or J RM/JR) ratio was <0.6 or >3.5 then only unstable ordering occurred on 1D MEMSD. Here , \nthe requirement of JLM/JL (or J RM/JR) ratio  to be  >0.6 suggested  that for long range ordering , the \nmagnitude of OMC exchange coupling strength should be at least 60% of the magnitude of \nexchange coupling strength between the two atoms of the left (or right) FM electrode. We \nstudied 1D MEMSD system up to 1000 F M atoms, each FM electrodes having 500 atoms (Fig. \n16e). In conclusion, m agnetic coupling with one OMC remained effective in maintaining \nantiparallel arrangement between the two FM electrodes.  \nOur Monte Carlo studies provide d deeper insight about the exch ange coupling of a n \nOMC with the two FM electrodes. Interestingly, the magnitude of OMC’s exchange coupling wit h \nthe two FM electrodes can  be very different . For instance if J RM/JR was in  0.6 to 3.5 range,  then \nJLM/JL could be as small as -0.01 to produce a long range antiferromagnetic ordering on to the \ntwo FM electrodes. T his simulation result is consistent with the experimental observation of  \nsignificant change in magnetic moment of MTJ with one FM electrode. These MTJs possessed \nTa/Co/NiFe/AlOx/Pd and N iFe/AlOx/Pd configurations. Our magnetization studies with these \nMTJs represented the case when OMC exhibited a strong magnetic coupling with one electrode \n(FM electrode) and weakly connected to another electrode (nonmagnetic or may be weakly \nmagnetic Pd e lectrode  [41]. Pd has exhibited ferromagnetism w hen treated with thiol ended \nmolecules like the one used in the present study [8, 19] ; hence Pd may be considered as weak \nferromagnet whose couling strength with OMC is expectedly weak . OMCs increased \n(decreased) the magnetic moment of the MTJ with  Ta/Co/ NiFe (NiFe) magnetic electrode  (Fig. \n4). In summary,  Monte Carlo simul ation and magnetization studies suggested that  a OMC \nestablished ferromagnetic and antiferromagnetic exchange coupling with the Ta/Co/NiFe and \nNiFe FM electrodes, respectively.  \n1D MEMSD simulation s were also used for estimating  the magnitude of OMC  induced \ninter-electrode coupling strength. According to the model developed by Wong et al. [10] the \nOMC i nduced exchange coupling depended  on the magnitude of an OMC exchange coupling \nwith the two FM electrodes (equation 6). Assuming, J LM/JL = J RM/JR =0.6, JR = JL = J NiFe and JLM \nand J RM >> J AlOx, the strength of molecule induced exchange coupling  (J) is = 0.6 JNiFe. Here,  32 JNiFe is the interatomic exchange coupling strength in NiFe FM electrode.  Interatomic exchange \ncoupling strength of a ferromagnetic material is equiva lent to  its Curie temperature; C urie \ntemperature governs  the amount of thermal energy required to disrupt the ferromagnetic \nordering due to inter -atomic ferromagnetic exchange coupling. Hence, J NiFe can be directly \nestimated from the Curie temperature of NiFe met al. It is noteworthy that an OMC is only \nconnected to Ni Fe electrode; hence utilization  of Curie temperature for the Ni Fe will provide the \norder of its exchange coupling (J NiFe). Utilizing NiFe ’s Curie temperature to be ~900 K [23, 28]  \nthe OMC induced exchange coupling turned out to be ~540 K.  \nThe magnitude of OMC induced exchange coupling from Monte Carlo simulation is in \nagreement with the experimentally observed exchange cou pling in our magnetization study. \nMagneti c moment versus temperature study  of MEMSD -A exhibited a sharp trans ition \ncorresponding to  ~330 K Neel temperature (Fig. 3). Neel temperature of MEMSD represent ed \nthe magnitude of OMCs induced AFC between the two FM  electrodes. Antiferromagnetic \nsurface oxides of NiFe can also exhibit similar transition;  however we argue against this \npossibility because Neel temperature of the oxides of NiFe and Co are much higher than ~330 \nK, the temperature which is believed to rep resent OMC induced exchange coupling strength . \nNeel temperature of MEMSD is close r to the magnitude of theoretically calculated coupling \nstrength from our 1D MEMSD Monte Carlo simulations. For more accurate analysis similar MC \nsimulations with higher dimen sionality and inclusion of the other potential factors like dipolar \ncoupling, biquadratic coupling, and magnetic  anisotropies of the individual FM electrodes will be \nrequired.     \n \nConclusion:  \nManeuvering inter -layer coupling has strong potential to produ ce novel magnetic \nmaterials out of existing magnetic materials. Here we discussed MEMSD system  in which \nmolecular coupler produced dramatic effect on the magnetic properties of the ferromagnetic \nelectrodes. MEMSD paradigm offers  enormous flexibility and opportunities  to develop novel \nmagnetic metamaterials. MEMSD can produce spin valves and molecular spin  devices for the \napplication in quantum computation.  MEMSD is also suitable for fundamental level \nunderstanding because of the possibility of performing extensive t ransport and magnetic \nstudies. 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Jeong (/uni110C.9/uni1165.46/uni11BC.2/uni1106.35 /uni1175.2/uni11AB.2/uni1100.34 ᅵ),1,‡J. O. Piatek,1G. Boero,2and H. M. Rønnow1\n1Laboratory for Quantum Magnetism, Institute of Physics,\nEcole Polytechnique F´ ederale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\n2Microsystems Laboratory, Ecole Polytechnique F´ ederale d e Lausanne (EPFL), CH-1015 Lausanne, Switzerland\nWe present direct local-probe evidence for strongly hybrid ized nuclear-electronic spin states of an\nIsing ferromagnet LiHoF 4in a transverse magnetic field. The nuclear-electronic stat es are addressed\nviaa magnetic resonance in the GHz frequency range using cop lanar resonators and avector network\nanalyzer. The magnetic resonance spectrum is successfully traced over the entire field-temperature\nphase diagram, which is remarkably well reproduced by mean- field calculations. Our method can be\ndirectly applied to a broad class of materials containing ra re-earth ions for probing the substantially\nmixed nature of the nuclear and electronic moments.\nThe compound LiHoF 4is widely regarded as a proto-\ntypical system realizing the transverse-field Ising model\n[1]. The groundstate in zero field is ferromagnetically or-\ndered, while applying a relatively small transverse field\ninduces a zero-temperature quantum phase transition at\nHc= 4.95 T into a quantum paramagnet [2], as shown in\nFig. 1. Meanwhile, the hyperfine coupling strength of a\nHo3+ion is exceptionally large with a coupling constant\nA= 39(1) mK [3, 4]. The resulting strong hybridization\nbetween the electronic and nuclear magnetic moments [5]\nleads to two dramatic effects close to the quantum critical\npoint: (i) significant modification of the low-temperature\nmagnetic phase boundary (see Fig. 1) [2]; (ii) incomplete\nmode softening of the low energy electronic excitations\nat the critical point [6]. Therefore, this system provides a\nrare opportunity to explore the quantum phase transition\nof a magnet coupled to a nuclear spin bath [2, 6–8].\nThe impact of strong hybridization has also been\nhighlighted for magnetic-ion diluted insulators, such as\nLiYF4:Ho3+using magnetic resonance [9, 10]. A similar\nline of effort has achieved more recently single-molecule\nmagnetic resonance with a rare-earth ion [11]. Further-\nmore, strong hybridization is of great interest in quan-\ntum information science [12–14]. As much as these ex-\namples focus on the single-ion limit, the other limiting\ncase of many-body systems, such as LiHoF 4, provides\na very different and complementary perspective. While\nin the long-range-ordered state the hybridization is sup-\npressed, an applied transverse field introduces quantum\nfluctuations enhancing the hybridization towards Hc.\nHowever, probing directly the strongly hybridized\nstates in LiHoF 4using spectroscopic methods, at the low-\nest energy scale, has so far been restricted to the thermal\nparamagnetic phase in the single-ion limit. The involved\nenergy scale is too low to be resolved by the neutron scat-\ntering [6, 7]. Magnetic resonance on165Ho nuclei would\nprovide a direct way of probing the hybridized nuclear-\nelectronic states. However, the resonance in the ordered\nphase is expected around the frequency of 4.5 GHz in zero\nfield, which does not fall into the operating frequencies\nof conventional nuclear magnetic resonance (NMR) nor\nelectron spin resonance (ESR) instrumentation. Some\nFIG. 1. Phase diagram of LiHoF 4in a transverse magnetic\nfield with the experimental data taken from Ref. [2]. Solid\nline represents a mean-field calculation following Ref. [7] tak-\ning into account strong hyperfine interaction, while dashed\nline is calculated without hyperfine interaction. Inset sho ws\nschematicenergylevels for theIsingspins intheorderedph ase\n(left) and its modification by hyperfine interactions with th e\nnuclear spins (right).\nstudies have reported a hyperfine structure in ESR [3, 15],\nbut all in the paramagnetic regime above the ordering\ntemperature Tc= 1.53 K [2]. To date, microscopic evi-\ndence for the realization of the unique nuclear-electronic\nIsing model [16, 17] is absent.\nHere we demonstrate experimentally nuclear-electronic\nmagnetic resonance in LiHoF 4using coplanar microwave\nresonators and a vector network analyzer (VNA). We suc-\ncessfully trace the temperature and field evolution of the\nspectrum over the entire phase diagram, and show that\nit is remarkably well reproduced by a mean-field calcu-\nlation with parameters set by independent spectroscopic\nmeasurements [3, 4, 7, 8].\nWe begin with a description of our experimental setup\nshown in Fig. 2. A series of microwave coplanar res-\nonators with different fundamental frequencies from 1.7\nto 5.6 GHz were prepared. The impedance of the res-2\nFIG. 2. Schematic drawing of the setup showing the sample\nmounted on top of the microwave coplanar resonator inside\nthe vertical field magnet.\nonator is matched to the rest of the system by optimiz-\ning the gap size between the conductors. The oscillating\nmagnetic field, B(t), generated at the sample position is\nparallel to the surface. A cube shaped sample of 2 ×2×2\nmm3was placed at the center of the active strip, with a\nsub-millimeter gap in-between to avoid unwanted heat-\ning. The measurement geometry was chosen such that\nthe applied magnetic field, B0, is along the crystallo-\ngraphicbaxis of the tetragonal Scheelite structure, and\nB(t) is perpendicular to both B0and thecaxis to sat-\nisfy the magnetic resonance condition. We measured the\nS11parameter, which is defined as the ratio of the re-\nflected to the input power, using a VNA which is con-\nnected through a low-loss cryogenic coaxial cable to the\ncoplanar resonator. The coaxial cable was thermally an-\nchored at each stage of the dilution refrigerator including\nthe 1 K pot, Still, and mixing chamber to ensure ther-\nmalisation. The sample thermometer was located only 5\nmm away from the sample which was thermally anchored\nto the mixing chamber. With an input power of -16 dBm\napplied by the network analyzer, the sample base tem-\nperature was 0.15 K to within 0.01 K.\nTo guide and interpret our experimental investigation,\nwe perform a model calculation using a mean-field ap-\nproximation. The full Hamiltonian has been well charac-\nterized through a number of different experiments [3, 7]and is given by,\nH=/summationdisplay\ni/bracketleftBig\nHCF(ˆJi) +AˆJi·ˆIi−gLµBˆJi·B0/bracketrightBig\n−1\n2/summationdisplay\nij/summationdisplay\nαβJDDαβˆJiαˆJjβ−1\n2/summationdisplay\n/angbracketleftij/angbracketrightJexˆJi·ˆJj(1)\nwhereˆJi(J= 8) and ˆIi(I= 7/2) are the electronic\nand nuclear angular momentum operators at site i, the\ndipolar coupling constant JD=n(gLµB)2= 13.5 mK,\nDαβis the dimensionless coupling parameter for the\ndipole-dipole interaction [18], and the negligible exchange\nconstant Jex=−1.2 mK. The nuclear Zeeman and\nquadrupole interactions are assumed to be negligible [9].\nThe crystal field interaction HCFwith the surrounding\nions splits the electronic states resulting in a groundstate\nwhich is a non-Kramers doublet with a strong Ising-like\nanisotropy and the first excited state 11 K above. In the\nordered state, dipolar coupling lifts the groundstate de-\ngeneracy resulting in pseudo-spins up and down which we\nlabel as|↑∝angbracketrightand|↓∝angbracketrightstates. Each state is further split into\n8 nuclear-electronic states by the hyperfine interaction\n(Fig. 1(a), inset).\nThe total Hamiltonian can be diagonalized in the ba-\nsis of (2 J+ 1)×(2I+ 1) = 136 nuclear-electronic\n|α∝angbracketright=|mJ,mI∝angbracketrightstates. The evolution of the lowest states\nwith the applied transverse field is shown in Fig. 3(a).\nThe energy level difference between consecutive states,\n∆E, changes dramatically with the field as illustrated\nin Fig. 3(b). In the first approximation ∆ Eis propor-\ntional to A|∝angbracketleftJ∝angbracketright|, where|∝angbracketleftJ∝angbracketright|is the magnitude of the total\nangular momentum, hence ∆ Edecreases with the field\nand reaches a minimum at Hc. The diagram shown in\nFig. 3(b) allows us to predict at which field the magnetic\nresonance occurs for a given frequency.\nExperimentally we observed magnetic transitions be-\ntween the adjacent nuclear-electronic levels through reso-\nnant absorption of continuous microwaves by the sample\non a coplanar resonator. Figure 3(c) presents a typical\nfrequency-field map at 0.3 K of the S11parameter using a\nresonator with the unloaded frequency of 3.4 GHz. The\nmap shows a clear anomaly around 3.6 T indicative of\nmagnetic resonance. This field value indeed agrees with\nthe one predicted by mean-field calculations, which can\nbe seen in Fig. 3(b) by taking an intersect of blue dashed\nline for 3.4 GHz with the solids lines for the energy level\ndifference.\nFor an in-depth comparison between experiments and\ncalculations, we proceed to directly calculate the imag-\ninary part of the frequency-dependent susceptibility\nχ′′(f) which is responsible for magnetic resonance ab-\nsorption [19, 20]. The calculations were performed within\nthe linear-response framework [18] using the mean-field\nwavefunctions |α∝angbracketrightand|α′∝angbracketright,3\nFIG. 3. (a) Calculated energy levels for the nuclear-electr onic groundstate in the mean-field approximation as a functi on of\ntransverse magnetic field, and (b) the field evolution of the c orresponding energy difference between the adjacent levels . The\ncalculations were performed for 0.3 K to compare with the exp eriments. Magnetic resonance would occur when the excitati on\nfrequency (dashed lines) intersects with the energy differe nce. (c) Frequency-field map of the experimental S11parameter,\nwhere the anomaly around 3.6 T corresponds to the expected re sonance field for the frequency of 3.4 GHz. Inset shows\nconstant-field cuts of the S11map at (dashed) and away (solid) from the resonance. (d) Freq uency-field map of the calculated\nχ′′intensity. (e) Experimental magnetic resonance spectra ob tained for several different frequencies, and (f) the calcul atedχ′′\nfor the corresponding frequencies.\nχ′′(f) =/summationdisplay\nαα′∝angbracketleftα|(gLµBˆJy+gNµNˆIy)|α′∝angbracketright∝angbracketleftα′|(gLµBˆJy+gNµNˆIy)|α∝angbracketright\n(Eα′−Eα−hf)2+ Γ2\nα′αΓα′α(nα−nα′) +χ′(0), (2)\nwhereEαis the energy of the hybridized nuclear-\nelectronic eigenstates in the presence of the mean-field,\nnα= exp(−βEα)/Zis the thermal population factor\nandZ=/summationtext\nα′exp(−βEα′) is the partition function. The\nsubscript yrefers to the oscillating field direction. The\nlifetime in the linear-response calculation of the states\nis assumed to be independent of field and temperature,\nand was fixed to 40 ns, corresponding to a damping of\nΓα′α= 0.17 GHz, which provided the best match to our\ndata. The lifetime broadening may result from direct or\nindirect contributions from the electronic dipolar and ex-\nchange or nuclear dipolar couplings [20], which we leave\nfor future study. We note that the contribution to suscep-\ntibility from electronic moments, ˆJy, is 500 times larger\nthan the contribution from nuclear moments ˆIy. There-\nfore, despite the predominantly nuclear-spin nature of the\n|↑∝angbracketrightlevels, the response we measure comes mainly from\nthe electrons. This gives a tremendous enhancement ofthe signal from the nuclear states amplified by electronic\nmoments. Figure 3(d) presents the calculated frequency-\nfield map of χ′′intensity at 0.3 K, which shows a drastic\nchange upon approaching Hcfrom below. Resonant ab-\nsorption is expected from our calculations to be in the 2\nto 4.5 GHz bandwidth.\nThe absorptive part of the susceptibility is experimen-\ntally estimated as χ′′∝∆(1/Q) [19], where the qual-\nity factor Qis defined as the loaded frequency divided\nby the full-width-half-maximum in the absorption pro-\nfile in frequency as shown in the inset of Fig. 3(c). In\nFig. 3(e) we show the experimental magnetic resonance\nspectra at 0.3 K for several different frequencies by plot-\nting ∆(1/Q) = 1/Q−b, wherebis a uniform background,\nwhich can be compared to the calculated spectra at 0.3 K\nin Fig. 3(f). Both calculations and measurements at fre-\nquencies of 3.4 and 3.9 GHz show resonant peaks around\n3.6 and 3.0 T, respectively. Conversely, no resonance4\nfeatures are visible for the frequency of 1.7 GHz in both\ncalculations and experiments. The predicted transitions\nbetween second-nearest neighbouring levels at 5.6 GHz\nis too weak to be observed experimentally. The calcu-\nlated spectrum for 4.5 GHz appears as a broad hump at\nfields below 2 T, which can be expected from Fig. 3(d)\nwhere the frequency line cuts along the strongest χ′′in-\ntensity. For a better comparison the Avalue was slightly\nreduced by 3%, which is nearly within the uncertainty\nfrom the reported one [3]. In principle, the uncertainty\nin the crystal field parameters can influence our calcu-\nlations [8]. Nevertheless, excellent agreement with the\nexperiments is remarkable considering that the model\nis essentially parameter-free. Some minor discrepancies\nsuch as the fine structure in the 4.5 GHz experimental\nspectrum are likely due to fixed lifetime of all levels in\nour model. However, since the modes around 4.5 GHz\nlie very close in the relevant field range, that structure\nwould depend critically on the tiny variations of param-\neters. We therefore consider it more prudent to use a\nconstant damping. The high-field tails in 3.4 and 3.9\nGHz spectra are possibly due to the neglected effects of\nfluctuations.\nFurthermore, we investigate the temperature evolution\nof the spectrum for 3.4 GHz from 0.15 to 2.5 K as shown\nin Fig. 4(a). At base temperature a resonance peak ap-\npears around 3.7 T, which on warming decreases in am-\nplitude and shifts to lower fields. The former is due to re-\ndistribution of the thermal population of states at higher\ntemperatures. The latter reflects the decreasing size of\nthe ordered electronic moment with increasing temper-\nature, sensed by the nuclei through the hyperfine inter-\nactions. In Fig. 4(b) we track the resonance field as a\nfunction of temperature. Our measurements are shown\nto be very sensitive to small variations of the hyperfine\ncoupling as depicted by the bands.\nAs shown in Fig. 3 and 4, all the salient features\nof the experimental results are well reproduced by the\nmodel calculations, thereby validating the transverse-\nfield nuclear-electronic Ising model [16, 17]. The ex-\ncellent description of the experimental results by our\nmodel implies that the probed states have a strongly hy-\nbridized character of both nuclear and electronic degrees\nof freedom. While this has been only hinted by previous\nbulk measurements [2] and neutron spectroscopy [6], here\nwe show directly the transitions between the strongly\nhybridized nuclear-electronic states. Likewise, the pre-\nsented magnetic resonance should be distinguished from\nconventional NMR and ESR where the electronic and nu-\nclear moments are approximated to product states [19–\n21].\nTo highlight qualitative difference in the hybridized\nstates between those in the many-body system and in\nthe single-ion limit, we calculate the groundstate entan-\nglement entropy [22, 23] between the electronic and nu-\nclear moments as a measure of the hybridization. We em-\nFIG. 4. (a) Temperature evolution of the spectra from 0.15\nK to 2.5 K, from experiments (blue) and calculations (red),\nusingexcitation frequencyof3.4 GHz. (b)The resonance fiel d\nas afunctionoftemperaturefor twodifferentfrequencies. T he\ncolored bands are calculations using hyperfine constant in t he\nrange of ±3 % from the value used in Fig. 3(f). Black line\nreproduces the calculated phase boundary.\nploy the Schmidt decomposition of the mean-field wave-\nfunction, |Ψ∝angbracketright=/summationtext\nncn|mJ∝angbracketright ⊗ |mI∝angbracketright, wherecn≥0 and/summationtext\nnc2\nn= 1, where the entanglement entropy is given\nby the von Neumann entropy S=−/summationtext\nn|cn|2ln|cn|2.\nThe calculated entropy in the absence of dipolar interac-\ntions decreases smoothly with a transverse field (Fig. 5)5\n00.010.020.030.04Entanglement entropy × 0.05\n0 1 2 3 4 5 6 7 8 9\nMagnetic field (T) \nFIG. 5. Entanglement entropy calculated for the groundstat e\nof LiHoF 4as a function of transverse magnetic field (solid\nline). Dashed line is the calculation without dipolar inter ac-\ntions.\nin agreement with those reported by Ref. [17]. How-\never, by turning on dipolar coupling the model produces\na cusp-like peak at Hc, that is, the hybridization in the\nordered state of LiHoF 4increases with the applied field\nuntil it reaches a peak at the critical point. The field es-\nsentially mixes the higher excited states into the ground-\nstate, thereby enhancing the hybridization. Increasingly\nlarger field, H > H c, magnetizes the electronic and nu-\nclear moments along the field direction such that the\ngroundstate approaches a product state.\nTo summarize, we have demonstrated Ho nuclear-\nelectronic magnetic resonance of LiHoF 4in a trans-\nverse magnetic field over the entire field-temperature\nphase diagram. The spectral evolution is remarkably\nwell reproduced by mean-field calculations, validating\nthe transverse-field nuclear-electronic Ising model. Tak-\ning advantage of the well-characterized model nature of\nLiHoF 4, we have successfully probed the strongly hy-\nbridized states and their evolution in the long-range-\nordered state. Our experimental scheme will find direct\napplications not only in the Li RF4(R=rare earth) fam-\nily [8, 24, 25], but also other Rcontaining compounds\nincluding spin glass [16, 17, 26–28] and spin ice [29, 30].\nWe are grateful to M. Graf, S. S. Kim and P. Jorba\nCabre for their contribution in building experimental\nsetup at initial stage, B. Dalla Piazza for sharing his\ninsight into the mean-field and linear-response theory.\nWe also thank J. Jensen, A. Feofanov and D. Yoon for\nhelpful discussions. M.J. is grateful to support by Euro-\npean Commission through Marie Sk/suppress lodowska-Curie Ac-\ntion COFUND (EPFL Fellows). This work was sup-\nported by the Swiss National Science Foundation, the\nMPBH network and European Research Council grant\nCONQUEST. 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Lett. 108, 217203 (2012)."    },    {        "title": "0706.1662v1.Temperature_dependence_of_magnetic_anisotropy_in_bulk_and_nanoparticles_of_Pr_0_5Sr_0_5MnO_3.pdf",        "content": " 1 \n \nTemperature dependence of magnetic anisotropy in bulk and nanoparticles \nof Pr 0.5Sr0.5MnO 3 \nS.S.Rao and S.V.Bhat*\nDepartment of Physics, Indian Institute of Science, Bangalore-560012, India \n \n           Nanoparticles  (size 20, 40 and 60 nm) of Pr 0.5Sr0.5MnO 3 are prepared by sol-gel \ntechnique and their magnetic properties are studied using ferromagnetic resonance and \nmagnetization measurements. A comparison with the properties of the bulk material \nshows that the ferromagnetic transition at 265 K remains unaffected but the anti-\nferromagnetic transition at T N = 150 K disappears in the nanoparticles. Further, the \ntemperature dependence of magn etic anisotropy shows a comp lex behavior, being higher \nin the nanoparticles at high te mperatures and lower at lower temperatures in comparison \nwith the bulk. \n \n \n*svbhat@physics.iisc.ernet.in  ; for correspondence \n \n \n \n \n \n  2 \nIntroduction: \n        Nanoparticles and nanowires of doped rare -earth manganites [1-4] have recently \nbeen shown to exhibit strikingly different phases and properties compared to those of \ntheir bulk counterparts. For example, we have recently shown that there is either a \nweakening or a complete suppression of the charge ordered (CO) phase and a switch over \nfrom anti-ferromagnetic (AF) phase to ferromagnetic (FM) phase in nanowires of  \nPr0.5Ca0.5MnO 3 and Pr 0.57Ca0.41Ba0.02MnO 3 and nanoparticles of Nd 0.5Ca0.5MnO 3 [5-7]. An \nissue still unresolved is the nature of magnetic anisotropy (MA) in the materials at nanoscale. For example, Shames et al observed that the MA  is smaller in nanoparticles of \nLa\n0.9Ca0.1MnO 3 compared to its value in the bulk [8].  This result is in conflict with the \nexpectations and observations in literature, where it is reported that the MA in nanoparticles is larger than that  in the bulk [9, 10]. Another aspect of MA that is being \nactively investigated presently is the temp erature dependence of MA in nanoparticles. \nThere is recent evidence [9] that MA in na noparticles is a temperat ure dependent quantity \neven though it was earlier treated to be indepe ndent of temperature. MA happens to be an \nimportant parameter with respect to the application of magnetic materials for various \nmagnetic data storage devices. Therefore, we  have undertaken a detailed study of MA in \ndoped rare earth manganites using ferromagn etic resonance (FMR) and magnetization \nmeasurements. In this report, we present our results on the bulk and nanoparticles of Pr\n0.5Sr0.5MnO 3 (PSMO). Our FMR results show that close to the FM transition \ntemperature, MA in the bulk is considerably  smaller than that of the nanoparticles. \nHowever, as the samples are cooled, it increases faster and becomes more than that in the nanoparticles below a certain temperature. Such  a crossover in MA has not been reported \nso far to the best of our knowledge.  3 \n             PSMO in the bulk is known to undergo a ferromagnetic transition at T C = 265 K \nand when cooled further, transforms to an antiferromagnetic phase at T N = 150 K [11, \n12]. The composition dependence of MA in Pr 0.5+xSr0.5-xMnO 3 single crystals has recently \nbeen reported using torque magnetometry , magnetization and FMR measurements at \nroom temperature [13]. It is found that even  in the paramagnetic and AFM phases, there \nis evidence for the presence of MA indi cating the occurrence of phase separation. \nHowever, a quantitative study of the temperature dependence of MA and the comparison \nof FMR results with those of magnetization ar e yet to be carried out. Here we report such \na study for the bulk and nanoparticles (sizes 20, 40 and 60 nm) of PSMO. \nExperimental results and discussion:  \n The PSMO nanoparticles of size  20 nm were prepared by polymer assisted sol-gel \nmethod [7]. The as prepared nanoparticles were heated further to obtain particles of larger (40 and 60 nm) sizes.  PSMO bulk is prepared  by crushing single crystals of PSMO \ngrown using float zone technique. X -ray diffraction and transmission electron \nmicroscopy are used to examine the phase purity, particle size and crystalline nature of the nanoparticles. The Rietveld  refined XRD pattern of PSMO 20 is shown in figure 1; \nthe inset shows its TEM micrograph. From the XRD pattern it is found that PSMO 20 \ncrystallizes in the orthorhombic phase in the space group Pbnm with the unit cell parameters a = 5. 44325 A\no, b = 5.44758 Ao, c = 7.72473 Ao and volume V = 229.058 \nAo3 ; The corresponding bulk values are a = 5.443 Ao, b = 5.423 Ao and c = 7.644 Ao, V = \n225.63 Ao3  [14]. \n             Magnetization measuremen ts were carried out using a commercial vibrating \nsample magnetometer (VSM) attached to a physical property measurement system \n(PPMS) working in the temperature range fro m 2 K to 300 K. Magnetic field could be \nswept between –9 T and + 9 T. Figure 2 (a) presents the results of magnetization  4 \nmeasurements as a function of temperature on bulk and nano PSMO 20 in the \ntemperature range 4 K – 300 K measured in the presence of a magnetic field of 100 G. \nFigure 2 (b) shows the isothermal magnetizati on vs magnetic field plots at 10 K. It is \nclear from figure 2(a) that the bulk PSMO undergoes a paramagnetic (PM) -FM transition \nat ~ 265 K and a broad AFM transition at ~ 150 K in conformity with earlier reports. In \ncontrast, in the case of PSMO 20 it is seen that while the transition to the FM phase remains practically unchanged (approximately the same T\nc, slightly broader width), the \ntransition to the AFM phase has completely disappeared. The latter behavior is similar to \nour earlier reports on nanowires of PCMO [5] and nanoparticles of NCMO [7]. The \nmagnetization of bulk PSMO is  found to be linearly dependent on H and is found to be \ncompletely reversible as expected for an anti-ferromagnet. The nanoparticles, instead, show hysteresis and saturation at relatively low magnetic field values. The inset shows \nthe hysteresis loop in an expanded view . The saturation magn etization of the \nnanoparticles is found to be considerably smaller (1.53 µ\nB/f.u, obtained at 10 K, 5T) than \nthat of the bulk value (3.5 µB/f.u) similar to the observations in other systems.  \n            In figures 3 (a) and (b), we present M vs H plots at 186 K in PSMO bulk and PSMO20 respectively. The temperature was chosen such that proper comparison can be \nmade between the results of the two samples (both are ferromagnetic at this temperature). \nInsets to the figures 3 (a) and 3 (b) show th e hysteresis behavior on an expanded scale. \nCoercive fields were determined from these plots. Similar experiments were carried out \nat 200 K and also on nanoparticles of size s 40 and 60 nm. The values of MA were \nestimated from the hysteresis loops like the ones depicted in the insets of figure 3. The \nshapes of the hysteresis loops indicated that the magnetic anisotropy is of uniaxial type \nand the axes of different gr ains are randomly oriented. For such a system, the uniaxial  5 \nanisotrop y field H u can be calculated from  Hu = 0.479 H c [15] . The ani sotropy fields fo r \nPSMO bulk and PSMO 2 0 were obtain ed by follo wing  this proc edure. \n               F MR measure ments were carried out using  a Bruker EMX  X-band ESR \nspectrometer b y swe eping the ma gnetic fields from 0 – 14900 G on loosel y pack ed \nfreest anding particles. DPPH was used  as the  field marker. In figure 4, the FMR  signals \nare p resented fo r bulk and nano PSMO20 at  186 K and 200  K. For the bu lk sample, the  \nsignals are characteristic of uni axial anisotropy. In the signals from  the nan o sam ple, due  \nto the bro adenin g caused  by the disord er, the two  featur es co rrespondin g to   and H r IIH r⊥, \nwhere   r IIH  and H r⊥ are the resonan ce fields fo r H paral lel and p erpendicular to the \nanisotropy axis, are not  resol ved. Neverth eless, the values of  and Hr IIH r⊥ can be \nmeasured, and ar e used to estimate the unia xial anisotr opy usin g the formula [16] H u(FMR)  \n=  2/3 (H r⊥ -  ). The  values of uniax ial anisotro py thus obtain ed as a function of  \ntemperatur e for th e bul k as well as th e nanosi zed samples (20, 40 an d 60 nm) ar e \npresent ed in figure 5. It is seen t hat the nano sa mples show m uch weak er temperature \ndependen ce of Hr IIH\nu(FMR) compared to tha t of the bulk sa mple . The  values of Hu f or 40 a nd \n60 nm pa rticles are nearly identical and a re larger than those  for the 20 nm pa rticle \nthroug hout the temper ature ran ge. The bul k values are smaller than those for \nnanoparticles for T > 19 5 K. Around this temperature, there is a cross ov er in the values,  \nbelow whic h temperature the  bulk v alues are higher than those  of the na no particles. The \nbulk MA is se en to  go throug h a peak around  185 K b elow whi ch it d ecreases, most \nprobabl y due to the impe nding  antife rroma gnetic transition. \n                As mentioned earlier [8], in nanoparticles of LCMO, MA was reported to be \nsmaller than that of the bulk. However, in a number of reports [ 9, 10] , MA fo r \nnanoparticles is shown to  be larger than in the bulk. This  is also  expected on theoretical \ngrounds sin ce, due  to the surfa ce disord er th e symmetr y of small clusters  is lower than   6 \nthat predicted from the crystal structure of the bulk. In fact, a random anisotropy \nanalysis leads to the scaling of the contribution of the surface anisotropy as 1/R2, where R \n= average cluster radius. Though this is strictly true for cubic anisotropy, qualitatively \nsimilar conclusions can be drawn for uniaxial anisotropy as well. \n          The results reported in this work point towards a more complex scenario where the \nvariation of relative magnitudes of anisotropy in nano and bulk samples as a function of \ntemperature is seen to be quite different. As is seen from figure 5, the bulk H u is a strong  \nfunction of temperature. For the nanosamples H u is found to increase with decreasing  \ntemperature though at a considerably lower rate than in the bulk. This property is expected to be useful for practical applications in magnetic data storage devices where \nmagnetic anisotropy needs to be  independent of temperature.  \n      The anisotropy values estimated from th e hysteresis plots at two temperatures (186 K \nand 200 K) for the bulk and the PSMO20 sample s are also shown in the figure 5. These \nare seen to be significantly lower than the corresponding values obtained from FMR. Earlier reports have shown that the anisotropies determined from static measurements e.g., from coercive fields could be quite di fferent from those at microwave frequencies \n[17,18]. This difference could arise either fr om surface relaxation effects or magneto \nelastic effects or both. All the same, for a sample of an assembly of grains with the randomly oriented axes of anisotropy, FMR is  known to be a more reliable technique for \nthe estimation of magnetic anisotropy. \nConclusions:   In summary, we find that the antiferromagnetic phase observed in the \nbulk form of Pr 0.5Sr0.5MnO 3 completely disappears in th e nano-sized samples. The high \ntemperature anisotropy of nanoparticles is much larger than that of the bulk sample. The \nMA of the nanosamples is a much less sensitive function of temperature than that of the \nbulk.  7 \nAcknowledgements:  The authors gratefully acknowledge the help of Prof. C. N. R. \nRao, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore for providing \nthe facilities for magnetization measurements. The Department of Science and \nTechnology and the Council of Scientific and I ndustrial Research are thanked for \nfinancial support.  \n  \n1. M. A. Lopez-Quintela, L. E. Hueso, J.  Rivas and F. Rivadulla. Nanotechnology 14     \n     ,212, 2003. \n2. A. Dutta, N. Gayathri a nd R. Ranganathan, Phys. Rev.  B 68 , 054432, 2003 \n3. S Han, C Li, Z Liu, B Lei, D Zhang, W Jin, X Liu, T Tang, and C Zhou , Nano  \n           Lett, 4, 1241, (2004). \n4. Colossal Magnetoresistance, Charge  Ordering and Related Properties , edited by  \n       C. N.R. Rao and B. Raveau , World Scientific, Singapore, 1998.  \n \n5. S. S. Rao, K. N. Anuradha, S. Sarang i, and S. V. Bhat, Appl. Phys. Lett. 87,  \n        182503, (2005).  \n6. K.N. Anuradha, S.S. Rao and S. V. Bhat;  J. Nanosci. Nanotech. 7, 1775, (2007).    \n 7. S. S. Rao, S. Tripathy, D. Pandey, and S. V. Bhat, Phys. Rev. B, 74, 144416, (2006) \n8. A.I. Shames , E. Rozenberg, G. Gorodetsky, I. Felner, E. Sominski ,G. Pang and A.  \n          Gedanken, Current Applied Physics 7, 21, 2007. \n9.  C. Antoniak, J. Lindner and M. Farle, Europhys. Lett, 70, 250,  2005.   \n10.  P. Gambardella, S. Rusponi, M. Veronese, S. S. Dhesi, C. Grazioli, A. Dallmeyer, I.  \n     Cabria, R. Zeller, P. H. Dederichs, K. Kern, C. Carbone and H. Brune, Science, 300,    \n    1130, 2003.  8 \n \n11. M. M. Savosta , V. A. Borodin , M. Ma ryško, Z. Jirák, J. Hejtmánek, and P. Novák,  \n        Pys. Rev. B, 65, 224418, 2002. \n12.Y. Tomioka, A. Asamitsu, Y. Moritom o, H. Kuwahara and Y. Tokura,   \n         Phys.Rev.Lett, 74, 5108, 1995. \n13. S Cao, B Kang, J Zhang and S Yuan, Appl.Phys.Lett, 88,172503, 2006. \n14. K. Kniiek, Z. Jirak, E. Polle rt, F. Zounova and S. Vratisla , J. Solid State Chem.  100,      \n        292, (1992). \n15. J Das, Sangeeta S, Kalarickal, K- S Kim and C .E. Pa tton, Cond-mat/0610160. \n16. V. Likodimos and M. Pissas , Phys.Rev.B, 73, 214417, 2006.  \n17. G. F. Goya, T. S. Berquo,  F. C. Fonseca and M. P. Morales, J. Appl. Phys, 94, 3520,  \n        2003  \n18. R. Lopusnik, J. P. Nibarger, T. J. Silva, and Z. Celinski, Appl. Phys. Lett, 83, 96,  \n       2003.  9 \nFigure Captions: \n1. Figure 1 shows the observed (dots) and Rietveld fitted (continuous lines) XRD \npatterns of PSMO 20. The inset of this figure depicts the TEM micrograph of \nPSMO 20, non spherical and slightly elongate d particles of diameter 20 nm are \nclearly seen. \n2. Figure 2a) shows the variation of ma gnetization of PSMO 20 and PSMO BULK \nwith the temperature (M-T) measured at  100 G while warming after the samples \nwere cooled in zero field to the lowest temperature from room temperature (ZFC). \nFigure 2b) describes the isothermal magnetization variation with the magnetic field (M-H) ranges between –5T to + 5T at 10 K for both PSMO 20 and PSMO \nBULK. The inset of this figure shows the enlarged version of M-H of PSMO 20 measured at 10 K. The coer cive field value is indicat ed as 930 G by an arrow. \n3. Figures 3a) and 3b) show the M-H loops of PSMO BULK and PSMO20 at 186 K \nrespectively. The expanded views of both figures are shown as insets in the \nrespective figures, the coercive fields of PSMO BULK and PSMO20 are 81G and 181G  are indicated in the inset figures by an arrow. \n4. The FMR signals of PSMO BULK at 186 K and 200 K are shown in the figures \n4a) and 4b) respectively. The sharp line which is seen at around 3300 G in these \ntwo figures is due to DPPH. Figures 4c) and 4d) show the FMR signals of \nPSMO20 at 190 K and 200 K respectively. \n5. This figure shows the temperature variation of magnetic anisotropy (H u) obtained \nfrom the FMR measurements performed on PSMO BULK ( \u0004), PSMO20 ( \u000f), \nPSMO40 ( \u001a) and PSMO 60 ( %). This figure also contains the magnetic \nanisotropy values obtained from M-H measurements on PSMO bulk ( 0) and \nPSMO 20 (< ) at 186 K and 200 K.  10 \n \n \n \n  \n \n \n \n \n \n \n \n \n \nFigure 1  Rao et al. \n  11 \n0501001502002503000246810\n  Magnetization M  (emu/g)\nTempe ratur e T (K)PSMO BULK\nPSMO 20100 G\n   (a)\n-60000-40000-20000 0 200004000060000-40-30-20-10010203040\n  \n-4000 0 4000-30030Magnetization (emu/g)\nMagnetic Field (G)HC = 929.6 G10 K\n20 nmMagnetization M  (emu/g)\nMagn etic Field H (G)PSMO BUL KPSMO 2010 K\n    (b)\n \n \n \nFigure 2  Rao et al. \n \n \n  12 \n-20000 0 2000 0-20020\n-2000 0 20000Magnetization (emu/g)\nMagnetic Field (G)Hc = 181 G186 K\nPSM O 20Magnetization (emu/g)\nMagnet ic Field (G )186 K\nPSMO  20\n(b)-20000 0 20000-50050\n-2000 0 20000Magnetization (emu/g)\nMagnetic Field (G)Hc = 81 G186 K\nPSMO BULKMagn etization (emu/g)\nMagnetic Field (G)186 K\nPSMO BULK\n(a)\n \n \n \n \n \n \n \n \nFigure 3  Rao et al. \n \n \n  \n  13 \n0 6000 12000 0 6000 12000  \n dP/dH (a.u)(a)\n186 K\nPSMO BUL K200 K\nPSMO BUL K \n  \n (b)\n190 K\nPSMO 20 \n  \nMagnetic Field (G)(c)\n200 K\nPSMO 20 \n  \nMagnetic Field (G)(d)\n \n \n \n \n \n \n \n \n \n \nFigure 4  Rao et al. \n \n \n \n \n \n \n \n \n \n \n \n \n  14 \n100 120 140 160 180 200 220 240 260050010001500200025003000\n  Magnetic ani sotropy (Hu,G)\nTemperature (K)  FMR\n bulk \n 20 nm \n 40 nm \n 60 nm \nHysteresis \n bulk \n 20 nm\n \n \n \n \n \n \n \n \nFigure 5  Rao et al. \n \n \n "    },    {        "title": "1112.0610v1.Microwave_induced_DC_Signal_in_a_Permalloy_Thin_Strip_at_Low_Applied_Magnetic_Field.pdf",        "content": " 1Microwave-induced DC Signal in a Permalloy Thin Strip at Low \nApplied Magnetic Field \nZiqian Wang, 1 Lujun Huang,1  Xiaofeng Zhu,1,2 Xiaoshuang Chen,1  \nWei Lu 1* \n1 National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, \nChinese Academy of Sciences, Shanghai 200083, China  \n2 Department of Physics and Astronomy, University of Manitoba, Winnipeg, R3T 2N2, \nCanada  \nAbstract \nWe investigated the ferromagnetic resonance signals in a polycrystalline permalloy \nthin strip under in-plane low static magnetic field. A series of DC voltages, which contain ferromagnetic resonance or spin wave resonance signals, were measured by \ninducing microwave frequencies greater th an 10 gigahertz. The resonant signals \nmeasured in low magnetic field show different properties from those detected in high field condition. Based on the theory of DC effects in ferromagnetic resonance and the experimental data of anisotropic magn etoresistance, a quantitative model was \nproposed. We found that the shape anisotropy significantly affects magnetization, and distorts the resonant signals in low field condition.  \n \n       ____________________________ \n* Corresponding author. Email address: luwei@mail.sitp.ac.cn   2Anisotropic Magnetoresistance (AMR) effect 1 plays a crucial role in the detection of \nthe electrical properties of  ferromagnetic materials 2, 3. The direct current (DC) effect, \nwhich originates from the coupling of micr owave-induced radio frequency (rf) current \nand the time-dependent AMR, have been widely researched and discussed for a long \ntime with growing interest 2-4. The AMR is given as R=R 0+RA-RAsin2θ, where R 0 is \nthe resistance with the magnetization being perpendicular to the rf current, R A is the \nadditional resistance and θ is the intersection angle between the magnetization M and \nthe rf current j 1,4. Permalloy (Py) is an ideal ferromagnetic material with significant \nAMR and has been widely investigated for its microwave-induced DC signal 4-9. In \nthe previous theory for Py, M precesses in phase with the rf current under a constant \nmagnetic field when the microwave is applied to the sample, and the material \nresistance changes with time due to the variation of θ. During a signal period, the rf \ncurrent and the alternate resistance generate a nonzero voltage, which can be recorded \nby lock-in technology in a time-averaging DC form 4. The DC voltage spectrum \nmeasured by sweeping the magnetic fiel d shows pronounced resonance of around 1 \nμV, including ferromagnetic resonance (F MR), spin wave resonance (SWR) or \nperpendicular standing spin wave (PSSW) resonance, as which the SWR performs in boundary conditions due to an extremely thin thickness \n4,9-11.  \n Most analysis of those DC voltage signals requires large equipment to provide a static \nmagnetic field H greater than 1000 Oe \n12, because the magnetization and spin \nprecession in a high field condition can be  easily analyzed. Nevertheless, without \nrelevant research, the complicated low-field feature of signal makes the utilization of \nmicrowave-induced DC effect in weak field condition a difficult task. This largely \nunexplored region limits the miniaturization an d energy cost of relational production. \nSince it is widely pointed out that the anisotropy, which is determined by the shape of a ferromagnetic material sample, produces the assignable influence on the magnetization behavior \n13, 14, we provide a physical model in this letter to explain the \nlow field spectrum in terms of shape anisotropy. Two phenomena, the signal distortion and incompletion, are discussed as a general case.   3The experimental configuration is shown in Fig.1(a). For the detection of AMR and DC \nvoltage, a thin Py (Ni 80Fe20) strip with polycrystalline structure has been prepared by \nUniversity of Manitoba 4. Our sample’s thickness is thinner than the skin depth in this \nexperiment15. The sample’s geometry parameters are : thickness=49nm, width=200 μm, \nlength=2400 μm, with bond pads on each side. A rectangular waveguide is used to \ntransform the microwave energy into the sample’s plane normally. Our Py sample is fixed between two poles of an electromagnet. The electromagnet provides static \nmagnetic field H (maximal ≈1.5T)\n parallel to the sample’s plane. All data are \nmeasured at room temperature. To better understand the signal of our experiment, a \ncoordinate system is defined, as shown in Fig.1( b). In order to quantify the shape \nanisotropy of our sample, AMR is measured first.  \n From the result of AMR measurement, the angle dependence matches \nR=R\n0+RA-RAsin2θ, while the magnetic field dependence is complicated. With the \nweakening of H, the resistance of our sample is gradually increased to R 0+RA. That is \nto say, M reverts to the z-axis. Since the direction of H is fixed, other magnetic field \naffects M. Regarding the shape anisotropy of ferromagnetic material with thin strip \nstructure, an anisotropic magnetic field HA is defined with its direction of the long \naxis of our sample (z-axis). The physical meaning of HA is explained as follows. The \nmagnetization intends to align along the z-axis in this thin strip structure because of \nlower magnetic energy, as if a static magnetic field parallel to the z-axis, pushing the \nmagnetization to its direction. The magnetization precesses around H+HA, which can \nbe treated as the same value and direction as H in the high field condition. In the low \nfield condition ( H is less than 200Oe), HA cannot be neglected. Hence, the \nmagnetization is affected by both shape anis otropy and the external field. Considering \nof HA, the magnetization is situated between H and HA, and we find:  \nLR-R+R =R(t)A A 0                    ( 1 )   4with 2\n2sin =L⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+AH HHGGθ . Here L describes the angle dependence of AMR in low \nfield condition. With increasing external field H, L approaches to 1 and M turns to the \ndirection of H, with an intersection angle θ between z-axis. As demonstrated in \nFig.2(a), the resistance R 0 of our sample for perpendicular magnetization is 89.2 Ω, \nwhile R A is 1.48 Ω. The fitted anisotropic field HA in our sample is 4.8 Oe.  \n \nAfter the measurement of shape anisotropy we further pursue a quantitative description of microwave-induced DC vol tage. The resonant DC voltage signal \n(PSSW and FMR) of microwave response in a different frequency is illustrated in \nFig.2(c).  Although the FMR is not excited in the PSSW resonant frequency (in our \nsample, the frequency is about 2 GHz higher than FMR’s), an additional \nmagnetization circumrotating around the equilibrium axis is induced. Such additional magnetization precession also results in DC effect by coupling the rf current. The PSSW acquires higher energy than FMR so that it can be easily distinguished, as \nshown in Fig.2(c). The magnetic field dependence of FMR signal lines exhibits symmetry and asymmetry shapes\n4. In PSSW line shape, the symmetry part is much \nsmaller than that of FMR's in this sample . In low field conditions, the PSSW modes' \ncurves in the embedded figure in Fig.2(c) are distorted with regard to the PSSW signal in high field condition. Furthermore, as the frequency reduces to 10.2GHz, fewer peaks are detected and the signal becomes incomplete.  It is thus reasonable to speculate that the observed signal distortion and incompletion \nare caused by \nHA. Next, the DC signals are investigated in low field conditions (less \nthan 500Oe). As for a Py thin strip, the DC voltage U MW with respect to the \ntime-dependent AMR and the microwave at the frequency of ω/2π can be described as\n〈UMW〉=〈jcos( ωt+Φ)RAsin2θ(t)〉 (〈〉denotes the time-averaging), here Φ represents \nthe phase difference between the rf current j(t) and time-dependent AMR R(t) By \ncombining with Eq. (1) and separating the contributions from microwave frequency  5ω/2π and H, 〈UMW〉takes the form: \n2\nA MW ) cos(  jR>= U< LL− Φ β .      (2) \nHere β is a frequency-depend coefficient, which represents the intensity of precession. \nWhen the microwave is at the resonant frequency, β is maximized. 2LL−  \ndescribes the range of resistance vibration in a certain H. Noted that a resonant \ncondition is not a prerequisite to the DC signal 8, DC voltage can also be measured in \nnon-resonant condition. The matching of the theory based on Eq.2 is verified in Fig.3.   \nThe direction of \nH is first set as 45 °to show the detailed signal with microwave \nfrequency in the range between 10.3GHz and 10.7GHz, and the signal is presented in \nFig.3(a). At different frequencies, the signal curves are significantly different. The complete PSSW signal curve shown in Fig.2(c) is recorded with the form of a quasi-centrosymmetry. Furthermore, the signal curve is distorted when the applied field is reduced. Such phenomenon can named as “signal distortion”. As the assisted \nmicrowave frequency in the recorded signal curve down to 10.5GHz, the PSSW \nresonance signal becomes distorted. In the low field, \nM is strongly influenced by HA. \nWhen the Py sample is assisted with the microwave energy of 10.3GHz, the required \nmagnetic field for PSSW resonance at this frequency is weaker than 4.8Oe. Such \nresonant requirement cannot be fulfilled because HA always exists and H+HA is still \nlarger than 4.8Oe, resulting in the incomplete PSSW signal curves. This condition is \nnamed as “signal incompletion”. Also, as the frequency continues to decrease to 10.3GHz, the signal strength is further reduced.   According to Eq.2, the signal curves nearby zero field are different when the distortion occurs, as shown in Fig.3(b) while the frequency is set as 10.8GHz. Such \ndifference depends on the intersection angle \nθ. When θ is smaller than 45 °and the \napplied field is minimized, the intensit y of DC signal decreases monotonically. \nReferring again to the Fig.2(b), we may conclude that the signal intensity also  6depends on 2LL− , which characterizes the intensity of oscillation of R(t) and is \nalso reduced monotonically. When θ is larger than 45 °, 2LL−  does not \nmonotonically decrease or increase with H. The signal intensity first increases and \nthen vanishes with decreasing applied field. We simulate these signal distortion trends \nin Fig.3(b). While the applied field is parallel to the length of the sample, no DC \nsignal is recorded because of the minimum 2LL− .  \n \nThe DC signal corresponding to 10.2GHz microwave is a nonresonant spectrum \nbecause the resonant magnetic field is smaller than HA. The spectrum nearby \nzero-field performs as an asymmetric type with two non-resonant peaks on each side, \nas shown in Fig.3(c). From the discussion above, no signal is recorded in the parallel \nconfiguration because 2LL− =0. In the non-parallel configuration, the types of \nsignal curves are similar with different mechanisms. When θ＜45°, the DC signal \nincreases (or decreases if the applied field is reversed) is mainly contributed by β  in \nEq.2. When H approaches to the resonant field, the intensity of magnetization \nprecession is strengthened and thus DC voltage signal is enhanced. However, when H \nis perpendicular to z-axis, 2LL−  becomes the major factor affecting 〈UMW〉; in \nother words, with minimized H, the oscillation of R intensified with M deviating from \nthe x-axis, and conducing to the strengthened signal. \n  In conclusion, we discussed the unusual microwave-induced DC voltage signals of an in-plane magnetized permalloy thin strip in low field condition with different \nconfigurations and microwave frequencies. The shape anisotropy of this sample is \ncharacterized by an additional, static magnetic field with certain magnitude and direction through the measurement of AMR.  The detection of a microwave-induced \nDC voltage shows that this additional magnetic field is strongly involved in the ferromagnetic resonance when the applied field is weak. First, with the decreasing of \nH, the magnitude of  H+HA is not approximately equal to H and the direction of  7magnetic is deviated while H is weak. As a result, the DC signal curve is distorted. \nSecond, the additional field contributes a minimum magnitude to H+HA. If the \nmagnetic field required for resonance is weaker than the minimum H+HA, the \nresonant peak disappears, resulting in the incomplete DC signal curve. In view of the \nresults of the measurement and simulation of AMR and DC voltage signal for the Py thin strip, this low-field model is reasonable to explain the unusual signals in in-plane \nmagnetization. Further research on how to control the shape anisotropy would be meaningful for the detection and utili zation of the electrical properties of \nferromagnetic materials.  \nAcknowledgment s \nThe authors appreciate James Torley from University of Colorado at Colorado Springs for critical reading of the manuscript. This work was supported in part by the State Key Program for Basic Research of Ch ina grants (2007CB613206), the National \nNatural Science Foundation of China grants (10725418, 10734090, 10990104, and \n60976092), the Fund of Shanghai Science and Technology Foundation grants \n(09DJ1400203, 09ZR1436100, 10JC1416100, and 10510704700).              8 \n   \nReferences \n1. T. McGuire, and R. Potter, IEEE Trans. on Magn. 11,1018 (1975). \n2. H. J. Juretschke, J. Appl. Phys. 31, 1401 (1960). \n3. W. G. Egan and H. J. Juretschke, J. Appl. Phys. 34, 1477 (1963). \n4. N. Mecking, Y . S. Gui, and C. -M. Hu, Phys. Rev. B. 76, 224430(2007). \n5. M. V . Costache, S. M. Watts, M. Sladkov, C. H. Van der wal, and B. J. van Wees, Appl. Phys. \nLett. 89, 232115 (2006). \n6. M.V . Costache, M. Sladkov, S. M. Watts, C. H. Van der wal, and B. J. van Wees,Phys. Rev. \nLett. 97, 216603 (2006). \n7. E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). \n8. X. F. Zhu, M. Harder, J. Tayler, A. Wirthmann, B. Zhang, W. Lu, Y . S. Gui and C.-M.Hu, \nPhys. Rev. B. 83, 140402 (2011). \n9. Ursula Ebels, et,al, in Spin Dynamics in Co nfined Magnetic Structures I,  edited by B. \nHillebrands and K. Ounadjela, Sp ringer, Berlin, 20 02, pp.185-219. \n10. C. Herring, C. Kittel, Phys. Rev 81, 869, (1951). \n11. W. S. Ament, G. T. Rado, Phys. Rev 97,1558, (1955). \n12. Y . S. Gui, N. Mecking, and C. -M. Hu, Phys. Rev. Lett, 98,217603 (2007). \n13. J. A. Osborn, Phys. Rev. 67, 351 (1945). \n14. B. A. Kalinikos and A. N. Slavin, J. Phys. C: Solid State Phys. 19, 7013 (1986). \n15. M. Xu, Trifon. M. Liakopoulos, Chong H. Ahn,  Suk Hee Han and Hi Jung Kim, IEEE Trans. \non Magn. 34, 1369 (1998). \n     9 \n     \nFigure Captions: \n \nFIG.1. (a) The schematic experimental setup and the structure of our sample. The microwave power is injected to the Py stri p directly through a rectangular waveguide, \nand the signal is measured by the Lock-in amplifier. (b) The defined coordinate \nsystem in this paper and the illustration of two resonance mode: z-axis is parallel with \nthe length of the Py sample, x-axis is parallel with the width and y-axis is \nperpendicular with the plane of this thin strip. \nH under different directions θ is due to \nthe applied field or external field, HA is due to the shape of the anisotropic field.   \n \nFIG.2. (a)The red data show that the measured AMR depends on the applied magnetic \nfield at different intersection angles θ and the blue data is the simulation used by \nequation (1). (b) The red data indicate that the measured AMR depends on the \ndirections of H EX at different external field and the blue data is the simulation used by \nequation (1). (c) The measured microwave DC signal at different frequencies and \nfixed intersection angle θ between sample and  H at 45°. From bottom to top, the \nsignal curves correspond to microwave frequencies of 10.2GHz, 10.8GHz, 11.7GHz, \n12GHz, 15GHz, 16.5GHz, 17.1Hz, 17.7GHz, respect ively. The detail information of \nsignal curves in low field condition (-200 Oe to 200 Oe) with 10.2GHz and 10.8GHz is also presented. \n \nFIG.3. (a) Fitting (red lines) of the PSSW sp ectra (gray dots) for different frequencies \nf=\nω/2π at θ=45°. (b) Fitting of the PSSW spectra for different angles θ at \nf=10.8GHz.(c) .Fitting of the PSSW spectra for different angles θ at f=10.2GHz. \n \n       10 \n  \n "    },    {        "title": "0903.3813v1.Multi_terminal_spin_dependent_transport_in_ballistic_carbon_nanotubes.pdf",        "content": "arXiv:0903.3813v1  [cond-mat.mes-hall]  23 Mar 2009Multi-terminal spin-dependent transport in ballistic car bon nanotubes\nAudrey Cottet, Ch´ eryl Feuillet-Palma and Takis Kontos\nEcole Normale Sup´ erieure, Laboratoire Pierre Aigrain,\n24 rue Lhomond, 75231 Paris Cedex 05, France\n(Dated: November 8, 2018)\nWe study theoretically non-local spin-transport in a balli stic carbon nanotube contacted to two\nferromagnetic leads and two normal metal leads. When the mag netizations of the two ferromagnets\nare changed from a parallel to an antiparallel configuration , the circuit shows an hysteretic behavior\nwhich is specific to the few-channels regime. In the coherent limit, the amplitude of the magnetic\nsignals is strongly enhanced due to resonance effects occuri ng inside the nanotube. Our calcula-\ntions pave the way to new experiments on low-dimensional non -local spin-transport, which should\ngive results remarkably different from the experiments real ized so far in the multichannel diffusive\nincoherent regime.\nPACS numbers: 73.23.-b, 73.23.Ad, 85.75.-d\nI. INTRODUCTION\nNon-local electric effects have been observed since\nthe early days of mesoscopic physics, e.g. in metal-\nlic circuits1,2. This fact is related to the primarily\nnon-local nature of electronic wave functions in quan-\ntum coherent conductors. The spin-degree of freedom\nhas raised little attention in this context, although its\ncontrol and detection is one of the major challenges\nof nanophysics, nowadays. Non-local spin signals have\nbeenstudiedformulti-terminalmetallicconductors3,4,5,6,\nsemiconductors7and graphene8, in the multichannel dif-\nfusive incoherent (MDI) regime. It has been found that\na non-equilibrium spin accumulation induced by a fer-\nromagnet into a given conductor can be detected as a\nvoltage across the interface between this conductor and\nanother ferromagnet9. However, to our knowledge, spin-\ndependent non local effects have not been investigated in\nthe coherent regime, so far.\nCarbon-nanotubes-based circuits are appealing candi-\ndates for observing a non-local, spin-dependent, and co-\nherent behavior of electrons. First, electronic transport\nin carbon nanotubes (CNTs) can reach the few-channels\nballisticregime,assuggestedbytheobservationofFabry-\nPerot-like interference patterns10. Secondly, spin injec-\ntion has already been demonstrated in CNTs connected\nto two ferromagnetic leads (see Ref. 11 for a review).\nThirdly, non local voltages have been observed in CNTs\ncontacted to four normal-metal leads12, which suggests\nthat electrons can propagate in the nanotube sections\nbelow the contacts. The study of non-local spin trans-\nport in CNTs has recently triggered some experimental\nefforts13,14,15. However,atheoreticalinsightonthistopic\nis lacking. Some major questions to address are what are\nthe signatures of a non-local and spin-dependent behav-\nior of electrons in a nanoconductor, and to which extent\nthese signaturesarespecific to the coherent regimeor the\nfew-channels case.\nIn this paper, we study the behavior of a CNT with\ntwo normal metal ( N) leads and two ferromagnetic ( F)\nleads magnetized in colinear directions. Two leads areused as source and drain to define a local conductance\nGcand the other two are used to probe a non-local volt-\nageVcoutside the classical current path. We consider\ntwo different setups which differ on the positions of the\nFleads. Setup (a) corresponds to the standard geome-\ntry used for the study of the MDI limit. In setup (b),\nthe two Fleads play the role of the voltage probes, so\nthat no magnetic response is allowed in the MDI limit.\nWe mainly focus on the coherent regime, using a scat-\ntering description with two transverse modes, to account\nfor the twofold orbital degeneracy commonly observed in\nCNTs16. This minimal description is appropriate at low\ntemperatures and bias voltages. We take into account\nboth the spin-polarization of the tunneling probabilities\nat the ferromagnetic contacts and the Spin-Dependence\nofInterfacialPhaseShifts (SDIPS) whichhasbeenshown\nto affect significantly spin-dependent transport in the\ntwo-terminals case17,18,19. This approach leads to strong\nqualitative differences with the MDI case. In particular,\nwefind a magneticsignalinthe conductance Gcofsetups\n(a) and (b), which would not occur in the MDI limit. We\nalso predict an unprecedented magnetic signal in Vcfor\nsetup (b). We find that these effects already arise in the\nincoherent few channels regime. However, they are much\nstronger in the coherent case, due to resonances which\noccur inside the CNT. These resonances make the circuit\nsensitive to the SDIPS, which can furthermore enhance\nthe amplitude of the magnetic signals.\nThis paper is organized as follows: section II defines\nsetups (a) and (b), section III discusses the multichannel\ndiffusive incoherent (MDI) limit, section IV focuses on\nthe coherent four-channels (CFC) scattering description,\nsection V presents an incoherent four-channels (IFC) de-\nscription, section VI discusses the experimental results\npresently available, and section VII concludes.\nII. DEFINITION OF SETUPS (A) AND (B)\nIn this article, we consider a central conductor (CC)\nconnectedtoanensemble Loftwoferromagnetic( F)and2\nFIG. 1: Left: The two types of circuits [setups (a) and (b)]\nstudied in this article. The central conductor (represente d\nwith the grey bar) is contacted to two normal metal leads\nNand two ferromagnetic leads F, which can be magnetized\nin parallel ( c=P) or antiparallel ( c=AP) configurations.\nThe only difference between setups (a) and (b) is the posi-\ntion of the two Fleads. Contacts 1 and 2 are used as source\nand drain to measure a local conductance Gcand contacts 3\nand 4 are used to probe a non-local voltage Vcoutside the\nclassical current path. Right: Tables presenting the behav -\niors of setups (a) and (b) in various regimes. We study the\nexistence of the signals MG= (GP−GAP)/GP,VPand\nMV= (VP−VAP)/Vb. We compare the predictions of the\ncoherent four-channels model (CFC) of section IV, the inco-\nherent four-channels model (IFC) of section V and the multi-\nchannel diffusive incoherent (MDI) model of section III.\ntwo normal metal ( N) reservoirs. We study the two con-\nfigurations presented in Fig. 1. In both cases, lead 1 is\nconnected to a bias voltage source Vb, lead 2 is connected\nto ground, whereas leads 3 and 4 are left floating. The\nonly difference between setups (a) and (b) is the position\nof the two Fleads. These Fleads can be magnetized\nin parallel ( c=P) or antiparallel ( c=AP) configu-\nrations. We will study the conductance Gc=∂Ic\n1/∂Vb\nbetween contacts 1 and 2 and the voltage drop Vcbe-\ntween leads 3 and 4. The dependence of these quanti-\nties on the magnetic configuration cof the ferromagnetic\nelectrodes can be characterizedwith the magnetic signals\nMG= (GP−GAP)/GPandMV= (VP−VAP)/Vb.III. MULTICHANNEL DIFFUSIVE\nINCOHERENT LIMIT\nWe first briefly discuss the behavior of setups (a)\nand (b) in the multichannel diffusive incoherent (MDI)\nregime. This case has been thoroughly investigated, in\nrelation with experiments in which the CC is a metallic\nisland3,4,5,6. For a theoretical description of this regime,\none can use spin-currents and a spin-dependent electro-\nchemical potential µσwhich obey a local spin-dependent\nOhm’s law, provided the mean free path in the sample\nis much shorter than the spin-flip length. We refer the\nreader to Ref. 20 for a detailed justification of this ap-\nproach from the Boltzmann equations, and to Ref. 21 for\nan overview of this field of research. In this section, we\nsummarize the behaviors expected for setups (a) and (b)\nin the MDI limit (see Appendix A for a short derivation\nof these results from a resistors model). A finite current\nbetween leads 1 and 2 can lead to a spin accumulation\n(i.e.µ↑/ne}ationslash=µ↓) in the CC if lead 1 or 2 is ferromagnetic,\nbecause spins are injected into and extracted from the\nCC with different rates in this case. The spin accumu-\nlation diffuses along the CC beyond lead 2, and reaches\nleads 3 and 4, provided the spin-flip length is sufficiently\nlong. Then, leads 3 and 4 can be used to detect the spin\naccumulation provided one of them is ferromagnetic. In-\ndeed, a local unbalance µ↑/ne}ationslash=µ↓in the CC will produce\na voltage drop between the floating lead j∈ {3,4}and\nthe CC if jis ferromagnetic (this voltage drops aims at\nequilibrating the spin currents between the CC and the\nferromagnetic contact). One can thus conclude that in\nsetup(a), aspinaccumulationoccurswhen Vb/ne}ationslash= 0, which\nleads toVc=V3−V4/ne}ationslash= 0. In contrast, one finds Vc= 0\nin setup (b) because a currentflow between the Nleads 1\nand 2 cannot produce any spin accumulation. For com-\npleteness, we also mention that in the MDI limit, one\nfindsGP=GAPfor both setups (a) and (b), due to the\nfact that leads 3 and 4 are left floating (see Appendix A).\nThe table in Fig. 1 summarizes these results.\nIV. COHERENT FOUR-CHANNELS LIMIT\nA. General scattering description\nInthissection,westudythecasewheretheCCisabal-\nlistic carbon nanotube (CNT) allowing coherent trans-\nport. The observation of Fabry-Perot like interference\npatterns10suggests that it is possible, with certain type\nof metallic contacts, to neglect electronic interactions in-\nside CNTs. We thus use a Landauer-B¨ uttiker scatter-\ning description22. We take into account two transverse\nmodesp∈ {K,K′}, to account for the twofold orbital\ndegeneracy commonly observed in CNTs16. Each trans-\nversemodeonehastwospinsubmodes σ∈ {↑,↓},defined\ncolinearly to the polarization of the Fleads. This gives\nfour channels m= (p,σ) in total. We assume that spin\nis conserved upon scattering by the CNT/lead interfaces3\nand upon propagation along the CNT. This requires, in\nparticular, that the magnetization direction can be con-\nsidered as uniform in the four Fleads, and that spin-\norbitcouplingandspin-flipeffectscanbeneglectedinside\nthe CNT and upon interfacial scattering. For simplicity,\nwe also assume that the transverse index pis conserved.\nIn the linear regime, the average current through lead j\nwrites\nIc\nj=/summationdisplay\nkGjkVc\nk (1)\nwith\nGjk=GK[4δjk−/summationdisplay\nm/vextendsingle/vextendsingleSm\njk/vextendsingle/vextendsingle2] (2)\nGK=e2/h, andSm\njkthe scattering amplitude from lead\nkto leadjfor electrons of channel m. Equation (1) in-\nvolves the electrostatic potential Vc\nkof leadk(we assume\nthat the leads are in local equilibrium, so that each one\nhas a single chemical potential for both spin directions).\nNote that GjkandSm\njkimplicitly depend on the configu-\nrationcof the ferromagnetic electrodes. In this section,\nwe calculate GcandVcby using the general notations\nof Fig. 2 for the scattering amplitudes. The phase shift\nδjkacquired by electrons along the CNT from contacts\njtokcan be considered as independent from m, with\nδjk=δkj23. Inpractice, δ12,δ23,andδ34canbetunedus-\ning local gate voltage electrodes to change the electronic\nwavevector in the different CNT sections24. We will thus\nstudy the signals Gc,Vc,MGandMVas a function\nof these phases. The calculation of the voltage drop Vc\nrequires to determine Vc\n3andVc\n4from/an}bracketle{tI3/an}bracketri}ht=/an}bracketle{tI4/an}bracketri}ht= 0.\nThis yields:\nVc\nVb=G41G32−G42G31\nG34G43−G33G44(3)\nand\nGc/GK=G11+[G13(G44G31−G41G34)\n+G14(G41G33−G31G43)]/[G34G43−G33G44]\n(4)\nUsingthenotationsofFig.2, theelements |Sm\njk|occurring\nin Eq. (3) through the coefficients Gjkof Eq. (2) can be\ncalculated as\n|Sm\n41|=/vextendsingle/vextendsingleD−1\nmt1mt2mt3mt4m/vextendsingle/vextendsingle (5)\n|Sm\n31|=/vextendsingle/vextendsingleD−1\nmt1mt2m/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingle/vextendsingleu′\n3m+r4m(t3mv′\n3m−r′\n3mu′\n3m)ei2δ34/vextendsingle/vextendsingle/vextendsingle(6)\n|Sm\n42|=/vextendsingle/vextendsingleD−1\nmt3mt4m/vextendsingle/vextendsingle\n×/vextendsingle/vextendsinglev2m+r1m(t2mu2m−r2mv2m)ei2δ12/vextendsingle/vextendsingle(7)FIG. 2: Scheme representing the notations used for the scat-\ntering amplitudes of channel min setups (a) and (b). The\nrectangles with full lines represent the different leads and the\ndashed rectangles represent the nanotube sections between\nthose leads. We note δjkthe phase shift acquired by electrons\nwhile propagating along the nanotube between contacts jand\nk. At this stage, we use δjk=δkj, but we keep tjm/ne}ationslash=t′\njm,\nujm/ne}ationslash=u′\njm, andvjm/ne}ationslash=v′\njmfor transparency of the calcula-\ntion.\nSm\n32=Sm\n31Sm\n42/Sm\n41 (8)\n|Sm\n34|=/vextendsingle/vextendsingleD−1\nmt′\n4m/vextendsingle/vextendsingle×/vextendsingle/vextendsinglev′\n3m(1−r1mr2mei2δ12)\n+ei2δ23(r′\n2m+r1mei2δ12[t2mt′\n2m−r2mr′\n2m])\n×(t′\n3mu′\n3m−r3mv′\n3m)| (9)\nand\n|Sm\n43|=/vextendsingle/vextendsingleD−1\nmt4m/vextendsingle/vextendsingle×/vextendsingle/vextendsinglev3m[1−r1mr2mei2δ12]\n+ei2δ23[r′\n2m+r1mei2δ12(t2mt′\n2m−r2mr′\n2m)]\n×[t3mu3m−r3mv3m]| (10)\nwith\nDm= [/parenleftbig\n1−r1mr2mei2δ12/parenrightbig/parenleftbig\n1−r′\n2mr3mei2δ23/parenrightbig\n×/parenleftBig\n1−r′\n3mr4mei2δ34/parenrightBig\n]\n−t2mt′\n2mr1mr3m/parenleftBig\n1−r′\n3mr4mei2δ34/parenrightBig\nei2(δ12+δ23)\n−t3mt′\n3mr′\n2mr4m/parenleftbig\n1−r1mr2mei2δ12/parenrightbig\nei2(δ23+δ34)\n−t2mt′\n2mt3mt′\n3mr1mr4mei2(δ12+δ23+δ34)(11)\nThe missing coefficients G33andG44can be obtained\nfrom the above Eqs. using G33=−(G34+G31+G32)4\nandG44=−(G43+G41+G42). For calculating Gc, one\nfurthermore needs\n|Sm\n14|=/vextendsingle/vextendsingleD−1\nmt′\n1mt′\n2mt′\n3mt′\n4m/vextendsingle/vextendsingle (12)\n|Sm\n13|=/vextendsingle/vextendsingleD−1\nmt′\n1mt′\n2m/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingleu3m+r4mei2δ34(t′\n3mv3m−r′\n3mu3m)/vextendsingle/vextendsingle(13)\nand\nSm\n11−r′\n1m=D−1\nmt1mt′\n1mei2δ12{r2m/parenleftbig\n1−r′\n3mr4mei2δ34/parenrightbig\n+ei2δ23[r3m+r4mei2δ34(t3mt′\n3m−r3mr′\n3m)]\n×[t2mt′\n2m−r2mr′\n2m]} (14)\nThe denominator Dmaccounts for multiple resonances\ninside the CNT. Figure 3 depicts some resonances Am\nn,\nwithn∈[1,6], which can occur in limiting cases where\nt2[3],m=t′\n2[3],m= 0 or 1. In the general case, Eq. (11)\nFIG. 3: Scheme representing different resonances (noted Am\ni)\nwhich can occur in setups (a) and (b) when t2[3],m=t′\n2[3],m=\n0 or 1. The upper numbers indicate the position of contacts\n1, 2, 3 and 4.\nindicates that these different resonances are coupled. For\nt2m=t′\n2m= 0,Gccorresponds to the conductance of a\ntwo-terminals device, independent from δ23andδ34, and\nVcvanishes. For t2m=t′\n2m/ne}ationslash= 0 and t3m=t′\n3m= 0,Gc\ndepends on δ12andδ23, but not on δ34, andVcstill van-\nishes. Having a non-local signal Vc/ne}ationslash= 0 requires a direct\nCNT-CNT transmission at both contacts 2 and 3. It also\nrequires that the four channels mare not coupled to the\nleads in the same way. Indeed, from Eqs. (3) and (8), one\ncan check that if all the Sm\njkcoefficients are independent\nfromm, one finds Vc= 0 due to the series structure\nof the device25. Interestingly, a finite Vchas already\nbeen obtained in a CNT connected to four normal-metal\nleads12, which suggests that the KandK′modes were\nnot similarly coupled to those leads. In principle, such an\nasymmetry is also possible with ferromagnetic contacts.\nB. Parametrization of the lead/nanotube contacts\nIn the following, we assume that the top and bottom\nhalves of the three terminals contacts j∈ {2,3}in Fig. 1are symmetric. We furthermore take into account that\nthe scattering matrix associated to each contact is invari-\nant upon transposition, due to spin-conservation26. This\ngivestjm=t′\njm,rjm=r′\njm, andujm=u′\njm=vjm=\nv′\njmforj∈ {2,3}. In this case, on can check from Eqs.\n(5-14) that GcandVcdepend onlyonsixinterfacialscat-\ntering phases, i.e. those of r1m,t2m,r2m,t3m,r3mand\nr4m, which correspond to processes during which elec-\ntrons remain inside the CNT27. For contacts j∈ {2,3},\nit is thus convenient to use the parametrization\ntj(p,σ)=/radicalBig\nTj,p(1+σPj,p)ei(ϕT\nj,p+σ\n2∆ϕT\nj,p)(15)\nrj(p,σ)=/bracketleftBigg/radicalbigg\n1−/vextendsingle/vextendsingle/vextendsingletj(p,σ)sin[φ(p,σ)\nj]/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingletj(p,σ)/vextendsingle/vextendsinglecos[φ(p,σ)\nj]/bracketrightBig\nei(ϕR\nj,p+σ\n2∆ϕR\nj,p)(16)\n|ujm|=/radicalBig\n1−|rjm|2−|tjm|2(17)\nwith\nφ(p,σ)\nj=ϕR\nj,p−ϕT\nj,p+σ\n2(∆ϕR\nj,p−∆ϕT\nj,p)\nThe above expressions depend on six real parameters\nTj,p,Pj,p,ϕT\nj,p,ϕR\nj,p, ∆ϕR\nj,pand ∆ϕT\nj,p28. In order to\nhave unitary lead/CNT scattering matrices, on must use\n0≤Tj,p(1 +σPj,p)≤1 and29π/2≤φm\nj[2π]≤3π/2.\nThese conditions imply 0 ≤ |tjm|2≤1, 0≤ |rjm|2≤1\nand300≤ |ujm|2≤1/2. For contacts j∈ {1,4}, one can\nuse\n/vextendsingle/vextendsingle/vextendsinglet(p,σ)\nj/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsinglet′(p,σ)\nj/vextendsingle/vextendsingle/vextendsingle=/radicalBig\nTj,p(1+σPj,p) (18)\nand\narg(r(p,σ)\nj) =cpϕR\nj+σ\n2∆ϕR\nj,p (19)\nwithcK(K′)=±1. In Eq. (19), we have assumed/summationtext\np,σarg(r(p,σ)\n1) = 0 and/summationtext\np,σarg(r(p,σ)\n4) = 0, because,\nfrom Eqs. (5-14), these quantities only shift the varia-\ntionsofGc,Vc,MGandMVwith respectto δ12andδ34,\nrespectively. The parameters Pj,p, withj∈ {1,2,3,4},\nproduce a spin-polarization of the transmission probabil-\nities|tm\nj|2. The parameters ∆ ϕR(T)\nj,pallow to take into\naccount the Spin-Dependence of Interfacial Phase Shifts\n(SDIPS), which has already been shown to affect signif-\nicantly the behavior of CNT spin-valves17,18,19. We will\nshow below that the SDIPS also modifies the behavior of\nmulti-terminal setups. Note that for j∈ {2,3}, the pa-\nrameters ∆ ϕR(T)\np,jalso contribute to the spin dependence\nof|rjm|2and|ujm|2: the SDIPS and the spin-dependence\nof interfacial scattering probabilities are not independent\nin three-terminal contacts, due to the unitarity of scat-\ntering processes.5\nFIG. 4: Signals GP(top left panel), VP(top right panel),\nMG(bottom left panel), and MV(bottom right panel) as\na function of δ12(horizontal axes) and δ34(vertical axes)\nfor a setup (a) with symmetric KandK′channels. We\nhave used T1,K[K′]= 0.6,T2,K[K′]= 0.1,T(3)4,K[K′]= 0.3,\nP2[3],K[K′]= 0.4,ϕR\n1(4)= 0,ϕR\n2(3),K[K′]=π,ϕT\n2[3],K[K′]= 0,\n∆ϕR{T}\n2(3),K[K′]= 0 and δ23=π. We have indicated the position\nof the resonances Am\n1(4)andAm\n3(5)with red and blue dashed\nlines, respectively.\nC. Behavior of setup (a)\nWe now consider setup (a), which has been frequently\nused in the MDI regime, for studying the spin accumu-\nlation effect3,4,5,6,7,8. We first assume that the Kand\nK′channels are coupled identically to the leads. This\ncase is illustrated by Fig. 4, which shows the varia-\ntions ofGP(top left panel), VP(top right panel), MG\n(bottom left panel) and MV(bottom right panel) ver-\nsusδ12(horizontal axes) and δ34(vertical axes). One\ncan first notice that all these signals present strong vari-\nations with δ12andδ34, due to quantum interferences\noccurring inside the CNT. In Fig. 4, GP(δ12) presents\npeaks which correspond to the resonances Am\n1(see e.g.\nred dashed line), because we consider a case where T2m\nis weak (these peaks also correspond accidentally to the\nresonances Am\n4, which are much broader). A more re-\nmarkable result is that GP(δ34) presents antiresonances\nwhich correspond to Am\n3andAm\n5(see e.g. blue dashed\nline). This is a signature of the strongly non-local nature\nof current transport in this circuit: the electric signal\nmeasured in a given section of the CNT can be sensitive\nto resonances occurringin other sections of the CNT. We\nnote that in Fig. 4,/vextendsingle/vextendsingleVP/vextendsingle/vextendsinglepresents the same type of vari-\nations as GPwithδ12andδ34. In the general case, the\nresonances or antiresonances shown by the electric sig-\nnals will not necessarily correspond to those defined in\nFig. 3, due to the strong coupling between these differ-\nent types of resonances. Importantly, we find that the\nMGsignal can be finite, contrarily to what happens in\nthe MDI limit. Indeed, in Fig. 4, MGcan exceed 8%.\nWe note that in Fig. 4, MGpresents minima approx-imately correlated with the maxima of GPin theδ12\ndirection, and with the minima of GPin theδ34direc-\ntion. In the case of a Smmatrix independent from m,\nonefinds Vc= 0(see section IV). By continuity, since we\nhave used in Fig. 4 relatively low values for P2[3],K[K′],\nno SDIPS and symmetric KandK′channels, we find/vextendsingle/vextendsingleVP/vextendsingle/vextendsingle≪Vb. More precisely, a lowest order development\nwith respect to P2andP3yieldsVP∼ −VAP∼λP2P3,\nwithλ≪1 a function of the different system parame-\nters. In these conditions, MVpresents the same type\nof variations as VP(one has MV∼2VP). When the\nKandK′modes are strongly asymmetric, it is possi-\nble to obtain a strong/vextendsingle/vextendsingleVP/Vb/vextendsingle/vextendsingleratio for relatively low\npolarizations P2[3],p. This case is illustrated by Fig. 5,\nwhere we have used T1[4],K/ne}ationslash=T1[4],K′andϕR\n1(4)/ne}ationslash= 0, so\nthatA(K,σ)\n1[4]/ne}ationslash=A(K′,σ)\n1[4]andA(K,σ)\n3[5]/ne}ationslash=A(K′,σ)\n3[5]. In this\ncase, the variations shown by the different electric sig-\nnals are more complicated than previously. However, we\nfindVP∼VAP, so that the amplitude of MVremains\ncomparable to that of Fig. 4.\nFIG. 5: Signals GP,VP,MG, andMVas a function of δ12\nandδ34, for asetup(a) with dissymmetric KandK′channels.\nWe have used T1[4],K= 0.5,T1[4],K′= 0.3,T2,K[K′]= 0.1,\nT3,K[K′]= 0.3,P2[3],K[K′]= 0.4,ϕR\n2(3),K[K′]=π,ϕT\n2[3],K[K′]=\n0,ϕR\n1(4)=π/2, ∆ϕR{T}\n2(3),K[K′]= 0 and δ23=π. We have\nindicated the position of the resonances Am\n1(4)andAm\n3(5)with\nred and blue dashed lines, respectively.\nWe now discuss the signs of the different signals. We\nhave already seen above that with the parameters of Fig.\n4, one has VP>0 andVAP<0. In other conditions,\nit is possible to have VP<0 andVAP>0, orVPand\nVAPboth positive, or both negative (not shown). In the\nCFC model, the signs of VPandVAPare thus indepen-\ndent, whereas MDI models usually give opposite signs\nforVPandVAP(see e.g. Eq. (21) of Appendix A and\nRefs. 35,36). Figure 5 illustrates that there exists sets of\nparameters such that the non-local voltage VPchanges\nsign while sweeping δ12orδ34(this result is also true for\nδ23)31. It is also possible to find sets of parameters such\nthatMV(not shown) and MG(see Fig. 6, bottom left\npanel, full lines) change sign with δ12,δ23orδ34.6\nWe nowbrieflydiscussthe effects ofthe contactspolar-\nizations. One can generally increase the amplitude of the\nmagnetic signals by increasing Pj,p(not shown),/vextendsingle/vextendsingle∆ϕR\nj,p/vextendsingle/vextendsingle\n(see Fig. 6, red full lines) and/vextendsingle/vextendsingle∆ϕT\nj,p/vextendsingle/vextendsingle(not shown). A\nstrong SDIPS can split the resonances or antiresonances\nof the electric signals (not shown), like already found in\nthe two-terminals F/CNT/Fcase17. Interestingly, in the\ncase of a two-terminals F/CNT/Fdevice with a K−K′\ndegeneracy and no SDIPS (using 1 = F, 2 =Fand no\nleads 3 and 4), Ref. 17 has found that the oscillations of\nMGwithδ12are symmetric, and a finite SDIPS is nec-\nessary to break this symmetry. In contrast, in setup (a),\nthe oscillations of MG(δ12) can be asymmetric in spite\nof theK−K′degeneracy and the absence of a SDIPS\n(see Fig. 6, bottom left panel).\nFIG. 6: Signals GP,VP,MG, andMVas a function of\nδ12for setup (a). We consider a case with no SDIPS (black\nlines) and a case with a finite SDIPS (red lines correspond-\ning to ∆ ϕR\n2(3),K[K′]= 0.15π). We have used T1[4],K= 0.2,\nT1[4],K′= 0.6,T2,K[K′]= 0.5,T3,K[K′]= 0.3,P2[3],K[K′]= 0.2,\nϕR\n1(4)= 0.12π,ϕR\n2(3),K[K′]=π,ϕT\n2[3],K[K′]= ∆ϕT\n2(3),K[K′]= 0,\nδ23=πandδ34= 0.12π. Thefull linescorrespond totheCFC\nprediction (section IV), and the dotted lines to the IFC pre-\ndiction (section V). The second does not depend on δ12. In\nthe IFC case, the MG signal is hardly visible in this figure\nbecause it is of the order of 0.06%.\nD. Behavior of setup (b)\nIn setup (b), the types of resonances or antiresonances\nshownbythe electricsignalsdepend againonthe valueof\nthe coupling between the different CNT sections. We will\nonly highlight the most interesting specificities of setup\n(b), because it has many common properties with setup\n(a). Fig. 7 shows, with black [red] full lines, examples\nofGP(δ12),MG(δ12),VP(δ12), andMV(δ12) curves, for\nsymmetric [asymmetric] KandK′channels . Strikingly,\nin both cases, the magnetoconductance MGbetween the\nNleads 1 and 2 can be finite, although the two Fleadsare located outside the classical current path. This is in\nstrongcontrastwith the MDI limit. From Eqs. (3-8), the\nvoltage difference Vcvanishes if the scattering properties\nof contacts 1 or 2 are independent from the transversein-\ndexp, regardless of the scattering properties of contacts\n3 and 432. This leads to the paradoxical situation where\na magnetic signal can be measured between the two N\nleads but not between the two Fleads (see black full\nlines in Fig. 7). By continuity, when the K−K′asym-\nmetry is not large at contacts 1 and 2, the amplitude\nof the signals VPandMVmeasured between contacts\n3 and 4 will remain very small. It is possible to obtain\nstrongeramplitudes for VcandMVin the opposite limit\nof strongly asymmetric KandK′channels (see red full\nlines in Fig. 7, for which we have used ϕ2,K/ne}ationslash=ϕ2,K′).\nWith setup (b), is thus also possible to obtain magnetic\nsignals in both GcandVc, whereas MGandMVwould\nvanish in the MDIlimit.\nFIG. 7: Signals GP,VP,MG, andMVas a function of\nδ12, for setup (b). The black (red) lines correspond to the\ncases of KandK′channels coupled identically (differently)\nto the contacts, i.e. ϕR\n2,K=ϕR\n2,K′=π(ϕR\n2,K= 1.2π,\nϕR\n2,K′= 0.6π). Wehaveused T1[2],K(K′)= 0.4,T3,K[K′]= 0.3,\nT4,K[K′]= 0.5,P3[4],K[K′]= 0.4,ϕR\n1(4)=ϕT\n2(3),K[K′]=\n∆ϕR(T)\n3,K[K′]= ∆ϕR\n4,K[K′]= 0,δ23=π/2 andδ34=π. The\nfull lines correspond to the CFC prediction and the dotted\nlines to the IFC prediction. In the black case, the MGsignal\nvanishes in the IFC limit, and the VPandMVsignals vanish\nin both the IFC and CFC limits. The MG signal of the red\ncase is hardly visible in the IFC limit because it is of the ord er\nof 0.03%.\nE. Comparison with the MDI limit\nIn this section, we summarize the most striking differ-\nences between the coherent four-channels (CFC) model\nof section IV and the MDI model of section III. For\nsetup (b), the CFC model allows VP/ne}ationslash= 0 and MV/ne}ationslash= 07\nwhereas one finds VP= 0 and MV= 0 with the MDI\nmodel. Another remarkable result is that for both setups\n(a) and (b), the CFC model gives GP/ne}ationslash=GAPwhereas\nthe MDI model imposes GP=GAP. The table in Fig. 1\nsummarizes these results.\nV. INCOHERENT FOUR-CHANNELS LIMIT\nIn order to determine whether the specific spin-\ndependent behavior of the CFC model is due to co-\nherence or to the low number of channels, it is inter-\nesting to consider the incoherent four-channels (IFC)\nlimit. If the phase relaxation length of the CNT is\nmuchshorterthanthedistancebetweenthedifferentcon-\ntacts, the global transmission and reflection probabilities\nof setups (a) and (b) can be calculated by composing\nthe scattering probabilities of the different contacts in-\nstead of the scattering amplitudes33. We have checked\nthat this leads to replacing the scattering probabilities/vextendsingle/vextendsingle/vextendsingleSm\nαβ({rjm,tjm,vjm,ujm,δij})/vextendsingle/vextendsingle/vextendsingle2\noccurringin Eqs.(1) and\n(2) bySm\nαβ({|rjm|2,|tjm|2,|vjm|2,|ujm|2,0}). Impor-\ntantly, this description remains intrinsically quantum\nsince the channel quantization is taken into account. In\nFigs. 6 and 7, we show with black and red dotted lines\ntheIFCvaluescorrespondingtothedifferentCFCcurves.\nWe find that Gc,Vc,MGandMVdo not depend any-\nmore on the phases δij. However, GP/ne}ationslash=GAPis still\npossible for setups (a) and (b). More precisely, we have\ncheckedanalyticallythatusingidentical KandK′modes\nleads to32GP=GAP, and we have checked numerically\nthatGP/ne}ationslash=GAPoccurs in case of a K/K′asymmetry\nat one of the four contacts for setup (a), and at contacts\n1 or 2 for setup (b). We can also obtain VP/ne}ationslash= 0 and\nMV/ne}ationslash= 0 for setup (b) [and, more trivially, for setup (a)],\nwith the same symmetry restrictions as for the CFC case\n(see table in Fig. 1). Therefore, having VP/ne}ationslash= 0 and\nMV/ne}ationslash= 0 for setup (b), and MG/ne}ationslash= 0 for setups (a) and\n(b) is not a specificity the coherent case: using a very\nsmall number of transport channels already allows these\nproperties. It is nevertheless important to notice that\nthe values of MGandMVare strongly enhanced in the\nCFC case, due to resonanceeffects. Moreover,in the IFC\ncase, the circuit is insensitive to the SDIPS, whereas in\nthe coherent case, the SDIPS furthermore increases the\namplitude of MVandMG34. At last, the coherent case\npresents the interest of allowing strong variations of the\nelectric signals with the gate-controlled phases δ12,δ23\nandδ34.\nVI. DISCUSSION ON FIRST EXPERIMENTS\nReference 13 reports on a Single Wall Carbon Nan-\notube (SWNT) circuit biased like in Fig. 1, but with\nfour ferromagnetic leads. A hysteretic Vchas been mea-\nsured by flipping sequentially the magnetizations of thetwoinnercontacts. However,no conclusioncanbe drawn\nfrom this experiment, due to the lack of information on\nthe conduction regime followed by the device. Reference\n14reportson Vcmeasurementsfora setup(a) made with\na SWNT. The authors of this Ref. have observed a fi-\nniteVcwhich oscillates around zero while the back gate\nvoltage of the sample is swept. This suggests that this\nexperimentwasinthecoherentregime. However,theam-\nplitude of Vcwas very low (max( |Vc/Vb|)∼0.01), which\nindicates, in the framework of the scattering model, that\ntheKandK′modes were very close and the spin polar-\nizationofthe contactsscatteringpropertiesveryweak. It\nis therefore not surprising that these authors did not ob-\ntain a measurable MVsignal. Although setup (a) seems\nvery popular in the nanospintronics community for his-\ntorical reasons3, we have shown abovethat setup (b) also\nhighly deserves an experimental effort, as well as MG\nmeasurements in general.\nIn this article, we have chosen to focus on the case\nof double mode quantum wires because this is adapted\nfor describing CNT based devices, which are presently\namongthe most advanced nanospintronicsdevices. How-\never, technological progress might offer the opportunity\nto observe the effects depicted in this article in other\ntypes of nanowires, like e.g. semiconducting nanowires.\nIndeed, quantum interferences have already been ob-\nserved in Si37and InAs quantum wires38, and spin-\ninjection has already been demonstrated in Si layers39\nand InAs quantum dots40. One majordifficulty may con-\nsist in reaching the few modes and fully ballistic41regime\nwith these devices.\nVII. CONCLUSION\nIn this work, we have studied theoretically various cir-\ncuitsconsistingofacarbonnanotube with twotransverse\nmodes, contacted to two normal metal leads and two fer-\nromagnetic leads. Two contacts are used as source and\ndrain to define a local conductance, and the two other\ncontacts are left floating, to define a non-local voltage\noutside the classical current path. When the magneti-\nzations of the two ferromagnetic contacts are changed\nfrom a parallel to an antiparallel configuration, we pre-\ndict, in the local conductance and the non-local voltage,\nmagneticsignalswhicharespecifictothecaseofasystem\nwith a low number of channels. In particular, we propose\nan arrangement of the normal and ferromagnetic leads\n[setup (b)] which would give no magnetic response in the\nmultichannel diffusive incoherent (MDI) limit, but which\nallows magnetic responses in both the local conductance\nand the non-local voltage in the two-modes regime. The\nmore traditional arrangement [setup (a)] used for the\nstudy of the MDI limit also shows a qualitatively new\nbehavior, i.e. a magnetic response in the local conduc-\ntance. These specific magnetic behaviors are strongly\nreinforced in the coherent case, due to resonance effects\noccurring inside the nanotube, and also, possibly, due to8\ntheSpin-Dependence ofInterfacialPhaseShifts. Ourcal-\nculations pave the way to new experiments on non-local\nspin-transport in low-dimensional conductors.\nWe acknowledge discussions with H. U. Baranger and\nG. E. W. Bauer. This work was financially supported by\nthe ANR-05-NANO-055 contract, the European Union\ncontractFP6-IST-021285-2and the C’NanoIlede France\ncontract SPINMOL.\nVIII. APPENDIX A: DISCUSSION OF THE MDI\nLIMIT WITH A RESISTORS NETWORK\nIn this appendix, we discuss the MDI regime with\nan elementary but insightful resistors network model.\nWhen the electronic mean free path is much shorter\nthan the spin-flip length, it is possible to define a spin-\ndependent electrochemical potential which obeys a local\nspin-dependent Ohm’s law21. Thus, neglecting spin-flip\nscattering inside the CC, one can use the effective resis-\ntor network of Fig. 8 to describe the behaviors of setups\n(a) and (b) in the MDI limit13,20. For completeness, we\nallow the four leads j∈ {1,2,3,4}to be ferromagnetic,\nwith colinear magnetizations. The left (right) part of\nthe resistors network corresponds to the up (down) spin\nchannels. Due to intra-lead spin-flip scattering, electrons\nFIG. 8: Resistors network used to describe the behavior of a\none dimensional Central Conductor (CC) connected to four\nleadsj∈ {1,2,3,4}in the MDI limit. The contact between\nleadjand the CC is represented by the resistors Rσ\nj,rσ\nj, and\nerσ\nj. The section j−kof the CC is modeled with the two\nresistors Rjk. When lead jis not ferromagnetic, one must\nuseR↑\nj=R↓\nj,r↑\nj=r↓\nj, and er↑\nj=er↓\nj\nare in local equilibrium in lead j. This equilibration is\ndescribed by the electrical connection of ↑and↓chan-\nnels at node j, which has an electric potential Vc\nj. The\nsectionj−kof the CC is modeled with the two resistors\nRjk. The contact between lead jand the CC is repre-\nsented by the resistors Rσ\nj,rσ\nj, and/tildewiderσ\nj. When lead jis\nnot ferromagnetic, one must use R↑\nj=R↓\nj,r↑\nj=r↓\nj, and\n/tildewider↑\nj=/tildewider↓\nj. The current flowing from lead 1 to lead 2 isIc\n1=Ic\n1↑+Ic\n1↓. Since leads 3 and 4 are floating, they\nsupply the CC with spin currents which are perfectly\nequilibrated, i.e. Ij↑=−Ij↓forα∈ {3,4}. We find\nGc=HVb/D (20)\nand\nVc=/parenleftBig\nA↑\n34−A↓\n34/parenrightBig/parenleftBig\nB↑\n12−B↓\n12/parenrightBig\nVb/R23H(21)\nwith\nR23=R↑\n3+R↓\n3/summationtext\nσ(Rσ\n34−Rσ\n3)+/summationdisplay\nσ(/tildewiderσ\n2+rσ\n3+R23) (22)\nH=/summationdisplay\nσ(Rσ\n12+ασ,σ+ασ,−σ) (23)\nD=/parenleftBig\nR↑\n12+α↑,↑/parenrightBig/parenleftBig\nR↓\n12+α↓,↓/parenrightBig\n−α↑,↓α↓,↑(24)\n2Aσ\n34=Rσ\n34/bracketleftBig\n(R↑\n3+R↓\n3)/(R↑\n34+R↓\n34)/bracketrightBig\n−Rσ\n3(25)\nBσ\n12= (Rσ\n12+ασ,σ+α−σ,σ)/parenleftbig\nR−σ\n12−R−σ\n2/parenrightbig\n(26)\nασ,σ′=Rσ\n2(Rσ′\n12−Rσ′\n2)/R23 (27)\nand\nRσ\njk=Rσ\nj+/tildewiderσ\nj+Rσ\nk+rσ\nk+Rjk (28)\nfor (j,k)∈ {1,2,3,4}2. The value of His independent\nfrom the contacts magneticconfiguration, but Ddepends\non the relative configuration of leads 1 and 2, so that\nGP/ne}ationslash=GAPis possible provided leads 1 and 2 are fer-\nromagnetic. In contrast, the value of Gcis independent\nfromthe magnetizationdirections ofleads 3or4 because,\ndue toI3(4)↑=−I3(4)↓, the resistors Rσ\n3,rσ\n3,/tildewiderσ\n3,Rσ\n4and\nrσ\n4of Fig. 8 are connected in series with R−σ\n3,r−σ\n3,/tildewider−σ\n3,\nR−σ\n4andr−σ\n4respectively. We conclude that for setups\n(a) and (b), one has GP=GAPin the MDI limit. From\nEq. (21), having VP/ne}ationslash= 0 requires that at least one of\nthe biased leads 1 or 2 is ferromagnetic (for the gener-\nation of a spin-accumulation), and at least one of the\nfloating leads 3 or 4 is ferromagnetic (for the detection\nof this spin-accumulation). These conditions are fulfilled\nfor setup (a), but not for setup (b). 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B 42, 11479 (1990).\n26Spin-conservation allows to map the scattering descriptio n\nof each spin component σ∈ {↑,↓}onto a spinless problem.\nTime reversal symmetry in each of these spinless problems\nimpliestjm=t′\njm,ujm=u′\njmandvjm=v′\njm.\n27From Eq. (14), the phase of t1mt′\n1m/r′\n1mcan also af-\nfect the signals GcandVc. However, since the scattering\nmatrix of contact 1 is unitary, one finds t1mt′\n1m/r′\n1m=\n−r1m|t1m/r1m|2. It is thus sufficient to use arg( r1m) as a\nparameter.\n28The complete parametrization of the scattering matrices\nof contacts j∈ {2,3}requires extra parameters ϕU\nj,p, and\n∆ϕU\nj,psuch that arg( uj(p,σ)) =ϕU\nj,p+σ∆ϕU\nj,p/2 andr′′\njm=\n−(1−2|ujm|2)1/2eiarg[(ujm)2/(Am\nj+Bm\nj)]. However, one can\ncheck that, with the assumptions made in part IV.B, the\nsignalsGcandVcare independent from ϕU\nj,pand ∆ϕU\nj,p\nfor both setups (a) and (b).\n29For completeness, we point out that the condition on φm\nj\ncan be relaxed if tm\nj= 0.\n30Forj∈ {2,3}, one has |ujm|2≤1/2 because we assume\nthat an electron arriving from contact jenters the top and\nbottom portions of the CNT with the same probabilities.\n31We find that the sign changes of VPcan already occur in\nthe spin-degenerate case. Accordingly, a change of sign in\nVPhas already been observed for a CNT with four normal\nmetal leads, by sweeping the CNT back-gate voltage12.\n32We have checked analytically that this property is true\neven when the contacts 2 and 3 are spatially asymmetric\ni.e.u(2)3m/ne}ationslash=v(2)3m.\n33S. Datta, Electronic Transport in Mesoscopic Systems ,\nCambridge University Press, Cambridge (1995).\n34In the IFC case, MVandMGare sensitive to the ∆ ϕR[T]\n2(3),p\nparameters through the reflection probabilities of contact s\n2 and 3, but not through the scattering phases.\n35M. Johnson and R.H. Silsbee, Phys. Rev. B 37, 5312\n(1988).\n36S. Takahashi and S. Maekawa, Phys. Rev. B 67, 052409\n(2003).\n37A. T. Tilke, F. C. Simmel, H. Lorenz, R. H. Blick , and J.\nP. Kotthaus, Phys. Rev. B 68, 075311 (2003).\n38Y.-J. Doh, A. L. Roest, E. P. A. M. Bakkers,\nS. De Franceschi, L. P. Kouwenhoven, cond-\nmat/arXiv:0712.4298\n39BerendT. Jonker, George Kioseoglou, AubreyT.Hanbicki,\nConnie H. Li, Phillip E. Thompson, Nature Physics 3, 542\n(2007).\n40K. Hamaya, M. Kitabatake, K. Shibata, M. Jung, M.\nKawamura, K. Hirakawa, T. Machida, S. Ishida, Y.\nArakawa, Appl.Phys.Lett. 91, 022107 (2007); K.Hamaya,\nS. Masubuchi, M. Kawamura, T. Machida, M. Jung, K.\nShibata, K.Hirakawa, T.Taniyama, S.Ishida, Y.Arakawa,\nAppl. Phys. Lett. 90, 053108 (2007).\n41X. Zhou, S. A. Dayeh, D. Aplin, D. Wang, and E. T. Yu,\nAppl. Phys. Lett. 89, 053113 (2006)."    },    {        "title": "1204.2254v2.Complex_itinerant_ferromagnetism_in_noncentrosymmetric_Cr11Ge19.pdf",        "content": "Complex itinerant ferromagnetism in noncentrosymmetric\nCr11Ge19\nN. J. Ghimire,1, 2M. A. McGuire,2D. S. Parker,2B. C. Sales,2J.-Q.\nYan,2, 3V. Keppens,3M. Koehler,3R. M. Latture,3and D. Mandrus1, 2, 3\n1Department of Physics and Astronomy,\nThe University of Tennessee, Knoxville, Tennessee 37996, USA\n2Materials Science and Technology Division,\nOak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n3Department of Materials Science and Engineering,\nThe University of Tennessee, Knoxville, Tennessee, 37996, USA\n(Dated: August 27, 2018)\nAbstract\nThe noncentrosymmetric ferromagnet Cr 11Ge19has been investigated by electrical transport,\nAC and DC magnetization, heat capacity, x-ray di\u000braction, resonant ultrasound spectroscopy, and\n\frst principles electronic structure calculations. Complex itinerant ferromagnetism in this material\nis indicated by nonlinearity in conventional Arrott plots, unusual behavior of AC susceptibility,\nand a weak heat capacity anomaly near the Curie temperature (88 K). The inclusion of spin\nwave excitations was found to be important in modeling the low temperature heat capacity. The\ntemperature dependence of the elastic moduli and lattice constants, including negative thermal\nexpansion along the c axis at low temperatures, indicates strong magneto-elastic coupling in this\nsystem. Calculations show strong evidence for itinerant ferromagnetism and suggest a noncollinear\nground state may be expected.\nPACS numbers: 75.30.Kz, 75.10.-b\n1arXiv:1204.2254v2  [cond-mat.str-el]  14 Jun 2012I. INTRODUCTION\nItinerant ferromagnets crystallizing in noncentrosymmetric space groups have attracted\nmuch attention recently. The key factor is that the lack of an inversion center in the crystal\nlattice means that Dzyaloshinsky-Moriya (DM) spin-orbit interactions are allowed.1,2These\ninteractions add a term to the free energy\u0000 !m\u0001r\u0002\u0000 !mfavoring perpendicular orientation\nof the spins. In metallic systems the DM term can lead to helimagnetism3,4and complex\nspin textures resembling liquid crystal phases when the helimagnetism is destabilized.5{7\nMnSi is the most heavily studied system in this class of materials. MnSi orders at 29.5\nK forming a long period helimagnet with a wavelength \u0015h\u0018180\u0017A weakly pinned along\nthe<111>direction. It crystallizes in the noncentrosymmetric space group P213 with the\nB20 structure. At low temperatures and in applied magnetic \felds above 6 kOe a \feld\npolarized phase appears. Just below the transition temperature and for magnetic \felds\napplied along <100>a phase known as the Aphase is stabilized. Recently this phase has\nbeen identi\fed as a skyrmion lattice.6,8Skyrmions are a direct result of DM interactions\nallowed in the crystal lattice with no inversion symmetry. Similar spin textures have been\nobserved in two other materials so far, FeGe9and Fe 1\u0000xCoxSi,10both having the same\nstructure as MnSi. Cr 1=3NbS 2is another example where the DM interaction is responsible\nfor the helimagnetism.11,12In this material the period of the helix is \u0018480\u0017A, and the DM\ninteraction is thought to be stabilized by the lack of an inversion center between the two\nchromium atoms along the c axis as it crystallizes in a noncentrosymmetric space group\nP6322. Recently, Lorentz microscopy and small-angle electron di\u000braction studies showed an\nemergence in the applied magnetic \feld of a periodic and nonlinear magnetic order called a\nchiral magnetic soliton lattice in addition to the the zero-\feld chiral helimagnetic structure.13\nIndeed, there are many other ferromagnets that crystallize in space groups lacking an\ninversion center. However, until now, both the theoretical and experimental studies have\nbeen concentrated mostly within the materials with the cubic B20 crystal structure. Thus\ninvestigation of noncentrosymmetric magnets with di\u000berent structures is important for un-\nderstanding the consequences of DM interactions in a wide variety of compounds. Cr 11Ge19\nis such a material14{16. It crystallizes in the noncentrosymmetric space group P4n2 belong-\ning to the point group D8\n2dand orders ferromagnetically below about 90 K. Interestingly, D2d\nis one of the crystallographic classes in which a ferromagnet is expected to have a thermody-\n2namically stable magnetic vortex phase in a certain range of applied magnetic \feld.17,18This\nphase is reminiscent of an Abrikosov vortex lattice in a type II superconductor. The possi-\nbility of such a structure was also predicted19for MnSi, FeGe, Fe xCo1\u0000xSi, and Co xMn1\u0000xSi.\nSkyrmion lattices have since been identi\fed in the \frst three of these compounds, as stated\nabove. Relatively few studies have appeared on Cr 11Ge19. In early work, Zagryazhskii et\nal.15reported Cr 11Ge19to be a semimetallic ferromagnet with a transition temperature of \u0018\n86 K. Intriguingly, they point out the lack of a lambda anomaly at the ferromagnetic tran-\nsition temperature in their speci\fc heat measurements. A linear mu\u000en tin orbital (LMTO)\ncalculation of electronic density of states20indicated it to be a low moment itinerant ferro-\nmagnet. A study of thermoelectric properties on a single crystal above room temperature\nreported the material to have a metallic behavior with dominant p-type conductivity and a\nrelatively low resistivity.21\nIn this paper we report magnetization, transport, and thermodynamic properties of\nCr11Ge19together with results obtained from electronic structure calculations. Both the\nexperimental results and calculations indicate that Cr 11Ge19is a good example of an itin-\nerant electron ferromagnet, with signatures of both spin wave excitations and magnetic\n\ructuations apparent in the data. Although no direct evidence for a helimagnetic or other\nexotic magnetic ground state has been found in the polycrystalline samples we have studied,\nthe behavior of this material is unusual in several respects and deserves further study in\nsingle crystal form.\nII. EXPERIMENTAL DETAILS\nPolycrystalline samples were prepared and studied. Stoichiometric amounts of high purity\nCr pieces (99.999 %) and Ge pieces (99.9999 %) were arc-melted in an argon atmosphere.\nThe resulting ingot was then sealed in a quartz tube and annealed at 900oC for one week.\nThe annealed ingot was then ground into \fne powder inside a He-\flled glove box and pressed\ninto a pellet, which was again sealed in an evacuated quartz tube and annealed at 900oC\nfor another week.\nSingle crystal growth was also attempted using two di\u000berent techniques: a \rux method\nusing Ge as a self \rux and a modi\fed Bridgman method. Both growths were carried out\nusing a molar ratio of Cr : Ge = 20 : 80 of the starting materials. In the \rux method,\n3a total charge consisting of 7 g of Ge pieces (99.9999 % pure) and Cr powders (99.99 %\npure) were loaded in a 5 ml alumina crucible. A catch crucible containing quartz wool\nwas mounted on top of a growth crucible and both were sealed in a silica ampoule under\nvacuum. The sealed ampoule was heated to 1100oC over 10 hours and homogenized at\n1100oC for 30 hours, furnace cooled to 1000oC and then slowly cooled to 910oC at the\nrate of 2oC per hour. Once the furnace reached 910oC, the excess \rux was decanted from\nthe crystals. Single crystals with cubic shape and of an average dimension of about 0.5 mm\nwere obtained. The so called modi\fed Bridgman method was employed by \frst melting a\ntotal charge of 10 g of Ge pieces (99.9999 % pure) and Cr pieces (99.999 %) in an argon\natmosphere. The arc-melted ingot was broken into pieces and loaded into a well-cleaned\nquartz tube of 14 mm inner diameter with a pointed bottom forming a Bridgman crucible.\nThe tube was placed in an upright position inside a box furnace and \frst heated to 1100oC\nover 10 hours and homogenized for 30 hours. It was then cooled quickly to 1000oC and then\nslowly cooled to 900oC at the rate of 2oC per hour which was subsequently furnace-cooled\nto room temperature. Tiny cube-shaped crystals with a typical dimension of 0.1 mm were\nalways observed in the middle of the resulting boule.\nRoom temperature x-ray di\u000braction on powders from pulverized single crystals con\frmed\nsingle phase for the crystals obtained in both techniques. The atomic ratio was studied using\na Hitachi bench-top scanning electron microscope (SEM) with a Bruker energy dispersive x-\nray spectrometer (EDS). The atomic percentages of Cr and Ge observed are 42 at.% and 58\nat.%, respectively, which is within the expected uncertainty for standardless measurements\non irregular surfaces. Crystals obtained from modi\fed Bridgman method were too small\nfor convenient characterization, while the relatively larger crystals grown by \rux method\ntypically had some residual Ge \rux on the crystal surface. Therefore in this study we\ncharacterized polycrystalline material, which was \u001598% pure based on powder di\u000braction\nand EDS measurements.\nAll the measurements were carried out in pieces cut from the same compact polycrys-\ntalline pellet that was determined to be 82% of the theoretical density. X-ray powder di\u000brac-\ntion was performed at room temperature using a PANalytical X'Pert powder di\u000bractometer\nfor phase identi\fcation and structural re\fnement. X-ray powder patterns were also obtained\nevery 20 K on cooling from room temperature down to 11 K. DC magnetization measure-\nments were performed using a Quantum Design magnetic property measurement system\n4(MPMS). AC susceptibility, speci\fc heat, and resistivity measurements were conducted in\na Quantum Design physical property measurement system (PPMS). AC susceptibility was\nmeasured by using a drive coil frequency of 85 Hz and an excitation \feld of 10 Oe at di\u000berent\napplied DC magnetic \felds from 0 to 10 kOe. Speci\fc heat measurements were performed\non a small piece of 27.5 mg. Resistivity was measured using platinum wires and Epotek\nH20E silver epoxy in a four-wire con\fguration on a 1.9 \u00021.2\u00021.4 mm3rectangular bar.\nThe temperature dependence of the elastic moduli was obtained using resonant ultrasound\nspectroscopy (RUS) on a 1.261 x 2.122 x 3.899 mm3polycrystalline pellet using a custom\ndesigned probe inserted into a Quantum Design Versalab.22\nIII. RESULTS AND DISCUSSION\nA. Crystal chemistry\n/s48 \n/s49 \n/s50 \n/s51 \n/s67/s114/s71/s101/s97\n/s98/s99\nFIG. 1: (Color online) The tetragonal Cr 11Ge19structure. (a) Arrangement of Cr and Ge atoms\nin the complex Nowotny chimney ladder structure emphasizing the long c axis (52.321 \u0017A). (b) A\nview down the c axis. One turn of the Cr helix is emphasized on moving from 0, 1, 2, and 3\ncounterclockwise. (c) A perpendicular view showing a Ge helix within a Cr helix. The Cr atoms\nare shown as black (larger) balls, and the Ge atoms are shown as orange (smaller) balls.\n5Cr11Ge19crystallizes in the Mn 11Si19structure type in a family of compounds known\nas Nowotny chimney ladders (NCLs). These are a series of intermetallic compounds with\ncomposition TnXm, where 2>m/n>1.25.21Here Tis a transition metal element and Xis\na main group metal. These compounds have a complex structure in which Tatoms form\n4-fold helices inside of which Xatoms form separate helices.23NCLs have been found to\nfollow the 14 electron rule, according to which a NCL compound having 14 valence electrons\n(VEL) per main group metal atom should be semiconductor and one with VEL less than 14\nshould be metal.21The rule holds for Cr 11Ge19as it has total of 12.9 valence electrons per\nGe atom and is known to have metallic behavior. Figure 1 shows the structure of Cr 11Ge19.\nIt has a very long c axis (52.321 \u0017A) as shown in Fig. 1(a). Figure 1(b) shows the view\ndown the c axis. The Cr atoms (black) form helices, shaped like chimneys, within which\nthe helices of Ge atoms (orange) reside. In Fig. 1(c) we emphasize the helices of Cr and\nGe. The Cr-Cr distance is the shortest along the helix (3.124 - 3.138 \u0017A),14and hence,\nsubstantial Cr-Cr interaction can be expected in the direction of the helix. However, the\nstructure is much more complex due to the presence of large number of atoms (120) in the\nunit cell. It has 12 inequivalent Cr sites and 10 inequivalent Ge sites. We used the reported\nstructure14for the Rietveld re\fnement of the room temperature x-ray powder pattern. The\n\ft is reasonably good considering the complex structure as shown in Fig. 2. Atomic positions\nand occupancies were not re\fned because of the di\u000eculty introduced by the large number\nof atoms in the unit cell. The inset in Fig. 2 shows a magni\fed part of the \ft at higher\nangles in which indexed peaks are seen more clearly. Lattice constants obtained from the \ft\nare a = 5.805 \u0017A and c = 52.321 \u0017A, which are in good agreement with previously reported\nvalues.15,20\nB. DC magnetization\nFigure 3 shows the temperature dependence of magnetization of Cr 11Ge19in an applied\n\feld of 10 kOe. As the sample is cooled the transition from a paramagnetic state to an\nordered ferromagnetic state is clearly visible. The inset shows the Curie-Weiss \ft of \u001f\u00001\n= (T\u0000\u0012CW)=Cto the high temperature part of the inverse susceptibility above 220 K.\nThe parameters obtained are Curie constant C= 5.34 K cm3mol\u00001F.U.\u00001and the Curie-\nWeiss temperature \u0012CW= 128.6 K. The e\u000bective moment per mole of chromium atom peff\n6/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48 /s49/s50/s48 /s49/s51/s48/s48/s53/s48/s48/s48/s49/s48/s48/s48/s48/s49/s53/s48/s48/s48/s50/s48/s48/s48/s48/s50/s53/s48/s48/s48\n/s55/s49 /s55/s50 /s55/s51 /s55/s52 /s55/s53 /s55/s54 /s55/s55 /s55/s56\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s50 /s32/s40/s100/s101/s103/s41/s40/s52/s32/s49/s32/s49/s52/s41\n/s40/s52/s32/s48/s32/s49/s54/s41/s40/s51/s32/s50/s32/s50/s55/s41\n/s40/s52/s32/s50/s32/s49/s50/s41\n/s40/s51/s32/s51/s32/s49/s54/s41/s40/s52/s32/s50/s32/s51/s41/s40/s52/s32/s48/s32/s49/s56/s41\n/s32/s32\n/s50 /s32/s40/s100/s101/s103/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41FIG. 2: (Color online) Rietveld re\fnement of x-ray powder pattern of Cr 11Ge19collected at room\ntemperature.\nFIG. 3: (Color online) M=H as a function of temperature measured at an applied \feld of H= 10\nkOe. Inset shows the \ft to the Curie-Weiss law.\ncalculated from the Curie constant is 1.97 \u0016B.\nFigure 4(a) shows the magnetization Mof Cr 11Ge19as a function of \feld at several\ntemperatures. At low temperatures Msaturates above 20 kOe. The saturation is suppressed\nwith increasing temperature, and MversusHbecomes a straight line at higher temperatures.\nThe saturated magnetic moment obtained in the ordered state is 0.49 \u0016B/Cr. Within the\n7Stoner model, itinerant ferromagnets obey the relation\nM(H;T)2=\u0000A\nB+1\nB(H\nM(H;T)); (1)\nwhereAandBare independent of H.24{26Ais a temperature dependent term and vanishes at\nTC. This should give straight lines on an Arrott plot,27M2(H;T) versusH=M (H;T), with a\nstraight line passing through the origin at the transition temperature . In Fig. 4(b) we show\nArrott plots for Cr 11Ge19. These Arrott plots are not perfectly straight lines as expected\nand observed in itinerant ferromagnets like ZrZn 2, Ni 3Al, and NiPt alloys.24However, the\nArrott plots for Cr 11Ge19are similar to those observed in the case of MnSi28and the layered\nitinerant ferromagnet LaCoAsO.29Such behavior was explained by Takahashi,30who in his\ntheory added zero point local spin \ructuations, which were previously neglected. This theory\npredicts\nh= [TA\n3(2 +p\n5)Tc]2m5; (2)\nwhere h = 2 \u0016BH and m = 2 M(T)=Nomagnetization per magnetic site. The parameter TA\ncharacterizes the dispersion of the static magnetic susceptibility in wave-vector ( q) space.\nFrom Eq. (2) it can be seen that M4versusH=M should be a straight line at TC. Such a\nlinear relation has been con\frmed in MnSi and Fe 1\u0000xCoxSi.30,31Figure 4(c) shows the M4\nversusH=M curve of Cr 11Ge19which shows straight line behavior. The plots are almost a\nstraight line between 85 and 90 K. We estimate the Curie temperature to be 88 K.\nC. AC susceptibility\nFigure 5 shows the temperature dependance of the real part of the AC susceptibility\nmeasured in the applied DC \felds as indicated. At zero applied DC bias \feld (not shown)\nthe AC susceptibility increases quickly with decreasing temperature in the vicinity of the\ntransition temperature and decreases slightly upon further cooling. The e\u000bect of external\n\felds is quite striking. First, the \feld remarkably suppresses the ac susceptibility. Second,\na shoulder appears at lower \felds near TCwhich is de\fned by two peaks, one sharp peak\nin the vicinity of the transition temperature and the other broader peak below TC. With\nthe increase in \feld, the peak near TCgets suppressed and shifts slightly towards higher\ntemperature, whereas the broader peak below TCbecomes broader and shifts towards lower\ntemperature and is almost completely suppressed at H = 10 kOe. Similar behavior has\n8/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s49/s50/s51/s52/s53/s54\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s99\n/s49/s49/s48/s32/s75 /s49/s48/s53/s32/s75 /s49/s48/s48/s32/s75 /s57/s53/s32/s75 /s57/s48/s32/s75 /s56/s53/s32/s75 /s56/s48/s32/s75 /s55/s53/s32/s75 \n/s32/s32/s77/s52\n/s32/s40/s52 /s66 /s32/s70/s46/s85/s46/s45/s52\n/s41\n/s72/s47/s77 /s32/s40/s109/s111/s108/s32/s70/s46/s85/s46/s32/s99/s109/s45/s51\n/s41/s55/s48/s32/s75 \n/s32/s32/s77/s32/s40\n/s66 /s70/s46/s85/s46/s45/s49\n/s41\n/s72/s32/s40/s107/s79/s101/s41/s50/s32/s75 \n/s55/s48/s32/s75 \n/s49/s49/s48/s32/s75 \n/s51/s48/s48/s32/s75 /s97\n/s98\n/s32/s32/s77/s50\n/s32/s40 /s32/s70/s46/s85/s46/s45/s50\n/s41\n/s72/s47/s77 /s32/s40/s109/s111/s108/s32/s70/s46/s85/s46/s32/s99/s109/s45/s51\n/s41/s55/s48/s32/s75 \n/s55/s53/s32/s75 \n/s56/s48/s32/s75 \n/s56/s53/s32/s75 \n/s57/s48/s32/s75 FIG. 4: (Color online) (a) Mversus HforCr11Ge19at indicated temperatures. The plots from 70\nK to 110 K are in the interval of every 5 K. (b) M2versus H=M (Arrott plot) and (c) M4versus\nH=M forCr11Ge19at indicated temperatures.\nbeen observed in a PdMn alloy32, GdFe 2Zn2033, MnSi34and FeGe.35This AC susceptibility\nbehavior in GdFe 2Zn20has been interpreted as a manifestation of both the itinerant and the\nlocal moments in the material as it contains both 4f (local) and 3d (itinerant) moments. In\nthis material, the peak observed near TCshows the behavior of local moments as observed in\nCeAgSb 2,33and the broader peak at lower temperature is reminiscent of itinerant behavior\nas observed in ZnZr 2.33MnSi and FeGe show similar AC susceptibility behavior, but are\nknown to have no local moments. In these later two materials, the Aphase has been tracked\nout by AC susceptibility measurements conducted on single crystals.34,35\n9χFIG. 5: (Color online) AC susceptibility of Cr 11Ge19as a function of temperature at indicated\napplied \felds.\nD. Heat capacity\n γ β δ\nFIG. 6: (Color online) Temperature dependence of molar heat capacity of Cr 11Ge19. Inset shows\nthe low temperature \ft of the heat capacity.\nThe molar heat capacity of Cr 11Ge19from 2 to 200 K is shown in Fig. 6. There is no\nobvious lambda anomaly near TC, but upon closer inspection TCis marked by a small kink as\nshown by the arrow in Fig. 6. Zagryazhskii et al. reported that they observed a monotonic\nincrease in the speci\fc heat capacity from 55 to 300 K with no lambda anomaly.15The small\n10kink observed near TCis suppressed upon application of the magnetic \feld. Figure. 7 shows\nthe speci\fc heat capacity as a function of temperature between 50 and 110 K measured in\nzero \feld and at 50 kOe. The inset shows the di\u000berence curve obtained by plotting \u0001C p\n(the di\u000berence between C pmeasured in a 50 kOe \feld and in a zero applied \feld) as a\nfunction of temperature giving a clear peak near TC. Mohn and Hilscher36have discussed\nthe in\ruence of spin \ructuations on the speci\fc heat of itinerant ferromagnets. In Stoner\ntheory, the magnetic contribution vanishes above TC. In contrast, in systems with spin\n\ructuations, it is only the macroscopic moment that disappears at TCas spin \ructuations\npersist above the ordering temperature. The magnetic contribution to the discontinuity in\nthe speci\fc heat at the transition temperature in case of pure single particle excitations is\ngiven by \u0001Cm=M2\no\n\u001foTC. When spin \ructuations are taken into account, the discontinuity is\ngiven by \u0001C m=M2\no\n2\u001foTC(1\n2t4\nc+t2\nc+1\n2), whereMois the spontaneous magnetization, \u001fois the\ninitial ferromagnetic susceptibility, TCis the transition temperature, and tc=TC=Ts\nCwith\nTs\nCbeing the Curie temperature derived from the pure Stoner type behavior.36In Cr 11Ge19\nthe discontinuity in the speci\fc heat at TCcalculated for pure Stoner type excitations is\n1.2J (mol-atom)\u00001K\u00001. This value is small enough to explain the absence of a well-\nde\fned lambda anomaly in the speci\fc heat capacity near the ferromagnetic transition.\nBut, there is considerable uncertainty in the calculation. The spontaneous magnetization\nMois calculated by using the theory applicable for an itinerant ferromagnet37by \ftting\nstraight lines obtained at higher \felds in the Arrott plot even though the Arrott plot in this\nmaterial does not behave perfectly as in the case of systems like ZrZn 2.38There can also\nbe appreciable uncertainty introduced by \u001fo, which might include other components than\nonly the spin susceptibility (e.g., a diamagnetic component). It should be noted that we\nhave not included the contribution due to spin \ructuations in the calculation because of the\ndi\u000eculty introduced by the large number of electrons (1680 per unit cell) in estimating Ts\nC\nfrom band structure calculations. Inclusion of spin \ructuations further decreases \u0001 Cm. In\ncase of maximum spin \ructuations \u0001 Cmis reduced by a factor of 4. Thus, the presence\nof spin \ructuations in the material reduces the possibility of getting a sizable discontinuity\nin speci\fc heat at the transition temperature even if some uncertainty might have been\nintroduced in the calculation of \u0001 Cm.\nThe low temperature speci\fc heat data could not be well modeled by Cp=\rT+\fT3. This\nindicates additional excitations may be contributing to the heat capacity at low temperature.\n11∆FIG. 7: (Color online) Heat capacity of Cr 11Ge19in ambient \feld and 50 kOe. The inset shows\n\u0001Cp=Cp(H= 0) - Cp(H = 50 kOe) .\nSince this material is magnetically ordered below 88 K, magnetic excitations were considered\nby inclusion of a term in C pproportional to T3\n2.39This resulted in a good \ft to the data\nas shown in the inset of Fig. 6. The \ft yields the electronic heat capacity coe\u000ecient \r=\n7.26mJ=molK2, the phonon speci\fc heat coe\u000ecient \f= 0.06mJ=molK4, and the magnetic\nspeci\fc heat coe\u000ecient \u000e= 2.18mJ=molK5=2. The Debye temperature determined from \fis\n319 K. Forcing a \ft without the magnetic term gives a much lower value for the \ftted Debye\ntemperature (\u0018240 K). Elastic constant data presented below give a Debye temperature of\n340 K, further justifying the inclusion of spin excitations in modeling the heat capacity.\nE. Resistivity and magnetoresistance\nElectrical resistivity of Cr 11Ge19vs temperature is plotted in Fig. 8. The temperature\ndependence of the resistivity is metallic over the whole temperature range. A slope change\nis observed at s90 K, which is consistent with a signi\fcant loss of spin-disorder scattering\nupon magnetic ordering. The TCinferred from resistivity is in good agreement with the value\nofTCobtained from magnetization measurements. The room temperature value of electrical\nresistivity, 0.35 m\ncm, is in good agreement with the value (0.345 m\ncm) reported by\nZagryazhskii et al.15measured on a polycrystal sample and is about a factor of 2 higher than\nthe value reported by Caillat et al.21on a single crystal sample. The observed excess value\n12ρ10 −6 Ω\n∆ρ ρFIG. 8: (Color online) Resistivity of Cr 11Ge19as a function of temperature. Inset shows the\nmagnetoresistance.\nof resistivity can be attributed to grain boundary scattering in the polycrystalline sample.\nThe residual resistance ratio ( \u001a300K=\u001a2K) is large, having a value of 89. The inset in Fig.\n8 shows the magnetoresistance de\fned as \u0001 \u001a=\u001a, where \u0001\u001a=\u001aH-\u001awith\u001aHand\u001abeing\nthe resistivity measured at 50 kOe and zero applied magnetic \feld, respectively. Negative\nmagnetoresistance is observed below 150 K with the largest e\u000bect in the vicinity of TCwhere\n\ructuations are the strongest.\nF. Thermal expansion and elastic moduli\nThe temperature dependence of the lattice parameters is plotted in Fig. 9. Both a(T)\nandc(T) are normalized by dividing with the corresponding room temperature values. No\nstructural phase transition is observed on decreasing the temperature down to 11 K. How-\never, the temperature dependence of the lattice parameters a(T) andc(T) show dramatic\ndi\u000berences below the magnetic ordering temperature. Below TC,a(T) decreases continu-\nously down to 11 K, whereas c(T) shows a region of negative thermal expansion. This\nbehavior shows the presence of magneto-elastic coupling. This coupling is also evident in\nthe temperature dependence of elastic constants as discussed below.\nResonant ultrasound spectroscopy (RUS) measurements were conducted to obtain the\ntemperature dependence of the longitudinal ( C11) and shear ( C44) elastic moduli. C11and\n13FIG. 9: (Color online) Temperature dependence of lattice constants of Cr 11Ge19. The lattice\nconstants are normalized by dividing with values at 300 K.\nC44were used to obtain the shear and longitudinal sound velocities. For polycrystalline\nsamples, the shear velocity is given by40vs=q\nC44\ndand the longitudinal velocity is given\nby vl=q\nC11\nd, wheredis the density of the sample. Anderson's formula41was then used to\ncalculate the Debye temperature from the average sound velocity just above the transition\ntemperature (90 K). The Debye temperature obtained is 340 K, which is consistent with the\nDebye temperature estimated from Cp(T) above.\nThe temperature dependence of the longitudinal C11and shear C44elastic moduli of\nCr11Ge19is plotted in Figs. 10(a) and 10(b), respectively. In both \fgures the insets focus on\nthe region from about 50 to 200 K so as to show the behavior near the Curie temperature.\nBetween 220 and 250 K the ultrasonic absorption in the sample became so great that for\nseveral temperatures not enough resonances were observed to allow for an accurate determi-\nnation of both elastic moduli. However, a few resonances that depend almost exclusively on\nC44remained visible throughout this region, which allowed us to follow the shear modulus\nover the entire temperature range. The typical temperature dependence of elastic moduli\nis that at higher temperatures they increase linearly with decrease in temperature and ap-\nproach absolute zero with zero slope.42,43In Cr 11Ge19, deviation from the normal behavior\nis observed in both the longitudinal and shear elastic constants. C11starts softening well\nabove the Curie temperature without showing any remarkable feature at the transition tem-\nperature. C44, on the other hand, increases with decreasing temperature down to TCand\n14/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s57/s56/s49/s48/s48/s49/s48/s50/s49/s48/s52/s49/s48/s54/s49/s48/s56/s49/s49/s48/s49/s49/s50/s49/s49/s52/s49/s49/s54\n/s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53 /s49/s53/s48 /s49/s55/s53 /s50/s48/s48/s49/s49/s51/s46/s48/s49/s49/s51/s46/s53/s49/s49/s52/s46/s48/s49/s49/s52/s46/s53/s49/s49/s53/s46/s48/s49/s49/s53/s46/s53\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s51/s57/s52/s48/s52/s49/s52/s50/s52/s51/s52/s52/s52/s53\n/s53/s48 /s56/s48 /s49/s49/s48 /s49/s52/s48 /s49/s55/s48 /s50/s48/s48/s52/s51/s46/s54/s52/s51/s46/s55/s52/s51/s46/s56/s52/s51/s46/s57/s52/s52/s46/s48/s52/s52/s46/s49/s52/s52/s46/s50/s52/s52/s46/s51/s52/s52/s46/s52\n/s32/s32/s67\n/s49/s49/s32/s40/s71/s80/s97/s41\n/s84/s32/s40/s75/s41/s97\n/s32/s32/s67\n/s49 /s49 /s32/s40/s71/s80 /s97/s41\n/s84/s32/s40/s75/s41\n/s32/s32/s67\n/s52/s52/s32/s40/s71/s80/s97/s41\n/s84/s32/s40/s75/s41/s98\n/s32/s32/s67\n/s52 /s52 FIG. 10: Variation of elastic moduli (a) C11and (b) C44as a function of temperature.\nthen softens upon further cooling. This demonstrates the interaction between the magnetic\nordering and the crystal lattice in Cr 11Ge19. This behavior is reminiscent of the Invar e\u000bect\nin ferromagnetic materials and is in accord with the prediction of Landau's theory of second\norder magneto-elastic coupling.44\nG. Electronic structure calculations\nExperimental results have indicated that some of the properties of Cr 11Ge19deviate\nfrom what is expected in an ordinary ferromagnet. In an attempt to understand these\nunusual behaviors we have performed \frst principles calculations of Cr 11Ge19in both a\n15FIG. 11: (Color online) Electronic density of states of Cr 11Ge19in (a) nonmagnetic state and (b)\nmagnetic state.\nnonmagnetic state and a collinear ferromagnetic state, using the all-electron code WIEN2K45\nin the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof.46Atomic\nsphere radii of 2.41 and 2.13 Bohr radii were used for the Cr and Ge atoms, respectively,\nand an RK maxof 7, where R is the minimum atomic sphere radius and K is the largest\nplane-wave vector used in the expansion. Calculations proceeded slowly due to the large\nnumber (1680) of electrons in the unit cell; \fnal results were converged to within 2 meV\nper unit cell, a small value considering the large unit cell. We \fnd strong evidence for a\nmagnetic ground state of Cr 11Ge19, with the ferromagnetic ordering some 87 meV per Cr\natom lower in energy than the nonmagnetic ground state. The ordered moment averages\napproximately 1 \u0016Bper Cr atom, but is unevenly distributed amongst the 12 inequivalent\nCr sites in the unit cell, with moment per site ranging from 0.3 to 1.7 \u0016B, which we interpret\nas indirect evidence of the propensity of the system towards a noncollinear ground state.\nThis interpretation of a noncollinear ground state is based upon the DFT constraining of the\nmoments to be collinear, which will necessarily reduce the value of the calculated moment\n16for those spins which in the actual physical system are not collinear. This interpretation\nis strengthened by the fact that the reported14Cr-Cr nearest neighbor distances vary only\nfrom 3.124 to 3.138 \u0017A amongst the inequivalent Cr sites, so that the di\u000berences in calculated\nmoment are more likely to be an artifact of the collinearity assumed rather than indicative\nof physically distinct moment values. The calculated average moment is somewhat higher\nthan the observed experimental value of 0.5 \u0016B; this overestimation sometimes occurs with\nthe GGA. Due to the time-consuming nature of the calculations we have not carried out\nadditional local density approximation (LDA) calculations, which may better match the\nactual ordered moment.\nTo better understand the electronic structure we have calculated the electronic density\nof states (DOS) in both the magnetic and nonmagnetic states, using the \frst principles\ncalculated band structure with approximately 300 kpoints in the full Brillouin zone. These\nDOS are depicted in Fig. 11. The nonmagnetic state [Fig. 11(a)] shows a huge peak\nin the density of states exactly at the Fermi energy, highly favorable towards a Stoner-\ntype ferromagnetic instability (recall the Stoner criterion IN0>1, where I is the exchange\ncorrelation integral and N 0the Fermi level density of states). This DOS is in rough agreement\nwith the non-self-consistent band structure calculations of Ref. 20. With I for Cr taken from\nRef. 47 as 0.38 eV and the Fermi level DOS of approximately 5.7/Cr/eV, the Stoner criterion\nis well satis\fed, and as described earlier this \fts with the magnetic ground state we \fnd.\nWe turn now to the magnetic state DOS [Fig. 11(b)]. The majority spin-up DOS still has\na peak very near EF, but this peak is much lower than in the nonmagnetic case. Substantial\nspectral weight for the spin-up states is transferred below E F, as is expected for the majority\nspin, while the spin-down DOS is somewhat more equally distributed above and below E F.\nWe note also that the strongly magnetic nature of this system is paralleled by the spin-up\nand spin-down DOS not coinciding until several eV from the Fermi level. We note also that\nthe band gap has disappeared, with instead a deep minimum in the spin-up DOS just above\nEFand a somewhat less deep minimum in the spin-down DOS. This band gap absence, to\nbe compared with its existence in the nonmagnetic state, is again indicative of the strong\nmagnetism present in this material.\n17IV. CONCLUSION\nThe results presented here, both from experiment and \frst principles calculations, indi-\ncate unusual magnetism in Cr 11Ge19. The behavior of the magnetization and heat capacity\nsuggest itinerant, noncollinear ferromagnetism with a Curie temperature near 88 K, and\nthis description is supported by \frst principles calculations. The magnetism appears to be\nstrongly coupled to the crystal lattice, as indicated by anomalous behavior of the lattice\nparameters and the elastic moduli at and below TC. The in\ruence of spin-wave excitations\nis observed in the heat capacity at low temperature. Interestingly, some of the properties\nare similar to those of MnSi and other itinerant ferromagnets. It is interesting to speculate\nabout possible helimagnetism in Cr 11Ge19, based on the observed properties and the nature\nof the crystal structure. However, the present data cannot con\frm the magnetic structure,\nand single crystals of suitable size for neutron di\u000braction are not yet available. Our observa-\ntions clearly point to complex and interesting magnetism in this compound, and show that\nfurther study would be of interest.\nACKNOWLEDGEMENTS\nWe thank S. E. Nagler for stimulating discussions and U. K. R o\u0019ler for pointing out\nthat magnetic vortices have been predicted to form in the space group to which Cr 11Ge19\nbelongs. Research was supported by the U.S. Department of Energy, O\u000ece of Basic Energy\nSciences, Materials Sciences and Engineering Division. M.A.M. acknowledges support from\nthe U.S. Department of Energy, Energy E\u000eciency and Renewable Energy, O\u000ece of Vehicle\nTechnologies, Propulsion Materials Program. D.P. supported by the ORNL LDRD SEED\nfunding project S12-006, \\Rare Earth Free Magnets: Compute, Create, Characterize\". V.K.\nand M.K. acknowledge support from DOD DEPSCoR Grant No. N00014-08-1-0783 and\nNSF-DMR-0804719. We are grateful for the technical assistance of Douglas E. Fielden at\nthe University of Tennessee.\n1I. 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B 16, 255 (1977).\n20"    },    {        "title": "0810.0389v1.Voltage_controlled_inversion_of_magnetic_anisotropy_in_a_ferromagnetic_thin_film_at_room_temperature.pdf",        "content": "arXiv:0810.0389v1  [cond-mat.mtrl-sci]  2 Oct 2008\n/CE /D3/D0/D8/CP/CV/CT \r/D3/D2 /D8/D6/D3/D0/D0/CT/CS /CX/D2 /DA /CT/D6/D7/CX/D3/D2 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CX/D2/CP /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D8/CW/CX/D2 /AS/D0/D1 /CP/D8 /D6/D3 /D3/D1 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/C5 /CF /CT/CX/D0/CT/D61/B8 /BT /BU/D6/CP/D2/CS/D0/D1/CP/CX/CT/D61/B8 /CB /BZ/CT/D4/D6/GC/CV/D71/B8 /C5 /BT/D0/D8/CW/CP/D1/D1/CT/D61/B8 /C5/C7/D4 /CT/D01/B8 /BV /BU/CX/CW/D0/CT/D62/B8 /C0 /C0/D9/CT/CQ/D02,4/B8 /C5 /CB /BU/D6/CP/D2/CS/D82/B8 /CA /BZ/D6/D3/D7/D71,3/CP/D2/CS /CB /CC/BU /BZ/D3 /CT/D2/D2/CT/D2 /DB /CT/CX/D21,3\n1/CF /CP/D0/D8/CW/CT/D6/B9/C5/CT/CX/H3/D2/CT/D6/B9/C1/D2/D7/D8/CX/D8/D9/D8/B8 /BU/CP /DD /CT/D6/CX/D7\r /CW/CT /BT/CZ /CP/CS/CT/D1/CX/CT /CS/CT/D6 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/D3/CU5mm2/D3/D2 /D8/CW/CTˆ x /B9ˆ y /CU/CP\r/CT /D3/CU /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6/D7/BA /CC /D3 /D4/D6/CT/DA /CT/D2 /D8/D3 /DC/CX/CS/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C6/CX /AS/D0/D1/B8 /CP10nm /D8/CW/CX\r /CZ /BT/D9 /AS/D0/D1 /DB /CP/D7 /CS/CT/D4 /D3/D7/CX/D8/CT/CS /CX/D2/B9/D7/CX/D8/D9 /D3/D2 /D8/D3/D4 /D3/CU /D8/CW/CT /C6/CX/D0/CP /DD /CT/D6/BA /CC/CW/CT /D1 /D9/D0/D8/CX/CU/D9/D2\r/D8/CX/D3/D2/CP/D0 /CW /DD/CQ/D6/CX/CS /D3/CQ/D8/CP/CX/D2/CT/CS /CP/CU/D8/CT/D6 /C6/CX /CS/CT/D4 /D3/D7/CX/D8/CX/D3/D2 /CX/D7 /D7/CZ /CT/D8\r /CW/CT/CS /D7\r /CW/CT/D1/CP/D8/CX\r/CP/D0/D0/DD/CX/D2 /BY/CX/CV/BA /BD/B4/CP/B5 /D8/D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /D8/CW/CT /CS/CT/AS/D2/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CP/D2/CV/D0/CT/D7 /D8/CW/CP/D8 /CS/CT/D7\r/D6/CX/CQ /CT /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU/CE /D3/D0/D8/CP/CV/CT \r /D3/D2/D8/D6 /D3/D0 /D0/CT /CS /CX/D2/DA/CT/D6/D7/CX/D3/D2 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6 /D3/D4/DD /BG/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 M= (M,Θ,Φ) /CP/D2/CS /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CSH= (H,θ,φ) /CX/D2 /D8/CW/CT/D7/CP/D1/D4/D0/CT/B9/CPꜶ/DC/CT/CS \r/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1/BA/CC /D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /D7/D8/CP/D8/CX\r /D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D7/D4 /D3/D2/D7/CT /D3/CU /D8/CW/CT /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D8/CW/CX/D2/AS/D0/D1/BB/D4/CX/CT/DE/D3 /CT/D0/CT\r/D8/D6/CX\r /CP\r/D8/D9/CP/D8/D3/D6 /CW /DD/CQ/D6/CX/CS /DB /CT /CT/D1/D4/D0/D3 /DD /D7/D9/D4 /CT/D6\r/D3/D2/CS/D9\r/D8/CX/D2/CV /D5/D9/CP/D2 /D8/D9/D1 /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D2\r/CT /CS/CT/B9/DA/CX\r/CT /B4/CB/C9/CD/C1/BW/B5 /D1/CP/CV/D2/CT/D8/D3/D1/CT/D8/D6/DD /BA /CC/CW/CT /C9/D9/CP/D2 /D8/D9/D1 /BW/CT/D7/CX/CV/D2 /C5/C8/C5/CB/B9/CG/C4/B9/BJ /CB/C9/CD/C1/BW /D1/CP/CV/D2/CT/D8/D3/D1/CT/B9/D8/CT/D6 /CX/D7 /D7/CT/D2/D7/CX/D8/CX/DA /CT /D8/D3 /D8/CW/CT /D4/D6/D3 /CY/CT\r/D8/CX/D3/D2 m=mˆH /D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /D1/D3/D1/CT/D2 /D8 m /D3/D2 /D8/D3 /D8/CW/CT/D9/D2/CX/D8 /DA /CT\r/D8/D3/D6ˆH=H\nH\n/BA /CF /CT \r/D3/D6/D6/CT\r/D8/CT/CS m /CU/D3/D6 /D8/CW/CT /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6/CP/D2/CS /D9/D7/CT/CS /D8/CW/CT /C6/CX /AS/D0/D1 /DA /D3/D0/D9/D1/CTV= 3.5×10−13m3/D8/D3 \r/CP/D0\r/D9/D0/CP/D8/CT /D8/CW/CT /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT /D4/D6/D3 /CY/CT\r/D8/CX/D3/D2\nM=m/V /D3/CU /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D3/D2 /D8/D3ˆH /BA /BT/D0/D0 /D1/CP/CV/D2/CT/D8/D3/D1/CT/D8/D6/DD /CS/CP/D8/CP /D7/CW/D3 /DB/D2 /CX/D2 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV/DB /CT/D6/CT /D6/CT\r/D3/D6/CS/CT/CS /CP/D8 /CP /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT T= 300K /BA/CC/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /D3/CU /D8/CW/CT /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D8/CW/CX/D2 /AS/D0/D1/BB/D4/CX/CT/DE/D3 /CT/D0/CT\r/D8/D6/CX\r /CP\r/D8/D9/CP/D8/D3/D6/CW /DD/CQ/D6/CX/CS 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/CT/DC/CW/CX/CQ/CX/D8/D7 /CP \r/D3/D1/D4/D6/CT/D7/D7/CX/DA /CT /D7/D8/D6/CP/CX/D2 /CP/D0/D3/D2/CVˆ y /CP/D2/CS /CP /D8/CT/D2/D7/CX/D0/CT /D7/D8/D6/CP/CX/D2 /CP/D0/D3/D2/CV\nˆ x /CJ\r/CU/BA /BY/CX/CV/BA /BD/B4/CS/B5℄/BA/CC /D3 /CS/CT/D7\r/D6/CX/CQ /CT /D8/CW/CT /CX/D1/D4/CP\r/D8 /D3/CU /D8/CW/CX/D7 /D0/CP/D8/D8/CX\r/CT /D7/D8/D6/CP/CX/D2 /D3/D2 /D8/CW/CT /C6/CX /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/DB /CT /D9/D7/CT /CP /D1/CP/CV/D2/CT/D8/CX\r /CU/D6/CT/CT /CT/D2/CT/D6/CV/DD /CS/CT/D2/D7/CX/D8 /DD /CP/D4/D4/D6/D3/CP\r /CW/BA /CC/CW/CT /CU/D6/CT/CT /CT/D2/CT/D6/CV/DDFtot\n/CX/D7 /CP /D1/CT/CP/D7/D9/D6/CT /CU/D3/D6/D8/CW/CT /CP/D2/CV/D9/D0/CP/D6 /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CW/CP/D6/CS/D2/CT/D7/D7/B8 /DB/CX/D8/CW /D1/CP/DC/CX/D1/CP /CX/D2Ftot\n\r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV/D8/D3 /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CW/CP/D6/CS /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /CP/D2/CS /D1/CX/D2/CX/D1/CP /D8/D3 /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CT/CP/D7/DD /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA /C1/D2/CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1/B8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D0/DB /CP /DD/D7 /D6/CT/D7/CX/CS/CT/D7 /CX/D2 /CP /D0/D3 \r/CP/D0 /D1/CX/D2/CX/D1 /D9/D1 /D3/CUFtot\n/BA /BV/D3/D2 /D8/D6/CP/D6/DD/D8/D3 /D7/CX/D2/CV/D0/CT/B9\r/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /AS/D0/D1/D7/B8 /D8/CW/CT /CT/DA /CP/D4 /D3/D6/CP/D8/CT/CS /C6/CX /AS/D0/D1/D7 /CP/D6/CT /D4 /D3/D0/DD\r/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /CP/D2/CS /D8/CW /D9/D7 /D7/CW/D3 /DB /D2/D3/D2/CT/D8 \r/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /DB/CW/CX\r /CW /D1/CP /DD \r/D3/D1/D4 /CT/D8/CT /DB/CX/D8/CW /D7/D8/D6/CP/CX/D2/B9/CX/D2/CS/D9\r/CT/CS /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX/CT/D7/CP/D2/CS /CW/CT/D6/CT/CQ /DD /D6/CT/CS/D9\r/CT /D8/CW/CT /CP\r /CW/CX/CT/DA /CP/CQ/D0/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /CT/AR/CT\r/D8 /CJ/BD/BL℄/BAFtot\n/CX/D7 /D8/CW /D9/D7 /CV/CX/DA /CT/D2/CQ /DD\nFtot=Fstat+Fdemag+Fmagel. /B4/BD/B5/CC/CW/CT /AS/D6/D7/D8 /D8/CT/D6/D1Fstat=−µ0MH(sinΘsinΦsin θsinφ+cosΘcos θ+sinΘcosΦsin θcosφ)/CX/D2 /BX/D5/BA /B4/BD/B5 /CX/D7 /D8/CW/CT /CI/CT/CT/D1/CP/D2/D2 /D8/CT/D6/D1 /CP/D2/CS /CS/CT/D7\r/D6/CX/CQ /CT/D7 /D8/CW/CT /CX/D2/AT/D9/CT/D2\r/CT /D3/CU /CP/D2 /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r/CE /D3/D0/D8/CP/CV/CT \r /D3/D2/D8/D6 /D3/D0 /D0/CT /CS /CX/D2/DA/CT/D6/D7/CX/D3/D2 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6 /D3/D4/DD /BH/AS/CT/D0/CSH /D3/D2 /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CUM /BA /CC/CW/CT /D9/D2/CX/CP/DC/CX/CP/D0 /CS/CT/D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CT/D6/D1Fdemag=\nµ0\n2M2sin2Θcos2Φ /CX/D7 /D8/CW/CT /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD \r/CP/D9/D7/CT/CS /CQ /DD /D8/CW/CT /D8/CW/CX/D2/B9/AS/D0/D1 /D7/CW/CP/D4 /CT /D3/CU /D8/CW/CT /D7/CP/D1/D4/D0/CT /CJ/BE/BL℄/BA/CC/CW/CT /D0/CP/D7/D8 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D8/D3 /BX/D5/BA /B4/BD/B5\nFmagel=3\n2λ/parenleftbig\ncNi\n12−cNi\n11/parenrightbig\n[ε1(sin2Θsin2Φ−1/3) /B4/BE/B5\n+ε2(cos2Θ−1/3)+ε3(sin2Θcos2Φ−1/3)]/CS/CT/D7\r/D6/CX/CQ /CT/D7 /D8/CW/CT /CX/D2/AT/D9/CT/D2\r/CT /D3/CU /D8/CW/CT /D0/CP/D8/D8/CX\r/CT /D7/D8/D6/CP/CX/D2/D7 /D3/D2 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CJ/BE/BJ℄/BA /CC/CW/CT /D7/D8/D6/CP/CX/D2/D7/CP/D0/D3/D2/CV /D8/CW/CTˆ x /B9/B8ˆ y /B9 /CP/D2/CSˆ z /B9/CP/DC/CX/D7 /CP/D6/CT /CS/CT/D2/D3/D8/CT/CS /CX/D2 /CE /D3/CX/CV/D8 /B4/D1/CP/D8/D6/CX/DC/B5 /D2/D3/D8/CP/D8/CX/D3/D2 /CJ/BF/BC℄ /CP/D7ε1,ε2\n/CP/D2/CSε3\n/B8/D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA /BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT/B8 cNi\n11= 2.5×1011N/m2/CP/D2/CScNi\n12= 1.6×1011N/m2/CP/D6/CT /D8/CW/CT/CT/D0/CP/D7/D8/CX\r /D1/D3 /CS/D9/D0/CX /D3/CU /C6/CX /CJ/BE/BK℄/BA /CC/CW/CT /CT/AR/CT\r/D8/D7 /D3/CU /D7/CW/CT/CP/D6 /D7/D8/D6/CP/CX/D2/D7 /B4εi,i∈ {4,5,6} /B5 /CP /DA /CT/D6/CP/CV/CT /D3/D9/D8 /CX/D2 /D3/D9/D6/D4 /D3/D0/DD\r/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /AS/D0/D1 /CP/D2/CS /D8/CW /D9/D7 /CP/D6/CT /D2/CT/CV/D0/CT\r/D8/CT/CS/BA/CD/D7/CX/D2/CVsin2Θsin2Φ+cos2Θ+sin2Θcos2Φ = 1 /CP/D2/CS /D3/D1/CX/D8/D8/CX/D2/CV /CX/D7/D3/D8/D6/D3/D4/CX\r /D8/CT/D6/D1/D7/B8 /BX/D5/BA /B4/BE/B5\r/CP/D2 /CQ /CT /D6/CT/DB/D6/CX/D8/D8/CT/D2 /CP/D7\nFmagel=Ku,magel,ycos2Θ+Ku,magel,zsin2Θcos2Φ /B4/BF/B5/DB/CX/D8/CW\nKu,magel,y=3\n2λ(cNi\n12−cNi\n11)(ε2−ε1) /B4/BG/B5\nKu,magel,z=3\n2λ(cNi\n12−cNi\n11)(ε3−ε1)./BW/D9/CT /D8/D3 /D8/CW/CT /CT/D0/CP/D7/D8/CX\r/CX/D8 /DD /D3/CU /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6 /CP/D2/CS /D8/CW/CT /C6/CX /AS/D0/D1/B8 /D8/CW/CT /D7/D8/D6/CP/CX/D2/D7εi\n/CP/D6/CT /D2/D3/D8/CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CU /CT/CP\r /CW /D3/D8/CW/CT/D6/BA /CC/CW/CT /D7/D8/D6/CP/CX/D2/D7ε1\n/CP/D2/CSε2\n/CX/D2 /D8/CW/CTˆ x /B9ˆ y /D4/D0/CP/D2/CT /CP/D6/CT /D0/CX/D2/CZ /CT/CS /CQ /DD /D8/CW/CT/C8 /D3/CX/D7/D7/D3/D2 /D6/CP/D8/CX/D3ν= 0.45 /D3/CU /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6 /CJ/BE/BG ℄ /CP\r\r/D3/D6/CS/CX/D2/CV /D8/D3\nε1=−νε2. /B4/BH/B5/BY /D9/D6/D8/CW/CT/D6/D1/D3/D6/CT/B8 /CS/D9/CT /D8/D3 /D8/CW/CT /CT/D0/CP/D7/D8/CX\r/CX/D8 /DD /D3/CU /D8/CW/CT /C6/CX /AS/D0/D1/B8 /D8/CW/CT /D3/D9/D8/B9/D3/CU/B9/D4/D0/CP/D2/CT /D7/D8/D6/CP/CX/D2ε3\n\r/CP/D2 /CQ /CT/CT/DC/D4/D6/CT/D7/D7/CT/CS /CP/D7 /CJ/BD/BL℄\nε3=−cNi\n12\ncNi\n11(ε1+ε2). /B4/BI/B5/BT/D7/D7/D9/D1/CX/D2/CV /CP /D0/CX/D2/CT/CP/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D3/CU /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6 /DB/CX/D8/CW /D0/CP/D8/CT/D6/CP/D0 /CS/CX/D1/CT/D2/D7/CX/D3/D2 L /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /CX/D8/D7/CS/D3/D1/CX/D2/CP/D2 /D8 /CT/D0/D3/D2/CV/CP/D8/CX/D3/D2 /CP/DC/CX/D7/B8 /DB /CT \r/CP/D2 \r/CP/D0\r/D9/D0/CP/D8/CT /D8/CW/CT /D7/D8/D6/CP/CX/D2 /CX/D2 /D8/CW/CT /C6/CX /AS/D0/D1 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /D8/CW/CT/CP\r/D8/D9/CP/D8/D3/D6/B3/D7 /CS/D3/D1/CX/D2/CP/D2 /D8 /CT/D0/D3/D2/CV/CP/D8/CX/D3/D2 /CP/DC/CX/D7 /CP/D7\nε2=δL\nLVp\n180V, /B4/BJ/B5/DB/CW/CT/D6/CTδL/L= 1.3×10−3/CX/D7 /D8/CW/CT /D2/D3/D1/CX/D2/CP/D0 /CU/D9/D0/D0 /CP\r/D8/D9/CP/D8/D3/D6 /D7/D8/D6/D3/CZ /CT /CU/D3/D6 /D8/CW/CT /CU/D9/D0/D0 /DA /D3/D0/D8/CP/CV/CT /D7/DB/CX/D2/CV\n−30V≤Vp≤+150V /BA /CF/CX/D8/CW /BX/D5/D7/BA /B4/BH/B5/B8 /B4/BI/B5 /CP/D2/CS /B4/BJ/B5 /CP/D0/D0 /D7/D8/D6/CP/CX/D2/D7 /CX/D2 /D8/CW/CT /C6/CX /AS/D0/D1 \r/CP/D2/D8/CW /D9/D7 /CS/CX/D6/CT\r/D8/D0/DD /CQ /CT \r/CP/D0\r/D9/D0/CP/D8/CT/CS /CU/D3/D6 /CP/D2 /DD /CV/CX/DA /CT/D2 /DA /D3/D0/D8/CP/CV/CTVp\n/BA /CC/CW/CX/D7 /CP/D0/D0/D3 /DB/D7 /D8/D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT/CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CUVp\n/CQ /DD /D1/CX/D2/CX/D1/CX/DE/CX/D2/CV /D8/CW/CT /D8/D3/D8/CP/D0/D1/CP/CV/D2/CT/D8/D3 /CT/D0/CP/D7/D8/CX\r /CU/D6/CT/CT /CT/D2/CT/D6/CV/DD /CS/CT/D2/D7/CX/D8 /DDFtot\n/CJ/BX/D5/BA /B4/BD /B5℄/BA/BW/D9/CT /D8/D3 /D8/CW/CT /D2/CT/CV/CP/D8/CX/DA /CT /D1/CP/CV/D2/CT/D8/D3/D7/D8/D6/CX\r/D8/CX/D3/D2 /B4λ <0 /B5 /D3/CU /C6/CX /CP/D2/CScNi\n11> cNi\n12\n/B8 /D8/CW/CT /CX/D2/B9/D4/D0/CP/D2/CT /CT/CP/D7/DD/CP/DC/CX/D7 /D3/CUFtot\n/CX/D7 /D3/D6/CX/CT/D2 /D8/CT/CS /D3/D6/D8/CW/D3/CV/D3/D2/CP/D0 /D8/D3 /D8/CT/D2/D7/CX/D0/CT /CP/D2/CS /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 \r/D3/D1/D4/D6/CT/D7/D7/CX/DA /CT /D7/D8/D6/CP/CX/D2/D7 /CX/D2 /D8/CW/CT/CP/CQ/D7/CT/D2\r/CT /D3/CU /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/D7/BA /BW/D9/CT /D8/D3 /D8/CW/CT /D7/D8/D6/D3/D2/CV /D9/D2/CX/CP/DC/CX/CP/D0 /D3/D9/D8/B9/D3/CU/B9/D4/D0/CP/D2/CT /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD/CE /D3/D0/D8/CP/CV/CT \r /D3/D2/D8/D6 /D3/D0 /D0/CT /CS /CX/D2/DA/CT/D6/D7/CX/D3/D2 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6 /D3/D4/DD /BI\r/CP/D9/D7/CT/CS /CQ /DDFdemag\n/B8 /D8/CW/CT /CX/D2/B9/D4/D0/CP/D2/CT /CT/CP/D7/DD /CP/DC/CX/D7 /CX/D7 /D8/CW/CT /CV/D0/D3/CQ/CP/D0 /CT/CP/D7/DD /CP/DC/CX/D7/BA /BY /D3/D6Vp>0V /B8 /D8/CW/CT/C6/CX /AS/D0/D1 /CT/DC/CW/CX/CQ/CX/D8/D7 /CP /D8/CT/D2/D7/CX/D0/CT /D7/D8/D6/CP/CX/D2 /CP/D0/D3/D2/CVˆ y /B4ε2>0 /B5 /CP/D2/CS /CP \r/D3/D1/D4/D6/CT/D7/D7/CX/DA /CT /D7/D8/D6/CP/CX/D2 /CP/D0/D3/D2/CVˆ x/B4ε1<0 /B5/BA /CC/CW/CT /CT/CP/D7/DD /CP/DC/CX/D7 /D3/CUFtot\n/CX/D7 /D8/CW /D9/D7 /D3/D6/CX/CT/D2 /D8/CT/CS /CP/D0/D3/D2/CV /D8/CW/CTˆ x /B9/CS/CX/D6/CT\r/D8/CX/D3/D2 /CJ\r/CU/BA /BY/CX/CV/BA /BD/B4\r/B5℄/BA/BT \r\r/D3/D6/CS/CX/D2/CV/D0/DD /B8 /CU/D3/D6Vp<0V /B8 /D8/CW/CT /CT/CP/D7/DD /CP/DC/CX/D7 /D3/CUFtot\n/CX/D7 /D3/D6/CX/CT/D2 /D8/CT/CS /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /D8/CW/CT \r/D3/D1/D4/D6/CT/D7/D7/CX/DA /CT/D7/D8/D6/CP/CX/D2 /CP/D0/D3/D2/CVˆ y /CP/D2/CS /D3/D6/D8/CW/D3/CV/D3/D2/CP/D0 /D8/D3 /D8/CW/CT /D8/CT/D2/D7/CX/D0/CT /D7/D8/D6/CP/CX/D2 /CP/D0/D3/D2/CVˆ x /CJ\r/CU/BA /BY/CX/CV/BA /BD/B4/CT/B5℄/BA /CF /CT /D8/CW /D9/D7/CT/DC/D4 /CT\r/D8 /CP /BL/BC◦/D6/D3/D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CX/D2/B9/D4/D0/CP/D2/CT /CT/CP/D7/DD /CP/DC/CX/D7 /D3/CUFtot\n/D9/D4 /D3/D2 \r /CW/CP/D2/CV/CX/D2/CV /D8/CW/CT /D4 /D3/D0/CP/D6/CX/D8 /DD /D3/CUVp\n/BA/C6/D3/D8/CT /D8/CW/CP/D8 /CX/D2 /D8/CW/CT /CP/CQ/D7/CT/D2\r/CT /D3/CU /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/D7 /D8/CW/CT/D6/CT /CP/D6/CT /CP/D0/DB /CP /DD/D7 /D8 /DB /D3 /CT/D2/CT/D6/CV/CT/D8/CX\r/CP/D0/D0/DD/CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /CP/D2 /D8/CX/D4/CP/D6/CP/D0/D0/CT/D0 /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /D3/CUM /CP/D0/D3/D2/CV /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CT/CP/D7/DD /CP/DC/CX/D7/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CU/D3/D6/D7/CX/D1/D4/D0/CX\r/CX/D8 /DD /B8 /DB /CT /D3/D2/D0/DD /D7/CW/D3 /DB /D3/D2/CT /D3/CU /D8/CW/CT /D8 /DB /D3 /D6/CT/D7/D9/D0/D8/CX/D2/CV /D4 /D3/D7/D7/CX/CQ/D0/CT /CX/D2/B9/D4/D0/CP/D2/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /D3/CUM /CX/D2/BY/CX/CV/D7/BA /BD/B4/CQ/B5 /CP/D2/CS /BD/B4/CS/B5/BA /C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7 /D8/CW/CT/D7/CT /D4/CP/D2/CT/D0/D7 /D7/CW/D3 /DB /D8/CW/CP/D8 /CX/D8 /D7/CW/D3/D9/D0/CS /CQ /CT /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 \r /CW/CP/D2/CV/CT/D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CUM /CU/D6/D3/D1M||ˆ x /D8/D3M||ˆ y /DA/CX/CP /D8/CW/CT /CP/D4/D4/D0/CX\r/CP/D8/CX/D3/D2 /D3/CU /CP/D4/D4/D6/D3/D4/D6/CX/CP/D8/CT /DA /D3/D0/D8/CP/CV/CT/D7 Vp/D8/D3 /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6/BA/CC/CW/CT /D8/D3/D8/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /CU/D6/CT/CT /CT/D2/CT/D6/CV/DD /CS/CT/D2/D7/CX/D8 /DDFtot\n/CX/D2 /BX/D5/BA /B4/BD/B5 /CX/D7 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D0/DD /CP\r\r/CT/D7/D7/CX/CQ/D0/CT/CQ /DD /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D7/D3/D2/CP/D2\r/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/BA /CC/CW/CT /BY/C5/CA /CT/D5/D9/CP/D8/CX/D3/D2/D7 /D3/CU /D1/D3/D8/CX/D3/D2 /CJ/BF/BD /B8 /BF/BE /B8 /BF/BF ℄\n/parenleftbiggω\nγ/parenrightbigg2\n=1\nM2\nssin2Θ/parenleftBig/parenleftbig\n∂2\nΦFtot/parenrightbig/parenleftbig\n∂2\nΘFtot/parenrightbig\n−(∂Φ∂ΘFtot)2/parenrightBig\n|Θ0,Φ0\n/B4/BK/B5\nand∂ΘFtot|Θ=Θ0=∂ΦFtot|Φ=Φ0= 0/D0/CX/D2/CZ /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CSµ0Hres\n/D8/D3 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r/CU/D6/CT/CT /CT/D2/CT/D6/CV/DD /CS/CT/D2/D7/CX/D8 /DDFtot\n/BA /C1/D2 /BX/D5/BA /B4/BK/B5/B8γ /CX/D7 /D8/CW/CT /CV/DD/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D6/CP/D8/CX/D3 /CP/D2/CSω= 2πνMW\n/DB/CX/D8/CW /D8/CW/CT/D1/CX\r/D6/D3 /DB /CP /DA /CT /CU/D6/CT/D5/D9/CT/D2\r/DD νMW\n/BA /CC/CW/CT /CT/AR/CT\r/D8 /D3/CU /CS/CP/D1/D4/CX/D2/CV /CX/D7 /D2/CT/CV/D0/CT\r/D8/CT/CS /CX/D2 /BX/D5/BA /B4/BK/B5/B8 /CP/D7 /CS/CP/D1/D4/CX/D2/CV/D3/D2/D0/DD /CP/AR/CT\r/D8/D7 /D8/CW/CT /D0/CX/D2/CT/D7/CW/CP/D4 /CT /DB/CW/CX\r /CW /DB/CX/D0/D0 /D2/D3/D8 /CQ /CT /CS/CX/D7\r/D9/D7/D7/CT/CS /CX/D2 /D8/CW/CX/D7 /CP/D6/D8/CX\r/D0/CT/BA /CF /CT /D2/D3/D8/CT /D8/CW/CP/D8/D8/CW/CT /BY/C5/CA /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CSµ0Hres\n/D1/CT/CP/D7/D9/D6/CT/CS /CX/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 \r/CP/D2 /CQ /CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /CP /CS/CX/D6/CT\r/D8/CX/D2/CS/CX\r/CP/D8/D3/D6 /D3/CU /D8/CW/CT /D6/CT/D0/CP/D8/CX/DA /CT /D1/CP/CV/D2/CT/D8/CX\r /CW/CP/D6/CS/D2/CT/D7/D7 /CP/D0/D3/D2/CV /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /CS\r /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /CS/CX/D6/CT\r/D8/CX/D3/D2/BA/BY /D6/D3/D1 /BX/D5/D7/BA /B4/BD /B5 /CP/D2/CS /B4/BK/B5 /D3/D2/CT /AS/D2/CS/D7 /D8/CW/CP/D8 /D7/D1/CP/D0/D0/CT/D6 /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS/D7 \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD/CT/CP/D7/CX/CT/D6 /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /CP/D2/CS /D0/CP/D6/CV/CT/D6 /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS/D7 /D8/D3 /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CW/CP/D6/CS/CT/D6 /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA/BG/BA /BY /CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D7/D3/D2/CP/D2\r/CT/C1/D2 /D8/CW/CX/D7 /CB/CT\r/D8/CX/D3/D2/B8 /DB /CT /D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /D3/CU /D3/D9/D6 /CU/CT/D6/D6/D3/D1/CP/CV/B9/D2/CT/D8/CX\r /D8/CW/CX/D2 /AS/D0/D1/BB/D4/CX/CT/DE/D3 /CT/D0/CT\r/D8/D6/CX\r /CP\r/D8/D9/CP/D8/D3/D6 /CW /DD/CQ/D6/CX/CS /D7/D8/D6/D9\r/D8/D9/D6/CT /D9/D7/CX/D2/CV /BY/C5/CA /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/BA /BY/CX/CV/B9/D9/D6/CT /BE/B4/CP/B5 /D7/CW/D3 /DB/D7 /CU/D3/D9/D6 /BY/C5/CA /D7/D4 /CT\r/D8/D6/CP /D6/CT\r/D3/D6/CS/CT/CS /CP/D8 /D6/D3 /D3/D1 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /DB/CX/D8/CWH||ˆ y /CP/D2/CS /CP /DA /D3/D0/D8/B9/CP/CV/CTVp∈ {−30V,0V,30V,90V} /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA /BX/CP\r /CW /D7/D4 /CT\r/D8/D6/D9/D1/D7/CW/D3 /DB/D7 /D3/D2/CT /D7/D8/D6/D3/D2/CV /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D7/D3/D2/CP/D2\r/CT /DB/CW/CX\r /CW /AL /CS/D9/CT /D8/D3 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D1/D3 /CS/D9/D0/CP/B9/D8/CX/D3/D2 /CP/D2/CS /D0/D3 \r /CZ/B9/CX/D2 /CS/CT/D8/CT\r/D8/CX/D3/D2 /AL /CW/CP/D7 /CP /D0/CX/D2/CT/D7/CW/CP/D4 /CT \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CT /AS/D6/D7/D8 /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT /D3/CU /CP/C4/D3/D6/CT/D2 /D8/DE/CX/CP/D2 /D0/CX/D2/CT /CJ/BF/BG℄/BA /CC/CW/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CSµ0Hres\n/D3/CU /CP /CV/CX/DA /CT/D2 /BY/C5/CA /D0/CX/D2/CT /CX/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CP/D7/D8/CW/CT /CP/D6/CX/D8/CW/D1/CT/D8/CX\r /D1/CT/CP/D2 /D3/CU /CX/D8/D7 /D1/CP/DC/CX/D1 /D9/D1 /CP/D2/CS /D1/CX/D2/CX/D1 /D9/D1 /CP/D2/CS /CS/CT/D4/CX\r/D8/CT/CS /CQ /DD /D8/CW/CT /CU/D9/D0/D0 /D7/D5/D9/CP/D6/CT/D7/CX/D2 /BY/CX/CV/BA /BE/B4/CP/B5/BA /CC/CW/CT /AS/CV/D9/D6/CT /D7/CW/D3 /DB/D7 /D8/CW/CP/D8/B8 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CSH||ˆ y /B8µ0Hres\n/CX/D7/D7/CW/CX/CU/D8/CT/CS /D8/D3 /CW/CX/CV/CW/CT/D6 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/D7 /CU/D3/D6 /CX/D2\r/D6/CT/CP/D7/CX/D2/CV Vp\n/B8 /DB/CW/CX/D0/CT /D8/CW/CT /D0/CX/D2/CT/D7/CW/CP/D4 /CT /CX/D7 /D2/D3/D8 /D7/CX/CV/D2/CX/AS\r/CP/D2 /D8/D0/DD\r /CW/CP/D2/CV/CT/CS/BA /BT \r\r/D3/D6/CS/CX/D2/CV /D8/D3 /BX/D5/D7/BA /B4/BD/B5 /CP/D2/CS /B4/BK/B5/B8 /D8/CW/CX/D7 /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CP/D8 /D8/CW/CTˆ y /CS/CX/D6/CT\r/D8/CX/D3/D2 /CQ /CT\r/D3/D1/CT/D7 /CX/D2/B9\r/D6/CT/CP/D7/CX/D2/CV/D0/DD /CW/CP/D6/CS/CT/D6 /CP/D7Vp\n/CX/D7 /CX/D2\r/D6/CT/CP/D7/CT/CS/BA/CC /D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /CT/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CUµ0Hres\n/DB/CX/D8/CWVp\n/CX/D2 /D1/D3/D6/CT /CS/CT/D8/CP/CX/D0/B8 /DB /CT /D6/CT\r/D3/D6/CS/CT/CS /BY/C5/CA/D7/D4 /CT\r/D8/D6/CP /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D8/CW/D3/D7/CT /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BE/B4/CP/B5 /CU/D3/D6Vp\n/CX/D2\r/D6/CT/CP/D7/CX/D2/CV /CU/D6/D3/D1−30V /D8/D3+90V/CE /D3/D0/D8/CP/CV/CT \r /D3/D2/D8/D6 /D3/D0 /D0/CT /CS /CX/D2/DA/CT/D6/D7/CX/D3/D2 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6 /D3/D4/DD /BJ\n100 150 200 250(a)\n+90V\n0VFMR Signal (arb. units)\nµ0H (mT)Vp= -30V+30V\nµ0Hres\n-40 -20 0 20 40 60 80 100120130140150160170180190µ0Hres (mT)\nVp (V)Ni on actuator, 300 K(b)\nH || xH || yH || y/BY/CX/CV/D9/D6/CT /BE/BA /B4/CP/B5 /BY/C5/CA /D7/D4 /CT\r/D8/D6/CP /D6/CT\r/D3/D6/CS/CT/CS /DB/CX/D8/CWH||ˆ y /CP/D8 /CS/CX/AR/CT/D6/CT/D2 /D8 /DA /D3/D0/D8/CP/CV/CT/D7 Vp\n/BA /BT/D2/CX/D2\r/D6/CT/CP/D7/CX/D2/CV /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CSµ0Hres\n/B4/D7/D3/D0/CX/CS /D7/D5/D9/CP/D6/CT/D7/B5 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS /CU/D3/D6 /CX/D2\r/D6/CT/CP/D7/CX/D2/CV Vp\n/DB/CW/CX/D0/CT/D8/CW/CT /D0/CX/D2/CT/D7/CW/CP/D4 /CT /CX/D7 /D2/D3/D8 /D7/CX/CV/D2/CX/AS\r/CP/D2 /D8/D0/DD /CP/D0/D8/CT/D6/CT/CS/BA /B4/CQ/B5 /CC/CW/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CUµ0Hres\n/D3/D2Vp\n/CX/D7/D5/D9/CP/D0/CX/D8/CP/D8/CX/DA /CT/D0/DD /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D3/D6H||ˆ y /CP/D2/CSH||ˆ x /BA /BY /D9/D0/D0 /D7/DD/D1 /CQ /D3/D0/D7 \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CX/D2\r/D6/CT/CP/D7/CX/D2/CV Vp/CP/D2/CS /D3/D4 /CT/D2 /D7/DD/D1 /CQ /D3/D0/D7 /D8/D3 /CS/CT\r/D6/CT/CP/D7/CX/D2/CV Vp\n/BA /CC/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8 /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS/D7\r/CP/D0\r/D9/D0/CP/D8/CT/CS /CU/D6/D3/D1 /D1/CP/CV/D2/CT/D8/D3 /CT/D0/CP/D7/D8/CX\r /D8/CW/CT/D3/D6/DD /B4\r/CU/BA /CB/CT\r/BA /BF /B5 /DD/CX/CT/D0/CS/CX/D2/CV /DA /CT/D6/DD /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW/D8/CW/CT /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/BA /BY /D3/D6H||ˆ y /B8 /CX/D2\r/D6/CT/CP/D7/CX/D2/CV Vp\n/CX/D2\r/D6/CT/CP/D7/CT/D7 µ0Hres\n/CP/D2/CS /D8/CW /D9/D7 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r/CW/CP/D6/CS/D2/CT/D7/D7 /D3/CU /D8/CW/CX/D7 /CS/CX/D6/CT\r/D8/CX/D3/D2/B8 /DB/CW/CX/D0/CT /CU/D3/D6H||ˆ x /B8 /CX/D2\r/D6/CT/CP/D7/CX/D2/CV Vp\n/CS/CT\r/D6/CT/CP/D7/CT/D7 µ0Hres\n/CP/D2/CS /D8/CW /D9/D7 /D8/CW/CT/D1/CP/CV/D2/CT/D8/CX\r /CW/CP/D6/CS/D2/CT/D7/D7 /D3/CU /D8/CW/CX/D7 /CS/CX/D6/CT\r/D8/CX/D3/D2/BA /CC/CW/CT /CW /DD/D7/D8/CT/D6/CT/D7/CX/D7 /D3/CUµ0Hres\n/CX/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /CW /DD/D7/D8/CT/D6/CT/D8/CX\r/D7/D8/D6/CT/D7/D7/B9/D7/D8/D6/CP/CX/D2 \r/D9/D6/DA /CT /D3/CU /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6/BA/B4/D9/D4/D7/DB /CT/CT/D4/B5 /CP/D2/CS /CS/CT\r/D6/CT/CP/D7/CX/D2/CV /CQ/CP\r /CZ /D8/D3−30V /B4/CS/D3 /DB/D2/D7/DB /CT/CT/D4/B5 /CX/D2 /D7/D8/CT/D4/D7 /D3/CU∆Vp= 10V /BA /BY /D3/D6\nH||ˆ y /B8 /DB /CT /D3/CQ/D8/CP/CX/D2 /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS/D7 /D7/CW/D3 /DB/D2 /CQ /DD /D8/CW/CT /D7/D5/D9/CP/D6/CT/D7 /CX/D2 /BY/CX/CV/BA /BE/B4/CQ/B5/BA /C0/CT/D6/CT/B8 /CU/D9/D0/D0/D7/D5/D9/CP/D6/CT/D7 /CS/CT/D4/CX\r/D8 /D8/CW/CT /D9/D4/D7/DB /CT/CT/D4 /CP/D2/CS /D3/D4 /CT/D2 /D7/D5/D9/CP/D6/CT/D7 /D8/CW/CT /CS/D3 /DB/D2/D7/DB /CT/CT/D4 /D3/CUVp\n/BA /BT/D7 /CS/CX/D7\r/D9/D7/D7/CT/CS /CX/D2 /D8/CW/CT\r/D3/D2 /D8/CT/DC/D8 /D3/CU /BY/CX/CV/BA /BE/B4/CP/B5/B8 /BY/CX/CV/BA /BE/B4/CQ/B5 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT /BY/C5/CA /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS /CU/D3/D6H||ˆ y /CX/D2\r/D6/CT/CP/D7/CT/D7/DB/CX/D8/CW /CX/D2\r/D6/CT/CP/D7/CX/D2/CV Vp\n/CP/D2/CS /CS/CT\r/D6/CT/CP/D7/CT/D7 /DB/CX/D8/CW /CS/CT\r/D6/CT/CP/D7/CX/D2/CV Vp\n/BA /CC/CW/CT /D7/D1/CP/D0/D0 /CW /DD/D7/D8/CT/D6/CT/D7/CX/D7 /CQ /CT/D8 /DB /CT/CT/D2/D9/D4/B9 /CP/D2/CS /CS/D3 /DB/D2/D7/DB /CT/CT/D4 /CX/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /CW /DD/D7/D8/CT/D6/CT/D8/CX\r /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D3/CU /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6 /CJ/BE/BG ℄/BA /BV/CP/D6/D6/DD/CX/D2/CV/D3/D9/D8 /D8/CW/CT /D7/CP/D1/CT /D7/CT/D6/CX/CT/D7 /D3/CU /BY/C5/CA /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /DB/CX/D8/CWH||ˆ x /DD/CX/CT/D0/CS/D7 /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS/D7 /D7/CW/D3 /DB/D2/CQ /DD /D8/CW/CT /D8/D6/CX/CP/D2/CV/D0/CT/D7 /CX/D2 /BY/CX/CV/BA /BE/B4/CQ/B5/BA /BY /D3/D6 /D8/CW/CX/D7 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/B8 µ0Hres\n/CS/CT\r/D6/CT/CP/D7/CT/D7 /CU/D3/D6/CX/D2\r/D6/CT/CP/D7/CX/D2/CV Vp\n/CP/D2/CS /DA/CX\r/CT /DA /CT/D6/D7/CP/BA /C1/D2 /D8/CT/D6/D1/D7 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /DB /CT /D8/CW /D9/D7 \r/CP/D2 \r/D3/D2\r/D0/D9/CS/CT /D8/CW/CP/D8/D8/CW/CTˆ x /CS/CX/D6/CT\r/D8/CX/D3/D2 /CQ /CT\r/D3/D1/CT/D7 /CT/CP/D7/CX/CT/D6 /CU/D3/D6 /CX/D2\r/D6/CT/CP/D7/CX/D2/CV Vp\n/DB/CW/CX/D0/CT /CP/D8 /D8/CW/CT /D7/CP/D1/CT /D8/CX/D1/CT /D8/CW/CTˆ y /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /CQ /CT\r/D3/D1/CT/D7 /CW/CP/D6/CS/CT/D6/BA/BY /D3/D6 /CP /D1/D3/D6/CT /D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT /CS/CX/D7\r/D9/D7/D7/CX/D3/D2 /DB /CT /CW/CP /DA /CT /CP/D0/D7/D3 /D4/D0/D3/D8/D8/CT/CS /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /CT/DC/D4 /CT\r/D8/CT/CS /CU/D6/D3/D1/BX/D5/D7/BA /B4/BD/B5 /CP/D2/CS /B4/BK/B5 /CP/D7 /D7/D3/D0/CX/CS /D0/CX/D2/CT/D7 /CX/D2 /BY/CX/CV/BA /BE/B4/CQ/B5/BA /CC/CW/CT/D7/CT /D0/CX/D2/CT/D7 /D7/CW/D3 /DB /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS/D7 /D3/CQ/D8/CP/CX/D2/CT/CS/CQ /DD /CP/D7/D7/D9/D1/CX/D2/CV /CP /D0/CX/D2/CT/CP/D6/B8 /D2/D3/D2/B9/CW /DD/D7/D8/CT/D6/CT/D8/CX\r /DA /D3/D0/D8/CP/CV/CT/B9/D7/D8/D6/CP/CX/D2 /D6/CT/D0/CP/D8/CX/D3/D2 /CJ\r/CU/BA /BX/D5/BA /B4/BJ /B5℄ /CP/D2/CS /D7/D3/D0/DA/CX/D2/CV/BX/D5/BA /B4/BK/B5 /DB/CX/D8/CWFtot\n/CU/D6/D3/D1 /BX/D5/BA /B4/BD /B5/BA /CF /CT /D9/D7/CT /CP /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 Ms= 370kA /m/CP/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /CB/C9/CD/C1/BW /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /CP/D2/CS /CPg /B9/CU/CP\r/D8/D3/D6 /D3/CU /BE/BA/BD/BI/BH /CJ/BF/BH℄/BA /BV/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D8/CW/CT/CU/CP\r/D8 /D8/CW/CP/D8 /D8/CW/CT /CP\r/D8/D9/CP/D8/D3/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D7/CP/D8/D9/D6/CP/D8/CT/D7 /CP/D8 /CW/CX/CV/CW /DA /D3/D0/D8/CP/CV/CT/D7 /CP/D2/CS /D8/CW/CT /CS/CP/D8/CP /CX/D2 /BY/CX/CV/BA /BE/B4/CQ/B5/D3/D2/D0/DD /D7/CW/D3 /DB /CP /D1/CX/D2/D3/D6 /D0/D3 /D3/D4 /D3/CU /D8/CW/CT /CU/D9/D0/D0 /CP\r/D8/D9/CP/D8/D3/D6 /D7/DB/CX/D2/CV/B8 /D8/CW/CT /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /CX/D7 /CX/D2 /CU/D9/D0/D0 /CP/CV/D6/CT/CT/D1/CT/D2 /D8/D8/D3 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7/BA /CC/CW/CX/D7 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT/D7 /D8/CW/CP/D8 /CU/D3/D6Vp<0V /D8/CW/CTˆ y /B9/CS/CX/D6/CT\r/D8/CX/D3/D2 /CX/D7/D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CT/CP/D7/CX/CT/D6 /D8/CW/CP/D2 /D8/CW/CTˆ x /B9/CS/CX/D6/CT\r/D8/CX/D3/D2/B8 /DB/CW/CX/D0/CT /CU/D3/D6Vp>0V /D8/CW/CTˆ x /B9/CS/CX/D6/CT\r/D8/CX/D3/D2 /CX/D7 /CT/CP/D7/CX/CT/D6/D8/CW/CP/D2 /D8/CW/CTˆ y /B9/CS/CX/D6/CT\r/D8/CX/D3/D2/BA /C5/D3/D6/CT/D3 /DA /CT/D6/B8 /CP/D8Vp= 0V /B8 /CQ /D3/D8/CW /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8 /CP/D2/CS /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /DD/CX/CT/D0/CS/CE /D3/D0/D8/CP/CV/CT \r /D3/D2/D8/D6 /D3/D0 /D0/CT /CS /CX/D2/DA/CT/D6/D7/CX/D3/D2 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6 /D3/D4/DD /BK\n/s49/s52/s53/s49/s53/s48/s49/s53/s53/s49/s54/s48θ/s61/s48/s176\n/s52/s53/s176\n/s57/s48/s176\n/s49/s51/s53/s176\n/s49/s56/s48/s176/s50/s50/s53/s176/s50/s55/s48/s176/s51/s49/s53/s176\n/s49/s52/s53\n/s49/s53/s48\n/s49/s53/s53\n/s49/s54/s48φ/s61/s57/s48/s176\n/s86/s112/s61/s32 /s43/s50/s48/s32 /s86µ0Hres (mT)/s40/s97/s41\n/s86/s112/s61/s32 /s45/s51/s48/s32 /s86\nHy\nxθ/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48φ/s61/s48/s176\n/s52/s53/s176\n/s57/s48/s176\n/s49/s51/s53/s176\n/s49/s56/s48/s176/s50/s50/s53/s176/s50/s55/s48/s176/s51/s49/s53/s176\n/s50/s48/s48\n/s52/s48/s48\n/s54/s48/s48\n/s56/s48/s48µ0Hres (mT)/s40/s98/s41\n/s32/s86/s112/s61/s32 /s43/s50/s48/s32 /s86\n/s32/s86/s112/s61/s32 /s45/s51/s48/s32 /s86θ/s61/s57/s48/s176\nHz\nxφ\nθ=90° /s50/s55/s48/s176/s49/s53/s48/s32 /s109/s84/s49/s54/s48/s32 /s109/s84/s49/s55/s48/s32 /s109/s84\n/s32/s49/s56/s48/s32 /s109/s84\n/s57/s48/s176\nφ=90° /BY/CX/CV/D9/D6/CT /BF/BA /B4/CP/B5 /CC/CW/CT /D7/DD/D1 /CQ /D3/D0/D7 /D7/CW/D3 /DB /D8/CW/CT /BY/C5/CA /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CSµ0Hres(θ) /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6/CP \r/D3/D2/D7/D8/CP/D2 /D8 /CP\r/D8/D9/CP/D8/D3/D6 /DA /D3/D0/D8/CP/CV/CTVp=−30V /B4/D3/D4 /CT/D2 /D8/D6/CX/CP/D2/CV/D0/CT/D7/B5 /CP/D2/CSVp= +20V /B4/D7/D3/D0/CX/CS\r/CX/D6\r/D0/CT/D7/B5 /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/D6/CX/CT/D2 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/D8/CW/CT /C6/CX /AS/D0/D1/BA/C1/D2 /D7/D9/D1/D1/CP/D6/DD /B8 /D8/CW/CT /BY/C5/CA /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 \r/D3/D2\r/D0/D9/D7/CX/DA /CT/D0/DD /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /D8/CW/CP/D8 /CX/D8 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT/D8/D3 /CX/D2 /DA /CT/D6/D8 /D8/CW/CT /CX/D2/B9/D4/D0/CP/D2/CT /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /D3/CU /D3/D9/D6 /CW /DD/CQ/D6/CX/CS /D7/D8/D6/D9\r/D8/D9/D6/CT/B8 /CX/BA/CT/BA /D8/D3 /CX/D2 /DA /CT/D6/D8 /D8/CW/CT/D7/CX/CV/D2 /D3/CUKu,magel,y\n/D7/D3/D0/CT/D0/DD /CQ/DD \r/CW/CP/D2/CV/CX/D2/CV Vp\n/BA /BT/D7 /D8/CW/CT /BY/C5/CA /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /CU/D9/D6/D8/CW/CT/D6/D1/D3/D6/CT /CP/D0/D0/D3 /DB/D7 /D8/D3/D5/D9/CP/D2 /D8/CX/D8/CP/D8/CX/DA /CT/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT /CP/D0/D0 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /D8/D3 /D8/CW/CT /CU/D6/CT/CT /CT/D2/CT/D6/CV/DD /B8 /BX/D5/BA /B4/BD /B5/B8 /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /DA /CT\r/D8/D3/D6M /CX/D2 /D3/D9/D6 /D7/CP/D1/D4/D0/CT \r/CP/D2 /CQ /CT \r/CP/D0\r/D9/D0/CP/D8/CT/CS /CP /D4/D6/CX/D3/D6/CX /CU/D3/D6 /CP/D6/CQ/CX/D8/D6/CP/D6/DD H/CP/D2/CSVp\n/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CX/D8 /CX/D7 /D2/D3/D8 /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /CS/CX/D6/CT\r/D8/D0/DD /D1/CT/CP/D7/D9/D6/CT /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /CP/D7/CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CUVp\n/DB/CX/D8/CW /BY/C5/CA/BA /CC /D3 /CS/CT/D1/D3/D2/D7/D8/D6/CP/D8/CT /D8/CW/CP/D8 /D8/CW/CT /D4/CX/CT/DE/D3/B9/DA /D3/D0/D8/CP/CV/CT \r/D3/D2 /D8/D6/D3/D0 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r/CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CX/D2/CS/CT/CT/CS /CP/D0/D0/D3 /DB/D7 /CU/D3/D6 /CP /DA /D3/D0/D8/CP/CV/CT \r/D3/D2 /D8/D6/D3/D0 /D3/CUM /B8 /DB /CT /D2/D3 /DB /D8/D9/D6/D2 /D8/D3 /D1/CP/CV/D2/CT/D8/D3/D1/CT/D8/D6/DD /BA/BH/BA /C5/CP/CV/D2/CT/D8/D3/D1/CT/D8/D6/DD/C1/D2 /D8/CW/CX/D7 /CB/CT\r/D8/CX/D3/D2/B8 /DB /CT /D7/CW/D3 /DB /D8/CW/CP/D8 /CX/D8 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /D2/D3/D8 /D3/D2/D0/DD \r /CW/CP/D2/CV/CT /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD/CQ/D9/D8 /D8/D3 /CS/CT/D0/CX/CQ /CT/D6/CP/D8/CT/D0/DD /CX/D6/D6 /CT/DA/CT/D6/D7/CX/CQ/D0/DD /CP/D2/CS/BB/D3/D6 /D6 /CT/DA/CT/D6/D7/CX/CQ/D0/DD /D6/D3/D8/CP/D8/CTM /D7/CX/D1/D4/D0/DD /CQ /DD \r /CW/CP/D2/CV/CX/D2/CV Vp\n/BA/CC /D3 /D8/CW/CX/D7 /CT/D2/CS/B8 /DB /CT /D2/CT/CT/CS /D8/D3 /CT/D1/D4/D0/D3 /DD /CP/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D8/CT\r /CW/D2/CX/D5/D9/CT /D8/CW/CP/D8 /CX/D7 /CS/CX/D6/CT\r/D8/D0/DD /D7/CT/D2/D7/CX/D8/CX/DA /CT/D8/D3 /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CUM /D6/CP/D8/CW/CT/D6 /D8/CW/CP/D2 /D8/D3 /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX/CT/D7/BA /C0/CT/D6/CT /DB /CT /D9/D7/CT/CS /CB/C9/CD/C1/BW/D1/CP/CV/D2/CT/D8/D3/D1/CT/D8/D6/DD /D8/D3 /D6/CT\r/D3/D6/CS /D8/CW/CT /D4/D6/D3 /CY/CT\r/D8/CX/D3/D2 m /D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /D1/D3/D1/CT/D2 /D8 m /D3/D2 /D8/D3 /D8/CW/CT/CS/CX/D6/CT\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CSH /BA/C1/D2 /CP /AS/D6/D7/D8 /D7/CT/D6/CX/CT/D7 /D3/CU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /DB /CT /D6/CT\r/D3/D6/CS/CT/CS m /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r/AS/CT/D0/CS /D1/CP/CV/D2/CX/D8/D9/CS/CT µ0H /CP/D8 /AS/DC/CT/CS /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /D3/CUH /CP/D2/CS /AS/DC/CT/CS /DA /D3/D0/D8/CP/CV/CT/D7 Vp\n/CP/D8T= 300K /BA/BY/CX/CV/D9/D6/CT /BG/B4/CP/B5 /D7/CW/D3 /DB/D7M=m/V /D1/CT/CP/D7/D9/D6/CT/CS /DB/CX/D8/CWH||ˆ y /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r/AS/CT/D0/CS /D7/D8/D6/CT/D2/CV/D8/CW /CP/D8 \r/D3/D2/D7/D8/CP/D2 /D8 /DA /D3/D0/D8/CP/CV/CTVp= +30V /B4/CU/D9/D0/D0 \r/CX/D6\r/D0/CT/D7/B5 /CP/D2/CSVp=−30V /B4/D3/D4 /CT/D2/D8/D6/CX/CP/D2/CV/D0/CT/D7/B5/BA /CC/CW/CT /C6/CX /AS/D0/D1 /D7/CW/D3 /DB/D7 /CP /D6/CT\r/D8/CP/D2/CV/D9/D0/CP/D6 M(H) /D0/D3 /D3/D4 /CU/D3/D6Vp=−30V /B8 /DB/CW/CX/D0/CT /CU/D3/D6\nVp= +30V /D8/CW/CT /D6/CT/D1/CP/D2/CT/D2 /D8 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /D0/D3 /DB /CT/D6/CT/CS /CQ /DD /CP /CU/CP\r/D8/D3/D6 /D3/CU /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/D0/DD /D8/CW/D6/CT/CT/BA/BT \r\r/D3/D6/CS/CX/D2/CV /D8/D3/B8 /CT/BA/CV/BA/B8 /C5/D3/D6/D6/CX/D7/CW /CJ/BE/BL℄/B8 /D8/CW/CT /D6/CT\r/D8/CP/D2/CV/D9/D0/CP/D6 /D0/D3 /D3/D4 /CU/D3/D6Vp=−30V /CX/D2/CS/CX\r/CP/D8/CT/D7 /CP/D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CT/CP/D7/DD /CP/DC/CX/D7/B8 /DB/CW/CX/D0/CT /D8/CW/CT /D7/D1/D3 /D3/D8/CW/B8 /D7/B9/D7/CW/CP/D4 /CT/CS /D0/D3 /D3/D4 /CU/D3/D6Vp= +30V /CX/D2/CS/CX\r/CP/D8/CT/D7 /CP/CE /D3/D0/D8/CP/CV/CT \r /D3/D2/D8/D6 /D3/D0 /D0/CT /CS /CX/D2/DA/CT/D6/D7/CX/D3/D2 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6 /D3/D4/DD /BD/BC\nH || y\nH || x\n-20 -15 -10 -5 0 5 10 15 20-400-2000200400\nNi on actuator, 300 K-30 V(b)\nµ0H (mT)M (kA/m) Vp = +30 V-400-2000200400\n+30 VVp = -30 V (a)/BY/CX/CV/D9/D6/CT /BG/BA /B4/CP/B5M(H) /B9/D0/D3 /D3/D4/D7 /D6/CT\r/D3/D6/CS/CT/CS /DB/CX/D8/CWH||ˆ y /D7/CW/D3 /DB /CP /CW/CX/CV/CW/CT/D6 /D6/CT/D1/CP/D2/CT/D2 /D8 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/CU/D3/D6Vp=−30V /D8/CW/CP/D2 /CU/D3/D6Vp= +30V /B8 /D8/CW /D9/D7 /D8/CW/CTˆ y /B9/CP/DC/CX/D7 /CX/D7 /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CT/CP/D7/CX/CT/D6 /CU/D3/D6\nVp=−30V /D8/CW/CP/D2 /CU/D3/D6Vp= +30V /BA /B4/CQ/B5 /CC/CW/CTˆ x /B9/CP/DC/CX/D7 /CX/D7 /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CT/CP/D7/CX/CT/D6 /CU/D3/D6Vp= +30V/D8/CW/CP/D2 /CU/D3/D6Vp=−30V /BA/D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CW/CP/D6/CS/CT/D6 /CP/DC/CX/D7/BA /CC/CW /D9/D7 /BY/CX/CV/BA /BG/B4/CP/B5 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CTˆ y /B9/CS/CX/D6/CT\r/D8/CX/D3/D2 /CX/D7 /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD/CT /CP/D7/CX/CT/D6 /CU/D3/D6Vp=−30V /CP/D2/CS /CW/CP/D6 /CS/CT/D6 /CU/D3/D6Vp= +30V /AL /CP/D7 /CT/DC/D4 /CT\r/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT /BY/C5/CA/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/BA /BV/CW/CP/D2/CV/CX/D2/CV /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CUH /D8/D3H||ˆ x /DD/CX/CT/D0/CS/D7 /D8/CW/CTM(H) /B9/D0/D3 /D3/D4/D7 /D7/CW/D3 /DB/D2/CX/D2 /BY/CX/CV/BA /BG/B4/CQ/B5/BA /BY /D3/D0/D0/D3 /DB/CX/D2/CV /D8/CW/CT /D7/CP/D1/CT /D0/CX/D2/CT /D3/CU /CP/D6/CV/D9/D1/CT/D2 /D8 /DB /CT \r/CP/D2 \r/D3/D2\r/D0/D9/CS/CT /D8/CW/CP/D8 /D8/CW/CTˆ x /B9/CS/CX/D6/CT\r/D8/CX/D3/D2/CX/D7 /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CT /CP/D7/CX/CT/D6 /CU/D3/D6Vp= +30V /CP/D2/CS /CW/CP/D6 /CS/CT/D6 /CU/D3/D6Vp=−30V /BA /BT/D0/D8/D3/CV/CT/D8/CW/CT/D6 /D8/CW/CT/D7/CT/D6/CT/D7/D9/D0/D8/D7 /D7/CW/D3 /DB /D8/CW/CP/D8 /DB /CT /D3/CQ/D7/CT/D6/DA /CT /CP/D2 /CX/D2/B9/D4/D0/CP/D2/CT /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /B8 /D8/CW/CT /CT/CP/D7/DD /CP/DC/CX/D7 /D3/CU /DB/CW/CX\r /CW /CX/D7 /D4/CP/D6/CP/D0/D0/CT/D0/D8/D3ˆ y /CU/D3/D6Vp=−30V /CP/D2/CS /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3ˆ x /CU/D3/D6Vp= +30V /BA /CC/CW/CT/D7/CT /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2/D7 /CP/D6/CT /CU/D9/D0/D0/DD\r/D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /BY/C5/CA /D6/CT/D7/D9/D0/D8/D7/B8 /CP/D2/CS \r/D3/D6/D6/D3/CQ /D3/D6/CP/D8/CT /D8/CW/CT /D7/CX/D1/D4/D0/CT /D1/D3 /CS/CT/D0 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BD/BA/BU/CT/CU/D3/D6/CT /CS/CX/D7\r/D9/D7/D7/CX/D2/CV /CU/D9/D6/D8/CW/CT/D6 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D1/CP/CV/D2/CT/D8/D3/D1/CT/D8/D6/DD /D6/CT/D7/D9/D0/D8/D7/B8 /DB /CT /D2/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /CU/D6/CT/CT/CT/D2/CT/D6/CV/DD /D1/CX/D2/CX/D1/CP /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BD/B4\r/B8/CT/B5 /CP/D7 /DB /CT/D0/D0 /CP/D7 /D8/CW/CT /CX/D2/B9/D4/D0/CP/D2/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS/D7 /CX/D2 /BY/CX/CV/BA /BF /B4/CP/B5/CP/D6/CT /CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /CQ /DD /BD/BK/BC◦/BA /CC/CW/CX/D7 /CS/CT/CV/CT/D2/CT/D6/CP\r/DD /D1/CP /DD /D0/CT/CP/CS /D8/D3 /CS/CT/D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CS/D9/CT /D8/D3 /CS/D3/D1/CP/CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CX/CU /D8/CW/CT /D4 /D3/D0/CP/D6/CX/D8 /DD /D3/CUVp\n/CX/D7 /D6/CT/D4 /CT/CP/D8/CT/CS/D0/DD /CX/D2 /DA /CT/D6/D8/CT/CS /CP/D7 /D8/CW/CT/D6/CT /CP/D6/CT /CP/D0/DB /CP /DD/D7 /D8 /DB /D3 /CT/D2/CT/D6/CV/CT/D8/CX\r/CP/D0/D0/DD/CT/D5/D9/CP/D0/D0/DD /CU/CP /DA /D3/D6/CP/CQ/D0/CT /CQ/D9/D8 /D3/D4/D4 /D3/D7/CX/D8/CT /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /D3/CUM /BA /CC/CW /D9/D7/B8 /D8/D3 /CP\r /CW/CX/CT/DA /CT /CP /D6/CT/DA /CT/D6/D7/CX/CQ/D0/CT \r/D3/D2 /D8/D6/D3/D0/D3/CUM /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CS/CT/CV/CT/D2/CT/D6/CP\r/DD /D3/CUFtot\n/D2/CT/CT/CS/D7 /D8/D3 /CQ /CT /D0/CX/CU/D8/CT/CS/BA /CC/CW/CX/D7 \r/CP/D2 /CQ /CT /CP\r /CW/CX/CT/DA /CT/CS /CX/CU /CP/D9/D2/CX/CS/CX/D6/CT\r/D8/CX/D3/D2/CP/D0 /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CX/D7 /D7/D9/D4 /CT/D6/CX/D1/D4 /D3/D7/CT/CS /D3/D2 /D8/CW/CT /D9/D2/CX/CP/DC/CX/CP/D0 /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /BA /CA/CT/CV/CP/D6/CS/CX/D2/CV Ftot/CX/D2 /BX/D5/BA /B4/BD /B5/B8 /D3/D2/D0/DDFstat\n/D4 /D3/D7/D7/CT/D7/D7/CT/D7 /D8/CW/CT /CS/CT/D7/CX/D6/CP/CQ/D0/CT /D9/D2/CX/CS/CX/D6/CT\r/D8/CX/D3/D2/CP/D0 /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /BA /C0/CT/D2\r/CT/B8 /CP /D7/D1/CP/D0/D0/CQ/D9/D8 /AS/D2/CX/D8/CT /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CSH/ne}ationslash= 0 \r/CP/D2 /CQ /CT /D9/D7/CT/CS /D8/D3 /D0/CX/CU/D8 /D8/CW/CT /BD/BK/BC◦/CS/CT/CV/CT/D2/CT/D6/CP\r/DD /BA /CC/CW/CX/D7/CP/D4/D4/D6/D3/CP\r /CW /DB /D3/D6/CZ/D7 /CU/D3/D6 /CP/D0/D0H /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/D7/B8 /CT/DC\r/CT/D4/D8 /CU/D3/D6H /CT/DC/CP\r/D8/D0/DD /D4/CP/D6/CP/D0/D0/CT/D0ˆ x /D3/D6ˆ y /BA /BY /D3/D6H||ˆ x/B4/DB/CX/D8/CWH >0 /B5/B8 /D8/CW/CT /CT/CP/D7/DD /CP/DC/CX/D7 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3ˆ x /CX/D2 /BY/CX/CV/BA /BD/B4\r/B5 /CX/D7 /D6/CT/D4/D0/CP\r/CT/CS /CQ /DD /CP/D2 /CT/CP/D7/CX/CT/D6 /D4 /D3/D7/CX/D8/CX/DA /CT/CE /D3/D0/D8/CP/CV/CT \r /D3/D2/D8/D6 /D3/D0 /D0/CT /CS /CX/D2/DA/CT/D6/D7/CX/D3/D2 /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CX/D7/D3/D8/D6 /D3/D4/DD /BD/BD\n-40 -20 0 20 40 60 80 100 120-300-200-300-200-1000\n M (kA/m)(b)\nµ0H=-5 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\r/D0/CT/CP/D6/D0/DD /D7/CW/D3 /DB/D7 /D8/CW/CP/D8/D9/D2/CS/CT/D6 /D8/CW/CT/D7/CT \r/D3/D2/CS/CX/D8/CX/D3/D2/D7 /D8/CW/CT /D0/D3 \r/CP/D0 /D1/CX/D2/CX/D1/CP /D3/CUFtot\n/CP/D6/CT /D2/D3/D2/B9/CS/CT/CV/CT/D2/CT/D6/CP/D8/CT /CU/D3/D6Vp=−30V /CP/D2/CS\nVp= +120V /BA/CC /D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CUM /B8 /DB /CT /D8/D6/CP\r/CT/CS /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /D3/CU /D8/CW/CT /D8/D3/D8/CP/D0 /CU/D6/CT/CT /CT/D2/CT/D6/CV/DD\nFtot\n/CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CUVp\n/BA /CC/CW/CX/D7 /DB /CP/D7 /CS/D3/D2/CT /CQ /DD /D1/CX/D2/CX/D1/CX/DE/CX/D2/CV /BX/D5/BA /B4/BD /B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT/CX/D2/B9/D4/D0/CP/D2/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 Θ /D3/CUM /B8 /DB/CW/CX/D0/D7/D8 /CP/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8Φ = 90◦/CS/D9/CT /D8/D3 /D8/CW/CT /D7/D8/D6/D3/D2/CV /D7/CW/CP/D4 /CT/CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /BA/CC /D3 /D7/CX/D1 /D9/D0/CP/D8/CT /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 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\r /CW/D3/D7/CT/D2 /D8/D3 /CV/CX/DA /CT /CP /CV/D3 /D3 /CS /AS/D8 /D8/D3 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /CS/CP/D8/CP /CP/D8 /D4 /D3/CX/D2 /D8/D7/BT /CP/D2/CS /CX/D7 /CP/D7/D7/D9/D1/CT/CS /D8/D3 /D6/CT/D1/CP/CX/D2 \r/D3/D2/D7/D8/CP/D2 /D8 /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D3/CUVp\n/BA/CC/CW/CT /D6/CT/D7/D9/D0/D8/CX/D2/CV /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/D7 /D3/CUM /CP/D6/CT /D7/CW/D3 /DB/D2 /CQ /DD /D8/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT/D7 /CX/D2 /BY/CX/CV/BA /BH/BA /CC/CW/CT /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/DD/CX/CT/D0/CS/D7 /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D0/DD /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D6/D6 /CT/DA/CT/D6/D7/CX/CQ/D0/CT M /D6/D3/D8/CP/D8/CX/D3/D2 /CS/D9/D6/CX/D2/CV /D8/CW/CT /AS/D6/D7/D8 /DA /D3/D0/D8/CP/CV/CT /D7/DB /CT/CT/D4/CX/D2 /BY/CX/CV/BA /BH /B4/CP/B5 /CP/D7 /DB /CT/D0/D0 /CP/D7 /D8/CW/CT /D6 /CT/DA/CT/D6/D7/CX/CQ/D0/CT M /D6/D3/D8/CP/D8/CX/D3/D2/D7 /CX/D2 /D8/CW/CT /D7/CT\r/D3/D2/CS /DA /D3/D0/D8/CP/CV/CT /D7/DB /CT/CT/D4 /CX/D2 /BY/CX/CV/BA /BH /B4/CP/B5/CP/D2/CS /CX/D2 /BY/CX/CV/D7/BA /BH /B4/CQ/B8\r/B5/BA /CC/CW/CT /D7/CX/D1 /D9/D0/CP/D8/CT/CS /D8/D3/D8/CP/D0 /D7/DB/CX/D2/CV /D3/CUM /CX/D7 /CX/D2 /CT/DC\r/CT/D0/D0/CT/D2 /D8 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW/D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6H||ˆ y /CP/D2/CS /CX/D2 /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CU/D3/D6H||ˆ x /BA /CC/CW/CT /CU/CP\r/D8 /D8/CW/CP/D8 /D8/CW/CT/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /CT/DC/CW/CX/CQ/CX/D8 /CP /D1/D3/D6/CT /D6/D3/D9/D2/CS/CT/CS /D7/CW/CP/D4 /CT /D8/CW/CP/D2 /D8/CW/CT /D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2 /CX/D7 /CP/D8/D8/D6/CX/CQ/D9/D8/CT/CS/D8/D3 /CS/D3/D1/CP/CX/D2 /CU/D3/D6/D1/CP/D8/CX/D3/D2 /CS/D9/D6/CX/D2/CV /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D6/CT/D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/B8 /DB/CW/CX\r /CW /CX/D7 /D2/CT/CV/D0/CT\r/D8/CT/CS /CX/D2 /D8/CW/CT/D7/CX/D1 /D9/D0/CP/D8/CX/D3/D2/BA/CC /CP/CZ /CT/D2 /D8/D3/CV/CT/D8/CW/CT/D6/B8 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the MgO -based magnetic tunnel junction using \nfield-induced ferromagnetic resonance  \n \nK. Konishi1, D. K. Dixit2, A. A. Tulapurkar2, S. Miwa1, T. Nozaki3, H. Kubota3, A. \nFukushima3, S. Yuasa3 and Y . Suzuki1 \n \n1 Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka \n560-8531, Japan  \n2 Department of Physics, Indian Institute of Technology , Bombay , Powai, Mumbai \n400076, India  \n3 National Institute  of Advanced Industrial Science and Technology (AIST), Spintronics \nResearch Center, Ibaraki 305-8568, Japan  \n \nAbstract  \n The r adio-frequency (RF)  voltage amplification property of a tunnel magnetoresistance \ndevice  driven by an RF external -magnetic -field-induced ferromagnetic resonance was \nstudied . The proposed device consists  of a magnetic tunnel junction  (MTJ ) and an \nelectrically  isolated coplanar waveguide. The i nput RF voltage applied to the waveguide \ncan excite the resonant dynamics in the free layer magnetization, leading to the  \ngeneration of an  output RF voltage under a DC bias current. The dependences of the RF \nvoltage gain on the s tatic e xternal magnetic field strength and angle were systematically \ninvestigated . The design principle s for the enhanc ement of the gain factor are also \ndiscussed.  2 \n The discovery of giant tunneling magneto -resistance (TMR) in MgO -based magnetic \ntunnel junctions (MTJ s) [1,2]  accelerates the development of spin device s using MTJ s, \nsuch as reading head s for hard disk drives (HDD s) and spin-transfer torque magnetic \nrandom access memor ies (STT-MRAM s) [3]. Further,  spin-torque oscillator s [4] and \nspin-torque diode effect  [5] using MgO -based MTJ s have attractive characteristics  \nbecause of  their large magneto -resistance effect. In addition to these devices, a \nMgO -based MTJ can be used to amplify  radio -frequency (RF)  signals  with frequency \ntunability. Thus  far, several RF amplification properties of an MTJ structure have been \nproposed and demonstrated experimentally. The study by Slonczewski  [6] was the first \nto propose the above mentioned properties . The o ther concepts of RF amplification have \nbeen proposed using negative differential resistance  [7], vortex -core resonance  [8,9], \nand spin -torque -induced ferromagnetic resonance in an MTJ  [10]. These devices  for RF \namplification  were combined with the spin-transfer torque effect [ 11,12]. On the other \nhand,  in this study,  we proposed a spin transistor using an MTJ , which is driven by a  \ncurrent -induced magnetic field , and we demonstrated a direct current (DC) power gain \nof more than 1 at room temperature  [13]. In this paper , we present the  RF amplification \nproperties  in the current -induced magnetic field driven spin transistor . \nA schematic structure of the proposed device  with the measurement circuit  is shown in \nFig. 1(a). When  a coplanar waveguide (CPW) is combined with an MTJ, an RF \nmagnetic field is generated  around the CPW under an application of RF current to the \nCPW. Further, f erromagnetic resonance (FMR) is induced when  the frequency of the RF \nmagnetic field is tuned to the resonant frequency of magnetization of the free layer in \nthe MTJ. Generally, magnetization precession can be excited efficiently under the \nresonance condition, leading  to the  generation of a  large RF output voltage from the 3 \n MTJ under  a large  DC bias current . If the output voltage exceeds  the input voltage, the \nproposed device can be used for the amplification of  RFs. \n MTJ films with a structure of buffer layers /PtMn/CoFe/Ru/CoFeB(3)/MgO  \n(1.1)/CoFeB(3)/Ru(1.5)/NiFe(2)/cap ping layers  (nm in thickness)  were deposited on \nSi/SiO 2 substrates using a magnetron sputtering method (Canon ANELVA C7100). \nFrom  the current -in-plane tunneling  (CIPT)  measurements, the resistance -area product \nand the magn eto-resistance ratio were evaluated to be 3.8 m2 and 110%, respectively. \nThe multilayer film was patterned into junctions (0.3  × 0.6 m2) with an electrically \nisolated coplanar waveguide . The RF power , which generates the RF magnetic field,  \nwas applied at port-1 (Vin) of the vector network analyzer (VNA). Further , a bias current \nwas applied to the MTJ  through a DC port of the bias Tee. The output signal from the \nMTJ  (Vout) was detected  by port -2 of the VNA. The RF amplification property was \nevaluated b y monitoring  the S21 (= Vout/Vin) parameter. The m agnetization  \nconfigurations  of the pinned and free layers and the magnetic field are shown in Fig. \n1(b). The x-axis (direction perpendicular to the CPW)  is defined to be parallel to the \neasy axis of the free layer. The e xternal magnetic field , Hext, is applied along  the \ndirection with angle  field from the x-axis. The S 21 parameter was measured under \nvarious  Hext and field conditions.  The input RF power and the DC bias current were \nfixed at 20 W and  -8 mA in all the measurement s. Here, the positive current is defined \nas one in which  the electrons flow  from  the free layer to  the pinned layer .  \nIn order  to extract the FMR signal clean ly, the background signal, which originates  \nfrom the transmission property of the CPW, was subtracted from the raw S 21 data. The \nbackground signal was obtained under the sufficiently  large Hext = 1 kOe applied along  \nthe direction of the x-axis, becaus e the FMR could not be induced in  this configura tion. 4 \n In this paper, S21 indicates the measured S 21 parameter without the background.  \n  Fig. 2 (a) shows the typical MR curve when field is 0°. The obtained MR ratio \nbecame three times smaller than that obtained by the CIPT measurement , owing  to the \ninfluence of  the parasitic series resistance.  A shift field of 50 Oe  was obtained , which is \ncaused by orange -peel coupling ( a parallel state is preferred under zero magnetic field). \nThe measured gain factor ( i.e., the absolute value [S21], which is hereafter  simply  \ncalled S21) as a function of frequency is shown in Fig. 2 (b) , where field was fixed at \n55°. Depending on the strength of the external magnetic field, the amplitude of S21 and \nthe resonant frequency were changed , indicat ing that the FMR excitation was induced \nby the application of the RF signal to the CPW. The value of S21 becomes small  when  \nHext = 50 Oe , possibly  because of  the inhomogeneous precession motion attributed  to \nmulti -domain formation. The observed resonant frequency was plotted as a function of \nthe external magnetic field with field = 55 ° in Fig. 2 (c) . The observed shift was well \nreproduced by Kittel’s formula (red curv e in the same figure) . From the fitting, the \nin-plane and out -of-plane anisotrop y field s were evaluated to be  15 Oe and 7500 Oe, \nrespectively.  \n Fig. 3 (a) shows S21_max as functions of the strength and angle of Hext. Here , we define \nS21_max  as the peak height  of S21 at each angle and external magnetic field . The red \narea indicates the high amplification  gain, which  reaches the maximum value of \napproximately  0.07.  \n The output voltage from  the MTJ  (Vout()) for small precession angles is described \nby eq. (1)  [see supple ment in ref. 4]. \n )1( sin\n24'tilt prec 0P AP  iR RVout\n \nwhere '() is the efficiency of the RF circuit ; RP and RAP are the resistances in parallel 5 \n and anti -parallel states,  respectively ; i0 is the bias current ; prec is the precession angle ; \nand tilt denotes  the relative angle between  the pinned and free layer magnetization . \nWhen field = 55o and Hext = 100 Oe, a relative angle  of tilt = 83 ° was obtained by \nresistance monitoring . The largest RF gain factor was obtained at this field and angle . \nThis behavior is consistent with eq. (1).  Furthermore, the resonant frequency decreases  \nwith a decrease in  the external magnetic field is small , which in turn results in an \nincrease in the  precession angle. From  the above discussion, it can be concluded that it \nis important to reduce  the resonan t frequency and obtain  a relative angle  of 90°to obtain \na large output.  \n Here , we compare the  experiment al results  with the  simulation  results . Using the \nparameters  obtained from our experiments and  from  Kittel’s fitting , we simulated the \nRF gain  by using  a macro -spin model  based on the Landau -Lifshitz -Gilbert equation . \nFig. 3(b) shows the simulated result  as functions of Hext and field. We assume  that the  \ndamping factor  = 0.01 and temperature T = 300 K. Here, the influence of  the thermal \neffect is treated as a random magnetic field [15] . The experimental result s were well \nreproduced qualitatively by this simple simulation . However , in the simulation, the \nmaximum gain was obtained for both  a small  magnetic field and a small tilt angle at \naround Hext = 50 Oe and field = 15°. At this field and angle , the shift field and in -plane \nmagnetic anisotropy are cancelled simultaneously by the external magnetic field . \nSubsequently , the resonant frequency becomes very low and  the RF gain increases . \nHowever, this condition is difficult to achieve in the experiment, because  a \nmulti -domain  structure is formed  in the low magnetic field range . Further,  the simulated \ngain is approximately  5 times greater than that obtained by the  experimental result. In \nthe experiment, the linewidth of the FMR peak is broaden ed because of non-uniform 6 \n precession attributed  to a relatively large sample size. This leads  to a small Q -value  (~2) \nin the experiment, and consequently , the gain factor becomes smaller  than expected . \nThe RF gain does not exceed 1 even in the simulation. One possible approach to \nrealize high RF amplification  is to introduce perpendicular magnetic anisotropy  [16]. \nWhen  the out -of-plane anisotropy is cancelled by the external magnetic field, a high \nQ-value of 50 can be obtained. This value is 25 times greater than that obtained by  our \nresult s. Thus , the RF gain factor can exceed 1 if we apply their condition  (i.e., using \nFe-rich CoFeB  free layer for perpendicular magnetic anisotropy and applying external \nmagnetic field to cancel  demagnetization field ) to our experiment . \nAnother possible approach to enhance the RF gain factor is to utilize the spin transfer \neffect. In the present measur ement condition s, the spin transfer torque has  only a  small  \ninfluence on the RF gain , because tilt is nearly 90 degree when the maximum S21 was \nobtained  and the electric breakdown voltage of the sample was relatively low. However, \nif the effect of  the spin torque is  sufficiently  large, the effective damping factor will be  \nreduced [14],  inducing  a large precession angle ; hence, the output voltage from the MTJ \ncan be increased , leading to  the realization of  a higher RF gain.  This technique can be  \nadapted  to a magnetic field feedback oscillator [15].  \n In summary, the amplification property of MTJ s afforded by  the \nmagnetic -field-induced FMR was proposed and demonstrated. A maximum voltage gain \nof 0.07 was achieved under the optimized external magnetic field condition. In addition, \nthe static external magnetic field strength and angle dependences of the voltage gain \nwere systematically investigated and reproduced qualitatively  by a simple macro -spin \nmodel simulation. The i mprovement of the uniform precession in the element , as well as \nthe introduction of perpendicular magnetic anisotropy or the spin transfer effect , were 7 \n shown to be  effective in achieving RF voltage gains greater than 1.  \nAcknowledgement  \n This work was mainly supported by the New Energy and Industrial Technology \nDevelopment Organization (NEDO) Spintronics Nonvolatile Device Project. K. Konishi \nis supported by a Research Fellowship of the JSPS for young scien tists. \n \n  8 \n [1] S. Yuasa et al., Nat. Mater. 3, 868 (2004).  \n[2] S. S. Parkin et al., Nat. Mater. 3, 862 (2004).  \n[3] T. Kishi et al., in Tech. Dig. – Int. Electron Devices Meet., 2008, 309.  \n[4] A. M. Deac et al., Nat. Phys. 4, 803 (2008).  \n[5] A. Tulapurkar et al., Nature 438, 339 (2005)  \n[6] J. C. Slonczewski, U. S. patent 5,  695, 864 (9 Dec. 1997).  \n[7] H. Tomita et al, J. Magnetism 16, 140 (2011).  \n[8] S. Kasai et al., Appl. Phys. Express 1, 091302 (2008).  \n[9] T. Nozaki et al., Appl. Phys. Lett. 95, 022513 (2009).  \n[10] L. Xue et al., Appl. Phys. Lett. 99, 022505 (2011).  \n[11] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).  \n[12] L. Berger, Phys. Rev. B 54, 9353 (1996).  \n[13] K. Konishi et al., Appl. Phys. Express 2, 063004 (2009).  \n[13] S. Petit et al., Phys. Rev. Lett. 98, 077203 (2007).  \n[14] S. Ishibashi et al., Appl. Phys. Express 1, 091302 (2008).  \n[15] R. H. Koch et al., Phys. Rev. Lett. 92, 088302 (2004).  \n[16] D. Dixit et. al ., Appl. Phys. Lett. 101, 122410 (2012).  \n \n \n  9 \n Figure 1  \n \n \n \n \n(c)  \n \n \n  \n10 \n Figure 2  \n \n \n \n \n \n \n  \n(a) \n(b) 11 \n Figure 3  \n \n \n \n \n \n  \n(b) (a) 12 \n Figure caption s \n \nFigure 1  \n(a) Concept of the proposed device with the measurement circuit. The FMR in the free \nlayer of  the MTJ is induced by  an RF magnetic field due to the application of  an RF \nsignal from port -1 of the VNA. Under DC bias, the MTJ produces an RF output, which \nis applied  to port -2 of the VNA. The VNA measures  the S21 (= V out/Vin) parameter , \nwhich  represents  the gain factor.  \n \n(b) Schematic image of  the configuration of magnetization and magnetic fields. M free \nand M pin denote  the magnetization of the free and pinned layer ; tilt and field indicate the \nrelative angle and the angle of the external magnetic field ; and hRF, Hext, and Hshift \nindicate the RF magnetic field, the external magnetic field , and shifted field, \nrespectively.  \n \nFigure 2  \n(a) Typical MR curve ( field = 0°). \n \n(b) Gain factor ( absolute value [S21]) as a function of frequency ( field = 55°).  \n \n(c) Resonant frequency as a function of the external magnetic field ( field = 55°). The \nblack dots and red curve show the experimental results and the fitting curve, \nrespectively.  \n \nFigure 3  \nS21_max  as functions of field and Hext. (a) Experimental result  and (b) simulated result.  \n \n "    },    {        "title": "0707.0703v1.Detection_of_the_magneto_structural_phase_coexistence_in_MnAs_epilayers_at_a_very_early_stage.pdf",        "content": "arXiv:0707.0703v1  [cond-mat.mtrl-sci]  4 Jul 2007Detectionof the magneto-structuralphase coexistencein M nAs epilayers at\navery early stage\nJ. Milano,∗L. B. Steren, and A. H. V. Repetto Llamazares\nCNEA - Centro At ´omico Bariloche and Instituto Balseiro, UNCu,\n(R8402AGP) San Carlos de Bariloche, R ´ıo Negro, Argentina.\nV. Garcia, M. Marangolo, M. Eddrief, and V. H. Etgens\nInstitut des NanoSciences de Paris, INSP,\nUniversit ´e Pierre et Marie Curie-Paris 6,\nUniversit ´e Denis Diderot-Paris 7, CNRS UMR 7588,\nCampus Boucicaut, 140 rue de Lourmel, 75015 Paris, France.\n(Dated: November 8, 2018)\nAbstract\nWe report on the appearance of magnetic stripes in MnAs/GaAs (100) epilayers at temperatures well be-\nlow the ferromagnetic transition of the system. The study ha s been performed by ferromagnetic resonance\nexperiments (FMR) on MnAs epilayers grown on (100) and (111) GaAs substrates. The FMR spectra of\nthe MnAs/GaAs(100) samples at 180 K reveal theappearance of zones of different magnetic behavior with\nrespect tothelow-temperature homogeneous ferromagnetic phase. Theangular andthetemperature depen-\ndence of the spectra serve us to detect the inter-growth of th e non-magnetic phase into the ferromagnetic\nphase at a very early stage of the process. The experimental d ata show that the new phase nucleates in\na self-arranged array of stripes in MnAs/GaAs(100) thin film s while it grows randomly in the same films\ngrown onGaAs(111).\n1Due to their physical and chemical properties, the MnAs comp ound is a promising material\nfor spintronicdevelopments1. Indeed, with room temperature ferromagnetism and extreme ly low\ninterface reactivity with GaAs, MnAs is a strong candidate f or spin injection devices. Since the\nearlier studies of MnAs epilayers, the coexistence of two ma gneto-structural phases in a finite\ntemperature range has been one of the most interesting topic of research.2One of these phases\nis ferromagnetic ( α-phase) and the other one is non-magnetic ( β-phase).3The bulk compound\ndoesnotshowsuchphasecoexistence;onthecontrary,itpre sentsamagneto-structuralfirst-order\ntransition at 313 K from the low temperature αphase to the βone above this temperature4,5,6.\nThe origin of the phase coexistence in MnAs thin films has been well addressed and is due to\nstrain induced by the substrate7,8,9. The phase coexistence has been experimentally observed\nusing several and very different techniques,9,13,14,15,16,17,18,19,20. All of the experimental works\nreferred previously show that the phase coexistence starts around 273 K. However, in a recent\npaper21, weobserved, above200 K, an increase ofthe coercivefield wi th increasing temperature,\nwhich is very unusual for a ferromagnet. The most plausible e xplanation for this effect is the\nappearance of pinning centers around 200 K, associated with a very early stage of the growth of\ntheβphase. With the aim of assessing unambiguously the appearan ce of phase coexistence and\nto give a quantitative description of its temperature evolu tion, we have investigated the magnetic\nbehavior of MnAs epilayers grown over GaAs(100) and GaAs(11 1) by FMR experiments. This\ntechnique allows us to follow the changes of the ferromagnet ic domain shapes with temperature\nthatprovideusnecessary informationto establishthetemp eraturerangeofthephasecoexistence.\nMnAs epilayers were grown by molecular beam epitaxy (MBE) as described elsewhere21.\nMnAs thin films of 66 nm thick were grown on (100) and (111) GaAs substrates, named M100-\n66 and M111-66 respectively. The ferromagnetic resonance m easurements were performed at ν\n∼9.45 GHz(X-band).\nInFig. 1(a), weshowtheFMRspectracorrespondingtoaM100- 66filmmeasured at different\ntemperaturesintheout-of-planegeometry, ΘH=0[seeFig. 1(c)]. Inthe130K-272Ktemperature\nrange, we observe a main resonance line, mM. At 180 K, the FMR spectra show two additional\nmodes at lower resonance fields, bMandtM, respectively. From 272 K to 278 K, the mMand\ntMmodes become weaker and the intensity of the bMmode increases. The latter is the only one\nwhichis presentat temperatures higherthan 278K.\nPerformingsimilarFMRexperiments,A.Neyetal.19observedalsotwocoexistingmodesbutonly\nin a temperature range of 5 K (283K-288K) and they attributet his effect to the phase coexistance\n20.5 1.0 1.5\nH [T]Absorption derivative [arb. units]\n150200250300\nT [K]00.250.500.751IrmM\ntM\nbM0.9 1.1 1.3\nH [T]A. d. [arb. units]\nT = 130KT = 200KT = 240KT = 272KT = 276KT = 278K\nM100-6601\n0.250.500.75Iri\n150 200 250 300mMtMbMT = 290K\nA. d. [arb. units]\nH [T]T = 240K\n0 0.025 0.050 0.075 0.100\nH [T]Absorption derivative [arb. units]T = 130KT = 200KT = 240KT = 270KT = 290KT = 310K\n         M111-66/c81/c81Hz\nMH\ny x/c70/c70Hcb\na/c97phase\n/c98phase\nFIG. 1: (a) FMR spectra in the out-of-plane geometry measure d at different temperatures for the M100-66\nsample. The lower inset shows the intensity ratios of the thr ee modes, normalized to the total intensity of\nthespectra. Theupper insetshowsthebestfittedcurveforth espectrum atT=240K.(b)FMRspectrainthe\nin-plane geometry, measured at different temperatures for the M111-66 sample. (c) Axis convention and\nscheme of the stripe-like domains inMnAs/GaAs(100) sample s.\nonset. However, our measurements show that the appearance o f the additional modes occurs 100\nK lower than the starting phase coexistance temperature. In order to quantify the fraction of the\nsample associated to each mode, we calculate the relative in tensity of the resonance peaks. The\nmodesintensity Ii(i:m,b)isdefinedasthedoubleintegraloftheFMRsignal22andisproportional\nto the magnetic moment of the sample fraction associated to e ach mode. We fit the FMR spectra\nwithasumofthreeLorentziancurves. Therelativeintensit yofeachmode, Iri=Ii/It(Itstandsfor\n3thetotalintensity),wasadjustedinordertogetthebestfit ofthewholespectra. IntheinsetofFig.\n1(a), we plot thetemperature dependence of Irifor thethree modes. Two remarkable features can\nbe observed from this figure: i) the intensity of the tMandbMmodes remain almost frozen from\ntheironsetand up to272 K and ii)thetemperatureat which Irbincreases abruptlyagrees with the\nonsetofthephasecoexistencereported bytheotherauthors14,19,20.\nIn Fig. 1(b) we show the FMR spectra as a function of temperatu re for the M111-66 film in\nthe in-plane geometry, ΘH=π/2. We observe a totally different behavior with respect to th e\nother substrate orientation. All the FMR spectra present on ly one resonance mode in the whole\ntemperaturewindowofthestudy. Weattributethisdifferen tbehavior,withrespecttothatreported\nforM100-66film,toadifferenttopologicalarrangementoft heαandβdomainsduringthephase\ncoexistence. Theabsence of additionalmodesin the FMR spec traof M111-66films is associated\nto an homogeneous distribution of the αand theβphases within the sample. In these conditions\ntheFMRsignalwillnotpresentchangescomparedtothatofth epureαphase. Theseresultsagree\nwiththoseobtainedby othertechniques12,23.\nTheexistenceofadditionalmodesintheFMRspectraoftheM1 00-66filmssuggeststhatthere\nare regions within the sample with different magnetic behav ior than the usual αphase. These\nregions can differ in their magnetization, anisotropy and/ or demagnetizing factor due to differ-\nentdomainsshape. Wediscardchangesinanisotropyduringt hephasecoexistencebasedonRefs.\n6,24. Inaddition,wecanneglectchangesinmagnetization, duetothefactthatmagneto-crystalline\nanisotropyis moresensitiveto structural changes than mag netization;then wecan expect that the\nmagnetization of the ferromagnetic domains during the coex istence is similar to the bulk one as\nwell. Taking into account these aspects, we can assure that t he shift of the tMandbMmodes\ncompared to the mMone is due to the existence of zones with different demagneti zing factors.\nTheseresultsrevealthatthephasecoexistencestartsself -orderedandthat,theappearanceofthe β\nphasedrivestheferromagneticdomainstochangetheirshap esin thenew zones. We alsoobserve\nthat from 180 K to 278 K, the pure αphase regions and the zones where α-βphases coexist are\npresent simultaneouslyinthesample.\nTheαandβphase coexistence has been always modelled to be set in the wh ole sample from\nthe onset temperature. This idea leads the authors of Ref.10to propose as possibleexplanation of\nthephenomenathatglidingmisfitdislocationsmovealong th eMnAs/GaAsinterface torearrange\nthe strains along sample. However, our results contradict p artially this model. The coexistence\nof pureαphase and α-βphase regions below 278 K indicates that the strain spatial d istribution\n4is not homogeneous. Therefore, we suggest that the dislocat ions are not free to move along the\ninterfaces dueto thefact thattheirthermal energy is small erthan thepinningdefects potentialsat\ntheintermediatetemperaturerange.\nBeyond 278 K, the fact that the peaks merge into a single mode i ndicates that above this temper-\nature there is no more than one magnetic domains-type behavi or in the samples, then the phase\ncoexistenceis extendedalongthewholesample.\nInordertogiveaquantitativedescriptionofthemagneticb ehavioroftheM100-66sample,we\nmodel the different contributions to the free energy densit y,F, of our system. Ffor the pure α\nphaseregion,associatedtothe mMmode,isgivenby:\nFmM=−M·H+/parenleftBigµo\n2NmM\nzM2+Kn/parenrightBig\ncos2Θ+Ksin2Θsin2Φ, (1)\nwhere the first and second terms account for the Zeeman intera ction and the demagnetizing con-\ntributionofacontinuousthinfilmwhichliesinthethe xyplane(NmM\nz=1)25,respectively. Misthe\nsaturationmagnetizationand Kis themagneto-crystallineanisotropyforMnAs,whichhas a hard\naxis perpendicular to thehexagonal basal plane6. In ouraxis convention[Fig. 1(c)], thehard axis\nis parallel to the yaxis.Θis the polar and Φthe azimuthal angle of the M, respect to the [100]\ndirection. Knis an uniaxial anisotropy, that favors an in-plane magnetiz ation and arises from the\nsurface anisotropy and the intrinsicanisotropycaused by a slightdistortionofthe hexagonalbase\noftheMnAsunitcell11,24.\nThe energy density of the αphase in the coexistence zones, FbM, takes into account the available\ninformation about the magnetic domain shapes for T >273 K obtained from pictures of scanning\ntunneling and atomic force microscopies11,16. Then, in order to model the self-ordered arrange-\nment of the αandβphases we will assume that the magnetic domains have a stripe -like shape\nfor thewholetemperaturerange. We considerthat theirshap e is arectangular prism and that side\nb≫a,casshowninFig. 1(c)25. Then,FbMis expressedby24,\nFbM=−M·H+/parenleftBigµo\n2NbM\nzM2+Kn/parenrightBig\ncos2Θ−\n/parenleftBigµo\n2NbM\nxM2−K/parenrightBig\nsin2Θsin2Φ,(2)\nwhereNbM\nx=NbM\nz−125. The free energy of the tMmode is also described by an expression\nsimilartotheEq. (2)but,withdifferent demagnetizingpar ameters.\nTo estimate the magnetic constants corresponding to our sys tem, we perform a self-consistent\n50.080.100.120.14\nc/a\n200 250\nT [K]0.840.860.880.90Nz\nFIG.2: Demagnetization factor and c/aratio as afunction of temperature.\ncalculationoftheresonancefields usingtheequationderiv edbyRef. 26:\nω2=γ2\nM2sin2Θ/bracketleftBigg\n∂2F\n∂Θ2∂2F\n∂Φ2−/parenleftbigg∂2F\n∂Θ∂Φ/parenrightbigg2/bracketrightBigg\n, (3)\nevaluated at the equilibrium angles Θeq,Φeq, whereγ=gµB/¯his the gyromagnetic ratio. The\nequilibrium conditions for M,Θeq,Φeqare obtained by minimizing the total free energy of the\nsystem (FmM+FbM+FtM) for each orientationof the applied magneticfield. Taking i nto account\nonlyFmMwe determine KandKnby fitting the resonance field as a function of the polar angle,\nΘH, and keeping ΦHfixed (see Fig. 1(c) for each temperature; then, we are able to determine the\ndemagnetizing factor from FbmandFtm.Mis obtained from magnetometric measurements. The\nresonance field shift of the bMmode in comparison with the mMone is defined by the change of\nthe demagnetizing factors; the thinner are the magnetic str ipes, the larger is NbM\nx. In Fig 2, we\nshowthetemperaturedependence ofthecalculated NbM\nzvalueforthe bMmode. In this figurewe\ncan identify three different regimes: i) 180 K<T<200 K,NbM\nzdecreases from 0.90 to 0.865; ii)\nNbM\nzremains almost constant up to 272 K and iii) forT>272 K,NbM\nzdecreases down to 0.85 at\n278 K. We are unable to determine the NbM\nzvalue beyond 278 K, because the resonance peaks\nmerge into a single one. Lindner et al.24deduced a demagnetizing factor of NbM\nz=0.84 for the\n6stripes at room temperature. Our results indicate that for T<200 K and T>272 K the magnetic\nstripes become thinner [side a, as referred in Fig 1(c)]; and between 200 K and 272 K the strip es\nwidthremainsalmostconstant.\nFollowingRef. 27,itispossibletodeterminethe c/aratiofrom NbM\nzfordomainswhoseshapes\nare rectangular prisms. In Fig. 2 the calculated c/aratio as a function of temperature is plotted.\nIfweassumethat theheight(side c)ofthestripesisthefilmthickness( c=66 nm)wefind that the\nstripes width (side a) varies from ∼950 nm to ∼500 nm when we increase the temperature from\n180 K to 278 K. In the frame of this model, the higher resonance field of the tMmode indicates\nthat there are zones of ferromagnetic stripes of different s izes in the films, i.e. thoseassociated to\nmodetMare wider than those related to bMmode. Due to the similarities of the ferromagnetic\ndomainsrelatedtoboth,the bMandtMmodes,wewillonlypresenttheanalysisofthedataofthe\nfirst mode.\nSummarizing,weinvestigatedthecoexistenceofthe αandβphasesinMnAsthinfilms,grown\noverGaAs(100)and GaAs(111)substratesby performingFMRe xperimentsas afunctionoftem-\nperature and angles. The spectra behavior indicates that th e phase coexistence is self-ordered for\nMnAs/GaAs(100) and it starts ∼100 K below the onset temperature detected by other experi-\nmental techniques. We show that the α+βregions are distributed along the samples in islands\nembedded in the αhomogeneous phase for temperatures lowerthan 278 K. From 27 2 K, the vol-\nume of the α+βregions suffers an abrupt increase and, from 278 K, the coexi stence is extended\nto the whole sample. The temperature which the strongest cha nge is detected matches with the\nonset of phase coexistence reported in the literature. On th e other hand, the FMR measurements\nin MnAs/GaAs(111) thin films agree with the fact that the βphase inter-grows randomly within\ntheαmatrix.\nWe acknowledgeJ. P´ erez and R. Ben´ avidez for technical ass istance. The authors thank partial\nfinancialsupportfromFundaci´ onAntorchas,FONCyT03-132 97,UniversidadNacionaldeCuyo,\nCONICET, ANR-MOMESand theECOS-SUD. J.Mand L.B.Sare membe rsofCONICET.\n∗Electronic address: milano@cab.cnea.gov.ar\n1I.ˇZuti´ c, et al., Rev.Mod. Phys. 76, 323 (2004).\n2L.D¨ aweritz, Rep. Prog. Phys. 69, 2581 (2006).\n73R.H. Wilson, et al., Acta Cryst. 17, 95 (1964).\n4B.T. M.Willis, et al., Proc. Phys. Soc. B 67, 290 (1954).\n5J. B.Goodenough, et al., Phys.Rev. 157, 389 (1967).\n6R.W.D.Blois, et al., Phys. Rev. 130, 1347 (1963).\n7C.Adriano, et al., Appl. Phys. Lett. 88, 151906 (2006).\n8I. Rungger, et al., Phys. Rev. B 74, 024429 (2006).\n9V. Garcia, et al., Submitted to Phys.Rev. Lett..\n10V. M.Kaganer, et al.,Phys. Rev. Lett. 85, 341 (2000).\n11A.K. Das,et al., Phys.Rev. Lett. 91, 087203 (2003).\n12Y. Takagaki, et al., Appl. Phys. Lett. 85, 1505 (2004).\n13V. M.Kaganer, et al.,Phys. Rev. B 66, 045305 (2002).\n14N.Mattoso, et al., Phys. Rev. B 70, 115324 (2004).\n15F.Iikawa, et al., Phys. Rev.B 65, 205328 (2002).\n16M. K¨ astner, et al.,J. Appl. Phys. 92, 5711 (2002).\n17J. Mohanty, et al., Appl. Phys. Lett. 83, 2829 (2003).\n18T. Plake, et al.,Appl. Phys.Lett. 80, 2523 (2002).\n19A.Ney, et al., Phys. Rev. B 69, 081306(R) (2004).\n20F.Vidal, et al., Phys. Rev. B 74, 115330 (2006).\n21L.B.Steren, et al.,Phys. Rev. B 74, 144402 (2006).\n22A.Layadi, et al., J.Magn. Magn. Mater. 92, 143 (1990).\n23L.D¨ aweritz, et al., J. Vac. Sci.Technol. B 23, 1759 (2005).\n24J. Lindner, et al., J. Magn. Magn. Mater. 227, 159 (2004).\n25B.D. Cullity, Introduction tomagnetic materials (Addison-Wesley, Massachusetts, 1972).\n26J. Smit,et al., Philips Res. Rep. 10, 113 (1955).\n27A.Aharoni, J. Appl. Phys. 83, 3432 (1998).\n8"    },    {        "title": "0710.2172v1.Direct_current_voltage_induced_by_microwave_signal_in_a_ferromagnetic_wire.pdf",        "content": "  1/1  Direct current voltage induced by microwave signal in a \nferromagnetic wire \n \nA.Yamaguchi1, K. Motoi1, H. Miyajima1, and Y . Nakatani2 \n \n1Department of Physics, Keio University, Hiyoshi 3-14-1, Yokohama, Kanagawa 223-8522, \nJapan  \n2University of Electro-communications, Chof ugaoka 1-5-1, Chofu, Tokyo 182-8585, Japan   2/2  [ABSTRACT] \nExperimental results of rectification of a constant wave radio frequency (RF) current \nflowing in a single-layered ferromagnetic wire ar e presented. We show that a detailed external \nmagnetic field dependence of the RF current in duced a direct-current voltage spectrum. The \nmechanism of the rectification is discussed in a term of the spin transfer torque, and the \nrectification is closely related to resonant sp in wave excitation with the assistant of the \nspin-polarized RF current. The micromagnetic si mulation taking into account the spin transfer \ntorque provides strong evidence which supports th e generation of spin wave excitation by the \nRF current. \n \n                    \n   3/3  I. INTRODUCTION \nRecently it is found that the spin and elect ric charge of electrons in nano-artificial \nmagnetic systems exhibit peculiar behaviors in the radio-frequency (RF) region through \nferromagnetic resonance (FMR) which has led to th e spin-polarized current in the RF region. \nFurthermore, spin wave excitation and magnetizatio n reversal is induced by the spin-polarized \nRF current. From the view points of fundamental nano-scale magnetism and technical spintronics, many theoretical [1, 2] and experi mental [3 - 16] studies have been progressed and \nfocused on a system with spatial distribution of magnetization such as magnetic multilayers [3 \n– 10], magnetic tunnel junctions (MTJ) [1 1], spin-valves [12], and single-layered \nferromagnetic wires [13 – 15]. These systems have  numerous possible technical applications in \nmagnetoresistive random access memory (MRAM), microwave generators [9, 10], and so on. \nThe physical mechanism, as revealed by the experi mental results, is described in term of the \nspin angular momentum transferring which o ccurs due to the interaction between the \nspin-polarized current and the magnetic moment. One of the most interesting properties is the \nrectification of RF current. When the resonant RF current flows through a system with magnetization distribution, it resonantly excit es spin wave in the system and the magnetic \nmoment precesses around the effective magnetic fiel d consisting of magnetic anisotropy field, \ndemagnetizing field and external field. As a result, the direct-current (DC) voltage is produced by the magnetoresistance oscillation due to FMR generated by the spin-transfer effect or RF \nfield [11 - 19]. \nIn the previous studies of pillar structur e systems consisting of two ferromagnetic \nlayers (a spin-polarizer layer and a free layer) ha ve been adopted to originate the spin-transfer \neffect. On the other hand, it has been clarified that the spin dynamics is  excited by the spin- \ntransfer torque due to the spin-polarized curre nt flowing through a system with domain wall or   4/4  magnetic vortex core [13 - 16]. \nIn this paper, we present detail experimental  results of the rectifying effect observed \nin a single-layered ferromagnetic nanowire [13]. As mentioned above, the rectifying effect is \nattributable to the resonant excitati on of spin waves by the RF current. ()01cos ItI tω= at \ntime t. Once the spin wave is excited in the wire, the magnetic moment precesses around the \neffective magnetic field with the angular frequency ω1, the resistance change ∆R(t) will \noscillate as a function of 2 cosωt, originating from the anisotropic magnetoresistance (AMR) \neffect. The DC voltage given by ()()() Vt It R t=⋅ ∆  is generated by a term involving \nmixing between RF current and magnetoresistance; \n() ( ) ()01 2 1 2cos cos 2 Vt V t t ωω ωω⎡ ⎤ =− + +⎣ ⎦    (1) \nIf 12ωωω== , ()( )01c o s 2 2 Vt V t ω =+ ; that is, the nanowire acts as a detector which \nturns the RF signal to DC voltage with second harmonic oscillation. \nII. RECTIFYING EFFECT OF RF CURRENT \nWe focus on a single layered Ni 81Fe19 wires [Fig. 1]. The wires are fabricated on \nMgO substrates using the procedure described in Ref. 13. We examine 30 and 50-nm-thick Ni\n81Fe19 nanowires and a 20-nm-thick Au nanowire contacted with Au electrode, of the 100 nm \nthick, as shown in Fig. 1. The width of the 50-nm-thick Ni 81Fe19 nanowires is 300, 650, and \n2200 nm and that of 30-nm-thick Ni 81Fe19 and Au nanowires is 5 µm and 300 nm, respectively.   5/5  A ground-signal-ground (GSG) type microwave prove is connected to the nanowire, and the \nbias-tee circuit detects the DC voltage differen ce induced by the RF current flowing through \nthe nanowire, as schematically illustrated in Fig. 1. The sinusoidal constant wave RF current is \ninjected into the nanowire by a signal genera tor with a frequency range from 10 MHz to 40 \nGHz. The external magnetic field extH is applied in the substrate plane as a function of angle \nθ to the longitudinal axis of the nanowire, while the Oersted field produced by the RF currents \nflowing in the loop of the Au electrodes genera tes an additional field at the sample. However, \nthe field is canceled out at the nanowire position because of the symmetrical arrangement of the Au electrode. All experiments are performed at room temperature.  \nFigure 5 shows the magnetic field dependence of the DC voltage difference generated \nin the wires with widths of 5 µm. The RF current density injected into the wire is \nabout\n102.3 10× A/m2, the value of which is relatively low for the Joule heating to be applied. \nAccording to our recent estimation of the curre nt density from the measurement of the RF \ncharacteristic impedance of the wire, the density is found to be almost same to the value. An \nexternal magnetic field is applied in th e substrate plane at the angle of (a) 0D, (b) 30D, (c) \n60D, and (d) 90D to the wire-axis. As seen in the figure, the peak position of resonance \nspectrum shifts to the higher frequency region with  the external magnetic field. The sign of the \nDC voltage reverses when the angle of th e applied magnetic field is changed form θ to   6/6  180θ+D. On the other hand, no similar effects are observed in the 20-nm-thick Au nanowires, \nwhich strongly indicates that the resonance is of magnetic origin and generated by the RF \ncurrent. \nFigure 3 shows the magnetic field dependence of the resonance frequency of the \nnanowire with widths of 300, 650, 2200, and 5000 nm, where the magnetic field is applied at \nthe angle of D45 to the wire-axis. The resonan ce frequency increases with increasing \nmagnetic field but decreasing wire-width, whic h means that the resonance mode possibly \nreflects the shape magnetic anisotropy. \nAccording to the extended Kittel’s e quation [20], the ferromagnetic resonance \nfrequency at the magnetic field H is approximately given by \n() ( )1\n02 ()B\nAS AgHH H M H Hµµω ⎡⎤ =⋅ + + ⋅ +⎣⎦=,        ( 2 )  \nwhere MS denotes the saturation magnetization, g the Lande factor, = Planck’s constant, µ0 \nthe permeability of free space, Bµ the Bohr magneton, and AH the effective anisotropy \nfield including demagnetizing and anisotropy fields  in the substrate plane. The experimentally \ndetermined AH for the nanowires with widths of 300, 650, 2200, and 5000 nm is 1160, 830, \n260, and 100 Oe, respectively. The resonance freque ncy is in excellent agreement with that of \ncalculation using the extended Kittel’s eq.(2), indi cating the existence of resonant spin wave \nexcitation. As the RF current induces the pr ecession of the magnetic moments in the nanowires   7/7  and the excitation of the spin wave, the resist ance due to the AMR effect oscillates at the \nresonance frequency. This resistance oscillation generates the DC voltage by combining with \nthe RF current, as observed in the case of the spin -torque diode effect in the pillar structure \nsystems [11, 12]. \nIII. MICROMAGNETICS CALCULATION \nThe current-induced dynamics of the magnetic moment are analytically calculated by \nthe micromagnetic simulations based on the Landau-Lifshitz-Gilbert (LLG) equation including \nthe spin-transfer term [16, 21]. The modified LLG equation is given by \n()0 eff sttγα∂∂=− × + × − ⋅∇\n∂∂mmmH m u m ,    (3) \nwhere m denotes the unit vector along the local magnetization, 0γ the gyromagnetic ratio, \neffH the effective magnetic field including th e exchange and demagnetizing fields, and α \nthe Gilbert damping constant, respectively. The l ast term in eq. (3) represents the spin-transfer \ntorque which describes the spin transferred fro m conduction electrons to localized spins. The \nspin-transfer effect is a combined effect of the spatial non-uniformity of magnetization \ndistribution and the current flowing. The vector with dimension of velocity, \n()/2s BsPg eMµ=−uj , is essentially the spin current asso ciated with the electric current, \nwhere j is the current density, P the spin polarization of the current, and e the electronic \ncharge, respectively. The spatial non-uniformity of the magnetization distribution in a nanowire   8/8  is mainly caused by the spatial dispersion of the demagnetizing field. \nA micromagnetic simulation of the system c onsidering the spin transfer torque is \nperformed to confirm the generation of spin wave  excitation by the constant wave RF current \n[16, 21]. The nanowire adopted in the simula tion has a width and length of 300 nm and 4 µm, \nrespectively. The parameters used for the calculation are; the unit cell size of 4 nm × 4 nm \nwith constant thickness 50 nm, the magnetiza tion 1.08 T, the spin current density u = 3.25 m/s, \nand the spin polarization p = 0.7 [16]. In the simulation, the external magnetic field of 200 Oe \nis applied in the substrate plane at the angle of 45D from the wire-axis, where the magnitude \nof the field is not large enough to direct perfectly the magnetization along the direction of the \napplied magnetic field. The local spins in the nanowire are almost parallel to the effective \nanisotropy field.  \nFigure 4 (a) shows a snapshot of the map and the cross-section of the z component of \nthe magnetization in the middle of the simulati on. The black and white stripes correspond to \nthe –z and +z components of the magnetization, respectively, and the stripe magnetization pattern due to the spin wave excitation induced by  the RF current is clearly observed. It should \nbe noted that the period of the stripe pattern of  approximately 100 nm is consistent with the \nspin wave length estimated by \nAK, where A and K are the exchange stiffness constant and \nthe anisotropy energy written by K =0B C g Hµµ , respectively. The magnetization response is   9/9  simulated as a function of the RF current frequency. The mean values of the y and z \ncomponents of magnetization are shown in Fig. 4 (b) as a function of the RF frequency. The mean value of x component is not suitable fo r the analysis as it has large background because \nof the uniaxial magnetic anisotropy. As shown in  Fig. 4 (b), both y and z components peak at \n10.4 GHz, which is consistent with the spin wa ve resonance frequency. The simulation result is \nin accord with the experimental result.  \nIV . SIMPLE ANYLYTICAL MODEL AND EXPERIMENTAL RESULT \nHere, we derive an analytical expression for the DC voltage produced by the spin \nwave excitation.  Suppose we have an ( x, y, z) coordinate system: each component \ncorresponds to the short wire axis, longitudinal wire axis, and normal to the substrate plane, \nrespectively, as schematically shown in Fig. 5 (a). The external magnetic field \nextH is applied \nalong the angle θ from the y-coordinate axis. Then, let us define a new coordinate system ( a, \nb, c) where the b direction corresponds to the equilib rium direction of unit magnetization 0m \nalong an effective magnetic fieldeffH  including extH  and AH (the shape magnetic \nanisotropy field). The unit vector 0m inclines at an angle ψ to the y-coordinate axis, and \nAH is parallel to the y-coordinate axis. The magnetization precession around effH only \nresults in a small time-dependent compone nt of the magnetization perpendicular to 0m. The \nassumption is valid whenever the spin wave exc itation is small.  Then, we decompose the unit   10/10  vector m in the ( a, b, c) coordinate system as () ()0 ttδ=+ mm m , considering a small \nvariation () ()() ,0,acmt mtδ=m  and ()00,1,0=m . The vector m is obtained from \neq.(3): \n00 effttδγα∂∂≈− × + × −∂∂mmHm s ,                                   ( 4 )  \nwhere  \n(),,abcmmm=m , ()221ba cmm m=− + ,         ( 5 )  \nand the spin transfer torque s almost perpendicular to 0m in the ( a, b, c) coordinate system \nis defined as \n() ( ) cos , sin ,s ab cjs s sψψ =⋅ ∇= ⋅ −su m ,         ( 6 )  \nThe effective field effH is expressed in the ( a, b, c) coordinate system as \n()\n()sin sin\ncos cos\n00ae x t A\neff b ext AHH H\nHH Hθψψ\nθψψ−− ⎛⎞⎛⎞\n⎜⎟⎜⎟== − +⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠H .        ( 7 )  \nSubstituting eq. (5), (6), and (7) into eq. (3), we obtain the magnetization dynamics; \n0cos\n0s i nac bc a\nbc a b\nca b b a acmm Hm s\nmm H j sttmm H m H m sψ\nγ αψ− ⎛⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞\n∂∂⎜⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟≈− + − −⎜⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂∂⎜⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟−− ⎝⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (8) \nUnder the conditions 1=m  and 1bm≈, eq. (8) is reduced to,  \n0cosac bc a\nca b a a cmm Hm sjmm H H m sttψγα− ⎛⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞∂∂≈− + −⎜⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟−− ∂∂⎝⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠,  (9) \nassuming that the time-dependent magnetization precession angle around the direction of \n0mis parallel to effH and that the angle is very small. The solution of eq. (9) can be   11/11  presented by the Fourier transformation;  \n() ()2 2\n0cos 1 k a a\nk c ca kii mj s\nii mj s H iωω ω α ψ\nωω α ω γ ωωω α+ ⎛⎞ ⎛⎞ ⎛ ⎞= ⎜⎟ ⎜⎟ ⎜ ⎟−+ − −+ ⎝⎠ ⎝ ⎠ ⎝⎠,       ( 1 0 )  \nwhere 0 kb Hωγ=  denotes the spin wave resonant angular frequency corresponding to ()Hω  \ngiven by eq.(2). \nDue to the spin wave excitation with precession angle of lφ, the small variation in the \nmagnetic moment around the direction of effH induces the time-dependent AMR effect. \nWhen ()2cos 1lφ≈ and ()2sin 0lφ≈ for small lφ, we can approximately derive the \naverage time-dependent resistance ()Rt in each cell;  \n() ()() ()22 1cos cos sin 2 sin 22RtR t R t∆ψ φ ∆ψ ψ φ⎡ ⎤=+ ≈ −⎢ ⎥⎣ ⎦∑ AA\nA ,   (11) \nwhere ∆R denotes the resistance change due to the AMR effect, the number l the index of the \ndomain of the spin wave excitation, and () sin 2 2\naφ∆≈A mm by using the a component \nofδm. When the RF current given by ()()0cos ItI t ω=  flows in the wire, the frequency \nspectrum of the induced voltage is expr essed by the Fourier transformation of \n() ()()Vt R tIt= ; \n()()2\n0sin 2 cos VA Iωωψ ψ= ,           ( 1 2 )  \nwhere ()Aω is written by \n()() ( )\n()22 2 1\n0\n222 2 2 22/ c o s\n4ka k k c a\nkkss H j RAbdωα ω ω ω ω γ ψ\nω\nωω ω α ω−⋅+ − ⋅ − ∆≈⋅⋅ −+,                ( 1 3 )    12/12  where b and d are the width and thickness of the nanowire, respectively. \nIn the case that extH  is higher than AH, the magnetic moment directs almost \nalong ext eff≈ HH . Then, we obtain ψθ≈ ( see Fig. 5 (a)), which well corresponds to the \nresult described in Ref 13. The relation ψθ≈ gives the frequency variation of the induced \nvoltage ()Vω proportional to 2\n0sin 2 cosIθθ.  On the contrary, extH  is smaller than \nAH , the magnetic moment is almost parallel to the direction of effH . Then, \n0 const.ψ θ <= <D, and the ()Vω is almost proportional to sinextHθ. This means that the \nout-of-plane component of magnetization leads to the dispersion shape.  \nAs a result, the field and angle de pendences of the induced DC voltage ()ωV  are \nsummarized as follows: \n(i) () θθω cos2sin∝ V  in ,ext A>> HH  \n(ii) () θ ω sinextH V∝ in ext A< HH   \nFigures 5 (b), (c), and (d) show the angle dependence of the induced DC voltage at \nthe resonance frequency in the magnetic field of 400, 100, and 30 Oe, respectively.  The inset \nof Fig.5 (b) shows the experimental result of the magnetoresistance for 30 nm-thick Ni 81Fe19 \nwire with 5 µm width in the magnetic field perpendi cular to the longitudinal wire axis \n(D90=θ ). \nThe result of the magnetoresistance measureme nt in the inset of Fig. 5 (b) offers   13/13  evidence that the magnetization is almost parallel to the direction of the field extH  at \n400ext= H Oe. Then, according to the analytical model described above, the angle \ndependence of the induced DC voltage ()ωV  should be proportional to θθcos2sin , \ncorresponding to the case (i). In Fig. 5 (b), the dashed-line represents the θθcos2sin  curve, \nand it is in good agreement with the experime ntal results. The magnetization is almost along \nthe direction of AH at 30ext= H  Oe. Then, the angle depe ndence of the induced DC \nvoltage()ωV  corresponds to the case (ii), that is () θ ω sinextH V∝ . In Fig. 5 (d), the \nexperimental result is consistent with the sinθ curve. As shown in Figs. 5 (b) and (d), we \nfound that the simple analytical model expression given by eqs. (12) and (13) qualitatively fits \nwell to the experimental resu lts. The angle dependence of ()ωV  at 100ext= H  Oe is \nintermediate with the angle dependence between the cases (i) and (ii) as shown in Fig. 5 (c). \nThe spectra shape is determined by ()Aω of eq. (13). ()Aω is given by the \nsuperposition of the Lozentzian function with a maximum at kωω=  and the dispersion \nfunction. The component of the Lozentzian function basically governs ()Aω, indicating that \nthe in-plane spin transfer component determ ines the spectrum shape. However, as the angle θ \nincreases, the component of the dispersion function in ()Aω increases. This is consistent \nwith the actual spectra shapes observed in the experimental results shown in Fig. 2. The \nadditional wiggles in the trace of Fig. 2 corres pond to the additional spin wave modes derived   14/14  from the complicated spin wave excitation. Wh ile, if the in-plane RF field induce the \nmagnetization dynamics, the spectrum shape is described only the dispersion function [19]. \nThe solution of eqs. (12) and (13) can be also represented as follows: \n() () sin 2 cos VI Rω ωψ ψ =⋅ ∆⋅ Γ ⋅ ,      ( 1 4 )  \n()()()\n()22 2 1\n0\n222 2 2 22c o s\n4ka k k c a\nkkjs js Hωωα ω ω ω γ ψ\nω\nωω α ω ω−⋅+ −⋅ −\nΓ≈\n−+.   (15) \nHere, ()ωΓ  gives the dimensionless value and the spect rum shape. We can fit the change in \nthe normalized DC voltage () RI V∆ω  observed using the eq.(14). With the in-plane and the \nout-of-plane component of the spin transfer torque ajs, cjs, and the Gilbert damping \nparameter α as fit parameters, the typical signals measured in the wire with width of 5 µm in \nthe application of the external field 400ext= H Oe could be fitted as shown in Fig. 6. These \nfits allow us to determine those pa rameters, which were found to be \n()65, , (4.45 10 ,6.79 10 ,0.016)acjs jsα=× ×  at 15θ=Dand \n()66, , (1.45 10 ,3.13 10 ,0.010)acjs jsα=× ×  at 30θ=D in the wire with width of 5 µm. \nThe Gilbert damping constants determined by the fittings agree with the values reported by \nCostache et al. [18], and they are larger than the va lue commonly accepted for a thin film of \nNi81Fe19 [22]. In addition, we note here that the DC voltage spectrum shape, comprising of the \nsuperposition of the Lorentzian function and the dispersion function, changes to a more \ncomplex and wiggle shape as the frequency and a ngle applied external field change. These can   15/15  be due to the spatial inhomogeneities of the inc oherent spin wave excitation and out-of-plane \ncomponent neglected in the time-averaging resistance.  Electrical conduction in a ferromagnetic meta l generally depends on the direction of \nthe magnetization. The phenomenologi cal description of electrical field \nEgiven by the two \ncurrent model takes the following form \n()()0H ρρ ρ ρ⊥⊥=+⋅ ⋅−+ ×E jmjmm j& .                              ( 1 6 )  \nHere, j denotes the electrical current, m denotes the unit vector along the local \nmagnetization, ρ⊥, ρ& the resistivity components refer to the parallel and perpendicular \ndirections of j with respect to m, \n0Hρ the extraordinary Hall resistivity component, \nrespectively. Our result derived from the analytical  model given by eq. (3) is consistent with \nthat of the eq. (16) extended by Juretschke [17]. Juretschke introduced the phenomenological \noscillating component of the magnetization to eq . (16) and obtained the generation of the DC \nvoltage, assuming the small angle of magneti zation precession caused by the RF field or \nsomething [17]. Here, what is the origin of th e magnetization dynamics? The RF field in the \nsubstrate plane and perpendicular to the plane causes the magnetization precession. In the \nformer case, the angle dependence of the induced  DC voltage agrees with the case (i) or (ii). \nHowever, in the later case, it is proportional to sin 2θ [19]. In our case of this study, the RF \nfield is canceled out at the wire position due to the symmetrical arrangement of the electrodes.   16/16  Instead, the spin-polarized RF current flowing through the ferromagnetic wire, with the spatial \nnon-uniformity of the magnetization distribution caused by the spatial dispersion of the demagnetizing field, produces the spin transfer torque, and it generates the magnetization dynamics. \n The DC voltage originates from the magn etoresistance oscillation due to the \nmagnetization precession generated by the spin-transfer effect at the FMR frequency. We measure the DC voltage induced by RF signal along the major and minor axes of the single-layered ferromagnetic wire. The DC\nL and DC S voltage, which respectively corresponds \nto that induced along the major-axis and minor-axi s of the wire, is shown in Fig. 7 in the case \nof H = 500 Oe and θ =30D, where the solid and dashed lines indicate the DC L and DC S voltage \nspectra, respectively. The rectifying effect is  interpreted in terms of the magnetoresistance \noscillation and the planer Hall effect, as introduced by eq. (16).  \nV . CONCLUSION \n (1) The resistance oscillation due  to the spin wave excitation caused by spin-polarized RF \ncurrent generates the DC voltage by combing with the RF current in the single layered \nferromagnetic wires. (2) The experimental results for the rectificati on of RF current flowing in the wire in the   17/17  presence of an external magnetic field show th e existence of a regime: with the precession of \nthe average magnetization around the effective anis otropy magnetic field including the external \nand demagnetizing magnetic field. \n(3) The applied magnetic field angle θ dependence of the induced DC voltage is proportional \nto sin 2 cosθθ in the high magnetic field and sinθ in the low filed, respectively. \nThe rectifying effect of the RF current resu lts in a highly-sensitive measurement of spin \ndynamics in nano-scale magnets. This effect f acilitates the development of spintronics devices \nsuch as radio-frequency detectors in tel ecommunication [23], magnetic sensors, and \ncoplanar-type high-frequency band-pass filters.  \nACKNOWLEDGMENTS  \nThe present work was partly supported by a Grants-in-Aid from the Ministry of \nEducation, Culture, Sports, Science, and T echnology, Japan, by the Keio Leading-edge \nLaboratory of Science and Technology proj ect 2007, and by the Foundation Advanced \nTechnology Institute, Japan. \n \n        18/18  Reference \n[1] J. C. Slonczewski, J. Magn. Magn. Mater ., vol. 159, pp. L1-L7, 1996. \n[2] L. Berger, Phys. Rev. B , vol. 54, pp. 9353-9358, 1996. \n[3] E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science,  vol. 285, \npp. 867-870 1999.; J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, \nPhys. Rev. Lett ., vol. 84, pp. 3149-3152, 2000. \n[4] M. Tsoi, A. G. M. Jansen, J. Bass, W. –C . Chiang, M. Seck, V . Tsoi, and P. 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Buhrman, \nScience,  vol. 307, pp. 228-231, 2005. \n[9] S. Kaka, M. R. Pufall, W . H. Rippard, T . J. Silva, S. E. Russek, and J. A. Katine, Nature,  \nvol. 437, 389-392, 2005. \n[10] F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature , vol. 437, pp. 393-395, \n2005. \n[11] A. A. Tulapurkar, Y . Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. \nDjayaprawira, N. Watanabe, and S. Yuasa, Nature , vol. 438, pp. 339-342, 2005.   19/19  [12] J. C. Sankey, P . M. Braganca, A. G . F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. \nRalph, Phys. Rev. Lett ., vol. 96, pp. 227601-1-227601-4, 2006. \n[13] A. Yamaguchi, H. Miyajima, T. Ono, S. Yuasa, A. Tulapurkar, and Y . Nakatani, Appl. Phys. \nLett., vol. 90, pp. 182507-1-182507-3, 2007. \n[14] Y . S. Gui, N. Mecking, X. Zhou, Gwyn Williams, and C. –M. Hu, Phys. Rev. Lett.,  vol. 98, \npp.107602-1-107602-4, 2007. \n[15] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinjo, Phys. Rev. Lett ., vol. 92, \npp. 077205-1-077205-4, 2004. \n[16] S. Kasai, Y . Nakatani, K. Kobayashi, H. Kohno, and T. Ono, Phys. Rev. Lett ., vol. 97, pp. \n107204-1-107204-4, 2006. \n[17] H. J. Juretschke, J. Appl. Phys ., vol. 31, pp. 1401-1406, 1960.; W. M. Moller and H. J. \nJuretschke, Phys. Rev. B , vol. 2, pp. 2651-2660, 1970. \n[18] M. V . Costache, M. Sladkov, C. H. van der Wal, and B. J. van Wees , Appl. Phys. Lett ., vol. \n89, pp. 192506-1-192506-3, 2006. \n[19] A. Yamaguchi and H. Miyajima, unpublished. [20] C. Kittel, Introduction to Solid State Physics Ch. 16 (John Wiley, New York, 1986), 6\nth ed. \n[21] A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys. Lett ., vol. 69, pp. 990-996, \n2005. [ 2 2 ]  T .  J .  S i l v a ,  C .  S .  L e e ,  T .  M .  C r a w f o r d ,  a n d  C .  T .  R o g e r s ,  J. Appl. Phys ., vol. 85, pp. \n7849-7862, 1999.  \n[23] A. Yamaguchi, H. Miyajima, S. Kasai, and T. Ono, Appl. Phys. Lett ., vol.  90, pp. \n212505-1-212505-3, 2007.  \n  \n \n   20/20  Figure caption \n \nFigure 1  \nThe radio frequency (RF) electric circuit and the optical micrograph (top view) of the system \nconsisting of electrodes and the magnetic nanow ire. The magnetic field is applied in the \nsubstrate plane at an angle of θ to the longitudinal axis of the nanowire. The direct-current \n(DC) voltage across the wire is measured by using a bias tee. \n \nFigure 2  \nThe DC voltage response generated in the 30 nm-thick Ni 81Fe19 wire with 5 µm width in \nresponse to the RF current as a function of RF  current frequency when the external magnetic \nfield is applied at an angle of (a) 0θ=D, (b) 30θ=D, (c) 60θ=D, and (d) 90θ=D. Each \nDC response is vertically shifted for clarity. \n \nFigure 3  \nResonance frequency as a function of the magnetic field. The circle, square, up-triangle, and \ndown-triangle marks denote the experimental data points for the nanowires with widths of 300, \n650, 2200, and 5000 nm, respectively. The solid-l ine curve represents the best fitting to the \nextended Kittel’s equation. \n \n \n   21/21  Figure 4 \n(a) Typical micromagnetic simulation including the spin transfer and field-like torques (shown in inset). The simulated pattern (black and white  stripes) shows the oscillating magnetization in \na magnetic nanowire. The black stripe corresponds  to the –z component of the magnetization; \nthe white stripe, the +z component. (b) The y and z components of the magnetic moment are shown as functions of the RF current frequency.  The red and black lines correspond to the y \nand z components, respectively.  \n Figure 5  (a) Schematic coordinate system adopted in the simple analytical calculation model. The \nlongitudinal wire axis is paralle l to the y-axis. Angle dependence \nθ of the DC voltage \namplitude when the external magnetic field of (b) 400 Oe, (c) 100 Oe, and (d) 30 Oe was \napplied in the substrate plane, respectiely. Th e inset shows the result of magnetoresistance \nmeasurement when the field was applied at 90θ=D. The dashed-line in (b) and (c) represents \na sin 2 cosθθcurve, and the dashed-lin e in (d) corresponds to a sinθ curve. \n \nFigure 6 \nMeasured DC voltage as a function of the RF cu rrent frequency in the wire with width of 5 µm. \nThe solid lines are the fit to the data using eq. (15). The dotted and dashed lines correspond to \nthe in-plane and out-of-plane component, respectively.   22/22  Figure 7 \nThe DC L and DC S voltage as a function of RF current frequency in H= 500 Oe at θ = 30D. \n \n   \n \n   \n \n    \n \n     23/23  \nBias tee\nVSignal\ngenerator V olt meter\n10 µmNi81Fe19 wire\nMgOHAu\nG\nGS\nθ\n \nFig. 1 A. Yamaguchi et al. \n \n \n \n     24/24  DC voltage ( mV)\n10 5 0(a)\nq = 00.5\n10 5 0(b)\nq = 305\n10 5 0(c)\nq = 605\n10 5 0(d)\nq = 905\n400 Oe\n200 Oe\n100 Oe\n30 Oe\nRF current frequency (GHz) \nFig. 2 A. Yamaguchi et al. \n \n   \n \n    \n \n \n \n   25/25  15\n10\n5\n0Frequency (GHz)\n600 400 200 0\nMagnetic field (Oe)300 nm\n650 nm\n2200 nm\n5000 nm\n \nFig. 3 A. Yamaguchi et al. \n \n  \n \n   \n \n     26/26  2.0\n1.0\n0Direction consine (x10-6  rad.)\n20 15 10 5 0\nRF current frequenc y (GHz)y\nz\nz y\nx+z\n-z\nw = 300 nm\n-1.0-0.500.51.0Mz/MS (x10-3  arb. unit)\n-1 0 1\nPosition ( µm)(a)\n(b)\n \nFig. 4 A. Yamaguchi et al. \n \n  \n   27/27   \n-30-20-100102030\n360 270 180 90 0\nAngle (Deg.)30 Oe(d)-30-20-100102030∆ DC volt.  (µV)\n100 Oe(c)-30-20-100102030\n400 Oe(b)\n27.5\n27.0Resistance ( Ω)\n-500 0 500\nMagnetic field (Oe)θ = 90ψm m0 conduction \nelectron spinNanowire\nzx\nya\nbcHext\nRF currentθ(a)\n \nFig. 5 A. Yamaguchi et al.   28/28  V/I∆R (arb. unit)\n10 8 6 4\nRF current frequency (GHz)15301\nw = 5 µm\n400 Oe\n \nFig. 6 A. Yamaguchi et al. \n  \n \n \n \n  \n \n    29/29  -20-10010DCL volt. ( µV)\n15 10 5 0\nRF current frequenc y (GHz)-2-1012\nDCS volt. ( µV)\nRF DCS\nDCL500 Oe\nθ = 30∆V\n \nFig. 7 A. Yamaguchi et al. "    },    {        "title": "1506.05631v1.Dispersive_read_out_of_ferromagnetic_resonance_for_strongly_coupled_magnons_and_microwave_photons.pdf",        "content": "arXiv:1506.05631v1  [cond-mat.mtrl-sci]  18 Jun 2015Dispersive read-out of ferromagnetic resonance for strong ly coupled magnons and\nmicrowave photons\nJ. A. Haigh,1N. J. Lambert,2A. C. Doherty,3and A. J. Ferguson2\n1Hitachi Cambridge Laboratory, J. J. Thomson Ave., Cambridg e CB3 0HE, United Kingdom\n2Microelectronics Group, Cavendish Laboratory, Universit y of Cambridge,\nJ. J. Thomson Ave., Cambridge CB3 0HE, United Kingdom\n3Centre for Engineered Quantum Systems, School of Physics,\nThe University of Sydney, Sydney, NSW 2006, Australia\n(Dated: October 1, 2018)\nWe demonstrate the dispersive measurement of ferromagneti c resonance in a yttrium iron garnet\nsphere embedded within a microwave cavity. The reduction in the longitudinal magnetization at\nresonance is measured as a frequency shift in the cavity mode coupled to the sphere. This mea-\nsurement is a result of the intrinsic non-linearity in magne tization dynamics, indicating a promising\nroute towards experiments in magnon cavity quantum electro -dynamics.\nI. INTRODUCTION\nThe dispersive limit of interaction between two cou-\npled oscillators occurs when they are significantly de-\ntuned from one another, but their amplitudes and fre-\nquencies are still linked contingent on a non-linearity in\nsome part of the system. In this coupling regime the\nprobing of one oscillator can be used to infer the prop-\nerties and state of the other1with a well defined and\nlimited backaction2. The use of dispersive measurement\ntechniques has allowed photon number measurement3–5\nin cavityquantumelectro-dynamics(QED), beenapplied\nto mechanical resonators6coupled to both optical7and\nmicrowave cavities8, and has been central to the mea-\nsurement of individual9,10and coupled11superconduct-\ning qubits in circuit QED.\nIn this letter we demonstrate the dispersive mea-\nsurement of ferromagnetic resonance (FMR) in a fer-\nrimagnetic insulator strongly coupled to a microwave\ncavity12. This system has been the subject of re-\ncent interest in the dynamics of hybrid magnon-photon\nmodes13, with the prospect of coherent control of long-\nlived single magnon states at low temperatures14and\nthe coupling via the resonator to qubits15or other mag-\nnetic elements16. While conventional FMR measures\nhow the complex susceptibility tensor depends on fre-\nquency and magnetic field17, here we measure the change\nin longitudinal magnetization, a quantity proportional\nto the magnon number. The experiment demonstrates\nthat the methods of cavity QED can be applied in\nthis system at room temperature, albeit in the classi-\ncal limit of many magnons and photons, differentiating\nthis work from previous measurements of longitudinal\nmagnetization by inductive coupling18, magneto optical\nmethods19,20, Brillouin light scattering21and magnetic\nresonance force microscopy22,23. We note that dilute\nparamagnetic spin ensembles have previously been mea-\nsured dispersively24,25motivated by the prospect of stor-\ning quantum information in a spin ensemble26. However,\ndue to the absence of exchange coupling in these param-\nagnetic systems low temperatures and large static fieldsare needed to polarize the system to allow access to dis-\ntinct magnon modes27.\nII. EXPERIMENTAL SETUP\nOur microwave cavity is constructed from a 28mm\nlength of semi-rigid copper coaxial cable of diameter\n3.5mm (fig. 1(a)), chosen to give a resonance at around\n3.5GHz for the fundamental mode. Capacitive couplings\nto the input and output of the resonator are achieved\nvia a<1mm gap between the inner conductor of the\ncavity and two cables which connect the device to the\nmeasurement instruments. To ensure that the cavity is\nsymmetric and undercoupled the capacitor gaps are ad-\njusted at each end while measuring the reflection coef-\nficientsS22andS11, and we achieve a loaded Q-factor\nof 260 which can be compared to an internal Q-factor of\n515. The 1mm diameter YIG sphere is embedded within\nthe dielectric of the semi-rigid cable, at the antinode of\nmagnetic field. The cavity is mounted in an electromag-\nnet, with its long axis parallel to the applied field direc-\ntion. Transmissionmeasurementsaremade with avector\nnetwork analyzer, with an additional microwave source\ncoupled in via a power combiner to allow two-frequency\nmeasurements (fig.1(b)).\nStrong coupling of the uniform ferromagnetic reso-\nnance mode to the fundamental mode of the electro-\nmagnetic resonator is demonstrated by a measurement\nof the transmission amplitude |S21|as a function of mag-\nnetic field and probe frequency ωp(fig.1(c)). The char-\nacteristic frequency splitting of the coupled modes ob-\nserved at the degeneracy point gives a coupling rate of\ng/2π= 130MHz, greater than the linewidth of both\nthe cavity mode and the FMR mode ( κ/2π= 14MHz\nandγ/2π= 6MHz respectively). This indicates that the\nstrong coupling regime g≫κ,γis achieved, with a co-\noperativity ( g2/κγ) of around 200. In addition to the\nuniform mode, other non-uniform magneto-static modes\nin the sphere28,29couple to the resonator due to the ra-\ndial dependence of microwave field in the cavity. One of2\nFIG. 1. a) Schematic of the semi-rigid coaxial resonator wit h\nYIG sphere inside. b) Circuit equivalent diagram of the mea-\nsurement setup and cavity. The probe tone ωpand mix-\ning are provided by a vector network analyzer (blue) while\nFMR drive tone ωdis from a separate microwave source (red).\nc)Transmission |S21|through cavity as a function of magnetic\nfield, demonstrating the strong coupling of the uniform FMR\nmode (blue arrow) to the λ/2 cavity mode (red arrow). In\naddition a number of other spin wave modes in the sphere\ncouple to the cavity; the strongest is indicated by the green\narrow. Inset shows a reference map of the measurement re-\ngion over the modes of the system (only the uniform FMR\nmode is shown).\nthese, indicated by the green arrow in the figure, with\na coupling rate of ≈35MHz, is also strongly coupled to\nthe resonator.\nIII. DISPERSIVE MEASUREMENTS\nTo perform the dispersive measurement we move away\nfrom the degeneracy point by tuning the static magnetic\nfield so that the FMR mode and cavity mode are not\nresonantly coupled. Two microwave tones are applied to\nthe cavity. We monitor the phase of a transmitted probe\ntoneωpclosetothe cavityresonance ωcasadrivetone ωd\nis swept acrossthe resonant frequency of the FMR mode.\nAs shown in fig.2(a), when the drive tone is resonant\nwith the FMR mode there is an increase in the phase of\nprobe tone of nearly 1mrad, corresponding to a cavity\nfrequency shift of +5kHz. This measurement is repeated\nfor different external magnetic fields (fig. 2(b)). The two\nFMR lines which couple most strongly to the cavity are\nvisible and the field dependence of the uniform mode30\nyields the gyromagnetic ratio γ= 28.6GHz T−1.\nFIG. 2. Dispersive measurement: a) Phase shift of the probe\ntoneωp(-10dBm) at the resonant frequency of the cavity\nas a function of the drive frequency ωd, for input drive power\n25dBm (offresonance with thecavityalarge amountof power\nis reflected). The field is fixed at µH0= 1800G. A slow time\ndependent dependent phase drift and field independent back-\nground have been subtracted. The dashed line is a fit to the\nsquare of a Lorentzian (square of eqn.2), appropriate since\nwe are sensitive to the square of the transverse magnetiza-\ntion. The fit gives the magnon decay rate γ/2π= 6MHz.\nb) Phase shift as a function of magnetic field and drive fre-\nquency. Probe frequency tracks the resonance frequency of\nthe cavity, ≈3.55GHz, as a function of field.\nThe variation with static magnetic field is used to\nstudy the dispersive measurement as a function of de-\ntuning between cavity and FMR mode frequencies, ∆ =\nωf−ωc. The change in the cavity frequency when the\ndrive tone is applied, ∆ ωc, is computed from the mea-\nsured phase change in ωpand follows a 1 /∆ dependence\nas indicated by the dashed fitted line in fig.3(a). We also\nmeasure the equilibrium cavity frequency shift with the\ndrive tone turned off, obtained from the peak in trans-\nmission amplitude as a function of the probe frequency,\nalso well fitted by 1 /∆ (fig.3(b)).\nThe dispersive measurement can be understood by\nconsideringthe susceptibility χ=χ′+iχ′′transverseto a\nlargestatic field ofa uniformly magnetized ferromagnetic\nsphere. In the small amplitude limit31,\nχ′=γM0(ωf−ω)\n(ωf−ω)2+ω2\nfα2(1)3\nχ′′=αγM0ωf\n(ωf−ω)2+ω2\nfα2(2)\nwhereωf=γH0is the resonant frequency, γis the gyro-\nmagnetic ratio and αis the Gilbert damping coefficient.\nFor large detuning ∆ = ωf−ωc≫ωfαthe susceptibility\nis dominated by the dispersive (real) part, and reduces\nto\nχ∼γM0\n∆. (3)\nThe change in susceptibility during magnetization pre-\ncession is taken in to account through the change in the\nprojection of the magnetization along the static field di-\nrection. The condition that |Mz|2+|My|2+|Mx|2=M2\n0\ngives|Mz| ≈M0/parenleftBig\n1−|Mx|2+|My|2\n2M2\n0/parenrightBig\nforMz≫Mx,My32.\nReplacing M0withMzin eqn.3 results in a reduction in\nthe dispersive susceptibility by a factor Mz/M0. Cavity\nperturbation theory33indicates that a small change in\nsome part of the material properties of the cavity results\ninaproportionalchangeinfrequency,inourcasetheonly\nmodification is that of the permeability µ=µ0(1+χ) of\nthe sphere34. The equilibrium cavity shift from its bare\nvalueωc0forχ= 0 is then proportional to the suscepti-\nbility given by eqn. 3. When we additionally drive FMR,\nthe reduction of this susceptibility by Mz/M0gives rise\nto the frequency shift ∆ ωc, which then alsofollows a 1 /∆\ndependence as observed in the experiment.\nFrom our measurement we obtain an estimate of the\nchange in Mzthrough comparison of the two frequency\nshifts discussed:\n∆ωc\nωc−ωc0≈∆χ\nχ≈∆Mz\nM0(4)\nThis results in the data plotted in fig.4, where it is seen\nthat the amplitude of the change in Mzis independent\nof detuning.\nFigure 5(a) shows the peak frequency shift as a func-\ntion of the power of the FMR drive tone. The shift is lin-\nearly proportional to the power, and therefore quadratic\nin the magnetic field amplitude. This is due to the fact\nthat we are sensitive to the square of the transversemag-\nnetizations, whicharethemselveslinearin theappliedr.f.\nfield as given by the susceptibility relation. Figure5(b)\nshows the dispersive measurement as a function of the\npower of the probe tone. Here there is no dependence of\nthe amplitude of the frequency shift over a wide range of\npowers, with the only change being in the signal to noise\nratio improving as the probe field is increased.\nIV. DISCUSSION\nRecent work on magnon-photon interactions13,16have\nanalyzed this system in terms of a fully quantum me-\nchanical Hamiltonian35. As such, it is useful to discuss\na description of our measurements based on this model./s48/s48/s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48\n/s32/s32\n/s32/s99/s45\n/s99/s48/s47/s50 /s40/s77/s72/s122/s41/s97/s41\n/s98/s41/s48/s53\n/s32/s32\n/s32/s99/s47/s50 /s32/s40/s107/s72/s122/s41\n/s32/s32/s40\n/s102/s32/s45/s32\n/s99/s41 /s32/s40/s71/s72/s122/s41\n/s99/s41\n/s99/s32/s32/s40/s97/s46/s117/s46/s41\n/s102/s32/s45/s32 /s32/s32 /s39\n/s32 /s39/s39\nFIG. 3. Dependence of measurements on detuning between\ncavity and FMR mode frequencies. a) change in cavity fre-\nquency due to FMR, taken from peak amplitude in phase\nchange plotted in fig. 2(a). b) change in cavity frequency\ndue to the equilibrium susceptibility of the sphere from tha t\nwhenχ= 0. The value of ωc0 is obtained from the offset\nin the 1/∆ fit. c) For reference, the calculated general form\nof dispersive and dissipative susceptibility of the sphere as a\nfunction of detuning from the FMR resonance, from eqn.1,2,\nfor low damping α.\n/s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49\n/s32/s32\n/s40\n/s102/s32/s45/s32\n/s99/s41 /s32/s40/s71/s72/s122/s41/s77\n/s122/s47/s77\n/s115/s32/s40/s37/s41\nFIG. 4. Percentage change in longitudinal magnetization Mz\nas a function of detuning.\nIn the dispersive limit ∆ ≫g, degenerate second order\nperturbation theory gives an effective Hamiltonian anal-\nogous to circuit QED36. In the rotating frame at the4\n/s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s46/s49/s49\n/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s48/s49/s50/s51/s52\n/s32/s32/s99/s47/s50 /s32/s40/s107/s72/s122/s41\n/s100/s114/s105/s118/s101/s32/s112/s111/s119/s101/s114/s32/s40/s100/s66/s109/s41/s97/s41\n/s53\n/s32/s32/s99/s47/s50 /s32/s40/s107/s72/s122/s41\n/s112/s114/s111/s98/s101/s32/s112/s111/s119/s101/s114/s32/s40/s100/s66/s109/s41/s98/s41\nFIG. 5. a) Dependence of the cavity frequency shift on the\nFMR driving microwave power at fixed static magnetic field\nµ0H0= 180mT. A linear fit to the data is shown. b) Power\ndependenceofthecavityfrequencyshift as afunctionofpro be\npower at the cavity resonance frequency for the same static\nfield.\ndrive frequency ωdwe obtain\nHeff= ∆fJz+¯h∆ca†a+g0(βJ++β∗J−)\n+¯h2g2\n0\n∆a†aJz+g2\n0\n∆(Jz+1\n¯h(J2\nx+J2\ny)) (5)\nwhere ∆ f=ωf−ωdis the drive detuning from the FMR\nmode, ∆ c=ωc−ωdis the drive detuning from the cavity\nandJzis the large angular momentum associated with\nthe magnetization Mz.J+andJ−are the correspond-\ning angular momentum raising and lowering operators\nandβis proportional to the amplitude of the microwave\ndrive at frequency ωd. Here we use the single spin cou-\npling rate g0related to the measured total coupling byg=g0√\nN. The first two terms are the bare uniform\nferromagnetic mode and cavity mode with lowering op-\neratora. The third represents the microwave driving of\nthe FMR mode with amplitude βand frequency ωd. The\nlast two terms are the ones that we are interested in: a\nmodification of the cavity frequency proportional to Jz\nand a magnon non-linearity respectively. It is clear that,\nas the cavity frequency shift term commutes with the\nFMR mode Hamiltonian, this measurement of Jzis in\nprinciple quantum non-demolition37and as in the classi-\ncal analysis depends inversely on the detuning.\nIn our measurements, the change in the number of\nmagnons38nfrom the equilibrium thermal occupation\n(Mz=M0−µBn) when the system is driven is n≈\n3×1015, where we have used an effective value for the\nnumber of spins at room temperature to calculate M039.\nWhile this magnon number is large, micro-fabricated on-\nchip devices, enabling substantial reduction in volume\nwhile maintaining strong coupling16could, at low tem-\nperatures, make dispersive measurements in the low n\nlimit and the prospect of the generation of magnon num-\nber states through measurement40a possibility. 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These resonances appear because the ferromagnets are half-\nmetallic, so that the Fermi level can be placed above the majority bu t below the minority\nspin band. In that case, the majority spins will be propagating, but the minority spins will\nbe evanescent. At low temperatures, the Fano resonances can b e exploited to implement a\ndigital binary switch that can be turned on or off with a very small gat e voltage swing of\nfew tens of µV, leading to extremely small dynamic power dissipation during switchin g. An\narray of 500,000 ×500,000 such transistors can detect ultrasmall changes in a magne tic field\nwith a sensitivity of 1 femto-Tesla/√\nHz, if each transistor is biased near a Fano resonance.\n∗Corresponding author. E-mail: marc.cahay@uc.edu\n11 Introduction\nDespite the fact that the first Spin Field Effect Transistor (SPINFE T) was proposed almost\ntwo decades ago [1], and numerous clones have appeared since then [2, 3, 4], no SPINFET\nhas ever been experimentally demonstrated. The primary obstacle to experimental demon-\nstration is the inability to achieve high spin injection efficiency at the int erface between the\nsource and channel, and high spin detection efficiency at the interfa ce between the channel\nand drain. The spin injection efficiency is critical in determining the tra nsistor performance.\nFor example, it can be easily shown that the maximum ratio of the on-c onductance to off-\nconductance of the transistor is [5]\nGon\nGoff=1+ζSζD\n1−ζsζD(1)\nwhereζSis the spin injection efficiency at the source/channel interface and ζDis the spin\ndetection efficiency at the drain/channel interface. Consequent ly, in order to achieve a on-\noff conductance ratio of 105, typical of modern transistors, the spin injection and detection\nefficiencies have to be as large as 99.9995%, which is an extremely tall o rder.\nThe maximum spin injection efficiency demonstrated to date is 90% at lo w temperatures\n[6]. With that value, the conductance on-off ratio is a mere 9.5 - a far c ry from 105. In\nother words, the conductance modulation is critically dependent on spin injection efficiency.\nTherefore, well-engineered SPINFETs should have excellent spin inj ection and detection\nefficiencies.\nHalf metallic ferromagnets [7, 8, 9], in which carriers at the Fermi leve l have only one\nspin (majority spin), are the optimum electrical spin injectors. The y are often invoked\nas the most promising route to achieving nearly 100% spin injection effi ciency. Here, we\nreport our investigation of Spin Field Effect Transistors with half met allic source and drain\ncontacts. An example could be an InAs channel with Lanthanum Str ontium Manganate\n(LSMO) contacts, which are known to have very high degree of spin polarization at low\ntemperatures and approximate ideal half metals. We restrict ours elves to a quantum wire\n2channel since it is known to produce maximum conductance modulatio n [1].\nWehave studied ballisticspintransportthroughthisdevice by solving thePauli equation,\nwhich yields the transmission amplitudes of both majority and minority spins in the source\ncontact. Assumptionofballistictransportisrealisticsincemobilityinq uantumwirechannels\ncan be reasonably high. We also neglect spin relaxation in the channel since the Elliott-Yafet\nmechanism [10] requires carrier scattering (which is assumed to be a bsent in ballsitic models)\nand the D’yakonov-Perel’ mechanism [11] is absent in a quantum wire c hannel where only\none subband is occupied. Finally, we neglect any scattering at the int erface between the\nferromagnet and semiconductor, which can cause spin relaxation.\nWith the above model, we have studied spin dependent transmission s pectra of electrons\nthrough the SPINFET. Our study revealed the existence of Fano r esonances in the trans-\nmission spectra since the Fermi level in the device can be located abo ve the majority spin\nband in the source (which contributes propagating electrons), bu t below the minority spin\nband (which is evanescent). It is well known that such a situation ca uses Fano resonances\n[12]. We found that the Fano resonances are narrow and well-resolv ed at sufficiently low\ntemperatures. We can therefore bias the transistor around suc h a resonance (by applying an\nappropriate dc gate voltage), so that a small change in the gate vo ltage will vary the electron\ninjection energy around the Fano resonance and switch the trans mission through the device\nfrom maximum to minimum. This is the basis of a digital ‘switch’ which can b e switched\non and off with a very small gate voltage swing, resulting in extremely s mall dynamic power\ndissipation during switching.\nWe also considered the situation when the transistor is placed in a sta tic magnetic field\ndirected along the channel. As long as the transistor is biased near a Fano resonance, a small\nchange in the magnetic field can change the transmission through th e device (and therefore\nthe channel conductance) from maximum to nearly zero. Thus, th is device is capable of\ndetecting minute changes in a magnetic field. We show that an approp riately designed\nsystem can detect magnetic field changes with a sensitivity of ∼1 femtoTesla/√\nHz.\n3This paper is organized as follows. In the next Section, we present t he theory. In Section\n3, we present results and in Section 4, we present calculations of th e sensitivity of a magnetic\nfield sensor based on this principle. The conclusions are presented in Section 5.\n2 Theory\nConsider a Spin Field Effect Transistor (Spin FET) with half-metallic fer romagnetic source\nand drain contacts, as shown in Fig. 1(a). The channel is a quantum wire in which only the\nlowest subband is occupied by carriers. The conduction band discon tinuity at the heteroin-\nterface between the channel and the gate insulator results in an e ffective electric field along\nthe y-axis, which induces a Rashba spin orbit interaction in the chann el. The ferromagnetic\ncontacts induce a static magnetic field in the x-direction. This field ca n also be applied with\nexternal sources.\nThe equilibrium energy band diagram along the channel (x-direction) is shown in Fig.\n1(b). We adopt the Stoner model and assume that the majority an d minority spin bands in\nthe ferromagnets are exchange split by an amount ∆. The Fermi lev el is below the bottom\nof the minority spin band so that the spin polarization of carriers at t he Fermi energy in the\nferromagnets is 100% (hence “half metallic”). The majority spins fr om the ferromagnet will\nbepropagating and the minority spins will be evanescent . For the sake of simplicity, we will\nneglect space charge effects and assume that the conduction ban d edge in the channel is flat\nandinvariantinthex-coordinate. Furthermore,thecontactpot entialsatthecontact/channel\ninterface will be represented by two delta barriers of height Γ.\nIf we assume that the confinement potential in the z-direction is pa rabolic and given by\nV(z) = (1/2)m∗ω2\n0z2, then the approximate energy dispersion relations of the lowest sp in\nsplit subbands in the channel (derived using perturbation theory) are given by [13, 14, 15]\nE1(kx) =1\n2¯hω+∆Ec+¯h2k2\nx\n2m∗−/radicalBigg\n(ηkx)2+/parenleftbiggg∗µBB\n2/parenrightbigg2\n, (2)\nE2(kx) =1\n2¯hω+∆Ec+¯h2k2\nx\n2m∗+/radicalBigg\n(ηkx)2+/parenleftbiggg∗µBB\n2/parenrightbigg2\n, (3)\n4where the subscripts ‘1’ and ‘2’ refer to the lower and upper subba nds,ω=/radicalBig\nω2\n0+ω2\nc,\nωc=eB/m∗,g∗istheeffective Land´ eg-factorinthechannel, kxisthewavevector component\nin the x-direction, m∗is the effective mass of carriers, and ηis the strength of the Rashba\ninteraction. The quantity ∆ Ecis the potential barrier between the ferromagnets and the\nchannel, and includes the confinement energy due to confinement in the y-direction (see Fig.\n1(b)).\nIf the axial magnetic field Bis below a critical field Bc(= 4ηm∗/(¯h2gµB)), then the\ndispersion relation E1(kx) will have a camel-back shape with a local maximum at kx= 0\nand two local minima at kx=±(/radicalBig\n4η2(m∗)2/¯h4−(g∗µBB)2/4)/η[15]. Above Bc, the shape\nofE1(kx) is approximately parabolic with a global minimum at kx= 0. The dispersion\nrelationE2(kx), on the other hand, always has an approximately parabolic shape w ith a\nglobal minimum at kx= 0.\nThe eigenspinors in the two subbands E1(kx) andE2(kx) are [15]\nΨ1(B,kx) =/bracketleftBigg\ncos(θkx)\nsin(θkx)/bracketrightBigg\nΨ2(B,kx) =/bracketleftBigg\nsin(θkx)\n−cos(θkx)/bracketrightBigg\n(4)\nwhereθkx= -(1/2)arctan[(g∗µBB)/(2ηkx)].\nThefactthattheeigenspinorsarewavevector ( kx)dependent tellsusthatneithersubband\nhas a fixed spin quantization axis (the spin quantization axis changes with the wavevector).\nAt any arbitrary kx, each of the two spin eigenspinors in the channel will be mutually\northogonal, but each will be a mixture (or superposition) of both ma jority and minority\nspins in the ferromagnetic contacts. Moreover, at any fixed incide nt energy, the eigenspinors\nin the two spin split subbands E1andE2will not be orthogonal, so that there will be\ncoupling between them. As a result, majority spins injected from th e ferromagnet (with any\narbitrary injection energy) will be mixed with minority spins in the chan nel. This is termed\n“spin mixing”. Note that spin mixing occurs because of the simultaneous presence of the\naxial magnetic field and spin orbit interaction. If the former were ab sent,θkx= 0, and the\n5eigenspinors will be z-polarized states/bracketleftBigg\n1\n0/bracketrightBigg\nand/bracketleftBigg\n0\n1/bracketrightBigg\nwhich are wavevector independent,\nwhile if the latter were absent, θkx=π/2 and the eigenspinors will be x-polarized states\n(1/√\n2)/bracketleftBigg\n1\n1/bracketrightBigg\nand (1/√\n2)/bracketleftBigg\n1\n−1/bracketrightBigg\nwhich are also wavevector independent. In both of the\nabove cases, the eigenspinors in the two subbands will be always ort hogonal (at any energy)\nand there will be no mixing or coupling between them. Therefore, simu ltaneous presence of\nthe axial magnetic field and spin orbit interaction are required to indu ce spin mixing.\nIn order to study ballistic transport in the Spin FET, we used the mod el developed in\nrefs. [13, 14] to calculate the (spin-dependent) quantum mechan ical transmission amplitude\ntof an electron through the channel (in the presence of “spin mixing ”) as function of the\ninjectionenergy E. Forthese calculations, we have used theparameters listed inTable I. The\ntransmission amplitude t(E) also depends on the axial magnetic field Band the exchange\nsplitting ∆ in the contacts. In [13, 14], we assumed that the bottom s of both majority and\nminority spin bands in the ferromagnet are below the Fermi level, so t hat both spins are\npropagating modes. In the present case, we modified the formalism to account for the fact\nthat minority spins are evanescent.\nFrom the calculated transmission amplitude as a function of energy, we calculate the\nlinear response conductance using the finite temperature Landau er formula:\nG↑(∆Ec,B,∆) = ( e2/4hkT)/integraldisplay∞\n∆EcdE|t↑(E,B,∆)|2sech2/parenleftbiggE−EF\n2kT/parenrightbigg\nG↓(∆Ec,B,∆) = ( e2/4hkT)/integraldisplay∞\n∆EcdE|t↓(E,B,∆)|2sech2/parenleftbiggE−EF\n2kT/parenrightbigg\nGtotal(∆Ec,B,∆) =G↑(∆Ec,B,∆)+G↓(∆Ec,B,∆) (5)\nwhere↑and↓refer to majority and minority spins in the contact, respectively. T he last\nequality is a consequence of the fact that in the ferromagnetic con tacts, the two eigen-\nspinors are orthogonal at any given energy. However, since the m inority spin is evanescent,\nG↓(∆Ec,B,∆)≡0, andGtotal(∆Ec,B,∆) =G↑(∆Ec,B,∆). We emphasize that although\nthe evanescent modes do not directly contribute to the total con ductance, they nonetheless\nhave anindirectinfluence because they renormalize the transmission probability of t he prop-\n6agating modes [16, 17], and therefore affect G↑(∆Ec,B,∆) and ultimately Gtotal(∆Ec,B,∆).\nBecause one incident mode from the contact (majority spin) is prop agating, while the\nother (minority spin) is evanescent, the transmission spectrum |t↑↓|versusEwill contain\nresonance/antiresonance pairs closely spaced in energy, a featu re known as Fano resonances\n[12, 18, 19, 20]. From Equation (5), it is obvious that these Fano res onances will show up\nin the plot of Gtotalversus ∆ Ec, at sufficiently low temperatures, for a given value of Band\n∆. Note that we can vary ∆ Ecwith the gate voltage; therefore, a plot of Gtotalas function\nof gate voltage will show Fano resonances at low enough temperatu res.\n3 Results\nIn Fig. 2, we plot Gtotalversus ∆ Ecat a temperature of 0 K, for different values of the\nexchange splitting energy ∆ in the contacts. Since ∆ Echas an approximately linear depen-\ndence on the gate voltage Vg(over small ranges of gate voltage variation), this plot can also\nbe viewed as a plot of GtotalversusVg. The magnetic field Balong the channel is assumed\nto be 0.6 Tesla which is above the critical field Bc. Although ηshould vary with Vg, this\nvariation is negligible over the gate voltage range we consider, since ηhas a weak dependence\nonVg[21]. Therefore, we assume that ηis constant over the entire range of ∆ Ec(orVg) and\nhas the value given in Table I.\nIn Fig. 2, we clearly see the Fano resonance-antiresonance pairs. Additionally, there are\nisolated resonances where the conductance reaches the maximum value ofe2/h. These are\ndue to Ramsauer resonances discussed inref. [13]. The locationsof the Ramsauer resonances\n(on the ∆ Ecaxis) are fairly insensitive to the exchange splitting energy ∆, but th e locations\nof the Fano resonance-antiresonance pairs are strongly depend ent on ∆. This is due to\nthe fact that the decay length of the evanescent minority spin ban d in the contact has a\nstrong dependence on ∆. The value of ∆ affects the amount of coup ling/mixing between\nthe two propagating modes in the semiconductor channel via the bo undary conditions at the\ncontacts. That, in turn, affect the Fano resonances, but not th e Ramsauer resonances.\n7The energy separation between the Fermi level and the subband b ottoms depends on\nthe axial magnetic field B. Therefore, we expect the magnetic field to influence the Fano\nresonances, and it does. This is illustrated in Fig. 3 which is a plot of Gtotalversus ∆ Ecfor\ndifferent values of the axial magnetic field Bfor a fixed value of ∆ = 6 eV. For the range\nof ∆Ecconsidered here, there are two Ramsauer resonances. In additio n, there is a Fano\nresonance preceding each Ramsauer resonance. The Ramsauer r esonances correspond to\nperfect transmission of the majority spin band, while each Fano res onance appears in place\nof a second Ramsauer resonance that would have appeared owing t o perfect transmission of\nthe minority spin band, were it propagating instead of being evanesc ent.\nAround a Fano resonance, a very small change in ∆ Ecwill switch the conductance from\nthe maximum value of e2/hto zero. Therefore, we can realize a binary switch by biasing\nthe device close to a Fano resonance. This can be achieved by applyin g an appropriate fixed\ndc voltage on the gate. Around the leftmost Fano resonance, the change in ∆ Ec(δ(∆Ec))\nrequired to switch thedevice onor off, is ∼20µeV and is nearly independent of the strength\nof the magnetic field B. Sinceδ(∆Ec) =qVg−qVins(whereVgis the gate voltage, qis the\nelectronic charge and Vinsis the gate voltage dropped across the gate insulator), the gate\nvoltage swing Vtrequired to switch the device from on-to-off is 20 µV +Vins. For acceptable\nerror rate, we want this voltage to exceed the thermal noise volta ge on the gate capacitor,\nwhich isUth=/radicalBig\nkT/Cg, whereCgisthe gatecapacitance and Tisthe operatingtemperature\n[22]. We will later show that the Fano resonances begin to wash out at temperatures above\n0.1 K, so that T≤0.1 K. Assuming Cg= 1 fF (including interconnects), Uth≤37µV.\nTherefore, we must choose the gate insulator such that Vins≥17µV.\nIf the gate capacitance is 1 fF (including interconnects), then the energy dissipated in\nswitching this transistor is (1/2) CV2\nt= 7×10−25Joules≈kTln2 (at the temperature of\n0.1 K). If switched adiabatically, the energy dissipation can be much le ss [23]. Consequently,\nthe binary switch that we propose here is an extremely low power switch.\nNear a Ramsauer resonance (perfect transmission of the majorit y spin band), the con-\n8ductance versus ∆ Eccurve is fairly insensitive to the axial magnetic field B. However, near\na Fano resonance, the conductance curve is extremely sensitive t oB. For instance, at zero\ntemperature, when the gate is biased close to a Fano resonance (a t a voltage such that ∆ Ec\n= 4.1922 eV), there is a 100% change in the conductance (from a max imum value of e2/h\nto zero) when the magnetic field Bis changed from 0.4 T to 0.372 T, a mere change of 28\nmT. Therefore, at very low temperatures, this device can operat e as a magnetic field sensor.\nWe show below that it can actually detect fields with a sensitivity of ∼1 femto-Tesla/√\nHz,\nusing standard analysis as in refs. [24, 25].\n4 Magnetic field sensor\nConsider a single device. The noise current has two components - th ermal noise and shot\nnoise. The thermal noise current is given by Ithermal\nn=√4kTG∆fζ, whereGis the source-\nto-drain conductance, ∆ fis the frequency bandwidth, and ζis the noise suppression fac-\ntor due to carrier confinement in quantum wires [26]. The shot noise c urrent is Ishot\nn=\n/radicalBig\n(2/3)eGV∆f[27]. Therefore the total noise current in a transistor is In=C√\nG, where\nC=/radicalBig\n[4kTζ2+(2/3)eV]∆f.\nWhenusedasamagneticfieldsensor, thechangeinthecurrentthr oughasingletransistor\ninamagneticfield Histhesignalcurrent Isandisgivenby Is=SHwhereSisthesensitivity.\nTo keep the analysis tractable, we will assume that current change s linearly in a magnetic\nfield, so that Sis independent of H. Therefore, the signal to noise ratio (SNR) for a single\ntransistor is\nSNR=SH\nIthermaln+Ishotn=SH\nC√\nG(6)\nThe conductance of Ntransistors operating in parallel is NG. Therefore, the noise current\nofNtransistors in parallel is√\nNIn, whereas the signal current is NIs. Thus, the SNR\nincreases as√\nN. This allows us to detect very small magnetic fields by using a large\nnumber of transistors in parallel, each acting as a sensor.\nFor a change of magnetic field of 28 mTesla ≡280 Oe, the conductance of a transistor\n9changes by e2/h. If the source-to-drain voltage is 100 mV, then the drain current changes by\n39µA. Therefore S= 0.14µA/Oe. Fora field of 1 femto-Tesla = 10−11Oe, thesignal current\nIsin a single transistor is 1.4 ×10−18Amperes. The noise current is dominated by shot\nnoise at the operating temperature of 0.1 K and is given by 6.4 ×10−13Amperes/√\nHz.\nTherefore, the SNR for a single transistor is 2.2 ×10−6/√\nHz. The SNR for 2.5 ×1011\ntransistor sensors in parallel will be√\n2.5×1011times larger and exceed unity/√\nHz(0 db),\nwhich makes the signal measurable against the background of noise . Therefore with an array\nof 500,000 ×500,000 transistors, we should be able to detect a magnetic field of 1 femto-\nTesla with a bandwidth of 1 Hz. With a transistor density of 2.5 ×109/cm2, this sensor\ncan be implemented in a 10 cm ×10 cm chip. The dynamic power dissipated to detect 1\nfemto-Tesla (in a bandwidth of 1 Hz) will be only 35 nWatts.\nIn Fig. 4, we show that the Fano resonances wash out very quickly w ith increasing\ntemperature. The Ramsauer resonances are more robust agains t temperature. The Fano\nresonances are essentially indiscernible at temperatures above 0.3 K, so that we should\nrestrict all device operation to below 0.1 K.\n5 Conclusion\nIn conclusion, we have shown that a Spin FET with half metallic contacts has both Ram-\nsauer and Fano resonances in the transmission spectrum. In ref. [13], we found only the\nRamsauer resonances, and not the Fano resonances, since we did not consider half metallic\nferromagnetic contacts, so that both majority and minority spins in the contact were propa-\ngating modes. Here, we have considered the case where the major ity spins are propagating,\nbut the minority spins are evanescent. This causes the Fano reson ances. The locations of the\nFano resonances are very sensitive to an axial magnetic field and th e exchange splitting in\nthe contacts. We can bias the transistor near a Fano resonance u sing a dc gate voltage and,\nat low temperatures, realize very low power binary switches (dissipa ting∼kTln2 energy\nper switching event), as well as sensitive magnetic field detectors t hat can detect ultrasmall\n10fields with a sensitivity ∼1 femto-Tesla/√\nHz.\n11References\n[1] S. Datta and B. Das, Appl. Phys. Lett., 56, 665 (1990).\n[2] J. Schliemann, J. C. Egues and D. Loss, Phys. Rev. Lett., 90, 146801 (2003).\n[3] X. Cartoix´ a, D. Z-Y Ting and Y-C Chang, Appl. Phys. Lett., 83, 1462 (2003).\n[4] K. C. Hall and M. E. Flatt´ e, Appl. Phys. Lett., 88, 162503 (2006).\n[5] A. R. Trivedi, S. Bandyopadhyay and M. Cahay (unpublished).\n[6] R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waa g and L. W.\nMolenkamp, Nature (London), 402, 787 (1999).\n[7] R. A. de Groot, F. M. Mueller, P. G. van Engen and K. H. J. Buscho w, Phys. Rev.\nLett.,50, 2024 (1983).\n[8] W. E. Pickett and J. S. Moodera, Phys. Today, 54, 39 (2001).\n[9] S. Sanvito and N. A. Hill, Phys. Rev. Lett., 87, 267202 (2001).\n[10] R. J. Elliott, Phys. Rev., 96, 266 (1954).\n[11] M. I. D’yakonov and V. I. Perel’, Sov. Phys. Solid State ,13, 3023 (1972).\n[12] U. Fano, Phys. Rev. 124, 1866 (1961).\n[13] M. Cahay and S. Bandyopadhyay, Phys. Rev. B, 68, 115316 (2003).\n[14] M. Cahay and S. Bandyopadhyay, Phys. Rev. B, 69, 045303 (2004).\n[15] S. Bandyopadhyay, S. Pramanik and M. Cahay, Superlat. Micro struct.,35, 67 (2004).\n[16] P. F. Bagwell, Phys. Rev. B, 41, 10354 (1990).\n[17] M. Cahay, S. Bandyopadhyay, M. A. Osman and H. L. Grubin, Su rface Science, 228,\n301 (1990).\n12[18] P.F. Bagwell and R.K. Lake, Phys. Rev. B, 46, 15329 (1992).\n[19] C. Kunze and P.F. Bagwell, Physical Review B, 51, 13410, (1995).\n[20] L. Serra and D. S´ anchez, www.arXiv.org/cond-mat/0610147 .\n[21] J. Nitta, T. Akazaki, H. Takayanagi and T. Enoki, Phys. Rev. L ett.,78, 1335 (1997).\n[22] Laszlo B. Kish, Phys. Lett. A, 305, 144 (2002).\n[23] R. K. Cavin, V. V. Zhirnov, J. A. Hutchby and G. I. Bourianoff, F luctuations and Noise\nLetters,5, C29 (2005).\n[24] M. Tondra, J. M. Daughton, D. Wang, R. S. Beech, A. Fink and J . A. Taylor, J. Appl.\nPhys.,83, 6688 (1998).\n[25] S. Bandyopadhyay and M. Cahay, Physica E, 27, 98 (2005).\n[26] A. Svizhenko, S. Bandyopadhyay and M. A. Stroscio, J. Phys.: Condens. Matt., 11,\n3697 (1999).\n[27] C. W. J. Beenakker and M. B¨ uttiker, Phys. Rev. B, 46, 1889 (1992).\n13Table I: Parameters of the spin interferometer\nFermi Energy EF(eV) 4.2\nRashba spin-orbit coupling constant αR(10−11eVcm) 1.\nLande Factor g∗-14.9\nEffective mass mf∗in Fe contact ( m0) 1.\nEffective mass ms∗in InAs channel ( m0) 0.023\nLength of the channel( µm) 0.15\nStrength of delta scatterer at the contact/channel interface (ev˚A) 2.0\n14Figure 1 : (a) Structure of a Spin Field Effect Transistor with half metallic (HM) source\nand drain contacts. (b) Energy band diagram along the channel: Als o shown as dashed lines\nare the resonant energy states above ∆ Ec. The barriers at the ferromagnet/semiconductor\ninterface are modeled as simple one-dimensional delta-potentials.\nFigure 2 : Zero temperature conductance Gtotal(=G↑) as a function of ∆ Ecfor various\nvalues of the exchange energy ∆ in the half-metallic contacts. The a xial magnetic field B=\n0.6 Tesla. Other parameters used to obtain these curves are listed in Table I.\nFigure 3 : Zero temperature conductance Gtotal(=G↑) as a function of ∆ Ecfor various\nvalues of the axial magnetic field B. The exchange energy ∆ in the half-metallic contacts is\nassumed to be 6 eV. The Fano and Ramsauer resonances are indicat ed.\nFigure 4 : (a) Temperature dependence of the conductance modulation of the Spin FET\nwhen the exchange energy ∆ in the half-metallic contacts is 6 eV. The Fano resonances\nare washed out much faster than the Ramsauer resonances as th e temperature rises. (b)\nTemperature dependence of the leftmost Fano resonance shown in higher resolution. From\ntop to bottom, the curves correspond to a temperature of 0, 0.0 5, 0.1, 0.15, 0.2, 0.25, and\n0.3 K, respectively.\n15HM HM\nSemiconductorInsulatorGate\nSource Drain\nx\nyz\nFigure 1(a)\n16∆)(xδΓ\n∆FE) ( Lx− Γδ\nFE\ncE∆\nLx= 0=x xI II III \nFigure 1(b)\n174191.5 41924192.5 41934193.5 41944194.5 419500.10.20.30.40.50.60.70.80.91\n∆E\nc (meV)Conductance(e2/h)\n  \n6eV\n4.5eV5eV\nFigure 2\n184191.5 41924192.5 41934193.5 41944194.5 419500.10.20.30.40.50.60.70.80.91\n∆E\nc (meV)Conductance(e2/h)\n  \nB=0.4T\n0.2T0.05T\nFigure 3\n194191.5 41924192.5 41934193.5 41944194.5 419500.10.20.30.40.50.60.70.80.91\n∆E\nc (meV)Conductance(e2/h)\n2K1K0K\nFigure 4(a)\n204191.85 4191.9 4191.95 4192 4192.0500.10.20.30.40.50.60.70.80.91\n∆Ec (meV)Conductance(e2/h)\nFigure 4(b)\n21"    },    {        "title": "1108.1286v1.Thermoelectric_detection_of_ferromagnetic_resonance_of_a_nanoscale_ferromagnet.pdf",        "content": "Thermoelectric detection of ferromagnetic resonance of a nanoscale ferromagnet\nF. L. Bakker,\u0003J. Flipse, A. Slachter, D. Wagenaar, and B. J. van Wees\nPhysics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, The Netherlands\n(Dated: August 10, 2021)\nWe present thermoelectric measurements of the heat dissipated due to ferromagnetic resonance of\na Permalloy strip. A microwave magnetic \feld, produced by an on-chip coplanar strip waveguide,\nis used to drive the magnetization precession. The generated heat is detected via Seebeck mea-\nsurements on a thermocouple connected to the ferromagnet. The observed resonance peak shape is\nin agreement with the Landau-Lifshitz-Gilbert equation and is compared with thermoelectric \fnite\nelement modeling. Unlike other methods, this technique is not restricted to electrically conductive\nmedia and is therefore also applicable to for instance ferromagnetic insulators.\nPACS numbers: 76.50.+g, 75.78.-n, 72.15.Jf, 85.80.Fi\nThermal e\u000bects in ferromagnetic materials are sub-\nject to extensive research since the discovery of the\nspin-Seebeck e\u000bect [1{3]. Recently, spin dynamics and\n(spin-) caloritronics, two popular branches of spintron-\nics, started to come together as spin pumping induced\nby spin dynamics has been proposed as the origin of the\nspin-Seebeck e\u000bect [4, 5]. Magnetization dynamics has\nbeen studied thoroughly in magnetic systems as it is an\nimportant mechanism for future spintronic applications,\ne.g. for microwave generators [6, 7] and spin sources via\nspin pumping [8, 9]. However, dissipation mechanisms\nthat accompany magnetization dynamics, and cause lo-\ncal heating, are still not fully understood [10, 11].\nHere, we focus on a new aspect, the coupling between\nmagnetization dynamics and the generation of heat. We\ndeduce from thermoelectric measurements on a Permal-\nloy (Py) island the heat dissipation during ferromagnetic\nresonance. This on-chip detection technique, based on\nthe Seebeck e\u000bect, o\u000bers a novel method for characteriz-\ning ferromagnetic resonance and hence is distinctly di\u000ber-\nent from other techniques, such as scanning thermal mi-\ncroscopy [12, 13]. Due to the thermal detection, electrical\ncontact to the ferromagnet is in principle not required.\nHence, this method allows for FMR measurements on\nnon-conductive materials like ferromagnetic insulators.\nWhen a ferromagnet is brought into resonance, energy\nis absorbed from the applied microwave \feld. This en-\nergy causes the magnetization ~Mto precess around an\ne\u000bective \feld ~Hand the motion is well described by\nthe Landau-Lifshitz-Gilbert (LLG) equation d~M=dt =\n\u0000\r~M\u0002~H+(\u000b=M s)~M\u0002d~M=dt with\r= 176 GHz/T the\ngyromagnetic ratio. The last term in the LLG equation\ndescribes the damping of the magnetization towards the\ndirection of the e\u000bective \feld ~H, using the phenomeno-\nlogical damping parameter \u000b. This process is purely\ndissipative and converts magnetostatic energy into heat.\nDuring ferromagnetic resonance (FMR) this continuous\ndissipation leads to heating of the ferromagnetic mate-\nrial.\nIn this experiment, we measured the temperature of a\nferromagnet while subject to a microwave magnetic \feld.A ferromagnetic strip is placed close to the shortened\nend of a coplanar strip waveguide (CSW) as shown in\nFig. 1. Microwave power is applied to the CSW, leading\nto an out of plane rf magnetic \feld. A static magnetic\n\feldh0is applied along the easy axis of the magnet. In\naddition, a thermocouple consisting of a NiCu and Pt\nwire is connected to the ferromagnet by a Au bridge. In\nthis way, the temperature can be measured by making\nuse of the Seebeck e\u000bect. The Seebeck e\u000bect describes\nthe generation of a voltage due to a temperature gradi-\nent,rV=\u0000SrT, withSthe material dependent See-\nbeck coe\u000ecient. The voltage that develops across the\nPt wire is di\u000berent than the voltage that develops across\nthe NiCu wire, leading to a nonzero voltage between the\ntwo wires. This thermovoltage scales with the Seebeck\ncoe\u000ecients ( SPt\u0000SNiCu) and the temperature di\u000berence\n(T1\u0000T0). Note that NiCu is chosen because of its rel-\natively high Seebeck coe\u000ecient ( SNiCu = -40\u0016V=K and\nSPt= -5\u0016V=K ). In the following, we calculate the tem-\nT1VPt\nNiCu\nT0\nCSW\nIrfhrf\nh0Pyx\ny\nFIG. 1. (color online) Concept of the thermoelectric detec-\ntion of ferromagnetic resonance. A coplanar strip waveguide\ngenerates a microwave magnetic \feld hrfin the ^zdirection\nacting on a small ferromagnetic strip. A static magnetic \feld\nh0is applied along the ^ xaxis. When the strip is brought into\nresonance, it absorbs energy from the \feld which is dissipated\nas heat. The dissipation is detected by a Pt-NiCu thermocou-\nple which probes the temperature T1of the ferromagnet with\nrespect to a reference temperature T0via the Seebeck e\u000bect.arXiv:1108.1286v1  [cond-mat.mes-hall]  5 Aug 20112\nperature rise during FMR from the dissipated power.\nThe energy of a ferromagnetic particle in a magnetic\n\feld, the Zeeman energy, is given by:\nE=\u0000Z\nV~M\u0001~BdV (1)\nwith~Mthe magnetization, ~B=\u00160\u0010\n~Hext+~HD=2\u0011\nthe\nsum of the externally applied magnetic \feld and the de-\nmagnetizing \feld and Vthe volume of the particle [14].\nHere,~HDis divided by two to compensate for counting\neach volume twice in the integral. For this experiment, a\nstatic magnetic \feld, h0, is applied in the ^ xdirection and\na driving rf \feld, hrfcos!t, in the ^zdirection, making\n~B= (h0\u0000Nxmx=2;\u0000Nymy=2;hrfcos!t\u0000Nzmz=2)\nwithNx,NyandNzthe demagnetization factors. The\ndissipation energy can now be calculated from the time\nderivative of E, assuming a uniform ~Mand~B:\ndE\ndt=\u0000V \nd~M\ndt\u0001~B+~M\u0001d~B\ndt!\n(2)\nwhere the \frst part of Eq. 2 expresses the dissipation due\nto the magnetization motion and the second part the en-\nergy absorbed from the microwave \feld. In equilibrium,\nthe absorption of energy from the microwave \feld equals\nthe dissipation,hdE=dti= 0, leading to heating of the\nferromagnet. In order to \fnd an expression for the dis-\nsipated power, we use a procedure similar to Ref. [15]\nwhere the magnetization dynamics is described by the\nlinearized LLG equation. We assume that for small angle\nprecessional motion dmx=dt= 0 such that mxis constant\nand the solution to the Landau-Lifshitz-Gilbert (LLG)\nequation can be written in terms of the sum of in-phase\nand out-of-phase susceptibilities. The components my\nandmzare now de\fned as my=\u001f0\ny!1cos!t+\u001f00\ny!1sin!t\nandmz=\u001f0\nz!1cos!t+\u001f00\nz!1sin!twith:\n\u001f0\ny=\u000b!2(!y+!z)\n(!2\u0000!y!z)2+ (\u000b!)2(!y+!z)2\n\u001f00\ny=!(!2\u0000!y!z)\n(!2\u0000!y!z)2+ (\u000b!)2(!y+!z)2(3)\nHere,!is the frequency of the driving rf \feld, !y=\n\r(h0\u0000(Nx\u0000Ny)ms),!z=\r(h0\u0000(Nx\u0000Nz)ms),\n!1=\rhrfandmsthe saturation magnetization. The\n\u001f0\nzand\u001f00\nzare related via \u001f0\nz=\u000b\u001f0\ny\u0000!y=!\u001f00\nyand\n\u001f00\nz=\u000b\u001f00\ny+!y=!\u001f0\ny, respectively. With these expres-\nsions formx,my,mzandBx,By,Bzone can easily \fnd\nits time derivatives and calculate the relevant dot prod-\nuct of Eq. 2. We are not interested in high frequency\nvariations in the dissipated power and hence, average out\nall contributions that vary with a frequency !or 2!and\n\fnd:\n\u001cdE\ndt\u001d\ndissipation=\u001f00\nz!2\n1!V\n2\r(4)\n1 /uni03BCmIrf\n1 2NiCu PtFMI+V+V-I-Irfa) b)FIG. 2. (color online) a) Scanning electron microscope (SEM)\nimage of the thermoelectric FMR device. b) Image of a device\nwith four contacts for dc AMR detection of FMR.\nFrom this expression we can deduce that the resonance\npeak shape of the dissipated power is determined by\n\u001f00\nzand scales with the applied microwave \feld and fre-\nquency. In order to convert this power into a tempera-\nture rise, we make use of 3D \fnite element thermoelectric\nmodeling. For details about the modeling we refer to ear-\nlier publications [16, 17].\nThe samples are fabricated using a three-step electron\nbeam lithography process on top of a thermally oxidized\nSi substrate. A scanning electron microscope (SEM) im-\nage of the investigated devices is shown in Fig. 2. The\ndevices consist of a 50nm thick Py strip (2 \u0016m\u0002400 nm)\nclose to a 100 nm thick Au coplanar strip waveguide. The\nCSW is made using an optical lithography process. In the\nthermoelectric device (Fig. 2a), there are two 40 nm thick\ncontacts (Pt and NiCu) forming a thermocouple. The\nPy island is connected to the thermocouple by a highly\nthermal conductive Au contact (thickness: 120 nm). The\nother side of the thermocouple is connected by 120 nm\nthick Au contacts to the bonding pads. In the case of the\nanisotropic magnetoresistance (AMR) device (Fig. 2b),\nfour 120 nm thick Au contacts directly connect to the Py\nstrip. The NiCu is deposited by DC sputtering to pre-\nserve the original alloy composition (45% Ni, 55% Cu).\nTo avoid lift-o\u000b problems a double-layer resist technique\nwith a large undercut (PMMA-MA and PMMA 950K) is\nused. The Au, Pt and Py are deposited using an e-beam\nevaporator (base pressure 1 \u000210\u00007mbar) and a single\nlayer resist (PMMA 950K). Prior to the Au deposition,\nthe NiCu, Pt and Py surfaces are cleaned with Ar ion\nmilling.\nFor the measurements, we have used a frequency mod-\nulation method to obtain a better signal to noise ratio\nand to remove background voltages due to heating of\nthe CSW short. The microwave \feld frequency is al-\nternated between two di\u000berent values with a separation\nof 5 GHz. A lock-in ampli\fer, tuned to the same fre-\nquency (17 Hz), measures the di\u000berence in dc voltage\nacross contacts 1 and 2 (Fig. 2a) between the two fre-\nquencies (V=Vf=high\u0000Vf=low ). Because of the large3\nseparation between Vf=low andVf=high , they can not both\nful\fll the resonance condition at a speci\fc magnetic \feld.\nWith this method, one e\u000bectively measures the di\u000berence\nin the Seebeck or AMR voltage when the ferromagnet is\nin- and o\u000b-resonance. All measurements were performed\nat room temperature.\nFig. 3a shows the measured Seebeck voltage as a func-\ntion of magnetic \feld for di\u000berent rf \feld frequencies (10\n- 20 GHz) for 12 dBm rf power. The position of the peaks\nand dips correspond to the resonance \feld for flowand\nfhigh, respectively. We have plotted the peak position\nas a function of the applied rf frequency in Fig. 3b and\nfound peak heights ranging from 46 nV at 10 GHz to\n105 nV at 17 GHz. For a uniform precessional mode, the\nresonance \feld is related to !by the Kittel equation [18]:\n!2=\r2(h0\u0000(Nx\u0000Ny)ms)(h0\u0000(Nx\u0000Nz)ms) (5)\nThe line corresponds to a \ft of Eq. 5 and the almost\nperfect \ft con\frms the uniform precessional motion. We\nobtained the following \ftting parameters: Nx= 0:01,\nNy= 0:09,Nz= 0:90 and\u00160ms= 1:11 T. These param-\neters have been used to calculate the dissipated power of\nEq. 4 for di\u000berent frequencies. However, in order to do\nthis accurately one \frst need to determine the magnitude\nof the rf magnetic \feld experimentally.\nTo obtain the correct experimental value for the mag-\nnitude of the rf magnetic \feld, we have measured the dc\nanisotropic magnetoresistace (AMR) in a dedicated de-\nvice with a four-terminal geometry (shown in Fig. 2b).\nThe AMR e\u000bect describes the dependence of the resis-\ntance on the angle \u0012between the current ~Iand the di-\nrection of magnetization ~MbyR=R0\u0000\u0001Rsin2\u0012, where\nR0is the resistance of the strip when ~Iand~Mare par-\nallel, and \u0001 Rthe di\u000berence in resistance between the\nparallel and perpendicular alignment of ~Iand~M. A mea-\nsurement of Ras a function of a perpendicular applied\nmagnetic \feld is plotted in Fig. 4c. From this measure-\nment, we determined the magnitude of the AMR e\u000bect\nand found \u0001 R=R 0= 1:5 %. For a steady resonant pre-\ncession, the average cone angle \u0012cof the precession can\nnow be extracted from the observed AMR voltage.\nFig. 4a displays the AMR voltage versus magnetic \feld\nfor di\u000berent microwave \feld frequencies. For this mea-\nsurement, we have used a dc current Idcof 300\u0016A.\nThe obtained voltage now corresponds to the dc cur-\nrent multiplied with the resistance change, being V=\nIdc\u0001Rsin2\u0012c, and the precession angle can be extracted.\nUsing Eq. 3 and an expression for the average cone angle:\n\n\u00122\nc\u000b\n=!2\n1(\u001f02\ny+\u001f002\ny+\u001f02\nz+\u001f002\nz)=2 (6)\none can deduce hrf=!1=\rfrom a \ft of the measured\npeak height, and the result is plotted in Fig. 4b for a\nmicrowave power of 12 dBm. The \feld strength is found\nto decrease twofold when the frequency is increased from\n10 to 20 GHz. We attribute this to frequency dependent\nSeebeck Voltage [ /uni03BCV ]\nMagnetic Field [ mT ]10 GHz20 GHz\n12 GHz\n11 GHz13 GHz14 GHz15 GHz16 GHz17 GHz18 GHz19 GHz\n0 50 100 150 200 250 300 3508101214161820\nMagnetic Field [mT]Frequency [GHz]a)\nb)\nSeebeck voltage [nV]\nMagnetic Field [mT]−400 −200 0 200 40000.20.40.60.811.21.41.6\n15 GHz17 GHz\n16 GHz18 GHz19 GHz20 GHz\n0 50 100 150 200 250 300 350050100150200\n010203040506070\nPower [nW]c)\n10 GHz\n20 GHz15 GHzFIG. 3. (color online) a) Series of Seebeck voltage versus\nmagnetic \feld measurements for 11 di\u000berent frequencies. The\ntraces are o\u000bset by 150 nV for clarity. Due to the modula-\ntion technique using two driving frequencies that are 5 GHz\napart, peaks and dips are observed at the resonance \felds for\nboth frequencies. b) Frequency versus the magnetic \feld at\nthe center of the resonance peak. The line corresponds to\na \ft of the Kittel equation. c) Generated power and corre-\nsponding Seebeck voltage calculated using Eq. 4 and thermo-\nelectric \fnite-element modeling for multiple frequencies. The\nmeasured peak heights of a) are indicated by the black dots.\nattenuation of the microwave signal, leading to smaller rf\n\felds at higher frequencies.\nNow we can calculate, using \fnite element modeling in\nComsol Multiphysics, the Seebeck voltage that is gener-\nated due to the heating of the ferromagnet. In this model\nwe impose the constant heat \rux, given by Eq. 4, through\nthe top layer of the ferromagnet and solve the thermo-\nelectric model [16, 17]. Both, the heat \rux and the calcu-\nlated Seebeck voltage are plotted in Fig. 3c (solid lines)\nfor multiple frequencies. Peak heights ranging from 98\ntill 197 nV are calculated. For comparison, the observed\npeak height of Fig. 3a is replotted in Fig. 3c as black\ndots.\nFor a \fxed rf \feld strength, the Seebeck voltage should\nincrease monotonically with frequency due to an increas-\ning dissipation (see Eq. 4). However, because of the ex-\nperimental variation in rf \feld strength for di\u000berent fre-\nquencies, a speci\fc relation between the calculated See-\nbeck voltage and the frequency is found (Fig. 3c). The\nexperimental data is in agreement with the calculations\nwithin a factor of two and follows partially the same4\n−400 −200 0 200 4000123456V [/uni03BCV ] \nMagnetic Field [mT]10 GHz20 GHz\n12 GHz\n11 GHz13 GHz14 GHz15 GHz16 GHz17 GHz18 GHz19 GHz\n10 12 14 16 18 20 220.511.5\nFrequency [GHz]\n /uni03BC0h1 [mT]\n−200 −100 0 100 20015.615.715.8\nR [/uni03A9] \nMagnetic Field [mT]h0     Ia) b)\nc)\nFIG. 4. (color online) a) AMR voltage vs. magnetic \feld.\nPeaks and dips are observed for the resonance \felds of the\ntwo driving frequencies (5 GHz apart). The di\u000berent traces\nare o\u000bset for clarity reasons. b) The magnitude of the\nrf \feld is extracted from the peak height of the resonance\n(V=Idc\u0001Rsin2\u0012c). c) Anisotropic magnetoresistance mea-\nsurement with ~Iand~Bperpendicularly aligned.\ntrend. This discrepancy is attributed to small sample\nto sample variations in the rf \feld strength. Since the\nAMR measurements are performed on a separate device,\nthe \felds can di\u000ber for the thermoelectric device. More-\nover, small shifts in the contact area of the thermocouple\ncan lead to changes in the heat transport and hence, dif-\nferent thermo voltages.\nFurthermore, circulating rf currents combined with an\noscillating magnetoresistance at the same frequency can\ncause dc voltages via a rectifying e\u000bect and mimic the\nobserved thermal behavior in our devices [15]. We have\nexcluded these e\u000bects by using a similar device with an\nAu-Au thermocouple such that the Seebeck e\u000bect van-\nishes. For this device we observed a \rat background\nvoltage without peaks and dips. We note that thermal\nvoltages can be of importance in other device geometries\nwhere ferromagnets are electrically connected to nonmag-\nnetic metals. For example, detection of interface voltages\nthat arise due to spin-pumping [19, 20] cannot be easily\ndistinguished from generated Seebeck voltages. Thor-\nough temperature or material dependent measurements\nmight o\u000ber a solution to discriminate between both mech-\nanisms.\nIn conclusion, we have demonstrated a new thermo-\nelectric detection technique for ferromagnetic resonance.\nThe observed resonance peaks are in good agreement\nwith the LLG equation and thermoelectric \fnite element\nmodeling. Additionally, this technique can be applied on\nthe nanoscale and is not limited to conductive ferromag-\nnetic media. Thermal detection o\u000bers a valid, alterna-\ntive method for studying the dissipation, i.e. the Gilbert\ndamping term, in nanoscale ferromagnetic islands. We\nhope that these results stimulate research for new physi-\ncal e\u000bects that arises from coupling between magnetiza-tion dynamics and caloritronics.\nWe would like to acknowledge B. Wolfs, M. de Roosz\nand J.G. Holstein for technical assistance. This work is\npart of the research program of the Foundation for Fun-\ndamental Research on Matter (FOM) and supported by\nNanoLab and the Zernike Institute for Advanced Mate-\nrials.\n\u0003f.l.bakker@rug.nl\n[1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n[2] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom,\nJ. P. Heremans, and R. C. Myers, Nature Materials 9,\n898 (2010).\n[3] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi,\nJ. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai,\nG. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature\nMaterials 9, 894 (2010).\n[4] J. Xiao, G. E. W. Bauer, K. C. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 81, 214418 (2010).\n[5] H. Adachi, J.-i. Ohe, S. Takahashi, and S. Maekawa,\nPhys. Rev. B 83, 094410 (2011).\n[6] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature 425, 380 (2003).\n[7] D. Houssameddine, U. Ebels, B. Dela et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-\nP. Michel, L. Prejbeanu-Buda, M.-C. Cyrille, O. Redon,\nand B. Dieny, Nature materials 6, 441 (2007).\n[8] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and\nB. I. Halperin, Physical Review B 66, 060404 (2002).\n[9] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B 66, 224403 (2002).\n[10] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[11] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[12] R. Meckenstock, Rev. Sci. Instrum. 79, 041101 (2008).\n[13] F. Sakran, A. Copty, M. Golosovsky, D. Davidov, and\nP. Monod, Applied Physics Letters 84, 4499 (2004).\n[14] J. Coey, Magnetism and magnetic materials (Cambridge\nUniversity Press, 2010).\n[15] M. V. Costache, S. M. Watts, M. Sladkov, C. H. van der\nWal, and B. J. van Wees, Applied Physics Letters 89,\n232115 (2006).\n[16] F. L. Bakker, A. Slachter, J.-P. Adam, and B. J. van\nWees, Physical Review Letters 105, 136601 (2010).\n[17] A. Slachter, F. L. Bakker, and B. J. van Wees, ArXiv\ne-prints (2011), arXiv:1107.3290 [cond-mat.mes-hall].\n[18] C. Kittel, Introduction to Solid State Physics -7th ed.\n(John Wiley & Sons, Inc., New York, 1995).\n[19] X. Wang, G. E. W. Bauer, B. J. van Wees, A. Brataas,\nand Y. Tserkovnyak, Phys. Rev. Lett. 97, 216602 (2006).\n[20] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Physical Review Letters 97,\n216603 (2006)."    },    {        "title": "2006.00348v1.Magnetization_dynamics_in_proximity_coupled_superconductor_ferromagnet_superconductor_multilayers.pdf",        "content": "Magnetization dynamics in proximity-coupled\nsuperconductor/ferromagnet/superconductor multilayers\nI. A. Golovchanskiy1;2;3, N. N. Abramov2, V. S. Stolyarov1;3, V. I. Chichkov2, M. Silayev1;4,\nI. V. Shchetinin2, A. A. Golubov1;5, ,V. V. Ryazanov2;6, A. V. Ustinov2;7;8, M. Yu. Kupriyanov1;9\n1Moscow Institute of Physics and Technology, State University,\n9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia;\n2National University of Science and Technology MISIS, 4 Leninsky prosp., Moscow, 119049, Russia;\n3Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia;\n4Department of Physics and Nanoscience Center, University of Jyv askyl a,\nP.O. Box 35 (YFL), Jyv askyl a FI-40014, Finland;\n5Faculty of Science and Technology and MESA+ Institute for Nanotechnology,\nUniversity of Twente, 7500 AE Enschede, The Netherlands;\n6Institute of Solid State Physics (ISSP RAS), Chernogolovka, 142432, Moscow region, Russia;\n7Physikalisches Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany;\n8Russian Quantum Center, Skolkovo, Moscow 143025, Russia;\n9Skobeltsyn Institute of Nuclear Physics, MSU, Moscow, 119991, Russia\nIn this work, magnetization dynamics is studied in superconductor/ferromagnet/superconductor\nthree-layered \flms in a wide frequency, \feld, and temperature ranges using the broad-band ferro-\nmagnetic resonance measurement technique. It is shown that in presence of both superconducting\nlayers and of superconducting proximity at both superconductor/ferromagnet interfaces a massive\nshift of the ferromagnetic resonance to higher frequencies emerges. The phenomenon is robust and\nessentially long-range: it has been observed for a set of samples with the thickness of ferromagnetic\nlayer in the range from tens up to hundreds of nanometers. The resonance frequency shift is char-\nacterized by proximity-induced magnetic anisotropies: by the positive in-plane uniaxial anisotropy\nand by the drop of magnetization. The shift and the corresponding uniaxial anisotropy grow with\nthe thickness of the ferromagnetic layer. For instance, the anisotropy reaches 0.27 T in experiment\nfor a sample with 350 nm thick ferromagnetic layer, and about 0.4 T in predictions, which makes\nit a ferromagnetic \flm structure with the highest anisotropy and the highest natural resonance fre-\nquency ever reported. Various scenarios for the superconductivity-induced magnetic anisotropy are\ndiscussed. As a result, the origin of the phenomenon remains unclear. Application of the proximity-\ninduced anisotropies in superconducting magnonics is proposed as a way for manipulations with a\nspin-wave spectrum.\nI. INTRODUCTION\nLast two decades can be associated with a remark-\nable progress in areas of spin condensed matter physics,\nnamely, in spintronics1,2and magnonics3,4. Develop-\nments in spin physics have also advanced research in\nsuperconducting systems: by hybridizing superconduct-\ning and ferromagnetic orders intriguing physics emerges\nand new device functionality can be achieved, which\nis inaccessible in conventional systems. Thus, su-\nperconducting spintronics5can be viewed as a way\nfor manipulation with spin states employing an inter-\nplay between ferromagnetic an superconducting spin\norders. A long list of examples includes supercon-\nductor/ferromagnet/superconductor (S/F/S) josephson\njunctions6that can be employed as phase pi-shifters7\nand memory elements8,9, F/S/F-based spin valves10, and\nmore complex long-range spin-triplet superconducting\nsystems11{14. Superconducting spintronics necessarily\ninvolves the superconducting proximity15between fer-\nromagnetic and superconducting subsystems. On the\nother hand, superconducting magnonics can be viewed\nas manipulation with eigen-states of collective spin ex-\ncitations via their interaction with a superconductingsubsystem16{18. In contrast to superconducting spintron-\nics, in superconducting magnonics the proximity e\u000bect\nappears to be undesirable due to a possible suppression\nof fundamental characteristics of superconducting sub-\nsystem and consequently, degradation of the magnonic\nspectrum19.\nRecently, a qualitatively new manifestation of super-\nconductor/ferromagnet hybridization has been reported,\nwhich in a way merges both areas the superconduct-\ning spintronics and the superconducting magnonics. In\nRefs.20,21a drastic increase of the ferromagnetic res-\nonance frequency has been observed in superconduc-\ntor/ferromagnet/superconductor three-layers in presence\nof superconducting proximity between superconducting\nand ferromagnetic layers. The origin of the phenomenon\nremains unclear. Possible explanations that has been\nproposed so far are attributed to incorporation of the\nspin-triplet superconducting pairing mechanism20or to\nan interplay of magnetization dynamics with the vor-\ntex/Meissner state of superconducting layers21. No con-\nvincing explanation has been provided so far.\nIn this paper, we report a detailed experimental study\nof the e\u000bect of superconducting proximity in S/F/S\nheterostructures on magnetization dynamics in the F-\nlayer. Experiments are performed using a broad-bandarXiv:2006.00348v1  [cond-mat.supr-con]  30 May 20202\nferromagnetic resonance (FMR) measurement technique\nin magnetic \feld, frequency and temperature domains.\nThis work is organized as follows. Section II gives exper-\nimental details. Section III provides experimental results:\nmicrowave ferromagnetic resonance absorption spectra\nat \feld-frequency domain at di\u000berent temperatures and\ntheir quantitative analysis. For a complete picture, we\nalso suggest to review previous research studies on sim-\nilar systems (see Refs.20{22). Section IV is devoted to\ndiscussion of experimental results where we state that\nthe e\u000bect of superconducting proximity in S/F/S sys-\ntems can not be explained employing concepts of the su-\nperconducting Meissner screening or of the vortex phase.\nWhile the origin of the phenomena remains unclear at\nthis stage, the authors suspect a contribution of spin-\ntriplet superconductivity. Section V demonstrates capa-\nbilities of the e\u000bect for manipulation of the spin-wave\nspectrum in S/F/S-based continuous \flms and magnonic\ncrystals.\nII. EXPERIMENTAL DETAILS\nFIG. 1. Schematic illustration of the investigated chip-\nsample. A series of S/F/S \flm rectangles is placed directly\non top of the central transmission line of the co-planar waveg-\nuide. Magnetic \feld His applied in-plane along the x-axis.\nMagnetization dynamics is studied by measuring\nthe ferromagnetic resonance absorption spectrum us-\ning the VNA-FMR approach23{25. A schematic il-\nlustration of the investigated chip-sample is shown in\nFig. 1. The chip consists of 150 nm thick supercon-\nducting niobium (Nb) co-planar waveguide with 50 Ohm\nimpedance and 82-150-82 \u0016m center-gap-center dimen-\nsions. The waveguide is fabricated on top of Si/SiO x\nsubstrate using magnetron sputtering of Nb, optical\nlithography and plasma-chemical etching techniques.\nA series of niobium/permalloy(Py=Fe 20Ni80)/niobium\n(Nb/Py/Nb) \flm structures with lateral dimensions X\u0002\nY= 50\u0002140\u0016m and spacing of 25 \u0016m along the x\u0000axis\nis placed directly on top of the central transmission line\nof the waveguide using optical lithography, magnetronsputtering and the lift-o\u000b technique. Importantly, depo-\nsition of Nb/Py/Nb three-layers is performed in a single\nvacuum cycle ensuring an electron-transparent metallic\nNb/Py interfaces. A 20-nm-thick Si spacing is deposited\nbetween Nb co-planar and Nb/Py/Nb threelayers in or-\nder to ensure electrical insulation of the studied samples\nfrom the waveguide. Five di\u000berent samples has been fab-\nricated and measured with di\u000berent thickness of super-\nconducting (S) and ferromagnetic (F) layers (see Tab. I).\nOne of samples was fabricated with an additional insu-\nlating (I) layer at one of S/F interfaces.\nSample ID S(Nb) F(Py) I(AlO x)S(Nb)\nS1 110 19 0 110\nS2 110 19 0 7\nS3 85 22 10 115\nS4 140 45 0 140\nS5 110 350 0 110\nTABLE I. Parameters of studied samples.\nThe experimental chip was installed in a copper sample\nholder and wire bonded to PCB with SMP RF connec-\ntors. A thermometer and a heater were attached directly\nto the holder for precise temperature control. The holder\nwas placed in a superconducting solenoid inside a closed-\ncycle cryostat (Oxford Instruments Triton, base temper-\nature 1.2 K). The response of experimental samples was\nstudied by analyzing the transmitted microwave signal\nS21(f;H) with the VNA Rohde & Schwarz ZVB20. For\nexclusion of parasitic box resonance modes from consid-\neration, all measured spectra S21(f;H) have been \frst\nnormalized with S21(f) at\u00160H= 0:3 T, and then dif-\nferentiated numerically in respect to H. The response of\nexperimental samples was studied in the \feld range from\n-0.22 T to 0.22 T, in the frequency range from 0 up to\n18 GHz, and in the temperature range from 1.7 to 11 K.\nIII. EXPERIMENTAL RESULTS:\nFERROMAGNETIC RESONANCE IN\nPROXIMITY-COUPLED S/F/S SYSTEMS\nFigure 2 illustrates the studied phenomenon using\nS(Nb)/F(Py)/S(Nb) sample with 110 nm thick Nb lay-\ners and 19 nm thick Py layer. This sample is referred\nto as S1. Thickness of Py layer is selected for direct\ncomparison of obtained results with previous research\nstudies20,21. Figures 2a,b show FMR absorption spec-\ntradS21(f;H)=dH atT= 2 K (a), which is far below\nthe superconducting critical temperature Tcof Nb, and\natT= 9 K (b), which corresponds to Tc. Both spec-\ntra contain a single \feld-dependent spectral line, i.e., the\nFMR absorption line. FMR absorption spectra at dif-\nferent temperatures have been \ftted with the Lorentz\ncurve and the dependencies of the resonance frequency\non magnetic \feld fr(H) have been extracted. Figure 2c3\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s40/s97/s41/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s45 /s50/s46 /s48/s120 /s49/s48/s45 /s52\n/s45 /s49/s46 /s54/s120 /s49/s48/s45 /s52\n/s45 /s49/s46 /s50/s120 /s49/s48/s45 /s52\n/s45 /s56/s46 /s48/s120 /s49/s48/s45 /s53\n/s45 /s52/s46 /s48/s120 /s49/s48/s45 /s53\n/s45 /s50/s46 /s55/s120 /s49/s48/s45 /s50/s48\n/s52/s46 /s48/s120 /s49/s48/s45 /s53\n/s56/s46 /s48/s120 /s49/s48/s45 /s53\n/s49/s46 /s50/s120 /s49/s48/s45 /s52\n/s49/s46 /s54/s120 /s49/s48/s45 /s52\n/s50/s46 /s48/s120 /s49/s48/s45 /s52/s84 /s32/s61/s32/s49/s46/s55/s32/s75\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s84 /s32/s61/s32/s57/s46/s48/s32/s75\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s45 /s51/s46 /s48/s48/s120 /s49/s48/s45 /s52\n/s45 /s50/s46 /s52/s48/s120 /s49/s48/s45 /s52\n/s45 /s49/s46 /s56/s48/s120 /s49/s48/s45 /s52\n/s45 /s49/s46 /s50/s48/s120 /s49/s48/s45 /s52\n/s45 /s54/s46 /s48/s48/s120 /s49/s48/s45 /s53\n/s48/s46 /s48/s48\n/s54/s46 /s48/s48/s120 /s49/s48/s45 /s53\n/s49/s46 /s50/s48/s120 /s49/s48/s45 /s52\n/s49/s46 /s56/s48/s120 /s49/s48/s45 /s52\n/s50/s46 /s52/s48/s120 /s49/s48/s45 /s52\n/s51/s46 /s48/s48/s120 /s49/s48/s45 /s52\n/s40/s98/s41\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s32/s49/s46/s55/s32/s75\n/s32/s53/s46/s57/s32/s75\n/s32/s55/s46/s49/s32/s75\n/s32/s55/s46/s57/s32/s75\n/s32/s56/s46/s53/s32/s75\n/s32/s57/s46/s48/s32/s75/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102\n/s114/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84\n/s40/s99/s41\nFIG. 2. a,b) FMR absorption spectra dS21(f;H)=dH for S1 sample measured at T= 2 K>T c(a) andT= 9 K.Tc(b). The\ngrayscale is coded in absolute units. c) Dependencies of the FMR frequency on magnetic \feld fr(H) at di\u000berent temperatures\nfor S1 sample.\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48\n/s32/s83/s49\n/s32/s83/s49/s32/s102/s105/s116\n/s32/s83/s50\n/s32/s83/s51/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s102/s105/s101/s108/s100/s32 /s72\n/s97/s44/s32/s109/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\n/s40/s97/s41\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s49/s46/s48/s50/s49/s46/s48/s52/s49/s46/s48/s54/s49/s46/s48/s56/s49/s46/s49/s48\n/s32/s83/s49\n/s32/s83/s50\n/s32/s83/s51/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s32/s109/s97/s103/s110/s101/s116/s105/s115/s97/s116/s105/s111/s110/s32 /s77\n/s101/s102/s102/s44/s32/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\n/s40/s98/s41\nFIG. 3. The dependence of the anisotropy \feld Ha(a) and e\u000bective magnetization Meff(b) on temperature. Black square\ndots correspond to S1 S/F/S sample, red circular dots correspond to S2 S/F/s' sample, and blue diamond dots correspond to\nS3 S/F/I/S sample. Green curve in (a) is the \ft of Ha(T) with Eq. 2, which yields the following parameters: \u00160Ha0= 77 mT,\nTc= 9:0 K,p= 3:7.\ncollects resonance curves fr(H) that are measured at dif-\nferent temperatures. Basically, Fig. 2 demonstrates the\nessence of the phenomenon: it shows that upon decreas-\ning the temperature below Tcthe resonance curve fr(H)\nshifts gradually to higher frequencies. For instance, upon\ndecreasing the temperature the frequency of the natu-\nral FMRfr(H= 0) increases from about 0.5 GHz at\nT\u00159 K to about 8.5 GHz at T= 1:7 K.\nFMR curves fr(H) in Fig. 2c follow the typical Kittel\ndependence for thin in-plane-magnetized ferromagnetic\n\flms at in-plane magnetic \feld:\n(2\u0019fr=\u00160\r)2= (H+Ha) (H+Ha+Meff) (1)\nwhere\u00160is the vacuum permeability, \r= 1:856\u0002\n1011Hz/T is the gyromagnetic ratio for permalloy, Ha\nis the uniaxial anisotropy \feld that is aligned with theexternal \feld, and Meff=Ms+Mais the e\u000bective\nsaturation magnetization, which includes the saturation\nmagnetization Msand the out-of-plane anisotropy \feld\nMa. The \ft of FMR curves in Fig. 2c with Eq. 1 yields\nthe dependence of superconducting proximity-induced\nanisotropy \felds HaandMeffon temperature given in\nFig. 3 with black squares.\nFigure 3 shows that at T > T cthe anisotropy \feld is\nnegligible\u00160Ha\u0018 \u00002\u000210\u00004T and the e\u000bective magne-\ntization is\u00160Meff\u00191:1 T. These parameters are typical\nfor permalloy thin \flms. Also, at T >T cno dependence\nofHaandMeffon temperature is observed. At T <T c\nupon cooling the anisotropy \feld Haincreases gradually\nand reaches \u00160Ha\u001978 mT atT= 2 K. This value is\nwell consistent with previous studies on samples with the\nsame thickness of Py layer20,21. The dependence Ha(T)\ncan be characterized by \ftting it with the following ex-4\npression\nHa=Ha0(1\u0000(T=Tc)p) (2)\nwhereHa0is the e\u000bective anisotropy \feld at zero temper-\nature,Tcis the critical temperature, and pis a free expo-\nnent parameter. The \ft of Ha(T) with Eq. 2 is shown in\nFig. 3a with blue curve and yields the zero-temperature\nanisotropy \u00160Ha0= 77 mT.\nImportantly, the e\u000bective magnetization also demon-\nstrates a temperature dependence: upon cooling \u00160Meff\ndrops by about 70 mT. Such e\u000bect was has not been ob-\ntained in previous studies20,21due to instrumental limi-\ntations. The drop \u0000\u0001Meffand the uniaxial anisotropy\n\feldHaat 2 K are roughly equal. Thus, we state that su-\nperconductivity in S/F/S structure a\u000bects the magneti-\nzation dynamics by inducing positive in-plane anisotropy\nand by the drop of e\u000bective magnetization.\nAs the next step, following Ref.20, we con\frm that\nboth superconducting layers are required for development\nof the e\u000bect of superconducting proximity on magnetiza-\ntion dynamics, and that electrical conductivity, i.e. the\nproximity, also is required to take place at both S/F in-\nterfaces. The following S(Nb)/F(Py)/s'(Nb') sample is\nstudied with 110 nm thick S(Nb) layer, 19 nm thick Py\nlayer, which are similar to S1 sample, and thin 7 nm\nthick s'(Nb') layer . This sample is referred to as S2 (see\nTab. I). The upper s'(Nb') layer of S2 sample is argued to\nbe non-superconducting due to its small thickness, below\nthe superconducting coherence length and the London\npenetration depth, and due to the action of the inverse\nproximity e\u000bect. Yet, the upper layer is expected to re-\nproduce the microstructure of the upper Nb/Py inter-\nface. Basically, S2 sample represent S1 S/F/S sample\nwith a removed superconducting layer. FMR absorption\nspectra of S2 sample show no noticeable temperature de-\npendence, which is consistent with previous studies20,\nand practically match with the spectrum of S1 sample\natT&Tc(Fig. 2b). Fitting procedures of FMR spec-\ntra and of resonance curves for S2 sample yield Ha(T)\nandMeff(T) dependencies that a shown in Fig. 3 with\nred circular dots. The anisotropy \feld Ha(T) in Fig. 3a\nis negligible, it varies in the range from \u00005\u000210\u00004T\nto\u00003\u000210\u00004T and shows no dependence on tempera-\nture. The e\u000bective magnetization curve Meff(T), being\nat\u00160Meff\u00191:072 T atT >T c, shows a minor increase\nby\u00160\u0001Meff\u00193 mT upon decreasing temperature and\ncrossingTc. Note that variation of Meffwith tempera-\nture for S2 sample is opposite to one for S1 sample.\nNext, the following S(Nb)/F(Py)/I(AlO x)/S(Nb) sam-\nple is studied with thicknesses of Nb and Py layers similar\nto S1 and S2 samples, and additional insulating layer at\none of S/F interfaces. The sample is refereed to S3 (see\nTab. I). Basically, S3 sample represent S1 S/F/S sample\nwith suppressed conductivity at one of S/F interfaces.\nFMR absorption spectra of S3 sample shows no notice-\nable temperature dependence, which is consistent with\nprevious studies20. Blue diamond dots in Fig. 3 show\nHa(T) andMeff(T) dependencies for S3 sample. Theanisotropy \feld Ha(T) in Fig. 3a is negligible, though is\nslightly higher than one for S1 and S2 samples. It varies\nin the range from 3 to 5 mT and shows insigni\fcant de-\npendence on temperature. The e\u000bective magnetization\ncurveMeff(T), varies in the range from 1.1 up to 1.2 T\nand shows a minor drop by \u00160\u0001Meff\u001910 mT in vicinity\ntoTc. Therefore, with S2 and S3 samples we con\frm that\nboth superconducting layers are required for development\nof the e\u000bect of superconducting proximity on magneti-\nzation dynamics and that superconducting proximity is\nrequired to take place at both S/F interfaces.\nAs a crucial step, the dependence of phenomenon\non the thickness of the F-layer is revealed. Fig-\nure 4 demonstrates this dependence with a di\u000berent\nS(Nb)/F(Py)/S(Nb) sample with 140 nm thick Nb layers\nand 45 nm thick Py layer. This sample is referred to as\nS4 (see Tab. I). Figure 4a collects resonance curves fr(H)\nthat are measured at di\u000berent temperatures. It shows\nthat upon decreasing the temperature below Tcthe reso-\nnance curve fr(H) shifts gradually to higher frequencies\nfollowing the same trend as for S1 sample. Comparison\nof Fig. 4a with Fig. 2a immediately indicates that the ef-\nfect of the superconducting proximity in S/F/S systems\non magnetization dynamics is substantially stronger for\nthe thicker S4 sample: upon decreasing the temperature\nthe frequency of the natural FMR increases from about\n1 GHz atT= 10 K up to about 14.5 GHz at T= 3 K.\nIn other terms, by increasing the thickness of the F layer\nby a factor of 2.3 the enhancement of the natural FMR\nfrequency of S/F/S sample in superconducting state at\nT\u001cTchas increased by a factor of 1.6.\nThe \ft of FMR curves in Fig. 4a with Eq. 1 yields\nthe dependence of superconducting proximity-induced\nanisotropy \felds HaandMeffon temperature that are\ngiven in Fig. 4b,c with black squares. Figure 4b shows\nthat atT > T cthe anisotropy \feld is negligible as in\ncase of S1, S2 and S3 samples. At T <T cupon cooling\nthe anisotropy \feld Haincreases gradually and reaches\n\u00160Ha\u0019200 mT at T= 2 K.\nThe temperature dependence of the e\u000bective magneti-\nzationMeff(T) given in Figure 4c is more complex and\nis qualitatively di\u000berent from one for S1 sample. Upon\ncooling\u00160Meff\frst drops from 1.2 T at T >T cto about\n0.6 T atT.Tcand than increases gradually up to\nabout 1.03 T at T= 2K. We argue that such tempera-\nture dependence can be explained by \feld dependence of\nproximity-induced parameters. Indeed, at \fxed T < T c\nat upper-right section of a resonance absorption spec-\ntrumS21(f;H) superconductivity is partially suppressed\nby external \feld and microwave radiation, and therefore\nHais expected to be reduced while Meffis expected\nto be increased as compared to lower-left section of the\nspectrum. This phenomenon can be illustrated by \ftting\nof FMR curves in Fig. 4a with Eq. 1 in the limited \feld\nrange. Red circular dots in Fig. 4b,c show temperature\ndependencies of HaandMeffobtained by \ftting only\npart of FMR curves at \u00160H < 90 mT. Figure 4c shows\nthat the drop of MeffatT.Tcis signi\fcantly reduced:5\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102\n/s114/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84/s32/s51/s46/s48/s32/s75\n/s32/s53/s46/s54/s32/s75\n/s32/s54/s46/s57/s32/s75\n/s32/s55/s46/s53/s32/s75\n/s32/s55/s46/s57/s32/s75\n/s32/s56/s46/s51/s32/s75\n/s32/s49/s48/s32/s75\n/s40/s97/s41\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48\n/s40/s98/s41/s32/s111/s98/s116/s97/s105/s110/s101/s100/s32/s117/s115/s105/s110/s103/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s101/s110/s116/s105/s114/s101/s32/s102/s105/s101/s108/s100/s32/s114/s97/s110/s103/s101\n/s32/s111/s98/s116/s97/s105/s110/s101/s100/s32/s117/s115/s105/s110/s103/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s99/s117/s116/s111/s102/s102/s32\n/s48/s72 /s60/s57/s48/s32/s109 /s84\n/s32/s102/s105/s116/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s102/s105/s101/s108/s100/s32\n/s48/s72\n/s97/s44/s32/s109/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s49/s46/s50/s32/s111/s98/s116/s97/s105/s110/s101/s100/s32/s117/s115/s105/s110/s103/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s101/s110/s116/s105/s114/s101/s32/s102/s105/s101/s108/s100/s32/s114/s97/s110/s103/s101\n/s32/s32/s111/s98/s116/s97/s105/s110/s101/s100/s32/s117/s115/s105/s110/s103/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s99/s117/s116/s111/s102/s102/s32\n/s48/s72 /s60/s57/s48/s32/s109 /s84/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s32/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32\n/s48/s77\n/s101/s102/s102/s44/s32/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\n/s40/s99/s41\nFIG. 4. a) Dependencies of the FMR frequency on magnetic \feld fr(H) at di\u000berent temperatures for S4 sample. b,c) The\ndependence of the anisotropy \feld Ha(b) and e\u000bective magnetization (c) on temperature. The data in b,c that is shown with\nblack square dots was obtained by \ftting fr(H) in the entire \feld range from 0 up to 200 mT. The data in b,c that is shown with\nred circular dots was obtained by \ftting fr(H) in the cut-o\u000b \feld range from 0 up to 90 mT. Green curve in (b) shows the \ft\nofHa(T), which is obtained using the cut-o\u000b \feld range, with Eq. 2, which yields the following parameters: \u00160Ha0= 196 mT,\nTc= 9:0 K,p= 7:7.\nupon cooling \u00160Meff\frst drops from 1.2 T at T > T c\nto about 0.8 T at T.Tcand than increases gradually\nup to about 1.03 T at T= 2K. Green curve in Fig-\nure 4b shows the \ft of Ha(T), which is obtained using\nthe cut-o\u000b \feld range, with Eq. 2. The \ft yields the\nzero-temperature anisotropy \u00160Ha0= 196 mT. Overall,\nthe drop \u0000\u0001Meffand the induced Haat 2 K are are\nroughly equal as in case of S1 sample: the anisotropy\n\feld\u00160Ha0= 196 mT while the drop of the e\u000bective\nmagnetization \u00160\u0001Meff\u0019 \u0000170 mT.\nImportantly, FMR parameters of the S1 sample, Ha(T)\nandMeff(T) in Fig. 3, are mostly unchanged when ob-\ntained using the same limited range of magnetic \felds\n\u00160H < 90 mT. This fact can be explained by frequency\ndependence of proximity-induced anisotropy \felds. In-\ndeed, resonance frequencies for S1 sample are typically\nby a factor of 2 lower than for S4 sample. Therefore,\nthe superconducting state of S-layers in S1 sample is less\na\u000bected by microwave radiation than in S4 sample.\nFigure 5 demonstrates the e\u000bect of the superconduct-\ning proximity in S/F/S systems on magnetization dy-\nnamics for a di\u000berent S(Nb)/F(Py)/S(Nb) sample with\na radically thicker 350 nm thick Py layer. This sam-\nple is referred to as S5 (see Tab. I). Figure 5a collects\nresonance curves fr(H) that are measured at di\u000berent\ntemperatures; it shows that upon decreasing the temper-\nature below Tcthe resonance curve fr(H) shifts gradually\nto higher frequencies following the same trend as for S1\nand S4 samples. However, the enhancement of the FMR\nfrequency upon decreasing temperature at T < T cis so\nintense that the FMR curve approaches the instrumen-\ntal frequency band limit already at T\u00188 K (note the\ntemperature range in legend of Fig. 5a). Comparison of\nFig. 5a with Figs. 2a and 4a con\frms that the e\u000bect of\nthe superconducting proximity in S/F/S systems on mag-netization dynamics enhances with growing thickness of\nthe F-layer. Upon decreasing the temperature the fre-\nquency of the natural FMR of S5 sample increases from\nabout 1 GHz at T > T cup to about 17 GHz already\natT= 8 K. Proximity to the superconducting critical\ntemperature, insu\u000ecient signal-to-noise ratio, parasitic\nbox modes, did not allow to \ft resonance curves consid-\nering both HaandMeffin Eq. 1 as \ftting parameters.\nTherefore, the \ftting routine was modi\fed for S5 sam-\nple as follows. First, fr(H) curves have been \ftted at\nT > T cwith Eq. 1. The \ft yields \u00160Meff\u00191:076 T\nand\u00160Ha\u00181 mT. Next, fr(H) curves at T < T chave\nbeen \ftted with Eq. 1 considering magnetization \fxed at\n\u00160Meff= 1:076 T and considering Haas the only \ftting\nparameter. The dependence Ha(T) is given in Fig. 5b\nwith black squares. It shows that the e\u000bective anisotropy\n\feld reaches \u00160Ha\u00190:27 T at 8 K. Note that by \fxing\nMeffthe so-obtained anisotropy \feld Hais expected to\nbe underestimated since according to Meff(T) depen-\ndencies for S1 and S4 samples Meffshould actually drop\natT < T c. Green curve in Figure 5b shows the \ft of\nHa(T) with Eq. 2. The \ft yields the extrapolated zero-\ntemperature anisotropy \u00160Ha0= 375 mT, which is also\nexpected to be underestimated.\nSummarizing experiential \fndings, superconductivity\nin S/F/S three-layers shifts the FMR to higher frequen-\ncies. The shift can be quanti\fed by the proximity-\ninduced positive in-plane anisotropy Haand by a drop\nof e\u000bective magnetization Meff. BothHaand the drop\nofMeffare roughly equal and are \feld-, frequency- and\ntemperature-dependent. The phenomenon requires both\nsuperconducting layers of S/F/S and presence of super-\nconducting proximity at both S/F interfaces. The phe-\nnomenon shows a dependence on the thickness of the F-\nlayer: for thicker F-layer the shift of the FMR frequency6\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48\n/s32/s56/s46/s48/s32/s75\n/s32/s56/s46/s49/s32/s75\n/s32/s56/s46/s50/s32/s75\n/s32/s56/s46/s51/s32/s75\n/s32/s56/s46/s52/s32/s75\n/s32/s56/s46/s53/s32/s75\n/s32/s56/s46/s54/s32/s75\n/s32/s56/s46/s55/s32/s75\n/s32/s57/s46/s53/s32/s75/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102\n/s114/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s109 /s84\n/s40/s97/s41\n/s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s40/s98/s41/s32/s72\n/s97/s32/s100/s97/s116/s97\n/s32/s102/s105/s116/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s102/s105/s101/s108/s100/s32\n/s48/s72\n/s97/s44/s32/s84\n/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75\nFIG. 5. a) Dependencies of the FMR frequency on magnetic \feld fr(H) at di\u000berent temperatures for S5 sample. b) The\ndependence of the anisotropy \feld Haon temperature. Green curve in (b) shows the \ft of Ha(T) with Eq. 2, which yields the\nfollowing parameters: \u00160Ha0= 375 mT, Tc= 8:74 K,p= 13:9\nis substantially stronger. In addition, it should be noted\nthat (i) no dependence of the FMR spectrum on the in-\nput power has been observed in the range of input power\nfrom -15 dB to 0 dB; (ii) all measured spectra for all\nsamples are \feld-reversible; and (iii) no dependence of\nthe FMR linewidth on experimental parameters could be\nnoted owing partially to insu\u000ecient signal-to-noise ratio.\nAs a \fnal remark it should be noted that, technically,\nsamples S4 and S5 demonstrate the highest natural FMR\nfrequencies and corresponding in-plane anisotropies for\nin-plane magnetized ferromagnetic \flm systems ever re-\nported (see, for instance, Ref.26for comparison).\nIV. DISCUSSIONS: POSSIBLE ORIGIN OF\nPROXIMITY-INDUCED ANISOTROPIES IN\nS/F/S SYSTEMS\nA natural initial guess for the origin of the e\u000bect of\nsuperconducting proximity in S/F/S systems on mag-\nnetization dynamics is the Meissner screening of exter-\nnal \feld, the so-called lensing e\u000bect18,27. For instance,\none could employ \ruxometric or magnetometric demag-\nnetizing factors28,29of the system for estimation of a\nhypothetical diamagnetic moment in Nb layers that in-\nduces magnetostatic \feld Ha. However, this estimation\nis not required since the following set of unful\flled con-\nditions points towards irrelevance of the lensing e\u000bect in\ndiscussed experiments: (i) In case of the lensing e\u000bect\nthe induced Hais not a constant but a \feld-dependent\nquantity18. (ii) In case of the lensing e\u000bect the induced\nHashould decrease with increasing thickness of the F-\nlayer. (iii) The lensing e\u000bect should hold for S/F/I/S\nstructure (S3 sample) and should be only halved for S/F\nstructure (S2 sample) . (iv) The \feld that is induced\nby the lensing e\u000bect can not exceed the \frst critical\n\feld, which in Nb is about 100 mT (see values of Ha\nin Figs. 4b and 5b). None of the above hypothetical ef-fects does take place. In addition, consideration of the\nlensing e\u000bect does not clarify possible origin of the drop\nof magnetization \u0001 MeffatT <T cin Figs. 3b and 4c.\nIn fact, S/F (S2) and S/F/I/S (S3) structures may\nevidence the e\u000bect of Meissner screening on precessing\nmagnetization in thin \flm geometry. Meissner screening\nis expected to show itself in the absence of the in-plane\nanisotropy and in the presence of small negative out-of-\nplane uniaxial anisotropy. The later might be indicated\nby a small variation of MeffatT\u0018Tcin Figs. 3c and 4c.\nThe next hypothetical candidate for impact on mag-\nnetostatic state of the F-layer is the vortex phase. The\nfollowing set of unful\flled conditions evidence that the\nvortex phase can not have any e\u000bect on magnetization\ndynamics: (i) The e\u000bect of the vortex phase should hold\nfor S/F/I/S structure (S3 sample). (ii) Presence of the\nvortex phase that is induced by the external magnetic\n\feld should, in the \frst place, lead to hysteresis in the\nabsorption spectrum due to pinning30. (ii) The density of\na vortex phase that is induced by the external \feld is ex-\npected to be \feld-dependent leading to \feld-dependence\nof hypothetical vortex-phase-induced anisotropies. (iii)\nPresence of the vortex phase in superconducting thin\n\flms induces only insigni\fcant total magnetic moments\nand corresponding stray \felds. In addition, low expected\ndensity and arbitrary nature of the out-of-plane vortex\nphase unfavor its possible contribution.\nMechanisms that are considered above are limited to\nmagnetostatic interactions between F- and S-subsystems.\nAlternative explanations imply electronic correlations be-\ntween superconducting and ferromagnetic subsystems.\nFor instance, in Refs.31{33the electromagnetic proxim-\nity e\u000bect and spin polarization in planar superconductor-\nferromagnet structures are discussed. The electromag-\nnetic proximity e\u000bect implies presence of the supercon-\nducting condensate in the ferromagnetic layer and induc-\ntion of screening currents in the S/F system as a response7\non magnetic moment34rather than on magnetic \feld.\nWhile in general the electromagnetic proximity e\u000bect is\ndiamagnetic and induces magnetic \feld that counteracts\nthe magnetization, at certain thicknesses of the F-layer\nthe so-called paramagnetic electromagnetic proximity ef-\nfect can take place, which induces magnetic \feld along\nthe magnetization31. However, large thickness of F-layers\nin our experiments of 20, 40 and 350 nm in comparison to\nthe typical electron correlation length of singlet pairs in\nferromagnets35{37\u0018F\u00181 nm, and predicted oscillating\nbehaviour of the sign of induced \feld with the thickness of\nthe F-layer rule-out contribution of the electromagnetic\nproximity e\u000bect on magnetization dynamics in considered\nS/F/S systems.\nAlso, one can rule-out possible contribution of the\nspin-inverse proximity e\u000bect or the so-called spin-\nscreening38,39. The spin-screening considers accu-\nmulation of spins with polarization opposite to F-\nmagnetization in a thin layer of the S-subsystem of the\norder of the coherence length in vicinity to the S/F in-\nterface. Such spin orientation could possibly produce\nstray \felds of a required direction along magnetization in\nthe F-layer. Yet, owing to thin \flm geometry and small\ndemagnetizing factors28,29of the system an implausibly\nlarge magnetization of the spin-polarized area is required\nfor induction of the observed Ha, which is far above su-\nperconducting critical \felds.\nAnother possible explanation for the e\u000bect of supercon-\nducting proximity in S/F/S systems on magnetization\ndynamics is provided in the very \frst report of the ef-\nfect. In Ref.20it is proposed that the e\u000bective anisotropy\n\feld is produced due to interaction of magnetization with\nspin-polarized spin-triplet superconducting electrons via\nthe spin-transfer torque mechanism40{43. This mecha-\nnism requires presence of spin-triplet superconducting\npairs as a necessary ingredient. In Ref.20it is proposed\nthat the spin-triplet superconductivity is induced by the\ndynamically precessing magnetization in accordance with\nthe Ref.44. However, such mechanism requires large fre-\nquency of magnetization precession that should be com-\nparable to the depairing frequency and is inconsistent\nwith the frequency range of reported results.\nThus, we state that at this stage even a qualitative\nexplanation of the e\u000bect of superconducting proximity\nin S/F/S systems on magnetization dynamics is unavail-\nable. Yet, long-range nature of the phenomenon and the\nmandatory S/F/S symmetry of the phenomenon are sig-\nnatures for a role of spin-triplet superconductivity35.\nV. PROSPECTS OF THE PROXIMITY EFFECT\nFOR APPLICATION IN MAGNONICS\nThe e\u000bect of the superconducting proximity in S/F/S\nsystems on magnetization dynamics can be e\u000bective in\nmagnonics for variation of the FMR frequency or for\nmodulation of the spin-wave velocity. In this section,\nmicromagnetic simulations are employed45for calcula-tion of spin-wave spectra for S/F/S-based continuous\n\flms and periodic structures in the magnetostatic sur-\nface wave (MSSW) geometry46,47, following Refs.48{50.\nThe following micromagnetic parameters of studied F-\nlayers are considered, which correspond to S1 sample:\nthickness of F-layer d= 20 nm, the saturation magne-\ntization\u00160Ms= 1 T, the anisotropy \feld Ha= 0, the\napplied \feld \u00160H= 0:02 T, the exchange sti\u000bness con-\nstantA= 1:3\u000210\u000011J/m, and the gyromagnetic ratio\n\u00160\r= 2:21\u0002105m/A/s. The excitation \feld pulse has\nthe maximum frequency fmax= 30 GHz, the gaussian\nspatial pro\fle with the width at half-maximum of 200\nnm, and the amplitude of 0.001Ms. In simulations, the\ndiamagnetic (Meissner) contribution of S-subsystem on\nmagnetization dynamics was accounted via the method of\nimages17,51in case of continuous S-layers and via the dia-\nmagnetic representation of superconductors18,19in case\nof a \fnite-size S-elements. The e\u000bect of the supercon-\nducting proximity in S/F/S is represented by a local uni-\naxial anisotropy \feld \u00160Hs= 0:07 T that corresponds to\nS1 sample (see Fig. 3a).\nFigure 6a collects simulation results for continuous\nthin \flms. Blue solid curves show a typical dispersion\ncurve for MSSW in the plain F-\flm that is obtained\nwith simulations in absence of any contribution from S-\nsubsystem. Simulation results are well con\frmed by the\nanalytical dispersion relation51, shown with blue dashed\ncurves. Red solid and dashed lines show dispersion curve\nof MSSW in the S/F bilayer in presence of magnetostatic\ninteraction between the S and the F subsystems. The\nmagnetostatic interaction is accounted using the method\nof images17,51. It shows that in presence of magneto-\nstatic interaction the dispersion is nonreciprocal: the fre-\nquency pass-band for positive wavenumbers is approxi-\nmately doubled as compared to the pass-band for nega-\ntive wavenumbers. The nonreciprocity is a known prop-\nerty of MSSWs, which emerges due to asymmetry of the\nferromagnetic \flm across its thickness or due to asymme-\ntry of its surrounding (see Ref.51for details). Purple solid\nand dashed lines show dispersion curve of MSSW in the\nS/F/S three-layer in presence of the proximity-induced\nuniaxial anisotropy Hsbut absence of magnetostatic in-\nteraction between the S and the F subsystems. It shows\nthat at zero wavenumber the di\u000berence in frequencies be-\ntween the plain \flm and the \flm with uniaxial anisotropy\nis maximum and corresponds to the di\u000berence in FMR\nfrequencies. Upon increasing the wavenumper the dif-\nference in frequencies reduces. Black solid and dashed\nlines show dispersion curve of MSSW in the S/F/S three-\nlayer in presence of both the proximity-induced uniaxial\nanisotropy Hsand of the magnetostatic interaction be-\ntween the S and the F subsystems. Comparison of these\ncurves with dispersions in plain F \flm, in S/F bilayer\nand in F \flm with proximity-induced uniaxial anisotropy\nHsindicates that both the proximity-induced anisotropy\nand the magnetostatic screening a\u000bect the kinetics of\nspin waves. The magnetostatic screening is the domi-\nnating e\u000bect on spin-wave velocity at the range of higher8\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s53/s49/s48/s49/s53/s50/s48/s50/s53/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s87/s97/s118/s101/s110/s117/s109 /s98/s101/s114/s32/s49/s47 /s44/s32 /s109 /s45 /s49/s32/s80/s108/s97 /s105/s110 /s32/s70 /s77/s32/s102/s105/s108/s109/s44/s32/s115/s105/s109/s117 /s108/s97 /s116/s105/s111/s110 /s115\n/s32/s70 /s77/s47/s83 /s67/s32/s98/s105/s108/s97 /s121 /s101 /s114/s44/s32/s115/s105/s109/s117 /s108/s97 /s116/s105/s111/s110 /s115\n/s32/s83 /s67/s47/s70 /s77/s47/s83 /s67/s32/s116/s104 /s114/s101 /s101 /s108/s97 /s121 /s101 /s114/s44/s32/s115/s105/s109/s117 /s108/s97 /s116/s105/s111/s110 /s115\n/s32/s70 /s77/s32/s102/s105/s108/s109/s32/s119 /s105/s116/s104 /s32/s83 /s67/s45/s105/s110 /s100/s117 /s99 /s101 /s100/s32/s97 /s110 /s105/s115/s111/s116/s114/s111/s112/s121 /s44/s32/s115/s105/s109/s117 /s108/s97 /s116/s105/s111/s110 /s115\n/s32/s80/s108/s97 /s105/s110 /s32/s70 /s77/s32/s102/s105/s108/s109/s44/s32/s116/s104 /s101 /s111/s114/s121 \n/s32/s70 /s77/s47/s83 /s67/s32/s98/s105/s108/s97 /s121 /s101 /s114/s44/s32/s116/s104 /s101 /s111/s114/s121 \n/s32/s83 /s67/s47/s70 /s77/s47/s83 /s67/s32/s116/s104 /s114/s101 /s101 /s108/s97 /s121 /s101 /s114/s44/s32/s116/s104 /s101 /s111/s114/s121 \n/s32/s70 /s77/s32/s102/s105/s108/s109/s32/s119 /s105/s116/s104 /s32/s83 /s67/s45/s105/s110 /s100/s117 /s99 /s101 /s100/s32/s97 /s110 /s105/s115/s111/s116/s114/s111/s112/s121 /s44/s32/s116/s104 /s101 /s111/s114/s121 \n/s40/s97/s41\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s48/s46/s48/s48/s48/s50/s48/s48/s46/s48\n/s87/s97/s118/s101/s110/s117/s109 /s98/s101/s114/s32/s49/s47 /s44/s32 /s109 /s45 /s49/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s109/s97/s120\n/s109/s105/s110\n/s40/s98/s41\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s87/s97/s118/s101/s110/s117/s109 /s98/s101/s114/s32/s49/s47 /s44/s32 /s109 /s45 /s49/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102/s44/s32/s71/s72/s122\n/s48/s46/s48/s48/s48/s50/s48/s48/s46/s48\n/s109 /s105/s110/s109 /s97/s120\n/s40/s99/s41\nFIG. 6. a) Dispersion curves for MSSW that propagate in continuous F, S/F, and S/F/S \flms. Dispersion curves that are\nobtained numerically are shown with solid lines. Dispersion curves that are obtained using analytical expression are shown with\ndashed lines. b,c) Spin-wave spectra of S/F/S based magnonic crystals that are formed in MSSW geometry with the lattice\nparametera= 1\u0016m. Inserts in (b,c) show schematic illustrations of the considered magnonic crystals.\nwavenumbers, while the proximity-induced anisotropy is\ndominating in vicinity to 0 wavenumbers.\nFigures 6b,c show the spin-wave spectrum of S/F/S-\nbased magnonic crystals where periodicity of the dis-\npersion is reached by periodic location of S/F/S-three-\nlayered areas. Figure 6b shows the spectrum of the hy-\nbrid magnonic crystal that consists of alternating F and\nS/F/S sections (see the inset) with the lattice period\na= 1\u0016m, the width of F-section 0.5 \u0016m, and the thick-\nness of S-layers 120 nm. Calculating this spectrum both\nthe diamagnetic representation of S-stripes18,19and local\nS/F/S-induced anisotropy are considered. The spectrum\ncan be characterized as conventional one: it consist of\nallowed and forbidden bands, the forbidden bands are\nopened at Brillouin wavenumbers 1 =2a. The width of\nband gaps reduces at higher frequencies. For instance,\nthe \frst (lower-frequency) band gap is of width about\n1.8 GHz, and the second band gap is of width 1 GHz.\nFigure 6c shows the spectrum for an alternative re-\nalization of the hybrid magnonic crystal, which consists\nof alternating F/S and S/F/S sections (see the inset).\nLower S-subsystem forms a continuous layer, so the struc-\nture is spatially asymmetric in respect to the z-axis. For\nthis structure similar geometrical parameters are consid-\nered: the lattice period a= 1\u0016m, the width of F/S-\nsection 0.5 \u0016m, and the thickness of the upper S-layers\n120 nm. Calculating this spectrum the diamagnetic rep-\nresentation of S-stripes18,19, the image method17,51have\nbeen used for \fnite-size and continuous superconduct-\ning elements, respectively. The e\u000bect of the proxim-\nity in S/F/S sections is represented by the same local\nanisotropyHs. The spectrum for this spatially asymmet-\nric structure is di\u000berent. The spectrum consist of allowed\nand forbidden bands. The forbidden bands are of similar\nwidth as in Fig. 6b: the \frst (lower-frequency) band gap\nis of width about 1.7 GHz, and the second band gap isof width 0.9 GHz. However, spatial asymmetry induces\nnonreciprocity of the spectrum and indirect location of\nband gaps away from Brillouin wavenumbers.\nIt should be noted that in both cases the e\u000bect of the\nproximity in S/F/S sections is dominating for formation\nof band gaps: in absence of this e\u000bect forbidden bands\nare not obtained. This can be explained by a rather weak\ndiamagnetic response of S-subsystems on spin waves with\nconsidered wavelength. However, diamagnetic response\nof S-subsystems does a\u000bect frequency and wavenumber\nposition of allowed and forbidden bands .\nAs a \fnal remark we should note that for magnonic\ncrystals with thicker F-layers the bandwidth of the for-\nbidden bands is expected to increase correlating with\nthe zero-temperature anisotropy \feld. In particular, the\nbandwidth of the forbidden bands for a S/F/S-based\nmagnonic crystal with F-layer of thickness of a few hun-\ndreds of nm is expected to be comparable with values for\nbi-componental magnonic crystals52,53.\nVI. CONCLUSION\nSummarizing, magnetization dynamics is studied\nin superconductor/ferromagnet/superconductor multi-\nlayers in presence of superconducting proximity. It\nis shown that superconductivity in S/F/S three-layers\nshifts the FMR to higher frequencies. Presence of\nboth S-layers and proximity at both S/F interfaces are\nmandatory for the phenomenon. The frequency shift\nis quanti\fed by the proximity-induced positive in-plane\nanisotropy Haand by a drop of e\u000bective magnetization\nMeff. BothHaand the drop of Meffare comparable.\nThe phenomenon shows a dependence on the thickness of\nthe F-layer: for thicker F-layer the shift of the FMR fre-\nquency is substantially stronger. For two studied samples9\nwith thickness of the F-layer 45 and 350 nm the highest\nnatural FMR frequencies and corresponding anisotropies\nare reached among in-plane magnetized ferromagnetic\nsystems. At the current stage even a qualitative explana-\ntion of the e\u000bect of superconducting proximity in S/F/S\nsystems on magnetization dynamics is unavailable.\nApplication of the proximity-induced anisotropies for\nmanipulation with the spin-wave spectrum is demon-\nstrated for continuous \flms and periodic magnonic\ncrystals. 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Phys.\n111, 064326 (2012)."    },    {        "title": "1312.3663v1.Microwave_manipulation_of_electrically_injected_spin_polarized_electrons_in_silicon.pdf",        "content": "arXiv:1312.3663v1  [cond-mat.mes-hall]  12 Dec 2013Microwave manipulation of electrically injected spin pola rized electrons in silicon\nC. C. Lo∗and J. J. L. Morton\nLondon Centre for Nanotechnology, University College Lond on, London WC1H 0AH, U.K. and\nDepartment of Electronic and Electrical Engineering,\nUniversity College London, London WC1E 7JE, U.K.\nJ. Li and I. Appelbaum\nDepartment of Physics and the Center for Nanophysics and Adv anced Materials,\nUniversity of Maryland, College Park, Maryland 20742, USA\n(Dated: November 2, 2017)\nWe demonstrate microwave manipulation of the spin states of electrically injected spin-polarized\nelectrons in silicon. Although the silicon channel is bound ed by ferromagnetic metal films, we\nshow that moderate microwave power can be applied to the devi ces without altering the device\noperation significantly. Resonant microwave irradiation i s used to induce spin rotation of spin-\npolarized electrons as they travel across a silicon channel , and the resultant spin polarization is\nsubsequently detected by a ferromagnetic Schottky barrier spin detector. These results demonstrate\nthe potential for combining advanced electron spin resonan ce techniques to complement the study\nof semiconductor spintronic devices beyond standard magne totransport measurements.\nThe spin degree-of-freedom of charge carriers in bulk\nsemiconductors has been studied by electron spin reso-\nnance (ESR) for decades, yielding invaluable spin relax-\nation information [ 1,2]. These measurements demon-\nstrated that the spin relaxation times in group IV semi-\nconductors, notably in silicon, are quite long. Silicon-\nbased spintronic devices are especially relevant due to\nthe widespread use ofthe material in conventionalmicro-\nelectronics, and hence using silicon as the basis for next\ngeneration spintronic devices is even more attractive [ 3].\nHowever,whilethespindegreeoffreedominsemiconduc-\ntor spintronic devices is easily accessible in many group\nIII-V and II-VI compound materials with optical tech-\nniques, these approaches are difficult to implement with\ngroup IV devices due to the indirect nature of the band\ngap [4], although optical detection of spin injection in\nsilicon has been demonstrated [ 5]. It is because of this\ndifficulty that most group IV-based semiconductor spin-\ntronic devices are studied by quasistatic electrical mag-\nnetotransport techniques. These approaches include the\nnon-local 4-terminal detection of spin accumulation with\nopen-circuit voltage [ 6,7] or spin current with hot elec-\ntron techniques [ 8–11], both of which reveal spin-valve\nand spin precession effects, and can be used to determine\nspin-related properties such as spin diffusion length and\nrelaxation times in prototypical spintronic devices.\nIt is thus interesting to consider incorporating elec-\ntron spin resonance techniques to aid the study of the\nunderlying physics, as advanced pulse sequences will al-\nlow the elucidation of the spin dynamics in these devices.\nWe note that ferromagnetic resonance, a closely related\ntechnique, hasbeen recently utilized to induce dynamical\nspin injection (spin pumping) in degenerately doped sil-\nicon [12]. In the present work, we operate at conditions\nresonant to the electrons in nearly intrinsic silicon, far\n∗cheuk.lo@ucl.ac.uk.awayfrom the ferromagnetic resonancesof the metal lay-\ners in the device. We show that resonant microwave ra-\ndiation can be used to induce spin rotation in the silicon\nchannel without severely affecting the device operation,\nthus paving the way toward pulsed ESR spin manipula-\ntion.\nThe four-terminal devices studied here consist of two\nferromagnetic (FM) thin films sandwiching a silicon (Si)\nlayer in between, forming a spin-valve [ 8]. A schematic\nof the device and definitions of the voltages and currents\nare shown in Fig. 1. The three components of the de-\nChannel DetectorEmitter\nCu/\nCoFeAl 2O3\nAl/Cu NiFe/Cu\nVE VSi n-SiID\nV/AsignalIBFMIC\nIAi-Si\nL = 225μm\nB BSpin-valve Hanle\nFIG. 1. Schematics and energy band diagram of the four\nterminal silicon spintronic device consisting of the emitt er,\nchannel and detector. The ferromagnetic layers are shaded\nin gray. Definitions of bias voltages, voltage polarities an d\ncurrent components (labelled in red) are also indicated. In set:\nOrientation of the ferromagnetic layers relative to the app lied\nmagnetic field Bin the spin-valve and Hanle measurement\ngeometries.2\nvice include the emitter, the Si channel and the detector.\nThe emitter consists of a CoFe FM cathode, Al 2O3tun-\nnel barrier and Al/Cu anode for injecting spin-polarized\nelectrons into the channel. The emitter covers an area\nof approximately 0 .02 mm2, and is activated by a bias\nvoltage of VE/lessorsimilar−0.7V reflecting the Schottky barrier\nheight, with the tunnel current ( IC) increasing quasi-\nexponentially for more negative voltages. At VEsev-\neral hundred mV above this threshold, only ∼0.1% of\nthe emitter current is ballistically injected into the sil-\nicon channel layer, which is nominally undoped Si and\nhas a thickness of L= 225µm. By applying different\nbias voltages across the silicon channel ( VSi), the aver-\nage electron transit time to the detector can be varied.\nThe spin detector consists of a buried FM (BFM) layer\nof NiFe, and a Schottky junction with an n-doped (phos-\nphorus) Si substrate. The current measured from the\ndetector ( ID, “signal” in Fig. 1) has a magnitude which\nis∼0.1% of the total injected current. IDdepends on\nthe polarization of the electrons reaching the detector\ndue to spin-dependent inelastic scattering in the BFM,\nand hence it is a measure of spin current. More details\nof the device fabrication and operation can be found in\nRef.13.\nAll measurements were carried out in an X-band ( ∼\n9.3GHz) ESR spectrometer (Bruker Elexsys 580) with a\nhelium flow cryostat and cylindrical dielectric resonator.\nThe devices were mounted and wire bonded on thin\nprinted circuit boards for establishing electrical connec-\ntions, and then loaded into the microwave resonator in\nthe cryostat for measurements. The inset of Fig. 1shows\nthe principle measurement configurations used in this\nstudy: We will refer to the in-plane magnetic field config-\nuration as the spin-valve geometry, and the out-of-plane\nconfiguration as the Hanle geometry.\nThe devices were first characterized by standard mag-\nnetotransportmeasurementstoextractthespinfliptimes\n(τs) and carrier transit times ( τt) across the silicon chan-\nnel. In all the measurements reported we used an emitter\nbias ofVE=−1.5 V (IC≈ −55 mA) as it was found\nto give the best signal-to-noise ratio. Fig. 2(a) shows the\nnormalized spin-valve measurements of IDfor various VSi\nat a cryostat temperature of T= 20 K (the actual de-\nvice temperature will be discussed shortly). The inset\nof Fig.2(b) shows precession oscillations observed when\nmonitoring IDin the Hanle configuration. These oscilla-\ntions unambiguously show that IDdoes indeed provide a\nspin-polarization signal [ 9], and are well described by a\nspin drift-diffusion model using current-sensing spin de-\ntectors [14,15]:\n∆ID∝/integraldisplay1√\n4πDtL\nte−L2(1−t/τt)2\n4Dte−t/τscos(ωt)dt,(1)\nwhereDis the electron carrier(and spin) diffusion coeffi-\ncient and ω=gµBB//planckover2pi1the Larmor precession frequency,\nwithgbeing the Land´ e g-factor,µBthe Bohr magne-\nton and /planckover2pi1the reduced Planck’s constant. The transitB (mT)ID(B)/ IPD\n−10 −5 0 5 10 0.800.850.900.951.00 (a)\n(b)\n(c)VSi  = 0V\n-2V\n-6V\n-15V\n-20V-10VP\nAP P\nAP P P\nτt (ns)IDP−IDAP /IDP+IDAP \n0 50 100 150 200 2500.040.050.060.070.080.09E = |VSi | / L (V/cm)τt1/  (ns −1 )\n   0  500 1000 1500 20000.000.050.100.150.200.250.300.35\n Δ ID (nA) \nτs = 140 ± 37 nsB (mT)−80 −40 0 40 80 −0.1   0 0.1\nFIG. 2. (a) Spin-valve measurements carried out at a cryo-\nstat temperature of 20K. (b, inset) Precession oscillation s\nmeasured in the Hanle configuration for VSi=−40V (red),\nwith the fitted model shown in dashed lines (gray), and the\nextracted peak electron transit times shown in (b). (c) Spin\nrelaxation time τsextraction based on the correlation between\nspin-valve signal amplitude and peak electron transit time s.\nThe blue horizontal lines indicate the FWHM of the transit\ntime distribution.3\ntime distribution across the channel can be found by ex-\namining the Fourier transform of the precession oscilla-\ntions [16]. Fig.2(b) shows the peak transit time as the\napplied electric field, E=|VSi|/L, is varied. In a per-\nfectly intrinsic silicon channel where no band bending is\nexpected, τtshould simply scale inversely with the ap-\nplied electric field in drift dominated carrier transport.\nThe two distinct regimes observed indicate that resid-\nual band bending and diffusive transport plays an in-\ncreasingly important role in the low electric field regime\nofE <500 V/cm. We combine the spin-valve and τt\ndata to deduce τsof electrons in the silicon channel as\n(IP\nD−IAP\nD)/(IP\nD+IAP\nD)∝e−τt/τs, where the superscripts\nP and AP correspond to the magnetic alignment of the\ntwo FM layers in the parallel and anti-parallel configu-\nrations, respectively. The results are shown in Fig. 2(c),\nand we obtain τs= 140±37ns. The blue horizontal bars\nateachdatapointindicatetheFWHM ofthetransittime\ndistribution, illustrating the significant diffusion-induced\nbroadening of carrier transit times for small drift fields.\nThe value of τs, which is equivalent to the spin-lattice\nrelaxation time T1in ESR literature, is significantly\nshorter than would be expected at the cryostat tem-\nperature of T= 20K [ 17], and indicate heating in\nthe device up to ≈120 K. Sample heating is also con-\nfirmed from standard ESR measurements of the back-\nground phosphorus signal in the n-Si detector, as we\nfound significant reduction in the hyperfine splitting of\nthe phosphorus donor-bound electrons when the emit-\nter bias was increased[ 18]. In fact, the hyperfine-split\ndonors can no longer be resolvedin ESR for VE/lessorsimilar−1.0V\n(IC≈ −16mA), indicating the sample temperature has\nalready risen to above the carrier freeze-out temperature\nof≈70K. We attribute this large amount of heating to\nthe inefficient thermal anchoring of the sample on the\nprinted circuit board in the flow cryostat, which is espe-\ncially important when large currents are passed through\nthe emitter for hot electron injection.\nToinvestigatetheeffectsofresonantmicrowavesonthe\ndevice, we turn on and fix the microwave excitation to\n9.73GHz at 80mW, and increase the in-plane magnetic\nfieldBto around 350mT, which correspondsto the reso-\nnance condition for g≈2 conduction electrons in silicon.\nWe note that at these magnetic fields the magnetization\nof the FM layers are always in the parallel configuration,\nhenceIDprovides a signal which is proportional to the\nprojection of the spin polarization along the magnetiza-\ntion of the injecting FM. Fig. 3(a) shows the variations\ninIDasBis swept, with VSi=−5 V and the sam-\nple oriented in the spin-valve configuration. A resonant\ndecrease in IDis observed at approximately 348 mT.\nTo confirm that the origin of the resonance signal\ncomes from microwave-induced variations in the electron\nspin polarization reaching the detector, and not due to\nother spin-dependent transport mechanisms or bolomet-\nric detection in either the Si channel or n-Si detector, we\nexamine the orientation and source of spin resonance sig-\nnalmorecarefully. We alsoemploylock-indetection withB (mT)∂I/∂B (pA/mT) \n346 347 348 349 350−10  0 10 20 30 40 50 60 70\nIBFM  (x10-2 )\nID Spin-valve Hanle ID (pA) \n415420\n(b)(a)\ng = 2.004(2) \nFIG. 3. Electrically detected spin resonance signal of the s ili-\ncon spintronic device. (a) Resonant signal detected in the D C\ncurrent change in detector current IDin the spin valve config-\nuration. (b) Lock-in detection of the buried ferromagnet cu r-\nrentIBFM(red) and detector current ID(blue, middle trace)\nin the spin-valve configuration. When configured in the Hanle\ngeometry, the resonance signal in IDvanishes (blue, bottom\ntrace).\nmagnetic field modulation, with 0 .2−0.3mT amplitude\nat 1kHz to improve the signal-to-noise ratio. The mid-\ndle trace in Fig. 3(b) shows the lock-in detected signal of\nID. The current collected by the BFM, shown as the top\ntrace in Fig. 3(b), does not reveal any resonance signal\nat all, implying that the origin of the signal is not due\nto spin-dependent transport mechanisms in the channel\nlayer,butratherdirectspinmanipulationbytheresonant\noscillating magnetic field.\nTo further eliminate the possibility that spin-\ndependent transport processesin the n-doped Si detector\nare responsible for the resonance signal in ID, we rotate\nthe sample to the Hanle geometry. In this configuration,\nthe signal disappears as shown in the bottom trace of\nFig.3(b), ruling out spin-dependent transport in the n-Si\nor bolometric detection as the source of the resonant sig-\nnal as no anisotropy is expected from such mechanisms.\nIndeed, most spin-dependent scattering processes, such\nas those among donor-conduction electron or conduction\nelectron-conduction electron, are expected to vanish at\ndevice temperatures of T >20K [19,20]. The only\nstrong spin-dependent transport mechanisms in silicon\nat these temperatures would be spin-dependent hopping\nor recombination [ 21], neither of which are applicable\nfor the present (undoped) devices. In fact, the van-\nishing signal in the Hanle geometry is to be expected,4−Δ IDP,res /( IDP−IDAP )\n  0 200 400 6000.000.010.020.030.040.050.06\nE = |VSi | / L (V/cm)\nFIG. 4. Electrically detected spin resonance signal amplit ude\nfor various applied electric fields at a cryostat temperatur e\nof 20K. The dashed line corresponds to fits to the data as\ndescribed in the main text.\nsince in this case the injected spin polarized electrons are\nquickly depolarized by the strong out-of-plane magnetic\nfield due to rapid Larmor precession, making the reso-\nnant microwave ineffective in inducing any signal vari-\nation. These measurements confirm that the resonance\nsignal in IDin the spin-valve configuration is indeed due\nto microwave-induced rotation of the electron spins in\nthe silicon channel, and its subsequent projection to the\nBFM magnetization, as detected by ID.\nWe can now examine the effect of the applied electric\nfield on the resonant change in detector current. The\nresonant microwave for a given power induces rotation of\nthe electron spins at the Rabi frequency ω1=gµBB1//planckover2pi1,\nwhereB1is the amplitude of the microwave magnetic\nfieldcomponent. As theaverage τtincreases(byreducing\nE), so should the electron rotation angle, and hence an\nincreasedspin resonancesignal. Neglecting spin diffusion\neffects, we can show that:\n∆IP,res\nD\nIP\nD−IAP\nD≈sin2(ω1τt/2), (2)\nwhere ∆ IP,res\nD=IP,on−res\nD−IP,off−res\nDis the integrated\nelectrically detected spin resonance signal amplitude af-ter lock-in detection. Fig. 4shows the evolution of\n∆IP,res\nDnormalized in this way for various applied elec-\ntric fields across the channel. The dashed line is a fit\nwith Eq. 2assuming B1= 30µT. We note that due to\nthe relatively small B1field in continuous wave measure-\nments,ω1≈5×106rad/s, which is slow compared with\nthe transit times achievable in the device, hence no Rabi-\nlike oscillations are expected. In addition, the significant\nbroadening of the transit time distribution (Fig. 2(c)) di-\nminishes the achievable oscillation amplitudes. We also\nfound increased significanceof microwaverectification ef-\nfects at low bias electric fields; improved sample designs\noptimized for microwave resonator-based ESR measure-\nments can be used to reduce such detrimental rectifica-\ntion effects [ 22], and allow high microwave power pulsed\nESR measurements to be performed on the devices.\nIn summary, we have applied standard magnetotrans-\nport measurements and combined it with electrical de-\ntection of spin resonance to study prototypical silicon\nspintronic devices. We have shown that the applied spin\nresonance microwaves can be used to manipulate the in-\njectedelectronspinstatesinthesilicontransportchannel\nin the spin valve geometry. The vanishing resonance sig-\nnals in the Hanle geometry clearly rules out other spin-\ndependent transport mechanisms as the source of the\nresonance signal. This combination of electron spin res-\nonance and magnetotransport measurements can be an\ninvaluable tool for investigating other spintronic mate-\nrials and devices, as electron spin resonance techniques\ncan provide precise spectroscopic information and unam-\nbiguous interpretation of the temporal spin dynamics in\nthese systems.\nACKNOWLEDGMENTS\nWork at UCL is supported by the European Re-\nsearchCouncil underthe EuropeanCommunitysSeventh\nFramework Programme (FP7/20072013)/ERC (grant\nagreementno. 279781). C.C.L.issupportedbytheRoyal\nCommission for the Exhibition of 1851. J.L. and I.A. ac-\nknowledge the support of the Maryland NanoCenter and\nits FabLab at U. of Maryland, and funding by the Of-\nfice of Naval Research under contract N000141110637,\nthe National Science Foundation under contracts ECCS-\n0901941 and ECCS-1231855, and the Defense Threat\nReduction Agency under contract HDTRA1-13-1-0013.\nJ.J.L.M. is supported by the Royal Society.\n[1] J. H. Pifer. 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Physical Review Letters ,\n110:127201, 2013.\n[13] Jing Li, Lan Qing, Hanan Dery, and Ian Appelbaum.\nField-induced negative differential spin lifetime in silic on.\nPhys. Rev. Lett. , 108:157201, 2012.[14] I. Appelbaum and D. J. Monsma. Transit-time spin\nfield-effect transistor. Applied Physics Letters , 90:262501,\n2007.\n[15] Jing Li and Ian Appelbaum. Modeling spin transport\nwith current-sensing spin detectors. Appl. Phys. Lett. ,\n95:152501, 2009.\n[16] H. J. Jang and I. Appelbaum. Spin polarized electron\ntransport near the si/sio 2interface. Physical Review Let-\nters, 103:117202, 2009.\n[17] J. L. Cheng, M. W. Wu, and J. Fabian. Theory of the\nspinrelaxationofconductionelectrons insilicon. Physical\nReview Letters , 104(1):016601, 2010.\n[18] D. E. Dugdale, S. D. Lacey, and G. Lancaster. A temper-\nature dependent hyperfine interaction in n-type silicon.\nJ. Phys. C: Solid St. Phys. , 4:654, 1971.\n[19] C. C. Lo, V. Lang, R. E. George, J. J. L. Morton, A. M.\nTyryshkin, S. A. Lyon, J. Bokor, and T. Schenkel. Elec-\ntrically detected magnetic resonance of neutral donors\ninteracting with a two-dimensional electron gas. Physi-\ncal Review Letters , 106(20):207601, 2011.\n[20] C. C. Lo. Electrical detection of spin-dependent transport\nin silicon . PhD thesis, University of California, Berkeley,\n2011.\n[21] M. S. Brandt, S. T. B. Goennenwein, T. Graf, H. Huebl,\nS. Lauterbach, and M. Stutzmann. Spin-dependent\ntransport in elemental and compound semiconductors\nand nanostructures. Physica Status Solidi (C) , 1(8):2056,\n2004.\n[22] C. C. Lo, F. R. Bradbury, A. M. Tyryshkin, C. D. Weis,\nJ. Bokor, T. Schenkel, andS. A. Lyon. Suppressionof mi-\ncrowave rectification effects in electrically detected mag-\nnetic resonance measurements. Applied Physics Letters ,\n100(6):063510, 2012."    },    {        "title": "1310.7996v1.Spin_torque_ferromagnetic_resonance_with_magnetic_field_modulation.pdf",        "content": "Spin torque ferromagnetic resonance with magnetic \feld modulation\nA. M. Gon\u0018 calves,1, 2I. Barsukov,1Y.-J. Chen,1L. Yang,1J. A. Katine,3and I. N. Krivorotov1\n1Physics and Astronomy, University of California, Irvine, CA 92697\n2Centro Brasileiro de Pesquisas F\u0013 \u0010sicas, Rua Dr. Xavier Sigaud, 150, Rio de Janeiro 22.290-180, RJ, Brazil\n3HGST Research Center, San Jose, CA 95135\nWe demonstrate a technique of broadband spin torque ferromagnetic resonance (ST-FMR) with\nmagnetic \feld modulation for measurements of spin wave properties in magnetic nanostructures.\nThis technique gives great improvement in sensitivity over the conventional ST-FMR measure-\nments, and application of this technique to nanoscale magnetic tunnel junctions (MTJs) reveals\na rich spectrum of standing spin wave eigenmodes. Comparison of the ST-FMR measurements\nwith micromagnetic simulations of the spin wave spectrum allows us to explain the character of\nlow-frequency magnetic excitations in nanoscale MTJs.\nThe following article appeared in Appl. Phys. Lett. 103, 172406 (2013) and may be found\nat http://dx.doi.org/10.1063/1.4826927\nCopyright (2013) American Institute of Physics. This article may be downloaded for personal use\nonly. Any other use requires prior permission of the author and the American Institute of Physics.\nNanoscale magnetic tunnel junctions are promising\ncandidates for non-volatile spin torque memory (STT-\nRAM) [1{5] and current-controlled microwave oscillators\n[6{11]. Optimization of MTJ performance for these ap-\nplications requires quantitative evaluation of the spin\ntorque (ST) vector [12], magnetic damping[13], voltage-\ncontrolled magnetic anisotropy [14{16], and the spectrum\nof magnetic excitations of the MTJ [17, 18]. A common\ntechniques for quantitative measurements of these prop-\nerties is ST ferromagnetic resonance (ST-FMR) [19, 20].\nThere are three main realizations of ST-FMR: (i) time\ndomain [21], (ii) network analyzer [22], and (iii) recti\f-\ncation [19] methods. The \frst two methods employ non-\nzero bias current while the recti\fcation technique allows\nstudies of MTJ magneto-dynamics at zero bias. In the\nrecti\fcation ST-FMR, a microwave current Iaccos(2\u0019ft)\n\rowing across the MTJ applies ST to the magnetization\nand thereby induces resistance oscillations Raccos(2\u0019ft+\n ). Mixing of the current and resistance oscillations re-\nsults in a direct voltage Vmix=1\n2IacRaccos( ) generated\nby the sample. Measurements of the ST-FMR spectrum\nVmix(f) give a series of spectral peaks arising from spin\nwave resonances of the MTJ [19, 20, 25], which can give\ncomprehensive information on voltage-driven magneto-\ndynamics in the sample. Despite its popularity, the\nrecti\fcation technique su\u000bers from frequency-dependent\nbackground signals due to non-linearities and impedance\nmismatches within the microwave circuit. For example,\nmeasurements of MTJ with collinear free and pinned\nlayer magnetizations (the STT-RAM geometry) are chal-\nlenging with the recti\fcation ST-FMR because magnetic\nsignals are typically weaker than the backgrounds.\nIn this Letter we present a simple technique that cir-\ncumvents the drawbacks of the conventional ST-FMR\nand gives reliable data for an arbitrary magnetic state\nof the MTJ, including the collinear con\fguration. We\nsolve the problem of the parasitic backgrounds by using\nmagnetic \feld modulation. The modulation \feld of a\nfew Oersteds is applied to the sample by augmenting aconventional ST-FMR setup with a copper wire that car-\nries kHz-range sinusoidal current of a few Amperes and\nis placed directly above the MTJ sample as shown in\nFig. 1(a). The modulation \feld from the wire is collinear\nwith the dc applied magnetic \feld at the sample loca-\ntion. A continuous microwave current is applied to the\nsample via a bias tee, and a recti\fed voltage generated\nby the sample is measured by a lock-in ampli\fer at the\n\feld modulation frequency. For reference, we also make\nconventional ST-FMR measurements in which the sinu-\nsoidal magnetic \feld modulation is replaced by a square\nwave amplitude modulation of the microwave current.\nWe make ST-FMR measurements of magnetization\ndynamics in a 170 \u000290 nm2elliptical MTJ nanopil-\nlar with in-plane magnetizations of the pinned and\nfree layers. The nanopillar is patterned by ion\nmilling from a Ta(5)/SAF/MgO(1.024)/FL/Ta(5) mul-\ntilayer (thicknesses in nm) with resistance-area prod-\nuct of 14 \n \u0001\u0016m2deposited by magnetron sputter-\ning in a Singulus TIMARIS system. Here SAF =\nPtMn(15)/ Co 70Fe30(2.5)/ Ru(0.85)/ Co 40Fe40B20(2.4)\nis the synthetic antiferromagnet pinned layer and FL =\nCo60Fe20B20(1.8) is the free layer. Prior to patterning,\nthe multilayers are annealed for 2 hours at 300\u000eC in a 1\nTesla in-plane magnetic \feld that sets the pinned layer\nexchange bias direction parallel to the long axis of the\nnanopillar. For evaluation of the \feld-modulated ST-\nFMR method, we employ the collinear con\fguration, in\nwhich an external magnetic \feld of 30 mT is applied along\nthe easy axis of the nanopillar. This con\fguration is\nparticularly unfavorable for ST-FMR measurements be-\ncause (i) ST proportional to the sine of the angle be-\ntween the pinned and the free layer magnetizations is\nsmall and (ii) resistance oscillations at the frequency of\nthe microwave drive are small. Not surprisingly, the ST-\nFMR spectrum for this con\fguration measured with the\namplitude modulation technique (Fig. 1(c)) is dominated\nby a frequency-dependent non-magnetic background. In\ncontrast, \feld-modulated ST-FMR spectrum shown inarXiv:1310.7996v1  [cond-mat.mes-hall]  30 Oct 20132\nFIG. 1: (Color online) (a) Sketch of the \feld-modulation ST-FMR setup. Comparison of ST-FMR spectra measured by (b)\n\feld and (c) amplitude modulation techniques for external magnetic \feld of 30 mT applied along the easy axis of a 170 \u000290 nm2\nelliptical MTJ nanopillar. Field modulation e\u000eciently removes large frequency-dependent background of non-magnetic origin\nand reveals four spin wave eigenmodes of the MTJ.\nFig. 1(b) is nearly background-free and reveals four res-\nonances arising from excitation of spin wave eigenmodes\nof the MTJ.\nWe next employ the \feld-modulated ST-FMR for stud-\nies of MTJ spin wave eigenmodes as a function of external\nmagnetic \feld applied in the plane of the sample parallel\nto the hard axis of the MTJ nanopillar. Fig. 2(a) gives\na color plot summary of ST-FMR spectra measured for\nan 85\u000250 nm2elliptical MTJ nanopillar as a function of\nhard-axis magnetic \feld. As the resonance signals are\npredominantly antisymmetric (similar to Fig. 1(b)), the\nresonance frequencies are found in the vicinity of zero-\ncrossing of the recti\fed voltage. Three prominent spin\nwave resonances labeled as 1, 2, and 3 are seen in the\nentire magnetic \feld range. The \frst mode exhibits the\nhighest intensity, lowest frequency, and a \feld dispersion\nwith a frequency minimum, which is characteristic of the\nhard-axis quasi-uniform mode. The nature of the higher\nfrequency spin wave modes is not immediately clear. The\nfrequency of the second mode monotonically increases\nwith the \feld, while the third mode shows a frequency\nminimum at a \feld higher than that of the \frst mode.We also observe a resonance labeled 0.5 in Fig. 2(a) for\nwhich frequency and linewidth are exactly half of those of\nthe \frst mode. Therefore, the 0.5 resonance is not a real\nspin wave mode but an artifact arising from frequency\ndoubling of the current by nonlinearities of the circuit.\nNext, we derive an expression describing the spec-\ntral line shape measured by the \feld-modulated ST-\nFMR. The line shape Vmix(f) without \feld modulation\nis a sum of symmetric S(f) and antisymmetric A(f)\nLorentzians [24] Vmix(f) =VsS(f) +VaA(f), where\nS(f) =1\n1+(f\u0000fr)2=\u001b2r,A(f) =(f\u0000fr)=\u001br\n1+(f\u0000fr)2=\u001b2r,fris the reso-\nnance frequency and \u001bris the linewidth. The amplitudes\nof the Lorentzians, VsandVa, are functions of the ST vec-\ntor, magnetic parameters of the system [24], and voltage\ncontrolled magnetic anisotropy [16]. When the modula-\ntion \feld is small compared to the resonance linewidth in\nthe \feld domain, the RMS voltage signal ~Vmix(f) mea-\nsured by the lock-in ampli\fer is proportional to the \frst\nderivative of the recti\fed voltage Vmix(f) with respect to\nthe modulated variable{the external magnetic \feld B:\n~Vmix(f) =BmdVmix(f)\ndB=Bm\u0014dVs\ndBS(f) +dVa\ndBA(f) +1\n\u001brd\u001br\ndB\u0000\n2VsA2(f) +Va\u0002\n2A3(f)=S(f)\u0000A(f)\u0003\u0001\n(1)\n+1\n\u001brdfr\ndB\u0000\n2VsS(f)A(f) +Va\u0002\nA2(f)\u0000S2(f)\u0003\u0001\u0015\n;\nwhereBmis the RMS amplitude of the modulation \feld,\nand the last term proportional to d fr=dBis usually dom-\ninant. IfVsandVaare weak functions of magnetic \feld\nthen the symmetric part of ~Vmix(f) is proportional toVaand the antisymmetric part is proportional to Vs. For\nour samples, the resonances are nearly antisymmetric and\nthusVs\u001dVa, which means that the \feld-like compo-\nnent of ST is much smaller than the in-plane component3\nFIG. 2: (Color online) (a) Field-modulated ST-FMR spectra measured for an 85 \u000250 nm2elliptical MTJ nanopillar as a function\nof hard-axis magnetic \feld. (b) MTJ spin wave spectra versus hard-axis magnetic \feld derived from micromagnetic simulations.\nThe color plot shows FFT amplitude of the MTJ out-of-plane magnetic moment excited by a sinc-shaped current pulse.\n[12, 24].\nTo understand the nature of the observed spin wave\nexcitations, we perform OOMMF [23, 25] micromagnetic\nsimulations of magnetization dynamics. To fully account\nfor all magnetic interactions in the MTJ, we employ a\nthree dimensional model with three ferromagnetic layers:\nfree, SAF top and SAF bottom. We use material param-\neters obtained from independent measurements and/or\ntheir accepted literature values [26]. In the simulations,\nspin wave dynamics is excited by a combined pulse of\nST and Oersted \feld, both resulting from a sinc-shaped\nJcsin (2\u0019fct0)\n2\u0019fct0spatially uniform current pulse [27] with\nthe amplitude Jc= 9\u0002106A/cm2, the cut-o\u000b frequency\nfc= 20 GHz and the time variable t0=t\u0000500 ps. The\nspatial pro\fle of the Oersted \feld is assumed to be that\nof a long wire with elliptical cross section. The direction\nof the ST vector acting on the free layer is determined\nby the magnetization orientation of the SAF top layer.\nThe spectrum of spin wave eigenmodes is obtained via\nthe Fast-Fourier-transform (FFT) of the time dependent\nout-of-plane component of the MTJ net magnetic mo-\nment. The simulated FFT spectra are shown in Fig. 2(b)\nas a function of external magnetic \feld. Several spin\nwave modes are clearly seen in Fig. 2(b), of which three\nmodes stand out due to their large amplitude. The \feld\ndispersion of these modes is in a good agreement with\nthe experimental data shown in Fig. 2(a). The \frst and\nthird modes have a frequency minimum at approximately\n50 mT and 75 mT { values similar to those observed in\nthe experiment. The absence of the 0.5 resonance in the\nsimulation con\frms that this feature is an artifact.\nTo identify the character of the observed spin wavemodes, we perform layer- and spatially resolved analy-\nsis of the mode amplitude. The frequency dependence\nof the FFT precession amplitude at 50 mT \feld, aver-\naged over all micromagnetic cells of each magnetic layer\nis shown in Fig. 3(a). While static magnetic moments of\nall three layers are similar to each other, the amplitude\nof the free layer precession is signi\fcantly higher than\nthat of the SAF layers. Thus we classify the observed\nspin wave excitations as the free layer modes. Analysis\nof spatial distribution of the amplitude and phase of the\nexcitations in the free layer, shown in Fig. 3(b), reveals\nthe nature of these spin wave modes. The amplitude and\nthe phase of the \frst mode are homogeneous across the\nentire free layer, which means that this mode is the free\nlayer quasi-uniform mode. The second mode has maxi-\nmum excitation amplitude at the edges of the free layer\nwith a node along the short axis. The phase of the oscil-\nlations undergoes a 180\u000eshift across the node indicating\nthat the second mode is antisymmetric. The spatial pro-\n\fle of the third mode is qualitatively similar to that of\nthe second mode, but the node is along the long axis of\nthe ellipse. Deviations of the nodes from the symmetry\naxes of the ellipse are due to a stray \feld from the SAF\nlayer. Given the imperfections of the real MTJ structure\nand uncertainties in the MTJ material parameters, the\nenergies of the spin wave eigenmodes found in simulations\nare in good agreement with the experimental results.\nIn conclusion, we developed a method of \feld-\nmodulated ST-FMR for characterization of magnetiza-\ntion dynamics in nanostructures and compared it to\nthe conventional amplitude-modulated ST-FMR. The\n\feld-modulated ST-FMR technique suppresses large non-4\nFIG. 3: (Color online) Micromagnetic simulations of spin wave eigenmodes of 85 \u000250 nm2MTJ nanopillar at 50 mT hard axis\nmagnetic \feld: (a) Precession amplitude of the free layer, SAF top and bottom layers as a function of frequency. (b) Color-coded\nspatial maps of precession amplitude and phase of the three dominant spin wave modes in the MTJ free layer.\nmagnetic background signals and thereby reveals a rich\nspectrum of spin wave eigenmodes. Using the \feld-\nmodulated ST-FMR, we measured the spectrum of spin\nwaves in nanoscale elliptical MTJs as a function of hard-\naxis magnetic \feld. Comparison of the ST-FMR data\nwith micromagnetic simulations shows that the observed\nmodes are low-energy eigenmodes of the free layer.We thank J. Langer for magnetic multilayer deposition\nand R. Arias for helpful discussions. This work was sup-\nported by SRC and Intel through grant No. 2011-IN-2152\nas well as by NSF through Grants No. DMR-1210850,\nNo. DMR-0748810, and No. ECCS-1002358. A. M. G.\nthanks CAPES Foundation, Ministry of Education of\nBrazil for funding through process number 4977-11-4.\n[1] J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989)\n[2] Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. Valet,\nAppl. Phys. Lett. 84, 3118 (2004)\n[3] G. D. Fuchs, N. C. Emley, I. N. Krivorotov, P. M. Bra-\nganca, E. M. Ryan, S. I. Kiselev, J. C. Sankey, D. C.\nRalph, R. A. Buhrman, and J. A. Katine, Appl. 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Phys. 97,\n10J901 (2005)\n[26] Gilbert damping \u000b=0.01, spectroscopic split-ting\r=1.759 \u00021011rad/s \u0001T, SAF coupling\n\u001bAF=-3.88 \u000210\u00004J/m2, free layer to SAF cou-\npling\u001bF=5.0 \u000210\u00006J/m2,MS;free=855 kA/m,\nMS;SAF-top =1050 kA/m, MS;SAF-bot =1650 kA/m,\nexchange bias \feld Bex. bias = 0:11 T\n[27] G. Venkat, D.Kumar, M. Franchin, O. Dmytriiev, M.\nMruczkiewicz, H. Fangohr, A.Barman, M. Krawczyk,\nand A. Prabhakar, IEEE Trans. Magn. 49, 524 (2013)"    },    {        "title": "1410.3781v1.Coherent_coupling_between_ferromagnetic_magnon_and_superconducting_qubit.pdf",        "content": "Coherent coupling between ferromagnetic magnon and superconducting qubit\nY. Tabuchi1, S. Ishino1, A. Noguchi1, T. Ishikawa1, R. Yamazaki1, K. Usami1, and Y. Nakamura1;2\n1Research Center for Advanced Science and Technology (RCAST),\nThe University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan and\n2Center for Emergnent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan\n(Dated: October 15, 2014)\nRigidity of an ordered phase in condensed mat-\nter results in collective excitation modes spatially\nextending in macroscopic dimensions1. Magnon\nis a quantum of an elementary excitation in the\nordered spin system, such as ferromagnet. Be-\ning low dissipative, dynamics of magnons in fer-\nromagnetic insulators has been extensively stud-\nied and widely applied for decades in the con-\ntexts of ferromagnetic resonance2,3, and more re-\ncently of Bose-Einstein condensation4as well as\nspintronics5,6. Moreover, towards hybrid systems\nfor quantum memories and transducers, coupling\nof magnons and microwave photons in a resonator\nhave been investigated7{10. However, quantum-\nstate manipulation at the single-magnon level has\nremained elusive because of the lack of anhar-\nmonic element in the system. Here we demon-\nstrate coherent coupling between a magnon exci-\ntation in a millimetre-sized ferromagnetic sphere\nand a superconducting qubit, where the interac-\ntion is mediated by the virtual photon excitation\nin a microwave cavity. We obtain the coupling\nstrength far exceeding the damping rates, thus\nbringing the hybrid system into the strong cou-\npling regime. Furthermore, we \fnd a tunable\nmagnon-qubit coupling scheme utilising a para-\nmetric drive with a microwave. Our approach\nprovides a versatile tool for quantum control and\nmeasurement of the magnon excitations and thus\nopens a new discipline of quantum magnonics.\nSingle electron spins, being a natural and genuine two-\nlevel system, play crucial roles in numerous applications\nin quantum information processing. The intrinsic draw-\nbacks, however, are its small magnetic moment \u0016B, the\nBohr magneton, and the limited spatial extension of the\nelectron wavefunction, making coherent coupling with an\nelectromagnetic \feld rather weak. To circumvent the\nproblems, paramagnetic spin ensembles have been ac-\ntively studied using atoms11, NV centres12,13, and rare-\nearth ions in a crystal14,15. The coupling strength is\nlargely enhanced by the square-root of the number of\nspins involved. At the same time, a collective spin excita-\ntion mode, which matches the input electromagnetic-\feld\nmode, is spanned in the spatially and spectrally extended\nensemble. However, with an increased spin density for\nstronger coupling, the spin-spin interactions among the\nensemble drastically degrade the coherence of the system\nand thus make a trade-o\u000b.\nWe move one-step further by introducing ferromag-nets. Even though they typically have a spin density sev-\neral orders of magnitude higher, the strong exchange and\ndipolar interactions among the spins dominate their dy-\nnamics and form narrow-linewidth magnetostatic modes.\nThe simplest mode has the uniform spin precessions of\nthe rigid spins in the whole volume, called the Kit-\ntel mode. Coherent coupling between the Kittel-mode\nmagnons and microwave photons in a cavity was recently\ndemonstrated in the quantum regime8.\nSuperconducting qubits are also an excellent example\nof quantized collective excitations in macroscopic-scale\nelectrical circuits, where the nonlinearity of Josephson\njunctions plays a crucial role for the realisation of the\nqubit, i.e., an e\u000bective two-level system. The progress\nin the last decade has made the qubits and their inte-\ngrated circuits one of the most advanced technologies for\nquantum information processing16{18. In the setups of\ncircuit quantum electrodynamics, a qubit as an arti\fcial\natom is coupled strongly to a microwave resonator19or\na waveguide20. They allow precise control and readout\nof the qubit states as well as synthesis and characterisa-\ntion of arbitrary quantum states in the microwave modes\ncoupled to the qubits21. These techniques can readily\nbe applied to quantum engineering of other physical sys-\ntems.\nIn this Letter, we demonstrate a hybrid quantum\nsystem which combines two heterogeneous collective-\nexcitation modes, i.e., the Kittel mode in a ferromagnetic\ncrystal and a superconducting qubit. The nonlinearity\nbrought by the qubit is the critical element to allow non-\nclassical control of magnon excitations.\nOur experimental setup is illustrated in Fig. 1. A\ntransmon-type superconducting qubit and a single-\ncrystalline yttrium-iron-garnet (YIG) sphere are\nmounted in a microwave cavity. The qubit with a 0.7-\nmm-long dipole antenna has a resonant frequency !q=2\u0019\nof 8.136 GHz. It strongly couples to the electric \felds\nof the cavity modes; e.g., the coupling strength gq=2\u0019\nbetween the qubit and the TE 102(transverse electric)\nmode at!102=2\u0019= 8:488 GHz is 121 MHz (See Sup-\nplementary Information for other detailed parameters).\nThe YIG sphere with a diameter of 0.5 mm is glued to\nan aluminium-oxide rod and mounted near the anti-node\nof the magnetic \feld of the TE 102mode. We also apply\na local static \feld Bstatic\u00180.29 T which makes the YIG\nsphere a single-domain ferromagnet (Ext. Fig. 1). The\nsphere now has an enormous magnetic dipole moment\nN\u0016Bwhich couples strongly to the magnetic \feld of\nthe cavity mode. The large enhancement factor N=arXiv:1410.3781v1  [quant-ph]  14 Oct 20142\n1:4\u00021018is the number of net electron spins in the\nsphere; we take an advantage of the high spin density\nas compared to that of previously studied paramagnetic\nsystems.\nWe perform a series of spectroscopic measurements in a\ndilution refrigerator at T= 10 mK. All the data are taken\nin the quantum regime where very few thermally-excited\nphotons and magnons exist. The average probe-photon\nnumber in the cavity is also kept below one.\nTo characterise the coupling between the magnon and\nphoton, we perform spectroscopy with the qubit fre-\nquency for detuned. Figure 1b shows the normal-mode\nsplitting between TE 102mode and the Kittel mode in the\nYIG sphere. The pronounced anticrossing indicates the\nstrong coupling between the two systems8. We obtain the\ncoupling strength, gm=2\u0019=21.0 MHz, and the linewidths\nof the TE 102and Kittel modes, \u0014102=2\u0019=2.5 MHz and\n\rm=2\u0019=1.4 MHz, from the \ft. The additional splitting\nseen in the upper branch originates from another mag-\nnetostatic mode which is detuned from the Kittel mode\nby 4.3 MHz and is coupled to the cavity mode with a\nstrength of 4.2 MHz.\nWhile the qubit and the magnon electrically and mag-\nnetically couple to the cavity mode respectively, they\nhave negligibly small direct interaction. Therefore, we\n\frst establish a static coupling scheme between the\nmagnon and the qubit by utilizing the presence of the\ncavity mode (Fig. 2a)22. We tune the qubit and the\nmagnon frequencies, !qand!FMR, while both are far\ndetuned from the cavity frequency !c(\u0011!102). When\n!q'!FMR, coherent exchange of the qubit excitation\nand a magnon is mediated by the virtual-photon excita-\ntion in the cavity mode. The interaction is described by\na Jaynes-Cummings-type Hamiltonian, which is written\nas\n^Hqm,s=~=gqm,s^\u001b+^c+g\u0003\nqm,s^\u001b\u0000^cy; (1)\nwheregqm,s=gqgm=\u0001 is the e\u000bective qubit-magnon cou-\npling strength, and ^\u001b\u0000[= (^\u001b+)y] and ^care annihilation\noperators of the qubit and the magnon, respectively. The\ndetuning \u0001 is the di\u000berence between the bare frequencies\nof the qubit and the cavity mode (See Supplementary In-\nformation). The \frst and the second excited states of the\nhybridised system are bonding and antibonding states\nbetween the qubit and the magnon excitation.\nTo demonstrate the qubit-magnon coupling, we per-\nform qubit excitation spectroscopy by using the qubit\nreadout through the cavity TE 103mode. Despite\nthe large detuning, the TE 103mode at!103=2\u0019=\n10:461 GHz is subject to a dispersive frequency shift:\nthe change in the cavity re\rection coe\u000ecient, Re(\u0001 r), at\n!103re\rects the qubit state. Figure 2b shows Re(\u0001 r) as\na function of the excitation microwave frequency and the\nstatic magnetic \feld. The anticrossing is a manifestation\nof the magnon-vacuum-induced Rabi splitting, which in-\ndicates coherent coupling between the qubit and the Kit-\ntel mode. From the size of the splitting, the e\u000bective cou-\npling strength gqm,s=2\u0019=10.0 MHz is obtained. The cou-pling strength far exceeds the linewidths of the qubit and\nthe magnon, \rq=2\u0019= 1:2 MHz and \rm=2\u0019= 1:3 MHz,\nrespectively. The value is also in a good agreement with\nthe calculated value of 11.8 MHz based on the lowest-\norder approximation (See Supplementary Information).\nFor dynamical control of the magnon quantum state,\nfast on-o\u000b switching of the interaction with the qubit\nwould be useful. In the static coupling scheme, however,\nit is technically quite challenging to sweep the magnetic\n\feld in the time scale faster than the coupling strength.\nTo obtain a tunable coupling between the qubit and the\nKittel mode, we adopt a parametrically-induced interac-\ntion which has been proposed and demonstrated in su-\nperconducting circuits23,24(Fig. 3a). In order to suppress\nthe static coupling, a large detuning of 274 MHz between\nthe qubit and the Kittel mode ( !FMR=2\u0019= 8:410 GHz)\nis chosen. Next, we introduce a drive microwave at the\nmean of the qubit and the Kittel-mode frequencies, i.e.,\n!d= (!FMR+!q)=2, to induce the parametric coupling.\nThis is basically a third-order nonlinear process; the sys-\ntem absorbs two drive photons and excites the qubit and\na magnon simultaneously. The substantial third-order\nnonlinearity stems from the anharmonicity of the qubit\nas well as the large coupling strengths gqandgm. The\ninteraction Hamiltonian for the parametrically-induced\ncoupling is written as\n^Hqm,p=~=gqm,p^\u001b+^cy+g\u0003\nqm,p^\u001b\u0000^c; (2)\nwhere the e\u000bective coupling strength gqm,p is propor-\ntional to the drive microwave power Pd[Eq. (S7) in Sup-\nplementary Information].\nTo understand how the continuous-wave spectroscopy\nreveals the parametrically-induced coupling, we consider\nthe inset of Fig. 3a which illustrates the energy levels\nof the driven hybrid system in the bases of the qubit\nstatesfjgi;jeigand the Kittel-mode magnon-number\nstatesfjnmi=j0i;j1i;j2i;\u0001\u0001\u0001g. The two-photon drive\ninduces the Rabi splitting 2 gqm,ppnm+ 1 between the\nstatesjg;nmiandje;nm+ 1i. Using a probe microwave\nwith frequency near !FMR, the change of the magnon\nexcitation spectrum in the driven system is monitored.\nThe colour intensity plots in Fig. 3b show the re\rec-\ntion coe\u000ecient as a function of the probe frequency and\nthe parametric drive frequency for several drive powers.\nWhen the drive frequency hits the two-photon resonance\ncondition (yellow dashed lines), the dip in the spectrum\nsplits into two. As the drive power increases, an an-\nticrossing feature grows in the spectra, along with an\nabrupt shift of the magnon frequency (ii !iii). The\nshift from 8.410 GHz to 8.405 GHz is identi\fed as a\nqubit-state-dependent shift, corresponding to the tran-\nsitionjg;0i$j g;1ifor smallPd(i, ii), andje;0i$j e;1i\nfor largePd(iii-vi). This is due to the population transfer\nfromjg;0itoje;0iat largePdas well as the residual static\ncoupling between the qubit and the Kittel mode (See\nSupplementary Information). The anticrossing observed\nin theje;0i$j e;1itransition manifests parametrically-\ninduced coupling between the qubit and the magnon.3\nAs seen in the cross sections presented in Fig. 3c, the\nspacing between the dips, corresponding to 2 gqm,p=2\u0019,\nincreases linearly with the drive power. This indicates\nthe capability of arbitral and dynamical coupling be-\ntween qubit and the Kittel mode via tailoring the para-\nmetric drive. The maximum coupling gqm,p=2\u0019obtained\nis 3.4 MHz, which again exceeds the decoherence rates of\nthe qubit and the magnon and assures coherent coupling\nbetween them.\nThe tunable coupling scheme opens a door to quantum-\nstate engineering of magnon. Starting from the ground\nstatejg;0i, for example, a pulsed parametric drive would\ngenerate the single-magnon Fock state (together with\na qubit excitation) je;1iwith a pulse duration Tp=\n\u0019=2gqm,p. It would also create the qubit-magnon Bell\nstate (1=p\n2)fjg;0i+je;1igwithTp=\u0019=4gqm,p. Such\nan entangling gate, together with readily applicablesingle-qubit gates, magnon-state displacement, and qubit\nreadout, enables synthesis and a characterization of ar-\nbitrary quantum states of the magnetostatic mode.\nMagnons in a macroscopic-scale ferromagnetic crystal\nare now ready to be controlled in the quantum regime.\nThis leads us to investigate the ultimate limit of spin-\ntronics and magnonics at the single-quantum level. It\nwould also be of particular interest to consider an anal-\nogy with the recently advancing optoelectromechanics25:\nPhonons in nanomechanical devices, yet another exam-\nple of spatially-extended collective excitations in solid,\ncoherently interact both with microwave and optical de-\ngrees of freedom, and thus are studied as a candidate\nfor realising quantum transducers between two largely-\nseparated frequency domains26{28. 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Synthesizing arbitrary quantum states\nin a superconducting resonator. Nature 459, 546{549\n(2009).\n22Imamo\u0015 glu, A. Cavity QED based on collective magnetic\ndipole coupling: spin ensembles as hybrid two-level systems.\nPhys. Rev. Lett. 102, 083602 (2009).\n23Leek, P.J. et al. Using sideband transitions for two-qubit\noperations in superconducting circuits. Phys. Rev. B 79,\n180511 (2009).\n24Zakka-Bajjani, E. et al. Quantum superposition of a single\nmicrowave photon in two di\u000berent `colour' states. Nat. Phys.\n7, 599 (2011).\n25Aspelmeyer, M., Kippenberg, T.J. & Marquardt, F. Cavity\noptomechanics. Preprint at http://arxiv.org/abs/1303.0733\n(2013).\n26Bochmann, J., Vainsencher, A., Awschalom, D. D. & Cle-\nland, A. N. Nanomechanical coupling between microwave\nand optical photons. Nat. Phys. 9, 712{716 (2013).\n27Bagci, T. et al. Optical detection of radio waves through a\nnanomechanical transducer. Nature 507, 81{85 (2014).\n28Andrews, R.W. et al. Bidirectional and e\u000ecient conversion\nbetween microwave and optical light. Nat. Phys. 10, 321{\n326 (2014).4\n-2 -1 0 1 2\nCoil current δIc (mA)8.408.458.508.55Probe frequency (GHz)\nTransmission coefficient Re(  )\n0.00.10.20.3a b\nt\n0.5 mm 0.5  µm1 mm\n5 mm\nElectric field\nMagnetic field\nStatic field\nYIG sphereQubit\nFigure 1. Qubit-magnon hybrid quantum system in a microwave cavity .a, Simulated microwave \feld distribution\nof the TE 102(transverse electric) mode in the cavity. The upper half of the cavity displays the electric \feld which couples\nto a transmon-type superconducting qubit (right insets: optical image of the antenna pads and false-colour scanning-electron\nmicrograph of the Josephson junction bridging them). The lower half shows the magnetic \feld which couples to the spins\nin a single-crystalline sphere of yttrium iron garnet (YIG; photo in the left inset). A static magnetic \feld Bstatic around\n0.29 T is applied locally to the sphere with a compact magnetic circuit consisting of a pair of permanent magnets and a\nsuperconducting coil (Ext. Fig. 1). b, Magnon-photon normal-mode splitting. Amplitude of the microwave transmission\ncoe\u000ecient, Re( t), is measured through the TE 102mode as a function of the probe frequency and the static magnetic \feld. The\n\feld is represented by the relative coil current \u000eIcwhich is de\fned to be zero at the anticrossing. Here, we use a probe power\nof\u0000141 dBm corresponding to the single-photon average occupancy of the TE 102mode. The anticrossing indicates coherent\ncoupling between the cavity mode and the uniformly precessing magnetostatic mode (Kittel mode) in the sphere. The dashed\nlines show the TE 102-mode (white) and Kittel-mode (green) frequencies obtained from \ftting.5\n-1.0 -0.5 0 0.5 1.0 8.12 8.13 8.14 8.15\n0.000.050.100.150.20\nCoil current  δI   (mA) q Excitation frequency (GHz)Excitation frequency (GHz)\nReflection coefficient  Re(∆r\u0001)\n0.00.10.20.3\n8.12 8.13 8.14 8.150 mA-0.2 mA-0.4 mAδI   =\n-1.15 mAEnergy\nGround stateEffective \ncouplingTE103\nQubit\nQubit ReadoutCavity\nTE102Cavity\nKittel modea b c\nExcitation \nmicrowaveq\nFigure 2. Magnon-vacuum-induced Rabi splitting of the superconducting qubit .a, Cavity-mediated coupling scheme.\nThe energy diagram illustrates the frequencies, !c(\u0011!102) and!103, of the cavity TE 102and TE 103modes, the Kittel-mode\nfrequency!FMR, and the qubit frequency !q. The qubit couples to the electric \feld of the TE 102mode with a coupling\nstrengthgq, while the Kittel mode in the YIG sphere magnetically couples to the same mode with a coupling strength gm.\nWhen!q'!FMR, a qubit excitation is transformed to a Kittel-mode magnon and vice versa via the virtual-photon excitation\nin the TE 102mode. The qubit also couples to the TE 103mode weakly, which results in the dispersive frequency shift of\nthe cavity mode depending on the states of the qubit. The qubit state can be measured through the change in the cavity\nresponse to the probe microwave. b, Magnon-vacuum-induced Rabi splitting. Change in the cavity re\rection coe\u000ecient \u0001 rat\n!103is measured as a function of the qubit excitation frequency and the static magnetic \feld represented by the relative coil\ncurrent\u000eIq. We measure the re\rection coe\u000ecient with a probe power of \u0000135 dBm corresponding to the single-photon average\noccupancy of the TE 103mode, and with a qubit excitation power of \u0000130 dBm corresponding to the qubit Rabi frequency of\n1.7 MHz. Another magnetostatic mode weakly couples to the qubit and gives rise to an additional splitting around 8.125 GHz.\nc, Cross sections of bat various static magnetic \felds. For clarity, the individual curves are o\u000bset vertically by 0.06 each from\nbottom to top.6\n0.940.950.960.970.98\n 68 70 8.396 8.400Probe frequency (GHz) 8.404 8.408 8.412\n 70  68  66\nDrive frequency (MHz) offset from 8200 MHz 68  60  64  52  56  60  64\nReflection coefficient  Re( r)0.80\n0.750.850.900.951.00\n8.395 8.400\nProbe frequency (GHz)Reflection coefficient Re( r)\n8.405 8.410 8.415iiiiiiivvvi\ni ii iii iv v viReflection\nmeasurementEnergy\nGround stateTunable\ncouplingParametric\ndriveTE102Cavity\nKittel modea\nbc\nQubitDrive\n24\n0\n0 0.5 1 (MHz)\n(pW)\nFigure 3. Tunable coupling between superconducting qubit and Kittel mode .a, Parametrically-induced coupling\nscheme. The Kittel mode is detuned from the qubit so that the static coupling via the TE 102mode is negligible. An alternative\ncoupling between the qubit and the Kittel mode is parametrically induced by applying a drive microwave at the frequency\n!d= (!FMR+!q)=2. The coupling strength gqm,p is controlled by the drive amplitude. The inset shows energy levels labelled\nwith the qubit state ( jgiorjei) and the magnon number ( nm= 0;1;\u0001\u0001\u0001). The parametric drive (red dashed arrows) induces the\ntwo-photon transitions, e.g., jg;0i,j e;1i, which results in the Rabi splitting. The blue arrows depict the Kittel-mode decay,\nand the red circle at je;0iillustrates the dominant steady-state population at higher drive powers. b, Kittel-mode spectra\nindicating tunable coupling with the qubit. The amplitude of the cavity re\rection coe\u000ecient Re( r) is plotted as a function of\nthe probe and the parametric-drive frequencies. The spectra are measured at the probe power of \u0000135 dBm. Small but \fnite\nmixing of the Kittel mode with the TE 102mode allows the direct magnon-mode spectroscopy by the probe microwave. Panels\ni-vi correspond to the drive power Pdof\u0000104,\u0000103,\u000099,\u000096.5,\u000093.5, and\u000090.5 dBm, respectively. c, Cross sections of\nthe plots in bat the drive frequencies corresponding to ( !FMR +!q)=2 (yellow dashed lines in b). Note that the frequency\ndepends on the drive power through the ac-Stark shift of the qubit. For clarity, the individual curves are o\u000bset vertically by\n\u00000:03 each from top to bottom. Inset: Coupling strength gqm,p as a function of the drive power Pd. Also plotted are the linear\n\ft (red dashed line) and the theoretical expectation (green dashed line; See Supplementary Information for the details).7\nA\nGB\nD\n10 mm\nEH\nF\nCa b\nI\nBD\nE\nA\nI\nExtended Figure 1. Experimental apparatus .a,Cut model of the cavity. The elongated rectangular cavity (A) made of\noxygen-free copper has dimensions of 24 \u00023\u000253 mm. Two SMA connectors (B; another is on the opposite side) are attached\nfor the microwave transmission and re\rection spectroscopy. Couplings of the ports to cavity modes are tuned appropriately\nby adjusting the length of the connector centre pin protruding into the cavity. A pair of disc-shape neodymium permanent\nmagnets (C; another is behind the cavity), with a diameter of 10 mm and a thickness of 1.0 mm each, are placed at the ends\nof a magnetic yoke (D) made of pure iron. The magnets produce a static \feld of about 0.29 T at the 4-mm gap in between.\nThe magnetic \feld can be additionally tuned by the current along a 104-turn superconducting coil (E). The \feld-to-current\nconversion ratio is approximately 1.7 T/A. An YIG sphere (F) glued to an aluminium-oxide rod along the crystal axis h110i\nis mounted in the cavity at the centre of the gap between the magnets. The static \feld is applied in parallel with the crystal\naxish100i. A transmon-type superconducting qubit (G), consisting of two large-area aluminium pads and a single Josephson\njunction (Al/Al 2O3/Al), is fabricated on a silicon substrate (See the inset of Fig. 1a) and is mounted inside the cavity. The qubit\nand the YIG sphere are separated by 35 mm in the horizontal direction. A double-layer magnetic shield made of aluminium (H)\nand pure iron (I) covers a half of the cavity to protect the qubit from the stray \feld of the magnet. b,Picture of the apparatus\nhanging beneath the mixing-chamber plate of a dilution refrigerator. The cavity is cooled down to 10 mK.8\nVector\nNetworkAnalyzer\n300 K\n4 K\n10 mK 26dB\n1dB\nCryogenic\namplifierAmplifier\nNbTi wire  Attenurators\nCu wire\nMagnet\n& CoilMW\ncavityMicrowave \nsource\nIsolators\nQubit YIG Yoke16dBCurrent \nsource20dB\nLow-pass filter\nExtended Figure 2. Measurement setup . The attenuation on the input lines, which at 10 GHz amounts to 49 dB including\nlosses at cables (phosphor-bronze coaxial cables; Coax Corp. SC-119/50-PBC-PBC) and connectors, is enough to prevent the\nroom-temperature thermal noise from reaching the sample space. To further diminish noise above 12 GHz at input port, we\nintroduce a low-pass \flter (RLC F-30-12.4-R). For the output line, we place a circulator and isolators (Quinstar XTE0812KCS\nand XTE0812K), which give the isolation ratio of more than 60 dB. All measurements in this Letter are done with a vector\nnetwork analyser (VNA; Agilent E5071C). For the static- and parametric-coupling experiments, the excitation microwave\ngenerated by a microwave source (Agilent E8247C) is combined with the probe microwave from VNA in a directional coupler\n(Krytar 120420) and is introduced to the input port. The re\rected or transmitted signal from the cavity is ampli\fed in a series\nof ampli\fers at 4 K (Caltech CRYO4-12) and at room temperature (MITEQ AFS4-08001200-09-10P-4). A current source\n(Yokogawa GS200) is used to tune the static \feld applied to the YIG sphere.9\nDrive frequency (GHz)\nCoil current  δI  (mA)\nReflection coefficient   Re(    )∆rb aEnergy\nGround stateTE102Cavity\nKittel modeTE102\nQubitTE103\nQubit ReadoutCavity\nDrive microwaveKittel mode\nParametric coupling \n4 8\nq12 168.28.38.48.58.6\n0.00.10.20.3\nExtended Figure 3. Spectroscopy of parametrically-induced transitions .a,Method of the spectroscopy. The qubit\ndispersively couples to the cavity TE 103mode, giving rise to the qubit-state-dependent frequency shift \u001f103. To investigate\nthe energy-level structure of the hybrid system consisting of the qubit, the TE 102mode and the Kittel mode, we search for\ntransitions by driving the system and by simultaneously monitoring the qubit excitation via the frequency shift of the TE 103\nmode. A few allowed transitions are observed in the low-drive-power limit, while many dipole-forbidden transitions appear\nwhen the system is driven strongly. b,Spectrum taken with a strong drive power. Change in the cavity re\rection \u0001 rat\n!103is measured as a function of the drive frequency and the static magnetic \feld represented by the coil current \u000eIq. For\na reference, \u000eIqis de\fned to be zero when the Kittel-mode frequency !FMR coincides with the qubit frequency !q. We use\nthe probe and drive microwave power of \u0000135 dBm and\u000096.7 dBm, respectively. The line observed at 8.47 GHz, broadened\nand shifted due to the high-power drive, corresponds to the TE 102mode. Note that the Kittel mode is also detected in the\nindirect measurement via the qubit because of the residual interaction. The two-photon parametric transition is observed at\nthe frequency ( !FMR+!q)=2. Other qubit-related multi-photon transitions, e.g., the qubit-TE 102-mode parametric transition\nat [(!q+!c)=2]=2\u0019= 8:29 GHz, are seen as horizontal lines.\n-12 -10 -8 -6 -4 -2 0 2-12-8-404\nDrive frequency (MHz)Probe frequency (MHz) \n0.01.0\n0.5\nSpectral density (arb. unit)\nExtended Figure 4. Numerical simulation of parametrically-induced coupling . A simulated Kittel-mode spectrum is\nplotted as a function of the probe frequency and the parametric-drive frequency. Parameters used for the simulation correspond\nto those in panel v of Fig. 3b (See Supplementary Information). The residual static coupling \u0018shifts the Kittel-mode frequency\ndepending on the qubit states ( jgiandjei). The diagonal dashed line depicts the detuning of the parametric drive. Away from\nthe two-photon parametric resonance jg;0i,j e;1i, only the Kittel-mode excitation jg;0i$j g;1i(upper horizontal dashed\nline) appears since the qubit is in the ground state. As the drive frequency approaches the resonance, parametric drive excites\nthe qubit together with a Kittel-mode magnon, leading to a shift of the Kittel-mode spectrum to the line je,0i$j e;1i(lower\nhorizontal dashed line). The dressed states between jg;0iandje;1igive rise to an avoided crossing at the resonance, indicating\nthe parametrically-induced coherent coupling (Inset of Fig. 3).10\nSupplementary Information for \\Coherent coupling between ferromagnetic magnon\nand superconducting qubit\"\nY. Tabuchi1, S. Ishino1, A. Noguchi1, T. Ishikawa1, R. Yamazaki1, K. Usami1, and Y. Nakamura1;2\n1Research Center for Advanced Science and Technology (RCAST),\nThe University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan and\n2Center for Emergnent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan\nEXPERIMENTAL APPARARTUS\nAn illustrative drawing and a photo of the experimen-\ntal apparatus are shown in Ext. Fig. 1. The wiring, cir-\ncuit components and instruments used in the experiments\nare shown in Ext. Fig. 2. Each of the microwave powers\npresented in the paper refers to the one at the relevant\ncavity port.\nQUBIT AND CAVITY MODES\nWe use a transmon-type qubit with the bare qubit\nfrequency!q,bare=2\u0019and the anharmonicity \u000b=2\u0019of\n8.204 GHz and\u0000158 MHz, respectively. The anhar-\nmonicity\u000bis de\fned as the di\u000berence !ef\u0000!gebetween\nthe bare transition frequencies of the ground state to\nthe \frst excited state !ge(\u0011!q,bare ) and the \frst ex-\ncited state to the second excited state of the qubit !ef.\nThe qubit couples to the cavity modes TE 101, TE 102and\nTE103, with coupling strengths of 85 MHz, 121 MHz\nand 142 MHz, respectively. Owing to the coupling to\nthe cavity \feld, the qubit frequency !q=2\u0019is shifted by\n\u000071 MHz (Lamb shift) from !q,bare=2\u0019.\nThe cavity has the bare mode frequencies\n(!101;bare=2\u0019,!102;bare=2\u0019and!103;bare=2\u0019) of\n6.993 GHz, 8.420 GHz and 10.452 GHz. They are\nsubject to a shift due to coupling with the qubit by\n\u00006 MHz, +68 MHz, +9 MHz, respectively. We tune the\ncoupling of the cavity modes to the input and output\nports by adjusting the length of the centre pin protrud-\ning into the cavity. The input and output couplings for\nthe modes TE 102and TE 103are\u0014in;102=2\u0019= 0:68 MHz,\n\u0014out;102=2\u0019= 0:53 MHz,\u0014in;103=2\u0019= 0:15 MHz and\n\u0014out;103=2\u0019= 2:72 MHz. The internal loss of the\nmodes TE 102and TE 103are 1.26 MHz and 1.24 MHz,\nrespectively.\nThroughout the paper, we use the renormalized fre-\nquencies for the qubit !qand the cavity modes ( !101,\n!102\u0011!c, and!103), taking into account the fre-\nquency shifts due to the interaction. Namely, !q=2\u0019=\n8.136 GHz, !101=2\u0019= 6.987 GHz, !102=2\u0019(\u0011!c=2\u0019) =\n8.488 GHz and !103=2\u0019= 10.461 GHz.\nYIG SAMPLE\nWe purchased the YIG single crystal from Ferrisphere\nInc.29The spherical sample is polished to the surface\nroughness below 50 nm and glued to an aluminium-oxiderod at the factory. We apply the static magnetic \feld\nBstatic to theh100icrystal axis.\nQUBIT-MAGNON COUPLING STRENGTH\nThe e\u000bective coupling between the qubit and the Kit-\ntel modegqm,s =gqgm=\u0001 should be calculated using\nthe bare qubit !q,bare and cavity !102,bare frequencies to\navoid double-counting of the e\u000bect of interactions in the\nperturbative treatment. Therefore, we use the detuning\n\u0001=2\u0019= (!q,bare\u0000!102;bare)=2\u0019of 215 MHz, and evaluate\nthe coupling to be gqm,s=2\u0019= 11:8 MHz.\nPARAMETRICALLY INDUCED COUPLING\na. Spectroscopy To \fnd parametrically-induced\ntransitions, we perform spectroscopy of the hybrid\nsystem. Extended Figure 3a illustrates the method. We\nuse the TE 103mode for the qubit readout; when the\ndrive microwave hits transitions related to the qubit, the\ncavity re\rection coe\u000bcient at !103changes.\nIn Ext. Fig. 3b, the upper diagonal dashed line cor-\nresponds to the Kittel-mode frequency, which linearly\nmoves along with the coil current and splits when the fre-\nquency coincides with the TE 102-mode frequency. (Note\nthat the Kittel mode is also observed by the direct mea-\nsurement through the TE 102mode as shown in Fig. 1b.)\nOn the other hand, the lower dashed line is ascribed\nto the parametrically-induced transition. The transition\nfrequency is equal to ( !q+!FMR)=2. The transition fre-\nquency and the Kittel mode frequency !FMR meet each\nother at the qubit frequency !q.\nb. Coupling strength The parametrically-induced\ncoupling strength gqm,p is derived from a Hamiltonian\ntaking into account the TE 102mode, the qubit and the\nKittel mode, which is expressed as\n^H=~=!102;bare^ay^a+!FMR,bare ^cy^c\n+\u0010\n!q,bare\u0000\u000b\n2\u0011\n^by^b+\u000b\n2\u0010\n^by^b\u00112\n+gq(^ay^b+ ^a^by) +gm(^ay^c+ ^a^cy)\n+r\n\u0014102Pd\n~!d\u0002\n^ayexp (\u0000i!d) + ^aexp (i!d)\u0003\n;(S3)\nwhere ^aand ^bare the annihilation operators of the\nphoton in the TE 102mode and a transmon qubit, \u000b\nis the anharmonicity of the qubit, and \u0014102=\u0014out;10211\nsince we introduce the microwave drive through the cav-\nity output port. Here, we de\fne the lowest two lev-\nels of the anharmonic oscillator as the qubit subspace.\nProvided we explicitly write the lowering operator as\n^b=P\nn\u00150pn+ 1jnihn+ 1j, the qubit lowering operator\nis de\fned as ^ \u001b\u0000= (^\u001b+)y=j0ih1j. By applying successive\nunitary transformations to diagonalise the Hamiltonian30\nand using approximation in which higher order terms are\nsafely truncated, we obtain the e\u000bective Hamiltonian\n^He\u000b=^Hqm,p+^Hres; (S4)\nwhere ^Hqm,pand ^Hresrepresent a parametrically-induced\ntunable coupling and a residual static coupling between\nthe qubit and the Kittel mode, respectively. They are\nwritten as\n^Hqm,p=~=gqm,pei(!q+!FMR+2\u0018\u00002!d)t^\u001b+^cy\n+g\u0003\nqm,pe\u0000i(!q+!FMR+2\u0018\u00002!d)t^\u001b\u0000^c(S5)\n^Hres=~= 2\u0018^cy^c^\u001b+^\u001b\u0000: (S6)\nThe coupling strength gqm,p is expressed as\ngqm,p'2gqg\u0003\nm\u0012\u0014102Pd\n~!d\u0013\u0012gq\n\u0001d\u00132\n\u0002\"\n5=6\n(\u0001q\u0000\u0001m)2\u0001m\n\u00001\n(\u0001q\u0000\u0001m)(\u0001q\u0000\u0001m\u0000\u000b)\u0001m\n+2=3\n(\u0001q\u0000\u0001m)(\u0001q+ \u0001 m)\u0001m\n+4=3\n(\u0001q\u0000\u0001m)2(\u0001q+ \u0001 m)\n+1=3\n\u0001q(\u0001q+ \u0001 m)\u0001m\n+1=2\n(\u0001q\u0000\u0001m\u0000\u000b)(\u0001q\u0000\u0001m\u00002\u000b)\u0001m\n+1=6\n\u0001q(\u0001q\u0000\u0001m)\u0001m+1=2\n\u0001q(\u0001q\u0000\u0001m)2\n+2=3\n\u0001q(\u0001q\u0000\u0001m)(\u0001q+ \u0001 m)\n\u00001=6\n(\u0001q\u0000\u0001m)(\u0001q\u0000\u0001m\u00002\u000b)\u0001m\n\u00001=2\n(\u0001q\u0000\u000b)(\u0001q\u0000\u0001m)(\u0001q\u0000\u0001m\u0000\u000b)\n\u00001=12\n(\u0001q\u0000\u000b)(\u0001q\u0000\u0001m)\u0001m\n+1=12\n(\u0001q\u0000\u000b)(\u0001q\u0000\u0001m\u00002\u000b)\u0001m\n\u00001=6\n(\u0001q\u0000\u000b)(\u0001q\u0000\u0001m)(\u0001q\u0000\u0001m\u00002\u000b)#\n;(S7)and the residual coupling strength \u0018is denoted as\n\u0018'\u0000jgqj2jgmj2\n\u0002\"\n5=12\n\u0001q(\u0001q\u0000\u0001m)\u0001m+5=12\n(\u0001q\u0000\u0001m)\u00012m\n+1=4\n\u0001q\u00012m+1=12\n\u00012q\u0001m+1=12\n\u00012q(\u0001q\u0000\u0001m)\n+5=12\n(\u0001q\u0000\u0001m)2\u0001m\u00001=12\n(\u0001q\u0000\u000b)\u00012m\n\u00005=12\n(\u0001q\u0000\u000b)(\u0001q\u0000\u0001m\u0000\u000b)\u0001m\n\u00005=12\n(\u0001q\u0000\u0001m\u0000\u000b)\u00012m\u00001=12\n(\u0001q\u0000\u000b)2\u0001m\n\u00001=12\n(\u0001q\u0000\u000b)2(\u0001q\u0000\u0001m\u0000\u000b)\n\u00005=12\n(\u0001q\u0000\u0001m\u0000\u000b)2\u0001m#\n: (S8)\nHere, we de\fne \u0001 q=!102;bare\u0000!q,bare +2gq=(!102;bare\u0000\n!q,bare ), \u0001 m=!102;bare\u0000!FMR+gq=(!102;bare\u0000!q,bare ),\nand \u0001 d=!102;bare\u0000!d+gq=(!102;bare\u0000!q,bare ). The\nconversion ratio gqm,p=Pdbetween the drive power and\nthe coupling strength in our experiment is estimated\nto be 4.9 MHz/pW, which is indicated in the inset of\nFig. 3c as a green dashed line. The discrepancy be-\ntween the theoretical (4.9 MHz/pW) and the experimen-\ntal (4.0\u00060.4 MHz/pW) values is attributed to higher-\norder terms dropped in the secular approximation.\nc. Residual coupling In the tunable-coupling exper-\niment, we suppress the static coupling between the qubit\nand the Kittel mode, gqm,s, by setting a relatively large\ndetuning of 274 MHz. However, the residual coupling\n^Hresremains \fnite, which shifts the Kittel-mode fre-\nquency depending on the qubit state ( jgiorjei). We\nobtain the residual coupling strength \u0018=2\u0019of\u00002.5 MHz\nfrom the observation of the Kittel-mode doublet in the\npresence of qubit excitation, whereas the calculated value\nfrom Eq. (S8) is\u00001.2 MHz.\nIn order to understand the e\u000bect of the residual cou-\npling, we perform a numerical simulation of the Kittel-\nmode spectrum under the parametric drive. Parameters\nare chosen such that they reproduce the result shown\nin panel v of Fig. 3b: the residual coupling \u0018=2\u0019=\n\u00002:5 MHz, the coupling strength gqm,p=2\u0019= 2.2 MHz,\nthe Kittel mode linewidth \rm=2\u0019= 1.3 MHz, the qubit\nrelaxation rate \rq=2\u0019= 0.13 MHz and the qubit dephas-\ning rate\r\u001e=2\u0019= 0.3 MHz.\nExtended Figure 4 shows the simulated Kittel-mode\nspectrum as a function of the probe frequency and the\ndrive frequency. The simulation reveals that the residual\ncoupling ^Hresshifts the Kittel-mode frequency depending\non the qubit state. As the parametric drive excites the\nqubit and the Kittel mode simultaneously, the Kittel-\nmode frequency is shifted by 2 \u0018. At the resonance it\nfurther shows two-photon Rabi splitting. The simulation12\nreproduces the characteristic features in Fig. 3b pretty well.\n29www.ferrisphere.com/\n30Scholz, I., van Beek, J.D. & Ernst, M. Operator-based Flo-quet theory in solid-state NMR. Solid State Nucl. Mag. 37,\n39-59 (2010)."    }]
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